1. Kinematics of Mobile Robots

2. Forward Kinematics of Differential drive

3. Where can we find differential drives?
Braitenberg Vehicles
Micromouse
Most of our robots
Unicycle
Segway
Many many others

4. Kinematics of Differential drive
Differential Drive is the most common kinematic choice
- difference in wheels’ speeds determines its turning angle
All of the miniature robots…
Pioneer, Rug warrior
Questions (forward kinematics)
Given the wheel’s velocities or positions, what is the robot’s velocity/position ? VL
Are there any inherent system constraints? VR
1) Specify system measurements
2) Determine the point (the radius) around which the robot is turning.
3) Determine the speed at which the robot is turning to obtain the robot velocity.
4) Integrate to find position.

5. Kinematics of Differential drive
1) Specify system measurements y
- consider possible coordinate systems VL x q 2d VR
(assume a wheel radius of 1)

6. Kinematics of Differential drive – radius of turning
1) Specify system measurements y
- consider possible coordinate systems
2) Determine the point (the radius) around which the robot is turning. VL x q 2d VR
ICC “instantaneous center of curvature”
(assume a wheel radius of 1)

7. Kinematics of Differential drive – angular velocity
1) Specify system measurements y
- consider possible coordinate systems
2) Determine the point (the radius) around which the robot is turning.
- to minimize wheel slippage, this point (the ICC) must lie at the intersection of the wheels’ axles VL x
- each wheel must be traveling at the same angular velocity q 2d VR
ICC “instantaneous center of curvature”
(assume a wheel radius of 1)
 = angular velocity

8. Kinematics of Differential drive
1) Specify system measurements y
- consider possible coordinate systems
2) Determine the point (the radius) around which the robot is turning. w
- to minimize wheel slippage, this point (the ICC) must lie at the intersection of the wheels’ axles VL x
- each wheel must be traveling at the same angular velocity around the ICC q 2d VR
ICC “instantaneous center of curvature”
(assume a wheel radius of 1)

9. Kinematics of Differential drive
1) Specify system measurements x y
- consider possible coordinate systems
2) Determine the point (the radius) around which the robot is turning. w
- each wheel must be traveling at the same angular velocity around the ICC VL
3) Determine the robot’s speed around the ICC and its linear velocity 2d VR
ICC
w(R+d) = VL R
w(R-d) = VR
robot’s turning radius
(assume a wheel radius of 1)

10. Kinematics of Differential drive
1) Specify system measurements x y
- consider possible coordinate systems
2) Determine the point (the radius) around which the robot is turning. w
- each wheel must be traveling at the same angular velocity around the ICC VL
3) Determine the robot’s speed around the ICC and then linear velocity 2d VR
ICC “instantaneous center of curvature”
ICC
w(R+d) = VL R
w(R-d) = VR
robot’s turning radius
Thus,
w = ( VR - VL ) / 2d
R = 2d ( VR + VL ) / ( VR - VL )
(assume a wheel radius of 1)

11. Kinematics of Differential drive – robot’s velocity
1) Specify system measurements x y
- consider possible coordinate systems
2) Determine the point (the radius) around which the robot is turning. w
- each wheel must be traveling at the same angular velocity around the ICC VL
3) Determine the robot’s speed around the ICC and then linear velocity 2d VR
ICC
w(R+d) = VL R
w(R-d) = VR
robot’s turning radius
Thus,
w = ( VR - VL ) / 2d
R = 2d ( VR + VL ) / ( VR - VL )
So, the robot’s velocity is
V = wR = ( VR + VL ) / 2

12. Kinematics of Differential drive – integrate to obtain positionx y
Vx = V(t) cos(q(t))
w(t)
V(t)
Vy = V(t) sin(q(t))
q(t) VL Vx 2d VR
ICC “instantaneous center of curvature”
ICC
R(t)
robot’s turning radius
with
w = ( VR - VL ) / 2d
R = 2d ( VR + VL ) / ( VR - VL )
V = wR = ( VR + VL ) / 2
What has to happen to change the ICC ?

