1

Cooperative Robot Control and Synchronization ofLagrangian Systems

Soon-Jo Chung and Jean-Jacques E. Slotine

Abstract

This article presents a simple synchronization framework that can be directly applied to cooperative control ofmulti-vehicle systems and oscillation synchronization in robotic manipulation and locomotion. A dynamical networkof multiple Lagrangian systems is constructed by adding diffusive couplings to otherwise freely moving robots orflying vehicles. The proposed decentralized tracking control law synchronizes an arbitrary number of robots intoa common trajectory with global exponential convergence, and is shown to be a generalization of the averageconsensus problem. Exact nonlinear stability results, derived by contraction analysis, provide a fresh perspectiveon the multi-agent coordination and control problem. The proposed decentralized strategy with local couplings canbe systematically applied to Lagrangian systems, and is further extended to time-delayed communications, adaptivesynchronization, partial-joint coupling, and concurrent synchronization of heterogeneous networks.

I. INTRODUCTION

Distributed and decentralized synchronization phenomena are areas of intense research. Group synchro-nization and cooperative control are topics that are currently receiving a lot of interest in a variety ofresearch communities, including biology [15]; artificial intelligence [1], [5]; neuroscience [11], [20], [3];chaotic synchronization for communications [34]; robotics [13], [17], [41]; wireless sensor networks [36],[43]; spacecraft formation flight [10]; automatic control of multi-agent systems such as mobile robots [18],[27], [37]; and distributed systems [2], [30].

In this article, synchronization is defined as a complete match of all configuration variables of eachdynamical system such that x1 = x2 = · · · = xp and p denotes the number of sub-systems in the network.While such a definition directly concerns the attitude tracking problem, this paper further addressessynchronization of biased variables with application to the position coordination problem. In the lattercase, synchronization corresponds to y1 = y2 = · · · = yp where yi, 1 ≤ i ≤ p connotes a vector of biasedvariables constructed from the configuration vector xi such that xi(t) = yi(t) + bi(t) and the separationvector bi(t) is independent of the dynamics.

The objective of this paper is to introduce a unified synchronization framework that can be directlyapplied to cooperative control of multi-robot systems or formation flying aerospace vehicles to tracka common desired trajectory. Although an uncoupled trajectory control law, in the absence of externaldisturbances, would achieve synchronization to a common desired trajectory, non-identical disturbancesjustify the mutual trajectory synchronization. For example, a large swarm of robots can first synchronizetheir attitudes and positions to form a certain formation pattern, then track the common trajectory toaccomplish the given mission. In production processes, such as manufacturing and automotive applications,where high flexibility, manipulability, and maneuverability cannot be achieved by a single system [41],there has been widespread interest in cooperative schemes for multiple robot manipulators that track apredefined trajectory. A stellar formation flight interferometer [10] is another example where precisioncontrol of relative spacecraft motions is indispensable. The proposed synchronization tracking control lawcan be implemented for such purposes, where a common desired trajectory can be explicitly given. Theproposed strategy can achieve more efficient and robust performance through local interactions, especiallyin the presence of non-identical external disturbances. We also generalize our proposed control law such

Assistant Professor of Aerospace Engineering, Iowa State University (e-mail: sjchung@alum.mit.edu).Professor of Mechanical Engineering & Information Sciences, Professor of Brain & Cognitive Sciences, MIT (e-mail: jjs@mit.edu).

arX

iv:0

711.

1709

v2 [

mat

h.O

C]

11

Dec

200

7

2

that multiple dynamic systems can synchronize without the need for a common reference trajectory. Asa result, other potential applications include oscillation synchronization of robotic locomotion [17], [40],[42], and tele-manipulation of robots [33], [23], [6].

Another benefit of synchronization is its implication for model reduction. The exponential synchro-nization of multiple nonlinear dynamics allows us to reduce the dimensionality of the stability analysisof a large network. The model reduction aspect of synchronization, also introduced for spatially inter-connected systems in [10], is further generalized and strengthened in this article. This implies that oncethe network is proven to synchronize, we can regard a network as a single set of synchronized dynamics,which simplifies any additional stability analysis.

The main contributions of this work can be stated as follows.• In contrast with prior work on consensus and flocking problems using graphs, the proposed strategy

primarily deals with dynamical networks consisting of nonlinear time-varying dynamics. While manymechanical systems exhibit nonlinear dynamics that cannot be captured by linearization, determiningstability of nonlinear dynamic network systems is much more complex.

• We introduce contraction analysis [28], [45], [51] as our main nonlinear stability tool for reducingthe complexity and dimensionality associated with multi-agent systems, thereby deriving exact andglobal results with exponential convergence (see the Appendix for the further treatment of contractiontheory).

• The proposed control laws are of a decentralized form requiring only local velocity/position couplingfeedback for global exponential convergence, thereby facilitating implementation in real systems.As opposed to some previous work requiring all-to-all coupling [41], this will significantly reducecommunication burdens.

• The theory is generalized and extended to multi-robot systems with non-identical dynamics, linearPD coupling, partial joint coupling, uni-directional coupling, and adaptive control. The trackingsynchronization can also reduce to the state synchronization problem without a common desiredtrajectory.

• Time-delayed communications and robustness properties are also discussed.

A. Comparison with Related WorkThe consensus problems on graph [36] and the coordination of multi-agent systems [18], [27], [37],

[38] are closely related with the synchronization problem. In particular, the use of graph theory andLaplacian produced many interesting results [24], [18], [27], [32], [36]. However, synchronization to theaverage of the initial conditions might not be applicable to multi-robot and multi-vehicle systems, wherea desired trajectory is explicitly defined. Furthermore, the aforementioned work mainly deals with simpledynamic models such as linear systems and single or double integrator models without nonlinear couplingmatrices. In contrast, we aim at addressing highly nonlinear systems (e.g. helicopters, attitude dynamics ofspacecraft, walking robots and manipulator robots). As shall be seen later, the proof of the synchronizationfor network systems that possess nonlinearly coupled inertia matrices is more involved. This paper focuseson dynamical networks consisting of highly nonlinear systems.

One recent representative work on the mutual trajectory synchronization of multi-link robots is [41].The paper [41] proposes a nonlinear tracking control law to synchronize multiple robot manipulatorsin order to track a common desired trajectory. In essence, [41] proves the stability of the control lawwith estimated velocity and acceleration states from a set of nonlinear observers, which were used foran experimental demonstration of a two-robot system. The following difficulties can be identified. Thenumber of variables to be estimated increases with the number of robots to be synchronized, whichimposes a significant communication burden. Additionally, the feedback of estimated acceleration errorsrequires unnecessary information and complexity. Thus, a method to eliminate both the all-to-all couplingand the feedback of the acceleration terms is explored in this paper.

Another notable approach to synchronization of robot networks is to exploit the passivity of the input-output dynamics [6], [7], [23]. Its property of robustness to time-delays is particularly attractive. However,

3

robot dynamics are passive only with velocity outputs unless composite variables are employed. In addition,the mutual synchronization problem, which not only synchronizes the sub-members but also enforcesthem to follow a common reference trajectory, is not addressed. In particular, it shall be shown that ourproposed control law is a generalized version of the control law presented in [7]. Another recent workusing the passive decomposition [22], [24] is interesting in the sense that it shares the same philosophy ofmutual synchronization. The passive decomposition describes a strategy of decoupling into two dynamics:the same system representing the internal group formation shape, and the locked system describing thetotal group maneuver. One drawback of [22] is its dependency on a centralized control architecture;the decoupling is not generally ensured under the decentralized control. [24] only considers linear doubleintegrator models, which can degenerate into a simpler problem (see Section IV). We believe our approachusing contraction analysis has a clear advantage in its broad applications to a larger class of identical ornonidentical nonlinear systems even with time-delays, non-passive input-output, and complex couplinggeometry including concurrent synchronization and partial degrees-of-freedom coupling, while ensuringa simple decentralized coupling control law (see Figure 1 for network structures permitted here).

B. OrganizationThe organization of this paper is as follows: modeling of robots based on the Lagrangian formulation

is described in Section II. Section III summarizes the main theorems of this paper; the new trackingcontrol law, which synchronizes each robot to track the same desired trajectory, is proposed. The proof ofexponential synchronization is more involved than that of the tracking stability and treated separately inSection IV. The remainder of the paper further highlights the unique contributions of this work. The mainidea of this paper is extended to networks with linear Proportional-Derivative (PD) coupling in SectionV-A. A few examples of dynamics networks are given in Section V for validating the effectiveness ofthe proposed synchronization framework. The same section also introduces the partial joint couplings.Additionally, the properties of robustness to transmission delays and disturbances are discussed in SectionVI. An adaptive control version of the proposed synchronization strategy is presented in Section VII whileSection VIII offers a potential application of the proposed scheme to the concurrent synchronization andleader-follower problem.

II. MODELING OF MULTI-ROBOT NETWORK

An exponentially stabilizing nonlinear control law that can synchronize an arbitrary number of robots totrack a common reference trajectory is introduced in this paper. The proposed controller is much simplerthan earlier work, such as the synchronization law in [41], in terms both of the computational load andthe required signals. This paper is devoted to the use of the Lagrangian formulation for its simplicity indealing with complex systems involving multiple dynamics. The equations of motion for a robot withmultiple joints (qi ∈ Rn) can be derived by exploiting the Euler-Lagrange equations:

Li(qi, qi) = Ki(qi, qi)− Vi(qi) =1

2qTi Mi(qi)qi − Vi(qi)

d

dt

∂Li(qi, qi)

∂qi− ∂Li(qi, qi)

∂qi= τi (1)

where i, (1 ≤ i ≤ p) denotes the index of robots or dynamical systems comprising a network, and p isthe total number of the individual elements. Equation (1) can be represented as

Mi(qi)qi + Ci(qi, qi)qi + gi(qi) = τi (2)

where gi(qi) = ∂Vi

qi, and, τi is a generalized force or torque acting on the i-th robot.

Note that we define Ci(qi, qi) such that (Mi − 2Ci) is skew-symmetric [44], and this property playsa central role in our stability analysis using contraction theory [9].

4

1

54 3

4

21212

3

(a) identical robots

4

1

4 3

12

5

31

22

(b) non-identical robots

Fig. 1. Multi-agent networks of identical or nonidentical robots using local couplings. They are on balanced bi-directional graphs.

The following key assumptions are used throughout this paper. The robot system in (2) is fully actuated.In other words, the number of control inputs is equal to the dimension of their configuration manifold(= n). The mass-inertia matrix M is assumed to be uniformly positive definite, for all positions q in therobot workspace [44].

III. A NEW APPROACH TO SYNCHRONIZATION TRACKING CONTROL

We consider the synchronization of multiple robots following a common time-varying trajectory. Atracking controller introduced in this section achieves not only global and exponential synchronization ofthe configuration variables, but also global exponential convergence to the desired trajectory.

