Mobile Robotics:11. Kinematics 2

Dr. B

rian Mac N

amee (w

ww

.comp.dit.ie/bm

acnamee)

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2of25 Acknowledgments

These notes are based (heavily) on those provided by the authors to accompany “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 and Amazon.com

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3of25 More KinematicsToday we will continue our discussion of kinematics and movement of robots through a workspace

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Wheel 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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Wheel Kinematic Constraints: Fixed Standard Wheel

Robot Chassis

XR

YR

P

l

Aβ

vα

The fixed standard wheel has a fixed angle to the robot chassis

Motion is limited to: – Back and forth

along the wheel plane

– Rotation around the contact point with the ground plane

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Wheel Kinematic Constraints: Fixed Standard Wheel (cont…)

The first constraint states that all motion along the wheel plane is accompanied by the appropriate amount of wheel spin

Which, through some maths jiggery-pokery we can write as:

0coscossin rRl I

0 spin wheel

toduemovement

plane wheel

alongmovement

movement along wheel plane movement due to wheel spin

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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:

0plane wheel theto

anglesright at movement

0cossincos IRl

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8of25 Wheel Kinematic ConstraintsSimilar equations can be determined for steerable standard wheels, but we won’t worry about those

There are no constraints for Swedish wheels, castor wheels or spherical wheels - why?

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9of25 Robot Kinematic Constraints

Given a robot with M wheels– 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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10of25 Instantaneous Center of RotationEach wheel has a zero motion line through its horizontal axis perpendicular to the wheel planeAt any moment wheel motion through this line must be zeroSo the wheel must be moving along some circle of radius R such that the centre of this circle is on the zero motion lineThe centre point is called the instantaneous centre of rotation (ICR)

When R is at infinity the wheel moves in a straight line

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Instantaneous Center of Rotation (cont…)

Zero motion lines

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Instantaneous Center of Rotation (cont…)

What about these configurations?

Differential Drive Tricycle

13of25 Mobile Robot ManeuverabilityManeuverability 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 steerability

Maneuverability is simply the sum of mobility and steerability

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Maneuverability Of Three-Wheel Configurations

Where δM is manoeuvrability, δm is mobility and δs is steerability

15of25 Holonomic RobotsIn robotics the concept of holonomy is often used

The term holonomic is used in many branches of mathematics

In mobile robotics holonomic refers to the kinematic constraints of a robot chassis

A holonomic has zero kinematic constraints

A non-holonomic robot has some constraints

Fixed and steered standard wheels impose non-holonomic constraints

16of25 Robots In Their WorkspaceWhen we think about the degrees of freedom of a robot we are not telling the whole story

Not only do we have to think about the arrangement of the robot, but also the robot’s pose within its environment

So it is very important to consider the robot within its workspace

17of25 Paths & TrajectoriesIt is easy to talk about the paths we expect robots to take through their environment

A path is specified in three dimensions as the robot’s x coordinate, y coordinate and rotation (θ)A trajectory involves a fourth dimension - time

18of25 Path/Trajectory ConsiderationsSuppose we want to perform the following:

– Move along XI axis at a constant speed of 1m/s for 1

second– Change orientation clockwise 90° in 1 second

– Move along YI axis at 1 m/s for 1 second

Let’s see how a holonomic robot and then a non-holonomic robot would achieve this

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Path/Trajectory Considerations: Holonomic Robot

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Path/Trajectory Considerations: Non-Holonomic Robot

21of25 Motion Control (Kinematic Control)The objective of a kinematic controller is to follow a trajectory described by its position and/or velocity profiles as function of time

Motion control is not straight forward because mobile robots are non-holonomic systems

However, it has been studied by various research groups and some adequate solutions for (kinematic) motion control of a mobile robot system are available

22of25 Motion Control: Open Loop ControlTrajectory divided in motion segments of defined shape:

– Straight lines and segments of a circle

Control problem:– Pre-compute a smooth trajectory

based on line and circle segments

Disadvantages:– It is not at all an easy task to pre-

compute a feasible trajectory – Limitations and constraints of the

robot’s velocities and accelerations– Does not adapt or correct the

trajectory if changes of the environment occur

– The resulting trajectories are usually not smooth

y I

x I

goal

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yR

x R

goal

v(t)

(t)

starte

Motion Control: Feedback ControlMotion control becomes a closed-loop problem where we try to minimise the error between the robot’s current position and the position of its goal

24of25 SummaryToday we looked at:

– Kinematic constraints imposed by robot wheel arrangments

– Paths & trajectories– Kinematic motion control

Next time we will start to look at localisation and mapping

25of25 Questions

?