SIDRA 2012 talk
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The document presents a decentralized controller-observer scheme designed for multi-robot weighted centroid tracking, focusing on local information communication to achieve global task performance. It details the formulation and proof of convergence for a collective state estimation under various communication topologies and control law designs. The study includes theoretical analysis, simulations, and experiments demonstrating the robustness and effectiveness of the proposed method in real-world scenarios.
SIDRA 2012 talk
- 1. A decentralized controller-observer scheme for multi-robot weighted centroid tracking Gianluca Antonelli† , Filippo Arrichiello† , Fabrizio Caccavale⊕ , Alessandro Marino‡ in alphabetical order † University of Cassino and Southern Lazio, Italy http://webuser.unicas.it/lai/robotica ⊕ University of Basilicata, Italy http://www.difa.unibas.it ‡ University of Salerno, Italy http://www.unisa.it Antonelli, Arrichiello, Caccavale, Marino Benevento, 12 September 2012
- 2. General objective In a multi-robot scenario local information local communication ⇒ global task local controller time-varying topology Antonelli, Arrichiello, Caccavale, Marino Benevento, 12 September 2012
- 3. Sketch Decentralized controller-observer for weighted centroid tracking Time-varying reference for weighted centroid (formation as centroid+displacement) Each robot estimates the collective state (i.e., robots positions) Convergence proof for first-order dynamics continuous-time fixed/switching communication topologies directed/undirected graphs saturated inputs Antonelli, Arrichiello, Caccavale, Marino Benevento, 12 September 2012
- 4. Sketch Decentralized controller-observer for weighted centroid tracking Time-varying reference for weighted centroid (formation as centroid+displacement) Each robot estimates the collective state (i.e., robots positions) Convergence proof for first-order dynamics continuous-time fixed/switching communication topologies directed/undirected graphs saturated inputs Antonelli, Arrichiello, Caccavale, Marino Benevento, 12 September 2012
- 5. Sketch Decentralized controller-observer for weighted centroid tracking Time-varying reference for weighted centroid (formation as centroid+displacement) Each robot estimates the collective state (i.e., robots positions) Convergence proof for first-order dynamics continuous-time fixed/switching communication topologies directed/undirected graphs saturated inputs Antonelli, Arrichiello, Caccavale, Marino Benevento, 12 September 2012
- 6. Modeling N robots with n DOFs each: Single state: xi ∈ Rn ˙ Individual dynamics: xi = ui (single-integrator dynamics) T 1 T Collective state: x = xT . . . xN ∈ RN n ˙ Collective dynamics: x = u Global estimate computed by robot i: i x ∈ RN n ˆ 1 x − 1x ˜ x ˆ 2x x − 2x ˜ ˆ 2 ˜ Collective estimation error: x = . = . ∈ RN n . . . . Nx ˜ x − Nx ˆ Antonelli, Arrichiello, Caccavale, Marino Benevento, 12 September 2012
