Dynamic and Adversarial Reach- avoid Symbolic Planning Laya Shamgah Advisor: Dr. Karimoddini July 21 st 2017 Thrust 1: Modeling, Analysis and Control of Large-scale Autonomous Vehicles (MACLAV) Sub-trust 1-2: Cooperative Localization, Navigation and Control of LSASVs 1
Motivation Reach-avoid Problem: Traveling from an initial point to a desired location while avoiding obstacles Dynamic Adversarial Static Environment Dynamic Environment Environment Challenge: Autonomous Coordination of autonomous vehicles to achieve their sophisticated goals in an dynamic and adversarial environment Dynamic and Adversarial Reach-avoid Symbolic Planning Laya Shamgah 2
Objective Objective of research: To develop a computationally effective reactive planning method for autonomous vehicles in a dynamic adversarial environment. Dynamic Adversarial Reach-avoid scenario: • attacker: tries to reach the target while avoiding of capture. • defender: tries to capture the attacker before reaching the defending area. 3 Dynamic and Adversarial Reach-avoid Symbolic Planning Laya Shamgah 3
Challenges and Gaps Existing methods Pursuit-evasion games Probabilistic approaches Differential games [Bhadauria et al. 2012] [Vitus et sl. 2011] [Tomlin et al.2011,2015] Solving only the avoidance problem Challenges Assuming limitations on the vehicle’s movements Requiring information about the opponent vehicle High computational cost Lack of Reactiveness 4 Dynamic and Adversarial Reach-avoid Symbolic Planning Laya Shamgah 4
Proposed approach To reduce the complexity: 1- Using Symbolic Control Techniques for abstraction of the (infinitely) large original problem to a (finite) small abstracted environment, 2- Designing a DES supervisor to achieve a complex task over an abstract environment 3- Projecting back the solution to the original domain. Remark: This is the first result in the literature that uses symbolic control techniques for the reach-avoid problem. attacker Target Abstraction ? 𝐐 defender 5 Dynamic and Adversarial Reach-avoid Symbolic Planning Laya Shamgah 5
Proposed Hybrid Structure DES supervisor Abstraction of Vehicle Dynamics Bisumulation-based Discrete Signal abstraction Vehicle Dynamics Continuous Signals Interface 6 Dynamic and Adversarial Reach-avoid Symbolic Planning Laya Shamgah 6
Proposed Implementation Approach Hierarchical Control Supervisor, operator,… Supervisor Temporal Logic high-level Controller Symbolic Planning Planner Low-level Controller Real-time low-level controller 7 Dynamic and Adversarial Reach-avoid Symbolic Planning Laya Shamgah 7
