1. UNC Chapel Hill M. C. Lin Introduction to Motion Planning Applications Overview of the Problem Basics – Planning for Point Robot –Visibility Graphs –Roadmap –Cell Decomposition –Potential Field

2. UNC Chapel Hill M. C. Lin Goals Compute motion strategies, e.g., –Geometric paths –Time-parameterized trajectories –Sequence of sensor-based motion commands Achieve high-level goals, e.g., –Go to the door and do not collide with obstacles –Assemble/disassemble the engine –Build a map of the hallway –Find and track the target (an intruder, a missing pet, etc.)

3. UNC Chapel Hill M. C. Lin Fundamental Question Are two given points connected by a path?

4. UNC Chapel Hill M. C. Lin Basic Problem Problem statement: Compute a collision-free pathfor a rigid or articulated moving object among static obstacles. Input –Geometry of a moving object (a robot, a digital actor, or a molecule) and obstacles –How does the robot move? –Kinematics of the robot (degrees of freedom) –Initial and goal robot configurations (positions & orientations) Output Continuous sequence of collision-free robot configurations connecting the initial and goal configurations

5. UNC Chapel Hill M. C. Lin Example: Rigid Objects

6. UNC Chapel Hill M. C. Lin Example: Articulated Robot

7. UNC Chapel Hill M. C. Lin Is it easy?

8. UNC Chapel Hill M. C. Lin Hardness Results Several variants of the path planning problem have been proven to be PSPACE-hard. A complete algorithm may take exponential time. –A complete algorithm finds a path if one exists and reports no path exists otherwise. Examples –Planar linkages [Hopcroftet al., 1984] –Multiple rectangles [Hopcroftet al., 1984]

9. UNC Chapel Hill M. C. Lin Tool: Configuration Space Difficulty –Number of degrees of freedom (dimension of configuration space) –Geometric complexity

10. UNC Chapel Hill M. C. Lin Extensions of the Basic Problem More complex robots –Multiple robots –Movable objects –Nonholonomic& dynamic constraints –Physical models and deformable objects –Sensorlessmotions (exploiting task mechanics) –Uncertainty in control

11. UNC Chapel Hill M. C. Lin Extensions of the Basic Problem More complex environments –Moving obstacles –Uncertainty in sensing More complex objectives –Optimal motion planning –Integration of planning and control –Assembly planning –Sensing the environment Model building Target finding, tracking

12. UNC Chapel Hill M. C. Lin Next Few Lectures Present a coherent framework for motion planning problems: –Configuration space and related concepts –Algorithms based on random sampling and algorithms based on processing critical geometric events Emphasize “practical” algorithms with some guarantees of performance over “theoretical” or purely “heuristic” algorithms

13. UNC Chapel Hill M. C. Lin Practical Algorithms A complete motion planner always returns a solution when one exists and indicates that no such solution exists otherwise. Most motion planning problems are hard, meaning that complete planners take exponential time in the number of degrees of freedom, moving objects, etc.

14. UNC Chapel Hill M. C. Lin Practical Algorithms Theoretical algorithms strive for completeness and low worst-case complexity –Difficult to implement –Not robust Heuristic algorithms strive for efficiency in commonly encountered situations. –No performance guarantee Practical algorithms with performance guarantees –Weaker forms of completeness –Simplifying assumptions on the space: “exponential time” algorithms that work in practice

15. UNC Chapel Hill M. C. Lin Problem Formulation for Point Robot Input –Robot represented as a point in the plane –Obstacles represented as polygons –Initial and goal positions Output –A collision-free path between the initial and goal positions

16. UNC Chapel Hill M. C. Lin Framework

17. UNC Chapel Hill M. C. Lin Visibility Graph Method Observation: If there is a a collision-free path between two points, then there is a polygonal path that bends only at the obstacles vertices. Why? –Any collision-free path can be transformed into a polygonal path that bends only at the obstacle vertices. A polygonal path is a piecewise linear curve.

18. UNC Chapel Hill M. C. Lin Visibility Graph A visibility graphis a graph such that –Nodes: q init, q goal, or an obstacle vertex. –Edges: An edge exists between nodes u and v if the line segment between u and v is an obstacle edge or it does not intersect the obstacles.

19. UNC Chapel Hill M. C. Lin A Simple Algorithm for Building Visibility Graphs

20. UNC Chapel Hill M. C. Lin Computational Efficiency Simple algorithm O(n 3 ) time More efficient algorithms –Rotational sweep O(n 2 log n) time –Optimal algorithm O(n 2 ) time –Output sensitive algorithms O(n 2 ) space

21. UNC Chapel Hill M. C. Lin Framework

22. UNC Chapel Hill M. C. Lin Breadth-First Search

23. UNC Chapel Hill M. C. Lin Breadth-First Search

24. UNC Chapel Hill M. C. Lin Breadth-First Search

25. UNC Chapel Hill M. C. Lin Breadth-First Search

26. UNC Chapel Hill M. C. Lin Breadth-First Search

27. UNC Chapel Hill M. C. Lin Breadth-First Search

28. UNC Chapel Hill M. C. Lin Breadth-First Search

29. UNC Chapel Hill M. C. Lin Breadth-First Search

30. UNC Chapel Hill M. C. Lin Breadth-First Search

31. UNC Chapel Hill M. C. Lin Breadth-First Search

32. UNC Chapel Hill M. C. Lin Other Search Algorithms Depth-First Search Best-First Search, A*

33. UNC Chapel Hill M. C. Lin Framework

34. UNC Chapel Hill M. C. Lin Summary Discretize the space by constructing visibility graph Search the visibility graph with breadth- first search Q: How to perform the intersection test?

