Incremental Path Planning - PDF Free Download

Incremental Path Planning Dynamic and Incremental A* Prof. Brian Williams (help from Ihsiang Shu) 6./6.8 Cognitive Robotics February 7 th,

Outline Review: Optimal Path Planning in Partially Known Environments. Continuous Optimal Path Planning Dynamic A* Incremental A* (LRTA*)

[Zellinsky, 9]. Generate global path plan from initial map.. Repeat until goal reached or failure: Execute next step in current global path plan Update 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 h = J M N O E I L G h = B D H K h = h = h = h = h = h = h = S A C F Robot moves along backpointers towards goal. Uses sensors to detect discrepancies along way.

Obstacle Encountered! h = J M N O E I L G h = B D H K h = h = h = h = h = h = S A C F At state A, robot discovers edge from D to H is blocked (cost, units). Update map and reinvoke planner.

Continue Path Execution h = J M N O E I L G h = B D H K h = h = h = h = h = h = S A C F A s previous path is still optimal. Continue moving robot along back pointers.

Second Obstacle, Replan! h = J M N O E I L G h = B D H K h = h = h = h = h = h = S A C F At C robot discovers blocked edges from C to F and H (cost, units). Update map and reinvoke planner.

Path Execution Achieves Goal h = J M N O E I L G h = B D H K h = h = h = h = h = h = h = S A C F 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, 9]. Generate global path plan from initial map.. 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. to orders of magnitude speedup relative to a non-incremental path planner.

Map and Path Concepts c(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 Concepts list : States with estimates to be propagated to other states. States on list tagged Sorted by key function k. State tag t(x) : : has no estimate h. : estimate needs to be propagated. : estimate propagated.

D* Fundamental Search Concepts k(g,x) : key function minimum of h(g,x) before modification, and all values assumed by h(g,x) since X was placed on the 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 Initially Mark G Open and Queue Mark all other states New Run 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 (no cost determined yet).

Use D* to Compute Initial Path J M N O List (,G) E I L G h = B D H K S A C F 8: if kold = h(x) then 9: for each neighbor Y of X: : if t(y) = or : (b(y) = X and h(y) h(x) + c(x,y)) or : (b(y) X and h(y) > h(x) + c(x,y)) then : b(y) = X; Insert(Y,h(X) + c(x,y)) Add Goal node to the list. Process list until the robot s current state is.

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 List (,G) E I L G h = B D H K S A C F 8: if kold = h(x) then 9: for each neighbor Y of X: : if t(y) = or : (b(y) = X and h(y) h(x) + c(x,y)) or : (b(y) X and h(y) > h(x) + c(x,y)) then : b(y) = X; Insert(Y,h(X) + c(x,y)) Add new neighbors of G onto the list Create backpointers to G.

Use D* to Compute Initial Path J M N O E I L G h = B D H K S A C F List (,G) (,K) (,L) (,O) 8: if kold = h(x) then 9: for each neighbor Y of X: : if t(y) = or : (b(y) = X and h(y) h(x) + c(x,y)) or : (b(y) X and h(y) > h(x) + c(x,y)) then : b(y) = X; Insert(Y,h(X) + c(x,y)) Add new neighbors of G onto the list Create backpointers to G.

Use D* to Compute Initial Path J M N O E I L G h = h = B D H K h = S A C F List (,G) (,K) (,L) (,O) (,L) (,O) (,F) (,H) 8: if kold = h(x) then 9: for each neighbor Y of X: : if t(y) = or : (b(y) = X and h(y) h(x) + c(x,y)) or : (b(y) X and h(y) > h(x) + c(x,y)) then : b(y) = X; Insert(Y,h(X) + c(x,y)) Add new neighbors of K on to the list and create backpointers.

Use D* to Compute Initial Path J M N O E I L G h = h = h = h = B D H K h = S A C F List (,G) (,K) (,L) (,O) (,L) (,O) (,F) (,H) (,F) (,H) (,I) (,N) 8: if kold = h(x) then 9: for each neighbor Y of X: : if t(y) = or : (b(y) = X and h(y) h(x) + c(x,y)) or : (b(y) X and h(y) > h(x) + c(x,y)) then : b(y) = X; Insert(Y,h(X) + c(x,y)) Add new neighbors of L, then O on to the list and create backpointers.

