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Incremental Path Planning
Dynamic and Incremental A*
Prof. Brian Williams (help from Ihsiang Shu)
16.412/6.834 Cognitive RoboticsFebruary 17th, 2004
Outline
Review: Optimal Path Planning in Partially Known Environments.Continuous Optimal Path Planning
Dynamic A*Incremental A* (LRTA*)
[Zellinsky, 92]
1. Generate global path plan from initial map.
2. Repeat until goal reached or failure:Execute next step in current global path planUpdate map based on sensors.If map changed generate new global path from map.
Compute Optimal Path
J M N O
E I L G
B D H K
S A C F
Begin Executing Optimal Path
J M N O
E I L G
B D H K
S A C Fh = 5
h = 4
h = 3
h = 4
h = 4
h = 3
h = 2
h = 3
h = 3
h = 2
h = 1
h = 2
h = 2
h = 1
h = 0
h = 1
Robot moves along backpointers towards goal.Uses sensors to detect discrepancies along way.
Obstacle Encountered!
J M N O
E I L G
B D H K
S A C Fh = 5
h = 4
h = 3
h = 4
h = 4
h = 3
h = 2
h = 3
h = 3
h = 2
h = 1
h = 1
h = 2
h = 1
h = 0
h = 1
At state A, robot discovers edge from D to H is blocked (cost 5,000 units).Update map and reinvoke planner.
Continue Path Execution
J M N O
E I L G
B D H K
S A C Fh = 5
h = 4
h = 3
h = 4
h = 4
h = 3
h = 2
h = 3
h = 3
h = 2
h = 1
h = 1
h = 2
h = 1
h = 0
h = 1
A’s previous path is still optimal.Continue moving robot along back pointers.
Second Obstacle, Replan!
J M N O
E I L G
B D H K
S A C Fh = 5
h = 4
h = 3
h = 4
h = 4
h = 3
h = 2
h = 3
h = 3
h = 2
h = 1
h = 1
h = 2
h = 1
h = 0
h = 1
At C robot discovers blocked edges from C to F and H (cost 5,000 units).Update map and reinvoke planner.
Path Execution Achieves Goal
h = 5
J M N O
E I L G
B D H K
S A C Fh = 5
h = 4
h = 3
h = 4
h = 4
h = 3
h = 2
h = 3
h = 2
h = 1
h = 1
h = 2
h = 1
h = 0
h = 1
Follow back pointers to goal.No further discrepancies detected; goal achieved!
Outline
Review: Optimal Path Planning in Partially Known Environments.Continuous Optimal Path Planning
Dynamic A*Incremental A* (LRTA*)
What is Continuous Optimal Path Planning?
Supports search as a repetitive online process.Exploits similarities between a series of searches to solve much faster than solving each search starting from scratch.Reuses the identical parts of the previous search tree, while updating differences.Solutions guaranteed to be optimal.On the first search, behaves like traditional algorithms.
D* behaves exactly like Dijkstra’s.Incremental A* A* behaves exactly like A*.
Dynamic A* (aka D*) [Stenz, 94]1. Generate global path plan from initial map.
2. Repeat until Goal reached, or failure.Execute next step of current global path plan.Update map based on sensor information.Incrementally update global path plan from map changes.
1 to 3 orders of magnitude speedup relative to a non-incremental path planner.
Map and Path Conceptsc(X,Y) : Cost to move from Y to X. c(X,Y) is undefined if move disallowed.
Neighbors(X) : Any Y such that c(X,Y) or c(Y,X) is defined.
o(G,X) : Optimal path cost to Goal from X.
h(G,X) : Estimate of optimal path cost to goal from X.
b(X) = Y : backpointer from X to Y. Y is the first state on path from X to G.
D* Search ConceptsOPEN list : States with estimates to be propagated to other states.
States on list tagged OPEN Sorted by key function k.
State tag t(X) : NEW : has no estimate h. OPEN : estimate needs to be propagated.CLOSED : estimate propagated.
D* Fundamental Search Conceptsk(G,X) : key function minimum of
h(G,X) before modification, andall values assumed by h(G,X) since X was placed on the OPEN list.
Lowered state : k(G,X) = current h(G,X), Propagate decrease to descendants and other nodes.
Raised state : k(G,X) < current h(G,X), Propagate increase to dscendants and other nodes.Try to find alternate shorter paths.
Running D* First Time on Graph
InitiallyMark G Open and QueueMark all other states NewRun Process_States on queue until path found or empty.
When edge cost c(X,Y) changes If X is marked Closed, then
Update h(X)Mark X open and queue with key h(X).
Use D* to Compute Initial Path
J M N O
E I L G
B D H K
S A C F
NEW
NEW
NEW
NEW
NEW
NEW
NEW
NEW
NEW
NEW
NEW
NEW
NEW
NEW
NEW
NEW
States initially tagged NEW (no cost determined yet).
Use D* to Compute Initial Path
J M N O
E I L G
B D H K
S A C F
NEW
NEW
NEW
NEW
NEW
NEW
NEW
NEW
NEW
NEW
NEW
NEW
NEW
OPEN
NEW
NEW
OPEN List1 (0,G)
h = 0
8: if kold = h(X) then9: for each neighbor Y of X:10: if t(Y) = NEW or11: (b(Y) = X and h(Y) ≠ h(X) + c(X,Y)) or12: (b(Y) ≠ X and h(Y) > h(X) + c(X,Y)) then13: b(Y) = X; Insert(Y,h(X) + c(X,Y))
Add Goal node to the OPEN list.Process OPEN list until the robot’s current state is CLOSED.
Process_State: New or Lowered State
Remove from Open list , state X with lowest k If X is a new/lowered state, its path cost is optimal!Then propagate to each neighbor Y
If Y is New, give it an initial path cost and propagate.If Y is a descendant of X, propagate any change.Else, if X can lower Y’s path cost, Then do so and propagate.
Use D* to Compute Initial Path
J M N O
E I L G
B D H K
S A C F
NEW
NEW
NEW
NEW
NEW
NEW
NEW
NEW
NEW
NEW
NEW
NEW
NEW
OPEN
NEW
NEW
OPEN List1 (0,G)
h = 0
8: if kold = h(X) then9: for each neighbor Y of X:10: if t(Y) = NEW or11: (b(Y) = X and h(Y) ≠ h(X) + c(X,Y)) or12: (b(Y) ≠ X and h(Y) > h(X) + c(X,Y)) then13: b(Y) = X; Insert(Y,h(X) + c(X,Y))
Add new neighbors of G onto the OPEN listCreate backpointers to G.
Use D* to Compute Initial Path
J M N O
E I L G
B D H K
S A C F
h = 1
h = 1
h = 1
h = 0
h = 1
NEW
NEW
NEW
NEW
NEW
NEW
NEW
NEW
NEW
OPEN
NEW
NEW
OPEN
CLOSED
OPEN
NEW
OPEN List1 (0,G)2 (1,K) (1,L) (1,O)
8: if kold = h(X) then9: for each neighbor Y of X:10: if t(Y) = NEW or11: (b(Y) = X and h(Y) ≠ h(X) + c(X,Y)) or12: (b(Y) ≠ X and h(Y) > h(X) + c(X,Y)) then13: b(Y) = X; Insert(Y,h(X) + c(X,Y))
Add new neighbors of G onto the OPEN listCreate backpointers to G.