Reach-avoid Problem Description Assumptions: attacker 𝑦 𝑢 = 𝑔(𝑦 𝑢 , 𝑣(𝑢)) • Defender vehicle dynamics: Target • Environment (P) is a bounded convex set • Target is in a fixed position ? 𝐐 • The initial position of the attacker and the defender are within P • Defender vehicle has full observability over the position of the defender attacker other Problem: Design a controller to obtain trajectory 𝑘=1,…,𝑛 𝑄 𝑗𝑘 , which satisfies 𝑦 𝑢 ∈ P = 𝑗=1,…,𝑜 the objective of the defender. 8 Dynamic and Adversarial Reach-avoid Symbolic Planning Laya Shamgah 8
Proposed Framework Design Steps: 1.Extracting decision-making strategies 2.Construction of LTL Specification 𝝌 = 𝝌 𝒃 → 𝝌 𝒆 3.Checking realizability of 𝝌 4.Synthesizing the supervisor automaton G which satisfies 𝝌 5.Designing the hybrid controller 9 Dynamic and Adversarial Reach-avoid Symbolic Planning Laya Shamgah 9
Step1: Optimal Decision-making Strategies 𝐐 𝟐𝟐 𝐐 𝟐𝟑 𝐐 𝟐𝟒 • Modeled as a finite two-player zero-sum game in matrix form 𝐐 𝟑𝟐 𝐐 𝟑𝟑 𝐐 𝟑𝟒 • Attacker is the maximizer player and Defender is the minimizer ○ a * 𝐐 𝟒𝟐 𝐐 𝟒𝟑 𝐐 𝟒𝟒 ● d Objective Function ′ ∈ 𝑄 𝑗𝑘 ′ , 𝑦 𝑒 0 𝑗𝑔 𝑦 𝑏 ′ , 𝑦 𝑢 ∈ 𝑄 𝑗𝑘 ∞ 𝑗𝑔𝑦 𝑏 ′ , 𝑦 𝑒 ′ 𝑀 𝑦 𝑏 = 𝛾 ′ − 𝑦 𝑒 ′ ∥ + ′ − 𝑦 𝑢 ∥ 𝑝𝑢ℎ𝑓𝑠𝑥𝑗𝑡𝑓 𝛽 ∥ 𝑦 𝑏 ′ −𝑦 𝑢 ∥ + 𝛿 ∥ 𝑦 𝑒 ∥ 𝑦 𝑏 Distance between the vehicles Distance between the defender and Distance between the the target attacker and the target 10 Dynamic and Adversarial Reach-avoid Symbolic Planning Laya Shamgah 10
Step1: Decision-making Example: attacker 𝐐 𝟐𝟐 𝐐 𝟐𝟑 𝐐 𝟐𝟒 Optimization Parameters: 𝑸 𝟑𝟑 𝑸 𝟒𝟐 𝑸 𝟐𝟐 𝐐 𝟑𝟐 𝐐 𝟑𝟑 𝐐 𝟑𝟒 𝛽 = 1 ○ a 𝑸 𝟒𝟑 * 3.414 4.650 4.650 𝛾 = 1 defender 𝐐 𝟒𝟐 𝐐 𝟒𝟑 𝐐 𝟒𝟒 𝛿 = 0.5 𝑸 𝟑𝟒 2 3.236 3.236 ● d • Defender : min 𝑛𝑏𝑦 3.414 , 4.650 , 𝑛𝑗𝑜 2 , 3.236 = 3.236 → 𝑄 23 • Attacker : max 𝑛𝑗𝑜 3.414 , 2 , 4.650 , 3.236 = 3.236 → 𝑄 11 Nash Equilibrium decision : (𝒃 𝟐𝟐 , 𝒆 𝟑𝟒 ) Temporal formula 𝒐𝒏(𝒐𝒏 − 𝟑) games should be solved to calculate all the temporal transition rules. ⃞(𝒃 𝟑𝟐 ∧ 𝒆 𝟒𝟒 → ⃝𝒆 𝟑𝟒 ) 11 Dynamic and Adversarial Reach-avoid Symbolic Planning Laya Shamgah 11
Step 2: Construction of LTL Specification Cla lassical log logic: Tem emporal log logic • “I am hungry ry ” • "I am always hungry ry “ • "I will eventually be hungry ry “ • "I will be hungry ry until I eat something" Temporal logic: Linear Temporal Logic (LTL) is a formal high-level language to describe many complex missions and a wide class of properties can be expressed by LTL: • Coverage : eventually visit all regions • Sequencing : visit P2 before you go to P3 • Avoidance : until you go to P2 avoid P1 and P3 Dynamic and Adversarial Reach-avoid Symbolic Planning Laya Shamgah 12
