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A decentralized controller-observer scheme for multi-robot weighted centroid tracking, Gianluca Antonelli, Filippo Arrichiello, Fabrizio Caccavale, Alessandro Marino
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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
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  • 1. A decentralized controller-observer scheme for multi-robot weighted centroid trackingGianluca Antonelli , Filippo Arrichiello ,Fabrizio Caccavale , Alessandro Marino in alphabetical order University of Cassino and Southern Lazio, Italyhttp://webuser.unicas.it/lai/robotica University of Basilicata, Italyhttp://www.difa.unibas.it University of Salerno, Italy http://www.unisa.it Antonelli, Arrichiello, Caccavale, Marino Benevento, 12 September 2012

2. General objectiveIn a multi-robot scenario local information local communication global task local controller time-varying topology Antonelli, Arrichiello, Caccavale, Marino Benevento, 12 September 2012 3. SketchDecentralized controller-observer for weighted centroid trackingTime-varying reference for weighted centroid(formation as centroid+displacement)Each robot estimates the collective state(i.e., robots positions)Convergence proof forrst-order dynamicscontinuous-timexed/switching communication topologiesdirected/undirected graphssaturated inputs Antonelli, Arrichiello, Caccavale, Marino Benevento, 12 September 2012 4. SketchDecentralized controller-observer for weighted centroid trackingTime-varying reference for weighted centroid(formation as centroid+displacement)Each robot estimates the collective state(i.e., robots positions)Convergence proof forrst-order dynamicscontinuous-timexed/switching communication topologiesdirected/undirected graphssaturated inputs Antonelli, Arrichiello, Caccavale, Marino Benevento, 12 September 2012 5. SketchDecentralized controller-observer for weighted centroid trackingTime-varying reference for weighted centroid(formation as centroid+displacement)Each robot estimates the collective state(i.e., robots positions)Convergence proof forrst-order dynamicscontinuous-timexed/switching communication topologiesdirected/undirected graphssaturated inputs Antonelli, Arrichiello, Caccavale, Marino Benevento, 12 September 2012 6. ModelingN robots with n DOFs each:Single state: xi Rn Individual dynamics: xi = ui (single-integrator dynamics) T 1TCollective state: x = xT . . . xN RN n Collective dynamics: x = uGlobal estimate computed by robot i: i x RN n 1 x 1xx 2x x 2x 2 Collective estimation error: x = . = . RN n . . .. Nx x Nx Antonelli, Arrichiello, Caccavale, Marino Benevento, 12 September 2012 7. Problem statementTask (weighted centroid)N(x) = i xi = T I n x Rn i=1Design goals, for each robot:state observer providing an estimate, i x RN n , asymptoticallyconvergent to the collective state xfeedback control law, ui = ui (xi , i x, Ni ) Rn , such that (x)asymptotically converges to a time-varying reference, d (t)Each robot knows in advance: d (t), d (t) Antonelli, Arrichiello, Caccavale, Marino Benevento, 12 September 2012 8. Proposed approach -1-i th control law: iui = ui (i x) = d + kc d (i x) 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: iui = ui (i x) = d + kc d (i x) 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 feedbackconsensus-like term i th state observer: ix = ko j x ix + i x ix + iu jNi i = diag O n I n O n u1 (i x) . = j + k (i x) . , uj ( )iu =. ix 2d cdi x) uN ( collective input estimated by robot i Antonelli, Arrichiello, Caccavale, Marino Benevento, 12 September 2012 11. Proposed approach -2-local feedbackconsensus-like term i th state observer: ix = ko j x ix + i x ix + iu jNi i = diag O n I n O n u1 (i x) . = j + k (i x) . , uj ( )iu =. ix 2d cdi x) uN ( collective input estimated by robot i Antonelli, Arrichiello, Caccavale, Marino Benevento, 12 September 2012 12. Proposed approach -2-local feedbackconsensus-like term i th state observer: ix = ko j x ix + i x ix + iu jNi i = diag O n I n O n u1 (i x) . = j + k (i x) . , uj ( )iu =. ix 2d cdi x) uN ( collective input estimated by robot i Antonelli, Arrichiello, Caccavale, Marino Benevento, 12 September 2012 13. Proposed approach -2-local feedbackconsensus-like term i th state observer: ix = ko j x ix + i x ix + iu jNi i = diag O n I n O n u1 (i x) . = j + k (i x) . , uj ( )iu =. ix 2d cdi x) uN ( collective input estimated by robot i Antonelli, Arrichiello, Caccavale, Marino Benevento, 12 September 2012 14. Collective dynamicsEstimation error:x = ko (L I N n + ) x + (1N I N n ) u u with L Laplacian matrix embedding the topologyTracking error: N kc = kc 2 T I n i xi 2 i=1 Antonelli, Arrichiello, Caccavale, Marino Benevento, 12 September 2012 15. Stability proof for undirected connected topologiesLyapunov function: 11V ( , ) = xT x + T x 22after straightforward computations. . . N kc n ko m N kc n x x V ( , ) x 2 N kc n kc2 Antonelli, Arrichiello, Caccavale, Marino Benevento, 12 September 2012 16. Stability proof for undirected connected topologiesV is negative denite with a proper choice of the design gains ko and kc : 2kcNn ko > Nn+m4comments:N , n and are known parametersthe control gain kc is free (altough positive)the term m 0 is embedding the connection properties(null for unconnected graphs)(not surprisingly) the observer gain ko is lower bounded Antonelli, Arrichiello, Caccavale, MarinoBenevento, 12 September 2012 17. Extensions -1- Directed topologies Switching topologiesconvergence for balancedproof by the concept ofand strongly connectedCommon LyapunovgraphsFunctionproof by resorting to the gains tuned on the worstconcept of mirror graph caseAll the case studies above analyzed also for saturated inputs Antonelli, Arrichiello, Caccavale, Marino Benevento, 12 September 2012 18. Extensions -2-Centroid and formation N11 (x) =xiNi=1T2 (x) = (x2 x1 )T (x3 x2 )T . . . (xN xN 1 )TSolved and analized by resorting to a similar controller-observer schemeand 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 inputs7 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 localizationobstacle avoidancevarious topologiesinitial errorAntonelli, Arrichiello, Caccavale, Marino Benevento, 12 September 2012 22. Experiments - estimation errors estimate errors w.r.t. real pos rob 0estimate errors w.r.t. real pos rob 1 48 36 24 12 00 02040 6080 020 406080 estimate errors w.r.t. real pos rob 2estimate errors w.r.t. real pos rob 310 15 10 55 00 02040 6080 020 406080 estimate errors w.r.t. real pos rob 4 8 6 4 2 0 020 406080100 Antonelli, Arrichiello, Caccavale, Marino Benevento, 12 September 2012 23. Experiments - task error 0.6centroid error 0.5 0.4 0.3 0.2 0.100.1 0 10 2030 405060 70 80 1.5 formation error1 0.500.5 11.5 0 10 2030 405060 70 80 Antonelli, Arrichiello, Caccavale, MarinoBenevento, 12 September 2012 24. Experiments - path estimate (thin) and real path seen from 45 4.54 3.5intentional large3initial error in the 2.5state estimate2 1.51 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 chapterLa Locanda dei MestieriPiazza 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 trackingGianluca Antonelli , Filippo Arrichiello ,Fabrizio Caccavale , Alessandro Marino in alphabetical order University of Cassino and Southern Lazio, Italyhttp://webuser.unicas.it/lai/robotica University of Basilicata, Italyhttp://www.difa.unibas.it University of Salerno, Italy http://www.unisa.it Antonelli, Arrichiello, Caccavale, Marino Benevento, 12 September 2012


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