Feb 17, 2002Robotics 1 Copyright Martin P. Aalund, Ph.D. Kinematics Kinematics is the science of motion without regard to forces. We study the position,

Feb 17, 2002 Robotics 1 Copyright Martin P. Aalund, Ph.D.

Kinematics

• Kinematics is the science of motion without regard to forces.

• We study the position, velocity, acceleration, jerk etc of objects

• Concerned with the location of Objects

• We will define coordinate systems or frames to define there location


Definitions

• Velocity: The derivative of position with respect to time.

• Acceleration: The derivative of velocity with respect to time.

• Jerk: The derivative of acceleration with respect to time.

• Link: Nearly rigid structure between joints.

• Joint: Allow relative motion between links.

• Joint Angle: Measurement of the relative position of two links

t

axvaj

t

vxva

t

xxv

txx

)(


Definitions Continued

• Joint Space: Relative coordinates that are referenced to coordinate frames at the robot joints.

• Cartesian Space or Task Space. Global or base coordinate frame

• Jacobian: Specifies a mapping of Velocities in joint space to velocity in Cartesian or Task Space.

• Singularity: Region or point at which the Jacobian is singular.


Mathematical Background

sin ( ) sin 0( ) 0 cos 0( ) 1 sin 45 deg( ) 0.707

90 deg 1.571 rad 3.142 rad 180 deg 3.142 rad

xsin x( )

d

dcos x( )

d

dxcos x( ) sin x( )

Chain Rule

tF u( )

d

d

F

u

u

t

Example

Let F u( ) sin u( ) u x t( ) ThentF

d

dcos x( )

x

t


Forward Kinematics

X

Y

R

31°

x

y

• Lets look at a simple link (1DOF)


Forward Kinematics

X

Y

R

31°

x

y

• Want to know the end point of link in terms of X and Y

• We have R and Theta

• From Geometry we candetermine the position:)cos(Rx

)sin(Ry

)sin(Rx

)cos(Ry

• and then the velocity


Two Link Example

x

Yx

y

R1

R2


Two Link Example

X

Yx

y

1 1 2 1 2

x R 1 cos 1 R 2 cos 2

y R 1 sin 2 R 2 sin 2

2

1

21

21

)sin()sin(

)cos()cos(

R

R

y

x


Coordinate Frames

R

X0

Y0

X 1

Y 1


Coordinate Frames

• Interested in Some Point described by a Vector R

• We need to express R which we know in Frame 1 in Frame 0

)cos(

)sin(

1

1

Ry

Rx

)cos()sin(

)sin()cos(

11110

11110

yxy

yxx

X0

Y0

X 1

Y 1R

1

Equation is independent of values


Coordinate Frames

1

1

11

11

0

0

)cos()sin(

)sin()cos(

y

x

y

x

• Now lets put it in matrix form

• So what if we want to map the other way?

• What is the inverse of T? Why?

)cos()sin(

)sin()cos(

11110

11110

yxy

yxx

1

1

0

0

y

xT

y

x

0

01

1

1

y

xT

y

x


Coordinate Frames

• If we look at the columns and rows of T we see that they have a norm of one.

• Also if we take the dot product of the columns we find they are orthogonal to each other.

• So T is an ortho-normal Matrices. Thus its transpose is its inverse.

•

)cos()sin(

)sin()cos(

11

111

T

)cos()sin(

)sin()cos(

11

11

T

• This was a simple 2DOF example what about 3.

• If we project a Z axes out the plane generated by the X and Y axes, then a rotation around the Z axes will not affect the Z position of the vector R.


Coordinate Frames

100

0)cos()sin(

0)sin()cos(

zT

)cos()sin(0

)sin()cos(0

001

xT

)cos(0)sin(

010

)sin(0)cos(

YT

• The 3D transformation axes about the Z axes is:

• Similarly for rotations around the X or Y axes we get

• Shorthand notation is often usedcos()=c sin()=s


Coordinate Transformations

• For multiple transformation we simply multiply by more matrices.

• If we have multiple rotations about the same axes we can just add the angles of the matrices. Does this make sense.