Use D* to Compute Initial Path
J M N O
E I L G
B D H K
S A C F
h = 2
h = 1
h = 2
h = 1
h = 0
h = 1
NEW
NEW
NEW
NEW
NEW
NEW
NEW
NEW
NEW
OPEN
OPEN
NEW
OPEN
CLOSED
CLOSED
OPEN
OPEN List1 (0,G)2 (1,K) (1,L) (1,O)3 (1,L) (1,O) (2,F) (2,H)
8: if kold = h(X) then9: for each neighbor Y of X:10: if t(Y) = NEW or11: (b(Y) = X and h(Y) ≠ h(X) + c(X,Y)) or12: (b(Y) ≠ X and h(Y) > h(X) + c(X,Y)) then13: b(Y) = X; Insert(Y,h(X) + c(X,Y))
Add new neighbors of K on to the OPEN list and create backpointers.
Use D* to Compute Initial Path
J M N O
E I L G
B D H K
S A C F
h = 2
h = 2
h = 1
h = 2
h = 2
h = 1
h = 0
h = 1
NEW
NEW
NEW
NEW
NEW
OPEN
NEW
NEW
OPEN
CLOSED
OPEN
NEW
CLOSED
CLOSED
CLOSED
OPEN
OPEN List1 (0,G)2 (1,K) (1,L) (1,O)3 (1,L) (1,O) (2,F) (2,H)4 (2,F) (2,H) (2,I) (2,N)8: if kold = h(X) then9: for each neighbor Y of X:10: if t(Y) = NEW or11: (b(Y) = X and h(Y) ≠ h(X) + c(X,Y)) or12: (b(Y) ≠ X and h(Y) > h(X) + c(X,Y)) then13: b(Y) = X; Insert(Y,h(X) + c(X,Y))
Add new neighbors of L, then O on to the OPEN list and create backpointers.
Use D* to Compute Initial Path
J M N O
E I L G
B D H K
S A C F
h = 2
h = 3
h = 2
h = 1
h = 2
h = 2
h = 1
h = 0
h = 1
NEW
NEW
NEW
NEW
NEW
OPEN
NEW
NEW
OPEN
CLOSED
OPEN
OPEN
CLOSED
CLOSED
CLOSED
CLOSED
OPEN List1 (0,G)2 (1,K) (1,L) (1,O)3 (1,L) (1,O) (2,F) (2,H)4 (2,F) (2,H) (2,I) (2,N)5 (2,H) (2,I) (2,N) (3,C)
Continue until current state S is closed.
Use D* to Compute Initial Path
J M N O
E I L G
B D H K
S A C F
h = 3
h = 2
h = 3
h = 2
h = 1
h = 2
h = 2
h = 1
h = 0
h = 1
NEW
NEW
NEW
NEW
NEW
OPEN
OPEN
NEW
OPEN
CLOSED
CLOSED
OPEN
CLOSED
CLOSED
CLOSED
CLOSED
OPEN List1 (0,G)2 (1,K) (1,L) (1,O)3 (1,L) (1,O) (2,F) (2,H)4 (2,F) (2,H) (2,I) (2,N)5 (2,H) (2,I) (2,N) (3,C)6 (2,I) (2,N) (3,C) (3,D)
Continue until current state S is closed.
Use D* to Compute Initial Path
J M N O
E I L G
B D H K
S A C F
h = 3
h = 3
h = 2
h = 3
h = 3
h = 2
h = 1
h = 2
h = 2
h = 1
h = 0
h = 1
NEW
OPEN
NEW
NEW
OPEN
CLOSED
OPEN
NEW
OPEN
CLOSED
CLOSED
OPEN
CLOSED
CLOSED
CLOSED
CLOSED
OPEN List1 (0,G)2 (1,K) (1,L) (1,O)3 (1,L) (1,O) (2,F) (2,H)4 (2,F) (2,H) (2,I) (2,N)5 (2,H) (2,I) (2,N) (3,C)6 (2,I) (2,N) (3,C) (3,D)7 (2,N) (3,C) (3,D) (3,E) (3,M)
Continue until current state S is closed.
Use D* to Compute Initial Path
J M N O
E I L G
B D H K
S A C F
h = 3
h = 3
h = 2
h = 3
h = 3
h = 2
h = 1
h = 2
h = 2
h = 1
h = 0
h = 1
NEW
OPEN
NEW
NEW
OPEN
CLOSED
OPEN
NEW
CLOSED
CLOSED
CLOSED
OPEN
CLOSED
CLOSED
CLOSED
CLOSED
OPEN List1 (0,G)2 (1,K) (1,L) (1,O)3 (1,L) (1,O) (2,F) (2,H)4 (2,F) (2,H) (2,I) (2,N)5 (2,H) (2,I) (2,N) (3,C)6 (2,I) (2,N) (3,C) (3,D)7 (2,N) (3,C) (3,D) (3,E) (3,M)8 (3,C) (3,D) (3,E) (3,M)
Continue until current state S is closed.
Use D* to Compute Initial Path
J M N O
E I L G
B D H K
S A C F
h = 3
h = 4
h = 3
h = 2
h = 3
h = 3
h = 2
h = 1
h = 2
h = 2
h = 1
h = 0
h = 1
NEW
OPEN
NEW
NEW
OPEN
CLOSED
OPEN
OPEN
CLOSED
CLOSED
CLOSED
CLOSED
CLOSED
CLOSED
CLOSED
CLOSED
OPEN List1 (0,G)2 (1,K) (1,L) (1,O)3 (1,L) (1,O) (2,F) (2,H)4 (2,F) (2,H) (2,I) (2,N)5 (2,H) (2,I) (2,N) (3,C)6 (2,I) (2,N) (3,C) (3,D)7 (2,N) (3,C) (3,D) (3,E) (3,M)8 (3,C) (3,D) (3,E) (3,M)9 (3,D) (3,E) (3,M) (4,A)
Continue until current state S is closed.
Use D* to Compute Initial Path
J M N O
E I L G
B D H K
S A C F
h = 4
h = 3
h = 4
h = 3
h = 2
h = 3
h = 3
h = 2
h = 1
h = 2
h = 2
h = 1
h = 0
h = 1
NEW
OPEN
OPEN
NEW
OPEN
CLOSED
CLOSED
OPEN
CLOSED
CLOSED
CLOSED
CLOSED
CLOSED
CLOSED
CLOSED
CLOSED
OPEN List1 (0,G)2 (1,K) (1,L) (1,O)3 (1,L) (1,O) (2,F) (2,H)4 (2,F) (2,H) (2,I) (2,N)5 (2,H) (2,I) (2,N) (3,C)6 (2,I) (2,N) (3,C) (3,D)7 (2,N) (3,C) (3,D) (3,E) (3,M)8 (3,C) (3,D) (3,E) (3,M)9 (3,D) (3,E) (3,M) (4,A)10 (3,E) (3,M) (4,A) (4,B)Continue until current state S is closed..