13. Kinematics of Differential drive
4) Integrate to obtain position x y
Vx = V(t) cos(q(t))
w(t)
Vy = V(t) sin(q(t))
Thus,
x(t) = ∫ V(t) cos(q(t)) dt VL
y(t) = ∫ V(t) sin(q(t)) dt 2d
q(t) = ∫ w(t) dt VR
ICC
R(t)
robot’s turning radius
with
w = ( VR - VL ) / 2d
R = 2d ( VR + VL ) / ( VR - VL )
V = wR = ( VR + VL ) / 2

14. Kinematics of Differential drive – velocity components
speed
Vx = V(t) cos(q(t))
Vy = V(t) sin(q(t))
w(t)
Thus,
x(t) =  V(t) cos(q(t)) dt VL x
y(t) =  V(t) sin(q(t)) dt 2d
q(t) =  w(t) dt VR
ICC
Kinematics
R(t)
robot’s turning radius
with
w = ( VR - VL ) / 2d
R = 2d ( VR + VL ) / ( VR - VL )
V = wR = ( VR + VL ) / 2
What has to happen to change the ICC ?

15. Inverse Kinematics of Differential Drive

16. Inverse Kinematics – the problem
Given a desired position or velocity, what can we do to achieve it?
Key question: y
VL (t) x
VR(t)
starting position
final position

17. Inverse Kinematics – one solution
Given a desired position or velocity, what can we do to achieve it?
Key question: y
VL (t) x
VR(t)
starting position
final position

18. Inverse Kinematics – another solution
Given a desired position or velocity, what can we do to achieve it?
Key question: y
VL (t) x
VR(t)
starting position
final position

19. Inverse Kinematics – many numerical solutions to equations
Given a desired position or velocity, what can we do to achieve it?
Key question: y
Need to solve these equations:
x =  V(t) cos(q(t)) dt
y =  V(t) sin(q(t)) dt
VL (t) x
q =  w(t) dt
VR(t)
w = ( VR - VL ) / 2d
V = wR = ( VR + VL ) / 2
starting position
final position
for VL (t) and VR(t) .
There are lots of solutions...

20. Inverse Kinematics – finding the best solution
Given a desired position or velocity, what can we do to achieve it?
Key question: y
Finding some solution is not hard, but finding the “best” solution is very difficult...
VL (t) x
quickest time
most energy efficient
smoothest velocity
profiles
VR(t)
starting position
final position
VL (t) t
VL (t)
It all depends on who gets to define “best”...

21. Inverse Kinematics - decomposition
Usual approach: decompose the problem and control only a few DOF at a time
Differential Drive y
VL (t) x
VR(t)
starting position
final position

22. Inverse Kinematics – decomposition for Differential Drive
Usual approach: decompose the problem and control only a few DOF at a time
Differential Drive
(1) turn so that the wheels are parallel to the line between the original and final position of the robot origin. y
-VL (t) = VR (t) = Vmax
VL (t) x
VR(t)
starting position
final position

23. Inverse Kinematics
Usual approach: decompose the problem and control only a few DOF at a time
Differential Drive
(1) turn so that the wheels are parallel to the line between the original and final position of the robot origin. y
-VL (t) = VR (t) = Vmax
(2) drive straight until the robot’s origin coincides with the destination
VL (t) x
VL (t) = VR (t) = Vmax
VR(t)
starting position
final position

24. Inverse Kinematics
Usual approach: decompose the problem and control only a few DOF at a time
Differential Drive
(1) turn so that the wheels are parallel to the line between the original and final position of the robot origin. y
-VL (t) = VR (t) = Vmax
(2) drive straight until the robot’s origin coincides with the destination
VL (t) x
VL (t) = VR (t) = Vmax
VR(t)
(3) rotate again in order to achieve the desired final orientation
starting position
final position
-VL (t) = VR (t) = Vmax
VL (t)
VR (t) t

25. Problem
Represent the forward and inverse kinematics for this robot using notation from the previous lecture.

26. Sources Prof. Maja Mataric Dr. Fred Martin
Bryce Tucker and former PSU students
A. Ferworn,
Prof. Gaurav Sukhatme, USC Robotics Research Laboratory
Paul Hannah
Reuven Granot, Technion
Dodds, Harvey Mudd College