Decomposition of planning for multi-agent systems under LTL specifications

Transcription

1 Decomposition of planning for multi-agent systems under LTL specifications Jana Tumova and Dimos V. Dimarogonas KTH Royal Institute of Technology R E C O N F I G December 14, 2015

2 General Motivation and Approach Correct-by-design mission and motion planning for multi-agent systems under complex high-level specifications Decomposition of planning for multi-agent systems under LTL specifications 1/22

3 General Motivation and Approach Correct-by-design mission and motion planning for multi-agent systems under complex high-level specifications Approach Linear Temporal Logic (LTL) as a specification language Expressive power Rigorousness Some resemblance to natural language Hierarchical approach 1. Abstraction of a dynamical system 2. Discrete planning 3. Implementation Decomposition of planning for multi-agent systems under LTL specifications 1/22

4 Challenge: Computational Feasibility of Step 2 Correct-by-design mission and motion planning for multi-agent systems under complex high-level specifications Decomposition of planning for multi-agent systems under LTL specifications 2/22

5 Challenge: Computational Feasibility of Step 2 Correct-by-design mission and motion planning for multi-agent systems under complex high-level specifications Example A straightforward extension of the automata-based approach Decomposition of planning for multi-agent systems under LTL specifications 2/22

6 Related Work: A Top-Down Approach E.g., [CDSB12] Chen, Ding, Stefanescu, Belta, Formal Approach to the Deployment of Distributed Robotic Teams, TRO, Split the team formula into independent ones Works for a subclass of LTL only Decomposition of planning for multi-agent systems under LTL specifications 3/22

7 Aim: A Bottom-Up Approach A team N = {1,..., N} of agents Abstraction of action and motion capabilities: a finite discrete transition system T i Synchronization capabilities High-level behavior specification for each agent Independent motion specification over the states Dependent task specification over the inputs/actions Efficiently synthesize plans fulfilling the tasks A satisfying trace of each T i Necessary synchronizations Decomposition of planning for multi-agent systems under LTL specifications 4/22

8 Model and Specification An agent model A finite discrete transition system Ti = (S i, s init,i, A i, T i, Π i, L i ) α j A transition s j s j+1 has an unknown execution duration Services Σi and a labeling function L i : A i 2 Π i 2 E i E i = {ɛ i } is a special silent service Synchronization requests Synci = {sync i (I ) {i} I N } Behavior β i = (τ i = s i,1 α i,1 s i,2 α i,2..., γ i = r i,1 r i,2..., T i = t si,1 t αi,1...) Strategy (τ i, γ i ) A collection of strategies induce a set of collective behaviors Specification Motion specification: an LTL\X formula φ i over Π i Task specification: an LTL formula ψ i over Σ = i N Σ i Decomposition of planning for multi-agent systems under LTL specifications 5/22

9 Specification Semantics Motion specification: an LTL \X formula φ i over Π i Interpreted on Li (s 1 )L i (s 2 )... of a behavior of a single agent β i = (τ i = s i,1 α i,1 s i,2 α i,2..., γ i = r i,1 r i,2..., T i = t si,1 t αi,1...) Task specification: an LTL formula ψ i over Σ = i N Σ i Special notion of local LTL satisfaction interpreted on a collective behavior B = β 1,..., β N from agent i s viewpoint wτi = L i,ι1 L i,ι2... is the subsequence of non-silent elements of v τi = L i (α i,1 )L i (α i,2 )... Ti (w τi ) = t i,ι1 t i,ι2... is the subsequence of T i corresponding to w τi v τi (t) is the service set that agent i provides at time t wbi = ω i,ι1 ω i,ι2..., where ω i,ιj = ( i N v τ i (t i,ιj ) ) Σ Interpreted on w Bi Decomposition of planning for multi-agent systems under LTL specifications 6/22

10 Local LTL Satisfaction Example Behavior β 1 τ 1 = s 1,1α 1,1s 1,2α 1,2s 1,3α 1,3... γ 1 = sync 1({1, 2})sync 1({1}) sync 1({1, 2})... Behavior β 2 τ 2 = s 2,1α 2,1s 2,2α 2,2s 2,3α 2,3... γ 2 = sync 2({1, 2})sync 2({2}) sync 2({1, 2})... Decomposition of planning for multi-agent systems under LTL specifications 7/22

