Formation Control with Virtual Leaders and Reduced Communications

1. Formation Control with Virtual Leaders and Reduced Communications • Xiaorui Xi and Eyad H. Abed • Department of Electrical and Computer Engineering • and the Institute for Systems Research • University of Maryland, College Park, MD, USA abed@umd.edu Workshop on Swarming in Natural and Engineered Systems Napa Valley, California August 3-4, 2005

2. Outline • Introduction to flocking • Background • Objective and model • Pre-specified formation design • Emergent formation design • Bifurcation and adaptation • Conclusions and future work

3. Background • Rules for moving together - Okubo’85, Reynolds’87 • Separation: avoiding collision with neighboring agents • Alignment: matching velocity with neighboring agents • Cohesion: staying close to neighboring agents • Possibilities for agent motion protocols: • Leader / followers • Virtual leaders / followers (Leonard/Fiorelli ’01) • No - leader Craig W. Reynolds (1987)

4. Background, cont’d. Virtual leaders (“virtual beacons”) • Introduced by Leonard/Fiorelli (IEEE CDC ’01) • Includes inter-agent long range attractive, short range repulsive forces • Virtual leaders are moving reference points • They are used to manipulate group geometry and direct the motion of the group

5. Objectives • Achieve distributed control laws for a group of agents for: • formation maintenance • path following • Study emergent formations and pre-specified formations • Reduce inter-agent communications • Equilibrium and stability analysis for designs • Use bifurcations for task changing

6. Model • N identical mobile autonomous agents in the plane • The dynamics of each agent: • The neighborhood Ni(t) of agent i is defined as a circle of radius daround agent i • Pre-define a desired path p(t)with end point at the position of the target: p(t) = rt , for t ≥ T

7. Pre-specified formation design – introduction • N virtual leaders: rid = p(t) +bi, i = 1, … , N, where • Pre-define the values of bi s to specify the desired formation • Desired formation can rotate and translate • The values of bi s must • satisfy the following two • conditions: • If agent i and j are • neighbors, then • (2) Two pre-specified desired formations

8. Pre-specified formations – introduction (cont’d) • The dynamics of each agent is • f : the magnitude of the attractive force between the agent and its virtual leader. • g : the magnitude of the repulsive force between two neighboring agents • Ni(t) : the label set of agents i’s neighbors at time t • Change the system states to , and analyze the following new system d (1)

9. Pre-specified formation design – analysis Proposition : Consider system (1) with , where q is a constant vector, and assume f is a linear function, i.e. f (||z||) = k||z|| , where k is a positive constant. Then every solution of this system converges asymptotically to an equilibrium of this system. Proof: Use the following Lyapunov function and the LaSalle’s theorem. Details omitted. Remark: from a large number of simulations and analysis of many special cases, we found that there is only one equilibrium point which is stable. It is for all i, where is the solution of

10. Pre-specified formation design – 3 agents • Example: Suppose we have a 3-agent group with the desired formation is an equilateral triangle which is specified by , and . In equation (1), we let d = 10, • Numerically solving equation (1) for i = 1,2,3, we can get eight equilibria. Linearization of equation (1) near these equilibria tells us that this system has only one (locally) stable equilibrium point -- all others are saddle points.

11. Pre-specified formation: Equilibria and eigenvalues Evals: [-1 -1 -1 -1 -1 -1] Evals: [-13.6375 3.4971 -1.0996 -1 -0.9968 -1] Evals: [-13.6375 3.4971 -1.0996 -1 -0.9968 -1] Evals: [-13.6375 3.4971 -1.0996 -1 -0.9968 -1]

12. Pre-specified formation: Equilibria and eigenvalues (cont’d) Evals: [-18.6569 -11.8229 4.3782 2.5550 -1 -1] Evals: [-18.6569 -11.8229 4.3782 2.5550 -1 -1] Evals: [-18.6569 -11.8229 4.3782 2.5550 -1 -1] Evals: [-13.1048 -5.0025 -5.0024 -1 -1 3.0998]

13. Pre-specified formation design – analysis (cont’d) Path tracking:Consider system (1) with p(t) a smooth slowly varying curve with , and f (||z||) = k||z||. Divide the time interval (0,T) into a union of short intervals along each of which q(t) can be taken as a constant vector qj. Also, let k and the function g be chosen such that in each interval [tj-1, tj], →aj as t→tj , for all i, where aj solves . Then the trajectory of every solution of this system converges to the trajectory a(t), where a(t)solves

14. Pre-specified formation design – simulation • A 6-agent group: 6 virtual leaders, , linear f function (with change in two virtual leaders’ paths at a certain time). * agents  virtual leaders o point on desired path + target

15. Pre-specified formation design – simulation Six (6) agents follow their virtual leaders to a target and rotate

16. Pre-specified formation design – simulation Speeds of the 6 agents Trajectories of the 6 agents

17. Pre-specified formation design – simulation • Simulation of 6-agent group with 6 virtual leaders, and nonlinear f function. * agents  virtual leaders - desired path

