1. Car-Like Robot: How to Park a Car? (Nonholonomic Planning)

2. Types of Path Constraints
Local constraints: e.g., lie in free space
Differential constraints: e.g., have bounded curvature
Global constraints: e.g., have minimal length

3. Types of Path Constraints
Local constraints: e.g., lie in free space
Differential constraints: e.g., have bounded curvature
Global constraints: e.g., have minimal length

4. Car-Like Robot y x Configuration space is 3-dimensional: q = (x, y, q)

5. Example: Car-Like Robotq f
dx/dt = v cosq
dy/dt = v sinq
dq/dt = (v/L) tan f
|f| < F
dx sinq – dy cosq = 0 L q y x
Configuration space is 3-dimensional: q = (x, y, q)
But control space is 2-dimensional: (v, f) with |v| = sqrt[(dx/dt)2+(dy/dt)2]

6. Example: Car-Like Robotq f
dx/dt = v cosq
dy/dt = v sinq
dq/dt = (v/L) tan f
|f| < F
dx sinq – dy cosq = 0 L q y x
Lower-bounded turning radius

7. How Can This Work? Tangent Space/Velocity Spacex L q f x y q
(x,y,q)
(dx,dy,dq)
(dx,dy)
dx/dt = v cosq
dy/dt = v sinq
dq/dt = (v/L) tan f
|f| < F
dx sinq – dy cosq = 0 q

8. How Can This Work? Tangent Space/Velocity Spacex L q f x y q
(x,y,q)
(dx,dy,dq)
(dx,dy)
dx/dt = v cosq
dy/dt = v sinq
dq/dt = (v/L) tan f
|f| < F q

9. Type 1 Maneuver  Allows sidewise motion CYL(x,y,dq,h) h = 2r tandq
d = 2r(1/cosdq - 1) > 0
(x,y,q0) dq h h
(x,y) q r dq dq r
When dq  0, so does d and the cylinder becomes arbitrarily small
 Allows sidewise motion

10. Type 2 Maneuver
 Allows pure rotation

11. Combination

12. Combination

13. Coverage of a Path by Cylindersq + q q’ y x
Path created ignoring the car constraints

14. Path Examples

15. Drawbacks Final path can be far from optimal
Not applicable to car that can only move forward (e.g., think of an airplane)

16. Reeds and Shepp Paths

17. Reeds and Shepp Paths
CC|C0
CC|C
C|CS0C|C
Given any two configurations, the shortest RS paths between them is also the shortest path

18. Example of Generated Path
Holonomic
Nonholonomic

19. Other Technique: Control-Based Sampling
dx/dt = v cos q
dy/dt = v sin q
dq/dt = (v/L) tan f
|f| < F
dx sinq – dy cosq = 0
1. Select a node m
2. Pick v, f, and dt
3. Integrate motion from m
new configuration

20. Other Technique: Control-Based Sampling
Indexing array: A 3-D grid is placed over the configuration space. Each milestone falls into one cell of the grid. A maximum number of milestones is allowed in each cell (e.g., 2 or 3).
Asymptotic completeness: If a path exists, the planner is guaranteed to find one if the resolution of the grid is fine enough.

21. Computed Paths Tractor-trailer Car That Can Only Turn Left
jmax=45o, jmin=22.5o
jmax=45o

22. Application

23. Architectural Design: Verification of Building Code
C. Han

24. Other “Similar” Robots/Moving Objects (Nonholonomic)
Rolling-with-no-sliding contact (friction), e.g.: car, bicycle, roller skate
Submarine, airplane
Conservation of angular momentum: satellite robot, under-actuated robot, cat Why is it useful?
- Fewer actuators: design simplicity, less weight
- Convenience (think about driving a car with 3 controls!)