A. Proposed Synchronization Control StrategyWe introduce a simple alternative control law for the mutual synchronization problem of multiple robots,

as seen in [41]. The following tracking control law with two-way-ring symmetry is proposed for the i-throbot in the network comprised of p identical robots (see Figure 1(a)):

τi = M(qi)qir + C(qi, qi)qir + g(qi) (3)−K1(qi − qir) + K2(qi−1 − qi−1,r) + K2(qi+1 − qi+1,r)

where a positive-definite matrix K1 ∈ Rn×n is a feedback gain for the i-th robot, and another positive-definite matrix K2 ∈ Rn×n is a coupling gain with the adjacent members (i − 1 and i + 1). The abovecontrol law can also be applied to a network consisting of p non-identical robots (Figure 1(b)), as shallbe seen in Section V-C.

While the common desired time-varying trajectory (or the virtual leader dynamics) is denoted by qd(t),the reference velocity vector, qir is given by shifting the common desired velocity qd with the positionerror:

qir = qd −Λqi = qd −Λ(qi − qd) (4)

where Λ is a positive diagonal matrix.In contrast with [41], the proposed control law requires only the coupling feedback of the most adjacentrobots (i − 1 and i + 1) for exponential convergence (see Figure 1). Note that the last (p-th) robot isconnected with the first robot to form a ring network as suggested in [51]. Moreover, estimates of q areno longer required.

The closed-loop dynamics using (2) and (3) become

M(qi)si + C(qi, qi)si + K1si −K2si−1 −K2si+1 = 0 (5)

where si denotes the composite variable si = qi − qir.This system is equivalent to

M(qi)si + C(qi, qi)si + K1si −K2(si−1 + si+1) + K2

p∑j=1

sj = K2

p∑j=1

sj (6)

5

Hence, the closed-loop dynamics of each robot have the same excitation input u(t) = K2

∑pj=1 sj , which

facilitates the use of contraction analysis.Let us define the following p× p block square matrices:

[LpA,B] =

A B 0 0 · · · BB A B 0 · · · 0... . . . . . . . . . ...0 B A B 0B · · · 0 0 B A

p×p

, [UpA] =

A A · · · AA A · · · A...

... . . . ...A A · · · A

p×p

(7)

By the definition of the controller in (3), [LpA,B] has only three nonzero matrix elements in each row (i.e.,

A,B,B).Then, we can write the closed-loop dynamics in (6) in a block matrix form

[M]x + [C]x +([Lp

K1,−K2] + [Up

K2])x = [Up

K2]x (8)

where

[M] =

M(q1) · · · 0... . . . ...0 · · · M(qp)

, [C] =

C(q1, q1) · · · 0... . . . ...0 · · · C(qp, qp)

, x =

s1...sp

(9)

Note that [LpK1,−K2

] can be viewed as the modified Laplacian of the network in the context of graphtheory. In other words, [Lp

K1,−K2] indicates the connectivity with adjacent systems as well as the strength

of the coupling by K2. Note that there are only three nonzero elements in each row of the matrix, whichimplies that there exist diffusive couplings only between adjacent members. The network graphs illustratedin Figure 1 are balanced due to bi-directional coupling [36]. However, it should be noted that the matrix[Lp

K1,−K2] is different from the standard Laplacian found in [36]. By definition, every row sum of the

Laplacian matrix is zero. Hence, the Laplacian matrix always has a zero eigenvalue corresponding to aright eigenvector, 1 = (1, 1, · · · , 1)T [36]. In contrast, a strictly positive definite [Lp

K1,−K2] is required

for exponential convergence for the proposed control law in this paper. In other words, unless otherwisenoted, [Lp

K1,−K2] is assumed to have no zero eigenvalue. For example, the block matrix for p = 4 becomes

[L4K1,−K2

] =

+K1 −K2 0 −K2

−K2 +K1 −K2 00 −K2 +K1 −K2

−K2 0 −K2 +K1

, (10)

which is positive definite with K1 − 2K2 > 0. For a two-robot network, it is straightforward to verify

[L2K1,−K2

] =

[+K1 −K2

−K2 +K1

](11)

requiring the condition K1 −K2 > 0 for positive definite [L2K1,−K2

].We are well poised to introduce the main theorems of the present paper. The following condition should

be true for exponential convergence to the common desired trajectory qd.

Theorem 3.1: Global Exponential Convergence to the Desired TrajectoryIf [Lp

K1,−K2] is positive definite, then every member of the network follows the desired trajectory qd

exponentially fast regardless of initial conditions.

[LpK1,−K2

] > 0

In other words, if K1−2K2 > 0, then qi, (i = 1, 2, · · · , p, p ≥ 3) converges to qd exponentially fast fromany initial conditions. For two-robot systems (p = 2), K1 −K2 > 0 needs to be true instead.

6

Proof: We can cancel out the [UpK2

] matrix term in (8) to obtain

[M]x + [C]x + [LpK1,−K2

]x = 0. (12)

Equation (12) corresponds to a conventional tracking problem. We use contraction theory (see the Ap-pendix) to prove that x tends to zero exponentially with [Lp

K1,−K2] > 0. For example, consider the virtual

system of y obtained by replacing x with y in (12).

[M]y + [C]y + [LpK1,−K2

]y = 0 (13)

This virtual y system has two particular solutions: x = (s1, · · · , sp)T and 0. The squared-length analysiswith respect to the positive-definite metric [M] yields

d

dt

(δyT [M]δy

)= 2δyT [M]δy + δyT [M]δy (14)

= −2δyT([C]δy + [Lp

K1,−K2]δy)

+ δyT [M]δy = −2δyT [LpK1,−K2

]δy

where we used the skew-symmetric property of [M]− 2[C].Accordingly, [Lp

K1,−K2] > 0 will make the system contracting (δy→ 0), thus all solutions of y converge

to a single trajectory exponentially fast. This in turn indicates that the composite variable of each robottends to zero exponentially (s → 0). By the definition of si = q − qd + Λ(qi − qd), the exponentialconvergence of qi to the common reference trajectory qd is proven (see the hierarchical combination inTheorem A.2). The positive-definiteness of [Lp

K1,−K2] corresponds to K1−K2 > 0 for two-robot systems

(p = 2). For a network consisting of more than two robots (p ≥ 3), it can be shown that K1 − 2K2 is asufficient condition of the positive-definiteness of [Lp

K1,−K2] given K1 > 0,K2 > 0.

The next question to be addressed is how to guarantee the synchronization of the individual dynamics.The following theorem and lemma are derived in Section IV.

Theorem 3.2: Synchronization of Multiple RobotsSuppose the conditions in Theorem 3.1 are true, thus the individual dynamics are exponentially trackingthe common desired trajectory. A swarm of p robots synchronize exponentially from any initial conditionsif ∃ diagonal matrices K1 > 0, K2 > 0 such that

[LpK1,−K2

] + [UpK2

] > 0

In addition, Λ is a positive diagonal matrix defining a stable composite variable si = ˙qi + Λqi.

This theorem corresponds to synchronization with stable tracking. The proof is expanded in Section IVby separating the two different time scales of the closed-loop dynamics. As shall be seen in Section V-C,multiple dynamics need not be identical to achieve stable synchronization.

It is useful to note that the above condition corresponds to K1 + K2 > 0 for two-robot and three-robotnetworks (p = 2, 3). A four-robot network (p = 4) would require K1 + 2K2 > 0. We can also constructa network of multiple robots that can synchronize even without stable tracking. In this case, the followlemma can be used.

Lemma 3.3: Synchronization of Identical Robots Without Trajectory TrackingSuppose the conditions in Theorem 3.1 are not true, thus the individual systems are indifferent or unstable.Nevertheless, a swarm of p identical robots synchronize from any initial conditions if ∃ diagonal matricesK1 > 0, K2 such that

[LpK1,−K2

] + [UpK2

] > 0

For indifferent tracking, a common desired trajectory qd(t) is no longer required. For unstable tracking,Λ should be sufficiently large such that ‖Λ‖ > ‖K1−K2‖

σ(M(q))for p = 2 or ‖Λ‖ > ‖K1−2K2‖

σ(M(q))for p ≥ 3, where

7

σ(·) denotes the smallest singular value. In contrast with Theorem 3.2, the individual dynamics must beidentical in the unstable tracking case.

This lemma describes synchronization with indifferent dynamics (K1 − K2 = 0) or unstable tracking(K1 −K2 < 0) for p = 2; indifferent (K1 − 2K2 = 0) or unstable (K1 − 2K2 < 0) for p ≥ 3. SectionIV provides a unified treatment of both stable tracking (Theorem 3.2) and indifferent tracking (Lemma3.3) while tracking with instability has been proven in [9]. Note that indifferent tracking reduces the mainproblem into the standard Laplacian problem that does not deal with a common time-varying desiredtrajectory.

Note that we can render the system synchronized first, then follow the common trajectory by tuning thegains properly. For an example of a two-robot network, K2 > 0 ensures that the two robots synchronizefaster than they follow the common desired trajectory, since K1 + K2 > K1 −K2 for ∀ K2 > 0. Thisindicates that there exist two different time scales in the closed-loop systems constructed with the proposedcontrollers. For two-robot systems, the convergence of exponential tracking is proportional to K1 −K2

whereas the synchronization has a convergence rate of K1 + K2. This multi-time-scale behavior will beexploited in the subsequent sections.

IV. PROOF OF EXPONENTIAL SYNCHRONIZATION

We prove Theorem 3.2 and Lemma 3.3 for the exponential synchronization of multiple nonlineardynamics in this section. First, we describe the difficulties inherent in proving the synchronization ofLagrangian systems that possess nonlinearly coupled inertia matrices in Section IV-A. We then presentthe synchronization proof Section IV-C. We also show that our method is more general than prior work byreducing our control law to the standard synchronization problem without trajectory tracking in SectionIV-D. The main idea about model reduction via synchronization is emphasized in Section IV-F.

A. Challenges with Nonlinear Inertia MatrixThe difficulties associated with nonlinear time-varying inertia matrices can be easily demonstrated with

the following two-robot example. The closed-loop dynamics of two identical robots from (6) becomes

M(q1)s1 + C(q1, q1)s1 + (K1 + K2)s1 = u(t)

M(q2)s2 + C(q2, q2)s2 + (K1 + K2)s2 = u(t)

u(t) = K2(s1 + s2)

(15)

which is equivalent to[M2]{s}+ [C2]{s}+ [K]{s} = [U2

K2]{s}

where

[M2] =

[M(q1) 0

0 M(q2)

], [C2] =

[C(q1, q1) 0

0 C(q2, q2)

], {s} =

(s1

s2

)[K] = [L2

K1,−K2] + [U2

K2] =

[K1 + K2 0

0 K1 + K2

], [U2

K2] =

[K2 K2

K2 K2

] (16)

Note that s1 and s2 are the composite variables defined in (5):

s1 = q1 − q1r = q1 + Λq1 − (qd + Λqd)

s2 = q2 − q2r = q2 + Λq2 − (qd + Λqd)(17)

Direct application of synchronization (Theorem A.4) appears elusive because the original dynamicsin (2) with the control law (3) is in general not contracting. Since (2) is a second-order differentialequation, this can also be viewed as a high-order contraction problem [31]. If we transform the dynamicsinto a first-order canonical form, we have to prove that they are contracting in the same metric while

8

preserving the input symmetry [39]. For example, multiplying (15) by M−1 breaks the input symmetry: i.e.,M−1(q1)u(t) 6= M−1(q2)u(t). In essence, M(q1) 6= M(q2) makes this problem intractable in general.