Substituting we get

100

0)cos()sin(

0)sin()cos(

)(

)sin()sin()cos()cos()cos(

)sin()cos()cos()sin()sin(

100

0)cos()cos()sin()sin()sin()cos()cos()sin(

0)cos()sin()sin()cos()sin()sin()cos()cos(

100

0)cos()sin(

0)sin()cos(

100

0)cos()sin(

0)sin()cos(

)(

z

z

T

T


Coordinate Transformation

• So now we can express a vector in a reference frame rotated to an arbitrary angle in space.

• What if we want to express it in a translated frame


Translating Coordinate Frames

X0

Y0

R

X1

Y1


Translating Coordinate Frames

X0

Y0

R

X1

Y1

• Now we can translate and rotate.

• Or in terms of vectors

yyy

xxx

10

10

RRR 10



What if we have a rotation and a translation

• First we can rotate the frames

• then we can translate

y

x

y

xT

y

x

1

1

0

0

X0’

Y0’

X 1

Y 1R

1

X0

Y0


Homogeneous Transformation Matrices

• 3x3 Rotation Matrix

• 3x1 Displacement Vector

• For a displacement and a rotation

100

011

011

1 CS

SC

T

1

1

1

1

z

y

x

R

RRTR 10


4x4 Homogeneous Matrix

• If we want to perform a rotation and a translation with one operation

• We can create a homogeneous Transformation Matrix

1000

100

011

011

0

0

0

z

yCS

xSC

TH

z

y

x

z

y

x

T

z

y

x

1

1

1

0

0

0

001

1

1

0

0

0

z

y

x

Tz

y

x

H

100

011

011

CS

SC

T


Homogeneous Transformation Matrices

• What does a pure translation look like

• What does a pure rotation look like

• Multiply to move from one coordinate to another.

1000

100

010

001

z

y

x

TT

1000

0100

0011

0011

CS

SC

TR


D-H Parameters

• Denavit-Hartenberg (D-H) are used to describe a robot link.

• One Coordinate System is created for each link.– Each Axes is orthogonal

– Use the right hand rule

• Link are assumed to be rigid

• Four Parameters are used– dn Link Offset

– an Link Length (Common Normal Distance)

– n Joint Angle

– n Link Twist Angle


Notation

• Coordinate systems are represented with brackets{B}, {0}, etc.

• Vectors– Lets Look at a Vector P

Described in Frame A

– Leading Subscript describes the framein which the Vector is described or Referenced

– Individual Elements of a vectorare described by a trailing subscript

z

y

xA

p

p

p

P

Frame of Reference


Matrix Notation

RAB

TAB

New Frame of Reference

Original Frame of Reference


Homogenous Transformations Represent 3 Things

• Describe a Frame

• Map from one Frame to another

• Act as an Operator to move within a Frame

TAB


Transforms Describe Frames

• Frames can be described byA Homogenous TransformationMatrices

• Description of Frame– Columns of are the Unit Vectors defining the directions of the

principle axes of {B} in terms of {A}

– Rows of are the Unit Vectors defining the directions of the principle axes of {A} in terms of {B}

– is the location of the origin of {B} in terms or {A}

RAB

BorgAP

AB

AB

AB

BA

BA

BAA

B

Z

Y

X

ZYXRˆ

ˆ

ˆ

ˆˆˆ

1000

100

011

011

BorgZA

BorgYA

BorgXA

AB

P

PCS

PSC

T RAB

RAB


Mapping Between Frames

• Maps vector from Frame {B} to Frame {A}

• will rotate a vector to project its components originally described in {B} in the {A} of Frame

• will translate the vector to adjust its origin from frame {B} to its new origin in {A}

BorgAP

1000

100

011

011

BorgZA

BorgYA

BorgXA

AB

P

PCS

PSC

T RAB

RAB

PP AB


Acts as an Operator within a coordinate frame

• Operator will rotate and translate a vector

• Defines the angle to rotate about

• Defines the distance to translate

• Usually drop subscripts

• Operates on to create

R

BorgP

1000

100

011

011

2

2

2

orgZ

orgY

orgX

P

PCS

PSC

T RT

1PA2PA


an

• Link Length (Common Normal)

• Measured from zn to zn+1 along xn

• The Common perpendicular is shortest distance between two lines in space.– Does not necessarily lie in the physical link.

