Op#mal Control of Nonlinear Systems with Temporal Logic Specifica#ons

Transcription

1 Op#mal Control of Nonlinear Systems with Temporal Logic Specifica#ons Eric M. Wolff 1 Ufuk Topcu 2 and Richard M. Murray 1 1 Caltech and 2 UPenn University of Michigan October 1, 2013

2 Autonomous Systems in the Field 2

3 Mo#va#on How do we specify tasks for autonomous systems? How do we compute op#mal solu#ons? How do we handle high- dimensional con#nuous dynamics? 3

4 An Example Problem Spec: Avoid obstacles, pick- up supplies at region A and then do surveillance on regions B and C. 4

5 Main Contribu#ons Trajectory genera#on techniques for high- dimensional (10+ dim) and nonlinear systems with temporal logic specifica#ons Automata- guided temporal logic planning Wolff and Murray [ISRR 2013] Mixed- integer linear encoding of LTL Wolff, Topcu, Murray [IROS 2013, ICRA sub] Improve on discrete abstrac#on techniques 5

6 Related Work Discrete abstrac#ons (Alur00, Belta06, Habets06, Kloetzer08, Pappas06, Tabuada06, Wongpiromsarn10) Low dimensional systems (<= 6) Mathema#cal programming: Constrained trajectory genera#on (Bemporad99, Earl06, Richards02) Finite- horizon LTL proper#es (Karaman08, Kwon08) Simple tasks 6

7 These Problems are Hard! Dynamic constraints - > undecidable Task specifica#on - > PSPACE 7

8 Outline Preliminaries System model Linear temporal logic (LTL) Automata- guided temporal logic planning Constrained reachability Examples Mixed- integer linear encoding of LTL A finite- dimensional encoding Examples 8

9 Outline Preliminaries System model Linear temporal logic (LTL) Automata- guided temporal logic planning Constrained reachability Examples Mixed- integer linear encoding of LTL A finite- dimensional encoding Examples 9

10 System Model Discrete- #me nonlinear system x t+1 = f(x t, u t ) x X R n u U R m Labels L : X 2 AP Trajectory: x = x(x 0,u) = x 0 x 1 x 2 x i+1 = f(x i,u) for some u U for i = 0,1, Word: L(x) = L(x 0 )L(x 1 )L(x 2 ) 10

11 Linear Temporal Logic (LTL) Want to specify properties such as: Response: always SIGNAL after a REQUEST arrives Liveness: always eventually PICKUP Safety: always remain SAFE Priority: do JOB1 until JOB2 Guarantee: eventually reach GOAL Linear temporal logic (LTL): A logic for reasoning about how properties change over time Reason about infinite sequences σ = s 0 s 1 s 2... of states Propositional logic: (and), (or), (implies), (not) Temporal operators: U (until), (next), (always), (eventually) 11

12 Linear Temporal Logic (LTL) Want to specify properties such as: Response: ( REQUEST SIGNAL ) Liveness: PICKUP Safety: SAFE Priority: JOB1 U JOB2 Guarantee: GOAL Linear temporal logic (LTL): A logic for reasoning about how properties change over time Reason about infinite sequences σ = s 0 s 1 s 2... of states Propositional logic: (and), (or), (implies), (not) Temporal operators: U (until), (next), (always), (eventually) 12

13 Outline Preliminaries System model Linear temporal logic (LTL) Automata- guided temporal logic planning Constrained reachability Examples Mixed- integer linear encoding of LTL A finite- dimensional encoding Examples 13

14 Problem Overview Spec: Avoid obstacles, pick- up supplies at region A and then do surveillance on regions B and C. 14

15 From Logic to Automaton Informal spec: Avoid obstacles, pick- up supplies at region A and then do surveillance on regions B and C. LTL Specifica#on Büchi Automaton ϕ = <> A & []<> B & []<> C & [] S Automa#c transla#on from logic to automaton! GasMn,Oddoux: hqp:// cachan.fr/~gas#n/ltl2ba/ 15

