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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

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

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

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

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

J M N O

E I L G

B D H K

S A C Fh = 5

Follow back pointers to goal.No further discrepancies detected; goal achieved!

Outline

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

States initially tagged NEW (no cost determined yet).

J M N O

E I L G

B D H K

S A C F

OPEN List1 (0,G)

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.

J M N O

E I L G

B D H K

S A C F

OPEN List1 (0,G)

Add new neighbors of G onto the OPEN listCreate backpointers to G.

J M N O

E I L G

B D H K

S A C F

CLOSED

OPEN List1 (0,G)2 (1,K) (1,L) (1,O)

Add new neighbors of G onto the OPEN listCreate backpointers to G.

J M N O

E I L G

B D H K

S A C F

CLOSED

OPEN List1 (0,G)2 (1,K) (1,L) (1,O)3 (1,L) (1,O) (2,F) (2,H)

Add new neighbors of K on to the OPEN list and create backpointers.

J M N O

E I L G

B D H K

S A C F

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)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.

J M N O

E I L G

B D H K

S A C F

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.

J M N O

E I L G

B D H K

S A C F

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)

J M N O

E I L G

B D H K

S A C F

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)

J M N O

E I L G

B D H K

S A C F

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)

J M N O

E I L G

B D H K

S A C F

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)

J M N O

E I L G

B D H K

S A C F

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..

J M N O

E I L G

B D H K

S A C F

CLOSED

OPEN List10 (3,E) (3,M) (4,A) (4,B)11 (3,M) (4,A) (4,B) (4,J)121314

J M N O

E I L G

B D H K

S A C F

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

J M N O

E I L G

B D H K

S A C Fh = 5

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

J M N O

E I L G

B D H K

S A C Fh = 5

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

J M N O

E I L G

B D H K

S A C Fh = 5

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)

D* Completed Initial Path

J M N O

E I L G

B D H K

S A C Fh = 5

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

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

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

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

J M N O

E I L G

B D H K

S A C Fh = 5

h = 5002

CLOSED

OPEN List1 (2,H)2 (3,D)34

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.

J M N O

E I L G

B D H K

S A C Fh = 5

CLOSED

OPEN List1 (2,H)2 (3,D)34

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.

J M N O

E I L G

B D H K

S A C Fh = 5

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

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

CLOSED

OPEN List1 (2,F) (2,H)234

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 = 5002

CLOSED

OPEN List1 (2,F) (2,H)2 (3,C)34

Processing F raises descendant C’s cost, and propagates.Processing H does nothing.

J M N O

E I L G

B D H K

S A C Fh = 5

CLOSED

Closed

CLOSED

OPEN List1 (2,F) (2,H)2 (3,C)34

C may be suboptimal, check neighbors; Better path through A!However, A may be suboptimal, and updating creates a loop!

J M N O

E I L G

B D H K

S A C Fh = 5

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.

J M N O

E I L G

B D H K

S A C Fh = 5

h = 5003

CLOSED

OPEN List1 (2,F) (2,H)2 (3,C)3 (4,A)4

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.

J M N O

E I L G

B D H K

S A C Fh = 5

CLOSED

OPEN List1 (2,F) (2,H)2 (3,C)3 (4,A)4

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.

J M N O

E I L G

B D H K

S A C Fh = 5

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))

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.

J M N O

E I L G

B D H K

S A C Fh = 5

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

Bug? How does C getpointed to A?

Complete Path Execution

J M N O

E I L G

B D H K

S A C Fh = 5

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: 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

Incremental A* On Gridworld[Koenig and Likhachev, 01]

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])

∞∞

Initialize all values and add start to queue. Only start is locally inconsistent .

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])

∞∞

Check local consistency of smallest node in queue,and update g-value.

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])

∞∞

Update successor nodes.

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

(A, [4;2])(S, [0;0])

∞∞

Add changed successor node A to priority queue.

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

(A, [4;2])(S, [0;0])

∞∞

Update change to successor node B.

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

(A, [4;2]) (B, [9;6])(S, [0;0])

∞∞

Add successor node B to priority queue.

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

(A, [4;2]) (B, [9;6])(S, [0;0])

∞∞

Check local consistency of smallest key, A, in the priority queue.

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

(A, [4;2]) (B, [9;6])(S, [0;0])

3 (B, [9;6])

∞∞

Update change to successor node C.

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

(A, [4;2]) (B, [9;6])(S, [0;0])

3 (C, [5;4]) (B, [9;6])

∞∞

Add successor C to priority queue.

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

(A, [4;2]) (B, [9;6])(S, [0;0])

3 (C, [5;4]) (B, [9;6])

Update change to successor node D.

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

(A, [4;2]) (B, [9;6])(S, [0;0])

3 (C, [5;4]) (D, [7;6]) (B, [9;6])

Add successor D to priority queue.

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

(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])

Check local consistency of smallest key, C, in queue.

Node C has no successors, so nothing is added to the queue.

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

(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])

Check local consistency of smallest key, D.

Update successor G, and add to queue.

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

(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])

Check local consistency of smallest key, G.

(G, [8;8]) (B, [9;6])5

U - Priority Queue

(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])

The goal is locally consistent, hence search terminates.

(G, [8;8]) (B, [9;6])5

U - Priority Queue

(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])

The search does not necessarily make all vertices locally consistent, only those expanded.

0rhs(B) = 6

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, [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

U - Priority Queue1

(B, [9;6])

We discover that the cost of edge S-B decreases to 0.

rhs(B) = 0

U - Priority Queue1 (B, [3;0])

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])

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])

Update rhs value of successor node D and add to queue.

U - Priority Queue

(D, [2;1])(B, [3;0])(D, [2;1]) (G, [5;5])

Update rhs value of successor node G and add to queue.

U - Priority Queue

(D, [2;1]) (G, [5;5])(B, [3;0])

3 (G, [3;3])

Check local consistency of D, and expand successors.

U - Priority Queue

(D, [2;1]) (G, [5;5])(B, [3;0])

3 (G, [3;3])

Check local consistency of smallest key, G, in queue.

U - Priority Queue

(D, [2;1]) (G, [5;5])(B, [3;0])

3 (G [3;3])

Goal is locally consistent, thus finished.At the end of search, if g(sgoal) = ∞, then there is no path from sstart to sgoal.

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

We discover that the cost of edge B-D increases to 4.

rhs(B) = 0

rhs(D) = 1

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])

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])

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])

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

(D, [5;4])(B, [2;1])

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

(D, [5;4])(B, [2;1])

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*.

incremental path planningdspace.mit.edu/bitstream/handle/1721.1/36832/16... · incremental path...

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