Use D* to Compute Initial Path
J M N O
E I L G
B D H K
S A C F
h = 4
h = 3
h = 4
h = 4
h = 3
h = 2
h = 3
h = 3
h = 2
h = 1
h = 2
h = 2
h = 1
h = 0
h = 1
OPEN
CLOSED
OPEN
NEW
OPEN
CLOSED
CLOSED
OPEN
CLOSED
CLOSED
CLOSED
CLOSED
CLOSED
CLOSED
CLOSED
CLOSED
OPEN List10 (3,E) (3,M) (4,A) (4,B)11 (3,M) (4,A) (4,B) (4,J)121314
Continue until current state S is closed.
Use D* to Compute Initial Path
J M N O
E I L G
B D H K
S A C F
h = 4
h = 3
h = 4
h = 4
h = 3
h = 2
h = 3
h = 3
h = 2
h = 1
h = 2
h = 2
h = 1
h = 0
h = 1
OPEN
CLOSED
OPEN
NEW
CLOSED
CLOSED
CLOSED
OPEN
CLOSED
CLOSED
CLOSED
CLOSED
CLOSED
CLOSED
CLOSED
CLOSED
OPEN List10 (3,E) (3,M) (4,A) (4,B)11 (3,M) (4,A) (4,B) (4,J)12 (3,M) (4,A) (4,B)1314
Continue until current state S is closed.
Use D* to Compute Initial Path
J M N O
E I L G
B D H K
S A C Fh = 5
h = 4
h = 3
h = 4
h = 4
h = 3
h = 2
h = 3
h = 3
h = 2
h = 1
h = 2
h = 2
h = 1
h = 0
h = 1
OPEN
CLOSED
OPEN
OPEN
CLOSED
CLOSED
CLOSED
CLOSED
CLOSED
CLOSED
CLOSED
CLOSED
CLOSED
CLOSED
CLOSED
CLOSED
OPEN List10 (3,E) (3,M) (4,A) (4,B)11 (3,M) (4,A) (4,B) (4,J)12 (3,M) (4,A) (4,B)13 (4,B) (4,J) (5,S)1415
Continue until current state S is closed.
Use D* to Compute Initial Path
J M N O
E I L G
B D H K
S A C Fh = 5
h = 4
h = 3
h = 4
h = 4
h = 3
h = 2
h = 3
h = 3
h = 2
h = 1
h = 2
h = 2
h = 1
h = 0
h = 1
OPEN
CLOSED
CLOSED
OPEN
CLOSED
CLOSED
CLOSED
CLOSED
CLOSED
CLOSED
CLOSED
CLOSED
CLOSED
CLOSED
CLOSED
CLOSED
OPEN List10 (3,E) (3,M) (4,A) (4,B)11 (3,M) (4,A) (4,B) (4,J)12 (3,M) (4,A) (4,B)13 (4,B) (4,J) (5,S)14 (4,J) (5,S)15
Continue until current state S is closed.
Use D* to Compute Initial Path
J M N O
E I L G
B D H K
S A C Fh = 5
h = 4
h = 3
h = 4
h = 4
h = 3
h = 2
h = 3
h = 3
h = 2
h = 1
h = 2
h = 2
h = 1
h = 0
h = 1
CLOSED
CLOSED
CLOSED
OPEN
CLOSED
CLOSED
CLOSED
CLOSED
CLOSED
CLOSED
CLOSED
CLOSED
CLOSED
CLOSED
CLOSED
CLOSED
OPEN List10 (3,E) (3,M) (4,A) (4,B)11 (3,M) (4,A) (4,B) (4,J)12 (3,M) (4,A) (4,B)13 (4,B) (4,J) (5,S)14 (4,J) (5,S)15 (5,S)
Continue until current state S is closed.
D* Completed Initial Path
J M N O
E I L G
B D H K
S A C Fh = 5
h = 4
h = 3
h = 4
h = 4
h = 3
h = 2
h = 3
h = 3
h = 2
h = 1
h = 2
h = 2
h = 1
h = 0
h = 1
CLOSED
CLOSED
CLOSED
CLOSED
CLOSED
CLOSED
CLOSED
CLOSED
CLOSED
CLOSED
CLOSED
CLOSED
CLOSED
CLOSED
CLOSED
CLOSED
OPEN List10 (3,E) (3,M) (4,A) (4,B)11 (3,M) (4,A) (4,B) (4,J)12 (3,M) (4,A) (4,B)13 (4,B) (4,J) (5,S)14 (4,J) (5,S)15 (5,S)16 NULL
Done: Current state S is closed, and Open list is empty.
Begin Executing Optimal Path
J M N O
E I L G
B D H K
S A C Fh = 5
h = 4
h = 3
h = 4
h = 4
h = 3
h = 2
h = 3
h = 3
h = 2
h = 1
h = 2
h = 2
h = 1
h = 0
h = 1
CLOSED
CLOSED
CLOSED
CLOSED
CLOSED
CLOSED
CLOSED
CLOSED
CLOSED
CLOSED
CLOSED
CLOSED
CLOSED
CLOSED
CLOSED
CLOSED
Robot moves along backpointers towards goalUses sensors to detect discrepancies along way.
Obstacle Encountered!
J M N O
E I L G
B D H K
S A C Fh = 5
h = 4
h = 3
h = 4
h = 4
h = 3
h = 2
h = 3
h = 3
h = 2
h = 1
h = 1
h = 2
h = 1
h = 0
h = 1
CLOSED
CLOSED
CLOSED
CLOSED
CLOSED
CLOSED
CLOSED
CLOSED
CLOSED
CLOSED
OPEN
CLOSED
CLOSED
CLOSED
CLOSED
CLOSED
At state A, robot discovers edge D to H is blocked off (cost 5,000 units).Update map and reinvoke D*
Running D* After Edge Cost Change
When edge cost c(Y,X) changes If X is marked Closed, then
Update h(X)Mark X open and queue, key is new h(X).
Run Process_State on queue until path to current state is shown optimal, or queue Open List is empty.
D* Update From First Obstacle
J M N O
E I L G
B D H K
S A C Fh = 5
h = 4
h = 3
h = 4
h = 4
h = 3
h = 2
h = 3
h = 3
h = 2
h = 1
h = 1
h = 2
h = 1
h = 0
h = 1
CLOSED
CLOSED
CLOSED
CLOSED
CLOSED
CLOSED
CLOSED
CLOSED
CLOSED
CLOSED
OPEN
CLOSED
CLOSED
CLOSED
CLOSED
CLOSED
OPEN List1 (2,H)234
Function: Modify-Cost(X,Y,eval)1: c(X,Y) = eval2: if t(X) = CLOSED
then Insert(X,h(X))3: return Get-Kmin()
Propagate changes starting at H
D* Update From First Obstacle
J M N O
E I L G
B D H K
S A C Fh = 5
h = 4
h = 3
h = 4
h = 4
h = 5002
h = 2
h = 3
h = 3
h = 2
h = 1
h = 1
h = 2
h = 1
h = 0
h = 1
CLOSED
CLOSED
CLOSED
CLOSED
CLOSED
CLOSED
OPEN
CLOSED
CLOSED
CLOSED
CLOSED
CLOSED
CLOSED
CLOSED
CLOSED
CLOSED
OPEN List1 (2,H)2 (3,D)34
8: if kold = h(X) then9: for each neighbor Y of X:10: if t(Y) = NEW or11: (b(Y) = X and h(Y) ≠ h(X) + c(X,Y)) or12: (b(Y) ≠ X and h(Y) > h(X) + c(X,Y)) then13: b(Y) = X; Insert(Y,h(X) + c(X,Y))
Raise cost of H’s descendant D, and propagate.
