IROS 2011 talk 2 (Filippo's file)

574 views

Published on

G. Antonelli and F. Arrichiello and F. Caccavale and A. Marino, A decentralized controller-observer scheme for weighted centroid tracking, 2011 IEEE/RSJ International Conference on Intelligent Robots and Systems, San Franscisco, CA, pp. 2778--2783, 2011.

Published in: Technology
0 Comments
0 Likes
Statistics
Notes
  • Be the first to comment

  • Be the first to like this

No Downloads
Views
Total views
574
On SlideShare
0
From Embeds
0
Number of Embeds
1
Actions
Shares
0
Downloads
4
Comments
0
Likes
0
Embeds 0
No embeds

No notes for slide

IROS 2011 talk 2 (Filippo's file)

  1. 1. A decentralized controller-observer scheme for multi-robot weighted centroid tracking Gianluca Antonelli1 , Filippo Arrichiello1 , Fabrizio Caccavale2 , Alessandro Marino1 1.University of Cassino, Italy Robotics Research Group of the DAEIMI http://webuser.unicas.it/lai/robotica 2.University of Basilicata, Italy AREA Lab of the DIFA http://www.difa.unibas.itG. Antonelli, F. Arrichiello, F. Caccavale, A. Marino IEEE/RSJ IROS, San Francisco, 28 September 2011
  2. 2. Scenario ◮ Each robot uses only local information from its sensors and ◮ Each robot can communicate with its only neighbors Aim ◮ Make the multi-robot system able to achieve a global task using a decentralized control approach ◮ Approach robust to time-varying communication topologies G. Antonelli, F. Arrichiello, F. Caccavale, A. Marino IEEE/RSJ IROS, San Francisco, 28 September 2011
  3. 3. Outline Decentralized controller-observer for weighted centroid tracking ◮ Develop for each robot a local observer of collective system state ◮ Local estimations used by local control laws ◮ Cooperatively track an assigned time-varying reference for weighted centroid ◮ Convergence proof for fixed/switching communication topologies, as well as for directed/undirected graphs ◮ Validation via numerical simulations G. Antonelli, F. Arrichiello, F. Caccavale, A. Marino IEEE/RSJ IROS, San Francisco, 28 September 2011
  4. 4. Modeling Modeling for a team of N robots ◮ Individual robot state: xi ∈ IRn T ◮ Collective state: x = xT 1 . . . xT N ∈ IRNn ◮ Global estimate computed by agent i: i x ∈ IRNn ˆ 1   1  ˜ x x− ˆ x  2˜   x − 2ˆ   x  x ◮ Estimation error: ˜ =  .  =  .  x  .   .  . . N ˜ x x − Nˆ x ◮ ˙ Individual dynamics: xi = ui (single-integrator dynamics) ◮ ˙ Collective dynamics: x = u G. Antonelli, F. Arrichiello, F. Caccavale, A. Marino IEEE/RSJ IROS, San Francisco, 28 September 2011
  5. 5. Problem statement The task for the system of agents is encoded by the smooth function σ ∈ IRn (weighted centroid) N ◮ σ(x) = i =1 αi xi = αT ⊗ In x Main design goals: ◮ design, for each agent, a state observer providing an estimate, x ∈ IRNn , asymptotically convergent to the collective state x iˆ ◮ to design, for each agent, a feedback control law, ui = ui (xi , i ˆ, Ni ) , such that σ(x) asymptotically converges to a x given (in general time-varying) reference, σ d (t) Each agent knows in advance the goal: σ d (t), σ d (t) ˙ G. Antonelli, F. Arrichiello, F. Caccavale, A. Marino IEEE/RSJ IROS, San Francisco, 28 September 2011
  6. 6. Proposed approach Control law for the ith agent: αi ◮ ui = ui (i ˆ) = x α 2 σ d + kc σ d − σ(i ˆ) ˙ x State observer for the ith agent: ˙ ◮ ix ˆ = ko jˆ x − i ˆ + Πi x − i ˆ x x + iu ˆ j∈Ni u1 (i ˆ)   x where i u =  . , uj (i ˆ) = αj σ d + kc σ d − σ(i ˆ)  .  ˆ . x α 2 ˙ x uN (i ˆ) x and Π i = diag On ··· In ··· On ko , kc > 0 gain to be selected to ensure convergence G. Antonelli, F. Arrichiello, F. Caccavale, A. Marino IEEE/RSJ IROS, San Francisco, 28 September 2011
  7. 7. Proposed approach The estimation error dynamics: ◮ ˙ x = −ko (L ⊗ INn + Π) ˜ + (1N ⊗ INn ) u − u ˜ x ˆ L=Laplacian matrix of the communication graph The task tracking error dynamics: ˙ kc N 2 ◮ σ = −kc σ − ˜ ˜ α 2 i =1 αi αT ⊗ In i x ˜ G. Antonelli, F. Arrichiello, F. Caccavale, A. Marino IEEE/RSJ IROS, San Francisco, 28 September 2011
  8. 8. Stability proof for undirected connected topologies Candidate Lyapunov function: ◮ V (˜, σ) = Vo + Vc = 1 ˜T ˜ + 1 σ T σ x ˜ 2x x 2˜ ˜ Time derivative of Vo along the system’s trajectories: ◮ ˙ Vo = −ko ˜T (L ⊗ INn + Π) ˜ + ˜T ((1N ⊗ INn ) u − u) x x x ˆ (L ⊗ INn + Π) positive definite for undirected and connected topologies ˙ 2 N i T ◮ Vo ≤ −ko λm x ˜ + i =1 ˜ (u x − i u) ˆ where λm is the smallest eigenvalue of (L ⊗ INn + Π) √ ◮ Vo ≤ − ko λm − 2Nkc n ˜ 2 ˙ x G. Antonelli, F. Arrichiello, F. Caccavale, A. Marino IEEE/RSJ IROS, San Francisco, 28 September 2011