11 Local LTL Satisfaction Example Behavior β 1 τ 1 = s 1,1α 1,1s 1,2α 1,2s 1,3α 1,3... γ 1 = sync 1({1, 2})sync 1({1}) sync 1({1, 2})... α1,1 = 1, α1,2 = 3,... Behavior β 2 τ 2 = s 2,1α 2,1s 2,2α 2,2s 2,3α 2,3... γ 2 = sync 2({1, 2})sync 2({2}) sync 2({1, 2})... α2,1 = 2, α2,2 = 1,... Decomposition of planning for multi-agent systems under LTL specifications 7/22

12 Local LTL Satisfaction Example Behavior β 1 τ 1 = s 1,1α 1,1s 1,2α 1,2s 1,3α 1,3... γ 1 = sync 1({1, 2})sync 1({1}) sync 1({1, 2})... α1,1 = 1, α1,2 = 3,... T 1 = Behavior β 2 τ 2 = s 2,1α 2,1s 2,2α 2,2s 2,3α 2,3... γ 2 = sync 2({1, 2})sync 2({2}) sync 2({1, 2})... α2,1 = 2, α2,2 = 1,... T 2 = Decomposition of planning for multi-agent systems under LTL specifications 7/22

13 Local LTL Satisfaction Example Behavior β 1 τ 1 = s 1,1α 1,1s 1,2α 1,2s 1,3α 1,3... γ 1 = sync 1({1, 2})sync 1({1}) sync 1({1, 2})... α1,1 = 1, α1,2 = 3,... T 1 = v τ1 = E 1{σ 1,1}{σ 1,1, σ 1,2}... w τ1 = {σ 1,1}{σ 1,1, σ 1,2}... T i (w τ1 ) = w B1 = {σ 1,1}{σ 1,1, σ 1,2, σ 2,1}... Behavior β 2 τ 2 = s 2,1α 2,1s 2,2α 2,2s 2,3α 2,3... γ 2 = sync 2({1, 2})sync 2({2}) sync 2({1, 2})... α2,1 = 2, α2,2 = 1,... T 2 = v τ2 = {σ 2,1}E 2{σ 2,1}... w τ2 = {σ 2,1}{σ 2,1}... T i (w τ1 ) = w B2 = {σ 2,1}{σ 1,1, σ 1,2, σ 2,1}... Decomposition of planning for multi-agent systems under LTL specifications 7/22

14 Example I Agent 1 is a ground vehicle and has to avoid walls and obstacles. Agent 2 and Agent 3 are UAVs and their environment is obstacle-free except for the walls. Motion specifications Agent 1: Keep avoiding R1, φ 1 = G R1 1. Agent 2: Keep avoiding R2, φ 2 = G R2 2. Agent 3: Periodically survey R1 and R2, φ 3 = G F R1 3 G F R2 3. R1 R2 R3 R4 Task specifications Agent 1: periodically load ( ) with the help of agent 2 ( ) and the assistance of agent 3 ( ), then unload ( ) with the help of agent 2 ( ) or the assistance of agent 3 ( ) ψ 1 = load help assist G (load X (unload (help assist))) G (unload X (load help assist)) Agent 2: Periodically provide inform service ( ), ψ 2 = GFinform. Agent 3: Nothing specific, ψ 3 = true. Decomposition of planning for multi-agent systems under LTL specifications 8/22

15 Problem Formulation For each i N, synthesize τ i, γ i so that the set of induced behaviors is nonempty the motion specification φ i is satisfied the task specification ψ i is locally satisfied Decomposition of planning for multi-agent systems under LTL specifications 9/22

16 Our Hierarchical Approach I Each φ i is translated to a Büchi automaton B φ i N motion products P i = T i B φ i are built Each motion product is reduced to P i by systematic removal of states, where no services of interest are available Each ψ i is translated to a Büchi automaton B ψ i N task and motion products P i = P i B ψ i Each motion and task product is reduced to P i by systematic removal of states, where no dependent services are available A global product P = P 1... P N containing only states relevant for planning of dependent tasks is constructed Decomposition of planning for multi-agent systems under LTL specifications 10/22

