k -Color Multi-Robot Motion Planning Kiril Solovey Tel-Aviv University, Israel WAFR, 2012 * Joint work with Dan Halperin Kiril Solovey (TAU) k -Color Motion Planning WAFR, 2012 1 / 23
Multi-Robot Problems Classic : Every robot has start and target positions Kiril Solovey (TAU) k -Color Motion Planning WAFR, 2012 2 / 23
Multi-Robot Problems Classic : Every robot has start and target positions Kiril Solovey (TAU) k -Color Motion Planning WAFR, 2012 2 / 23
Multi-Robot Problems Classic : Every robot has start and target positions Kiril Solovey (TAU) k -Color Motion Planning WAFR, 2012 2 / 23
Multi-Robot Problems Classic : Every robot has start and target positions Unlabeled : ◮ Identical robots ◮ Interchangeable target positions ◮ All target positions are occupied Kiril Solovey (TAU) k -Color Motion Planning WAFR, 2012 2 / 23
Multi-Robot Problems Classic : Every robot has start and target positions Unlabeled : ◮ Identical robots ◮ Interchangeable target positions ◮ All target positions are occupied Kiril Solovey (TAU) k -Color Motion Planning WAFR, 2012 2 / 23
Multi-Robot Problems Classic : Every robot has start and target positions Unlabeled : ◮ Identical robots ◮ Interchangeable target positions ◮ All target positions are occupied Kiril Solovey (TAU) k -Color Motion Planning WAFR, 2012 2 / 23
Multi-Robot Problems Classic : Every robot has start and target positions Unlabeled : ◮ Identical robots ◮ Interchangeable target positions ◮ All target positions are occupied Kiril Solovey (TAU) k -Color Motion Planning WAFR, 2012 2 / 23
Multi-Robot Problems Classic : Every robot has start and target positions Unlabeled : ◮ Identical robots ◮ Interchangeable target positions ◮ All target positions are occupied Kiril Solovey (TAU) k -Color Motion Planning WAFR, 2012 2 / 23
Multi-Robot Problems Classic : Every robot has start and target positions Unlabeled : ◮ Identical robots ◮ Interchangeable target positions ◮ All target positions are occupied k-Color : ◮ Several groups of identical robots ◮ Interchangeable positions in each group Kiril Solovey (TAU) k -Color Motion Planning WAFR, 2012 2 / 23
Multi-Robot Problems Classic : Every robot has start and target positions Unlabeled : ◮ Identical robots ◮ Interchangeable target positions ◮ All target positions are occupied k-Color : ◮ Several groups of identical robots ◮ Interchangeable positions in each group Unlabeled = 1 -Color Classic = Fully-Colored Kiril Solovey (TAU) k -Color Motion Planning WAFR, 2012 2 / 23
Contribution UPUMP : novel algorithm for the unlabeled problem Tailor-made for multi-robot General Simple Technique: ◮ Samples of amplified configurations ◮ Unlabeled problem reduced to several discrete problems Kiril Solovey (TAU) k -Color Motion Planning WAFR, 2012 3 / 23
Contribution UPUMP : novel algorithm for the unlabeled problem Tailor-made for multi-robot General Simple Technique: ◮ Samples of amplified configurations ◮ Unlabeled problem reduced to several discrete problems KPUMP : A straightforward extension of UPUMP to the k -color case Kiril Solovey (TAU) k -Color Motion Planning WAFR, 2012 3 / 23
Previous Related Work (Partial List) Schwartz & Sharir - Piano Movers III ’83 ◮ First complete multi-robot algorithm ◮ Disk robots Hopcroft et al. - Hardness of the warehouse problem ’84 ◮ Rectangular robot in the plane is PSPACE-hard van den Berg et al. - Optimal decoupling into sequential plans ’09 ◮ Problem decomposed into sequential subproblems Sampling based methods ◮ Svestka & Overmars - Coordinated path planning ’98 ◮ Hirsch & Halperin - Hybrid motion planning ’02 Composite robot approach (see next slide) Kiril Solovey (TAU) k -Color Motion Planning WAFR, 2012 4 / 23
Composite Robot Approach Group of robots considered as a single robot Apply known sampling-based techniques ◮ PRM - Kavraki et al. ◮ RRT - Kuffner & LaValle ◮ EST - Hsu et al. Each sample is a collection of positions—one for every robot Disadvantages: ◮ Ignores properties of the multi-robot problem ◮ Performs all operations in high-dimensional configuration space ◮ Increase in the number of robots drastically increases running time Kiril Solovey (TAU) k -Color Motion Planning WAFR, 2012 5 / 23
Observation Movements of individual robots can be easily produced if the other robots are considered as obstacles × 3 Kiril Solovey (TAU) k -Color Motion Planning WAFR, 2012 6 / 23
Observation Movements of individual robots can be easily produced if the other robots are considered as obstacles × 3 Kiril Solovey (TAU) k -Color Motion Planning WAFR, 2012 6 / 23
