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

𝑥𝑖 − 𝑥𝑗

Distance-Based Formation Control

• Interaction rule for follower agent i :

2

: pair-wise edge-tension energy weighted consensus protocol

( depends on )

: state (position)

Underlying graph

𝑑𝑖𝑗 W

eig

ht

𝑥𝑖 − 𝑥𝑗

: desired distance

between i and j

ℰ𝑖𝑗 𝑥𝑖 − 𝑥𝑗 = 0 if 𝑥𝑖 − 𝑥𝑗 = 𝑑𝑖𝑗

𝑥 𝑖 𝑡 = − 𝜕ℰ𝑖𝑗 𝑥𝑖 − 𝑥𝑗

𝜕𝑥𝑖

𝑇

𝑗∈𝒩 𝑖

= − 𝑤𝑖𝑗 𝑥𝑖 − 𝑥𝑗 (𝑥𝑖 − 𝑥𝑗)

𝑗∈𝒩 𝑖

Motivation

• Assume some agents (leaders) can be arbitrarily controlled – Remaining agents (followers) obey the original interaction rule

– Leaders’ motion is considered as the inputs to the network

3

How the network can be adaptively changed to

maximize the effectiveness of leader’s inputs?

How can we measure the impact of leader’s input to the network?

(𝑁𝑓 followers and 𝑁ℓ leaders)

ℰ𝑖𝑗 = 0 if i, j ∉ 𝐄

𝑥𝑓 ∈ ℝ𝑁𝑓𝑑

𝑥ℓ ∈ ℝ𝑁ℓ𝑑

ℰ 𝑥(𝑡) =1

2 ℰ𝑖𝑗 𝑥𝑖 − 𝑥𝑗

𝑁

𝑗=1

𝑁

𝑖=1

𝑥 𝑓 𝑡 = −

𝜕ℰ 𝑥𝑓, 𝑥ℓ

𝜕𝑥𝑓

𝑇

𝑥 ℓ 𝑡 = 𝑢(𝑡)

Evaluate Leader’s Influence on the Network

• Reachability/controllability [Rahmani,Mesbahi&Egerstedt 2009]

– Global point-to point property (arbitrary configurations)

– Long-term index

• Instantaneous network response

– Local property (depends on a particular configuration)

– Short-term index

4

The instantaneous index can be more useful under a dynamic situation (e.g., change the topology adaptively to maximize the leader’s 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

leaders’ vel.

followers’ vel.

angular velocity of joints

end-effector velocity

We need integral action w.r.t. time to get the followers’ response

5

𝑟 𝑇𝑊𝑟𝑟

𝜃 𝑇𝑊𝜃𝜃

𝑚 =𝑥 𝑓

𝑇𝑄𝑓𝑥 𝑓

𝑥 ℓ𝑇𝑄ℓ𝑥 ℓ

𝑟 = 𝑓 𝜃 , 𝑟 =𝜕𝑓

𝜕𝜃 𝜃

𝜃

𝑟 : states of end-effector 𝜃 : joint angles 𝑊𝑟 ,𝑊𝜃 ≻ 0 : weight matrices

Can we get more direct connection between 𝑥ℓ 𝑡 and 𝑥𝑓 𝑡 ?

𝑥𝑓 (𝑡) = −𝜕ℰ 𝑥𝑓, 𝑥ℓ

𝜕𝑥𝑓

𝑇

𝑥ℓ 𝑡 = 𝑢 𝑡 : given

Dynamics of agents

Approximation of the Dynamics

Def: Rigid-link approximation [Kawashima&Egerstedt CDC2011]

– Followers are fast enough to perfectly maintain the desired distance 𝑑𝑖𝑗 ∀ 𝑖, 𝑗 ∈ 𝐄 ∀𝑡

Constraint on each

connected agent pair

6

∀ 𝑖, 𝑗 ∈ 𝐄

(𝐄: edge set)

∀ 𝑖, 𝑗 ∈ 𝐄

• Rewrite with the rigidity matrix [B.Roth 1981]

• Under this approximation,

Manipulability under the Rigid-link Approximation

• Theorem [Kawashima&Egerstedt CDC2011]

Approximate manipulability Manipulability of leader-

follower networks

Given the rigidity matrix

7

𝑚 can evaluate the instantaneous effectiveness leader’s input 𝑥ℓ by considering current config.(𝑥) and network topology (𝐄).

