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
Energ
y
π₯π β π₯π
Distance-Based Formation Control
β’ Interaction rule for follower agent i :
2
: pair-wise edge-tension energy weighted consensus protocol
( depends on )
: state (position)
Underlying graph
πππ W
eig
ht
π₯π β π₯π
: desired distance
between i and j
β°ππ π₯π β π₯π = 0 if π₯π β π₯π = πππ
π₯ π π‘ = β πβ°ππ π₯π β π₯π
ππ₯π
π
πβπ© π
= β π€ππ π₯π β π₯π (π₯π β π₯π)
πβπ© π
Motivation
β’ Assume some agents (leaders) can be arbitrarily controlled β Remaining agents (followers) obey the original interaction rule
β Leadersβ motion is considered as the inputs to the network
3
How the network can be adaptively changed to
maximize the effectiveness of leaderβs inputs?
How can we measure the impact of leaderβs input to the network?
(ππ followers and πβ leaders)
β°ππ = 0 if i, j β π
π₯π β βπππ
π₯β β βπβπ
β° π₯(π‘) =1
2 β°ππ π₯π β π₯π
π
π=1
π
π=1
π₯ π π‘ = β
πβ° π₯π, π₯β
ππ₯π
π
π₯ β π‘ = π’(π‘)
Evaluate Leaderβs Influence on the Network
β’ Reachability/controllability [Rahmani,Mesbahi&Egerstedt 2009]
β Global point-to point property (arbitrary configurations)
β Long-term index
β’ Instantaneous network response
β Local property (depends on a particular configuration)
β Short-term index
4
The instantaneous index can be more useful under a dynamic situation (e.g., change the topology adaptively to maximize the leaderβs effect)
Manipulability index of leader-follower networks [Kawashima&Egerstedt, CDC2011]
Responsiveness
r
Robot-arm manipulability [Yoshikawa 1985, Bicchi, et al. 2000]
Leader-follower manipulability
Kinematic relation
Velocity of end-effector is directly connected with the angular velocity
leadersβ vel.
followersβ vel.
angular velocity of joints
end-effector velocity
We need integral action w.r.t. time to get the followersβ response
5
π ππππ
π ππππ
π =π₯ π
ππππ₯ π
π₯ βππβπ₯ β
π = π π , π =ππ
ππ π
π
π : states of end-effector π : joint angles ππ ,ππ β» 0 : weight matrices
Can we get more direct connection between π₯β π‘ and π₯π π‘ ?
π₯π (π‘) = βπβ° π₯π, π₯β
ππ₯π
π
π₯β π‘ = π’ π‘ : given
Dynamics of agents
Approximation of the Dynamics
Def: Rigid-link approximation [Kawashima&Egerstedt CDC2011]
β Followers are fast enough to perfectly maintain the desired distance πππ β π, π β π βπ‘
Constraint on each
connected agent pair
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β π, π β π
(π: edge set)
β π, π β π
β’ Rewrite with the rigidity matrix [B.Roth 1981]
β’ Under this approximation,
Manipulability under the Rigid-link Approximation
β’ Theorem [Kawashima&Egerstedt CDC2011]
Approximate manipulability Manipulability of leader-
follower networks
Given the rigidity matrix
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π can evaluate the instantaneous effectiveness leaderβs input π₯β by considering current config.(π₯) and network topology (π).
