1. B659: Principles of Intelligent Robot Motion Kris Hauser

2. Agenda  Skimming through Principles Ch. 2, 5.1, 6.1

3. Fundamental question of motion planning  Are the two given points connected by a path? Forbidden region Feasible space

4. Tool: Configuration Space

5. Problem free space s g free path obstacle

6. Problem g s Semi-free path

7.  Local constraints: lie in free space  Differential constraints: have bounded curvature  Global constraints: have minimal length Types of Path Constraints

8. Homotopy of Free Paths

9. Motion-Planning Framework Continuous representation Discretization Graph searching (blind, best-first, A*)

10. Path-Planning Approaches 1. Roadmap Represent the connectivity of the free space by a network of 1-D curves 2. Cell decomposition Decompose the free space into simple cells and represent the connectivity of the free space by the adjacency graph of these cells 3. Potential field Define a function over the free space that has a global minimum at the goal configuration and follow its steepest descent

11. Roadmap Methods  Visibility graph Introduced in the Shakey project at SRI in the late 60s. Can produce shortest paths in 2-D configuration spaces

12. Simple (Naïve) Algorithm 1. Install all obstacles vertices in VG, plus the start and goal positions 2. For every pair of nodes u, v in VG 3. If segment(u,v) is an obstacle edge then 4. insert (u,v) into VG 5. else 6. for every obstacle edge e 7. if segment(u,v) intersects e 8. then goto 2 9. insert (u,v) into VG 10. Search VG using A*

13. Complexity  Simple algorithm: O(n 3 ) time  Rotational sweep: O(n 2 log n)  Optimal algorithm: O(n 2 )  Space: O(n 2 )

14. Rotational Sweep

19. Reduced Visibility Graph tangent segments  Eliminate concave obstacle vertices can’t be shortest path

20. Generalized (Reduced) Visibility Graph tangency point

21. Three-Dimensional Space Computing the shortest collision-free path in a polyhedral space is NP-hard Shortest path passes through none of the vertices locally shortest path homotopic to globally shortest path

22. Roadmap Methods  Voronoi diagram Introduced by Computational Geometry researchers. Generate paths that maximizes clearance. O(n log n) time O(n) space

23. Roadmap Methods  Visibility graph  Voronoi diagram  Silhouette First complete general method that applies to spaces of any dimension and is singly exponential in # of dimensions [Canny, 87]  Probabilistic roadmaps

24. Cell-Decomposition Methods Two classes of methods:  Exact cell decomposition The free space F is represented by a collection of non-overlapping cells whose union is exactly F Example: trapezoidal decomposition

25. Path-Planning Approaches 1. Roadmap Represent the connectivity of the free space by a network of 1-D curves 2. Cell decomposition Decompose the free space into simple cells and represent the connectivity of the free space by the adjacency graph of these cells 3. Potential field Define a function over the free space that has a global minimum at the goal configuration and follow its steepest descent

26. Trapezoidal decomposition

30. critical events  criticality-based decomposition …

31. Trapezoidal decomposition Planar sweep  O(n log n) time, O(n) space

32. Cell-Decomposition Methods Two classes of methods:  Exact cell decomposition  Approximate cell decomposition F is represented by a collection of non-overlapping cells whose union is contained in F Examples: quadtree, octree, 2 n -tree

33. Octree Decomposition

34. Sketch of Algorithm 1. Compute cell decomposition down to some resolution 2. Identify start and goal cells 3. Search for sequence of empty/mixed cells between start and goal cells 4. If no sequence, then exit with no path 5. If sequence of empty cells, then exit with solution 6. If resolution threshold achieved, then exit with failure 7. Decompose further the mixed cells 8. Return to 2

35. Path-Planning Approaches 1. Roadmap Represent the connectivity of the free space by a network of 1-D curves 2. Cell decomposition Decompose the free space into simple cells and represent the connectivity of the free space by the adjacency graph of these cells 3. Potential field Define a function over the free space that has a global minimum at the goal configuration and follow its steepest descent

36.  Approach initially proposed for real-time collision avoidance [Khatib, 86]. Hundreds of papers published on it. Potential Field Methods Goal Robot

37. Attractive and Repulsive fields

38. Local-Minimum Issue  Perform best-first search (possibility of combining with approximate cell decomposition)  Alternate descents and random walks  Use local-minimum-free potential (navigation function)

39. Sketch of Algorithm (with best-first search) 1. Place regular grid G over space 2. Search G using best-first search algorithm with potential as heuristic function

40. Simple Navigation Function 0 11 1 2 22 23 3 3 44 5

41. 1 122 2 3 3 3 45 21 0 4

42. 1 122 2 3 3 3 45 21 0 4

43. Completeness of Planner  A motion planner is complete if it finds a collision-free path whenever one exists and return failure otherwise.  Visibility graph, Voronoi diagram, exact cell decomposition, navigation function provide complete planners  Weaker notions of completeness, e.g.: - resolution completeness (PF with best-first search) - probabilistic completeness (PF with random walks)

44.  A probabilistically complete planner returns a path with high probability if a path exists. It may not terminate if no path exists.  A resolution complete planner discretizes the space and returns a path whenever one exists in this representation.

45. Preprocessing / Query Processing  Preprocessing: Compute visibility graph, Voronoi diagram, cell decomposition, navigation function  Query processing: - Connect start/goal configurations to visibility graph, Voronoi diagram - Identify start/goal cell - Search graph

46. Issues for Future Lectures on Planning  Space dimensionality  Geometric complexity of the free space  Constraints other than avoiding collision  The goal is not just a position to reach  Etc …

47. Semester Project  Topic of your choosing, advised and approved by instructor  Groups of 1-3 students  Schedule Proposal (Feb.)Proposal (Feb.) Mid term report / discussion (March)Mid term report / discussion (March) Final presentation (end of April)Final presentation (end of April)

48. Semester Project Discussion  Three types of project: Make a robot or virtual character perform a taskMake a robot or virtual character perform a task Implement and analyze an algorithm (empirically or mathematically)Implement and analyze an algorithm (empirically or mathematically) Analyze natural motion using principles from roboticsAnalyze natural motion using principles from robotics

49. Project Ideas  Robot chess  Finding and tracking people indoors  UI for assistive robot arms  Analysis for psychological observation studies (Prof. Yu)  Outdoor vehicle navigation (Prof. Johnson)  Motion in social contexts (Profs. Scheutz and Sabanovic)

50. Platforms Available  GUIs, simulations: OpenGL (basic 3D GUI)OpenGL (basic 3D GUI) Player/Stage/Gazebo (mobile robot simulation)Player/Stage/Gazebo (mobile robot simulation) Open Dynamics Engine (rigid body simulation)Open Dynamics Engine (rigid body simulation)  Toolkits: KrisLibrary: math, optimization, kinematics, geometry, motion planningKrisLibrary: math, optimization, kinematics, geometry, motion planning Motion Planning Toolkit (MPK)Motion Planning Toolkit (MPK) OOPSMPOOPSMP  Robots: Lynxmotion AL5D hobbyist armLynxmotion AL5D hobbyist arm Segway RMP with laser sensorSegway RMP with laser sensor Staubli TX90L industrial robot armsStaubli TX90L industrial robot arms ERA-MOBI mobile robotERA-MOBI mobile robot

51. Readings  Principles Ch. 3.1-3.4  Lozano-Perez (1986)  *Lozano-Perez (1983)