LaiOffer Self-Driving-Car-Lecture 3: Any-Angle Search

1. ANY-ANGLE SEARCH
AI Frontiers 2018

2. Source: Courtesy of Alex Nash

3. Source: Courtesy of Alex Nash

4. Source: Courtesy of Alex Nash

5. Source: Courtesy of Alex Nash

6. Source: Courtesy of Alex Nash

7. Source: Courtesy of Alex Nash

8. Source: Courtesy of Alex Nash

9. Source: Courtesy of Alex Nash

10. ANY-ANGLE TRADEOFFS
Path Length
ComputationTime

11. goal
start
A* (GRID GRAPHS)

12. goal
start
A* (GRID GRAPHS)

13. goal
start
A* (GRID GRAPHS)

14. goal
start
A* (GRID GRAPHS)

15. goal
start
A* (GRID GRAPHS)

16. goal
start
A* (GRID GRAPHS)

17. goal
start
A* (GRID GRAPHS)

18. goal
start
A* (GRID GRAPHS)
Pros
• Fast. Search is performed on a simple graph
• Graph edges are implied by the grid
• Each vertex has up to 8 neighbors
Cons
• Paths are long and unrealistic looking

19. ANY-ANGLE TRADEOFFS
Path Length
ComputationTime
A* (grid graphs)

20. ANY-ANGLE TRADEOFFS
Path Length
ComputationTime
A* (grid graphs)
Can we do better?
Hint: “Graphs don’t have to be grids”

21. goal
start
A* (VISIBILITY GRAPHS)

22. goal
start
A* (VISIBILITY GRAPHS)

23. goal
start
A* (VISIBILITY GRAPHS)
Pros
• Guaranteed to find a shortest path
Cons
• Slow. Search is performed on a complex graph
• Determining graph edges requires line-of-sight checks
• Each vertex can have lots of edges

24. goal
start
A* (VISIBILITY GRAPHS)
Pros
• Guaranteed to find a shortest path
Cons
• Slow. Search is performed on a complex graph
• Determining graph edges requires line-of-sight checks
• Each vertex can have lots of edges

25. ANY-ANGLE TRADEOFFS
Path Length
ComputationTime
A* (grid graphs)
A* (visibility graphs)

26. ANY-ANGLE TRADEOFFS
Path Length
ComputationTime
A* (grid graphs)
A* (visibility graphs)
Can we do better?
Hint: “Tighten slack ropes”

27. goal
start
A* (POST-SMOOTHING)

28. goal
start
A* (POST-SMOOTHING)

29. goal
start
A* (POST-SMOOTHING)

30. goal
start
A* (POST-SMOOTHING)

31. goal
start
A* (POST-SMOOTHING)
Pros
• Simple and fast
Cons
• Is often ineffective because search is decoupled from post-processing
• A* often finds paths that cannot be smoothed

32. ANY-ANGLE TRADEOFFS
Path Length
ComputationTime
A* (grid graphs)
A* (visibility graphs)
A* with post-smoothing

33. ANY-ANGLE TRADEOFFS
Path Length
ComputationTime
A* (grid graphs)
A* (visibility graphs)
A* with post-smoothing
Any other ideas?

34. FIELD D*
• Field D* is an extension of A*
• Propagates information along grid edges
• Does not constrain paths to be formed by grid edges
• Widely used by roboticists
Dave Ferguson,Anthony Stentz: Using interpolation to improve path planning:The Field D* algorithm. J. Field Robotics
23(2): 79-101 (2006)

35. FIELD D*
• Key idea:
• The choice of a node to expand does not have to be grid nodes
• It can be an intermediate node between grid nodes
Dave Ferguson,Anthony Stentz: Using interpolation to improve path planning:The Field D* algorithm. J. Field Robotics
23(2): 79-101 (2006)

36. FIELD D*
goal
start
Dave Ferguson,Anthony Stentz: Using interpolation to improve path planning:The Field D* algorithm. J. Field Robotics
23(2): 79-101 (2006)

37. FIELD D*
goal
start
Dave Ferguson,Anthony Stentz: Using interpolation to improve path planning:The Field D* algorithm. J. Field Robotics
23(2): 79-101 (2006)

38. FIELD D*
goal
start
Dave Ferguson,Anthony Stentz: Using interpolation to improve path planning:The Field D* algorithm. J. Field Robotics
23(2): 79-101 (2006)
f = 1 + sqrt(5) = 3.24
f = sqrt(2) + 2 = 3.41

