03_wheeledlocomotion

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ETH Master Course: 151-0854-00L

Autonomous Mobile Robots

Wheeled Locomotion

Roland Siegwart

Margarita Chli

Martin RufliDavide Scaramuzza


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Lecture OverviewMobile Robot Control Scheme

2

raw data

positionglobal map

Sensing Acting

InformationExtraction

PathExecution

CognitionPath Planning

knowledge,data base

missioncommands

Real World

Environment

LocalizationMap Building

MotionControl

Perce

ption

actuator

commands

environment modellocal map

path

3 - Wheeled Locomotion


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Lecture OverviewReview: Efficiency of Locomotion Types

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Wheeled motion is highly efficient on

hard and flat surfaces (usually man-made)

generally restricted to man-made structures

the de-facto standard for mobile robotics


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Lecture OverviewReview: Dimensionality

The Degree of Freedom (DOF) of a workspaceis

its overall dimensionality On (flat) ground,

In the air or below water,

For a robotic system

is its ability to achieve various poses

is its ability to achieve various velocities

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DOFDDOF M

3DOF

6DOF

DOFDDOF


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Lecture OverviewReview: Kinematics Terminology

Kinematics Origin: kinein(Greek)to move

The subfield of Mechanics dealing with motions of bodies

Forward kinematics

Given is a set of actuator positions

Determine corresponding reference pose

Inverse kinematics

Given is a desired reference pose

Determine corresponding actuator positions

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Lecture OverviewPreview: Wheeled Kinematics

Wheels Are often subject to motion constraints

Often do not allow to compute kinematics directly

Consequently, for most wheeled robots, actuator positions do notmap to unique reference poses

There is no direct(i.e., instantaneous) way to measure a robots position

Position must be integrated over time, depends on the path taken

Understanding mobile robotic motion requires an understanding ofwheel constraints placed on the robots mobility

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Wheeled LocomotionHolonomic System

Holonomic systems Diff. eqn. of are integrableto the final position

the measure of the traveled distance of each wheel issufficient to calculate the final position of the robot

Examples Ballbot

Robots composed out of (multiple) wheels that do notconstrain motion (i.e., Castor, Swedish and Omni-wheels)

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0y

0x

I


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Iy

Ix

Wheeled LocomotionNon-Holonomic Systems

Non-holonomic systems Diff. eqn. of are not integrableto the final position

The measure of the traveled distance sof each wheelis not sufficient to calculate the final robot position

Knowledge of the movement as a function of time becomes necessary

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2s

1s

I


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Representation of differential forward kinematics Robot pose

Mapping velocities between two frames

Wheeled LocomotionHomogeneous Transformation

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Tyx 0101010

100

0cossin

0sincos10

R

T

yxRR

111101100

0y

0x

1y

1x

1

2


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For robots containing (several) actuated wheels,find , with the actuator velocities

Assumptions

Movement on a horizontal plane

Point contact of the wheels Wheels not deformable

Pure rolling, no slipping, skidding or sliding

No friction for rotation around contact point

Wheels connected to a rigid frame (chassis)

Wheeled LocomotionAssumptions

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),,(1

0

nf i

0y

0x

1y

1x

1

2


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Wheeled LocomotionStandard and Castor Wheels

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Standard wheel a)

two degrees of freedom

rotation around the (motorized) wheelaxle and the contact point

Castor wheel b)

three degrees of freedom

rotation around the wheel axle, the

contact point and the castor axle


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Wheeled LocomotionSwedish and Spherical Wheels

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Swedish wheel c)


rotation around the (motorized) wheelaxle, around the rollers and aroundthe contact point

Ball (spherical wheel) d)


suspension technically not solved


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Ry

Rx

Differential Forward KinematicsFixed Standard Wheel

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TR yx 111

0)(cos)cos()sin( rRl IT

0)(sin)sin()cos( ITRl


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Differential Forward KinematicsSteered Standard Wheel

