Kinematics and Dynamics of Mobile Robots · Kinematics and Dynamics of Mobile Robots. rrlab.cs.uni-kl.de Kaiserslautern, Germany Kaiserslautern Frankfurt Berlin Hamburg Bremen Hanover

rrlabcsuni-klde

Karsten Berns

Robotics Research Lab

Department of Computer Science

University of Kaiserslautern Germany

Kinematics and Dynamics of Mobile Robots

rrlabcsuni-klde

KaiserslauternGermany

Kaiserslautern

Frankfurt

Berlin

Hamburg

Bremen

Hanover

Cologne

Stuttgart

Munich

Braunschweig

Federal Republic of Germany

bull Deutschland

bull 16 constituent states

bull area of 357386 square

kilometres

bull (137988 sq mi)

bull 83 million inhabitants

bull Capital Berlin

(52deg31primeN 13deg23primeE)

rrlabcsuni-klde

University of Kaiserslautern ndash Statistics

gt 14500 Students

gt 2000 Staff

gt 150 Full Professors

Third party funds gt80 Mio euro

rrlabcsuni-klde

Science Alliance Kaiserslautern

rrlabcsuni-klde

Scientific Researchers ~ 140

Technical Staff ~ 15

Professors 12

Senior Engineers 5

ZNT Statistics

Center for Commercial Vehicles Technology

rrlabcsuni-klde

Research at Robotics Research Lab (RRLab)

Methodology and Algorithms

Autonomous Mobile Robots

Hu

ma

no

id

Ro

bo

ts

Off-r

oa

d R

obo

ts

Serv

ic

e R

obo

ts

rrlabcsuni-klde

Off-road Projekte

BauwesenLandwirtschaftTest-Fahrzeuge Rescue Roboter

rrlabcsuni-klde

Autonomous Navigation of a Tractor - Windrow Guidance 18kmh

rrlabcsuni-klde

rrlabcsuni-klde

Contents

Modeling of Robots and Space

Vehicle Kinematics

Analytical Solution

Geometric Solution

Bicycle Model

Maneuverability of AMRs

Rigid Vehicle Dynamics

AMR ndash Modeling 9

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Modeling of AMRs

and its Operational Environment

rrlabcsuni-kldeAMR ndash Modeling 11

Modeling of AMRs and the Operational Environment

Reduction of the robot complexityAMR movements as trajectories in a 3D Euclidian space

Objects are represented by boarder lines or geometric objects eg

polygon lines grids

boxes

AMR position can be

represented by a single

point in space

center of mass

kinematic center

Orientation (heading) represented as a vector due to the world frame


Configuration Space

Configuration space (C-space) spanned over the degrees of freedom that should be considered

Eg a mobile robot in a 2D space contains all possible positions and orientations (119909 119910 120593) and maps each point to reachable or not reachable

2D robot C-space


Configuration Space

Trajectory planning must take into account the AMR shape

=gt widened obstacles by the

reflected shape of the AMR

=gt if point representation

can reach a configuration the

same holds for the real robot

2D robot c-space


Configuration Space

Extension of C-Space considering the derivatives of each degree of freedom

=gt Extension space (X-space)

Eg X-Space for considering velocity

Widen obstacles so that collision avoidance remainspossible (brakingdeceleration) 2D robot X-space

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Coordinate Systems


Modeling of a movement of an AMR by transformation of

the robot coordinate systems in a 3D Euclidian space


Description of AMR Pose as Six-Tuple

In general the pose of an arbitrary object in the Cartesian space can be described as a six-tuple 119909 119910 119911 120593 120595 120579

The position vector 119874119906 in an object frame 119874 can be presented

in base frame coordinates 119861119906 by

119861119906 =119909119910119911

+ 119874119861119877 120593120595 120579 119874 119906

with 119909 119910 119911 119879 being the translation vector between the origin of the two frames and 119877 (3x3 matrix) the respective (combined) rotation matrix


Homogeneous Transformation Matrix

Another possibility to express the same as above are homogeneous transformation matrices Those 4 times 4 matrices(for 3D space) are composed as shown below

119877 119906Ԧ119901119879 119904

119877 3 times 3 rotation (orientation) matrix

119906 Translation (position) - 119906 = 119906119909 119906119910 119906119911119879

Ԧ119901 Perspective transformation in general Ԧ119901 = 000 119879

119904 Scaling factor in general 119904 = 1


3D Rotation Matrices

119877 is usually a combination of several rotations around the elementary axis

119877119909 120593 describes a rotation around the x-axis of an arbitrary coordinate system via the angle 120593 In an analog matter the other two required matrices are defined

119877119909 120593 =1 0 00 cos120593 minus sin1205930 sin120593 cos120593

119877119910 120595 =cos120595 0 sin1205950 1 0

minus sin120595 0 cos120595

119877119911 θ =cos θ minus sin θ 0sin θ cos θ 00 0 1 119909 119906

120593 119910

119907

119911119908


Expression of Linked Rotations

Two possible ways of expressing linked rotations

Euler angles (rotation around variable axes)

119877 = 119877119911 120593 sdot 119877119909 120595 sdot 119877119911prime θ

Tait-Bryan angles ldquoRoll Pitch Yawrdquo (rotation around fixed axes)

119877 = 119877119911 θ sdot 119877119910 120595 sdot 119877119909 120593


Roll Pitch Yaw

Euler rotations along ldquonewrdquo axes

Roll-Pitch-Yaw rotations along fixed axes

Terrestrial robotics the roll pitch yaw system is used (change of AMR pose according to world coordinate system)

Yaw θ

Pitchψ

Roll 120593

119911

119910119909

20

Roll

Pitch Yaw

119909

119911119910


Combined Rotational Matrix (Roll Pitch Yaw)

Arbitrary orientation in 3D space can be described using three rotations

119877119904 as resulting matrix

(abbreviations 119904120593 for sin120593 and 119888120593 for cos120593)

119877119904 =

119888θ 119888120595 119888θ 119904120595 119904120593 minus 119904θ 119888120593 119888θ 119904120595 119888120593 + 119904θ 119904120593119904θ 119888120595 119904θ 119904120595 119904120593 + 119888θ 119888120593 119904θ 119904120595 119888120593 minus 119888θ 119904120593minus119904120595 119888120595 119904120593 119888120595 119888120593


Linear Velocity

Transformation velocities between two frames is similar to pose transformation

Transformation of a linear velocity vector 119861 Ԧ119907119902 of an arbitrary

point Ԧ119902 presented in frame 119861 to frame 119860119860 Ԧ119907119902 = 119861

119860119877119861 Ԧ119907119902

If the origin of frame 119861 has also a linear velocity relative to frame 119860 then

119860 Ԧ119907119902 =119860 Ԧ119907119874119861 + 119861

119860119877119861 Ԧ119907119902

If in addition point Ԧ119902 is rotating around an arbitrary axis with

the rotational velocity 119860Ω119861 then the linear velocity can be

calculated with

119860 Ԧ119907119876 = 119860 Ԧ119907119874119861 + 119861119860119877119861 Ԧ119907119876 +

119860ω119861 times 119861119860119877119861 Ԧ119902


Rotational Velocity

A rotational vector 119861120596 related to frame 119861 can be transferred

to frame 119860 with

119860120596 = 119861119860119877119861120596


Linear and Rotational Velocity

Rotational and linear velocity of a sequence of segments connected by rotational or prismatic joints can stepwise be calculated

Rotational velocity 119894+1120596119894+1 and the linear velocity 119894+1 Ԧ119907119894+1relative to frame 119894 + 1 can be determined as

119894+1120596119894+1 = 119894119894+1119877 sdot 119894120596119894 + ሶθ119894+1

119894+1 Ԧ119890119911119894+1

119894+1 Ԧ119907119894+1 = 119894119894+1119877 119894 Ԧ119907119894 +

119894120596119894 times119894 Ԧ119901119894+1

with 119894 Ԧ119901119894+1 vector in direction of the segment 119894 120596119894 the rotational

velocity and 120595119894 the rotation of segment 119894 around the elementary 119911-axis



119894119894+1119877 is the inverse of the orientation matrix 119894+1

119894119877 from frame 119894 to

119894 + 1

119894+1119894119877 is an orthonormal matrix

its inverse 119894+1119894119877minus1 = 119894

119894+1119877 is the transposed matrix 119894+1119894119877119879

(example in 2D space)

119894119894+1119877 θ = 119894+1

119894119877minus1 θ = 119894+1119894119877119879 θ =

119888θ 119904θ 119909119908minus119904θ 119888θ 1199101199080 0 1

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World (Global) and Robot (Local) Coordinate System


YW

XW

θ

YR

XR

P


World and Robot Coordinate System

119875 describes the location of the kinematic center relative to the world coordinate system

120579 is the mathematically positive angle between 119883119882 and 119883119877 119874 119883119882 119884119882 ෝ= origin of the world coordinate system

119874 119883119877 119884119877 ෝ= 119875 ෝ= origin of the robotrsquos coordinate system

119883119877-axis is longitudinal axis of the robot through its kinematic center

119884119877-axis is lateral axis of the robot through its kinematics center

120585119882 = 119909 119910 120579 119879 coordinates of the kineamtic center in world

coordinates



Orientation via rotation around 119911-axis

119877 θ =cos θ sin θ 0minussin θ cos θ 0

0 0 1

Speed transformation from world- to robot coordinate system

ሶԦ120585119877 = 119877 θሶԦ120585119882

Coordinate transformation

YW

XW

θ

YR

XR



Rotation around 119911-axis withθ = 90 deg

119877120587

2=

0 1 0minus1 0 00 0 1

Speed calculation in robot coordinates

ሶԦ120585119877 = 119877120587

2ሶ

120585119882 =0 1 0minus1 0 00 0 1

sdotሶ119909ሶ119910ሶθ

=ሶ119910

minus ሶ119909ሶθ

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Vehicle Kinematics -

Analytical Solution

rrlabcsuni-kldeAMR ndash Kinematics 31

Control of the Test Vehicle GATOR


Forward kinematics for wheel driven robot

What is the robotrsquos trajectory if wheel geometry and speed are known

YW

XW

θ

V(t)

120596(t)

rrlabcsuni-kldeAMR ndash kinematics 33

Example Differential Drive

119903 radius of the wheel

119889 distance wheel center to kinematic center

ሶ120595119903 ሶ120595119897 Angular velocity of leftright wheel

ሶԦ120585119882 speed calculation in world coordinates

ሶԦ120585119882 = ሶ119909 ሶ119910 ሶ120579119879

= 119891 119889 119903 120579 ሶ120595119897 ሶ120595119903

= 119877 120579 minus1 ሶԦ120585119877

passive wheel

active wheel

d



Calculation of the contribution of the left and right wheel to the speed along the longitudinal axis (119883119877)

Calculation of the contribution of the left and right wheel to the angular velocity of kinematic center

Summation of both results in the velocity and angular velocity along the 119883119877-axis where

ሶ119909119877119903 =119903 ሶ1205951199032

ሶ119909119877119897 =119903 ሶ120595119897

2

120596119903 =119903 ሶ1205951199032119889

120596119897 =minus119903 ሶ120595119897

2119889

ሶԦ120585119882 = 119877 120579 minus1 ሶԦ120585119903ሶԦ120585119903 =

119903 ሶ1205951199032

+119903 ሶ120595119897

2 0

119903 ሶ1205951199032119889

minus119903 ሶ120595119897

2119889

119879



Since 119877 120579 is orthonormal

119877 120579 minus1 =cos 120579 minus sin 120579 0sin 120579 cos 120579 00 0 1

Given 120579 =120587

2 119903 = 1 119889 = 1 ሶ120595119903 = 4 ሶ120595119897 = 2 and wheel speed

notated in rounds per second 2120587

119904

ሶԦ120585119882 =ሶ119909ሶ119910ሶ120579

=0 minus1 01 0 00 0 1

sdot301

=031


Calculation of Wheel kinematics

Stepwise calculation of the linear and the rotational velocity for wheeled vehicles in 2D environment

Kinematic center moved with ሶ119909 ሶ119910 ሶ120579119879

Determine rotational velocity of each wheel

Collect all equations and solve the equation system

Consider

Equation system must no be solvable

No dynamic aspects will be considered

Calculation of the speed of the kinematic center based on thewheel speed can also be determined out of the equationsystem (inverse problem)

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Basic Wheel Types

AMR ndash Kinematics 37

Standardwheel

Steerablestandard

wheel

Castorwheel

Swedish orMecanum

wheel

Ball orspherical

wheel


Wheel Properties

Standard and steerable wheel

Linear velocity along wheel plane ሶ119909 = 119903 ሶ120595

No sliding orthogonal to wheel plane ሶ119910 = 0

Castor wheel


Linear perpendicular velocity ሶ119910 = minus119889119888 ሶ120573

SwedishMecanum wheel

Linear velocity along wheel plane ሶ119909 = 119903 ሶ120595 cos 120574

120574 angle of passive rollers (45 deg or 90 deg)

Linear perpendicular velocity ሶ119910 = 119903 ሶ120595 sin 120574 + 119903119901119903 ሶ120595119901119903


Kinematic Parameters



120572 Angle between 119909-axis and wheel mount point

120573 Angle between the straight line through the kinematic center and the fixing point of the wheel and the 119910-axis of the wheel frame

119889 Distance from the kinematic center to the wheel fixingpoint

119889119888 Distance from the wheel fixing point to the wheel supporting point (Castor wheel only)

120574 Angle between the 119909-axis of the wheel and rolling direction of the rollers (Mecanum wheel only)


Kinematic Parameters Standard Wheel

ሶ120595 119903


Kinematic Parameters Steerable Wheel

ሶ120595 119903


Kinematic Parameters Castor Wheel

ሶ120595 119903


Kinematic Parameters Mecanum Wheel

ሶ120595 119903


Kinematic Parameters Ball Wheel

ሶ120595 119903


Calculation of Wheel Velocity

Given velocity vector Ԧ119907 = ሶ119909 ሶ119910 ሶ120579119879

of the kinematic center

Calculation of linear velocity of standard wheel due to the speed of the kinematic center

Stepwise apply the following equations

Initial parameters 11205961 = 00 ሶ120579119879

and 1 Ԧ1199071 = ሶ119909 ሶ119910 0 119879

No additional rotational speed all 119894120596119894 = 00 ሶ120579119879

119894+1120596119894+1 = 119894119894+1119877 sdot 119894120596119894 + ሶ120579119894+1

119894+1 Ԧ119890119911119894+1

119894+1 Ԧ119907119894+1 = 119894119894+1119877 119894 Ԧ119907119894 +

119894120596119894 times119894 Ԧ119901119894+1



After the rotation around the 119911-axis with angle 120572 2 Ԧ1199072 is

calculated as

2 Ԧ1199072 =119888120572 ሶ119909 + 119904120572 ሶ119910minus119904120572 ሶ119909 + 119888120572 ሶ119910

0

Due to the translation 119889 3 Ԧ1199073 is

3 Ԧ1199073 =

119888120572 ሶ119909 + 119904120572 ሶ119910

minus119904120572 ሶ119909 + 119888120572 ሶ119910 + 119889 ሶ1205790



The last rotation around the 119911-axis with angle 120573 minus 90degtransfers the 119909-axis of the last frame to the rolling direction of the wheel

For the calculation of 4 Ԧ1199074 in the next equation

sin 120573 minus 90deg = minus cos120573 and cos 120573 minus 90deg = sin 120573 is used

4 Ԧ1199074 =119904 120572 + 120573 ሶ119909 minus 119888 120572 + 120573 ሶ119910 minus 119888120573 119889 ሶ120579

119888 120572 + 120573 ሶ119909 + 119904 120572 + 120573 ሶ119910 + 119904120573 119889 ሶ1205790

rrlabcsuni-kldeAMR ndash 4 Modeling 49


Last step is to equalize the linear velocity vector of the standard wheel to the velocity of the kinematic center represented in the wheel frame

119904 120572 + 120573 minus119888 120572 + 120573 minus119888120573 119889

119888 120572 + 120573 119904 120572 + 120573 119904120573 1198890 0 0

ሶ119909119910 ሶሶ120579

=119903 ሶ12059500

1st equation 119903 ሶ120595 = 119888 120572 + 120573 ሶ119909 + 119904 120572 + 120573 ሶ119910 + 119904120573 119889 ሶ120579 is called the

rolling constraint of the wheel because it describes the speed in the rolling direction of the wheel

2nd equation 119888 120572 + 120573 ሶ119909 + 119904 120572 + 120573 ሶ119910 + 119904120573 119889 ሶ120579 = 0 is called the

sliding constraint which describes the speed perpendicular to wheel plane

Sliding constraints are collected in Matrix 119878 ndash add row (119888(120572 +


Wheel Type-specific Velocity Vector

In case of the steerable standard wheel the fixed angle 120573 is replaced by a function 120573 119905

Equation could also be applied to the spherical wheel (based on the forces which affect the wheel and change 120573 119905 only a linear velocity in the rolling direction exists)

In case of castor wheels the y-component of the velocity

vector is depending on the angular velocity ሶ120573 and the length of the rod

119904 120572 + 120573(119905) minus119888 120572 + 120573(119905) minus119888120573(119905)119889

119888 120572 + 120573(119905) 119904 120572 + 120573(119905) 119904120573(119905)1198890 0 0

ሶ119909ሶ119910ሶ120579

=119903 ሶ120595

minus119889119888 ሶ120573(119905)0

Sliding constraints are collected in Matrix 119878 ndash add row 119888 120572 + 120573(119905) 119904 120572 + 120573(119905) 119904120573(119905)119889 (is used in section

maneuverability)



Swedish or Mecanum wheel is able to move in an omnidirectional way

119904 120572 + 120573 + 120574 minus119888 120572 + 120573 + 120574 minus119888 120573 + 120574 119889

119888 120572 + 120573 + 120574 119904 120572 + 120573 + 120574 119904 120573 + 120574 1198890 0 0

ሶ119909ሶ119910ሶ120579

=119903 ሶ120595 cos 120574

119903 ሶ120595 sin 120574 + 119903119901119903 ሶ1205951199011199030


Vehicle Motion Calculation

Transformation of wheel velocities to velocity of kinematic center

Determine parameters 120572 120573 120574d indicating the wheel pose

Insert 120572 120573 120574 into appropriate wheel equation (as shown before) for each wheel

Solve system of equations


Differential Drive Kinematics

Two fixed standard wheels mounted on one axis

Kinematic center located in the middle of the axis

Distance between the kinematic center and each wheel 119889

Kinematic center is origin of coordinate frame

One solution for modeling the wheel configuration is to place the wheels on the 119910-axis

Kinematic parameters

120572119897 = 90 deg

120573119897 = 0 deg

120572119903 = minus90 deg

120573119903 = 180 deg



Apply equations for standard wheels (for the left and the right wheel)

119904 120572119897 + 120573119897 ሶ119909 minus 119888 120572119897 + 120573119897 ሶ119910 minus 119888120573119897 119889 ሶ120579 = 119903119897 ሶ120595119897

119888 120572119897 + 120573119897 ሶ119909 + 119904 120572119897 + 120573119897 ሶ119910 + 119904120573119897 119889 ሶ120579 = 0

119904 120572119903 + 120573119903 ሶ119909 minus 119888 120572119903 + 120573119903 ሶ119910 minus 119888120573119903 119889 ሶ120579 = 119903119903 ሶ120595119903119888 120572119903 + 120572119903 ሶ119909 + 119904 120572119903 + 120573119903 ሶ119910 + 119904120573119903 119889 ሶ120579 = 0

If the above mentioned parameters are inserted the following equations will result

ሶ119909 minus 119889 ሶ120579 = 119903119897 ሶ120595119897

ሶ119910 = 0

ሶ119909 + 119889 ሶ120579 = 119903119903 ሶ120595119903119910 = 0



Solving the system of equations yields

ሶ119909ሶ119910ሶ120579

=

1

2119903119897 ሶ120595119897 + 119903119903 ሶ120595119903

01

2119889minus119903119897 ሶ120595119897 + 119903119903 ሶ120595119903


Differential Drive Kinematics of MARVIN

rrlabcsuni-klde

CROMSCI ndash Equipped with 3 Steerable Standard Wheels


Climbing robot CROMSCI Wheel settings of CROMSCI

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Omnidirectional Drive Kinematics


Typical orientations of the 3 steerable wheels of an omnidirectional vehicle

rrlabcsuni-kldeAMR ndash Kinematicsg 59


Three steerable standard wheels

Kinematic center located in the middle of the robot


Origin of coordinates frame is kinematic center


1205721 = 0 deg

1205722 = 120 deg

1205723 = minus120 deg



Applying equation for steerable standard wheel for each wheel leads to the following system of equations

119904 1205721 + 1205731 ሶ119909 minus 119888 1205721 + 1205731 ሶ119910 minus 119889 sdot 119888 1205731 ሶ120579 = 1199031 ሶ1205951



119888 1205721 + 1205731 ሶ119909 + 119904 1205721 + 1205731 ሶ119910 + 119889 sdot 119904 1205731 ሶ120579 = 0

119888 1205722 + 1205732 ሶ119909 + 119904 1205722 + 1205732 ሶ119910 + 119889 sdot 119904 1205732 ሶ120579 = 0

119888 1205723 + 1205733 ሶ119909 + 119904 1205723 + 1205733 ሶ119910 + 119889 sdot 119904 1205733 ሶ120579 = 0



Based on these equation the steering angles 120573119894 119894 = 123 can be determined

119888 120572119894 + 120573119894 sdot ሶ119909 + 119904 120572119894 + 120573119894 sdot ሶ119910 + 119889 sdot 119904 120573119894 sdot ሶ120579 = 0

119888 120572119894 sdot 119888 120573119894 sdot ሶ119909 minus 119904 120572119894 sdot 119904 120573119894 sdot ሶ119909 + 119904 120572119894 sdot 119888 120573119894 sdot ሶ119910 + 119888 120572119894 sdot 119904 120573119894 sdot ሶ119910 + 119889 sdot 119904 120573119894 sdot ሶ120579 = 0

119888 120573119894 sdot 119888 120572119894 sdot ሶ119909 + 119904 120572119894 sdot ሶ119910 minus 119904 120573119894 sdot 119904 120572119894 sdot ሶ119909 minus 119888 120572119894 sdot ሶ119910 minus 119889 sdot ሶ120579 = 0

119888 120572119894 sdot ሶ119909 + 119904 120572119894 sdot ሶ119910

119904 120572119894 sdot ሶ119909 minus 119888 120572119894 sdot ሶ119910 minus 119889 sdot ሶ120579=119904 120573119894119888 120573119894

atan2 119888 120572119894 sdot ሶ119909 + 119904 120572119894 sdot ሶ119910 119904 120572119894 sdot ሶ119909 minus 119888 120572119894 sdot ሶ119910 minus 119889 sdot ሶ120579 = 120573119894

Also the angular velocity of the wheel ሶ120595119894 can now be calculated

ሶ120595119894 =1

119903119894119904 120572119894 + 120573119894 ሶ119909 minus 119888 120572119894 + 120573119894 ሶ119910 minus 119889 sdot 119888 120573119894 ሶ120579

rrlabcsuni-klde

The Climbing Robot CROMSCI


rrlabcsuni-klde

Mecanum Vehicle


Schematic configurationof a Mecanum wheel(Viewed from above)

The mobile robot PRIAMOS of the University of Karlsruhe driven by Mecanum wheels


Mecanum Kinematics

To set up the kinematic equation first the parameters 120572 120573 120574for each wheel must be determined

The parameters for four wheels are

1205721 = 45 deg 1205731 = 45 deg 1205741 = minus45 deg

1205722 = 135 deg 1205732 = minus45 deg 1205742 = 45 deg

1205723 = minus135 deg 1205733 = 225 deg 1205743 = minus45 deg

1205724 = minus45 deg 1205734 = 135 deg 1205744 = 45 deg


Mecanum Kinematics

Assume all driven wheels have the same radius 119903 the same distance 119889 from the kinematic center and the above mentioned parameters 120572 120573 120574

Using equation for Mecanum wheels leads to equations

119904 45 deg ሶ119909 minus 119888 45 deg ሶ119910 minus 119889 ሶ120579 = 119903 sdot 119888 minus45 deg ሶ1205951

119904 135 deg ሶ119909 minus 119888 135 deg ሶ119910 minus 119889 ሶ120579 = 119903 sdot 119888 45 deg ሶ1205952




Mecanum Kinematics

The velocities ሶ119909 ሶ119910 ሶ120579 of the kinematic center can be calculated

ሶ119909 =119903

4ሶ1205951 + ሶ1205952 + ሶ1205953 + ሶ1205954

ሶ119910 =119903

4minus ሶ1205951 + ሶ1205952 minus ሶ1205953 + ሶ1205954

ሶ120579 =119903

119889 2ሶ1205951 minus ሶ1205952 minus ሶ1205953 + ሶ1205954


Mecanum Kinematics

The velocity vector of the kinematic center can be determined as

ሶ119909ሶ119910ሶ120579

=119903119908ℎ1198901198901198974

sdot1 1 1 1minus1 1 minus1 1119886 minus119886 minus119886 119886

sdot

ሶ1205951

ሶ1205952

ሶ1205953

ሶ1205954

with a =2 2

119889

Assuming the absence of slip the following must hold

ሶ1205954 = ሶ1205951 + ሶ1205952 minus ሶ1205953

rrlabcsuni-klde


rrlabcsuni-kldeAMR ndash Maneuverability 69

Choice of Wheel Configuration

Choice of wheels and their configuration depends on hellip

Ability to maneuver

Controllability

Stability

An optimum can only be achieved concerning a specific application


Stability

Minimal number of wheels for static stability is 2 if center of gravity is located under the wheel axis (e g robot Cye)

Under standard conditions itrsquos 3

If more than 3 wheels are present adjustment to rough terrain becomes necessary

Typical setups

2 driven wheels on a single axis with 1 or 2 passive castor wheels

Single driven and steered wheel with two passive fixed wheels


Maneuverability

Robotrsquos degree of freedom (DOF)

Outstanding ability to maneuver when the only driven wheel can be rotated in an active way

Example Ackermann steering

Poor ability to maneuver

5 m radius required for single rotation of a car

Lateral movement only possible troughforwardbackward movement


Controllability

Costs for locomotion of the robot on a given trajectory

Contradiction to the robotrsquos ability to maneuver

Examples

Ackermann steering easy to control

Mecanum wheels very costly if high precision is required

Costs for controllability depend on hellip

Wheel type

Drive geometry

Sensors to determine driving situation

Motortransmission


Degree of Mobility

The degree of mobility 119891119898 isin 123 points out how the sliding constraints (defined in Matrix 119878 see slide 42) affect the

possibilities for robot movement 119877 120579 ሶ120585119897 The mobility can be calculated by using the rank of the

matrix of sliding constraints of all standard wheels (steerable and fixed wheels) = number of independent constraints

119891119898 = 3 minus rank 119878


Degree of Mobility


Given only two fixed wheels geometrically we can set

1198891 = 1198892 1205731 = 1205732 = 0 1205721 + 120587 = 1205722

Matrix 119878 has two sliding constraints of the form (119888(120572 +


Degree of Steerability

The degree of steerability 119891119904 isin 012 quantifies the degree of controllable freedom and is determined via the number of independently controllable steering parameters

Define 119878prime Part of 119878 defining sliding constraints of steerable wheels

119891119904 = rank 119878prime

More possibilities in steering will result in lower mobility

119891119904 = 0 No steered wheel (differential drive)

119891119904 = 1 At least one steered wheel (Ackermann steering)

119891119904 = 2 At least two steered wheels no fixed wheel


Degree of Maneuverability

The degree of maneuverability 119891119872 isin 23 points out the number of DOF the robot can manipulate

119891119872 = 119891119898 + 119891119904

rrlabcsuni-klde

Vehicle Kinematics

Geometrical Solution


Tricycle Drive

Given 120575 119907

Unknown 119907prime 119907119897 119907119903 Solution

119903 = 119897 sdot cot 120575

119907prime =119907

cos 120575

119907119897 = 119907 sdot119903 + 119889

119903

119907119903 = 119907 sdot119903 minus 119889

119903 Tricycle drive kinematics

120575

rrlabcsuni-kldeAMR ndash 4 Kinematics 79

Ackermann Steering

Given 119907119863 120575 119897 119889

Unknown 119907119903119903 119907119897119903 119907119903119891 119907119897119891

Solution

120602 =119897

tan 120575

119907119897119903 =120602 minus 1198892 sdot 119907119863

120602

119907119903119903 =120602 + 1198892 sdot 119907119863

120602

119907119897119891 =120602 minus 1198892 2 + 1198972 sdot tan 120575

119897sdot 119907119863

119907119903119891 =1206022 + 1198972 sdot tan 120575

119897sdot 119907119863

Kinematics ofAckermann steering

120575

120602

rrlabcsuni-klde

Omnidrive Kinematics


Given 120602 120593 119907 119889

Unknown 1205931 1205932 1199071 1199072 Solution

120549119909 = 120602 sdot cos120593120549119910 = 120602 sdot sin1205931205931 = arctan120549119910 minus 1198891205491199091206021 = 120602 sdot cos120593 cos12059311199071 = 119907 sdot 1206021120602

1205932 = arctan120549119910 + 119889

1205491199091206022 = 120602 sdot cos120593 cos1205932

1199072 = 119907 sdot 1206022120602

120549119910 = 120602 ∙ 119904119894119899 120593

1205931

1205932

1206021

1206022

120593 120549119909 = 120602 ∙ cos120593

120602

119889

rrlabcsuni-klde

Bicycle Model


119907


Bicycle Model

Circular road of radius 120602

Low speed motion

Slip angles at both wheels are zero

Small total lateral force from both tires

119865119910 =1198981199072

120602

Point 119875 is instantaneous rolling center

Velocity at 119862 is perpendicular to line 119875119862

Heading angle 120579

Slip angle 120575119907 Course angle 120574 = 120579 + 120575119907


Bicycle Model

Given 120575119891 120575119903 119907 120579

Unknown ሶ119883 ሶ119884 ሶ120579

Solution

120575119907 = tanminus1119897119891 tan 120575119903+119897119903 tan 120575119891

119897119891+119897119903

ሶ120579 =119907

120602

120602 =119897119891+119897119903

tan 120575119891 minustan 120575119903 cos 120575119907

ሶ119883 = 119907 cos 120579 + 120575119907ሶ119884 = 119907 sin 120579 + 120575119907

ሶ120579 =119907 cos 120575119907

119897119891+119897119903tan 120575119891 minus tan 120575119903

119907

rrlabcsuni-klde

Ackermann Steering -gt Bicycle Model


119862 = 119874 120575119903 = 0 120575119891 = 120575

119897119903 = 0 119897119891 = 119897

120575119907 = 0

ሶ120579 =119907

120602

120602 =119897

tan 120575119891

ሶ119883 = 119907 cos 120579ሶ119884 = 119907 sin 120579ሶ120579 =

119907

119897119891tan 120575119891

119907

rrlabcsuni-klde

KaiserslauternGermany

Kaiserslautern

Frankfurt

Berlin

Hamburg

Bremen

Hanover

Cologne

Stuttgart

Munich

Braunschweig

Federal Republic of Germany

bull Deutschland

bull 16 constituent states

bull area of 357386 square

kilometres

bull (137988 sq mi)

bull 83 million inhabitants

bull Capital Berlin

(52deg31primeN 13deg23primeE)

rrlabcsuni-klde


gt 14500 Students

gt 2000 Staff



rrlabcsuni-klde


rrlabcsuni-klde



Professors 12

Senior Engineers 5

ZNT Statistics


rrlabcsuni-klde




Hu

ma

no

id

Ro

bo

ts

Off-r

oa

d R

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Serv

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rrlabcsuni-klde

Off-road Projekte


rrlabcsuni-klde


rrlabcsuni-klde

rrlabcsuni-klde

Contents


Vehicle Kinematics

Analytical Solution

Geometric Solution

Bicycle Model




rrlabcsuni-klde

Modeling of AMRs






polygon lines grids

boxes

AMR position can be


point in space

center of mass

kinematic center



Configuration Space



2D robot C-space


Configuration Space







2D robot c-space


Configuration Space





rrlabcsuni-klde

Coordinate Systems









119861119906 =119909119910119911

+ 119874119861119877 120593120595 120579 119874 119906





119877 119906Ԧ119901119879 119904










119877119910 120595 =cos120595 0 sin1205950 1 0



120593 119910

119907

119911119908





119877 = 119877119911 120593 sdot 119877119909 120595 sdot 119877119911prime θ


119877 = 119877119911 θ sdot 119877119910 120595 sdot 119877119909 120593


Roll Pitch Yaw




Yaw θ

Pitchψ

Roll 120593

119911

119910119909

20

Roll

Pitch Yaw

119909

119911119910






119877119904 =



Linear Velocity




119860119877119861 Ԧ119907119902


119860 Ԧ119907119902 =119860 Ԧ119907119874119861 + 119861

119860119877119861 Ԧ119907119902



calculated with

119860 Ԧ119907119876 = 119860 Ԧ119907119874119861 + 119861119860119877119861 Ԧ119907119876 +

119860ω119861 times 119861119860119877119861 Ԧ119902


Rotational Velocity



119860120596 = 119861119860119877119861120596





119894+1120596119894+1 = 119894119894+1119877 sdot 119894120596119894 + ሶθ119894+1

119894+1 Ԧ119890119911119894+1

119894+1 Ԧ119907119894+1 = 119894119894+1119877 119894 Ԧ119907119894 +

119894120596119894 times119894 Ԧ119901119894+1






119894119877 from frame 119894 to

119894 + 1





119894119894+1119877 θ = 119894+1

119894119877minus1 θ = 119894+1119894119877119879 θ =

119888θ 119904θ 119909119908minus119904θ 119888θ 1199101199080 0 1

rrlabcsuni-klde



YW

XW

θ

YR

XR

P









coordinates





0 0 1


ሶԦ120585119877 = 119877 θሶԦ120585119882


YW

XW

θ

YR

XR




119877120587

2=

0 1 0minus1 0 00 0 1


ሶԦ120585119877 = 119877120587

2ሶ

120585119882 =0 1 0minus1 0 00 0 1


=ሶ119910


rrlabcsuni-klde


Analytical Solution






YW

XW

θ

V(t)

120596(t)







ሶԦ120585119882 = ሶ119909 ሶ119910 ሶ120579119879

= 119891 119889 119903 120579 ሶ120595119897 ሶ120595119903

= 119877 120579 minus1 ሶԦ120585119877

passive wheel

active wheel

d






ሶ119909119877119903 =119903 ሶ1205951199032

ሶ119909119877119897 =119903 ሶ120595119897

2

120596119903 =119903 ሶ1205951199032119889

120596119897 =minus119903 ሶ120595119897

2119889

ሶԦ120585119882 = 119877 120579 minus1 ሶԦ120585119903ሶԦ120585119903 =

119903 ሶ1205951199032

+119903 ሶ120595119897

2 0

119903 ሶ1205951199032119889

minus119903 ሶ120595119897

2119889

119879





Given 120579 =120587



119904

ሶԦ120585119882 =ሶ119909ሶ119910ሶ120579

=0 minus1 01 0 00 0 1

sdot301

=031







Consider




rrlabcsuni-klde

Basic Wheel Types


Standardwheel

Steerablestandard

wheel

Castorwheel

Swedish orMecanum

wheel

Ball orspherical

wheel


Wheel Properties




Castor wheel


















ሶ120595 119903



ሶ120595 119903



ሶ120595 119903



ሶ120595 119903



ሶ120595 119903








and 1 Ԧ1199071 = ሶ119909 ሶ119910 0 119879


119894+1120596119894+1 = 119894119894+1119877 sdot 119894120596119894 + ሶ120579119894+1

119894+1 Ԧ119890119911119894+1

119894+1 Ԧ119907119894+1 = 119894119894+1119877 119894 Ԧ119907119894 +

119894120596119894 times119894 Ԧ119901119894+1




calculated as

2 Ԧ1199072 =119888120572 ሶ119909 + 119904120572 ሶ119910minus119904120572 ሶ119909 + 119888120572 ሶ119910

0


3 Ԧ1199073 =

119888120572 ሶ119909 + 119904120572 ሶ119910

minus119904120572 ሶ119909 + 119888120572 ሶ119910 + 119889 ሶ1205790






4 Ԧ1199074 =119904 120572 + 120573 ሶ119909 minus 119888 120572 + 120573 ሶ119910 minus 119888120573 119889 ሶ120579

119888 120572 + 120573 ሶ119909 + 119904 120572 + 120573 ሶ119910 + 119904120573 119889 ሶ1205790




119904 120572 + 120573 minus119888 120572 + 120573 minus119888120573 119889

119888 120572 + 120573 119904 120572 + 120573 119904120573 1198890 0 0

ሶ119909119910 ሶሶ120579

=119903 ሶ12059500












119904 120572 + 120573(119905) minus119888 120572 + 120573(119905) minus119888120573(119905)119889

119888 120572 + 120573(119905) 119904 120572 + 120573(119905) 119904120573(119905)1198890 0 0

ሶ119909ሶ119910ሶ120579

=119903 ሶ120595

minus119889119888 ሶ120573(119905)0


maneuverability)




119904 120572 + 120573 + 120574 minus119888 120572 + 120573 + 120574 minus119888 120573 + 120574 119889

119888 120572 + 120573 + 120574 119904 120572 + 120573 + 120574 119904 120573 + 120574 1198890 0 0

ሶ119909ሶ119910ሶ120579

=119903 ሶ120595 cos 120574

119903 ሶ120595 sin 120574 + 119903119901119903 ሶ1205951199011199030















120572119897 = 90 deg

120573119897 = 0 deg

120572119903 = minus90 deg

120573119903 = 180 deg




119904 120572119897 + 120573119897 ሶ119909 minus 119888 120572119897 + 120573119897 ሶ119910 minus 119888120573119897 119889 ሶ120579 = 119903119897 ሶ120595119897

119888 120572119897 + 120573119897 ሶ119909 + 119904 120572119897 + 120573119897 ሶ119910 + 119904120573119897 119889 ሶ120579 = 0



ሶ119909 minus 119889 ሶ120579 = 119903119897 ሶ120595119897

ሶ119910 = 0

ሶ119909 + 119889 ሶ120579 = 119903119903 ሶ120595119903119910 = 0




ሶ119909ሶ119910ሶ120579

=

1

2119903119897 ሶ120595119897 + 119903119903 ሶ120595119903

01

2119889minus119903119897 ሶ120595119897 + 119903119903 ሶ120595119903



rrlabcsuni-klde




rrlabcsuni-klde











1205721 = 0 deg

1205722 = 120 deg

1205723 = minus120 deg







119888 1205721 + 1205731 ሶ119909 + 119904 1205721 + 1205731 ሶ119910 + 119889 sdot 119904 1205731 ሶ120579 = 0

119888 1205722 + 1205732 ሶ119909 + 119904 1205722 + 1205732 ሶ119910 + 119889 sdot 119904 1205732 ሶ120579 = 0

119888 1205723 + 1205733 ሶ119909 + 119904 1205723 + 1205733 ሶ119910 + 119889 sdot 119904 1205733 ሶ120579 = 0







119888 120572119894 sdot ሶ119909 + 119904 120572119894 sdot ሶ119910




ሶ120595119894 =1


rrlabcsuni-klde



rrlabcsuni-klde

Mecanum Vehicle





Mecanum Kinematics



1205721 = 45 deg 1205731 = 45 deg 1205741 = minus45 deg

1205722 = 135 deg 1205732 = minus45 deg 1205742 = 45 deg


1205724 = minus45 deg 1205734 = 135 deg 1205744 = 45 deg


Mecanum Kinematics








Mecanum Kinematics


ሶ119909 =119903

4ሶ1205951 + ሶ1205952 + ሶ1205953 + ሶ1205954

ሶ119910 =119903

4minus ሶ1205951 + ሶ1205952 minus ሶ1205953 + ሶ1205954

ሶ120579 =119903

119889 2ሶ1205951 minus ሶ1205952 minus ሶ1205953 + ሶ1205954


Mecanum Kinematics


ሶ119909ሶ119910ሶ120579

=119903119908ℎ1198901198901198974


sdot

ሶ1205951

ሶ1205952

ሶ1205953

ሶ1205954

with a =2 2

119889


ሶ1205954 = ሶ1205951 + ሶ1205952 minus ሶ1205953

rrlabcsuni-klde





Ability to maneuver

Controllability

Stability



Stability




Typical setups




Maneuverability








Controllability



Examples




Wheel type

Drive geometry


Motortransmission


Degree of Mobility




119891119898 = 3 minus rank 119878


Degree of Mobility



1198891 = 1198892 1205731 = 1205732 = 0 1205721 + 120587 = 1205722






119891119904 = rank 119878prime








119891119872 = 119891119898 + 119891119904

rrlabcsuni-klde

Vehicle Kinematics



Tricycle Drive

Given 120575 119907


119903 = 119897 sdot cot 120575

119907prime =119907

cos 120575

119907119897 = 119907 sdot119903 + 119889

119903

119907119903 = 119907 sdot119903 minus 119889


120575


Ackermann Steering

Given 119907119863 120575 119897 119889

Unknown 119907119903119903 119907119897119903 119907119903119891 119907119897119891

Solution

120602 =119897

tan 120575

119907119897119903 =120602 minus 1198892 sdot 119907119863

120602

119907119903119903 =120602 + 1198892 sdot 119907119863

120602

119907119897119891 =120602 minus 1198892 2 + 1198972 sdot tan 120575

119897sdot 119907119863

119907119903119891 =1206022 + 1198972 sdot tan 120575

119897sdot 119907119863


120575

120602

rrlabcsuni-klde



Given 120602 120593 119907 119889



1205932 = arctan120549119910 + 119889

1205491199091206022 = 120602 sdot cos120593 cos1205932

1199072 = 119907 sdot 1206022120602

120549119910 = 120602 ∙ 119904119894119899 120593

1205931

1205932

1206021

1206022

120593 120549119909 = 120602 ∙ cos120593

120602

119889

rrlabcsuni-klde

Bicycle Model


119907


Bicycle Model


Low speed motion



119865119910 =1198981199072

120602






Bicycle Model

Given 120575119891 120575119903 119907 120579

Unknown ሶ119883 ሶ119884 ሶ120579

Solution

120575119907 = tanminus1119897119891 tan 120575119903+119897119903 tan 120575119891

119897119891+119897119903

ሶ120579 =119907

120602

120602 =119897119891+119897119903

tan 120575119891 minustan 120575119903 cos 120575119907

ሶ119883 = 119907 cos 120579 + 120575119907ሶ119884 = 119907 sin 120579 + 120575119907

ሶ120579 =119907 cos 120575119907

119897119891+119897119903tan 120575119891 minus tan 120575119903

119907

rrlabcsuni-klde



119862 = 119874 120575119903 = 0 120575119891 = 120575

119897119903 = 0 119897119891 = 119897

120575119907 = 0

ሶ120579 =119907

120602

120602 =119897

tan 120575119891

ሶ119883 = 119907 cos 120579ሶ119884 = 119907 sin 120579ሶ120579 =

119907

119897119891tan 120575119891

119907

rrlabcsuni-klde


gt 14500 Students

gt 2000 Staff



rrlabcsuni-klde


rrlabcsuni-klde



Professors 12

Senior Engineers 5

ZNT Statistics


rrlabcsuni-klde




Hu

ma

no

id

Ro

bo

ts

Off-r

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d R

obo

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Serv

ic

e R

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rrlabcsuni-klde

Off-road Projekte


rrlabcsuni-klde


rrlabcsuni-klde

rrlabcsuni-klde

Contents


Vehicle Kinematics

Analytical Solution

Geometric Solution

Bicycle Model




rrlabcsuni-klde

Modeling of AMRs






polygon lines grids

boxes

AMR position can be


point in space

center of mass

kinematic center



Configuration Space



2D robot C-space


Configuration Space







2D robot c-space


Configuration Space





rrlabcsuni-klde

Coordinate Systems









119861119906 =119909119910119911

+ 119874119861119877 120593120595 120579 119874 119906





119877 119906Ԧ119901119879 119904










119877119910 120595 =cos120595 0 sin1205950 1 0



120593 119910

119907

119911119908





119877 = 119877119911 120593 sdot 119877119909 120595 sdot 119877119911prime θ


119877 = 119877119911 θ sdot 119877119910 120595 sdot 119877119909 120593


Roll Pitch Yaw




Yaw θ

Pitchψ

Roll 120593

119911

119910119909

20

Roll

Pitch Yaw

119909

119911119910






119877119904 =



Linear Velocity




119860119877119861 Ԧ119907119902


119860 Ԧ119907119902 =119860 Ԧ119907119874119861 + 119861

119860119877119861 Ԧ119907119902



calculated with

119860 Ԧ119907119876 = 119860 Ԧ119907119874119861 + 119861119860119877119861 Ԧ119907119876 +

119860ω119861 times 119861119860119877119861 Ԧ119902


Rotational Velocity



119860120596 = 119861119860119877119861120596





119894+1120596119894+1 = 119894119894+1119877 sdot 119894120596119894 + ሶθ119894+1

119894+1 Ԧ119890119911119894+1

119894+1 Ԧ119907119894+1 = 119894119894+1119877 119894 Ԧ119907119894 +

119894120596119894 times119894 Ԧ119901119894+1






119894119877 from frame 119894 to

119894 + 1





119894119894+1119877 θ = 119894+1

119894119877minus1 θ = 119894+1119894119877119879 θ =

119888θ 119904θ 119909119908minus119904θ 119888θ 1199101199080 0 1

rrlabcsuni-klde



YW

XW

θ

YR

XR

P









coordinates





0 0 1


ሶԦ120585119877 = 119877 θሶԦ120585119882


YW

XW

θ

YR

XR




119877120587

2=

0 1 0minus1 0 00 0 1


ሶԦ120585119877 = 119877120587

2ሶ

120585119882 =0 1 0minus1 0 00 0 1


=ሶ119910


rrlabcsuni-klde


Analytical Solution






YW

XW

θ

V(t)

120596(t)







ሶԦ120585119882 = ሶ119909 ሶ119910 ሶ120579119879

= 119891 119889 119903 120579 ሶ120595119897 ሶ120595119903

= 119877 120579 minus1 ሶԦ120585119877

passive wheel

active wheel

d






ሶ119909119877119903 =119903 ሶ1205951199032

ሶ119909119877119897 =119903 ሶ120595119897

2

120596119903 =119903 ሶ1205951199032119889

120596119897 =minus119903 ሶ120595119897

2119889

ሶԦ120585119882 = 119877 120579 minus1 ሶԦ120585119903ሶԦ120585119903 =

119903 ሶ1205951199032

+119903 ሶ120595119897

2 0

119903 ሶ1205951199032119889

minus119903 ሶ120595119897

2119889

119879





Given 120579 =120587



119904

ሶԦ120585119882 =ሶ119909ሶ119910ሶ120579

=0 minus1 01 0 00 0 1

sdot301

=031







Consider




rrlabcsuni-klde

Basic Wheel Types


Standardwheel

Steerablestandard

wheel

Castorwheel

Swedish orMecanum

wheel

Ball orspherical

wheel


Wheel Properties




Castor wheel


















ሶ120595 119903



ሶ120595 119903



ሶ120595 119903



ሶ120595 119903



ሶ120595 119903








and 1 Ԧ1199071 = ሶ119909 ሶ119910 0 119879


119894+1120596119894+1 = 119894119894+1119877 sdot 119894120596119894 + ሶ120579119894+1

119894+1 Ԧ119890119911119894+1

119894+1 Ԧ119907119894+1 = 119894119894+1119877 119894 Ԧ119907119894 +

119894120596119894 times119894 Ԧ119901119894+1




calculated as

2 Ԧ1199072 =119888120572 ሶ119909 + 119904120572 ሶ119910minus119904120572 ሶ119909 + 119888120572 ሶ119910

0


3 Ԧ1199073 =

119888120572 ሶ119909 + 119904120572 ሶ119910

minus119904120572 ሶ119909 + 119888120572 ሶ119910 + 119889 ሶ1205790






4 Ԧ1199074 =119904 120572 + 120573 ሶ119909 minus 119888 120572 + 120573 ሶ119910 minus 119888120573 119889 ሶ120579

119888 120572 + 120573 ሶ119909 + 119904 120572 + 120573 ሶ119910 + 119904120573 119889 ሶ1205790




119904 120572 + 120573 minus119888 120572 + 120573 minus119888120573 119889

119888 120572 + 120573 119904 120572 + 120573 119904120573 1198890 0 0

ሶ119909119910 ሶሶ120579

=119903 ሶ12059500












119904 120572 + 120573(119905) minus119888 120572 + 120573(119905) minus119888120573(119905)119889

119888 120572 + 120573(119905) 119904 120572 + 120573(119905) 119904120573(119905)1198890 0 0

ሶ119909ሶ119910ሶ120579

=119903 ሶ120595

minus119889119888 ሶ120573(119905)0


maneuverability)




119904 120572 + 120573 + 120574 minus119888 120572 + 120573 + 120574 minus119888 120573 + 120574 119889

119888 120572 + 120573 + 120574 119904 120572 + 120573 + 120574 119904 120573 + 120574 1198890 0 0

ሶ119909ሶ119910ሶ120579

=119903 ሶ120595 cos 120574

119903 ሶ120595 sin 120574 + 119903119901119903 ሶ1205951199011199030















120572119897 = 90 deg

120573119897 = 0 deg

120572119903 = minus90 deg

120573119903 = 180 deg




119904 120572119897 + 120573119897 ሶ119909 minus 119888 120572119897 + 120573119897 ሶ119910 minus 119888120573119897 119889 ሶ120579 = 119903119897 ሶ120595119897

119888 120572119897 + 120573119897 ሶ119909 + 119904 120572119897 + 120573119897 ሶ119910 + 119904120573119897 119889 ሶ120579 = 0



ሶ119909 minus 119889 ሶ120579 = 119903119897 ሶ120595119897

ሶ119910 = 0

ሶ119909 + 119889 ሶ120579 = 119903119903 ሶ120595119903119910 = 0




ሶ119909ሶ119910ሶ120579

=

1

2119903119897 ሶ120595119897 + 119903119903 ሶ120595119903

01

2119889minus119903119897 ሶ120595119897 + 119903119903 ሶ120595119903



rrlabcsuni-klde




rrlabcsuni-klde











1205721 = 0 deg

1205722 = 120 deg

1205723 = minus120 deg







119888 1205721 + 1205731 ሶ119909 + 119904 1205721 + 1205731 ሶ119910 + 119889 sdot 119904 1205731 ሶ120579 = 0

119888 1205722 + 1205732 ሶ119909 + 119904 1205722 + 1205732 ሶ119910 + 119889 sdot 119904 1205732 ሶ120579 = 0

119888 1205723 + 1205733 ሶ119909 + 119904 1205723 + 1205733 ሶ119910 + 119889 sdot 119904 1205733 ሶ120579 = 0







119888 120572119894 sdot ሶ119909 + 119904 120572119894 sdot ሶ119910




ሶ120595119894 =1


rrlabcsuni-klde



rrlabcsuni-klde

Mecanum Vehicle





Mecanum Kinematics



1205721 = 45 deg 1205731 = 45 deg 1205741 = minus45 deg

1205722 = 135 deg 1205732 = minus45 deg 1205742 = 45 deg


1205724 = minus45 deg 1205734 = 135 deg 1205744 = 45 deg


Mecanum Kinematics








Mecanum Kinematics


ሶ119909 =119903

4ሶ1205951 + ሶ1205952 + ሶ1205953 + ሶ1205954

ሶ119910 =119903

4minus ሶ1205951 + ሶ1205952 minus ሶ1205953 + ሶ1205954

ሶ120579 =119903

119889 2ሶ1205951 minus ሶ1205952 minus ሶ1205953 + ሶ1205954


Mecanum Kinematics


ሶ119909ሶ119910ሶ120579

=119903119908ℎ1198901198901198974


sdot

ሶ1205951

ሶ1205952

ሶ1205953

ሶ1205954

with a =2 2

119889


ሶ1205954 = ሶ1205951 + ሶ1205952 minus ሶ1205953

rrlabcsuni-klde





Ability to maneuver

Controllability

Stability



Stability




Typical setups




Maneuverability








Controllability



Examples




Wheel type

Drive geometry


Motortransmission


Degree of Mobility




119891119898 = 3 minus rank 119878


Degree of Mobility



1198891 = 1198892 1205731 = 1205732 = 0 1205721 + 120587 = 1205722






119891119904 = rank 119878prime








119891119872 = 119891119898 + 119891119904

rrlabcsuni-klde

Vehicle Kinematics



Tricycle Drive

Given 120575 119907


119903 = 119897 sdot cot 120575

119907prime =119907

cos 120575

119907119897 = 119907 sdot119903 + 119889

119903

119907119903 = 119907 sdot119903 minus 119889


120575


Ackermann Steering

Given 119907119863 120575 119897 119889

Unknown 119907119903119903 119907119897119903 119907119903119891 119907119897119891

Solution

120602 =119897

tan 120575

119907119897119903 =120602 minus 1198892 sdot 119907119863

120602

119907119903119903 =120602 + 1198892 sdot 119907119863

120602

119907119897119891 =120602 minus 1198892 2 + 1198972 sdot tan 120575

119897sdot 119907119863

119907119903119891 =1206022 + 1198972 sdot tan 120575

119897sdot 119907119863


120575

120602

rrlabcsuni-klde



Given 120602 120593 119907 119889



1205932 = arctan120549119910 + 119889

1205491199091206022 = 120602 sdot cos120593 cos1205932

1199072 = 119907 sdot 1206022120602

120549119910 = 120602 ∙ 119904119894119899 120593

1205931

1205932

1206021

1206022

120593 120549119909 = 120602 ∙ cos120593

120602

119889

rrlabcsuni-klde

Bicycle Model


119907


Bicycle Model


Low speed motion



119865119910 =1198981199072

120602






Bicycle Model

Given 120575119891 120575119903 119907 120579

Unknown ሶ119883 ሶ119884 ሶ120579

Solution

120575119907 = tanminus1119897119891 tan 120575119903+119897119903 tan 120575119891

119897119891+119897119903

ሶ120579 =119907

120602

120602 =119897119891+119897119903

tan 120575119891 minustan 120575119903 cos 120575119907

ሶ119883 = 119907 cos 120579 + 120575119907ሶ119884 = 119907 sin 120579 + 120575119907

ሶ120579 =119907 cos 120575119907

119897119891+119897119903tan 120575119891 minus tan 120575119903

119907

rrlabcsuni-klde



119862 = 119874 120575119903 = 0 120575119891 = 120575

119897119903 = 0 119897119891 = 119897

120575119907 = 0

ሶ120579 =119907

120602

120602 =119897

tan 120575119891

ሶ119883 = 119907 cos 120579ሶ119884 = 119907 sin 120579ሶ120579 =

119907

119897119891tan 120575119891

119907

rrlabcsuni-klde


rrlabcsuni-klde



Professors 12

Senior Engineers 5

ZNT Statistics


rrlabcsuni-klde




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rrlabcsuni-klde

Off-road Projekte


rrlabcsuni-klde


rrlabcsuni-klde

rrlabcsuni-klde

Contents


Vehicle Kinematics

Analytical Solution

Geometric Solution

Bicycle Model




rrlabcsuni-klde

Modeling of AMRs






polygon lines grids

boxes

AMR position can be


point in space

center of mass

kinematic center



Configuration Space



2D robot C-space


Configuration Space







2D robot c-space


Configuration Space





rrlabcsuni-klde

Coordinate Systems









119861119906 =119909119910119911

+ 119874119861119877 120593120595 120579 119874 119906





119877 119906Ԧ119901119879 119904










119877119910 120595 =cos120595 0 sin1205950 1 0



120593 119910

119907

119911119908





119877 = 119877119911 120593 sdot 119877119909 120595 sdot 119877119911prime θ


119877 = 119877119911 θ sdot 119877119910 120595 sdot 119877119909 120593


Roll Pitch Yaw




Yaw θ

Pitchψ

Roll 120593

119911

119910119909

20

Roll

Pitch Yaw

119909

119911119910






119877119904 =



Linear Velocity




119860119877119861 Ԧ119907119902


119860 Ԧ119907119902 =119860 Ԧ119907119874119861 + 119861

119860119877119861 Ԧ119907119902



calculated with

119860 Ԧ119907119876 = 119860 Ԧ119907119874119861 + 119861119860119877119861 Ԧ119907119876 +

119860ω119861 times 119861119860119877119861 Ԧ119902


Rotational Velocity



119860120596 = 119861119860119877119861120596





119894+1120596119894+1 = 119894119894+1119877 sdot 119894120596119894 + ሶθ119894+1

119894+1 Ԧ119890119911119894+1

119894+1 Ԧ119907119894+1 = 119894119894+1119877 119894 Ԧ119907119894 +

119894120596119894 times119894 Ԧ119901119894+1






119894119877 from frame 119894 to

119894 + 1





119894119894+1119877 θ = 119894+1

119894119877minus1 θ = 119894+1119894119877119879 θ =

119888θ 119904θ 119909119908minus119904θ 119888θ 1199101199080 0 1

rrlabcsuni-klde



YW

XW

θ

YR

XR

P









coordinates





0 0 1


ሶԦ120585119877 = 119877 θሶԦ120585119882


YW

XW

θ

YR

XR




119877120587

2=

0 1 0minus1 0 00 0 1


ሶԦ120585119877 = 119877120587

2ሶ

120585119882 =0 1 0minus1 0 00 0 1


=ሶ119910


rrlabcsuni-klde


Analytical Solution






YW

XW

θ

V(t)

120596(t)







ሶԦ120585119882 = ሶ119909 ሶ119910 ሶ120579119879

= 119891 119889 119903 120579 ሶ120595119897 ሶ120595119903

= 119877 120579 minus1 ሶԦ120585119877

passive wheel

active wheel

d






ሶ119909119877119903 =119903 ሶ1205951199032

ሶ119909119877119897 =119903 ሶ120595119897

2

120596119903 =119903 ሶ1205951199032119889

120596119897 =minus119903 ሶ120595119897

2119889

ሶԦ120585119882 = 119877 120579 minus1 ሶԦ120585119903ሶԦ120585119903 =

119903 ሶ1205951199032

+119903 ሶ120595119897

2 0

119903 ሶ1205951199032119889

minus119903 ሶ120595119897

2119889

119879





Given 120579 =120587



119904

ሶԦ120585119882 =ሶ119909ሶ119910ሶ120579

=0 minus1 01 0 00 0 1

sdot301

=031







Consider




rrlabcsuni-klde

Basic Wheel Types


Standardwheel

Steerablestandard

wheel

Castorwheel

Swedish orMecanum

wheel

Ball orspherical

wheel


Wheel Properties




Castor wheel


















ሶ120595 119903



ሶ120595 119903



ሶ120595 119903



ሶ120595 119903



ሶ120595 119903








and 1 Ԧ1199071 = ሶ119909 ሶ119910 0 119879


119894+1120596119894+1 = 119894119894+1119877 sdot 119894120596119894 + ሶ120579119894+1

119894+1 Ԧ119890119911119894+1

119894+1 Ԧ119907119894+1 = 119894119894+1119877 119894 Ԧ119907119894 +

119894120596119894 times119894 Ԧ119901119894+1




calculated as

2 Ԧ1199072 =119888120572 ሶ119909 + 119904120572 ሶ119910minus119904120572 ሶ119909 + 119888120572 ሶ119910

0


3 Ԧ1199073 =

119888120572 ሶ119909 + 119904120572 ሶ119910

minus119904120572 ሶ119909 + 119888120572 ሶ119910 + 119889 ሶ1205790






4 Ԧ1199074 =119904 120572 + 120573 ሶ119909 minus 119888 120572 + 120573 ሶ119910 minus 119888120573 119889 ሶ120579

119888 120572 + 120573 ሶ119909 + 119904 120572 + 120573 ሶ119910 + 119904120573 119889 ሶ1205790




119904 120572 + 120573 minus119888 120572 + 120573 minus119888120573 119889

119888 120572 + 120573 119904 120572 + 120573 119904120573 1198890 0 0

ሶ119909119910 ሶሶ120579

=119903 ሶ12059500












119904 120572 + 120573(119905) minus119888 120572 + 120573(119905) minus119888120573(119905)119889

119888 120572 + 120573(119905) 119904 120572 + 120573(119905) 119904120573(119905)1198890 0 0

ሶ119909ሶ119910ሶ120579

=119903 ሶ120595

minus119889119888 ሶ120573(119905)0


maneuverability)




119904 120572 + 120573 + 120574 minus119888 120572 + 120573 + 120574 minus119888 120573 + 120574 119889

119888 120572 + 120573 + 120574 119904 120572 + 120573 + 120574 119904 120573 + 120574 1198890 0 0

ሶ119909ሶ119910ሶ120579

=119903 ሶ120595 cos 120574

119903 ሶ120595 sin 120574 + 119903119901119903 ሶ1205951199011199030















120572119897 = 90 deg

120573119897 = 0 deg

120572119903 = minus90 deg

120573119903 = 180 deg




119904 120572119897 + 120573119897 ሶ119909 minus 119888 120572119897 + 120573119897 ሶ119910 minus 119888120573119897 119889 ሶ120579 = 119903119897 ሶ120595119897

119888 120572119897 + 120573119897 ሶ119909 + 119904 120572119897 + 120573119897 ሶ119910 + 119904120573119897 119889 ሶ120579 = 0



ሶ119909 minus 119889 ሶ120579 = 119903119897 ሶ120595119897

ሶ119910 = 0

ሶ119909 + 119889 ሶ120579 = 119903119903 ሶ120595119903119910 = 0




ሶ119909ሶ119910ሶ120579

=

1

2119903119897 ሶ120595119897 + 119903119903 ሶ120595119903

01

2119889minus119903119897 ሶ120595119897 + 119903119903 ሶ120595119903



rrlabcsuni-klde




rrlabcsuni-klde











1205721 = 0 deg

1205722 = 120 deg

1205723 = minus120 deg







119888 1205721 + 1205731 ሶ119909 + 119904 1205721 + 1205731 ሶ119910 + 119889 sdot 119904 1205731 ሶ120579 = 0

119888 1205722 + 1205732 ሶ119909 + 119904 1205722 + 1205732 ሶ119910 + 119889 sdot 119904 1205732 ሶ120579 = 0

119888 1205723 + 1205733 ሶ119909 + 119904 1205723 + 1205733 ሶ119910 + 119889 sdot 119904 1205733 ሶ120579 = 0







119888 120572119894 sdot ሶ119909 + 119904 120572119894 sdot ሶ119910




ሶ120595119894 =1


rrlabcsuni-klde



rrlabcsuni-klde

Mecanum Vehicle





Mecanum Kinematics



1205721 = 45 deg 1205731 = 45 deg 1205741 = minus45 deg

1205722 = 135 deg 1205732 = minus45 deg 1205742 = 45 deg


1205724 = minus45 deg 1205734 = 135 deg 1205744 = 45 deg


Mecanum Kinematics








Mecanum Kinematics


ሶ119909 =119903

4ሶ1205951 + ሶ1205952 + ሶ1205953 + ሶ1205954

ሶ119910 =119903

4minus ሶ1205951 + ሶ1205952 minus ሶ1205953 + ሶ1205954

ሶ120579 =119903

119889 2ሶ1205951 minus ሶ1205952 minus ሶ1205953 + ሶ1205954


Mecanum Kinematics


ሶ119909ሶ119910ሶ120579

=119903119908ℎ1198901198901198974


sdot

ሶ1205951

ሶ1205952

ሶ1205953

ሶ1205954

with a =2 2

119889


ሶ1205954 = ሶ1205951 + ሶ1205952 minus ሶ1205953

rrlabcsuni-klde





Ability to maneuver

Controllability

Stability



Stability




Typical setups




Maneuverability








Controllability



Examples




Wheel type

Drive geometry


Motortransmission


Degree of Mobility




119891119898 = 3 minus rank 119878


Degree of Mobility



1198891 = 1198892 1205731 = 1205732 = 0 1205721 + 120587 = 1205722






119891119904 = rank 119878prime








119891119872 = 119891119898 + 119891119904

rrlabcsuni-klde

Vehicle Kinematics



Tricycle Drive

Given 120575 119907


119903 = 119897 sdot cot 120575

119907prime =119907

cos 120575

119907119897 = 119907 sdot119903 + 119889

119903

119907119903 = 119907 sdot119903 minus 119889


120575


Ackermann Steering

Given 119907119863 120575 119897 119889

Unknown 119907119903119903 119907119897119903 119907119903119891 119907119897119891

Solution

120602 =119897

tan 120575

119907119897119903 =120602 minus 1198892 sdot 119907119863

120602

119907119903119903 =120602 + 1198892 sdot 119907119863

120602

119907119897119891 =120602 minus 1198892 2 + 1198972 sdot tan 120575

119897sdot 119907119863

119907119903119891 =1206022 + 1198972 sdot tan 120575

119897sdot 119907119863


120575

120602

rrlabcsuni-klde



Given 120602 120593 119907 119889



1205932 = arctan120549119910 + 119889

1205491199091206022 = 120602 sdot cos120593 cos1205932

1199072 = 119907 sdot 1206022120602

120549119910 = 120602 ∙ 119904119894119899 120593

1205931

1205932

1206021

1206022

120593 120549119909 = 120602 ∙ cos120593

120602

119889

rrlabcsuni-klde

Bicycle Model


119907


Bicycle Model


Low speed motion



119865119910 =1198981199072

120602






Bicycle Model

Given 120575119891 120575119903 119907 120579

Unknown ሶ119883 ሶ119884 ሶ120579

Solution

120575119907 = tanminus1119897119891 tan 120575119903+119897119903 tan 120575119891

119897119891+119897119903

ሶ120579 =119907

120602

120602 =119897119891+119897119903

tan 120575119891 minustan 120575119903 cos 120575119907

ሶ119883 = 119907 cos 120579 + 120575119907ሶ119884 = 119907 sin 120579 + 120575119907

ሶ120579 =119907 cos 120575119907

119897119891+119897119903tan 120575119891 minus tan 120575119903

119907

rrlabcsuni-klde



119862 = 119874 120575119903 = 0 120575119891 = 120575

119897119903 = 0 119897119891 = 119897

120575119907 = 0

ሶ120579 =119907

120602

120602 =119897

tan 120575119891

ሶ119883 = 119907 cos 120579ሶ119884 = 119907 sin 120579ሶ120579 =

119907

119897119891tan 120575119891

119907

rrlabcsuni-klde



Professors 12

Senior Engineers 5

ZNT Statistics


rrlabcsuni-klde




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rrlabcsuni-klde

Off-road Projekte


rrlabcsuni-klde


rrlabcsuni-klde

rrlabcsuni-klde

Contents


Vehicle Kinematics

Analytical Solution

Geometric Solution

Bicycle Model




rrlabcsuni-klde

Modeling of AMRs






polygon lines grids

boxes

AMR position can be


point in space

center of mass

kinematic center



Configuration Space



2D robot C-space


Configuration Space







2D robot c-space


Configuration Space





rrlabcsuni-klde

Coordinate Systems









119861119906 =119909119910119911

+ 119874119861119877 120593120595 120579 119874 119906





119877 119906Ԧ119901119879 119904










119877119910 120595 =cos120595 0 sin1205950 1 0



120593 119910

119907

119911119908





119877 = 119877119911 120593 sdot 119877119909 120595 sdot 119877119911prime θ


119877 = 119877119911 θ sdot 119877119910 120595 sdot 119877119909 120593


Roll Pitch Yaw




Yaw θ

Pitchψ

Roll 120593

119911

119910119909

20

Roll

Pitch Yaw

119909

119911119910






119877119904 =



Linear Velocity




119860119877119861 Ԧ119907119902


119860 Ԧ119907119902 =119860 Ԧ119907119874119861 + 119861

119860119877119861 Ԧ119907119902



calculated with

119860 Ԧ119907119876 = 119860 Ԧ119907119874119861 + 119861119860119877119861 Ԧ119907119876 +

119860ω119861 times 119861119860119877119861 Ԧ119902


Rotational Velocity



119860120596 = 119861119860119877119861120596





119894+1120596119894+1 = 119894119894+1119877 sdot 119894120596119894 + ሶθ119894+1

119894+1 Ԧ119890119911119894+1

119894+1 Ԧ119907119894+1 = 119894119894+1119877 119894 Ԧ119907119894 +

119894120596119894 times119894 Ԧ119901119894+1






119894119877 from frame 119894 to

119894 + 1





119894119894+1119877 θ = 119894+1

119894119877minus1 θ = 119894+1119894119877119879 θ =

119888θ 119904θ 119909119908minus119904θ 119888θ 1199101199080 0 1

rrlabcsuni-klde



YW

XW

θ

YR

XR

P









coordinates





0 0 1


ሶԦ120585119877 = 119877 θሶԦ120585119882


YW

XW

θ

YR

XR




119877120587

2=

0 1 0minus1 0 00 0 1


ሶԦ120585119877 = 119877120587

2ሶ

120585119882 =0 1 0minus1 0 00 0 1


=ሶ119910


rrlabcsuni-klde


Analytical Solution






YW

XW

θ

V(t)

120596(t)







ሶԦ120585119882 = ሶ119909 ሶ119910 ሶ120579119879

= 119891 119889 119903 120579 ሶ120595119897 ሶ120595119903

= 119877 120579 minus1 ሶԦ120585119877

passive wheel

active wheel

d






ሶ119909119877119903 =119903 ሶ1205951199032

ሶ119909119877119897 =119903 ሶ120595119897

2

120596119903 =119903 ሶ1205951199032119889

120596119897 =minus119903 ሶ120595119897

2119889

ሶԦ120585119882 = 119877 120579 minus1 ሶԦ120585119903ሶԦ120585119903 =

119903 ሶ1205951199032

+119903 ሶ120595119897

2 0

119903 ሶ1205951199032119889

minus119903 ሶ120595119897

2119889

119879





Given 120579 =120587



119904

ሶԦ120585119882 =ሶ119909ሶ119910ሶ120579

=0 minus1 01 0 00 0 1

sdot301

=031







Consider




rrlabcsuni-klde

Basic Wheel Types


Standardwheel

Steerablestandard

wheel

Castorwheel

Swedish orMecanum

wheel

Ball orspherical

wheel


Wheel Properties




Castor wheel


















ሶ120595 119903



ሶ120595 119903



ሶ120595 119903



ሶ120595 119903



ሶ120595 119903








and 1 Ԧ1199071 = ሶ119909 ሶ119910 0 119879


119894+1120596119894+1 = 119894119894+1119877 sdot 119894120596119894 + ሶ120579119894+1

119894+1 Ԧ119890119911119894+1

119894+1 Ԧ119907119894+1 = 119894119894+1119877 119894 Ԧ119907119894 +

119894120596119894 times119894 Ԧ119901119894+1




calculated as

2 Ԧ1199072 =119888120572 ሶ119909 + 119904120572 ሶ119910minus119904120572 ሶ119909 + 119888120572 ሶ119910

0


3 Ԧ1199073 =

119888120572 ሶ119909 + 119904120572 ሶ119910

minus119904120572 ሶ119909 + 119888120572 ሶ119910 + 119889 ሶ1205790






4 Ԧ1199074 =119904 120572 + 120573 ሶ119909 minus 119888 120572 + 120573 ሶ119910 minus 119888120573 119889 ሶ120579

119888 120572 + 120573 ሶ119909 + 119904 120572 + 120573 ሶ119910 + 119904120573 119889 ሶ1205790




119904 120572 + 120573 minus119888 120572 + 120573 minus119888120573 119889

119888 120572 + 120573 119904 120572 + 120573 119904120573 1198890 0 0

ሶ119909119910 ሶሶ120579

=119903 ሶ12059500












119904 120572 + 120573(119905) minus119888 120572 + 120573(119905) minus119888120573(119905)119889

119888 120572 + 120573(119905) 119904 120572 + 120573(119905) 119904120573(119905)1198890 0 0

ሶ119909ሶ119910ሶ120579

=119903 ሶ120595

minus119889119888 ሶ120573(119905)0


maneuverability)




119904 120572 + 120573 + 120574 minus119888 120572 + 120573 + 120574 minus119888 120573 + 120574 119889

119888 120572 + 120573 + 120574 119904 120572 + 120573 + 120574 119904 120573 + 120574 1198890 0 0

ሶ119909ሶ119910ሶ120579

=119903 ሶ120595 cos 120574

119903 ሶ120595 sin 120574 + 119903119901119903 ሶ1205951199011199030















120572119897 = 90 deg

120573119897 = 0 deg

120572119903 = minus90 deg

120573119903 = 180 deg




119904 120572119897 + 120573119897 ሶ119909 minus 119888 120572119897 + 120573119897 ሶ119910 minus 119888120573119897 119889 ሶ120579 = 119903119897 ሶ120595119897

119888 120572119897 + 120573119897 ሶ119909 + 119904 120572119897 + 120573119897 ሶ119910 + 119904120573119897 119889 ሶ120579 = 0



ሶ119909 minus 119889 ሶ120579 = 119903119897 ሶ120595119897

ሶ119910 = 0

ሶ119909 + 119889 ሶ120579 = 119903119903 ሶ120595119903119910 = 0




ሶ119909ሶ119910ሶ120579

=

1

2119903119897 ሶ120595119897 + 119903119903 ሶ120595119903

01

2119889minus119903119897 ሶ120595119897 + 119903119903 ሶ120595119903



rrlabcsuni-klde




rrlabcsuni-klde











1205721 = 0 deg

1205722 = 120 deg

1205723 = minus120 deg







119888 1205721 + 1205731 ሶ119909 + 119904 1205721 + 1205731 ሶ119910 + 119889 sdot 119904 1205731 ሶ120579 = 0

119888 1205722 + 1205732 ሶ119909 + 119904 1205722 + 1205732 ሶ119910 + 119889 sdot 119904 1205732 ሶ120579 = 0

119888 1205723 + 1205733 ሶ119909 + 119904 1205723 + 1205733 ሶ119910 + 119889 sdot 119904 1205733 ሶ120579 = 0







119888 120572119894 sdot ሶ119909 + 119904 120572119894 sdot ሶ119910




ሶ120595119894 =1


rrlabcsuni-klde



rrlabcsuni-klde

Mecanum Vehicle





Mecanum Kinematics



1205721 = 45 deg 1205731 = 45 deg 1205741 = minus45 deg

1205722 = 135 deg 1205732 = minus45 deg 1205742 = 45 deg


1205724 = minus45 deg 1205734 = 135 deg 1205744 = 45 deg


Mecanum Kinematics








Mecanum Kinematics


ሶ119909 =119903

4ሶ1205951 + ሶ1205952 + ሶ1205953 + ሶ1205954

ሶ119910 =119903

4minus ሶ1205951 + ሶ1205952 minus ሶ1205953 + ሶ1205954

ሶ120579 =119903

119889 2ሶ1205951 minus ሶ1205952 minus ሶ1205953 + ሶ1205954


Mecanum Kinematics


ሶ119909ሶ119910ሶ120579

=119903119908ℎ1198901198901198974


sdot

ሶ1205951

ሶ1205952

ሶ1205953

ሶ1205954

with a =2 2

119889


ሶ1205954 = ሶ1205951 + ሶ1205952 minus ሶ1205953

rrlabcsuni-klde





Ability to maneuver

Controllability

Stability



Stability




Typical setups




Maneuverability








Controllability



Examples




Wheel type

Drive geometry


Motortransmission


Degree of Mobility




119891119898 = 3 minus rank 119878


Degree of Mobility



1198891 = 1198892 1205731 = 1205732 = 0 1205721 + 120587 = 1205722






119891119904 = rank 119878prime








119891119872 = 119891119898 + 119891119904

rrlabcsuni-klde

Vehicle Kinematics



Tricycle Drive

Given 120575 119907


119903 = 119897 sdot cot 120575

119907prime =119907

cos 120575

119907119897 = 119907 sdot119903 + 119889

119903

119907119903 = 119907 sdot119903 minus 119889


120575


Ackermann Steering

Given 119907119863 120575 119897 119889

Unknown 119907119903119903 119907119897119903 119907119903119891 119907119897119891