Instead, suppose that M(q) remains constant, thereby making C(q, q) zero. Then, we can easily proves1 and s2 tend to each other from

Ms1 + (K1 + K2)s1 = K2(s1 + s2)

Ms2 + (K1 + K2)s2 = K2(s1 + s2)(18)

Since the virtual system with the common input u(t) = K2(s1 + s2)

My + (K1 + K2)y = u(t) (19)

is contracting with K1 +K2 > 0. Hence, its particular solutions s1 and s2 tend to each other exponentiallyfast according to the synchronization theorem (Theorem A.4). Without loss of generality, this result caneasily be extended to arbitrarily large networks. The synchronization of a large network with a constantmetric, as seen in (18), is already discussed in [51] using contraction analysis.

We now turn to a more difficult problem focused on the synchronization of two robots with non-constantnonlinear metrics (M(q1) 6= M(q2)).

B. Existence of Flow-Invariant SubmanifoldWe define and prove the existence of a flow-invariant manifold for any system given in (15), regardless

of [K], prior to finding the condition of [K] for exponential synchronization. Let us start with the definitionof flow-invariant manifold.Consider a flow-invariant differentiable submanifold of the configuration space for a two-robot system,

Mq = {(q1; q1; q2; q2)|q1 = q2, q1 = q2} (20)

Note that Mq ia a flow-invariant submanifold for the closed-loop system (15). This follows from thefact that the initial condition, q1(t0) = q2(t0), q1(t0) = q2(t0) for an arbitrary t0 ∈ R+ yields q1(·) =q2(·), which implies that any trajectory starting in Mq remains in Mq. The flow-invariant manifold,Mq corresponds to the synchronization of the two dynamical systems. Define Vsync as a matrix of theorthonormal vectors such that Vsync is a projection on M⊥

q . Then, it verifies [39]

Vsyncx = 0⇐⇒ x ∈Mq (21)

C. Contraction with Multiple Time ScalesWe present the proof of synchronization for a multi-robot network first, followed by an example of a

two-robot network (p = 2) in Section IV-E. Recall the closed-loop dynamics given in (8):

[M]x + [C]x + [LpK1,−K2

]x = 0 (22)

Since [LpK1,−K2

] is a constant real symmetric matrix, we can perform the spectral decomposition [8].This is a special case of the concurrent synchronization [39] that corresponds to convergence to a flowinvariant subspace (the eigenspace).

[LpK1,−K2

] = V[D]VT (23)

where [D] is a block diagonal matrix, and the square matrix V is composed of the orthonormal eigenvectorssuch that VTV = VVT = Ipn, since the symmetry of [Lp

K1,−K2] gives rise to real eigenvalues and

orthonormal eigenvectors [48].Pre-multiplying (22) by VT and setting x = VVTx result in(

VT [M]V)VT x +

(VT [C]V

)VTx +

(VT [Lp

K1,−K2]V)VTx = 0 (24)

9

By setting VTx = z, (24) becomes(VT [M]V

)z +

(VT [C]V

)z + [D]z = 0 (25)

Then, we can develop the squared-length analysis. Notice that(VT [M]V

)is always symmetric positive

definite since [M] is symmetric positive definite. Also, note that

d

dt

(VT [M]V

)− 2

(VT [C]V

)(26)

is skew-symmetric, which facilitates our stability analysis.Since the modified Laplacian [Lp

K1,−K2] represents a regular graph, where each member has the same

number of neighbors (= 2 for p ≥ 3),

[1] =1√p[In, In, · · · , In]T (27)

is the pn×n block column matrix of eigenvectors associated with the eigenvalues λ(K1−K2) for p = 2or λ(K1 − 2K2) for p ≥ 3. Note that In denotes the n × n identity matrix, and the [1] matrix consistsof p matrices of In. The eigenvector matrix [1] represents the common trajectory tracking state (e.g.,s1 + s2 + · · ·+ sp).

Similar to (21), we can define a pn×(p−1)n matrix Vsync that consists of the orthonormal eigenvectorsother than [1] such that

VTV =

([1]T

VTsync

)([1] Vsync

)=

[[1]T [1] [1]TVsync

VTsync[1] VT

syncVsync

]=

[In 0n×(p−1)n

0(p−1)n×n Ip(n−1)

](28)

where we used the orthogonality between [1] and Vsync.Accordingly, the block diagonal matrix [D], which represents the eigenvalues of [Lp

K1,−K2], can be

partitioned from (23)

VT [LpK1,−K2

]V =

[[1]T [Lp

K1,−K2][1] [1]T [Lp

K1,−K2]Vsync

VTsync[L

pK1,−K2

][1] VTsync[L

pK1,−K2

]Vsync

]=

[D1 0n×(p−1)n

0(p−1)n×n D2

](29)

It should be emphasized that D1 = K1 − 2K2 for p ≥ 3 (or D1 = K1 −K2 for p = 2) represents thetracking gain while D2 corresponds to the synchronization gain. We can choose the diagonal control gainmatrices K1 and K2 such that

D2 = VTsync[L

pK1,−K2

]Vsync > D1 = [1]T [LpK1,−K2

][1], (30)

thereby ensuring that the robots synchronize faster than they follow the common desired trajectory. Thecondition in (30) can be written as D2 = K1 + K2 > D1 = K1 −K2 for two robots (p = 2).This multi timescale behavior is graphically illustrated in Figure 2. The figure depicts that s1 and s2

synchronize first, then they converge to the desired trajectory while staying together. This observationmotivates separation of the two different time scales, namely K1 + K2 and K1 −K2.

Consider the virtual system of y = (yt,ys)T which has the following two particular solutions:(

yt = [1]Tx ys = VTsyncx

)T and(yt = 0 ys = 0

)T (31)(VT [M]V

)y +

(VT [C]V

)y + [D]y = 0 (32)

For [D] > 0, we can show that the above virtual system is contracting (i.e., δy → 0 globally andexponentially). We take the symmetric positive definite block matrix VT [M]V as our contraction metric.

10

S1S2

S1=S2

S1=S2=0

synchronization

tracking

Fig. 2. Multiple timescales of synchronization (faster) and tracking (slower). The dashed line indicates the desired trajectory, and arrowsindicate increasing time. The drawing is conceptual, since strictly speaking s1 and s2 synchronize exponentially.

Performing the squared-length analysis with respect to this metric yields

d

dtδyT

(VT [M]V

)δy = 2δyT

(VT [M]V

)δy + δyT

(VT [M]V

)δy (33)

= −2δyT((

VT [C]V)δy + [D]δy

)+ δyT

(VT [M]V

)δy

= −2δyT [D]δy

where we used the skew-symmetric property of(VT [M]V

)− 2

(VT [C]V

).

The above equation can be rewritten in terms of two different time scales

d

dt

(δytδys

)T [[1]T [M][1] [1]T [M]Vsync

VTsync[M][1] VT

sync[M]Vsync

](δytδys

)= −2

(δytδys

)T [D1 00 D2

](δytδys

)(34)

If [D] > 0, which is in fact equivalent to [LpK1,−K2

] > 0, the combined virtual system in (32) iscontracting. In other words, δy→ 0 exponentially fast. This in turn implies all solutions of y tend to thesingle trajectory. As a result, [1]Tx = 1/

√p(s1 + · · · + sp) and VT

syncx tend to zero exponentially. Notethat s1, · · · , sp → 0 has already been proven for Theorem 3.1, which is a sufficient condition to makethe sum of the composite variables also tend to zero (i.e., [1]Tx → 0). The new result in this section isthe synchronization VT

syncx→ 0.It is straightforward to show that VT

syncx → 0 and Λ > 0 also hierarchically make q1, · · · ,qpsynchronize exponentially (see Theorem A.2). This can be verified by the following contracting dynamicsconstructed from (17)

VTsync{q}+

(VTsync[Λ]Vsync

)VTsync{q} = VT

syncx → 0 (35)

where {q} = (q1, · · · ,qp)T . Note that the orthonormal vectors Vsync cancel the common input termqd + Λqd. Also, [Λ] is a block diagonal matrix of Λ > 0, thereby yielding VT

sync[Λ][1] = 0 from

VsyncVTsync{q}+ [1][1]T{q} = {q} (36)

Consequently, VTsync{q} → 0 implies the synchronization of the original state variable qi, i = 1, · · · , p.

This also implies that the diagonal terms of the metric, VTsync[M][1] tend to zero exponentially, thereby

eliminating the coupling of the inertia term VT [M]V in (33,34). Note that we assume here that qd(t)is identical for each member. If qd(t) were different for each dynamics, si → sj would imply thesynchronization of qi − qj to the difference of the desired trajectories, which is useful to constructphase synchronization.

11

Now let us relate the condition in Theorem 3.2 with [D] > 0. The block matrix [UpK2

] also has [1] asits eigenvector. We multiply [Lp

K1,−K2] + [Up

K2] by its orthonormal eigenvectors other than [1]:

[LpK1,−K2

]Vsync + [UpK2

]Vsync = [LpK1,−K2

]Vsync = VsyncD2 (37)

which shows that Vsync also represents the orthonormal eigenvectors of [K] = [LpK1,−K2

] + [UpK2

]. Inother words, D2 corresponds to the eigenvalues of [K].

The remaining eigenvalue of [K], associated with [1], is K1 + (p − 2)K2, which is greater than thetracking eigenvalue D1 = K1−2K2 for p ≥ 3. Hence, the synchronization occurs if [Lp

K1,−K2]+[U2

K2] > 0:

[D] > 0⇐⇒ D1 > 0 and [LpK1,−K2

] + [U2K2

] > 0. (38)

This completes the proof of Theorem 3.2.

D. Synchronization Without a Common Reference TrajectoryWe now turn into the more standard synchronization problem where the tracking gain is zero D1 =

K1−K2 = 0 for p = 2 or D1 = K1−2K2 = 0 for p ≥ 3, which fails the exponential stability condition inTheorem 3.1. In this case, our modified Laplacian [Lp

K1,−K2] reduces to the standard weighted Laplacian

whose row sum is zero. For synchronization to the weighted average of the initial conditions, we do notrequire the common desired trajectory qd, and qd can simply be set to zero as follows:

qir = −Λqi, si = qi + Λqi (39)

In other words, our control strategy represents a more generalized framework for the synchronizationof multiple Lagrangian systems. The combined virtual system per se is then semi-contracting ([28], [29])since the squared-length analysis in (33) yields the negative semi-definite matrix:

d

dt

(δytδys

)T [[1]T [M][1] [1]T [M]Vsync

VTsync[M][1] VT

sync[M]Vsync

](δytδys

)= −2

(δytδys

)T [0 00 D2

](δytδys

)≤ 0 (40)

While δyt, representing the tracking dynamics, remains in a finite ball due to D1 = 0, δys tends to zeroasymptotically due to D2 > 0. This result can be proven as follows. V is uniformly continuous since abounded δys from (32) leads to a bounded V since

V = −4δyTs D2δys (41)

Due to V ≤ 0, the use of Barbalat’s lemma [44] verifies that V → 0 as t → ∞. This implies that δystends to zero asymptotically fast. As a result, VT

syncx tends to zero asymptotically. From the hierarchicalcombination discussed in (35), this also implies VT

sync{q} = 0. This completes the proof of Lemma 3.3with indifferent tracking.