– If axes intersect then it is zero.

– Not defined for Prismatic, set to zero

z1z2

a1


n

• Link Twist Angle

• Measured from zn to zn+1 around xn

• Usually a multiple of 90 degrees

1

z1

z2

x1


dn

• Link Offset– Measured from xn-1 to xn along zn

– Joint variable for Prismatic joint

z1

z2

y1

x1

y2

x2

d1


n

• Joint Angle

• The angle between xn-1 and xn

• Measured around zn from xn-1 and xn

z1

z2

y1

x1

x2

2

x2

x1


Assigning Coordinate Frames

1. First place the Z axes along the joint axes of rotation, or translation. Imagine them of infinite length

2. Identify the Common Perpendiculars or point of intersection between each pair of Z axes Place the Xn axes along the common perpendicular between axes n and n+1. If the axes intersect Xn is assigned normal to the plane containing the intersecting axes.

3. Use the right hand rule to define the Y axes. 4. The base frame (0) is fixed and does not move.

It is usually picked to be coincident with frame 1 so that when the joint variable is zero:a0=0 0=0 Link Variables (Both are constants)

d1=0 n=0 Joint Variables (One is always a constant)

5. Chose the origin and direction of {N} freely, but in general assign it to make as many parameter zero as possible


Special Cases

• If an=0 (the Z axes intersect)

– then pick Xn to be normal to the plane defined by Zn and Zn+1. This leads to two possible Definitions of X and thus Y.

– If Zn and Zn+1 are parallel and coincident, then the assignment of Xn is arbitrary.

• If Zn-1 and Zn are parallel then pick dn is 0


DH Parameters

• Make a Matrix of the DH parameters for all the links in the robot.

• This will make defining the transformation from one axes to another much easier.

i i-1 ai-1 di

i 1

0 a0 d1 1

2 1 a1 d2

2 … … … n

n-1 an-1 dn n


Forward Kinematics

• Used to move from Joint coordinates to end-effector coordinates

• We will use DH parameters to Define Frames– Each Link will be assigned a frame

– Transformation will be used to move between links

• We will start at the bottom and work our way out the chain of links


Transformation between Frames

• A transformation between Frame {i-1} and {i) can be defined as having 4 steps each containing only one link parameter.1. translate along zi, by di

2. rotate around zi, by i-

3. translate along xi-1, by ai-1,

4. rotate around xi-1, by i-1

• We can perform this with two transforms– First a translation and rotation with regards to zi

– Second a translation and rotation with regards to xi-1

• Remember the operations will be performed from Left to Right.


Transformations

1000

100

00

00

1000

00

00

001

11

11

1

11

i

ii

ii

ii

ii

i

i

d

cs

sc

cs

sc

a

T


DH Based Transformation Matrices

1000

0

1111

1111

1

11

iiiiiii

iiiiiii

iii

i

dcacscss

dsscccs

asc

T


Forward Kinematics

• Used to mover from Joint coordinates to end-effector coordinates

• Just Multiply the individual T matrices.

• For a robot with N joints we will get N Transformations

• If we are interested in a point beyond the last joint such as an end-effector or tool. We define a coordinate system for the Tool point of interest and create a transform from the last joint to the point.

• We will find it helpful to define other points in the robot work space with coordinate systems relative to the robot base system.

TTTTT NNN12

312

01

0


Standard Frames include

• Wrist Frame {W} Same as the last robot frame I.e. Frame {N}

• Base frame {B} Located at the base of the manipulator same as frame {0}

• Station frame {S} Usually used as the base point for robot motions. Calculated relative to the Base Frame

• Tool Frame {T} Could be tip of a torch, point between finger of a gripper or the lens of a camera. specified relative to the {W} Frame

• Goal Frame {G} Location where the robot is desired to move. At the end of motion the {G} and {T} should coincide.

