Optimal Multi-Robot Path Planning on Graphs: Structure, Complexity, Algorithms, and Applications

Transcription

1 Optimal Multi-Robot Path Planning on Graphs: Structure, Complexity, Algorithms, and Applications Jingjin Yu Computer Science Rutgers CASE 2016 Multi-Robot Workshop

2 Outline Problem Statement Structural Properties of Optimal Formulations Complexity of Non-Optimal Fomulations Complexity of Optimal Formulations General Graphs Planar Graphs An ILP Based Novel Solution Application to the Continuous Domain Conclusion The micromvp Platform

3 Problem Statement 1 2 X I 4 X G Forbidden moves G = (V, E) MPP Problem: (G, X I, X G ), solution: collision free P = {p 1,, 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 Applications

5 Structural Properties of Optimal Formulations Y-LaValle, Arxiv Clockwise Counterclockwise Makespan Total time x + 1 2x + 3 x + 4 x x Left path only Using right path Total distance 4x + 8 4x + 10 Total time 4x x x 4 4 x Theorem. A pair of the four MPP objectives on makespan, total time, max distance, and total distance demonstrates a Pareto-optimal structure.

6 Complexity of Non-Optimal Formulation Feasibility is not guaranteed Y-Rus, WAFR ? Theorem. Feasibility of MPP can be decided in linear time. Moreover, a solution for a feasible instance can be computed in cubic time. What about finding optimal solutions?

7 Intractability of Time Optimal MPP Theorem. Min Makespan MPP is NP-hard. Y-LaValle, AAAI 13 3SAT x 1 x 3 x 4 x 1 x 2 x 4 ( x 2 x 3 x 4 ) c 1 c 2 c 3 m = 3 clauses n = 4 variables Min Makespan MPP c 1 c 2 c 3 x 4 x 4 x 4 x 3 x 3 x 3 c 1 c 2 c 3 x 2 x 2 x 2 x 1 x 1 x 1 m

8 Intractability Distance Optimal MPP Y-LaValle, Arxiv NP-hardness of distance optimal MPP is slightly more tricky Theorem. MPP is NP-hard when optimizing min makespan, min total time, min max distance, and min total distance.

9 The Planar Case (G is Planar) Y, IEEE RA-L, 2016 Planar Monotone 3-SAT Planar MPP x 1 x 4 x 5 (x 2 x 3 ) x 1 x 2 x 3 x 3 x 4 x 5 c 1 c 2 c 3 c 4 c 3 c 4 c 2 c 1 c 3 c 2 c 4 c 1 Theorem. Optimal Planar MPP (PMPP) is NP-hard for min makespan, min total time, min max distance, and min total distance objectives.

10 Practical Implications Optimal MPP and PMPP are often NP-hard c 3 c 4 c 1 Polynomial time exact solution Polynomial time suboptimal solutions? Engineering the environment helps Two way, multi-lane roads Elevated intersections c 2 We offer rigorous, quantitative justifications of these phenomena through complexity theory c 3 c 2 c 1 c 4

11 Approaches Based on Discrete Search Discrete approaches are mostly A* based The global search space is G n Differ on the handling of robot interactions n independent robots: n G Local Repair A* (LRA*) [Zelinsky, IEEE TRA 92] Planning for each robot and resolving conflicts locally as they appear Windowed Hierarchical Cooperative A* (WHCA*) [Sliver, AIIDE 05] Local space-time window for handling multi-robot interactions IDA* based [Sharon-Stern-Felner-Sturtevant, AAAI 12] Iteratively handle more robot-robot interaction Maximum Group Size (MGS n ) [Standley-Korf, IJCAI 11] Grouping robots into larger bundles as necessary The authors pushed a highly effective implementation joint search space: G n Many additional approaches: [Ryan 08], [van Den Berg et al. 09], [Wagner- Choset 11], [Boyarski et al. 15],

12 An Integer Programming Based Novel Approach Key idea: time expansion Y-LaValle, WAFR 12 2 r 1 r 2 3, r 1 r 2 1 t = T=4 4 Theorem. Fixing a natural number T, a MPP instance admits a solution with at most T time steps if and only if the corresponding time-expanded network with T periods admits a solution consisting of disjoint paths.

13 ILP Approach: The Constraints Meet-on-vertex u v w 1 2 Meet-on-edge u v 1 2 u u v w v t t + 1 t t + 1 i (x uv,t,t+1 i + x vv,t,t+1 i + x wv,t,t+1 ) 1 i (x uv,t,t+1 i + x vu,t,t+1 ) 1 1 i n 1 i n

14 Algorithm for Min Makespan Y-LaValle, ICRA 13, TRO-16, in press, online n max x i,i, subject to i=1 e j, x i,j 1 i=1 v, x i,j = n e j δ + (v) e j δ (v) x i,j Pick an initial T Build the timeexpanded network No T = T + 1 Set up an ILP model Feasible? Run optimizer Yes Return the path set Additional heuristics Reachability analysis Divide and conquer Other objectives (total time, max distance, total distance)

15 Performance Makespan 24x18 grid, with some vertices randomly removed to simulate obstacles 44% robot density Exact solution Near-optimal solution

16 Performance Total Time and Total Distance Min Total Time Min Total Distance

17 We Can Solve Some Tough Problems states > 10 4 branching factor A 7-step min makespan plan

18 Generalization to Continuous Domain Y-Rus, ISRR 15 Lattice overlay Restore connectivity Snapping Trajectory planning Path smoothing

19 Conclusion Contributions Structure and complexity MPP and PMPP appear NP-hard in general Algorithmic solution Effective integer programming based solution approach Extensible to continuous problems Future work Algorithmic solutions for more realistic setup Continuously appearing start and goal locations Various constraints Effective environment design Almost planar design with guaranteed traffic throughput Planar graph for doing the same? Remove the ILP dependency

20 micromvp (micro Multi-Vehicle Platform) An open platform targeting robotics research and education Low cost (currently <$100 per vehicle, $100 for the sensing platform ) Easy assembly 20min to build a vehicle enabled by 3D printing 1 minute to setup the platform Small scale (in my backpack!) Suitable for deployment everywhere Open (very soon)