Solution

120602 =119897

tan 120575

119907119897119903 =120602 minus 1198892 sdot 119907119863

120602

119907119903119903 =120602 + 1198892 sdot 119907119863

120602

119907119897119891 =120602 minus 1198892 2 + 1198972 sdot tan 120575

119897sdot 119907119863

119907119903119891 =1206022 + 1198972 sdot tan 120575

119897sdot 119907119863


120575

120602

rrlabcsuni-klde



Given 120602 120593 119907 119889



1205932 = arctan120549119910 + 119889

1205491199091206022 = 120602 sdot cos120593 cos1205932

1199072 = 119907 sdot 1206022120602

120549119910 = 120602 ∙ 119904119894119899 120593

1205931

1205932

1206021

1206022

120593 120549119909 = 120602 ∙ cos120593

120602

119889

rrlabcsuni-klde

Bicycle Model


119907


Bicycle Model


Low speed motion



119865119910 =1198981199072

120602






Bicycle Model

Given 120575119891 120575119903 119907 120579

Unknown ሶ119883 ሶ119884 ሶ120579

Solution

120575119907 = tanminus1119897119891 tan 120575119903+119897119903 tan 120575119891

119897119891+119897119903

ሶ120579 =119907

120602

120602 =119897119891+119897119903

tan 120575119891 minustan 120575119903 cos 120575119907

ሶ119883 = 119907 cos 120579 + 120575119907ሶ119884 = 119907 sin 120579 + 120575119907

ሶ120579 =119907 cos 120575119907

119897119891+119897119903tan 120575119891 minus tan 120575119903

119907

rrlabcsuni-klde



119862 = 119874 120575119903 = 0 120575119891 = 120575

119897119903 = 0 119897119891 = 119897

120575119907 = 0

ሶ120579 =119907

120602

120602 =119897

tan 120575119891

ሶ119883 = 119907 cos 120579ሶ119884 = 119907 sin 120579ሶ120579 =

119907

119897119891tan 120575119891

119907

rrlabcsuni-klde




Hu

ma

no

id

Ro

bo

ts

Off-r

oa

d R

obo

ts

Serv

ic

e R

obo

ts

rrlabcsuni-klde

Off-road Projekte


rrlabcsuni-klde


rrlabcsuni-klde

rrlabcsuni-klde

Contents


Vehicle Kinematics

Analytical Solution

Geometric Solution

Bicycle Model




rrlabcsuni-klde

Modeling of AMRs






polygon lines grids

boxes

AMR position can be


point in space

center of mass

kinematic center



Configuration Space



2D robot C-space


Configuration Space







2D robot c-space


Configuration Space





rrlabcsuni-klde

Coordinate Systems









119861119906 =119909119910119911

+ 119874119861119877 120593120595 120579 119874 119906





119877 119906Ԧ119901119879 119904










119877119910 120595 =cos120595 0 sin1205950 1 0



120593 119910

119907

119911119908





119877 = 119877119911 120593 sdot 119877119909 120595 sdot 119877119911prime θ


119877 = 119877119911 θ sdot 119877119910 120595 sdot 119877119909 120593


Roll Pitch Yaw




Yaw θ

Pitchψ

Roll 120593

119911

119910119909

20

Roll

Pitch Yaw

119909

119911119910






119877119904 =



Linear Velocity




119860119877119861 Ԧ119907119902


119860 Ԧ119907119902 =119860 Ԧ119907119874119861 + 119861

119860119877119861 Ԧ119907119902



calculated with

119860 Ԧ119907119876 = 119860 Ԧ119907119874119861 + 119861119860119877119861 Ԧ119907119876 +

119860ω119861 times 119861119860119877119861 Ԧ119902


Rotational Velocity



119860120596 = 119861119860119877119861120596





119894+1120596119894+1 = 119894119894+1119877 sdot 119894120596119894 + ሶθ119894+1

119894+1 Ԧ119890119911119894+1

119894+1 Ԧ119907119894+1 = 119894119894+1119877 119894 Ԧ119907119894 +

119894120596119894 times119894 Ԧ119901119894+1






119894119877 from frame 119894 to

119894 + 1





119894119894+1119877 θ = 119894+1

119894119877minus1 θ = 119894+1119894119877119879 θ =

119888θ 119904θ 119909119908minus119904θ 119888θ 1199101199080 0 1

rrlabcsuni-klde



YW

XW

θ

YR

XR

P









coordinates





0 0 1


ሶԦ120585119877 = 119877 θሶԦ120585119882


YW

XW

θ

YR

XR




119877120587

2=

0 1 0minus1 0 00 0 1


ሶԦ120585119877 = 119877120587

2ሶ

120585119882 =0 1 0minus1 0 00 0 1


=ሶ119910


rrlabcsuni-klde


Analytical Solution






YW

XW

θ

V(t)

120596(t)







ሶԦ120585119882 = ሶ119909 ሶ119910 ሶ120579119879

= 119891 119889 119903 120579 ሶ120595119897 ሶ120595119903

= 119877 120579 minus1 ሶԦ120585119877

passive wheel

active wheel

d






ሶ119909119877119903 =119903 ሶ1205951199032

ሶ119909119877119897 =119903 ሶ120595119897

2

120596119903 =119903 ሶ1205951199032119889

120596119897 =minus119903 ሶ120595119897

2119889

ሶԦ120585119882 = 119877 120579 minus1 ሶԦ120585119903ሶԦ120585119903 =

119903 ሶ1205951199032

+119903 ሶ120595119897

2 0

119903 ሶ1205951199032119889

minus119903 ሶ120595119897

2119889

119879





Given 120579 =120587



119904

ሶԦ120585119882 =ሶ119909ሶ119910ሶ120579

=0 minus1 01 0 00 0 1

sdot301

=031







Consider




rrlabcsuni-klde

Basic Wheel Types


Standardwheel

Steerablestandard

wheel

Castorwheel

Swedish orMecanum

wheel

Ball orspherical

wheel


Wheel Properties




Castor wheel


















ሶ120595 119903



ሶ120595 119903



ሶ120595 119903



ሶ120595 119903



ሶ120595 119903








and 1 Ԧ1199071 = ሶ119909 ሶ119910 0 119879


119894+1120596119894+1 = 119894119894+1119877 sdot 119894120596119894 + ሶ120579119894+1

119894+1 Ԧ119890119911119894+1

119894+1 Ԧ119907119894+1 = 119894119894+1119877 119894 Ԧ119907119894 +

119894120596119894 times119894 Ԧ119901119894+1




calculated as

2 Ԧ1199072 =119888120572 ሶ119909 + 119904120572 ሶ119910minus119904120572 ሶ119909 + 119888120572 ሶ119910

0


3 Ԧ1199073 =

119888120572 ሶ119909 + 119904120572 ሶ119910

minus119904120572 ሶ119909 + 119888120572 ሶ119910 + 119889 ሶ1205790






4 Ԧ1199074 =119904 120572 + 120573 ሶ119909 minus 119888 120572 + 120573 ሶ119910 minus 119888120573 119889 ሶ120579

119888 120572 + 120573 ሶ119909 + 119904 120572 + 120573 ሶ119910 + 119904120573 119889 ሶ1205790




119904 120572 + 120573 minus119888 120572 + 120573 minus119888120573 119889

119888 120572 + 120573 119904 120572 + 120573 119904120573 1198890 0 0

ሶ119909119910 ሶሶ120579

=119903 ሶ12059500












119904 120572 + 120573(119905) minus119888 120572 + 120573(119905) minus119888120573(119905)119889

119888 120572 + 120573(119905) 119904 120572 + 120573(119905) 119904120573(119905)1198890 0 0

ሶ119909ሶ119910ሶ120579

=119903 ሶ120595

minus119889119888 ሶ120573(119905)0


maneuverability)




119904 120572 + 120573 + 120574 minus119888 120572 + 120573 + 120574 minus119888 120573 + 120574 119889

119888 120572 + 120573 + 120574 119904 120572 + 120573 + 120574 119904 120573 + 120574 1198890 0 0

ሶ119909ሶ119910ሶ120579

=119903 ሶ120595 cos 120574

119903 ሶ120595 sin 120574 + 119903119901119903 ሶ1205951199011199030















120572119897 = 90 deg

120573119897 = 0 deg

120572119903 = minus90 deg

120573119903 = 180 deg




119904 120572119897 + 120573119897 ሶ119909 minus 119888 120572119897 + 120573119897 ሶ119910 minus 119888120573119897 119889 ሶ120579 = 119903119897 ሶ120595119897

119888 120572119897 + 120573119897 ሶ119909 + 119904 120572119897 + 120573119897 ሶ119910 + 119904120573119897 119889 ሶ120579 = 0



ሶ119909 minus 119889 ሶ120579 = 119903119897 ሶ120595119897

ሶ119910 = 0

ሶ119909 + 119889 ሶ120579 = 119903119903 ሶ120595119903119910 = 0




ሶ119909ሶ119910ሶ120579

=

1

2119903119897 ሶ120595119897 + 119903119903 ሶ120595119903

01

2119889minus119903119897 ሶ120595119897 + 119903119903 ሶ120595119903



rrlabcsuni-klde




rrlabcsuni-klde











1205721 = 0 deg

1205722 = 120 deg

1205723 = minus120 deg







119888 1205721 + 1205731 ሶ119909 + 119904 1205721 + 1205731 ሶ119910 + 119889 sdot 119904 1205731 ሶ120579 = 0

119888 1205722 + 1205732 ሶ119909 + 119904 1205722 + 1205732 ሶ119910 + 119889 sdot 119904 1205732 ሶ120579 = 0

119888 1205723 + 1205733 ሶ119909 + 119904 1205723 + 1205733 ሶ119910 + 119889 sdot 119904 1205733 ሶ120579 = 0







119888 120572119894 sdot ሶ119909 + 119904 120572119894 sdot ሶ119910




ሶ120595119894 =1


rrlabcsuni-klde



rrlabcsuni-klde

Mecanum Vehicle





Mecanum Kinematics



1205721 = 45 deg 1205731 = 45 deg 1205741 = minus45 deg

1205722 = 135 deg 1205732 = minus45 deg 1205742 = 45 deg


1205724 = minus45 deg 1205734 = 135 deg 1205744 = 45 deg


Mecanum Kinematics








Mecanum Kinematics


ሶ119909 =119903

4ሶ1205951 + ሶ1205952 + ሶ1205953 + ሶ1205954

ሶ119910 =119903

4minus ሶ1205951 + ሶ1205952 minus ሶ1205953 + ሶ1205954

ሶ120579 =119903

119889 2ሶ1205951 minus ሶ1205952 minus ሶ1205953 + ሶ1205954


Mecanum Kinematics


ሶ119909ሶ119910ሶ120579

=119903119908ℎ1198901198901198974


sdot

ሶ1205951

ሶ1205952

ሶ1205953

ሶ1205954

with a =2 2

119889


ሶ1205954 = ሶ1205951 + ሶ1205952 minus ሶ1205953

rrlabcsuni-klde





Ability to maneuver

Controllability

Stability



Stability




Typical setups




Maneuverability








Controllability



Examples




Wheel type

Drive geometry


Motortransmission


Degree of Mobility




119891119898 = 3 minus rank 119878


Degree of Mobility



1198891 = 1198892 1205731 = 1205732 = 0 1205721 + 120587 = 1205722






119891119904 = rank 119878prime








119891119872 = 119891119898 + 119891119904

rrlabcsuni-klde

Vehicle Kinematics



Tricycle Drive

Given 120575 119907


119903 = 119897 sdot cot 120575

119907prime =119907

cos 120575

119907119897 = 119907 sdot119903 + 119889

119903

119907119903 = 119907 sdot119903 minus 119889


120575


Ackermann Steering

Given 119907119863 120575 119897 119889

Unknown 119907119903119903 119907119897119903 119907119903119891 119907119897119891

Solution

120602 =119897

tan 120575

119907119897119903 =120602 minus 1198892 sdot 119907119863

120602

119907119903119903 =120602 + 1198892 sdot 119907119863

120602

119907119897119891 =120602 minus 1198892 2 + 1198972 sdot tan 120575

119897sdot 119907119863

119907119903119891 =1206022 + 1198972 sdot tan 120575

119897sdot 119907119863


120575

120602

rrlabcsuni-klde



Given 120602 120593 119907 119889



1205932 = arctan120549119910 + 119889

1205491199091206022 = 120602 sdot cos120593 cos1205932

1199072 = 119907 sdot 1206022120602

120549119910 = 120602 ∙ 119904119894119899 120593

1205931

1205932

1206021

1206022

120593 120549119909 = 120602 ∙ cos120593

120602

119889

rrlabcsuni-klde

Bicycle Model


119907


Bicycle Model


Low speed motion



119865119910 =1198981199072

120602






Bicycle Model

Given 120575119891 120575119903 119907 120579

Unknown ሶ119883 ሶ119884 ሶ120579

Solution

120575119907 = tanminus1119897119891 tan 120575119903+119897119903 tan 120575119891

119897119891+119897119903

ሶ120579 =119907

120602

120602 =119897119891+119897119903

tan 120575119891 minustan 120575119903 cos 120575119907

ሶ119883 = 119907 cos 120579 + 120575119907ሶ119884 = 119907 sin 120579 + 120575119907

ሶ120579 =119907 cos 120575119907

119897119891+119897119903tan 120575119891 minus tan 120575119903

119907

rrlabcsuni-klde



119862 = 119874 120575119903 = 0 120575119891 = 120575

119897119903 = 0 119897119891 = 119897

120575119907 = 0

ሶ120579 =119907

120602

120602 =119897

tan 120575119891

ሶ119883 = 119907 cos 120579ሶ119884 = 119907 sin 120579ሶ120579 =

119907

119897119891tan 120575119891

119907

rrlabcsuni-klde

Off-road Projekte


rrlabcsuni-klde


rrlabcsuni-klde

rrlabcsuni-klde

Contents


Vehicle Kinematics

Analytical Solution

Geometric Solution

Bicycle Model




rrlabcsuni-klde

Modeling of AMRs






polygon lines grids

boxes

AMR position can be


point in space

center of mass

kinematic center



Configuration Space



2D robot C-space


Configuration Space







2D robot c-space


Configuration Space





rrlabcsuni-klde

Coordinate Systems









119861119906 =119909119910119911

+ 119874119861119877 120593120595 120579 119874 119906





119877 119906Ԧ119901119879 119904










119877119910 120595 =cos120595 0 sin1205950 1 0



120593 119910

119907

119911119908





119877 = 119877119911 120593 sdot 119877119909 120595 sdot 119877119911prime θ


119877 = 119877119911 θ sdot 119877119910 120595 sdot 119877119909 120593


Roll Pitch Yaw




Yaw θ

Pitchψ

Roll 120593

119911

119910119909

20

Roll

Pitch Yaw

119909

119911119910






119877119904 =



Linear Velocity




119860119877119861 Ԧ119907119902


119860 Ԧ119907119902 =119860 Ԧ119907119874119861 + 119861

119860119877119861 Ԧ119907119902



calculated with

119860 Ԧ119907119876 = 119860 Ԧ119907119874119861 + 119861119860119877119861 Ԧ119907119876 +

119860ω119861 times 119861119860119877119861 Ԧ119902


Rotational Velocity



119860120596 = 119861119860119877119861120596





119894+1120596119894+1 = 119894119894+1119877 sdot 119894120596119894 + ሶθ119894+1

119894+1 Ԧ119890119911119894+1

119894+1 Ԧ119907119894+1 = 119894119894+1119877 119894 Ԧ119907119894 +

119894120596119894 times119894 Ԧ119901119894+1






119894119877 from frame 119894 to

119894 + 1





119894119894+1119877 θ = 119894+1

119894119877minus1 θ = 119894+1119894119877119879 θ =

119888θ 119904θ 119909119908minus119904θ 119888θ 1199101199080 0 1

rrlabcsuni-klde



YW

XW

θ

YR

XR

P









coordinates





0 0 1


ሶԦ120585119877 = 119877 θሶԦ120585119882


YW

XW

θ

YR

XR




119877120587

2=

0 1 0minus1 0 00 0 1


ሶԦ120585119877 = 119877120587

2ሶ

120585119882 =0 1 0minus1 0 00 0 1


=ሶ119910


rrlabcsuni-klde


Analytical Solution






YW

XW

θ

V(t)

120596(t)







ሶԦ120585119882 = ሶ119909 ሶ119910 ሶ120579119879

= 119891 119889 119903 120579 ሶ120595119897 ሶ120595119903

= 119877 120579 minus1 ሶԦ120585119877

passive wheel

active wheel

d






ሶ119909119877119903 =119903 ሶ1205951199032

ሶ119909119877119897 =119903 ሶ120595119897

2

120596119903 =119903 ሶ1205951199032119889

120596119897 =minus119903 ሶ120595119897

2119889

ሶԦ120585119882 = 119877 120579 minus1 ሶԦ120585119903ሶԦ120585119903 =

119903 ሶ1205951199032

+119903 ሶ120595119897

2 0

119903 ሶ1205951199032119889

minus119903 ሶ120595119897

2119889

119879





Given 120579 =120587



119904

ሶԦ120585119882 =ሶ119909ሶ119910ሶ120579

=0 minus1 01 0 00 0 1

sdot301

=031







Consider




rrlabcsuni-klde

Basic Wheel Types


Standardwheel

Steerablestandard

wheel

Castorwheel

Swedish orMecanum

wheel

Ball orspherical

wheel


Wheel Properties




Castor wheel


















ሶ120595 119903



ሶ120595 119903



ሶ120595 119903



ሶ120595 119903



ሶ120595 119903








and 1 Ԧ1199071 = ሶ119909 ሶ119910 0 119879


119894+1120596119894+1 = 119894119894+1119877 sdot 119894120596119894 + ሶ120579119894+1

119894+1 Ԧ119890119911119894+1

119894+1 Ԧ119907119894+1 = 119894119894+1119877 119894 Ԧ119907119894 +

119894120596119894 times119894 Ԧ119901119894+1




calculated as

2 Ԧ1199072 =119888120572 ሶ119909 + 119904120572 ሶ119910minus119904120572 ሶ119909 + 119888120572 ሶ119910

0


3 Ԧ1199073 =

119888120572 ሶ119909 + 119904120572 ሶ119910

minus119904120572 ሶ119909 + 119888120572 ሶ119910 + 119889 ሶ1205790






4 Ԧ1199074 =119904 120572 + 120573 ሶ119909 minus 119888 120572 + 120573 ሶ119910 minus 119888120573 119889 ሶ120579

119888 120572 + 120573 ሶ119909 + 119904 120572 + 120573 ሶ119910 + 119904120573 119889 ሶ1205790




119904 120572 + 120573 minus119888 120572 + 120573 minus119888120573 119889

119888 120572 + 120573 119904 120572 + 120573 119904120573 1198890 0 0

ሶ119909119910 ሶሶ120579

=119903 ሶ12059500












119904 120572 + 120573(119905) minus119888 120572 + 120573(119905) minus119888120573(119905)119889

119888 120572 + 120573(119905) 119904 120572 + 120573(119905) 119904120573(119905)1198890 0 0

ሶ119909ሶ119910ሶ120579

=119903 ሶ120595

minus119889119888 ሶ120573(119905)0


maneuverability)




119904 120572 + 120573 + 120574 minus119888 120572 + 120573 + 120574 minus119888 120573 + 120574 119889

119888 120572 + 120573 + 120574 119904 120572 + 120573 + 120574 119904 120573 + 120574 1198890 0 0

ሶ119909ሶ119910ሶ120579

=119903 ሶ120595 cos 120574

119903 ሶ120595 sin 120574 + 119903119901119903 ሶ1205951199011199030















120572119897 = 90 deg

120573119897 = 0 deg

120572119903 = minus90 deg

120573119903 = 180 deg




119904 120572119897 + 120573119897 ሶ119909 minus 119888 120572119897 + 120573119897 ሶ119910 minus 119888120573119897 119889 ሶ120579 = 119903119897 ሶ120595119897

119888 120572119897 + 120573119897 ሶ119909 + 119904 120572119897 + 120573119897 ሶ119910 + 119904120573119897 119889 ሶ120579 = 0



ሶ119909 minus 119889 ሶ120579 = 119903119897 ሶ120595119897

ሶ119910 = 0

ሶ119909 + 119889 ሶ120579 = 119903119903 ሶ120595119903119910 = 0




ሶ119909ሶ119910ሶ120579

=

1

2119903119897 ሶ120595119897 + 119903119903 ሶ120595119903

01

2119889minus119903119897 ሶ120595119897 + 119903119903 ሶ120595119903



rrlabcsuni-klde




rrlabcsuni-klde











1205721 = 0 deg

1205722 = 120 deg

1205723 = minus120 deg







119888 1205721 + 1205731 ሶ119909 + 119904 1205721 + 1205731 ሶ119910 + 119889 sdot 119904 1205731 ሶ120579 = 0

119888 1205722 + 1205732 ሶ119909 + 119904 1205722 + 1205732 ሶ119910 + 119889 sdot 119904 1205732 ሶ120579 = 0

119888 1205723 + 1205733 ሶ119909 + 119904 1205723 + 1205733 ሶ119910 + 119889 sdot 119904 1205733 ሶ120579 = 0







119888 120572119894 sdot ሶ119909 + 119904 120572119894 sdot ሶ119910




ሶ120595119894 =1


rrlabcsuni-klde



rrlabcsuni-klde

Mecanum Vehicle





Mecanum Kinematics



1205721 = 45 deg 1205731 = 45 deg 1205741 = minus45 deg

1205722 = 135 deg 1205732 = minus45 deg 1205742 = 45 deg


1205724 = minus45 deg 1205734 = 135 deg 1205744 = 45 deg


Mecanum Kinematics








Mecanum Kinematics


ሶ119909 =119903

4ሶ1205951 + ሶ1205952 + ሶ1205953 + ሶ1205954

ሶ119910 =119903

4minus ሶ1205951 + ሶ1205952 minus ሶ1205953 + ሶ1205954

ሶ120579 =119903

119889 2ሶ1205951 minus ሶ1205952 minus ሶ1205953 + ሶ1205954


Mecanum Kinematics


ሶ119909ሶ119910ሶ120579

=119903119908ℎ1198901198901198974


sdot

ሶ1205951

ሶ1205952

ሶ1205953

ሶ1205954

with a =2 2

119889


ሶ1205954 = ሶ1205951 + ሶ1205952 minus ሶ1205953

rrlabcsuni-klde





Ability to maneuver

Controllability

Stability



Stability




Typical setups




Maneuverability








Controllability



Examples




Wheel type

Drive geometry


Motortransmission


Degree of Mobility




119891119898 = 3 minus rank 119878


Degree of Mobility



1198891 = 1198892 1205731 = 1205732 = 0 1205721 + 120587 = 1205722






119891119904 = rank 119878prime








119891119872 = 119891119898 + 119891119904

rrlabcsuni-klde

Vehicle Kinematics



Tricycle Drive

Given 120575 119907


119903 = 119897 sdot cot 120575

119907prime =119907

cos 120575

119907119897 = 119907 sdot119903 + 119889

119903

119907119903 = 119907 sdot119903 minus 119889


120575


Ackermann Steering

Given 119907119863 120575 119897 119889

Unknown 119907119903119903 119907119897119903 119907119903119891 119907119897119891

Solution

120602 =119897

tan 120575

119907119897119903 =120602 minus 1198892 sdot 119907119863

120602

119907119903119903 =120602 + 1198892 sdot 119907119863

120602

119907119897119891 =120602 minus 1198892 2 + 1198972 sdot tan 120575

119897sdot 119907119863

119907119903119891 =1206022 + 1198972 sdot tan 120575

119897sdot 119907119863


120575

120602

rrlabcsuni-klde



Given 120602 120593 119907 119889



1205932 = arctan120549119910 + 119889

1205491199091206022 = 120602 sdot cos120593 cos1205932

1199072 = 119907 sdot 1206022120602

120549119910 = 120602 ∙ 119904119894119899 120593

1205931

1205932

1206021

1206022

120593 120549119909 = 120602 ∙ cos120593

120602

119889

rrlabcsuni-klde

Bicycle Model


119907


Bicycle Model


Low speed motion



119865119910 =1198981199072

120602






Bicycle Model

Given 120575119891 120575119903 119907 120579

Unknown ሶ119883 ሶ119884 ሶ120579

Solution

120575119907 = tanminus1119897119891 tan 120575119903+119897119903 tan 120575119891

119897119891+119897119903

ሶ120579 =119907

120602

120602 =119897119891+119897119903

tan 120575119891 minustan 120575119903 cos 120575119907

ሶ119883 = 119907 cos 120579 + 120575119907ሶ119884 = 119907 sin 120579 + 120575119907

ሶ120579 =119907 cos 120575119907

119897119891+119897119903tan 120575119891 minus tan 120575119903

119907

rrlabcsuni-klde



119862 = 119874 120575119903 = 0 120575119891 = 120575

119897119903 = 0 119897119891 = 119897

120575119907 = 0

ሶ120579 =119907

120602

120602 =119897

tan 120575119891

ሶ119883 = 119907 cos 120579ሶ119884 = 119907 sin 120579ሶ120579 =

119907

119897119891tan 120575119891

119907

rrlabcsuni-klde


rrlabcsuni-klde

rrlabcsuni-klde

Contents


Vehicle Kinematics

Analytical Solution

Geometric Solution

Bicycle Model




rrlabcsuni-klde

Modeling of AMRs






polygon lines grids

boxes

AMR position can be


point in space

center of mass

kinematic center



Configuration Space



2D robot C-space


Configuration Space







2D robot c-space


Configuration Space





rrlabcsuni-klde

Coordinate Systems









119861119906 =119909119910119911

+ 119874119861119877 120593120595 120579 119874 119906





119877 119906Ԧ119901119879 119904










119877119910 120595 =cos120595 0 sin1205950 1 0



120593 119910

119907

119911119908





119877 = 119877119911 120593 sdot 119877119909 120595 sdot 119877119911prime θ


119877 = 119877119911 θ sdot 119877119910 120595 sdot 119877119909 120593


Roll Pitch Yaw




Yaw θ

Pitchψ

Roll 120593

119911

119910119909

20

Roll

Pitch Yaw

119909

119911119910






119877119904 =



Linear Velocity




119860119877119861 Ԧ119907119902


119860 Ԧ119907119902 =119860 Ԧ119907119874119861 + 119861

119860119877119861 Ԧ119907119902



calculated with

119860 Ԧ119907119876 = 119860 Ԧ119907119874119861 + 119861119860119877119861 Ԧ119907119876 +

119860ω119861 times 119861119860119877119861 Ԧ119902


Rotational Velocity



119860120596 = 119861119860119877119861120596





119894+1120596119894+1 = 119894119894+1119877 sdot 119894120596119894 + ሶθ119894+1

119894+1 Ԧ119890119911119894+1

119894+1 Ԧ119907119894+1 = 119894119894+1119877 119894 Ԧ119907119894 +

119894120596119894 times119894 Ԧ119901119894+1






119894119877 from frame 119894 to

119894 + 1





119894119894+1119877 θ = 119894+1

119894119877minus1 θ = 119894+1119894119877119879 θ =

119888θ 119904θ 119909119908minus119904θ 119888θ 1199101199080 0 1

rrlabcsuni-klde



YW

XW

θ

YR

XR

P









coordinates





0 0 1


ሶԦ120585119877 = 119877 θሶԦ120585119882


YW

XW

θ

YR

XR




119877120587

2=

0 1 0minus1 0 00 0 1


ሶԦ120585119877 = 119877120587

2ሶ

120585119882 =0 1 0minus1 0 00 0 1


=ሶ119910


rrlabcsuni-klde


Analytical Solution






YW

XW

θ

V(t)

120596(t)







ሶԦ120585119882 = ሶ119909 ሶ119910 ሶ120579119879

= 119891 119889 119903 120579 ሶ120595119897 ሶ120595119903

= 119877 120579 minus1 ሶԦ120585119877

passive wheel

active wheel

d






ሶ119909119877119903 =119903 ሶ1205951199032

ሶ119909119877119897 =119903 ሶ120595119897

2

120596119903 =119903 ሶ1205951199032119889

120596119897 =minus119903 ሶ120595119897

2119889

ሶԦ120585119882 = 119877 120579 minus1 ሶԦ120585119903ሶԦ120585119903 =

119903 ሶ1205951199032

+119903 ሶ120595119897

2 0

119903 ሶ1205951199032119889

minus119903 ሶ120595119897

2119889

119879





Given 120579 =120587



119904

ሶԦ120585119882 =ሶ119909ሶ119910ሶ120579

=0 minus1 01 0 00 0 1

sdot301

=031







Consider




rrlabcsuni-klde

Basic Wheel Types


Standardwheel

Steerablestandard

wheel

Castorwheel

Swedish orMecanum

wheel

Ball orspherical

wheel


Wheel Properties




Castor wheel


















ሶ120595 119903



ሶ120595 119903



ሶ120595 119903



ሶ120595 119903



ሶ120595 119903








and 1 Ԧ1199071 = ሶ119909 ሶ119910 0 119879


119894+1120596119894+1 = 119894119894+1119877 sdot 119894120596119894 + ሶ120579119894+1

119894+1 Ԧ119890119911119894+1

119894+1 Ԧ119907119894+1 = 119894119894+1119877 119894 Ԧ119907119894 +

119894120596119894 times119894 Ԧ119901119894+1




calculated as

2 Ԧ1199072 =119888120572 ሶ119909 + 119904120572 ሶ119910minus119904120572 ሶ119909 + 119888120572 ሶ119910

0


3 Ԧ1199073 =

119888120572 ሶ119909 + 119904120572 ሶ119910

minus119904120572 ሶ119909 + 119888120572 ሶ119910 + 119889 ሶ1205790






4 Ԧ1199074 =119904 120572 + 120573 ሶ119909 minus 119888 120572 + 120573 ሶ119910 minus 119888120573 119889 ሶ120579

119888 120572 + 120573 ሶ119909 + 119904 120572 + 120573 ሶ119910 + 119904120573 119889 ሶ1205790




119904 120572 + 120573 minus119888 120572 + 120573 minus119888120573 119889

119888 120572 + 120573 119904 120572 + 120573 119904120573 1198890 0 0

ሶ119909119910 ሶሶ120579

=119903 ሶ12059500












119904 120572 + 120573(119905) minus119888 120572 + 120573(119905) minus119888120573(119905)119889

119888 120572 + 120573(119905) 119904 120572 + 120573(119905) 119904120573(119905)1198890 0 0

ሶ119909ሶ119910ሶ120579

=119903 ሶ120595

minus119889119888 ሶ120573(119905)0


maneuverability)




119904 120572 + 120573 + 120574 minus119888 120572 + 120573 + 120574 minus119888 120573 + 120574 119889

119888 120572 + 120573 + 120574 119904 120572 + 120573 + 120574 119904 120573 + 120574 1198890 0 0

ሶ119909ሶ119910ሶ120579

=119903 ሶ120595 cos 120574

119903 ሶ120595 sin 120574 + 119903119901119903 ሶ1205951199011199030















120572119897 = 90 deg

120573119897 = 0 deg

120572119903 = minus90 deg

120573119903 = 180 deg




119904 120572119897 + 120573119897 ሶ119909 minus 119888 120572119897 + 120573119897 ሶ119910 minus 119888120573119897 119889 ሶ120579 = 119903119897 ሶ120595119897

119888 120572119897 + 120573119897 ሶ119909 + 119904 120572119897 + 120573119897 ሶ119910 + 119904120573119897 119889 ሶ120579 = 0



ሶ119909 minus 119889 ሶ120579 = 119903119897 ሶ120595119897

ሶ119910 = 0

ሶ119909 + 119889 ሶ120579 = 119903119903 ሶ120595119903119910 = 0




ሶ119909ሶ119910ሶ120579

=

1

2119903119897 ሶ120595119897 + 119903119903 ሶ120595119903

01

2119889minus119903119897 ሶ120595119897 + 119903119903 ሶ120595119903



rrlabcsuni-klde




rrlabcsuni-klde











1205721 = 0 deg

1205722 = 120 deg

1205723 = minus120 deg







119888 1205721 + 1205731 ሶ119909 + 119904 1205721 + 1205731 ሶ119910 + 119889 sdot 119904 1205731 ሶ120579 = 0

119888 1205722 + 1205732 ሶ119909 + 119904 1205722 + 1205732 ሶ119910 + 119889 sdot 119904 1205732 ሶ120579 = 0

119888 1205723 + 1205733 ሶ119909 + 119904 1205723 + 1205733 ሶ119910 + 119889 sdot 119904 1205733 ሶ120579 = 0







119888 120572119894 sdot ሶ119909 + 119904 120572119894 sdot ሶ119910




ሶ120595119894 =1


rrlabcsuni-klde



rrlabcsuni-klde

Mecanum Vehicle





Mecanum Kinematics



1205721 = 45 deg 1205731 = 45 deg 1205741 = minus45 deg

1205722 = 135 deg 1205732 = minus45 deg 1205742 = 45 deg


1205724 = minus45 deg 1205734 = 135 deg 1205744 = 45 deg


Mecanum Kinematics








Mecanum Kinematics


ሶ119909 =119903

4ሶ1205951 + ሶ1205952 + ሶ1205953 + ሶ1205954

ሶ119910 =119903

4minus ሶ1205951 + ሶ1205952 minus ሶ1205953 + ሶ1205954

ሶ120579 =119903

119889 2ሶ1205951 minus ሶ1205952 minus ሶ1205953 + ሶ1205954


Mecanum Kinematics


ሶ119909ሶ119910ሶ120579

=119903119908ℎ1198901198901198974


sdot

ሶ1205951

ሶ1205952

ሶ1205953

ሶ1205954

with a =2 2

119889


ሶ1205954 = ሶ1205951 + ሶ1205952 minus ሶ1205953

rrlabcsuni-klde





Ability to maneuver

Controllability

Stability



Stability




Typical setups




Maneuverability








Controllability



Examples




Wheel type

Drive geometry


Motortransmission


Degree of Mobility




119891119898 = 3 minus rank 119878


Degree of Mobility



1198891 = 1198892 1205731 = 1205732 = 0 1205721 + 120587 = 1205722






119891119904 = rank 119878prime








119891119872 = 119891119898 + 119891119904

rrlabcsuni-klde

Vehicle Kinematics



Tricycle Drive

Given 120575 119907


119903 = 119897 sdot cot 120575

119907prime =119907

cos 120575

119907119897 = 119907 sdot119903 + 119889

119903

119907119903 = 119907 sdot119903 minus 119889


120575


Ackermann Steering

Given 119907119863 120575 119897 119889

Unknown 119907119903119903 119907119897119903 119907119903119891 119907119897119891

Solution

120602 =119897

tan 120575

119907119897119903 =120602 minus 1198892 sdot 119907119863

120602

119907119903119903 =120602 + 1198892 sdot 119907119863

120602

119907119897119891 =120602 minus 1198892 2 + 1198972 sdot tan 120575

119897sdot 119907119863

119907119903119891 =1206022 + 1198972 sdot tan 120575

119897sdot 119907119863


120575

120602

rrlabcsuni-klde



Given 120602 120593 119907 119889



1205932 = arctan120549119910 + 119889

1205491199091206022 = 120602 sdot cos120593 cos1205932

1199072 = 119907 sdot 1206022120602

120549119910 = 120602 ∙ 119904119894119899 120593

1205931

1205932

1206021

1206022

120593 120549119909 = 120602 ∙ cos120593

120602

119889

rrlabcsuni-klde

Bicycle Model


119907


Bicycle Model


Low speed motion



119865119910 =1198981199072

120602






Bicycle Model

Given 120575119891 120575119903 119907 120579

Unknown ሶ119883 ሶ119884 ሶ120579

Solution

120575119907 = tanminus1119897119891 tan 120575119903+119897119903 tan 120575119891

119897119891+119897119903

ሶ120579 =119907

120602

120602 =119897119891+119897119903

tan 120575119891 minustan 120575119903 cos 120575119907

ሶ119883 = 119907 cos 120579 + 120575119907ሶ119884 = 119907 sin 120579 + 120575119907

ሶ120579 =119907 cos 120575119907

119897119891+119897119903tan 120575119891 minus tan 120575119903

119907

rrlabcsuni-klde



119862 = 119874 120575119903 = 0 120575119891 = 120575

119897119903 = 0 119897119891 = 119897

120575119907 = 0

ሶ120579 =119907

120602

120602 =119897

tan 120575119891

ሶ119883 = 119907 cos 120579ሶ119884 = 119907 sin 120579ሶ120579 =

119907

119897119891tan 120575119891

119907

rrlabcsuni-klde

Contents


Vehicle Kinematics

Analytical Solution

Geometric Solution

Bicycle Model




rrlabcsuni-klde

Modeling of AMRs






polygon lines grids

boxes

AMR position can be


point in space

center of mass

kinematic center



Configuration Space



2D robot C-space


Configuration Space







2D robot c-space


Configuration Space





rrlabcsuni-klde

Coordinate Systems









119861119906 =119909119910119911

+ 119874119861119877 120593120595 120579 119874 119906





119877 119906Ԧ119901119879 119904










119877119910 120595 =cos120595 0 sin1205950 1 0



120593 119910

119907

119911119908





119877 = 119877119911 120593 sdot 119877119909 120595 sdot 119877119911prime θ


119877 = 119877119911 θ sdot 119877119910 120595 sdot 119877119909 120593


Roll Pitch Yaw




Yaw θ

Pitchψ

Roll 120593

119911

119910119909

20

Roll

Pitch Yaw

119909

119911119910






119877119904 =



Linear Velocity




119860119877119861 Ԧ119907119902


119860 Ԧ119907119902 =119860 Ԧ119907119874119861 + 119861

119860119877119861 Ԧ119907119902



calculated with

119860 Ԧ119907119876 = 119860 Ԧ119907119874119861 + 119861119860119877119861 Ԧ119907119876 +

119860ω119861 times 119861119860119877119861 Ԧ119902


Rotational Velocity



119860120596 = 119861119860119877119861120596





119894+1120596119894+1 = 119894119894+1119877 sdot 119894120596119894 + ሶθ119894+1

119894+1 Ԧ119890119911119894+1

119894+1 Ԧ119907119894+1 = 119894119894+1119877 119894 Ԧ119907119894 +

119894120596119894 times119894 Ԧ119901119894+1






119894119877 from frame 119894 to

119894 + 1





119894119894+1119877 θ = 119894+1

119894119877minus1 θ = 119894+1119894119877119879 θ =

119888θ 119904θ 119909119908minus119904θ 119888θ 1199101199080 0 1

rrlabcsuni-klde



YW

XW

θ

YR

XR

P









coordinates





0 0 1


ሶԦ120585119877 = 119877 θሶԦ120585119882


YW

XW

θ

YR

XR




119877120587

2=

0 1 0minus1 0 00 0 1


ሶԦ120585119877 = 119877120587

2ሶ

120585119882 =0 1 0minus1 0 00 0 1


=ሶ119910


rrlabcsuni-klde


Analytical Solution






YW

XW

θ

V(t)

120596(t)







ሶԦ120585119882 = ሶ119909 ሶ119910 ሶ120579119879

= 119891 119889 119903 120579 ሶ120595119897 ሶ120595119903

= 119877 120579 minus1 ሶԦ120585119877

passive wheel

active wheel

d






ሶ119909119877119903 =119903 ሶ1205951199032

ሶ119909119877119897 =119903 ሶ120595119897

2

120596119903 =119903 ሶ1205951199032119889

120596119897 =minus119903 ሶ120595119897

2119889

ሶԦ120585119882 = 119877 120579 minus1 ሶԦ120585119903ሶԦ120585119903 =

119903 ሶ1205951199032

+119903 ሶ120595119897

2 0

119903 ሶ1205951199032119889

minus119903 ሶ120595119897

2119889

119879





Given 120579 =120587



119904

ሶԦ120585119882 =ሶ119909ሶ119910ሶ120579

=0 minus1 01 0 00 0 1

sdot301

=031







Consider




rrlabcsuni-klde

Basic Wheel Types


Standardwheel

Steerablestandard

wheel

Castorwheel

Swedish orMecanum

wheel

Ball orspherical

wheel


Wheel Properties




Castor wheel


















ሶ120595 119903



ሶ120595 119903



ሶ120595 119903



ሶ120595 119903



ሶ120595 119903








and 1 Ԧ1199071 = ሶ119909 ሶ119910 0 119879


119894+1120596119894+1 = 119894119894+1119877 sdot 119894120596119894 + ሶ120579119894+1

119894+1 Ԧ119890119911119894+1

119894+1 Ԧ119907119894+1 = 119894119894+1119877 119894 Ԧ119907119894 +

119894120596119894 times119894 Ԧ119901119894+1




calculated as

2 Ԧ1199072 =119888120572 ሶ119909 + 119904120572 ሶ119910minus119904120572 ሶ119909 + 119888120572 ሶ119910

0


3 Ԧ1199073 =

119888120572 ሶ119909 + 119904120572 ሶ119910

minus119904120572 ሶ119909 + 119888120572 ሶ119910 + 119889 ሶ1205790






4 Ԧ1199074 =119904 120572 + 120573 ሶ119909 minus 119888 120572 + 120573 ሶ119910 minus 119888120573 119889 ሶ120579

119888 120572 + 120573 ሶ119909 + 119904 120572 + 120573 ሶ119910 + 119904120573 119889 ሶ1205790




119904 120572 + 120573 minus119888 120572 + 120573 minus119888120573 119889

119888 120572 + 120573 119904 120572 + 120573 119904120573 1198890 0 0

ሶ119909119910 ሶሶ120579

=119903 ሶ12059500












119904 120572 + 120573(119905) minus119888 120572 + 120573(119905) minus119888120573(119905)119889

119888 120572 + 120573(119905) 119904 120572 + 120573(119905) 119904120573(119905)1198890 0 0

ሶ119909ሶ119910ሶ120579

=119903 ሶ120595

minus119889119888 ሶ120573(119905)0


maneuverability)




119904 120572 + 120573 + 120574 minus119888 120572 + 120573 + 120574 minus119888 120573 + 120574 119889

119888 120572 + 120573 + 120574 119904 120572 + 120573 + 120574 119904 120573 + 120574 1198890 0 0

ሶ119909ሶ119910ሶ120579

=119903 ሶ120595 cos 120574

119903 ሶ120595 sin 120574 + 119903119901119903 ሶ1205951199011199030















120572119897 = 90 deg

120573119897 = 0 deg

120572119903 = minus90 deg

120573119903 = 180 deg




119904 120572119897 + 120573119897 ሶ119909 minus 119888 120572119897 + 120573119897 ሶ119910 minus 119888120573119897 119889 ሶ120579 = 119903119897 ሶ120595119897

119888 120572119897 + 120573119897 ሶ119909 + 119904 120572119897 + 120573119897 ሶ119910 + 119904120573119897 119889 ሶ120579 = 0



ሶ119909 minus 119889 ሶ120579 = 119903119897 ሶ120595119897

ሶ119910 = 0

ሶ119909 + 119889 ሶ120579 = 119903119903 ሶ120595119903119910 = 0




ሶ119909ሶ119910ሶ120579

=

1

2119903119897 ሶ120595119897 + 119903119903 ሶ120595119903

01

2119889minus119903119897 ሶ120595119897 + 119903119903 ሶ120595119903



rrlabcsuni-klde




rrlabcsuni-klde











1205721 = 0 deg

1205722 = 120 deg

1205723 = minus120 deg







119888 1205721 + 1205731 ሶ119909 + 119904 1205721 + 1205731 ሶ119910 + 119889 sdot 119904 1205731 ሶ120579 = 0

119888 1205722 + 1205732 ሶ119909 + 119904 1205722 + 1205732 ሶ119910 + 119889 sdot 119904 1205732 ሶ120579 = 0

119888 1205723 + 1205733 ሶ119909 + 119904 1205723 + 1205733 ሶ119910 + 119889 sdot 119904 1205733 ሶ120579 = 0







119888 120572119894 sdot ሶ119909 + 119904 120572119894 sdot ሶ119910




ሶ120595119894 =1


rrlabcsuni-klde



rrlabcsuni-klde

Mecanum Vehicle





Mecanum Kinematics



1205721 = 45 deg 1205731 = 45 deg 1205741 = minus45 deg

1205722 = 135 deg 1205732 = minus45 deg 1205742 = 45 deg


1205724 = minus45 deg 1205734 = 135 deg 1205744 = 45 deg


Mecanum Kinematics








Mecanum Kinematics


ሶ119909 =119903

4ሶ1205951 + ሶ1205952 + ሶ1205953 + ሶ1205954

ሶ119910 =119903

4minus ሶ1205951 + ሶ1205952 minus ሶ1205953 + ሶ1205954

ሶ120579 =119903

119889 2ሶ1205951 minus ሶ1205952 minus ሶ1205953 + ሶ1205954


Mecanum Kinematics


ሶ119909ሶ119910ሶ120579

=119903119908ℎ1198901198901198974


sdot

ሶ1205951

ሶ1205952

ሶ1205953

ሶ1205954

with a =2 2

119889


ሶ1205954 = ሶ1205951 + ሶ1205952 minus ሶ1205953

rrlabcsuni-klde





Ability to maneuver

Controllability

Stability



Stability




Typical setups




Maneuverability








Controllability



Examples




Wheel type

Drive geometry


Motortransmission


Degree of Mobility




119891119898 = 3 minus rank 119878


Degree of Mobility



1198891 = 1198892 1205731 = 1205732 = 0 1205721 + 120587 = 1205722






119891119904 = rank 119878prime








119891119872 = 119891119898 + 119891119904

rrlabcsuni-klde

Vehicle Kinematics



Tricycle Drive

Given 120575 119907


119903 = 119897 sdot cot 120575

119907prime =119907

cos 120575

119907119897 = 119907 sdot119903 + 119889

119903

119907119903 = 119907 sdot119903 minus 119889


120575


Ackermann Steering

Given 119907119863 120575 119897 119889

Unknown 119907119903119903 119907119897119903 119907119903119891 119907119897119891

Solution

120602 =119897

tan 120575

119907119897119903 =120602 minus 1198892 sdot 119907119863

120602

119907119903119903 =120602 + 1198892 sdot 119907119863

120602

119907119897119891 =120602 minus 1198892 2 + 1198972 sdot tan 120575

119897sdot 119907119863

119907119903119891 =1206022 + 1198972 sdot tan 120575

119897sdot 119907119863


120575

120602

rrlabcsuni-klde



Given 120602 120593 119907 119889



1205932 = arctan120549119910 + 119889

1205491199091206022 = 120602 sdot cos120593 cos1205932

1199072 = 119907 sdot 1206022120602

120549119910 = 120602 ∙ 119904119894119899 120593

1205931

1205932

1206021

1206022

120593 120549119909 = 120602 ∙ cos120593

120602

119889

rrlabcsuni-klde

Bicycle Model


119907


Bicycle Model


Low speed motion



119865119910 =1198981199072

120602






Bicycle Model

Given 120575119891 120575119903 119907 120579

Unknown ሶ119883 ሶ119884 ሶ120579

Solution

120575119907 = tanminus1119897119891 tan 120575119903+119897119903 tan 120575119891

119897119891+119897119903

ሶ120579 =119907

120602

120602 =119897119891+119897119903

tan 120575119891 minustan 120575119903 cos 120575119907

ሶ119883 = 119907 cos 120579 + 120575119907ሶ119884 = 119907 sin 120579 + 120575119907

ሶ120579 =119907 cos 120575119907

119897119891+119897119903tan 120575119891 minus tan 120575119903

119907

rrlabcsuni-klde



119862 = 119874 120575119903 = 0 120575119891 = 120575

119897119903 = 0 119897119891 = 119897

120575119907 = 0

ሶ120579 =119907

120602

120602 =119897

tan 120575119891

ሶ119883 = 119907 cos 120579ሶ119884 = 119907 sin 120579ሶ120579 =

119907

119897119891tan 120575119891

119907

rrlabcsuni-klde

Modeling of AMRs






polygon lines grids

boxes

AMR position can be


point in space

center of mass

kinematic center



Configuration Space



2D robot C-space


Configuration Space







2D robot c-space


Configuration Space





rrlabcsuni-klde

Coordinate Systems









119861119906 =119909119910119911

+ 119874119861119877 120593120595 120579 119874 119906





119877 119906Ԧ119901119879 119904










119877119910 120595 =cos120595 0 sin1205950 1 0



120593 119910

119907

119911119908





119877 = 119877119911 120593 sdot 119877119909 120595 sdot 119877119911prime θ


119877 = 119877119911 θ sdot 119877119910 120595 sdot 119877119909 120593


Roll Pitch Yaw




Yaw θ

Pitchψ

Roll 120593

119911

119910119909

20

Roll

Pitch Yaw

119909

119911119910






119877119904 =



Linear Velocity




119860119877119861 Ԧ119907119902


119860 Ԧ119907119902 =119860 Ԧ119907119874119861 + 119861

119860119877119861 Ԧ119907119902



calculated with

119860 Ԧ119907119876 = 119860 Ԧ119907119874119861 + 119861119860119877119861 Ԧ119907119876 +

119860ω119861 times 119861119860119877119861 Ԧ119902


Rotational Velocity



119860120596 = 119861119860119877119861120596





119894+1120596119894+1 = 119894119894+1119877 sdot 119894120596119894 + ሶθ119894+1

119894+1 Ԧ119890119911119894+1

119894+1 Ԧ119907119894+1 = 119894119894+1119877 119894 Ԧ119907119894 +

119894120596119894 times119894 Ԧ119901119894+1






119894119877 from frame 119894 to

119894 + 1





119894119894+1119877 θ = 119894+1

119894119877minus1 θ = 119894+1119894119877119879 θ =

119888θ 119904θ 119909119908minus119904θ 119888θ 1199101199080 0 1

rrlabcsuni-klde



YW

XW

θ

YR

XR

P









coordinates





0 0 1


ሶԦ120585119877 = 119877 θሶԦ120585119882


YW

XW

θ

YR

XR




119877120587

2=

0 1 0minus1 0 00 0 1


ሶԦ120585119877 = 119877120587

2ሶ

120585119882 =0 1 0minus1 0 00 0 1


=ሶ119910


rrlabcsuni-klde


Analytical Solution






YW

XW

θ

V(t)

120596(t)







ሶԦ120585119882 = ሶ119909 ሶ119910 ሶ120579119879

= 119891 119889 119903 120579 ሶ120595119897 ሶ120595119903

= 119877 120579 minus1 ሶԦ120585119877

passive wheel

active wheel

d






ሶ119909119877119903 =119903 ሶ1205951199032

ሶ119909119877119897 =119903 ሶ120595119897

2

120596119903 =119903 ሶ1205951199032119889

120596119897 =minus119903 ሶ120595119897

2119889

ሶԦ120585119882 = 119877 120579 minus1 ሶԦ120585119903ሶԦ120585119903 =

119903 ሶ1205951199032

+119903 ሶ120595119897

2 0

119903 ሶ1205951199032119889

minus119903 ሶ120595119897

2119889

119879





Given 120579 =120587



119904

ሶԦ120585119882 =ሶ119909ሶ119910ሶ120579

=0 minus1 01 0 00 0 1

sdot301

=031







Consider




rrlabcsuni-klde

Basic Wheel Types


Standardwheel

Steerablestandard

wheel

Castorwheel

Swedish orMecanum

wheel

Ball orspherical

wheel


Wheel Properties




Castor wheel


















ሶ120595 119903



ሶ120595 119903



ሶ120595 119903



ሶ120595 119903



ሶ120595 119903








and 1 Ԧ1199071 = ሶ119909 ሶ119910 0 119879


119894+1120596119894+1 = 119894119894+1119877 sdot 119894120596119894 + ሶ120579119894+1

119894+1 Ԧ119890119911119894+1

119894+1 Ԧ119907119894+1 = 119894119894+1119877 119894 Ԧ119907119894 +

119894120596119894 times119894 Ԧ119901119894+1




calculated as

2 Ԧ1199072 =119888120572 ሶ119909 + 119904120572 ሶ119910minus119904120572 ሶ119909 + 119888120572 ሶ119910

0


3 Ԧ1199073 =

119888120572 ሶ119909 + 119904120572 ሶ119910

minus119904120572 ሶ119909 + 119888120572 ሶ119910 + 119889 ሶ1205790






4 Ԧ1199074 =119904 120572 + 120573 ሶ119909 minus 119888 120572 + 120573 ሶ119910 minus 119888120573 119889 ሶ120579

119888 120572 + 120573 ሶ119909 + 119904 120572 + 120573 ሶ119910 + 119904120573 119889 ሶ1205790




119904 120572 + 120573 minus119888 120572 + 120573 minus119888120573 119889

119888 120572 + 120573 119904 120572 + 120573 119904120573 1198890 0 0

ሶ119909119910 ሶሶ120579

=119903 ሶ12059500












119904 120572 + 120573(119905) minus119888 120572 + 120573(119905) minus119888120573(119905)119889

119888 120572 + 120573(119905) 119904 120572 + 120573(119905) 119904120573(119905)1198890 0 0

ሶ119909ሶ119910ሶ120579

=119903 ሶ120595

minus119889119888 ሶ120573(119905)0


maneuverability)




119904 120572 + 120573 + 120574 minus119888 120572 + 120573 + 120574 minus119888 120573 + 120574 119889

119888 120572 + 120573 + 120574 119904 120572 + 120573 + 120574 119904 120573 + 120574 1198890 0 0

ሶ119909ሶ119910ሶ120579

=119903 ሶ120595 cos 120574

119903 ሶ120595 sin 120574 + 119903119901119903 ሶ1205951199011199030















120572119897 = 90 deg

120573119897 = 0 deg

120572119903 = minus90 deg

120573119903 = 180 deg




119904 120572119897 + 120573119897 ሶ119909 minus 119888 120572119897 + 120573119897 ሶ119910 minus 119888120573119897 119889 ሶ120579 = 119903119897 ሶ120595119897

119888 120572119897 + 120573119897 ሶ119909 + 119904 120572119897 + 120573119897 ሶ119910 + 119904120573119897 119889 ሶ120579 = 0



ሶ119909 minus 119889 ሶ120579 = 119903119897 ሶ120595119897

ሶ119910 = 0

ሶ119909 + 119889 ሶ120579 = 119903119903 ሶ120595119903119910 = 0




ሶ119909ሶ119910ሶ120579

=

1

2119903119897 ሶ120595119897 + 119903119903 ሶ120595119903

01

2119889minus119903119897 ሶ120595119897 + 119903119903 ሶ120595119903



rrlabcsuni-klde




rrlabcsuni-klde











1205721 = 0 deg

1205722 = 120 deg

1205723 = minus120 deg







119888 1205721 + 1205731 ሶ119909 + 119904 1205721 + 1205731 ሶ119910 + 119889 sdot 119904 1205731 ሶ120579 = 0

119888 1205722 + 1205732 ሶ119909 + 119904 1205722 + 1205732 ሶ119910 + 119889 sdot 119904 1205732 ሶ120579 = 0

119888 1205723 + 1205733 ሶ119909 + 119904 1205723 + 1205733 ሶ119910 + 119889 sdot 119904 1205733 ሶ120579 = 0







119888 120572119894 sdot ሶ119909 + 119904 120572119894 sdot ሶ119910




ሶ120595119894 =1


rrlabcsuni-klde



rrlabcsuni-klde

Mecanum Vehicle





Mecanum Kinematics



1205721 = 45 deg 1205731 = 45 deg 1205741 = minus45 deg

1205722 = 135 deg 1205732 = minus45 deg 1205742 = 45 deg


1205724 = minus45 deg 1205734 = 135 deg 1205744 = 45 deg


Mecanum Kinematics








Mecanum Kinematics


ሶ119909 =119903

4ሶ1205951 + ሶ1205952 + ሶ1205953 + ሶ1205954

ሶ119910 =119903

4minus ሶ1205951 + ሶ1205952 minus ሶ1205953 + ሶ1205954

ሶ120579 =119903

119889 2ሶ1205951 minus ሶ1205952 minus ሶ1205953 + ሶ1205954


Mecanum Kinematics


ሶ119909ሶ119910ሶ120579

=119903119908ℎ1198901198901198974


sdot

ሶ1205951

ሶ1205952

ሶ1205953

ሶ1205954

with a =2 2

119889


ሶ1205954 = ሶ1205951 + ሶ1205952 minus ሶ1205953

rrlabcsuni-klde





Ability to maneuver

Controllability

Stability



Stability




Typical setups




Maneuverability








Controllability



Examples




Wheel type

Drive geometry


Motortransmission


Degree of Mobility




119891119898 = 3 minus rank 119878


Degree of Mobility



1198891 = 1198892 1205731 = 1205732 = 0 1205721 + 120587 = 1205722






119891119904 = rank 119878prime








119891119872 = 119891119898 + 119891119904

rrlabcsuni-klde

Vehicle Kinematics



Tricycle Drive

Given 120575 119907


119903 = 119897 sdot cot 120575

119907prime =119907

cos 120575

119907119897 = 119907 sdot119903 + 119889

119903

119907119903 = 119907 sdot119903 minus 119889


120575


Ackermann Steering

Given 119907119863 120575 119897 119889

Unknown 119907119903119903 119907119897119903 119907119903119891 119907119897119891

Solution

120602 =119897

tan 120575

119907119897119903 =120602 minus 1198892 sdot 119907119863

120602

119907119903119903 =120602 + 1198892 sdot 119907119863

120602

119907119897119891 =120602 minus 1198892 2 + 1198972 sdot tan 120575

119897sdot 119907119863

119907119903119891 =1206022 + 1198972 sdot tan 120575

119897sdot 119907119863


120575

120602

rrlabcsuni-klde



Given 120602 120593 119907 119889



1205932 = arctan120549119910 + 119889

1205491199091206022 = 120602 sdot cos120593 cos1205932

1199072 = 119907 sdot 1206022120602

120549119910 = 120602 ∙ 119904119894119899 120593

1205931

1205932

1206021

1206022

120593 120549119909 = 120602 ∙ cos120593

120602

119889

rrlabcsuni-klde

Bicycle Model


119907


Bicycle Model


Low speed motion



119865119910 =1198981199072

120602






Bicycle Model

Given 120575119891 120575119903 119907 120579

Unknown ሶ119883 ሶ119884 ሶ120579

Solution

120575119907 = tanminus1119897119891 tan 120575119903+119897119903 tan 120575119891

119897119891+119897119903

ሶ120579 =119907

120602

120602 =119897119891+119897119903

tan 120575119891 minustan 120575119903 cos 120575119907

ሶ119883 = 119907 cos 120579 + 120575119907ሶ119884 = 119907 sin 120579 + 120575119907

ሶ120579 =119907 cos 120575119907

119897119891+119897119903tan 120575119891 minus tan 120575119903

119907

rrlabcsuni-klde



119862 = 119874 120575119903 = 0 120575119891 = 120575

119897119903 = 0 119897119891 = 119897

120575119907 = 0

ሶ120579 =119907

120602

120602 =119897

tan 120575119891

ሶ119883 = 119907 cos 120579ሶ119884 = 119907 sin 120579ሶ120579 =

119907

119897119891tan 120575119891

119907





polygon lines grids

boxes

AMR position can be


point in space

center of mass

kinematic center



Configuration Space



2D robot C-space


Configuration Space







2D robot c-space


Configuration Space





rrlabcsuni-klde

Coordinate Systems









119861119906 =119909119910119911

+ 119874119861119877 120593120595 120579 119874 119906





119877 119906Ԧ119901119879 119904










119877119910 120595 =cos120595 0 sin1205950 1 0



120593 119910

119907

119911119908





119877 = 119877119911 120593 sdot 119877119909 120595 sdot 119877119911prime θ


119877 = 119877119911 θ sdot 119877119910 120595 sdot 119877119909 120593


Roll Pitch Yaw




Yaw θ

Pitchψ

Roll 120593

119911

119910119909

20

Roll

Pitch Yaw

119909

119911119910






119877119904 =



Linear Velocity




119860119877119861 Ԧ119907119902


119860 Ԧ119907119902 =119860 Ԧ119907119874119861 + 119861

119860119877119861 Ԧ119907119902



calculated with

119860 Ԧ119907119876 = 119860 Ԧ119907119874119861 + 119861119860119877119861 Ԧ119907119876 +

119860ω119861 times 119861119860119877119861 Ԧ119902


Rotational Velocity



119860120596 = 119861119860119877119861120596





119894+1120596119894+1 = 119894119894+1119877 sdot 119894120596119894 + ሶθ119894+1

119894+1 Ԧ119890119911119894+1

119894+1 Ԧ119907119894+1 = 119894119894+1119877 119894 Ԧ119907119894 +

119894120596119894 times119894 Ԧ119901119894+1






119894119877 from frame 119894 to

119894 + 1





119894119894+1119877 θ = 119894+1

119894119877minus1 θ = 119894+1119894119877119879 θ =

119888θ 119904θ 119909119908minus119904θ 119888θ 1199101199080 0 1

rrlabcsuni-klde



YW

XW

θ

YR

XR

P









coordinates





0 0 1


ሶԦ120585119877 = 119877 θሶԦ120585119882


YW

XW

θ

YR

XR




119877120587

2=

0 1 0minus1 0 00 0 1


ሶԦ120585119877 = 119877120587

2ሶ

120585119882 =0 1 0minus1 0 00 0 1


=ሶ119910


rrlabcsuni-klde


Analytical Solution






YW

XW

θ

V(t)

120596(t)







ሶԦ120585119882 = ሶ119909 ሶ119910 ሶ120579119879

= 119891 119889 119903 120579 ሶ120595119897 ሶ120595119903

= 119877 120579 minus1 ሶԦ120585119877

passive wheel

active wheel

d






ሶ119909119877119903 =119903 ሶ1205951199032

ሶ119909119877119897 =119903 ሶ120595119897

2

120596119903 =119903 ሶ1205951199032119889

120596119897 =minus119903 ሶ120595119897

2119889

ሶԦ120585119882 = 119877 120579 minus1 ሶԦ120585119903ሶԦ120585119903 =

119903 ሶ1205951199032

+119903 ሶ120595119897

2 0

119903 ሶ1205951199032119889

minus119903 ሶ120595119897

2119889

119879





Given 120579 =120587



119904

ሶԦ120585119882 =ሶ119909ሶ119910ሶ120579

=0 minus1 01 0 00 0 1

sdot301

=031







Consider




rrlabcsuni-klde

Basic Wheel Types


Standardwheel

Steerablestandard

wheel

Castorwheel

Swedish orMecanum

wheel

Ball orspherical

wheel


Wheel Properties




Castor wheel


















ሶ120595 119903



ሶ120595 119903



ሶ120595 119903



ሶ120595 119903



ሶ120595 119903








and 1 Ԧ1199071 = ሶ119909 ሶ119910 0 119879


119894+1120596119894+1 = 119894119894+1119877 sdot 119894120596119894 + ሶ120579119894+1

119894+1 Ԧ119890119911119894+1

119894+1 Ԧ119907119894+1 = 119894119894+1119877 119894 Ԧ119907119894 +

119894120596119894 times119894 Ԧ119901119894+1




calculated as

2 Ԧ1199072 =119888120572 ሶ119909 + 119904120572 ሶ119910minus119904120572 ሶ119909 + 119888120572 ሶ119910

0


3 Ԧ1199073 =

119888120572 ሶ119909 + 119904120572 ሶ119910

minus119904120572 ሶ119909 + 119888120572 ሶ119910 + 119889 ሶ1205790






4 Ԧ1199074 =119904 120572 + 120573 ሶ119909 minus 119888 120572 + 120573 ሶ119910 minus 119888120573 119889 ሶ120579

119888 120572 + 120573 ሶ119909 + 119904 120572 + 120573 ሶ119910 + 119904120573 119889 ሶ1205790




119904 120572 + 120573 minus119888 120572 + 120573 minus119888120573 119889

119888 120572 + 120573 119904 120572 + 120573 119904120573 1198890 0 0

ሶ119909119910 ሶሶ120579

=119903 ሶ12059500












119904 120572 + 120573(119905) minus119888 120572 + 120573(119905) minus119888120573(119905)119889

119888 120572 + 120573(119905) 119904 120572 + 120573(119905) 119904120573(119905)1198890 0 0

ሶ119909ሶ119910ሶ120579

=119903 ሶ120595

minus119889119888 ሶ120573(119905)0


maneuverability)




119904 120572 + 120573 + 120574 minus119888 120572 + 120573 + 120574 minus119888 120573 + 120574 119889

119888 120572 + 120573 + 120574 119904 120572 + 120573 + 120574 119904 120573 + 120574 1198890 0 0

ሶ119909ሶ119910ሶ120579

=119903 ሶ120595 cos 120574

119903 ሶ120595 sin 120574 + 119903119901119903 ሶ1205951199011199030















120572119897 = 90 deg

120573119897 = 0 deg

120572119903 = minus90 deg

120573119903 = 180 deg




119904 120572119897 + 120573119897 ሶ119909 minus 119888 120572119897 + 120573119897 ሶ119910 minus 119888120573119897 119889 ሶ120579 = 119903119897 ሶ120595119897

119888 120572119897 + 120573119897 ሶ119909 + 119904 120572119897 + 120573119897 ሶ119910 + 119904120573119897 119889 ሶ120579 = 0



ሶ119909 minus 119889 ሶ120579 = 119903119897 ሶ120595119897

ሶ119910 = 0

ሶ119909 + 119889 ሶ120579 = 119903119903 ሶ120595119903119910 = 0




ሶ119909ሶ119910ሶ120579

=

1

2119903119897 ሶ120595119897 + 119903119903 ሶ120595119903

01

2119889minus119903119897 ሶ120595119897 + 119903119903 ሶ120595119903



rrlabcsuni-klde




rrlabcsuni-klde











1205721 = 0 deg

1205722 = 120 deg

1205723 = minus120 deg







119888 1205721 + 1205731 ሶ119909 + 119904 1205721 + 1205731 ሶ119910 + 119889 sdot 119904 1205731 ሶ120579 = 0

119888 1205722 + 1205732 ሶ119909 + 119904 1205722 + 1205732 ሶ119910 + 119889 sdot 119904 1205732 ሶ120579 = 0

119888 1205723 + 1205733 ሶ119909 + 119904 1205723 + 1205733 ሶ119910 + 119889 sdot 119904 1205733 ሶ120579 = 0







119888 120572119894 sdot ሶ119909 + 119904 120572119894 sdot ሶ119910




ሶ120595119894 =1


rrlabcsuni-klde



rrlabcsuni-klde

Mecanum Vehicle





Mecanum Kinematics



1205721 = 45 deg 1205731 = 45 deg 1205741 = minus45 deg

1205722 = 135 deg 1205732 = minus45 deg 1205742 = 45 deg


1205724 = minus45 deg 1205734 = 135 deg 1205744 = 45 deg


Mecanum Kinematics








Mecanum Kinematics


ሶ119909 =119903

4ሶ1205951 + ሶ1205952 + ሶ1205953 + ሶ1205954

ሶ119910 =119903

4minus ሶ1205951 + ሶ1205952 minus ሶ1205953 + ሶ1205954

ሶ120579 =119903

119889 2ሶ1205951 minus ሶ1205952 minus ሶ1205953 + ሶ1205954


Mecanum Kinematics


ሶ119909ሶ119910ሶ120579

=119903119908ℎ1198901198901198974


sdot

ሶ1205951

ሶ1205952

ሶ1205953

ሶ1205954

with a =2 2

119889


ሶ1205954 = ሶ1205951 + ሶ1205952 minus ሶ1205953

rrlabcsuni-klde





Ability to maneuver

Controllability

Stability



Stability




Typical setups




Maneuverability








Controllability



Examples




Wheel type

Drive geometry


Motortransmission


Degree of Mobility




119891119898 = 3 minus rank 119878


Degree of Mobility



1198891 = 1198892 1205731 = 1205732 = 0 1205721 + 120587 = 1205722






119891119904 = rank 119878prime








119891119872 = 119891119898 + 119891119904

rrlabcsuni-klde

Vehicle Kinematics



Tricycle Drive

Given 120575 119907


119903 = 119897 sdot cot 120575

119907prime =119907

cos 120575

119907119897 = 119907 sdot119903 + 119889

119903

119907119903 = 119907 sdot119903 minus 119889


120575


Ackermann Steering

Given 119907119863 120575 119897 119889

Unknown 119907119903119903 119907119897119903 119907119903119891 119907119897119891

Solution

120602 =119897

tan 120575

119907119897119903 =120602 minus 1198892 sdot 119907119863

120602

119907119903119903 =120602 + 1198892 sdot 119907119863

120602

119907119897119891 =120602 minus 1198892 2 + 1198972 sdot tan 120575

119897sdot 119907119863

119907119903119891 =1206022 + 1198972 sdot tan 120575

119897sdot 119907119863


120575

120602

rrlabcsuni-klde



Given 120602 120593 119907 119889



1205932 = arctan120549119910 + 119889

1205491199091206022 = 120602 sdot cos120593 cos1205932

1199072 = 119907 sdot 1206022120602

120549119910 = 120602 ∙ 119904119894119899 120593

1205931

1205932

1206021

1206022

120593 120549119909 = 120602 ∙ cos120593

120602

119889

rrlabcsuni-klde

Bicycle Model


119907


Bicycle Model


Low speed motion



119865119910 =1198981199072

120602






Bicycle Model

Given 120575119891 120575119903 119907 120579

Unknown ሶ119883 ሶ119884 ሶ120579

Solution

120575119907 = tanminus1119897119891 tan 120575119903+119897119903 tan 120575119891

119897119891+119897119903

ሶ120579 =119907

120602

120602 =119897119891+119897119903

tan 120575119891 minustan 120575119903 cos 120575119907

ሶ119883 = 119907 cos 120579 + 120575119907ሶ119884 = 119907 sin 120579 + 120575119907

ሶ120579 =119907 cos 120575119907

119897119891+119897119903tan 120575119891 minus tan 120575119903

119907

rrlabcsuni-klde



119862 = 119874 120575119903 = 0 120575119891 = 120575

119897119903 = 0 119897119891 = 119897

120575119907 = 0

ሶ120579 =119907

120602

120602 =119897

tan 120575119891

ሶ119883 = 119907 cos 120579ሶ119884 = 119907 sin 120579ሶ120579 =

119907

119897119891tan 120575119891

119907


Configuration Space



2D robot C-space


Configuration Space







2D robot c-space


Configuration Space





rrlabcsuni-klde

Coordinate Systems









119861119906 =119909119910119911

+ 119874119861119877 120593120595 120579 119874 119906





119877 119906Ԧ119901119879 119904










119877119910 120595 =cos120595 0 sin1205950 1 0



120593 119910

119907

119911119908





119877 = 119877119911 120593 sdot 119877119909 120595 sdot 119877119911prime θ


119877 = 119877119911 θ sdot 119877119910 120595 sdot 119877119909 120593


Roll Pitch Yaw




Yaw θ

Pitchψ

Roll 120593

119911

119910119909

20

Roll

Pitch Yaw

119909

119911119910






119877119904 =



Linear Velocity




119860119877119861 Ԧ119907119902


119860 Ԧ119907119902 =119860 Ԧ119907119874119861 + 119861

119860119877119861 Ԧ119907119902



calculated with

119860 Ԧ119907119876 = 119860 Ԧ119907119874119861 + 119861119860119877119861 Ԧ119907119876 +

119860ω119861 times 119861119860119877119861 Ԧ119902


Rotational Velocity



119860120596 = 119861119860119877119861120596





119894+1120596119894+1 = 119894119894+1119877 sdot 119894120596119894 + ሶθ119894+1

119894+1 Ԧ119890119911119894+1

119894+1 Ԧ119907119894+1 = 119894119894+1119877 119894 Ԧ119907119894 +

119894120596119894 times119894 Ԧ119901119894+1






119894119877 from frame 119894 to

119894 + 1





119894119894+1119877 θ = 119894+1

119894119877minus1 θ = 119894+1119894119877119879 θ =

119888θ 119904θ 119909119908minus119904θ 119888θ 1199101199080 0 1

rrlabcsuni-klde



YW

XW

θ

YR

XR

P









coordinates





0 0 1


ሶԦ120585119877 = 119877 θሶԦ120585119882


YW

XW

θ

YR

XR




119877120587

2=

0 1 0minus1 0 00 0 1


ሶԦ120585119877 = 119877120587

2ሶ

120585119882 =0 1 0minus1 0 00 0 1


=ሶ119910


rrlabcsuni-klde


Analytical Solution






YW

XW

θ

V(t)

120596(t)







ሶԦ120585119882 = ሶ119909 ሶ119910 ሶ120579119879

= 119891 119889 119903 120579 ሶ120595119897 ሶ120595119903

= 119877 120579 minus1 ሶԦ120585119877

passive wheel

active wheel

d






ሶ119909119877119903 =119903 ሶ1205951199032

ሶ119909119877119897 =119903 ሶ120595119897

2

120596119903 =119903 ሶ1205951199032119889

120596119897 =minus119903 ሶ120595119897

2119889

ሶԦ120585119882 = 119877 120579 minus1 ሶԦ120585119903ሶԦ120585119903 =

119903 ሶ1205951199032

+119903 ሶ120595119897

2 0

119903 ሶ1205951199032119889

minus119903 ሶ120595119897

2119889

119879





Given 120579 =120587



119904

ሶԦ120585119882 =ሶ119909ሶ119910ሶ120579

=0 minus1 01 0 00 0 1

sdot301

=031







Consider




rrlabcsuni-klde

Basic Wheel Types


Standardwheel

Steerablestandard

wheel

Castorwheel

Swedish orMecanum

wheel

Ball orspherical

wheel


Wheel Properties




Castor wheel


















ሶ120595 119903



ሶ120595 119903



ሶ120595 119903



ሶ120595 119903



ሶ120595 119903








and 1 Ԧ1199071 = ሶ119909 ሶ119910 0 119879


119894+1120596119894+1 = 119894119894+1119877 sdot 119894120596119894 + ሶ120579119894+1

119894+1 Ԧ119890119911119894+1

119894+1 Ԧ119907119894+1 = 119894119894+1119877 119894 Ԧ119907119894 +

119894120596119894 times119894 Ԧ119901119894+1




calculated as

2 Ԧ1199072 =119888120572 ሶ119909 + 119904120572 ሶ119910minus119904120572 ሶ119909 + 119888120572 ሶ119910

0


3 Ԧ1199073 =

119888120572 ሶ119909 + 119904120572 ሶ119910

minus119904120572 ሶ119909 + 119888120572 ሶ119910 + 119889 ሶ1205790






4 Ԧ1199074 =119904 120572 + 120573 ሶ119909 minus 119888 120572 + 120573 ሶ119910 minus 119888120573 119889 ሶ120579

119888 120572 + 120573 ሶ119909 + 119904 120572 + 120573 ሶ119910 + 119904120573 119889 ሶ1205790




119904 120572 + 120573 minus119888 120572 + 120573 minus119888120573 119889

119888 120572 + 120573 119904 120572 + 120573 119904120573 1198890 0 0

ሶ119909119910 ሶሶ120579

=119903 ሶ12059500












119904 120572 + 120573(119905) minus119888 120572 + 120573(119905) minus119888120573(119905)119889

119888 120572 + 120573(119905) 119904 120572 + 120573(119905) 119904120573(119905)1198890 0 0

ሶ119909ሶ119910ሶ120579

=119903 ሶ120595

minus119889119888 ሶ120573(119905)0


maneuverability)




119904 120572 + 120573 + 120574 minus119888 120572 + 120573 + 120574 minus119888 120573 + 120574 119889

119888 120572 + 120573 + 120574 119904 120572 + 120573 + 120574 119904 120573 + 120574 1198890 0 0

ሶ119909ሶ119910ሶ120579

=119903 ሶ120595 cos 120574

119903 ሶ120595 sin 120574 + 119903119901119903 ሶ1205951199011199030















120572119897 = 90 deg

120573119897 = 0 deg

120572119903 = minus90 deg

120573119903 = 180 deg




119904 120572119897 + 120573119897 ሶ119909 minus 119888 120572119897 + 120573119897 ሶ119910 minus 119888120573119897 119889 ሶ120579 = 119903119897 ሶ120595119897

