Responsiveness and Manipulability of Formation of Multi-Robot Networks Hiroaki Kawashima *1 , Guangwei Zhu *2 , Jianghai Hu *2 , Magnus Egerstedt *3 *1 Kyoto University *2 Purdue University *3 Georgia Institute of Technology CDC2012 Session: Network Identification and Analysis 1
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Responsiveness and Manipulability of Formation of Multi-Robot Networks
Hiroaki Kawashima*1, Guangwei Zhu*2, Jianghai Hu*2, Magnus Egerstedt*3
*1 Kyoto University *2 Purdue University *3 Georgia Institute of Technology
CDC2012
Session: Network Identification and Analysis
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Energ
y
π₯π β π₯π
Distance-Based Formation Control
β’ Interaction rule for follower agent i :
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: pair-wise edge-tension energy weighted consensus protocol
β’ Given leaderβs input direction, what is the most effective network topology in terms of maximizing the manipulability?
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Topology optimization argmaxπ
π (π₯β , π₯, π) =π₯β
ππ½ππππ½π₯β
π₯β ππβπ₯β
(The number of edges,
π , is constrained)
After t=2
After t=2
Application Example: Online Leader Selection
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Online leader selection
β’ To move the network toward a target point, which is the most effective leader
[Kawashima&Egerstedt, ACC2012]
πβ π‘ = argmaxπ
π π(π, π₯ π‘ , π)
Limitation of Manipulability
1. Manipulability cannot compare βrigid formationsβ with the same agent configurations.
2. Manipulability cannot deal with βedge weightsβ (i.e., gains of the pair-wised edge-tension energies).
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Why? In the rigid-link approximation, we only considered the convergence point of the followers when the leader is moved.
π = 3.0
π = 3.0
π = 1.3
How can we overcome these limitations?
Stiffness and Rigidity Indices
β’ Stiffness matrix is defined based on the spring-mass analogy, and it coincides with the Hessian of the edge-tension energy β°, in the formation-control context:
β’ Rigidity indices are defined by the eigenvalues of the Hessian
β Worst-case rigidity index (WRI): smallest eigenvalue of π»: ππ π» , π = rank(π»)