Use D* to Compute Initial Path J M N O h = h = E I L G h = h = B D H K List (,G) (,K) (,L) (,O) (,L) (,O) (,F) (,H) (,F) (,H) (,I) (,N) (,H) (,I) (,N) (,C) h = S A C F Continue until current state S is closed.

Use D* to Compute Initial Path J M N O h = E I L G h = h = h = B D H K List (,G) (,K) (,L) (,O) (,L) (,O) (,F) (,H) (,F) (,H) (,I) (,N) (,H) (,I) (,N) (,C) 6 (,I) (,N) (,C) (,D) h = S A C F Continue until current state S is closed.

Use D* to Compute Initial Path J M N O h = E I L G h = h = B D H K h = h = S A C F List (,G) (,K) (,L) (,O) (,L) (,O) (,F) (,H) (,F) (,H) (,I) (,N) (,H) (,I) (,N) (,C) 6 (,I) (,N) (,C) (,D) 7 (,N) (,C) (,D) (,E) (,M) Continue until current state S is closed.

Use D* to Compute Initial Path J M N O h = E I L G B D H K h = h = h = h = h = S A C F List (,G) (,K) (,L) (,O) (,L) (,O) (,F) (,H) (,F) (,H) (,I) (,N) (,H) (,I) (,N) (,C) 6 (,I) (,N) (,C) (,D) 7 (,N) (,C) (,D) (,E) (,M) 8 (,C) (,D) (,E) (,M) 9 (,D) (,E) (,M) (,A) Continue until current state S is closed.

Use D* to Compute Initial Path J M N O E I L G h = h = B D H K h = h = h = h = h = S A C F Continue until current state S is closed.. List (,G) (,K) (,L) (,O) (,L) (,O) (,F) (,H) (,F) (,H) (,I) (,N) (,H) (,I) (,N) (,C) 6 (,I) (,N) (,C) (,D) 7 (,N) (,C) (,D) (,E) (,M) 8 (,C) (,D) (,E) (,M) 9 (,D) (,E) (,M) (,A) (,E) (,M) (,A) (,B)

Use D* to Compute Initial Path h = J M N O E I L G h = h = h = h = h = B D H K List (,E) (,M) (,A) (,B) (,M) (,A) (,B) (,J) h = h = S A C F Continue until current state S is closed.

Use D* to Compute Initial Path h = J M N O E I L G h = h = h = h = h = B D H K List (,E) (,M) (,A) (,B) (,M) (,A) (,B) (,J) (,M) (,A) (,B) h = h = S A C F Continue until current state S is closed.

Use D* to Compute Initial Path h = J M N O E I L G h = h = h = h = h = B D H K List (,E) (,M) (,A) (,B) (,M) (,A) (,B) (,J) (,M) (,A) (,B) (,B) (,J) (,S) h = h = h = S A C F Continue until current state S is closed.

Use D* to Compute Initial Path h = J M N O E I L G h = h = h = h = h = B D H K List (,E) (,M) (,A) (,B) (,M) (,A) (,B) (,J) (,M) (,A) (,B) (,B) (,J) (,S) (,J) (,S) h = h = h = S A C F Continue until current state S is closed.

Use D* to Compute Initial Path h = J M N O E I L G h = h = h = h = h = B D H K List (,E) (,M) (,A) (,B) (,M) (,A) (,B) (,J) (,M) (,A) (,B) (,B) (,J) (,S) (,J) (,S) (,S) h = h = h = S A C F Continue until current state S is closed.

D* Completed Initial Path h = J M N O E I L G h = B D H K h = h = h = h = h = h = h = S A C F List (,E) (,M) (,A) (,B) (,M) (,A) (,B) (,J) (,M) (,A) (,B) (,B) (,J) (,S) (,J) (,S) (,S) 6 NULL Done: Current state S is closed, and Open list is empty.