Process_State: Raised State
If X is a raise state its cost might be suboptimal.Try reducing cost of X using an optimal neighbor Y.
h(Y) ≤ [h(X) before it was raised]propagate X’s cost to each neighbor Y
If Y is New, Then give it an initial path cost and propagate.If Y is a descendant of X, Then propagate ANY change.If X can lower Y’s path cost,
Postpone: Queue X to propagate when optimal (reach current h(X))If Y can lower X’s path cost, and Y is suboptimal,
Postpone: Queue Y to propagate when optimal (reach current h(Y)).Postponement avoids creating cycles.
D* Update From First Obstacle
h =
J M N O
E I L G
B D H K
S A C Fh = 5
h = 4
h = 3
h = 4
h = 4
3
h = 2
h = 3
h = 3
h = 2
h = 1
h = 1
h = 2
h = 1
h = 0
h = 1
CLOSED
CLOSED
CLOSED
CLOSED
CLOSED
CLOSED
CLOSED
CLOSED
CLOSED
CLOSED
CLOSED
CLOSED
CLOSED
CLOSED
CLOSED
OPEN List1 (2,H)2 (3,D)34
OPEN
4: if kold < h(X) then5: for each neighbor Y of X:6: if h(Y) ≤ kold and h(X) > h(Y) + C(Y,X) then7: b(X) = Y; h(X) = h(Y) + c(Y,X);
D may not be optimal, check neighbors for better path.Transitioning to I is better, and I’s path is optimal, so update D.
D* Update From First Obstacle
J M N O
E I L G
B D H K
S A C Fh = 5
h = 4
h = 3
h = 4
h = 4
h = 3
h = 2
h = 3
h = 3
h = 2
h = 1
h = 1
h = 2
h = 1
h = 0
h = 1
CLOSED
CLOSED
CLOSED
CLOSED
CLOSED
CLOSED
CLOSED
CLOSED
CLOSED
CLOSED
CLOSED
CLOSED
CLOSED
CLOSED
CLOSED
CLOSED
OPEN List12 (3,D)3 NULL4
All neighbors of D have consistent h-values.No further propagation needed.
Continue Path Execution
J M N O
E I L G
B D H K
S A C Fh = 5
h = 4
h = 3
h = 4
h = 4
h = 3
h = 2
h = 3
h = 3
h = 2
h = 1
h = 1
h = 2
h = 1
h = 0
h = 1
CLOSED
CLOSED
CLOSED
CLOSED
CLOSED
CLOSED
CLOSED
CLOSED
CLOSED
CLOSED
CLOSED
CLOSED
CLOSED
CLOSED
CLOSED
CLOSED
OPEN List12 (3,D)3 NULL4
A’s path optimal.Continue moving robot along backpointers.
Second Obstacle!
J M N O
E I L G
B D H K
S A C Fh = 5
h = 4
h = 3
h = 4
h = 4
h = 3
h = 2
h = 3
h = 3
h = 2
h = 1
h = 1
h = 2
h = 1
h = 0
h = 1
CLOSED
CLOSED
CLOSED
CLOSED
CLOSED
CLOSED
CLOSED
CLOSED
CLOSED
CLOSED
OPEN
CLOSED
CLOSED
CLOSED
CLOSED
OPEN
OPEN List1 (2,F) (2,H)234
Function: Modify-Cost(X,Y,eval)1: c(X,Y) = eval2: if t(X) = CLOSED
then Insert(X,h(X))3: return Get-Kmin()
At C robot discovers blocked edges C to F and H (cost 5,000 units).Update map and reinvoke D* until H(current position optimal).
D* Update From Second Obstacle
J M N O
E I L G
B D H K
S A C Fh = 5
h = 4
h = 3
h = 4
h = 4
h = 3
h = 2
h = 3
h = 5002
h = 2
h = 1
h = 1
h = 2
h = 1
h = 0
h = 1
CLOSED
CLOSED
CLOSED
CLOSED
CLOSED
CLOSED
CLOSED
CLOSED
CLOSED
CLOSED
CLOSED
OPEN
CLOSED
CLOSED
CLOSED
CLOSED
OPEN List1 (2,F) (2,H)2 (3,C)34
8: if kold = h(X) then9: for each neighbor Y of X:10: if t(Y) = NEW or11: (b(Y) = X and h(Y) ≠ h(X) + c(X,Y)) or12: (b(Y) ≠ X and h(Y) > h(X) + c(X,Y)) then13: b(Y) = X; Insert(Y,h(X) + c(X,Y))
Processing F raises descendant C’s cost, and propagates.Processing H does nothing.
D* Update From Second Obstacle
h = 4
J M N O
E I L G
B D H K
S A C Fh = 5
h = 4
h = 3
h = 4
h = 3
h = 2
h = 3
h = 2
h = 1
h = 1
h = 2
h = 1
h = 0
h = 1
CLOSED
CLOSED
CLOSED
CLOSED
CLOSED
CLOSED
CLOSED
Closed
CLOSED
CLOSED
CLOSED
Open
CLOSED
CLOSED
CLOSED
CLOSED
OPEN List1 (2,F) (2,H)2 (3,C)34
h = 5
4: if kold < h(X) then5: for each neighbor Y of X:6: if h(Y) ≤ kold and h(X) > h(Y) + C(Y,X) then7: b(X) = Y; h(X) = h(Y) + c(Y,X);
C may be suboptimal, check neighbors; Better path through A!However, A may be suboptimal, and updating creates a loop!
D* Update From Second Obstacle
J M N O
E I L G
B D H K
S A C Fh = 5
h = 4
h = 3
h = 4
h = 3
h = 2
h = 3
h = 2
h = 1
h = 1
h = 2
h = 1
h = 0
h = 1
CLOSED
CLOSED
CLOSED
CLOSED
CLOSED
CLOSED
CLOSED
OPEN
CLOSED
CLOSED
CLOSED
CLOSED
CLOSED
CLOSED
CLOSED
CLOSED
OPEN List1 (2,F) (2,H)2 (3,C)3 (4,A)4
h = 5002h = 5003
15: for each neighbor Y of X:16: if t(Y) = NEW or17: (b(Y) = X and h(Y) ≠ h(X) + c(X,Y)) then18: b(Y) = X; Insert(Y,h(X) + c(X,Y))
Don’t change C’s path to A (yet).Instead, propagate increase to A.
Process_State: Raised State
If X is a raise state its cost might be suboptimal.Try reducing cost of X using an optimal neighbor Y.
h(Y) ≤ [h(X) before it was raised]propagate X’s cost to each neighbor Y
If Y is New, Then give it an initial path cost and propagate.If Y is a descendant of X, Then propagate ANY change.If X can lower Y’s path cost,
Postpone: Queue X to propagate when optimal (reach current h(X))If Y can lower X’s path cost, and Y is suboptimal,
Postpone: Queue Y to propagate when optimal (reach current h(Y)).Postponement avoids creating cycles.
