3 Read More for Chapter 3-2 and 3-3x

Wheel constrants

c thm chng 3.2 v 3.3

Under these assumptions, we present two constraints for every wheel type. The first constraint enforces the concept of rolling contact that the wheel must roll when motion takes place in the appropriate direction. The second constraint enforces the concept of no lateral slippage that the wheel must not slide orthogonal to the wheel plane.

The fixed standard wheel has no vertical axis of rotation for steering. Its angle to the chassis is thus fixed, and it is limited to motion back and forth along the wheel plane and rotation around its contact point with

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A fixed standard wheel and its parameters.

and rotation around its contact point with the ground plane.

a fixed standard wheel A and indicates its position pose relative to the robots local reference frame {XR, YR}

The angle of the wheel plane relative to the chassis is denoted by , which is fixed since the fixed standard wheel is not steerable. The wheel, which has radius r , can spin over time, and so its rotational position around its horizontal axle is a function of time t:: (t).

The rolling constraint for this wheel enforces that all motion along the direction of the

wheel plane must be accompanied by the appropriate amount of wheel spin so that there is

pure rolling at the contact point:

The first term of the sum denotes the total motion along the wheel plane. The three

elements of the vector on the left represent mappings from each of to their

contributions for motion along the wheel plane.

The sliding constraint for this wheel enforces that the component of the wheels motion

orthogonal to the wheel plane must be zero:

(2.12)

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orthogonal to the wheel plane must be zero:

This constrains the component of motion

along XI to be zero and since XI and XR are

parallel in this example, the wheel is

constrained from sliding sideways, as

expected.

(2.13)

(2.14)

The steered standard wheel differs from the fixed standard wheel only in that there is an additional degree of freedom: the wheel may rotate around a vertical axis passing through the center of the wheel and the ground contact point. The equations of position for the steered standard wheel (figure 3.5) are identical to that of the fixed standard wheel shown in the figure with one exception. The orientation of the wheel to the robot chassis is no longer a single fixed value, , but instead varies as a function of time: (t) .

(2.15)

(2.16)

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4A steered standard wheel and its parameters.

Castor wheels are able to steer around a vertical axis. However, unlike the steered standard wheel, the vertical axis of rotation in a castor wheel does not pass through the ground contact point.The wheel contact point is now at position B, which is connected by a rigid rod of ABfixed length d to point A fixes the location of the vertical axis about which B steers, and this point A has a position specified in the robots reference frame, as in the figure. We assume that the plane of the wheel is aligned with AB at all times. Similar to the steered standard wheel, the castor wheel has two parameters that vary as a function of time, (t), represents the wheel spin over time as before. (t) denotes the steering angle and orientation of AB over time.

(2.17)

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5A castor wheel and its parameters.

(2.18)

Swedish wheels have no vertical axis of rotation, yet are able to move omnidirectionallylike the castor wheel. This is possible by adding a degree of freedom to the fixed standard wheel.Swedish wheels consist of a fixed standard wheel with rollers attached to the wheelperimeter with axes that are antiparallel to the main axis of the fixed wheel component. The exact angle between the roller axes and the main axis can vary, as shown in figure.

(2.19)

(2.20)

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6A Swedish wheel and its parameters.

Spherical wheel, a ball or spherical wheel, places no direct constraints on motion .Such a mechanism has no principal axis of rotation, and therefore no appropriaterolling or sliding constraints exist. As with castor wheels, the spherical wheel is clearly omnidirectional and places no constraints on the robot chassis kinematics.Therefore equation simply describes the roll rate of the ball in the direction of motionvA of point A of the robot.

By definition the wheel rotation orthogonal to this direction

is zero.

(2.21)

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A spherical wheel and its parameters.

is zero.

(2.22)

3.6. Working Space of mobile robot

Identifying a robots space of possible configurations is important because surprisingly

it can exceed .

In addition to workspace, we care about how the robot is able to move between various

configurations: what are the types of paths that it can follow and, furthermore, what are its

possible trajectories through this configuration space?

In defining the workspace of a robot, it is useful to first examine its admissible velocityspace.

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3.7. Degree of Freedom (DOF) of robot

The Degree of Freedom (DOF) is the robots ability to achieve various poses.

Bicycle: DOF=3

Differential wheel robot DOF=3

A robot is holonomic if the controllable degrees of freedom are equal to the total degrees of freedom.

3.8. Non-Holonomic Robots

In robotics a system is non-holonomic if the controllable degrees of freedom are lessthan the total degrees of freedom

yI x1, y1

s1=s2 ; s1R=s2R ; s1L=s2L

but: x1 x2 ; y1 y2

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Non-holonomic systems

differential equations are not integrable to the final position.

the measure of the traveled distance of each wheel is not sufficient to calculate the final position of the robot. One has also to know how this movement was executed as a function of time.

s1L s1R

s2L

s2R

xI

x2, y2

s1

s2

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Kinematic Position Control: Coordinates Transformation

y

Coordinates transformation into polar coordinates

with its origin at goal position:

System description, in the new polar coordinates( ) = xyxa arctangent, 2tan :Note

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System description, in the new polar coordinates

for for

( )

= yxyxa arctangent, 2tan :Note

Kinematic Position Control: Remarks

The coordinates transformation is not defined at x = y = 0; as in such a point the determinant of

the Jacobian matrix of the transformation is not defined, i.e. it is unbounded

For the forward direction

of the robot points toward the goal,

for it is the backward direction.

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for it is the backward direction.

By properly defining the forward direction of the robot at its initial configuration, it is always

possible to have at t=0. However this does not mean that a remains in I1 for all time t.

y

Kinematic Position Control: The Control Law

It can be shown, that with

the feedback controlled system

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will drive the robot to

The control signal v has always constant sign,

the direction of movement is kept positive or negative during movement

parking maneuver is performed always in the most natural way and without ever inverting

its motion.

( ) ( )000 ,,,, =

Kinematic Position Control: Stability Issue

It can further be shown, that the closed loop control system is locally exponentially stable if

Proof:

for small x > cosx = 1, sinx = x

0 ; 0 ; 0 > kkkk

1.5)(3,8,)k,k,(kk ==

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and the characteristic polynomial of the matrix A of all roots

have negative real parts.

Additional Text

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(2.38)

(2.39)

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(2.40)

Kinematic model,

(2..41)

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(2.42)

(2.43)

(2.44)

(2.45)

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(2.46)

(2.47)

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(2.48)

The control law

(2.49)

(2.50)

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(2.51)

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3 Read More for Chapter 3-2 and 3-3x