Begin Executing Optimal Path h = J M N O E I L G h = B D H K h = h = h = h = h = h = h = S A C F Robot moves along backpointers towards goal Uses sensors to detect discrepancies along way.

Obstacle Encountered! h = J M N O E I L G h = B D H K h = h = h = h = h = h = S A C F At state A, robot discovers edge D to H is blocked off (cost, 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 h = J M N O E I L G h = B D H K h = h = h = h = h = h = S A C F List (,H) Function: Modify-Cost(X,Y,eval) : c(x,y) = eval : if t(x) = then Insert(X,h(X)) : return Get-Kmin() Propagate changes starting at H

D* Update From First Obstacle h = J M N O E I L G h = B D H K h = h = h = h = h = h = h = S A C F List (,H) (,D) 8: if kold = h(x) then 9: for each neighbor Y of X: : if t(y) = or : (b(y) = X and h(y) h(x) + c(x,y)) or : (b(y) X and h(y) > h(x) + c(x,y)) then : 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 h = h = B D H K h = h = h = h = h = h = S A C F List (,H) (,D) : if kold < h(x) then : for each neighbor Y of X: 6: if h(y) kold and h(x) > h(y) + C(Y,X) then 7: 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 h = J M N O E I L G h = h = h = h = B D H K List (,D) NULL h = h = h = S A C F All neighbors of D have consistent h-values. No further propagation needed.

Continue Path Execution h = J M N O E I L G h = h = h = h = B D H K List (,D) NULL h = h = h = S A C F A s path optimal. Continue moving robot along backpointers.

Second Obstacle! h = J M N O E I L G h = B D H K h = h = h = h = h = h = S A C F List (,F) (,H) Function: Modify-Cost(X,Y,eval) : c(x,y) = eval : if t(x) = then Insert(X,h(X)) : return Get-Kmin() At C robot discovers blocked edges C to F and H (cost, units). Update map and reinvoke D* until H(current position optimal).

D* Update From Second Obstacle h = J M N O E I L G h = B D H K h = h = h = h = h = h = h = S A C F List (,F) (,H) (,C) 8: if kold = h(x) then 9: for each neighbor Y of X: : if t(y) = or : (b(y) = X and h(y) h(x) + c(x,y)) or : (b(y) X and h(y) > h(x) + c(x,y)) then : 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 = J M N O E I L G h = B D H K h = h = h = h = h = h = S A C F Closed h = Open List (,F) (,H) (,C) : if kold < h(x) then : for each neighbor Y of X: 6: if h(y) kold and h(x) > h(y) + C(Y,X) then 7: 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 h = J M N O E I L G h = B D H K h = h = h = h = h = S A C F h = h = List (,F) (,H) (,C) (,A) : for each neighbor Y of X: 6: if t(y) = or 7: (b(y) = X and h(y) h(x) + c(x,y)) then 8: b(y) = X; Insert(Y,h(X) + c(x,y)) Don t change C s path to A (yet). Instead, propagate increase to A.

D* Update From Second Obstacle h = J M N O E I L G h = B D H K h = h = h = h = h = h = S A C F h = List (,F) (,H) (,C) (,A) : if kold < h(x) then : for each neighbor Y of X: 6: if h(y) kold and h(x) > h(y) + C(Y,X) then 7: b(x) = Y; h(x) = h(y) + c(y,x); 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 h = J M N O E I L G h = B D H K h = h = h = h = h = h = S A C F h = List (,F) (,H) (,C) (,A) (,C) for each neighbor Y of X: 7: if (b(y) = X and h(y) h(x) + c(x,y)) then 8: b(y) = X; Insert(Y,h(X) + c(x,y)) 9: else : if b(y) X and h(y) > h(x) + c(x,y) then : 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.

