Using progress sets on non-deterministic transition systems for - PowerPoint PPT Presentation

Using progress sets on non-deterministic transition systems for multiple UAV motion planning Paul Rousse, Pierre-Jean Meyer , Dimos Dimarogonas KTH, Royal Institute of Technology July 14 th 2017

Outline Context and motivation Progress sets Experimental results Rousse, Meyer & Dimarogonas Non deterministic motion planning 2/15

Motivation example High-level control objective: ◮ get some water to be dropped on the fire ◮ while avoiding the obstacle Rousse, Meyer & Dimarogonas Non deterministic motion planning 3/15

Discrete representation fire obstacle water Partition of the environment ◮ relevant cells labeled with water, fire and obstacle ◮ other cells unlabeled Rousse, Meyer & Dimarogonas Non deterministic motion planning 4/15

Discrete representation fire obstacle water Model the system as a Finite Transition System Non-determinism caused by: ◮ disturbances ◮ unknown initial state within a cell Rousse, Meyer & Dimarogonas Non deterministic motion planning 5/15

High-level specifications fire obstacle water Linear Temporal Logic formula ◮ ϕ = ( � ✸ water ) ∧ ( � ✸ fire ) ∧ ( � ¬ obstacle ) ◮ goal: find a controller such that the formula is satisfied by the sequence of labels generated by the controlled system ◮ label sequence: {∅} , ..., {∅} , { water } , {∅} , ..., {∅} , { fire } , ... Rousse, Meyer & Dimarogonas Non deterministic motion planning 6/15

Control synthesis approach Abstract model Robotic system (Finite Transition System) Product Search Controller algorithm Automaton High level specification B¨ uchi Automaton (Linear Temporal Logic formula) Rousse, Meyer & Dimarogonas Non deterministic motion planning 7/15

Control synthesis approach Abstract model Robotic system (Finite Transition System) Product Search Controller algorithm Automaton High level specification B¨ uchi Automaton (Linear Temporal Logic formula) For non-deterministic transition systems, finding a controller may not be possible. ◮ Consider augmented transition systems with progress sets that represent guarantees of progress towards satisfaction of the specification Rousse, Meyer & Dimarogonas Non deterministic motion planning 7/15

Outline Context and motivation Progress sets Experimental results Rousse, Meyer & Dimarogonas Non deterministic motion planning 8/15

Progress set: intuition g 0 g 1 g 0 g 1 i 0 i 1 i 0 i 1 Rousse, Meyer & Dimarogonas Non deterministic motion planning 9/15

Progress set: intuition g 0 g 1 g 0 g 1 i 0 i 1 i 0 i 1 Possible transitions from i 0 with control input Rousse, Meyer & Dimarogonas Non deterministic motion planning 9/15

Progress set: intuition g 0 g 1 g 0 g 1 i 0 i 1 i 0 i 1 Possible transitions from i 1 with control input Rousse, Meyer & Dimarogonas Non deterministic motion planning 9/15

Progress set: intuition g 0 g 1 g 0 g 1 i 0 i 1 i 0 i 1 Possible infinite behavior on the transition system: i 0 → i 1 → i 0 → i 1 → i 0 → i 1 → . . . Rousse, Meyer & Dimarogonas Non deterministic motion planning 9/15

Progress set: intuition g 0 g 1 g 0 g 1 i 0 i 1 i 0 i 1 This infinite behavior may not be actually feasible by the continuous dynamics if i 0 and i 1 are considered together Rousse, Meyer & Dimarogonas Non deterministic motion planning 9/15

Progress set: intuition g 0 g 1 g 0 g 1 { ( i 0 , ) , ( i 1 , ) } i 0 i 1 Progress set : set { ( q 1 , u 1 ) , . . . , ( q m , u m ) } ∈ 2 Q × U of pairs (state,control) whose combined action is guaranteed to eventually leave the corresponding set of states { q 1 , . . . , q m } . Nilsson and Ozay, Incremental synthesis of switching protocols via abstraction refinement , CDC 2014. Rousse, Meyer & Dimarogonas Non deterministic motion planning 9/15

