Responsiveness

Responsiveness and Manipulability of Formation of Multi-Robot NetworksHiroaki Kawashima*1, Guangwei Zhu*2, Jianghai Hu*2, Magnus Egerstedt*3*1 Kyoto University*2 Purdue University*3 Georgia Institute of TechnologyCDC2012Session: Network Identification and Analysis 11

Energy

Distance-Based Formation ControlInteraction rule for follower agent i :

2

: pair-wise edge-tension energy

weighted consensus protocol ( depends on )

: state (position)Underlying graph

Weight: desired distance between i and j2MotivationAssume some agents (leaders) can be arbitrarily controlledRemaining agents (followers) obey the original interaction ruleLeaders motion is considered as the inputs to the network

3

How the network can be adaptively changed to maximize the effectiveness of leaders inputs?

How can we measure the impact of leaders input to the network?3Evaluate Leaders Influence on the NetworkReachability/controllability [Rahmani,Mesbahi&Egerstedt 2009]Global point-to point property (arbitrary configurations)Long-term indexInstantaneous network responseLocal property (depends on a particular configuration)Short-term index4The instantaneous index can be more useful under a dynamic situation (e.g., change the topology adaptively to maximize the leaders effect)Manipulability index of leader-follower networks[Kawashima&Egerstedt, CDC2011]Responsiveness4

r

Robot-arm manipulability[Yoshikawa 1985, Bicchi, et al. 2000]

Leader-follower manipulabilityKinematic relationVelocity of end-effector is directly connected with the angular velocityleaders vel.followers vel.angular velocityof jointsend-effectorvelocityWe need integral action w.r.t. time to get the followers response5

Dynamics of agents5Approximation of the Dynamics

Constraint on each connected agent pair6

Rewrite with the rigidity matrix [B.Roth 1981]Under this approximation, 6Manipulability under the Rigid-link ApproximationTheorem [Kawashima&Egerstedt CDC2011]Approximate manipulabilityManipulability of leader-follower networksGiven the rigidity matrix 7

7Application Example: Effective TopologyGiven leaders input direction, what is the most effective network topology in terms of maximizing the manipulability?8

Topology optimizationAfter t=2After t=28Application Example: Online Leader Selection9Online leader selection

To move the network toward a target point, which is the most effective leader[Kawashima&Egerstedt, ACC2012]9Limitation of Manipulability1. Manipulability cannot compare rigid formations with the same agent configurations.

2. Manipulability cannot deal with edge weights (i.e., gains of the pair-wised edge-tension energies).10Why? In the rigid-link approximation, we only considered the convergence point of the followers when the leader is moved.

How can we overcome these limitations?10Stiffness and Rigidity Indices11[Zhu&Hu, CDC09, CDC11]It dictates how fast the formation can be recoveredComplementary with the manipulability11ResponsivenessUnifies the notions of stiffness and manipulabilityAnalyze the convergence process of the followers

12non-zero eigenvaluest is often finite (leader cannot wait followers for infinite time)12Responsiveness13Manipuability (convergence point)Stiffness(rate of convergence)Non-zero eigs.J(t)Comparison of Responsiveness14

ttResponsivenessResponsivenessG4G5G1G2G41. Responsiveness can be used to compare rigid formations14Optimal Edge Weights (Link Resource Allocation)ij

Optimal Edge Weights (Link Resource Allocation)Numerical Results

optimizeoptimizeoptimize2. Responsiveness can deal with edge weights16ConclusionWe proposed the responsiveness of leader-follower networks, which unifies the previously defined notions of stiffness and manipulability.It measures the impact of leaders input in terms ofhow fast / how large (finally) followers respond.

Future workApplication to human-swarm interaction (e.g., networked robots adaptively changes gains and topologies to maximize human inputs injected through the leader).17ManipulabilityStiffnessThank you for your attention!17