1. Randomized Motion Planning
Jean-Claude Latombe
Computer Science Department Stanford University

2. Goal of Motion Planning
Answer queries about connectivity of a space
Classical example: find a collision-free path in robot configuration space among static obstacles
Examples of additional constraints:
Kinodynamic constraints
Visibility constraints

3. Outline Bits of history Approaches Probabilistic Roadmaps Applications
Conclusion

4. Shakey (Nilsson, 1969): Visibility graph
Early Work
Shakey (Nilsson, 1969): Visibility graph

5. Mathematical Foundations
Lozano-Perez, 1980: Configuration Space
C = S1 x S1

6. Computational Analysis
Reif, 1979: Hardness (lower-bound results)

7. Exact General-Purpose Path Planners
- Schwarz and Sharir, 1983: Exact cell decomposition based on Collins technique
- Canny, 1987: Silhouette method

8. Heuristic Planners
Khatib, 1986:
Potential Fields

9. Other Types of Constraints
E.g., Visibility-Based Motion Planning
Guibas, Latombe, LaValle, Lin, and Motwani, 1997

10. Outline Bits of history Approaches Probabilistic Roadmaps Applications
Conclusion

11. Criticality-Based Motion Planning
Principle:
Select a property P over the space of interest
Compute an arrangement of cells such that P stays constant over each cell
Build a search graph based on this arrangement
Example: Wilson’s Non-Directional Blocking Graphs for assembly planning
Other examples:
Schwartz-Sharir’s cell decomposition
Canny’s roadmap

12. Criticality-Based Motion Planning
Advantages:
Completeness
Insight
Drawbacks:
Computational complexity
Difficult to implement

13. Sampling-Based Motion Planning
Principle:
Sample the space of interest
Connect sampled points by simple paths
Search the resulting graph
Example: Probabilistic Roadmaps (PRM’s)
Other example: Grid-based methods (deterministic sampling)

14. Sampling-Based Motion Planning
Advantages:
Easy to implement
Fast, scalable to many degrees of freedom and complex constraints
Drawbacks:
Probabilistic completeness
Limited insight

15. Outline Bits of history Approaches Probabilistic Roadmaps Applications
Conclusion

16. Motivation Computing an explicit representation of the admissible
space is hard, but checking that a point lies in the
admissible space is fast

17. Probabilistic Roadmap (PRM)
admissible space
milestone mb mg
[Kavraki, Svetska, Latombe,Overmars, 95]

18. Sampling Strategies Multi vs. single query strategies
Multi-stage strategies
Obstacle-sensitive strategies
Lazy collision checking
Probabilistic biases (e.g., potential fields)

19. PRM With Dynamic Constraints in State x Time Spacemb mg
endgame region
m’ = f(m,u)
[Hsu, Kindel, Latombe, and Rock, 2000]

20. Relation to Art-Gallery Problems
[Kavraki, Latombe, Motwani, Raghavan, 95]

21. Narrow Passage Issue

22. Desirable Properties of a PRM
Coverage: The milestones should see most of the admissible space to guarantee that the initial and goal configurations can be easily connected to the roadmap
Connectivity: There should be a 1-to-1 map between the components of the admissible space and those of the roadmap

23. Complexity Measures
e-goodness [Kavraki, Latombe, Motwani, and Raghavan, 1995]
Path clearance [Kavraki, Koulountzakis, and Latombe, 1996]
e-complexity [Overmars and Svetska, 1998]
Expansiveness [Hsu, Latombe, and Motwani, 1997]

24. Expansiveness of Admissible Space

25. Expansiveness of Admissible Space
The admissible space is
expansive if each of its subsets has a large lookout
Prob[failure] = K exp(-r)
Lookout of F1

26. Two Very Different Cases
Expansive
Poorly expansive

27. A Few Remarks
Big computational saving is achieved at the cost of slightly reduced completeness
Computational complexity is a function of the shape of the admissible space, not the size needed to describe it
Randomization is not really needed; it is a convenient incremental scheme

