Distributed hybrid control synthesis for multi-agent systems from - PowerPoint PPT Presentation

Distributed hybrid control synthesis for multi-agent systems from high level specifcations OptHySYS Workshop, Trento, January 2017 Dimos V. Dimarogonas (joint work with Jana Tumova, Dimitris Boskos and Meng Guo) Automatic Control, KTH Royal Institute of Technology, Sweden R E C O N F I G

Table of Contents Introduction Multi-Agent Hybrid Control under Local LTL Tasks and Relative-Distance Constraints Abstractions for Constrained Multi-Agent Systems Multi-Agent Planning from Local LTL Specifications

Background • Multi-agent control: motivated by a large variety of engineering applications: transportation systems, robotics, smart grids • Multi-agent control objectives: simple/control type (consensus, formation control, ...) • Formal methods based planning: higher level objectives for single agent • Based on discrete represenations (aka abstractions) of control systems

State of the Art Single Agent-Single Task • High-level task specs using formal languages • Planning on discrete abstraction of agent dynamics • Implemented by continuous control sequence Multiple Agents-Multiple Tasks • Need for distributed, bottom-up solutions to deal with: • Distributed tasks and abstractions • Couplings, limited communication

Proposed approach • Multi-agent control layer: distributed control through continuous state information • Formal methods based planning: distributed task planning based on discrete information exchange • Hybrid control: blending continuous and discrete information, need for abstractions of multi-agent control systems

Today’s talk 40 R1 R3 R2 R4 35 • Task planning and control through 30 25 specification-based abstraction y (m) 20 15 10 5 • Abstractions of dynamically coupled 0 0 5 10 15 20 25 30 35 40 x (m) multi-agent systems • Distributed task planning for task-level dependencies

Table of Contents Introduction Multi-Agent Hybrid Control under Local LTL Tasks and Relative-Distance Constraints Abstractions for Constrained Multi-Agent Systems Multi-Agent Planning from Local LTL Specifications

Problem Formulation • A team of N mobile agents, x i ( t ) , u i ( t ) ∈ R 2 : x i ( t ) = u i ( t ) , ˙ i ∈ N = { 1 , · · · , N } . • Agents i can observe agent j ’s position x j ( t ) only if: � x i ( t ) − x j ( t ) � ≤ r . Initial network G 0 . • Sphere regions of interest: R i = { R i ℓ , ℓ ∈ { 1 , · · · , M i }} . R i ℓ = ( c i ℓ , r i ℓ ). • Assumptions on the workspace. • Services Σ i available at each region in R i .

Problem Formulation, cont’d • Local LTL task specification ϕ i , over Σ i . • Note that ϕ i can be co-safe or general LTL formulas. • ϕ i specifies the sequences at which the services should be done at certain regions. Problem How to synthesize the control input u i ( t ) and the discrete plan S i such that ∀ i ∈ N ϕ i is satisfied , and � x i ( t ) − x j ( t ) � ≤ r , ∀ ( i , j ) ∈ E 0 , ∀ t ∈ [0 , ∞ ).

Challenges • Discrete task planning • Continuous motion constraints • Sensing limitations Solution: three main steps. • High-level discrete plan synthesis. • Distributed potential-field-based motion control. • Hybrid control strategy.

Step1. Discrete Plan Synthesis Aim Each agent synthesizes a local discrete plan that satisfies ϕ i and minimizes a cost function. • Automata-based model-checking algorithm 1 • Discrete plan synthesized locally by each agent i ∈ N : S i = σ i 1 · · · σ is i ( σ i ( s i +1) · · · σ iN i ) ω , σ is i = ( R is i , Σ is i ) . • Our algorithm minimizes the maximal distance between two consecutive regions along the plan 2 . 1 C. Baier, J.-P Katoen. Principles of model checking , 2008. 2 S. L. Smith, J. Tumova, C. Belta, D. Rus. Optimal Path Planning for Surveillance with Temporal Logic Constraints. The International Journal of Robotics Research, 2011.

Step2. Distributed Motion Control • Setup for motion control: • Each agent has its goal region σ ig = ( R ig , Σ ig ), but only known locally. • Relative-distance constraints. Goal Design a distributed control law u i ( t ) such that one agent arrives at its goal region in finite time, given the relative-distance constraints. • Time-varying connectivity graph G ( t ) = ( N , E ( t )), where E ( t ) ⊆ N × N . • initially G (0) = G 0 ; dynamically add new edges.

• Solution: the two-mode control law (1) the active mode: � u i ( t ) � − d i p i − C act : h ij x ij , j ∈N i ( t ) (2) the passive mode: u i ( t ) � − � C pas : h ij x ij , j ∈N i ( t ) where x ij � x i − x j ; p i � x i − c i g ; R ig = ( c i g , r i g ). ε 3 ε 2 r 2 d i � h ij � ( � p i � 2 + ε ) 2 + 2 ( � p i � 2 + ε ); ( r 2 − � x ij � 2 ) 2 • ε > 0 is a key design parameter. • u i is local w.r.t. N i ( t ).

Convergence results Considering a potential-field like Lyapunov function it can be shown that: • G ( t ) remains connected. • There exists a finite time T f and one active agent i ⋆ ∈ N a , such that x j ( T f ) ∈ R i ⋆ g , ∀ j ∈ N . • All agents will enter R i ⋆ g , i.e., x j ∈ R i ⋆ g , ∀ j ∈ N . • The above holds for any number of active agents that 1 ≤ N a ≤ N .

