ONR MURI AIRFOILS: Animal Inspired Robust Flight with Outer and Inner Loop Strategies. Calin Belta

Transcription

1 ONR MURI AIRFOILS: Animal Inspired Robust Flight with Outer and Inner Loop Strategies Provable safety for animal inspired agile flight Calin Belta Hybrid and Networked Systems (HyNeSs) Lab Department of Mechanical Engineering Boston University

2 Hybrid Automaton (Thrust III) Outer-loop low-bandwidth navigation principles (Thrust II) q q dx = f q (x, u)dt + g q (x, u)dw q y(t) =h q (x)+v q q Inner-loop high-bandwidth modes (Thrust I) Two main goals - formal verification from user-defined safety specifications - control synthesis from animal-inspired navigation principles (Thrust II) and user-defined motion and safety specifications Specs: always avoid collisions, the probability of flying at speeds larger than x at altitudes lower than y should be lower than 0.1,

3 Approach: formal verification Model checking P max=? P 4 ((P 1 P 2 ) P 3 ) Property-based abstraction P1 P2 P4 ẋ = f q (x) y = h q (x) P3 Always avoid P4. Visit P1 or P2 and then P3 infinitely often.

4 Approach: formal verification Probabilistic model checking P max=? P 4 ((P 1 P 2 ) P 3 ) Property-based abstraction P1 P2 P4 dx = f q (x)dt + g q (x)dw q y(t) =h q (x)+v q P3 What is the probability of: Always avoid P4. Visit P1 or P2 and then P3 infinitely often.

5 Approach: formal synthesis Control strategy P max=? P 4 ((P 1 P 2 ) P 3 ) refinement Bisimulation, language equivalence abstraction Control strategy P1 P2 P4 ẋ = f q (x, u) y = h q (x) P3 Always avoid P4. Visit P1 or P2 and then P3 infinitely often.

6 Approach: formal synthesis Control strategy P max=? P 4 ((P 1 P 2 ) P 3 ) refinement Bisimulation, language equivalence abstraction Control strategy P1 P2 P4 dx = f q (x, u)dt + g q (x, u)dw q y(t) =h q (x)+v q P3 Satisfy with max probability: Always avoid P4. Visit P1 or P2 and then P3 infinitely often.

7 Outline Formal verification for continuous and hybrid systems Main idea Preliminary results Proposed work (3 year plan) Formal synthesis for continuous and hybrid systems Main idea Preliminary results Proposed work (3 year plan) Plan for the next year

8 Outline Formal verification for continuous and hybrid systems Main idea Preliminary results Proposed work (3 year plan) Formal synthesis for continuous and hybrid systems Main idea Preliminary results Proposed work (3 year plan) Plan for the next year

9 Formal verification for continuous and hybrid systems: main idea

10 Formal verification for continuous and hybrid systems: main idea Is there a trajectory reaching from green to red? green red

11 Formal verification for continuous and hybrid systems: main idea Very hard problem: - Numerical integration does not work (infinite sets) - Checking the emptiness of the intersection of the image of the green set with the red set is in general an undecidable problem (e.g., t :(e At S 1 ) S 2 is undecidable!) Is there a trajectory reaching from green to red? green red

12 Formal verification for continuous and hybrid systems: main idea Is there a trajectory reaching from green to red? green red

13 Formal verification for continuous and hybrid systems: main idea 1 st iteration Is there a trajectory reaching from green to red? Is there a trajectory reaching from green to red? green red green red

14 Formal verification for continuous and hybrid systems: main idea 1 st iteration Is there a trajectory reaching from green to red? Is there a trajectory reaching from green to red? green red green red

15 Formal verification for continuous and hybrid systems: main idea 2nd iteration Is there a trajectory reaching from green to red? Is there a trajectory reaching from green to red? green red green red

16 Formal verification for continuous and hybrid systems: main idea Bisimulation algorithm While there exist states X i, X j in the quotient such that remove X i from the set of states of the quotient add the states X i,1 and X i,2 Challenges: Set representation Computation of image and pre-image of sets through flows of vector fields Intersection and feasibility of sets

