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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

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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 14th 2017

SLIDE 2

Outline

Context and motivation Progress sets Experimental results

Rousse, Meyer & Dimarogonas Non deterministic motion planning 2/15

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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

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Discrete representation

fire water

bstacle

Partition of the environment

◮ relevant cells labeled with water, fire and obstacle ◮ other cells unlabeled

Rousse, Meyer & Dimarogonas Non deterministic motion planning 4/15

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Discrete representation

fire water

bstacle

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

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High-level specifications

fire water

bstacle

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

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Control synthesis approach

Robotic system High level specification (Linear Temporal Logic formula) Controller B¨ uchi Automaton Abstract model (Finite Transition System) Product Automaton Search algorithm

Rousse, Meyer & Dimarogonas Non deterministic motion planning 7/15

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Control synthesis approach

Robotic system High level specification (Linear Temporal Logic formula) Controller B¨ uchi Automaton Abstract model (Finite Transition System) Product Automaton Search algorithm

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

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Outline

Context and motivation Progress sets Experimental results

Rousse, Meyer & Dimarogonas Non deterministic motion planning 8/15

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Progress set: intuition

i0 i1 g0 g1

Rousse, Meyer & Dimarogonas Non deterministic motion planning 9/15

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Progress set: intuition

i0 i1 g0 g1

Possible transitions from i0 with control input

Rousse, Meyer & Dimarogonas Non deterministic motion planning 9/15

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Progress set: intuition

i0 i1 g0 g1

Possible transitions from i1 with control input

Rousse, Meyer & Dimarogonas Non deterministic motion planning 9/15

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Progress set: intuition

i0 i1 g0 g1

Possible infinite behavior on the transition system: i0 → i1 → i0 → i1 → i0 → i1 → . . .

Rousse, Meyer & Dimarogonas Non deterministic motion planning 9/15

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Progress set: intuition

i0 i1 g0 g1

This infinite behavior may not be actually feasible by the continuous dynamics if i0 and i1 are considered together

Rousse, Meyer & Dimarogonas Non deterministic motion planning 9/15

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Progress set: intuition

i0 i1 g0 g1

{(i0, ),(i1, )}

g0 g1

Progress set: set {(q1, u1), . . . , (qm, um)} ∈ 2Q×U of pairs (state,control) whose combined action is guaranteed to eventually leave the corresponding set of states {q1, . . . , qm}.

Nilsson and Ozay, Incremental synthesis of switching protocols via abstraction refinement, CDC 2014. Rousse, Meyer & Dimarogonas Non deterministic motion planning 9/15

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Control synthesis approach

Robotic system High level specification (Linear Temporal Logic formula) Controller B¨ uchi Automaton Finite Transition System augmented with progress sets Product Automaton Search algorithm

◮ 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

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Search algorithm using progress sets

x2 x1 G G init

Expand goal set G with the progress sets guaranteed to go to G Goal set: G

Rousse, Meyer & Dimarogonas Non deterministic motion planning 11/15

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Search algorithm using progress sets

x2 x1 G q1 G init

Expand goal set G with the progress sets guaranteed to go to G Goal set: G∪q1 = G

Rousse, Meyer & Dimarogonas Non deterministic motion planning 11/15

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Search algorithm using progress sets

x2 x1 G q1 q2 q3 q4 G init

Expand goal set G with the progress sets guaranteed to go to G Goal set: G∪q2 ∪ q3 ∪ q4 = G

Rousse, Meyer & Dimarogonas Non deterministic motion planning 11/15

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Search algorithm using progress sets

x2 x1 G q1 q2 q3 q4 q5 G init

Expand goal set G with the progress sets guaranteed to go to G Goal set: G∪q5

Rousse, Meyer & Dimarogonas Non deterministic motion planning 11/15

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Search algorithm using progress sets

x2 x1 G q1 q2 q3 q4 q5 q6 G init

Expand goal set G with the progress sets guaranteed to go to G Goal set: G∪q5∪q6

Rousse, Meyer & Dimarogonas Non deterministic motion planning 11/15

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Search algorithm using progress sets

x2 x1 G q1 q2 q3 q4 q5 q6 q7 G init

Expand goal set G with the progress sets guaranteed to go to G Goal set: G∪q5∪q6∪q7

Rousse, Meyer & Dimarogonas Non deterministic motion planning 11/15

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Search algorithm using progress sets

x2 x1 G q1 q2 q3 q4 q5 q6 q7 q8 G init

Expand goal set G with the progress sets guaranteed to go to G Goal set: G∪q5∪q6∪q7∪q8 Algorithm terminates: init ⊆ G∪q5 ∪q6 ∪q7 ∪q8

Rousse, Meyer & Dimarogonas Non deterministic motion planning 11/15

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Outline

Context and motivation Progress sets Experimental results

Rousse, Meyer & Dimarogonas Non deterministic motion planning 12/15

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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

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Experiment - video

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

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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

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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 = {(q0, u0), (q1, u1), ..., (qn, un)} ∈ 2Q×U ◮ get the set of controls U = {u0, u1, ..., un} involved in p ◮ compute the convex hull CU of U