D* Lite

1. D* Lite an incremental version of A* for navigating in unknown terrain It implements the same behavior as Stentz’ Focussed Dynamic A* but is algorithmically different. n.h.reyes@massey.ac.nz

2. Mars Rover

3. Incremental search + heuristic search How to search efficiently using heuristic to guide the search How to search efficiently by using re-using information from previous search results

4. Focussed Dynamic A* (D*) • Stentz 1995 • Clever heuristic method that achieves a speedup of one to two orders of magnitudes over repeated A* searches • The improvement is achieved by modifying previous search results locally • Extensively used on real robots, including outdoor high mobility multi-wheeled vehicle (HMMWV) • Integrated into Mars Rover prototypes and tactical mobile robot prototypes for urban reconnaissance

5. D* Litevs.D* • D* Liteimplements the same navigation strategy as D*, but is algorithmically different • Substantially shorter than D* • Uses only one tie-breaking criterion when comparing priorities (simplified maintenance) • No nested if statements with complex conditions • Simplifies the analysis of program flow • Easier to extend • At least as efficient as D*

6. Previously, we learned LPA* LPA* repeatedly determines shortest paths between Sstart and Sgoal as the edge costs of a graph change. original eight-connected gridworld

7. Path Planning LPA* repeatedly determines shortest paths between Sstart and Sgoal as the edge costs of a graph change. original eight-connected gridworld

8. Path Planning LPA* repeatedly determines shortest paths between Sstart and Sgoal as the edge costs of a graph change. changed eight-connected gridworld

9. Path Planning LPA* repeatedly determines shortest paths between Sstart and Sgoal as the edge costs of a graph change. changed eight-connected gridworld

10. LPA* • LPA* is an incremental version of A* that applies to the same finite path-planning problems as A*. • It shares with A*the fact that it uses non-negativeand consistentheuristicsh(s)that approximate the goal distances of the vertices sto focus its search. • Consistent heuristics obey the triangle inequality: • h(sgoal) = 0 • h(s) ≤c(s, s’) + h(s’); for all vertices s∈ S and s’∈ succ(s) with s≠ sgoal.

11. D* Lite LPA* repeatedly determines shortest paths between Sstart and Sgoal as the edge costs of a graph change. D* Literepeatedly determines shortest paths between the current vertex Scurrentof the robot and Sgoal as the edge costs of a graph change, while the robot moves towards Sgoal.

12. D* Lite LPA* repeatedly determines shortest paths between Sstart and Sgoal as the edge costs of a graph change. D* Literepeatedly determines shortest paths between the current vertex Scurrentof the robot and Sgoal as the edge costs of a graph change, while the robot moves towards Sgoal. D* Liteis suitable for solving goal-directed navigation problems in unknown terrains.

13. Free space Assumption

14. Free space Assumption

17. Free space assumption Move the robot on a shortest potentially unblocked path towards the goal.

18. Using the free space assumption in path planning • Search from the Goal towards the robot’s current location • This allows one to re-use parts of the search tree after the robot has moved. • This allows one to use heuristics to focus the search; thereby not requiring to search the entire graph.

19. Using the free space assumption in path planning • Search from the Goal towards the robot’s current location • This makes incremental search efficient.

20. Variables • S finite set of vertices • set of successorsof s • set of predecessors of s • Cost of moving from vertex s to vertex s’ • Start vertex – has no predecessors • Goal vertex– has no successors

21. LPA* Variables • Start distance = length of the shortest path from Sstart to S • g(s) = estimate of the Start distance g*(s)

22. D* Lite Variables • Goal distance = length of the shortest path from StoSgoal • g(s) = estimate of the Goal distance g*(s)

23. LPA*Rhs-value The rhs-values are one-step look-ahead values, based on the g-values; and thus, potentially better informed than the g-values

24. D* LiteRhs-value The rhs-values are one-step look-ahead values, based on the g-values; and thus, potentially better informed than the g-values