119888 120572119897 + 120573119897 ሶ119909 + 119904 120572119897 + 120573119897 ሶ119910 + 119904120573119897 119889 ሶ120579 = 0



ሶ119909 minus 119889 ሶ120579 = 119903119897 ሶ120595119897

ሶ119910 = 0

ሶ119909 + 119889 ሶ120579 = 119903119903 ሶ120595119903119910 = 0




ሶ119909ሶ119910ሶ120579

=

1

2119903119897 ሶ120595119897 + 119903119903 ሶ120595119903

01

2119889minus119903119897 ሶ120595119897 + 119903119903 ሶ120595119903



rrlabcsuni-klde




rrlabcsuni-klde











1205721 = 0 deg

1205722 = 120 deg

1205723 = minus120 deg







119888 1205721 + 1205731 ሶ119909 + 119904 1205721 + 1205731 ሶ119910 + 119889 sdot 119904 1205731 ሶ120579 = 0

119888 1205722 + 1205732 ሶ119909 + 119904 1205722 + 1205732 ሶ119910 + 119889 sdot 119904 1205732 ሶ120579 = 0

119888 1205723 + 1205733 ሶ119909 + 119904 1205723 + 1205733 ሶ119910 + 119889 sdot 119904 1205733 ሶ120579 = 0







119888 120572119894 sdot ሶ119909 + 119904 120572119894 sdot ሶ119910




ሶ120595119894 =1


rrlabcsuni-klde



rrlabcsuni-klde

Mecanum Vehicle





Mecanum Kinematics



1205721 = 45 deg 1205731 = 45 deg 1205741 = minus45 deg

1205722 = 135 deg 1205732 = minus45 deg 1205742 = 45 deg


1205724 = minus45 deg 1205734 = 135 deg 1205744 = 45 deg


Mecanum Kinematics








Mecanum Kinematics


ሶ119909 =119903

4ሶ1205951 + ሶ1205952 + ሶ1205953 + ሶ1205954

ሶ119910 =119903

4minus ሶ1205951 + ሶ1205952 minus ሶ1205953 + ሶ1205954

ሶ120579 =119903

119889 2ሶ1205951 minus ሶ1205952 minus ሶ1205953 + ሶ1205954


Mecanum Kinematics


ሶ119909ሶ119910ሶ120579

=119903119908ℎ1198901198901198974


sdot

ሶ1205951

ሶ1205952

ሶ1205953

ሶ1205954

with a =2 2

119889


ሶ1205954 = ሶ1205951 + ሶ1205952 minus ሶ1205953

rrlabcsuni-klde





Ability to maneuver

Controllability

Stability



Stability




Typical setups




Maneuverability








Controllability



Examples




Wheel type

Drive geometry


Motortransmission


Degree of Mobility




119891119898 = 3 minus rank 119878


Degree of Mobility



1198891 = 1198892 1205731 = 1205732 = 0 1205721 + 120587 = 1205722






119891119904 = rank 119878prime








119891119872 = 119891119898 + 119891119904

rrlabcsuni-klde

Vehicle Kinematics



Tricycle Drive

Given 120575 119907


119903 = 119897 sdot cot 120575

119907prime =119907

cos 120575

119907119897 = 119907 sdot119903 + 119889

119903

119907119903 = 119907 sdot119903 minus 119889


120575


Ackermann Steering

Given 119907119863 120575 119897 119889

Unknown 119907119903119903 119907119897119903 119907119903119891 119907119897119891

Solution

120602 =119897

tan 120575

119907119897119903 =120602 minus 1198892 sdot 119907119863

120602

119907119903119903 =120602 + 1198892 sdot 119907119863

120602

119907119897119891 =120602 minus 1198892 2 + 1198972 sdot tan 120575

119897sdot 119907119863

119907119903119891 =1206022 + 1198972 sdot tan 120575

119897sdot 119907119863


120575

120602

rrlabcsuni-klde



Given 120602 120593 119907 119889



1205932 = arctan120549119910 + 119889

1205491199091206022 = 120602 sdot cos120593 cos1205932

1199072 = 119907 sdot 1206022120602

120549119910 = 120602 ∙ 119904119894119899 120593

1205931

1205932

1206021

1206022

120593 120549119909 = 120602 ∙ cos120593

120602

119889

rrlabcsuni-klde

Bicycle Model


119907


Bicycle Model


Low speed motion



119865119910 =1198981199072

120602






Bicycle Model

Given 120575119891 120575119903 119907 120579

Unknown ሶ119883 ሶ119884 ሶ120579

Solution

120575119907 = tanminus1119897119891 tan 120575119903+119897119903 tan 120575119891

119897119891+119897119903

ሶ120579 =119907

120602

120602 =119897119891+119897119903

tan 120575119891 minustan 120575119903 cos 120575119907

ሶ119883 = 119907 cos 120579 + 120575119907ሶ119884 = 119907 sin 120579 + 120575119907

ሶ120579 =119907 cos 120575119907

119897119891+119897119903tan 120575119891 minus tan 120575119903

119907

rrlabcsuni-klde



119862 = 119874 120575119903 = 0 120575119891 = 120575

119897119903 = 0 119897119891 = 119897

120575119907 = 0

ሶ120579 =119907

120602

120602 =119897

tan 120575119891

ሶ119883 = 119907 cos 120579ሶ119884 = 119907 sin 120579ሶ120579 =

119907

119897119891tan 120575119891

119907


Configuration Space







2D robot c-space


Configuration Space





rrlabcsuni-klde

Coordinate Systems









119861119906 =119909119910119911

+ 119874119861119877 120593120595 120579 119874 119906





119877 119906Ԧ119901119879 119904










119877119910 120595 =cos120595 0 sin1205950 1 0



120593 119910

119907

119911119908





119877 = 119877119911 120593 sdot 119877119909 120595 sdot 119877119911prime θ


119877 = 119877119911 θ sdot 119877119910 120595 sdot 119877119909 120593


Roll Pitch Yaw




Yaw θ

Pitchψ

Roll 120593

119911

119910119909

20

Roll

Pitch Yaw

119909

119911119910






119877119904 =



Linear Velocity




119860119877119861 Ԧ119907119902


119860 Ԧ119907119902 =119860 Ԧ119907119874119861 + 119861

119860119877119861 Ԧ119907119902



calculated with

119860 Ԧ119907119876 = 119860 Ԧ119907119874119861 + 119861119860119877119861 Ԧ119907119876 +

119860ω119861 times 119861119860119877119861 Ԧ119902


Rotational Velocity



119860120596 = 119861119860119877119861120596





119894+1120596119894+1 = 119894119894+1119877 sdot 119894120596119894 + ሶθ119894+1

119894+1 Ԧ119890119911119894+1

119894+1 Ԧ119907119894+1 = 119894119894+1119877 119894 Ԧ119907119894 +

119894120596119894 times119894 Ԧ119901119894+1






119894119877 from frame 119894 to

119894 + 1





119894119894+1119877 θ = 119894+1

119894119877minus1 θ = 119894+1119894119877119879 θ =

119888θ 119904θ 119909119908minus119904θ 119888θ 1199101199080 0 1

rrlabcsuni-klde



YW

XW

θ

YR

XR

P









coordinates





0 0 1


ሶԦ120585119877 = 119877 θሶԦ120585119882


YW

XW

θ

YR

XR




119877120587

2=

0 1 0minus1 0 00 0 1


ሶԦ120585119877 = 119877120587

2ሶ

120585119882 =0 1 0minus1 0 00 0 1


=ሶ119910


rrlabcsuni-klde


Analytical Solution






YW

XW

θ

V(t)

120596(t)







ሶԦ120585119882 = ሶ119909 ሶ119910 ሶ120579119879

= 119891 119889 119903 120579 ሶ120595119897 ሶ120595119903

= 119877 120579 minus1 ሶԦ120585119877

passive wheel

active wheel

d






ሶ119909119877119903 =119903 ሶ1205951199032

ሶ119909119877119897 =119903 ሶ120595119897

2

120596119903 =119903 ሶ1205951199032119889

120596119897 =minus119903 ሶ120595119897

2119889

ሶԦ120585119882 = 119877 120579 minus1 ሶԦ120585119903ሶԦ120585119903 =

119903 ሶ1205951199032

+119903 ሶ120595119897

2 0

119903 ሶ1205951199032119889

minus119903 ሶ120595119897

2119889

119879





Given 120579 =120587



119904

ሶԦ120585119882 =ሶ119909ሶ119910ሶ120579

=0 minus1 01 0 00 0 1

sdot301

=031







Consider




rrlabcsuni-klde

Basic Wheel Types


Standardwheel

Steerablestandard

wheel

Castorwheel

Swedish orMecanum

wheel

Ball orspherical

wheel


Wheel Properties




Castor wheel


















ሶ120595 119903



ሶ120595 119903



ሶ120595 119903



ሶ120595 119903



ሶ120595 119903








and 1 Ԧ1199071 = ሶ119909 ሶ119910 0 119879


119894+1120596119894+1 = 119894119894+1119877 sdot 119894120596119894 + ሶ120579119894+1

119894+1 Ԧ119890119911119894+1

119894+1 Ԧ119907119894+1 = 119894119894+1119877 119894 Ԧ119907119894 +

119894120596119894 times119894 Ԧ119901119894+1




calculated as

2 Ԧ1199072 =119888120572 ሶ119909 + 119904120572 ሶ119910minus119904120572 ሶ119909 + 119888120572 ሶ119910

0


3 Ԧ1199073 =

119888120572 ሶ119909 + 119904120572 ሶ119910

minus119904120572 ሶ119909 + 119888120572 ሶ119910 + 119889 ሶ1205790






4 Ԧ1199074 =119904 120572 + 120573 ሶ119909 minus 119888 120572 + 120573 ሶ119910 minus 119888120573 119889 ሶ120579

119888 120572 + 120573 ሶ119909 + 119904 120572 + 120573 ሶ119910 + 119904120573 119889 ሶ1205790




119904 120572 + 120573 minus119888 120572 + 120573 minus119888120573 119889

119888 120572 + 120573 119904 120572 + 120573 119904120573 1198890 0 0

ሶ119909119910 ሶሶ120579

=119903 ሶ12059500












119904 120572 + 120573(119905) minus119888 120572 + 120573(119905) minus119888120573(119905)119889

119888 120572 + 120573(119905) 119904 120572 + 120573(119905) 119904120573(119905)1198890 0 0

ሶ119909ሶ119910ሶ120579

=119903 ሶ120595

minus119889119888 ሶ120573(119905)0


maneuverability)




119904 120572 + 120573 + 120574 minus119888 120572 + 120573 + 120574 minus119888 120573 + 120574 119889

119888 120572 + 120573 + 120574 119904 120572 + 120573 + 120574 119904 120573 + 120574 1198890 0 0

ሶ119909ሶ119910ሶ120579

=119903 ሶ120595 cos 120574

119903 ሶ120595 sin 120574 + 119903119901119903 ሶ1205951199011199030















120572119897 = 90 deg

120573119897 = 0 deg

120572119903 = minus90 deg

120573119903 = 180 deg




119904 120572119897 + 120573119897 ሶ119909 minus 119888 120572119897 + 120573119897 ሶ119910 minus 119888120573119897 119889 ሶ120579 = 119903119897 ሶ120595119897

119888 120572119897 + 120573119897 ሶ119909 + 119904 120572119897 + 120573119897 ሶ119910 + 119904120573119897 119889 ሶ120579 = 0



ሶ119909 minus 119889 ሶ120579 = 119903119897 ሶ120595119897

ሶ119910 = 0

ሶ119909 + 119889 ሶ120579 = 119903119903 ሶ120595119903119910 = 0




ሶ119909ሶ119910ሶ120579

=

1

2119903119897 ሶ120595119897 + 119903119903 ሶ120595119903

01

2119889minus119903119897 ሶ120595119897 + 119903119903 ሶ120595119903



rrlabcsuni-klde




rrlabcsuni-klde











1205721 = 0 deg

1205722 = 120 deg

1205723 = minus120 deg







119888 1205721 + 1205731 ሶ119909 + 119904 1205721 + 1205731 ሶ119910 + 119889 sdot 119904 1205731 ሶ120579 = 0

119888 1205722 + 1205732 ሶ119909 + 119904 1205722 + 1205732 ሶ119910 + 119889 sdot 119904 1205732 ሶ120579 = 0

119888 1205723 + 1205733 ሶ119909 + 119904 1205723 + 1205733 ሶ119910 + 119889 sdot 119904 1205733 ሶ120579 = 0







119888 120572119894 sdot ሶ119909 + 119904 120572119894 sdot ሶ119910




ሶ120595119894 =1


rrlabcsuni-klde



rrlabcsuni-klde

Mecanum Vehicle





Mecanum Kinematics



1205721 = 45 deg 1205731 = 45 deg 1205741 = minus45 deg

1205722 = 135 deg 1205732 = minus45 deg 1205742 = 45 deg


1205724 = minus45 deg 1205734 = 135 deg 1205744 = 45 deg


Mecanum Kinematics








Mecanum Kinematics


ሶ119909 =119903

4ሶ1205951 + ሶ1205952 + ሶ1205953 + ሶ1205954

ሶ119910 =119903

4minus ሶ1205951 + ሶ1205952 minus ሶ1205953 + ሶ1205954

ሶ120579 =119903

119889 2ሶ1205951 minus ሶ1205952 minus ሶ1205953 + ሶ1205954


Mecanum Kinematics


ሶ119909ሶ119910ሶ120579

=119903119908ℎ1198901198901198974


sdot

ሶ1205951

ሶ1205952

ሶ1205953

ሶ1205954

with a =2 2

119889


ሶ1205954 = ሶ1205951 + ሶ1205952 minus ሶ1205953

rrlabcsuni-klde





Ability to maneuver

Controllability

Stability



Stability




Typical setups




Maneuverability








Controllability



Examples




Wheel type

Drive geometry


Motortransmission


Degree of Mobility




119891119898 = 3 minus rank 119878


Degree of Mobility



1198891 = 1198892 1205731 = 1205732 = 0 1205721 + 120587 = 1205722






119891119904 = rank 119878prime








119891119872 = 119891119898 + 119891119904

rrlabcsuni-klde

Vehicle Kinematics



Tricycle Drive

Given 120575 119907


119903 = 119897 sdot cot 120575

119907prime =119907

cos 120575

119907119897 = 119907 sdot119903 + 119889

119903

119907119903 = 119907 sdot119903 minus 119889


120575


Ackermann Steering

Given 119907119863 120575 119897 119889

Unknown 119907119903119903 119907119897119903 119907119903119891 119907119897119891

Solution

120602 =119897

tan 120575

119907119897119903 =120602 minus 1198892 sdot 119907119863

120602

119907119903119903 =120602 + 1198892 sdot 119907119863

120602

119907119897119891 =120602 minus 1198892 2 + 1198972 sdot tan 120575

119897sdot 119907119863

119907119903119891 =1206022 + 1198972 sdot tan 120575

119897sdot 119907119863


120575

120602

rrlabcsuni-klde



Given 120602 120593 119907 119889



1205932 = arctan120549119910 + 119889

1205491199091206022 = 120602 sdot cos120593 cos1205932

1199072 = 119907 sdot 1206022120602

120549119910 = 120602 ∙ 119904119894119899 120593

1205931

1205932

1206021

1206022

120593 120549119909 = 120602 ∙ cos120593

120602

119889

rrlabcsuni-klde

Bicycle Model


119907


Bicycle Model


Low speed motion



119865119910 =1198981199072

120602






Bicycle Model

Given 120575119891 120575119903 119907 120579

Unknown ሶ119883 ሶ119884 ሶ120579

Solution

120575119907 = tanminus1119897119891 tan 120575119903+119897119903 tan 120575119891

119897119891+119897119903

ሶ120579 =119907

120602

120602 =119897119891+119897119903

tan 120575119891 minustan 120575119903 cos 120575119907

ሶ119883 = 119907 cos 120579 + 120575119907ሶ119884 = 119907 sin 120579 + 120575119907

ሶ120579 =119907 cos 120575119907

119897119891+119897119903tan 120575119891 minus tan 120575119903

119907

rrlabcsuni-klde



119862 = 119874 120575119903 = 0 120575119891 = 120575

119897119903 = 0 119897119891 = 119897

120575119907 = 0

ሶ120579 =119907

120602

120602 =119897

tan 120575119891

ሶ119883 = 119907 cos 120579ሶ119884 = 119907 sin 120579ሶ120579 =

119907

119897119891tan 120575119891

119907


Configuration Space





rrlabcsuni-klde

Coordinate Systems









119861119906 =119909119910119911

+ 119874119861119877 120593120595 120579 119874 119906





119877 119906Ԧ119901119879 119904










119877119910 120595 =cos120595 0 sin1205950 1 0



120593 119910

119907

119911119908





119877 = 119877119911 120593 sdot 119877119909 120595 sdot 119877119911prime θ


119877 = 119877119911 θ sdot 119877119910 120595 sdot 119877119909 120593


Roll Pitch Yaw




Yaw θ

Pitchψ

Roll 120593

119911

119910119909

20

Roll

Pitch Yaw

119909

119911119910






119877119904 =



Linear Velocity




119860119877119861 Ԧ119907119902


119860 Ԧ119907119902 =119860 Ԧ119907119874119861 + 119861

119860119877119861 Ԧ119907119902



calculated with

119860 Ԧ119907119876 = 119860 Ԧ119907119874119861 + 119861119860119877119861 Ԧ119907119876 +

119860ω119861 times 119861119860119877119861 Ԧ119902


Rotational Velocity



119860120596 = 119861119860119877119861120596





119894+1120596119894+1 = 119894119894+1119877 sdot 119894120596119894 + ሶθ119894+1

119894+1 Ԧ119890119911119894+1

119894+1 Ԧ119907119894+1 = 119894119894+1119877 119894 Ԧ119907119894 +

119894120596119894 times119894 Ԧ119901119894+1






119894119877 from frame 119894 to

119894 + 1





119894119894+1119877 θ = 119894+1

119894119877minus1 θ = 119894+1119894119877119879 θ =

119888θ 119904θ 119909119908minus119904θ 119888θ 1199101199080 0 1

rrlabcsuni-klde



YW

XW

θ

YR

XR

P









coordinates





0 0 1


ሶԦ120585119877 = 119877 θሶԦ120585119882


YW

XW

θ

YR

XR




119877120587

2=

0 1 0minus1 0 00 0 1


ሶԦ120585119877 = 119877120587

2ሶ

120585119882 =0 1 0minus1 0 00 0 1


=ሶ119910


rrlabcsuni-klde


Analytical Solution






YW

XW

θ

V(t)

120596(t)







ሶԦ120585119882 = ሶ119909 ሶ119910 ሶ120579119879

= 119891 119889 119903 120579 ሶ120595119897 ሶ120595119903

= 119877 120579 minus1 ሶԦ120585119877

passive wheel

active wheel

d






ሶ119909119877119903 =119903 ሶ1205951199032

ሶ119909119877119897 =119903 ሶ120595119897

2

120596119903 =119903 ሶ1205951199032119889

120596119897 =minus119903 ሶ120595119897

2119889

ሶԦ120585119882 = 119877 120579 minus1 ሶԦ120585119903ሶԦ120585119903 =

119903 ሶ1205951199032

+119903 ሶ120595119897

2 0

119903 ሶ1205951199032119889

minus119903 ሶ120595119897

2119889

119879





Given 120579 =120587



119904

ሶԦ120585119882 =ሶ119909ሶ119910ሶ120579

=0 minus1 01 0 00 0 1

sdot301

=031







Consider




rrlabcsuni-klde

Basic Wheel Types


Standardwheel

Steerablestandard

wheel

Castorwheel

Swedish orMecanum

wheel

Ball orspherical

wheel


Wheel Properties




Castor wheel


















ሶ120595 119903



ሶ120595 119903



ሶ120595 119903



ሶ120595 119903



ሶ120595 119903








and 1 Ԧ1199071 = ሶ119909 ሶ119910 0 119879


119894+1120596119894+1 = 119894119894+1119877 sdot 119894120596119894 + ሶ120579119894+1

119894+1 Ԧ119890119911119894+1

119894+1 Ԧ119907119894+1 = 119894119894+1119877 119894 Ԧ119907119894 +

119894120596119894 times119894 Ԧ119901119894+1




calculated as

2 Ԧ1199072 =119888120572 ሶ119909 + 119904120572 ሶ119910minus119904120572 ሶ119909 + 119888120572 ሶ119910

0


3 Ԧ1199073 =

119888120572 ሶ119909 + 119904120572 ሶ119910

minus119904120572 ሶ119909 + 119888120572 ሶ119910 + 119889 ሶ1205790






4 Ԧ1199074 =119904 120572 + 120573 ሶ119909 minus 119888 120572 + 120573 ሶ119910 minus 119888120573 119889 ሶ120579

119888 120572 + 120573 ሶ119909 + 119904 120572 + 120573 ሶ119910 + 119904120573 119889 ሶ1205790




119904 120572 + 120573 minus119888 120572 + 120573 minus119888120573 119889

119888 120572 + 120573 119904 120572 + 120573 119904120573 1198890 0 0

ሶ119909119910 ሶሶ120579

=119903 ሶ12059500












119904 120572 + 120573(119905) minus119888 120572 + 120573(119905) minus119888120573(119905)119889

119888 120572 + 120573(119905) 119904 120572 + 120573(119905) 119904120573(119905)1198890 0 0

ሶ119909ሶ119910ሶ120579

=119903 ሶ120595

minus119889119888 ሶ120573(119905)0


maneuverability)




119904 120572 + 120573 + 120574 minus119888 120572 + 120573 + 120574 minus119888 120573 + 120574 119889

119888 120572 + 120573 + 120574 119904 120572 + 120573 + 120574 119904 120573 + 120574 1198890 0 0

ሶ119909ሶ119910ሶ120579

=119903 ሶ120595 cos 120574

119903 ሶ120595 sin 120574 + 119903119901119903 ሶ1205951199011199030















120572119897 = 90 deg

120573119897 = 0 deg

120572119903 = minus90 deg

120573119903 = 180 deg




119904 120572119897 + 120573119897 ሶ119909 minus 119888 120572119897 + 120573119897 ሶ119910 minus 119888120573119897 119889 ሶ120579 = 119903119897 ሶ120595119897

119888 120572119897 + 120573119897 ሶ119909 + 119904 120572119897 + 120573119897 ሶ119910 + 119904120573119897 119889 ሶ120579 = 0



ሶ119909 minus 119889 ሶ120579 = 119903119897 ሶ120595119897

ሶ119910 = 0

ሶ119909 + 119889 ሶ120579 = 119903119903 ሶ120595119903119910 = 0




ሶ119909ሶ119910ሶ120579

=

1

2119903119897 ሶ120595119897 + 119903119903 ሶ120595119903

01

2119889minus119903119897 ሶ120595119897 + 119903119903 ሶ120595119903



rrlabcsuni-klde




rrlabcsuni-klde











1205721 = 0 deg

1205722 = 120 deg

1205723 = minus120 deg







119888 1205721 + 1205731 ሶ119909 + 119904 1205721 + 1205731 ሶ119910 + 119889 sdot 119904 1205731 ሶ120579 = 0

119888 1205722 + 1205732 ሶ119909 + 119904 1205722 + 1205732 ሶ119910 + 119889 sdot 119904 1205732 ሶ120579 = 0

119888 1205723 + 1205733 ሶ119909 + 119904 1205723 + 1205733 ሶ119910 + 119889 sdot 119904 1205733 ሶ120579 = 0







119888 120572119894 sdot ሶ119909 + 119904 120572119894 sdot ሶ119910




ሶ120595119894 =1


rrlabcsuni-klde



rrlabcsuni-klde

Mecanum Vehicle





Mecanum Kinematics



1205721 = 45 deg 1205731 = 45 deg 1205741 = minus45 deg

1205722 = 135 deg 1205732 = minus45 deg 1205742 = 45 deg


1205724 = minus45 deg 1205734 = 135 deg 1205744 = 45 deg


Mecanum Kinematics








Mecanum Kinematics


ሶ119909 =119903

4ሶ1205951 + ሶ1205952 + ሶ1205953 + ሶ1205954

ሶ119910 =119903

4minus ሶ1205951 + ሶ1205952 minus ሶ1205953 + ሶ1205954

ሶ120579 =119903

119889 2ሶ1205951 minus ሶ1205952 minus ሶ1205953 + ሶ1205954


Mecanum Kinematics


ሶ119909ሶ119910ሶ120579

=119903119908ℎ1198901198901198974


sdot

ሶ1205951

ሶ1205952

ሶ1205953

ሶ1205954

with a =2 2

119889


ሶ1205954 = ሶ1205951 + ሶ1205952 minus ሶ1205953

rrlabcsuni-klde





Ability to maneuver

Controllability

Stability



Stability




Typical setups




Maneuverability








Controllability



Examples




Wheel type

Drive geometry


Motortransmission


Degree of Mobility




119891119898 = 3 minus rank 119878


Degree of Mobility



1198891 = 1198892 1205731 = 1205732 = 0 1205721 + 120587 = 1205722






119891119904 = rank 119878prime








119891119872 = 119891119898 + 119891119904

rrlabcsuni-klde

Vehicle Kinematics



Tricycle Drive

Given 120575 119907


119903 = 119897 sdot cot 120575

119907prime =119907

cos 120575

119907119897 = 119907 sdot119903 + 119889

119903

119907119903 = 119907 sdot119903 minus 119889


120575


Ackermann Steering

Given 119907119863 120575 119897 119889

Unknown 119907119903119903 119907119897119903 119907119903119891 119907119897119891

Solution

120602 =119897

tan 120575

119907119897119903 =120602 minus 1198892 sdot 119907119863

120602

119907119903119903 =120602 + 1198892 sdot 119907119863

120602

119907119897119891 =120602 minus 1198892 2 + 1198972 sdot tan 120575

119897sdot 119907119863

119907119903119891 =1206022 + 1198972 sdot tan 120575

119897sdot 119907119863


120575

120602

rrlabcsuni-klde



Given 120602 120593 119907 119889



1205932 = arctan120549119910 + 119889

1205491199091206022 = 120602 sdot cos120593 cos1205932

1199072 = 119907 sdot 1206022120602

120549119910 = 120602 ∙ 119904119894119899 120593

1205931

1205932

1206021

1206022

120593 120549119909 = 120602 ∙ cos120593

120602

119889

rrlabcsuni-klde

Bicycle Model


119907


Bicycle Model


Low speed motion



119865119910 =1198981199072

120602






Bicycle Model

Given 120575119891 120575119903 119907 120579

Unknown ሶ119883 ሶ119884 ሶ120579

Solution

120575119907 = tanminus1119897119891 tan 120575119903+119897119903 tan 120575119891

119897119891+119897119903

ሶ120579 =119907

120602

120602 =119897119891+119897119903

tan 120575119891 minustan 120575119903 cos 120575119907

ሶ119883 = 119907 cos 120579 + 120575119907ሶ119884 = 119907 sin 120579 + 120575119907

ሶ120579 =119907 cos 120575119907

119897119891+119897119903tan 120575119891 minus tan 120575119903

119907

rrlabcsuni-klde



119862 = 119874 120575119903 = 0 120575119891 = 120575

119897119903 = 0 119897119891 = 119897

120575119907 = 0

ሶ120579 =119907

120602

120602 =119897

tan 120575119891

ሶ119883 = 119907 cos 120579ሶ119884 = 119907 sin 120579ሶ120579 =

119907

119897119891tan 120575119891

119907

rrlabcsuni-klde

Coordinate Systems









119861119906 =119909119910119911

+ 119874119861119877 120593120595 120579 119874 119906





119877 119906Ԧ119901119879 119904










119877119910 120595 =cos120595 0 sin1205950 1 0



120593 119910

119907

119911119908





119877 = 119877119911 120593 sdot 119877119909 120595 sdot 119877119911prime θ


119877 = 119877119911 θ sdot 119877119910 120595 sdot 119877119909 120593


Roll Pitch Yaw




Yaw θ

Pitchψ

Roll 120593

119911

119910119909

20

Roll

Pitch Yaw

119909

119911119910






119877119904 =



Linear Velocity




119860119877119861 Ԧ119907119902


119860 Ԧ119907119902 =119860 Ԧ119907119874119861 + 119861

119860119877119861 Ԧ119907119902



calculated with

119860 Ԧ119907119876 = 119860 Ԧ119907119874119861 + 119861119860119877119861 Ԧ119907119876 +

119860ω119861 times 119861119860119877119861 Ԧ119902


Rotational Velocity



119860120596 = 119861119860119877119861120596





119894+1120596119894+1 = 119894119894+1119877 sdot 119894120596119894 + ሶθ119894+1

119894+1 Ԧ119890119911119894+1

119894+1 Ԧ119907119894+1 = 119894119894+1119877 119894 Ԧ119907119894 +

119894120596119894 times119894 Ԧ119901119894+1






119894119877 from frame 119894 to

119894 + 1





119894119894+1119877 θ = 119894+1

119894119877minus1 θ = 119894+1119894119877119879 θ =

119888θ 119904θ 119909119908minus119904θ 119888θ 1199101199080 0 1

rrlabcsuni-klde



YW

XW

θ

YR

XR

P









coordinates





0 0 1


ሶԦ120585119877 = 119877 θሶԦ120585119882


YW

XW

θ

YR

XR




119877120587

2=

0 1 0minus1 0 00 0 1


ሶԦ120585119877 = 119877120587

2ሶ

120585119882 =0 1 0minus1 0 00 0 1


=ሶ119910


rrlabcsuni-klde


Analytical Solution






YW

XW

θ

V(t)

120596(t)







ሶԦ120585119882 = ሶ119909 ሶ119910 ሶ120579119879

= 119891 119889 119903 120579 ሶ120595119897 ሶ120595119903

= 119877 120579 minus1 ሶԦ120585119877

passive wheel

active wheel

d






ሶ119909119877119903 =119903 ሶ1205951199032

ሶ119909119877119897 =119903 ሶ120595119897

2

120596119903 =119903 ሶ1205951199032119889

120596119897 =minus119903 ሶ120595119897

2119889

ሶԦ120585119882 = 119877 120579 minus1 ሶԦ120585119903ሶԦ120585119903 =

119903 ሶ1205951199032

+119903 ሶ120595119897

2 0

119903 ሶ1205951199032119889

minus119903 ሶ120595119897

2119889

119879





Given 120579 =120587



119904

ሶԦ120585119882 =ሶ119909ሶ119910ሶ120579

=0 minus1 01 0 00 0 1

sdot301

=031







Consider




rrlabcsuni-klde

Basic Wheel Types


Standardwheel

Steerablestandard

wheel

Castorwheel

Swedish orMecanum

wheel

Ball orspherical

wheel


Wheel Properties




Castor wheel


















ሶ120595 119903



ሶ120595 119903



ሶ120595 119903



ሶ120595 119903



ሶ120595 119903








and 1 Ԧ1199071 = ሶ119909 ሶ119910 0 119879


119894+1120596119894+1 = 119894119894+1119877 sdot 119894120596119894 + ሶ120579119894+1

119894+1 Ԧ119890119911119894+1

119894+1 Ԧ119907119894+1 = 119894119894+1119877 119894 Ԧ119907119894 +

119894120596119894 times119894 Ԧ119901119894+1




calculated as

2 Ԧ1199072 =119888120572 ሶ119909 + 119904120572 ሶ119910minus119904120572 ሶ119909 + 119888120572 ሶ119910

0


3 Ԧ1199073 =

119888120572 ሶ119909 + 119904120572 ሶ119910

minus119904120572 ሶ119909 + 119888120572 ሶ119910 + 119889 ሶ1205790






4 Ԧ1199074 =119904 120572 + 120573 ሶ119909 minus 119888 120572 + 120573 ሶ119910 minus 119888120573 119889 ሶ120579

119888 120572 + 120573 ሶ119909 + 119904 120572 + 120573 ሶ119910 + 119904120573 119889 ሶ1205790




119904 120572 + 120573 minus119888 120572 + 120573 minus119888120573 119889

119888 120572 + 120573 119904 120572 + 120573 119904120573 1198890 0 0

ሶ119909119910 ሶሶ120579

=119903 ሶ12059500












119904 120572 + 120573(119905) minus119888 120572 + 120573(119905) minus119888120573(119905)119889

119888 120572 + 120573(119905) 119904 120572 + 120573(119905) 119904120573(119905)1198890 0 0

ሶ119909ሶ119910ሶ120579

=119903 ሶ120595

minus119889119888 ሶ120573(119905)0


maneuverability)




119904 120572 + 120573 + 120574 minus119888 120572 + 120573 + 120574 minus119888 120573 + 120574 119889

119888 120572 + 120573 + 120574 119904 120572 + 120573 + 120574 119904 120573 + 120574 1198890 0 0

ሶ119909ሶ119910ሶ120579

=119903 ሶ120595 cos 120574

119903 ሶ120595 sin 120574 + 119903119901119903 ሶ1205951199011199030















120572119897 = 90 deg

120573119897 = 0 deg

120572119903 = minus90 deg

120573119903 = 180 deg




119904 120572119897 + 120573119897 ሶ119909 minus 119888 120572119897 + 120573119897 ሶ119910 minus 119888120573119897 119889 ሶ120579 = 119903119897 ሶ120595119897

119888 120572119897 + 120573119897 ሶ119909 + 119904 120572119897 + 120573119897 ሶ119910 + 119904120573119897 119889 ሶ120579 = 0



ሶ119909 minus 119889 ሶ120579 = 119903119897 ሶ120595119897

ሶ119910 = 0

ሶ119909 + 119889 ሶ120579 = 119903119903 ሶ120595119903119910 = 0




ሶ119909ሶ119910ሶ120579

=

1

2119903119897 ሶ120595119897 + 119903119903 ሶ120595119903

01

2119889minus119903119897 ሶ120595119897 + 119903119903 ሶ120595119903



rrlabcsuni-klde




rrlabcsuni-klde











1205721 = 0 deg

1205722 = 120 deg

1205723 = minus120 deg







119888 1205721 + 1205731 ሶ119909 + 119904 1205721 + 1205731 ሶ119910 + 119889 sdot 119904 1205731 ሶ120579 = 0

119888 1205722 + 1205732 ሶ119909 + 119904 1205722 + 1205732 ሶ119910 + 119889 sdot 119904 1205732 ሶ120579 = 0

119888 1205723 + 1205733 ሶ119909 + 119904 1205723 + 1205733 ሶ119910 + 119889 sdot 119904 1205733 ሶ120579 = 0







119888 120572119894 sdot ሶ119909 + 119904 120572119894 sdot ሶ119910




ሶ120595119894 =1


rrlabcsuni-klde



rrlabcsuni-klde

Mecanum Vehicle





Mecanum Kinematics



1205721 = 45 deg 1205731 = 45 deg 1205741 = minus45 deg

1205722 = 135 deg 1205732 = minus45 deg 1205742 = 45 deg


1205724 = minus45 deg 1205734 = 135 deg 1205744 = 45 deg


Mecanum Kinematics








Mecanum Kinematics


ሶ119909 =119903

4ሶ1205951 + ሶ1205952 + ሶ1205953 + ሶ1205954

ሶ119910 =119903

4minus ሶ1205951 + ሶ1205952 minus ሶ1205953 + ሶ1205954

ሶ120579 =119903

119889 2ሶ1205951 minus ሶ1205952 minus ሶ1205953 + ሶ1205954


Mecanum Kinematics


ሶ119909ሶ119910ሶ120579

=119903119908ℎ1198901198901198974


sdot

ሶ1205951

ሶ1205952

ሶ1205953

ሶ1205954

with a =2 2

119889


ሶ1205954 = ሶ1205951 + ሶ1205952 minus ሶ1205953

rrlabcsuni-klde





Ability to maneuver

Controllability

Stability



Stability




Typical setups




Maneuverability








Controllability



Examples




Wheel type

Drive geometry


Motortransmission


Degree of Mobility




119891119898 = 3 minus rank 119878


Degree of Mobility



1198891 = 1198892 1205731 = 1205732 = 0 1205721 + 120587 = 1205722






119891119904 = rank 119878prime








119891119872 = 119891119898 + 119891119904

rrlabcsuni-klde

Vehicle Kinematics



Tricycle Drive

Given 120575 119907


119903 = 119897 sdot cot 120575

119907prime =119907

cos 120575

119907119897 = 119907 sdot119903 + 119889

119903

119907119903 = 119907 sdot119903 minus 119889


120575


Ackermann Steering

Given 119907119863 120575 119897 119889

Unknown 119907119903119903 119907119897119903 119907119903119891 119907119897119891

Solution

120602 =119897

tan 120575

119907119897119903 =120602 minus 1198892 sdot 119907119863

120602

119907119903119903 =120602 + 1198892 sdot 119907119863

120602

119907119897119891 =120602 minus 1198892 2 + 1198972 sdot tan 120575

119897sdot 119907119863

119907119903119891 =1206022 + 1198972 sdot tan 120575

119897sdot 119907119863


120575

120602

rrlabcsuni-klde



Given 120602 120593 119907 119889



1205932 = arctan120549119910 + 119889

1205491199091206022 = 120602 sdot cos120593 cos1205932

1199072 = 119907 sdot 1206022120602

120549119910 = 120602 ∙ 119904119894119899 120593

1205931

1205932

1206021

1206022

120593 120549119909 = 120602 ∙ cos120593

120602

119889

rrlabcsuni-klde

Bicycle Model


119907


Bicycle Model


Low speed motion



119865119910 =1198981199072

120602






Bicycle Model

Given 120575119891 120575119903 119907 120579

Unknown ሶ119883 ሶ119884 ሶ120579

Solution

120575119907 = tanminus1119897119891 tan 120575119903+119897119903 tan 120575119891

119897119891+119897119903

ሶ120579 =119907

120602

120602 =119897119891+119897119903

tan 120575119891 minustan 120575119903 cos 120575119907

ሶ119883 = 119907 cos 120579 + 120575119907ሶ119884 = 119907 sin 120579 + 120575119907

ሶ120579 =119907 cos 120575119907

119897119891+119897119903tan 120575119891 minus tan 120575119903

119907

rrlabcsuni-klde



119862 = 119874 120575119903 = 0 120575119891 = 120575

119897119903 = 0 119897119891 = 119897

120575119907 = 0

ሶ120579 =119907

120602

120602 =119897

tan 120575119891

ሶ119883 = 119907 cos 120579ሶ119884 = 119907 sin 120579ሶ120579 =

119907

119897119891tan 120575119891

119907






119861119906 =119909119910119911

+ 119874119861119877 120593120595 120579 119874 119906





119877 119906Ԧ119901119879 119904










119877119910 120595 =cos120595 0 sin1205950 1 0



120593 119910

119907

119911119908





119877 = 119877119911 120593 sdot 119877119909 120595 sdot 119877119911prime θ


119877 = 119877119911 θ sdot 119877119910 120595 sdot 119877119909 120593


Roll Pitch Yaw




Yaw θ

Pitchψ

Roll 120593

119911

119910119909

20

Roll

Pitch Yaw

119909

119911119910






119877119904 =



Linear Velocity




119860119877119861 Ԧ119907119902


119860 Ԧ119907119902 =119860 Ԧ119907119874119861 + 119861

119860119877119861 Ԧ119907119902



calculated with

119860 Ԧ119907119876 = 119860 Ԧ119907119874119861 + 119861119860119877119861 Ԧ119907119876 +

119860ω119861 times 119861119860119877119861 Ԧ119902


Rotational Velocity



119860120596 = 119861119860119877119861120596





119894+1120596119894+1 = 119894119894+1119877 sdot 119894120596119894 + ሶθ119894+1

119894+1 Ԧ119890119911119894+1

119894+1 Ԧ119907119894+1 = 119894119894+1119877 119894 Ԧ119907119894 +

119894120596119894 times119894 Ԧ119901119894+1






119894119877 from frame 119894 to

119894 + 1





119894119894+1119877 θ = 119894+1

119894119877minus1 θ = 119894+1119894119877119879 θ =

119888θ 119904θ 119909119908minus119904θ 119888θ 1199101199080 0 1

rrlabcsuni-klde



YW

XW

θ

YR

XR

P









coordinates





0 0 1


ሶԦ120585119877 = 119877 θሶԦ120585119882


YW

XW

θ

YR

XR




119877120587

2=

0 1 0minus1 0 00 0 1


ሶԦ120585119877 = 119877120587

2ሶ

120585119882 =0 1 0minus1 0 00 0 1


=ሶ119910


rrlabcsuni-klde


Analytical Solution






YW

XW

θ

V(t)

120596(t)







ሶԦ120585119882 = ሶ119909 ሶ119910 ሶ120579119879

= 119891 119889 119903 120579 ሶ120595119897 ሶ120595119903

= 119877 120579 minus1 ሶԦ120585119877

passive wheel

active wheel

d






ሶ119909119877119903 =119903 ሶ1205951199032

ሶ119909119877119897 =119903 ሶ120595119897

2

120596119903 =119903 ሶ1205951199032119889

120596119897 =minus119903 ሶ120595119897

2119889

ሶԦ120585119882 = 119877 120579 minus1 ሶԦ120585119903ሶԦ120585119903 =

119903 ሶ1205951199032

+119903 ሶ120595119897

2 0

119903 ሶ1205951199032119889

minus119903 ሶ120595119897

2119889

119879





Given 120579 =120587



119904

ሶԦ120585119882 =ሶ119909ሶ119910ሶ120579

=0 minus1 01 0 00 0 1

sdot301

=031







Consider




rrlabcsuni-klde

Basic Wheel Types


Standardwheel

Steerablestandard

wheel

Castorwheel

Swedish orMecanum

wheel

Ball orspherical

wheel


Wheel Properties




Castor wheel


















ሶ120595 119903



ሶ120595 119903



ሶ120595 119903



ሶ120595 119903



ሶ120595 119903








and 1 Ԧ1199071 = ሶ119909 ሶ119910 0 119879


119894+1120596119894+1 = 119894119894+1119877 sdot 119894120596119894 + ሶ120579119894+1

119894+1 Ԧ119890119911119894+1

119894+1 Ԧ119907119894+1 = 119894119894+1119877 119894 Ԧ119907119894 +

119894120596119894 times119894 Ԧ119901119894+1




calculated as

2 Ԧ1199072 =119888120572 ሶ119909 + 119904120572 ሶ119910minus119904120572 ሶ119909 + 119888120572 ሶ119910

0


3 Ԧ1199073 =

119888120572 ሶ119909 + 119904120572 ሶ119910

minus119904120572 ሶ119909 + 119888120572 ሶ119910 + 119889 ሶ1205790






4 Ԧ1199074 =119904 120572 + 120573 ሶ119909 minus 119888 120572 + 120573 ሶ119910 minus 119888120573 119889 ሶ120579

119888 120572 + 120573 ሶ119909 + 119904 120572 + 120573 ሶ119910 + 119904120573 119889 ሶ1205790




119904 120572 + 120573 minus119888 120572 + 120573 minus119888120573 119889

119888 120572 + 120573 119904 120572 + 120573 119904120573 1198890 0 0

ሶ119909119910 ሶሶ120579

=119903 ሶ12059500












119904 120572 + 120573(119905) minus119888 120572 + 120573(119905) minus119888120573(119905)119889

119888 120572 + 120573(119905) 119904 120572 + 120573(119905) 119904120573(119905)1198890 0 0

ሶ119909ሶ119910ሶ120579

=119903 ሶ120595

minus119889119888 ሶ120573(119905)0


maneuverability)




119904 120572 + 120573 + 120574 minus119888 120572 + 120573 + 120574 minus119888 120573 + 120574 119889

119888 120572 + 120573 + 120574 119904 120572 + 120573 + 120574 119904 120573 + 120574 1198890 0 0

ሶ119909ሶ119910ሶ120579

=119903 ሶ120595 cos 120574

119903 ሶ120595 sin 120574 + 119903119901119903 ሶ1205951199011199030















120572119897 = 90 deg

120573119897 = 0 deg

120572119903 = minus90 deg

120573119903 = 180 deg




119904 120572119897 + 120573119897 ሶ119909 minus 119888 120572119897 + 120573119897 ሶ119910 minus 119888120573119897 119889 ሶ120579 = 119903119897 ሶ120595119897

119888 120572119897 + 120573119897 ሶ119909 + 119904 120572119897 + 120573119897 ሶ119910 + 119904120573119897 119889 ሶ120579 = 0



ሶ119909 minus 119889 ሶ120579 = 119903119897 ሶ120595119897

ሶ119910 = 0

ሶ119909 + 119889 ሶ120579 = 119903119903 ሶ120595119903119910 = 0




ሶ119909ሶ119910ሶ120579

=

1

2119903119897 ሶ120595119897 + 119903119903 ሶ120595119903

01

2119889minus119903119897 ሶ120595119897 + 119903119903 ሶ120595119903



rrlabcsuni-klde




rrlabcsuni-klde











1205721 = 0 deg

1205722 = 120 deg

1205723 = minus120 deg







119888 1205721 + 1205731 ሶ119909 + 119904 1205721 + 1205731 ሶ119910 + 119889 sdot 119904 1205731 ሶ120579 = 0

119888 1205722 + 1205732 ሶ119909 + 119904 1205722 + 1205732 ሶ119910 + 119889 sdot 119904 1205732 ሶ120579 = 0

119888 1205723 + 1205733 ሶ119909 + 119904 1205723 + 1205733 ሶ119910 + 119889 sdot 119904 1205733 ሶ120579 = 0







119888 120572119894 sdot ሶ119909 + 119904 120572119894 sdot ሶ119910




ሶ120595119894 =1


rrlabcsuni-klde



rrlabcsuni-klde

Mecanum Vehicle





Mecanum Kinematics



1205721 = 45 deg 1205731 = 45 deg 1205741 = minus45 deg

1205722 = 135 deg 1205732 = minus45 deg 1205742 = 45 deg


1205724 = minus45 deg 1205734 = 135 deg 1205744 = 45 deg


Mecanum Kinematics








Mecanum Kinematics


ሶ119909 =119903

4ሶ1205951 + ሶ1205952 + ሶ1205953 + ሶ1205954

ሶ119910 =119903

4minus ሶ1205951 + ሶ1205952 minus ሶ1205953 + ሶ1205954

ሶ120579 =119903

119889 2ሶ1205951 minus ሶ1205952 minus ሶ1205953 + ሶ1205954


Mecanum Kinematics


ሶ119909ሶ119910ሶ120579

=119903119908ℎ1198901198901198974


sdot

ሶ1205951

ሶ1205952

ሶ1205953

ሶ1205954

with a =2 2

119889


ሶ1205954 = ሶ1205951 + ሶ1205952 minus ሶ1205953

rrlabcsuni-klde





Ability to maneuver

Controllability

Stability



Stability




Typical setups




Maneuverability








Controllability



Examples




Wheel type

Drive geometry


Motortransmission


Degree of Mobility




119891119898 = 3 minus rank 119878


Degree of Mobility



1198891 = 1198892 1205731 = 1205732 = 0 1205721 + 120587 = 1205722






119891119904 = rank 119878prime








119891119872 = 119891119898 + 119891119904

rrlabcsuni-klde

Vehicle Kinematics



Tricycle Drive

Given 120575 119907


119903 = 119897 sdot cot 120575

119907prime =119907

cos 120575

119907119897 = 119907 sdot119903 + 119889

119903

119907119903 = 119907 sdot119903 minus 119889


120575


Ackermann Steering

Given 119907119863 120575 119897 119889

Unknown 119907119903119903 119907119897119903 119907119903119891 119907119897119891

Solution

120602 =119897

tan 120575

119907119897119903 =120602 minus 1198892 sdot 119907119863

120602

119907119903119903 =120602 + 1198892 sdot 119907119863

120602

119907119897119891 =120602 minus 1198892 2 + 1198972 sdot tan 120575

119897sdot 119907119863

119907119903119891 =1206022 + 1198972 sdot tan 120575

119897sdot 119907119863


120575

120602

rrlabcsuni-klde



Given 120602 120593 119907 119889



1205932 = arctan120549119910 + 119889

1205491199091206022 = 120602 sdot cos120593 cos1205932

1199072 = 119907 sdot 1206022120602

120549119910 = 120602 ∙ 119904119894119899 120593

1205931

1205932

1206021

1206022

120593 120549119909 = 120602 ∙ cos120593

120602

119889

rrlabcsuni-klde

Bicycle Model


119907


Bicycle Model


Low speed motion



119865119910 =1198981199072

120602






Bicycle Model

Given 120575119891 120575119903 119907 120579

Unknown ሶ119883 ሶ119884 ሶ120579

Solution

120575119907 = tanminus1119897119891 tan 120575119903+119897119903 tan 120575119891

119897119891+119897119903

ሶ120579 =119907

120602

120602 =119897119891+119897119903

tan 120575119891 minustan 120575119903 cos 120575119907

ሶ119883 = 119907 cos 120579 + 120575119907ሶ119884 = 119907 sin 120579 + 120575119907

ሶ120579 =119907 cos 120575119907

119897119891+119897119903tan 120575119891 minus tan 120575119903

119907

rrlabcsuni-klde



119862 = 119874 120575119903 = 0 120575119891 = 120575

119897119903 = 0 119897119891 = 119897

120575119907 = 0

ሶ120579 =119907

120602

120602 =119897

tan 120575119891

ሶ119883 = 119907 cos 120579ሶ119884 = 119907 sin 120579ሶ120579 =

119907

119897119891tan 120575119891

119907




119877 119906Ԧ119901119879 119904










119877119910 120595 =cos120595 0 sin1205950 1 0



120593 119910

119907

119911119908





119877 = 119877119911 120593 sdot 119877119909 120595 sdot 119877119911prime θ


119877 = 119877119911 θ sdot 119877119910 120595 sdot 119877119909 120593


Roll Pitch Yaw




Yaw θ

Pitchψ

Roll 120593

119911

119910119909

20

Roll

Pitch Yaw

119909

119911119910






119877119904 =



Linear Velocity




119860119877119861 Ԧ119907119902


119860 Ԧ119907119902 =119860 Ԧ119907119874119861 + 119861

119860119877119861 Ԧ119907119902



calculated with

119860 Ԧ119907119876 = 119860 Ԧ119907119874119861 + 119861119860119877119861 Ԧ119907119876 +

119860ω119861 times 119861119860119877119861 Ԧ119902


Rotational Velocity



119860120596 = 119861119860119877119861120596





119894+1120596119894+1 = 119894119894+1119877 sdot 119894120596119894 + ሶθ119894+1

119894+1 Ԧ119890119911119894+1

119894+1 Ԧ119907119894+1 = 119894119894+1119877 119894 Ԧ119907119894 +

119894120596119894 times119894 Ԧ119901119894+1






119894119877 from frame 119894 to

119894 + 1





119894119894+1119877 θ = 119894+1

119894119877minus1 θ = 119894+1119894119877119879 θ =

119888θ 119904θ 119909119908minus119904θ 119888θ 1199101199080 0 1

rrlabcsuni-klde



YW

XW

θ

YR

XR

P









coordinates





0 0 1


ሶԦ120585119877 = 119877 θሶԦ120585119882


YW

XW

θ

YR

XR




119877120587

2=

0 1 0minus1 0 00 0 1


ሶԦ120585119877 = 119877120587

2ሶ

120585119882 =0 1 0minus1 0 00 0 1


=ሶ119910


rrlabcsuni-klde


Analytical Solution






YW

XW

θ

V(t)

120596(t)







ሶԦ120585119882 = ሶ119909 ሶ119910 ሶ120579119879

= 119891 119889 119903 120579 ሶ120595119897 ሶ120595119903

= 119877 120579 minus1 ሶԦ120585119877

passive wheel

active wheel

d






ሶ119909119877119903 =119903 ሶ1205951199032

ሶ119909119877119897 =119903 ሶ120595119897

2

120596119903 =119903 ሶ1205951199032119889

120596119897 =minus119903 ሶ120595119897

2119889

ሶԦ120585119882 = 119877 120579 minus1 ሶԦ120585119903ሶԦ120585119903 =

119903 ሶ1205951199032

+119903 ሶ120595119897

2 0

119903 ሶ1205951199032119889

minus119903 ሶ120595119897

2119889

119879





Given 120579 =120587



119904

ሶԦ120585119882 =ሶ119909ሶ119910ሶ120579

=0 minus1 01 0 00 0 1

sdot301

=031







Consider




rrlabcsuni-klde

Basic Wheel Types


Standardwheel

Steerablestandard

wheel

Castorwheel

Swedish orMecanum

wheel

Ball orspherical

wheel


Wheel Properties




Castor wheel


















ሶ120595 119903



ሶ120595 119903



ሶ120595 119903



ሶ120595 119903



ሶ120595 119903








and 1 Ԧ1199071 = ሶ119909 ሶ119910 0 119879


119894+1120596119894+1 = 119894119894+1119877 sdot 119894120596119894 + ሶ120579119894+1

119894+1 Ԧ119890119911119894+1

119894+1 Ԧ119907119894+1 = 119894119894+1119877 119894 Ԧ119907119894 +

119894120596119894 times119894 Ԧ119901119894+1




calculated as

2 Ԧ1199072 =119888120572 ሶ119909 + 119904120572 ሶ119910minus119904120572 ሶ119909 + 119888120572 ሶ119910

0


3 Ԧ1199073 =

119888120572 ሶ119909 + 119904120572 ሶ119910

minus119904120572 ሶ119909 + 119888120572 ሶ119910 + 119889 ሶ1205790






4 Ԧ1199074 =119904 120572 + 120573 ሶ119909 minus 119888 120572 + 120573 ሶ119910 minus 119888120573 119889 ሶ120579

119888 120572 + 120573 ሶ119909 + 119904 120572 + 120573 ሶ119910 + 119904120573 119889 ሶ1205790




119904 120572 + 120573 minus119888 120572 + 120573 minus119888120573 119889

119888 120572 + 120573 119904 120572 + 120573 119904120573 1198890 0 0

ሶ119909119910 ሶሶ120579

=119903 ሶ12059500












119904 120572 + 120573(119905) minus119888 120572 + 120573(119905) minus119888120573(119905)119889

119888 120572 + 120573(119905) 119904 120572 + 120573(119905) 119904120573(119905)1198890 0 0

ሶ119909ሶ119910ሶ120579

=119903 ሶ120595

minus119889119888 ሶ120573(119905)0


maneuverability)




119904 120572 + 120573 + 120574 minus119888 120572 + 120573 + 120574 minus119888 120573 + 120574 119889

119888 120572 + 120573 + 120574 119904 120572 + 120573 + 120574 119904 120573 + 120574 1198890 0 0

ሶ119909ሶ119910ሶ120579

=119903 ሶ120595 cos 120574

119903 ሶ120595 sin 120574 + 119903119901119903 ሶ1205951199011199030















120572119897 = 90 deg

120573119897 = 0 deg

120572119903 = minus90 deg

120573119903 = 180 deg




119904 120572119897 + 120573119897 ሶ119909 minus 119888 120572119897 + 120573119897 ሶ119910 minus 119888120573119897 119889 ሶ120579 = 119903119897 ሶ120595119897

119888 120572119897 + 120573119897 ሶ119909 + 119904 120572119897 + 120573119897 ሶ119910 + 119904120573119897 119889 ሶ120579 = 0



ሶ119909 minus 119889 ሶ120579 = 119903119897 ሶ120595119897

ሶ119910 = 0

ሶ119909 + 119889 ሶ120579 = 119903119903 ሶ120595119903119910 = 0




ሶ119909ሶ119910ሶ120579

=

1

2119903119897 ሶ120595119897 + 119903119903 ሶ120595119903

01

2119889minus119903119897 ሶ120595119897 + 119903119903 ሶ120595119903



rrlabcsuni-klde




rrlabcsuni-klde











1205721 = 0 deg

1205722 = 120 deg

1205723 = minus120 deg







119888 1205721 + 1205731 ሶ119909 + 119904 1205721 + 1205731 ሶ119910 + 119889 sdot 119904 1205731 ሶ120579 = 0

119888 1205722 + 1205732 ሶ119909 + 119904 1205722 + 1205732 ሶ119910 + 119889 sdot 119904 1205732 ሶ120579 = 0

119888 1205723 + 1205733 ሶ119909 + 119904 1205723 + 1205733 ሶ119910 + 119889 sdot 119904 1205733 ሶ120579 = 0







119888 120572119894 sdot ሶ119909 + 119904 120572119894 sdot ሶ119910




ሶ120595119894 =1


rrlabcsuni-klde



rrlabcsuni-klde

Mecanum Vehicle





Mecanum Kinematics



1205721 = 45 deg 1205731 = 45 deg 1205741 = minus45 deg

1205722 = 135 deg 1205732 = minus45 deg 1205742 = 45 deg


1205724 = minus45 deg 1205734 = 135 deg 1205744 = 45 deg


Mecanum Kinematics








Mecanum Kinematics


ሶ119909 =119903

4ሶ1205951 + ሶ1205952 + ሶ1205953 + ሶ1205954

ሶ119910 =119903

4minus ሶ1205951 + ሶ1205952 minus ሶ1205953 + ሶ1205954

ሶ120579 =119903

119889 2ሶ1205951 minus ሶ1205952 minus ሶ1205953 + ሶ1205954


Mecanum Kinematics


ሶ119909ሶ119910ሶ120579

=119903119908ℎ1198901198901198974


sdot

ሶ1205951

ሶ1205952

ሶ1205953

ሶ1205954

with a =2 2

119889


ሶ1205954 = ሶ1205951 + ሶ1205952 minus ሶ1205953

rrlabcsuni-klde





Ability to maneuver

Controllability

Stability



Stability




Typical setups




Maneuverability








Controllability



Examples




Wheel type

Drive geometry


Motortransmission


Degree of Mobility




119891119898 = 3 minus rank 119878


Degree of Mobility



1198891 = 1198892 1205731 = 1205732 = 0 1205721 + 120587 = 1205722






119891119904 = rank 119878prime








119891119872 = 119891119898 + 119891119904

rrlabcsuni-klde

Vehicle Kinematics



Tricycle Drive

Given 120575 119907


119903 = 119897 sdot cot 120575

119907prime =119907

cos 120575

119907119897 = 119907 sdot119903 + 119889

119903

119907119903 = 119907 sdot119903 minus 119889


120575


Ackermann Steering

Given 119907119863 120575 119897 119889

Unknown 119907119903119903 119907119897119903 119907119903119891 119907119897119891

Solution

120602 =119897

tan 120575

119907119897119903 =120602 minus 1198892 sdot 119907119863

120602

119907119903119903 =120602 + 1198892 sdot 119907119863

120602

119907119897119891 =120602 minus 1198892 2 + 1198972 sdot tan 120575

119897sdot 119907119863

119907119903119891 =1206022 + 1198972 sdot tan 120575

119897sdot 119907119863


120575

120602

rrlabcsuni-klde



Given 120602 120593 119907 119889



1205932 = arctan120549119910 + 119889

1205491199091206022 = 120602 sdot cos120593 cos1205932

1199072 = 119907 sdot 1206022120602

120549119910 = 120602 ∙ 119904119894119899 120593

1205931

1205932

1206021

1206022

120593 120549119909 = 120602 ∙ cos120593

120602

119889

rrlabcsuni-klde

Bicycle Model


119907


Bicycle Model


Low speed motion



119865119910 =1198981199072

120602






Bicycle Model

Given 120575119891 120575119903 119907 120579

Unknown ሶ119883 ሶ119884 ሶ120579

Solution

120575119907 = tanminus1119897119891 tan 120575119903+119897119903 tan 120575119891

119897119891+119897119903

ሶ120579 =119907

120602

120602 =119897119891+119897119903

tan 120575119891 minustan 120575119903 cos 120575119907

ሶ119883 = 119907 cos 120579 + 120575119907ሶ119884 = 119907 sin 120579 + 120575119907

ሶ120579 =119907 cos 120575119907

119897119891+119897119903tan 120575119891 minus tan 120575119903

119907

rrlabcsuni-klde



119862 = 119874 120575119903 = 0 120575119891 = 120575

119897119903 = 0 119897119891 = 119897

120575119907 = 0

ሶ120579 =119907

120602

120602 =119897

tan 120575119891

ሶ119883 = 119907 cos 120579ሶ119884 = 119907 sin 120579ሶ120579 =

119907

119897119891tan 120575119891

119907






119877119910 120595 =cos120595 0 sin1205950 1 0



120593 119910

119907

119911119908





119877 = 119877119911 120593 sdot 119877119909 120595 sdot 119877119911prime θ


119877 = 119877119911 θ sdot 119877119910 120595 sdot 119877119909 120593


Roll Pitch Yaw




Yaw θ

Pitchψ

Roll 120593

119911

119910119909

20

Roll

Pitch Yaw

119909

119911119910






119877119904 =



Linear Velocity




119860119877119861 Ԧ119907119902


119860 Ԧ119907119902 =119860 Ԧ119907119874119861 + 119861

119860119877119861 Ԧ119907119902



calculated with

119860 Ԧ119907119876 = 119860 Ԧ119907119874119861 + 119861119860119877119861 Ԧ119907119876 +

119860ω119861 times 119861119860119877119861 Ԧ119902


Rotational Velocity



119860120596 = 119861119860119877119861120596





119894+1120596119894+1 = 119894119894+1119877 sdot 119894120596119894 + ሶθ119894+1

119894+1 Ԧ119890119911119894+1

119894+1 Ԧ119907119894+1 = 119894119894+1119877 119894 Ԧ119907119894 +

119894120596119894 times119894 Ԧ119901119894+1






119894119877 from frame 119894 to

119894 + 1





119894119894+1119877 θ = 119894+1

119894119877minus1 θ = 119894+1119894119877119879 θ =

119888θ 119904θ 119909119908minus119904θ 119888θ 1199101199080 0 1

rrlabcsuni-klde



YW

XW

θ

YR

XR

P









coordinates





0 0 1


ሶԦ120585119877 = 119877 θሶԦ120585119882


YW

XW

θ

YR

XR




119877120587

2=

0 1 0minus1 0 00 0 1


ሶԦ120585119877 = 119877120587

2ሶ

120585119882 =0 1 0minus1 0 00 0 1


=ሶ119910


rrlabcsuni-klde


Analytical Solution






YW

XW

θ

V(t)

120596(t)







ሶԦ120585119882 = ሶ119909 ሶ119910 ሶ120579119879

= 119891 119889 119903 120579 ሶ120595119897 ሶ120595119903

= 119877 120579 minus1 ሶԦ120585119877

passive wheel

active wheel

d






ሶ119909119877119903 =119903 ሶ1205951199032

ሶ119909119877119897 =119903 ሶ120595119897

2

120596119903 =119903 ሶ1205951199032119889

120596119897 =minus119903 ሶ120595119897

2119889

ሶԦ120585119882 = 119877 120579 minus1 ሶԦ120585119903ሶԦ120585119903 =

119903 ሶ1205951199032

+119903 ሶ120595119897

2 0

119903 ሶ1205951199032119889

minus119903 ሶ120595119897

2119889

119879





Given 120579 =120587



119904

ሶԦ120585119882 =ሶ119909ሶ119910ሶ120579

=0 minus1 01 0 00 0 1

sdot301

=031







Consider




rrlabcsuni-klde

Basic Wheel Types


Standardwheel

Steerablestandard

wheel

Castorwheel

Swedish orMecanum

wheel

Ball orspherical

wheel


Wheel Properties




Castor wheel


















ሶ120595 119903



ሶ120595 119903



ሶ120595 119903



ሶ120595 119903



ሶ120595 119903








and 1 Ԧ1199071 = ሶ119909 ሶ119910 0 119879


119894+1120596119894+1 = 119894119894+1119877 sdot 119894120596119894 + ሶ120579119894+1

119894+1 Ԧ119890119911119894+1

119894+1 Ԧ119907119894+1 = 119894119894+1119877 119894 Ԧ119907119894 +

119894120596119894 times119894 Ԧ119901119894+1




calculated as

2 Ԧ1199072 =119888120572 ሶ119909 + 119904120572 ሶ119910minus119904120572 ሶ119909 + 119888120572 ሶ119910

0


3 Ԧ1199073 =

119888120572 ሶ119909 + 119904120572 ሶ119910

minus119904120572 ሶ119909 + 119888120572 ሶ119910 + 119889 ሶ1205790






4 Ԧ1199074 =119904 120572 + 120573 ሶ119909 minus 119888 120572 + 120573 ሶ119910 minus 119888120573 119889 ሶ120579

119888 120572 + 120573 ሶ119909 + 119904 120572 + 120573 ሶ119910 + 119904120573 119889 ሶ1205790




119904 120572 + 120573 minus119888 120572 + 120573 minus119888120573 119889

119888 120572 + 120573 119904 120572 + 120573 119904120573 1198890 0 0

ሶ119909119910 ሶሶ120579

=119903 ሶ12059500












119904 120572 + 120573(119905) minus119888 120572 + 120573(119905) minus119888120573(119905)119889

119888 120572 + 120573(119905) 119904 120572 + 120573(119905) 119904120573(119905)1198890 0 0

ሶ119909ሶ119910ሶ120579

=119903 ሶ120595

minus119889119888 ሶ120573(119905)0


maneuverability)




119904 120572 + 120573 + 120574 minus119888 120572 + 120573 + 120574 minus119888 120573 + 120574 119889

119888 120572 + 120573 + 120574 119904 120572 + 120573 + 120574 119904 120573 + 120574 1198890 0 0

ሶ119909ሶ119910ሶ120579

=119903 ሶ120595 cos 120574

119903 ሶ120595 sin 120574 + 119903119901119903 ሶ1205951199011199030















120572119897 = 90 deg

120573119897 = 0 deg

120572119903 = minus90 deg

120573119903 = 180 deg




119904 120572119897 + 120573119897 ሶ119909 minus 119888 120572119897 + 120573119897 ሶ119910 minus 119888120573119897 119889 ሶ120579 = 119903119897 ሶ120595119897

119888 120572119897 + 120573119897 ሶ119909 + 119904 120572119897 + 120573119897 ሶ119910 + 119904120573119897 119889 ሶ120579 = 0



ሶ119909 minus 119889 ሶ120579 = 119903119897 ሶ120595119897

ሶ119910 = 0

ሶ119909 + 119889 ሶ120579 = 119903119903 ሶ120595119903119910 = 0




ሶ119909ሶ119910ሶ120579

=

1

2119903119897 ሶ120595119897 + 119903119903 ሶ120595119903

01

2119889minus119903119897 ሶ120595119897 + 119903119903 ሶ120595119903



rrlabcsuni-klde




rrlabcsuni-klde











1205721 = 0 deg

1205722 = 120 deg

1205723 = minus120 deg







119888 1205721 + 1205731 ሶ119909 + 119904 1205721 + 1205731 ሶ119910 + 119889 sdot 119904 1205731 ሶ120579 = 0

119888 1205722 + 1205732 ሶ119909 + 119904 1205722 + 1205732 ሶ119910 + 119889 sdot 119904 1205732 ሶ120579 = 0

119888 1205723 + 1205733 ሶ119909 + 119904 1205723 + 1205733 ሶ119910 + 119889 sdot 119904 1205733 ሶ120579 = 0







119888 120572119894 sdot ሶ119909 + 119904 120572119894 sdot ሶ119910




ሶ120595119894 =1


rrlabcsuni-klde



rrlabcsuni-klde

Mecanum Vehicle





Mecanum Kinematics



1205721 = 45 deg 1205731 = 45 deg 1205741 = minus45 deg

1205722 = 135 deg 1205732 = minus45 deg 1205742 = 45 deg


1205724 = minus45 deg 1205734 = 135 deg 1205744 = 45 deg


Mecanum Kinematics








Mecanum Kinematics


ሶ119909 =119903

4ሶ1205951 + ሶ1205952 + ሶ1205953 + ሶ1205954

ሶ119910 =119903

4minus ሶ1205951 + ሶ1205952 minus ሶ1205953 + ሶ1205954

ሶ120579 =119903

119889 2ሶ1205951 minus ሶ1205952 minus ሶ1205953 + ሶ1205954


Mecanum Kinematics


ሶ119909ሶ119910ሶ120579

=119903119908ℎ1198901198901198974


sdot

ሶ1205951

ሶ1205952

ሶ1205953

ሶ1205954

with a =2 2

119889


ሶ1205954 = ሶ1205951 + ሶ1205952 minus ሶ1205953

rrlabcsuni-klde





Ability to maneuver

Controllability

Stability



Stability




Typical setups




Maneuverability








Controllability



Examples




Wheel type

Drive geometry


Motortransmission


Degree of Mobility




119891119898 = 3 minus rank 119878


Degree of Mobility



1198891 = 1198892 1205731 = 1205732 = 0 1205721 + 120587 = 1205722






119891119904 = rank 119878prime








119891119872 = 119891119898 + 119891119904

rrlabcsuni-klde

Vehicle Kinematics



Tricycle Drive

Given 120575 119907


119903 = 119897 sdot cot 120575

119907prime =119907

cos 120575

119907119897 = 119907 sdot119903 + 119889

119903

119907119903 = 119907 sdot119903 minus 119889


120575


Ackermann Steering

Given 119907119863 120575 119897 119889

Unknown 119907119903119903 119907119897119903 119907119903119891 119907119897119891

Solution

120602 =119897

tan 120575

119907119897119903 =120602 minus 1198892 sdot 119907119863

120602

119907119903119903 =120602 + 1198892 sdot 119907119863

120602

119907119897119891 =120602 minus 1198892 2 + 1198972 sdot tan 120575

119897sdot 119907119863

119907119903119891 =1206022 + 1198972 sdot tan 120575

119897sdot 119907119863


120575

120602

rrlabcsuni-klde



Given 120602 120593 119907 119889



1205932 = arctan120549119910 + 119889

1205491199091206022 = 120602 sdot cos120593 cos1205932

1199072 = 119907 sdot 1206022120602

120549119910 = 120602 ∙ 119904119894119899 120593

1205931

1205932

1206021

1206022

120593 120549119909 = 120602 ∙ cos120593

120602

119889

rrlabcsuni-klde

Bicycle Model


119907


Bicycle Model


Low speed motion



119865119910 =1198981199072

120602






Bicycle Model

Given 120575119891 120575119903 119907 120579

Unknown ሶ119883 ሶ119884 ሶ120579

Solution

120575119907 = tanminus1119897119891 tan 120575119903+119897119903 tan 120575119891

119897119891+119897119903

ሶ120579 =119907

120602

120602 =119897119891+119897119903

tan 120575119891 minustan 120575119903 cos 120575119907

ሶ119883 = 119907 cos 120579 + 120575119907ሶ119884 = 119907 sin 120579 + 120575119907

ሶ120579 =119907 cos 120575119907

119897119891+119897119903tan 120575119891 minus tan 120575119903

119907

rrlabcsuni-klde



119862 = 119874 120575119903 = 0 120575119891 = 120575

119897119903 = 0 119897119891 = 119897

120575119907 = 0

ሶ120579 =119907

120602

120602 =119897

tan 120575119891

ሶ119883 = 119907 cos 120579ሶ119884 = 119907 sin 120579ሶ120579 =

119907

119897119891tan 120575119891

119907





119877 = 119877119911 120593 sdot 119877119909 120595 sdot 119877119911prime θ


119877 = 119877119911 θ sdot 119877119910 120595 sdot 119877119909 120593


Roll Pitch Yaw




Yaw θ

Pitchψ

Roll 120593

119911

119910119909

20

Roll

Pitch Yaw

119909

119911119910






119877119904 =



Linear Velocity




119860119877119861 Ԧ119907119902


119860 Ԧ119907119902 =119860 Ԧ119907119874119861 + 119861

119860119877119861 Ԧ119907119902



calculated with

119860 Ԧ119907119876 = 119860 Ԧ119907119874119861 + 119861119860119877119861 Ԧ119907119876 +

119860ω119861 times 119861119860119877119861 Ԧ119902


Rotational Velocity



119860120596 = 119861119860119877119861120596





119894+1120596119894+1 = 119894119894+1119877 sdot 119894120596119894 + ሶθ119894+1

119894+1 Ԧ119890119911119894+1

119894+1 Ԧ119907119894+1 = 119894119894+1119877 119894 Ԧ119907119894 +

119894120596119894 times119894 Ԧ119901119894+1






119894119877 from frame 119894 to

119894 + 1





119894119894+1119877 θ = 119894+1

119894119877minus1 θ = 119894+1119894119877119879 θ =

119888θ 119904θ 119909119908minus119904θ 119888θ 1199101199080 0 1

rrlabcsuni-klde



YW

XW

θ

YR

XR

P









coordinates





0 0 1


ሶԦ120585119877 = 119877 θሶԦ120585119882


YW

XW

θ

YR

XR




119877120587

2=

0 1 0minus1 0 00 0 1


ሶԦ120585119877 = 119877120587

2ሶ

120585119882 =0 1 0minus1 0 00 0 1


=ሶ119910


rrlabcsuni-klde


Analytical Solution






YW

XW

θ

V(t)

120596(t)







ሶԦ120585119882 = ሶ119909 ሶ119910 ሶ120579119879

= 119891 119889 119903 120579 ሶ120595119897 ሶ120595119903

= 119877 120579 minus1 ሶԦ120585119877

passive wheel

active wheel

d






ሶ119909119877119903 =119903 ሶ1205951199032

ሶ119909119877119897 =119903 ሶ120595119897

2

120596119903 =119903 ሶ1205951199032119889

120596119897 =minus119903 ሶ120595119897

2119889

ሶԦ120585119882 = 119877 120579 minus1 ሶԦ120585119903ሶԦ120585119903 =

119903 ሶ1205951199032

+119903 ሶ120595119897

2 0

119903 ሶ1205951199032119889

minus119903 ሶ120595119897

2119889

119879





Given 120579 =120587



119904

ሶԦ120585119882 =ሶ119909ሶ119910ሶ120579

=0 minus1 01 0 00 0 1

sdot301

=031







Consider




rrlabcsuni-klde

Basic Wheel Types


Standardwheel

Steerablestandard

wheel

Castorwheel

Swedish orMecanum

wheel

Ball orspherical

wheel


Wheel Properties




Castor wheel


















ሶ120595 119903



ሶ120595 119903



ሶ120595 119903



ሶ120595 119903



ሶ120595 119903








and 1 Ԧ1199071 = ሶ119909 ሶ119910 0 119879


119894+1120596119894+1 = 119894119894+1119877 sdot 119894120596119894 + ሶ120579119894+1

119894+1 Ԧ119890119911119894+1

119894+1 Ԧ119907119894+1 = 119894119894+1119877 119894 Ԧ119907119894 +

119894120596119894 times119894 Ԧ119901119894+1




calculated as

2 Ԧ1199072 =119888120572 ሶ119909 + 119904120572 ሶ119910minus119904120572 ሶ119909 + 119888120572 ሶ119910

0


3 Ԧ1199073 =

119888120572 ሶ119909 + 119904120572 ሶ119910

minus119904120572 ሶ119909 + 119888120572 ሶ119910 + 119889 ሶ1205790






4 Ԧ1199074 =119904 120572 + 120573 ሶ119909 minus 119888 120572 + 120573 ሶ119910 minus 119888120573 119889 ሶ120579

119888 120572 + 120573 ሶ119909 + 119904 120572 + 120573 ሶ119910 + 119904120573 119889 ሶ1205790




119904 120572 + 120573 minus119888 120572 + 120573 minus119888120573 119889

119888 120572 + 120573 119904 120572 + 120573 119904120573 1198890 0 0

ሶ119909119910 ሶሶ120579

=119903 ሶ12059500












119904 120572 + 120573(119905) minus119888 120572 + 120573(119905) minus119888120573(119905)119889

119888 120572 + 120573(119905) 119904 120572 + 120573(119905) 119904120573(119905)1198890 0 0

ሶ119909ሶ119910ሶ120579

=119903 ሶ120595

minus119889119888 ሶ120573(119905)0


maneuverability)




119904 120572 + 120573 + 120574 minus119888 120572 + 120573 + 120574 minus119888 120573 + 120574 119889