  9. 9. Stability proof for undirected connected topologies Time derivative of Vc along the system’s trajectories: √ ◮ Vc = σ T σ ≤ −kc σ 2 + 2Nkc n α /2 σ ˙ ˜ ˜˙ ˜ ˜ ˜ x For the overall derivative of the candidate Lyapunov function, yields: T ˙ λo − ρo −ρc ˜ x ˜ x ◮ V ≤− ˜ σ −ρc kc ˜ σ √ √ where ρo = Nkc n, λo = ko λm and ρc = Nkc n α /2 Negative definite with a proper choice of the design gains ko and kc : √ α 2 ◮ ko > N kc λm n + Nn 4 G. Antonelli, F. Arrichiello, F. Caccavale, A. Marino IEEE/RSJ IROS, San Francisco, 28 September 2011
  10. 10. Directed topologies Stability preserved for balanced and strongly connected directed graph ◮ The Laplacian L is not symmetric ◮ Mirror graph GS = same set of nodes and edges of G but undirected L + LT ◮ Symmetric part of the Laplacian, LS = , valid Laplacian for 2 GS , if and only if G is balanced. If G is strongly connected, then GS is connected Then, it yields: ◮ ˙ Vo = −ko ˜T (LS ⊗ INn + Π) ˜ + ˜T ((1N ⊗ INn ) u − u) x x x ˆ where (LS ⊗ INn + Π) is positive definite. The rest of the proof is analogous G. Antonelli, F. Arrichiello, F. Caccavale, A. Marino IEEE/RSJ IROS, San Francisco, 28 September 2011
  11. 11. Switching topologies Time-varying network topology ◮ Network topology described via finite collection of K graphs of order N, Γ = {G1 , . . . , GK }, each characterized by adjacency matrix Ak , k ∈ I = {1, . . . , K } ◮ Adjacency matrix modeled as function of time A = As(t) , where s(·) : t ∈ IR → I is a switching signal ◮ Previous Lyapunov function is a Common Lyapunov Function for any switching signal s(t), if each graph in Γ balanced and strongly (directed) or simply (undirected) connected ◮ Tuning of ko and kc according to the worst case scenario, i.e., considering minimum value of λm over the finite set of topologies G. Antonelli, F. Arrichiello, F. Caccavale, A. Marino IEEE/RSJ IROS, San Francisco, 28 September 2011
  12. 12. Numerical simulations First case study: fixed undirected topology ◮ As first case study, a fixed undirected topology with 4 vehicles (N = 4) moving in a plane (n = 2) G. Antonelli, F. Arrichiello, F. Caccavale, A. Marino IEEE/RSJ IROS, San Francisco, 28 September 2011
  13. 13. Numerical simulations First case study: fixed undirected topology Estimation error 6 5 4 m 4 2 3 0 1 2 3 4 5 6 7 time [s] 2 Task error m 1 1.5 Sigma 0 1 Sigmad m Start 0.5 −1 End −2 0 0 5 10 15 20 25 1 2 3 4 5 6 7 8 m time [s] G. Antonelli, F. Arrichiello, F. Caccavale, A. Marino IEEE/RSJ IROS, San Francisco, 28 September 2011
  14. 14. Numerical simulations Second case study: directed switching topology ◮ As second case study, a switching directed topology with 8 vehicles (N = 8) moving in the 3D-space (n = 3) t=2s t=4s G. Antonelli, F. Arrichiello, F. Caccavale, A. Marino IEEE/RSJ IROS, San Francisco, 28 September 2011
  15. 15. Numerical simulations Second case study: directed switching topology 7 Estimation error 4 6 5 3 4 m 3 2 2 1 z [m] 1 0 0 1 2 3 4 5 6 7 0 time [s] Sigma Task error −1 Sigmad 0.5 Start −2 0.4 6 End 0.3 m 4 0.2 30 2 25 0.1 20 15 0 0 10 5 0 −0.1 −2 −5 0 1 2 3 4 5 6 7 y [m] x [m] time [s] G. Antonelli, F. Arrichiello, F. Caccavale, A. Marino IEEE/RSJ IROS, San Francisco, 28 September 2011
  16. 16. Conclusions ◮ Decentralized controller-observer approach for a multi-robot system ◮ Each robot estimates the collective state of the system by using only local information ◮ The estimated state is used by the individual robots to cooperatively track a global assigned time-varying task function ◮ Stability proof presented for the cases of undirected connected topologies, strongly connected and balanced directed graphs, as well as for switching topologies ◮ Validated in numerical simulations G. Antonelli, F. Arrichiello, F. Caccavale, A. Marino IEEE/RSJ IROS, San Francisco, 28 September 2011
  17. 17. Future works ◮ Extend the class of achievable task functions ◮ Make robust to dynamic lost/addition of robots ◮ Improve the scalability of the approach ◮ Extend to the case of non-holonomic vehciles ◮ Experimental validation with a team of khepera III robots G. Antonelli, F. Arrichiello, F. Caccavale, A. Marino IEEE/RSJ IROS, San Francisco, 28 September 2011

×