25. D* LiteRhs-value The rhs-values are one-step look-ahead values, based on the g-values and thus potentially better informed than the g-values • g-value = rhs-value: cell is locally consistent • g-value ≠ rhs-value: cell is locally inconsistent • g-value > rhs-value: cell is locally overconsistent • g-value < rhs-value: cell is locally underconsistent • the priority queue contains exactly the locally inconsistentvertices • their priority is [min(g(s),rhs(s)) + h(s,sstart) + km; min(g(s),rhs(s))] • smaller priorities first, according to a lexicographic ordering

26. Shortest Path • If all vertices are locally consistent, • g(s) == g*(s) ; for all vertices s • one can trace back the shortest path from Sstart to Sgoal. from Sstart to Sgoal • From vertex s, find a successors’ that minimises g(s’) + c(s, s’). • Ties can be broken arbitrarily. • Repeat until Sgoal is reached.

27. Selective Update • D*Lite does not make all vertices locally consistent after some edge costs have changed • It uses heuristicsto focus the search • It updates only the g-values that are relevant for computing the shortest path

28. Priority • D*Lite maintains a priority queue for keeping track of locally inconsistent vertices – vertices that potentially needs their g-values updated to make them locally consistent • Priority of a vertex = key • Key – vector with 2 components k(s) = [ k1(s); k2(s) ] k1(s) = min(g(s), rhs(s)) + h(s, sstart) + km k2(s) = min(g(s), rhs(s))

29. Priority • Priority of a vertex = key • Key – vector with 2 components k(s) = [ k1(s); k2(s) ] k1(s) = min(g(s), rhs(s)) + + h(s, sstart) + km k2(s) = min(g(s), rhs(s)) The vertex with the smallest key is expanded first by D*Lite. Key comparison (lexicographic ordering): k(s) ≤ k’(s) iffeither(k1(s) <k1‘(s) ) or (k1(s) ==k1‘(s) ) and (k2(s) ≤k2‘(s) )

30. Heuristic Function (8-connectedGridworld) • As an approximation of thedistance between two cells, we use the maximum of the absolute differences of their x and y coordinates. • These heuristics are for eight-connected gridworlds what Manhattan distances are for four-connected gridworlds. D*Liteh(s, sstart) – calculated relative to the startposition Lifelong Planning A* h(s, sgoal) – calculated relative to the goalposition

31. Priority Queue Management The pseudocode uses the following functions to manage the priority queue: • U.TopKey() - returns the smallest priority of all vertices in priority queue U. (If U is empty, then U.TopKey() returns [∞; ∞].) • U.Pop() - deletes the vertex with the smallest priority in priority queue U and returns the vertex. • U.Insert(s; k) - inserts vertex s into priority queue U with priority k. • Update(s; k) - changes the priority of vertex s in priority queue U to k. (It does nothing if the current priority of vertex s already equals k.) • U.Remove(s) - removes vertex s from priority queue U.

32. LPA* Pseudocode(just for comparison)

33. D* LitePseudocode gets satisfied when one of the edge costs change

34. Route-planning example:4-connected gridworld Example, Step 1

35. Route-planning example:4-connected gridworld Example, Step 2

36. Route-planning example:4-connected gridworld Example, Step 3

37. Route-planning example:4-connected gridworld Example, Step 4

38. Route-planning example:4-connected gridworld Example, Step 5

39. After one cell gets blocked… Blocked cell: C3 The previous search results are carried over to help solve this re-planning problem.

41. After one cell gets blocked… Blocked cell: C3 Neighbours of C3: B3, C4, D3, C2 Among the neighbours, Cells B3 & C2 calculates their rhs-values based on C3; therefore, set their g-values to ∞

42. Route-planning example:4-connected gridworld Example, Step 6

43. Route-planning example:4-connected gridworld Example, Step 7

44. Route-planning example:4-connected gridworld Example, Step 8

45. Route-planning example:4-connected gridworld Example, Step 9

46. Route-planning example:4-connected gridworld Example, Step 10

47. Route-planning example:4-connected gridworld Example, Step 11

48. Route-planning example:4-connected gridworld Example, Step 12

49. Simulation Example D* Liteoperating in a 4-Connected Gridworld • demonstration of the use of the km variable

50. Route-planning example #2:4-connected gridworld Example, Step 1