D* Update From Second Obstacle
J M N O
E I L G
B D H K
S A C Fh = 5
h = 4
h = 3
h = 4
h = 5003
h = 3
h = 2
h = 3
h = 2
h = 1
h = 1
h = 2
h = 1
h = 0
h = 1
CLOSED
CLOSED
CLOSED
CLOSED
CLOSED
CLOSED
CLOSED
OPEN
CLOSED
CLOSED
CLOSED
CLOSED
CLOSED
CLOSED
CLOSED
CLOSED
OPEN List1 (2,F) (2,H)2 (3,C)3 (4,A)4
4: if kold < h(X) then5: for each neighbor Y of X:6: if h(Y) ≤ kold and h(X) > h(Y) + C(Y,X) then7: b(X) = Y; h(X) = h(Y) + c(Y,X);
h = 5002
A may not be optimal, check neighbors for better path.Transitioning to D is better, and D’s path is optimal, so update A.
D* Update From Second Obstacle
J M N O
E I L G
B D H K
S A C Fh = 5
h = 4
h = 3
h = 4
h = 4
h = 3
h = 2
h = 3
h = 2
h = 1
h = 1
h = 2
h = 1
h = 0
h = 1
CLOSED
CLOSED
CLOSED
CLOSED
CLOSED
CLOSED
CLOSED
OPEN
CLOSED
CLOSED
CLOSED
CLOSED
CLOSED
CLOSED
CLOSED
CLOSED
OPEN List1 (2,F) (2,H)2 (3,C)3 (4,A)4
4: if kold < h(X) then5: for each neighbor Y of X:6: if h(Y) ≤ kold and h(X) > h(Y) + C(Y,X) then7: b(X) = Y; h(X) = h(Y) + c(Y,X);
h = 5002
A may not be optimal, check neighbors for better path.Transitioning to D is better, and D’s path is optimal, so update A.
D* Update From Second Obstacle
J M N O
E I L G
B D H K
S A C Fh = 5
h = 4
h = 3
h = 4
h = 4
h = 3
h = 2
h = 3
h = 2
h = 1
h = 1
h = 2
h = 1
h = 0
h = 1
CLOSED
CLOSED
CLOSED
CLOSED
CLOSED
CLOSED
CLOSED
CLOSED
CLOSED
CLOSED
CLOSED
OPEN
CLOSED
CLOSED
CLOSED
CLOSED
OPEN List1 (2,F) (2,H)2 (3,C)3 (4,A)4 (5,C)
for each neighbor Y of X:17: if (b(Y) = X and h(Y) ≠ h(X) + c(X,Y)) then18: b(Y) = X; Insert(Y,h(X) + c(X,Y))19: else20: if b(Y) ≠ X and h(Y) > h(X) + c(X,Y) then21: Insert(X,h(X))
h = 5
A can improve neighbor C, so queue C.
Process_State: New or Lowered State
Remove from Open list , state X with lowest k If X is a new/lowered state its path cost is optimal,Then propagate to each neighbor Y
If Y is New, give it an initial path cost and propagate.If Y is a descendant of X, propagate any change.Else, if X can lower Y’s path cost, Then do so and propagate.
D* Update From Second Obstacle
J M N O
E I L G
B D H K
S A C Fh = 5
h = 4
h = 3
h = 4
h = 3
h = 2
h = 3
h = 2
h = 1
h = 1
h = 2
h = 1
h = 0
h = 1
CLOSED
CLOSED
CLOSED
CLOSED
CLOSED
CLOSED
CLOSED
CLOSED
CLOSED
CLOSED
CLOSED
OPEN
CLOSED
CLOSED
CLOSED
CLOSED
OPEN List1 (2,F) (2,H)2 (3,C)3 (4,A)4 (5,C)5
C lowered to optimal; no neighbors affected. Current state reached, so Process_State terminates.
h = 4 h = 5
8: if kold = h(X) then9: for each neighbor Y of X:10: if t(Y) = NEW or11: (b(Y) = X and h(Y) ≠ h(X) + c(X,Y)) or12: (b(Y) ≠ X and h(Y) > h(X) + c(X,Y)) then13: b(Y) = X; Insert(Y,h(X) + c(X,Y))
Bug? How does C getpointed to A?
Complete Path Execution
h = 5
J M N O
E I L G
B D H K
S A C Fh = 5
h = 4
h = 3
h = 4
h = 4
h = 3
h = 2
h = 3
h = 2
h = 1
h = 1
h = 2
h = 1
h = 0
h = 1
CLOSED
CLOSED
CLOSED
CLOSED
CLOSED
CLOSED
CLOSED
CLOSED
CLOSED
CLOSED
CLOSED
CLOSED
CLOSED
CLOSED
CLOSED
CLOSED
Follow back pointers to Goal.No further discrepancies detected; goal achieved!
D* Pseudo CodeFunction: Process-State()1: X = Min-State()2: if X=NULL then return -13: kold = Get-Kmin(); Delete(X)4: if kold < h(X) then5: for each neighbor Y of X:6: if h(Y) ≤ kold and h(X) > h(Y) + C(Y,X) then7: b(X) = Y; h(X) = h(Y) + c(Y,X);8: if kold = h(X) then9: for each neighbor Y of X:10: if t(Y) = NEW or11: (b(Y) = X and h(Y) ≠ h(X) + c(X,Y)) or12: (b(Y) ≠ X and h(Y) > h(X) + c(X,Y)) then13: b(Y) = X; Insert(Y,h(X) + c(X,Y))14: else15: for each neighbor Y of X:16: if t(Y) = NEW or17: (b(Y) = X and h(Y) ≠ h(X) + c(X,Y)) then18: b(Y) = X; Insert(Y,h(X) + c(X,Y))19: else20: if b(Y) ≠ X and h(X) > h(X) + c(X,Y) then21: Insert(X,h(X))22: else23: if b(Y) ≠ X and h(X) > h(Y) + c(Y,X) and24: t(Y) = CLOSED and h(Y) > kold then25: Insert(Y,h(Y))26: return Get-Kmin()
Function: Modify-Cost(X,Y,eval)1: c(X,Y) = eval2: if t(X) = CLOSED
then Insert(X,h(X))3: return Get-Kmin()
Function: Insert(X, hnew)1: if t(X) = NEW
then k(x) = hnew2: else if t(X) = OPEN
then k(X) = min(k(X), hnew)3: else
k(X) = min(h(X), hnew)4: h(X) = hnew;5: t(X) = OPEN
Outline
Review: Optimal Path Planning in Partially Known Environments.Continuous Optimal Path Planning
Dynamic A*Incremental A* (LRTA*)
Incremental A* On Gridworld[Koenig and Likhachev, 01]
Incremental A* Versus Other Searches
Incremental A* Versus Other Searches
Definitions: Graph and CostsSucc(s) : the set of successors of vertex s.Pred(s) : the set of predecessors of vertex s.c(s,s’) : the cost of moving from vertex s to vertex s’.
g*(s) : true start distance, from sstart to vertex s.g(s) : estimate of the start distance of vertex s.h(s) : heuristic estimate of goal distance of vertex s.
Definitions: Local Consistency & Updaterhs(s) : one-step lookahead estimate for g(s)
if s = sstart then 0 else mins’∈Pred(s)(g(s’) + c(s’,s))
locally consistent : g-value of a vertex equals it’s rhs value.Do nothing.
overconsistent : g-value of a vertex is greater than rhs value.Relax: set g-value to rhs value.
underconsistent : g-value of a vertex is less than rhs value.Upperbound: Set g-value to inf
Incremental A* in Brief1) Initialize start node with g estimates of 0 and enqueue.