39. FIELD D*
goal
start
Dave Ferguson,Anthony Stentz: Using interpolation to improve path planning:The Field D* algorithm. J. Field Robotics
23(2): 79-101 (2006)
f = 1 + sqrt(5) = 3.24
f = sqrt(2) + 2 = 3.41
f = sqrt(10) = 3.16

40. FIELD D*
goal
start
Dave Ferguson,Anthony Stentz: Using interpolation to improve path planning:The Field D* algorithm. J. Field Robotics
23(2): 79-101 (2006)
f = 1 + sqrt(5) = 3.24
f = sqrt(2) + 2 = 3.41
f = sqrt(10) = 3.16

41. goal
FIELD D*
start
Dave Ferguson,Anthony Stentz: Using interpolation to improve path planning:The Field D* algorithm. J. Field Robotics
23(2): 79-101 (2006)

42. goal
FIELD D*
start
Pros
• Interleaves smoothing with search
Cons
• Restricted to rectangular grids
• Subject to interpolation error
• Complex path extraction

43. FIELD D*
The Field D* Algorithm 9
f a path from the initial state to the goal has been calculated, the path is extracted
nitial position and iteratively computing the cell boundary point to move to next.
erpolation technique, it is possible to compute the path cost of any point inside a
he corners, which is useful for both extracting the path and getting back on track
perfect (which is usually the case for real robots). See Section 6 for more on path
a path planned using Field D* showing individual grid cells. Notice that the path is not
nd exiting cells at corner points.
10 D. Ferguson and A. Stentz
Fig. 11. Planning through a potential field of obstacles. At high grid resolutions, Field D* produces smoo
curves through both uniform and non-uniform cost environments; this is not generally true of standard gr
based planners.
amount of computation required when updating the neighbors of a popped state (lines {09 - 14})
only considering those states actually affected by the new value of the popped state and how the
states are affected.
To do this, we keep track of a backpointer for each state, specifying from which states it curren
derives its path cost. Since, in Field D*, the successor of each state is a point on an edge connecti
two of its neighboring states, this backpointer needs to specify the two states that form the endpoin
of this edge. We use bptr(s) to refer to the most clockwise of the two endpoint states relative
!
Dave Ferguson,Anthony Stentz: Using interpolation to improve path planning:The Field D* algorithm. J. Field Robotics
23(2): 79-101 (2006)

44. ANY-ANGLE TRADEOFFS
Path Length
ComputationTime
A* (grid graphs)
A* (visibility graphs)
A* with post-smoothing (grid graphs)
Field D*

45. THETA*
• Like Field D*,Theta* is also an extension of A*
• It is (arguably) a simpler extension of A*
• Like Field D*, it also propagates information along grid edges
• Like Field D*, it also does not constrain paths to be formed by
grid edges
Alex Nash, Kenny Daniel, Sven Koenig,Ariel Felner:Theta*:Any-Angle Path Planning on Grids.AAAI 2007: 1177-1183
Kenny Daniel,Alex Nash, Sven Koenig,Ariel Felner:Theta*:Any-Angle Path Planning on Grids. J.Artif. Intell. Res. (JAIR)
39: 533-579 (2010)

46. THETA*
• Key idea:
• The parent of a node does not need to be its neighbor
• When expanding a node s and generating a node s’,
typically, parent(s’) = s
• Now, parent(s’) can be parent(s) if this path is shorter
Alex Nash, Kenny Daniel, Sven Koenig,Ariel Felner:Theta*:Any-Angle Path Planning on Grids.AAAI 2007: 1177-1183
Kenny Daniel,Alex Nash, Sven Koenig,Ariel Felner:Theta*:Any-Angle Path Planning on Grids. J.Artif. Intell. Res. (JAIR)
39: 533-579 (2010)

47. goal
start
THETA*

48. goal
start
THETA*

49. goal
start
THETA*

50. goal
start
THETA*

51. goal
start
THETA*

52. goal
start
THETA*

53. goal
start
THETA*
Pros
• Interleaves smoothing with search
Cons
• Restricted to rectangular grids
• Simpler path extraction

54. THETA*
Field D* A* with post-smoothing Theta*
Shortest paths found by three algorithms

55. ANY-ANGLE TRADEOFFS
Path Length
ComputationTime
A* (grid graphs)
A* (visibility graphs)
A* with post-smoothing (grid graphs)
Theta*
Field D*