1. Mobile Robotics: 10. Kinematics 1
Dr. Brian Mac Namee (
Mobile Robotics: 10. Kinematics 1

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. Mobile Robot Kinematics
Kinematics is the study of how mechanical systems behave
We need to understand the mechanical behaviour of robots in order to:
Design appropriate robots for the tasks
To understand how to create control software for the robot hardware

4. Mobile Robot Kinematics
We will focus on mobile robot kinematics
Mobile robot kinematics is similar to robot manipulator kinematics
However, a mobile robot can move unbounded with respect to its environment
There is no direct way to measure the robot’s position
Position must be integrated over time
Leads to inaccuracies in position (motion) estimate
The number 1 challenge in mobile robotics!
Understanding mobile robot motion starts with understanding wheel constraints

5. Mobile Robot Kinematics
During our derivation of a kinematic model for a mobile robot we will model the robot as a rigid body on wheels, operating on a horizontal plane
The total dimensionality of this kinematic model on the plane is three:
Two for position in the plane
One for orientation along the vertical axis, which is orthogonal to the plane
In order to specify the position of the robot on the plane we establish a relationship between the global reference frame of the plane and the local reference frame of the robot

6. Representing Robot Position
The axes XI and YI define an arbitrary initial basis on the plane as the global reference frame from some origin O: {XI,YI}

7. Representing Robot Position (cont…)
To specify the position of the robot, choose a point P on the robot chassis as its position reference point
The basis {XR, YR} defines two axes relative to P on the robots chassis and is thus the robot’s local reference frame

8. Representing Robot Position (cont…)
The position of P in the global reference frame is specified by coordinates x and y, and the angular difference between the global and local reference frame is given by 

9. Representing Robot Position (cont…)
We can describe the pose of the robot as a vector with these three elements: I = [ x y  ]
I subscript indicates that the basis of this pose is the global reference frame.

10. Representing Robot Position (cont…)
To describe robot motion in terms of component motions, it is necessary to map motion along the axes of the global reference frame to motion along the axes of the robot’s local reference frame

11. Representing Robot Position (cont…)
This mapping is accomplished using the orthogonal rotation matrix:
This matrix can be used to map motion in the global reference frame {XI, YI} to motion in terms of the local reference frame {XR, YR}

12. Representing Robot Position (cont…)
This operation is denoted by R( ) I because the computation of this operation depends on the value of  :
R = R(/2) I

13. Representing Robot Position (cont…)
For example, consider the robot in the figure right
Because =/2 we can easily compute the rotation matrix R:

14. Representing Robot Position (cont…)
Given some velocity (x’, y’, ’) in the global reference frame we can compute the components of motion along this robot’s local axes XR and YR

15. Representing Robot Position (cont…)
In this case, due to the specific angle of the robot, motion along XR is equal to y’ and motion along YR is -x’

16. Representing Robot Position
Representing a robot within an arbitrary initial frame
Initial frame: {XI, YI}
Robot frame: {XR, YR}
Robot position: I =
Mapping between the two frames:
where

17. Kinematics Model
Goal: establish the robot speed as a function of the wheel speeds φi, steering angles βi and steering speeds βi and the geometric parameters of the robot (configuration coordinates)
Forward kinematics
Inverse kinematics
Why not ? ‘Cause it’s too hard!

18. Forward Kinematic Models
The forward kinematic model of a differential- drive robot is relatively straight-forward
The differential drive robot has two wheels each of diameter r
Given a point P centred between the two drive wheels, each wheel is a distance l from P

19. Forward Kinematic Models (cont…)
Given r, l, θ and the spinning speed of each wheel, φ1 and φ2 a forward kinematic model would predict the robots overall speed in the global reference frame as:

20. Forward Kinematic Models (cont…)
We know that we can compute a robot’s motion in the global reference frame from the motion in its local reference frame
I = R(θ)-1R
So, the strategy is to first compute the contribution of each of the two wheels in the local reference frame and then convert these to the global reference frame

21. Forward Kinematic Models (cont…)
The contribution of each wheel’s spinning speed to the translation speed at P in the direction of +XR is given as:
The total translation speed at P is given as:
The speed is even easier at it must always be zero

22. Forward Kinematic Models (cont…)
The contribution of each wheel’s spinning speed to the rotation speed at P is given as:
The total rotation speed at P is given as:

23. Forward Kinematic Models (cont…)
Combining the individual formulas yields a kinematic model for a differential drive robot as:
where

24. Forward Kinematic Models (cont…)
Suppose the robot is positioned such that θ = π/2, r = 1, and l = 1 and the robot engages its wheels unevenly with φ1 = 4 and φ2 = 2
We can compute the velocity in the global reference frame as:

25. Exercise
Suppose the robot is positioned such that θ = π/2, r = 2, and l = 1 and the robot engages its wheels unevenly with φ1 = 3 and φ2 = 6
Compute the velocity of the robot in the global reference frame.

26. Summary Kinematics is the study of how we get robots to move
Mobile robot kinematics are particularly interesting as the robot’s movement is relatively unbounded
A forward kinematic model takes the details of robots geometry and the speed of its motors and converts these into global speeds

27. Questions? ?