Domain-Dependent View of Multiple Robots Path Planning Pavel Surynek Charles University, Prague Czech Republic

Outline of the Talk Problem definition Problem definition Motivation by real-world problems Motivation by real-world problems Motivation for the domain-dependent view Motivation for the domain-dependent view –why don’t use domain-independent approach Difficulty of the problem Difficulty of the problem Domain-dependent polynomial-time solving algorithm Domain-dependent polynomial-time solving algorithm Complexity analysis Complexity analysis –polynomial-time Final remarks Final remarks STAIRS 2008Pavel Surynek

Multi-robot Path Planning Input: Graph G=(V,E), V={v 1,v 2,...,v n } and a set of robots R={r 1,r 2,...,r μ }, where μ<n Input: Graph G=(V,E), V={v 1,v 2,...,v n } and a set of robots R={r 1,r 2,...,r μ }, where μ<n –each robot is placed in a vertex (at most one robot in a vertex) –a robot can move into an unoccupied vertex through an edge (no other robot is allowed to enter the vertex) –initial positions of robots... simple function S 0 : R →V –goal positions of robots... simple function S + : R → V Task: Find a sequence of allowed moves for robots such that all the robots reach their goal positions starting from the given initial positions Task: Find a sequence of allowed moves for robots such that all the robots reach their goal positions starting from the given initial positions STAIRS 2008Pavel Surynek (Ryan, 2007)

Motivation Rearrangement of robots in tight space Rearrangement of robots in tight space Motion planning for a group of robots Motion planning for a group of robots Automated traffic control Automated traffic control STAIRS 2008Pavel Surynek

Motivating experiments (1) Problems motivated by heavy car traffic Problems motivated by heavy car traffic –ignoring traffic rules Several planners (SGPlan, SATPlan, LPG) were evaluated on these problems Several planners (SGPlan, SATPlan, LPG) were evaluated on these problems STAIRS 2008Pavel Surynek Graph Initial positions Goal positions

Motivating experiments (2) STAIRS 2008Pavel Surynek Problem Number of vertices/free Solvable Plan length/ optimal SGPlan (seconds) SATPlan (seconds) LPG (seconds) 017/6Yes1/ /7Yes6/ /6Yes25/ /5Yes216/ /10Yes12/ /8Yes178/ /7Yes176/ /6Yes64/ /5Yes72/560.04> /4Yes112/600.05> /3Yes170/980.17> /4Yes154/ > /3YesN/A> /10Yes10/ /8Yes30/ /4Yes124/N/A> /3Yes114/780.76> /2Yes208/ > /2NoN/A> /2NoN/A> /20Yes72/640.08>

Analysis of experiments All the problems are small - graphs of about 15 vertices All the problems are small - graphs of about 15 vertices SGPlan and SATPlan are trying to find shortest possible solutions - this complicates the task SGPlan and SATPlan are trying to find shortest possible solutions - this complicates the task SGPlan and SATPlan often do not solve the problem within the time limit of 2 minutes SGPlan and SATPlan often do not solve the problem within the time limit of 2 minutes LPG searches for a suboptimal solution LPG searches for a suboptimal solution LPG performs as best however still not able to solve several problems in the time limit of 2 minutes LPG performs as best however still not able to solve several problems in the time limit of 2 minutes Conclusion: Domain-independent planners do not perform well on the problem Conclusion: Domain-independent planners do not perform well on the problem STAIRS 2008Pavel Surynek

Difficult or easy? Is the problem difficult or easy (NP-complete or polynomial-time solvable)? Is the problem difficult or easy (NP-complete or polynomial-time solvable)? Answer: If at least two vertices are unoccupied the problem is polynomial-time solvable Answer: If at least two vertices are unoccupied the problem is polynomial-time solvable Outline of the algorithm: Outline of the algorithm: –decompose the graph into bi-connected components (result is a tree where vertices are represented by bi- connected components) –move robots into their goal bi-connected components or decide that it is not possible –solve the problem within the individual bi-connected components STAIRS 2008Pavel Surynek

Algorithm: decomposition into bi-connected components Decomposition of the graph G=(V,E) into bi-connected components Decomposition of the graph G=(V,E) into bi-connected components –an undirected G=(V,E) graph is bi-connected if |V|≥3 and for every v  V H=(V-{v},E’) is connected (E’ is E restricted on V-{v}) Result is a tree of singleton vertices and bi-connected components... denoted B 1,B 2,...,B k Result is a tree of singleton vertices and bi-connected components... denoted B 1,B 2,...,B k –(B i is a vertex or a bi-connected component) STAIRS 2008Pavel Surynek B1B1 B2B2 B3B3 B4B4 B5B5 B6B6 B6B6

Algorithm: moving robots into goal bi-connected components (1) Observation: Arbitrary vertex in the graph can be made unoccupied Observation: Arbitrary vertex in the graph can be made unoccupied Observation: A robot can be moved to arbitrary vertex within the bi-connected component Observation: A robot can be moved to arbitrary vertex within the bi-connected component STAIRS 2008Pavel Surynek G=(V,E) r1r1 r2r2 r3r3 r4r4 r4r4 r3r3 r2r2 r1r1 “Shift” robots along a path connecting unoccupied vertex and vertex to be freed r2r2 r4r4 r5r5 r1r1 r3r3 r6r6 r7r7 r8r8 r9r9 r 12 r 10 r 11 r 13 r 14 r 15 r 16 r 17 C1C1 C2C2 C3C3 C4C4

Algorithm: moving robots into goal bi-connected components (2) Observation: To move a robot from one bi-component to another bi-component we need to free a path between the components Observation: To move a robot from one bi-component to another bi-component we need to free a path between the components If the path cannot be freed the problem is unsolvable If the path cannot be freed the problem is unsolvable STAIRS 2008Pavel Surynek bi-connected component BjBj BiBi r1r1

Algorithm: solving problem within bi-connected component Observation: Bi-connected component can be constructed from a cycle by adding loops Observation: Bi-connected component can be constructed from a cycle by adding loops –We inductively place robots into loops starting with the last loop and proceeding to the original cycle –By placing robots in a loop we obtain the smaller problem –For the last cycle we need two unoccupied vertices STAIRS 2008Pavel Surynek r3r3 r4r4 r5r5 r1r1 r2r2 2-connected remainder r6r6 ? ?? ? ? ? ? ? r1r1 ? ? ? ?? r1r1 ? ? r2r2 ? ? ?

Complexity analysis Decomposition of the graph into bi-connected components: O(|V|+|E|) Decomposition of the graph into bi-connected components: O(|V|+|E|) Moving robots into goal bi-connected components: O(|V| 3 ) Moving robots into goal bi-connected components: O(|V| 3 ) Solving problem in the individual bi-connected component: O(|V| 3 ) Solving problem in the individual bi-connected component: O(|V| 3 ) The algorithm requires O(|V| 3 ) steps in total The algorithm requires O(|V| 3 ) steps in total STAIRS 2008Pavel Surynek

Conclusion and remarks We proposed a polynomial-time solving algorithm for multiple robots path planning problem in the case where there are at least two unoccupied vertices We proposed a polynomial-time solving algorithm for multiple robots path planning problem in the case where there are at least two unoccupied vertices The work on the specialized domain-dependent algorithm is partially motivated by inefficiency of domain-independent planners on the problem The work on the specialized domain-dependent algorithm is partially motivated by inefficiency of domain-independent planners on the problem Currently we are working on the generalization of the approach - some theoretical results are already known (Wilson, 1973) Currently we are working on the generalization of the approach - some theoretical results are already known (Wilson, 1973) STAIRS 2008Pavel Surynek