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Ry

Rx



Differential Forward KinematicsSteered Standard Wheel

TR yx 111


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Differential Forward KinematicsCastor Wheel

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Ry

Rx


0)(sin)sin()cos( dRld IT

v


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Differential Forward KinematicsSwedish Wheel

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Ry

Rx

0cos)()cos()cos()sin( rRl IT

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

rrRl

v


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Differential Forward KinematicsSpherical Wheel

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Ry

Rx




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Differential Forward KinematicsConcatenation of Constraints

Given a wheeled robot Each wheel imposes constraints on its motion

Only fixed and steerable standard wheels impose no-sliding constraints

Suppose the robot has standard wheels of radius , then the

individual wheel constraints can be concatenated in matrix form Rolling constraints

No-sliding constraints

Solving for results in an expression for Differential Forward Kinematics

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1)(

)()(

sf NN

s

f

t

tt

sf NN

31

11

)()(

sf NN

ss

fs

J

JJ

)( 12 NrrdiagJ

0)()(1

Is RC

31

1

1)(

)(

sf NN

ss

f

sC

CC

0

0)()(21

JRJ Is

I

ir


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Wheel ArrangementsStaticStability of Overall System

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Stability requires

At least 3 wheels in ground contact

That CoG lies within support triangle

Stability is improved by 4 and more wheels

Such arrangements are hyper static

Necessitates a flexible suspension system



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Wheel ArrangementsTwo and Three Wheeled Robots

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Two wheels:

Three wheels:

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Wheel ArrangementsFour Wheeled Robots

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Four wheels:

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Motion ConstraintsDegree of Maneuverability

Maneuverabiliy degree of mobility

degree of steerability

The maneuverability of a robot is composed of

Mobility (restricted by the no-sliding constraints)

Additional freedom contributed by the steering mechanisms

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m

s

smM

25


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Motion ConstraintsDegree of Mobility

To avoid any lateral slip the motion vectorneeds to satisfy:

Hence, to avoid lateral slip must belong to the null spaceNof

The null space of is the space where any vector verifies

Geometrically this can be shown by the Instantaneous Center of Rotation(ICR)

The larger the rank of , the more constrained a robots mobility

Consequently, , with

no standard wheels:

all directions constrained:

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IR )( )(1 sC

)(1 sC 0)(1 nC s

IR

)(

0)()(1

Is RC

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1

1)(

)(

sf NN

ss

f

sC

CC

n

)(1 s

C

)(3)(dim11 ssm

CrankCN 3)(0 1 sCrank

0)(1 sCrank 3)(1 sCrank

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Motion ConstraintsInstantaneous Center of Rotation

Instantaneous Center of Rotation (ICR) For any robot with the ICR is constrained to

lie on a line

For any robot with the ICR can be set to any point on the 2D plane

ExamplesAckermann Steering Bicycle

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3M

2M

27


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Motion ConstraintsDegree of Steerability

Degree of steerability Represents an indirectdegree of motion

The particular orientation at any instant imposes a kinematic constraint

The ability to change that orientation leads to an additional degree ofmaneuverability

Range of :

Examples:

One steered wheel: Tricycle,

Two steered wheels: Two-steer,

Two steered wheels on a common axis: Car (with Ackermann steering),

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)(1 sss Crank

20 ss

1s

1s

2s

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Motion ConstraintsDegree of Maneuverability

Degree of Maneuverability Two robots with the same may not be equivalent

instantaneously

Examples

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smM M

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Motion ConstraintsOmni-Drive Example

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Omni-Drive

0y

0x

30


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Motion ConstraintsTwo-Steer Example

Two-Steer

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0y

0x

31


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Differential Inverse KinematicsIterative Scheme Control Theory

In presence of non-holonomic constraints, diff-erential inverse kinematics must be considered

Transformation between velocities instead of positions

Solved via (state ) feedback control

Example: differential drive (kinematic unicycle), AMR book Chapter 3.6

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(nonintegrable)Robot Model

-

Control law

),,( yx),( 21

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