This will eventually decouple the metric matrix with Λ > 0, as seen in (34), since VTsync[M][1] tends

to zero simultaneously as VTsync{q} = 0. As a result, when M(qi)−M(qj) is sufficiently close to zero,

the convergence of δys → 0 turns exponential.We can also prove synchronization in the presence of tracking instability (D1 < 0) with sufficiently small‖D1‖ by decoupling the unstable dynamics from the stable synchronization dynamics. We refer the readersto [9] for details. In essence, we can show the synchronization can occur fast enough to overcome thetracking instability (see the simulation results in Section V-D).

12

Fast Inhibition: The dynamics of a large network with semi-contracting stability as in (40) can beinstantaneously transformed to contracting dynamics by the addition of a single inhibitory coupling link.In other words, a single inhibitory link will also make δyt → 0. For instance, we can add a single inhibitorylink between two arbitrary elements a and b while we keep the rest of the elements the same: [51]

τa = M(qa)qar + C(qa, qa)qar + g(qa)− 2K2sa + K2sa−1 + K2sa+1 −K(sa + sb) (42)τb = M(qb)qbr + C(qb, qb)qbr + g(qb)− 2K2sb + K2sb−1 + K2sb+1 −K(sa + sb)

where 2K2 is substituted for K1, and K > 0.Hence, we can easily show that [Lp

K1,−K2] is now strictly positive definite, in contrast with the original

semi-contracting system. As a result, the closed-loop system is contracting, resulting in si → 0 and qi → 0from (39). This fast inhibition is useful to rapidly destroy unwanted synchronized oscillation of robots.

E. ExamplesFor the case of a two-agent network, we can easily verify that

VT [M]V =

[M(q1)+M(q2)

2M(q1)−M(q2)

2M(q1)−M(q2)

2M(q1)+M(q2)

2

], z = VTx =

([1]T

VTsync

)x =

[ 1√2I 1√

2I

1√2I − 1√

2I

](s1

s2

),

(VT [C]V

)=

[C(q1,q1)+C(q2,q2)

2C(q1,q1)−C(q2,q2)

2C(q1,q1)−C(q2,q2)

2C(q1,q1)+C(q2,q2)

2

], [D] =

[K1 −K2 0

0 K1 + K2

](43)

If K1 + K2 > K1 −K2 > 0, the rate of the virtual length in (33) is uniformly negative definite fornonzero δyt and δys:

d

dt

(δytδys

)T [M(q1)+M(q2)2

M(q1)−M(q2)2

M(q1)−M(q2)2

M(q1)+M(q2)2

](δytδys

)(44)

=− 2

(δytδys

)T [K1 −K2 0

0 K1 + K2

](δytδys

)< 0.

Consequently, the combined virtual system in (32) is contracting. This in turn implies all solutions of ytand ys tend to the single trajectory. As a result, sp = s1 + s2 and sm = s1− s2 tend to zero exponentially.It is straightforward to show that sm → 0 also hierarchically makes q1 tend to q2 exponentially (seeTheorem A.2).From the definition of the composite variables in (17), we can find the following contracting dynamics,

(q1 − q2) + Λ(q1 − q2) = sm (45)

Note that y + Λy = 0 is contracting with Λ > 0. Consequently, Λ > 0 and sm → 0 make q1 → q2

exponentially fast. This also implies that the diagonal terms of the metric, M(q1)−M(q2)2

tend to zeroexponentially, thereby eliminating the coupling of the inertia term.[

M(q1)+M(q2)2

M(q1)−M(q2)2

M(q1)−M(q2)2

M(q1)+M(q2)2

]−→

[M(q1) δMδM M(q1)

](46)

where the perturbation term δM = M(q1)−M(q2)2

tends exponentially to zero, since ‖δM‖ ≤ m‖q2 −q1‖ → 0.

We can extend the method in this section to arbitrarily large networks. For example, a network of threerobots has the following V whose columns are orthonormal eigenvectors of [Lp=3

K1,−K2]:

V =[[1] Vsync

]=

1√3I − 2√

6I 0

1√3I 1√

6I − 1√

2I

1√3I 1√

6I 1√

2I

. (47)

13

Note that VTV = VVT = I3n.The new transformed inertia matrix, VT [M]V is written as

13M1 + 1

3M2 + 1

3M3 −

√2

3M1 + 1

3√

2M2 + 1

3√

2M3 − 1√

6M2 + 1√

6M3

−√

23

M1 + 13√

2M2 + 1

3√

2M3

23M1 + 1

6M2 + 1

6M3 − 1

2√

3M2 + 1

2√

3M3

− 1√6M2 + 1√

6M3 − 1

2√

3M2 + 1

2√

3M3

12M2 + 1

2M3

(48)

where M1 = M(q1), M2 = M(q2), and M3 = M(q3) for notational simplicity. This matrix is symmetricand positive definite. Also notice that its off-diagonal terms vanish as q1 → q2,q2 → q3.The diagonal matrix [D] is also computed asK1 − 2K2 0 0

0 K1 + K2 00 0 K1 + K2

(49)

For a four-robot network p = 4 (see Figure 1), [D] is given asK1 − 2K2 0 0 0

0 K1 + 2K2 0 00 0 K1 00 0 0 K1

(50)

where K1 represents the synchronization gain associated with the synchronization of diagonal members(q1 = q3,q2 = q4), while K1 + 2K2 represents the synchronization gain of direct couplings (e.g.,q1 = q2,q3 = q4). This is a percolation effect discussed in [39]. The percolation effect can be exploitedin order to prove the synchronization to the large unknown invariant set (e.g., VT

syncx = 0 from a largenetwork) by verifying the synchronization to a known, not necessarily invariant, subset of the globalflow-invariant set [14].

F. A Perspective on Model Reduction by SynchronizationIt is emphasized that the exponential synchronization of multiple nonlinear dynamics allows us to reduce

the dimensionality of the stability analysis of a large network. As noted earlier, the synchronization rate(D2) is faster than the tracking rate (D1). Assuming that the dynamics are synchronized, the augmenteddynamics in (12) reduces to

M(q)s + C(q, q)s + (K1 − 2K2)s = 0 (51)

where q = q1 = · · · = qp, and D1 = K1 − 2K2 is replaced by K1 −K2 for p = 2.This implies that once the network is proven to synchronize, we can regard a network as a single set ofsynchronized dynamics, which simplifies any additional stability analysis.

V. EXTENSIONS AND EXAMPLES

Let us examine the effectiveness of the proposed control law in a variety of nonlinear dynamics networks.

A. PD Synchronization of RobotsOne may consider the following Proportional and Derivative (PD) coupling control law for two identical

robots from (2) with p = 2,

τ1 = −K1(q1 + Λq1) + K2(q2 + Λq2)

τ2 = −K1(q2 + Λq2) + K2(q1 + Λq1)(52)

where qi = qi − qd and the bounded reference position qd has zero velocity such that ˙qi = qi.

14

For simplicity, the gravity term in (2) is assumed to be zero or canceled by a feed-forward control law.Then, the closed-loop dynamics satisfy

M(q1)q1 + C(q1, q1)q1 + Kq1 + KΛq1 = u(t)

M(q2)q2 + C(q2, q2)q2 + Kq2 + KΛq2 = u(t)(53)

where K = K1 + K2 and u(t) = K2(q1 + q2) + K2Λ(q1 + q2).Similar to Section IV-C, we can perform a spectral decomposition:(

VT [M]V)VT x +

(VT [C]V

)VT x

+(VT [Lp

K1,−K2]V)

VT x +(VT [Lp

K1Λ,−K2Λ]V)VT x = 0

(54)

Using the following Lyapunov function, it is straightforward to show that this PD coupling control lawdrives the system to the desired rest state qd globally and asymptotically while tending to the synchronizedflow-invariant manifold, Mq in (20):

V =1

2

(qpqm

)T [M1+M2

2M1−M2

2M1−M2

2M1+M2

2

](qpqm

)+

1

2

(qpqm

)T [(K1 −K2)Λ 0

0 (K1 + K2)Λ

](qpqm

). (55)

where

M1 = M(q1), M2 = M(q2), VTx =

(qpqm

)=

1√2

(q1 + q2

q1 − q2

), qp =

1√2(q1 + q2). (56)

The rate of V can be computed as

dV

dt= −

(qpqm

)T [K1 −K2 0

0 K1 + K2

](qpqm

)≤ 0 (57)

which implies that V is negative semi-definite with K1 + K2 > 0 and K1 −K2 > 0.By invoking LaSalle’s invariant set theorem [44], we can conclude that qp, qp, qm, and qm tend to

zero with global and asymptotic convergence. This implies that q1 and q2 will follow qd while q1 andq2 synchronize asymptotically.

Theorem 5.1: PD Synchronization of Two Identical RobotsTwo identical robots synchronize asymptotically and globally with the PD coupling law in (52) if ∃K2 > 0such that

K1 + K2 > 0 Λ > 0

In addition, K1 −K2 > 0 is required for asymptotic stability of the PD control.

B. Velocity Coupling, Λ = 0

Note that if Λ = 0, the PD coupling control law in (52) reduces to

τ1 = −K1q1 + K2q2

τ2 = −K1q2 + K2q1

(58)

This velocity coupling control can also be derived from the exponential tracking control law in (3) bysetting qd = 0, qd = 0 and Λ = 0. Therefore, the proof of Theorem 5.1 with Λ = 0 is the same asSection IV-C whereas the convergence rate is now exponential compared to the asymptotic convergenceof the PD control. On the other hand, we can find that positions do not synchronize in the absence of thegravity term even though the velocities synchronize exponentially fast.

15

C. Synchronization of Heterogeneous RobotsNotice that the proposed tracking and synchronization control law in (3) can easily be applied to a

network consisting of heterogenous robots in (2) if the stable tracking condition in Theorem 3.2 is true.

τi = Mi(qi)qir + Ci(qi, qi)qir + gi(qi)

−K1(qi − qir) + K2(qi−1 − qi−1,r) + K2(qi+1 − qi+1,r) (59)

where Mi, Ci, and gi(qi) represent the i-th robot dynamics, which can be different from robot to robot.Each robot has the same number of configuration variables (qi ∈ Rn).For example, the M matrix and C matrix notations used in the previous sections can be interpreted asM(q1) → M1(q1) and M(q2) → M2(q2) with M1(·) 6= M2(·) (the same for the C matrices). Hence,the assumption of non-identical dynamics does not alter the proof of exponential tracking in Theorem3.1 and stable synchronization in Section IV. However, the synchronization with indifferent or unstabletracking (Lemma 3.3) is no longer true for the case of non-identical robots since VT

syncx = 0 is not aflow-invariant manifold, and q1 = q2 does not cancel the off-diagonal terms M1(q1)−M2(q2)

2in the metric

matrix.The above discussion is summarized in the following lemma.Lemma 5.2: Synchronization of Multiple Non-Identical Robots

If [LpK1,−K2

] is positive definite, then every member of the network follows the desired trajectory qdexponentially fast.