Step 2: Construction of LTL Specification The LTL formulas ( 𝜒) are constructed over ( Σ) using Boolean operators and temporal operators. • 𝜯 : A finite set of atomic proposition: 𝑞 ∈ 𝛵 ( 𝑞 𝑑𝑏𝑜 𝑐𝑓 𝑓𝑗𝑢ℎ𝑓𝑠 𝑈 𝑝𝑠 𝐺) • Boolean operators: negation (¬), disjunction ( ∨ ), conjunction ( ∧ ), implication ( → ) • Modal temporal operators : next ( 𝑃 ), until ( 𝒱 ), eventually ( ◊ ) and always ( ⎕ ) Dynamic and Adversarial Reach-avoid Symbolic Planning Laya Shamgah 13
Step 2: Construction of LTL Specification Temporal Operators: Operators Definition Diagram ○ 𝛘 𝜒 is true in the next moment of time □ 𝛘 𝜒 is true in all future moments ◊ 𝛘 𝜒 is true in some future moment 𝛘𝐯𝛚 𝜒 is true until ψis true Dynamic and Adversarial Reach-avoid Symbolic Planning Laya Shamgah 14
Step 2: Construction of LTL Specification Reactive to changes in Static Environment Dynamic Environment Dynamic Environment Vehicle φ vehicle φ = (𝜒 𝑓 → 𝜒 𝑡 ) Dynamic and Adversarial Reach-avoid Symbolic Planning Laya Shamgah 15
Step 2: Construction of LTL Specification Reach-avoid Specification: attacker Target 𝝌 = 𝝌 𝒃 → 𝝌 𝒆 ? 𝝌 𝒃 : all assumptions on the attacker 𝐐 𝝌 𝒆 : all assumptions on the defender and its desired behavior 𝒘 ⋀ 𝝌 𝒑𝒄𝒌 defender 𝒘 𝒘 𝒘 𝒘 𝝌 𝒘 = 𝝌 𝒋𝒐𝒋𝒖 ⋀ 𝝌 𝒕𝒋𝒐𝒉 ⋀ 𝝌 𝒖𝒇𝒔𝒏 ⋀ 𝝌 𝒔𝒗𝒎 𝒘 𝝌 𝒋𝒐𝒋𝒖 Boolean ( 𝑪 ) 1 Initial position of vehicle 𝒘 𝝌 𝒕𝒋𝒐𝒉 Temporal ( □𝑼 ) Singularity constraint: At each time the vehicle can be in only one region 2 𝒘 𝝌 𝒖𝒇𝒔𝒏 Temporal ( □𝑼 ) 3 Termination of the game 𝒘 𝝌 𝒔𝒗𝒎 Temporal ( □𝑼 ) 4 Transitions rules over the partitioned area 𝒘 𝝌 𝒑𝒄𝒌 Temporal ( □ ◊ 𝑪 ) Objective of the vehicle 5 Dynamic and Adversarial Reach-avoid Symbolic Planning Laya Shamgah 16
Step 3-4: Discrete Design Procedure Step 3: Checking realizability of 𝝌 Check if there exists any admissible behavior of the attacker such that no behavior of adapter can satisfy 𝝌 𝒆 . Step 4: Synthesis of automaton G • If 𝝌 is realizable then G = Q, q 0 , A, D, 𝜀, ℎ • Synthesis Process: 𝐻𝑇 =< 𝑊, 𝐵, 𝐸, Θ, ρ 𝑏 , 𝜍 𝑒 , 𝜒 > • 𝐻 ⊨ 𝜒 • Every path on G is a behavior of the attacker and the corresponding behavior of the defender, which ends when the defender will win 17 Dynamic and Adversarial Reach-avoid Symbolic Planning Laya Shamgah 17
Step 5: Hybrid Control Design Online implementation: • Heading angle( 𝜾 ) Velocity( 𝒗 ) • Attacker’s Continuous behavior Discrete path path 𝒚(𝒖) Interface 𝒃 𝒋 → 𝒃 𝒋+𝟐 𝒆 𝒋 → 𝒆 𝒋+𝟐 18 Dynamic and Adversarial Reach-avoid Symbolic Planning Laya Shamgah 18
Example: Description 𝐐 𝟐𝟐 𝐐 𝟐𝟑 𝐐 𝟐𝟒 attacker Operation region Initial positions Target 𝐐 𝟑𝟐 𝐐 𝟑𝟑 𝐐 𝟑𝟒 3 attacker: 𝑄 11 𝑄 23 𝑄 = 𝑄 𝑗𝑘 defender: 𝑄 31 defender 𝑗,𝑘=1 𝐐 𝟒𝟐 𝐐 𝟒𝟑 𝐐 𝟒𝟒 Problem : Design a controller to obtain trajectory 𝑦(𝑢) which satisfies 𝜒 = 𝜒 𝑏 → 𝜒 𝑒 19 Dynamic and Adversarial Reach-avoid Symbolic Planning Laya Shamgah 19
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