All other g estimates (g,rhs values) are initialized to INF. Only the start node is locally inconsistent.
2) Take node with smallest key from priority queue. Update g-value, and add all successors to the priority queue.
3) Loop step 2 until either the goal is locally consistent or the priority becomes empty.
4) To find shortest path: Start with s’ = goal, working to start,next s’ = predecessor (s) of s’ with best g(s) + c(s, s’),
5) Wait for edge costs to change. Update edge costs and add edge’s destination node to the queue.
6) Go to step 2.
Definitions: Incremental A* Search QueueU : priority queue, contains all
locally inconsistent vertices.
U ordered by key k(s)k(s) : [k1(s), k2(s)]k1(s) : min(g(s), rhs(s)) + h(s)k2(s) : min(g(s), rhs(s))
k1(s) corresponds directly to f(s) of A*. First time through IA*, min(g(s), rhs(s)) = true g*(s).
k2(s) breaks ties, using minimum start distance.
First Call To Incremental A* Search
rhs(S) = 0 g(S) = ∞rhs(A) = ∞ g(A) = ∞rhs(B) = ∞ g(B) = ∞rhs(C) = ∞ g(C) = ∞rhs(D) = ∞ g(D) = ∞rhs(G) = ∞ g(G) = ∞
U - Priority Queue, (X, [k1(s);k2(s)])
1 (S, [0;0])
0
C
S
B
G
A
D
∞
∞
∞2
61
5
2
3
4
2
∞
∞∞
∞∞
∞
∞
∞
0
3
1
1
0
2
Initialize all values and add start to queue. Only start is locally inconsistent .
First Call To Incremental A* Search
rhs(S) = 0 g(s) = 0rhs(A) = ∞ g(A) = ∞rhs(B) = ∞ g(B) = ∞rhs(C) = ∞ g(C) = ∞rhs(D) = ∞ g(D) = ∞rhs(G) = ∞ g(G) = ∞
U - Priority Queue1 (S, [0;0])
0
C
S
B
G
A
D
∞
∞
∞2
61
5
2
3
4
2
∞
∞∞
∞∞
∞
∞
0
0
3
1
1
0
2
Check local consistency of smallest node in queue,and update g-value.
First Call To Incremental A* Search
rhs(S) = 0 g(s) = 0rhs(A) = 2 g(A) = ∞rhs(B) = ∞ g(B) = ∞rhs(C) = ∞ g(C) = ∞rhs(D) = ∞ g(D) = ∞rhs(G) = ∞ g(G) = ∞
U - Priority Queue
21 (S, [0;0])
0
C
S
B
G
A
D
∞
∞
∞2
61
5
2
3
4
2
∞
∞∞
∞∞
∞
2
0
0
3
1
1
0
2
Update successor nodes.
First Call To Incremental A* Search
rhs(S) = 0 g(s) = 0rhs(A) = 2 g(A) = ∞rhs(B) = ∞ g(B) = ∞rhs(C) = ∞ g(C) = ∞rhs(D) = ∞ g(D) = ∞rhs(G) = ∞ g(G) = ∞
U - Priority Queue
21
(A, [4;2])(S, [0;0])
0
C
S
B
G
A
D
∞
∞
∞2
61
5
2
3
4
2
∞
∞∞
∞∞
∞
2
0
0
3
1
1
0
2
Add changed successor node A to priority queue.
First Call To Incremental A* Search
rhs(S) = 0 g(s) = 0rhs(A) = 2 g(A) = ∞rhs(B) = 6 g(B) = ∞rhs(C) = ∞ g(C) = ∞rhs(D) = ∞ g(D) = ∞rhs(G) = ∞ g(G) = ∞
U - Priority Queue
21
(A, [4;2])(S, [0;0])
0
C
S
B
G
A
D
∞
∞
∞2
61
5
2
3
4
2
∞
∞6
∞∞
∞
2
0
0
3
1
1
0
2
Update change to successor node B.
First Call To Incremental A* Search
rhs(S) = 0 g(s) = 0rhs(A) = 2 g(A) = ∞rhs(B) = 6 g(B) = ∞rhs(C) = ∞ g(C) = ∞rhs(D) = ∞ g(D) = ∞rhs(G) = ∞ g(G) = ∞
U - Priority Queue
21
(A, [4;2]) (B, [9;6])(S, [0;0])
0
C
S
B
G
A
D
∞
∞
∞2
61
5
2
3
4
2
∞
∞6
∞∞
∞
2
0
0
3
1
1
0
2
Add successor node B to priority queue.
First Call To Incremental A* Search
rhs(S) = 0 g(s) = 0rhs(A) = 2 g(A) = 2rhs(B) = 6 g(B) = ∞rhs(C) = ∞ g(C) = ∞rhs(D) = ∞ g(D) = ∞rhs(G) = ∞ g(G) = ∞
U - Priority Queue
21
(A, [4;2]) (B, [9;6])(S, [0;0])
0
C
S
B
G
A
D
2
∞
∞2
61
5
2
3
4
2
∞
∞6
∞∞
∞
2
0
0
3
1
1
0
2
Check local consistency of smallest key, A, in the priority queue.
First Call To Incremental A* Search
rhs(S) = 0 g(s) = 0rhs(A) = 2 g(A) = 2rhs(B) = 6 g(B) = ∞rhs(C) = 4 g(C) = ∞rhs(D) = ∞ g(D) = ∞rhs(G) = ∞ g(G) = ∞
U - Priority Queue
21
(A, [4;2]) (B, [9;6])(S, [0;0])
3 (B, [9;6])
0
C
S
B
G
A
D
2
∞
∞2
61
5
2
3
4
2
∞
∞6
∞∞
4
2
0
0
3
1
1
0
2
Update change to successor node C.
First Call To Incremental A* Search
rhs(S) = 0 g(s) = 0rhs(A) = 2 g(A) = 2rhs(B) = 6 g(B) = ∞rhs(C) = 4 g(C) = ∞rhs(D) = ∞ g(D) = ∞rhs(G) = ∞ g(G) = ∞
U - Priority Queue
21
(A, [4;2]) (B, [9;6])(S, [0;0])
3 (C, [5;4]) (B, [9;6])
0
C
S
B
G
A
D
2
∞
∞2
61
5
2
3
4
2
∞
∞6
∞∞
4
2
0
0
3
1
1
0
2
Add successor C to priority queue.
First Call To Incremental A* Search
rhs(S) = 0 g(s) = 0rhs(A) = 2 g(A) = 2rhs(B) = 6 g(B) = ∞rhs(C) = 4 g(C) = ∞rhs(D) = 6 g(D) = ∞rhs(G) = ∞ g(G) = ∞
U - Priority Queue
21
(A, [4;2]) (B, [9;6])(S, [0;0])
3 (C, [5;4]) (B, [9;6])
0
C
S
B
G
A
D
2
∞
∞2
61
5
2
3
4
2
∞
∞6
6∞
4
2
0
0
3
1
1
0
2
Update change to successor node D.