16 Problem Statement Given: a determinis#c nonlinear dynamical system, ini#al state x 0, a Büchi automaton A (specifica#on) Goal: Find a control input sequence u such that the word L(x(x 0,u)) is accepted by A 16

17 Solu#on Main idea: Use specifica#on automaton to guide constrained reachability computa#ons Compute a word (from the trajectory) that is accepted by the automaton 17

18 Solu#on Environment Logic Automaton Spec: ϕ = <> A & []<> B & []<> C & [] S 18

19 Constrained Reachability Given: sets X 1, X 2 X, horizon length N Goal: Find a control input sequence u and a horizon length N such x 1,...,x N 1 X 1, x N X 2 such that x t+1 =f(x t,u t ) for t = 1,...,N 1 CstReach(X 1,X 2 ) 19

20 Solving Constrained Reachability CstReach(X 1,X 2 ) can be encoded as a mixed- integer linear program using the big- M formula#on Big- M Enforce that state is in union of polyhedra H i x <= K i + M(1- z i ), z i {0,1}, sum(z) = 1 Independent of dynamics 20

21 Examples Systems Quadrotor (10 dim) Chained integrators (4, 12, 20 dim) Car- like robot (nonlinear + driw) Specifica#ons Visit n goals Repeatedly visit n goals 21

22 Examples Model: 10- dim quadrotor Spec: ϕ = <> D1 & <> D2 & <> D3 & <> D4 & [] safe 22

23 Eventually (solver) 10/1/13 23

24 Eventually (total) 10/1/13 24

25 Repeatedly (solver) 10/1/13 25

26 Review (1/2) Environment Logic Automaton Spec: ϕ = <> A & []<> B & []<> C & [] S 26

27 Future Work (1/2) Stochas#c constrained reachability [Horowitz, Wolff, Murray ACC14- sub] Improved composi#on of subproblems Improved heuris#cs 27

28 Outline Preliminaries System model Linear temporal logic (LTL) Automata- guided temporal logic planning Constrained reachability Examples Mixed- integer linear encoding of LTL A finite- dimensional encoding Examples 28

29 Problem Descrip#on Discrete- #me nonlinear systems Piecewise affine Differen#ally flat Small- #me locally controllable min J(x(x 0,u)) s.t. x(x 0,u) = ϕ Cost func#on J Task specifica#on ϕ 29

30 A Fragment of LTL Core ϕ safe = ψ ϕ goal = ψ ϕ per = ψ ϕ live = ψ 30

31 A Fragment = of LTL Core ϕ safe = ψ ϕ goal = ψ ϕ per = ψ ϕ live = ψ Response ϕ 1 resp = (ψ φ) ϕ 2 resp = (ψ φ) ϕ 3 resp = (ψ φ) ϕ 4 resp = (ψ φ) 31

32 A Fragment = of LTL Core Response Fairness ϕ safe = ψ ϕ goal = ψ ϕ per = ψ ϕ live = ψ ϕ 1 resp = (ψ φ) ϕ 2 resp = (ψ φ) ϕ 3 resp = (ψ φ) ϕ 4 resp = (ψ φ) ϕ 1 fair = ψ m φ j j=1 ϕ 2 fair = ψ m φ j j=1 ϕ 3 fair = ψ m φ j j=1 32

33 A Fragment = of LTL Core Response Fairness ϕ safe = ψ ϕ goal = ψ ϕ per = ψ ϕ live = ψ ϕ 1 resp = (ψ φ) ϕ 2 resp = (ψ φ) ϕ 3 resp = (ψ φ) ϕ 4 resp = (ψ φ) ϕ 1 fair = ψ m φ j j=1 ϕ 2 fair = ψ m φ j j=1 ϕ 3 fair = ψ m φ j j=1 What is missing? Nested temporal operators Nega#ons 33

