1. Path Planning for a Point Robot
Motion Planning
Path Planning for a Point Robot
(This slides are based on Latombe and Hwang slides)

2. Configuration Space: Tool to Map a Robot to a Point

3. Problem
free space s
free path g

4. Problem
semi-free path

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

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

7. 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 function over the free space that has a global minimum at the goal configuration and follow its steepest descent

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

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

10. Complexity Simple algorithm: O(n3) time Rotational sweep: O(n2 log n)
Optimal algorithm: O(n2)
Space: O(n2)

11. Rotational Sweep Check the following page:
VISIBLE_VERTICES(PolygonSet S, Point p) 1. initialize balanced search tree T 2. initialize list L 3. Sort the obstacle vertices in set S according to the clockwise angle that the half-line from p to each vertex makes with the positive x-axis. In case of ties vertices closer to p should come before vertices farther from p. Let W1, ..., Wn be the sorted list. 4. Let r be the half-line parallel to the positive x-axis starting at p. Find the obstacle edges that are properly intersected by r, and store them in a balanced search tree T in the order in which they are intersected by r. 5. i = 1 // rotate the ray to the first vertex 6. while not (i > n) 7. Delete from T the obstacle edges incident to Wi that lie on the counterclockwise side of the half-line from p to Wi. 8. if (VISIBLE(p, Wi, T)) 9. L.append(Wi) 10. Insert into T the obstacle edges incident to Wi that lie on the clockwise side of the half-line from p to Wi. 11. increment i // rotate the ray to the next vertex 12. return L
Check the following page:

12. Rotational Sweep

13. Rotational Sweep

14. Rotational Sweep

15. Rotational Sweep

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

17. Generalized (Reduced) Visibility Graph
tangency point

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

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

20. 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 function over the free space that has a global minimum at the goal configuration and follow its steepest descent

21. 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 Examples:
Polygonal decomposition
Trapezoidal decomposition

22. Polygonal Configuration Space
- represent the configuration space boundary and obstacles as polygons
- polygons may be concave
- decompose the free space into convex cells

23. Polygonal Configuration Space
cells form channels instead of line paths
channels give the robot more information to avoid obstacles encountered at run time
- cells form channels instead of line paths
- the interior of the channel lies entirely in free space
- an easy way to generate a path is to connect the midpoint of every cell boundary

24. Polygonal Configuration Space
qfinal
qinit
- to do a query, identify the cells where the initial and final configurations lie
- look for a sequence of cells that connect the initial and final cells
- connect the midpoints together
- can apply spline curve fitting techniques if we want a smooth path

25. Trapezoidal decomposition

26. Trapezoidal decomposition

27. Trapezoidal decomposition

28. Trapezoidal decomposition

29. Trapezoidal decomposition
critical events  criticality-based decomposition …

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

31. Generalization to 3 Dimensions
- the sweeping method can be generalized to 3D quite easily
- instead of sweeping lines, we sweep planes, the result are parallelpiped cells
- the figure here is crude because it is just lifted from the 2D case. In general, we put a plane when we come to a vertex in the sweep
- the connectivity graph is generated much in the same way as the 2D case by connecting points on the cell boundaries

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, 2n-tree

33. Octree Decomposition

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

35. 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 function over the free space that has a global minimum at the goal configuration and follow its steepest descent

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

40. 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.

41. 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

42. Summary: Exact Cell Decomposition

43. Summary: Approximate Cell Decomposition

44. Summary: Potential Fields