1. Mobile Robotics: 11. Kinematics 2
Dr. Brian Mac Namee (
Mobile Robotics: 11. Kinematics 2

2. The book can be bought at: The MIT Press and Amazon.com
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:
The book can be bought at: The MIT Press and Amazon.com

3. More Kinematics
Today we will continue our discussion of kinematics and movement of robots through a workspace

4. 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)

5. Wheel Kinematic Constraints: Fixed Standard Wheel
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
Robot Chassis XR YR P l A β v α
Example:
For example, suppose that wheel A is in a position such that a=0, b=0.
This would place the contact point of the wheel on XI with the plane of the wheel oriented parallel to YI. If q=0 then sliding constraint (3.6) reduces to:

6. 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:
movement along wheel plane
movement due to wheel spin

7. 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:

8. Wheel Kinematic Constraints
Similar 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?

9. 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?

10. Instantaneous Center of Rotation
Each wheel has a zero motion line through its horizontal axis perpendicular to the wheel plane
At any moment wheel motion through this line must be zero
So the wheel must be moving along some circle of radius R such that the centre of this circle is on the zero motion line
The centre point is called the instantaneous centre of rotation (ICR)
When R is at infinity the wheel moves in a straight line

11. Instantaneous Center of Rotation (cont…)
Zero motion lines

12. Instantaneous Center of Rotation (cont…)
What about these configurations?
Differential Drive
Tricycle

13. Mobile 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 steerability
Maneuverability is simply the sum of mobility and steerability

14. Maneuverability Of Three-Wheel Configurations
Where δM is manoeuvrability, δm is mobility and δs is steerability

15. Holonomic Robots In 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

16. Robots In Their Workspace
When 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

17. Paths & Trajectories
It 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

18. Path/Trajectory Considerations
Suppose 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

19. Path/Trajectory Considerations: Holonomic Robot

20. Path/Trajectory Considerations: Non-Holonomic Robot

21. 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

22. Motion Control: Open Loop Control
Trajectory 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

23. Motion Control: Feedback Control
Motion 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

24. Summary Today 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

25. Questions ?