D* Update From Second Obstacle h = J M N O E I L G h = B D H K h = h = h = h = h = S A C F h = h = List (,F) (,H) (,C) (,A) (,C) 8: if kold = h(x) then 9: for each neighbor Y of X: : if t(y) = or : (b(y) = X and h(y) h(x) + c(x,y)) or : (b(y) X and h(y) > h(x) + c(x,y)) then : b(y) = X; Insert(Y,h(X) + c(x,y)) C lowered to optimal; no neighbors affected. Current state reached, so Process_State terminates. Bug? How does C get pointed to A?

Complete Path Execution h = J M N O E I L G h = B D H K h = h = h = h = h = h = h = S A C F Follow back pointers to Goal. No further discrepancies detected; goal achieved!

D* Pseudo Code Function: Process-State() : X = Min-State() : if X=NULL then return - : kold = Get-Kmin(); Delete(X) : if kold < h(x) then : for each neighbor Y of X: 6: if h(y) kold and h(x) > h(y) + C(Y,X) then 7: b(x) = Y; h(x) = h(y) + c(y,x); 8: if kold = h(x) then 9: for each neighbor Y of X: : if t(y) = or : (b(y) = X and h(y) h(x) + c(x,y)) or : (b(y) X and h(y) > h(x) + c(x,y)) then : b(y) = X; Insert(Y,h(X) + c(x,y)) : else : for each neighbor Y of X: 6: if t(y) = or 7: (b(y) = X and h(y) h(x) + c(x,y)) then 8: b(y) = X; Insert(Y,h(X) + c(x,y)) 9: else : if b(y) X and h(x) > h(x) + c(x,y) then : Insert(X,h(X)) : else : if b(y) X and h(x) > h(y) + c(y,x) and : t(y) = and h(y) > kold then : Insert(Y,h(Y)) 6: return Get-Kmin() Function: Modify-Cost(X,Y,eval) : c(x,y) = eval : if t(x) = then Insert(X,h(X)) : return Get-Kmin() Function: Insert(X, h new ) : if t(x) = then k(x) = h new : else if t(x) = then k(x) = min(k(x), h new ) : else k(x) = min(h(x), h new ) : h(x) = h new ; : t(x) =

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, ]

Incremental A* Versus Other Searches

Definitions: Graph and Costs Succ(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 s start 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 & Update rhs(s) : one-step lookahead estimate for g(s) if s = s start then else min s 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 Brief ) Initialize start node with g estimates of and enqueue. All other g estimates (g,rhs values) are initialized to INF. Only the start node is locally inconsistent. ) Take node with smallest key from priority queue. Update g-value, and add all successors to the priority queue. ) Loop step until either the goal is locally consistent or the priority becomes empty. ) To find shortest path: Start with s = goal, working to start, next s = predecessor (s) of s with best g(s) + c(s, s ), ) Wait for edge costs to change. Update edge costs and add edge s destination node to the queue. 6) Go to step.

Definitions: Incremental A* Search Queue U : priority queue, contains all locally inconsistent vertices. U ordered by key k(s) k(s) : [k (s), k (s)] k (s) : min(g(s), rhs(s)) + h(s) k (s) : min(g(s), rhs(s)) k(s) corresponds directly to f(s) of A*. First time through IA*, min(g(s), rhs(s)) = true g*(s). k(s) breaks ties, using minimum start distance.

First Call To Incremental A* Search U - Priority Queue, (X, [k(s);k(s)]) (S, [;]) S 6 A B C D G rhs(s) = rhs(a) = rhs(b) = rhs(c) = rhs(d) = g(s) = g(a) = g(b) = g(c) = g(d) = rhs(g) = g(g) = Initialize all values and add start to queue. Only start is locally inconsistent.

First Call To Incremental A* Search U - Priority Queue (S, [;]) S 6 A B C D G rhs(s) = g(s) = rhs(a) = rhs(b) = rhs(c) = rhs(d) = g(a) = g(b) = g(c) = g(d) = rhs(g) = g(g) = Check local consistency of smallest node in queue, and update g-value.

First Call To Incremental A* Search S 6 A B C D G U - Priority Queue (S, [;]) rhs(s) = g(s) = rhs(a) = rhs(b) = rhs(c) = rhs(d) = g(a) = g(b) = g(c) = g(d) = rhs(g) = g(g) = Update successor nodes.