17 Our Hierarchical Approach II An accepting run in the global product projected onto the original system gives a motion plan a task execution plan a synchronization plan for each agent i, that is correct-by-design with respect to φ i and ψ i. Decomposition of planning for multi-agent systems under LTL specifications 11/22

18 Reduction Step Motion product P i = T i B φ i A state p of P i is significant if it is either the initial state or if there exists a transition (p, σ, p ) δ i, such that σ E i Insignificant states can be iteratively removed and replaced with transitions leading from all its predecessors to all its successors. The new transition is labeled with sequence of actions. Decomposition of planning for multi-agent systems under LTL specifications 12/22

19 Example I Revisited Agent 1 is a ground vehicle and has to avoid walls and obstacles. Agent 2 and Agent 3 are UAVs and their environment is obstacle-free except for the walls. Motion specifications Agent 1: Keep avoiding R1, φ 1 = G R1 1. Agent 2: Keep avoiding R2, φ 2 = G R2 2. Agent 3: Periodically survey R1 and R2, φ 3 = G F R1 3 G F R2 3. 2a R1 R2 R3 R4 2b Task specifications 1 3 Agent 1: periodically load ( ) with the help of agent 2 ( ) and the assistance of agent 3 ( ), then unload ( ) with the help of agent 2 ( ) or the assistance of agent 3 ( ) ψ 1 = load help assist G (load X (unload (help assist))) G (unload X (load help assist)) Agent 2: Periodically provide inform service ( ), ψ 2 = GFinform. Agent 3: Nothing specific, ψ 3 = true. Decomposition of planning for multi-agent systems under LTL specifications 13/22

20 Example I Revisited Centralized approach Each TS: 100 states Product TS: states B φ 1, Bφ 2, Bφ 3, Bψ 1, Bψ 2, Bψ 3 : 2, 2, 3, 2, 2, 1 states, respectively respectively Intersection BA: = 330 states The overall product P: 330 mil. states Our approach: P 1, P 2, P 3 : 200, 200, 300 states, respectively P1, P 2, P 3 : 27,17,8 states, respectively The largest structure handled has cca states. Decomposition of planning for multi-agent systems under LTL specifications 14/22

21 Remarks The worst-case complexity meets the complexity of the centralized solution Suitable for sparsely distributed services of interest and occasional needs for collaboration The bottleneck is still the product P Decomposition of planning for multi-agent systems under LTL specifications 15/22

22 Event-triggered Receding Horizon Approach Each φ i is translated to a Büchi automaton B φ i N motion products P i = T i B φ i are built Each motion product is reduced to P i by systematic removal of states, where no services of interest are available Each ψ i is translated to a Büchi automaton B ψ i N task and motion products P i = P i B ψ i Each motion and task product is reduced to P i by systematic removal of states, where no dependent services are available A global product P = P 1... P N containing only states relevant for planning of dependent tasks is constructed Decomposition of planning for multi-agent systems under LTL specifications 16/22

23 Stepwise Receding Horizon Decomposition of planning for multi-agent systems under LTL specifications 17/22

24 Event-triggered Receding Horizon Decomposition of planning for multi-agent systems under LTL specifications 18/22

25 Example II Agent 1 can load (l H, l A, l B ), carry, and unload (u H, u A, u B ) a heavy object H or a light object A, B, in the green cells. ψ 1 = F(l H h H X u H GF (l i X u i ))) i {A,B} Agent 2 is capable of helping the agent 1 to load object H (h H ), and to execute simple tasks in the purple regions (t 1 t 5 ). ψ 2 = GF (t 1 X (t 2 X (t 3 X (t 4 X t 5 s 4 )))))) Agent 3 is capable of taking a snapshot of the rooms (s 1 s 5 ) when being present in there. ψ 3 = i {2,4,5} GF s i Decomposition of planning for multi-agent systems under LTL specifications 19/22

26 Example II cca 3 mil. vs. hundreds to thousands of states Decomposition of planning for multi-agent systems under LTL specifications 20/22

27 Remarks The worst-case complexity still the same as for the centralized case Suitable for collaborations executed in small (dynamically changing) subgroups Decomposition of planning for multi-agent systems under LTL specifications 21/22

28 Future Work More interesting models and specifications Collision avoidance Decomposition of formulas Thank you! Decomposition of planning for multi-agent systems under LTL specifications 22/22