17 Outline Formal verification for continuous and hybrid systems Main idea Preliminary results Proposed work (3 year plan) Formal synthesis for continuous and hybrid systems Main idea Preliminary results Proposed work (3 year plan) Plan for the next year

18 Formal verification for continuous and hybrid systems: prelim results Approach: exploit properties induced by the geometry of sets and the structure of the vector fields ẋ = A q x + b q x P q,q Q ẋ = i 1,...,i n {0,1} c q i 1,...,i n x i x i n n x P q,q Q x k+1 = A l x k + b l x k P q,q Q

19 Formal verification for continuous and hybrid systems: prelim results Multi-affine dynamics Newton-Euler Equations m v = mv ω G ω = Gω ω ẋ = i 1,...,i n {0,1} c q i 1,...,i n x i x i n n

20 Formal verification for continuous and hybrid systems: prelim results Reachability computation for multi-affine vector fields

21 Formal verification for continuous and hybrid systems: prelim results Temporal logic analysis of continuous-space discrete-time PWA systems Region from which all trajectories satisfy Region from which all trajectories satisfy Find the maximal region such that all trajectories originating from it satisfy Find the maximal region such that all trajectories originating from it satisfy

22 Outline Formal verification for continuous and hybrid systems Main idea Preliminary results Proposed work (3 year plan) Formal synthesis for continuous and hybrid systems Main idea Preliminary results Proposed work (3 year plan) Plan for the next year

23 Formal verification for continuous and hybrid systems: proposed work Proposed work (3 year plan) Verification for piecewise affine systems with noise in continuous time; specs given as LTL and CTL formulas over linear predicates in the state dx = A q xdt + b q dt + G q dw x P q,q Q Verification for piecewise affine systems with noise in discrete time; specs given as LTL and CTL formulas over linear predicates in the state x k+1 = A q x k + b q + G q w k x k P q,q Q Verification for polynomial hybrid systems with rectangular partitions in continuous time ẋ = f q (x) x R q,q Q Verification for polynomial hybrid systems with rectangular partitions and noise in continuous time dx = f q (x)dt + g q (x)dw x R q,q Q

24 Outline Formal verification for continuous and hybrid systems Main idea Preliminary results Proposed work (3 year plan) Formal synthesis for continuous and hybrid systems Main idea Preliminary results Proposed work (3 year plan) Plan for the next year

25 Formal synthesis for continuous and hybrid systems: main idea state Feedback automaton control Refinement Language equivalence Abstraction region Feedback hybrid automaton Feedback controller Avoid the grey region for all times. Visit the blue region, then the green region, and then keep surveying the striped blue and green regions, in this order.

26 Formal synthesis for continuous and hybrid systems: main idea state Feedback automaton control Refinement Language equivalence Abstraction region Feedback hybrid automaton Feedback controller Avoid the grey region for all times. Visit the blue region, then the green region, and then keep surveying the striped blue and green regions, in this order.

27 Outline Formal verification for continuous and hybrid systems Main idea Preliminary results Proposed work (3 year plan) Formal synthesis for continuous and hybrid systems Main idea Preliminary results Proposed work (3 year plan) Plan for the next year

28 Formal synthesis for continuous and hybrid systems: prelim results state Feedback automaton control Refinement Language equivalence Abstraction region Feedback hybrid automaton Feedback controller Avoid the grey region for all times. Visit the blue region, then the green region, and then keep surveying the striped blue and green regions, in this order.

29 Formal synthesis for continuous and hybrid systems: prelim results Library of controllers for polytopes polyhedral Control-to-facet in simplex Stay-inside simplex Control-to-set-offacets in simplex Control-to-facet in polytope Stay-inside polytope Control-to-facet in rectangle Stay-inside rectangle Control-to-set-of-facets in rectangle checking for existence of controllers amounts to checking the non-emptiness of polyhedral sets in U if controllers exist, they can be constructed everywhere in the polytopes by using simple formulas

30 Formal synthesis for continuous and hybrid systems: prelim results Multi-affine dynamics Controlled Newton-Euler Equations e.g.,aircraft with gas-jet actuators, quadrotors, etc.