First Call To Incremental A* Search
rhs(S) = 0 g(s) = 0rhs(A) = 2 g(A) = 2rhs(B) = 6 g(B) = ∞rhs(C) = 4 g(C) = ∞rhs(D) = 6 g(D) = ∞rhs(G) = ∞ g(G) = ∞
U - Priority Queue
21
(A, [4;2]) (B, [9;6])(S, [0;0])
3 (C, [5;4]) (D, [7;6]) (B, [9;6])
0
C
S
B
G
A
D
2
∞
∞2
61
5
2
3
4
2
∞
∞6
6∞
4
2
0
0
3
1
1
0
2
Add successor D to priority queue.
First Call To Incremental A* Search
rhs(S) = 0 g(s) = 0rhs(A) = 2 g(A) = 2rhs(B) = 6 g(B) = ∞rhs(C) = 4 g(C) = 4rhs(D) = 6 g(D) = ∞rhs(G) = ∞ g(G) = ∞
U - Priority Queue
21
(A, [4;2]) (B, [9;6])(S, [0;0])
3 (C, [5;4]) (D, [7;6]) (B, [9;6])4 (D, [7;6]) (B, [9;6])
0
C
S
B
G
A
D
2
4
∞2
61
5
2
3
4
2
∞
∞6
6∞
4
2
0
0
3
1
1
0
2
Check local consistency of smallest key, C, in queue.
Node C has no successors, so nothing is added to the queue.
0
C
S
B
G
A
D
2
4
∞2
61
5
2
3
4
2
6
∞6
68
4
2
0
0
3
1
1
0
2
First Call To Incremental A* Search
rhs(S) = 0 g(s) = 0rhs(A) = 2 g(A) = 2rhs(B) = 6 g(B) = ∞rhs(C) = 4 g(C) = 4rhs(D) = 6 g(D) = 6rhs(G) = 8 g(G) = ∞
(G, [8;8]) (B, [9;6])5
U - Priority Queue
21
(A, [4;2]) (B, [9;6])(S, [0;0])
3 (C, [5;4]) (D, [7;6]) (B, [9;6])4 (D, [7;6]) (B, [9;6])
1
Check local consistency of smallest key, D.
Update successor G, and add to queue.
First Call To Incremental A* Search
g(G) = 8rhs(G) = 8g(D) = 6rhs(D) = 6g(C) = 4rhs(C) = 4g(B) = ∞rhs(B) = 6g(A) = 2rhs(A) = 2g(s) = 0rhs(S) = 0
(G, [8;8]) (B, [9;6])5
U - Priority Queue
21
(A, [4;2]) (B, [9;6])(S, [0;0])
3 (C, [5;4]) (D, [7;6]) (B, [9;6])4 (D, [7;6]) (B, [9;6])
0
C
S
B
G
A
D
2
4
82
61
5
2
3
4
2
6
∞6
68
4
2
0
0
3
1
1
0
2
Check local consistency of smallest key, G.
First Call To Incremental A* Search
g(G) = 8rhs(G) = 8g(D) = 6rhs(D) = 6g(C) = 4rhs(C) = 4g(B) = ∞rhs(B) = 6g(A) = 2rhs(A) = 2g(s) = 0rhs(S) = 0
(G, [8;8]) (B, [9;6])5
U - Priority Queue
21
(A, [4;2]) (B, [9;6])(S, [0;0])
3 (C, [5;4]) (D, [7;6]) (B, [9;6])4 (D, [7;6]) (B, [9;6])
0
C
S
B
G
A
D
2
4
82
61
5
2
3
4
2
6
∞6
68
4
2
0
0
3
1
1
0
2
The goal is locally consistent, hence search terminates.
First Call To Incremental A* Search
g(G) = 8rhs(G) = 8g(D) = 6rhs(D) = 6g(C) = 4rhs(C) = 4g(B) = ∞rhs(B) = 5g(A) = 2rhs(A) = 2g(s) = 0rhs(S) = 0
(G, [8;8]) (B, [9;6])5
U - Priority Queue
21
(A, [4;2]) (B, [9;6])(S, [0;0])
3 (C, [5;4]) (D, [7;6]) (B, [9;6])4 (D, [7;6]) (B, [9;6])
2
6 1 5
2
3
4
2
0
C
S
B
G
A
D
2
4
8
6
∞6
68
4
2
0
0
3
1
1
0
2
The search does not necessarily make all vertices locally consistent, only those expanded.
0rhs(B) = 6
First Call To Incremental A* Search
g(G) = 6rhs(G) = 6g(D) = 6rhs(D) = 6g(C) = 4rhs(C) = 4g(B) = ∞g(A) = 2rhs(A) = 2g(s) = 0rhs(S) = 0
U - Priority Queue1
2
15
2
3
4
2
0
C
S
B
G
A
D
2
4
8
6
6
68
4
2
0
0
3
1
1
0
2
(B, [9;6])
∞
For next search, we keep rhs and g-values so that we know which nodes not to traverse again.
06rhs(B) = 6
Incremental A* After Edge Decrease
g(G) = 6rhs(G) = 6g(D) = 6rhs(D) = 6g(C) = 4rhs(C) = 4g(B) = ∞g(A) = 2rhs(A) = 2g(s) = 0rhs(S) = 0
U - Priority Queue1
2
15
2
3
4
2
0
C
S
B
G
A
D
2
4
8
6
6
68
4
2
0
0
3
1
1
0
2
(B, [9;6])
∞
We discover that the cost of edge S-B decreases to 0.
rhs(B) = 0
Incremental A* After Edge Decrease
g(G) = 6rhs(G) = 6g(D) = 6rhs(D) = 6g(C) = 4rhs(C) = 4g(B) = ∞g(A) = 2rhs(A) = 2g(s) = 0rhs(S) = 0
U - Priority Queue1 (B, [3;0])
0
2
15
2
3
4
2
0
C
S
B
G
A
D
2
4
8
6
∞0
68
4
2
0
0
3
1
1
0
2
The affected nodes, B in this case, are updated and added to the queue.
g(G) = 6rhs(G) = 6g(D) = 6rhs(D) = 6g(C) = 4rhs(C) = 4g(B) = 0rhs(B) = 0g(A) = 2rhs(A) = 2g(s) = 0rhs(S) = 0
U - Priority Queue1 (B, [3;0])
Incremental A* After Edge Decrease
0
2
15
2
3
4
2
0
C
S
B
G
A
D
2
4
8
6
00
68
4
2
0
0
3
1
1
0
2
Check local consistency of smallest key, B, in queue.
rhs(S) = 0 g(s) = 0rhs(A) = 2 g(A) = 2rhs(B) = 0 g(B) = 0rhs(C) = 4 g(C) = 4rhs(D) = 1 g(D) = 6rhs(G) = 6 g(G) = 6
U - Priority Queue
21 (B, [3;0])
(D, [2;1])
Incremental A* After Edge Decrease
0
2
15
2
3
4
2
0
C
S
B
G
A
D
2
4
8
6
00
18
4
2
0
0
3
1
1
0
2
Update rhs value of successor node D and add to queue.
rhs(S) = 0 g(s) = 0rhs(A) = 2 g(A) = 2rhs(B) = 0 g(B) = 0rhs(C) = 4 g(C) = 4rhs(D) = 1 g(D) = 6rhs(G) = 5 g(G) = 6
U - Priority Queue
21
(D, [2;1])(B, [3;0])(D, [2;1]) (G, [5;5])
Incremental A* After Edge Decrease
0
2
15
2
3
4
2
0
C
S
B
G
A
D
2
4
8
6
00
15
4
2
0
0
3
1
1
0
2
Update rhs value of successor node G and add to queue.