18. Pre-specified formation design – simulation Trajectories of the 6 agents, the desired path (dash-dot line) and four snapshots of the agents’ positions at t = 0, 15, 30, 50. Speeds of the 6 agents and virtual leaders (dash-dot line)

19. Emergent formation design – introduction Connected formation aroundr0: A formation is called a connected formation around r0 = (x0,y0)T, If it is a connected formation, and for any straight defined by y = kx + b passing r0, if there is an agent i at ri = (xi,yi)T that satisfies yi≥kxi + bi, then there must exist at least one agent, say agent j at rj = (xj,yj)T, that satisfies yi≤ kxi+bi. Connected formation around r0 Quasi-connected formation aroundr0: A formation is called a quasi-connected formation around r0 = (x0,y0)T, if it contains multiple connected sub-formations, and each of them is a connected formation round r0. Quasi-connected formation around r0

20. Emergent formations • A single virtual leader: . • The dynamics of each agent is • Here, an achieved formation is an emergent formation aroundr0 . • Theorem: A formation emergent from (2) with p(t) = constant satisfies the following three conditions: • There is no collision between any pair of agents. • Every agent has at least one neighbor, except possibly the one at r0. • The formation is a “connected formation aroundr0 ,” or a “quasi-connected formation aroundr0.” (2)

21. Emergent formations, cont’d. • Note: Here, there is no pre-specified desired formation in the design. Rather, all agents tend to squeeze toward the position r0 under the attractive forces. However, the distances between neighboring agents are controlled through the repulsive forces so that a balanced emergent formation results.

22. Emergent formations– simulation • Simulation of a 30-agent group with one virtual leader, , and linear f function.  agent o virtual leader (point on desired path) + target

23. Emergent formations - trajectories Trajectories of the 30 agents Speeds of the agents

24. Pre-specified formation design with blind areas • Now use the control laws where . • Change the system states to , giving the system (3)

25. Pre-specified formation design with blind areas (cont’d) • Define Bi(t), agent i’s blind area at time t, as the region inside the circle of radius α around the point : where is a constant, and is the unique solution of (Note that is also an equilibrium point of equation (3).)

26. Pre-specified formation design with blind areas (cont’d) * Virtual leader • Agent When agent i is inside its blind area: = 0 and When agent i is outside its blind area: = 1 and (Repulsive forces OFF.) (Repulsive forces ON.)

27. Pre-specified formation design with blind areas, cont’d. • Proposition:Considersystem (3) with , where q is a constant vector, suppose function fis a convex function and agent i is inside its blind area. Then every solution of this system converges asymptotically to the equilibrium point of the system. Proof:By Lyapunov stability theory. • Proposition:With this design, for each agent with where q is a constant vector, once the agent enters its blind area, it will never exit. Proof: Inside the blind area, the only force acting on the agent is the attractive force exerted by its virtual leader. Note: Path following for slowly varying paths can be achieved as before, by using short time intervals along which the path is approximated as constant, and ensuring convergence at each such point.

28. Pre-specified formation design with blind areas, cont’d. P vs. α (for ω= 0.7) P vs. ω (for α = d/4) P: Percentage savings in sensor power using blind area design over the pre-specified formation design without blind areas. (Here sensor power savings is measured by savings in time with sensors ON.) (Note: These figures based on a closely related blind area design, allowing modulation of attractive as well as repulsive forces. Simulation of 1000 random initial conditions and 6 agents.)

29. Pre-specified formation design with blind areas, cont’d. P vs. α (for ω= 0.7) P vs. ω (for α = d/4) Note: These simulation results assume uniformly distributed initial conditions of all the agents at time 0. If the initial positions are known to be close to the virtual leaders but still randomly distributed, the power savings are found to be even greater (95%).

30. Bifurcation and adaptation • Marginally stable swarms are much more agile than very stable configurations (Bonabeau, 1996). • Thus, operation close to the stability boundary facilitates switching behavior through small changes in control parameters. • Design of bifurcations into swarm dynamics facilitates easy transition from straight flight to hovering to circling for reconnaisance, etc.

31. Bifurcation and adaptation, cont’d. Pitchfork bifurcation induces swarm splitting

32. Bifurcation and adaptation, cont’d. Stable Hopf bifurcation  Stable allows swarm hovering and settling

33. Conclusions and future work • Two formation control laws with virtual leaders are designed for the flocking and path following of a group of mobile autonomous agents. • In the pre-specified formation design, the configuration of the virtual leaders group coincides with a desired formation. • The absence of attractive forces between neighboring agents and repulsive forces within blind areas is a new feature. This may reduce sensing requirements in future designs aiming at a reduced level of communication among agents. • Bifurcations offer a systematic means for task changing for a group. • We have also begun a influence of noise on swarm behavior to study resilience to possible jamming by an adversary. Noise-induced transitions? • A theory of intermittent communications and control for swarms is needed.