β Provides us the direct connection between π₯β π‘ and π₯π π‘
π π₯ = π π π₯ π β(π₯)
πππ
π₯π π‘ = βπβ° π₯
ππ₯π
π
π₯π π‘ = βπ πβ π β π₯β π‘ π½ π₯, πΈ β βπ π
β π β
πβπ
βΌ
π =π₯π
ππππ₯π
π₯β ππβπ₯β
π (π₯β , π₯, π) =π₯β
ππ½ππππ½π₯β
π₯β ππβπ₯β
βΌ
Application Example: Effective Topology
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»)
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πΏπ₯ (π‘) = βπ»πΏπ₯(π‘), πΏπ₯ 0 = πΏπ₯0 where π» βπ2β°
ππ₯2 π₯=π₯β
[Zhu&Hu, CDC09, CDC11]
It dictates how fast the formation can be recovered
(ππ β₯ ππ+1)
Complementary with the manipulability
Responsiveness
β’ Unifies the notions of stiffness and manipulability β Analyze the convergence process of the followers
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πΏπ₯π (π‘) = βπ»πππΏπ₯π(π‘) β π»πβπΏπ₯β,
π π‘, π₯, πΊ, πΏπ₯β =πΏπ₯π(π‘)
2
πΏπ₯β2
Solve the following zero-state response with πΏπ₯π 0 = 0:
using the fact π»πβ = βπ»ππ πππβ πΌπ and the decomposition π»ππ = ππ¬ππ
πΏπ₯π π‘ = πDiag 1 β πβπππ‘ ππ πππβ πΌπ πΏπ₯β
π¬ = ππ’ππ π1, β¦ , ππ :
π = [π£1, β¦ , π£π]: eigen vec.
non-zero eigenvalues
where π»ππ βπ2β°
ππ₯π2
π₯β, π»πβ β
π2β°
ππ₯πππ₯β π₯β
t is often finite (leader cannot wait
followers for infinite time)
π‘ βΆ β Manipulability π
Responsiveness
β’ Responsiveness takes into account the rate of convergence and the convergence point simultaneously.
β How fast followers respond? (evaluated by given time π‘)
β How much followers finally respond? (evaluated by given input, i.e., leaderβs motion, πΏπ₯β)
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Manipuability
(convergence point)
Stiffness
(rate of convergence)
π π‘, πΏπ₯β, π₯, π =πΏπ₯π(π‘)
2
πΏπ₯β2
=πΏπ₯β
ππ½ π‘ ππ½ π‘ πΏπ₯β
πΏπ₯βππΏπ₯β
= 1 β πβπππ‘ 2π£π
π πππβ πΌπ πΏπ₯β
2
π
π=1
π»ππ =π2β°
ππ₯π2
π₯β
= ππ¬ππ
π¬ = ππ’ππ π1, β¦ , ππ
π = [π£1, β¦ , π£π]
Non-zero eigs.
πΏπ₯π π‘ = πDiag 1 β πβπππ‘ ππ πππβ πΌπ πΏπ₯β
J(t)
Comparison of Responsiveness
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π π‘, πΏπ₯β, π₯, π = 1 β πβπππ‘ 2π£π
π πππβ πΌπ πΏπ₯β
2
π
π=1
t
t
Resp
on
siv
eness
Resp
on
siv
eness
G4
G5
G1
G2
G4
1. Responsiveness can be used to compare rigid formations
Optimal Edge Weights (Link Resource Allocation)
β’ Link weights can be viewed as the amount of resource.
β’ Given limited amount of total resource, find the optimal link resource allocation in terms of maximizing the convergence rate
of the responsiveness ππ π»ππ .
maximizeπππ
ππ π»ππ
subject to πππ2
π,π βπ€
β€ πΆ and πππ = 0 for π, π β π€
β’ π β rank of π»ππ
β’ πΆ β total amount of resource
β’ πππ β weight for link {π, π}
i
j
πππ2
β°ππ
π₯π β π₯π
(ππ β₯ ππ+1)
Optimal Edge Weights (Link Resource Allocation) Numerical Results
optimize optimize optimize
2. Responsiveness can deal with edge weights
Conclusion
β’ We proposed the responsiveness of leader-follower networks, which unifies the previously defined notions of stiffness and manipulability.
β’ It measures the impact of leaderβs input in terms of β how fast / how large (finally) followers respond.
β’ Future work β Application to human-swarm interaction (e.g., networked
robots adaptively changes gains and topologies to maximize human inputs injected through the leader).
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Manipulability Stiffness
π π‘, πΏπ₯β, π₯, π = 1 β πβπππ‘ 2π£π
π πππβ πΌπ πΏπ₯β
2
π
π=1
Thank you for your attention!