– Provides us the direct connection between 𝑥ℓ 𝑡 and 𝑥𝑓 𝑡

𝑅 𝑥 = 𝑅𝑓 𝑥 𝑅ℓ(𝑥)

𝑁𝑓𝑑

𝑥𝑓 𝑡 = −𝜕ℰ 𝑥

𝜕𝑥𝑓

𝑇

𝑥𝑓 𝑡 = −𝑅𝑓†𝑅ℓ 𝑥ℓ 𝑡 𝐽 𝑥, 𝐸 ≜ −𝑅𝑓

†𝑅ℓ

𝑁ℓ𝑑

∼

𝑚 =𝑥𝑓

𝑇𝑄𝑓𝑥𝑓

𝑥ℓ 𝑇𝑄ℓ𝑥ℓ

𝑚 (𝑥ℓ , 𝑥, 𝐄) =𝑥ℓ

𝑇𝐽𝑇𝑄𝑓𝐽𝑥ℓ

𝑥ℓ 𝑇𝑄ℓ𝑥ℓ

∼

Application Example: Effective Topology

• Given leader’s input direction, what is the most effective network topology in terms of maximizing the manipulability?

8

Topology optimization argmax𝐄

𝑚 (𝑥ℓ , 𝑥, 𝐄) =𝑥ℓ

𝑇𝐽𝑇𝑄𝑓𝐽𝑥ℓ

𝑥ℓ 𝑇𝑄ℓ𝑥ℓ

(The number of edges,

𝐄 , is constrained)

After t=2

After t=2

Application Example: Online Leader Selection

9

Online leader selection

• To move the network toward a target point, which is the most effective leader

[Kawashima&Egerstedt, ACC2012]

𝑖ℓ 𝑡 = argmax𝑖

𝑚 𝑒(𝑖, 𝑥 𝑡 , 𝐄)

Limitation of Manipulability

1. 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).

10

Why? In the rigid-link approximation, we only considered the convergence point of the followers when the leader is moved.

𝑚 = 3.0

𝑚 = 3.0

𝑚 = 1.3

How can we overcome these limitations?

Stiffness and Rigidity Indices

• Stiffness matrix is defined based on the spring-mass analogy, and it coincides with the Hessian of the edge-tension energy ℰ, in the formation-control context:

• Rigidity indices are defined by the eigenvalues of the Hessian

– Worst-case rigidity index (WRI): smallest eigenvalue of 𝐻: 𝜆𝑟 𝐻 , 𝑟 = rank(𝐻)

11

𝛿𝑥 (𝑡) = −𝐻𝛿𝑥(𝑡), 𝛿𝑥 0 = 𝛿𝑥0 where 𝐻 ≜𝜕2ℰ

𝜕𝑥2 𝑥=𝑥∗

[Zhu&Hu, CDC09, CDC11]

It dictates how fast the formation can be recovered

(𝜆𝑘 ≥ 𝜆𝑘+1)

Complementary with the manipulability

Responsiveness

• Unifies the notions of stiffness and manipulability – Analyze the convergence process of the followers

12

𝛿𝑥𝑓 (𝑡) = −𝐻𝑓𝑓𝛿𝑥𝑓(𝑡) − 𝐻𝑓ℓ𝛿𝑥ℓ,

𝜈 𝑡, 𝑥, 𝐺, 𝛿𝑥ℓ =𝛿𝑥𝑓(𝑡)

2

𝛿𝑥ℓ2

Solve the following zero-state response with 𝛿𝑥𝑓 0 = 0:

using the fact 𝐻𝑓ℓ = −𝐻𝑓𝑓 𝟏𝑁𝑓⊗ 𝐼𝑑 and the decomposition 𝐻𝑓𝑓 = 𝑉𝛬𝑉𝑇

𝛿𝑥𝑓 𝑡 = 𝑉Diag 1 − 𝑒−𝜆𝑘𝑡 𝑉𝑇 𝟏𝑁𝑓⊗ 𝐼𝑑 𝛿𝑥ℓ

𝛬 = 𝐃𝐢𝐚𝐠 𝜆1, … , 𝜆𝑟 :

𝑉 = [𝑣1, … , 𝑣𝑟]: eigen vec.

non-zero eigenvalues

where 𝐻𝑓𝑓 ≜𝜕2ℰ

𝜕𝑥𝑓2

𝑥∗, 𝐻𝑓ℓ ≜

𝜕2ℰ

𝜕𝑥𝑓𝜕𝑥ℓ 𝑥∗

t is often finite (leader cannot wait

followers for infinite time)

𝑡 ⟶ ∞ Manipulability 𝑚

Responsiveness

• Responsiveness takes into account the rate of convergence and the convergence point simultaneously.