Control synthesis approach Finite Transition System Robotic system augmented with progress sets Product Search Controller algorithm Automaton High level specification B¨ uchi Automaton (Linear Temporal Logic formula) ◮ Provided: augmented transition system with progress sets ◮ Main contribution : how to use these progress sets for the control synthesis Rousse, Meyer & Dimarogonas Non deterministic motion planning 10/15

Search algorithm using progress sets x 2 G Expand goal set G with G the progress sets guaranteed to go to G x 1 Goal set : init G Rousse, Meyer & Dimarogonas Non deterministic motion planning 11/15

Search algorithm using progress sets x 2 G Expand goal set G with G the progress sets guaranteed to go to G q1 x 1 Goal set : init G ∪ q 1 = G Rousse, Meyer & Dimarogonas Non deterministic motion planning 11/15

Search algorithm using progress sets x 2 G Expand goal set G with q3 q4 G the progress sets guaranteed to go to G q1 q2 x 1 Goal set : init G ∪ q 2 ∪ q 3 ∪ q 4 = G Rousse, Meyer & Dimarogonas Non deterministic motion planning 11/15

Search algorithm using progress sets x 2 G Expand goal set G with q3 q4 G the progress sets guaranteed to go to G q1 q2 x 1 Goal set : init G ∪ q 5 q5 Rousse, Meyer & Dimarogonas Non deterministic motion planning 11/15

Search algorithm using progress sets x 2 G Expand goal set G with q3 q4 G the progress sets guaranteed to go to G q1 q2 x 1 Goal set : init G ∪ q 5 ∪ q 6 q5 q6 Rousse, Meyer & Dimarogonas Non deterministic motion planning 11/15

Search algorithm using progress sets x 2 G Expand goal set G with q3 q4 G the progress sets guaranteed to go to G q1 q2 x 1 Goal set : init G ∪ q 5 ∪ q 6 ∪ q 7 q5 q7 q6 Rousse, Meyer & Dimarogonas Non deterministic motion planning 11/15

Search algorithm using progress sets x 2 G Expand goal set G with the progress sets q3 q4 guaranteed to go to G G Goal set : q1 q2 G ∪ q 5 ∪ q 6 ∪ q 7 ∪ q 8 x 1 init Algorithm terminates: q5 init ⊆ G ∪ q 5 ∪ q 6 ∪ q 7 ∪ q 8 q8 q7 q6 Rousse, Meyer & Dimarogonas Non deterministic motion planning 11/15

Outline Context and motivation Progress sets Experimental results Rousse, Meyer & Dimarogonas Non deterministic motion planning 12/15

Experiment System constituted of 2 quadrotors: quad 1 and quad 2 Control objective: surveillance and safety ◮ ϕ = ( � ✸ a ) ∧ ( � ✸ b ) ∧ ( � ¬ collide ) ∧ ( � ¬ out ) Rousse, Meyer & Dimarogonas Non deterministic motion planning 13/15

Experiment - video Rousse, Meyer & Dimarogonas Non deterministic motion planning 14/15

Conclusion Proposed approach ◮ Consider non-deterministic finite transition systems augmented with progress sets ◮ Propose a new planning algorithm for the control synthesis under LTL specifications Future work ◮ Computation of progress sets for general classes of systems Rousse, Meyer & Dimarogonas Non deterministic motion planning 15/15

Progress set identification Single integrator system with disturbances: ◮ ˙ x = u + w Abstract continuous dynamics into a Finite Transition System ◮ Q : finite set of states ◮ U : finite set of control inputs Check a candidate progress set ◮ candidate: p = { ( q 0 , u 0 ) , ( q 1 , u 1 ) , ..., ( q n , u n ) } ∈ 2 Q × U ◮ get the set of controls U = { u 0 , u 1 , ..., u n } involved in p ◮ compute the convex hull C U of U p is a progress set ⇔ 0 / ∈ C U Rousse, Meyer & Dimarogonas Non deterministic motion planning 16/15