Potential-field-based Design Consider the following potential-field function: V ( x ( t )) � 1 � � � φ c ( x ij ) + b i φ g ( x i ) 2 i ∈N j ∈N i ( t ) i ∈N • φ c ( x ij ) is an attractive potential to agent i ’s neighbors. • φ g ( · ) is an attractive force to agent i ’s goal: • b i = 1, ∀ i ∈ N a and b i = 0, ∀ i ∈ N p . N = N a ∪ N p . Connectivity Results G ( t ) remains connected. No existing edges within E ( T s ) will be lost. • Proof shows that V ( t ) remains bounded for t ∈ [ T s , ∞ ). New edges might be added but no existing edges will be lost.

Convergence (non-switching case) Constant sets of passive and active agents. Analysis of the critical points of V : • Regions around the critical points: S i � { x ∈ R 2 N | � x − 1 N ⊗ c i g � ≤ r S ( ε ) } , ∀ i ∈ N a . Let S � ∪ i ∈N a S i and S ¬ � R 2 N \ S . • Lemma 1: There exists ε 1 > 0 such that if ε < ε 1 , all critical points of V in S ¬ are non-degenerate saddle points. • Lemma 2,3: There exists ε < min { ε 2 , ε 6 } such that regions {S i } are sufficiently far. Critical points are close to the region center.

Lemma 4: There exists ε min > 0 such that if ε < ε min , all critical points of V within S are local minima. Convergence Results There exists a finite time T f ∈ [ T s , ∞ ) and one active agent i ⋆ ∈ N a , such that x j ( T f ) ∈ R i ⋆ g , ∀ j ∈ N , while � x i ( t ) − x j ( t ) � < r , ∀ ( i , j ) ∈ E ( T s ) and ∀ t ∈ [ T s , T f ]. • The system converge to the set of local minima within S i ⋆ for one active agent i ⋆ ∈ N a . • All agents would enter R i ⋆ g , i.e., x j ∈ R i ⋆ g , ∀ j ∈ N . • All edges within E ( T s ) will be preserved for all t > T s The above theorem holds for any number of active agents that 1 ≤ N a ≤ N .

Step3. Hybrid Control: sc-safe LTL task case Case one All tasks { ϕ i } are given as sc-safe LTL formulas. • If ϕ i is sc-safe, every agent has a finite plan τ i = ( R i 1 , Σ i 1 )( R i 2 , Σ i 2 ) · · · ( R iN i , Σ iN i ) . Local switching policy • When R ik is reached, provide the services Σ ik and then set goal to R i ( k +1) . • After ( R iN i , Σ iN i ), set b i = 0 and be passive. • Guaranteed that ∀ i ∈ N , ϕ i is eventually satisfied, and � x i ( t ) − x j ( t ) � < r , ∀ ( i , j ) ∈ E (0) and ∀ t ≥ 0.

Step3. Hybrid Control: general LTL task case • LTL and mixed sc-safe LTL/LTL tasks can be also tackled under different switching policies • Account for infiniteness of satisfying plans • Further ongoing extension to double-integrator dynamics with collision avoidance and quantified specs (MITL and STL formulas)

Four agents with co-safe or general LTL tasks: Workspace • Π 1 = { π 1 tl , π 1 tr , π 1 br , π 1 bl } . Σ 1 = { σ 11 , σ 12 } . • Π 2 = { π 2 tl , π 2 tr , π 2 bl } . Σ 2 = { σ 21 , σ 22 , σ 23 } . • Π 3 = { π 3 tr , π 3 br , π 3 bl } . Σ 3 = { σ 31 , σ 32 , σ 33 } . • Π 4 = { π 4 tl , π 4 tr , π 4 br , π 4 bl } . Σ 4 = { σ 41 , σ 42 , σ 43 } . Sc-safe LTL task General LTL task • ϕ 1 = ♦ ( σ 12 ∧ ♦ ( σ 11 ∧ ♦ σ 12 )). • ϕ 1 = �♦ σ 11 ∧ �♦ σ 12 . • ϕ 2 = ♦ ( σ 21 ∨ σ 22 ) ∧ ♦ σ 23 . • ϕ 2 = �♦ ( σ 21 ∨ σ 22 ∨ σ 23 ) • ϕ 3 = ♦ ( σ 31 ∨ σ 32 ) ∧ ♦ σ 33 . • ϕ 3 = �♦ ( σ 31 ∨ σ 32 ∨ σ 33 ) • ϕ 4 = ♦ ( σ 42 ∧ ♦ ( σ 41 ∧ ♦ σ 42 )). • ϕ 4 = �♦ σ 41 ∧ �♦ σ 42

Scenario one 40 8 d12 R1 R3 d23 7 d34 R2 R4 35 6 5 Distance (m) 30 4 3 2 25 1 y (m) 0 20 1 0 2000 4000 6000 8000 10000 Time (s) 5 15 R1 R2 R3 4 R4 10 Reach events 3 5 2 1 0 0 5 10 15 20 25 30 35 40 0 x (m) 0 2000 4000 6000 8000 10000 Time (ms)

Scenario two 40 8 d12 R1 R3 7 d23 d34 R2 R4 35 6 5 Distance (m) 4 30 3 2 25 1 y (m) 0 20 1 0 5000 10000 15000 20000 Time (ms) 5 15 R1 R2 R3 4 R4 10 Reach events 3 5 2 1 0 0 5 10 15 20 25 30 35 40 0 x (m) 0 5000 10000 15000 20000 Time (ms)