35. UNC Chapel Hill M. C. Lin Summary Represent the connectivity of the configuration space in the visibility graph Running time O(n3) –Compute the visibility graph –Search the graph –An optimal O(n2) time algorithm exists. Space O(n2) Can we do better?

36. UNC Chapel Hill M. C. Lin Classic Path Planning Approaches Roadmap – Represent the connectivity of the free space by a network of 1-D curves Cell decomposition – Decompose the free space into simple cells and represent the connectivity of the free space by the adjacency graph of these cells Potential field – Define a potential function over the free space that has a global minimum at the goal and follow the steepest descent of the potential function

37. UNC Chapel Hill M. C. Lin Classic Path Planning Approaches Roadmap – Represent the connectivity of the free space by a network of 1-D curves Cell decomposition – Decompose the free space into simple cells and represent the connectivity of the free space by the adjacency graph of these cells Potential field – Define a potential function over the free space that has a global minimum at the goal and follow the steepest descent of the potential function

38. UNC Chapel Hill M. C. Lin Roadmap Visibility graph Shakey Project, SRI [Nilsson, 1969] Voronoi Diagram Introduced by computational geometry researchers. Generate paths that maximizes clearance. Applicable mostly to 2-D configuration spaces.

39. UNC Chapel Hill M. C. Lin Voronoi Diagram Space O(n) Run time O(n log n)

40. UNC Chapel Hill M. C. Lin Other Roadmap Methods Silhouette First complete general method that applies to spaces of any dimensions and is singly exponential in the number of dimensions [Canny 1987] Probabilistic roadmaps

41. UNC Chapel Hill M. C. Lin Classic Path Planning Approaches Roadmap – Represent the connectivity of the free space by a network of 1-D curves Cell decomposition – Decompose the free space into simple cells and represent the connectivity of the free space by the adjacency graph of these cells Potential field – Define a potential function over the free space that has a global minimum at the goal and follow the steepest descent of the potential function

42. UNC Chapel Hill M. C. Lin Cell-decomposition Methods Exact cell decomposition The free space F is represented by a collection of non-overlapping simple cells whose union is exactly F Examples of cells: trapezoids, triangles

43. UNC Chapel Hill M. C. Lin Trapezoidal Decomposition

44. UNC Chapel Hill M. C. Lin Computational Efficiency Running time O(n log n) by planar sweep Space O(n) Mostly for 2-D configuration spaces

45. UNC Chapel Hill M. C. Lin Adjacency Graph Nodes: cells Edges: There is an edge between every pair of nodes whose corresponding cells are adjacent.

46. UNC Chapel Hill M. C. Lin Summary Discretize the space by constructing an adjacency graph of the cells Search the adjacency graph

47. UNC Chapel Hill M. C. Lin Cell-decomposition Methods Exact cell decomposition Approximate cell decomposition –F is represented by a collection of non- overlapping cells whose union is contained in F. –Cells usually have simple, regular shapes, e.g., rectangles, squares. –Facilitate hierarchical space decomposition

48. UNC Chapel Hill M. C. Lin Quadtree Decomposition

49. UNC Chapel Hill M. C. Lin Octree Decomposition

50. UNC Chapel Hill M. C. Lin Algorithm Outline

51. UNC Chapel Hill M. C. Lin Classic Path Planning Approaches Roadmap – Represent the connectivity of the free space by a network of 1-D curves Cell decomposition – Decompose the free space into simple cells and represent the connectivity of the free space by the adjacency graph of these cells Potential field – Define a potential function over the free space that has a global minimum at the goal and follow the steepest descent of the potential function

52. UNC Chapel Hill M. C. Lin Potential Fields Initially proposed for real-time collision avoidance [Khatib 1986]. Hundreds of papers published. A potential field is a scalar function over the free space. To navigate, the robot applies a force proportional to the negated gradient of the potential field. A navigation function is an ideal potential field that –has global minimum at the goal –has no local minima –grows to infinity near obstacles –is smooth

53. UNC Chapel Hill M. C. Lin Attractive & Repulsive Fields

54. UNC Chapel Hill M. C. Lin How Does It Work?

55. UNC Chapel Hill M. C. Lin Algorithm Outline Place a regular grid G over the configuration space Compute the potential field over G Search G using a best-first algorithm with potential field as the heuristic function

56. UNC Chapel Hill M. C. Lin Local Minima What can we do? –Escape from local minima by taking random walks –Build an ideal potential field – navigation function – that does not have local minima

57. UNC Chapel Hill M. C. Lin Question Can such an ideal potential field be constructed efficiently in general?

58. UNC Chapel Hill M. C. Lin Completeness A complete motion planner always returns a solution when one exists and indicates that no such solution exists otherwise. –Is the visibility graph algorithm complete? Yes. –How about the exact cell decomposition algorithm and the potential field algorithm?