119888 120572 + 120573 + 120574 119904 120572 + 120573 + 120574 119904 120573 + 120574 1198890 0 0

ሶ119909ሶ119910ሶ120579

=119903 ሶ120595 cos 120574

119903 ሶ120595 sin 120574 + 119903119901119903 ሶ1205951199011199030















120572119897 = 90 deg

120573119897 = 0 deg

120572119903 = minus90 deg

120573119903 = 180 deg




119904 120572119897 + 120573119897 ሶ119909 minus 119888 120572119897 + 120573119897 ሶ119910 minus 119888120573119897 119889 ሶ120579 = 119903119897 ሶ120595119897

119888 120572119897 + 120573119897 ሶ119909 + 119904 120572119897 + 120573119897 ሶ119910 + 119904120573119897 119889 ሶ120579 = 0



ሶ119909 minus 119889 ሶ120579 = 119903119897 ሶ120595119897

ሶ119910 = 0

ሶ119909 + 119889 ሶ120579 = 119903119903 ሶ120595119903119910 = 0




ሶ119909ሶ119910ሶ120579

=

1

2119903119897 ሶ120595119897 + 119903119903 ሶ120595119903

01

2119889minus119903119897 ሶ120595119897 + 119903119903 ሶ120595119903



rrlabcsuni-klde




rrlabcsuni-klde











1205721 = 0 deg

1205722 = 120 deg

1205723 = minus120 deg







119888 1205721 + 1205731 ሶ119909 + 119904 1205721 + 1205731 ሶ119910 + 119889 sdot 119904 1205731 ሶ120579 = 0

119888 1205722 + 1205732 ሶ119909 + 119904 1205722 + 1205732 ሶ119910 + 119889 sdot 119904 1205732 ሶ120579 = 0

119888 1205723 + 1205733 ሶ119909 + 119904 1205723 + 1205733 ሶ119910 + 119889 sdot 119904 1205733 ሶ120579 = 0







119888 120572119894 sdot ሶ119909 + 119904 120572119894 sdot ሶ119910




ሶ120595119894 =1


rrlabcsuni-klde



rrlabcsuni-klde

Mecanum Vehicle





Mecanum Kinematics



1205721 = 45 deg 1205731 = 45 deg 1205741 = minus45 deg

1205722 = 135 deg 1205732 = minus45 deg 1205742 = 45 deg


1205724 = minus45 deg 1205734 = 135 deg 1205744 = 45 deg


Mecanum Kinematics








Mecanum Kinematics


ሶ119909 =119903

4ሶ1205951 + ሶ1205952 + ሶ1205953 + ሶ1205954

ሶ119910 =119903

4minus ሶ1205951 + ሶ1205952 minus ሶ1205953 + ሶ1205954

ሶ120579 =119903

119889 2ሶ1205951 minus ሶ1205952 minus ሶ1205953 + ሶ1205954


Mecanum Kinematics


ሶ119909ሶ119910ሶ120579

=119903119908ℎ1198901198901198974


sdot

ሶ1205951

ሶ1205952

ሶ1205953

ሶ1205954

with a =2 2

119889


ሶ1205954 = ሶ1205951 + ሶ1205952 minus ሶ1205953

rrlabcsuni-klde





Ability to maneuver

Controllability

Stability



Stability




Typical setups




Maneuverability








Controllability



Examples




Wheel type

Drive geometry


Motortransmission


Degree of Mobility




119891119898 = 3 minus rank 119878


Degree of Mobility



1198891 = 1198892 1205731 = 1205732 = 0 1205721 + 120587 = 1205722






119891119904 = rank 119878prime








119891119872 = 119891119898 + 119891119904

rrlabcsuni-klde

Vehicle Kinematics



Tricycle Drive

Given 120575 119907


119903 = 119897 sdot cot 120575

119907prime =119907

cos 120575

119907119897 = 119907 sdot119903 + 119889

119903

119907119903 = 119907 sdot119903 minus 119889


120575


Ackermann Steering

Given 119907119863 120575 119897 119889

Unknown 119907119903119903 119907119897119903 119907119903119891 119907119897119891

Solution

120602 =119897

tan 120575

119907119897119903 =120602 minus 1198892 sdot 119907119863

120602

119907119903119903 =120602 + 1198892 sdot 119907119863

120602

119907119897119891 =120602 minus 1198892 2 + 1198972 sdot tan 120575

119897sdot 119907119863

119907119903119891 =1206022 + 1198972 sdot tan 120575

119897sdot 119907119863


120575

120602

rrlabcsuni-klde



Given 120602 120593 119907 119889



1205932 = arctan120549119910 + 119889

1205491199091206022 = 120602 sdot cos120593 cos1205932

1199072 = 119907 sdot 1206022120602

120549119910 = 120602 ∙ 119904119894119899 120593

1205931

1205932

1206021

1206022

120593 120549119909 = 120602 ∙ cos120593

120602

119889

rrlabcsuni-klde

Bicycle Model


119907


Bicycle Model


Low speed motion



119865119910 =1198981199072

120602






Bicycle Model

Given 120575119891 120575119903 119907 120579

Unknown ሶ119883 ሶ119884 ሶ120579

Solution

120575119907 = tanminus1119897119891 tan 120575119903+119897119903 tan 120575119891

119897119891+119897119903

ሶ120579 =119907

120602

120602 =119897119891+119897119903

tan 120575119891 minustan 120575119903 cos 120575119907

ሶ119883 = 119907 cos 120579 + 120575119907ሶ119884 = 119907 sin 120579 + 120575119907

ሶ120579 =119907 cos 120575119907

119897119891+119897119903tan 120575119891 minus tan 120575119903

119907

rrlabcsuni-klde



119862 = 119874 120575119903 = 0 120575119891 = 120575

119897119903 = 0 119897119891 = 119897

120575119907 = 0

ሶ120579 =119907

120602

120602 =119897

tan 120575119891

ሶ119883 = 119907 cos 120579ሶ119884 = 119907 sin 120579ሶ120579 =

119907

119897119891tan 120575119891

119907


Roll Pitch Yaw




Yaw θ

Pitchψ

Roll 120593

119911

119910119909

20

Roll

Pitch Yaw

119909

119911119910






119877119904 =



Linear Velocity




119860119877119861 Ԧ119907119902


119860 Ԧ119907119902 =119860 Ԧ119907119874119861 + 119861

119860119877119861 Ԧ119907119902



calculated with

119860 Ԧ119907119876 = 119860 Ԧ119907119874119861 + 119861119860119877119861 Ԧ119907119876 +

119860ω119861 times 119861119860119877119861 Ԧ119902


Rotational Velocity



119860120596 = 119861119860119877119861120596





119894+1120596119894+1 = 119894119894+1119877 sdot 119894120596119894 + ሶθ119894+1

119894+1 Ԧ119890119911119894+1

119894+1 Ԧ119907119894+1 = 119894119894+1119877 119894 Ԧ119907119894 +

119894120596119894 times119894 Ԧ119901119894+1






119894119877 from frame 119894 to

119894 + 1





119894119894+1119877 θ = 119894+1

119894119877minus1 θ = 119894+1119894119877119879 θ =

119888θ 119904θ 119909119908minus119904θ 119888θ 1199101199080 0 1

rrlabcsuni-klde



YW

XW

θ

YR

XR

P









coordinates





0 0 1


ሶԦ120585119877 = 119877 θሶԦ120585119882


YW

XW

θ

YR

XR




119877120587

2=

0 1 0minus1 0 00 0 1


ሶԦ120585119877 = 119877120587

2ሶ

120585119882 =0 1 0minus1 0 00 0 1


=ሶ119910


rrlabcsuni-klde


Analytical Solution






YW

XW

θ

V(t)

120596(t)







ሶԦ120585119882 = ሶ119909 ሶ119910 ሶ120579119879

= 119891 119889 119903 120579 ሶ120595119897 ሶ120595119903

= 119877 120579 minus1 ሶԦ120585119877

passive wheel

active wheel

d






ሶ119909119877119903 =119903 ሶ1205951199032

ሶ119909119877119897 =119903 ሶ120595119897

2

120596119903 =119903 ሶ1205951199032119889

120596119897 =minus119903 ሶ120595119897

2119889

ሶԦ120585119882 = 119877 120579 minus1 ሶԦ120585119903ሶԦ120585119903 =

119903 ሶ1205951199032

+119903 ሶ120595119897

2 0

119903 ሶ1205951199032119889

minus119903 ሶ120595119897

2119889

119879





Given 120579 =120587



119904

ሶԦ120585119882 =ሶ119909ሶ119910ሶ120579

=0 minus1 01 0 00 0 1

sdot301

=031







Consider




rrlabcsuni-klde

Basic Wheel Types


Standardwheel

Steerablestandard

wheel

Castorwheel

Swedish orMecanum

wheel

Ball orspherical

wheel


Wheel Properties




Castor wheel


















ሶ120595 119903



ሶ120595 119903



ሶ120595 119903



ሶ120595 119903



ሶ120595 119903








and 1 Ԧ1199071 = ሶ119909 ሶ119910 0 119879


119894+1120596119894+1 = 119894119894+1119877 sdot 119894120596119894 + ሶ120579119894+1

119894+1 Ԧ119890119911119894+1

119894+1 Ԧ119907119894+1 = 119894119894+1119877 119894 Ԧ119907119894 +

119894120596119894 times119894 Ԧ119901119894+1




calculated as

2 Ԧ1199072 =119888120572 ሶ119909 + 119904120572 ሶ119910minus119904120572 ሶ119909 + 119888120572 ሶ119910

0


3 Ԧ1199073 =

119888120572 ሶ119909 + 119904120572 ሶ119910

minus119904120572 ሶ119909 + 119888120572 ሶ119910 + 119889 ሶ1205790






4 Ԧ1199074 =119904 120572 + 120573 ሶ119909 minus 119888 120572 + 120573 ሶ119910 minus 119888120573 119889 ሶ120579

119888 120572 + 120573 ሶ119909 + 119904 120572 + 120573 ሶ119910 + 119904120573 119889 ሶ1205790




119904 120572 + 120573 minus119888 120572 + 120573 minus119888120573 119889

119888 120572 + 120573 119904 120572 + 120573 119904120573 1198890 0 0

ሶ119909119910 ሶሶ120579

=119903 ሶ12059500












119904 120572 + 120573(119905) minus119888 120572 + 120573(119905) minus119888120573(119905)119889

119888 120572 + 120573(119905) 119904 120572 + 120573(119905) 119904120573(119905)1198890 0 0

ሶ119909ሶ119910ሶ120579

=119903 ሶ120595

minus119889119888 ሶ120573(119905)0


maneuverability)




119904 120572 + 120573 + 120574 minus119888 120572 + 120573 + 120574 minus119888 120573 + 120574 119889

119888 120572 + 120573 + 120574 119904 120572 + 120573 + 120574 119904 120573 + 120574 1198890 0 0

ሶ119909ሶ119910ሶ120579

=119903 ሶ120595 cos 120574

119903 ሶ120595 sin 120574 + 119903119901119903 ሶ1205951199011199030















120572119897 = 90 deg

120573119897 = 0 deg

120572119903 = minus90 deg

120573119903 = 180 deg




119904 120572119897 + 120573119897 ሶ119909 minus 119888 120572119897 + 120573119897 ሶ119910 minus 119888120573119897 119889 ሶ120579 = 119903119897 ሶ120595119897

119888 120572119897 + 120573119897 ሶ119909 + 119904 120572119897 + 120573119897 ሶ119910 + 119904120573119897 119889 ሶ120579 = 0



ሶ119909 minus 119889 ሶ120579 = 119903119897 ሶ120595119897

ሶ119910 = 0

ሶ119909 + 119889 ሶ120579 = 119903119903 ሶ120595119903119910 = 0




ሶ119909ሶ119910ሶ120579

=

1

2119903119897 ሶ120595119897 + 119903119903 ሶ120595119903

01

2119889minus119903119897 ሶ120595119897 + 119903119903 ሶ120595119903



rrlabcsuni-klde




rrlabcsuni-klde











1205721 = 0 deg

1205722 = 120 deg

1205723 = minus120 deg







119888 1205721 + 1205731 ሶ119909 + 119904 1205721 + 1205731 ሶ119910 + 119889 sdot 119904 1205731 ሶ120579 = 0

119888 1205722 + 1205732 ሶ119909 + 119904 1205722 + 1205732 ሶ119910 + 119889 sdot 119904 1205732 ሶ120579 = 0

119888 1205723 + 1205733 ሶ119909 + 119904 1205723 + 1205733 ሶ119910 + 119889 sdot 119904 1205733 ሶ120579 = 0







119888 120572119894 sdot ሶ119909 + 119904 120572119894 sdot ሶ119910




ሶ120595119894 =1


rrlabcsuni-klde



rrlabcsuni-klde

Mecanum Vehicle





Mecanum Kinematics



1205721 = 45 deg 1205731 = 45 deg 1205741 = minus45 deg

1205722 = 135 deg 1205732 = minus45 deg 1205742 = 45 deg


1205724 = minus45 deg 1205734 = 135 deg 1205744 = 45 deg


Mecanum Kinematics








Mecanum Kinematics


ሶ119909 =119903

4ሶ1205951 + ሶ1205952 + ሶ1205953 + ሶ1205954

ሶ119910 =119903

4minus ሶ1205951 + ሶ1205952 minus ሶ1205953 + ሶ1205954

ሶ120579 =119903

119889 2ሶ1205951 minus ሶ1205952 minus ሶ1205953 + ሶ1205954


Mecanum Kinematics


ሶ119909ሶ119910ሶ120579

=119903119908ℎ1198901198901198974


sdot

ሶ1205951

ሶ1205952

ሶ1205953

ሶ1205954

with a =2 2

119889


ሶ1205954 = ሶ1205951 + ሶ1205952 minus ሶ1205953

rrlabcsuni-klde





Ability to maneuver

Controllability

Stability



Stability




Typical setups




Maneuverability








Controllability



Examples




Wheel type

Drive geometry


Motortransmission


Degree of Mobility




119891119898 = 3 minus rank 119878


Degree of Mobility



1198891 = 1198892 1205731 = 1205732 = 0 1205721 + 120587 = 1205722






119891119904 = rank 119878prime








119891119872 = 119891119898 + 119891119904

rrlabcsuni-klde

Vehicle Kinematics



Tricycle Drive

Given 120575 119907


119903 = 119897 sdot cot 120575

119907prime =119907

cos 120575

119907119897 = 119907 sdot119903 + 119889

119903

119907119903 = 119907 sdot119903 minus 119889


120575


Ackermann Steering

Given 119907119863 120575 119897 119889

Unknown 119907119903119903 119907119897119903 119907119903119891 119907119897119891

Solution

120602 =119897

tan 120575

119907119897119903 =120602 minus 1198892 sdot 119907119863

120602

119907119903119903 =120602 + 1198892 sdot 119907119863

120602

119907119897119891 =120602 minus 1198892 2 + 1198972 sdot tan 120575

119897sdot 119907119863

119907119903119891 =1206022 + 1198972 sdot tan 120575

119897sdot 119907119863


120575

120602

rrlabcsuni-klde



Given 120602 120593 119907 119889



1205932 = arctan120549119910 + 119889

1205491199091206022 = 120602 sdot cos120593 cos1205932

1199072 = 119907 sdot 1206022120602

120549119910 = 120602 ∙ 119904119894119899 120593

1205931

1205932

1206021

1206022

120593 120549119909 = 120602 ∙ cos120593

120602

119889

rrlabcsuni-klde

Bicycle Model


119907


Bicycle Model


Low speed motion



119865119910 =1198981199072

120602






Bicycle Model

Given 120575119891 120575119903 119907 120579

Unknown ሶ119883 ሶ119884 ሶ120579

Solution

120575119907 = tanminus1119897119891 tan 120575119903+119897119903 tan 120575119891

119897119891+119897119903

ሶ120579 =119907

120602

120602 =119897119891+119897119903

tan 120575119891 minustan 120575119903 cos 120575119907

ሶ119883 = 119907 cos 120579 + 120575119907ሶ119884 = 119907 sin 120579 + 120575119907

ሶ120579 =119907 cos 120575119907

119897119891+119897119903tan 120575119891 minus tan 120575119903

119907

rrlabcsuni-klde



119862 = 119874 120575119903 = 0 120575119891 = 120575

119897119903 = 0 119897119891 = 119897

120575119907 = 0

ሶ120579 =119907

120602

120602 =119897

tan 120575119891

ሶ119883 = 119907 cos 120579ሶ119884 = 119907 sin 120579ሶ120579 =

119907

119897119891tan 120575119891

119907






119877119904 =



Linear Velocity




119860119877119861 Ԧ119907119902


119860 Ԧ119907119902 =119860 Ԧ119907119874119861 + 119861

119860119877119861 Ԧ119907119902



calculated with

119860 Ԧ119907119876 = 119860 Ԧ119907119874119861 + 119861119860119877119861 Ԧ119907119876 +

119860ω119861 times 119861119860119877119861 Ԧ119902


Rotational Velocity



119860120596 = 119861119860119877119861120596





119894+1120596119894+1 = 119894119894+1119877 sdot 119894120596119894 + ሶθ119894+1

119894+1 Ԧ119890119911119894+1

119894+1 Ԧ119907119894+1 = 119894119894+1119877 119894 Ԧ119907119894 +

119894120596119894 times119894 Ԧ119901119894+1






119894119877 from frame 119894 to

119894 + 1





119894119894+1119877 θ = 119894+1

119894119877minus1 θ = 119894+1119894119877119879 θ =

119888θ 119904θ 119909119908minus119904θ 119888θ 1199101199080 0 1

rrlabcsuni-klde



YW

XW

θ

YR

XR

P









coordinates





0 0 1


ሶԦ120585119877 = 119877 θሶԦ120585119882


YW

XW

θ

YR

XR




119877120587

2=

0 1 0minus1 0 00 0 1


ሶԦ120585119877 = 119877120587

2ሶ

120585119882 =0 1 0minus1 0 00 0 1


=ሶ119910


rrlabcsuni-klde


Analytical Solution






YW

XW

θ

V(t)

120596(t)







ሶԦ120585119882 = ሶ119909 ሶ119910 ሶ120579119879

= 119891 119889 119903 120579 ሶ120595119897 ሶ120595119903

= 119877 120579 minus1 ሶԦ120585119877

passive wheel

active wheel

d






ሶ119909119877119903 =119903 ሶ1205951199032

ሶ119909119877119897 =119903 ሶ120595119897

2

120596119903 =119903 ሶ1205951199032119889

120596119897 =minus119903 ሶ120595119897

2119889

ሶԦ120585119882 = 119877 120579 minus1 ሶԦ120585119903ሶԦ120585119903 =

119903 ሶ1205951199032

+119903 ሶ120595119897

2 0

119903 ሶ1205951199032119889

minus119903 ሶ120595119897

2119889

119879





Given 120579 =120587



119904

ሶԦ120585119882 =ሶ119909ሶ119910ሶ120579

=0 minus1 01 0 00 0 1

sdot301

=031







Consider




rrlabcsuni-klde

Basic Wheel Types


Standardwheel

Steerablestandard

wheel

Castorwheel

Swedish orMecanum

wheel

Ball orspherical

wheel


Wheel Properties




Castor wheel


















ሶ120595 119903



ሶ120595 119903



ሶ120595 119903



ሶ120595 119903



ሶ120595 119903








and 1 Ԧ1199071 = ሶ119909 ሶ119910 0 119879


119894+1120596119894+1 = 119894119894+1119877 sdot 119894120596119894 + ሶ120579119894+1

119894+1 Ԧ119890119911119894+1

119894+1 Ԧ119907119894+1 = 119894119894+1119877 119894 Ԧ119907119894 +

119894120596119894 times119894 Ԧ119901119894+1




calculated as

2 Ԧ1199072 =119888120572 ሶ119909 + 119904120572 ሶ119910minus119904120572 ሶ119909 + 119888120572 ሶ119910

0


3 Ԧ1199073 =

119888120572 ሶ119909 + 119904120572 ሶ119910

minus119904120572 ሶ119909 + 119888120572 ሶ119910 + 119889 ሶ1205790






4 Ԧ1199074 =119904 120572 + 120573 ሶ119909 minus 119888 120572 + 120573 ሶ119910 minus 119888120573 119889 ሶ120579

119888 120572 + 120573 ሶ119909 + 119904 120572 + 120573 ሶ119910 + 119904120573 119889 ሶ1205790




119904 120572 + 120573 minus119888 120572 + 120573 minus119888120573 119889

119888 120572 + 120573 119904 120572 + 120573 119904120573 1198890 0 0

ሶ119909119910 ሶሶ120579

=119903 ሶ12059500












119904 120572 + 120573(119905) minus119888 120572 + 120573(119905) minus119888120573(119905)119889

119888 120572 + 120573(119905) 119904 120572 + 120573(119905) 119904120573(119905)1198890 0 0

ሶ119909ሶ119910ሶ120579

=119903 ሶ120595

minus119889119888 ሶ120573(119905)0


maneuverability)




119904 120572 + 120573 + 120574 minus119888 120572 + 120573 + 120574 minus119888 120573 + 120574 119889

119888 120572 + 120573 + 120574 119904 120572 + 120573 + 120574 119904 120573 + 120574 1198890 0 0

ሶ119909ሶ119910ሶ120579

=119903 ሶ120595 cos 120574

119903 ሶ120595 sin 120574 + 119903119901119903 ሶ1205951199011199030















120572119897 = 90 deg

120573119897 = 0 deg

120572119903 = minus90 deg

120573119903 = 180 deg




119904 120572119897 + 120573119897 ሶ119909 minus 119888 120572119897 + 120573119897 ሶ119910 minus 119888120573119897 119889 ሶ120579 = 119903119897 ሶ120595119897

119888 120572119897 + 120573119897 ሶ119909 + 119904 120572119897 + 120573119897 ሶ119910 + 119904120573119897 119889 ሶ120579 = 0



ሶ119909 minus 119889 ሶ120579 = 119903119897 ሶ120595119897

ሶ119910 = 0

ሶ119909 + 119889 ሶ120579 = 119903119903 ሶ120595119903119910 = 0




ሶ119909ሶ119910ሶ120579

=

1

2119903119897 ሶ120595119897 + 119903119903 ሶ120595119903

01

2119889minus119903119897 ሶ120595119897 + 119903119903 ሶ120595119903



rrlabcsuni-klde




rrlabcsuni-klde











1205721 = 0 deg

1205722 = 120 deg

1205723 = minus120 deg







119888 1205721 + 1205731 ሶ119909 + 119904 1205721 + 1205731 ሶ119910 + 119889 sdot 119904 1205731 ሶ120579 = 0

119888 1205722 + 1205732 ሶ119909 + 119904 1205722 + 1205732 ሶ119910 + 119889 sdot 119904 1205732 ሶ120579 = 0

119888 1205723 + 1205733 ሶ119909 + 119904 1205723 + 1205733 ሶ119910 + 119889 sdot 119904 1205733 ሶ120579 = 0







119888 120572119894 sdot ሶ119909 + 119904 120572119894 sdot ሶ119910




ሶ120595119894 =1


rrlabcsuni-klde



rrlabcsuni-klde

Mecanum Vehicle





Mecanum Kinematics



1205721 = 45 deg 1205731 = 45 deg 1205741 = minus45 deg

1205722 = 135 deg 1205732 = minus45 deg 1205742 = 45 deg


1205724 = minus45 deg 1205734 = 135 deg 1205744 = 45 deg


Mecanum Kinematics








Mecanum Kinematics


ሶ119909 =119903

4ሶ1205951 + ሶ1205952 + ሶ1205953 + ሶ1205954

ሶ119910 =119903

4minus ሶ1205951 + ሶ1205952 minus ሶ1205953 + ሶ1205954

ሶ120579 =119903

119889 2ሶ1205951 minus ሶ1205952 minus ሶ1205953 + ሶ1205954


Mecanum Kinematics


ሶ119909ሶ119910ሶ120579

=119903119908ℎ1198901198901198974


sdot

ሶ1205951

ሶ1205952

ሶ1205953

ሶ1205954

with a =2 2

119889


ሶ1205954 = ሶ1205951 + ሶ1205952 minus ሶ1205953

rrlabcsuni-klde





Ability to maneuver

Controllability

Stability



Stability




Typical setups




Maneuverability








Controllability



Examples




Wheel type

Drive geometry


Motortransmission


Degree of Mobility




119891119898 = 3 minus rank 119878


Degree of Mobility



1198891 = 1198892 1205731 = 1205732 = 0 1205721 + 120587 = 1205722






119891119904 = rank 119878prime








119891119872 = 119891119898 + 119891119904

rrlabcsuni-klde

Vehicle Kinematics



Tricycle Drive

Given 120575 119907


119903 = 119897 sdot cot 120575

119907prime =119907

cos 120575

119907119897 = 119907 sdot119903 + 119889

119903

119907119903 = 119907 sdot119903 minus 119889


120575


Ackermann Steering

Given 119907119863 120575 119897 119889

Unknown 119907119903119903 119907119897119903 119907119903119891 119907119897119891

Solution

120602 =119897

tan 120575

119907119897119903 =120602 minus 1198892 sdot 119907119863

120602

119907119903119903 =120602 + 1198892 sdot 119907119863

120602

119907119897119891 =120602 minus 1198892 2 + 1198972 sdot tan 120575

119897sdot 119907119863

119907119903119891 =1206022 + 1198972 sdot tan 120575

119897sdot 119907119863


120575

120602

rrlabcsuni-klde



Given 120602 120593 119907 119889



1205932 = arctan120549119910 + 119889

1205491199091206022 = 120602 sdot cos120593 cos1205932

1199072 = 119907 sdot 1206022120602

120549119910 = 120602 ∙ 119904119894119899 120593

1205931

1205932

1206021

1206022

120593 120549119909 = 120602 ∙ cos120593

120602

119889

rrlabcsuni-klde

Bicycle Model


119907


Bicycle Model


Low speed motion



119865119910 =1198981199072

120602






Bicycle Model

Given 120575119891 120575119903 119907 120579

Unknown ሶ119883 ሶ119884 ሶ120579

Solution

120575119907 = tanminus1119897119891 tan 120575119903+119897119903 tan 120575119891

119897119891+119897119903

ሶ120579 =119907

120602

120602 =119897119891+119897119903

tan 120575119891 minustan 120575119903 cos 120575119907

ሶ119883 = 119907 cos 120579 + 120575119907ሶ119884 = 119907 sin 120579 + 120575119907

ሶ120579 =119907 cos 120575119907

119897119891+119897119903tan 120575119891 minus tan 120575119903

119907

rrlabcsuni-klde



119862 = 119874 120575119903 = 0 120575119891 = 120575

119897119903 = 0 119897119891 = 119897

120575119907 = 0

ሶ120579 =119907

120602

120602 =119897

tan 120575119891

ሶ119883 = 119907 cos 120579ሶ119884 = 119907 sin 120579ሶ120579 =

119907

119897119891tan 120575119891

119907


Linear Velocity




119860119877119861 Ԧ119907119902


119860 Ԧ119907119902 =119860 Ԧ119907119874119861 + 119861

119860119877119861 Ԧ119907119902



calculated with

119860 Ԧ119907119876 = 119860 Ԧ119907119874119861 + 119861119860119877119861 Ԧ119907119876 +

119860ω119861 times 119861119860119877119861 Ԧ119902


Rotational Velocity



119860120596 = 119861119860119877119861120596





119894+1120596119894+1 = 119894119894+1119877 sdot 119894120596119894 + ሶθ119894+1

119894+1 Ԧ119890119911119894+1

119894+1 Ԧ119907119894+1 = 119894119894+1119877 119894 Ԧ119907119894 +

119894120596119894 times119894 Ԧ119901119894+1






119894119877 from frame 119894 to

119894 + 1





119894119894+1119877 θ = 119894+1

119894119877minus1 θ = 119894+1119894119877119879 θ =

119888θ 119904θ 119909119908minus119904θ 119888θ 1199101199080 0 1

rrlabcsuni-klde



YW

XW

θ

YR

XR

P









coordinates





0 0 1


ሶԦ120585119877 = 119877 θሶԦ120585119882


YW

XW

θ

YR

XR




119877120587

2=

0 1 0minus1 0 00 0 1


ሶԦ120585119877 = 119877120587

2ሶ

120585119882 =0 1 0minus1 0 00 0 1


=ሶ119910


rrlabcsuni-klde


Analytical Solution






YW

XW

θ

V(t)

120596(t)







ሶԦ120585119882 = ሶ119909 ሶ119910 ሶ120579119879

= 119891 119889 119903 120579 ሶ120595119897 ሶ120595119903

= 119877 120579 minus1 ሶԦ120585119877

passive wheel

active wheel

d






ሶ119909119877119903 =119903 ሶ1205951199032

ሶ119909119877119897 =119903 ሶ120595119897

2

120596119903 =119903 ሶ1205951199032119889

120596119897 =minus119903 ሶ120595119897

2119889

ሶԦ120585119882 = 119877 120579 minus1 ሶԦ120585119903ሶԦ120585119903 =

119903 ሶ1205951199032

+119903 ሶ120595119897

2 0

119903 ሶ1205951199032119889

minus119903 ሶ120595119897

2119889

119879





Given 120579 =120587



119904

ሶԦ120585119882 =ሶ119909ሶ119910ሶ120579

=0 minus1 01 0 00 0 1

sdot301

=031







Consider




rrlabcsuni-klde

Basic Wheel Types


Standardwheel

Steerablestandard

wheel

Castorwheel

Swedish orMecanum

wheel

Ball orspherical

wheel


Wheel Properties




Castor wheel


















ሶ120595 119903



ሶ120595 119903



ሶ120595 119903



ሶ120595 119903



ሶ120595 119903








and 1 Ԧ1199071 = ሶ119909 ሶ119910 0 119879


119894+1120596119894+1 = 119894119894+1119877 sdot 119894120596119894 + ሶ120579119894+1

119894+1 Ԧ119890119911119894+1

119894+1 Ԧ119907119894+1 = 119894119894+1119877 119894 Ԧ119907119894 +

119894120596119894 times119894 Ԧ119901119894+1




calculated as

2 Ԧ1199072 =119888120572 ሶ119909 + 119904120572 ሶ119910minus119904120572 ሶ119909 + 119888120572 ሶ119910

0


3 Ԧ1199073 =

119888120572 ሶ119909 + 119904120572 ሶ119910

minus119904120572 ሶ119909 + 119888120572 ሶ119910 + 119889 ሶ1205790






4 Ԧ1199074 =119904 120572 + 120573 ሶ119909 minus 119888 120572 + 120573 ሶ119910 minus 119888120573 119889 ሶ120579

119888 120572 + 120573 ሶ119909 + 119904 120572 + 120573 ሶ119910 + 119904120573 119889 ሶ1205790




119904 120572 + 120573 minus119888 120572 + 120573 minus119888120573 119889

119888 120572 + 120573 119904 120572 + 120573 119904120573 1198890 0 0

ሶ119909119910 ሶሶ120579

=119903 ሶ12059500












119904 120572 + 120573(119905) minus119888 120572 + 120573(119905) minus119888120573(119905)119889

119888 120572 + 120573(119905) 119904 120572 + 120573(119905) 119904120573(119905)1198890 0 0

ሶ119909ሶ119910ሶ120579

=119903 ሶ120595

minus119889119888 ሶ120573(119905)0


maneuverability)




119904 120572 + 120573 + 120574 minus119888 120572 + 120573 + 120574 minus119888 120573 + 120574 119889

119888 120572 + 120573 + 120574 119904 120572 + 120573 + 120574 119904 120573 + 120574 1198890 0 0

ሶ119909ሶ119910ሶ120579

=119903 ሶ120595 cos 120574

119903 ሶ120595 sin 120574 + 119903119901119903 ሶ1205951199011199030















120572119897 = 90 deg

120573119897 = 0 deg

120572119903 = minus90 deg

120573119903 = 180 deg




119904 120572119897 + 120573119897 ሶ119909 minus 119888 120572119897 + 120573119897 ሶ119910 minus 119888120573119897 119889 ሶ120579 = 119903119897 ሶ120595119897

119888 120572119897 + 120573119897 ሶ119909 + 119904 120572119897 + 120573119897 ሶ119910 + 119904120573119897 119889 ሶ120579 = 0



ሶ119909 minus 119889 ሶ120579 = 119903119897 ሶ120595119897

ሶ119910 = 0

ሶ119909 + 119889 ሶ120579 = 119903119903 ሶ120595119903119910 = 0




ሶ119909ሶ119910ሶ120579

=

1

2119903119897 ሶ120595119897 + 119903119903 ሶ120595119903

01

2119889minus119903119897 ሶ120595119897 + 119903119903 ሶ120595119903



rrlabcsuni-klde




rrlabcsuni-klde











1205721 = 0 deg

1205722 = 120 deg

1205723 = minus120 deg







119888 1205721 + 1205731 ሶ119909 + 119904 1205721 + 1205731 ሶ119910 + 119889 sdot 119904 1205731 ሶ120579 = 0

119888 1205722 + 1205732 ሶ119909 + 119904 1205722 + 1205732 ሶ119910 + 119889 sdot 119904 1205732 ሶ120579 = 0

119888 1205723 + 1205733 ሶ119909 + 119904 1205723 + 1205733 ሶ119910 + 119889 sdot 119904 1205733 ሶ120579 = 0







119888 120572119894 sdot ሶ119909 + 119904 120572119894 sdot ሶ119910




ሶ120595119894 =1


rrlabcsuni-klde



rrlabcsuni-klde

Mecanum Vehicle





Mecanum Kinematics



1205721 = 45 deg 1205731 = 45 deg 1205741 = minus45 deg

1205722 = 135 deg 1205732 = minus45 deg 1205742 = 45 deg


1205724 = minus45 deg 1205734 = 135 deg 1205744 = 45 deg


Mecanum Kinematics








Mecanum Kinematics


ሶ119909 =119903

4ሶ1205951 + ሶ1205952 + ሶ1205953 + ሶ1205954

ሶ119910 =119903

4minus ሶ1205951 + ሶ1205952 minus ሶ1205953 + ሶ1205954

ሶ120579 =119903

119889 2ሶ1205951 minus ሶ1205952 minus ሶ1205953 + ሶ1205954


Mecanum Kinematics


ሶ119909ሶ119910ሶ120579

=119903119908ℎ1198901198901198974


sdot

ሶ1205951

ሶ1205952

ሶ1205953

ሶ1205954

with a =2 2

119889


ሶ1205954 = ሶ1205951 + ሶ1205952 minus ሶ1205953

rrlabcsuni-klde





Ability to maneuver

Controllability

Stability



Stability




Typical setups




Maneuverability








Controllability



Examples




Wheel type

Drive geometry


Motortransmission


Degree of Mobility




119891119898 = 3 minus rank 119878


Degree of Mobility



1198891 = 1198892 1205731 = 1205732 = 0 1205721 + 120587 = 1205722






119891119904 = rank 119878prime








119891119872 = 119891119898 + 119891119904

rrlabcsuni-klde

Vehicle Kinematics



Tricycle Drive

Given 120575 119907


119903 = 119897 sdot cot 120575

119907prime =119907

cos 120575

119907119897 = 119907 sdot119903 + 119889

119903

119907119903 = 119907 sdot119903 minus 119889


120575


Ackermann Steering

Given 119907119863 120575 119897 119889

Unknown 119907119903119903 119907119897119903 119907119903119891 119907119897119891

Solution

120602 =119897

tan 120575

119907119897119903 =120602 minus 1198892 sdot 119907119863

120602

119907119903119903 =120602 + 1198892 sdot 119907119863

120602

119907119897119891 =120602 minus 1198892 2 + 1198972 sdot tan 120575

119897sdot 119907119863

119907119903119891 =1206022 + 1198972 sdot tan 120575

119897sdot 119907119863


120575

120602

rrlabcsuni-klde



Given 120602 120593 119907 119889



1205932 = arctan120549119910 + 119889

1205491199091206022 = 120602 sdot cos120593 cos1205932

1199072 = 119907 sdot 1206022120602

120549119910 = 120602 ∙ 119904119894119899 120593

1205931

1205932

1206021

1206022

120593 120549119909 = 120602 ∙ cos120593

120602

119889

rrlabcsuni-klde

Bicycle Model


119907


Bicycle Model


Low speed motion



119865119910 =1198981199072

120602






Bicycle Model

Given 120575119891 120575119903 119907 120579

Unknown ሶ119883 ሶ119884 ሶ120579

Solution

120575119907 = tanminus1119897119891 tan 120575119903+119897119903 tan 120575119891

119897119891+119897119903

ሶ120579 =119907

120602

120602 =119897119891+119897119903

tan 120575119891 minustan 120575119903 cos 120575119907

ሶ119883 = 119907 cos 120579 + 120575119907ሶ119884 = 119907 sin 120579 + 120575119907

ሶ120579 =119907 cos 120575119907

119897119891+119897119903tan 120575119891 minus tan 120575119903

119907

rrlabcsuni-klde



119862 = 119874 120575119903 = 0 120575119891 = 120575

119897119903 = 0 119897119891 = 119897

120575119907 = 0

ሶ120579 =119907

120602

120602 =119897

tan 120575119891

ሶ119883 = 119907 cos 120579ሶ119884 = 119907 sin 120579ሶ120579 =

119907

119897119891tan 120575119891

119907


Rotational Velocity



119860120596 = 119861119860119877119861120596





119894+1120596119894+1 = 119894119894+1119877 sdot 119894120596119894 + ሶθ119894+1

119894+1 Ԧ119890119911119894+1

119894+1 Ԧ119907119894+1 = 119894119894+1119877 119894 Ԧ119907119894 +

119894120596119894 times119894 Ԧ119901119894+1






119894119877 from frame 119894 to

119894 + 1





119894119894+1119877 θ = 119894+1

119894119877minus1 θ = 119894+1119894119877119879 θ =

119888θ 119904θ 119909119908minus119904θ 119888θ 1199101199080 0 1

rrlabcsuni-klde



YW

XW

θ

YR

XR

P









coordinates





0 0 1


ሶԦ120585119877 = 119877 θሶԦ120585119882


YW

XW

θ

YR

XR




119877120587

2=

0 1 0minus1 0 00 0 1


ሶԦ120585119877 = 119877120587

2ሶ

120585119882 =0 1 0minus1 0 00 0 1


=ሶ119910


rrlabcsuni-klde


Analytical Solution






YW

XW

θ

V(t)

120596(t)







ሶԦ120585119882 = ሶ119909 ሶ119910 ሶ120579119879

= 119891 119889 119903 120579 ሶ120595119897 ሶ120595119903

= 119877 120579 minus1 ሶԦ120585119877

passive wheel

active wheel

d






ሶ119909119877119903 =119903 ሶ1205951199032

ሶ119909119877119897 =119903 ሶ120595119897

2

120596119903 =119903 ሶ1205951199032119889

120596119897 =minus119903 ሶ120595119897

2119889

ሶԦ120585119882 = 119877 120579 minus1 ሶԦ120585119903ሶԦ120585119903 =

119903 ሶ1205951199032

+119903 ሶ120595119897

2 0

119903 ሶ1205951199032119889

minus119903 ሶ120595119897

2119889

119879





Given 120579 =120587



119904

ሶԦ120585119882 =ሶ119909ሶ119910ሶ120579

=0 minus1 01 0 00 0 1

sdot301

=031







Consider




rrlabcsuni-klde

Basic Wheel Types


Standardwheel

Steerablestandard

wheel

Castorwheel

Swedish orMecanum

wheel

Ball orspherical

wheel


Wheel Properties




Castor wheel


















ሶ120595 119903



ሶ120595 119903



ሶ120595 119903



ሶ120595 119903



ሶ120595 119903








and 1 Ԧ1199071 = ሶ119909 ሶ119910 0 119879


119894+1120596119894+1 = 119894119894+1119877 sdot 119894120596119894 + ሶ120579119894+1

119894+1 Ԧ119890119911119894+1

119894+1 Ԧ119907119894+1 = 119894119894+1119877 119894 Ԧ119907119894 +

119894120596119894 times119894 Ԧ119901119894+1




calculated as

2 Ԧ1199072 =119888120572 ሶ119909 + 119904120572 ሶ119910minus119904120572 ሶ119909 + 119888120572 ሶ119910

0


3 Ԧ1199073 =

119888120572 ሶ119909 + 119904120572 ሶ119910

minus119904120572 ሶ119909 + 119888120572 ሶ119910 + 119889 ሶ1205790






4 Ԧ1199074 =119904 120572 + 120573 ሶ119909 minus 119888 120572 + 120573 ሶ119910 minus 119888120573 119889 ሶ120579

119888 120572 + 120573 ሶ119909 + 119904 120572 + 120573 ሶ119910 + 119904120573 119889 ሶ1205790




119904 120572 + 120573 minus119888 120572 + 120573 minus119888120573 119889

119888 120572 + 120573 119904 120572 + 120573 119904120573 1198890 0 0

ሶ119909119910 ሶሶ120579

=119903 ሶ12059500












119904 120572 + 120573(119905) minus119888 120572 + 120573(119905) minus119888120573(119905)119889

119888 120572 + 120573(119905) 119904 120572 + 120573(119905) 119904120573(119905)1198890 0 0

ሶ119909ሶ119910ሶ120579

=119903 ሶ120595

minus119889119888 ሶ120573(119905)0


maneuverability)




119904 120572 + 120573 + 120574 minus119888 120572 + 120573 + 120574 minus119888 120573 + 120574 119889

119888 120572 + 120573 + 120574 119904 120572 + 120573 + 120574 119904 120573 + 120574 1198890 0 0

ሶ119909ሶ119910ሶ120579

=119903 ሶ120595 cos 120574

119903 ሶ120595 sin 120574 + 119903119901119903 ሶ1205951199011199030















120572119897 = 90 deg

120573119897 = 0 deg

120572119903 = minus90 deg

120573119903 = 180 deg




119904 120572119897 + 120573119897 ሶ119909 minus 119888 120572119897 + 120573119897 ሶ119910 minus 119888120573119897 119889 ሶ120579 = 119903119897 ሶ120595119897

119888 120572119897 + 120573119897 ሶ119909 + 119904 120572119897 + 120573119897 ሶ119910 + 119904120573119897 119889 ሶ120579 = 0



ሶ119909 minus 119889 ሶ120579 = 119903119897 ሶ120595119897

ሶ119910 = 0

ሶ119909 + 119889 ሶ120579 = 119903119903 ሶ120595119903119910 = 0




ሶ119909ሶ119910ሶ120579

=

1

2119903119897 ሶ120595119897 + 119903119903 ሶ120595119903

01

2119889minus119903119897 ሶ120595119897 + 119903119903 ሶ120595119903



rrlabcsuni-klde




rrlabcsuni-klde











1205721 = 0 deg

1205722 = 120 deg

1205723 = minus120 deg







119888 1205721 + 1205731 ሶ119909 + 119904 1205721 + 1205731 ሶ119910 + 119889 sdot 119904 1205731 ሶ120579 = 0

119888 1205722 + 1205732 ሶ119909 + 119904 1205722 + 1205732 ሶ119910 + 119889 sdot 119904 1205732 ሶ120579 = 0

119888 1205723 + 1205733 ሶ119909 + 119904 1205723 + 1205733 ሶ119910 + 119889 sdot 119904 1205733 ሶ120579 = 0







119888 120572119894 sdot ሶ119909 + 119904 120572119894 sdot ሶ119910




ሶ120595119894 =1


rrlabcsuni-klde



rrlabcsuni-klde

Mecanum Vehicle





Mecanum Kinematics



1205721 = 45 deg 1205731 = 45 deg 1205741 = minus45 deg

1205722 = 135 deg 1205732 = minus45 deg 1205742 = 45 deg


1205724 = minus45 deg 1205734 = 135 deg 1205744 = 45 deg


Mecanum Kinematics








Mecanum Kinematics


ሶ119909 =119903

4ሶ1205951 + ሶ1205952 + ሶ1205953 + ሶ1205954

ሶ119910 =119903

4minus ሶ1205951 + ሶ1205952 minus ሶ1205953 + ሶ1205954

ሶ120579 =119903

119889 2ሶ1205951 minus ሶ1205952 minus ሶ1205953 + ሶ1205954


Mecanum Kinematics


ሶ119909ሶ119910ሶ120579

=119903119908ℎ1198901198901198974


sdot

ሶ1205951

ሶ1205952

ሶ1205953

ሶ1205954

with a =2 2

119889


ሶ1205954 = ሶ1205951 + ሶ1205952 minus ሶ1205953

rrlabcsuni-klde





Ability to maneuver

Controllability

Stability



Stability




Typical setups




Maneuverability








Controllability



Examples




Wheel type

Drive geometry


Motortransmission


Degree of Mobility




119891119898 = 3 minus rank 119878


Degree of Mobility



1198891 = 1198892 1205731 = 1205732 = 0 1205721 + 120587 = 1205722






119891119904 = rank 119878prime








119891119872 = 119891119898 + 119891119904

rrlabcsuni-klde

Vehicle Kinematics



Tricycle Drive

Given 120575 119907


119903 = 119897 sdot cot 120575

119907prime =119907

cos 120575

119907119897 = 119907 sdot119903 + 119889

119903

119907119903 = 119907 sdot119903 minus 119889


120575


Ackermann Steering

Given 119907119863 120575 119897 119889

Unknown 119907119903119903 119907119897119903 119907119903119891 119907119897119891

Solution

120602 =119897

tan 120575

119907119897119903 =120602 minus 1198892 sdot 119907119863

120602

119907119903119903 =120602 + 1198892 sdot 119907119863

120602

119907119897119891 =120602 minus 1198892 2 + 1198972 sdot tan 120575

119897sdot 119907119863

119907119903119891 =1206022 + 1198972 sdot tan 120575

119897sdot 119907119863


120575

120602

rrlabcsuni-klde



Given 120602 120593 119907 119889



1205932 = arctan120549119910 + 119889

1205491199091206022 = 120602 sdot cos120593 cos1205932

1199072 = 119907 sdot 1206022120602

120549119910 = 120602 ∙ 119904119894119899 120593

1205931

1205932

1206021

1206022

120593 120549119909 = 120602 ∙ cos120593

120602

119889

rrlabcsuni-klde

Bicycle Model


119907


Bicycle Model


Low speed motion



119865119910 =1198981199072

120602






Bicycle Model

Given 120575119891 120575119903 119907 120579

Unknown ሶ119883 ሶ119884 ሶ120579

Solution

120575119907 = tanminus1119897119891 tan 120575119903+119897119903 tan 120575119891

119897119891+119897119903

ሶ120579 =119907

120602

120602 =119897119891+119897119903

tan 120575119891 minustan 120575119903 cos 120575119907

ሶ119883 = 119907 cos 120579 + 120575119907ሶ119884 = 119907 sin 120579 + 120575119907

ሶ120579 =119907 cos 120575119907

119897119891+119897119903tan 120575119891 minus tan 120575119903

119907

rrlabcsuni-klde



119862 = 119874 120575119903 = 0 120575119891 = 120575

119897119903 = 0 119897119891 = 119897

120575119907 = 0

ሶ120579 =119907

120602

120602 =119897

tan 120575119891

ሶ119883 = 119907 cos 120579ሶ119884 = 119907 sin 120579ሶ120579 =

119907

119897119891tan 120575119891

119907





119894+1120596119894+1 = 119894119894+1119877 sdot 119894120596119894 + ሶθ119894+1

119894+1 Ԧ119890119911119894+1

119894+1 Ԧ119907119894+1 = 119894119894+1119877 119894 Ԧ119907119894 +

119894120596119894 times119894 Ԧ119901119894+1






119894119877 from frame 119894 to

119894 + 1





119894119894+1119877 θ = 119894+1

119894119877minus1 θ = 119894+1119894119877119879 θ =

119888θ 119904θ 119909119908minus119904θ 119888θ 1199101199080 0 1

rrlabcsuni-klde



YW

XW

θ

YR

XR

P









coordinates





0 0 1


ሶԦ120585119877 = 119877 θሶԦ120585119882


YW

XW

θ

YR

XR




119877120587

2=

0 1 0minus1 0 00 0 1


ሶԦ120585119877 = 119877120587

2ሶ

120585119882 =0 1 0minus1 0 00 0 1


=ሶ119910


rrlabcsuni-klde


Analytical Solution






YW

XW

θ

V(t)

120596(t)







ሶԦ120585119882 = ሶ119909 ሶ119910 ሶ120579119879

= 119891 119889 119903 120579 ሶ120595119897 ሶ120595119903

= 119877 120579 minus1 ሶԦ120585119877

passive wheel

active wheel

d






ሶ119909119877119903 =119903 ሶ1205951199032

ሶ119909119877119897 =119903 ሶ120595119897

2

120596119903 =119903 ሶ1205951199032119889

120596119897 =minus119903 ሶ120595119897

2119889

ሶԦ120585119882 = 119877 120579 minus1 ሶԦ120585119903ሶԦ120585119903 =

119903 ሶ1205951199032

+119903 ሶ120595119897

2 0

119903 ሶ1205951199032119889

minus119903 ሶ120595119897

2119889

119879





Given 120579 =120587



119904

ሶԦ120585119882 =ሶ119909ሶ119910ሶ120579

=0 minus1 01 0 00 0 1

sdot301

=031







Consider




rrlabcsuni-klde

Basic Wheel Types


Standardwheel

Steerablestandard

wheel

Castorwheel

Swedish orMecanum

wheel

Ball orspherical

wheel


Wheel Properties




Castor wheel


















ሶ120595 119903



ሶ120595 119903



ሶ120595 119903



ሶ120595 119903



ሶ120595 119903








and 1 Ԧ1199071 = ሶ119909 ሶ119910 0 119879


119894+1120596119894+1 = 119894119894+1119877 sdot 119894120596119894 + ሶ120579119894+1

119894+1 Ԧ119890119911119894+1

119894+1 Ԧ119907119894+1 = 119894119894+1119877 119894 Ԧ119907119894 +

119894120596119894 times119894 Ԧ119901119894+1




calculated as

2 Ԧ1199072 =119888120572 ሶ119909 + 119904120572 ሶ119910minus119904120572 ሶ119909 + 119888120572 ሶ119910

0


3 Ԧ1199073 =

119888120572 ሶ119909 + 119904120572 ሶ119910

minus119904120572 ሶ119909 + 119888120572 ሶ119910 + 119889 ሶ1205790






4 Ԧ1199074 =119904 120572 + 120573 ሶ119909 minus 119888 120572 + 120573 ሶ119910 minus 119888120573 119889 ሶ120579

119888 120572 + 120573 ሶ119909 + 119904 120572 + 120573 ሶ119910 + 119904120573 119889 ሶ1205790




119904 120572 + 120573 minus119888 120572 + 120573 minus119888120573 119889

119888 120572 + 120573 119904 120572 + 120573 119904120573 1198890 0 0

ሶ119909119910 ሶሶ120579

=119903 ሶ12059500












119904 120572 + 120573(119905) minus119888 120572 + 120573(119905) minus119888120573(119905)119889

119888 120572 + 120573(119905) 119904 120572 + 120573(119905) 119904120573(119905)1198890 0 0

ሶ119909ሶ119910ሶ120579

=119903 ሶ120595

minus119889119888 ሶ120573(119905)0


maneuverability)




119904 120572 + 120573 + 120574 minus119888 120572 + 120573 + 120574 minus119888 120573 + 120574 119889

119888 120572 + 120573 + 120574 119904 120572 + 120573 + 120574 119904 120573 + 120574 1198890 0 0

ሶ119909ሶ119910ሶ120579

=119903 ሶ120595 cos 120574

119903 ሶ120595 sin 120574 + 119903119901119903 ሶ1205951199011199030















120572119897 = 90 deg

120573119897 = 0 deg

120572119903 = minus90 deg

120573119903 = 180 deg




119904 120572119897 + 120573119897 ሶ119909 minus 119888 120572119897 + 120573119897 ሶ119910 minus 119888120573119897 119889 ሶ120579 = 119903119897 ሶ120595119897

119888 120572119897 + 120573119897 ሶ119909 + 119904 120572119897 + 120573119897 ሶ119910 + 119904120573119897 119889 ሶ120579 = 0



ሶ119909 minus 119889 ሶ120579 = 119903119897 ሶ120595119897

ሶ119910 = 0

ሶ119909 + 119889 ሶ120579 = 119903119903 ሶ120595119903119910 = 0




ሶ119909ሶ119910ሶ120579

=

1

2119903119897 ሶ120595119897 + 119903119903 ሶ120595119903

01

2119889minus119903119897 ሶ120595119897 + 119903119903 ሶ120595119903



rrlabcsuni-klde




rrlabcsuni-klde











1205721 = 0 deg

1205722 = 120 deg

1205723 = minus120 deg







119888 1205721 + 1205731 ሶ119909 + 119904 1205721 + 1205731 ሶ119910 + 119889 sdot 119904 1205731 ሶ120579 = 0

119888 1205722 + 1205732 ሶ119909 + 119904 1205722 + 1205732 ሶ119910 + 119889 sdot 119904 1205732 ሶ120579 = 0

119888 1205723 + 1205733 ሶ119909 + 119904 1205723 + 1205733 ሶ119910 + 119889 sdot 119904 1205733 ሶ120579 = 0







119888 120572119894 sdot ሶ119909 + 119904 120572119894 sdot ሶ119910




ሶ120595119894 =1


rrlabcsuni-klde



rrlabcsuni-klde

Mecanum Vehicle





Mecanum Kinematics



1205721 = 45 deg 1205731 = 45 deg 1205741 = minus45 deg

1205722 = 135 deg 1205732 = minus45 deg 1205742 = 45 deg


1205724 = minus45 deg 1205734 = 135 deg 1205744 = 45 deg


Mecanum Kinematics








Mecanum Kinematics


ሶ119909 =119903

4ሶ1205951 + ሶ1205952 + ሶ1205953 + ሶ1205954

ሶ119910 =119903

4minus ሶ1205951 + ሶ1205952 minus ሶ1205953 + ሶ1205954

ሶ120579 =119903

119889 2ሶ1205951 minus ሶ1205952 minus ሶ1205953 + ሶ1205954


Mecanum Kinematics


ሶ119909ሶ119910ሶ120579

=119903119908ℎ1198901198901198974


sdot

ሶ1205951

ሶ1205952

ሶ1205953

ሶ1205954

with a =2 2

119889


ሶ1205954 = ሶ1205951 + ሶ1205952 minus ሶ1205953

rrlabcsuni-klde





Ability to maneuver

Controllability

Stability



Stability




Typical setups




Maneuverability








Controllability



Examples




Wheel type

Drive geometry


Motortransmission


Degree of Mobility




119891119898 = 3 minus rank 119878


Degree of Mobility



1198891 = 1198892 1205731 = 1205732 = 0 1205721 + 120587 = 1205722






119891119904 = rank 119878prime








119891119872 = 119891119898 + 119891119904

rrlabcsuni-klde

Vehicle Kinematics



Tricycle Drive

Given 120575 119907


119903 = 119897 sdot cot 120575

119907prime =119907

cos 120575

119907119897 = 119907 sdot119903 + 119889

119903

119907119903 = 119907 sdot119903 minus 119889


120575


Ackermann Steering

Given 119907119863 120575 119897 119889

Unknown 119907119903119903 119907119897119903 119907119903119891 119907119897119891

Solution

120602 =119897

tan 120575

119907119897119903 =120602 minus 1198892 sdot 119907119863

120602

119907119903119903 =120602 + 1198892 sdot 119907119863

120602

119907119897119891 =120602 minus 1198892 2 + 1198972 sdot tan 120575

119897sdot 119907119863

119907119903119891 =1206022 + 1198972 sdot tan 120575

119897sdot 119907119863


120575

120602

rrlabcsuni-klde



Given 120602 120593 119907 119889



1205932 = arctan120549119910 + 119889

1205491199091206022 = 120602 sdot cos120593 cos1205932

1199072 = 119907 sdot 1206022120602

120549119910 = 120602 ∙ 119904119894119899 120593

1205931

1205932

1206021

1206022

120593 120549119909 = 120602 ∙ cos120593

120602

119889

rrlabcsuni-klde

Bicycle Model


119907


Bicycle Model


Low speed motion



119865119910 =1198981199072

120602






Bicycle Model

Given 120575119891 120575119903 119907 120579

Unknown ሶ119883 ሶ119884 ሶ120579

Solution

120575119907 = tanminus1119897119891 tan 120575119903+119897119903 tan 120575119891

119897119891+119897119903

ሶ120579 =119907

120602

120602 =119897119891+119897119903

tan 120575119891 minustan 120575119903 cos 120575119907

ሶ119883 = 119907 cos 120579 + 120575119907ሶ119884 = 119907 sin 120579 + 120575119907

ሶ120579 =119907 cos 120575119907

119897119891+119897119903tan 120575119891 minus tan 120575119903

119907

rrlabcsuni-klde



119862 = 119874 120575119903 = 0 120575119891 = 120575

119897119903 = 0 119897119891 = 119897

120575119907 = 0

ሶ120579 =119907

120602

120602 =119897

tan 120575119891

ሶ119883 = 119907 cos 120579ሶ119884 = 119907 sin 120579ሶ120579 =

119907

119897119891tan 120575119891

119907




119894119877 from frame 119894 to

119894 + 1





119894119894+1119877 θ = 119894+1

119894119877minus1 θ = 119894+1119894119877119879 θ =

119888θ 119904θ 119909119908minus119904θ 119888θ 1199101199080 0 1

rrlabcsuni-klde



YW

XW

θ

YR

XR

P









coordinates





0 0 1


ሶԦ120585119877 = 119877 θሶԦ120585119882


YW

XW

θ

YR

XR




119877120587

2=

0 1 0minus1 0 00 0 1


ሶԦ120585119877 = 119877120587

2ሶ

120585119882 =0 1 0minus1 0 00 0 1


=ሶ119910


rrlabcsuni-klde


Analytical Solution






YW

XW

θ

V(t)

120596(t)







ሶԦ120585119882 = ሶ119909 ሶ119910 ሶ120579119879

= 119891 119889 119903 120579 ሶ120595119897 ሶ120595119903

= 119877 120579 minus1 ሶԦ120585119877

passive wheel

active wheel

d






ሶ119909119877119903 =119903 ሶ1205951199032

ሶ119909119877119897 =119903 ሶ120595119897

2

120596119903 =119903 ሶ1205951199032119889

120596119897 =minus119903 ሶ120595119897

2119889

ሶԦ120585119882 = 119877 120579 minus1 ሶԦ120585119903ሶԦ120585119903 =

119903 ሶ1205951199032

+119903 ሶ120595119897

2 0

119903 ሶ1205951199032119889

minus119903 ሶ120595119897

2119889

119879





Given 120579 =120587



119904

ሶԦ120585119882 =ሶ119909ሶ119910ሶ120579

=0 minus1 01 0 00 0 1

sdot301

=031







Consider




rrlabcsuni-klde

Basic Wheel Types


Standardwheel

Steerablestandard

wheel

Castorwheel

Swedish orMecanum

wheel

Ball orspherical

wheel


Wheel Properties




Castor wheel


















ሶ120595 119903



ሶ120595 119903



ሶ120595 119903



ሶ120595 119903



ሶ120595 119903








and 1 Ԧ1199071 = ሶ119909 ሶ119910 0 119879


119894+1120596119894+1 = 119894119894+1119877 sdot 119894120596119894 + ሶ120579119894+1

119894+1 Ԧ119890119911119894+1

119894+1 Ԧ119907119894+1 = 119894119894+1119877 119894 Ԧ119907119894 +

119894120596119894 times119894 Ԧ119901119894+1




calculated as

2 Ԧ1199072 =119888120572 ሶ119909 + 119904120572 ሶ119910minus119904120572 ሶ119909 + 119888120572 ሶ119910

0


3 Ԧ1199073 =

119888120572 ሶ119909 + 119904120572 ሶ119910

minus119904120572 ሶ119909 + 119888120572 ሶ119910 + 119889 ሶ1205790






4 Ԧ1199074 =119904 120572 + 120573 ሶ119909 minus 119888 120572 + 120573 ሶ119910 minus 119888120573 119889 ሶ120579

119888 120572 + 120573 ሶ119909 + 119904 120572 + 120573 ሶ119910 + 119904120573 119889 ሶ1205790




119904 120572 + 120573 minus119888 120572 + 120573 minus119888120573 119889

119888 120572 + 120573 119904 120572 + 120573 119904120573 1198890 0 0

ሶ119909119910 ሶሶ120579

=119903 ሶ12059500












119904 120572 + 120573(119905) minus119888 120572 + 120573(119905) minus119888120573(119905)119889

119888 120572 + 120573(119905) 119904 120572 + 120573(119905) 119904120573(119905)1198890 0 0

ሶ119909ሶ119910ሶ120579

=119903 ሶ120595

minus119889119888 ሶ120573(119905)0


maneuverability)




119904 120572 + 120573 + 120574 minus119888 120572 + 120573 + 120574 minus119888 120573 + 120574 119889

119888 120572 + 120573 + 120574 119904 120572 + 120573 + 120574 119904 120573 + 120574 1198890 0 0

ሶ119909ሶ119910ሶ120579

=119903 ሶ120595 cos 120574

119903 ሶ120595 sin 120574 + 119903119901119903 ሶ1205951199011199030















120572119897 = 90 deg

120573119897 = 0 deg

120572119903 = minus90 deg

120573119903 = 180 deg




119904 120572119897 + 120573119897 ሶ119909 minus 119888 120572119897 + 120573119897 ሶ119910 minus 119888120573119897 119889 ሶ120579 = 119903119897 ሶ120595119897

119888 120572119897 + 120573119897 ሶ119909 + 119904 120572119897 + 120573119897 ሶ119910 + 119904120573119897 119889 ሶ120579 = 0



ሶ119909 minus 119889 ሶ120579 = 119903119897 ሶ120595119897

ሶ119910 = 0

ሶ119909 + 119889 ሶ120579 = 119903119903 ሶ120595119903119910 = 0




ሶ119909ሶ119910ሶ120579

=

1

2119903119897 ሶ120595119897 + 119903119903 ሶ120595119903

01

2119889minus119903119897 ሶ120595119897 + 119903119903 ሶ120595119903



rrlabcsuni-klde




rrlabcsuni-klde











1205721 = 0 deg

1205722 = 120 deg

1205723 = minus120 deg







119888 1205721 + 1205731 ሶ119909 + 119904 1205721 + 1205731 ሶ119910 + 119889 sdot 119904 1205731 ሶ120579 = 0

119888 1205722 + 1205732 ሶ119909 + 119904 1205722 + 1205732 ሶ119910 + 119889 sdot 119904 1205732 ሶ120579 = 0

119888 1205723 + 1205733 ሶ119909 + 119904 1205723 + 1205733 ሶ119910 + 119889 sdot 119904 1205733 ሶ120579 = 0







119888 120572119894 sdot ሶ119909 + 119904 120572119894 sdot ሶ119910




ሶ120595119894 =1


rrlabcsuni-klde



rrlabcsuni-klde

Mecanum Vehicle





Mecanum Kinematics



1205721 = 45 deg 1205731 = 45 deg 1205741 = minus45 deg

1205722 = 135 deg 1205732 = minus45 deg 1205742 = 45 deg


1205724 = minus45 deg 1205734 = 135 deg 1205744 = 45 deg


Mecanum Kinematics








Mecanum Kinematics


ሶ119909 =119903

4ሶ1205951 + ሶ1205952 + ሶ1205953 + ሶ1205954

ሶ119910 =119903

4minus ሶ1205951 + ሶ1205952 minus ሶ1205953 + ሶ1205954

ሶ120579 =119903

119889 2ሶ1205951 minus ሶ1205952 minus ሶ1205953 + ሶ1205954


Mecanum Kinematics


ሶ119909ሶ119910ሶ120579

=119903119908ℎ1198901198901198974


sdot

ሶ1205951

ሶ1205952

ሶ1205953

ሶ1205954

with a =2 2

119889


ሶ1205954 = ሶ1205951 + ሶ1205952 minus ሶ1205953

rrlabcsuni-klde





Ability to maneuver

Controllability

Stability



Stability




Typical setups




Maneuverability








Controllability



Examples




Wheel type

Drive geometry


Motortransmission


Degree of Mobility




119891119898 = 3 minus rank 119878


Degree of Mobility



1198891 = 1198892 1205731 = 1205732 = 0 1205721 + 120587 = 1205722






119891119904 = rank 119878prime








119891119872 = 119891119898 + 119891119904

rrlabcsuni-klde

Vehicle Kinematics



Tricycle Drive

Given 120575 119907


119903 = 119897 sdot cot 120575

119907prime =119907

cos 120575

119907119897 = 119907 sdot119903 + 119889

119903

119907119903 = 119907 sdot119903 minus 119889


120575


Ackermann Steering

Given 119907119863 120575 119897 119889

Unknown 119907119903119903 119907119897119903 119907119903119891 119907119897119891

Solution

120602 =119897

tan 120575

119907119897119903 =120602 minus 1198892 sdot 119907119863

120602

119907119903119903 =120602 + 1198892 sdot 119907119863

120602

119907119897119891 =120602 minus 1198892 2 + 1198972 sdot tan 120575

119897sdot 119907119863

119907119903119891 =1206022 + 1198972 sdot tan 120575

119897sdot 119907119863


120575

120602

rrlabcsuni-klde



Given 120602 120593 119907 119889



1205932 = arctan120549119910 + 119889

1205491199091206022 = 120602 sdot cos120593 cos1205932

1199072 = 119907 sdot 1206022120602

120549119910 = 120602 ∙ 119904119894119899 120593

1205931

1205932

1206021

1206022

120593 120549119909 = 120602 ∙ cos120593

120602

119889

rrlabcsuni-klde

Bicycle Model


119907


Bicycle Model


Low speed motion



119865119910 =1198981199072

120602






Bicycle Model

Given 120575119891 120575119903 119907 120579

Unknown ሶ119883 ሶ119884 ሶ120579

Solution

120575119907 = tanminus1119897119891 tan 120575119903+119897119903 tan 120575119891

119897119891+119897119903

ሶ120579 =119907

120602

120602 =119897119891+119897119903

tan 120575119891 minustan 120575119903 cos 120575119907

ሶ119883 = 119907 cos 120579 + 120575119907ሶ119884 = 119907 sin 120579 + 120575119907

ሶ120579 =119907 cos 120575119907

119897119891+119897119903tan 120575119891 minus tan 120575119903

119907

rrlabcsuni-klde



119862 = 119874 120575119903 = 0 120575119891 = 120575

119897119903 = 0 119897119891 = 119897

120575119907 = 0

ሶ120579 =119907

120602

120602 =119897

tan 120575119891

ሶ119883 = 119907 cos 120579ሶ119884 = 119907 sin 120579ሶ120579 =

119907

119897119891tan 120575119891

119907

rrlabcsuni-klde



YW

XW

θ

YR

XR

P









coordinates





0 0 1


ሶԦ120585119877 = 119877 θሶԦ120585119882


YW

XW

θ

YR

XR




119877120587

2=

0 1 0minus1 0 00 0 1


ሶԦ120585119877 = 119877120587

2ሶ

120585119882 =0 1 0minus1 0 00 0 1


=ሶ119910


rrlabcsuni-klde


Analytical Solution






YW

XW

θ

V(t)

120596(t)







ሶԦ120585119882 = ሶ119909 ሶ119910 ሶ120579119879

= 119891 119889 119903 120579 ሶ120595119897 ሶ120595119903

= 119877 120579 minus1 ሶԦ120585119877

passive wheel

active wheel

d






ሶ119909119877119903 =119903 ሶ1205951199032

ሶ119909119877119897 =119903 ሶ120595119897

2

120596119903 =119903 ሶ1205951199032119889

120596119897 =minus119903 ሶ120595119897

2119889

ሶԦ120585119882 = 119877 120579 minus1 ሶԦ120585119903ሶԦ120585119903 =

119903 ሶ1205951199032

+119903 ሶ120595119897

2 0

119903 ሶ1205951199032119889

minus119903 ሶ120595119897

2119889

119879





Given 120579 =120587



119904

ሶԦ120585119882 =ሶ119909ሶ119910ሶ120579

=0 minus1 01 0 00 0 1

sdot301

=031







Consider




rrlabcsuni-klde

Basic Wheel Types


Standardwheel

Steerablestandard

wheel

Castorwheel

Swedish orMecanum

wheel

Ball orspherical

wheel


Wheel Properties




Castor wheel


















ሶ120595 119903



ሶ120595 119903



ሶ120595 119903



ሶ120595 119903



ሶ120595 119903








and 1 Ԧ1199071 = ሶ119909 ሶ119910 0 119879


119894+1120596119894+1 = 119894119894+1119877 sdot 119894120596119894 + ሶ120579119894+1

119894+1 Ԧ119890119911119894+1

119894+1 Ԧ119907119894+1 = 119894119894+1119877 119894 Ԧ119907119894 +

119894120596119894 times119894 Ԧ119901119894+1




calculated as

2 Ԧ1199072 =119888120572 ሶ119909 + 119904120572 ሶ119910minus119904120572 ሶ119909 + 119888120572 ሶ119910

0


3 Ԧ1199073 =

119888120572 ሶ119909 + 119904120572 ሶ119910

minus119904120572 ሶ119909 + 119888120572 ሶ119910 + 119889 ሶ1205790






4 Ԧ1199074 =119904 120572 + 120573 ሶ119909 minus 119888 120572 + 120573 ሶ119910 minus 119888120573 119889 ሶ120579

119888 120572 + 120573 ሶ119909 + 119904 120572 + 120573 ሶ119910 + 119904120573 119889 ሶ1205790




119904 120572 + 120573 minus119888 120572 + 120573 minus119888120573 119889

119888 120572 + 120573 119904 120572 + 120573 119904120573 1198890 0 0

ሶ119909119910 ሶሶ120579

=119903 ሶ12059500












119904 120572 + 120573(119905) minus119888 120572 + 120573(119905) minus119888120573(119905)119889

119888 120572 + 120573(119905) 119904 120572 + 120573(119905) 119904120573(119905)1198890 0 0

ሶ119909ሶ119910ሶ120579

=119903 ሶ120595

minus119889119888 ሶ120573(119905)0


maneuverability)




119904 120572 + 120573 + 120574 minus119888 120572 + 120573 + 120574 minus119888 120573 + 120574 119889

119888 120572 + 120573 + 120574 119904 120572 + 120573 + 120574 119904 120573 + 120574 1198890 0 0

ሶ119909ሶ119910ሶ120579

=119903 ሶ120595 cos 120574

119903 ሶ120595 sin 120574 + 119903119901119903 ሶ1205951199011199030















120572119897 = 90 deg

120573119897 = 0 deg

120572119903 = minus90 deg

120573119903 = 180 deg




119904 120572119897 + 120573119897 ሶ119909 minus 119888 120572119897 + 120573119897 ሶ119910 minus 119888120573119897 119889 ሶ120579 = 119903119897 ሶ120595119897

119888 120572119897 + 120573119897 ሶ119909 + 119904 120572119897 + 120573119897 ሶ119910 + 119904120573119897 119889 ሶ120579 = 0



ሶ119909 minus 119889 ሶ120579 = 119903119897 ሶ120595119897

ሶ119910 = 0

ሶ119909 + 119889 ሶ120579 = 119903119903 ሶ120595119903119910 = 0




ሶ119909ሶ119910ሶ120579

=

1

2119903119897 ሶ120595119897 + 119903119903 ሶ120595119903

01

2119889minus119903119897 ሶ120595119897 + 119903119903 ሶ120595119903



rrlabcsuni-klde




rrlabcsuni-klde











1205721 = 0 deg

1205722 = 120 deg

1205723 = minus120 deg







119888 1205721 + 1205731 ሶ119909 + 119904 1205721 + 1205731 ሶ119910 + 119889 sdot 119904 1205731 ሶ120579 = 0

119888 1205722 + 1205732 ሶ119909 + 119904 1205722 + 1205732 ሶ119910 + 119889 sdot 119904 1205732 ሶ120579 = 0

119888 1205723 + 1205733 ሶ119909 + 119904 1205723 + 1205733 ሶ119910 + 119889 sdot 119904 1205733 ሶ120579 = 0







119888 120572119894 sdot ሶ119909 + 119904 120572119894 sdot ሶ119910




ሶ120595119894 =1


rrlabcsuni-klde



rrlabcsuni-klde

Mecanum Vehicle





Mecanum Kinematics



1205721 = 45 deg 1205731 = 45 deg 1205741 = minus45 deg

1205722 = 135 deg 1205732 = minus45 deg 1205742 = 45 deg


1205724 = minus45 deg 1205734 = 135 deg 1205744 = 45 deg


Mecanum Kinematics








Mecanum Kinematics


ሶ119909 =119903

4ሶ1205951 + ሶ1205952 + ሶ1205953 + ሶ1205954

ሶ119910 =119903

4minus ሶ1205951 + ሶ1205952 minus ሶ1205953 + ሶ1205954

ሶ120579 =119903

119889 2ሶ1205951 minus ሶ1205952 minus ሶ1205953 + ሶ1205954


Mecanum Kinematics


ሶ119909ሶ119910ሶ120579

=119903119908ℎ1198901198901198974


sdot

ሶ1205951

ሶ1205952

ሶ1205953

ሶ1205954

with a =2 2

119889


ሶ1205954 = ሶ1205951 + ሶ1205952 minus ሶ1205953

rrlabcsuni-klde





Ability to maneuver

Controllability

Stability



Stability




Typical setups




Maneuverability








Controllability



Examples




Wheel type

Drive geometry


Motortransmission


Degree of Mobility




119891119898 = 3 minus rank 119878


Degree of Mobility



1198891 = 1198892 1205731 = 1205732 = 0 1205721 + 120587 = 1205722






119891119904 = rank 119878prime








119891119872 = 119891119898 + 119891119904

rrlabcsuni-klde

Vehicle Kinematics



Tricycle Drive

Given 120575 119907


119903 = 119897 sdot cot 120575

119907prime =119907

cos 120575

119907119897 = 119907 sdot119903 + 119889

119903

119907119903 = 119907 sdot119903 minus 119889


120575


Ackermann Steering

Given 119907119863 120575 119897 119889

Unknown 119907119903119903 119907119897119903 119907119903119891 119907119897119891

Solution

120602 =119897

tan 120575

119907119897119903 =120602 minus 1198892 sdot 119907119863

120602

119907119903119903 =120602 + 1198892 sdot 119907119863

120602

119907119897119891 =120602 minus 1198892 2 + 1198972 sdot tan 120575

119897sdot 119907119863

119907119903119891 =1206022 + 1198972 sdot tan 120575

119897sdot 119907119863


120575

120602

rrlabcsuni-klde



Given 120602 120593 119907 119889



1205932 = arctan120549119910 + 119889

1205491199091206022 = 120602 sdot cos120593 cos1205932

1199072 = 119907 sdot 1206022120602

120549119910 = 120602 ∙ 119904119894119899 120593

1205931

1205932

1206021

1206022

120593 120549119909 = 120602 ∙ cos120593

120602

119889

rrlabcsuni-klde

Bicycle Model


119907


Bicycle Model


Low speed motion



119865119910 =1198981199072

120602






Bicycle Model

Given 120575119891 120575119903 119907 120579

Unknown ሶ119883 ሶ119884 ሶ120579

Solution

120575119907 = tanminus1119897119891 tan 120575119903+119897119903 tan 120575119891

119897119891+119897119903

ሶ120579 =119907

120602

120602 =119897119891+119897119903

tan 120575119891 minustan 120575119903 cos 120575119907

ሶ119883 = 119907 cos 120579 + 120575119907ሶ119884 = 119907 sin 120579 + 120575119907