– How fast followers respond? (evaluated by given time 𝑡)

– How much followers finally respond? (evaluated by given input, i.e., leader’s motion, 𝛿𝑥ℓ)

13

Manipuability

(convergence point)

Stiffness

(rate of convergence)

𝜈 𝑡, 𝛿𝑥ℓ, 𝑥, 𝐄 =𝛿𝑥𝑓(𝑡)

2

𝛿𝑥ℓ2

=𝛿𝑥ℓ

𝑇𝐽 𝑡 𝑇𝐽 𝑡 𝛿𝑥ℓ

𝛿𝑥ℓ𝑇𝛿𝑥ℓ

= 1 − 𝑒−𝜆𝑘𝑡 2𝑣𝑘

𝑇 𝟏𝑁𝑓⊗ 𝐼𝑑 𝛿𝑥ℓ

2

𝑟

𝑘=1

𝐻𝑓𝑓 =𝜕2ℰ

𝜕𝑥𝑓2

𝑥∗

= 𝑉𝛬𝑉𝑇

𝛬 = 𝐃𝐢𝐚𝐠 𝜆1, … , 𝜆𝑟

𝑉 = [𝑣1, … , 𝑣𝑟]

Non-zero eigs.

𝛿𝑥𝑓 𝑡 = 𝑉Diag 1 − 𝑒−𝜆𝑘𝑡 𝑉𝑇 𝟏𝑁𝑓⊗ 𝐼𝑑 𝛿𝑥ℓ

J(t)

Comparison of Responsiveness

14

𝜈 𝑡, 𝛿𝑥ℓ, 𝑥, 𝐄 = 1 − 𝑒−𝜆𝑘𝑡 2𝑣𝑘

𝑇 𝟏𝑁𝑓⊗ 𝐼𝑑 𝛿𝑥ℓ

2

𝑟

𝑘=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)

• Link weights can be viewed as the amount of resource.

• Given limited amount of total resource, find the optimal link resource allocation in terms of maximizing the convergence rate

of the responsiveness 𝜆𝑟 𝐻𝑓𝑓 .

maximize𝑐𝑖𝑗

𝜆𝑟 𝐻𝑓𝑓

subject to 𝑐𝑖𝑗2

𝑖,𝑗 ∈𝖤

≤ 𝐶 and 𝑐𝑖𝑗 = 0 for 𝑖, 𝑗 ∉ 𝖤

• 𝑟 – rank of 𝐻𝑓𝑓

• 𝐶 – total amount of resource

• 𝑐𝑖𝑗 – weight for link {𝑖, 𝑗}

i

j

𝑐𝑖𝑗2

ℰ𝑖𝑗

𝑥𝑖 − 𝑥𝑗

(𝜆𝑘 ≥ 𝜆𝑘+1)

Optimal Edge Weights (Link Resource Allocation) Numerical Results

optimize optimize optimize

2. Responsiveness can deal with edge weights

Conclusion

• We proposed the responsiveness of leader-follower networks, which unifies the previously defined notions of stiffness and manipulability.

• It measures the impact of leader’s input in terms of – how fast / how large (finally) followers respond.

• Future work – Application to human-swarm interaction (e.g., networked

robots adaptively changes gains and topologies to maximize human inputs injected through the leader).

17

Manipulability Stiffness

𝜈 𝑡, 𝛿𝑥ℓ, 𝑥, 𝐆 = 1 − 𝑒−𝜆𝑘𝑡 2𝑣𝑘

𝑇 𝟏𝑁𝑓⊗ 𝐼𝑑 𝛿𝑥ℓ

2

𝑟

𝑘=1

Thank you for your attention!