Analysis and Control of Multi-Robot Systems Formation Control of - PowerPoint PPT Presentation

Elective in Robotics 2014/2015 Analysis and Control of Multi-Robot Systems Formation Control of Multiple Robots Dr. Paolo Robuffo Giordano CNRS, Irisa/Inria ! Rennes, France

Formation Control of Multiple Robots 2 Robuffo Giordano P ., Multi-Robot Systems: Formation Control of Multiple Robots

Summary of Previous Lectures • What have we seen so far? • Graph Theory • Undirected graphs and Directed graphs G = ( V , E ) D = ( V , E ) v 3 v 3 v 5 v 5 v 1 v 4 v 1 v 4 v 2 v 2 • Connected graphs/Disconnected graphs weakly connected connected disconnected strongly connected 3 Robuffo Giordano P ., Multi-Robot Systems: Formation Control of Multiple Robots

Summary of Previous Lectures • Decentralization 1 1 2 2 3 3 10 Bytes/s 10 Bytes/s 4 4 5 5 6 ∆ ∈ R N × N A ∈ R N × N • Algebraic Graph Theory: Adjacency matrix , Degree matrix E ∈ R N × |E| L ∈ R N × N Incidence matrix and Laplacian matrix with L = ∆ − A = EE T 1 T L = 0 • Properties of the Laplacian: (and for undirected graphs) L 1 = 0 4 Robuffo Giordano P ., Multi-Robot Systems: Formation Control of Multiple Robots

Summary of Previous Lectures • Properties of the Laplacian: (all eigenvalues real and 0 = λ 1 ≤ λ 2 ≤ . . . ≤ λ N non-negative for undirected graphs) rank( L ) = N − 1 λ 2 > 0 • Graph connected if and only if ! and is the 1 eigenvector associated to λ 1 = 0 E T 1 = 0 rank( E ) = N − 1 • Also, and • Consensus protocol: u i = u i ( x i − x j ) ∀ j ∈ N i • agents with dynamics , find , , such that x i = u i ˙ N for some common but unspecified t →∞ x i ( t ) = ¯ lim x, ∀ i ¯ x X • Solution: equivalent to , yielding u i = ( x j − x i ) u = − Lx x = − Lx ˙ j ∈ N i 5 Robuffo Giordano P ., Multi-Robot Systems: Formation Control of Multiple Robots

Summary of Previous Lectures rank( L ) = N − 1 • Result: if the (undirected) graph is connected ( and/or λ 2 > 0 then the consensus converges to the average of the initial condition t →∞ x ( t ) = ( 1 T x 0 ) 1 lim N • The magnitude of dictates the rate of convergence λ 2 • For directed graphs, the conditions for the consensus convergence, i.e., rank( L ) = N − 1 0 < < ( λ 2 )  . . .  < ( λ N ) , , require presence of a rooted out-branching v 3 e 3 v 5 e 6 e 2 e 5 e 4 v 1 e 1 v 4 v 2 6 Robuffo Giordano P ., Multi-Robot Systems: Formation Control of Multiple Robots

Summary of Previous Lectures • For directed graphs, in general no convergence to the average of the initial condition, but just t →∞ x ( t ) = ( q T 1 x 0 ) p 1 = ( q T lim 1 x 0 ) 1 q 1 6 = 0 for some t →∞ x ( t ) = ( 1 T x 0 ) 1 lim • If the graph is balanced, then we re-obtain N • Consensus protocol: paradigm of many decentralized algorithms based on relative information • Can (and has been) extended to many variants (time-varying topologies, delays, more complex agent dynamics, etc.) 7 Robuffo Giordano P ., Multi-Robot Systems: Formation Control of Multiple Robots

Summary of Previous Lectures • Passivity: Σ • I/O characterization in “Energetic terms” • No internal production of energy • Passivity ingredients: ( ˙ = f ( x ) + g ( x ) u x • a dynamical system = h ( x ) y V ( x ) ∈ C 1 : R n → R + • a lower-bounded Storage function R t t 0 y T ( τ ) u ( τ )d τ ( V ( x ( t )) − V ( x ( t 0 )) ≤ • a passivity condition y T ( t ) u ( t ) ˙ V ( x ( t )) ≤ 8 Robuffo Giordano P ., Multi-Robot Systems: Formation Control of Multiple Robots

Summary of Previous Lectures • Passivity is w.r.t. an input/output pair and w.r.t. a Storage function ( u, y ) V ( x ) Current energy is at most equal to the initial energy + exchanged energy with outside • Passivity is strongly related to Lyapunov stability • Stable free evolution ( ) and stable zero-dynamics ( ) y ≡ 0 u ≡ 0 • “Easy” output feedback for (asympt.) stabilization e.g., • When possible, one can choose the “right” output for enforcing passivity • Many physical systems are passive, e.g., mechanical systems (robot manipulators) w.r.t. the pair force/velocity 9 Robuffo Giordano P ., Multi-Robot Systems: Formation Control of Multiple Robots