- 7. Problem statement Task (weighted centroid) N σ(x) = αi xi = αT ⊗ I n x ∈ Rn i=1 Design goals, for each robot: state observer providing an estimate, i x ∈ RN n , asymptotically ˆ convergent to the collective state x feedback control law, ui = ui (xi , i x, Ni ) ∈ Rn , such that σ(x) ˆ asymptotically converges to a time-varying reference, σ d (t) ˙ Each robot knows in advance: σ d (t), σ d (t) Antonelli, Arrichiello, Caccavale, Marino Benevento, 12 September 2012
- 8. Proposed approach -1- i th control law: αi ui = ui (i x) = ˆ σ d + kc σ d − σ(i x) ˙ ˆ ✏✏ α 2 ✏✏ ✏✏ ✏✏ ✏✏ ✏ ✮ each robot is feeding back its estimate of the collective state Antonelli, Arrichiello, Caccavale, Marino Benevento, 12 September 2012
- 9. Proposed approach -1- i th control law: αi ui = ui (i x) = ˆ σ d + kc σ d − σ(i x) ˙ ˆ ✏✏ α 2 ✏✏ ✏✏ ✏✏ ✏✏ ✏ ✮ each robot is feeding back its estimate of the collective state Antonelli, Arrichiello, Caccavale, Marino Benevento, 12 September 2012
- 10. Proposed approach -2- local feedback consensus-like term ✒ ✻ i th state observer: ˙ ix ˆ = ko j x − ix + Π i x − ix + iu ˆ ˆ ˆ ˆ j∈Ni ✟ ✟✟ ✟ ✟✟ Π i = diag O n · · · I n · · · O n u1 (i x) ✟✟ ˆ . ✟ ✟= αj σ + k σ − σ(i x) ˆ . , uj (✟ ) iu = . ✟ ix ˆ α 2 ˙d c d ˆ i x)✟ ✟ uN ( ✟ ˆ ✟ ✟ ✟✟ ✙ collective input estimated by robot i Antonelli, Arrichiello, Caccavale, Marino Benevento, 12 September 2012
- 11. Proposed approach -2- local feedback consensus-like term ✒ ✻ i th state observer: ˙ ix ˆ = ko j x − ix + Π i x − ix + iu ˆ ˆ ˆ ˆ j∈Ni ✟ ✟✟ ✟ ✟✟ Π i = diag O n · · · I n · · · O n u1 (i x) ✟✟ ˆ . ✟ ✟= αj σ + k σ − σ(i x) ˆ . , uj (✟ ) iu = . ✟ ix ˆ α 2 ˙d c d ˆ i x)✟ ✟ uN ( ✟ ˆ ✟ ✟ ✟✟ ✙ collective input estimated by robot i Antonelli, Arrichiello, Caccavale, Marino Benevento, 12 September 2012
- 12. Proposed approach -2- local feedback consensus-like term ✒ ✻ i th state observer: ˙ ix ˆ = ko j x − ix + Π i x − ix + iu ˆ ˆ ˆ ˆ j∈Ni ✟ ✟✟ ✟ ✟✟ Π i = diag O n · · · I n · · · O n u1 (i x) ✟✟ ˆ . ✟ ✟= αj σ + k σ − σ(i x) ˆ . , uj (✟ ) iu = . ✟ ix ˆ α 2 ˙d c d ˆ i x)✟ ✟ uN ( ✟ ˆ ✟ ✟ ✟✟ ✙ collective input estimated by robot i Antonelli, Arrichiello, Caccavale, Marino Benevento, 12 September 2012
- 13. Proposed approach -2- local feedback consensus-like term ✒ ✻ i th state observer: ˙ ix ˆ = ko j x − ix + Π i x − ix + iu ˆ ˆ ˆ ˆ j∈Ni ✟ ✟✟ ✟ ✟✟ Π i = diag O n · · · I n · · · O n u1 (i x) ✟✟ ˆ . ✟ ✟= αj σ + k σ − σ(i x) ˆ . , uj (✟ ) iu = . ✟ ix ˆ α 2 ˙d c d ˆ i x)✟ ✟ uN ( ✟ ˆ ✟ ✟ ✟✟ ✙ collective input estimated by robot i Antonelli, Arrichiello, Caccavale, Marino Benevento, 12 September 2012
- 14. Collective dynamics Estimation error: ˙ x = −ko (L ⊗ I N n + Π) x + (1N ⊗ I N n ) u − u ˜ ˜ ˆ with L Laplacian matrix embedding the topology Tracking error: N ˙ kc σ = −kc σ − ˜ ˜ α2 αT ⊗ I n i x i ˜ α 2 i=1 Antonelli, Arrichiello, Caccavale, Marino Benevento, 12 September 2012
- 15. Stability proof for undirected connected topologies Lyapunov function: 1 1 V (˜ , σ) = xT x + σT σ x ˜ ˜ ˜ ˜ ˜ 2 2 after straightforward computations. . . √ √ N kc n α ko λm − N kc n − ˜ x ˙ x ˜ V (˜ , σ) ≤ − x ˜ ˜ σ √ 2 N kc n α ˜ σ − kc 2 Antonelli, Arrichiello, Caccavale, Marino Benevento, 12 September 2012
- 16. Stability proof for undirected connected topologies ˙ V is negative definite with a proper choice of the design gains ko and kc : √ 2 kc Nn α ko > N n+ λm 4 comments: N , n and α are known parameters the control gain kc is free (altough positive) the term λm ≥ 0 is embedding the connection properties (null for unconnected graphs) (not surprisingly) the observer gain ko is lower bounded Antonelli, Arrichiello, Caccavale, Marino Benevento, 12 September 2012
