1. CS 326A: Motion Planning Criticality-Based Motion Planning: Target Finding

2. Trapezoidal decomposition

4. Criticality-Based Planning  Define a property P  Decompose the configuration space into “regular” regions (cells) over which P is constant.  Use this decomposition for planning  Issues: - What is P? It depends on the problem - How to use the decomposition?  Approach is practical only in low-dimensional spaces: - Complexity of the arrangement of cells - Sensitivity to floating point errors

5. Topics of this class and the next one  Target finding  Information (or belief) state/space  Assembly planning  Path space

6. 6 Example robot’s visibility region hiding region 1 cleared region 2 3 4 5 6 robot

7. 7 Problem  A target is hiding in an environment cluttered with obstacles  A robot or multiple robots with vision sensor must find the target  Compute a motion strategy with minimal number of robot(s)

8. 8 Assumptions  Target is unpredictable and can move arbitrarily fast  Environment is polygonal  Both the target and robots are modeled as points  A robot finds the target when the straight line joining them intersects no obstacles (omni-directional vision)

9. 9 Animated Target-Finding Strategy

10. 10 Does a solution always exist for a single robot? Hard to test: No “hole” in the workspace Easy to test: “Hole” in the workspace No !

11. 11 Effect of Geometry on the Number of Robots Two robots are needed

12. 12 Effect of Number n of Edges Minimal number of robots N =  (log n)

13. 13 Effect of Number h of Holes

14. 14 Information State  Example of an information state = (x,y,a=1,b=1,c=0)  An initial state is of the form (x,y,a=1,b=1,...,u=1)  A goal state is any state of the form (x,y,a=0,b=0,..., u=0) (x,y) a = 0 or 1 c = 0 or 1 b = 0 or 1 0  cleared region 1  contaminated region visibility region obstacle edge free edge

15. 15 Critical Line a=0 b=1 (x,y,a=0,b=1) Information state is unchanged (x,y,a=0,b=1) a=0 b=1 a=0 b=0 (x,y,a=0,b=0) Critical line contaminated area cleared area

16. 16 Grid-Based Discretization  Ignores critical lines  Visits many “equivalent” states  Many information states per grid point  Potentially very inefficient

17. 17 Discretization into Conservative Cells In each conservative cell, the “topology” of the visibility region remains constant, i.e., the robot keeps seeing the same obstacle edges

18. 18 Discretization into Conservative Cells In each conservative cell, the “topology” of the visibility region remains constant, i.e., the robot keeps seeing the same obstacle edges

19. 19 Discretization into Conservative Cells In each conservative cell, the “topology” of the visibility region remains constant, i.e., the robot keeps seeing the same obstacle edges

20. 20

21. 21 Search Graph  {Nodes} = {Conservative Cells} X {Information States}  Node (c,i) is connected to (c’,i’) iff: Cells c and c’ share an edge (i.e., are adjacent) Moving from c, with state i, into c’ yields state i’  Initial node (c,i) is such that: c is the cell where the robot is initially located i = (1, 1, …, 1)  Goal node is any node where the information state is (0, 0, …, 0)  Size is exponential in the number of edges

22. 22 Example A B CD E (C,a=1,b=1) (D,a=1)(B,b=1) a b

23. 23 A BC D E (C,a=1,b=1) (D,a=1)(B,b=1)(E,a=1)(C,a=1,b=0) Example

24. 24 A B CD E (C,a=1,b=1) (D,a=1)(B,b=1)(E,a=1)(C,a=1,b=0) (B,b=0)(D,a=1) Example

25. 25 A CD E (C,a=1,b=1) (D,a=1)(B,b=1)(E,a=1)(C,a=1,b=0) (B,b=0)(D,a=1) Much smaller search tree than with grid-based discretization ! B Example

26. 26 Visible Cleared Contaminated Example of Target-Finding Strategy

27. 27 More Complex Example 1 2 3

28. 28 Example with Recontaminations 6 54 3 21

29. 29 Example with Linear Number of Recontaminations Recontaminated area 43 21

30. 30 Example with Two Robots (Greedy algorithm)

31. 31 Example with Two Robots

32. 32 Example with Three Robots

33. 33 Robot with Cone of Vision

34. 34 Other Topics  Dimensioned targets and robots, three- dimensional environments  Non-guaranteed strategies  Concurrent model construction and target finding  Planning the escape strategy of the target