rhs(S) = 0 g(s) = 0rhs(A) = 2 g(A) = 2rhs(B) = 0 g(B) = 0rhs(C) = 4 g(C) = 4rhs(D) = 1 g(D) = 1rhs(G) = 5 g(G) = 6
U - Priority Queue
21
(D, [2;1]) (G, [5;5])(B, [3;0])
3 (G, [3;3])
Incremental A* After Edge Decrease
0
2
15
2
3
4
2
0
C
S
B
G
A
D
2
4
8
1
00
15
4
2
0
0
3
1
1
0
2
Check local consistency of D, and expand successors.
rhs(S) = 0 g(s) = 0rhs(A) = 2 g(A) = 2rhs(B) = 0 g(B) = 0rhs(C) = 4 g(C) = 4rhs(D) = 1 g(D) = 1rhs(G) = 5 g(G) = 5
U - Priority Queue
21
(D, [2;1]) (G, [5;5])(B, [3;0])
3 (G, [3;3])
Incremental A* After Edge Decrease
0
2
15
2
3
4
2
0
C
S
B
G
A
D
2
4
51
00
15
4
2
0
0
3
1
1
0
2
Check local consistency of smallest key, G, in queue.
rhs(S) = 0 g(s) = 0rhs(A) = 2 g(A) = 2rhs(B) = 0 g(B) = 0rhs(C) = 4 g(C) = 4rhs(D) = 1 g(D) = 1rhs(G) = 5 g(G) = 5
U - Priority Queue
21
(D, [2;1]) (G, [5;5])(B, [3;0])
3 (G [3;3])
Incremental A* After Edge Decrease
0
2
15
2
3
4
2
0
C
S
B
G
A
D
2
4
5
1
00
15
4
2
0
0
3
1
1
0
2
Goal is locally consistent, thus finished.At the end of search, if g(sgoal) = ∞, then there is no path from sstart to sgoal.
40
rhs(B) = 0
Incremental A* After Edge Increase
g(G) = 5rhs(G) = 5g(D) = 1rhs(D) = 1g(C) = 4rhs(C) = 4g(B) = 0g(A) = 2rhs(A) = 2g(s) = 0rhs(S) = 0
U - Priority Queue1
2
15
2
3
4
2
0
C
S
B
G
A
D
2
4
5
1
0
15
4
2
0
0
3
1
1
0
2
0
We discover that the cost of edge B-D increases to 4.
rhs(B) = 0
rhs(D) = 1
Incremental A* After Edge Increase
g(G) = 5rhs(G) = 5g(D) = 1rhs(D) = 4g(C) = 4rhs(C) = 4g(B) = 0g(A) = 2rhs(A) = 2g(s) = 0rhs(S) = 0
U - Priority Queue1 (D, [2;1])
0
2
45
2
3
4
2
0
C
S
B
G
A
D
2
4
5
1
00
45
4
2
0
0
3
1
1
0
2
The affected node, D, is updated and added to the queue.
g(G) = 5rhs(G) = 5g(D) = 1rhs(D) = 4g(C) = 4rhs(C) = 4g(B) = 0rhs(B) = 0g(A) = 2rhs(A) = 2g(s) = 0rhs(S) = 0
U - Priority Queue1 (D, [2;1])
Incremental A* After Edge Increase
0
2
45
2
3
4
2
0
C
S
B
G
A
D
2
4
5
1
00
45
4
2
0
0
3
1
1
0
2
Check local consistency of smallest key, D, in queue.
rhs(S) = 0 g(s) = 0rhs(A) = 2 g(A) = 2rhs(B) = 0 g(B) = 0rhs(C) = 4 g(C) = 4rhs(D) = 4 g(D) = INFrhs(G) = 5 g(G) = 5
U - Priority Queue
21 (D, [2;1])
(D, [5;4])
Incremental A* After Edge Increase
0
2
45
2
3
4
2
0
C
S
B
G
A
D
2
4
5
inf
00
45
4
2
0
0
3
1
1
0
2
Update g value of node D and add to queue.
rhs(S) = 0 g(s) = 0rhs(A) = 2 g(A) = 2rhs(B) = 0 g(B) = 0rhs(C) = 4 g(C) = 4rhs(D) = 4 g(D) = infrhs(G) = 5 g(G) = 5
U - Priority Queue
21
(D, [5;4])(B, [2;1])
Incremental A* After Edge Increase
0
2
45
2
3
4
2
0
C
S
B
G
A
D
2
4
5
inf
00
45
4
2
0
0
3
1
1
0
2
rhs values of successor nodes C and G are consistent, don’t add to queue.
rhs(S) = 0 g(s) = 0rhs(A) = 2 g(A) = 2rhs(B) = 0 g(B) = 0rhs(C) = 4 g(C) = 4rhs(D) = 4rhs(G) = 5 g(G) = 5
U - Priority Queue
21
(D, [5;4])(B, [2;1])
3
Incremental A* After Edge Increase
0
2
45
2
3
4
2
0
C
S
B
G
A
D
2
4
5
4
00
45
4
2
0
0
3
1
1
0
2
g(G) = 4
Check local consistency of D, and update; successors consistent.
Incremental A* Pseudo Codeprocedure Initialize(){02} U := ∅{03} for all s∈S rhs(s) = g(s) = ∞{04} rhs(sstart) = 0;{05} U.Insert(sstart, [h(sstart); 0]);
Procedure Main(){17} Initialize();{18} forever{19} ComputerShortestPath();{20} Wait for changes in edge costs;{21} for all directed edges (u,v) with changed costs{22} Update the edge cost c(u,v); {23} UpdateVertex(v);
Incremental A* Pseudo Codeprocedure ComputeShortestPath(){09} while (U.TopKey()<CalculateKey(sgoal)
OR rhs(sgoal) ≠ g(sgoal)){10} u = U.Pop();{11} if (g(u) > rhs(u)){12} g(u) = rhs(u);{13} for all s ∈ Succ(u) UpdateVertex(s){14} else{15} g(u) = ∞{16} for all s ∈ Succ(u) ∪ {u} UpdateVertex(s)
procedure UpdateVertex(u){06} if (u ≠ sstart)rhs(u)= min s’∈Pred(u)(g(s’)+ c(s’,u)){07} if (u ∈ U) U.Remove(u);{08} if (g(u) ≠ rhs(u)) U.Insert(u, CalculateKey(u));
procedure CalculateKey(s){01} return [min(g(s), rhs(s)) + h(s); min(g(s), rhs(s))];
Continuous Optimal Planning
1. Generate global path plan from initial map.
2. Repeat until Goal reached, or failure.Execute next step of current global path plan.Update map based on sensor information.Incrementally update global path plan from map changes.
1 to 3 orders of magnitude speedup relative to a non-incremental path planner.
Recap: Dynamic & Incremental A*
Supports search as a repetitive online process.Exploits similarities between a series of searches to solve much faster than solving each search starting from scratch.Reuses the identical parts of the previous search tree, while updating differences.Solutions guaranteed to be optimal.On the first search, behaves like traditional algorithms.
D* behaves exactly like Dijkstra’s.Incremental A* A* behaves exactly like A*.