[LpK1,−K2

] > 0 (60)

In addition, a swarm of p non-identical robots synchronize exponentially from any initial conditions if ∃diagonal matrices K1 > 0, K2 > 0 such that

[LpK1,−K2

] + [UpK2

] > 0

where the closed-loop dynamics in (8) can be rewritten to account for the non-identical dynamics of eachrobot:

[M]x + [C]x +([Lp

K1,−K2] + [Up

K2])x = [Up

K2]x (61)

where

[M] =

M1(q1) · · · 0... . . . ...0 · · · Mp(qp)

, [C] =

C1(q1, q1) · · · 0... . . . ...0 · · · Cp(qp, qp)

, x =

s1...sp

. (62)

D. Cooperative Control of Robot ManipulatorsThe proposed control law in (3) is simulated for two identical two-link robots shown in Figure 3(a). The

physical parameters of the robot are taken from the page 396 of [44] (see Table I). In order to comparewith the performance of the controller for a network consisting of non-identical robots, consider the twodifferent robots in Figure 3(b). The first robot has the same physical parameters as in Figure 3(a), andthe second robot’s parameters are given in Table I.

TABLE IPHYSICAL PARAMETERS OF THE ROBOTS SHOWN IN FIGURES 3.

Physical Parameters m1 `1 me I1 `c1 Ie `ce

Robots in Fig. 3(a) 1 kg 1 m 2 kg 0.12 kgm2 0.5 m 0.25 kgm2 0.6 mLeft robot in Fig. 3(b) 1 kg 1 m 2 kg 0.12 kgm2 0.5 m 0.25 kgm2 0.6 mRight robot in Fig. 3(b) 2 kg 1.2 m 2.5 kg 0.25 kgm2 0.6 m 0.4 kgm2 0.7 m

16

(a) Two identical two-link robots (b) Two different two-link robots

Fig. 3. Examples of cooperative control. The arrows indicate the coupling control law

0 1 2 3 4 5 6 7 8 9 100

0.5

1

time[s]

Join

t 1 [r

ad]

Identical Robot, Stable Tracking

0 1 2 3 4 5 6 7 8 9 100

0.2

0.4

0.6

0.8

1

time[s]

Join

t 2 [r

ad]

qd

q1

q2

qd

q1

q2

(a) Identical robots

0 1 2 3 4 5 6 7 8 9 100

0.5

1

time[s]

Join

t 1 [r

ad]

Non−Identical Robot, Stable Tracking

0 1 2 3 4 5 6 7 8 9 100

0.2

0.4

0.6

0.8

1

time[s]

Join

t 2 [r

ad]

qd

q1

q2

qd

q1

q2

(b) Non-identical robots

Fig. 4. Synchronization of two robots with stable tracking

The simulation result is presented in Figure 4. The control gains are defined as K1 = 5I, K2 = 1.5I,and Λ = 5I. The desired trajectory is qd =

(0.3(1− cos 2πt), 0.5(1− e−t)

)T . The lower joints areinitially at 0.3 rad with 1 rad/s (first robot) and 1.0 rad at rest (second robot). The upper joint of the firstrobot is initially at 0.4 rad. The rest of the initial conditions are zero. The same initial conditions areused for both the robots in Figures 3 (a) and (b). The synchronization gain of s1 and s2 corresponds toK1 +K2 = 6.5I, which is larger than the tracking convergence gain K1−K2 = 3.5I. As a result, we cansee in Figure 4 that the first and second robots exponentially synchronize first. Then, they exponentiallyconverge together to the desired trajectory.Let us now investigate the synchronization under the instability condition. For example, the control gainsare re-defined as K1 = 5I, K2 = 5.3I, and Λ = I. Now the tracking gain, K1−K2 = −0.3I is negative,thus instability occurs. Figures 5 shows the simulation result with the instability. For the network withtwo identical robots, both robots synchronize exponentially fast even though the synchronized trajectorybecomes unstable. In contrast, the exponential synchronization of the two different robots is no longerguaranteed in the presence of tracking instability, as seen in Figure 5(b).

17

0 2 4 6 8 10 12 14 16 18 200

5

10

15

time[s]

Join

t 1 [r

ad]

Identical Robot, Unstable Tracking

qd

q1

q2

0 2 4 6 8 10 12 14 16 18 20−10

−5

0

5

time[s]

Join

t 2 [r

ad]

qd

q1

q2

(a) Identical robots

0 2 4 6 8 10 12 14 16 18 200

2

4

6

8

time[s]

Join

t 1 [r

ad]

Non−Identical Robot, Unstable Tracking

0 2 4 6 8 10 12 14 16 18 20−4

−2

0

2

4

6

time[s]

Join

t 2 [r

ad]

qd

q1

q2

qd

q1

q2

(b) Non-identical robots

Fig. 5. Synchronization of two robots with unstable tracking

(a) A network of four robots (b) Modeling of manipulator robot on a cart [9]

Fig. 6. Synchronization of a four-robot network. The arrows indicate the coupling control law.

E. Tracking Synchronization of Four RobotsEven though the local coupling structure of the proposed control law in (3) has been emphasized,

the difference from all-to-all coupling is not evident in a network comprised of less than four members(p ≤ 3). To illustrate the effectiveness of the proposed scheme for a dynamical network comprised ofhighly nonlinear systems with p ≥ 4, a network of four identical 3-DOF robots is considered here (seeFigure 6). Each joint is assumed to be frictionless, and the dynamics modeling of the 3-DOF robot isbased upon the double inverted pendulum robot on a cart given in [9]. The physical parameters of eachrobot are given in Table II. The initial conditions are defined as in Table III.

The simulation result is presented in Figure 7. The four identical robots, initially at the conditions in

18

TABLE IIPHYSICAL PARAMETERS OF THE ROBOTS SHOWN IN FIGURES 6.

Physical Parameters M m1 `1 I1 `c1 m2 I2 `c2Values 4 kg 1 kg 1 m 0.12 kgm2 0.5 m 2 kg 0.25 kgm2 0.6 m

TABLE IIIPHYSICAL PARAMETERS OF THE ROBOTS SHOWN IN FIGURES 6.

Variable [unit] s [m] s [m/s] θ1 [rad] θ1 [rad/s] θ2 [rad] θ2 [rad/s]Robot 1 -0.5 1 0.3 0 -0.3 0Robot 2 -0.2 -0.5 1 0 0 0Robot 3 0.4 1 -0.7 0.4 2 0Robot 4 -0.3 0 -0.5 0 1.2 0.5

Table III, are driven to synchronize as well as to track the following desired trajectory:

sd = 0.2t, sd = 0.2, sd = 0

θ1d = cos(0.02πt), θ1d = − sin(0.02πt)(0.02π), θ1d = − cos(0.02πt)(0.02π)2 (63)

θ2d =π

4(1− cos(0.08πt)), θ2d =

π

4sin(0.08πt)(0.08π), θ2d =

π

4cos(0.08πt)(0.08π)2

Note that this simulation fully considers the nonlinear rotational dynamics of the robots, as opposed tosome earlier work on multi-agent coordination [36], [37].

For the control gains in the control law (3), we used K1 = I, K2 = 0.4I, and Λ = I. According toTheorems 3.1 and 3.2, the corresponding tracking gain K1−2K2 = 0.2 is smaller than the correspondingsynchronization gain K1 +2K2. Figure 7 shows that the system states synchronize exponentially fast andthe synchronized states converge to the desired trajectory together.

F. Synchronization with Partial Degrees-of-Freedom CouplingWe now consider multiple dynamics with partially coupled joints (or partially coupled variables). As

shown in Figure 8(a), we can assume that only the lower joint is coupled in a two-robot system havingtwo joint variables (q = (x1, x2)

T ) such that

τ1 = M(q1)q1r + C(q1, q1)q1r + g(q1)−K1s1 + K2

(˙x1

0

)q2

+ K2Λ

(x1

0

)q2

τ2 = M(q2)q2r + C(q2, q2)q2r + g(q2)−K1s2 + K2

(˙x1

0

)q1

+ K2Λ

(x1

0

)q1

(64)

Nevertheless, Theorems 3.1 and 3.2 are true with diagonal matrices, K1, K2 and Λ, which can beverified by writing the closed-loop system as in (15):

M(q1)s1 + C(q1, q1)s1 + (K1 + K2

[1 00 0

])s1 = u(t)

M(q2)s2 + C(q2, q2)s2 + (K1 + K2

[1 00 0

])s2 = u(t)

u(t) = K2

[1 00 0

](s1 + s2)

(65)

It is straightforward to prove that Theorems 3.1 and 3.2 still hold. This is because

(K1 + K2

[1 00 0

]) and (K1 −K2

[1 00 0

]) (66)

19

0 1 2 3 4 5 6 7 8

−1.5

−1

−0.5

0

0.5

1

1.5

2

s[m]

ROBOT 1

ROBOT 2

ROBOT 3

ROBOT 4

(a) Simulation result.

0 5 10 15 20 25 30 35 40

0

2

4

6

8

s [m

]

Synchronization of four−robot network

0 5 10 15 20 25 30 35 40

−0.5

0

0.5

1

Join

t 1 [r

ad]

0 5 10 15 20 25 30 35 40

0

0.5

1

1.5

2

time [s]

Join

t 2 [r

ad]

DesiredRobot1Robot2Robot3Robot4

DesiredRobot1Robot2Robot3Robot4

DesiredRobot1Robot2Robot3Robot4

(b) Time history of the states.

Fig. 7. Synchronization of a four-robot network

20

(a) The arrows indicate the coupling control law. (b) Simulation result.

Fig. 8. Two identical two-link robots with only one-joint coupling.

are still uniformly positive definite, enabling exponential synchronization and exponential convergenceto the desired trajectory, respectively. Hence, we did not break any assumption in the proof of Theorem3.2. Figure 8(b) shows exponential synchronization of two identical robots with the partial coupling lawabove. All simulation parameters are the same as in Figure 4. This partial-state coupling also works for thesemi-contracting case with (39), as presented in Section IV-D. While the coupled variables synchronizeto the weighted average of the initial conditions, the uncoupled variables synchronize to zero.

VI. EFFECTS OF TRANSMISSION DELAYS AND EXTERNAL DISTURBANCES

Now let us discuss the robustness properties of the proposed synchronization framework.

A. Synchronization with Time DelaysWang and Slotine [52] showed that contraction properties are conserved in time-delayed diffusion-like

couplings. We show herein that the same principle can be applied to the synchronization of multipleLagrangian systems discussed in this paper. In particular, the proposed synchronization coupling controllaw in (3) is proven to synchronize multiple robots as well as to track the common trajectory, regardlessof time delays in the communication.