ሶ120579 =119907 cos 120575119907

119897119891+119897119903tan 120575119891 minus tan 120575119903

119907

rrlabcsuni-klde



119862 = 119874 120575119903 = 0 120575119891 = 120575

119897119903 = 0 119897119891 = 119897

120575119907 = 0

ሶ120579 =119907

120602

120602 =119897

tan 120575119891

ሶ119883 = 119907 cos 120579ሶ119884 = 119907 sin 120579ሶ120579 =

119907

119897119891tan 120575119891

119907









coordinates





0 0 1


ሶԦ120585119877 = 119877 θሶԦ120585119882


YW

XW

θ

YR

XR




119877120587

2=

0 1 0minus1 0 00 0 1


ሶԦ120585119877 = 119877120587

2ሶ

120585119882 =0 1 0minus1 0 00 0 1


=ሶ119910


rrlabcsuni-klde


Analytical Solution






YW

XW

θ

V(t)

120596(t)







ሶԦ120585119882 = ሶ119909 ሶ119910 ሶ120579119879

= 119891 119889 119903 120579 ሶ120595119897 ሶ120595119903

= 119877 120579 minus1 ሶԦ120585119877

passive wheel

active wheel

d






ሶ119909119877119903 =119903 ሶ1205951199032

ሶ119909119877119897 =119903 ሶ120595119897

2

120596119903 =119903 ሶ1205951199032119889

120596119897 =minus119903 ሶ120595119897

2119889

ሶԦ120585119882 = 119877 120579 minus1 ሶԦ120585119903ሶԦ120585119903 =

119903 ሶ1205951199032

+119903 ሶ120595119897

2 0

119903 ሶ1205951199032119889

minus119903 ሶ120595119897

2119889

119879





Given 120579 =120587



119904

ሶԦ120585119882 =ሶ119909ሶ119910ሶ120579

=0 minus1 01 0 00 0 1

sdot301

=031







Consider




rrlabcsuni-klde

Basic Wheel Types


Standardwheel

Steerablestandard

wheel

Castorwheel

Swedish orMecanum

wheel

Ball orspherical

wheel


Wheel Properties




Castor wheel


















ሶ120595 119903



ሶ120595 119903



ሶ120595 119903



ሶ120595 119903



ሶ120595 119903








and 1 Ԧ1199071 = ሶ119909 ሶ119910 0 119879


119894+1120596119894+1 = 119894119894+1119877 sdot 119894120596119894 + ሶ120579119894+1

119894+1 Ԧ119890119911119894+1

119894+1 Ԧ119907119894+1 = 119894119894+1119877 119894 Ԧ119907119894 +

119894120596119894 times119894 Ԧ119901119894+1




calculated as

2 Ԧ1199072 =119888120572 ሶ119909 + 119904120572 ሶ119910minus119904120572 ሶ119909 + 119888120572 ሶ119910

0


3 Ԧ1199073 =

119888120572 ሶ119909 + 119904120572 ሶ119910

minus119904120572 ሶ119909 + 119888120572 ሶ119910 + 119889 ሶ1205790






4 Ԧ1199074 =119904 120572 + 120573 ሶ119909 minus 119888 120572 + 120573 ሶ119910 minus 119888120573 119889 ሶ120579

119888 120572 + 120573 ሶ119909 + 119904 120572 + 120573 ሶ119910 + 119904120573 119889 ሶ1205790




119904 120572 + 120573 minus119888 120572 + 120573 minus119888120573 119889

119888 120572 + 120573 119904 120572 + 120573 119904120573 1198890 0 0

ሶ119909119910 ሶሶ120579

=119903 ሶ12059500












119904 120572 + 120573(119905) minus119888 120572 + 120573(119905) minus119888120573(119905)119889

119888 120572 + 120573(119905) 119904 120572 + 120573(119905) 119904120573(119905)1198890 0 0

ሶ119909ሶ119910ሶ120579

=119903 ሶ120595

minus119889119888 ሶ120573(119905)0


maneuverability)




119904 120572 + 120573 + 120574 minus119888 120572 + 120573 + 120574 minus119888 120573 + 120574 119889

119888 120572 + 120573 + 120574 119904 120572 + 120573 + 120574 119904 120573 + 120574 1198890 0 0

ሶ119909ሶ119910ሶ120579

=119903 ሶ120595 cos 120574

119903 ሶ120595 sin 120574 + 119903119901119903 ሶ1205951199011199030















120572119897 = 90 deg

120573119897 = 0 deg

120572119903 = minus90 deg

120573119903 = 180 deg




119904 120572119897 + 120573119897 ሶ119909 minus 119888 120572119897 + 120573119897 ሶ119910 minus 119888120573119897 119889 ሶ120579 = 119903119897 ሶ120595119897

119888 120572119897 + 120573119897 ሶ119909 + 119904 120572119897 + 120573119897 ሶ119910 + 119904120573119897 119889 ሶ120579 = 0



ሶ119909 minus 119889 ሶ120579 = 119903119897 ሶ120595119897

ሶ119910 = 0

ሶ119909 + 119889 ሶ120579 = 119903119903 ሶ120595119903119910 = 0




ሶ119909ሶ119910ሶ120579

=

1

2119903119897 ሶ120595119897 + 119903119903 ሶ120595119903

01

2119889minus119903119897 ሶ120595119897 + 119903119903 ሶ120595119903



rrlabcsuni-klde




rrlabcsuni-klde











1205721 = 0 deg

1205722 = 120 deg

1205723 = minus120 deg







119888 1205721 + 1205731 ሶ119909 + 119904 1205721 + 1205731 ሶ119910 + 119889 sdot 119904 1205731 ሶ120579 = 0

119888 1205722 + 1205732 ሶ119909 + 119904 1205722 + 1205732 ሶ119910 + 119889 sdot 119904 1205732 ሶ120579 = 0

119888 1205723 + 1205733 ሶ119909 + 119904 1205723 + 1205733 ሶ119910 + 119889 sdot 119904 1205733 ሶ120579 = 0







119888 120572119894 sdot ሶ119909 + 119904 120572119894 sdot ሶ119910




ሶ120595119894 =1


rrlabcsuni-klde



rrlabcsuni-klde

Mecanum Vehicle





Mecanum Kinematics



1205721 = 45 deg 1205731 = 45 deg 1205741 = minus45 deg

1205722 = 135 deg 1205732 = minus45 deg 1205742 = 45 deg


1205724 = minus45 deg 1205734 = 135 deg 1205744 = 45 deg


Mecanum Kinematics








Mecanum Kinematics


ሶ119909 =119903

4ሶ1205951 + ሶ1205952 + ሶ1205953 + ሶ1205954

ሶ119910 =119903

4minus ሶ1205951 + ሶ1205952 minus ሶ1205953 + ሶ1205954

ሶ120579 =119903

119889 2ሶ1205951 minus ሶ1205952 minus ሶ1205953 + ሶ1205954


Mecanum Kinematics


ሶ119909ሶ119910ሶ120579

=119903119908ℎ1198901198901198974


sdot

ሶ1205951

ሶ1205952

ሶ1205953

ሶ1205954

with a =2 2

119889


ሶ1205954 = ሶ1205951 + ሶ1205952 minus ሶ1205953

rrlabcsuni-klde





Ability to maneuver

Controllability

Stability



Stability




Typical setups




Maneuverability








Controllability



Examples




Wheel type

Drive geometry


Motortransmission


Degree of Mobility




119891119898 = 3 minus rank 119878


Degree of Mobility



1198891 = 1198892 1205731 = 1205732 = 0 1205721 + 120587 = 1205722






119891119904 = rank 119878prime








119891119872 = 119891119898 + 119891119904

rrlabcsuni-klde

Vehicle Kinematics



Tricycle Drive

Given 120575 119907


119903 = 119897 sdot cot 120575

119907prime =119907

cos 120575

119907119897 = 119907 sdot119903 + 119889

119903

119907119903 = 119907 sdot119903 minus 119889


120575


Ackermann Steering

Given 119907119863 120575 119897 119889

Unknown 119907119903119903 119907119897119903 119907119903119891 119907119897119891

Solution

120602 =119897

tan 120575

119907119897119903 =120602 minus 1198892 sdot 119907119863

120602

119907119903119903 =120602 + 1198892 sdot 119907119863

120602

119907119897119891 =120602 minus 1198892 2 + 1198972 sdot tan 120575

119897sdot 119907119863

119907119903119891 =1206022 + 1198972 sdot tan 120575

119897sdot 119907119863


120575

120602

rrlabcsuni-klde



Given 120602 120593 119907 119889



1205932 = arctan120549119910 + 119889

1205491199091206022 = 120602 sdot cos120593 cos1205932

1199072 = 119907 sdot 1206022120602

120549119910 = 120602 ∙ 119904119894119899 120593

1205931

1205932

1206021

1206022

120593 120549119909 = 120602 ∙ cos120593

120602

119889

rrlabcsuni-klde

Bicycle Model


119907


Bicycle Model


Low speed motion



119865119910 =1198981199072

120602






Bicycle Model

Given 120575119891 120575119903 119907 120579

Unknown ሶ119883 ሶ119884 ሶ120579

Solution

120575119907 = tanminus1119897119891 tan 120575119903+119897119903 tan 120575119891

119897119891+119897119903

ሶ120579 =119907

120602

120602 =119897119891+119897119903

tan 120575119891 minustan 120575119903 cos 120575119907

ሶ119883 = 119907 cos 120579 + 120575119907ሶ119884 = 119907 sin 120579 + 120575119907

ሶ120579 =119907 cos 120575119907

119897119891+119897119903tan 120575119891 minus tan 120575119903

119907

rrlabcsuni-klde



119862 = 119874 120575119903 = 0 120575119891 = 120575

119897119903 = 0 119897119891 = 119897

120575119907 = 0

ሶ120579 =119907

120602

120602 =119897

tan 120575119891

ሶ119883 = 119907 cos 120579ሶ119884 = 119907 sin 120579ሶ120579 =

119907

119897119891tan 120575119891

119907





0 0 1


ሶԦ120585119877 = 119877 θሶԦ120585119882


YW

XW

θ

YR

XR




119877120587

2=

0 1 0minus1 0 00 0 1


ሶԦ120585119877 = 119877120587

2ሶ

120585119882 =0 1 0minus1 0 00 0 1


=ሶ119910


rrlabcsuni-klde


Analytical Solution






YW

XW

θ

V(t)

120596(t)







ሶԦ120585119882 = ሶ119909 ሶ119910 ሶ120579119879

= 119891 119889 119903 120579 ሶ120595119897 ሶ120595119903

= 119877 120579 minus1 ሶԦ120585119877

passive wheel

active wheel

d






ሶ119909119877119903 =119903 ሶ1205951199032

ሶ119909119877119897 =119903 ሶ120595119897

2

120596119903 =119903 ሶ1205951199032119889

120596119897 =minus119903 ሶ120595119897

2119889

ሶԦ120585119882 = 119877 120579 minus1 ሶԦ120585119903ሶԦ120585119903 =

119903 ሶ1205951199032

+119903 ሶ120595119897

2 0

119903 ሶ1205951199032119889

minus119903 ሶ120595119897

2119889

119879





Given 120579 =120587



119904

ሶԦ120585119882 =ሶ119909ሶ119910ሶ120579

=0 minus1 01 0 00 0 1

sdot301

=031







Consider




rrlabcsuni-klde

Basic Wheel Types


Standardwheel

Steerablestandard

wheel

Castorwheel

Swedish orMecanum

wheel

Ball orspherical

wheel


Wheel Properties




Castor wheel


















ሶ120595 119903



ሶ120595 119903



ሶ120595 119903



ሶ120595 119903



ሶ120595 119903








and 1 Ԧ1199071 = ሶ119909 ሶ119910 0 119879


119894+1120596119894+1 = 119894119894+1119877 sdot 119894120596119894 + ሶ120579119894+1

119894+1 Ԧ119890119911119894+1

119894+1 Ԧ119907119894+1 = 119894119894+1119877 119894 Ԧ119907119894 +

119894120596119894 times119894 Ԧ119901119894+1




calculated as

2 Ԧ1199072 =119888120572 ሶ119909 + 119904120572 ሶ119910minus119904120572 ሶ119909 + 119888120572 ሶ119910

0


3 Ԧ1199073 =

119888120572 ሶ119909 + 119904120572 ሶ119910

minus119904120572 ሶ119909 + 119888120572 ሶ119910 + 119889 ሶ1205790






4 Ԧ1199074 =119904 120572 + 120573 ሶ119909 minus 119888 120572 + 120573 ሶ119910 minus 119888120573 119889 ሶ120579

119888 120572 + 120573 ሶ119909 + 119904 120572 + 120573 ሶ119910 + 119904120573 119889 ሶ1205790




119904 120572 + 120573 minus119888 120572 + 120573 minus119888120573 119889

119888 120572 + 120573 119904 120572 + 120573 119904120573 1198890 0 0

ሶ119909119910 ሶሶ120579

=119903 ሶ12059500












119904 120572 + 120573(119905) minus119888 120572 + 120573(119905) minus119888120573(119905)119889

119888 120572 + 120573(119905) 119904 120572 + 120573(119905) 119904120573(119905)1198890 0 0

ሶ119909ሶ119910ሶ120579

=119903 ሶ120595

minus119889119888 ሶ120573(119905)0


maneuverability)




119904 120572 + 120573 + 120574 minus119888 120572 + 120573 + 120574 minus119888 120573 + 120574 119889

119888 120572 + 120573 + 120574 119904 120572 + 120573 + 120574 119904 120573 + 120574 1198890 0 0

ሶ119909ሶ119910ሶ120579

=119903 ሶ120595 cos 120574

119903 ሶ120595 sin 120574 + 119903119901119903 ሶ1205951199011199030















120572119897 = 90 deg

120573119897 = 0 deg

120572119903 = minus90 deg

120573119903 = 180 deg




119904 120572119897 + 120573119897 ሶ119909 minus 119888 120572119897 + 120573119897 ሶ119910 minus 119888120573119897 119889 ሶ120579 = 119903119897 ሶ120595119897

119888 120572119897 + 120573119897 ሶ119909 + 119904 120572119897 + 120573119897 ሶ119910 + 119904120573119897 119889 ሶ120579 = 0



ሶ119909 minus 119889 ሶ120579 = 119903119897 ሶ120595119897

ሶ119910 = 0

ሶ119909 + 119889 ሶ120579 = 119903119903 ሶ120595119903119910 = 0




ሶ119909ሶ119910ሶ120579

=

1

2119903119897 ሶ120595119897 + 119903119903 ሶ120595119903

01

2119889minus119903119897 ሶ120595119897 + 119903119903 ሶ120595119903



rrlabcsuni-klde




rrlabcsuni-klde











1205721 = 0 deg

1205722 = 120 deg

1205723 = minus120 deg







119888 1205721 + 1205731 ሶ119909 + 119904 1205721 + 1205731 ሶ119910 + 119889 sdot 119904 1205731 ሶ120579 = 0

119888 1205722 + 1205732 ሶ119909 + 119904 1205722 + 1205732 ሶ119910 + 119889 sdot 119904 1205732 ሶ120579 = 0

119888 1205723 + 1205733 ሶ119909 + 119904 1205723 + 1205733 ሶ119910 + 119889 sdot 119904 1205733 ሶ120579 = 0







119888 120572119894 sdot ሶ119909 + 119904 120572119894 sdot ሶ119910




ሶ120595119894 =1


rrlabcsuni-klde



rrlabcsuni-klde

Mecanum Vehicle





Mecanum Kinematics



1205721 = 45 deg 1205731 = 45 deg 1205741 = minus45 deg

1205722 = 135 deg 1205732 = minus45 deg 1205742 = 45 deg


1205724 = minus45 deg 1205734 = 135 deg 1205744 = 45 deg


Mecanum Kinematics








Mecanum Kinematics


ሶ119909 =119903

4ሶ1205951 + ሶ1205952 + ሶ1205953 + ሶ1205954

ሶ119910 =119903

4minus ሶ1205951 + ሶ1205952 minus ሶ1205953 + ሶ1205954

ሶ120579 =119903

119889 2ሶ1205951 minus ሶ1205952 minus ሶ1205953 + ሶ1205954


Mecanum Kinematics


ሶ119909ሶ119910ሶ120579

=119903119908ℎ1198901198901198974


sdot

ሶ1205951

ሶ1205952

ሶ1205953

ሶ1205954

with a =2 2

119889


ሶ1205954 = ሶ1205951 + ሶ1205952 minus ሶ1205953

rrlabcsuni-klde





Ability to maneuver

Controllability

Stability



Stability




Typical setups




Maneuverability








Controllability



Examples




Wheel type

Drive geometry


Motortransmission


Degree of Mobility




119891119898 = 3 minus rank 119878


Degree of Mobility



1198891 = 1198892 1205731 = 1205732 = 0 1205721 + 120587 = 1205722






119891119904 = rank 119878prime








119891119872 = 119891119898 + 119891119904

rrlabcsuni-klde

Vehicle Kinematics



Tricycle Drive

Given 120575 119907


119903 = 119897 sdot cot 120575

119907prime =119907

cos 120575

119907119897 = 119907 sdot119903 + 119889

119903

119907119903 = 119907 sdot119903 minus 119889


120575


Ackermann Steering

Given 119907119863 120575 119897 119889

Unknown 119907119903119903 119907119897119903 119907119903119891 119907119897119891

Solution

120602 =119897

tan 120575

119907119897119903 =120602 minus 1198892 sdot 119907119863

120602

119907119903119903 =120602 + 1198892 sdot 119907119863

120602

119907119897119891 =120602 minus 1198892 2 + 1198972 sdot tan 120575

119897sdot 119907119863

119907119903119891 =1206022 + 1198972 sdot tan 120575

119897sdot 119907119863


120575

120602

rrlabcsuni-klde



Given 120602 120593 119907 119889



1205932 = arctan120549119910 + 119889

1205491199091206022 = 120602 sdot cos120593 cos1205932

1199072 = 119907 sdot 1206022120602

120549119910 = 120602 ∙ 119904119894119899 120593

1205931

1205932

1206021

1206022

120593 120549119909 = 120602 ∙ cos120593

120602

119889

rrlabcsuni-klde

Bicycle Model


119907


Bicycle Model


Low speed motion



119865119910 =1198981199072

120602






Bicycle Model

Given 120575119891 120575119903 119907 120579

Unknown ሶ119883 ሶ119884 ሶ120579

Solution

120575119907 = tanminus1119897119891 tan 120575119903+119897119903 tan 120575119891

119897119891+119897119903

ሶ120579 =119907

120602

120602 =119897119891+119897119903

tan 120575119891 minustan 120575119903 cos 120575119907

ሶ119883 = 119907 cos 120579 + 120575119907ሶ119884 = 119907 sin 120579 + 120575119907

ሶ120579 =119907 cos 120575119907

119897119891+119897119903tan 120575119891 minus tan 120575119903

119907

rrlabcsuni-klde



119862 = 119874 120575119903 = 0 120575119891 = 120575

119897119903 = 0 119897119891 = 119897

120575119907 = 0

ሶ120579 =119907

120602

120602 =119897

tan 120575119891

ሶ119883 = 119907 cos 120579ሶ119884 = 119907 sin 120579ሶ120579 =

119907

119897119891tan 120575119891

119907




119877120587

2=

0 1 0minus1 0 00 0 1


ሶԦ120585119877 = 119877120587

2ሶ

120585119882 =0 1 0minus1 0 00 0 1


=ሶ119910


rrlabcsuni-klde


Analytical Solution






YW

XW

θ

V(t)

120596(t)







ሶԦ120585119882 = ሶ119909 ሶ119910 ሶ120579119879

= 119891 119889 119903 120579 ሶ120595119897 ሶ120595119903

= 119877 120579 minus1 ሶԦ120585119877

passive wheel

active wheel

d






ሶ119909119877119903 =119903 ሶ1205951199032

ሶ119909119877119897 =119903 ሶ120595119897

2

120596119903 =119903 ሶ1205951199032119889

120596119897 =minus119903 ሶ120595119897

2119889

ሶԦ120585119882 = 119877 120579 minus1 ሶԦ120585119903ሶԦ120585119903 =

119903 ሶ1205951199032

+119903 ሶ120595119897

2 0

119903 ሶ1205951199032119889

minus119903 ሶ120595119897

2119889

119879





Given 120579 =120587



119904

ሶԦ120585119882 =ሶ119909ሶ119910ሶ120579

=0 minus1 01 0 00 0 1

sdot301

=031







Consider




rrlabcsuni-klde

Basic Wheel Types


Standardwheel

Steerablestandard

wheel

Castorwheel

Swedish orMecanum

wheel

Ball orspherical

wheel


Wheel Properties




Castor wheel


















ሶ120595 119903



ሶ120595 119903



ሶ120595 119903



ሶ120595 119903



ሶ120595 119903








and 1 Ԧ1199071 = ሶ119909 ሶ119910 0 119879


119894+1120596119894+1 = 119894119894+1119877 sdot 119894120596119894 + ሶ120579119894+1

119894+1 Ԧ119890119911119894+1

119894+1 Ԧ119907119894+1 = 119894119894+1119877 119894 Ԧ119907119894 +

119894120596119894 times119894 Ԧ119901119894+1




calculated as

2 Ԧ1199072 =119888120572 ሶ119909 + 119904120572 ሶ119910minus119904120572 ሶ119909 + 119888120572 ሶ119910

0


3 Ԧ1199073 =

119888120572 ሶ119909 + 119904120572 ሶ119910

minus119904120572 ሶ119909 + 119888120572 ሶ119910 + 119889 ሶ1205790






4 Ԧ1199074 =119904 120572 + 120573 ሶ119909 minus 119888 120572 + 120573 ሶ119910 minus 119888120573 119889 ሶ120579

119888 120572 + 120573 ሶ119909 + 119904 120572 + 120573 ሶ119910 + 119904120573 119889 ሶ1205790




119904 120572 + 120573 minus119888 120572 + 120573 minus119888120573 119889

119888 120572 + 120573 119904 120572 + 120573 119904120573 1198890 0 0

ሶ119909119910 ሶሶ120579

=119903 ሶ12059500












119904 120572 + 120573(119905) minus119888 120572 + 120573(119905) minus119888120573(119905)119889

119888 120572 + 120573(119905) 119904 120572 + 120573(119905) 119904120573(119905)1198890 0 0

ሶ119909ሶ119910ሶ120579

=119903 ሶ120595

minus119889119888 ሶ120573(119905)0


maneuverability)




119904 120572 + 120573 + 120574 minus119888 120572 + 120573 + 120574 minus119888 120573 + 120574 119889

119888 120572 + 120573 + 120574 119904 120572 + 120573 + 120574 119904 120573 + 120574 1198890 0 0

ሶ119909ሶ119910ሶ120579

=119903 ሶ120595 cos 120574

119903 ሶ120595 sin 120574 + 119903119901119903 ሶ1205951199011199030















120572119897 = 90 deg

120573119897 = 0 deg

120572119903 = minus90 deg

120573119903 = 180 deg




119904 120572119897 + 120573119897 ሶ119909 minus 119888 120572119897 + 120573119897 ሶ119910 minus 119888120573119897 119889 ሶ120579 = 119903119897 ሶ120595119897

119888 120572119897 + 120573119897 ሶ119909 + 119904 120572119897 + 120573119897 ሶ119910 + 119904120573119897 119889 ሶ120579 = 0



ሶ119909 minus 119889 ሶ120579 = 119903119897 ሶ120595119897

ሶ119910 = 0

ሶ119909 + 119889 ሶ120579 = 119903119903 ሶ120595119903119910 = 0




ሶ119909ሶ119910ሶ120579

=

1

2119903119897 ሶ120595119897 + 119903119903 ሶ120595119903

01

2119889minus119903119897 ሶ120595119897 + 119903119903 ሶ120595119903



rrlabcsuni-klde




rrlabcsuni-klde











1205721 = 0 deg

1205722 = 120 deg

1205723 = minus120 deg







119888 1205721 + 1205731 ሶ119909 + 119904 1205721 + 1205731 ሶ119910 + 119889 sdot 119904 1205731 ሶ120579 = 0

119888 1205722 + 1205732 ሶ119909 + 119904 1205722 + 1205732 ሶ119910 + 119889 sdot 119904 1205732 ሶ120579 = 0

119888 1205723 + 1205733 ሶ119909 + 119904 1205723 + 1205733 ሶ119910 + 119889 sdot 119904 1205733 ሶ120579 = 0







119888 120572119894 sdot ሶ119909 + 119904 120572119894 sdot ሶ119910




ሶ120595119894 =1


rrlabcsuni-klde



rrlabcsuni-klde

Mecanum Vehicle





Mecanum Kinematics



1205721 = 45 deg 1205731 = 45 deg 1205741 = minus45 deg

1205722 = 135 deg 1205732 = minus45 deg 1205742 = 45 deg


1205724 = minus45 deg 1205734 = 135 deg 1205744 = 45 deg


Mecanum Kinematics








Mecanum Kinematics


ሶ119909 =119903

4ሶ1205951 + ሶ1205952 + ሶ1205953 + ሶ1205954

ሶ119910 =119903

4minus ሶ1205951 + ሶ1205952 minus ሶ1205953 + ሶ1205954

ሶ120579 =119903

119889 2ሶ1205951 minus ሶ1205952 minus ሶ1205953 + ሶ1205954


Mecanum Kinematics


ሶ119909ሶ119910ሶ120579

=119903119908ℎ1198901198901198974


sdot

ሶ1205951

ሶ1205952

ሶ1205953

ሶ1205954

with a =2 2

119889


ሶ1205954 = ሶ1205951 + ሶ1205952 minus ሶ1205953

rrlabcsuni-klde





Ability to maneuver

Controllability

Stability



Stability




Typical setups




Maneuverability








Controllability



Examples




Wheel type

Drive geometry


Motortransmission


Degree of Mobility




119891119898 = 3 minus rank 119878


Degree of Mobility



1198891 = 1198892 1205731 = 1205732 = 0 1205721 + 120587 = 1205722






119891119904 = rank 119878prime








119891119872 = 119891119898 + 119891119904

rrlabcsuni-klde

Vehicle Kinematics



Tricycle Drive

Given 120575 119907


119903 = 119897 sdot cot 120575

119907prime =119907

cos 120575

119907119897 = 119907 sdot119903 + 119889

119903

119907119903 = 119907 sdot119903 minus 119889


120575


Ackermann Steering

Given 119907119863 120575 119897 119889

Unknown 119907119903119903 119907119897119903 119907119903119891 119907119897119891

Solution

120602 =119897

tan 120575

119907119897119903 =120602 minus 1198892 sdot 119907119863

120602

119907119903119903 =120602 + 1198892 sdot 119907119863

120602

119907119897119891 =120602 minus 1198892 2 + 1198972 sdot tan 120575

119897sdot 119907119863

119907119903119891 =1206022 + 1198972 sdot tan 120575

119897sdot 119907119863


120575

120602

rrlabcsuni-klde



Given 120602 120593 119907 119889



1205932 = arctan120549119910 + 119889

1205491199091206022 = 120602 sdot cos120593 cos1205932

1199072 = 119907 sdot 1206022120602

120549119910 = 120602 ∙ 119904119894119899 120593

1205931

1205932

1206021

1206022

120593 120549119909 = 120602 ∙ cos120593

120602

119889

rrlabcsuni-klde

Bicycle Model


119907


Bicycle Model


Low speed motion



119865119910 =1198981199072

120602






Bicycle Model

Given 120575119891 120575119903 119907 120579

Unknown ሶ119883 ሶ119884 ሶ120579

Solution

120575119907 = tanminus1119897119891 tan 120575119903+119897119903 tan 120575119891

119897119891+119897119903

ሶ120579 =119907

120602

120602 =119897119891+119897119903

tan 120575119891 minustan 120575119903 cos 120575119907

ሶ119883 = 119907 cos 120579 + 120575119907ሶ119884 = 119907 sin 120579 + 120575119907

ሶ120579 =119907 cos 120575119907

119897119891+119897119903tan 120575119891 minus tan 120575119903

119907

rrlabcsuni-klde



119862 = 119874 120575119903 = 0 120575119891 = 120575

119897119903 = 0 119897119891 = 119897

120575119907 = 0

ሶ120579 =119907

120602

120602 =119897

tan 120575119891

ሶ119883 = 119907 cos 120579ሶ119884 = 119907 sin 120579ሶ120579 =

119907

119897119891tan 120575119891

119907

rrlabcsuni-klde


Analytical Solution






YW

XW

θ

V(t)

120596(t)







ሶԦ120585119882 = ሶ119909 ሶ119910 ሶ120579119879

= 119891 119889 119903 120579 ሶ120595119897 ሶ120595119903

= 119877 120579 minus1 ሶԦ120585119877

passive wheel

active wheel

d






ሶ119909119877119903 =119903 ሶ1205951199032

ሶ119909119877119897 =119903 ሶ120595119897

2

120596119903 =119903 ሶ1205951199032119889

120596119897 =minus119903 ሶ120595119897

2119889

ሶԦ120585119882 = 119877 120579 minus1 ሶԦ120585119903ሶԦ120585119903 =

119903 ሶ1205951199032

+119903 ሶ120595119897

2 0

119903 ሶ1205951199032119889

minus119903 ሶ120595119897

2119889

119879





Given 120579 =120587



119904

ሶԦ120585119882 =ሶ119909ሶ119910ሶ120579

=0 minus1 01 0 00 0 1

sdot301

=031







Consider




rrlabcsuni-klde

Basic Wheel Types


Standardwheel

Steerablestandard

wheel

Castorwheel

Swedish orMecanum

wheel

Ball orspherical

wheel


Wheel Properties




Castor wheel


















ሶ120595 119903



ሶ120595 119903



ሶ120595 119903



ሶ120595 119903



ሶ120595 119903








and 1 Ԧ1199071 = ሶ119909 ሶ119910 0 119879


119894+1120596119894+1 = 119894119894+1119877 sdot 119894120596119894 + ሶ120579119894+1

119894+1 Ԧ119890119911119894+1

119894+1 Ԧ119907119894+1 = 119894119894+1119877 119894 Ԧ119907119894 +

119894120596119894 times119894 Ԧ119901119894+1




calculated as

2 Ԧ1199072 =119888120572 ሶ119909 + 119904120572 ሶ119910minus119904120572 ሶ119909 + 119888120572 ሶ119910

0


3 Ԧ1199073 =

119888120572 ሶ119909 + 119904120572 ሶ119910

minus119904120572 ሶ119909 + 119888120572 ሶ119910 + 119889 ሶ1205790






4 Ԧ1199074 =119904 120572 + 120573 ሶ119909 minus 119888 120572 + 120573 ሶ119910 minus 119888120573 119889 ሶ120579

119888 120572 + 120573 ሶ119909 + 119904 120572 + 120573 ሶ119910 + 119904120573 119889 ሶ1205790




119904 120572 + 120573 minus119888 120572 + 120573 minus119888120573 119889

119888 120572 + 120573 119904 120572 + 120573 119904120573 1198890 0 0

ሶ119909119910 ሶሶ120579

=119903 ሶ12059500












119904 120572 + 120573(119905) minus119888 120572 + 120573(119905) minus119888120573(119905)119889

119888 120572 + 120573(119905) 119904 120572 + 120573(119905) 119904120573(119905)1198890 0 0

ሶ119909ሶ119910ሶ120579

=119903 ሶ120595

minus119889119888 ሶ120573(119905)0


maneuverability)




119904 120572 + 120573 + 120574 minus119888 120572 + 120573 + 120574 minus119888 120573 + 120574 119889

119888 120572 + 120573 + 120574 119904 120572 + 120573 + 120574 119904 120573 + 120574 1198890 0 0

ሶ119909ሶ119910ሶ120579

=119903 ሶ120595 cos 120574

119903 ሶ120595 sin 120574 + 119903119901119903 ሶ1205951199011199030















120572119897 = 90 deg

120573119897 = 0 deg

120572119903 = minus90 deg

120573119903 = 180 deg




119904 120572119897 + 120573119897 ሶ119909 minus 119888 120572119897 + 120573119897 ሶ119910 minus 119888120573119897 119889 ሶ120579 = 119903119897 ሶ120595119897

119888 120572119897 + 120573119897 ሶ119909 + 119904 120572119897 + 120573119897 ሶ119910 + 119904120573119897 119889 ሶ120579 = 0



ሶ119909 minus 119889 ሶ120579 = 119903119897 ሶ120595119897

ሶ119910 = 0

ሶ119909 + 119889 ሶ120579 = 119903119903 ሶ120595119903119910 = 0




ሶ119909ሶ119910ሶ120579

=

1

2119903119897 ሶ120595119897 + 119903119903 ሶ120595119903

01

2119889minus119903119897 ሶ120595119897 + 119903119903 ሶ120595119903



rrlabcsuni-klde




rrlabcsuni-klde











1205721 = 0 deg

1205722 = 120 deg

1205723 = minus120 deg







119888 1205721 + 1205731 ሶ119909 + 119904 1205721 + 1205731 ሶ119910 + 119889 sdot 119904 1205731 ሶ120579 = 0

119888 1205722 + 1205732 ሶ119909 + 119904 1205722 + 1205732 ሶ119910 + 119889 sdot 119904 1205732 ሶ120579 = 0

119888 1205723 + 1205733 ሶ119909 + 119904 1205723 + 1205733 ሶ119910 + 119889 sdot 119904 1205733 ሶ120579 = 0







119888 120572119894 sdot ሶ119909 + 119904 120572119894 sdot ሶ119910




ሶ120595119894 =1


rrlabcsuni-klde



rrlabcsuni-klde

Mecanum Vehicle





Mecanum Kinematics



1205721 = 45 deg 1205731 = 45 deg 1205741 = minus45 deg

1205722 = 135 deg 1205732 = minus45 deg 1205742 = 45 deg


1205724 = minus45 deg 1205734 = 135 deg 1205744 = 45 deg


Mecanum Kinematics








Mecanum Kinematics


ሶ119909 =119903

4ሶ1205951 + ሶ1205952 + ሶ1205953 + ሶ1205954

ሶ119910 =119903

4minus ሶ1205951 + ሶ1205952 minus ሶ1205953 + ሶ1205954

ሶ120579 =119903

119889 2ሶ1205951 minus ሶ1205952 minus ሶ1205953 + ሶ1205954


Mecanum Kinematics


ሶ119909ሶ119910ሶ120579

=119903119908ℎ1198901198901198974


sdot

ሶ1205951

ሶ1205952

ሶ1205953

ሶ1205954

with a =2 2

119889


ሶ1205954 = ሶ1205951 + ሶ1205952 minus ሶ1205953

rrlabcsuni-klde





Ability to maneuver

Controllability

Stability



Stability




Typical setups




Maneuverability








Controllability



Examples




Wheel type

Drive geometry


Motortransmission


Degree of Mobility




119891119898 = 3 minus rank 119878


Degree of Mobility



1198891 = 1198892 1205731 = 1205732 = 0 1205721 + 120587 = 1205722






119891119904 = rank 119878prime








119891119872 = 119891119898 + 119891119904

rrlabcsuni-klde

Vehicle Kinematics



Tricycle Drive

Given 120575 119907


119903 = 119897 sdot cot 120575

119907prime =119907

cos 120575

119907119897 = 119907 sdot119903 + 119889

119903

119907119903 = 119907 sdot119903 minus 119889


120575


Ackermann Steering

Given 119907119863 120575 119897 119889

Unknown 119907119903119903 119907119897119903 119907119903119891 119907119897119891

Solution

120602 =119897

tan 120575

119907119897119903 =120602 minus 1198892 sdot 119907119863

120602

119907119903119903 =120602 + 1198892 sdot 119907119863

120602

119907119897119891 =120602 minus 1198892 2 + 1198972 sdot tan 120575

119897sdot 119907119863

119907119903119891 =1206022 + 1198972 sdot tan 120575

119897sdot 119907119863


120575

120602

rrlabcsuni-klde



Given 120602 120593 119907 119889



1205932 = arctan120549119910 + 119889

1205491199091206022 = 120602 sdot cos120593 cos1205932

1199072 = 119907 sdot 1206022120602

120549119910 = 120602 ∙ 119904119894119899 120593

1205931

1205932

1206021

1206022

120593 120549119909 = 120602 ∙ cos120593

120602

119889

rrlabcsuni-klde

Bicycle Model


119907


Bicycle Model


Low speed motion



119865119910 =1198981199072

120602






Bicycle Model

Given 120575119891 120575119903 119907 120579

Unknown ሶ119883 ሶ119884 ሶ120579

Solution

120575119907 = tanminus1119897119891 tan 120575119903+119897119903 tan 120575119891

119897119891+119897119903

ሶ120579 =119907

120602

120602 =119897119891+119897119903

tan 120575119891 minustan 120575119903 cos 120575119907

ሶ119883 = 119907 cos 120579 + 120575119907ሶ119884 = 119907 sin 120579 + 120575119907

ሶ120579 =119907 cos 120575119907

119897119891+119897119903tan 120575119891 minus tan 120575119903

119907

rrlabcsuni-klde



119862 = 119874 120575119903 = 0 120575119891 = 120575

119897119903 = 0 119897119891 = 119897

120575119907 = 0

ሶ120579 =119907

120602

120602 =119897

tan 120575119891

ሶ119883 = 119907 cos 120579ሶ119884 = 119907 sin 120579ሶ120579 =

119907

119897119891tan 120575119891

119907






YW

XW

θ

V(t)

120596(t)







ሶԦ120585119882 = ሶ119909 ሶ119910 ሶ120579119879

= 119891 119889 119903 120579 ሶ120595119897 ሶ120595119903

= 119877 120579 minus1 ሶԦ120585119877

passive wheel

active wheel

d






ሶ119909119877119903 =119903 ሶ1205951199032

ሶ119909119877119897 =119903 ሶ120595119897

2

120596119903 =119903 ሶ1205951199032119889

120596119897 =minus119903 ሶ120595119897

2119889

ሶԦ120585119882 = 119877 120579 minus1 ሶԦ120585119903ሶԦ120585119903 =

119903 ሶ1205951199032

+119903 ሶ120595119897

2 0

119903 ሶ1205951199032119889

minus119903 ሶ120595119897

2119889

119879





Given 120579 =120587



119904

ሶԦ120585119882 =ሶ119909ሶ119910ሶ120579

=0 minus1 01 0 00 0 1

sdot301

=031







Consider




rrlabcsuni-klde

Basic Wheel Types


Standardwheel

Steerablestandard

wheel

Castorwheel

Swedish orMecanum

wheel

Ball orspherical

wheel


Wheel Properties




Castor wheel


















ሶ120595 119903



ሶ120595 119903



ሶ120595 119903



ሶ120595 119903



ሶ120595 119903








and 1 Ԧ1199071 = ሶ119909 ሶ119910 0 119879


119894+1120596119894+1 = 119894119894+1119877 sdot 119894120596119894 + ሶ120579119894+1

119894+1 Ԧ119890119911119894+1

119894+1 Ԧ119907119894+1 = 119894119894+1119877 119894 Ԧ119907119894 +

119894120596119894 times119894 Ԧ119901119894+1




calculated as

2 Ԧ1199072 =119888120572 ሶ119909 + 119904120572 ሶ119910minus119904120572 ሶ119909 + 119888120572 ሶ119910

0


3 Ԧ1199073 =

119888120572 ሶ119909 + 119904120572 ሶ119910

minus119904120572 ሶ119909 + 119888120572 ሶ119910 + 119889 ሶ1205790






4 Ԧ1199074 =119904 120572 + 120573 ሶ119909 minus 119888 120572 + 120573 ሶ119910 minus 119888120573 119889 ሶ120579

119888 120572 + 120573 ሶ119909 + 119904 120572 + 120573 ሶ119910 + 119904120573 119889 ሶ1205790




119904 120572 + 120573 minus119888 120572 + 120573 minus119888120573 119889

119888 120572 + 120573 119904 120572 + 120573 119904120573 1198890 0 0

ሶ119909119910 ሶሶ120579

=119903 ሶ12059500












119904 120572 + 120573(119905) minus119888 120572 + 120573(119905) minus119888120573(119905)119889

119888 120572 + 120573(119905) 119904 120572 + 120573(119905) 119904120573(119905)1198890 0 0

ሶ119909ሶ119910ሶ120579

=119903 ሶ120595

minus119889119888 ሶ120573(119905)0


maneuverability)




119904 120572 + 120573 + 120574 minus119888 120572 + 120573 + 120574 minus119888 120573 + 120574 119889

119888 120572 + 120573 + 120574 119904 120572 + 120573 + 120574 119904 120573 + 120574 1198890 0 0

ሶ119909ሶ119910ሶ120579

=119903 ሶ120595 cos 120574

119903 ሶ120595 sin 120574 + 119903119901119903 ሶ1205951199011199030















120572119897 = 90 deg

120573119897 = 0 deg

120572119903 = minus90 deg

120573119903 = 180 deg




119904 120572119897 + 120573119897 ሶ119909 minus 119888 120572119897 + 120573119897 ሶ119910 minus 119888120573119897 119889 ሶ120579 = 119903119897 ሶ120595119897

119888 120572119897 + 120573119897 ሶ119909 + 119904 120572119897 + 120573119897 ሶ119910 + 119904120573119897 119889 ሶ120579 = 0



ሶ119909 minus 119889 ሶ120579 = 119903119897 ሶ120595119897

ሶ119910 = 0

ሶ119909 + 119889 ሶ120579 = 119903119903 ሶ120595119903119910 = 0




ሶ119909ሶ119910ሶ120579

=

1

2119903119897 ሶ120595119897 + 119903119903 ሶ120595119903

01

2119889minus119903119897 ሶ120595119897 + 119903119903 ሶ120595119903



rrlabcsuni-klde




rrlabcsuni-klde











1205721 = 0 deg

1205722 = 120 deg

1205723 = minus120 deg







119888 1205721 + 1205731 ሶ119909 + 119904 1205721 + 1205731 ሶ119910 + 119889 sdot 119904 1205731 ሶ120579 = 0

119888 1205722 + 1205732 ሶ119909 + 119904 1205722 + 1205732 ሶ119910 + 119889 sdot 119904 1205732 ሶ120579 = 0

119888 1205723 + 1205733 ሶ119909 + 119904 1205723 + 1205733 ሶ119910 + 119889 sdot 119904 1205733 ሶ120579 = 0







119888 120572119894 sdot ሶ119909 + 119904 120572119894 sdot ሶ119910




ሶ120595119894 =1


rrlabcsuni-klde



rrlabcsuni-klde

Mecanum Vehicle





Mecanum Kinematics



1205721 = 45 deg 1205731 = 45 deg 1205741 = minus45 deg

1205722 = 135 deg 1205732 = minus45 deg 1205742 = 45 deg


1205724 = minus45 deg 1205734 = 135 deg 1205744 = 45 deg


Mecanum Kinematics








Mecanum Kinematics


ሶ119909 =119903

4ሶ1205951 + ሶ1205952 + ሶ1205953 + ሶ1205954

ሶ119910 =119903

4minus ሶ1205951 + ሶ1205952 minus ሶ1205953 + ሶ1205954

ሶ120579 =119903

119889 2ሶ1205951 minus ሶ1205952 minus ሶ1205953 + ሶ1205954


Mecanum Kinematics


ሶ119909ሶ119910ሶ120579

=119903119908ℎ1198901198901198974


sdot

ሶ1205951

ሶ1205952

ሶ1205953

ሶ1205954

with a =2 2

119889


ሶ1205954 = ሶ1205951 + ሶ1205952 minus ሶ1205953

rrlabcsuni-klde





Ability to maneuver

Controllability

Stability



Stability




Typical setups




Maneuverability








Controllability



Examples




Wheel type

Drive geometry


Motortransmission


Degree of Mobility




119891119898 = 3 minus rank 119878


Degree of Mobility



1198891 = 1198892 1205731 = 1205732 = 0 1205721 + 120587 = 1205722






119891119904 = rank 119878prime








119891119872 = 119891119898 + 119891119904

rrlabcsuni-klde

Vehicle Kinematics



Tricycle Drive

Given 120575 119907


119903 = 119897 sdot cot 120575

119907prime =119907

cos 120575

119907119897 = 119907 sdot119903 + 119889

119903

119907119903 = 119907 sdot119903 minus 119889


120575


Ackermann Steering

Given 119907119863 120575 119897 119889

Unknown 119907119903119903 119907119897119903 119907119903119891 119907119897119891

Solution

120602 =119897

tan 120575

119907119897119903 =120602 minus 1198892 sdot 119907119863

120602

119907119903119903 =120602 + 1198892 sdot 119907119863

120602

119907119897119891 =120602 minus 1198892 2 + 1198972 sdot tan 120575

119897sdot 119907119863

119907119903119891 =1206022 + 1198972 sdot tan 120575

119897sdot 119907119863


120575

120602

rrlabcsuni-klde



Given 120602 120593 119907 119889



1205932 = arctan120549119910 + 119889

1205491199091206022 = 120602 sdot cos120593 cos1205932

1199072 = 119907 sdot 1206022120602

120549119910 = 120602 ∙ 119904119894119899 120593

1205931

1205932

1206021

1206022

120593 120549119909 = 120602 ∙ cos120593

120602

119889

rrlabcsuni-klde

Bicycle Model


119907


Bicycle Model


Low speed motion



119865119910 =1198981199072

120602






Bicycle Model

Given 120575119891 120575119903 119907 120579

Unknown ሶ119883 ሶ119884 ሶ120579

Solution

120575119907 = tanminus1119897119891 tan 120575119903+119897119903 tan 120575119891

119897119891+119897119903

ሶ120579 =119907

120602

120602 =119897119891+119897119903

tan 120575119891 minustan 120575119903 cos 120575119907

ሶ119883 = 119907 cos 120579 + 120575119907ሶ119884 = 119907 sin 120579 + 120575119907

ሶ120579 =119907 cos 120575119907

119897119891+119897119903tan 120575119891 minus tan 120575119903

119907

rrlabcsuni-klde



119862 = 119874 120575119903 = 0 120575119891 = 120575

119897119903 = 0 119897119891 = 119897

120575119907 = 0

ሶ120579 =119907

120602

120602 =119897

tan 120575119891

ሶ119883 = 119907 cos 120579ሶ119884 = 119907 sin 120579ሶ120579 =

119907

119897119891tan 120575119891

119907




YW

XW

θ

V(t)

120596(t)







ሶԦ120585119882 = ሶ119909 ሶ119910 ሶ120579119879

= 119891 119889 119903 120579 ሶ120595119897 ሶ120595119903

= 119877 120579 minus1 ሶԦ120585119877

passive wheel

active wheel

d






ሶ119909119877119903 =119903 ሶ1205951199032

ሶ119909119877119897 =119903 ሶ120595119897

2

120596119903 =119903 ሶ1205951199032119889

120596119897 =minus119903 ሶ120595119897

2119889

ሶԦ120585119882 = 119877 120579 minus1 ሶԦ120585119903ሶԦ120585119903 =

119903 ሶ1205951199032

+119903 ሶ120595119897

2 0

119903 ሶ1205951199032119889

minus119903 ሶ120595119897

2119889

119879





Given 120579 =120587



119904

ሶԦ120585119882 =ሶ119909ሶ119910ሶ120579

=0 minus1 01 0 00 0 1

sdot301

=031







Consider




rrlabcsuni-klde

Basic Wheel Types


Standardwheel

Steerablestandard

wheel

Castorwheel

Swedish orMecanum

wheel

Ball orspherical

wheel


Wheel Properties




Castor wheel


















ሶ120595 119903



ሶ120595 119903



ሶ120595 119903



ሶ120595 119903



ሶ120595 119903








and 1 Ԧ1199071 = ሶ119909 ሶ119910 0 119879


119894+1120596119894+1 = 119894119894+1119877 sdot 119894120596119894 + ሶ120579119894+1

119894+1 Ԧ119890119911119894+1

119894+1 Ԧ119907119894+1 = 119894119894+1119877 119894 Ԧ119907119894 +

119894120596119894 times119894 Ԧ119901119894+1




calculated as

2 Ԧ1199072 =119888120572 ሶ119909 + 119904120572 ሶ119910minus119904120572 ሶ119909 + 119888120572 ሶ119910

0


3 Ԧ1199073 =

119888120572 ሶ119909 + 119904120572 ሶ119910

minus119904120572 ሶ119909 + 119888120572 ሶ119910 + 119889 ሶ1205790






4 Ԧ1199074 =119904 120572 + 120573 ሶ119909 minus 119888 120572 + 120573 ሶ119910 minus 119888120573 119889 ሶ120579

119888 120572 + 120573 ሶ119909 + 119904 120572 + 120573 ሶ119910 + 119904120573 119889 ሶ1205790




119904 120572 + 120573 minus119888 120572 + 120573 minus119888120573 119889

119888 120572 + 120573 119904 120572 + 120573 119904120573 1198890 0 0

ሶ119909119910 ሶሶ120579

=119903 ሶ12059500












119904 120572 + 120573(119905) minus119888 120572 + 120573(119905) minus119888120573(119905)119889

119888 120572 + 120573(119905) 119904 120572 + 120573(119905) 119904120573(119905)1198890 0 0

ሶ119909ሶ119910ሶ120579

=119903 ሶ120595

minus119889119888 ሶ120573(119905)0


maneuverability)




119904 120572 + 120573 + 120574 minus119888 120572 + 120573 + 120574 minus119888 120573 + 120574 119889

119888 120572 + 120573 + 120574 119904 120572 + 120573 + 120574 119904 120573 + 120574 1198890 0 0

ሶ119909ሶ119910ሶ120579

=119903 ሶ120595 cos 120574

119903 ሶ120595 sin 120574 + 119903119901119903 ሶ1205951199011199030















120572119897 = 90 deg

120573119897 = 0 deg

120572119903 = minus90 deg

120573119903 = 180 deg




119904 120572119897 + 120573119897 ሶ119909 minus 119888 120572119897 + 120573119897 ሶ119910 minus 119888120573119897 119889 ሶ120579 = 119903119897 ሶ120595119897

119888 120572119897 + 120573119897 ሶ119909 + 119904 120572119897 + 120573119897 ሶ119910 + 119904120573119897 119889 ሶ120579 = 0



ሶ119909 minus 119889 ሶ120579 = 119903119897 ሶ120595119897

ሶ119910 = 0

ሶ119909 + 119889 ሶ120579 = 119903119903 ሶ120595119903119910 = 0




ሶ119909ሶ119910ሶ120579

=

1

2119903119897 ሶ120595119897 + 119903119903 ሶ120595119903

01

2119889minus119903119897 ሶ120595119897 + 119903119903 ሶ120595119903



rrlabcsuni-klde




rrlabcsuni-klde











1205721 = 0 deg

1205722 = 120 deg

1205723 = minus120 deg







119888 1205721 + 1205731 ሶ119909 + 119904 1205721 + 1205731 ሶ119910 + 119889 sdot 119904 1205731 ሶ120579 = 0

119888 1205722 + 1205732 ሶ119909 + 119904 1205722 + 1205732 ሶ119910 + 119889 sdot 119904 1205732 ሶ120579 = 0

119888 1205723 + 1205733 ሶ119909 + 119904 1205723 + 1205733 ሶ119910 + 119889 sdot 119904 1205733 ሶ120579 = 0







119888 120572119894 sdot ሶ119909 + 119904 120572119894 sdot ሶ119910




ሶ120595119894 =1


rrlabcsuni-klde



rrlabcsuni-klde

Mecanum Vehicle





Mecanum Kinematics



1205721 = 45 deg 1205731 = 45 deg 1205741 = minus45 deg

1205722 = 135 deg 1205732 = minus45 deg 1205742 = 45 deg


1205724 = minus45 deg 1205734 = 135 deg 1205744 = 45 deg


Mecanum Kinematics








Mecanum Kinematics


ሶ119909 =119903

4ሶ1205951 + ሶ1205952 + ሶ1205953 + ሶ1205954

ሶ119910 =119903

4minus ሶ1205951 + ሶ1205952 minus ሶ1205953 + ሶ1205954

ሶ120579 =119903

119889 2ሶ1205951 minus ሶ1205952 minus ሶ1205953 + ሶ1205954


Mecanum Kinematics


ሶ119909ሶ119910ሶ120579

=119903119908ℎ1198901198901198974


sdot

ሶ1205951

ሶ1205952

ሶ1205953

ሶ1205954

with a =2 2

119889


ሶ1205954 = ሶ1205951 + ሶ1205952 minus ሶ1205953

rrlabcsuni-klde





Ability to maneuver

Controllability

Stability



Stability




Typical setups




Maneuverability








Controllability



Examples




Wheel type

Drive geometry


Motortransmission


Degree of Mobility




119891119898 = 3 minus rank 119878


Degree of Mobility



1198891 = 1198892 1205731 = 1205732 = 0 1205721 + 120587 = 1205722






119891119904 = rank 119878prime








119891119872 = 119891119898 + 119891119904

rrlabcsuni-klde

Vehicle Kinematics



Tricycle Drive

Given 120575 119907


119903 = 119897 sdot cot 120575

119907prime =119907

cos 120575

119907119897 = 119907 sdot119903 + 119889

119903

119907119903 = 119907 sdot119903 minus 119889


120575


Ackermann Steering

Given 119907119863 120575 119897 119889

Unknown 119907119903119903 119907119897119903 119907119903119891 119907119897119891

Solution

120602 =119897

tan 120575

119907119897119903 =120602 minus 1198892 sdot 119907119863

120602

119907119903119903 =120602 + 1198892 sdot 119907119863

120602

119907119897119891 =120602 minus 1198892 2 + 1198972 sdot tan 120575

119897sdot 119907119863

119907119903119891 =1206022 + 1198972 sdot tan 120575

119897sdot 119907119863


120575

120602

rrlabcsuni-klde



Given 120602 120593 119907 119889



1205932 = arctan120549119910 + 119889

1205491199091206022 = 120602 sdot cos120593 cos1205932

1199072 = 119907 sdot 1206022120602

120549119910 = 120602 ∙ 119904119894119899 120593

1205931

1205932

1206021

1206022

120593 120549119909 = 120602 ∙ cos120593

120602

119889

rrlabcsuni-klde

Bicycle Model


119907


Bicycle Model


Low speed motion



119865119910 =1198981199072

120602






Bicycle Model

Given 120575119891 120575119903 119907 120579

Unknown ሶ119883 ሶ119884 ሶ120579

Solution

120575119907 = tanminus1119897119891 tan 120575119903+119897119903 tan 120575119891

119897119891+119897119903

ሶ120579 =119907

120602

120602 =119897119891+119897119903

tan 120575119891 minustan 120575119903 cos 120575119907

ሶ119883 = 119907 cos 120579 + 120575119907ሶ119884 = 119907 sin 120579 + 120575119907

ሶ120579 =119907 cos 120575119907

119897119891+119897119903tan 120575119891 minus tan 120575119903

119907

rrlabcsuni-klde



119862 = 119874 120575119903 = 0 120575119891 = 120575

119897119903 = 0 119897119891 = 119897

120575119907 = 0

ሶ120579 =119907

120602

120602 =119897

tan 120575119891

ሶ119883 = 119907 cos 120579ሶ119884 = 119907 sin 120579ሶ120579 =

119907

119897119891tan 120575119891

119907







ሶԦ120585119882 = ሶ119909 ሶ119910 ሶ120579119879

= 119891 119889 119903 120579 ሶ120595119897 ሶ120595119903

= 119877 120579 minus1 ሶԦ120585119877

passive wheel

active wheel

d






ሶ119909119877119903 =119903 ሶ1205951199032

ሶ119909119877119897 =119903 ሶ120595119897

2

120596119903 =119903 ሶ1205951199032119889

120596119897 =minus119903 ሶ120595119897

2119889

ሶԦ120585119882 = 119877 120579 minus1 ሶԦ120585119903ሶԦ120585119903 =

119903 ሶ1205951199032

+119903 ሶ120595119897

2 0

119903 ሶ1205951199032119889

minus119903 ሶ120595119897

2119889

119879





Given 120579 =120587



119904

ሶԦ120585119882 =ሶ119909ሶ119910ሶ120579

=0 minus1 01 0 00 0 1

sdot301

=031







Consider




rrlabcsuni-klde

Basic Wheel Types


Standardwheel

Steerablestandard

wheel

Castorwheel

Swedish orMecanum

wheel

Ball orspherical

wheel


Wheel Properties




Castor wheel


















ሶ120595 119903



ሶ120595 119903



ሶ120595 119903



ሶ120595 119903



ሶ120595 119903








and 1 Ԧ1199071 = ሶ119909 ሶ119910 0 119879


119894+1120596119894+1 = 119894119894+1119877 sdot 119894120596119894 + ሶ120579119894+1

119894+1 Ԧ119890119911119894+1

119894+1 Ԧ119907119894+1 = 119894119894+1119877 119894 Ԧ119907119894 +

119894120596119894 times119894 Ԧ119901119894+1




calculated as

2 Ԧ1199072 =119888120572 ሶ119909 + 119904120572 ሶ119910minus119904120572 ሶ119909 + 119888120572 ሶ119910

0


3 Ԧ1199073 =

119888120572 ሶ119909 + 119904120572 ሶ119910

minus119904120572 ሶ119909 + 119888120572 ሶ119910 + 119889 ሶ1205790






4 Ԧ1199074 =119904 120572 + 120573 ሶ119909 minus 119888 120572 + 120573 ሶ119910 minus 119888120573 119889 ሶ120579

119888 120572 + 120573 ሶ119909 + 119904 120572 + 120573 ሶ119910 + 119904120573 119889 ሶ1205790




119904 120572 + 120573 minus119888 120572 + 120573 minus119888120573 119889

119888 120572 + 120573 119904 120572 + 120573 119904120573 1198890 0 0

ሶ119909119910 ሶሶ120579

=119903 ሶ12059500












119904 120572 + 120573(119905) minus119888 120572 + 120573(119905) minus119888120573(119905)119889

119888 120572 + 120573(119905) 119904 120572 + 120573(119905) 119904120573(119905)1198890 0 0

ሶ119909ሶ119910ሶ120579

=119903 ሶ120595

minus119889119888 ሶ120573(119905)0


maneuverability)




119904 120572 + 120573 + 120574 minus119888 120572 + 120573 + 120574 minus119888 120573 + 120574 119889

119888 120572 + 120573 + 120574 119904 120572 + 120573 + 120574 119904 120573 + 120574 1198890 0 0

ሶ119909ሶ119910ሶ120579

=119903 ሶ120595 cos 120574

119903 ሶ120595 sin 120574 + 119903119901119903 ሶ1205951199011199030















120572119897 = 90 deg

120573119897 = 0 deg

120572119903 = minus90 deg

120573119903 = 180 deg




119904 120572119897 + 120573119897 ሶ119909 minus 119888 120572119897 + 120573119897 ሶ119910 minus 119888120573119897 119889 ሶ120579 = 119903119897 ሶ120595119897

119888 120572119897 + 120573119897 ሶ119909 + 119904 120572119897 + 120573119897 ሶ119910 + 119904120573119897 119889 ሶ120579 = 0



ሶ119909 minus 119889 ሶ120579 = 119903119897 ሶ120595119897

ሶ119910 = 0

ሶ119909 + 119889 ሶ120579 = 119903119903 ሶ120595119903119910 = 0




ሶ119909ሶ119910ሶ120579

=

1

2119903119897 ሶ120595119897 + 119903119903 ሶ120595119903

01

2119889minus119903119897 ሶ120595119897 + 119903119903 ሶ120595119903



rrlabcsuni-klde




rrlabcsuni-klde











1205721 = 0 deg

1205722 = 120 deg

1205723 = minus120 deg







119888 1205721 + 1205731 ሶ119909 + 119904 1205721 + 1205731 ሶ119910 + 119889 sdot 119904 1205731 ሶ120579 = 0

119888 1205722 + 1205732 ሶ119909 + 119904 1205722 + 1205732 ሶ119910 + 119889 sdot 119904 1205732 ሶ120579 = 0

119888 1205723 + 1205733 ሶ119909 + 119904 1205723 + 1205733 ሶ119910 + 119889 sdot 119904 1205733 ሶ120579 = 0







119888 120572119894 sdot ሶ119909 + 119904 120572119894 sdot ሶ119910




ሶ120595119894 =1


rrlabcsuni-klde



rrlabcsuni-klde

Mecanum Vehicle





Mecanum Kinematics



1205721 = 45 deg 1205731 = 45 deg 1205741 = minus45 deg

1205722 = 135 deg 1205732 = minus45 deg 1205742 = 45 deg


1205724 = minus45 deg 1205734 = 135 deg 1205744 = 45 deg


Mecanum Kinematics








Mecanum Kinematics


ሶ119909 =119903

4ሶ1205951 + ሶ1205952 + ሶ1205953 + ሶ1205954

ሶ119910 =119903

4minus ሶ1205951 + ሶ1205952 minus ሶ1205953 + ሶ1205954

ሶ120579 =119903

119889 2ሶ1205951 minus ሶ1205952 minus ሶ1205953 + ሶ1205954


Mecanum Kinematics


ሶ119909ሶ119910ሶ120579

=119903119908ℎ1198901198901198974


sdot

ሶ1205951

ሶ1205952

ሶ1205953

ሶ1205954

with a =2 2

119889


ሶ1205954 = ሶ1205951 + ሶ1205952 minus ሶ1205953

rrlabcsuni-klde





Ability to maneuver

Controllability

Stability



Stability




Typical setups




Maneuverability








Controllability



Examples




Wheel type

Drive geometry


Motortransmission


Degree of Mobility




119891119898 = 3 minus rank 119878


Degree of Mobility



1198891 = 1198892 1205731 = 1205732 = 0 1205721 + 120587 = 1205722






119891119904 = rank 119878prime








119891119872 = 119891119898 + 119891119904

rrlabcsuni-klde

Vehicle Kinematics



Tricycle Drive

Given 120575 119907


119903 = 119897 sdot cot 120575

119907prime =119907

cos 120575

119907119897 = 119907 sdot119903 + 119889

119903

119907119903 = 119907 sdot119903 minus 119889


120575


Ackermann Steering

Given 119907119863 120575 119897 119889

Unknown 119907119903119903 119907119897119903 119907119903119891 119907119897119891

Solution

120602 =119897

tan 120575

119907119897119903 =120602 minus 1198892 sdot 119907119863

120602

119907119903119903 =120602 + 1198892 sdot 119907119863

120602

119907119897119891 =120602 minus 1198892 2 + 1198972 sdot tan 120575

119897sdot 119907119863

119907119903119891 =1206022 + 1198972 sdot tan 120575

119897sdot 119907119863


120575

120602

rrlabcsuni-klde



Given 120602 120593 119907 119889



1205932 = arctan120549119910 + 119889

1205491199091206022 = 120602 sdot cos120593 cos1205932

1199072 = 119907 sdot 1206022120602

120549119910 = 120602 ∙ 119904119894119899 120593

1205931

1205932

1206021

1206022

120593 120549119909 = 120602 ∙ cos120593

120602

119889

rrlabcsuni-klde

Bicycle Model


119907


Bicycle Model


Low speed motion



119865119910 =1198981199072

120602






Bicycle Model

Given 120575119891 120575119903 119907 120579

Unknown ሶ119883 ሶ119884 ሶ120579

Solution

120575119907 = tanminus1119897119891 tan 120575119903+119897119903 tan 120575119891

119897119891+119897119903

ሶ120579 =119907

120602

120602 =119897119891+119897119903

tan 120575119891 minustan 120575119903 cos 120575119907

ሶ119883 = 119907 cos 120579 + 120575119907ሶ119884 = 119907 sin 120579 + 120575119907

ሶ120579 =119907 cos 120575119907

119897119891+119897119903tan 120575119891 minus tan 120575119903

119907

rrlabcsuni-klde



119862 = 119874 120575119903 = 0 120575119891 = 120575

119897119903 = 0 119897119891 = 119897

120575119907 = 0

ሶ120579 =119907

120602

120602 =119897

tan 120575119891

ሶ119883 = 119907 cos 120579ሶ119884 = 119907 sin 120579ሶ120579 =

119907

119897119891tan 120575119891

119907






ሶ119909119877119903 =119903 ሶ1205951199032

ሶ119909119877119897 =119903 ሶ120595119897

2

120596119903 =119903 ሶ1205951199032119889

120596119897 =minus119903 ሶ120595119897

2119889

ሶԦ120585119882 = 119877 120579 minus1 ሶԦ120585119903ሶԦ120585119903 =

119903 ሶ1205951199032

+119903 ሶ120595119897

2 0

119903 ሶ1205951199032119889

minus119903 ሶ120595119897

2119889

119879





Given 120579 =120587



119904

ሶԦ120585119882 =ሶ119909ሶ119910ሶ120579

=0 minus1 01 0 00 0 1

sdot301

=031







Consider




rrlabcsuni-klde

Basic Wheel Types


Standardwheel

Steerablestandard

wheel

Castorwheel

Swedish orMecanum

wheel

Ball orspherical

wheel


Wheel Properties




Castor wheel


















ሶ120595 119903



ሶ120595 119903



ሶ120595 119903



ሶ120595 119903



ሶ120595 119903








and 1 Ԧ1199071 = ሶ119909 ሶ119910 0 119879


119894+1120596119894+1 = 119894119894+1119877 sdot 119894120596119894 + ሶ120579119894+1

119894+1 Ԧ119890119911119894+1

119894+1 Ԧ119907119894+1 = 119894119894+1119877 119894 Ԧ119907119894 +

119894120596119894 times119894 Ԧ119901119894+1




calculated as

2 Ԧ1199072 =119888120572 ሶ119909 + 119904120572 ሶ119910minus119904120572 ሶ119909 + 119888120572 ሶ119910

0


3 Ԧ1199073 =

119888120572 ሶ119909 + 119904120572 ሶ119910

minus119904120572 ሶ119909 + 119888120572 ሶ119910 + 119889 ሶ1205790






4 Ԧ1199074 =119904 120572 + 120573 ሶ119909 minus 119888 120572 + 120573 ሶ119910 minus 119888120573 119889 ሶ120579

119888 120572 + 120573 ሶ119909 + 119904 120572 + 120573 ሶ119910 + 119904120573 119889 ሶ1205790




119904 120572 + 120573 minus119888 120572 + 120573 minus119888120573 119889

119888 120572 + 120573 119904 120572 + 120573 119904120573 1198890 0 0

ሶ119909119910 ሶሶ120579

=119903 ሶ12059500












119904 120572 + 120573(119905) minus119888 120572 + 120573(119905) minus119888120573(119905)119889

119888 120572 + 120573(119905) 119904 120572 + 120573(119905) 119904120573(119905)1198890 0 0

ሶ119909ሶ119910ሶ120579

=119903 ሶ120595

minus119889119888 ሶ120573(119905)0


maneuverability)




119904 120572 + 120573 + 120574 minus119888 120572 + 120573 + 120574 minus119888 120573 + 120574 119889

119888 120572 + 120573 + 120574 119904 120572 + 120573 + 120574 119904 120573 + 120574 1198890 0 0

ሶ119909ሶ119910ሶ120579

=119903 ሶ120595 cos 120574

119903 ሶ120595 sin 120574 + 119903119901119903 ሶ1205951199011199030















120572119897 = 90 deg

120573119897 = 0 deg

120572119903 = minus90 deg

120573119903 = 180 deg




119904 120572119897 + 120573119897 ሶ119909 minus 119888 120572119897 + 120573119897 ሶ119910 minus 119888120573119897 119889 ሶ120579 = 119903119897 ሶ120595119897

119888 120572119897 + 120573119897 ሶ119909 + 119904 120572119897 + 120573119897 ሶ119910 + 119904120573119897 119889 ሶ120579 = 0



ሶ119909 minus 119889 ሶ120579 = 119903119897 ሶ120595119897

ሶ119910 = 0

ሶ119909 + 119889 ሶ120579 = 119903119903 ሶ120595119903119910 = 0




ሶ119909ሶ119910ሶ120579

=

1

2119903119897 ሶ120595119897 + 119903119903 ሶ120595119903

01

2119889minus119903119897 ሶ120595119897 + 119903119903 ሶ120595119903



rrlabcsuni-klde




rrlabcsuni-klde











1205721 = 0 deg

1205722 = 120 deg

1205723 = minus120 deg







119888 1205721 + 1205731 ሶ119909 + 119904 1205721 + 1205731 ሶ119910 + 119889 sdot 119904 1205731 ሶ120579 = 0

119888 1205722 + 1205732 ሶ119909 + 119904 1205722 + 1205732 ሶ119910 + 119889 sdot 119904 1205732 ሶ120579 = 0

119888 1205723 + 1205733 ሶ119909 + 119904 1205723 + 1205733 ሶ119910 + 119889 sdot 119904 1205733 ሶ120579 = 0







119888 120572119894 sdot ሶ119909 + 119904 120572119894 sdot ሶ119910




ሶ120595119894 =1


rrlabcsuni-klde



rrlabcsuni-klde

Mecanum Vehicle





Mecanum Kinematics



1205721 = 45 deg 1205731 = 45 deg 1205741 = minus45 deg

1205722 = 135 deg 1205732 = minus45 deg 1205742 = 45 deg


1205724 = minus45 deg 1205734 = 135 deg 1205744 = 45 deg


Mecanum Kinematics








Mecanum Kinematics


ሶ119909 =119903

4ሶ1205951 + ሶ1205952 + ሶ1205953 + ሶ1205954

ሶ119910 =119903

4minus ሶ1205951 + ሶ1205952 minus ሶ1205953 + ሶ1205954

ሶ120579 =119903

119889 2ሶ1205951 minus ሶ1205952 minus ሶ1205953 + ሶ1205954


Mecanum Kinematics


ሶ119909ሶ119910ሶ120579

=119903119908ℎ1198901198901198974


sdot

ሶ1205951

ሶ1205952

ሶ1205953

ሶ1205954

with a =2 2

119889


ሶ1205954 = ሶ1205951 + ሶ1205952 minus ሶ1205953

rrlabcsuni-klde





Ability to maneuver

Controllability

Stability



Stability




Typical setups




Maneuverability








Controllability



Examples




Wheel type

Drive geometry


Motortransmission


Degree of Mobility




119891119898 = 3 minus rank 119878


Degree of Mobility



1198891 = 1198892 1205731 = 1205732 = 0 1205721 + 120587 = 1205722






119891119904 = rank 119878prime








119891119872 = 119891119898 + 119891119904

rrlabcsuni-klde

Vehicle Kinematics



Tricycle Drive

Given 120575 119907


119903 = 119897 sdot cot 120575

119907prime =119907

cos 120575

119907119897 = 119907 sdot119903 + 119889

119903

119907119903 = 119907 sdot119903 minus 119889


120575


Ackermann Steering

Given 119907119863 120575 119897 119889

Unknown 119907119903119903 119907119897119903 119907119903119891 119907119897119891

Solution

120602 =119897

tan 120575

119907119897119903 =120602 minus 1198892 sdot 119907119863

120602

119907119903119903 =120602 + 1198892 sdot 119907119863

120602

119907119897119891 =120602 minus 1198892 2 + 1198972 sdot tan 120575

119897sdot 119907119863

119907119903119891 =1206022 + 1198972 sdot tan 120575

119897sdot 119907119863


120575

120602

rrlabcsuni-klde



Given 120602 120593 119907 119889



1205932 = arctan120549119910 + 119889

1205491199091206022 = 120602 sdot cos120593 cos1205932

1199072 = 119907 sdot 1206022120602

120549119910 = 120602 ∙ 119904119894119899 120593

1205931

1205932

1206021

1206022

120593 120549119909 = 120602 ∙ cos120593

120602

119889

rrlabcsuni-klde

Bicycle Model


119907


Bicycle Model


Low speed motion



119865119910 =1198981199072

120602






Bicycle Model

Given 120575119891 120575119903 119907 120579

Unknown ሶ119883 ሶ119884 ሶ120579

Solution

120575119907 = tanminus1119897119891 tan 120575119903+119897119903 tan 120575119891

119897119891+119897119903

ሶ120579 =119907

120602

120602 =119897119891+119897119903

tan 120575119891 minustan 120575119903 cos 120575119907

ሶ119883 = 119907 cos 120579 + 120575119907ሶ119884 = 119907 sin 120579 + 120575119907

ሶ120579 =119907 cos 120575119907

119897119891+119897119903tan 120575119891 minus tan 120575119903

119907

rrlabcsuni-klde



119862 = 119874 120575119903 = 0 120575119891 = 120575

119897119903 = 0 119897119891 = 119897

120575119907 = 0

ሶ120579 =119907

120602

120602 =119897

tan 120575119891

ሶ119883 = 119907 cos 120579ሶ119884 = 119907 sin 120579ሶ120579 =

119907

119897119891tan 120575119891

119907





Given 120579 =120587



119904

ሶԦ120585119882 =ሶ119909ሶ119910ሶ120579

=0 minus1 01 0 00 0 1

sdot301

=031







Consider




rrlabcsuni-klde

Basic Wheel Types


Standardwheel

Steerablestandard

wheel

Castorwheel

Swedish orMecanum

wheel

Ball orspherical

wheel


Wheel Properties




Castor wheel


















ሶ120595 119903



ሶ120595 119903



ሶ120595 119903



ሶ120595 119903



ሶ120595 119903








and 1 Ԧ1199071 = ሶ119909 ሶ119910 0 119879


119894+1120596119894+1 = 119894119894+1119877 sdot 119894120596119894 + ሶ120579119894+1

119894+1 Ԧ119890119911119894+1

119894+1 Ԧ119907119894+1 = 119894119894+1119877 119894 Ԧ119907119894 +

119894120596119894 times119894 Ԧ119901119894+1




calculated as

2 Ԧ1199072 =119888120572 ሶ119909 + 119904120572 ሶ119910minus119904120572 ሶ119909 + 119888120572 ሶ119910

0


3 Ԧ1199073 =

119888120572 ሶ119909 + 119904120572 ሶ119910

minus119904120572 ሶ119909 + 119888120572 ሶ119910 + 119889 ሶ1205790






4 Ԧ1199074 =119904 120572 + 120573 ሶ119909 minus 119888 120572 + 120573 ሶ119910 minus 119888120573 119889 ሶ120579

119888 120572 + 120573 ሶ119909 + 119904 120572 + 120573 ሶ119910 + 119904120573 119889 ሶ1205790




119904 120572 + 120573 minus119888 120572 + 120573 minus119888120573 119889

119888 120572 + 120573 119904 120572 + 120573 119904120573 1198890 0 0

ሶ119909119910 ሶሶ120579

=119903 ሶ12059500












119904 120572 + 120573(119905) minus119888 120572 + 120573(119905) minus119888120573(119905)119889

119888 120572 + 120573(119905) 119904 120572 + 120573(119905) 119904120573(119905)1198890 0 0

ሶ119909ሶ119910ሶ120579

=119903 ሶ120595

minus119889119888 ሶ120573(119905)0


maneuverability)




119904 120572 + 120573 + 120574 minus119888 120572 + 120573 + 120574 minus119888 120573 + 120574 119889

119888 120572 + 120573 + 120574 119904 120572 + 120573 + 120574 119904 120573 + 120574 1198890 0 0

ሶ119909ሶ119910ሶ120579

=119903 ሶ120595 cos 120574

119903 ሶ120595 sin 120574 + 119903119901119903 ሶ1205951199011199030















120572119897 = 90 deg

120573119897 = 0 deg

120572119903 = minus90 deg

120573119903 = 180 deg




119904 120572119897 + 120573119897 ሶ119909 minus 119888 120572119897 + 120573119897 ሶ119910 minus 119888120573119897 119889 ሶ120579 = 119903119897 ሶ120595119897

119888 120572119897 + 120573119897 ሶ119909 + 119904 120572119897 + 120573119897 ሶ119910 + 119904120573119897 119889 ሶ120579 = 0



ሶ119909 minus 119889 ሶ120579 = 119903119897 ሶ120595119897

ሶ119910 = 0

ሶ119909 + 119889 ሶ120579 = 119903119903 ሶ120595119903119910 = 0




ሶ119909ሶ119910ሶ120579

=

1

2119903119897 ሶ120595119897 + 119903119903 ሶ120595119903

01

2119889minus119903119897 ሶ120595119897 + 119903119903 ሶ120595119903



rrlabcsuni-klde




rrlabcsuni-klde











1205721 = 0 deg

1205722 = 120 deg

1205723 = minus120 deg







119888 1205721 + 1205731 ሶ119909 + 119904 1205721 + 1205731 ሶ119910 + 119889 sdot 119904 1205731 ሶ120579 = 0