First Call To Incremental A* Search S 6 A B C D G U - Priority Queue (S, [;]) (A, [;]) rhs(s) = g(s) = rhs(a) = rhs(b) = rhs(c) = rhs(d) = g(a) = g(b) = g(c) = g(d) = rhs(g) = g(g) = Add changed successor node A to priority queue.

First Call To Incremental A* Search S 6 A B 6 C D G U - Priority Queue (S, [;]) (A, [;]) rhs(s) = g(s) = rhs(a) = rhs(b) = 6 rhs(c) = rhs(d) = g(a) = g(b) = g(c) = g(d) = rhs(g) = g(g) = Update change to successor node B.

First Call To Incremental A* Search S 6 A B 6 C D G U - Priority Queue (S, [;]) (A, [;]) (B, [9;6]) rhs(s) = g(s) = rhs(a) = rhs(b) = 6 rhs(c) = rhs(d) = g(a) = g(b) = g(c) = g(d) = rhs(g) = g(g) = Add successor node B to priority queue.

First Call To Incremental A* Search S 6 A B 6 C D G U - Priority Queue (S, [;]) (A, [;]) (B, [9;6]) rhs(s) = g(s) = rhs(a) = g(a) = rhs(b) = 6 rhs(c) = rhs(d) = g(b) = g(c) = g(d) = rhs(g) = g(g) = Check local consistency of smallest key, A, in the priority queue.

First Call To Incremental A* Search S 6 A B 6 C D G U - Priority Queue (S, [;]) (A, [;]) (B, [9;6]) (B, [9;6]) rhs(s) = g(s) = rhs(a) = g(a) = rhs(b) = 6 rhs(c) = rhs(d) = g(b) = g(c) = g(d) = rhs(g) = g(g) = Update change to successor node C.

First Call To Incremental A* Search S 6 A B 6 C D G U - Priority Queue (S, [;]) (A, [;]) (B, [9;6]) (C, [;]) (B, [9;6]) rhs(s) = g(s) = rhs(a) = g(a) = rhs(b) = 6 rhs(c) = rhs(d) = g(b) = g(c) = g(d) = rhs(g) = g(g) = Add successor C to priority queue.

First Call To Incremental A* Search S 6 A B 6 C 6 D G U - Priority Queue (S, [;]) (A, [;]) (B, [9;6]) (C, [;]) (B, [9;6]) rhs(s) = g(s) = rhs(a) = g(a) = rhs(b) = 6 rhs(c) = rhs(d) = 6 g(b) = g(c) = g(d) = rhs(g) = g(g) = Update change to successor node D.

First Call To Incremental A* Search S 6 A B 6 C 6 D G U - Priority Queue (S, [;]) (A, [;]) (B, [9;6]) (C, [;]) (D, [7;6]) (B, [9;6]) rhs(s) = g(s) = rhs(a) = g(a) = rhs(b) = 6 rhs(c) = rhs(d) = 6 g(b) = g(c) = g(d) = rhs(g) = g(g) = Add successor D to priority queue.

First Call To Incremental A* Search S 6 A B 6 6 D C Check local consistency of smallest key, C, in queue. G U - Priority Queue (S, [;]) (A, [;]) (B, [9;6]) (C, [;]) (D, [7;6]) (B, [9;6]) (D, [7;6]) (B, [9;6]) rhs(s) = g(s) = rhs(a) = g(a) = rhs(b) = 6 g(b) = rhs(c) = g(c) = rhs(d) = 6 rhs(g) = Node C has no successors, so nothing is added to the queue. g(d) = g(g) =

Update successor G, and add to queue. First Call To Incremental A* Search S 6 A B 6 C 6 6 D 8 G U - Priority Queue (S, [;]) (A, [;]) (B, [9;6]) (C, [;]) (D, [7;6]) (B, [9;6]) (D, [7;6]) (B, [9;6]) (G, [8;8]) (B, [9;6]) rhs(s) = g(s) = rhs(a) = g(a) = rhs(b) = 6 g(b) = rhs(c) = g(c) = rhs(d) = 6 g(d) = 6 rhs(g) = 8 g(g) = Check local consistency of smallest key, D.