Figure 9 shows two two-link manipulators transmitting their state information to each other via time-delayed transmission channels. While T12 is a positive constant denoting the time delay in the communi-cation from the first robot to the second robot, T21 denotes the delay from the second robot to the firstrobot. Similar to [52], we can modify our original Lagrangian systems consisting of two identical robotsin (15) as follows

M(q1)s1 + C(q1, q1)s1 + (K1 −K2)s1 + G21τ21 = 0

M(q2)s2 + C(q2, q2)s2 + (K1 −K2)s2 + G12τ12 = 0(67)

where G21 and G12 are constant matrices (Rn×n).The communication between the two dynamics occurs by transmitting intermediate ”wave” variables,defined as [52]

u21 = GT21s1 + k21τ21 v12 = GT

21s1

u12 = GT12s2 + k12τ12 v21 = GT

12s2

(68)

21

Fig. 9. Synchronization of two identical robots with transmission delays

where k21 and k12 are strictly positive constants. Time delays of T21 and T12 result in

u12(t) = v12(t− T12) u21(t) = v21(t− T21) (69)

Notice that the original dynamics without any communication delays are contracting since K1−K2 > 0.Expanding (67), using the above relationships on the wave variables, yields

M(q1)s1 + C(q1, q1)s1 + (K1 −K2)s1 −1

k21

G21

(GT

12s2(t− T21) + GT21s1(t)

)= 0

M(q2)s2 + C(q2, q2)s2 + (K1 −K2)s2 −1

k12

G12

(GT

21s1(t− T12) + GT12s2(t)

)= 0

(70)

We can verify that (70) becomes equivalent to the original two-robot dynamics in (15) by setting k12 =k21 = 1 and G12 = G12 =

√K2. Note that

√K2 is the Cholesky decomposition of the positive-definite

symmetric matrix, K2.The resultant equations reflecting the time-delayed transmissions become

M(q1)s1 + C(q1, q1)s1 + K1s1 −K2s2(t− T21) = 0

M(q2)s2 + C(q2, q2)s2 + K1s2 −K2s1(t− T12) = 0(71)

which can be shown to be asymptotically contracting using the following differential length similar to[46], [52]:

V =1

2δsT1 M(q1)δs1 +

1

2δsT2 M(q2)δs2 +

1

2V1,2 (72)

where

V1,2 =

∫ t

t−T12

δvT12δv12dε+

∫ t

t−T21

δvT21δv21dε−∫ 0

−T12

δvT12δv12dε−∫ 0

−T21

δvT21δv21dε (73)

In conclusion, the robot network systems, individually contracting (exponentially converging) andinteracting through time-delayed diffusion-like coupling in (70) are asymptotically contracting regardlessof the values of the time delays. This in turn implies that individual robots still synchronize and follow thecommon trajectory asymptotically regardless of the time delays. We have exactly matched the analysis in[52] to allow the derivation of a time-delayed version of the proposed tracking control law in this paper.This shows that the proposed control law and its closed-loop system in (8) possess some robustnessproperties with respect to time delays.

22

B. Effect of Bounded DisturbancesDue to exponential tracking convergence of the proposed scheme, the property of robustness to bounded

deterministic disturbances can easily be determined. For example, consider the closed-loop system in (12),which is now subject to a vanishing perturbation [21] such that g(t,x = 0) = 0:

[M]x + [C]x + [LpK1,−K2

]x = g(t,x) (74)

The perturbation term g(t,x) vanishes at the equilibrium manifold x = 0. Let us further assume that itsatisfies the linear growth bound such that

‖g(t,x)‖ ≤ γ‖x‖, ∀t > 0. (75)

where γ is a positive constant.The squared-length analysis yields

d

dt

(δxT [M]δx

)= 2δxT [M]δx + δxT [M]δx

= 2δxT(−[C]δx− [Lp

K1,−K2]δx + δg(t,x)

)+ δxT [M]δx (76)

≤ −2δxT [LpK1,−K2

]δx + 2γδxT δx

where we used the skew-symmetric property of [M]− 2[C].Hence, the closed-loop system in (74) is contracting in the presence of the bounded disturbance if‖[Lp

K1,−K2]‖ > γ. This condition corresponds to ‖K1 −K2‖ > γ for two-robot systems. As a result, the

tracking gain also determines how robust the closed-loop system is with respect to a bounded disturbance.For a nonvanishing perturbation such that ‖g(t,x)‖ ≤ γ‖‖x‖ + δ, the comparison method [21] canstraightforwardly be developed to derive a bound on the solution. It should be emphasized that theexponential stability of the closed-loop system facilitates such a perturbation analysis, which highlightsanother benefit of contraction analysis. In contrast, the proof of robustness with asymptotic convergenceis more involved [21].

VII. ADAPTIVE SYNCHRONIZATION

First, we present the adaptive version of the proposed control law that adapts to the unknown parametricuncertainties of the robot dynamic models in Section VII-A. Second, similar to knowledge-based leader-follower networks [53], we present the synchronization of multiple adaptive dynamics that can adjust theirphysical parameters as well as their dynamic states in Sections VII-B and VII-C.

A. Synchronization with Adaptive Control LawConsider the following adaptive control law that has the same local coupling structure as the proposed

control law in (3):

τi = Mi(qi)qir + Ci(qi, qi)qir + gi(qi)−K1si + K2si−1 + K2si+1

= Yiai −K1si + K2si−1 + K2si+1 (77)

where si denotes the composite variable for the i-th robot such that si = qi − qir.The parameter estimate ai for the i-th member is updated by the correlation integral:

˙ai = −ΓYTi si (78)

where Γ is a symmetric positive definite matrix. Hence, the closed-loop system for a network comprisedof p non-identical robots can be written as[

[M] 00 [Γ−1]

](x

{ ˙a}

)+

[[C] 00 0

](x{a}

)+

[[Lp

K1,−K2] −[Y]

[Y]T 0

](x{a}

)= 0 (79)

23

where [M] and [C] are the block diagonal matrices of Mi(qi) and Ci(qi, qi), i = 1, · · · , p, as defined in(8). The additional block diagonal matrices are defined from (78) such that

[Γ−1] = diag(Γ−1,Γ−1 · · · ,Γ−1)p, [Y] = diag(Y1,Y2, · · · ,Yp) (80)

Also, x = (s1, s2, · · · , sp)T ,and {a} = (a1, a2, · · · , ap)T where ai denotes an error of the parameterestimate such that ai = ai− ai. Note that ai is a constant vector of the true parameter values for the i-throbot, resulting in ˙ai = ˙ai. If each robot is identical, ai = a for 1 ≤ i ≤ p.

Similar to Section IV-C, applying the spectral transformation using

Va =

[V 00 Ipn

], VT [Lp

K1,−K2]V = [D] (81)

to (79) leads to the following virtual system of (y1,y2)T[

VT [M]V 00 [Γ−1]

](y1

y2

)+

[VT [C]V 0

0 0

](y1

y2

)+

[[D] −VT [Y]

[Y]TV 0

](y1

y2

)= 0 (82)

Similar to [19], the virtual system has two particular solutions:

(y1 = VTx,y2 = {a})T , and (y1,y2)T = 0 (83)

The virtual length analysis indicates that (82) is semi-contracting by the negative semi-definite Jacobianwith [D] > 0:

d

dt

(δy1

δy2

)T [VT [M]V 0

0 [Γ−1]

](δy1

δy2

)= −2

(δy1

δy2

)T [[D] 00 0

](δy1

δy2

)(84)

Using Barbalat’s lemma (see Section IV-D), it is straightforward to show that δy1 tends asymptoticallyto zero from any initial condition. Consequently, the adaptive synchronization law in (77) synchronizesthe states of multiple dynamics in the presence of model uncertainties. Due to the semi-contracting virtualdynamics given in (82), the convergence result is now asymptotic instead of exponential.

B. Synchronization of Heterogeneous Adaptive Dynamics with Knowledge LeaderWe present the synchronization of multiple adaptive dynamics that attempt to mimic the physical

parameters of the leader, which is more analogous to evolutionary mutations of biological objects. Sucha leader is referred to as the knowledge leader of the adaptive network, which specifies “how to go” [53],as opposed to the desired trajectory qd(t) prescribing “where to go”. The physical parameters that can beactively adapted on include: the local control gains; and the mass and inertia by grabbing or discardingan object or perhaps by combining or dividing the robots (e.g., autonomous docking of space robots).We present the synchronization control law and the corresponding proof that are simpler than [53] whileachieving the same objective.Assume that the individual robot dynamics of the network are adaptive in the sense that their physicalparameters, connoted by ai in the previous section, are now adapting or mutating, according to the adaptionlaw

ai = ΓYTi si (85)

where the adaptive parameters are carefully selected such that the inertia matrix Mi(qi) is uniformlypositive definite. This is different from (78) where the actual physical parameters of the robot (ai) arefixed, but its estimates ai are updated in the adaptive control law (78). In this section, the actual valuesof ai are updated by (85) and the original dynamics in (2) are now called adaptive dynamics. Considerthe following control law for each dynamics.

τi = Yialeader −K1si + K2si−1 + K2si+1 (86)

24

where aleader, which corresponds to the physical parameters of the leader, is slowly varying or constant.By defining the parametric error ai = ai − aleader, we can derive the virtual dynamics, similar to

Section VII-A:[VT [M]V 0

0 [Γ−1]

](y1

y2

)+

[VT [C]V 0

0 0

](y1

y2

)+

[[D] VT [Y]

−[Y]TV 0

](y1

y2

)= 0 (87)

with the same two particular solutions given in the previous section. Consequently, we can conclude thatthe states of multiple adaptive dynamics synchronize by the knowledge feedback from the leader (Yialeader).While the synchronization of the physical parameters to the leader is not automatically guaranteed due tothe semi-contracting stability of (87), the additional condition of the persistency of excitation [44] suchas

∃α > 0, T > 0, ∀t ≥ 0

∫ t+T

t

Yi(t)TYi(t)dt ≥ αI (88)

leads to the convergence of ai to aleader.The convergence rate of ai → 0 can be improved by adding the diffusive coupling of the physical

parameters to the adaptation law in (86):

ai = ΓYTi si + ΓP(aleader − ai) (89)

where P is positive definite.Note that (86) is particularly attractive since it does not need the knowledge of its time-varying parameter

ai. If we regard qd(t) of the composite variable si in (86) as the power leader dynamics, this is the casewhen the knowledge leader and the power leader co-exist [53].