28. Outline Bits of history Approaches Probabilistic Roadmaps Applications
Conclusion

29. Design for Manufacturing and Servicing
General Motors
General Motors
General Electric
[Hsu, 2000]

30. Robot Programming and Placement
[Hsu, 2000]

31. Graphic Animation of Digital Actors
The Motion Factory
[Koga, Kondo, Kuffner, and Latombe, 1994]

32. Digital Actors With Visual Sensing
Simulated Vision
Kuffner, 1999
Segment environment
Render false-color scene offscreen
Scan pixels & record IDs
Actor camera image
Vision module image

33. [Kuffner and Inoue, 2000] (U. Tokyo)
Humanoid Robot
[Kuffner and Inoue, 2000] (U. Tokyo)

34. [Kindel, Hsu, Latombe, and Rock, 2000]
Space Robotics
robot
obstacles
air thrusters
gaz tank
air bearing
[Kindel, Hsu, Latombe, and Rock, 2000]

35. Total duration : 40 sec

36. Autonomous Helicopter
[Feron, 2000] (AA Dept., MIT)

37. Interacting Nonholonomic Robotsy1 x2 d x1 y2 q1 q2
(Grasp Lab - U. Penn)

38. Map Building
[Gonzalez, 2000]

39. Next-Best View Computation

40. Map Building
[Gonzalez, 2000]

41. Map Building
[Gonzalez, 2000]

42. Radiosurgical Planning
Cyberknife System (Accuray, Inc.) CARABEAMER Planner [Tombropoulos, Adler, and Latombe, 1997]

43. Radiosurgical Planning
• < Tumor < 2200
2000 < B2 + B4 < 2200
2000 < B4 < 2200
2000 < B3 + B4 < 2200
2000 < B3 < 2200
2000 < B1 + B3 + B4 < 2200
2000 < B1 + B4 < 2200
2000 < B1 + B2 + B4 < 2200
2000 < B1 < 2200
2000 < B1 + B2 < 2200
• 0 < Critical < 500
0 < B2 < 500 T C B1 B2 B3 B4

44. Sample Case 50% Isodose Surface 80% Isodose Surface
Conventional system’s plan
CARABEAMER’s plan

45. Reconfiguration Planning for Modular Robots
Casal and Yim, 1999
Xerox, Parc

46. Prediction of Molecular Motions
Ligand-protein binding
Protein folding
[Singh, Latombe, and Brutlag, 1999]
[Apaydin, 2000]

47. Capturing Energy Landscape
[Apaydin, 2000]

48. Outline Bits of history Approaches Probabilistic Roadmaps Applications
Conclusion

49. Conclusion
PRM planners have successfully solved many diverse complex motion problems with different constraints (obstacles, kinematics, dynamics, stability, visibility, energetic)
They are easy to implement
Fast convergence has been formally proven in expansive spaces. As computers get more powerful, PRM planners should allow us to solve considerably more difficult problems
Recent implementations solve difficult problems with many degrees of freedom at quasi-interactive rate

50. Issues Relatively large standard deviation of planning time
No rigorous termination criterion when no solution is found
New challenging applications …

51. Planning Minimally Invasive Surgery Procedures Amidst Soft-Tissue Structures

52. Planning Nice-Looking Motions for Digital Actors
A Bug’s Life (Pixar/Disney)
Toy Story (Pixar/Disney)
Antz (Dreamworks)
The Artist’s perspective on animated characters
Animation Variables (AVARS)
FEATURES:
Kinematic hierarchy of geometric primitives
Moving joints change shape and/or position
Situated in some scenario or storyline
Graphical representation of a “personality”
Top 3 movies and video games
Contrasts the different approaches to motion:
MOVIES: off-line ray-tracing hand-animated
GAMES: real-time polygon rend. Captured or hand-animated canned motions + real-time blending, warping, IK
If you have a good algorithm for generating motion for off-line animation, it will apply to real-time games in years (if you believe Moore’s law)
Tomb Raider 3 (Eidos Interactive)
The Legend of Zelda (Nintendo)
Final Fantasy VIII (SquareOne)

53. Dealing with 1,000s of Degrees of Freedom
Protein folding

54. Main Common Difficulty
Formulating motion constraints