Summary of Previous Lectures • Proper compositions of passive systems are passive • Example: parallel and feedback interconnection • In the feedback interconnection, the coupling is skew-symmetric (power-preserving interconnection) 10 Robuffo Giordano P ., Multi-Robot Systems: Formation Control of Multiple Robots

Summary of Previous Lectures • Port-Hamiltonian Systems (PHS): look at the network structure behind passivity • Passive systems are made of a “power-preserving interconnection” of: • Elements storing Energy • Elements dissipating Energy • Power ports with the external world 11 Robuffo Giordano P ., Multi-Robot Systems: Formation Control of Multiple Robots

Summary of Previous Lectures • In the general form, a PHS is with being the Hamiltonian (lower-bounded Storage function), and the passivity condition naturally embedded in the system structure H = − ∂ H T ∂ x + ∂ H T ∂ x R ( x ) ∂ H ∂ x g ( x ) u ≤ y T u ˙ • Among the many control techniques for PHS, we focused on the Energy Transfer control  u 1 �  �  y 1 � − α y 1 ( x 1 ) y T 0 2 ( x 2 ) = α ∈ R , α y 2 ( x 2 ) y T 1 ( x 1 ) 0 u 2 y 2 12 Robuffo Giordano P ., Multi-Robot Systems: Formation Control of Multiple Robots

Summary of Previous Lectures • This allows to transfer energy from a PHS to another PHS in a lossless way • The total does not change , but the individual energies may increase/ decrease depending on the value of the parameter α • This technique can be used in conjunction with the so-called Energy Tanks • In a PHS, there is an inherent passivity margin due to the 800 internal dissipation 600 400 H [J], E ext [J] Z t Z t 200 ∂ H T ∂ x R ( x ) ∂ H y T u d τ − H ( t ) − H ( t 0 ) = ∂ x d τ 0 − 200 t 0 t 0 | {z } − 400 ≤ 0 − 600 − 800 0 20 40 60 80 100 120 time [s] 13 Robuffo Giordano P ., Multi-Robot Systems: Formation Control of Multiple Robots

Summary of Previous Lectures • Idea: store back the dissipated energy, and use it to passively implement whatever action D ( x ) = ∂ H T ∂ x R ( x ) ∂ H • The PHS dissipated power is ∂ x T ( x t ) = 1 2 x 2 • Design a PHS Tank dynamics with energy function as t ≥ 0 1  ˙ = D ( x ) + ˜ x t u t  x t =  y t x t and then interconnect Tank and PHS by means of the skew-symmetric coupling 14 Robuffo Giordano P ., Multi-Robot Systems: Formation Control of Multiple Robots

Summary of Previous Lectures • The PHS becomes and the Tank dynamics is D ( x ) − w T x t = 1 g T ( x ) ∂ H ˙ ∂ x x t x t • Singularity for : Tank empty, therefore the action cannot be (passively) implemented • Anyway: the Tank is • continuously refilled by the term D ( x ) • possibly refilled by the action • and complete freedom in choosing the initial Tank energy level T ( x t ( t 0 )) 15 Robuffo Giordano P ., Multi-Robot Systems: Formation Control of Multiple Robots

Formation Control of Multiple UAVs • We are finally ready for some action (Formation Control of UAVs) 16 Robuffo Giordano P ., Multi-Robot Systems: Formation Control of Multiple Robots

Formation Control of Multiple UAVs • Let us then focus on the Passivity-based Decentralized Control of Multiple UAVs Σ • We start with a basic problem: formation control under sensing/comm. constraints 17 Robuffo Giordano P ., Multi-Robot Systems: Formation Control of Multiple Robots

Formation Control of Multiple UAVs • Formation Control with Time-varying graph topology • Robots are loosely coupled together • can gain/lose neighbors, but must show some form of cohesive behavior • Robots can decide to split or to join because of any constraint or task, e.g. • sensing and/or communication constraints • need to temporarily split for better maneuvering in cluttered environments • Overall motion controlled by selected robots (leaders) • Appropriate for “loose” tasks, e.g., coverage, persistent patrolling, exploration, etc. 18 Robuffo Giordano P ., Multi-Robot Systems: Formation Control of Multiple Robots

Formation Control of Multiple UAVs • Features: • decentralized design (local and 1-hop communication/sensing) • flexible formation: splits/joins due to • sensing/communication constraints • execution of extra tasks in parallel to the collective motion • Autonomy in avoiding obstacles and inter-agent collisions • Challenges: • Time-varying topology: ensure stability despite a switching dynamics • Guarantee passivity of the overall group behavior • Steady-state characteristics? (Velocity synchronization) • What if time delays are present in the communication links? • What about maintenance of group connectivity? 19 Robuffo Giordano P ., Multi-Robot Systems: Formation Control of Multiple Robots