C. Synchronization of Heterogeneous Adaptive Dynamics with Local Knowledge CouplingWhile (86) achieves the synchronization of multiple adaptive dynamics by the centralized knowledge

leader, it will be useful to achieve the same synchronization of adaptive dynamics using only localcouplings of the parameters (ai). Consider the adaptive dynamics with the following adaptation law:

ai = Γ(YTi si −YT

i−1si−1) (90)

Consider the following control law for each adaptive dynamics

τi = Yi(ai+1)−K1si + K2si−1 + K2si+1 (91)

which makes the closed-loop dynamics of the i-th member as

M(qi)si + C(qi, qi)si + K1si −K2si−1 −K2si+1 + Yi(ai − ai+1) = 0 (92)

Hence, the closed-loop system for a network comprised of p adaptive robots can be written as[[M] 00 [Γ−1]

](x{a}

)+

[[C] 00 0

](x{a}

)+

[[Lp

K1,−K2] [L]Y

−[L]TY 0

](x{a}

)= 0 (93)

where {a} = (a1, a2, · · · , ap)T , and

[L]Y =

Y1 −Y1 0 0 · · · 00 Y2 −Y2 0 · · · 0...

......

......

−Yp 0 0 0 · · · Yp

(94)

Using the same Va matrix as in (81), we can show that the virtual system derived from (93) issemi-contracting by the negative semi-definite Jacobian, thereby asymptotically synchronizing the statevariables: si → sj and qi → qj . If the persistency of excitation is present as in (88), the we can concludefrom (92) that the physical parameters synchronize as well (ai → ai+1). Without loss of generality, theabove synchronization control law and adaptation law can be represented in the context of adaptive controlusing the estimates of the physical parameters.

25

Potential Applications of Adaptive Synchronization: The benefit of adaptive synchronization is obviousin the case of indifferent tracking (Section IV-D), i.e., K1 = 2K2 in the adaptive control laws in (86)and (91). Since VT

syncx is not a flow-invariant manifold for a heterogeneous network, the heterogeneousdynamics do not synchronize in the absence of exponential tracking stability. Consequently, the knowledgecoupling introduced in this section provides a method of both the state and parameter synchronizationof heterogeneous adaptive dynamics. Examples of adaptive dynamics include autonomous docking ofhighly fractionated spacecraft or space robots. In another example, the physical parameters which varyadaptively could also be the local control terms, such as impedance control gains. For instance, a group ofrobot manipulators can share the load of some big object, and actively tune the impedance control gainsonce some external force is applied to the robot network. The synchronization of the local impedancecontrol gains can also be interpreted in the context of impedance matching of bilateral time-delayedtelemanipulation [33].

VIII. CONCURRENT SYNCHRONIZATION AND LEADER-FOLLOWER NETWORKS

We further generalize the proposed synchronization framework in the context of concurrent synchro-nization and leader-follower networks, which permit construction of complex network structures.

A. Leader-Follower Networks via Uni-Directional CouplingIf we interpret the common desired trajectory qd(t) as a reference command from the dynamics of

an independent leader or virtual leader, our synchronization framework reduces to the leader-followerproblem where a power leader is connected with every single member of the network. Motivated by [53],such a centralized power leader network can be simplified to permit the local uni-directional coupling ofthe leader dynamics. For example, we can add a uni-directional coupling from the single leader to theleader’s adjacent followers in the network structure that does not have any common desired trajectoryqd(t), as seen in Section IV-D. In this case, the leader dynamics or the virtual leader trajectory, denotedby q1(t) in lieu of qd(t), are connected locally with the adjacent followers and regarded as an activemember of the dynamic network.

The control law for the followers (2 ≤ j ≤ p) can be given as

τj = M(qj)qjr + C(qj, qj)qjr + g(qj)− 2Ksj + Ksj−1 + Ksj+1 + cjK(s1 − sj) (95)

where cj is 1 if the j-th member is connected with the leader. Otherwise, it is zero. Also, the compositevariable sj for the j-th member does not have a desired trajectory: qjr = −Λqj and sj = qj + Λqj .This can easily be converted to the closed-loop dynamics driven by the same input u(t) = Ks1. Note thatdirectional couplings yield unbalanced Laplacians in contrast with the symmetric [Lp

K1,−K2] of the previous

sections. The first n rows of this nonsymmetric Laplacian [Lp] are zero, which correspond to the leaderdynamics, whereas its first n columns have either 0 or −K depending on the existence of the coupling withthe leader. Similar to Section IV, we can perform a spectral decomposition with Vsync, that is, the matrixof the orthonormal eigenvalues of the balanced Laplacian from (23). If VT

sync([Lp]+ [Lp]T )Vsync > 0, for

sufficiently large K and Λ, we can show that every element of the network tends exponentially to eachother. In other words, the network synchronizes with the leader dynamics q1(t) when each dynamics donot have the common desired trajectory qd(t).

B. Concurrent Synchronization of Heterogeneous NetworksIn [39], concurrent synchronization is defined as a regime where the ensemble of dynamical elements

is divided into multiple groups of fully synchronized elements, but elements from different groups arenot necessarily synchronized and can exhibit entirely different dynamics. Mathematically, such a regimecorresponds to a flow invariant linear subspace of the global state space.

26

Leader

2nd Network

1st

Network

Fig. 10. Concurrent synchronization between two different groups. The desired trajectory inputs are denoted by the dashed-lines whereasthe solid lines indicate mutual diffusive couplings. The independent leader sends the same desired trajectory input to the first network group.

In the context of the synchronization of multiple Lagrangian dynamics, discussed in this paper, we areinterested in the concurrent synchronization of different aggregates of multiple identical or nonidenticaldynamics. As discussed in the previous sections, we pay particular attention to the fact that there existtwo different time scales of the proposed synchronization tracking control law. In the case of a two-robotnetwork, one rate is associated with the trajectory tracking (K1 − K2) while the other represents theconvergence rate of the synchronization (K1 +K2). This in turn implies that there are two different inputsto the system, namely, the common reference trajectory qd(t) and the diffusive couplings with the adjacentmembers.

Accordingly, we exploit a desired trajectory qd(t) to create multiple combinations of different dynamicsgroups. For instance, Figure 10 represents the concurrent synchronization of two different dynamicalnetworks. The first network, consisting of four different robots, has the diffusive coupling structureproposed by the tracking control law in (3). The independent leader sends a desired trajectory command qdto the first network. With an appropriate selection of gains, each dynamics in the first network synchronizewhile exponentially following the leader. Therefore, the proposed scheme can be interpreted in the contextof the leader-follower problem ([18], [25], [53]).

The second network consists of three heterogeneous dynamics, also different from those of the firstgroup. Each element receives a different desired trajectory input from the adjacent element of the firstnetwork. However, once the first network is synchronized, the second network also ends up receiving thesame desired trajectory to follow while they interact to synchronize exponentially fast. Accordingly, wecan achieve concurrent synchronization between two different network groups. This can be extended toarbitrarily large groups of synchronized dynamics by appropriately assigning the desired trajectory inputsand the diffusive couplings. Note that the number of agents in one layer can be different from that ofanother layer as seen in Figure 10.

IX. CONCLUSIONS

We have presented the new synchronization tracking control law that can be directly applied to co-operative control of multi-robot systems and oscillation synchronization in robotic manipulation andlocomotion. The proposed decentralized control law, which requires only local coupling feedback forglobal exponential convergence, eliminates both the all-to-all coupling and the feedback of the accelerationterms, thereby reducing communication burdens and complexity. Furthermore, in contrast with prior workwhich used simple single or double integrator models, the proposed method permits highly nonlinear

27

systems. Providing exact nonlinear stability results constitutes one of the main contributions of thispaper; global and exponential stability of the closed-loop system has been derived by using contractiontheory. Contraction analysis, overcoming a local result of Lyapunov’s indirect method, yields global resultsbased on differential stability analysis. While we have focused on the mutual synchronization problemwhere synchronization and trajectory following take place simultaneously, the proposed method has alsobeen shown to be a generalization of the average consensus problem that does not address trajectorytracking. It has been emphasized that there exist multiple time scales in the closed-loop systems: the fasterconvergence rates represents the transient boundary layer dynamics of synchronization while the slowerrate determines how fast the synchronized systems track the common reference trajectory. Exponentialsynchronization with a faster convergence rate enables reduction of multiple dynamics into a simplerform, thereby simplifying the stability analysis. It should be noted that the tracking convergence rate alsodetermines the robustness of the closed-loop system with respect to a bounded disturbance.

Simulation results show the effectiveness of the proposed control strategy. The proposed bi-directionalcoupling has also been generalized to permit partial-joint coupling and uni-directional coupling (leader-follower networks). Further extensions to PD coupling, time-delayed communications, adaptive syn-chronization, and concurrent synchronization of heterogeneous networks exemplify the benefit of thedifferential stability analysis based on contraction theory. For future work, we are interested in extendingthe proposed method to dynamical networks on unbalanced or open graphs. Since this paper is based onthe assumption of fully actuated dynamics, it would be useful to consider multi-agent systems consistingof underactuated mechanical systems.

ACKNOWLEDGEMENTS

The authors gratefully acknowledge reviewers’ comments for the previous and current versions of thepaper.

APPENDIXCONTRACTION THEORY

We exploit partial contraction theory [51] to prove the stability of coupled nonlinear dynamics. Lya-punov’s linearization method indicates that the local stability of the nonlinear system can be analyzed usingits differential approximation. What is new in contraction theory is that a differential stability analysiscan be made exact, thereby yielding global results on the nonlinear system. A brief review of the resultsfrom [28], [45], [51] is presented in this section. Readers are referred to these references for detaileddescriptions and proofs on the following theorems. Note that contraction theory is a generalization of theclassical Krasovskii’s theorem [44], and that approaches closely related to contraction, although not basedon differential analysis, can be traced back to [16], [12] and even to [26].

Consider a smooth nonlinear system

x(t) = f(x(t),u(x, t), t) (96)

where x(t) ∈ Rn, and f : Rn×Rm×R+ → Rn. A virtual displacement, δx is defined as an infinitesimaldisplacement at a fixed time– a common supposition in the calculus of variations.

Theorem A.1: For the system in (96), if there exists a uniformly positive definite metric,

M(x, t) = Θ(x, t)TΘ(x, t) (97)

where Θ is some smooth coordinate transformation of the virtual displacement, δz = Θδx, such that theassociated generalized Jacobian, F is uniformly negative definite, i.e., ∃λ > 0 such that

F =

(Θ(x, t) + Θ(x, t)

∂f

∂x

)Θ(x, t)−1 ≤ −λI, (98)

28

then all system trajectories converge globally to a single trajectory exponentially fast regardless of theinitial conditions, with a global exponential convergence rate of the largest eigenvalues of the symmetricpart of F.Such a system is said to be contracting. The proof is given in [28]. Equivalently, the system is contractingif ∃λ > 0 such that

M +

(∂f

∂x

)TM + M

∂f

∂x≤ −2λM (99)

It can also be shown that for a contracting autonomous system of the form x = f(x,u(x)), all trajectoriesconverge to an equilibrium point exponentially fast. In essence, contraction analysis implies that stabilityof nonlinear systems can be analyzed more simply by checking the negative definiteness of a propermatrix, rather than finding some implicit motion integral as in Lyapunov theory.

The following theorems are used to derive stability and synchronization of the coupled dynamicssystems.

Theorem A.2: Hierarchical combination [45], [51]Consider two contracting systems, of possibly different dimensions and metrics, and connect them inseries, leading to a smooth virtual dynamics of the form

d

dt

(δz1

δz2

)=

(F11 0F21 F22

)(δz1

δz2

)Then the combined system is contracting if F21 is bounded.

