Readiness in Formation Control of Multi-Robot System

Transcription

1 Readiness in Formation Control of Multi-Robot System Zhihao Xu (Univ. of Würzburg) Hiroaki Kawashima (Kyoto Univ.) Klaus Schilling (Univ. of Würzburg)

2 Motivation Scenario: Vehicles maintain an equally distributed formation on a circle with a moving object always at its centroid. Desired distances What motion of the vehicles is the best to track the target object in formation? Spiral motion? Target object (assume that movement is unpredictable) Which initial headings give the best response to the arbitrary movement of the target object? How much the formation is ready for any perturbation?

3 Outline Formation control Distance-based formation control Unicycle model for individual agents Readiness Definition from a general viewpoint Readiness optimization Case study Formation control with unicycle models Optimal readiness Conclusion

4 Outline Formation control Distance-based formation control Unicycle model for individual agents Readiness Definition from a general viewpoint Readiness optimization Case study Formation control with unicycle models Optimal readiness Conclusion

5 Distance-Based Formation Control Interaction rule for follower agent i : Underlying graph p i R 2 : position p i t = j N i E ij p i p j p i T = w ij p i p j (p i p j ) j N i E ij : pair-wise edge-tension energy E ij p i p j = 0 if p i p j = d ij weighted consensus protocol ( w ij depends on p i p j ) Energy E ij d ij : desired distance between i and j Weight w ij d ij p i p j p i p j 5

6 Leader-Follower Formation Control Assume one agent (leader) can be arbitrarily controlled Remaining agents (followers) obey the original interaction rule p = p 1 p 2 p N p f R 2N f p l R 2 (N f followers) Single integrator p f t = E p f, p l p f p l t : input to the network T N N E p(t) = 1 2 E ij p i p j i=1 j=1 E ij = 0 if i, j E (edgeset) 6

7 Unicycle Model Model i: follower s ID (i = 1,, N f ) p i : position θ i : heading direction v i : linear velocity ω i : angular velocity We want to realize Off-center point control Consider p i = μ i instead of p i = μ i p i ε θ i p i

8 Dynamics of Unicycle Formation Control Dynamics (stacked vector form) 8

9 Outline Formation control Distance-based formation control Unicycle model for individual agents Readiness Definition from a general viewpoint Readiness optimization Case study Formation control with unicycle models Optimal readiness Conclusion

10 Properties for Leader-Follower Formation Control Point-to-point (formation control) property Controllability [Rahmani,Mesbahi&Egerstedt SIAM09] Instantaneous/short-term response of formation Manipulability [Kawashima&Egerstedt CDC11] Responsiveness [Kawashima,Egerstedt,Zhu&Hu CDC12] Dependent on the single-integrator model of mobile agents. How to characterize the headings of nonholonomic (e.g. unicycles) agents in terms of shot-term response? Readiness of multi-robot formation

11 Response to the leader s perturbation Dynamics (stacked vector form) x = f(x(t), u) Initial condition p f (0) p l (0) = p f p l + δp l x 0 = x 0 p = p f p l satisfies desired distances Initial headings of the followers Perturbation on the leader (fixed over t [0, T]) x 0 = argmin x0 J x 0 We want to find best θ f0 that minimizes the edge-tension Jenergy x 0 : cost E(p functional T ) for arbitrary θ l 11

12 Readiness from general viewpoint Readiness is an index to describe how well the initial condition is prepared for a variety of possible disturbances (inputs) Given the dynamics of the agents: (Ex.) Unicycle formation case with the initial condition x 0 = x 0 R d, where u U is an exogenous input which is constant in short interval [0, T] Readiness for the response in interval [0, T] is characterized by (Ex.) Unicycle formation case,

13 Optimal initial condition Readiness Optimization Optimality Conditions (derived by the calculus of variations) Optimization via gradient descent: Costate equation with the costate

14 Outline Formation control Distance-based formation control Unicycle model for individual agents Readiness Definition from a general viewpoint Readiness optimization Case study Formation control with unicycle models Optimal readiness Conclusion

15 Case Study: Unicycle formation Perturbation (input) on leader, Optimization of followers headings Terminal cost Gradient descent started by tangential θ f

16 Cost Comparison (1): Cost J for Readiness Comparison of cost J with other headings Random Optimal

17 Cost Comparison (2): Distribution over θ l Uniform Headings Low Readiness Optimal Initial Headings High Readiness

18 Conclusion Readiness notion to characterize the initial condition of nonholonomic multi-agent systems Optimality condition (first-order necessary condition) Future Work Utilize the readiness to optimize both headings and positions Investigate optimization algorithm which can find global minima Acknowledgement: Dr. Egerstedt (Georgia Inst. of Tech.) for his valuable suggestions on the work.