119888 1205722 + 1205732 ሶ119909 + 119904 1205722 + 1205732 ሶ119910 + 119889 sdot 119904 1205732 ሶ120579 = 0

119888 1205723 + 1205733 ሶ119909 + 119904 1205723 + 1205733 ሶ119910 + 119889 sdot 119904 1205733 ሶ120579 = 0







119888 120572119894 sdot ሶ119909 + 119904 120572119894 sdot ሶ119910




ሶ120595119894 =1


rrlabcsuni-klde



rrlabcsuni-klde

Mecanum Vehicle





Mecanum Kinematics



1205721 = 45 deg 1205731 = 45 deg 1205741 = minus45 deg

1205722 = 135 deg 1205732 = minus45 deg 1205742 = 45 deg


1205724 = minus45 deg 1205734 = 135 deg 1205744 = 45 deg


Mecanum Kinematics








Mecanum Kinematics


ሶ119909 =119903

4ሶ1205951 + ሶ1205952 + ሶ1205953 + ሶ1205954

ሶ119910 =119903

4minus ሶ1205951 + ሶ1205952 minus ሶ1205953 + ሶ1205954

ሶ120579 =119903

119889 2ሶ1205951 minus ሶ1205952 minus ሶ1205953 + ሶ1205954


Mecanum Kinematics


ሶ119909ሶ119910ሶ120579

=119903119908ℎ1198901198901198974


sdot

ሶ1205951

ሶ1205952

ሶ1205953

ሶ1205954

with a =2 2

119889


ሶ1205954 = ሶ1205951 + ሶ1205952 minus ሶ1205953

rrlabcsuni-klde





Ability to maneuver

Controllability

Stability



Stability




Typical setups




Maneuverability








Controllability



Examples




Wheel type

Drive geometry


Motortransmission


Degree of Mobility




119891119898 = 3 minus rank 119878


Degree of Mobility



1198891 = 1198892 1205731 = 1205732 = 0 1205721 + 120587 = 1205722






119891119904 = rank 119878prime








119891119872 = 119891119898 + 119891119904

rrlabcsuni-klde

Vehicle Kinematics



Tricycle Drive

Given 120575 119907


119903 = 119897 sdot cot 120575

119907prime =119907

cos 120575

119907119897 = 119907 sdot119903 + 119889

119903

119907119903 = 119907 sdot119903 minus 119889


120575


Ackermann Steering

Given 119907119863 120575 119897 119889

Unknown 119907119903119903 119907119897119903 119907119903119891 119907119897119891

Solution

120602 =119897

tan 120575

119907119897119903 =120602 minus 1198892 sdot 119907119863

120602

119907119903119903 =120602 + 1198892 sdot 119907119863

120602

119907119897119891 =120602 minus 1198892 2 + 1198972 sdot tan 120575

119897sdot 119907119863

119907119903119891 =1206022 + 1198972 sdot tan 120575

119897sdot 119907119863


120575

120602

rrlabcsuni-klde



Given 120602 120593 119907 119889



1205932 = arctan120549119910 + 119889

1205491199091206022 = 120602 sdot cos120593 cos1205932

1199072 = 119907 sdot 1206022120602

120549119910 = 120602 ∙ 119904119894119899 120593

1205931

1205932

1206021

1206022

120593 120549119909 = 120602 ∙ cos120593

120602

119889

rrlabcsuni-klde

Bicycle Model


119907


Bicycle Model


Low speed motion



119865119910 =1198981199072

120602






Bicycle Model

Given 120575119891 120575119903 119907 120579

Unknown ሶ119883 ሶ119884 ሶ120579

Solution

120575119907 = tanminus1119897119891 tan 120575119903+119897119903 tan 120575119891

119897119891+119897119903

ሶ120579 =119907

120602

120602 =119897119891+119897119903

tan 120575119891 minustan 120575119903 cos 120575119907

ሶ119883 = 119907 cos 120579 + 120575119907ሶ119884 = 119907 sin 120579 + 120575119907

ሶ120579 =119907 cos 120575119907

119897119891+119897119903tan 120575119891 minus tan 120575119903

119907

rrlabcsuni-klde



119862 = 119874 120575119903 = 0 120575119891 = 120575

119897119903 = 0 119897119891 = 119897

120575119907 = 0

ሶ120579 =119907

120602

120602 =119897

tan 120575119891

ሶ119883 = 119907 cos 120579ሶ119884 = 119907 sin 120579ሶ120579 =

119907

119897119891tan 120575119891

119907







Consider




rrlabcsuni-klde

Basic Wheel Types


Standardwheel

Steerablestandard

wheel

Castorwheel

Swedish orMecanum

wheel

Ball orspherical

wheel


Wheel Properties




Castor wheel


















ሶ120595 119903



ሶ120595 119903



ሶ120595 119903



ሶ120595 119903



ሶ120595 119903








and 1 Ԧ1199071 = ሶ119909 ሶ119910 0 119879


119894+1120596119894+1 = 119894119894+1119877 sdot 119894120596119894 + ሶ120579119894+1

119894+1 Ԧ119890119911119894+1

119894+1 Ԧ119907119894+1 = 119894119894+1119877 119894 Ԧ119907119894 +

119894120596119894 times119894 Ԧ119901119894+1




calculated as

2 Ԧ1199072 =119888120572 ሶ119909 + 119904120572 ሶ119910minus119904120572 ሶ119909 + 119888120572 ሶ119910

0


3 Ԧ1199073 =

119888120572 ሶ119909 + 119904120572 ሶ119910

minus119904120572 ሶ119909 + 119888120572 ሶ119910 + 119889 ሶ1205790






4 Ԧ1199074 =119904 120572 + 120573 ሶ119909 minus 119888 120572 + 120573 ሶ119910 minus 119888120573 119889 ሶ120579

119888 120572 + 120573 ሶ119909 + 119904 120572 + 120573 ሶ119910 + 119904120573 119889 ሶ1205790




119904 120572 + 120573 minus119888 120572 + 120573 minus119888120573 119889

119888 120572 + 120573 119904 120572 + 120573 119904120573 1198890 0 0

ሶ119909119910 ሶሶ120579

=119903 ሶ12059500












119904 120572 + 120573(119905) minus119888 120572 + 120573(119905) minus119888120573(119905)119889

119888 120572 + 120573(119905) 119904 120572 + 120573(119905) 119904120573(119905)1198890 0 0

ሶ119909ሶ119910ሶ120579

=119903 ሶ120595

minus119889119888 ሶ120573(119905)0


maneuverability)




119904 120572 + 120573 + 120574 minus119888 120572 + 120573 + 120574 minus119888 120573 + 120574 119889

119888 120572 + 120573 + 120574 119904 120572 + 120573 + 120574 119904 120573 + 120574 1198890 0 0

ሶ119909ሶ119910ሶ120579

=119903 ሶ120595 cos 120574

119903 ሶ120595 sin 120574 + 119903119901119903 ሶ1205951199011199030















120572119897 = 90 deg

120573119897 = 0 deg

120572119903 = minus90 deg

120573119903 = 180 deg




119904 120572119897 + 120573119897 ሶ119909 minus 119888 120572119897 + 120573119897 ሶ119910 minus 119888120573119897 119889 ሶ120579 = 119903119897 ሶ120595119897

119888 120572119897 + 120573119897 ሶ119909 + 119904 120572119897 + 120573119897 ሶ119910 + 119904120573119897 119889 ሶ120579 = 0



ሶ119909 minus 119889 ሶ120579 = 119903119897 ሶ120595119897

ሶ119910 = 0

ሶ119909 + 119889 ሶ120579 = 119903119903 ሶ120595119903119910 = 0




ሶ119909ሶ119910ሶ120579

=

1

2119903119897 ሶ120595119897 + 119903119903 ሶ120595119903

01

2119889minus119903119897 ሶ120595119897 + 119903119903 ሶ120595119903



rrlabcsuni-klde




rrlabcsuni-klde











1205721 = 0 deg

1205722 = 120 deg

1205723 = minus120 deg







119888 1205721 + 1205731 ሶ119909 + 119904 1205721 + 1205731 ሶ119910 + 119889 sdot 119904 1205731 ሶ120579 = 0

119888 1205722 + 1205732 ሶ119909 + 119904 1205722 + 1205732 ሶ119910 + 119889 sdot 119904 1205732 ሶ120579 = 0

119888 1205723 + 1205733 ሶ119909 + 119904 1205723 + 1205733 ሶ119910 + 119889 sdot 119904 1205733 ሶ120579 = 0







119888 120572119894 sdot ሶ119909 + 119904 120572119894 sdot ሶ119910




ሶ120595119894 =1


rrlabcsuni-klde



rrlabcsuni-klde

Mecanum Vehicle





Mecanum Kinematics



1205721 = 45 deg 1205731 = 45 deg 1205741 = minus45 deg

1205722 = 135 deg 1205732 = minus45 deg 1205742 = 45 deg


1205724 = minus45 deg 1205734 = 135 deg 1205744 = 45 deg


Mecanum Kinematics








Mecanum Kinematics


ሶ119909 =119903

4ሶ1205951 + ሶ1205952 + ሶ1205953 + ሶ1205954

ሶ119910 =119903

4minus ሶ1205951 + ሶ1205952 minus ሶ1205953 + ሶ1205954

ሶ120579 =119903

119889 2ሶ1205951 minus ሶ1205952 minus ሶ1205953 + ሶ1205954


Mecanum Kinematics


ሶ119909ሶ119910ሶ120579

=119903119908ℎ1198901198901198974


sdot

ሶ1205951

ሶ1205952

ሶ1205953

ሶ1205954

with a =2 2

119889


ሶ1205954 = ሶ1205951 + ሶ1205952 minus ሶ1205953

rrlabcsuni-klde





Ability to maneuver

Controllability

Stability



Stability




Typical setups




Maneuverability








Controllability



Examples




Wheel type

Drive geometry


Motortransmission


Degree of Mobility




119891119898 = 3 minus rank 119878


Degree of Mobility



1198891 = 1198892 1205731 = 1205732 = 0 1205721 + 120587 = 1205722






119891119904 = rank 119878prime








119891119872 = 119891119898 + 119891119904

rrlabcsuni-klde

Vehicle Kinematics



Tricycle Drive

Given 120575 119907


119903 = 119897 sdot cot 120575

119907prime =119907

cos 120575

119907119897 = 119907 sdot119903 + 119889

119903

119907119903 = 119907 sdot119903 minus 119889


120575


Ackermann Steering

Given 119907119863 120575 119897 119889

Unknown 119907119903119903 119907119897119903 119907119903119891 119907119897119891

Solution

120602 =119897

tan 120575

119907119897119903 =120602 minus 1198892 sdot 119907119863

120602

119907119903119903 =120602 + 1198892 sdot 119907119863

120602

119907119897119891 =120602 minus 1198892 2 + 1198972 sdot tan 120575

119897sdot 119907119863

119907119903119891 =1206022 + 1198972 sdot tan 120575

119897sdot 119907119863


120575

120602

rrlabcsuni-klde



Given 120602 120593 119907 119889



1205932 = arctan120549119910 + 119889

1205491199091206022 = 120602 sdot cos120593 cos1205932

1199072 = 119907 sdot 1206022120602

120549119910 = 120602 ∙ 119904119894119899 120593

1205931

1205932

1206021

1206022

120593 120549119909 = 120602 ∙ cos120593

120602

119889

rrlabcsuni-klde

Bicycle Model


119907


Bicycle Model


Low speed motion



119865119910 =1198981199072

120602






Bicycle Model

Given 120575119891 120575119903 119907 120579

Unknown ሶ119883 ሶ119884 ሶ120579

Solution

120575119907 = tanminus1119897119891 tan 120575119903+119897119903 tan 120575119891

119897119891+119897119903

ሶ120579 =119907

120602

120602 =119897119891+119897119903

tan 120575119891 minustan 120575119903 cos 120575119907

ሶ119883 = 119907 cos 120579 + 120575119907ሶ119884 = 119907 sin 120579 + 120575119907

ሶ120579 =119907 cos 120575119907

119897119891+119897119903tan 120575119891 minus tan 120575119903

119907

rrlabcsuni-klde



119862 = 119874 120575119903 = 0 120575119891 = 120575

119897119903 = 0 119897119891 = 119897

120575119907 = 0

ሶ120579 =119907

120602

120602 =119897

tan 120575119891

ሶ119883 = 119907 cos 120579ሶ119884 = 119907 sin 120579ሶ120579 =

119907

119897119891tan 120575119891

119907

rrlabcsuni-klde

Basic Wheel Types


Standardwheel

Steerablestandard

wheel

Castorwheel

Swedish orMecanum

wheel

Ball orspherical

wheel


Wheel Properties




Castor wheel


















ሶ120595 119903



ሶ120595 119903



ሶ120595 119903



ሶ120595 119903



ሶ120595 119903








and 1 Ԧ1199071 = ሶ119909 ሶ119910 0 119879


119894+1120596119894+1 = 119894119894+1119877 sdot 119894120596119894 + ሶ120579119894+1

119894+1 Ԧ119890119911119894+1

119894+1 Ԧ119907119894+1 = 119894119894+1119877 119894 Ԧ119907119894 +

119894120596119894 times119894 Ԧ119901119894+1




calculated as

2 Ԧ1199072 =119888120572 ሶ119909 + 119904120572 ሶ119910minus119904120572 ሶ119909 + 119888120572 ሶ119910

0


3 Ԧ1199073 =

119888120572 ሶ119909 + 119904120572 ሶ119910

minus119904120572 ሶ119909 + 119888120572 ሶ119910 + 119889 ሶ1205790






4 Ԧ1199074 =119904 120572 + 120573 ሶ119909 minus 119888 120572 + 120573 ሶ119910 minus 119888120573 119889 ሶ120579

119888 120572 + 120573 ሶ119909 + 119904 120572 + 120573 ሶ119910 + 119904120573 119889 ሶ1205790




119904 120572 + 120573 minus119888 120572 + 120573 minus119888120573 119889

119888 120572 + 120573 119904 120572 + 120573 119904120573 1198890 0 0

ሶ119909119910 ሶሶ120579

=119903 ሶ12059500












119904 120572 + 120573(119905) minus119888 120572 + 120573(119905) minus119888120573(119905)119889

119888 120572 + 120573(119905) 119904 120572 + 120573(119905) 119904120573(119905)1198890 0 0

ሶ119909ሶ119910ሶ120579

=119903 ሶ120595

minus119889119888 ሶ120573(119905)0


maneuverability)




119904 120572 + 120573 + 120574 minus119888 120572 + 120573 + 120574 minus119888 120573 + 120574 119889

119888 120572 + 120573 + 120574 119904 120572 + 120573 + 120574 119904 120573 + 120574 1198890 0 0

ሶ119909ሶ119910ሶ120579

=119903 ሶ120595 cos 120574

119903 ሶ120595 sin 120574 + 119903119901119903 ሶ1205951199011199030















120572119897 = 90 deg

120573119897 = 0 deg

120572119903 = minus90 deg

120573119903 = 180 deg




119904 120572119897 + 120573119897 ሶ119909 minus 119888 120572119897 + 120573119897 ሶ119910 minus 119888120573119897 119889 ሶ120579 = 119903119897 ሶ120595119897

119888 120572119897 + 120573119897 ሶ119909 + 119904 120572119897 + 120573119897 ሶ119910 + 119904120573119897 119889 ሶ120579 = 0



ሶ119909 minus 119889 ሶ120579 = 119903119897 ሶ120595119897

ሶ119910 = 0

ሶ119909 + 119889 ሶ120579 = 119903119903 ሶ120595119903119910 = 0




ሶ119909ሶ119910ሶ120579

=

1

2119903119897 ሶ120595119897 + 119903119903 ሶ120595119903

01

2119889minus119903119897 ሶ120595119897 + 119903119903 ሶ120595119903



rrlabcsuni-klde




rrlabcsuni-klde











1205721 = 0 deg

1205722 = 120 deg

1205723 = minus120 deg







119888 1205721 + 1205731 ሶ119909 + 119904 1205721 + 1205731 ሶ119910 + 119889 sdot 119904 1205731 ሶ120579 = 0

119888 1205722 + 1205732 ሶ119909 + 119904 1205722 + 1205732 ሶ119910 + 119889 sdot 119904 1205732 ሶ120579 = 0

119888 1205723 + 1205733 ሶ119909 + 119904 1205723 + 1205733 ሶ119910 + 119889 sdot 119904 1205733 ሶ120579 = 0







119888 120572119894 sdot ሶ119909 + 119904 120572119894 sdot ሶ119910




ሶ120595119894 =1


rrlabcsuni-klde



rrlabcsuni-klde

Mecanum Vehicle





Mecanum Kinematics



1205721 = 45 deg 1205731 = 45 deg 1205741 = minus45 deg

1205722 = 135 deg 1205732 = minus45 deg 1205742 = 45 deg


1205724 = minus45 deg 1205734 = 135 deg 1205744 = 45 deg


Mecanum Kinematics








Mecanum Kinematics


ሶ119909 =119903

4ሶ1205951 + ሶ1205952 + ሶ1205953 + ሶ1205954

ሶ119910 =119903

4minus ሶ1205951 + ሶ1205952 minus ሶ1205953 + ሶ1205954

ሶ120579 =119903

119889 2ሶ1205951 minus ሶ1205952 minus ሶ1205953 + ሶ1205954


Mecanum Kinematics


ሶ119909ሶ119910ሶ120579

=119903119908ℎ1198901198901198974


sdot

ሶ1205951

ሶ1205952

ሶ1205953

ሶ1205954

with a =2 2

119889


ሶ1205954 = ሶ1205951 + ሶ1205952 minus ሶ1205953

rrlabcsuni-klde





Ability to maneuver

Controllability

Stability



Stability




Typical setups




Maneuverability








Controllability



Examples




Wheel type

Drive geometry


Motortransmission


Degree of Mobility




119891119898 = 3 minus rank 119878


Degree of Mobility



1198891 = 1198892 1205731 = 1205732 = 0 1205721 + 120587 = 1205722






119891119904 = rank 119878prime








119891119872 = 119891119898 + 119891119904

rrlabcsuni-klde

Vehicle Kinematics



Tricycle Drive

Given 120575 119907


119903 = 119897 sdot cot 120575

119907prime =119907

cos 120575

119907119897 = 119907 sdot119903 + 119889

119903

119907119903 = 119907 sdot119903 minus 119889


120575


Ackermann Steering

Given 119907119863 120575 119897 119889

Unknown 119907119903119903 119907119897119903 119907119903119891 119907119897119891

Solution

120602 =119897

tan 120575

119907119897119903 =120602 minus 1198892 sdot 119907119863

120602

119907119903119903 =120602 + 1198892 sdot 119907119863

120602

119907119897119891 =120602 minus 1198892 2 + 1198972 sdot tan 120575

119897sdot 119907119863

119907119903119891 =1206022 + 1198972 sdot tan 120575

119897sdot 119907119863


120575

120602

rrlabcsuni-klde



Given 120602 120593 119907 119889



1205932 = arctan120549119910 + 119889

1205491199091206022 = 120602 sdot cos120593 cos1205932

1199072 = 119907 sdot 1206022120602

120549119910 = 120602 ∙ 119904119894119899 120593

1205931

1205932

1206021

1206022

120593 120549119909 = 120602 ∙ cos120593

120602

119889

rrlabcsuni-klde

Bicycle Model


119907


Bicycle Model


Low speed motion



119865119910 =1198981199072

120602






Bicycle Model

Given 120575119891 120575119903 119907 120579

Unknown ሶ119883 ሶ119884 ሶ120579

Solution

120575119907 = tanminus1119897119891 tan 120575119903+119897119903 tan 120575119891

119897119891+119897119903

ሶ120579 =119907

120602

120602 =119897119891+119897119903

tan 120575119891 minustan 120575119903 cos 120575119907

ሶ119883 = 119907 cos 120579 + 120575119907ሶ119884 = 119907 sin 120579 + 120575119907

ሶ120579 =119907 cos 120575119907

119897119891+119897119903tan 120575119891 minus tan 120575119903

119907

rrlabcsuni-klde



119862 = 119874 120575119903 = 0 120575119891 = 120575

119897119903 = 0 119897119891 = 119897

120575119907 = 0

ሶ120579 =119907

120602

120602 =119897

tan 120575119891

ሶ119883 = 119907 cos 120579ሶ119884 = 119907 sin 120579ሶ120579 =

119907

119897119891tan 120575119891

119907


Wheel Properties




Castor wheel


















ሶ120595 119903



ሶ120595 119903



ሶ120595 119903



ሶ120595 119903



ሶ120595 119903








and 1 Ԧ1199071 = ሶ119909 ሶ119910 0 119879


119894+1120596119894+1 = 119894119894+1119877 sdot 119894120596119894 + ሶ120579119894+1

119894+1 Ԧ119890119911119894+1

119894+1 Ԧ119907119894+1 = 119894119894+1119877 119894 Ԧ119907119894 +

119894120596119894 times119894 Ԧ119901119894+1




calculated as

2 Ԧ1199072 =119888120572 ሶ119909 + 119904120572 ሶ119910minus119904120572 ሶ119909 + 119888120572 ሶ119910

0


3 Ԧ1199073 =

119888120572 ሶ119909 + 119904120572 ሶ119910

minus119904120572 ሶ119909 + 119888120572 ሶ119910 + 119889 ሶ1205790






4 Ԧ1199074 =119904 120572 + 120573 ሶ119909 minus 119888 120572 + 120573 ሶ119910 minus 119888120573 119889 ሶ120579

119888 120572 + 120573 ሶ119909 + 119904 120572 + 120573 ሶ119910 + 119904120573 119889 ሶ1205790




119904 120572 + 120573 minus119888 120572 + 120573 minus119888120573 119889

119888 120572 + 120573 119904 120572 + 120573 119904120573 1198890 0 0

ሶ119909119910 ሶሶ120579

=119903 ሶ12059500












119904 120572 + 120573(119905) minus119888 120572 + 120573(119905) minus119888120573(119905)119889

119888 120572 + 120573(119905) 119904 120572 + 120573(119905) 119904120573(119905)1198890 0 0

ሶ119909ሶ119910ሶ120579

=119903 ሶ120595

minus119889119888 ሶ120573(119905)0


maneuverability)




119904 120572 + 120573 + 120574 minus119888 120572 + 120573 + 120574 minus119888 120573 + 120574 119889

119888 120572 + 120573 + 120574 119904 120572 + 120573 + 120574 119904 120573 + 120574 1198890 0 0

ሶ119909ሶ119910ሶ120579

=119903 ሶ120595 cos 120574

119903 ሶ120595 sin 120574 + 119903119901119903 ሶ1205951199011199030















120572119897 = 90 deg

120573119897 = 0 deg

120572119903 = minus90 deg

120573119903 = 180 deg




119904 120572119897 + 120573119897 ሶ119909 minus 119888 120572119897 + 120573119897 ሶ119910 minus 119888120573119897 119889 ሶ120579 = 119903119897 ሶ120595119897

119888 120572119897 + 120573119897 ሶ119909 + 119904 120572119897 + 120573119897 ሶ119910 + 119904120573119897 119889 ሶ120579 = 0



ሶ119909 minus 119889 ሶ120579 = 119903119897 ሶ120595119897

ሶ119910 = 0

ሶ119909 + 119889 ሶ120579 = 119903119903 ሶ120595119903119910 = 0




ሶ119909ሶ119910ሶ120579

=

1

2119903119897 ሶ120595119897 + 119903119903 ሶ120595119903

01

2119889minus119903119897 ሶ120595119897 + 119903119903 ሶ120595119903



rrlabcsuni-klde




rrlabcsuni-klde











1205721 = 0 deg

1205722 = 120 deg

1205723 = minus120 deg







119888 1205721 + 1205731 ሶ119909 + 119904 1205721 + 1205731 ሶ119910 + 119889 sdot 119904 1205731 ሶ120579 = 0

119888 1205722 + 1205732 ሶ119909 + 119904 1205722 + 1205732 ሶ119910 + 119889 sdot 119904 1205732 ሶ120579 = 0

119888 1205723 + 1205733 ሶ119909 + 119904 1205723 + 1205733 ሶ119910 + 119889 sdot 119904 1205733 ሶ120579 = 0







119888 120572119894 sdot ሶ119909 + 119904 120572119894 sdot ሶ119910




ሶ120595119894 =1


rrlabcsuni-klde



rrlabcsuni-klde

Mecanum Vehicle





Mecanum Kinematics



1205721 = 45 deg 1205731 = 45 deg 1205741 = minus45 deg

1205722 = 135 deg 1205732 = minus45 deg 1205742 = 45 deg


1205724 = minus45 deg 1205734 = 135 deg 1205744 = 45 deg


Mecanum Kinematics








Mecanum Kinematics


ሶ119909 =119903

4ሶ1205951 + ሶ1205952 + ሶ1205953 + ሶ1205954

ሶ119910 =119903

4minus ሶ1205951 + ሶ1205952 minus ሶ1205953 + ሶ1205954

ሶ120579 =119903

119889 2ሶ1205951 minus ሶ1205952 minus ሶ1205953 + ሶ1205954


Mecanum Kinematics


ሶ119909ሶ119910ሶ120579

=119903119908ℎ1198901198901198974


sdot

ሶ1205951

ሶ1205952

ሶ1205953

ሶ1205954

with a =2 2

119889


ሶ1205954 = ሶ1205951 + ሶ1205952 minus ሶ1205953

rrlabcsuni-klde





Ability to maneuver

Controllability

Stability



Stability




Typical setups




Maneuverability








Controllability



Examples




Wheel type

Drive geometry


Motortransmission


Degree of Mobility




119891119898 = 3 minus rank 119878


Degree of Mobility



1198891 = 1198892 1205731 = 1205732 = 0 1205721 + 120587 = 1205722






119891119904 = rank 119878prime








119891119872 = 119891119898 + 119891119904

rrlabcsuni-klde

Vehicle Kinematics



Tricycle Drive

Given 120575 119907


119903 = 119897 sdot cot 120575

119907prime =119907

cos 120575

119907119897 = 119907 sdot119903 + 119889

119903

119907119903 = 119907 sdot119903 minus 119889


120575


Ackermann Steering

Given 119907119863 120575 119897 119889

Unknown 119907119903119903 119907119897119903 119907119903119891 119907119897119891

Solution

120602 =119897

tan 120575

119907119897119903 =120602 minus 1198892 sdot 119907119863

120602

119907119903119903 =120602 + 1198892 sdot 119907119863

120602

119907119897119891 =120602 minus 1198892 2 + 1198972 sdot tan 120575

119897sdot 119907119863

119907119903119891 =1206022 + 1198972 sdot tan 120575

119897sdot 119907119863


120575

120602

rrlabcsuni-klde



Given 120602 120593 119907 119889



1205932 = arctan120549119910 + 119889

1205491199091206022 = 120602 sdot cos120593 cos1205932

1199072 = 119907 sdot 1206022120602

120549119910 = 120602 ∙ 119904119894119899 120593

1205931

1205932

1206021

1206022

120593 120549119909 = 120602 ∙ cos120593

120602

119889

rrlabcsuni-klde

Bicycle Model


119907


Bicycle Model


Low speed motion



119865119910 =1198981199072

120602






Bicycle Model

Given 120575119891 120575119903 119907 120579

Unknown ሶ119883 ሶ119884 ሶ120579

Solution

120575119907 = tanminus1119897119891 tan 120575119903+119897119903 tan 120575119891

119897119891+119897119903

ሶ120579 =119907

120602

120602 =119897119891+119897119903

tan 120575119891 minustan 120575119903 cos 120575119907

ሶ119883 = 119907 cos 120579 + 120575119907ሶ119884 = 119907 sin 120579 + 120575119907

ሶ120579 =119907 cos 120575119907

119897119891+119897119903tan 120575119891 minus tan 120575119903

119907

rrlabcsuni-klde



119862 = 119874 120575119903 = 0 120575119891 = 120575

119897119903 = 0 119897119891 = 119897

120575119907 = 0

ሶ120579 =119907

120602

120602 =119897

tan 120575119891

ሶ119883 = 119907 cos 120579ሶ119884 = 119907 sin 120579ሶ120579 =

119907

119897119891tan 120575119891

119907












ሶ120595 119903



ሶ120595 119903



ሶ120595 119903



ሶ120595 119903



ሶ120595 119903








and 1 Ԧ1199071 = ሶ119909 ሶ119910 0 119879


119894+1120596119894+1 = 119894119894+1119877 sdot 119894120596119894 + ሶ120579119894+1

119894+1 Ԧ119890119911119894+1

119894+1 Ԧ119907119894+1 = 119894119894+1119877 119894 Ԧ119907119894 +

119894120596119894 times119894 Ԧ119901119894+1




calculated as

2 Ԧ1199072 =119888120572 ሶ119909 + 119904120572 ሶ119910minus119904120572 ሶ119909 + 119888120572 ሶ119910

0


3 Ԧ1199073 =

119888120572 ሶ119909 + 119904120572 ሶ119910

minus119904120572 ሶ119909 + 119888120572 ሶ119910 + 119889 ሶ1205790






4 Ԧ1199074 =119904 120572 + 120573 ሶ119909 minus 119888 120572 + 120573 ሶ119910 minus 119888120573 119889 ሶ120579

119888 120572 + 120573 ሶ119909 + 119904 120572 + 120573 ሶ119910 + 119904120573 119889 ሶ1205790




119904 120572 + 120573 minus119888 120572 + 120573 minus119888120573 119889

119888 120572 + 120573 119904 120572 + 120573 119904120573 1198890 0 0

ሶ119909119910 ሶሶ120579

=119903 ሶ12059500












119904 120572 + 120573(119905) minus119888 120572 + 120573(119905) minus119888120573(119905)119889

119888 120572 + 120573(119905) 119904 120572 + 120573(119905) 119904120573(119905)1198890 0 0

ሶ119909ሶ119910ሶ120579

=119903 ሶ120595

minus119889119888 ሶ120573(119905)0


maneuverability)




119904 120572 + 120573 + 120574 minus119888 120572 + 120573 + 120574 minus119888 120573 + 120574 119889

119888 120572 + 120573 + 120574 119904 120572 + 120573 + 120574 119904 120573 + 120574 1198890 0 0

ሶ119909ሶ119910ሶ120579

=119903 ሶ120595 cos 120574

119903 ሶ120595 sin 120574 + 119903119901119903 ሶ1205951199011199030















120572119897 = 90 deg

120573119897 = 0 deg

120572119903 = minus90 deg

120573119903 = 180 deg




119904 120572119897 + 120573119897 ሶ119909 minus 119888 120572119897 + 120573119897 ሶ119910 minus 119888120573119897 119889 ሶ120579 = 119903119897 ሶ120595119897

119888 120572119897 + 120573119897 ሶ119909 + 119904 120572119897 + 120573119897 ሶ119910 + 119904120573119897 119889 ሶ120579 = 0



ሶ119909 minus 119889 ሶ120579 = 119903119897 ሶ120595119897

ሶ119910 = 0

ሶ119909 + 119889 ሶ120579 = 119903119903 ሶ120595119903119910 = 0




ሶ119909ሶ119910ሶ120579

=

1

2119903119897 ሶ120595119897 + 119903119903 ሶ120595119903

01

2119889minus119903119897 ሶ120595119897 + 119903119903 ሶ120595119903



rrlabcsuni-klde




rrlabcsuni-klde











1205721 = 0 deg

1205722 = 120 deg

1205723 = minus120 deg







119888 1205721 + 1205731 ሶ119909 + 119904 1205721 + 1205731 ሶ119910 + 119889 sdot 119904 1205731 ሶ120579 = 0

119888 1205722 + 1205732 ሶ119909 + 119904 1205722 + 1205732 ሶ119910 + 119889 sdot 119904 1205732 ሶ120579 = 0

119888 1205723 + 1205733 ሶ119909 + 119904 1205723 + 1205733 ሶ119910 + 119889 sdot 119904 1205733 ሶ120579 = 0







119888 120572119894 sdot ሶ119909 + 119904 120572119894 sdot ሶ119910




ሶ120595119894 =1


rrlabcsuni-klde



rrlabcsuni-klde

Mecanum Vehicle





Mecanum Kinematics



1205721 = 45 deg 1205731 = 45 deg 1205741 = minus45 deg

1205722 = 135 deg 1205732 = minus45 deg 1205742 = 45 deg


1205724 = minus45 deg 1205734 = 135 deg 1205744 = 45 deg


Mecanum Kinematics








Mecanum Kinematics


ሶ119909 =119903

4ሶ1205951 + ሶ1205952 + ሶ1205953 + ሶ1205954

ሶ119910 =119903

4minus ሶ1205951 + ሶ1205952 minus ሶ1205953 + ሶ1205954

ሶ120579 =119903

119889 2ሶ1205951 minus ሶ1205952 minus ሶ1205953 + ሶ1205954


Mecanum Kinematics


ሶ119909ሶ119910ሶ120579

=119903119908ℎ1198901198901198974


sdot

ሶ1205951

ሶ1205952

ሶ1205953

ሶ1205954

with a =2 2

119889


ሶ1205954 = ሶ1205951 + ሶ1205952 minus ሶ1205953

rrlabcsuni-klde





Ability to maneuver

Controllability

Stability



Stability




Typical setups




Maneuverability








Controllability



Examples




Wheel type

Drive geometry


Motortransmission


Degree of Mobility




119891119898 = 3 minus rank 119878


Degree of Mobility



1198891 = 1198892 1205731 = 1205732 = 0 1205721 + 120587 = 1205722






119891119904 = rank 119878prime








119891119872 = 119891119898 + 119891119904

rrlabcsuni-klde

Vehicle Kinematics



Tricycle Drive

Given 120575 119907


119903 = 119897 sdot cot 120575

119907prime =119907

cos 120575

119907119897 = 119907 sdot119903 + 119889

119903

119907119903 = 119907 sdot119903 minus 119889


120575


Ackermann Steering

Given 119907119863 120575 119897 119889

Unknown 119907119903119903 119907119897119903 119907119903119891 119907119897119891

Solution

120602 =119897

tan 120575

119907119897119903 =120602 minus 1198892 sdot 119907119863

120602

119907119903119903 =120602 + 1198892 sdot 119907119863

120602

119907119897119891 =120602 minus 1198892 2 + 1198972 sdot tan 120575

119897sdot 119907119863

119907119903119891 =1206022 + 1198972 sdot tan 120575

119897sdot 119907119863


120575

120602

rrlabcsuni-klde



Given 120602 120593 119907 119889



1205932 = arctan120549119910 + 119889

1205491199091206022 = 120602 sdot cos120593 cos1205932

1199072 = 119907 sdot 1206022120602

120549119910 = 120602 ∙ 119904119894119899 120593

1205931

1205932

1206021

1206022

120593 120549119909 = 120602 ∙ cos120593

120602

119889

rrlabcsuni-klde

Bicycle Model


119907


Bicycle Model


Low speed motion



119865119910 =1198981199072

120602






Bicycle Model

Given 120575119891 120575119903 119907 120579

Unknown ሶ119883 ሶ119884 ሶ120579

Solution

120575119907 = tanminus1119897119891 tan 120575119903+119897119903 tan 120575119891

119897119891+119897119903

ሶ120579 =119907

120602

120602 =119897119891+119897119903

tan 120575119891 minustan 120575119903 cos 120575119907

ሶ119883 = 119907 cos 120579 + 120575119907ሶ119884 = 119907 sin 120579 + 120575119907

ሶ120579 =119907 cos 120575119907

119897119891+119897119903tan 120575119891 minus tan 120575119903

119907

rrlabcsuni-klde



119862 = 119874 120575119903 = 0 120575119891 = 120575

119897119903 = 0 119897119891 = 119897

120575119907 = 0

ሶ120579 =119907

120602

120602 =119897

tan 120575119891

ሶ119883 = 119907 cos 120579ሶ119884 = 119907 sin 120579ሶ120579 =

119907

119897119891tan 120575119891

119907










ሶ120595 119903



ሶ120595 119903



ሶ120595 119903



ሶ120595 119903



ሶ120595 119903








and 1 Ԧ1199071 = ሶ119909 ሶ119910 0 119879


119894+1120596119894+1 = 119894119894+1119877 sdot 119894120596119894 + ሶ120579119894+1

119894+1 Ԧ119890119911119894+1

119894+1 Ԧ119907119894+1 = 119894119894+1119877 119894 Ԧ119907119894 +

119894120596119894 times119894 Ԧ119901119894+1




calculated as

2 Ԧ1199072 =119888120572 ሶ119909 + 119904120572 ሶ119910minus119904120572 ሶ119909 + 119888120572 ሶ119910

0


3 Ԧ1199073 =

119888120572 ሶ119909 + 119904120572 ሶ119910

minus119904120572 ሶ119909 + 119888120572 ሶ119910 + 119889 ሶ1205790






4 Ԧ1199074 =119904 120572 + 120573 ሶ119909 minus 119888 120572 + 120573 ሶ119910 minus 119888120573 119889 ሶ120579

119888 120572 + 120573 ሶ119909 + 119904 120572 + 120573 ሶ119910 + 119904120573 119889 ሶ1205790




119904 120572 + 120573 minus119888 120572 + 120573 minus119888120573 119889

119888 120572 + 120573 119904 120572 + 120573 119904120573 1198890 0 0

ሶ119909119910 ሶሶ120579

=119903 ሶ12059500












119904 120572 + 120573(119905) minus119888 120572 + 120573(119905) minus119888120573(119905)119889

119888 120572 + 120573(119905) 119904 120572 + 120573(119905) 119904120573(119905)1198890 0 0

ሶ119909ሶ119910ሶ120579

=119903 ሶ120595

minus119889119888 ሶ120573(119905)0


maneuverability)




119904 120572 + 120573 + 120574 minus119888 120572 + 120573 + 120574 minus119888 120573 + 120574 119889

119888 120572 + 120573 + 120574 119904 120572 + 120573 + 120574 119904 120573 + 120574 1198890 0 0

ሶ119909ሶ119910ሶ120579

=119903 ሶ120595 cos 120574

119903 ሶ120595 sin 120574 + 119903119901119903 ሶ1205951199011199030















120572119897 = 90 deg

120573119897 = 0 deg

120572119903 = minus90 deg

120573119903 = 180 deg




119904 120572119897 + 120573119897 ሶ119909 minus 119888 120572119897 + 120573119897 ሶ119910 minus 119888120573119897 119889 ሶ120579 = 119903119897 ሶ120595119897

119888 120572119897 + 120573119897 ሶ119909 + 119904 120572119897 + 120573119897 ሶ119910 + 119904120573119897 119889 ሶ120579 = 0



ሶ119909 minus 119889 ሶ120579 = 119903119897 ሶ120595119897

ሶ119910 = 0

ሶ119909 + 119889 ሶ120579 = 119903119903 ሶ120595119903119910 = 0




ሶ119909ሶ119910ሶ120579

=

1

2119903119897 ሶ120595119897 + 119903119903 ሶ120595119903

01

2119889minus119903119897 ሶ120595119897 + 119903119903 ሶ120595119903



rrlabcsuni-klde




rrlabcsuni-klde











1205721 = 0 deg

1205722 = 120 deg

1205723 = minus120 deg







119888 1205721 + 1205731 ሶ119909 + 119904 1205721 + 1205731 ሶ119910 + 119889 sdot 119904 1205731 ሶ120579 = 0

119888 1205722 + 1205732 ሶ119909 + 119904 1205722 + 1205732 ሶ119910 + 119889 sdot 119904 1205732 ሶ120579 = 0

119888 1205723 + 1205733 ሶ119909 + 119904 1205723 + 1205733 ሶ119910 + 119889 sdot 119904 1205733 ሶ120579 = 0







119888 120572119894 sdot ሶ119909 + 119904 120572119894 sdot ሶ119910




ሶ120595119894 =1


rrlabcsuni-klde



rrlabcsuni-klde

Mecanum Vehicle





Mecanum Kinematics



1205721 = 45 deg 1205731 = 45 deg 1205741 = minus45 deg

1205722 = 135 deg 1205732 = minus45 deg 1205742 = 45 deg


1205724 = minus45 deg 1205734 = 135 deg 1205744 = 45 deg


Mecanum Kinematics








Mecanum Kinematics


ሶ119909 =119903

4ሶ1205951 + ሶ1205952 + ሶ1205953 + ሶ1205954

ሶ119910 =119903

4minus ሶ1205951 + ሶ1205952 minus ሶ1205953 + ሶ1205954

ሶ120579 =119903

119889 2ሶ1205951 minus ሶ1205952 minus ሶ1205953 + ሶ1205954


Mecanum Kinematics


ሶ119909ሶ119910ሶ120579

=119903119908ℎ1198901198901198974


sdot

ሶ1205951

ሶ1205952

ሶ1205953

ሶ1205954

with a =2 2

119889


ሶ1205954 = ሶ1205951 + ሶ1205952 minus ሶ1205953

rrlabcsuni-klde





Ability to maneuver

Controllability

Stability



Stability




Typical setups




Maneuverability








Controllability



Examples




Wheel type

Drive geometry


Motortransmission


Degree of Mobility




119891119898 = 3 minus rank 119878


Degree of Mobility



1198891 = 1198892 1205731 = 1205732 = 0 1205721 + 120587 = 1205722






119891119904 = rank 119878prime








119891119872 = 119891119898 + 119891119904

rrlabcsuni-klde

Vehicle Kinematics



Tricycle Drive

Given 120575 119907


119903 = 119897 sdot cot 120575

119907prime =119907

cos 120575

119907119897 = 119907 sdot119903 + 119889

119903

119907119903 = 119907 sdot119903 minus 119889


120575


Ackermann Steering

Given 119907119863 120575 119897 119889

Unknown 119907119903119903 119907119897119903 119907119903119891 119907119897119891

Solution

120602 =119897

tan 120575

119907119897119903 =120602 minus 1198892 sdot 119907119863

120602

119907119903119903 =120602 + 1198892 sdot 119907119863

120602

119907119897119891 =120602 minus 1198892 2 + 1198972 sdot tan 120575

119897sdot 119907119863

119907119903119891 =1206022 + 1198972 sdot tan 120575

119897sdot 119907119863


120575

120602

rrlabcsuni-klde



Given 120602 120593 119907 119889



1205932 = arctan120549119910 + 119889

1205491199091206022 = 120602 sdot cos120593 cos1205932

1199072 = 119907 sdot 1206022120602

120549119910 = 120602 ∙ 119904119894119899 120593

1205931

1205932

1206021

1206022

120593 120549119909 = 120602 ∙ cos120593

120602

119889

rrlabcsuni-klde

Bicycle Model


119907


Bicycle Model


Low speed motion



119865119910 =1198981199072

120602






Bicycle Model

Given 120575119891 120575119903 119907 120579

Unknown ሶ119883 ሶ119884 ሶ120579

Solution

120575119907 = tanminus1119897119891 tan 120575119903+119897119903 tan 120575119891

119897119891+119897119903

ሶ120579 =119907

120602

120602 =119897119891+119897119903

tan 120575119891 minustan 120575119903 cos 120575119907

ሶ119883 = 119907 cos 120579 + 120575119907ሶ119884 = 119907 sin 120579 + 120575119907

ሶ120579 =119907 cos 120575119907

119897119891+119897119903tan 120575119891 minus tan 120575119903

119907

rrlabcsuni-klde



119862 = 119874 120575119903 = 0 120575119891 = 120575

119897119903 = 0 119897119891 = 119897

120575119907 = 0

ሶ120579 =119907

120602

120602 =119897

tan 120575119891

ሶ119883 = 119907 cos 120579ሶ119884 = 119907 sin 120579ሶ120579 =

119907

119897119891tan 120575119891

119907



ሶ120595 119903



ሶ120595 119903



ሶ120595 119903



ሶ120595 119903



ሶ120595 119903








and 1 Ԧ1199071 = ሶ119909 ሶ119910 0 119879


119894+1120596119894+1 = 119894119894+1119877 sdot 119894120596119894 + ሶ120579119894+1

119894+1 Ԧ119890119911119894+1

119894+1 Ԧ119907119894+1 = 119894119894+1119877 119894 Ԧ119907119894 +

119894120596119894 times119894 Ԧ119901119894+1




calculated as

2 Ԧ1199072 =119888120572 ሶ119909 + 119904120572 ሶ119910minus119904120572 ሶ119909 + 119888120572 ሶ119910

0


3 Ԧ1199073 =

119888120572 ሶ119909 + 119904120572 ሶ119910

minus119904120572 ሶ119909 + 119888120572 ሶ119910 + 119889 ሶ1205790






4 Ԧ1199074 =119904 120572 + 120573 ሶ119909 minus 119888 120572 + 120573 ሶ119910 minus 119888120573 119889 ሶ120579

119888 120572 + 120573 ሶ119909 + 119904 120572 + 120573 ሶ119910 + 119904120573 119889 ሶ1205790




119904 120572 + 120573 minus119888 120572 + 120573 minus119888120573 119889

119888 120572 + 120573 119904 120572 + 120573 119904120573 1198890 0 0

ሶ119909119910 ሶሶ120579

=119903 ሶ12059500












119904 120572 + 120573(119905) minus119888 120572 + 120573(119905) minus119888120573(119905)119889

119888 120572 + 120573(119905) 119904 120572 + 120573(119905) 119904120573(119905)1198890 0 0

ሶ119909ሶ119910ሶ120579

=119903 ሶ120595

minus119889119888 ሶ120573(119905)0


maneuverability)




119904 120572 + 120573 + 120574 minus119888 120572 + 120573 + 120574 minus119888 120573 + 120574 119889

119888 120572 + 120573 + 120574 119904 120572 + 120573 + 120574 119904 120573 + 120574 1198890 0 0

ሶ119909ሶ119910ሶ120579

=119903 ሶ120595 cos 120574

119903 ሶ120595 sin 120574 + 119903119901119903 ሶ1205951199011199030















120572119897 = 90 deg

120573119897 = 0 deg

120572119903 = minus90 deg

120573119903 = 180 deg




119904 120572119897 + 120573119897 ሶ119909 minus 119888 120572119897 + 120573119897 ሶ119910 minus 119888120573119897 119889 ሶ120579 = 119903119897 ሶ120595119897

119888 120572119897 + 120573119897 ሶ119909 + 119904 120572119897 + 120573119897 ሶ119910 + 119904120573119897 119889 ሶ120579 = 0



ሶ119909 minus 119889 ሶ120579 = 119903119897 ሶ120595119897

ሶ119910 = 0

ሶ119909 + 119889 ሶ120579 = 119903119903 ሶ120595119903119910 = 0




ሶ119909ሶ119910ሶ120579

=

1

2119903119897 ሶ120595119897 + 119903119903 ሶ120595119903

01

2119889minus119903119897 ሶ120595119897 + 119903119903 ሶ120595119903



rrlabcsuni-klde




rrlabcsuni-klde











1205721 = 0 deg

1205722 = 120 deg

1205723 = minus120 deg







119888 1205721 + 1205731 ሶ119909 + 119904 1205721 + 1205731 ሶ119910 + 119889 sdot 119904 1205731 ሶ120579 = 0

119888 1205722 + 1205732 ሶ119909 + 119904 1205722 + 1205732 ሶ119910 + 119889 sdot 119904 1205732 ሶ120579 = 0

119888 1205723 + 1205733 ሶ119909 + 119904 1205723 + 1205733 ሶ119910 + 119889 sdot 119904 1205733 ሶ120579 = 0







119888 120572119894 sdot ሶ119909 + 119904 120572119894 sdot ሶ119910




ሶ120595119894 =1


rrlabcsuni-klde



rrlabcsuni-klde

Mecanum Vehicle





Mecanum Kinematics



1205721 = 45 deg 1205731 = 45 deg 1205741 = minus45 deg

1205722 = 135 deg 1205732 = minus45 deg 1205742 = 45 deg


1205724 = minus45 deg 1205734 = 135 deg 1205744 = 45 deg


Mecanum Kinematics








Mecanum Kinematics


ሶ119909 =119903

4ሶ1205951 + ሶ1205952 + ሶ1205953 + ሶ1205954

ሶ119910 =119903

4minus ሶ1205951 + ሶ1205952 minus ሶ1205953 + ሶ1205954

ሶ120579 =119903

119889 2ሶ1205951 minus ሶ1205952 minus ሶ1205953 + ሶ1205954


Mecanum Kinematics


ሶ119909ሶ119910ሶ120579

=119903119908ℎ1198901198901198974


sdot

ሶ1205951

ሶ1205952

ሶ1205953

ሶ1205954

with a =2 2

119889


ሶ1205954 = ሶ1205951 + ሶ1205952 minus ሶ1205953

rrlabcsuni-klde





Ability to maneuver

Controllability

Stability



Stability




Typical setups




Maneuverability








Controllability



Examples




Wheel type

Drive geometry


Motortransmission


Degree of Mobility




119891119898 = 3 minus rank 119878


Degree of Mobility



1198891 = 1198892 1205731 = 1205732 = 0 1205721 + 120587 = 1205722






119891119904 = rank 119878prime








119891119872 = 119891119898 + 119891119904

rrlabcsuni-klde

Vehicle Kinematics



Tricycle Drive

Given 120575 119907


119903 = 119897 sdot cot 120575

119907prime =119907

cos 120575

119907119897 = 119907 sdot119903 + 119889

119903

119907119903 = 119907 sdot119903 minus 119889


120575


Ackermann Steering

Given 119907119863 120575 119897 119889

Unknown 119907119903119903 119907119897119903 119907119903119891 119907119897119891

Solution

120602 =119897

tan 120575

119907119897119903 =120602 minus 1198892 sdot 119907119863

120602

119907119903119903 =120602 + 1198892 sdot 119907119863

120602

119907119897119891 =120602 minus 1198892 2 + 1198972 sdot tan 120575

119897sdot 119907119863

119907119903119891 =1206022 + 1198972 sdot tan 120575

119897sdot 119907119863


120575

120602

rrlabcsuni-klde



Given 120602 120593 119907 119889



1205932 = arctan120549119910 + 119889

1205491199091206022 = 120602 sdot cos120593 cos1205932

1199072 = 119907 sdot 1206022120602

120549119910 = 120602 ∙ 119904119894119899 120593

1205931

1205932

1206021

1206022

120593 120549119909 = 120602 ∙ cos120593

120602

119889

rrlabcsuni-klde

Bicycle Model


119907


Bicycle Model


Low speed motion



119865119910 =1198981199072

120602






Bicycle Model

Given 120575119891 120575119903 119907 120579

Unknown ሶ119883 ሶ119884 ሶ120579

Solution

120575119907 = tanminus1119897119891 tan 120575119903+119897119903 tan 120575119891

119897119891+119897119903

ሶ120579 =119907

120602

120602 =119897119891+119897119903

tan 120575119891 minustan 120575119903 cos 120575119907

ሶ119883 = 119907 cos 120579 + 120575119907ሶ119884 = 119907 sin 120579 + 120575119907

ሶ120579 =119907 cos 120575119907

119897119891+119897119903tan 120575119891 minus tan 120575119903

119907

rrlabcsuni-klde



119862 = 119874 120575119903 = 0 120575119891 = 120575

119897119903 = 0 119897119891 = 119897

120575119907 = 0

ሶ120579 =119907

120602

120602 =119897

tan 120575119891

ሶ119883 = 119907 cos 120579ሶ119884 = 119907 sin 120579ሶ120579 =

119907

119897119891tan 120575119891

119907



ሶ120595 119903



ሶ120595 119903



ሶ120595 119903



ሶ120595 119903








and 1 Ԧ1199071 = ሶ119909 ሶ119910 0 119879


119894+1120596119894+1 = 119894119894+1119877 sdot 119894120596119894 + ሶ120579119894+1

119894+1 Ԧ119890119911119894+1

119894+1 Ԧ119907119894+1 = 119894119894+1119877 119894 Ԧ119907119894 +

119894120596119894 times119894 Ԧ119901119894+1




calculated as

2 Ԧ1199072 =119888120572 ሶ119909 + 119904120572 ሶ119910minus119904120572 ሶ119909 + 119888120572 ሶ119910

0


3 Ԧ1199073 =

119888120572 ሶ119909 + 119904120572 ሶ119910

minus119904120572 ሶ119909 + 119888120572 ሶ119910 + 119889 ሶ1205790






4 Ԧ1199074 =119904 120572 + 120573 ሶ119909 minus 119888 120572 + 120573 ሶ119910 minus 119888120573 119889 ሶ120579

119888 120572 + 120573 ሶ119909 + 119904 120572 + 120573 ሶ119910 + 119904120573 119889 ሶ1205790




119904 120572 + 120573 minus119888 120572 + 120573 minus119888120573 119889

119888 120572 + 120573 119904 120572 + 120573 119904120573 1198890 0 0

ሶ119909119910 ሶሶ120579

=119903 ሶ12059500












119904 120572 + 120573(119905) minus119888 120572 + 120573(119905) minus119888120573(119905)119889

119888 120572 + 120573(119905) 119904 120572 + 120573(119905) 119904120573(119905)1198890 0 0

ሶ119909ሶ119910ሶ120579

=119903 ሶ120595

minus119889119888 ሶ120573(119905)0


maneuverability)




119904 120572 + 120573 + 120574 minus119888 120572 + 120573 + 120574 minus119888 120573 + 120574 119889

119888 120572 + 120573 + 120574 119904 120572 + 120573 + 120574 119904 120573 + 120574 1198890 0 0

ሶ119909ሶ119910ሶ120579

=119903 ሶ120595 cos 120574

119903 ሶ120595 sin 120574 + 119903119901119903 ሶ1205951199011199030















120572119897 = 90 deg

120573119897 = 0 deg

120572119903 = minus90 deg

120573119903 = 180 deg




119904 120572119897 + 120573119897 ሶ119909 minus 119888 120572119897 + 120573119897 ሶ119910 minus 119888120573119897 119889 ሶ120579 = 119903119897 ሶ120595119897

119888 120572119897 + 120573119897 ሶ119909 + 119904 120572119897 + 120573119897 ሶ119910 + 119904120573119897 119889 ሶ120579 = 0



ሶ119909 minus 119889 ሶ120579 = 119903119897 ሶ120595119897

ሶ119910 = 0

ሶ119909 + 119889 ሶ120579 = 119903119903 ሶ120595119903119910 = 0




ሶ119909ሶ119910ሶ120579

=

1

2119903119897 ሶ120595119897 + 119903119903 ሶ120595119903

01

2119889minus119903119897 ሶ120595119897 + 119903119903 ሶ120595119903



rrlabcsuni-klde




rrlabcsuni-klde











1205721 = 0 deg

1205722 = 120 deg

1205723 = minus120 deg







119888 1205721 + 1205731 ሶ119909 + 119904 1205721 + 1205731 ሶ119910 + 119889 sdot 119904 1205731 ሶ120579 = 0

119888 1205722 + 1205732 ሶ119909 + 119904 1205722 + 1205732 ሶ119910 + 119889 sdot 119904 1205732 ሶ120579 = 0

119888 1205723 + 1205733 ሶ119909 + 119904 1205723 + 1205733 ሶ119910 + 119889 sdot 119904 1205733 ሶ120579 = 0







119888 120572119894 sdot ሶ119909 + 119904 120572119894 sdot ሶ119910




ሶ120595119894 =1


rrlabcsuni-klde



rrlabcsuni-klde

Mecanum Vehicle





Mecanum Kinematics



1205721 = 45 deg 1205731 = 45 deg 1205741 = minus45 deg

1205722 = 135 deg 1205732 = minus45 deg 1205742 = 45 deg


1205724 = minus45 deg 1205734 = 135 deg 1205744 = 45 deg


Mecanum Kinematics








Mecanum Kinematics


ሶ119909 =119903

4ሶ1205951 + ሶ1205952 + ሶ1205953 + ሶ1205954

ሶ119910 =119903

4minus ሶ1205951 + ሶ1205952 minus ሶ1205953 + ሶ1205954

ሶ120579 =119903

119889 2ሶ1205951 minus ሶ1205952 minus ሶ1205953 + ሶ1205954


Mecanum Kinematics


ሶ119909ሶ119910ሶ120579

=119903119908ℎ1198901198901198974


sdot

ሶ1205951

ሶ1205952

ሶ1205953

ሶ1205954

with a =2 2

119889


ሶ1205954 = ሶ1205951 + ሶ1205952 minus ሶ1205953

rrlabcsuni-klde





Ability to maneuver

Controllability

Stability



Stability




Typical setups




Maneuverability








Controllability



Examples




Wheel type

Drive geometry


Motortransmission


Degree of Mobility




119891119898 = 3 minus rank 119878


Degree of Mobility



1198891 = 1198892 1205731 = 1205732 = 0 1205721 + 120587 = 1205722






119891119904 = rank 119878prime








119891119872 = 119891119898 + 119891119904

rrlabcsuni-klde

Vehicle Kinematics



Tricycle Drive

Given 120575 119907


119903 = 119897 sdot cot 120575

119907prime =119907

cos 120575

119907119897 = 119907 sdot119903 + 119889

119903

119907119903 = 119907 sdot119903 minus 119889


120575


Ackermann Steering

Given 119907119863 120575 119897 119889

Unknown 119907119903119903 119907119897119903 119907119903119891 119907119897119891

Solution

120602 =119897

tan 120575

119907119897119903 =120602 minus 1198892 sdot 119907119863

120602

119907119903119903 =120602 + 1198892 sdot 119907119863

120602

119907119897119891 =120602 minus 1198892 2 + 1198972 sdot tan 120575

119897sdot 119907119863

119907119903119891 =1206022 + 1198972 sdot tan 120575

119897sdot 119907119863


120575

120602

rrlabcsuni-klde



Given 120602 120593 119907 119889



1205932 = arctan120549119910 + 119889

1205491199091206022 = 120602 sdot cos120593 cos1205932

1199072 = 119907 sdot 1206022120602

120549119910 = 120602 ∙ 119904119894119899 120593

1205931

1205932

1206021

1206022

120593 120549119909 = 120602 ∙ cos120593

120602

119889

rrlabcsuni-klde

Bicycle Model


119907


Bicycle Model


Low speed motion



119865119910 =1198981199072

120602






Bicycle Model

Given 120575119891 120575119903 119907 120579

Unknown ሶ119883 ሶ119884 ሶ120579

Solution

120575119907 = tanminus1119897119891 tan 120575119903+119897119903 tan 120575119891

119897119891+119897119903

ሶ120579 =119907

120602

120602 =119897119891+119897119903

tan 120575119891 minustan 120575119903 cos 120575119907

ሶ119883 = 119907 cos 120579 + 120575119907ሶ119884 = 119907 sin 120579 + 120575119907

ሶ120579 =119907 cos 120575119907

119897119891+119897119903tan 120575119891 minus tan 120575119903

119907

rrlabcsuni-klde



119862 = 119874 120575119903 = 0 120575119891 = 120575

119897119903 = 0 119897119891 = 119897

120575119907 = 0

ሶ120579 =119907

120602

120602 =119897

tan 120575119891

ሶ119883 = 119907 cos 120579ሶ119884 = 119907 sin 120579ሶ120579 =

119907

119897119891tan 120575119891

119907



ሶ120595 119903



ሶ120595 119903



ሶ120595 119903








and 1 Ԧ1199071 = ሶ119909 ሶ119910 0 119879


119894+1120596119894+1 = 119894119894+1119877 sdot 119894120596119894 + ሶ120579119894+1

119894+1 Ԧ119890119911119894+1

119894+1 Ԧ119907119894+1 = 119894119894+1119877 119894 Ԧ119907119894 +

119894120596119894 times119894 Ԧ119901119894+1




calculated as

2 Ԧ1199072 =119888120572 ሶ119909 + 119904120572 ሶ119910minus119904120572 ሶ119909 + 119888120572 ሶ119910

0


3 Ԧ1199073 =

119888120572 ሶ119909 + 119904120572 ሶ119910

minus119904120572 ሶ119909 + 119888120572 ሶ119910 + 119889 ሶ1205790






4 Ԧ1199074 =119904 120572 + 120573 ሶ119909 minus 119888 120572 + 120573 ሶ119910 minus 119888120573 119889 ሶ120579

119888 120572 + 120573 ሶ119909 + 119904 120572 + 120573 ሶ119910 + 119904120573 119889 ሶ1205790




119904 120572 + 120573 minus119888 120572 + 120573 minus119888120573 119889

119888 120572 + 120573 119904 120572 + 120573 119904120573 1198890 0 0

ሶ119909119910 ሶሶ120579

=119903 ሶ12059500












119904 120572 + 120573(119905) minus119888 120572 + 120573(119905) minus119888120573(119905)119889

119888 120572 + 120573(119905) 119904 120572 + 120573(119905) 119904120573(119905)1198890 0 0

ሶ119909ሶ119910ሶ120579

=119903 ሶ120595

minus119889119888 ሶ120573(119905)0


maneuverability)




119904 120572 + 120573 + 120574 minus119888 120572 + 120573 + 120574 minus119888 120573 + 120574 119889

119888 120572 + 120573 + 120574 119904 120572 + 120573 + 120574 119904 120573 + 120574 1198890 0 0

ሶ119909ሶ119910ሶ120579

=119903 ሶ120595 cos 120574

119903 ሶ120595 sin 120574 + 119903119901119903 ሶ1205951199011199030















120572119897 = 90 deg

120573119897 = 0 deg

120572119903 = minus90 deg

120573119903 = 180 deg




119904 120572119897 + 120573119897 ሶ119909 minus 119888 120572119897 + 120573119897 ሶ119910 minus 119888120573119897 119889 ሶ120579 = 119903119897 ሶ120595119897

119888 120572119897 + 120573119897 ሶ119909 + 119904 120572119897 + 120573119897 ሶ119910 + 119904120573119897 119889 ሶ120579 = 0



ሶ119909 minus 119889 ሶ120579 = 119903119897 ሶ120595119897

ሶ119910 = 0

ሶ119909 + 119889 ሶ120579 = 119903119903 ሶ120595119903119910 = 0




ሶ119909ሶ119910ሶ120579

=

1

2119903119897 ሶ120595119897 + 119903119903 ሶ120595119903

01

2119889minus119903119897 ሶ120595119897 + 119903119903 ሶ120595119903



rrlabcsuni-klde




rrlabcsuni-klde











1205721 = 0 deg

1205722 = 120 deg

1205723 = minus120 deg







119888 1205721 + 1205731 ሶ119909 + 119904 1205721 + 1205731 ሶ119910 + 119889 sdot 119904 1205731 ሶ120579 = 0

119888 1205722 + 1205732 ሶ119909 + 119904 1205722 + 1205732 ሶ119910 + 119889 sdot 119904 1205732 ሶ120579 = 0

119888 1205723 + 1205733 ሶ119909 + 119904 1205723 + 1205733 ሶ119910 + 119889 sdot 119904 1205733 ሶ120579 = 0







119888 120572119894 sdot ሶ119909 + 119904 120572119894 sdot ሶ119910




ሶ120595119894 =1


rrlabcsuni-klde



rrlabcsuni-klde

Mecanum Vehicle





Mecanum Kinematics



1205721 = 45 deg 1205731 = 45 deg 1205741 = minus45 deg

1205722 = 135 deg 1205732 = minus45 deg 1205742 = 45 deg


1205724 = minus45 deg 1205734 = 135 deg 1205744 = 45 deg


Mecanum Kinematics








Mecanum Kinematics


ሶ119909 =119903

4ሶ1205951 + ሶ1205952 + ሶ1205953 + ሶ1205954

ሶ119910 =119903

4minus ሶ1205951 + ሶ1205952 minus ሶ1205953 + ሶ1205954

ሶ120579 =119903

119889 2ሶ1205951 minus ሶ1205952 minus ሶ1205953 + ሶ1205954


Mecanum Kinematics


ሶ119909ሶ119910ሶ120579

=119903119908ℎ1198901198901198974


sdot

ሶ1205951

ሶ1205952

ሶ1205953

ሶ1205954

with a =2 2

119889


ሶ1205954 = ሶ1205951 + ሶ1205952 minus ሶ1205953

rrlabcsuni-klde





Ability to maneuver

Controllability

Stability



Stability




Typical setups




Maneuverability








Controllability



Examples




Wheel type

Drive geometry


Motortransmission


Degree of Mobility




119891119898 = 3 minus rank 119878


Degree of Mobility



1198891 = 1198892 1205731 = 1205732 = 0 1205721 + 120587 = 1205722






119891119904 = rank 119878prime








119891119872 = 119891119898 + 119891119904

rrlabcsuni-klde

Vehicle Kinematics



Tricycle Drive

Given 120575 119907


119903 = 119897 sdot cot 120575

119907prime =119907

cos 120575

119907119897 = 119907 sdot119903 + 119889

119903

119907119903 = 119907 sdot119903 minus 119889


120575


Ackermann Steering

Given 119907119863 120575 119897 119889

Unknown 119907119903119903 119907119897119903 119907119903119891 119907119897119891

Solution

120602 =119897

tan 120575

119907119897119903 =120602 minus 1198892 sdot 119907119863

120602

119907119903119903 =120602 + 1198892 sdot 119907119863

120602

119907119897119891 =120602 minus 1198892 2 + 1198972 sdot tan 120575

119897sdot 119907119863

119907119903119891 =1206022 + 1198972 sdot tan 120575

119897sdot 119907119863


120575

120602

rrlabcsuni-klde



Given 120602 120593 119907 119889



1205932 = arctan120549119910 + 119889

1205491199091206022 = 120602 sdot cos120593 cos1205932

1199072 = 119907 sdot 1206022120602

120549119910 = 120602 ∙ 119904119894119899 120593

1205931

1205932

1206021

1206022

120593 120549119909 = 120602 ∙ cos120593

120602

119889

rrlabcsuni-klde

Bicycle Model


119907


Bicycle Model


Low speed motion



119865119910 =1198981199072

120602






Bicycle Model

Given 120575119891 120575119903 119907 120579

Unknown ሶ119883 ሶ119884 ሶ120579

Solution

120575119907 = tanminus1119897119891 tan 120575119903+119897119903 tan 120575119891

119897119891+119897119903

ሶ120579 =119907

120602

120602 =119897119891+119897119903

tan 120575119891 minustan 120575119903 cos 120575119907

ሶ119883 = 119907 cos 120579 + 120575119907ሶ119884 = 119907 sin 120579 + 120575119907

ሶ120579 =119907 cos 120575119907

119897119891+119897119903tan 120575119891 minus tan 120575119903

119907

rrlabcsuni-klde



119862 = 119874 120575119903 = 0 120575119891 = 120575

119897119903 = 0 119897119891 = 119897

120575119907 = 0

ሶ120579 =119907

120602

120602 =119897

tan 120575119891

ሶ119883 = 119907 cos 120579ሶ119884 = 119907 sin 120579ሶ120579 =

119907

119897119891tan 120575119891

119907



ሶ120595 119903



ሶ120595 119903








and 1 Ԧ1199071 = ሶ119909 ሶ119910 0 119879


119894+1120596119894+1 = 119894119894+1119877 sdot 119894120596119894 + ሶ120579119894+1

119894+1 Ԧ119890119911119894+1

119894+1 Ԧ119907119894+1 = 119894119894+1119877 119894 Ԧ119907119894 +

119894120596119894 times119894 Ԧ119901119894+1




calculated as

2 Ԧ1199072 =119888120572 ሶ119909 + 119904120572 ሶ119910minus119904120572 ሶ119909 + 119888120572 ሶ119910

0


3 Ԧ1199073 =

119888120572 ሶ119909 + 119904120572 ሶ119910

minus119904120572 ሶ119909 + 119888120572 ሶ119910 + 119889 ሶ1205790






4 Ԧ1199074 =119904 120572 + 120573 ሶ119909 minus 119888 120572 + 120573 ሶ119910 minus 119888120573 119889 ሶ120579

119888 120572 + 120573 ሶ119909 + 119904 120572 + 120573 ሶ119910 + 119904120573 119889 ሶ1205790




119904 120572 + 120573 minus119888 120572 + 120573 minus119888120573 119889

119888 120572 + 120573 119904 120572 + 120573 119904120573 1198890 0 0

ሶ119909119910 ሶሶ120579

=119903 ሶ12059500












119904 120572 + 120573(119905) minus119888 120572 + 120573(119905) minus119888120573(119905)119889

119888 120572 + 120573(119905) 119904 120572 + 120573(119905) 119904120573(119905)1198890 0 0

ሶ119909ሶ119910ሶ120579

=119903 ሶ120595

minus119889119888 ሶ120573(119905)0


maneuverability)




119904 120572 + 120573 + 120574 minus119888 120572 + 120573 + 120574 minus119888 120573 + 120574 119889

119888 120572 + 120573 + 120574 119904 120572 + 120573 + 120574 119904 120573 + 120574 1198890 0 0

ሶ119909ሶ119910ሶ120579

=119903 ሶ120595 cos 120574

119903 ሶ120595 sin 120574 + 119903119901119903 ሶ1205951199011199030















120572119897 = 90 deg

120573119897 = 0 deg

120572119903 = minus90 deg

120573119903 = 180 deg




119904 120572119897 + 120573119897 ሶ119909 minus 119888 120572119897 + 120573119897 ሶ119910 minus 119888120573119897 119889 ሶ120579 = 119903119897 ሶ120595119897

119888 120572119897 + 120573119897 ሶ119909 + 119904 120572119897 + 120573119897 ሶ119910 + 119904120573119897 119889 ሶ120579 = 0



ሶ119909 minus 119889 ሶ120579 = 119903119897 ሶ120595119897

ሶ119910 = 0

ሶ119909 + 119889 ሶ120579 = 119903119903 ሶ120595119903119910 = 0




ሶ119909ሶ119910ሶ120579

=

1

2119903119897 ሶ120595119897 + 119903119903 ሶ120595119903

01

2119889minus119903119897 ሶ120595119897 + 119903119903 ሶ120595119903



rrlabcsuni-klde




rrlabcsuni-klde











1205721 = 0 deg

1205722 = 120 deg

1205723 = minus120 deg







119888 1205721 + 1205731 ሶ119909 + 119904 1205721 + 1205731 ሶ119910 + 119889 sdot 119904 1205731 ሶ120579 = 0

119888 1205722 + 1205732 ሶ119909 + 119904 1205722 + 1205732 ሶ119910 + 119889 sdot 119904 1205732 ሶ120579 = 0

119888 1205723 + 1205733 ሶ119909 + 119904 1205723 + 1205733 ሶ119910 + 119889 sdot 119904 1205733 ሶ120579 = 0







119888 120572119894 sdot ሶ119909 + 119904 120572119894 sdot ሶ119910




ሶ120595119894 =1


rrlabcsuni-klde



rrlabcsuni-klde

Mecanum Vehicle





Mecanum Kinematics



1205721 = 45 deg 1205731 = 45 deg 1205741 = minus45 deg

1205722 = 135 deg 1205732 = minus45 deg 1205742 = 45 deg


1205724 = minus45 deg 1205734 = 135 deg 1205744 = 45 deg


Mecanum Kinematics








Mecanum Kinematics


ሶ119909 =119903

4ሶ1205951 + ሶ1205952 + ሶ1205953 + ሶ1205954

ሶ119910 =119903

4minus ሶ1205951 + ሶ1205952 minus ሶ1205953 + ሶ1205954

ሶ120579 =119903

119889 2ሶ1205951 minus ሶ1205952 minus ሶ1205953 + ሶ1205954


Mecanum Kinematics


ሶ119909ሶ119910ሶ120579

=119903119908ℎ1198901198901198974


sdot

ሶ1205951

ሶ1205952

ሶ1205953

ሶ1205954

with a =2 2

119889


ሶ1205954 = ሶ1205951 + ሶ1205952 minus ሶ1205953

rrlabcsuni-klde





Ability to maneuver

Controllability

Stability



Stability




Typical setups




Maneuverability








Controllability



Examples




Wheel type

Drive geometry


Motortransmission


Degree of Mobility




119891119898 = 3 minus rank 119878


Degree of Mobility



1198891 = 1198892 1205731 = 1205732 = 0 1205721 + 120587 = 1205722






119891119904 = rank 119878prime








119891119872 = 119891119898 + 119891119904

rrlabcsuni-klde

Vehicle Kinematics



Tricycle Drive

Given 120575 119907


119903 = 119897 sdot cot 120575

119907prime =119907

cos 120575

119907119897 = 119907 sdot119903 + 119889

119903

119907119903 = 119907 sdot119903 minus 119889


120575


Ackermann Steering

Given 119907119863 120575 119897 119889

Unknown 119907119903119903 119907119897119903 119907119903119891 119907119897119891

Solution

120602 =119897

tan 120575

119907119897119903 =120602 minus 1198892 sdot 119907119863

120602

119907119903119903 =120602 + 1198892 sdot 119907119863

120602

119907119897119891 =120602 minus 1198892 2 + 1198972 sdot tan 120575

119897sdot 119907119863

119907119903119891 =1206022 + 1198972 sdot tan 120575

119897sdot 119907119863


120575

120602

rrlabcsuni-klde



Given 120602 120593 119907 119889



1205932 = arctan120549119910 + 119889

1205491199091206022 = 120602 sdot cos120593 cos1205932

1199072 = 119907 sdot 1206022120602

120549119910 = 120602 ∙ 119904119894119899 120593

1205931

1205932

1206021

1206022

120593 120549119909 = 120602 ∙ cos120593

120602

119889

rrlabcsuni-klde

Bicycle Model


119907


Bicycle Model


Low speed motion



119865119910 =1198981199072

120602






Bicycle Model

Given 120575119891 120575119903 119907 120579

Unknown ሶ119883 ሶ119884 ሶ120579

Solution

120575119907 = tanminus1119897119891 tan 120575119903+119897119903 tan 120575119891

119897119891+119897119903

ሶ120579 =119907

120602

120602 =119897119891+119897119903

tan 120575119891 minustan 120575119903 cos 120575119907

ሶ119883 = 119907 cos 120579 + 120575119907ሶ119884 = 119907 sin 120579 + 120575119907

ሶ120579 =119907 cos 120575119907

119897119891+119897119903tan 120575119891 minus tan 120575119903

119907

rrlabcsuni-klde



119862 = 119874 120575119903 = 0 120575119891 = 120575

119897119903 = 0 119897119891 = 119897

120575119907 = 0

ሶ120579 =119907

120602

120602 =119897

tan 120575119891

ሶ119883 = 119907 cos 120579ሶ119884 = 119907 sin 120579ሶ120579 =

119907

119897119891tan 120575119891

119907



ሶ120595 119903








and 1 Ԧ1199071 = ሶ119909 ሶ119910 0 119879


119894+1120596119894+1 = 119894119894+1119877 sdot 119894120596119894 + ሶ120579119894+1

119894+1 Ԧ119890119911119894+1

119894+1 Ԧ119907119894+1 = 119894119894+1119877 119894 Ԧ119907119894 +

119894120596119894 times119894 Ԧ119901119894+1




calculated as

2 Ԧ1199072 =119888120572 ሶ119909 + 119904120572 ሶ119910minus119904120572 ሶ119909 + 119888120572 ሶ119910

0


3 Ԧ1199073 =

119888120572 ሶ119909 + 119904120572 ሶ119910

minus119904120572 ሶ119909 + 119888120572 ሶ119910 + 119889 ሶ1205790






4 Ԧ1199074 =119904 120572 + 120573 ሶ119909 minus 119888 120572 + 120573 ሶ119910 minus 119888120573 119889 ሶ120579

119888 120572 + 120573 ሶ119909 + 119904 120572 + 120573 ሶ119910 + 119904120573 119889 ሶ1205790




119904 120572 + 120573 minus119888 120572 + 120573 minus119888120573 119889

119888 120572 + 120573 119904 120572 + 120573 119904120573 1198890 0 0

ሶ119909119910 ሶሶ120579

=119903 ሶ12059500












119904 120572 + 120573(119905) minus119888 120572 + 120573(119905) minus119888120573(119905)119889

119888 120572 + 120573(119905) 119904 120572 + 120573(119905) 119904120573(119905)1198890 0 0