First Call To Incremental A* Search S 6 A B 6 C 6 6 D 8 8 G U - Priority Queue (S, [;]) (A, [;]) (B, [9;6]) (C, [;]) (D, [7;6]) (B, [9;6]) (D, [7;6]) (B, [9;6]) (G, [8;8]) (B, [9;6]) rhs(s) = rhs(a) = rhs(b) = 6 rhs(c) = rhs(d) = 6 g(s) = g(a) = g(b) = g(c) = g(d) = 6 rhs(g) = 8 g(g) = 8 Check local consistency of smallest key, G.

First Call To Incremental A* Search S 6 A B 6 C 6 6 D 8 8 G U - Priority Queue (S, [;]) (A, [;]) (B, [9;6]) (C, [;]) (D, [7;6]) (B, [9;6]) (D, [7;6]) (B, [9;6]) (G, [8;8]) (B, [9;6]) rhs(s) = rhs(a) = rhs(b) = rhs(c) = rhs(d) = 6 g(s) = g(a) = g(b) = g(c) = g(d) = 6 rhs(g) = 8 g(g) = 8 The search does not necessarily make all vertices locally consistent, only those expanded.

First Call To Incremental A* Search U - Priority Queue (B, [9;6]) S A B 6 C 6 6 D 8 8 G rhs(s) = rhs(a) = rhs(b) = 6 rhs(c) = rhs(d) = 6 g(s) = g(a) = g(b) = g(c) = g(d) = 6 rhs(g) = 6 g(g) = 6 For next search, we keep rhs and g-values so that we know which nodes not to traverse again.

Incremental A* After Edge Decrease U - Priority Queue (B, [9;6]) S 6 A B 6 C 6 6 D 8 8 G rhs(s) = rhs(a) = rhs(b) = 6 rhs(c) = rhs(d) = 6 g(s) = g(a) = g(b) = g(c) = g(d) = 6 rhs(g) = 6 g(g) = 6 We discover that the cost of edge S-B decreases to.

Incremental A* After Edge Decrease U - Priority Queue (B, [;]) S A B C 6 6 D 8 8 G rhs(s) = rhs(a) = rhs(b) = rhs(c) = rhs(d) = 6 g(s) = g(a) = g(b) = g(c) = g(d) = 6 rhs(g) = 6 g(g) = 6 The affected nodes, B in this case, are updated and added to the queue.

Incremental A* After Edge Decrease U - Priority Queue (B, [;]) S A B C 6 6 D 8 8 G rhs(s) = rhs(a) = rhs(b) = rhs(c) = rhs(d) = 6 g(s) = g(a) = g(b) = g(c) = g(d) = 6 rhs(g) = 6 g(g) = 6 Check local consistency of smallest key, B, in queue.

Incremental A* After Edge Decrease S A B C 6 D 8 8 G U - Priority Queue (B, [;]) (D, [;]) rhs(s) = g(s) = rhs(a) = g(a) = rhs(b) = g(b) = rhs(c) = g(c) = rhs(d) = g(d) = 6 rhs(g) = 6 g(g) = 6 Update rhs value of successor node D and add to queue.

Incremental A* After Edge Decrease S A B C 6 D 8 G U - Priority Queue (B, [;]) (D, [;]) (G, [;]) rhs(s) = g(s) = rhs(a) = g(a) = rhs(b) = g(b) = rhs(c) = g(c) = rhs(d) = g(d) = 6 rhs(g) = g(g) = 6 Update rhs value of successor node G and add to queue.

Incremental A* After Edge Decrease S A B C D 8 G U - Priority Queue (B, [;]) (D, [;]) (G, [;]) (G, [;]) rhs(s) = g(s) = rhs(a) = g(a) = rhs(b) = g(b) = rhs(c) = g(c) = rhs(d) = g(d) = rhs(g) = g(g) = 6 Check local consistency of D, and expand successors.

