Synthesis via Sampling-Based Abstractions
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1 Synthesis via Sampling-Based Abstractions Some Problems and Initial Ideas Matthias Rungger 2 Morteza Lahijanian 1 Lydia E Kavraki 1 Paulo Tabuada 2 Moshe Y Vardi 1 1 Department of Computer Science, Rice University 2 Cyber-Physical Systems Laboratory, UCLA
2 Problem statement Given a LTL specification ϕ and a control system S, find a controller C that enforces ϕ on S 2/6
3 2/6 Problem statement Given a LTL specification ϕ and a control system S, find a controller C that enforces ϕ on S Well-known abstraction/refinement approach 1 Compute a finite abstraction Ŝ of S 2 Synthesize controller Ĉ based on Ŝ 3 Refine solution Ĉ to C finite 1 Ŝ S infinite 2 abstract concrete Ĉ C 3
4 2/6 Problem statement Given a LTL specification ϕ and a control system S, find a controller C that enforces ϕ on S Well-known abstraction/refinement approach 1 Compute a finite abstraction Ŝ of S 2 Synthesize controller Ĉ based on Ŝ 3 Refine solution Ĉ to C = All done finite 1 Ŝ S infinite 2 abstract concrete Ĉ C 3
5 2/6 Problem statement Given a LTL specification ϕ and a control system S, find a controller C that enforces ϕ on S Well-known abstraction/refinement approach 1 Compute a finite abstraction Ŝ of S 2 Synthesize controller Ĉ based on Ŝ 3 Refine solution Ĉ to C = All done So what is the problem? finite 1 Ŝ S infinite 2 abstract concrete Ĉ C 3
6 3/6 Computing Abstractions 1D: Temperature T = c(t env T )
7 3/6 Computing Abstractions 1D: Temperature T = c(t env T )
8 3/6 Computing Abstractions 1D: Temperature T = c(t env T ) ˆX = 100
9 3/6 Computing Abstractions 1D: Temperature T = c(t env T ) 2D: Pendulum ˆX = 100
10 3/6 Computing Abstractions 1D: Temperature T = c(t env T ) 2D: Pendulum ˆX = 100
11 3/6 Computing Abstractions 1D: Temperature T = c(t env T ) 2D: Pendulum ˆX = 100 ˆX = 100 2
12 3/6 Computing Abstractions 1D: Temperature T = c(t env T ) 2D: Pendulum 3D: Unicycle Robot y x ϕ ˆX = 100 ˆX = 100 2
13 3/6 Computing Abstractions 1D: Temperature T = c(t env T ) 2D: Pendulum 3D: Unicycle Robot y x ϕ ˆX = 100 ˆX = 100 2
14 3/6 Computing Abstractions 1D: Temperature T = c(t env T ) 2D: Pendulum 3D: Unicycle Robot y x ϕ ˆX = 100 ˆX = ˆX = 100 3??
15 3/6 Computing Abstractions 1D: Temperature T = c(t env T ) 2D: Pendulum 3D: Unicycle Robot y ϕ 4D: Pendulum on a cart x ˆX = 100 ˆX = ˆX = 100 3??
16 3/6 Computing Abstractions 1D: Temperature T = c(t env T ) 2D: Pendulum 3D: Unicycle Robot y ϕ 4D: Pendulum on a cart x X ˆX = 100 ˆX = ˆX = 100 3?? ˆX = 100 4
17 4/6 Sampling-based Ideas to Compute Abstractions Synergistic approach for syntactically co-safe LTL Lower layer: use sampling-based methods to grow the abstraction Higher layer: use Büchi automaton (from ϕ) and environment geometry to guide the expansion Use synergistic layer to alternate between layers A Bhatia, L E Kavraki, and M Y Vardi Sampling-based motion planning with temporal goals In: ICRA IEEE, 2010 M R Maly, M Lahijanian, L E Kavraki, H Kress-Gazit, and M Y Vardi Iterative Temporal Motion Planning for Hybrid Systems in Partially Unknown Environments In: HSCC ACM, 2013
18 curse of dimensionality is no problem 4/6 Sampling-based Ideas to Compute Abstractions Synergistic approach for syntactically co-safe LTL Lower layer: use sampling-based methods to grow the abstraction Higher layer: use Büchi automaton (from ϕ) and environment geometry to guide the expansion Use synergistic layer to alternate between layers Solution (point-to-point) x init x end
19 curse of dimensionality is no problem 4/6 Sampling-based Ideas to Compute Abstractions Synergistic approach for syntactically co-safe LTL Lower layer: use sampling-based methods to grow the abstraction Higher layer: use Büchi automaton (from ϕ) and environment geometry to guide the expansion Use synergistic layer to alternate between layers Solution (point-to-point) x init x end Problem solved?
20 5/6 What if we have a set of initial states? X init
21 5/6 What if we have a set of initial states? Solve problem for some samples of X init X init
22 5/6 What if we have a set of initial states? X init Solve problem for some samples of X init Can we use local controllers to enlarge/robustify solutions?
23 5/6 What if we have a set of initial states? X init Solve problem for some samples of X init Can we use local controllers to enlarge/robustify solutions? safety specifications? (infinite behavior) X safe X init
24 5/6 What if we have a set of initial states? X init Solve problem for some samples of X init Can we use local controllers to enlarge/robustify solutions? safety specifications? (infinite behavior) X safe What are good heuristics to grow the abstraction? X init
25 5/6 What if we have a set of initial states? X init Solve problem for some samples of X init Can we use local controllers to enlarge/robustify solutions? safety specifications? (infinite behavior) X safe What are good heuristics to grow the abstraction? How to find loops? X init
26 5/6 What if we have a set of initial states? X init Solve problem for some samples of X init Can we use local controllers to enlarge/robustify solutions? safety specifications? (infinite behavior) X init X safe What are good heuristics to grow the abstraction? How to find loops? Can we merge close-by samples?
27 To answer those questions we combine Sampling-based planning (Rice) Morteza Lahijanian Lydia Kavraki 6/6
28 To answer those questions we combine Sampling-based planning (Rice) Morteza Lahijanian Lydia Kavraki Control theory (UCLA) Matthias Rungger Paulo Tabuada 6/6
29 To answer those questions we combine Sampling-based planning (Rice) Morteza Lahijanian Lydia Kavraki Control theory (UCLA) Matthias Rungger Paulo Tabuada Are we satisfied? 6/6
30 To answer those questions we combine Sampling-based planning (Rice) Control theory (UCLA) Matthias Rungger Morteza Lahijanian Lydia Kavraki Reactive synthesis (Rice) Moshe Vardi Paulo Tabuada 6/6
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