ሶ119909ሶ119910ሶ120579

=119903 ሶ120595

minus119889119888 ሶ120573(119905)0


maneuverability)




119904 120572 + 120573 + 120574 minus119888 120572 + 120573 + 120574 minus119888 120573 + 120574 119889

119888 120572 + 120573 + 120574 119904 120572 + 120573 + 120574 119904 120573 + 120574 1198890 0 0

ሶ119909ሶ119910ሶ120579

=119903 ሶ120595 cos 120574

119903 ሶ120595 sin 120574 + 119903119901119903 ሶ1205951199011199030















120572119897 = 90 deg

120573119897 = 0 deg

120572119903 = minus90 deg

120573119903 = 180 deg




119904 120572119897 + 120573119897 ሶ119909 minus 119888 120572119897 + 120573119897 ሶ119910 minus 119888120573119897 119889 ሶ120579 = 119903119897 ሶ120595119897

119888 120572119897 + 120573119897 ሶ119909 + 119904 120572119897 + 120573119897 ሶ119910 + 119904120573119897 119889 ሶ120579 = 0



ሶ119909 minus 119889 ሶ120579 = 119903119897 ሶ120595119897

ሶ119910 = 0

ሶ119909 + 119889 ሶ120579 = 119903119903 ሶ120595119903119910 = 0




ሶ119909ሶ119910ሶ120579

=

1

2119903119897 ሶ120595119897 + 119903119903 ሶ120595119903

01

2119889minus119903119897 ሶ120595119897 + 119903119903 ሶ120595119903



rrlabcsuni-klde




rrlabcsuni-klde











1205721 = 0 deg

1205722 = 120 deg

1205723 = minus120 deg







119888 1205721 + 1205731 ሶ119909 + 119904 1205721 + 1205731 ሶ119910 + 119889 sdot 119904 1205731 ሶ120579 = 0

119888 1205722 + 1205732 ሶ119909 + 119904 1205722 + 1205732 ሶ119910 + 119889 sdot 119904 1205732 ሶ120579 = 0

119888 1205723 + 1205733 ሶ119909 + 119904 1205723 + 1205733 ሶ119910 + 119889 sdot 119904 1205733 ሶ120579 = 0







119888 120572119894 sdot ሶ119909 + 119904 120572119894 sdot ሶ119910




ሶ120595119894 =1


rrlabcsuni-klde



rrlabcsuni-klde

Mecanum Vehicle





Mecanum Kinematics



1205721 = 45 deg 1205731 = 45 deg 1205741 = minus45 deg

1205722 = 135 deg 1205732 = minus45 deg 1205742 = 45 deg


1205724 = minus45 deg 1205734 = 135 deg 1205744 = 45 deg


Mecanum Kinematics








Mecanum Kinematics


ሶ119909 =119903

4ሶ1205951 + ሶ1205952 + ሶ1205953 + ሶ1205954

ሶ119910 =119903

4minus ሶ1205951 + ሶ1205952 minus ሶ1205953 + ሶ1205954

ሶ120579 =119903

119889 2ሶ1205951 minus ሶ1205952 minus ሶ1205953 + ሶ1205954


Mecanum Kinematics


ሶ119909ሶ119910ሶ120579

=119903119908ℎ1198901198901198974


sdot

ሶ1205951

ሶ1205952

ሶ1205953

ሶ1205954

with a =2 2

119889


ሶ1205954 = ሶ1205951 + ሶ1205952 minus ሶ1205953

rrlabcsuni-klde





Ability to maneuver

Controllability

Stability



Stability




Typical setups




Maneuverability








Controllability



Examples




Wheel type

Drive geometry


Motortransmission


Degree of Mobility




119891119898 = 3 minus rank 119878


Degree of Mobility



1198891 = 1198892 1205731 = 1205732 = 0 1205721 + 120587 = 1205722






119891119904 = rank 119878prime








119891119872 = 119891119898 + 119891119904

rrlabcsuni-klde

Vehicle Kinematics



Tricycle Drive

Given 120575 119907


119903 = 119897 sdot cot 120575

119907prime =119907

cos 120575

119907119897 = 119907 sdot119903 + 119889

119903

119907119903 = 119907 sdot119903 minus 119889


120575


Ackermann Steering

Given 119907119863 120575 119897 119889

Unknown 119907119903119903 119907119897119903 119907119903119891 119907119897119891

Solution

120602 =119897

tan 120575

119907119897119903 =120602 minus 1198892 sdot 119907119863

120602

119907119903119903 =120602 + 1198892 sdot 119907119863

120602

119907119897119891 =120602 minus 1198892 2 + 1198972 sdot tan 120575

119897sdot 119907119863

119907119903119891 =1206022 + 1198972 sdot tan 120575

119897sdot 119907119863


120575

120602

rrlabcsuni-klde



Given 120602 120593 119907 119889



1205932 = arctan120549119910 + 119889

1205491199091206022 = 120602 sdot cos120593 cos1205932

1199072 = 119907 sdot 1206022120602

120549119910 = 120602 ∙ 119904119894119899 120593

1205931

1205932

1206021

1206022

120593 120549119909 = 120602 ∙ cos120593

120602

119889

rrlabcsuni-klde

Bicycle Model


119907


Bicycle Model


Low speed motion



119865119910 =1198981199072

120602






Bicycle Model

Given 120575119891 120575119903 119907 120579

Unknown ሶ119883 ሶ119884 ሶ120579

Solution

120575119907 = tanminus1119897119891 tan 120575119903+119897119903 tan 120575119891

119897119891+119897119903

ሶ120579 =119907

120602

120602 =119897119891+119897119903

tan 120575119891 minustan 120575119903 cos 120575119907

ሶ119883 = 119907 cos 120579 + 120575119907ሶ119884 = 119907 sin 120579 + 120575119907

ሶ120579 =119907 cos 120575119907

119897119891+119897119903tan 120575119891 minus tan 120575119903

119907

rrlabcsuni-klde



119862 = 119874 120575119903 = 0 120575119891 = 120575

119897119903 = 0 119897119891 = 119897

120575119907 = 0

ሶ120579 =119907

120602

120602 =119897

tan 120575119891

ሶ119883 = 119907 cos 120579ሶ119884 = 119907 sin 120579ሶ120579 =

119907

119897119891tan 120575119891

119907








and 1 Ԧ1199071 = ሶ119909 ሶ119910 0 119879


119894+1120596119894+1 = 119894119894+1119877 sdot 119894120596119894 + ሶ120579119894+1

119894+1 Ԧ119890119911119894+1

119894+1 Ԧ119907119894+1 = 119894119894+1119877 119894 Ԧ119907119894 +

119894120596119894 times119894 Ԧ119901119894+1




calculated as

2 Ԧ1199072 =119888120572 ሶ119909 + 119904120572 ሶ119910minus119904120572 ሶ119909 + 119888120572 ሶ119910

0


3 Ԧ1199073 =

119888120572 ሶ119909 + 119904120572 ሶ119910

minus119904120572 ሶ119909 + 119888120572 ሶ119910 + 119889 ሶ1205790






4 Ԧ1199074 =119904 120572 + 120573 ሶ119909 minus 119888 120572 + 120573 ሶ119910 minus 119888120573 119889 ሶ120579

119888 120572 + 120573 ሶ119909 + 119904 120572 + 120573 ሶ119910 + 119904120573 119889 ሶ1205790




119904 120572 + 120573 minus119888 120572 + 120573 minus119888120573 119889

119888 120572 + 120573 119904 120572 + 120573 119904120573 1198890 0 0

ሶ119909119910 ሶሶ120579

=119903 ሶ12059500












119904 120572 + 120573(119905) minus119888 120572 + 120573(119905) minus119888120573(119905)119889

119888 120572 + 120573(119905) 119904 120572 + 120573(119905) 119904120573(119905)1198890 0 0

ሶ119909ሶ119910ሶ120579

=119903 ሶ120595

minus119889119888 ሶ120573(119905)0


maneuverability)




119904 120572 + 120573 + 120574 minus119888 120572 + 120573 + 120574 minus119888 120573 + 120574 119889

119888 120572 + 120573 + 120574 119904 120572 + 120573 + 120574 119904 120573 + 120574 1198890 0 0

ሶ119909ሶ119910ሶ120579

=119903 ሶ120595 cos 120574

119903 ሶ120595 sin 120574 + 119903119901119903 ሶ1205951199011199030















120572119897 = 90 deg

120573119897 = 0 deg

120572119903 = minus90 deg

120573119903 = 180 deg




119904 120572119897 + 120573119897 ሶ119909 minus 119888 120572119897 + 120573119897 ሶ119910 minus 119888120573119897 119889 ሶ120579 = 119903119897 ሶ120595119897

119888 120572119897 + 120573119897 ሶ119909 + 119904 120572119897 + 120573119897 ሶ119910 + 119904120573119897 119889 ሶ120579 = 0



ሶ119909 minus 119889 ሶ120579 = 119903119897 ሶ120595119897

ሶ119910 = 0

ሶ119909 + 119889 ሶ120579 = 119903119903 ሶ120595119903119910 = 0




ሶ119909ሶ119910ሶ120579

=

1

2119903119897 ሶ120595119897 + 119903119903 ሶ120595119903

01

2119889minus119903119897 ሶ120595119897 + 119903119903 ሶ120595119903



rrlabcsuni-klde




rrlabcsuni-klde











1205721 = 0 deg

1205722 = 120 deg

1205723 = minus120 deg







119888 1205721 + 1205731 ሶ119909 + 119904 1205721 + 1205731 ሶ119910 + 119889 sdot 119904 1205731 ሶ120579 = 0

119888 1205722 + 1205732 ሶ119909 + 119904 1205722 + 1205732 ሶ119910 + 119889 sdot 119904 1205732 ሶ120579 = 0

119888 1205723 + 1205733 ሶ119909 + 119904 1205723 + 1205733 ሶ119910 + 119889 sdot 119904 1205733 ሶ120579 = 0







119888 120572119894 sdot ሶ119909 + 119904 120572119894 sdot ሶ119910




ሶ120595119894 =1


rrlabcsuni-klde



rrlabcsuni-klde

Mecanum Vehicle





Mecanum Kinematics



1205721 = 45 deg 1205731 = 45 deg 1205741 = minus45 deg

1205722 = 135 deg 1205732 = minus45 deg 1205742 = 45 deg


1205724 = minus45 deg 1205734 = 135 deg 1205744 = 45 deg


Mecanum Kinematics








Mecanum Kinematics


ሶ119909 =119903

4ሶ1205951 + ሶ1205952 + ሶ1205953 + ሶ1205954

ሶ119910 =119903

4minus ሶ1205951 + ሶ1205952 minus ሶ1205953 + ሶ1205954

ሶ120579 =119903

119889 2ሶ1205951 minus ሶ1205952 minus ሶ1205953 + ሶ1205954


Mecanum Kinematics


ሶ119909ሶ119910ሶ120579

=119903119908ℎ1198901198901198974


sdot

ሶ1205951

ሶ1205952

ሶ1205953

ሶ1205954

with a =2 2

119889


ሶ1205954 = ሶ1205951 + ሶ1205952 minus ሶ1205953

rrlabcsuni-klde





Ability to maneuver

Controllability

Stability



Stability




Typical setups




Maneuverability








Controllability



Examples




Wheel type

Drive geometry


Motortransmission


Degree of Mobility




119891119898 = 3 minus rank 119878


Degree of Mobility



1198891 = 1198892 1205731 = 1205732 = 0 1205721 + 120587 = 1205722






119891119904 = rank 119878prime








119891119872 = 119891119898 + 119891119904

rrlabcsuni-klde

Vehicle Kinematics



Tricycle Drive

Given 120575 119907


119903 = 119897 sdot cot 120575

119907prime =119907

cos 120575

119907119897 = 119907 sdot119903 + 119889

119903

119907119903 = 119907 sdot119903 minus 119889


120575


Ackermann Steering

Given 119907119863 120575 119897 119889

Unknown 119907119903119903 119907119897119903 119907119903119891 119907119897119891

Solution

120602 =119897

tan 120575

119907119897119903 =120602 minus 1198892 sdot 119907119863

120602

119907119903119903 =120602 + 1198892 sdot 119907119863

120602

119907119897119891 =120602 minus 1198892 2 + 1198972 sdot tan 120575

119897sdot 119907119863

119907119903119891 =1206022 + 1198972 sdot tan 120575

119897sdot 119907119863


120575

120602

rrlabcsuni-klde



Given 120602 120593 119907 119889



1205932 = arctan120549119910 + 119889

1205491199091206022 = 120602 sdot cos120593 cos1205932

1199072 = 119907 sdot 1206022120602

120549119910 = 120602 ∙ 119904119894119899 120593

1205931

1205932

1206021

1206022

120593 120549119909 = 120602 ∙ cos120593

120602

119889

rrlabcsuni-klde

Bicycle Model


119907


Bicycle Model


Low speed motion



119865119910 =1198981199072

120602






Bicycle Model

Given 120575119891 120575119903 119907 120579

Unknown ሶ119883 ሶ119884 ሶ120579

Solution

120575119907 = tanminus1119897119891 tan 120575119903+119897119903 tan 120575119891

119897119891+119897119903

ሶ120579 =119907

120602

120602 =119897119891+119897119903

tan 120575119891 minustan 120575119903 cos 120575119907

ሶ119883 = 119907 cos 120579 + 120575119907ሶ119884 = 119907 sin 120579 + 120575119907

ሶ120579 =119907 cos 120575119907

119897119891+119897119903tan 120575119891 minus tan 120575119903

119907

rrlabcsuni-klde



119862 = 119874 120575119903 = 0 120575119891 = 120575

119897119903 = 0 119897119891 = 119897

120575119907 = 0

ሶ120579 =119907

120602

120602 =119897

tan 120575119891

ሶ119883 = 119907 cos 120579ሶ119884 = 119907 sin 120579ሶ120579 =

119907

119897119891tan 120575119891

119907




calculated as

2 Ԧ1199072 =119888120572 ሶ119909 + 119904120572 ሶ119910minus119904120572 ሶ119909 + 119888120572 ሶ119910

0


3 Ԧ1199073 =

119888120572 ሶ119909 + 119904120572 ሶ119910

minus119904120572 ሶ119909 + 119888120572 ሶ119910 + 119889 ሶ1205790






4 Ԧ1199074 =119904 120572 + 120573 ሶ119909 minus 119888 120572 + 120573 ሶ119910 minus 119888120573 119889 ሶ120579

119888 120572 + 120573 ሶ119909 + 119904 120572 + 120573 ሶ119910 + 119904120573 119889 ሶ1205790




119904 120572 + 120573 minus119888 120572 + 120573 minus119888120573 119889

119888 120572 + 120573 119904 120572 + 120573 119904120573 1198890 0 0

ሶ119909119910 ሶሶ120579

=119903 ሶ12059500












119904 120572 + 120573(119905) minus119888 120572 + 120573(119905) minus119888120573(119905)119889

119888 120572 + 120573(119905) 119904 120572 + 120573(119905) 119904120573(119905)1198890 0 0

ሶ119909ሶ119910ሶ120579

=119903 ሶ120595

minus119889119888 ሶ120573(119905)0


maneuverability)




119904 120572 + 120573 + 120574 minus119888 120572 + 120573 + 120574 minus119888 120573 + 120574 119889

119888 120572 + 120573 + 120574 119904 120572 + 120573 + 120574 119904 120573 + 120574 1198890 0 0

ሶ119909ሶ119910ሶ120579

=119903 ሶ120595 cos 120574

119903 ሶ120595 sin 120574 + 119903119901119903 ሶ1205951199011199030















120572119897 = 90 deg

120573119897 = 0 deg

120572119903 = minus90 deg

120573119903 = 180 deg




119904 120572119897 + 120573119897 ሶ119909 minus 119888 120572119897 + 120573119897 ሶ119910 minus 119888120573119897 119889 ሶ120579 = 119903119897 ሶ120595119897

119888 120572119897 + 120573119897 ሶ119909 + 119904 120572119897 + 120573119897 ሶ119910 + 119904120573119897 119889 ሶ120579 = 0



ሶ119909 minus 119889 ሶ120579 = 119903119897 ሶ120595119897

ሶ119910 = 0

ሶ119909 + 119889 ሶ120579 = 119903119903 ሶ120595119903119910 = 0




ሶ119909ሶ119910ሶ120579

=

1

2119903119897 ሶ120595119897 + 119903119903 ሶ120595119903

01

2119889minus119903119897 ሶ120595119897 + 119903119903 ሶ120595119903



rrlabcsuni-klde




rrlabcsuni-klde











1205721 = 0 deg

1205722 = 120 deg

1205723 = minus120 deg







119888 1205721 + 1205731 ሶ119909 + 119904 1205721 + 1205731 ሶ119910 + 119889 sdot 119904 1205731 ሶ120579 = 0

119888 1205722 + 1205732 ሶ119909 + 119904 1205722 + 1205732 ሶ119910 + 119889 sdot 119904 1205732 ሶ120579 = 0

119888 1205723 + 1205733 ሶ119909 + 119904 1205723 + 1205733 ሶ119910 + 119889 sdot 119904 1205733 ሶ120579 = 0







119888 120572119894 sdot ሶ119909 + 119904 120572119894 sdot ሶ119910




ሶ120595119894 =1


rrlabcsuni-klde



rrlabcsuni-klde

Mecanum Vehicle





Mecanum Kinematics



1205721 = 45 deg 1205731 = 45 deg 1205741 = minus45 deg

1205722 = 135 deg 1205732 = minus45 deg 1205742 = 45 deg


1205724 = minus45 deg 1205734 = 135 deg 1205744 = 45 deg


Mecanum Kinematics








Mecanum Kinematics


ሶ119909 =119903

4ሶ1205951 + ሶ1205952 + ሶ1205953 + ሶ1205954

ሶ119910 =119903

4minus ሶ1205951 + ሶ1205952 minus ሶ1205953 + ሶ1205954

ሶ120579 =119903

119889 2ሶ1205951 minus ሶ1205952 minus ሶ1205953 + ሶ1205954


Mecanum Kinematics


ሶ119909ሶ119910ሶ120579

=119903119908ℎ1198901198901198974


sdot

ሶ1205951

ሶ1205952

ሶ1205953

ሶ1205954

with a =2 2

119889


ሶ1205954 = ሶ1205951 + ሶ1205952 minus ሶ1205953

rrlabcsuni-klde





Ability to maneuver

Controllability

Stability



Stability




Typical setups




Maneuverability








Controllability



Examples




Wheel type

Drive geometry


Motortransmission


Degree of Mobility




119891119898 = 3 minus rank 119878


Degree of Mobility



1198891 = 1198892 1205731 = 1205732 = 0 1205721 + 120587 = 1205722






119891119904 = rank 119878prime








119891119872 = 119891119898 + 119891119904

rrlabcsuni-klde

Vehicle Kinematics



Tricycle Drive

Given 120575 119907


119903 = 119897 sdot cot 120575

119907prime =119907

cos 120575

119907119897 = 119907 sdot119903 + 119889

119903

119907119903 = 119907 sdot119903 minus 119889


120575


Ackermann Steering

Given 119907119863 120575 119897 119889

Unknown 119907119903119903 119907119897119903 119907119903119891 119907119897119891

Solution

120602 =119897

tan 120575

119907119897119903 =120602 minus 1198892 sdot 119907119863

120602

119907119903119903 =120602 + 1198892 sdot 119907119863

120602

119907119897119891 =120602 minus 1198892 2 + 1198972 sdot tan 120575

119897sdot 119907119863

119907119903119891 =1206022 + 1198972 sdot tan 120575

119897sdot 119907119863


120575

120602

rrlabcsuni-klde



Given 120602 120593 119907 119889



1205932 = arctan120549119910 + 119889

1205491199091206022 = 120602 sdot cos120593 cos1205932

1199072 = 119907 sdot 1206022120602

120549119910 = 120602 ∙ 119904119894119899 120593

1205931

1205932

1206021

1206022

120593 120549119909 = 120602 ∙ cos120593

120602

119889

rrlabcsuni-klde

Bicycle Model


119907


Bicycle Model


Low speed motion



119865119910 =1198981199072

120602






Bicycle Model

Given 120575119891 120575119903 119907 120579

Unknown ሶ119883 ሶ119884 ሶ120579

Solution

120575119907 = tanminus1119897119891 tan 120575119903+119897119903 tan 120575119891

119897119891+119897119903

ሶ120579 =119907

120602

120602 =119897119891+119897119903

tan 120575119891 minustan 120575119903 cos 120575119907

ሶ119883 = 119907 cos 120579 + 120575119907ሶ119884 = 119907 sin 120579 + 120575119907

ሶ120579 =119907 cos 120575119907

119897119891+119897119903tan 120575119891 minus tan 120575119903

119907

rrlabcsuni-klde



119862 = 119874 120575119903 = 0 120575119891 = 120575

119897119903 = 0 119897119891 = 119897

120575119907 = 0

ሶ120579 =119907

120602

120602 =119897

tan 120575119891

ሶ119883 = 119907 cos 120579ሶ119884 = 119907 sin 120579ሶ120579 =

119907

119897119891tan 120575119891

119907






4 Ԧ1199074 =119904 120572 + 120573 ሶ119909 minus 119888 120572 + 120573 ሶ119910 minus 119888120573 119889 ሶ120579

119888 120572 + 120573 ሶ119909 + 119904 120572 + 120573 ሶ119910 + 119904120573 119889 ሶ1205790




119904 120572 + 120573 minus119888 120572 + 120573 minus119888120573 119889

119888 120572 + 120573 119904 120572 + 120573 119904120573 1198890 0 0

ሶ119909119910 ሶሶ120579

=119903 ሶ12059500












119904 120572 + 120573(119905) minus119888 120572 + 120573(119905) minus119888120573(119905)119889

119888 120572 + 120573(119905) 119904 120572 + 120573(119905) 119904120573(119905)1198890 0 0

ሶ119909ሶ119910ሶ120579

=119903 ሶ120595

minus119889119888 ሶ120573(119905)0


maneuverability)




119904 120572 + 120573 + 120574 minus119888 120572 + 120573 + 120574 minus119888 120573 + 120574 119889

119888 120572 + 120573 + 120574 119904 120572 + 120573 + 120574 119904 120573 + 120574 1198890 0 0

ሶ119909ሶ119910ሶ120579

=119903 ሶ120595 cos 120574

119903 ሶ120595 sin 120574 + 119903119901119903 ሶ1205951199011199030















120572119897 = 90 deg

120573119897 = 0 deg

120572119903 = minus90 deg

120573119903 = 180 deg




119904 120572119897 + 120573119897 ሶ119909 minus 119888 120572119897 + 120573119897 ሶ119910 minus 119888120573119897 119889 ሶ120579 = 119903119897 ሶ120595119897

119888 120572119897 + 120573119897 ሶ119909 + 119904 120572119897 + 120573119897 ሶ119910 + 119904120573119897 119889 ሶ120579 = 0



ሶ119909 minus 119889 ሶ120579 = 119903119897 ሶ120595119897

ሶ119910 = 0

ሶ119909 + 119889 ሶ120579 = 119903119903 ሶ120595119903119910 = 0




ሶ119909ሶ119910ሶ120579

=

1

2119903119897 ሶ120595119897 + 119903119903 ሶ120595119903

01

2119889minus119903119897 ሶ120595119897 + 119903119903 ሶ120595119903



rrlabcsuni-klde




rrlabcsuni-klde











1205721 = 0 deg

1205722 = 120 deg

1205723 = minus120 deg







119888 1205721 + 1205731 ሶ119909 + 119904 1205721 + 1205731 ሶ119910 + 119889 sdot 119904 1205731 ሶ120579 = 0

119888 1205722 + 1205732 ሶ119909 + 119904 1205722 + 1205732 ሶ119910 + 119889 sdot 119904 1205732 ሶ120579 = 0

119888 1205723 + 1205733 ሶ119909 + 119904 1205723 + 1205733 ሶ119910 + 119889 sdot 119904 1205733 ሶ120579 = 0







119888 120572119894 sdot ሶ119909 + 119904 120572119894 sdot ሶ119910




ሶ120595119894 =1


rrlabcsuni-klde



rrlabcsuni-klde

Mecanum Vehicle





Mecanum Kinematics



1205721 = 45 deg 1205731 = 45 deg 1205741 = minus45 deg

1205722 = 135 deg 1205732 = minus45 deg 1205742 = 45 deg


1205724 = minus45 deg 1205734 = 135 deg 1205744 = 45 deg


Mecanum Kinematics








Mecanum Kinematics


ሶ119909 =119903

4ሶ1205951 + ሶ1205952 + ሶ1205953 + ሶ1205954

ሶ119910 =119903

4minus ሶ1205951 + ሶ1205952 minus ሶ1205953 + ሶ1205954

ሶ120579 =119903

119889 2ሶ1205951 minus ሶ1205952 minus ሶ1205953 + ሶ1205954


Mecanum Kinematics


ሶ119909ሶ119910ሶ120579

=119903119908ℎ1198901198901198974


sdot

ሶ1205951

ሶ1205952

ሶ1205953

ሶ1205954

with a =2 2

119889


ሶ1205954 = ሶ1205951 + ሶ1205952 minus ሶ1205953

rrlabcsuni-klde





Ability to maneuver

Controllability

Stability



Stability




Typical setups




Maneuverability








Controllability



Examples




Wheel type

Drive geometry


Motortransmission


Degree of Mobility




119891119898 = 3 minus rank 119878


Degree of Mobility



1198891 = 1198892 1205731 = 1205732 = 0 1205721 + 120587 = 1205722






119891119904 = rank 119878prime








119891119872 = 119891119898 + 119891119904

rrlabcsuni-klde

Vehicle Kinematics



Tricycle Drive

Given 120575 119907


119903 = 119897 sdot cot 120575

119907prime =119907

cos 120575

119907119897 = 119907 sdot119903 + 119889

119903

119907119903 = 119907 sdot119903 minus 119889


120575


Ackermann Steering

Given 119907119863 120575 119897 119889

Unknown 119907119903119903 119907119897119903 119907119903119891 119907119897119891

Solution

120602 =119897

tan 120575

119907119897119903 =120602 minus 1198892 sdot 119907119863

120602

119907119903119903 =120602 + 1198892 sdot 119907119863

120602

119907119897119891 =120602 minus 1198892 2 + 1198972 sdot tan 120575

119897sdot 119907119863

119907119903119891 =1206022 + 1198972 sdot tan 120575

119897sdot 119907119863


120575

120602

rrlabcsuni-klde



Given 120602 120593 119907 119889



1205932 = arctan120549119910 + 119889

1205491199091206022 = 120602 sdot cos120593 cos1205932

1199072 = 119907 sdot 1206022120602

120549119910 = 120602 ∙ 119904119894119899 120593

1205931

1205932

1206021

1206022

120593 120549119909 = 120602 ∙ cos120593

120602

119889

rrlabcsuni-klde

Bicycle Model


119907


Bicycle Model


Low speed motion



119865119910 =1198981199072

120602






Bicycle Model

Given 120575119891 120575119903 119907 120579

Unknown ሶ119883 ሶ119884 ሶ120579

Solution

120575119907 = tanminus1119897119891 tan 120575119903+119897119903 tan 120575119891

119897119891+119897119903

ሶ120579 =119907

120602

120602 =119897119891+119897119903

tan 120575119891 minustan 120575119903 cos 120575119907

ሶ119883 = 119907 cos 120579 + 120575119907ሶ119884 = 119907 sin 120579 + 120575119907

ሶ120579 =119907 cos 120575119907

119897119891+119897119903tan 120575119891 minus tan 120575119903

119907

rrlabcsuni-klde



119862 = 119874 120575119903 = 0 120575119891 = 120575

119897119903 = 0 119897119891 = 119897

120575119907 = 0

ሶ120579 =119907

120602

120602 =119897

tan 120575119891

ሶ119883 = 119907 cos 120579ሶ119884 = 119907 sin 120579ሶ120579 =

119907

119897119891tan 120575119891

119907




119904 120572 + 120573 minus119888 120572 + 120573 minus119888120573 119889

119888 120572 + 120573 119904 120572 + 120573 119904120573 1198890 0 0

ሶ119909119910 ሶሶ120579

=119903 ሶ12059500












119904 120572 + 120573(119905) minus119888 120572 + 120573(119905) minus119888120573(119905)119889

119888 120572 + 120573(119905) 119904 120572 + 120573(119905) 119904120573(119905)1198890 0 0

ሶ119909ሶ119910ሶ120579

=119903 ሶ120595

minus119889119888 ሶ120573(119905)0


maneuverability)




119904 120572 + 120573 + 120574 minus119888 120572 + 120573 + 120574 minus119888 120573 + 120574 119889

119888 120572 + 120573 + 120574 119904 120572 + 120573 + 120574 119904 120573 + 120574 1198890 0 0

ሶ119909ሶ119910ሶ120579

=119903 ሶ120595 cos 120574

119903 ሶ120595 sin 120574 + 119903119901119903 ሶ1205951199011199030















120572119897 = 90 deg

120573119897 = 0 deg

120572119903 = minus90 deg

120573119903 = 180 deg




119904 120572119897 + 120573119897 ሶ119909 minus 119888 120572119897 + 120573119897 ሶ119910 minus 119888120573119897 119889 ሶ120579 = 119903119897 ሶ120595119897

119888 120572119897 + 120573119897 ሶ119909 + 119904 120572119897 + 120573119897 ሶ119910 + 119904120573119897 119889 ሶ120579 = 0



ሶ119909 minus 119889 ሶ120579 = 119903119897 ሶ120595119897

ሶ119910 = 0

ሶ119909 + 119889 ሶ120579 = 119903119903 ሶ120595119903119910 = 0




ሶ119909ሶ119910ሶ120579

=

1

2119903119897 ሶ120595119897 + 119903119903 ሶ120595119903

01

2119889minus119903119897 ሶ120595119897 + 119903119903 ሶ120595119903



rrlabcsuni-klde




rrlabcsuni-klde











1205721 = 0 deg

1205722 = 120 deg

1205723 = minus120 deg







119888 1205721 + 1205731 ሶ119909 + 119904 1205721 + 1205731 ሶ119910 + 119889 sdot 119904 1205731 ሶ120579 = 0

119888 1205722 + 1205732 ሶ119909 + 119904 1205722 + 1205732 ሶ119910 + 119889 sdot 119904 1205732 ሶ120579 = 0

119888 1205723 + 1205733 ሶ119909 + 119904 1205723 + 1205733 ሶ119910 + 119889 sdot 119904 1205733 ሶ120579 = 0







119888 120572119894 sdot ሶ119909 + 119904 120572119894 sdot ሶ119910




ሶ120595119894 =1


rrlabcsuni-klde



rrlabcsuni-klde

Mecanum Vehicle





Mecanum Kinematics



1205721 = 45 deg 1205731 = 45 deg 1205741 = minus45 deg

1205722 = 135 deg 1205732 = minus45 deg 1205742 = 45 deg


1205724 = minus45 deg 1205734 = 135 deg 1205744 = 45 deg


Mecanum Kinematics








Mecanum Kinematics


ሶ119909 =119903

4ሶ1205951 + ሶ1205952 + ሶ1205953 + ሶ1205954

ሶ119910 =119903

4minus ሶ1205951 + ሶ1205952 minus ሶ1205953 + ሶ1205954

ሶ120579 =119903

119889 2ሶ1205951 minus ሶ1205952 minus ሶ1205953 + ሶ1205954


Mecanum Kinematics


ሶ119909ሶ119910ሶ120579

=119903119908ℎ1198901198901198974


sdot

ሶ1205951

ሶ1205952

ሶ1205953

ሶ1205954

with a =2 2

119889


ሶ1205954 = ሶ1205951 + ሶ1205952 minus ሶ1205953

rrlabcsuni-klde





Ability to maneuver

Controllability

Stability



Stability




Typical setups




Maneuverability








Controllability



Examples




Wheel type

Drive geometry


Motortransmission


Degree of Mobility




119891119898 = 3 minus rank 119878


Degree of Mobility



1198891 = 1198892 1205731 = 1205732 = 0 1205721 + 120587 = 1205722






119891119904 = rank 119878prime








119891119872 = 119891119898 + 119891119904

rrlabcsuni-klde

Vehicle Kinematics



Tricycle Drive

Given 120575 119907


119903 = 119897 sdot cot 120575

119907prime =119907

cos 120575

119907119897 = 119907 sdot119903 + 119889

119903

119907119903 = 119907 sdot119903 minus 119889


120575


Ackermann Steering

Given 119907119863 120575 119897 119889

Unknown 119907119903119903 119907119897119903 119907119903119891 119907119897119891

Solution

120602 =119897

tan 120575

119907119897119903 =120602 minus 1198892 sdot 119907119863

120602

119907119903119903 =120602 + 1198892 sdot 119907119863

120602

119907119897119891 =120602 minus 1198892 2 + 1198972 sdot tan 120575

119897sdot 119907119863

119907119903119891 =1206022 + 1198972 sdot tan 120575

119897sdot 119907119863


120575

120602

rrlabcsuni-klde



Given 120602 120593 119907 119889



1205932 = arctan120549119910 + 119889

1205491199091206022 = 120602 sdot cos120593 cos1205932

1199072 = 119907 sdot 1206022120602

120549119910 = 120602 ∙ 119904119894119899 120593

1205931

1205932

1206021

1206022

120593 120549119909 = 120602 ∙ cos120593

120602

119889

rrlabcsuni-klde

Bicycle Model


119907


Bicycle Model


Low speed motion



119865119910 =1198981199072

120602






Bicycle Model

Given 120575119891 120575119903 119907 120579

Unknown ሶ119883 ሶ119884 ሶ120579

Solution

120575119907 = tanminus1119897119891 tan 120575119903+119897119903 tan 120575119891

119897119891+119897119903

ሶ120579 =119907

120602

120602 =119897119891+119897119903

tan 120575119891 minustan 120575119903 cos 120575119907

ሶ119883 = 119907 cos 120579 + 120575119907ሶ119884 = 119907 sin 120579 + 120575119907

ሶ120579 =119907 cos 120575119907

119897119891+119897119903tan 120575119891 minus tan 120575119903

119907

rrlabcsuni-klde



119862 = 119874 120575119903 = 0 120575119891 = 120575

119897119903 = 0 119897119891 = 119897

120575119907 = 0

ሶ120579 =119907

120602

120602 =119897

tan 120575119891

ሶ119883 = 119907 cos 120579ሶ119884 = 119907 sin 120579ሶ120579 =

119907

119897119891tan 120575119891

119907







119904 120572 + 120573(119905) minus119888 120572 + 120573(119905) minus119888120573(119905)119889

119888 120572 + 120573(119905) 119904 120572 + 120573(119905) 119904120573(119905)1198890 0 0

ሶ119909ሶ119910ሶ120579

=119903 ሶ120595

minus119889119888 ሶ120573(119905)0


maneuverability)




119904 120572 + 120573 + 120574 minus119888 120572 + 120573 + 120574 minus119888 120573 + 120574 119889

119888 120572 + 120573 + 120574 119904 120572 + 120573 + 120574 119904 120573 + 120574 1198890 0 0

ሶ119909ሶ119910ሶ120579

=119903 ሶ120595 cos 120574

119903 ሶ120595 sin 120574 + 119903119901119903 ሶ1205951199011199030















120572119897 = 90 deg

120573119897 = 0 deg

120572119903 = minus90 deg

120573119903 = 180 deg




119904 120572119897 + 120573119897 ሶ119909 minus 119888 120572119897 + 120573119897 ሶ119910 minus 119888120573119897 119889 ሶ120579 = 119903119897 ሶ120595119897

119888 120572119897 + 120573119897 ሶ119909 + 119904 120572119897 + 120573119897 ሶ119910 + 119904120573119897 119889 ሶ120579 = 0



ሶ119909 minus 119889 ሶ120579 = 119903119897 ሶ120595119897

ሶ119910 = 0

ሶ119909 + 119889 ሶ120579 = 119903119903 ሶ120595119903119910 = 0




ሶ119909ሶ119910ሶ120579

=

1

2119903119897 ሶ120595119897 + 119903119903 ሶ120595119903

01

2119889minus119903119897 ሶ120595119897 + 119903119903 ሶ120595119903



rrlabcsuni-klde




rrlabcsuni-klde











1205721 = 0 deg

1205722 = 120 deg

1205723 = minus120 deg







119888 1205721 + 1205731 ሶ119909 + 119904 1205721 + 1205731 ሶ119910 + 119889 sdot 119904 1205731 ሶ120579 = 0

119888 1205722 + 1205732 ሶ119909 + 119904 1205722 + 1205732 ሶ119910 + 119889 sdot 119904 1205732 ሶ120579 = 0

119888 1205723 + 1205733 ሶ119909 + 119904 1205723 + 1205733 ሶ119910 + 119889 sdot 119904 1205733 ሶ120579 = 0







119888 120572119894 sdot ሶ119909 + 119904 120572119894 sdot ሶ119910




ሶ120595119894 =1


rrlabcsuni-klde



rrlabcsuni-klde

Mecanum Vehicle





Mecanum Kinematics



1205721 = 45 deg 1205731 = 45 deg 1205741 = minus45 deg

1205722 = 135 deg 1205732 = minus45 deg 1205742 = 45 deg


1205724 = minus45 deg 1205734 = 135 deg 1205744 = 45 deg


Mecanum Kinematics








Mecanum Kinematics


ሶ119909 =119903

4ሶ1205951 + ሶ1205952 + ሶ1205953 + ሶ1205954

ሶ119910 =119903

4minus ሶ1205951 + ሶ1205952 minus ሶ1205953 + ሶ1205954

ሶ120579 =119903

119889 2ሶ1205951 minus ሶ1205952 minus ሶ1205953 + ሶ1205954


Mecanum Kinematics


ሶ119909ሶ119910ሶ120579

=119903119908ℎ1198901198901198974


sdot

ሶ1205951

ሶ1205952

ሶ1205953

ሶ1205954

with a =2 2

119889


ሶ1205954 = ሶ1205951 + ሶ1205952 minus ሶ1205953

rrlabcsuni-klde





Ability to maneuver

Controllability

Stability



Stability




Typical setups




Maneuverability








Controllability



Examples




Wheel type

Drive geometry


Motortransmission


Degree of Mobility




119891119898 = 3 minus rank 119878


Degree of Mobility



1198891 = 1198892 1205731 = 1205732 = 0 1205721 + 120587 = 1205722






119891119904 = rank 119878prime








119891119872 = 119891119898 + 119891119904

rrlabcsuni-klde

Vehicle Kinematics



Tricycle Drive

Given 120575 119907


119903 = 119897 sdot cot 120575

119907prime =119907

cos 120575

119907119897 = 119907 sdot119903 + 119889

119903

119907119903 = 119907 sdot119903 minus 119889


120575


Ackermann Steering

Given 119907119863 120575 119897 119889

Unknown 119907119903119903 119907119897119903 119907119903119891 119907119897119891

Solution

120602 =119897

tan 120575

119907119897119903 =120602 minus 1198892 sdot 119907119863

120602

119907119903119903 =120602 + 1198892 sdot 119907119863

120602

119907119897119891 =120602 minus 1198892 2 + 1198972 sdot tan 120575

119897sdot 119907119863

119907119903119891 =1206022 + 1198972 sdot tan 120575

119897sdot 119907119863


120575

120602

rrlabcsuni-klde



Given 120602 120593 119907 119889



1205932 = arctan120549119910 + 119889

1205491199091206022 = 120602 sdot cos120593 cos1205932

1199072 = 119907 sdot 1206022120602

120549119910 = 120602 ∙ 119904119894119899 120593

1205931

1205932

1206021

1206022

120593 120549119909 = 120602 ∙ cos120593

120602

119889

rrlabcsuni-klde

Bicycle Model


119907


Bicycle Model


Low speed motion



119865119910 =1198981199072

120602






Bicycle Model

Given 120575119891 120575119903 119907 120579

Unknown ሶ119883 ሶ119884 ሶ120579

Solution

120575119907 = tanminus1119897119891 tan 120575119903+119897119903 tan 120575119891

119897119891+119897119903

ሶ120579 =119907

120602

120602 =119897119891+119897119903

tan 120575119891 minustan 120575119903 cos 120575119907

ሶ119883 = 119907 cos 120579 + 120575119907ሶ119884 = 119907 sin 120579 + 120575119907

ሶ120579 =119907 cos 120575119907

119897119891+119897119903tan 120575119891 minus tan 120575119903

119907

rrlabcsuni-klde



119862 = 119874 120575119903 = 0 120575119891 = 120575

119897119903 = 0 119897119891 = 119897

120575119907 = 0

ሶ120579 =119907

120602

120602 =119897

tan 120575119891

ሶ119883 = 119907 cos 120579ሶ119884 = 119907 sin 120579ሶ120579 =

119907

119897119891tan 120575119891

119907




119904 120572 + 120573 + 120574 minus119888 120572 + 120573 + 120574 minus119888 120573 + 120574 119889

119888 120572 + 120573 + 120574 119904 120572 + 120573 + 120574 119904 120573 + 120574 1198890 0 0

ሶ119909ሶ119910ሶ120579

=119903 ሶ120595 cos 120574

119903 ሶ120595 sin 120574 + 119903119901119903 ሶ1205951199011199030















120572119897 = 90 deg

120573119897 = 0 deg

120572119903 = minus90 deg

120573119903 = 180 deg




119904 120572119897 + 120573119897 ሶ119909 minus 119888 120572119897 + 120573119897 ሶ119910 minus 119888120573119897 119889 ሶ120579 = 119903119897 ሶ120595119897

119888 120572119897 + 120573119897 ሶ119909 + 119904 120572119897 + 120573119897 ሶ119910 + 119904120573119897 119889 ሶ120579 = 0



ሶ119909 minus 119889 ሶ120579 = 119903119897 ሶ120595119897

ሶ119910 = 0

ሶ119909 + 119889 ሶ120579 = 119903119903 ሶ120595119903119910 = 0




ሶ119909ሶ119910ሶ120579

=

1

2119903119897 ሶ120595119897 + 119903119903 ሶ120595119903

01

2119889minus119903119897 ሶ120595119897 + 119903119903 ሶ120595119903



rrlabcsuni-klde




rrlabcsuni-klde











1205721 = 0 deg

1205722 = 120 deg

1205723 = minus120 deg







119888 1205721 + 1205731 ሶ119909 + 119904 1205721 + 1205731 ሶ119910 + 119889 sdot 119904 1205731 ሶ120579 = 0

119888 1205722 + 1205732 ሶ119909 + 119904 1205722 + 1205732 ሶ119910 + 119889 sdot 119904 1205732 ሶ120579 = 0

119888 1205723 + 1205733 ሶ119909 + 119904 1205723 + 1205733 ሶ119910 + 119889 sdot 119904 1205733 ሶ120579 = 0







119888 120572119894 sdot ሶ119909 + 119904 120572119894 sdot ሶ119910




ሶ120595119894 =1


rrlabcsuni-klde



rrlabcsuni-klde

Mecanum Vehicle





Mecanum Kinematics



1205721 = 45 deg 1205731 = 45 deg 1205741 = minus45 deg

1205722 = 135 deg 1205732 = minus45 deg 1205742 = 45 deg


1205724 = minus45 deg 1205734 = 135 deg 1205744 = 45 deg


Mecanum Kinematics








Mecanum Kinematics


ሶ119909 =119903

4ሶ1205951 + ሶ1205952 + ሶ1205953 + ሶ1205954

ሶ119910 =119903

4minus ሶ1205951 + ሶ1205952 minus ሶ1205953 + ሶ1205954

ሶ120579 =119903

119889 2ሶ1205951 minus ሶ1205952 minus ሶ1205953 + ሶ1205954


Mecanum Kinematics


ሶ119909ሶ119910ሶ120579

=119903119908ℎ1198901198901198974


sdot

ሶ1205951

ሶ1205952

ሶ1205953

ሶ1205954

with a =2 2

119889


ሶ1205954 = ሶ1205951 + ሶ1205952 minus ሶ1205953

rrlabcsuni-klde





Ability to maneuver

Controllability

Stability



Stability




Typical setups




Maneuverability








Controllability



Examples




Wheel type

Drive geometry


Motortransmission


Degree of Mobility




119891119898 = 3 minus rank 119878


Degree of Mobility



1198891 = 1198892 1205731 = 1205732 = 0 1205721 + 120587 = 1205722






119891119904 = rank 119878prime








119891119872 = 119891119898 + 119891119904

rrlabcsuni-klde

Vehicle Kinematics



Tricycle Drive

Given 120575 119907


119903 = 119897 sdot cot 120575

119907prime =119907

cos 120575

119907119897 = 119907 sdot119903 + 119889

119903

119907119903 = 119907 sdot119903 minus 119889


120575


Ackermann Steering

Given 119907119863 120575 119897 119889

Unknown 119907119903119903 119907119897119903 119907119903119891 119907119897119891

Solution

120602 =119897

tan 120575

119907119897119903 =120602 minus 1198892 sdot 119907119863

120602

119907119903119903 =120602 + 1198892 sdot 119907119863

120602

119907119897119891 =120602 minus 1198892 2 + 1198972 sdot tan 120575

119897sdot 119907119863

119907119903119891 =1206022 + 1198972 sdot tan 120575

119897sdot 119907119863


120575

120602

rrlabcsuni-klde



Given 120602 120593 119907 119889



1205932 = arctan120549119910 + 119889

1205491199091206022 = 120602 sdot cos120593 cos1205932

1199072 = 119907 sdot 1206022120602

120549119910 = 120602 ∙ 119904119894119899 120593

1205931

1205932

1206021

1206022

120593 120549119909 = 120602 ∙ cos120593

120602

119889

rrlabcsuni-klde

Bicycle Model


119907


Bicycle Model


Low speed motion



119865119910 =1198981199072

120602






Bicycle Model

Given 120575119891 120575119903 119907 120579

Unknown ሶ119883 ሶ119884 ሶ120579

Solution

120575119907 = tanminus1119897119891 tan 120575119903+119897119903 tan 120575119891

119897119891+119897119903

ሶ120579 =119907

120602

120602 =119897119891+119897119903

tan 120575119891 minustan 120575119903 cos 120575119907

ሶ119883 = 119907 cos 120579 + 120575119907ሶ119884 = 119907 sin 120579 + 120575119907

ሶ120579 =119907 cos 120575119907

119897119891+119897119903tan 120575119891 minus tan 120575119903

119907

rrlabcsuni-klde



119862 = 119874 120575119903 = 0 120575119891 = 120575

119897119903 = 0 119897119891 = 119897

120575119907 = 0

ሶ120579 =119907

120602

120602 =119897

tan 120575119891

ሶ119883 = 119907 cos 120579ሶ119884 = 119907 sin 120579ሶ120579 =

119907

119897119891tan 120575119891

119907















120572119897 = 90 deg

120573119897 = 0 deg

120572119903 = minus90 deg

120573119903 = 180 deg




119904 120572119897 + 120573119897 ሶ119909 minus 119888 120572119897 + 120573119897 ሶ119910 minus 119888120573119897 119889 ሶ120579 = 119903119897 ሶ120595119897

119888 120572119897 + 120573119897 ሶ119909 + 119904 120572119897 + 120573119897 ሶ119910 + 119904120573119897 119889 ሶ120579 = 0



ሶ119909 minus 119889 ሶ120579 = 119903119897 ሶ120595119897

ሶ119910 = 0

ሶ119909 + 119889 ሶ120579 = 119903119903 ሶ120595119903119910 = 0




ሶ119909ሶ119910ሶ120579

=

1

2119903119897 ሶ120595119897 + 119903119903 ሶ120595119903

01

2119889minus119903119897 ሶ120595119897 + 119903119903 ሶ120595119903



rrlabcsuni-klde




rrlabcsuni-klde











1205721 = 0 deg

1205722 = 120 deg

1205723 = minus120 deg







119888 1205721 + 1205731 ሶ119909 + 119904 1205721 + 1205731 ሶ119910 + 119889 sdot 119904 1205731 ሶ120579 = 0

119888 1205722 + 1205732 ሶ119909 + 119904 1205722 + 1205732 ሶ119910 + 119889 sdot 119904 1205732 ሶ120579 = 0

119888 1205723 + 1205733 ሶ119909 + 119904 1205723 + 1205733 ሶ119910 + 119889 sdot 119904 1205733 ሶ120579 = 0







119888 120572119894 sdot ሶ119909 + 119904 120572119894 sdot ሶ119910




ሶ120595119894 =1


rrlabcsuni-klde



rrlabcsuni-klde

Mecanum Vehicle





Mecanum Kinematics



1205721 = 45 deg 1205731 = 45 deg 1205741 = minus45 deg

1205722 = 135 deg 1205732 = minus45 deg 1205742 = 45 deg


1205724 = minus45 deg 1205734 = 135 deg 1205744 = 45 deg


Mecanum Kinematics








Mecanum Kinematics


ሶ119909 =119903

4ሶ1205951 + ሶ1205952 + ሶ1205953 + ሶ1205954

ሶ119910 =119903

4minus ሶ1205951 + ሶ1205952 minus ሶ1205953 + ሶ1205954

ሶ120579 =119903

119889 2ሶ1205951 minus ሶ1205952 minus ሶ1205953 + ሶ1205954


Mecanum Kinematics


ሶ119909ሶ119910ሶ120579

=119903119908ℎ1198901198901198974


sdot

ሶ1205951

ሶ1205952

ሶ1205953

ሶ1205954

with a =2 2

119889


ሶ1205954 = ሶ1205951 + ሶ1205952 minus ሶ1205953

rrlabcsuni-klde





Ability to maneuver

Controllability

Stability



Stability




Typical setups




Maneuverability








Controllability



Examples




Wheel type

Drive geometry


Motortransmission


Degree of Mobility




119891119898 = 3 minus rank 119878


Degree of Mobility



1198891 = 1198892 1205731 = 1205732 = 0 1205721 + 120587 = 1205722






119891119904 = rank 119878prime








119891119872 = 119891119898 + 119891119904

rrlabcsuni-klde

Vehicle Kinematics



Tricycle Drive

Given 120575 119907


119903 = 119897 sdot cot 120575

119907prime =119907

cos 120575

119907119897 = 119907 sdot119903 + 119889

119903

119907119903 = 119907 sdot119903 minus 119889


120575


Ackermann Steering

Given 119907119863 120575 119897 119889

Unknown 119907119903119903 119907119897119903 119907119903119891 119907119897119891

Solution

120602 =119897

tan 120575

119907119897119903 =120602 minus 1198892 sdot 119907119863

120602

119907119903119903 =120602 + 1198892 sdot 119907119863

120602

119907119897119891 =120602 minus 1198892 2 + 1198972 sdot tan 120575

119897sdot 119907119863

119907119903119891 =1206022 + 1198972 sdot tan 120575

119897sdot 119907119863


120575

120602

rrlabcsuni-klde



Given 120602 120593 119907 119889



1205932 = arctan120549119910 + 119889

1205491199091206022 = 120602 sdot cos120593 cos1205932

1199072 = 119907 sdot 1206022120602

120549119910 = 120602 ∙ 119904119894119899 120593

1205931

1205932

1206021

1206022

120593 120549119909 = 120602 ∙ cos120593

120602

119889

rrlabcsuni-klde

Bicycle Model


119907


Bicycle Model


Low speed motion



119865119910 =1198981199072

120602






Bicycle Model

Given 120575119891 120575119903 119907 120579

Unknown ሶ119883 ሶ119884 ሶ120579

Solution

120575119907 = tanminus1119897119891 tan 120575119903+119897119903 tan 120575119891

119897119891+119897119903

ሶ120579 =119907

120602

120602 =119897119891+119897119903

tan 120575119891 minustan 120575119903 cos 120575119907

ሶ119883 = 119907 cos 120579 + 120575119907ሶ119884 = 119907 sin 120579 + 120575119907

ሶ120579 =119907 cos 120575119907

119897119891+119897119903tan 120575119891 minus tan 120575119903

119907

rrlabcsuni-klde



119862 = 119874 120575119903 = 0 120575119891 = 120575

119897119903 = 0 119897119891 = 119897

120575119907 = 0

ሶ120579 =119907

120602

120602 =119897

tan 120575119891

ሶ119883 = 119907 cos 120579ሶ119884 = 119907 sin 120579ሶ120579 =

119907

119897119891tan 120575119891

119907









120572119897 = 90 deg

120573119897 = 0 deg

120572119903 = minus90 deg

120573119903 = 180 deg




119904 120572119897 + 120573119897 ሶ119909 minus 119888 120572119897 + 120573119897 ሶ119910 minus 119888120573119897 119889 ሶ120579 = 119903119897 ሶ120595119897

119888 120572119897 + 120573119897 ሶ119909 + 119904 120572119897 + 120573119897 ሶ119910 + 119904120573119897 119889 ሶ120579 = 0



ሶ119909 minus 119889 ሶ120579 = 119903119897 ሶ120595119897

ሶ119910 = 0

ሶ119909 + 119889 ሶ120579 = 119903119903 ሶ120595119903119910 = 0




ሶ119909ሶ119910ሶ120579

=

1

2119903119897 ሶ120595119897 + 119903119903 ሶ120595119903

01

2119889minus119903119897 ሶ120595119897 + 119903119903 ሶ120595119903



rrlabcsuni-klde




rrlabcsuni-klde











1205721 = 0 deg

1205722 = 120 deg

1205723 = minus120 deg







119888 1205721 + 1205731 ሶ119909 + 119904 1205721 + 1205731 ሶ119910 + 119889 sdot 119904 1205731 ሶ120579 = 0

119888 1205722 + 1205732 ሶ119909 + 119904 1205722 + 1205732 ሶ119910 + 119889 sdot 119904 1205732 ሶ120579 = 0

119888 1205723 + 1205733 ሶ119909 + 119904 1205723 + 1205733 ሶ119910 + 119889 sdot 119904 1205733 ሶ120579 = 0







119888 120572119894 sdot ሶ119909 + 119904 120572119894 sdot ሶ119910




ሶ120595119894 =1


rrlabcsuni-klde



rrlabcsuni-klde

Mecanum Vehicle





Mecanum Kinematics



1205721 = 45 deg 1205731 = 45 deg 1205741 = minus45 deg

1205722 = 135 deg 1205732 = minus45 deg 1205742 = 45 deg


1205724 = minus45 deg 1205734 = 135 deg 1205744 = 45 deg


Mecanum Kinematics








Mecanum Kinematics


ሶ119909 =119903

4ሶ1205951 + ሶ1205952 + ሶ1205953 + ሶ1205954

ሶ119910 =119903

4minus ሶ1205951 + ሶ1205952 minus ሶ1205953 + ሶ1205954

ሶ120579 =119903

119889 2ሶ1205951 minus ሶ1205952 minus ሶ1205953 + ሶ1205954


Mecanum Kinematics


ሶ119909ሶ119910ሶ120579

=119903119908ℎ1198901198901198974


sdot

ሶ1205951

ሶ1205952

ሶ1205953

ሶ1205954

with a =2 2

119889


ሶ1205954 = ሶ1205951 + ሶ1205952 minus ሶ1205953

rrlabcsuni-klde





Ability to maneuver

Controllability

Stability



Stability




Typical setups




Maneuverability








Controllability



Examples




Wheel type

Drive geometry


Motortransmission


Degree of Mobility




119891119898 = 3 minus rank 119878


Degree of Mobility



1198891 = 1198892 1205731 = 1205732 = 0 1205721 + 120587 = 1205722






119891119904 = rank 119878prime








119891119872 = 119891119898 + 119891119904

rrlabcsuni-klde

Vehicle Kinematics



Tricycle Drive

Given 120575 119907


119903 = 119897 sdot cot 120575

119907prime =119907

cos 120575

119907119897 = 119907 sdot119903 + 119889

119903

119907119903 = 119907 sdot119903 minus 119889


120575


Ackermann Steering

Given 119907119863 120575 119897 119889

Unknown 119907119903119903 119907119897119903 119907119903119891 119907119897119891

Solution

120602 =119897

tan 120575

119907119897119903 =120602 minus 1198892 sdot 119907119863

120602

119907119903119903 =120602 + 1198892 sdot 119907119863

120602

119907119897119891 =120602 minus 1198892 2 + 1198972 sdot tan 120575

119897sdot 119907119863

119907119903119891 =1206022 + 1198972 sdot tan 120575

119897sdot 119907119863


120575

120602

rrlabcsuni-klde



Given 120602 120593 119907 119889



1205932 = arctan120549119910 + 119889

1205491199091206022 = 120602 sdot cos120593 cos1205932

1199072 = 119907 sdot 1206022120602

120549119910 = 120602 ∙ 119904119894119899 120593

1205931

1205932

1206021

1206022

120593 120549119909 = 120602 ∙ cos120593

120602

119889

rrlabcsuni-klde

Bicycle Model


119907


Bicycle Model


Low speed motion



119865119910 =1198981199072

120602






Bicycle Model

Given 120575119891 120575119903 119907 120579

Unknown ሶ119883 ሶ119884 ሶ120579

Solution

120575119907 = tanminus1119897119891 tan 120575119903+119897119903 tan 120575119891

119897119891+119897119903

ሶ120579 =119907

120602

120602 =119897119891+119897119903

tan 120575119891 minustan 120575119903 cos 120575119907

ሶ119883 = 119907 cos 120579 + 120575119907ሶ119884 = 119907 sin 120579 + 120575119907

ሶ120579 =119907 cos 120575119907

119897119891+119897119903tan 120575119891 minus tan 120575119903

119907

rrlabcsuni-klde



119862 = 119874 120575119903 = 0 120575119891 = 120575

119897119903 = 0 119897119891 = 119897

120575119907 = 0

ሶ120579 =119907

120602

120602 =119897

tan 120575119891

ሶ119883 = 119907 cos 120579ሶ119884 = 119907 sin 120579ሶ120579 =

119907

119897119891tan 120575119891

119907




119904 120572119897 + 120573119897 ሶ119909 minus 119888 120572119897 + 120573119897 ሶ119910 minus 119888120573119897 119889 ሶ120579 = 119903119897 ሶ120595119897

119888 120572119897 + 120573119897 ሶ119909 + 119904 120572119897 + 120573119897 ሶ119910 + 119904120573119897 119889 ሶ120579 = 0



ሶ119909 minus 119889 ሶ120579 = 119903119897 ሶ120595119897

ሶ119910 = 0

ሶ119909 + 119889 ሶ120579 = 119903119903 ሶ120595119903119910 = 0




ሶ119909ሶ119910ሶ120579

=

1

2119903119897 ሶ120595119897 + 119903119903 ሶ120595119903

01

2119889minus119903119897 ሶ120595119897 + 119903119903 ሶ120595119903



rrlabcsuni-klde




rrlabcsuni-klde











1205721 = 0 deg

1205722 = 120 deg

1205723 = minus120 deg







119888 1205721 + 1205731 ሶ119909 + 119904 1205721 + 1205731 ሶ119910 + 119889 sdot 119904 1205731 ሶ120579 = 0

119888 1205722 + 1205732 ሶ119909 + 119904 1205722 + 1205732 ሶ119910 + 119889 sdot 119904 1205732 ሶ120579 = 0

119888 1205723 + 1205733 ሶ119909 + 119904 1205723 + 1205733 ሶ119910 + 119889 sdot 119904 1205733 ሶ120579 = 0







119888 120572119894 sdot ሶ119909 + 119904 120572119894 sdot ሶ119910




ሶ120595119894 =1


rrlabcsuni-klde



rrlabcsuni-klde

Mecanum Vehicle





Mecanum Kinematics



1205721 = 45 deg 1205731 = 45 deg 1205741 = minus45 deg

1205722 = 135 deg 1205732 = minus45 deg 1205742 = 45 deg


1205724 = minus45 deg 1205734 = 135 deg 1205744 = 45 deg


Mecanum Kinematics








Mecanum Kinematics


ሶ119909 =119903

4ሶ1205951 + ሶ1205952 + ሶ1205953 + ሶ1205954

ሶ119910 =119903

4minus ሶ1205951 + ሶ1205952 minus ሶ1205953 + ሶ1205954

ሶ120579 =119903

119889 2ሶ1205951 minus ሶ1205952 minus ሶ1205953 + ሶ1205954


Mecanum Kinematics


ሶ119909ሶ119910ሶ120579

=119903119908ℎ1198901198901198974


sdot

ሶ1205951

ሶ1205952

ሶ1205953

ሶ1205954

with a =2 2

119889


ሶ1205954 = ሶ1205951 + ሶ1205952 minus ሶ1205953

rrlabcsuni-klde





Ability to maneuver

Controllability

Stability



Stability




Typical setups




Maneuverability








Controllability



Examples




Wheel type

Drive geometry


Motortransmission


Degree of Mobility




119891119898 = 3 minus rank 119878


Degree of Mobility



1198891 = 1198892 1205731 = 1205732 = 0 1205721 + 120587 = 1205722






119891119904 = rank 119878prime








119891119872 = 119891119898 + 119891119904

rrlabcsuni-klde

Vehicle Kinematics



Tricycle Drive

Given 120575 119907


119903 = 119897 sdot cot 120575

119907prime =119907

cos 120575

119907119897 = 119907 sdot119903 + 119889

119903

119907119903 = 119907 sdot119903 minus 119889


120575


Ackermann Steering

Given 119907119863 120575 119897 119889

Unknown 119907119903119903 119907119897119903 119907119903119891 119907119897119891

Solution

120602 =119897

tan 120575

119907119897119903 =120602 minus 1198892 sdot 119907119863

120602

119907119903119903 =120602 + 1198892 sdot 119907119863

120602

119907119897119891 =120602 minus 1198892 2 + 1198972 sdot tan 120575

119897sdot 119907119863

119907119903119891 =1206022 + 1198972 sdot tan 120575

119897sdot 119907119863


120575

120602

rrlabcsuni-klde



Given 120602 120593 119907 119889



1205932 = arctan120549119910 + 119889

1205491199091206022 = 120602 sdot cos120593 cos1205932

1199072 = 119907 sdot 1206022120602

120549119910 = 120602 ∙ 119904119894119899 120593

1205931

1205932

1206021

1206022

120593 120549119909 = 120602 ∙ cos120593

120602

119889

rrlabcsuni-klde

Bicycle Model


119907


Bicycle Model


Low speed motion



119865119910 =1198981199072

120602






Bicycle Model

Given 120575119891 120575119903 119907 120579

Unknown ሶ119883 ሶ119884 ሶ120579

Solution

120575119907 = tanminus1119897119891 tan 120575119903+119897119903 tan 120575119891

119897119891+119897119903

ሶ120579 =119907

120602

120602 =119897119891+119897119903

tan 120575119891 minustan 120575119903 cos 120575119907

ሶ119883 = 119907 cos 120579 + 120575119907ሶ119884 = 119907 sin 120579 + 120575119907

ሶ120579 =119907 cos 120575119907

119897119891+119897119903tan 120575119891 minus tan 120575119903

119907

rrlabcsuni-klde



119862 = 119874 120575119903 = 0 120575119891 = 120575

119897119903 = 0 119897119891 = 119897

120575119907 = 0

ሶ120579 =119907

120602

120602 =119897

tan 120575119891

ሶ119883 = 119907 cos 120579ሶ119884 = 119907 sin 120579ሶ120579 =

119907

119897119891tan 120575119891

119907




ሶ119909ሶ119910ሶ120579

=

1

2119903119897 ሶ120595119897 + 119903119903 ሶ120595119903

01

2119889minus119903119897 ሶ120595119897 + 119903119903 ሶ120595119903



rrlabcsuni-klde




rrlabcsuni-klde











1205721 = 0 deg

1205722 = 120 deg

1205723 = minus120 deg







119888 1205721 + 1205731 ሶ119909 + 119904 1205721 + 1205731 ሶ119910 + 119889 sdot 119904 1205731 ሶ120579 = 0

119888 1205722 + 1205732 ሶ119909 + 119904 1205722 + 1205732 ሶ119910 + 119889 sdot 119904 1205732 ሶ120579 = 0

119888 1205723 + 1205733 ሶ119909 + 119904 1205723 + 1205733 ሶ119910 + 119889 sdot 119904 1205733 ሶ120579 = 0







119888 120572119894 sdot ሶ119909 + 119904 120572119894 sdot ሶ119910




ሶ120595119894 =1


rrlabcsuni-klde



rrlabcsuni-klde

Mecanum Vehicle





Mecanum Kinematics



1205721 = 45 deg 1205731 = 45 deg 1205741 = minus45 deg

1205722 = 135 deg 1205732 = minus45 deg 1205742 = 45 deg


1205724 = minus45 deg 1205734 = 135 deg 1205744 = 45 deg


Mecanum Kinematics








Mecanum Kinematics


ሶ119909 =119903

4ሶ1205951 + ሶ1205952 + ሶ1205953 + ሶ1205954

ሶ119910 =119903

4minus ሶ1205951 + ሶ1205952 minus ሶ1205953 + ሶ1205954

ሶ120579 =119903

119889 2ሶ1205951 minus ሶ1205952 minus ሶ1205953 + ሶ1205954


Mecanum Kinematics


ሶ119909ሶ119910ሶ120579

=119903119908ℎ1198901198901198974


sdot

ሶ1205951

ሶ1205952

ሶ1205953

ሶ1205954

with a =2 2

119889


ሶ1205954 = ሶ1205951 + ሶ1205952 minus ሶ1205953

rrlabcsuni-klde





Ability to maneuver

Controllability

Stability



Stability




Typical setups




Maneuverability








Controllability



Examples




Wheel type

Drive geometry


Motortransmission


Degree of Mobility




119891119898 = 3 minus rank 119878


Degree of Mobility



1198891 = 1198892 1205731 = 1205732 = 0 1205721 + 120587 = 1205722






119891119904 = rank 119878prime








119891119872 = 119891119898 + 119891119904

rrlabcsuni-klde

Vehicle Kinematics



Tricycle Drive

Given 120575 119907


119903 = 119897 sdot cot 120575

119907prime =119907

cos 120575

119907119897 = 119907 sdot119903 + 119889

119903

119907119903 = 119907 sdot119903 minus 119889


120575


Ackermann Steering

Given 119907119863 120575 119897 119889

Unknown 119907119903119903 119907119897119903 119907119903119891 119907119897119891

Solution

120602 =119897

tan 120575

119907119897119903 =120602 minus 1198892 sdot 119907119863

120602

119907119903119903 =120602 + 1198892 sdot 119907119863

120602

119907119897119891 =120602 minus 1198892 2 + 1198972 sdot tan 120575

119897sdot 119907119863

119907119903119891 =1206022 + 1198972 sdot tan 120575

119897sdot 119907119863


120575

120602

rrlabcsuni-klde



Given 120602 120593 119907 119889



1205932 = arctan120549119910 + 119889

1205491199091206022 = 120602 sdot cos120593 cos1205932

1199072 = 119907 sdot 1206022120602

120549119910 = 120602 ∙ 119904119894119899 120593

1205931

1205932

1206021

1206022

120593 120549119909 = 120602 ∙ cos120593

120602

119889

rrlabcsuni-klde

Bicycle Model


119907


Bicycle Model


Low speed motion



119865119910 =1198981199072

120602






Bicycle Model

Given 120575119891 120575119903 119907 120579

Unknown ሶ119883 ሶ119884 ሶ120579

Solution

120575119907 = tanminus1119897119891 tan 120575119903+119897119903 tan 120575119891

119897119891+119897119903

ሶ120579 =119907

120602

120602 =119897119891+119897119903

tan 120575119891 minustan 120575119903 cos 120575119907

ሶ119883 = 119907 cos 120579 + 120575119907ሶ119884 = 119907 sin 120579 + 120575119907

ሶ120579 =119907 cos 120575119907

119897119891+119897119903tan 120575119891 minus tan 120575119903

119907

rrlabcsuni-klde



119862 = 119874 120575119903 = 0 120575119891 = 120575

119897119903 = 0 119897119891 = 119897

120575119907 = 0

ሶ120579 =119907

120602

120602 =119897

tan 120575119891

ሶ119883 = 119907 cos 120579ሶ119884 = 119907 sin 120579ሶ120579 =

119907

119897119891tan 120575119891

119907



rrlabcsuni-klde




rrlabcsuni-klde











1205721 = 0 deg

1205722 = 120 deg

1205723 = minus120 deg







119888 1205721 + 1205731 ሶ119909 + 119904 1205721 + 1205731 ሶ119910 + 119889 sdot 119904 1205731 ሶ120579 = 0

119888 1205722 + 1205732 ሶ119909 + 119904 1205722 + 1205732 ሶ119910 + 119889 sdot 119904 1205732 ሶ120579 = 0

119888 1205723 + 1205733 ሶ119909 + 119904 1205723 + 1205733 ሶ119910 + 119889 sdot 119904 1205733 ሶ120579 = 0







119888 120572119894 sdot ሶ119909 + 119904 120572119894 sdot ሶ119910




ሶ120595119894 =1


rrlabcsuni-klde



rrlabcsuni-klde

Mecanum Vehicle





Mecanum Kinematics



1205721 = 45 deg 1205731 = 45 deg 1205741 = minus45 deg

1205722 = 135 deg 1205732 = minus45 deg 1205742 = 45 deg


1205724 = minus45 deg 1205734 = 135 deg 1205744 = 45 deg


Mecanum Kinematics








Mecanum Kinematics


ሶ119909 =119903

4ሶ1205951 + ሶ1205952 + ሶ1205953 + ሶ1205954

ሶ119910 =119903

4minus ሶ1205951 + ሶ1205952 minus ሶ1205953 + ሶ1205954

ሶ120579 =119903

119889 2ሶ1205951 minus ሶ1205952 minus ሶ1205953 + ሶ1205954


Mecanum Kinematics


ሶ119909ሶ119910ሶ120579

=119903119908ℎ1198901198901198974


sdot

ሶ1205951

ሶ1205952

ሶ1205953

ሶ1205954

with a =2 2

119889


ሶ1205954 = ሶ1205951 + ሶ1205952 minus ሶ1205953

rrlabcsuni-klde





Ability to maneuver

Controllability

Stability



Stability




Typical setups




Maneuverability








Controllability



Examples




Wheel type

Drive geometry


Motortransmission


Degree of Mobility




119891119898 = 3 minus rank 119878


Degree of Mobility



1198891 = 1198892 1205731 = 1205732 = 0 1205721 + 120587 = 1205722






119891119904 = rank 119878prime








119891119872 = 119891119898 + 119891119904

rrlabcsuni-klde

Vehicle Kinematics



Tricycle Drive

Given 120575 119907


119903 = 119897 sdot cot 120575

119907prime =119907

cos 120575

119907119897 = 119907 sdot119903 + 119889

119903

119907119903 = 119907 sdot119903 minus 119889


120575


Ackermann Steering

Given 119907119863 120575 119897 119889

Unknown 119907119903119903 119907119897119903 119907119903119891 119907119897119891

Solution

120602 =119897

tan 120575

119907119897119903 =120602 minus 1198892 sdot 119907119863

120602

119907119903119903 =120602 + 1198892 sdot 119907119863

120602

119907119897119891 =120602 minus 1198892 2 + 1198972 sdot tan 120575

119897sdot 119907119863

119907119903119891 =1206022 + 1198972 sdot tan 120575

119897sdot 119907119863


120575

120602

rrlabcsuni-klde



Given 120602 120593 119907 119889



1205932 = arctan120549119910 + 119889

1205491199091206022 = 120602 sdot cos120593 cos1205932

1199072 = 119907 sdot 1206022120602

120549119910 = 120602 ∙ 119904119894119899 120593

1205931

1205932

1206021

1206022

120593 120549119909 = 120602 ∙ cos120593

120602

119889

rrlabcsuni-klde

Bicycle Model


119907


Bicycle Model


Low speed motion



119865119910 =1198981199072

120602






Bicycle Model

Given 120575119891 120575119903 119907 120579

Unknown ሶ119883 ሶ119884 ሶ120579

Solution

120575119907 = tanminus1119897119891 tan 120575119903+119897119903 tan 120575119891

119897119891+119897119903

ሶ120579 =119907

120602

120602 =119897119891+119897119903

tan 120575119891 minustan 120575119903 cos 120575119907

ሶ119883 = 119907 cos 120579 + 120575119907ሶ119884 = 119907 sin 120579 + 120575119907

ሶ120579 =119907 cos 120575119907

119897119891+119897119903tan 120575119891 minus tan 120575119903

119907

rrlabcsuni-klde



119862 = 119874 120575119903 = 0 120575119891 = 120575

119897119903 = 0 119897119891 = 119897

120575119907 = 0

ሶ120579 =119907

120602

120602 =119897

tan 120575119891

ሶ119883 = 119907 cos 120579ሶ119884 = 119907 sin 120579ሶ120579 =

119907

119897119891tan 120575119891

119907

rrlabcsuni-klde




rrlabcsuni-klde











1205721 = 0 deg

1205722 = 120 deg

1205723 = minus120 deg







119888 1205721 + 1205731 ሶ119909 + 119904 1205721 + 1205731 ሶ119910 + 119889 sdot 119904 1205731 ሶ120579 = 0

119888 1205722 + 1205732 ሶ119909 + 119904 1205722 + 1205732 ሶ119910 + 119889 sdot 119904 1205732 ሶ120579 = 0

119888 1205723 + 1205733 ሶ119909 + 119904 1205723 + 1205733 ሶ119910 + 119889 sdot 119904 1205733 ሶ120579 = 0







119888 120572119894 sdot ሶ119909 + 119904 120572119894 sdot ሶ119910




ሶ120595119894 =1


rrlabcsuni-klde



rrlabcsuni-klde

Mecanum Vehicle





Mecanum Kinematics



1205721 = 45 deg 1205731 = 45 deg 1205741 = minus45 deg

1205722 = 135 deg 1205732 = minus45 deg 1205742 = 45 deg


1205724 = minus45 deg 1205734 = 135 deg 1205744 = 45 deg


Mecanum Kinematics








Mecanum Kinematics


ሶ119909 =119903

4ሶ1205951 + ሶ1205952 + ሶ1205953 + ሶ1205954

ሶ119910 =119903

4minus ሶ1205951 + ሶ1205952 minus ሶ1205953 + ሶ1205954

ሶ120579 =119903

119889 2ሶ1205951 minus ሶ1205952 minus ሶ1205953 + ሶ1205954


Mecanum Kinematics


ሶ119909ሶ119910ሶ120579

=119903119908ℎ1198901198901198974


sdot

ሶ1205951

ሶ1205952

ሶ1205953

ሶ1205954

with a =2 2

119889


ሶ1205954 = ሶ1205951 + ሶ1205952 minus ሶ1205953

rrlabcsuni-klde





Ability to maneuver

Controllability

Stability



Stability




Typical setups




Maneuverability








Controllability



Examples




Wheel type

Drive geometry


Motortransmission


Degree of Mobility




119891119898 = 3 minus rank 119878


Degree of Mobility



1198891 = 1198892 1205731 = 1205732 = 0 1205721 + 120587 = 1205722






119891119904 = rank 119878prime








119891119872 = 119891119898 + 119891119904

rrlabcsuni-klde

Vehicle Kinematics



Tricycle Drive

Given 120575 119907


119903 = 119897 sdot cot 120575

119907prime =119907

cos 120575

119907119897 = 119907 sdot119903 + 119889

119903

119907119903 = 119907 sdot119903 minus 119889


120575


Ackermann Steering

Given 119907119863 120575 119897 119889

Unknown 119907119903119903 119907119897119903 119907119903119891 119907119897119891

Solution

120602 =119897

tan 120575

119907119897119903 =120602 minus 1198892 sdot 119907119863

120602

119907119903119903 =120602 + 1198892 sdot 119907119863

120602

119907119897119891 =120602 minus 1198892 2 + 1198972 sdot tan 120575

119897sdot 119907119863

119907119903119891 =1206022 + 1198972 sdot tan 120575

119897sdot 119907119863


120575

120602

rrlabcsuni-klde



Given 120602 120593 119907 119889



1205932 = arctan120549119910 + 119889

1205491199091206022 = 120602 sdot cos120593 cos1205932

1199072 = 119907 sdot 1206022120602

120549119910 = 120602 ∙ 119904119894119899 120593

1205931

1205932

1206021

1206022

120593 120549119909 = 120602 ∙ cos120593

120602

119889

rrlabcsuni-klde

Bicycle Model


119907


Bicycle Model


Low speed motion



119865119910 =1198981199072

120602






Bicycle Model

Given 120575119891 120575119903 119907 120579

Unknown ሶ119883 ሶ119884 ሶ120579

Solution

120575119907 = tanminus1119897119891 tan 120575119903+119897119903 tan 120575119891