34 A Fragment = of LTL Core Response Fairness ϕ safe = ψ ϕ goal = ψ ϕ per = ψ ϕ live = ψ ϕ 1 resp = (ψ φ) ϕ 2 resp = (ψ φ) ϕ 3 resp = (ψ φ) ϕ 4 resp = (ψ φ) ϕ 1 fair = ψ m φ j j=1 ϕ 2 fair = ψ m φ j j=1 ϕ 3 fair = ψ m φ j j=1 Recently extended to all of LTL [Wolff,Topcu,Murray ICRA14- sub] 34

35 Finite Parameteriza#on of Trajectory Let x = x pre (x suf ) ω x pre and x suf are finite x suf is a cycle Labels are disjunc#ons of polytopes Use a binary variable for each polyhedron every stage Big M: H x <= K + M(1- z), z {0,1} Convex hull X pre x suf 35

36 Linking the System and Logic Use a binary variable for each polyhedron every stage Big M: H x <= K + M(1- z), z {0,1} Convex hull Does system sa#sfy label ψ at #me t? P ψ t = i I ψ t z ψ i t 36

37 P ψ t, th Encoding LTL Constraints T = i I ψ t z ψ i t ϕ safe = ψ ϕ goal = ψ ϕ per = ψ P ψ t 1 t T pre, P ψ t 1 t T suf. t T pre P ψ t + t T suf P ψ t 1. P ψ t 1 t T suf ϕ live = ψ P ψ t 1. t T = suf 37

38 Add all constraints (dynamics + LTL) and solve MILP using off- the- shelf sowware NP- complete Branch + bound Solvers work well in prac#ce With H polyhedrons and T #me steps Safety = H T Goal = HT Complexity ϕ safe = ψ ϕ goal = ψ 38

39 Examples Systems Quadrotor (10- dim) Chained integrators Car- like robot Solu#on Simultaneous (Sim) Sequen#al (Seq) P D2 D1 Integrator (20 dim) SoluMon in 6 sec. 39

40 Examples Quadrotor Car- like robot 40

41 Quadrotor Example 10 dimensional quadrotor Linearized about hover (with fixed yaw) Feasible solu#ons found in seconds Cost func#on rewards staying lew 10/1/13 41

42 Results Feasible soln. (sec) Num. solved Model Dim. Sim. Seq. Sim. Seq. chain ± ± chain ± ± chain ± ±

43 Results Feasible soln. (sec) Num. solved Model Dim. Sim. Seq. Sim. Seq. chain ± ± chain ± ± chain ± ± quadrotor ±.66 ± 1.80 ±.15 ± quadrotor-flat ± ±

44 Results Feasible soln. (sec) Num. solved Model Dim. Sim. Seq. Sim. Seq. chain ± ± chain ± ± ± ± chain ± ± ± ± quadrotor ± ± quadrotor-flat ± ± car ± ± car ± ± car-flat ± ±

45 Improvement Over Abstrac#ons Prior Randomly generated `gridworld problems Beqer performance than finite abstrac#ons 45 Us

46 Future Work (2/2) Disturbances and reac#vity (RHC) Timed specifica#ons Using discrete solu#ons to simplify op#miza#on 46

47 Conclusions Two new LTL planning methods Automata- guided temporal logic planning [ISRR2013] Mixed- integer encoding of LTL [IROS 2013, ICRA sub] High dimensional and nonlinear systems Op#mal control 47

48 Conclusions Two new LTL planning methods Automata- guided temporal logic planning [ISRR2013] Mixed- integer encoding of LTL [IROS 2013, ICRA sub] High dimensional and nonlinear systems Op#mal control We can solve high dimensional, nonlinear LTL mo#on planning problems that were previously impossible. 48

49 Thank you! Contact: Eric M. Wolff Web: Funding: NDSEG fellowship, Boeing, AFOSR 49