Theorem A.3: Partial contraction [51]Consider a nonlinear system of the form x = f(x,x, t) and assume that the auxiliary system y = f(y,x, t)is contracting with respect to y. If a particular solution of the auxiliary y-system verifies a specific smoothproperty, then all trajectories of the original x-system verify this property exponentially. The original systemis said to be partially contracting.

Theorem A.4: Synchronization [51]Consider two coupled systems. If the dynamics equations verify

x1 − f(x1, t) = x2 − f(x2, t)

where the function f(x, t) is contracting in an input-independent metric, then x1 and x2 will converge toeach other exponentially, regardless of the initial conditions. Mathematically, stable concurrent synchro-nization corresponds to convergence to a flow-invariant linear subspace of the global state space [39].

REFERENCES

[1] Atkeson, C. G., Moore, A. W., and Schaal, S., “Locally Weighted Learning for Control,” Artificial Intelligence Review, 11, 1997.[2] Bamieh, B., Paganini, F., and Dahle, M.A., “Distributed Control of Spatially Invariant Systems,” IEEE Transactions on Automatic

Control, Vol. 47, No. 7., 2002.[3] Berthoz, A., “The Brain’s Sense of Movement,” Translated by Giselle Weiss, Harvard university press, Cambridge, MA, 2000.[4] Bizzi E., Giszter S.F., Loeb E., Mussa-Ivaldi F.A., Saltiel P., “Modular organization of motor behavior in the frog’s spinal cord,” Trends

in Neurosciences. Review 18:442-445, 1995.[5] Brooks, R., Cambrian Intelligence, M.I.T. Press, 1999.[6] Chopra, N., and Spong, M.W., “On Synchronization of Networked Passive Systems with Applications to Bilateral Teleoperation,”

Society of Instrumentation and Control Engineering of Japan Annual Conference, Okayama, Japan, August 8-10, 2005.[7] Chopra, N. and Spong, M.W., “Passivity-Based Control of Multi-Agent Systems,” in Advances in Robot Control: From Everyday

Physics to Human-Like Movements, Sadao Kawamura and Mikhail Svinin, Editors, pp. 107-134, Springer-Verlag, Berlin, 2006.[8] Chung, F., Spectral Graph Theory, Number 92 in CBMS Regional Conference Series in Mathematics, American Mathematical Society,

1997.[9] Chung, S.-J., Nonlinear Control and Synchronization of Multiple Lagrangian Systems with Application to Tethered Formation Flight

Spacecraft, Ph.D. thesis, MIT, 2007.[10] Chung, S.-J., Slotine, J.-J.E., and Miller, D.W., “Nonlinear Model Reduction and Decentralized Control of Tethered Formation Flight,”

Journal of Guidance, Control, and Dynamics, Vol. 30, No. 2, pp.390-400, 2007.[11] DAvella, A., Saltiel, P., Bizzi, E., “Combinations of Muscle Synergies in the Construction of a Natural Motor Behavior,” Nature

Neuroscience, 2003.

29

[12] Demidovich, B.P., “Dissipativity of a nonlinear system of differential equations,” ser. matematika mehanika, part I N.6, pp. 19–27;part II N.1, pp.3–8 Vestnik Moscow State University, 1962.

[13] Fierro, R., Song, P., Das, A., and Kumar, V., “Cooperative Control of Robot Formations,” in Cooperative Control and Optimization:Series on Applied Optimization, Kluwer Academic Press, 79-93, 2002.

[14] Gerard, Leonard and Slotine, J.J.-E., “Neuronal Networks And Controlled Symmetries, A Generic Framework,” arXiv:q-bio/0612049v2[q-bio.NC].

[15] Golubitsky, M., Stewart, I., “Synchrony versus Symmetry in Coupled Cells,” Equadiff 2003: Proceedings of the International Conferenceon Differential Equations, 2003.

[16] Hartmann, P., Ordinary differential equations John Wiley & Sons, New York, 1964.[17] Ijspeert, A. J., et al., “Simulation and Robotics Studies of Salamander Locmotion: Applying Neurobiological Principles to the Control

of Locomotion in Robotics,” Neuroinformatics. 3(3):171-95, 2005.[18] Jadbabaie, A., Lin, J., and Morse, A. S., “Coordination of Groups of Mobile Autonomous Agents Using Nearest Neighbor Rules,”

IEEE Transactions on Automatic Control, May 2003.[19] Jouffroy J. and Slotine,J.-J.E. “Methodological Remarks on Contraction Theory,” IEEE Conference on Decision and Control (CDC).

Atlantis, Paradise Island, Bahamas, 2004.[20] Kandel, E.R., Schwartz, J.H., and Jessel, T.M., Principles of Neural Science. 4th ed.McGraw-Hill, 2000.[21] Khalil, H.K., Nonlinear Systems, 3rd Ed.,Prentice Hall, Upper Saddle River, NJ, 2002.[22] Lee, D. and Li, P.Y., “Formation and Maneuver Control of Multiple Spacecraft ,” Proceedings of the 2003 American Control Conference,

Vol.1, pp. 278 - 283, June 2003.[23] Lee, D. and Spong, M.W., “Passive Bilateral Teleoperation with Constant Time-Delay,” Transactions on Robotics, vol. 22, no. 2, pp.

269-281, April 2006.[24] Lee, D. and Spong, M.W., “Stable Flocking of Multiple Inertial Agents on Balanced Graph,” Proceedings of the 2006 American Control

Conference, June 2006.[25] Leonard, N.E., and Fiorelli, E., “Virtual Leaders, Artificial Potentials and Coordinated Control of Groups,” 40th IEEE Conference on

Decision and Control, 2001.[26] Lewis, D.C., “Metric properties of differential equations,” American Journal of Mathematics, 71, pp. 294–312, 1949.[27] Lin, Z., Broucke, M., and Francis, B., “Local Control Strategies for Groups of Mobile Autonomous Agents. IEEE Trans. on Automatic

Control, 2004.[28] Lohmiller, W., and Slotine, J.J.E., “On Contraction Analysis for Nonlinear Systems,” Automatica, 34(6), 1998.[29] Lohmiller, W., and Slotine, J.J. E., “Nonlinear Process Control Using Contraction Theory,” A.I.Ch.E. Journal, March 2000.[30] Lohmiller, W., and Slotine, J.J.E., “Contraction Analysis of Nonlinear Distributed Systems,” International Journal of Control, 78(9),

2005.[31] Lohmiller, W., and Slotine, J.J.E., “High-Order Nonlinear Contraction Analysis,” MIT NSL Report, NSL-050901.[32] Mesbahi, M., and Hadaegh, F.Y., ”Formation Flying of Multiple Spacecraft via Graphs, Matrix Inequalities, and Switching,” AIAA

Journal of Guidance, Control, and Dynamics, (24) 2: 369-377, 2001.[33] Niemeyer, G., and Slotine J.J.E., “Telemanipulation with Time Delays,” International Journal of Robotics Research, 23(9), 2004.[34] Pecora, L.M., and Carroll, T.L., “Synchronisation in chaotic systems, Phys. Rev. Letter,” 64, 821-824, 1990.[35] Olfati-Saber, R., Nonlinear Control of Underactuated Mechanical Systems with Application to Robotics and Aerospace Vehicles, Ph.D.

thesis, Department of EECS, MIT, February 2001.[36] Olfati-Saber, R., and Murray, R.M., “Consensus Problems in Networks of Agents with Switching Topology and Time-Delays,” IEEE

Trans. on Automatic Control, vol. 49, no. 9, pp. 1520-1533, Sep. 2004.[37] Ogren, P., Egerstedt, M. and Hu, X., “A Control Lyapunov Function Approach to Multiagent Coordination,” IEEE Transactions on

Robotics and Automation, Vol. 18, No. 5, October 2002.[38] Ogren, P., Fiorelli, E., and Leonard, N.E., “Cooperative Control of Mobile Sensor Networks: Adaptive Gradient Climbing in a Distributed

Environment,” IEEE Trans. on Automatic Control, 49(8):1292.1302, 2004.[39] Pham, Q.-C., and Slotine, J.J. E., “Stable Concurrent Synchronization in Dynamic System Networks,” Neural Networks, 20(1), 2007.

65-87.[40] Pitti, A., Lungarella, M. and Kuniyoshi,Y. “Synchronization: Adaptive Mechanism Linking Internal and External Dynamics,” Proc. of

the 5th Int. Workshop on Epigenetic Robotics, 2006.[41] Rodriguez-Angeles, A., and Nijmeijer, H., “Mutual Synchronization of Robots via Estimated Sate Feedback: A Cooperative Approach,”

IEEE Transactions on Control Systems Technology, Vol. 12, No. 4, 2004.[42] Seo, K.-H., and Slotine, J.-J.E., “Models for Global Synchronization in CPG-based Locomotion,” Proceedings of IEEE Int. Conf. on

Robotics and Automation, Roma, Italy, April 2007.[43] Sinopoli, B., Sharp, C., Schaffert, S., Schenato, L., and Sastry, S., “Distributed control applications within sensor networks,” IEEE

Proceedings Special Issue on Distributed Sensor Networks, November 2003.[44] Slotine, J.-J.E., and Li, W., Applied Nonlinear Control, Prentice Hall,. New Jersey, 1991.[45] Slotine, J.-J.E., “Modular Stability Tools for Distributed Computation and Control,” Int. J. Adaptive Control and Signal Processing,

17(6), 2003.[46] Slotine, J.-J.E., and Lohmiller, W., “Modularity, Evolution, and the Binding Problem: A View from Stability Theory,” Neural Networks,

14(2), 2001.[47] Slotine, J.-J.E., Wang, W., and El Rifai, K., “Synchronization in Networks of Nonlinearly Coupled Continuous and Hybrid Oscillators,”

Sixteenth International Symposium on Mathematical Theory of Networks and Systems, July 2004.[48] Strang, G., Introduction to Applied Mathematics, Wellesley-Cambridge Press, Wellesey, MA, 1986.[49] Vidyasagar, M., Nonlinear Systems Analysis, 2nd ed., SIAM Classics in Mathematics, SIAM, Philadelphia, 1993.

30

[50] Wang, P.K.C., Hadaegh, F.Y., and Lau, K., “Synchronized Formation Rotation and Attitude Control of Multiple Free-Flying Spacecraft,”Journal of Guidance, Control, and Dynamics, Vol. 22, No. 1, 1999.

[51] Wang, W., and Slotine, J.J.E., “On Partial Contraction Analysis for Coupled Nonlinear Oscillators,” Biological Cybernetics, 92(1).2004.

[52] Wang, W., and Slotine, J.J.E., “Contraction Analysis of Time-Delayed Communications Using Simplified Wave Variables,” IEEE Trans.Aut. Control, 51(5), 2006.

[53] Wang, W., and Slotine, J.J.E., “A Theoretical Study of Different Leader Roles in Networks,” IEEE Transactions on Automatic Control,51(7), 2006.