ICRA 2013 talk 1

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In this paper, a decentralized control strategy for networked multi-robot systems that allows the tracking of the team centroid and the relative formation is presented. The proposed solution consists of a distributed observer-controller scheme where, based only on local information, each robot
estimates the collective state and tracks the two assigned control variables. We provide a formal stability analysis of the observer-controller scheme and we
relate convergence properties to the topology of the connectivity graph. Experiments are presented to validate the approach.

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ICRA 2013 talk 1

  1. 1. Decentralized control of dynamic centroid andformation for multi-robot systemsGianluca 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, Italyhttp://www.unisa.itAntonelli, Arrichiello, Caccavale, Marino Karlsruhe, 8 May 2013
  2. 2. General objectiveIn a multi-robot scenariolocal informationlocal communicationlocal controllertime-varying topology⇒ global taskAntonelli, Arrichiello, Caccavale, Marino Karlsruhe, 8 May 2013
  3. 3. SketchDecentralized controller-observer for dynamic centroid and formationTime-varying reference for weighted centroid and formation asdisplacementEach robot estimates the collective state(i.e., robots positions)Convergence proof forfirst-order dynamicscontinuous-timefixed/switching communication topologiesdirected/undirected graphssaturated inputsAntonelli, Arrichiello, Caccavale, Marino Karlsruhe, 8 May 2013
  4. 4. SketchDecentralized controller-observer for dynamic centroid and formationTime-varying reference for weighted centroid and formation asdisplacementEach robot estimates the collective state(i.e., robots positions)Convergence proof forfirst-order dynamicscontinuous-timefixed/switching communication topologiesdirected/undirected graphssaturated inputsAntonelli, Arrichiello, Caccavale, Marino Karlsruhe, 8 May 2013
  5. 5. SketchDecentralized controller-observer for dynamic centroid and formationTime-varying reference for weighted centroid and formation asdisplacementEach robot estimates the collective state(i.e., robots positions)Convergence proof forfirst-order dynamicscontinuous-timefixed/switching communication topologiesdirected/undirected graphssaturated inputsAntonelli, Arrichiello, Caccavale, Marino Karlsruhe, 8 May 2013
  6. 6. ModelingN robots with n DOFs each:Single state: xi ∈ RnIndividual dynamics: ˙xi = ui (single-integrator dynamics)Collective state: x = xT1 . . . xTNT∈ RNnCollective dynamics: ˙x = uGlobal estimate computed by robot i: i ˆx ∈ RNnCollective estimation error: ˜x⋆=1 ˜x2 ˜x...N ˜x=x − 1 ˆxx − 2 ˆx...x − N ˆx∈ RN2nAntonelli, Arrichiello, Caccavale, Marino Karlsruhe, 8 May 2013
  7. 7. Problem statementTasks (centroid and formation)σ1(x) =1NNi=1xi = J1xσ2(x) = (x2−x1)T (x3−x2)T . . . (xN −xN−1)T T= J2xDesign goals, for each robot:state observer providing an estimate, i ˆx ∈ RNn, asymptoticallyconvergent to the collective state xfeedback control law, ui = ui(xi, i ˆx, Ni) ∈ Rn , such that σ1(x),σ2(x) asymptotically converge to a time-varying reference, σ1,d(t),σ2,d(t)Each robot knows in advance the desired trajectoryAntonelli, Arrichiello, Caccavale, Marino Karlsruhe, 8 May 2013
  8. 8. Proposed approach -1-i th control law:ui(t, iˆx) = ˙σ1,d(t) + k1,c σ1,d(t) − σ1(iˆx) +J†2,i ˙σ2,d(t) + k2,c σ2,d(t) − σ2(iˆx)✏✏✏✏✏✏✏✏✏✏✏✮✟✟✟✟✟✟✟✟✟✙each robot is feeding back its estimate of the collective stateAntonelli, Arrichiello, Caccavale, Marino Karlsruhe, 8 May 2013
  9. 9. Proposed approach -1-i th control law:ui(t, iˆx) = ˙σ1,d(t) + k1,c σ1,d(t) − σ1(iˆx) +J†2,i ˙σ2,d(t) + k2,c σ2,d(t) − σ2(iˆx)✏✏✏✏✏✏✏✏✏✏✏✮✟✟✟✟✟✟✟✟✟✙each robot is feeding back its estimate of the collective stateAntonelli, Arrichiello, Caccavale, Marino Karlsruhe, 8 May 2013
  10. 10. Proposed approach -2-i th state observer:i ˙ˆx = koj∈Nijˆx − iˆx + Πi x − i ˆx + i ˆu✻consensus-like term      ✒local feedback           ✠collective input estimated by robot iΠi = diag On · · · In · · · Oni ˆu =u1(i ˆx)...uN (i ˆx), uj(t, i ˆx) = ˙σ1,d + k1,c σ1,d − 1N 1TN ⊗ Ini ˆx ++J†2,j ˙σ2,d + k2,c σ2,d − σ2(i ˆx)Antonelli, Arrichiello, Caccavale, Marino Karlsruhe, 8 May 2013
