Lecture 18.1 Multi-Robot Path Planning (3)
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1 CS 60/560 Introduction to Computational Robotics Fall 07, Rutgers University Lecture 8. Multi-Robot Path Planning () Instructor: Jingjin Yu
2 Outline Algorithms for optimal multi-robot path and motion planning Graph search based algorithm for MPP p Coupled search Decoupled search Integer linear programming models for MPP r
3 Optimal Formulations X I 5 6 X G 6 5 Given (G, X I, X G ), want collision free P = {p,, p n } Optimality objectives (minimization): Max time (makespan): min max time(p i) P P p i P Total time: min σ p P P i P time(p i ) Max distance: min max length(p i) P P p i P Total distance: min σ p P P i P length(p i )
4 Incompatibility of the Optimal Formulations Clockwise Counterclockwise Makespan Total time x + x + x + x + x Left path only Using right path Total distance x + 8 x + 0 Total time x + x + x x A pair of the four MPP r objectives on makespan, total time, max distance, and total distance demonstrates a Pareto-optimal structure.
5 Discrete Search Algorithms General methods Coupled search treat all robots as a single robot Decoupled search treat robots as individual ones as much as possible Recall the basic structure of search algorithms (still applies!) input: G = (V, E), x I, x G AddToQueue(x I, Queue); // Add x I to a queue of nodes to be expanded while(!isempty(queue)) x Front(Queue); // Retrieve the front of the queue if(x. expanded == true) continue; // Do not expand a node twice x. expanded = true; // Mark x as expanded if(x == x G ) return solution; // Return if goal is reached for each neighbor n i of x // Add all neighbors of to the queue if(n i. expanded == false) AddToQueue(n i, Queue) return failure;
6 Coupled Search Key: treat all robots as a whole For a single robot, # neighbors? Up to neighbors What about multi-robot case? Up to 5 neighbors per robot, including staying put The search works in a straightforward way For three robots, a neighboring node may be (east, north, stay) But, huge branching factor! For n robots, 5 n neighbors in the graph For robots, 5 neighbors per step Optimal, complete, but impractical (why exactly?)
7 Coupled Search Why Impractical? How large can a priority queue be? For a single robot, no more than V For n robots, V n Well, not exactly, a bit smaller, why? But close enough Suppose V = 0, n = 0 V n = 0 0! We cannot hope to even store the queue on hard disk So search will be extremely slow! S [f=0+7] S A 5 C G A [+6] B [+] C [6+] G [+0] B [+] B C [5+] G [8+0]
8 Decoupled Search Key idea: treat robots as individual ones as much as possible To start, plan optimal paths for individual robots This reduces branching factor: 5 n n Then, simulate the execution of the paths When there are conflicts, push all choices onto the priority queue Then continue the execution Initial paths may get updated/changed In the example One queue node corresponding to robot takes the junction first One queue node corresponding to robot takes the junction first Both will add an additional (makespan) cost of A rough sketch: practical implementations require lots of care-taking
9 Handling Different Objectives Different objectives cause the queue to be sorted differently x x Total distance Initially all choose left path Then - have conflict at t =, generating new nodes (robot i goes first) Then at t =, suppose we pick the node letting go first, three new nodes are created These three new nodes can be inserted into the front of the queue using a secondary heuristic After one more iteration, new nodes are generated Then one last iteration resolves all conflicts The total distance remains the same for all nodes, which is x + 8 Total time Initial node cost is x + 8 Here, at t =, new nodes, cost is now x + for all At t =, if goes through the right, cost is x +, otherwise, x +
10 Strengths and Weakness of Discrete Search Strengths of discrete search solutions When it works, the algorithm generally runs rather fast Because the overall algorithm is relatively simple due to its discrete nature Capable of solving large (sparse) problems Generally straightforward to implement and tweak Weaknesses As the interactions among the robots grow, performance degrades quickly As such, not suitable for solving very dense problems Not suitable for handling MPP r as the number of possible rotations can be very large; huge branching factor For 6-puzzle, >000 possible cycles Each cycle has two directions Enumerating becomes impossible
11 An Integer Programming Based Solver for MPP r Transform MPP into multiflow over time steps G Constraints u v w Each edge is a binary variable u v u u v w v t t + t t + i (x uv,t,t+ i + x vv,t,t+ i + x wv,t,t+ ) i (x uv,t,t+ i + x vu,t,t+ ) i n i n
12 An Example r r, r r t = 0 T=
13 Additional Splitting Heuristic The ILP-base algorithm can require big models 0 5 grid, 0 steps, 0 robots ~ million variables We can use a divide-and-conquer like heuristic through splitting over time The algorithm is no longer complete and may yield sub-optimal solutions
14 The Approach Can Solves Some Tough Problem states > 0 branching factor A 7-step min makespan plan
15 Some Examples in D Continuous Domain No static obstacles, 75 robots.7 seconds to compute,.6-optimal Random obstacles, 50 robots.0 seconds to compute,.9-optimal
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