- 17. Extensions -1- Directed topologies Switching topologies convergence for balanced proof by the concept of and strongly connected Common Lyapunov graphs Function proof by resorting to the gains tuned on the worst concept of mirror graph case All the case studies above analyzed also for saturated inputs Antonelli, Arrichiello, Caccavale, Marino Benevento, 12 September 2012
- 18. Extensions -2- Centroid and formation N 1 σ1 (x) = xi N i=1 T σ2 (x) = (x2 −x1 )T (x3 −x2 )T . . . (xN −xN −1 )T Solved and analized by resorting to a similar controller-observer scheme and Lyapunov approach Antonelli, Arrichiello, Caccavale, Marino Benevento, 12 September 2012
- 19. Comments Originality Estimating the whole state ⇒ is it really decentralized? Scalability (1000 robots, 10 neighbors ⇒ 8 ms on an Arduino) Need to know the desired trajectory in advance Robustness with respect to failure? Antonelli, Arrichiello, Caccavale, Marino Benevento, 12 September 2012
- 20. Simulations there is life beyond Lyapunov! Dozens of numerical simulations by changing the key parameters: 3 number of robots N 4 2 dimension n number of neighbors Ni 5 1 topology (un-directed, switching) 6 8 saturated inputs 7 Antonelli, Arrichiello, Caccavale, Marino Benevento, 12 September 2012
- 21. Experiments there is life beyond Matlab! 5 Khepera III by K-team real-time comm. real-time localization obstacle avoidance various topologies initial error Antonelli, Arrichiello, Caccavale, Marino Benevento, 12 September 2012
- 22. Experiments - estimation errors estimate errors w.r.t. real pos rob 0 estimate errors w.r.t. real pos rob 1 4 8 3 6 2 4 1 2 0 0 0 20 40 60 80 0 20 40 60 80 estimate errors w.r.t. real pos rob 2 estimate errors w.r.t. real pos rob 3 10 15 10 5 5 0 0 0 20 40 60 80 0 20 40 60 80 estimate errors w.r.t. real pos rob 4 8 6 4 2 0 0 20 40 60 80 100 Antonelli, Arrichiello, Caccavale, Marino Benevento, 12 September 2012
- 23. Experiments - task error 0.6 centroid error 0.5 0.4 0.3 0.2 0.1 0 −0.1 0 10 20 30 40 50 60 70 80 1.5 formation error 1 0.5 0 −0.5 −1 −1.5 0 10 20 30 40 50 60 70 80 Antonelli, Arrichiello, Caccavale, Marino Benevento, 12 September 2012
- 24. Experiments - path estimate (thin) and real path seen from 4 5 4.5 4 3.5 intentional large 3 initial error in the 2.5 state estimate 2 1.5 1 0.5 0.5 1 1.5 2 2.5 Antonelli, Arrichiello, Caccavale, Marino Benevento, 12 September 2012
- 25. Cena I-RAS Robotics & Automation Society, Italian chapter La Locanda dei Mestieri Piazza Piano di Corte, ore 20.30 Antonelli, Arrichiello, Caccavale, Marino Benevento, 12 September 2012
- 26. A decentralized controller-observer scheme for multi-robot weighted centroid tracking Gianluca Antonelli† , Filippo Arrichiello† , Fabrizio Caccavale⊕ , Alessandro Marino‡ in alphabetical order † University of Cassino and Southern Lazio, Italy http://webuser.unicas.it/lai/robotica ⊕ University of Basilicata, Italy http://www.difa.unibas.it ‡ University of Salerno, Italy http://www.unisa.it Antonelli, Arrichiello, Caccavale, Marino Benevento, 12 September 2012