1. Centre for Autonomous Systems Petter ÖgrenCAS talk1 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. Centre for Autonomous Systems Petter ÖgrenCAS talk2 Multi Agent Robotics Motivation: Flexibility Robustness Price Efficiency Feasibility Applications: Search and rescue missions Spacecraft inferometry Reconfigurable sensor array Carry large/awkward objects Formation flying

3. Centre for Autonomous Systems Petter ÖgrenCAS talk3 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. Centre for Autonomous Systems Petter ÖgrenCAS talk4 Quantifying Formation Keeping Will add Lyapunov like assumption satisfied by individual set-point controllers. => Think of as parameterized Lyapunov function. Definition: Formation Function

5. Centre for Autonomous Systems Petter ÖgrenCAS talk5 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. Centre for Autonomous Systems Petter ÖgrenCAS talk6 Main Assumption We can find a class K function  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) < V U Bounded completion time. Keeping formation velocity v 0, if V ¿ V U.

7. Centre for Autonomous Systems Petter ÖgrenCAS talk7 Speed along trajectory: How Do We Update s? Suggestion: s=v 0 t Problems: Bounded ctrl or local ass stability We want: V to be small Slowdown if V is large Speed v 0 if V is small Suggestion: Let s evolve with feedback from V.

8. Centre for Autonomous Systems Petter ÖgrenCAS talk8 Evolution of s Choosing to be: We can prove: Bounded V (error): V(x,s) < V U Bounded completion time. Keeping formation velocity v 0, if V ¿ V U.

9. Centre for Autonomous Systems Petter ÖgrenCAS talk9 Proof sketch: Formation error

10. Centre for Autonomous Systems Petter ÖgrenCAS talk10 Proof sketch: Finite Completion Time Find lower bound on ds/dt

11. Centre for Autonomous Systems Petter ÖgrenCAS talk11 The Unicycle Model, Dynamic and Kinematic Beard (2001) showed that the position of an off axis point x can be feedback linearized to:

12. Centre for Autonomous Systems Petter ÖgrenCAS talk12 Example: Formation Three unicycle robots along trajectory. V U =1, v 0 =0.1, then v 0 =0.3 ! 0.27 Stochastic measurement error in top robot at 12m mark.

13. Centre for Autonomous Systems Petter ÖgrenCAS talk13 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. Centre for Autonomous Systems Petter ÖgrenCAS talk14 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<=V U, s! s f and x! x f. But not: Completion time and v 0.

15. Centre for Autonomous Systems Petter ÖgrenCAS talk15 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 Edward Fiorelli and Naomi Ehrich Leonard eddie@princeton.edu, naomi@princeton.edu Mechanical and Aerospace Engineering Princeton University, USA Optimization and Systems Theory Royal Institute of Technology, Sweden Petter Ogren petter@math.kth.se Another extension:

16. Centre for Autonomous Systems Petter ÖgrenCAS talk16 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. Approach: Use artificial potentials and virtual body with dynamics.

17. Centre for Autonomous Systems Petter ÖgrenCAS talk17 Conclusions Moving formations by using Control Lyapunov Functions. Theoretical Properties: V <= V U, error T < T U, time v ¼ v 0 velocity Extension used for translation, rotation and expansion in gradient climbing mission

18. Centre for Autonomous Systems Petter ÖgrenCAS talk18 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