Incremental A* After Edge Decrease S A B C D G U - Priority Queue (B, [;]) (D, [;]) (G, [;]) (G, [;]) rhs(s) = g(s) = rhs(a) = g(a) = rhs(b) = g(b) = rhs(c) = g(c) = rhs(d) = g(d) = rhs(g) = g(g) = Check local consistency of smallest key, G, in queue.

Incremental A* After Edge Decrease S A B C D G U - Priority Queue (B, [;]) (D, [;]) (G, [;]) (G [;]) rhs(s) = g(s) = rhs(a) = g(a) = rhs(b) = g(b) = rhs(c) = g(c) = rhs(d) = g(d) = rhs(g) = g(g) = Goal is locally consistent, thus finished. At the end of search, if g(s goal ) =, then there is no path from s start to s goal.

Incremental A* After Edge Increase U - Priority Queue S A B C D G rhs(s) = rhs(a) = rhs(b) = rhs(c) = rhs(d) = g(s) = g(a) = g(b) = g(c) = g(d) = rhs(g) = g(g) = We discover that the cost of edge B-D increases to.

Incremental A* After Edge Increase U - Priority Queue (D, [;]) S A B C D G rhs(s) = rhs(a) = rhs(b) = rhs(c) = rhs(d) = g(s) = g(a) = g(b) = g(c) = g(d) = rhs(g) = g(g) = The affected node, D, is updated and added to the queue.

Incremental A* After Edge Increase U - Priority Queue (D, [;]) S A B C D G rhs(s) = rhs(a) = rhs(b) = rhs(c) = rhs(d) = g(s) = g(a) = g(b) = g(c) = g(d) = rhs(g) = g(g) = Check local consistency of smallest key, D, in queue.

Incremental A* After Edge Increase S A B C inf D G U - Priority Queue (D, [;]) (D, [;]) rhs(s) = g(s) = rhs(a) = g(a) = rhs(b) = g(b) = rhs(c) = g(c) = rhs(d) = g(d) = INF rhs(g) = g(g) = Update g value of node D and add to queue.

rhs values of successor nodes C and G are consistent, don t add to queue. Incremental A* After Edge Increase S A B C inf D G U - Priority Queue (B, [;]) (D, [;]) rhs(s) = g(s) = rhs(a) = g(a) = rhs(b) = g(b) = rhs(c) = g(c) = rhs(d) = g(d) = inf rhs(g) = g(g) =

Incremental A* After Edge Increase S A B C D G U - Priority Queue (B, [;]) (D, [;]) rhs(s) = g(s) = rhs(a) = g(a) = rhs(b) = g(b) = rhs(c) = g(c) = rhs(d) = g(g) = rhs(g) = g(g) = Check local consistency of D, and update; successors consistent.

Incremental A* Pseudo Code procedure Initialize() {} U := {} for all s S rhs(s) = g(s) = {} rhs(s start ) = ; {} U.Insert(s start, [h(s start ); ]); Procedure Main() {7} Initialize(); {8} forever {9} ComputerShortestPath(); {} Wait for changes in edge costs; {} for all directed edges (u,v) with changed costs {} Update the edge cost c(u,v); {} UpdateVertex(v);

Incremental A* Pseudo Code procedure ComputeShortestPath() {9} while (U.TopKey()<CalculateKey(sgoal) OR rhs(s goal ) g(s goal )) {} u = U.Pop(); {} if (g(u) > rhs(u)) {} g(u) = rhs(u); {} for all s Succ(u) UpdateVertex(s) {} else {} g(u) = {6} for all s Succ(u) {u} UpdateVertex(s) procedure UpdateVertex(u) {6} if (u sstart)rhs(u)= min s Pred(u)(g(s )+ c(s,u)) {7} if (u U) U.Remove(u); {8} if (g(u) rhs(u)) U.Insert(u, CalculateKey(u)); procedure CalculateKey(s) {} return [min(g(s), rhs(s)) + h(s); min(g(s), rhs(s))];

Continuous Optimal Planning. Generate global path plan from initial map.. 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. to 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*.