• So to get from the Station frame to the toll frame we simply calculate:

Where

TTTT wT

Bw

BS

ST

1

TT NB

W0


Example

• Book


Motor Vs Joint

• May have a gear reduction

• N is the gear ratio

• Power is conserved

• May be nonlinear through a four-bar linkage.

ooii

io

io

P

NN

1


Types of Gear Trains

• Harmonic Drive– High Gear Ratio

– Compact

– No Backlash

• Planetary– Robust

– Medium Gear Ration

• Worm Gear– Right Angle

– High Ratio

• Ball Nut– Rotational to Transnational

• Ball Screw– Ball nut with lower friction


Harmonic Drive

• A Harmonic Drive Consists of– Wave Generator

– Flex Spline

– Circular or Fixed Spline

• Back Drivable

• Output Reversing

• No Backlash

• As the Lobed Wave Generator Rotates– The Flex Spline is forced into the fixed

spline– Since there are more teeth on the fixed

spline than the flex, they will rotate relative to each other by the difference in teeth. (Equal to the number or lobes)

– Very High ratios 100 to 1 or more can easily be achieved.

Pictures From Harmonic Drive Inc.


Ball Screw or Nut

• Is really a form of a worm gear

• Has a gear of infinite diameter and a very long worm gear.

• As the worm rotates the nut moves up the worm (equivilent to rotation)

• Often described in pitch– Pitch is the distance the nut moves per screw revolution

– Units are distance or more precisely mm/rev or in/rev

• Convert Rotational motion to linear motion.

• An option to a rack and pinion.

• High Gear ration.

• Ball Screws are Back-drivable in

• Most ratios.


Worm Gear

• Worm Gears Can Provide a very high Gear Ratio.

• Each time the input shaft rotates one revolution the gear will move by one tooth.

• By using a large gear lets say 40 teeth we can get a high Gear ratio

• 40 to 1.

• Motor can be mounted at Right Angle to Outputshaft

• Usually not Back-Drivable

Picture from www.howstuffworks.com


Planetary

• The yellow gear (the sun) engages all three red gears (the planets) simultaneously.

• All three are attached to a plate (the planet carrier), and they engage the inside of the blue gear (the ring). Because there are three red gears, this gear train is extremely rugged.

• The Output Shaft is usually attached to the blue gear resulting in a 6 to 1 gear ratio.

• If we used the dark blue planet carrier as theout put we get a 7 to 1 gear ratio since the planets will orbit the sun.

• If we use the Planets carrier as the input, andthe ring as the output and fix the sun, we endup with a 1.17 (7/6) to 1 ratio.

• Planetary gears are often cascaded to get highgear ratios

the blue gear has six times the diameter of the yellow


Assigning Coordinate Frames Review

1. First place the Z axes along the joint axes of rotation, or translation. Imagine them of infinite length

2. Identify the Common Perpendiculars or point of intersection between each pair of Z axes Place the Xn axes along the common perpendicular between axes n and n+1. If the axes intersect Xn is assigned normal to the plane containing the intersecting axes.

3. Use the right hand rule to define the Y axes. 4. The base frame (0) is fixed and does not move.

It is usually picked to be coincident with frame 1 so that when the joint variable is zero:a0=0 0=0 Link Variables (Both are constants)

d1=0 n=0 Joint Variables (One is always a constant)

5. Chose the origin and direction of {N} freely, but in general assign it to make as many parameter zero as possible


Special Cases

• If an=0 (the Z axes intersect) then pick Xn to be normal to the plane defined by Zn and Zn-1. This leads to two possible Definitions of X and thus Y.

• If Zn and Zn+1 are parallel then dn is 0


DH Parameters

• Make a Matrix of the DH parameters for all the links in the robot.

• This will make defining the transformation from one axes to another much easier.

i i-1 ai-1 di

i 1

0 a0 d1 1

2 1 a1 d2

2 … … … n

n an dn n


DH Based Transformation Matrices

1000

0

111

1111

1

11

iiiiiii

iiiiiii

iii

i

dccscss

dsscccs

asc

T

Documents

Feb 17, 2002Robotics 1 Copyright Martin P. Aalund, Ph.D. Kinematics Kinematics is the science of motion without regard to forces. We study the position,