119897119891+119897119903

ሶ120579 =119907

120602

120602 =119897119891+119897119903

tan 120575119891 minustan 120575119903 cos 120575119907

ሶ119883 = 119907 cos 120579 + 120575119907ሶ119884 = 119907 sin 120579 + 120575119907

ሶ120579 =119907 cos 120575119907

119897119891+119897119903tan 120575119891 minus tan 120575119903

119907

rrlabcsuni-klde



119862 = 119874 120575119903 = 0 120575119891 = 120575

119897119903 = 0 119897119891 = 119897

120575119907 = 0

ሶ120579 =119907

120602

120602 =119897

tan 120575119891

ሶ119883 = 119907 cos 120579ሶ119884 = 119907 sin 120579ሶ120579 =

119907

119897119891tan 120575119891

119907

rrlabcsuni-klde











1205721 = 0 deg

1205722 = 120 deg

1205723 = minus120 deg







119888 1205721 + 1205731 ሶ119909 + 119904 1205721 + 1205731 ሶ119910 + 119889 sdot 119904 1205731 ሶ120579 = 0

119888 1205722 + 1205732 ሶ119909 + 119904 1205722 + 1205732 ሶ119910 + 119889 sdot 119904 1205732 ሶ120579 = 0

119888 1205723 + 1205733 ሶ119909 + 119904 1205723 + 1205733 ሶ119910 + 119889 sdot 119904 1205733 ሶ120579 = 0







119888 120572119894 sdot ሶ119909 + 119904 120572119894 sdot ሶ119910




ሶ120595119894 =1


rrlabcsuni-klde



rrlabcsuni-klde

Mecanum Vehicle





Mecanum Kinematics



1205721 = 45 deg 1205731 = 45 deg 1205741 = minus45 deg

1205722 = 135 deg 1205732 = minus45 deg 1205742 = 45 deg


1205724 = minus45 deg 1205734 = 135 deg 1205744 = 45 deg


Mecanum Kinematics








Mecanum Kinematics


ሶ119909 =119903

4ሶ1205951 + ሶ1205952 + ሶ1205953 + ሶ1205954

ሶ119910 =119903

4minus ሶ1205951 + ሶ1205952 minus ሶ1205953 + ሶ1205954

ሶ120579 =119903

119889 2ሶ1205951 minus ሶ1205952 minus ሶ1205953 + ሶ1205954


Mecanum Kinematics


ሶ119909ሶ119910ሶ120579

=119903119908ℎ1198901198901198974


sdot

ሶ1205951

ሶ1205952

ሶ1205953

ሶ1205954

with a =2 2

119889


ሶ1205954 = ሶ1205951 + ሶ1205952 minus ሶ1205953

rrlabcsuni-klde





Ability to maneuver

Controllability

Stability



Stability




Typical setups




Maneuverability








Controllability



Examples




Wheel type

Drive geometry


Motortransmission


Degree of Mobility




119891119898 = 3 minus rank 119878


Degree of Mobility



1198891 = 1198892 1205731 = 1205732 = 0 1205721 + 120587 = 1205722






119891119904 = rank 119878prime








119891119872 = 119891119898 + 119891119904

rrlabcsuni-klde

Vehicle Kinematics



Tricycle Drive

Given 120575 119907


119903 = 119897 sdot cot 120575

119907prime =119907

cos 120575

119907119897 = 119907 sdot119903 + 119889

119903

119907119903 = 119907 sdot119903 minus 119889


120575


Ackermann Steering

Given 119907119863 120575 119897 119889

Unknown 119907119903119903 119907119897119903 119907119903119891 119907119897119891

Solution

120602 =119897

tan 120575

119907119897119903 =120602 minus 1198892 sdot 119907119863

120602

119907119903119903 =120602 + 1198892 sdot 119907119863

120602

119907119897119891 =120602 minus 1198892 2 + 1198972 sdot tan 120575

119897sdot 119907119863

119907119903119891 =1206022 + 1198972 sdot tan 120575

119897sdot 119907119863


120575

120602

rrlabcsuni-klde



Given 120602 120593 119907 119889



1205932 = arctan120549119910 + 119889

1205491199091206022 = 120602 sdot cos120593 cos1205932

1199072 = 119907 sdot 1206022120602

120549119910 = 120602 ∙ 119904119894119899 120593

1205931

1205932

1206021

1206022

120593 120549119909 = 120602 ∙ cos120593

120602

119889

rrlabcsuni-klde

Bicycle Model


119907


Bicycle Model


Low speed motion



119865119910 =1198981199072

120602






Bicycle Model

Given 120575119891 120575119903 119907 120579

Unknown ሶ119883 ሶ119884 ሶ120579

Solution

120575119907 = tanminus1119897119891 tan 120575119903+119897119903 tan 120575119891

119897119891+119897119903

ሶ120579 =119907

120602

120602 =119897119891+119897119903

tan 120575119891 minustan 120575119903 cos 120575119907

ሶ119883 = 119907 cos 120579 + 120575119907ሶ119884 = 119907 sin 120579 + 120575119907

ሶ120579 =119907 cos 120575119907

119897119891+119897119903tan 120575119891 minus tan 120575119903

119907

rrlabcsuni-klde



119862 = 119874 120575119903 = 0 120575119891 = 120575

119897119903 = 0 119897119891 = 119897

120575119907 = 0

ሶ120579 =119907

120602

120602 =119897

tan 120575119891

ሶ119883 = 119907 cos 120579ሶ119884 = 119907 sin 120579ሶ120579 =

119907

119897119891tan 120575119891

119907








1205721 = 0 deg

1205722 = 120 deg

1205723 = minus120 deg







119888 1205721 + 1205731 ሶ119909 + 119904 1205721 + 1205731 ሶ119910 + 119889 sdot 119904 1205731 ሶ120579 = 0

119888 1205722 + 1205732 ሶ119909 + 119904 1205722 + 1205732 ሶ119910 + 119889 sdot 119904 1205732 ሶ120579 = 0

119888 1205723 + 1205733 ሶ119909 + 119904 1205723 + 1205733 ሶ119910 + 119889 sdot 119904 1205733 ሶ120579 = 0







119888 120572119894 sdot ሶ119909 + 119904 120572119894 sdot ሶ119910




ሶ120595119894 =1


rrlabcsuni-klde



rrlabcsuni-klde

Mecanum Vehicle





Mecanum Kinematics



1205721 = 45 deg 1205731 = 45 deg 1205741 = minus45 deg

1205722 = 135 deg 1205732 = minus45 deg 1205742 = 45 deg


1205724 = minus45 deg 1205734 = 135 deg 1205744 = 45 deg


Mecanum Kinematics








Mecanum Kinematics


ሶ119909 =119903

4ሶ1205951 + ሶ1205952 + ሶ1205953 + ሶ1205954

ሶ119910 =119903

4minus ሶ1205951 + ሶ1205952 minus ሶ1205953 + ሶ1205954

ሶ120579 =119903

119889 2ሶ1205951 minus ሶ1205952 minus ሶ1205953 + ሶ1205954


Mecanum Kinematics


ሶ119909ሶ119910ሶ120579

=119903119908ℎ1198901198901198974


sdot

ሶ1205951

ሶ1205952

ሶ1205953

ሶ1205954

with a =2 2

119889


ሶ1205954 = ሶ1205951 + ሶ1205952 minus ሶ1205953

rrlabcsuni-klde





Ability to maneuver

Controllability

Stability



Stability




Typical setups




Maneuverability








Controllability



Examples




Wheel type

Drive geometry


Motortransmission


Degree of Mobility




119891119898 = 3 minus rank 119878


Degree of Mobility



1198891 = 1198892 1205731 = 1205732 = 0 1205721 + 120587 = 1205722






119891119904 = rank 119878prime








119891119872 = 119891119898 + 119891119904

rrlabcsuni-klde

Vehicle Kinematics



Tricycle Drive

Given 120575 119907


119903 = 119897 sdot cot 120575

119907prime =119907

cos 120575

119907119897 = 119907 sdot119903 + 119889

119903

119907119903 = 119907 sdot119903 minus 119889


120575


Ackermann Steering

Given 119907119863 120575 119897 119889

Unknown 119907119903119903 119907119897119903 119907119903119891 119907119897119891

Solution

120602 =119897

tan 120575

119907119897119903 =120602 minus 1198892 sdot 119907119863

120602

119907119903119903 =120602 + 1198892 sdot 119907119863

120602

119907119897119891 =120602 minus 1198892 2 + 1198972 sdot tan 120575

119897sdot 119907119863

119907119903119891 =1206022 + 1198972 sdot tan 120575

119897sdot 119907119863


120575

120602

rrlabcsuni-klde



Given 120602 120593 119907 119889



1205932 = arctan120549119910 + 119889

1205491199091206022 = 120602 sdot cos120593 cos1205932

1199072 = 119907 sdot 1206022120602

120549119910 = 120602 ∙ 119904119894119899 120593

1205931

1205932

1206021

1206022

120593 120549119909 = 120602 ∙ cos120593

120602

119889

rrlabcsuni-klde

Bicycle Model


119907


Bicycle Model


Low speed motion



119865119910 =1198981199072

120602






Bicycle Model

Given 120575119891 120575119903 119907 120579

Unknown ሶ119883 ሶ119884 ሶ120579

Solution

120575119907 = tanminus1119897119891 tan 120575119903+119897119903 tan 120575119891

119897119891+119897119903

ሶ120579 =119907

120602

120602 =119897119891+119897119903

tan 120575119891 minustan 120575119903 cos 120575119907

ሶ119883 = 119907 cos 120579 + 120575119907ሶ119884 = 119907 sin 120579 + 120575119907

ሶ120579 =119907 cos 120575119907

119897119891+119897119903tan 120575119891 minus tan 120575119903

119907

rrlabcsuni-klde



119862 = 119874 120575119903 = 0 120575119891 = 120575

119897119903 = 0 119897119891 = 119897

120575119907 = 0

ሶ120579 =119907

120602

120602 =119897

tan 120575119891

ሶ119883 = 119907 cos 120579ሶ119884 = 119907 sin 120579ሶ120579 =

119907

119897119891tan 120575119891

119907







119888 1205721 + 1205731 ሶ119909 + 119904 1205721 + 1205731 ሶ119910 + 119889 sdot 119904 1205731 ሶ120579 = 0

119888 1205722 + 1205732 ሶ119909 + 119904 1205722 + 1205732 ሶ119910 + 119889 sdot 119904 1205732 ሶ120579 = 0

119888 1205723 + 1205733 ሶ119909 + 119904 1205723 + 1205733 ሶ119910 + 119889 sdot 119904 1205733 ሶ120579 = 0







119888 120572119894 sdot ሶ119909 + 119904 120572119894 sdot ሶ119910




ሶ120595119894 =1


rrlabcsuni-klde



rrlabcsuni-klde

Mecanum Vehicle





Mecanum Kinematics



1205721 = 45 deg 1205731 = 45 deg 1205741 = minus45 deg

1205722 = 135 deg 1205732 = minus45 deg 1205742 = 45 deg


1205724 = minus45 deg 1205734 = 135 deg 1205744 = 45 deg


Mecanum Kinematics








Mecanum Kinematics


ሶ119909 =119903

4ሶ1205951 + ሶ1205952 + ሶ1205953 + ሶ1205954

ሶ119910 =119903

4minus ሶ1205951 + ሶ1205952 minus ሶ1205953 + ሶ1205954

ሶ120579 =119903

119889 2ሶ1205951 minus ሶ1205952 minus ሶ1205953 + ሶ1205954


Mecanum Kinematics


ሶ119909ሶ119910ሶ120579

=119903119908ℎ1198901198901198974


sdot

ሶ1205951

ሶ1205952

ሶ1205953

ሶ1205954

with a =2 2

119889


ሶ1205954 = ሶ1205951 + ሶ1205952 minus ሶ1205953

rrlabcsuni-klde





Ability to maneuver

Controllability

Stability



Stability




Typical setups




Maneuverability








Controllability



Examples




Wheel type

Drive geometry


Motortransmission


Degree of Mobility




119891119898 = 3 minus rank 119878


Degree of Mobility



1198891 = 1198892 1205731 = 1205732 = 0 1205721 + 120587 = 1205722






119891119904 = rank 119878prime








119891119872 = 119891119898 + 119891119904

rrlabcsuni-klde

Vehicle Kinematics



Tricycle Drive

Given 120575 119907


119903 = 119897 sdot cot 120575

119907prime =119907

cos 120575

119907119897 = 119907 sdot119903 + 119889

119903

119907119903 = 119907 sdot119903 minus 119889


120575


Ackermann Steering

Given 119907119863 120575 119897 119889

Unknown 119907119903119903 119907119897119903 119907119903119891 119907119897119891

Solution

120602 =119897

tan 120575

119907119897119903 =120602 minus 1198892 sdot 119907119863

120602

119907119903119903 =120602 + 1198892 sdot 119907119863

120602

119907119897119891 =120602 minus 1198892 2 + 1198972 sdot tan 120575

119897sdot 119907119863

119907119903119891 =1206022 + 1198972 sdot tan 120575

119897sdot 119907119863


120575

120602

rrlabcsuni-klde



Given 120602 120593 119907 119889



1205932 = arctan120549119910 + 119889

1205491199091206022 = 120602 sdot cos120593 cos1205932

1199072 = 119907 sdot 1206022120602

120549119910 = 120602 ∙ 119904119894119899 120593

1205931

1205932

1206021

1206022

120593 120549119909 = 120602 ∙ cos120593

120602

119889

rrlabcsuni-klde

Bicycle Model


119907


Bicycle Model


Low speed motion



119865119910 =1198981199072

120602






Bicycle Model

Given 120575119891 120575119903 119907 120579

Unknown ሶ119883 ሶ119884 ሶ120579

Solution

120575119907 = tanminus1119897119891 tan 120575119903+119897119903 tan 120575119891

119897119891+119897119903

ሶ120579 =119907

120602

120602 =119897119891+119897119903

tan 120575119891 minustan 120575119903 cos 120575119907

ሶ119883 = 119907 cos 120579 + 120575119907ሶ119884 = 119907 sin 120579 + 120575119907

ሶ120579 =119907 cos 120575119907

119897119891+119897119903tan 120575119891 minus tan 120575119903

119907

rrlabcsuni-klde



119862 = 119874 120575119903 = 0 120575119891 = 120575

119897119903 = 0 119897119891 = 119897

120575119907 = 0

ሶ120579 =119907

120602

120602 =119897

tan 120575119891

ሶ119883 = 119907 cos 120579ሶ119884 = 119907 sin 120579ሶ120579 =

119907

119897119891tan 120575119891

119907







119888 120572119894 sdot ሶ119909 + 119904 120572119894 sdot ሶ119910




ሶ120595119894 =1


rrlabcsuni-klde



rrlabcsuni-klde

Mecanum Vehicle





Mecanum Kinematics



1205721 = 45 deg 1205731 = 45 deg 1205741 = minus45 deg

1205722 = 135 deg 1205732 = minus45 deg 1205742 = 45 deg


1205724 = minus45 deg 1205734 = 135 deg 1205744 = 45 deg


Mecanum Kinematics








Mecanum Kinematics


ሶ119909 =119903

4ሶ1205951 + ሶ1205952 + ሶ1205953 + ሶ1205954

ሶ119910 =119903

4minus ሶ1205951 + ሶ1205952 minus ሶ1205953 + ሶ1205954

ሶ120579 =119903

119889 2ሶ1205951 minus ሶ1205952 minus ሶ1205953 + ሶ1205954


Mecanum Kinematics


ሶ119909ሶ119910ሶ120579

=119903119908ℎ1198901198901198974


sdot

ሶ1205951

ሶ1205952

ሶ1205953

ሶ1205954

with a =2 2

119889


ሶ1205954 = ሶ1205951 + ሶ1205952 minus ሶ1205953

rrlabcsuni-klde





Ability to maneuver

Controllability

Stability



Stability




Typical setups




Maneuverability








Controllability



Examples




Wheel type

Drive geometry


Motortransmission


Degree of Mobility




119891119898 = 3 minus rank 119878


Degree of Mobility



1198891 = 1198892 1205731 = 1205732 = 0 1205721 + 120587 = 1205722






119891119904 = rank 119878prime








119891119872 = 119891119898 + 119891119904

rrlabcsuni-klde

Vehicle Kinematics



Tricycle Drive

Given 120575 119907


119903 = 119897 sdot cot 120575

119907prime =119907

cos 120575

119907119897 = 119907 sdot119903 + 119889

119903

119907119903 = 119907 sdot119903 minus 119889


120575


Ackermann Steering

Given 119907119863 120575 119897 119889

Unknown 119907119903119903 119907119897119903 119907119903119891 119907119897119891

Solution

120602 =119897

tan 120575

119907119897119903 =120602 minus 1198892 sdot 119907119863

120602

119907119903119903 =120602 + 1198892 sdot 119907119863

120602

119907119897119891 =120602 minus 1198892 2 + 1198972 sdot tan 120575

119897sdot 119907119863

119907119903119891 =1206022 + 1198972 sdot tan 120575

119897sdot 119907119863


120575

120602

rrlabcsuni-klde



Given 120602 120593 119907 119889



1205932 = arctan120549119910 + 119889

1205491199091206022 = 120602 sdot cos120593 cos1205932

1199072 = 119907 sdot 1206022120602

120549119910 = 120602 ∙ 119904119894119899 120593

1205931

1205932

1206021

1206022

120593 120549119909 = 120602 ∙ cos120593

120602

119889

rrlabcsuni-klde

Bicycle Model


119907


Bicycle Model


Low speed motion



119865119910 =1198981199072

120602






Bicycle Model

Given 120575119891 120575119903 119907 120579

Unknown ሶ119883 ሶ119884 ሶ120579

Solution

120575119907 = tanminus1119897119891 tan 120575119903+119897119903 tan 120575119891

119897119891+119897119903

ሶ120579 =119907

120602

120602 =119897119891+119897119903

tan 120575119891 minustan 120575119903 cos 120575119907

ሶ119883 = 119907 cos 120579 + 120575119907ሶ119884 = 119907 sin 120579 + 120575119907

ሶ120579 =119907 cos 120575119907

119897119891+119897119903tan 120575119891 minus tan 120575119903

119907

rrlabcsuni-klde



119862 = 119874 120575119903 = 0 120575119891 = 120575

119897119903 = 0 119897119891 = 119897

120575119907 = 0

ሶ120579 =119907

120602

120602 =119897

tan 120575119891

ሶ119883 = 119907 cos 120579ሶ119884 = 119907 sin 120579ሶ120579 =

119907

119897119891tan 120575119891

119907

rrlabcsuni-klde



rrlabcsuni-klde

Mecanum Vehicle





Mecanum Kinematics



1205721 = 45 deg 1205731 = 45 deg 1205741 = minus45 deg

1205722 = 135 deg 1205732 = minus45 deg 1205742 = 45 deg


1205724 = minus45 deg 1205734 = 135 deg 1205744 = 45 deg


Mecanum Kinematics








Mecanum Kinematics


ሶ119909 =119903

4ሶ1205951 + ሶ1205952 + ሶ1205953 + ሶ1205954

ሶ119910 =119903

4minus ሶ1205951 + ሶ1205952 minus ሶ1205953 + ሶ1205954

ሶ120579 =119903

119889 2ሶ1205951 minus ሶ1205952 minus ሶ1205953 + ሶ1205954


Mecanum Kinematics


ሶ119909ሶ119910ሶ120579

=119903119908ℎ1198901198901198974


sdot

ሶ1205951

ሶ1205952

ሶ1205953

ሶ1205954

with a =2 2

119889


ሶ1205954 = ሶ1205951 + ሶ1205952 minus ሶ1205953

rrlabcsuni-klde





Ability to maneuver

Controllability

Stability



Stability




Typical setups




Maneuverability








Controllability



Examples




Wheel type

Drive geometry


Motortransmission


Degree of Mobility




119891119898 = 3 minus rank 119878


Degree of Mobility



1198891 = 1198892 1205731 = 1205732 = 0 1205721 + 120587 = 1205722






119891119904 = rank 119878prime








119891119872 = 119891119898 + 119891119904

rrlabcsuni-klde

Vehicle Kinematics



Tricycle Drive

Given 120575 119907


119903 = 119897 sdot cot 120575

119907prime =119907

cos 120575

119907119897 = 119907 sdot119903 + 119889

119903

119907119903 = 119907 sdot119903 minus 119889


120575


Ackermann Steering

Given 119907119863 120575 119897 119889

Unknown 119907119903119903 119907119897119903 119907119903119891 119907119897119891

Solution

120602 =119897

tan 120575

119907119897119903 =120602 minus 1198892 sdot 119907119863

120602

119907119903119903 =120602 + 1198892 sdot 119907119863

120602

119907119897119891 =120602 minus 1198892 2 + 1198972 sdot tan 120575

119897sdot 119907119863

119907119903119891 =1206022 + 1198972 sdot tan 120575

119897sdot 119907119863


120575

120602

rrlabcsuni-klde



Given 120602 120593 119907 119889



1205932 = arctan120549119910 + 119889

1205491199091206022 = 120602 sdot cos120593 cos1205932

1199072 = 119907 sdot 1206022120602

120549119910 = 120602 ∙ 119904119894119899 120593

1205931

1205932

1206021

1206022

120593 120549119909 = 120602 ∙ cos120593

120602

119889

rrlabcsuni-klde

Bicycle Model


119907


Bicycle Model


Low speed motion



119865119910 =1198981199072

120602






Bicycle Model

Given 120575119891 120575119903 119907 120579

Unknown ሶ119883 ሶ119884 ሶ120579

Solution

120575119907 = tanminus1119897119891 tan 120575119903+119897119903 tan 120575119891

119897119891+119897119903

ሶ120579 =119907

120602

120602 =119897119891+119897119903

tan 120575119891 minustan 120575119903 cos 120575119907

ሶ119883 = 119907 cos 120579 + 120575119907ሶ119884 = 119907 sin 120579 + 120575119907

ሶ120579 =119907 cos 120575119907

119897119891+119897119903tan 120575119891 minus tan 120575119903

119907

rrlabcsuni-klde



119862 = 119874 120575119903 = 0 120575119891 = 120575

119897119903 = 0 119897119891 = 119897

120575119907 = 0

ሶ120579 =119907

120602

120602 =119897

tan 120575119891

ሶ119883 = 119907 cos 120579ሶ119884 = 119907 sin 120579ሶ120579 =

119907

119897119891tan 120575119891

119907

rrlabcsuni-klde

Mecanum Vehicle





Mecanum Kinematics



1205721 = 45 deg 1205731 = 45 deg 1205741 = minus45 deg

1205722 = 135 deg 1205732 = minus45 deg 1205742 = 45 deg


1205724 = minus45 deg 1205734 = 135 deg 1205744 = 45 deg


Mecanum Kinematics








Mecanum Kinematics


ሶ119909 =119903

4ሶ1205951 + ሶ1205952 + ሶ1205953 + ሶ1205954

ሶ119910 =119903

4minus ሶ1205951 + ሶ1205952 minus ሶ1205953 + ሶ1205954

ሶ120579 =119903

119889 2ሶ1205951 minus ሶ1205952 minus ሶ1205953 + ሶ1205954


Mecanum Kinematics


ሶ119909ሶ119910ሶ120579

=119903119908ℎ1198901198901198974


sdot

ሶ1205951

ሶ1205952

ሶ1205953

ሶ1205954

with a =2 2

119889


ሶ1205954 = ሶ1205951 + ሶ1205952 minus ሶ1205953

rrlabcsuni-klde





Ability to maneuver

Controllability

Stability



Stability




Typical setups




Maneuverability








Controllability



Examples




Wheel type

Drive geometry


Motortransmission


Degree of Mobility




119891119898 = 3 minus rank 119878


Degree of Mobility



1198891 = 1198892 1205731 = 1205732 = 0 1205721 + 120587 = 1205722






119891119904 = rank 119878prime








119891119872 = 119891119898 + 119891119904

rrlabcsuni-klde

Vehicle Kinematics



Tricycle Drive

Given 120575 119907


119903 = 119897 sdot cot 120575

119907prime =119907

cos 120575

119907119897 = 119907 sdot119903 + 119889

119903

119907119903 = 119907 sdot119903 minus 119889


120575


Ackermann Steering

Given 119907119863 120575 119897 119889

Unknown 119907119903119903 119907119897119903 119907119903119891 119907119897119891

Solution

120602 =119897

tan 120575

119907119897119903 =120602 minus 1198892 sdot 119907119863

120602

119907119903119903 =120602 + 1198892 sdot 119907119863

120602

119907119897119891 =120602 minus 1198892 2 + 1198972 sdot tan 120575

119897sdot 119907119863

119907119903119891 =1206022 + 1198972 sdot tan 120575

119897sdot 119907119863


120575

120602

rrlabcsuni-klde



Given 120602 120593 119907 119889



1205932 = arctan120549119910 + 119889

1205491199091206022 = 120602 sdot cos120593 cos1205932

1199072 = 119907 sdot 1206022120602

120549119910 = 120602 ∙ 119904119894119899 120593

1205931

1205932

1206021

1206022

120593 120549119909 = 120602 ∙ cos120593

120602

119889

rrlabcsuni-klde

Bicycle Model


119907


Bicycle Model


Low speed motion



119865119910 =1198981199072

120602






Bicycle Model

Given 120575119891 120575119903 119907 120579

Unknown ሶ119883 ሶ119884 ሶ120579

Solution

120575119907 = tanminus1119897119891 tan 120575119903+119897119903 tan 120575119891

119897119891+119897119903

ሶ120579 =119907

120602

120602 =119897119891+119897119903

tan 120575119891 minustan 120575119903 cos 120575119907

ሶ119883 = 119907 cos 120579 + 120575119907ሶ119884 = 119907 sin 120579 + 120575119907

ሶ120579 =119907 cos 120575119907

119897119891+119897119903tan 120575119891 minus tan 120575119903

119907

rrlabcsuni-klde



119862 = 119874 120575119903 = 0 120575119891 = 120575

119897119903 = 0 119897119891 = 119897

120575119907 = 0

ሶ120579 =119907

120602

120602 =119897

tan 120575119891

ሶ119883 = 119907 cos 120579ሶ119884 = 119907 sin 120579ሶ120579 =

119907

119897119891tan 120575119891

119907


Mecanum Kinematics



1205721 = 45 deg 1205731 = 45 deg 1205741 = minus45 deg

1205722 = 135 deg 1205732 = minus45 deg 1205742 = 45 deg


1205724 = minus45 deg 1205734 = 135 deg 1205744 = 45 deg


Mecanum Kinematics








Mecanum Kinematics


ሶ119909 =119903

4ሶ1205951 + ሶ1205952 + ሶ1205953 + ሶ1205954

ሶ119910 =119903

4minus ሶ1205951 + ሶ1205952 minus ሶ1205953 + ሶ1205954

ሶ120579 =119903

119889 2ሶ1205951 minus ሶ1205952 minus ሶ1205953 + ሶ1205954


Mecanum Kinematics


ሶ119909ሶ119910ሶ120579

=119903119908ℎ1198901198901198974


sdot

ሶ1205951

ሶ1205952

ሶ1205953

ሶ1205954

with a =2 2

119889


ሶ1205954 = ሶ1205951 + ሶ1205952 minus ሶ1205953

rrlabcsuni-klde





Ability to maneuver

Controllability

Stability



Stability




Typical setups




Maneuverability








Controllability



Examples




Wheel type

Drive geometry


Motortransmission


Degree of Mobility




119891119898 = 3 minus rank 119878


Degree of Mobility



1198891 = 1198892 1205731 = 1205732 = 0 1205721 + 120587 = 1205722






119891119904 = rank 119878prime








119891119872 = 119891119898 + 119891119904

rrlabcsuni-klde

Vehicle Kinematics



Tricycle Drive

Given 120575 119907


119903 = 119897 sdot cot 120575

119907prime =119907

cos 120575

119907119897 = 119907 sdot119903 + 119889

119903

119907119903 = 119907 sdot119903 minus 119889


120575


Ackermann Steering

Given 119907119863 120575 119897 119889

Unknown 119907119903119903 119907119897119903 119907119903119891 119907119897119891

Solution

120602 =119897

tan 120575

119907119897119903 =120602 minus 1198892 sdot 119907119863

120602

119907119903119903 =120602 + 1198892 sdot 119907119863

120602

119907119897119891 =120602 minus 1198892 2 + 1198972 sdot tan 120575

119897sdot 119907119863

119907119903119891 =1206022 + 1198972 sdot tan 120575

119897sdot 119907119863


120575

120602

rrlabcsuni-klde



Given 120602 120593 119907 119889



1205932 = arctan120549119910 + 119889

1205491199091206022 = 120602 sdot cos120593 cos1205932

1199072 = 119907 sdot 1206022120602

120549119910 = 120602 ∙ 119904119894119899 120593

1205931

1205932

1206021

1206022

120593 120549119909 = 120602 ∙ cos120593

120602

119889

rrlabcsuni-klde

Bicycle Model


119907


Bicycle Model


Low speed motion



119865119910 =1198981199072

120602






Bicycle Model

Given 120575119891 120575119903 119907 120579

Unknown ሶ119883 ሶ119884 ሶ120579

Solution

120575119907 = tanminus1119897119891 tan 120575119903+119897119903 tan 120575119891

119897119891+119897119903

ሶ120579 =119907

120602

120602 =119897119891+119897119903

tan 120575119891 minustan 120575119903 cos 120575119907

ሶ119883 = 119907 cos 120579 + 120575119907ሶ119884 = 119907 sin 120579 + 120575119907

ሶ120579 =119907 cos 120575119907

119897119891+119897119903tan 120575119891 minus tan 120575119903

119907

rrlabcsuni-klde



119862 = 119874 120575119903 = 0 120575119891 = 120575

119897119903 = 0 119897119891 = 119897

120575119907 = 0

ሶ120579 =119907

120602

120602 =119897

tan 120575119891

ሶ119883 = 119907 cos 120579ሶ119884 = 119907 sin 120579ሶ120579 =

119907

119897119891tan 120575119891

119907


Mecanum Kinematics








Mecanum Kinematics


ሶ119909 =119903

4ሶ1205951 + ሶ1205952 + ሶ1205953 + ሶ1205954

ሶ119910 =119903

4minus ሶ1205951 + ሶ1205952 minus ሶ1205953 + ሶ1205954

ሶ120579 =119903

119889 2ሶ1205951 minus ሶ1205952 minus ሶ1205953 + ሶ1205954


Mecanum Kinematics


ሶ119909ሶ119910ሶ120579

=119903119908ℎ1198901198901198974


sdot

ሶ1205951

ሶ1205952

ሶ1205953

ሶ1205954

with a =2 2

119889


ሶ1205954 = ሶ1205951 + ሶ1205952 minus ሶ1205953

rrlabcsuni-klde





Ability to maneuver

Controllability

Stability



Stability




Typical setups




Maneuverability








Controllability



Examples




Wheel type

Drive geometry


Motortransmission


Degree of Mobility




119891119898 = 3 minus rank 119878


Degree of Mobility



1198891 = 1198892 1205731 = 1205732 = 0 1205721 + 120587 = 1205722






119891119904 = rank 119878prime








119891119872 = 119891119898 + 119891119904

rrlabcsuni-klde

Vehicle Kinematics



Tricycle Drive

Given 120575 119907


119903 = 119897 sdot cot 120575

119907prime =119907

cos 120575

119907119897 = 119907 sdot119903 + 119889

119903

119907119903 = 119907 sdot119903 minus 119889


120575


Ackermann Steering

Given 119907119863 120575 119897 119889

Unknown 119907119903119903 119907119897119903 119907119903119891 119907119897119891

Solution

120602 =119897

tan 120575

119907119897119903 =120602 minus 1198892 sdot 119907119863

120602

119907119903119903 =120602 + 1198892 sdot 119907119863

120602

119907119897119891 =120602 minus 1198892 2 + 1198972 sdot tan 120575

119897sdot 119907119863

119907119903119891 =1206022 + 1198972 sdot tan 120575

119897sdot 119907119863


120575

120602

rrlabcsuni-klde



Given 120602 120593 119907 119889



1205932 = arctan120549119910 + 119889

1205491199091206022 = 120602 sdot cos120593 cos1205932

1199072 = 119907 sdot 1206022120602

120549119910 = 120602 ∙ 119904119894119899 120593

1205931

1205932

1206021

1206022

120593 120549119909 = 120602 ∙ cos120593

120602

119889

rrlabcsuni-klde

Bicycle Model


119907


Bicycle Model


Low speed motion



119865119910 =1198981199072

120602






Bicycle Model

Given 120575119891 120575119903 119907 120579

Unknown ሶ119883 ሶ119884 ሶ120579

Solution

120575119907 = tanminus1119897119891 tan 120575119903+119897119903 tan 120575119891

119897119891+119897119903

ሶ120579 =119907

120602

120602 =119897119891+119897119903

tan 120575119891 minustan 120575119903 cos 120575119907

ሶ119883 = 119907 cos 120579 + 120575119907ሶ119884 = 119907 sin 120579 + 120575119907

ሶ120579 =119907 cos 120575119907

119897119891+119897119903tan 120575119891 minus tan 120575119903

119907

rrlabcsuni-klde



119862 = 119874 120575119903 = 0 120575119891 = 120575

119897119903 = 0 119897119891 = 119897

120575119907 = 0

ሶ120579 =119907

120602

120602 =119897

tan 120575119891

ሶ119883 = 119907 cos 120579ሶ119884 = 119907 sin 120579ሶ120579 =

119907

119897119891tan 120575119891

119907


Mecanum Kinematics


ሶ119909 =119903

4ሶ1205951 + ሶ1205952 + ሶ1205953 + ሶ1205954

ሶ119910 =119903

4minus ሶ1205951 + ሶ1205952 minus ሶ1205953 + ሶ1205954

ሶ120579 =119903

119889 2ሶ1205951 minus ሶ1205952 minus ሶ1205953 + ሶ1205954


Mecanum Kinematics


ሶ119909ሶ119910ሶ120579

=119903119908ℎ1198901198901198974


sdot

ሶ1205951

ሶ1205952

ሶ1205953

ሶ1205954

with a =2 2

119889


ሶ1205954 = ሶ1205951 + ሶ1205952 minus ሶ1205953

rrlabcsuni-klde





Ability to maneuver

Controllability

Stability



Stability




Typical setups




Maneuverability








Controllability



Examples




Wheel type

Drive geometry


Motortransmission


Degree of Mobility




119891119898 = 3 minus rank 119878


Degree of Mobility



1198891 = 1198892 1205731 = 1205732 = 0 1205721 + 120587 = 1205722






119891119904 = rank 119878prime








119891119872 = 119891119898 + 119891119904

rrlabcsuni-klde

Vehicle Kinematics



Tricycle Drive

Given 120575 119907


119903 = 119897 sdot cot 120575

119907prime =119907

cos 120575

119907119897 = 119907 sdot119903 + 119889

119903

119907119903 = 119907 sdot119903 minus 119889


120575


Ackermann Steering

Given 119907119863 120575 119897 119889

Unknown 119907119903119903 119907119897119903 119907119903119891 119907119897119891

Solution

120602 =119897

tan 120575

119907119897119903 =120602 minus 1198892 sdot 119907119863

120602

119907119903119903 =120602 + 1198892 sdot 119907119863

120602

119907119897119891 =120602 minus 1198892 2 + 1198972 sdot tan 120575

119897sdot 119907119863

119907119903119891 =1206022 + 1198972 sdot tan 120575

119897sdot 119907119863


120575

120602

rrlabcsuni-klde



Given 120602 120593 119907 119889



1205932 = arctan120549119910 + 119889

1205491199091206022 = 120602 sdot cos120593 cos1205932

1199072 = 119907 sdot 1206022120602

120549119910 = 120602 ∙ 119904119894119899 120593

1205931

1205932

1206021

1206022

120593 120549119909 = 120602 ∙ cos120593

120602

119889

rrlabcsuni-klde

Bicycle Model


119907


Bicycle Model


Low speed motion



119865119910 =1198981199072

120602






Bicycle Model

Given 120575119891 120575119903 119907 120579

Unknown ሶ119883 ሶ119884 ሶ120579

Solution

120575119907 = tanminus1119897119891 tan 120575119903+119897119903 tan 120575119891

119897119891+119897119903

ሶ120579 =119907

120602

120602 =119897119891+119897119903

tan 120575119891 minustan 120575119903 cos 120575119907

ሶ119883 = 119907 cos 120579 + 120575119907ሶ119884 = 119907 sin 120579 + 120575119907

ሶ120579 =119907 cos 120575119907

119897119891+119897119903tan 120575119891 minus tan 120575119903

119907

rrlabcsuni-klde



119862 = 119874 120575119903 = 0 120575119891 = 120575

119897119903 = 0 119897119891 = 119897

120575119907 = 0

ሶ120579 =119907

120602

120602 =119897

tan 120575119891

ሶ119883 = 119907 cos 120579ሶ119884 = 119907 sin 120579ሶ120579 =

119907

119897119891tan 120575119891

119907


Mecanum Kinematics


ሶ119909ሶ119910ሶ120579

=119903119908ℎ1198901198901198974


sdot

ሶ1205951

ሶ1205952

ሶ1205953

ሶ1205954

with a =2 2

119889


ሶ1205954 = ሶ1205951 + ሶ1205952 minus ሶ1205953

rrlabcsuni-klde





Ability to maneuver

Controllability

Stability



Stability




Typical setups




Maneuverability








Controllability



Examples




Wheel type

Drive geometry


Motortransmission


Degree of Mobility




119891119898 = 3 minus rank 119878


Degree of Mobility



1198891 = 1198892 1205731 = 1205732 = 0 1205721 + 120587 = 1205722






119891119904 = rank 119878prime








119891119872 = 119891119898 + 119891119904

rrlabcsuni-klde

Vehicle Kinematics



Tricycle Drive

Given 120575 119907


119903 = 119897 sdot cot 120575

119907prime =119907

cos 120575

119907119897 = 119907 sdot119903 + 119889

119903

119907119903 = 119907 sdot119903 minus 119889


120575


Ackermann Steering

Given 119907119863 120575 119897 119889

Unknown 119907119903119903 119907119897119903 119907119903119891 119907119897119891

Solution

120602 =119897

tan 120575

119907119897119903 =120602 minus 1198892 sdot 119907119863

120602

119907119903119903 =120602 + 1198892 sdot 119907119863

120602

119907119897119891 =120602 minus 1198892 2 + 1198972 sdot tan 120575

119897sdot 119907119863

119907119903119891 =1206022 + 1198972 sdot tan 120575

119897sdot 119907119863


120575

120602

rrlabcsuni-klde



Given 120602 120593 119907 119889



1205932 = arctan120549119910 + 119889

1205491199091206022 = 120602 sdot cos120593 cos1205932

1199072 = 119907 sdot 1206022120602

120549119910 = 120602 ∙ 119904119894119899 120593

1205931

1205932

1206021

1206022

120593 120549119909 = 120602 ∙ cos120593

120602

119889

rrlabcsuni-klde

Bicycle Model


119907


Bicycle Model


Low speed motion



119865119910 =1198981199072

120602






Bicycle Model

Given 120575119891 120575119903 119907 120579

Unknown ሶ119883 ሶ119884 ሶ120579

Solution

120575119907 = tanminus1119897119891 tan 120575119903+119897119903 tan 120575119891

119897119891+119897119903

ሶ120579 =119907

120602

120602 =119897119891+119897119903

tan 120575119891 minustan 120575119903 cos 120575119907

ሶ119883 = 119907 cos 120579 + 120575119907ሶ119884 = 119907 sin 120579 + 120575119907

ሶ120579 =119907 cos 120575119907

119897119891+119897119903tan 120575119891 minus tan 120575119903

119907

rrlabcsuni-klde



119862 = 119874 120575119903 = 0 120575119891 = 120575

119897119903 = 0 119897119891 = 119897

120575119907 = 0

ሶ120579 =119907

120602

120602 =119897

tan 120575119891

ሶ119883 = 119907 cos 120579ሶ119884 = 119907 sin 120579ሶ120579 =

119907

119897119891tan 120575119891

119907

rrlabcsuni-klde





Ability to maneuver

Controllability

Stability



Stability




Typical setups




Maneuverability








Controllability



Examples




Wheel type

Drive geometry


Motortransmission


Degree of Mobility




119891119898 = 3 minus rank 119878


Degree of Mobility



1198891 = 1198892 1205731 = 1205732 = 0 1205721 + 120587 = 1205722






119891119904 = rank 119878prime








119891119872 = 119891119898 + 119891119904

rrlabcsuni-klde

Vehicle Kinematics



Tricycle Drive

Given 120575 119907


119903 = 119897 sdot cot 120575

119907prime =119907

cos 120575

119907119897 = 119907 sdot119903 + 119889

119903

119907119903 = 119907 sdot119903 minus 119889


120575


Ackermann Steering

Given 119907119863 120575 119897 119889

Unknown 119907119903119903 119907119897119903 119907119903119891 119907119897119891

Solution

120602 =119897

tan 120575

119907119897119903 =120602 minus 1198892 sdot 119907119863

120602

119907119903119903 =120602 + 1198892 sdot 119907119863

120602

119907119897119891 =120602 minus 1198892 2 + 1198972 sdot tan 120575

119897sdot 119907119863

119907119903119891 =1206022 + 1198972 sdot tan 120575

119897sdot 119907119863


120575

120602

rrlabcsuni-klde



Given 120602 120593 119907 119889



1205932 = arctan120549119910 + 119889

1205491199091206022 = 120602 sdot cos120593 cos1205932

1199072 = 119907 sdot 1206022120602

120549119910 = 120602 ∙ 119904119894119899 120593

1205931

1205932

1206021

1206022

120593 120549119909 = 120602 ∙ cos120593

120602

119889

rrlabcsuni-klde

Bicycle Model


119907


Bicycle Model


Low speed motion



119865119910 =1198981199072

120602






Bicycle Model

Given 120575119891 120575119903 119907 120579

Unknown ሶ119883 ሶ119884 ሶ120579

Solution

120575119907 = tanminus1119897119891 tan 120575119903+119897119903 tan 120575119891

119897119891+119897119903

ሶ120579 =119907

120602

120602 =119897119891+119897119903

tan 120575119891 minustan 120575119903 cos 120575119907

ሶ119883 = 119907 cos 120579 + 120575119907ሶ119884 = 119907 sin 120579 + 120575119907

ሶ120579 =119907 cos 120575119907

119897119891+119897119903tan 120575119891 minus tan 120575119903

119907

rrlabcsuni-klde



119862 = 119874 120575119903 = 0 120575119891 = 120575

119897119903 = 0 119897119891 = 119897

120575119907 = 0

ሶ120579 =119907

120602

120602 =119897

tan 120575119891

ሶ119883 = 119907 cos 120579ሶ119884 = 119907 sin 120579ሶ120579 =

119907

119897119891tan 120575119891

119907




Ability to maneuver

Controllability

Stability



Stability




Typical setups




Maneuverability








Controllability



Examples




Wheel type

Drive geometry


Motortransmission


Degree of Mobility




119891119898 = 3 minus rank 119878


Degree of Mobility



1198891 = 1198892 1205731 = 1205732 = 0 1205721 + 120587 = 1205722






119891119904 = rank 119878prime








119891119872 = 119891119898 + 119891119904

rrlabcsuni-klde

Vehicle Kinematics



Tricycle Drive

Given 120575 119907


119903 = 119897 sdot cot 120575

119907prime =119907

cos 120575

119907119897 = 119907 sdot119903 + 119889

119903

119907119903 = 119907 sdot119903 minus 119889


120575


Ackermann Steering

Given 119907119863 120575 119897 119889

Unknown 119907119903119903 119907119897119903 119907119903119891 119907119897119891

Solution

120602 =119897

tan 120575

119907119897119903 =120602 minus 1198892 sdot 119907119863

120602

119907119903119903 =120602 + 1198892 sdot 119907119863

120602

119907119897119891 =120602 minus 1198892 2 + 1198972 sdot tan 120575

119897sdot 119907119863

119907119903119891 =1206022 + 1198972 sdot tan 120575

119897sdot 119907119863


120575

120602

rrlabcsuni-klde



Given 120602 120593 119907 119889



1205932 = arctan120549119910 + 119889

1205491199091206022 = 120602 sdot cos120593 cos1205932

1199072 = 119907 sdot 1206022120602

120549119910 = 120602 ∙ 119904119894119899 120593

1205931

1205932

1206021

1206022

120593 120549119909 = 120602 ∙ cos120593

120602

119889

rrlabcsuni-klde

Bicycle Model


119907


Bicycle Model


Low speed motion



119865119910 =1198981199072

120602






Bicycle Model

Given 120575119891 120575119903 119907 120579

Unknown ሶ119883 ሶ119884 ሶ120579

Solution

120575119907 = tanminus1119897119891 tan 120575119903+119897119903 tan 120575119891

119897119891+119897119903

ሶ120579 =119907

120602

120602 =119897119891+119897119903

tan 120575119891 minustan 120575119903 cos 120575119907

ሶ119883 = 119907 cos 120579 + 120575119907ሶ119884 = 119907 sin 120579 + 120575119907

ሶ120579 =119907 cos 120575119907

119897119891+119897119903tan 120575119891 minus tan 120575119903

119907

rrlabcsuni-klde



119862 = 119874 120575119903 = 0 120575119891 = 120575

119897119903 = 0 119897119891 = 119897

120575119907 = 0

ሶ120579 =119907

120602

120602 =119897

tan 120575119891

ሶ119883 = 119907 cos 120579ሶ119884 = 119907 sin 120579ሶ120579 =

119907

119897119891tan 120575119891

119907


Stability




Typical setups




Maneuverability








Controllability



Examples




Wheel type

Drive geometry


Motortransmission


Degree of Mobility




119891119898 = 3 minus rank 119878


Degree of Mobility



1198891 = 1198892 1205731 = 1205732 = 0 1205721 + 120587 = 1205722






119891119904 = rank 119878prime








119891119872 = 119891119898 + 119891119904

rrlabcsuni-klde

Vehicle Kinematics



Tricycle Drive

Given 120575 119907


119903 = 119897 sdot cot 120575

119907prime =119907

cos 120575

119907119897 = 119907 sdot119903 + 119889

119903

119907119903 = 119907 sdot119903 minus 119889


120575


Ackermann Steering

Given 119907119863 120575 119897 119889

Unknown 119907119903119903 119907119897119903 119907119903119891 119907119897119891

Solution

120602 =119897

tan 120575

119907119897119903 =120602 minus 1198892 sdot 119907119863

120602

119907119903119903 =120602 + 1198892 sdot 119907119863

120602

119907119897119891 =120602 minus 1198892 2 + 1198972 sdot tan 120575

119897sdot 119907119863

119907119903119891 =1206022 + 1198972 sdot tan 120575

119897sdot 119907119863


120575

120602

rrlabcsuni-klde



Given 120602 120593 119907 119889



1205932 = arctan120549119910 + 119889

1205491199091206022 = 120602 sdot cos120593 cos1205932

1199072 = 119907 sdot 1206022120602

120549119910 = 120602 ∙ 119904119894119899 120593

1205931

1205932

1206021

1206022

120593 120549119909 = 120602 ∙ cos120593

120602

119889

rrlabcsuni-klde

Bicycle Model


119907


Bicycle Model


Low speed motion



119865119910 =1198981199072

120602






Bicycle Model

Given 120575119891 120575119903 119907 120579

Unknown ሶ119883 ሶ119884 ሶ120579

Solution

120575119907 = tanminus1119897119891 tan 120575119903+119897119903 tan 120575119891

119897119891+119897119903

ሶ120579 =119907

120602

120602 =119897119891+119897119903

tan 120575119891 minustan 120575119903 cos 120575119907

ሶ119883 = 119907 cos 120579 + 120575119907ሶ119884 = 119907 sin 120579 + 120575119907

ሶ120579 =119907 cos 120575119907

119897119891+119897119903tan 120575119891 minus tan 120575119903

119907

rrlabcsuni-klde



119862 = 119874 120575119903 = 0 120575119891 = 120575

119897119903 = 0 119897119891 = 119897

120575119907 = 0

ሶ120579 =119907

120602

120602 =119897

tan 120575119891

ሶ119883 = 119907 cos 120579ሶ119884 = 119907 sin 120579ሶ120579 =

119907

119897119891tan 120575119891

119907


Maneuverability








Controllability



Examples




Wheel type

Drive geometry


Motortransmission


Degree of Mobility




119891119898 = 3 minus rank 119878


Degree of Mobility



1198891 = 1198892 1205731 = 1205732 = 0 1205721 + 120587 = 1205722






119891119904 = rank 119878prime








119891119872 = 119891119898 + 119891119904

rrlabcsuni-klde

Vehicle Kinematics



Tricycle Drive

Given 120575 119907


119903 = 119897 sdot cot 120575

119907prime =119907

cos 120575

119907119897 = 119907 sdot119903 + 119889

119903

119907119903 = 119907 sdot119903 minus 119889


120575


Ackermann Steering

Given 119907119863 120575 119897 119889

Unknown 119907119903119903 119907119897119903 119907119903119891 119907119897119891

Solution

120602 =119897

tan 120575

119907119897119903 =120602 minus 1198892 sdot 119907119863

120602

119907119903119903 =120602 + 1198892 sdot 119907119863

120602

119907119897119891 =120602 minus 1198892 2 + 1198972 sdot tan 120575

119897sdot 119907119863

119907119903119891 =1206022 + 1198972 sdot tan 120575

119897sdot 119907119863


120575

120602

rrlabcsuni-klde



Given 120602 120593 119907 119889



1205932 = arctan120549119910 + 119889

1205491199091206022 = 120602 sdot cos120593 cos1205932

1199072 = 119907 sdot 1206022120602

120549119910 = 120602 ∙ 119904119894119899 120593

1205931

1205932

1206021

1206022

120593 120549119909 = 120602 ∙ cos120593

120602

119889

rrlabcsuni-klde

Bicycle Model


119907


Bicycle Model


Low speed motion



119865119910 =1198981199072

120602






Bicycle Model

Given 120575119891 120575119903 119907 120579

Unknown ሶ119883 ሶ119884 ሶ120579

Solution

120575119907 = tanminus1119897119891 tan 120575119903+119897119903 tan 120575119891

119897119891+119897119903

ሶ120579 =119907

120602

120602 =119897119891+119897119903

tan 120575119891 minustan 120575119903 cos 120575119907

ሶ119883 = 119907 cos 120579 + 120575119907ሶ119884 = 119907 sin 120579 + 120575119907

ሶ120579 =119907 cos 120575119907

119897119891+119897119903tan 120575119891 minus tan 120575119903

119907

rrlabcsuni-klde



119862 = 119874 120575119903 = 0 120575119891 = 120575

119897119903 = 0 119897119891 = 119897

120575119907 = 0

ሶ120579 =119907

120602

120602 =119897

tan 120575119891

ሶ119883 = 119907 cos 120579ሶ119884 = 119907 sin 120579ሶ120579 =

119907

119897119891tan 120575119891

119907


Controllability



Examples




Wheel type

Drive geometry


Motortransmission


Degree of Mobility




119891119898 = 3 minus rank 119878


Degree of Mobility



1198891 = 1198892 1205731 = 1205732 = 0 1205721 + 120587 = 1205722






119891119904 = rank 119878prime








119891119872 = 119891119898 + 119891119904

rrlabcsuni-klde

Vehicle Kinematics



Tricycle Drive

Given 120575 119907


119903 = 119897 sdot cot 120575

119907prime =119907

cos 120575

119907119897 = 119907 sdot119903 + 119889

119903

119907119903 = 119907 sdot119903 minus 119889


120575


Ackermann Steering

Given 119907119863 120575 119897 119889

Unknown 119907119903119903 119907119897119903 119907119903119891 119907119897119891

Solution

120602 =119897

tan 120575

119907119897119903 =120602 minus 1198892 sdot 119907119863

120602

119907119903119903 =120602 + 1198892 sdot 119907119863

120602

119907119897119891 =120602 minus 1198892 2 + 1198972 sdot tan 120575

119897sdot 119907119863

119907119903119891 =1206022 + 1198972 sdot tan 120575

119897sdot 119907119863


120575

120602

rrlabcsuni-klde



Given 120602 120593 119907 119889



1205932 = arctan120549119910 + 119889

1205491199091206022 = 120602 sdot cos120593 cos1205932

1199072 = 119907 sdot 1206022120602

120549119910 = 120602 ∙ 119904119894119899 120593

1205931

1205932

1206021

1206022

120593 120549119909 = 120602 ∙ cos120593

120602

119889

rrlabcsuni-klde

Bicycle Model


119907


Bicycle Model


Low speed motion



119865119910 =1198981199072

120602






Bicycle Model

Given 120575119891 120575119903 119907 120579

Unknown ሶ119883 ሶ119884 ሶ120579

Solution

120575119907 = tanminus1119897119891 tan 120575119903+119897119903 tan 120575119891

119897119891+119897119903

ሶ120579 =119907

120602

120602 =119897119891+119897119903

tan 120575119891 minustan 120575119903 cos 120575119907

ሶ119883 = 119907 cos 120579 + 120575119907ሶ119884 = 119907 sin 120579 + 120575119907

ሶ120579 =119907 cos 120575119907

119897119891+119897119903tan 120575119891 minus tan 120575119903

119907

rrlabcsuni-klde



119862 = 119874 120575119903 = 0 120575119891 = 120575

119897119903 = 0 119897119891 = 119897

120575119907 = 0

ሶ120579 =119907

120602

120602 =119897

tan 120575119891

ሶ119883 = 119907 cos 120579ሶ119884 = 119907 sin 120579ሶ120579 =

119907

119897119891tan 120575119891

119907


Degree of Mobility




119891119898 = 3 minus rank 119878


Degree of Mobility



1198891 = 1198892 1205731 = 1205732 = 0 1205721 + 120587 = 1205722






119891119904 = rank 119878prime








119891119872 = 119891119898 + 119891119904

rrlabcsuni-klde

Vehicle Kinematics



Tricycle Drive

Given 120575 119907


119903 = 119897 sdot cot 120575

119907prime =119907

cos 120575

119907119897 = 119907 sdot119903 + 119889

119903

119907119903 = 119907 sdot119903 minus 119889


120575


Ackermann Steering

Given 119907119863 120575 119897 119889

Unknown 119907119903119903 119907119897119903 119907119903119891 119907119897119891

Solution

120602 =119897

tan 120575

119907119897119903 =120602 minus 1198892 sdot 119907119863

120602

119907119903119903 =120602 + 1198892 sdot 119907119863

120602

119907119897119891 =120602 minus 1198892 2 + 1198972 sdot tan 120575

119897sdot 119907119863

119907119903119891 =1206022 + 1198972 sdot tan 120575

119897sdot 119907119863


120575

120602

rrlabcsuni-klde



Given 120602 120593 119907 119889



1205932 = arctan120549119910 + 119889

1205491199091206022 = 120602 sdot cos120593 cos1205932

1199072 = 119907 sdot 1206022120602

120549119910 = 120602 ∙ 119904119894119899 120593

1205931

1205932

1206021

1206022

120593 120549119909 = 120602 ∙ cos120593

120602

119889

rrlabcsuni-klde

Bicycle Model


119907


Bicycle Model


Low speed motion



119865119910 =1198981199072

120602






Bicycle Model

Given 120575119891 120575119903 119907 120579

Unknown ሶ119883 ሶ119884 ሶ120579

Solution

120575119907 = tanminus1119897119891 tan 120575119903+119897119903 tan 120575119891

119897119891+119897119903

ሶ120579 =119907

120602

120602 =119897119891+119897119903

tan 120575119891 minustan 120575119903 cos 120575119907

ሶ119883 = 119907 cos 120579 + 120575119907ሶ119884 = 119907 sin 120579 + 120575119907

ሶ120579 =119907 cos 120575119907

119897119891+119897119903tan 120575119891 minus tan 120575119903

119907

rrlabcsuni-klde



119862 = 119874 120575119903 = 0 120575119891 = 120575

119897119903 = 0 119897119891 = 119897

120575119907 = 0

ሶ120579 =119907

120602

120602 =119897

tan 120575119891

ሶ119883 = 119907 cos 120579ሶ119884 = 119907 sin 120579ሶ120579 =

119907

119897119891tan 120575119891

119907


Degree of Mobility



1198891 = 1198892 1205731 = 1205732 = 0 1205721 + 120587 = 1205722






119891119904 = rank 119878prime








119891119872 = 119891119898 + 119891119904

rrlabcsuni-klde

Vehicle Kinematics



Tricycle Drive

Given 120575 119907


119903 = 119897 sdot cot 120575

119907prime =119907

cos 120575

119907119897 = 119907 sdot119903 + 119889

119903

119907119903 = 119907 sdot119903 minus 119889


120575


Ackermann Steering

Given 119907119863 120575 119897 119889

Unknown 119907119903119903 119907119897119903 119907119903119891 119907119897119891

Solution

120602 =119897

tan 120575

119907119897119903 =120602 minus 1198892 sdot 119907119863

120602

119907119903119903 =120602 + 1198892 sdot 119907119863

120602

119907119897119891 =120602 minus 1198892 2 + 1198972 sdot tan 120575

119897sdot 119907119863

119907119903119891 =1206022 + 1198972 sdot tan 120575

119897sdot 119907119863


120575

120602

rrlabcsuni-klde



Given 120602 120593 119907 119889



1205932 = arctan120549119910 + 119889

1205491199091206022 = 120602 sdot cos120593 cos1205932

1199072 = 119907 sdot 1206022120602

120549119910 = 120602 ∙ 119904119894119899 120593

1205931

1205932

1206021

1206022

120593 120549119909 = 120602 ∙ cos120593

120602

119889

rrlabcsuni-klde

Bicycle Model


119907


Bicycle Model


Low speed motion



119865119910 =1198981199072

120602






Bicycle Model

Given 120575119891 120575119903 119907 120579

Unknown ሶ119883 ሶ119884 ሶ120579

Solution

120575119907 = tanminus1119897119891 tan 120575119903+119897119903 tan 120575119891

119897119891+119897119903

ሶ120579 =119907

120602

120602 =119897119891+119897119903

tan 120575119891 minustan 120575119903 cos 120575119907

ሶ119883 = 119907 cos 120579 + 120575119907ሶ119884 = 119907 sin 120579 + 120575119907

ሶ120579 =119907 cos 120575119907

119897119891+119897119903tan 120575119891 minus tan 120575119903

119907

rrlabcsuni-klde



119862 = 119874 120575119903 = 0 120575119891 = 120575

119897119903 = 0 119897119891 = 119897

120575119907 = 0

ሶ120579 =119907

120602

120602 =119897

tan 120575119891

ሶ119883 = 119907 cos 120579ሶ119884 = 119907 sin 120579ሶ120579 =

119907

119897119891tan 120575119891

119907





119891119904 = rank 119878prime








119891119872 = 119891119898 + 119891119904

rrlabcsuni-klde

Vehicle Kinematics



Tricycle Drive

Given 120575 119907


119903 = 119897 sdot cot 120575

119907prime =119907

cos 120575

119907119897 = 119907 sdot119903 + 119889

119903

119907119903 = 119907 sdot119903 minus 119889


120575


Ackermann Steering

Given 119907119863 120575 119897 119889

Unknown 119907119903119903 119907119897119903 119907119903119891 119907119897119891

Solution

120602 =119897

tan 120575

119907119897119903 =120602 minus 1198892 sdot 119907119863

120602

119907119903119903 =120602 + 1198892 sdot 119907119863

120602

119907119897119891 =120602 minus 1198892 2 + 1198972 sdot tan 120575

119897sdot 119907119863

119907119903119891 =1206022 + 1198972 sdot tan 120575

119897sdot 119907119863


120575

120602

rrlabcsuni-klde



Given 120602 120593 119907 119889



1205932 = arctan120549119910 + 119889

1205491199091206022 = 120602 sdot cos120593 cos1205932

1199072 = 119907 sdot 1206022120602

120549119910 = 120602 ∙ 119904119894119899 120593

1205931

1205932

1206021

1206022

120593 120549119909 = 120602 ∙ cos120593

120602

119889

rrlabcsuni-klde

Bicycle Model


119907


Bicycle Model


Low speed motion



119865119910 =1198981199072

120602






Bicycle Model

Given 120575119891 120575119903 119907 120579

Unknown ሶ119883 ሶ119884 ሶ120579

Solution

120575119907 = tanminus1119897119891 tan 120575119903+119897119903 tan 120575119891

119897119891+119897119903

ሶ120579 =119907

120602

120602 =119897119891+119897119903

tan 120575119891 minustan 120575119903 cos 120575119907

ሶ119883 = 119907 cos 120579 + 120575119907ሶ119884 = 119907 sin 120579 + 120575119907

ሶ120579 =119907 cos 120575119907

119897119891+119897119903tan 120575119891 minus tan 120575119903

119907

rrlabcsuni-klde



119862 = 119874 120575119903 = 0 120575119891 = 120575

119897119903 = 0 119897119891 = 119897

120575119907 = 0

ሶ120579 =119907

120602

120602 =119897

tan 120575119891

ሶ119883 = 119907 cos 120579ሶ119884 = 119907 sin 120579ሶ120579 =

119907

119897119891tan 120575119891

119907




119891119872 = 119891119898 + 119891119904

rrlabcsuni-klde

Vehicle Kinematics



Tricycle Drive

Given 120575 119907


119903 = 119897 sdot cot 120575

119907prime =119907

cos 120575

119907119897 = 119907 sdot119903 + 119889

119903

119907119903 = 119907 sdot119903 minus 119889


120575


Ackermann Steering

Given 119907119863 120575 119897 119889

Unknown 119907119903119903 119907119897119903 119907119903119891 119907119897119891

Solution

120602 =119897

tan 120575

119907119897119903 =120602 minus 1198892 sdot 119907119863

120602

119907119903119903 =120602 + 1198892 sdot 119907119863

120602

119907119897119891 =120602 minus 1198892 2 + 1198972 sdot tan 120575

119897sdot 119907119863

119907119903119891 =1206022 + 1198972 sdot tan 120575

119897sdot 119907119863


120575

120602

rrlabcsuni-klde



Given 120602 120593 119907 119889



1205932 = arctan120549119910 + 119889

1205491199091206022 = 120602 sdot cos120593 cos1205932

1199072 = 119907 sdot 1206022120602

120549119910 = 120602 ∙ 119904119894119899 120593

1205931

1205932

1206021

1206022

120593 120549119909 = 120602 ∙ cos120593

120602

119889

rrlabcsuni-klde

Bicycle Model


119907


Bicycle Model


Low speed motion



119865119910 =1198981199072

120602






Bicycle Model

Given 120575119891 120575119903 119907 120579

Unknown ሶ119883 ሶ119884 ሶ120579

Solution

120575119907 = tanminus1119897119891 tan 120575119903+119897119903 tan 120575119891

119897119891+119897119903

ሶ120579 =119907

120602

120602 =119897119891+119897119903

tan 120575119891 minustan 120575119903 cos 120575119907

ሶ119883 = 119907 cos 120579 + 120575119907ሶ119884 = 119907 sin 120579 + 120575119907

ሶ120579 =119907 cos 120575119907

119897119891+119897119903tan 120575119891 minus tan 120575119903

119907

rrlabcsuni-klde



119862 = 119874 120575119903 = 0 120575119891 = 120575

119897119903 = 0 119897119891 = 119897

120575119907 = 0

ሶ120579 =119907

120602

120602 =119897

tan 120575119891

ሶ119883 = 119907 cos 120579ሶ119884 = 119907 sin 120579ሶ120579 =

119907

119897119891tan 120575119891

119907

rrlabcsuni-klde

Vehicle Kinematics



Tricycle Drive

Given 120575 119907


119903 = 119897 sdot cot 120575

119907prime =119907

cos 120575

119907119897 = 119907 sdot119903 + 119889

119903

119907119903 = 119907 sdot119903 minus 119889


120575


Ackermann Steering

Given 119907119863 120575 119897 119889

Unknown 119907119903119903 119907119897119903 119907119903119891 119907119897119891

Solution

120602 =119897

tan 120575

119907119897119903 =120602 minus 1198892 sdot 119907119863

120602

119907119903119903 =120602 + 1198892 sdot 119907119863

120602

119907119897119891 =120602 minus 1198892 2 + 1198972 sdot tan 120575

119897sdot 119907119863

119907119903119891 =1206022 + 1198972 sdot tan 120575

119897sdot 119907119863


120575

120602

rrlabcsuni-klde



Given 120602 120593 119907 119889



1205932 = arctan120549119910 + 119889

1205491199091206022 = 120602 sdot cos120593 cos1205932

1199072 = 119907 sdot 1206022120602

120549119910 = 120602 ∙ 119904119894119899 120593

1205931

1205932

1206021

1206022

120593 120549119909 = 120602 ∙ cos120593

120602

119889

rrlabcsuni-klde

Bicycle Model


119907


Bicycle Model


Low speed motion



119865119910 =1198981199072

120602






Bicycle Model

Given 120575119891 120575119903 119907 120579

Unknown ሶ119883 ሶ119884 ሶ120579

Solution

120575119907 = tanminus1119897119891 tan 120575119903+119897119903 tan 120575119891

119897119891+119897119903

ሶ120579 =119907

120602

120602 =119897119891+119897119903

tan 120575119891 minustan 120575119903 cos 120575119907

ሶ119883 = 119907 cos 120579 + 120575119907ሶ119884 = 119907 sin 120579 + 120575119907

ሶ120579 =119907 cos 120575119907

119897119891+119897119903tan 120575119891 minus tan 120575119903

119907

rrlabcsuni-klde



119862 = 119874 120575119903 = 0 120575119891 = 120575

119897119903 = 0 119897119891 = 119897

120575119907 = 0

ሶ120579 =119907

120602

120602 =119897

tan 120575119891

ሶ119883 = 119907 cos 120579ሶ119884 = 119907 sin 120579ሶ120579 =

119907

119897119891tan 120575119891

119907


Tricycle Drive

Given 120575 119907


119903 = 119897 sdot cot 120575

119907prime =119907

cos 120575

119907119897 = 119907 sdot119903 + 119889

119903

119907119903 = 119907 sdot119903 minus 119889


120575


Ackermann Steering

Given 119907119863 120575 119897 119889

Unknown 119907119903119903 119907119897119903 119907119903119891 119907119897119891

Solution

120602 =119897

tan 120575

119907119897119903 =120602 minus 1198892 sdot 119907119863

120602

119907119903119903 =120602 + 1198892 sdot 119907119863

120602

119907119897119891 =120602 minus 1198892 2 + 1198972 sdot tan 120575

119897sdot 119907119863

119907119903119891 =1206022 + 1198972 sdot tan 120575

119897sdot 119907119863


120575

120602

rrlabcsuni-klde



Given 120602 120593 119907 119889



1205932 = arctan120549119910 + 119889

1205491199091206022 = 120602 sdot cos120593 cos1205932

1199072 = 119907 sdot 1206022120602

120549119910 = 120602 ∙ 119904119894119899 120593

1205931

1205932

1206021

1206022

120593 120549119909 = 120602 ∙ cos120593

120602

119889

rrlabcsuni-klde

Bicycle Model


119907


Bicycle Model


Low speed motion



119865119910 =1198981199072

120602






Bicycle Model

Given 120575119891 120575119903 119907 120579

Unknown ሶ119883 ሶ119884 ሶ120579

Solution

120575119907 = tanminus1119897119891 tan 120575119903+119897119903 tan 120575119891

119897119891+119897119903

ሶ120579 =119907

120602

120602 =119897119891+119897119903

tan 120575119891 minustan 120575119903 cos 120575119907

ሶ119883 = 119907 cos 120579 + 120575119907ሶ119884 = 119907 sin 120579 + 120575119907

ሶ120579 =119907 cos 120575119907

119897119891+119897119903tan 120575119891 minus tan 120575119903

119907

rrlabcsuni-klde



119862 = 119874 120575119903 = 0 120575119891 = 120575

119897119903 = 0 119897119891 = 119897

120575119907 = 0

ሶ120579 =119907

120602

120602 =119897

tan 120575119891

ሶ119883 = 119907 cos 120579ሶ119884 = 119907 sin 120579ሶ120579 =

119907

119897119891tan 120575119891

119907


Ackermann Steering

Given 119907119863 120575 119897 119889

Unknown 119907119903119903 119907119897119903 119907119903119891 119907119897119891

Solution

120602 =119897

tan 120575

119907119897119903 =120602 minus 1198892 sdot 119907119863

120602

119907119903119903 =120602 + 1198892 sdot 119907119863

120602

119907119897119891 =120602 minus 1198892 2 + 1198972 sdot tan 120575

119897sdot 119907119863

119907119903119891 =1206022 + 1198972 sdot tan 120575

119897sdot 119907119863


120575

120602

rrlabcsuni-klde



Given 120602 120593 119907 119889



1205932 = arctan120549119910 + 119889

1205491199091206022 = 120602 sdot cos120593 cos1205932

1199072 = 119907 sdot 1206022120602

120549119910 = 120602 ∙ 119904119894119899 120593

1205931

1205932

1206021

1206022

120593 120549119909 = 120602 ∙ cos120593

120602

119889

rrlabcsuni-klde

Bicycle Model


119907


Bicycle Model


Low speed motion



119865119910 =1198981199072

120602






Bicycle Model

Given 120575119891 120575119903 119907 120579

Unknown ሶ119883 ሶ119884 ሶ120579

Solution

120575119907 = tanminus1119897119891 tan 120575119903+119897119903 tan 120575119891

119897119891+119897119903

ሶ120579 =119907

120602

120602 =119897119891+119897119903

tan 120575119891 minustan 120575119903 cos 120575119907

ሶ119883 = 119907 cos 120579 + 120575119907ሶ119884 = 119907 sin 120579 + 120575119907

ሶ120579 =119907 cos 120575119907

119897119891+119897119903tan 120575119891 minus tan 120575119903

119907

rrlabcsuni-klde



119862 = 119874 120575119903 = 0 120575119891 = 120575

119897119903 = 0 119897119891 = 119897

120575119907 = 0

ሶ120579 =119907

120602

120602 =119897

tan 120575119891

ሶ119883 = 119907 cos 120579ሶ119884 = 119907 sin 120579ሶ120579 =

119907

119897119891tan 120575119891

119907

rrlabcsuni-klde



Given 120602 120593 119907 119889



1205932 = arctan120549119910 + 119889

1205491199091206022 = 120602 sdot cos120593 cos1205932

1199072 = 119907 sdot 1206022120602

120549119910 = 120602 ∙ 119904119894119899 120593

1205931

1205932

1206021

1206022

120593 120549119909 = 120602 ∙ cos120593

120602

119889

rrlabcsuni-klde

Bicycle Model


119907


Bicycle Model


Low speed motion



119865119910 =1198981199072

120602






Bicycle Model

Given 120575119891 120575119903 119907 120579

Unknown ሶ119883 ሶ119884 ሶ120579

Solution

120575119907 = tanminus1119897119891 tan 120575119903+119897119903 tan 120575119891

119897119891+119897119903

ሶ120579 =119907

120602

120602 =119897119891+119897119903

tan 120575119891 minustan 120575119903 cos 120575119907

ሶ119883 = 119907 cos 120579 + 120575119907ሶ119884 = 119907 sin 120579 + 120575119907

ሶ120579 =119907 cos 120575119907

119897119891+119897119903tan 120575119891 minus tan 120575119903

119907

rrlabcsuni-klde



119862 = 119874 120575119903 = 0 120575119891 = 120575

119897119903 = 0 119897119891 = 119897

120575119907 = 0

ሶ120579 =119907

120602

120602 =119897

tan 120575119891

ሶ119883 = 119907 cos 120579ሶ119884 = 119907 sin 120579ሶ120579 =

119907

119897119891tan 120575119891

119907

rrlabcsuni-klde

Bicycle Model


119907


Bicycle Model


Low speed motion



119865119910 =1198981199072

120602






Bicycle Model

Given 120575119891 120575119903 119907 120579

Unknown ሶ119883 ሶ119884 ሶ120579

Solution

120575119907 = tanminus1119897119891 tan 120575119903+119897119903 tan 120575119891

119897119891+119897119903

ሶ120579 =119907

120602

120602 =119897119891+119897119903

tan 120575119891 minustan 120575119903 cos 120575119907

ሶ119883 = 119907 cos 120579 + 120575119907ሶ119884 = 119907 sin 120579 + 120575119907

ሶ120579 =119907 cos 120575119907

119897119891+119897119903tan 120575119891 minus tan 120575119903

119907

rrlabcsuni-klde



119862 = 119874 120575119903 = 0 120575119891 = 120575

119897119903 = 0 119897119891 = 119897

120575119907 = 0

ሶ120579 =119907

120602

120602 =119897

tan 120575119891

ሶ119883 = 119907 cos 120579ሶ119884 = 119907 sin 120579ሶ120579 =

119907

119897119891tan 120575119891

119907


Bicycle Model


Low speed motion



119865119910 =1198981199072

120602






Bicycle Model

Given 120575119891 120575119903 119907 120579

Unknown ሶ119883 ሶ119884 ሶ120579

Solution

120575119907 = tanminus1119897119891 tan 120575119903+119897119903 tan 120575119891

119897119891+119897119903

ሶ120579 =119907

120602

120602 =119897119891+119897119903

tan 120575119891 minustan 120575119903 cos 120575119907

ሶ119883 = 119907 cos 120579 + 120575119907ሶ119884 = 119907 sin 120579 + 120575119907

ሶ120579 =119907 cos 120575119907

119897119891+119897119903tan 120575119891 minus tan 120575119903

119907

rrlabcsuni-klde



119862 = 119874 120575119903 = 0 120575119891 = 120575

119897119903 = 0 119897119891 = 119897

120575119907 = 0

ሶ120579 =119907

120602

120602 =119897

tan 120575119891

ሶ119883 = 119907 cos 120579ሶ119884 = 119907 sin 120579ሶ120579 =

119907

119897119891tan 120575119891

119907


Bicycle Model

Given 120575119891 120575119903 119907 120579

Unknown ሶ119883 ሶ119884 ሶ120579

Solution

120575119907 = tanminus1119897119891 tan 120575119903+119897119903 tan 120575119891

119897119891+119897119903

ሶ120579 =119907

120602

120602 =119897119891+119897119903

tan 120575119891 minustan 120575119903 cos 120575119907

ሶ119883 = 119907 cos 120579 + 120575119907ሶ119884 = 119907 sin 120579 + 120575119907

ሶ120579 =119907 cos 120575119907

119897119891+119897119903tan 120575119891 minus tan 120575119903

119907

rrlabcsuni-klde



119862 = 119874 120575119903 = 0 120575119891 = 120575

119897119903 = 0 119897119891 = 119897

120575119907 = 0

ሶ120579 =119907

120602

120602 =119897

tan 120575119891

ሶ119883 = 119907 cos 120579ሶ119884 = 119907 sin 120579ሶ120579 =

119907

119897119891tan 120575119891

119907

rrlabcsuni-klde



119862 = 119874 120575119903 = 0 120575119891 = 120575

119897119903 = 0 119897119891 = 119897

120575119907 = 0

ሶ120579 =119907

120602

120602 =119897

tan 120575119891

ሶ119883 = 119907 cos 120579ሶ119884 = 119907 sin 120579ሶ120579 =

119907

119897119891tan 120575119891

119907

Documents

Kinematics and Dynamics of Mobile Robots · Kinematics and Dynamics of Mobile Robots. rrlab.cs.uni-kl.de Kaiserslautern, Germany Kaiserslautern Frankfurt Berlin Hamburg Bremen Hanover