31 Formal synthesis for continuous and hybrid systems: prelim results state Feedback automaton control Refinement Language equivalence Abstraction region Feedback hybrid automaton Feedback controller Avoid the grey region for all times. Visit the blue region, then the green region, and then keep surveying the striped blue and green regions, in this order.

32 Formal synthesis for continuous and hybrid systems: prelim results Formal synthesis of control strategies for finite systems Control Observation u 1 Deterministic (D) u 2 u 1 Nondeterministic (N) u u Probabilistic (P) 0.7 u 2

33 Formal synthesis for continuous and hybrid systems: prelim results Formal synthesis of control strategies for finite systems Control Observation u 1 Deterministic (D) u 2 u 1 Nondeterministic (N) u u Probabilistic (P) 0.7 u 2 DD: LTL control strategies can be found by adapting existing LTL model checkers

34 Formal synthesis for continuous and hybrid systems: prelim results Formal synthesis of control strategies for finite systems Control Observation u 1 Deterministic (D) u 2 u 1 Nondeterministic (N) u u Probabilistic (P) 0.7 u 2 ND - LTL feedback control strategies can be found for an LTL fragment generated be deterministic Buchi automata

35 Formal synthesis for continuous and hybrid systems: prelim results Formal synthesis of control strategies for finite systems Control Observation u 1 Deterministic (D) u 2 u 1 Nondeterministic (N) u u Probabilistic (P) 0.7 u 2 ND - LTL feedback control strategies can be found for full LTL

36 Formal synthesis for continuous and hybrid systems: prelim results Formal synthesis of control strategies for finite systems Control Observation u 1 Deterministic (D) u 2 u 1 Nondeterministic (N) u u Probabilistic (P) 0.7 u 2 PD (Markov Decision Process, MDP) Feedback control strategies that maximize the probability of satisfying pctl formulas Feedback control strategies that maximize the probability of satisfying pltl formulas

37 Formal synthesis for continuous and hybrid systems: prelim results Formal synthesis of control strategies for finite systems Control Observation u 1 Deterministic (D) u 2 u 1 Nondeterministic (N) u u Probabilistic (P) 0.7 u 2 NN - feedback control strategies can be found for full LTL

38 Formal synthesis for continuous and hybrid systems: prelim results Formal synthesis of control strategies for finite systems Control Observation u 1 Deterministic (D) u 2 u 1 Nondeterministic (N) u u Probabilistic (P) 0.7 u 2 PP - POMDP open question (part of this project)

39 Outline Formal verification for continuous and hybrid systems Main idea Preliminary results Proposed work (3 year plan) Formal synthesis for continuous and hybrid systems Main idea Preliminary results Proposed work (3 year plan) Plan for the next year

40 Formal synthesis for continuous and hybrid systems: proposed work Proposed work (3 year plan) Establish the connection between actuator / sensor placement and computability of finite quotients for multi-affine and polynomial systems in continuous time (collaboration with Morgansen) ẋ = g(x)+bu y = h(x, u) Abstraction for control of continuous-time stochastic hybrid systems: piecewise affine, piecewise multi-affine, and polynomial dx = f q (x, u)dt + g q (x, u)dw q Control strategies for large MDPs and MDPs with uncertain parameters (collaboration with Paschalidis) Control strategies for POMDPs

41 Outline Formal verification for continuous and hybrid systems Main idea Preliminary results Proposed work (3 year plan) Formal synthesis for continuous and hybrid systems Main idea Preliminary results Proposed work (3 year plan) Plan for the next year

42 Plan for the next year Verification for piecewise affine systems with noise in continuous time; specs given as LTL and CTL formulas over linear predicates in the state dx = A q xdt + b q dt + G q dw x P q,q Q Verification for piecewise affine systems with noise in discrete time; specs given as LTL and CTL formulas over linear predicates in the state x k+1 = A q x k + b q + G q w k x k P q,q Q Establish the connection between actuator placement and computability of finite quotients for multi-affine systems in continuous time (collaboration with Morgansen) ẋ = g(x)+bu Control strategies for large MDPs (collaboration with Paschalidis)

43 Acknowledgements NSF CNS , CNS , CMMI , CNS ARO W911NF ONR MURI N AFOSR YIP FA