lecture11-kinematics2

7/27/2019 Lecture11-Kinematics2

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Mobile Robotics:

11. Kinematics 2

Dr.BrianMacN

amee(www.comp.dit.ie/bmacnamee)
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25Acknowledgments

These notes are based (heavily) on

those provided by the authors toaccompany Introduction to

Autonomous Mobile Robots by

Roland Siegwart and Illah R.Nourbakhsh

More information about the book is available at:http://autonomousmobilerobots.epfl.ch/

The book can be bought at:The MIT Press andAmazon.com
http://autonomousmobilerobots.epfl.ch/http://mitpress.mit.edu/catalog/item/default.asp?ttype=2&tid=10138http://www.amazon.com/Introduction-Autonomous-Mobile-Intelligent-Robotics/dp/026219502Xhttp://mitpress.mit.edu/images/products/books/026219502X-f30.jpghttp://www.amazon.com/Introduction-Autonomous-Mobile-Intelligent-Robotics/dp/026219502Xhttp://mitpress.mit.edu/catalog/item/default.asp?ttype=2&tid=10138http://autonomousmobilerobots.epfl.ch/


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25More Kinematics

Today we will continue our discussion of

kinematics and movement of robots through aworkspace


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25Wheel Kinematic Constraints: Assumptions

We will make the following assumptions about

wheels: Movement on a horizontal plane

Point contact of the wheels

Wheels are not deformable Pure rolling

v = 0 at contact point

No slipping, skidding or sliding

No friction for rotation around contact point

Steering axes orthogonal to the surface

Wheels connected by rigid frame (chassis)

r

v


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25Wheel Kinematic Constraints: Fixed Standard Wheel

Robot Chassis

XR

YR

P

l

A

v

The fixed standard

wheel has a fixedangle to the robot

chassis

Motion is limited to: Back and forth

along the wheel

plane

Rotation around thecontact point with

the ground plane


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Wheel Kinematic Constraints: Fixed Standard Wheel

(cont)

The second constraint is that motion at right angles

to the wheel plane must be zero

Which, through some maths jiggery-pokery we can

write as:

0planewheeltheto

anglesrightatmovement

0cossincos I

Rl


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25Wheel Kinematic Constraints

Similar equations can be determined for steerable

standard wheels, but we wont worry about those

There are no constraints for Swedish wheels,

castor wheels or spherical wheels - why?


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25Robot Kinematic Constraints

Given a robot withMwheels

Each wheel imposes zero or more constraints on the

robot motion

Only fixed and steerable standard wheels impose

constraints

What is the maneuverability of a robot considering

a combination of different wheels?


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25Instantaneous Center of Rotation

Each wheel has a zero motion line through its

horizontal axis perpendicular to the wheel planeAt any moment wheel motion through this line mustbe zero

So the wheel must be moving along some circle of

radiusR such that the centre of this circle is on thezero motion line

The centre point is called the instantaneous centre

of rotation (ICR)WhenR is at infinity the wheel moves in a straightline


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25Instantaneous Center of Rotation (cont)

Zero

motion

lines


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25Instantaneous Center of Rotation (cont)

What about these configurations?

Differential Drive Tricycle


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25Mobile Robot Maneuverability

Maneuverability can be considered a combination

of: The mobility available based on the sliding

constraints

The additional freedom contributed by the steering

(steerability)

Equations based on the constraints we spoke

about earlier can be derived to calculate mobility

and steerabilityManeuverability is simply the sum of mobility and

steerability

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25Maneuverability Of Three-Wheel Configurations

Where M is manoeuvrability, m is mobility and s is

steerability

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25Holonomic Robots

In robotics the concept ofholonomyis often used

The term holonomicis used in many branches ofmathematics

In mobile robotics holonomic refers to the kinematic

constraints of a robot chassisA holonomic has zero kinematic constraints

A non-holonomic robot has some constraints

Fixed and steered standard wheels impose non-holonomic constraints

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25Robots In Their Workspace

When we think about the degrees of freedom of a

robot we are not telling the whole storyNot only do we have to think about the

arrangement of the robot, but also the robots pose

within its environmentSo it is very important to consider the robot within

its workspace

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25Paths & Trajectories

It is easy to talk about the paths we expect robots

to take through their environmentA path is specified in three dimensions as the

robotsx coordinate,y coordinate and rotation ()

A trajectory involves a fourth dimension - time

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25Path/Trajectory Considerations

Suppose we want to perform the following:

Move alongXIaxis at a constant speed of 1m/s for 1second

Change orientation clockwise 90 in 1 second

Move along YIaxis at 1 m/s for 1 secondLets see how a holonomic robot and then a non-

holonomic robot would achieve this

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25Path/Trajectory Considerations: Holonomic Robot

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25Path/Trajectory Considerations: Non-Holonomic Robot

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M i C l (Ki i C l)


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25Motion Control (Kinematic Control)

The objective of a kinematic controller is to follow a

trajectory described by its position and/or velocityprofiles as function of time

Motion control is not straight forward because

mobile robots are non-holonomic systemsHowever, it has been studied by various research

groups and some adequate solutions for

(kinematic) motion control of a mobile robot system

are available

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M ti C t l O L C t l


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25Motion Control: Open Loop Control

Trajectory divided in motion segments of defined shape:

Straight lines and segments of a circleControl problem:

Pre-compute a smooth trajectorybased on line and circle segments

Disadvantages:

It is not at all an easy task to pre-compute a feasible trajectory

Limitations and constraints of therobots velocities and accelerations

Does not adapt or correct thetrajectory if changes of theenvironment occur

The resulting trajectories are usuallynot smooth

I

xI

goal

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M ti C t l F db k C t l


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yR

xR

goal

v(t)

(t)

start e

Motion Control: Feedback Control

Motion control becomes a closed-loop problem

where we try to minimise the error between therobots current position and the position of its goal

24

S


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25Summary

Today we looked at:

Kinematic constraints imposed by robot wheelarrangments

Paths & trajectories

Kinematic motion control

Next time we will start to look at localisation and

mapping

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Q ti


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25Questions

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