  11. 11. Proposed approach -2-i th state observer:i ˙ˆx = koj∈Nijˆx − iˆx + Πi x − i ˆx + i ˆu✻consensus-like term      ✒local feedback           ✠collective input estimated by robot iΠi = diag On · · · In · · · Oni ˆu =u1(i ˆx)...uN (i ˆx), uj(t, i ˆx) = ˙σ1,d + k1,c σ1,d − 1N 1TN ⊗ Ini ˆx ++J†2,j ˙σ2,d + k2,c σ2,d − σ2(i ˆx)Antonelli, Arrichiello, Caccavale, Marino Karlsruhe, 8 May 2013
  12. 12. Proposed approach -2-i th state observer:i ˙ˆx = koj∈Nijˆx − iˆx + Πi x − i ˆx + i ˆu✻consensus-like term      ✒local feedback           ✠collective input estimated by robot iΠi = diag On · · · In · · · Oni ˆu =u1(i ˆx)...uN (i ˆx), uj(t, i ˆx) = ˙σ1,d + k1,c σ1,d − 1N 1TN ⊗ Ini ˆx ++J†2,j ˙σ2,d + k2,c σ2,d − σ2(i ˆx)Antonelli, Arrichiello, Caccavale, Marino Karlsruhe, 8 May 2013
  13. 13. Proposed approach -2-i th state observer:i ˙ˆx = koj∈Nijˆx − iˆx + Πi x − i ˆx + i ˆu✻consensus-like term      ✒local feedback           ✠collective input estimated by robot iΠi = diag On · · · In · · · Oni ˆu =u1(i ˆx)...uN (i ˆx), uj(t, i ˆx) = ˙σ1,d + k1,c σ1,d − 1N 1TN ⊗ Ini ˆx ++J†2,j ˙σ2,d + k2,c σ2,d − σ2(i ˆx)Antonelli, Arrichiello, Caccavale, Marino Karlsruhe, 8 May 2013
  14. 14. Collective dynamics IEstimation error:˙˜x⋆= −ko (L ⊗ INn + Π) ˜x⋆+ (1N ⊗ INn) u − ˆu⋆with L Laplacian matrix embedding the topology and˜x⋆=1 ˜x2 ˜x...N ˜x=x − 1 ˆxx − 2 ˆx...x − N ˆx= 1N ⊗ x − ˆx⋆Antonelli, Arrichiello, Caccavale, Marino Karlsruhe, 8 May 2013
  15. 15. Collective dynamics IITracking error:˙˜σ1 = −k1,c ˜σ1 −k1,cNNi=1J1i˜x −k2,cNNi=1J†2,iJ2i˜x˙˜σ2 = −k2,c ˜σ2 − k2,cJ2Ni=1Γ Ti J†2,iJ2i˜x + −k1,cNJ2Ni=1Γ TiNj=1i˜xAntonelli, Arrichiello, Caccavale, Marino Karlsruhe, 8 May 2013
  16. 16. Stability proof for undirected connected topologiesLyapunov function:V (˜x, ˜σ) =12˜x∗T˜x∗ +12˜σT1 ˜σ1 +12˜σT2 ˜σ2after straightforward computations. . .˙V ≤ −˜x∗˜σ1˜σ2T λo − 2Nkc −kc/2 −Nkc−kc/2 k1,c 0−Nkc 0 k2,c˜x∗˜σ1˜σ2Antonelli, Arrichiello, Caccavale, Marino Karlsruhe, 8 May 2013
  17. 17. Stability proof for undirected connected topologies˙V is negative definite with a proper choice of the design gains ko and kc:ko >1λm2Nkc +k2c4k1,c+Nk2c2k2,ccomments:N is a known parameterthe control gains kc, k1,c, k2,c are free (altough positive)the term λm ≥ 0 is embedding the connection properties(null for unconnected graphs)(not surprisingly) the observer gain ko is lower boundedAntonelli, Arrichiello, Caccavale, Marino Karlsruhe, 8 May 2013
  18. 18. Extensions -1-Directed topologiesconvergence for balancedand strongly connectedgraphsproof by resorting to theconcept of mirror graphSwitching topologiesproof by the concept ofCommon LyapunovFunctiongains tuned on the worstcaseAll the case studies above analyzed also for saturated inputsAntonelli, Arrichiello, Caccavale, Marino Karlsruhe, 8 May 2013
  19. 19. Simulations there is life beyond Lyapunov!Dozens of numerical simulations by changing the key parameters:number of robots Ndimension nnumber of neighbors Nitopology (un-directed, switching)saturated inputs12345678Antonelli, Arrichiello, Caccavale, Marino Karlsruhe, 8 May 2013
  20. 20. Experiments there is life beyond Matlab!5 Khepera III by K-teamreal-time localizationvarious topologiesreal-time comm.obstacle avoidanceinitial errorAntonelli, Arrichiello, Caccavale, Marino Karlsruhe, 8 May 2013
  21. 21. Experiments - estimation errors0 20 40 60 8001234estimate errors w.r.t. real pos rob 00 20 40 60 8002468estimate errors w.r.t. real pos rob 10 20 40 60 800510estimate errors w.r.t. real pos rob 20 20 40 60 80051015estimate errors w.r.t. real pos rob 30 20 40 60 80 10002468estimate errors w.r.t. real pos rob 4Antonelli, Arrichiello, Caccavale, Marino Karlsruhe, 8 May 2013
  22. 22. Experiments - task error0 10 20 30 40 50 60 70 80−0.200.20.40.60 10 20 30 40 50 60 70 80−2−1012centroid errorformation errorAntonelli, Arrichiello, Caccavale, Marino Karlsruhe, 8 May 2013
  23. 23. Experiments - path0 0.5 1 1.5 2 2.5 300.511.522.533.544.555.5Path of all the robots from 0intentional largeinitial error in thestate estimateAntonelli, Arrichiello, Caccavale, Marino Karlsruhe, 8 May 2013
  24. 24. Decentralized control of dynamic centroid andformation for multi-robot systemsGianluca 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, Italyhttp://www.unisa.itAntonelli, Arrichiello, Caccavale, Marino Karlsruhe, 8 May 2013

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