A Control Lyapunov Function Approach to Multi Agent Coordination

1. A Control Lyapunov Function Approach to Multi Agent Coordination P. Ögren, M. Egerstedt* and X. Hu Royal Institute of Technology (KTH), Stockholm and Georgia Institute of Technology* IEEE Transactions on Robotics and Automation, Oct 2002

2. Motivation: Flexibility Robustness Price Efficiency Feasibility Applications: Search and rescue missions Spacecraft inferometry Reconfigurable sensor array Carry large/awkward objects Formation flying Multi Agent Robotics

3. Problem and Proposed Solution • Problem: How to make set-point controlled robots moving along trajectories in a formation ”wait” for eachother? • Idea: Combine Control Lyapunov Functions (CLF) with the Egerstedt&Hu virtual vehicle approach. • Under assumptions this will result in: • Bounded formation error (waiting) • Approx. of given formation velocity (if no waiting is nessesary). • Finite completion time (no 1-waiting).

4. Quantifying Formation Keeping Definition: Formation Function Will add Lyapunov like assumption satisfied by individual set-point controllers. => Think of as parameterized Lyapunov function.

5. Examples of Formation Function • Simple linear example ! • A CLF for the combined higher dimensional system: • Note that a,b, are design parameters. • The approach applies to any parameterized formation scheme with lyapunov stability results.

6. Main Assumption • We can find a class K function s such that the given set-point controllers satisfy: • This can be done when -dV/dt is lpd, V is lpd and decrescent. It allows us to prove: • Bounded V (error): V(x,s) < VU • Bounded completion time. • Keeping formation velocity v0, if V ¿ VU.

7. Speed along trajectory: How Do We Update s? • Suggestion: s=v0 t • Problems: Bounded ctrl or local ass stability We want: • V to be small • Slowdown if V is large • Speed v0 if V is small Suggestion: • Let s evolve with feedback from V.

8. Evolution of s Choosing to be: We can prove: • Bounded V (error): V(x,s) < VU • Bounded completion time. • Keeping formation velocity v0, if V ¿ VU.

9. Proof sketch: Formation error

10. Proof sketch: Finite Completion Time Find lower bound on ds/dt

11. The Unicycle Model, Dynamic and Kinematic Beard (2001) showed that the position of an off axis point x can be feedback linearized to:

12. Example: Formation • Three unicycle robots along trajectory. • VU=1, v0=0.1, then v0=0.3 ! 0.27 • Stochastic measurement error in top robot at 12m mark.

13. Extending Work by Beard et. al. ”Satisficing Control for Multi-Agent Formation Maneuvers”, in proc. CDC ’02 • It is shown how to find an explicit parameterization of the stabilizing controllers that fulfills the assumption • These controllers are also inverse optimal and have robustness properties to input disturbances • Implementation

14. What if dV/dt <= 0 ? • If we have semidefinite and stability by La Salle’s principle we choose as: • By a renewed La Salle argument we can still show: V<=VU , s! sf and x! xf. • But not: Completion time and v0.

15. Edward Fiorelli and Naomi Ehrich Leonard eddie@princeton.edu, naomi@princeton.edu Petter Ogren petter@math.kth.se Mechanical and Aerospace Engineering Princeton University, USA Optimization and Systems Theory Royal Institute of Technology, Sweden Another extension: Formations with a Mission: Stable Coordination of Vehicle Group Maneuvers Mathematical Theory of Networks and Systems (MTNS ‘02) Visit:http://graham.princeton.edu/for related information

16. Approach: Use artificial potentials and virtual body with dynamics. • Configuration space of virtual body is • for orientation, position and expansion factor: • Because of artificial potentials, vehicles in formation will translate, rotate, expand and contract with virtual body. • To ensure stability and convergence, prescribe virtual body • dynamics so that its speed is driven by a formation error. • Define direction of virtual body dynamics to satisfy mission. • Partial decoupling: Formation guaranteed independent of mission. • Prove convergence of gradient climbing.

17. Conclusions • Moving formations by using Control Lyapunov Functions. • Theoretical Properties: • V <= VU, error • T < TU, time • v ¼ v0 velocity • Extension used for translation, rotation and expansion in gradient climbing mission

18. Related Publications A Convergent DWA approach to Obstacle Avoidance • Formally validated • Merge of previous methods using new mathematical framework Obstacle Avoidance in Formation • Formally validated • Extending concept of Configuration Space Obstacle to formation case, thus decoupling formation keeping from obstacle avoidance