Exploiting the Observation Our approach: Sample a large collection of non-overlapping positions Construct a graph where an edge represents valid movement of individual robot V × 3 Kiril Solovey (TAU) k -Color Motion Planning WAFR, 2012 7 / 23
Exploiting the Observation Our approach: Sample a large collection of non-overlapping positions Construct a graph where an edge represents valid movement of individual robot V × 3 Kiril Solovey (TAU) k -Color Motion Planning WAFR, 2012 7 / 23
Exploiting the Observation Our approach: Sample a large collection of non-overlapping positions Construct a graph where an edge represents valid movement of individual robot G × 3 UPUMP considers multi-robot movements as well! Kiril Solovey (TAU) k -Color Motion Planning WAFR, 2012 7 / 23
Sampling Vertices Sample n non-overlapping single-robot positions where n > m m : number of robots n : number of samples V × 3 m = 3 , n = 0 Such V is called a pumped configuration Kiril Solovey (TAU) k -Color Motion Planning WAFR, 2012 8 / 23
Sampling Vertices Sample n non-overlapping single-robot positions where n > m m : number of robots n : number of samples V × 3 m = 3 , n = 1 Such V is called a pumped configuration Kiril Solovey (TAU) k -Color Motion Planning WAFR, 2012 8 / 23
Sampling Vertices Sample n non-overlapping single-robot positions where n > m m : number of robots n : number of samples V × 3 m = 3 , n = 2 Such V is called a pumped configuration Kiril Solovey (TAU) k -Color Motion Planning WAFR, 2012 8 / 23
Sampling Vertices Sample n non-overlapping single-robot positions where n > m m : number of robots n : number of samples V × 3 m = 3 , n = 3 Such V is called a pumped configuration Kiril Solovey (TAU) k -Color Motion Planning WAFR, 2012 8 / 23
Sampling Vertices Sample n non-overlapping single-robot positions where n > m m : number of robots n : number of samples V × 3 m = 3 , n = 4 Such V is called a pumped configuration Kiril Solovey (TAU) k -Color Motion Planning WAFR, 2012 8 / 23
Sampling Vertices Sample n non-overlapping single-robot positions where n > m m : number of robots n : number of samples V × 3 m = 3 , n = 5 Such V is called a pumped configuration Kiril Solovey (TAU) k -Color Motion Planning WAFR, 2012 8 / 23
Sampling Vertices Sample n non-overlapping single-robot positions where n > m m : number of robots n : number of samples V × 3 m = 3 , n = 5 Such V is called a pumped configuration Kiril Solovey (TAU) k -Color Motion Planning WAFR, 2012 8 / 23
Sampling Vertices Sample n non-overlapping single-robot positions where n > m m : number of robots n : number of samples V × 3 m = 3 , n = 5 Such V is called a pumped configuration Kiril Solovey (TAU) k -Color Motion Planning WAFR, 2012 8 / 23
Sampling Vertices Sample n non-overlapping single-robot positions where n > m m : number of robots n : number of samples V × 3 m = 3 , n = 6 Such V is called a pumped configuration Kiril Solovey (TAU) k -Color Motion Planning WAFR, 2012 8 / 23
Sampling Vertices Sample n non-overlapping single-robot positions where n > m m : number of robots n : number of samples V × 3 m = 3 , n = 7 Such V is called a pumped configuration Kiril Solovey (TAU) k -Color Motion Planning WAFR, 2012 8 / 23
Connection Construct a graph G = ( V , E ) ◮ Connect pairs of positions with a path ◮ Consider the rest of the positions as obstacles ◮ Add an edge if respective path does not collide with obstacles G × 3 G is a geometrically-embedded graph Kiril Solovey (TAU) k -Color Motion Planning WAFR, 2012 9 / 23
Connection Construct a graph G = ( V , E ) ◮ Connect pairs of positions with a path ◮ Consider the rest of the positions as obstacles ◮ Add an edge if respective path does not collide with obstacles G × 3 G is a geometrically-embedded graph Kiril Solovey (TAU) k -Color Motion Planning WAFR, 2012 9 / 23
Connection Construct a graph G = ( V , E ) ◮ Connect pairs of positions with a path ◮ Consider the rest of the positions as obstacles ◮ Add an edge if respective path does not collide with obstacles G × 3 G is a geometrically-embedded graph Kiril Solovey (TAU) k -Color Motion Planning WAFR, 2012 9 / 23
Connection Construct a graph G = ( V , E ) ◮ Connect pairs of positions with a path ◮ Consider the rest of the positions as obstacles ◮ Add an edge if respective path does not collide with obstacles G × 3 G is a geometrically-embedded graph Kiril Solovey (TAU) k -Color Motion Planning WAFR, 2012 9 / 23
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