Cooperative Robot Control and Synchronization of Lagrangian Systems

Transcription

1 Cooperative Robot Control and Synchronization of Lagrangian Systems Soon-Jo Chung and Jean-Jacques E. Slotine Abstract arxiv:7.79v [math.oc] Dec 7 This article presents a simple synchronization framework that can be directly applied to cooperative control of multi-vehicle systems and oscillation synchronization in robotic manipulation and locomotion. A dynamical network of multiple Lagrangian systems is constructed by adding diffusive couplings to otherwise freely moving robots or flying vehicles. The proposed decentralized tracking control law synchronizes an arbitrary number of robots into a common trajectory with global exponential convergence, and is shown to be a generalization of the average consensus problem. Exact nonlinear stability results, derived by contraction analysis, provide a fresh perspective on the multi-agent coordination and control problem. The proposed decentralized strategy with local couplings can be systematically applied to Lagrangian systems, and is further extended to time-delayed communications, adaptive synchronization, partial-joint coupling, and concurrent synchronization of heterogeneous networks. I. INTRODUCTION Distributed and decentralized synchronization phenomena are areas of intense research. Group synchronization and cooperative control are topics that are currently receiving a lot of interest in a variety of research communities, including biology [5]; artificial intelligence [], [5]; neuroscience [], [], [3]; chaotic synchronization for communications [34]; robotics [3], [7], [4]; wireless sensor networks [36], [43]; spacecraft formation flight []; automatic control of multi-agent systems such as mobile robots [8], [7], [37]; and distributed systems [], [3]. In this article, synchronization is defined as a complete match of all configuration variables of each dynamical system such that x = x = = x p and p denotes the number of sub-systems in the network. While such a definition directly concerns the attitude tracking problem, this paper further addresses synchronization of biased variables with application to the position coordination problem. In the latter case, synchronization corresponds to y = y = = y p where y i, i p connotes a vector of biased variables constructed from the configuration vector x i such that x i (t) = y i (t) + b i (t) and the separation vector b i (t) is independent of the dynamics. The objective of this paper is to introduce a unified synchronization framework that can be directly applied to cooperative control of multi-robot systems or formation flying aerospace vehicles to track a common desired trajectory. Although an uncoupled trajectory control law, in the absence of external disturbances, would achieve synchronization to a common desired trajectory, non-identical disturbances justify the mutual trajectory synchronization. For example, a large swarm of robots can first synchronize their attitudes and positions to form a certain formation pattern, then track the common trajectory to accomplish 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 [4], there has been widespread interest in cooperative schemes for multiple robot manipulators that track a predefined trajectory. A stellar formation flight interferometer [] is another example where precision control of relative spacecraft motions is indispensable. The proposed synchronization tracking control law can be implemented for such purposes, where a common desired trajectory can be explicitly given. The proposed strategy can achieve more efficient and robust performance through local interactions, especially in the presence of non-identical external disturbances. We also generalize our proposed control law such Assistant Professor of Aerospace Engineering, Iowa State University ( sjchung@alum.mit.edu). Professor of Mechanical Engineering & Information Sciences, Professor of Brain & Cognitive Sciences, MIT ( jjs@mit.edu).

2 that multiple dynamic systems can synchronize without the need for a common reference trajectory. As a result, other potential applications include oscillation synchronization of robotic locomotion [7], [4], [4], and tele-manipulation of robots [33], [3], [6]. Another benefit of synchronization is its implication for model reduction. The exponential synchronization of multiple nonlinear dynamics allows us to reduce the dimensionality of the stability analysis of a large network. The model reduction aspect of synchronization, also introduced for spatially interconnected systems in [], is further generalized and strengthened in this article. This implies that once the 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 many mechanical systems exhibit nonlinear dynamics that cannot be captured by linearization, determining stability of nonlinear dynamic network systems is much more complex. We introduce contraction analysis [8], [45], [5] as our main nonlinear stability tool for reducing the complexity and dimensionality associated with multi-agent systems, thereby deriving exact and global results with exponential convergence (see the Appendix for the further treatment of contraction theory). The proposed control laws are of a decentralized form requiring only local velocity/position coupling feedback for global exponential convergence, thereby facilitating implementation in real systems. As opposed to some previous work requiring all-to-all coupling [4], this will significantly reduce communication burdens. The theory is generalized and extended to multi-robot systems with non-identical dynamics, linear PD coupling, partial joint coupling, uni-directional coupling, and adaptive control. The tracking synchronization can also reduce to the state synchronization problem without a common desired trajectory. Time-delayed communications and robustness properties are also discussed. A. Comparison with Related Work The consensus problems on graph [36] and the coordination of multi-agent systems [8], [7], [37], [38] are closely related with the synchronization problem. In particular, the use of graph theory and Laplacian produced many interesting results [4], [8], [7], [3], [36]. However, synchronization to the average of the initial conditions might not be applicable to multi-robot and multi-vehicle systems, where a desired trajectory is explicitly defined. Furthermore, the aforementioned work mainly deals with simple dynamic models such as linear systems and single or double integrator models without nonlinear coupling matrices. In contrast, we aim at addressing highly nonlinear systems (e.g. helicopters, attitude dynamics of spacecraft, walking robots and manipulator robots). As shall be seen later, the proof of the synchronization for network systems that possess nonlinearly coupled inertia matrices is more involved. This paper focuses on dynamical networks consisting of highly nonlinear systems. One recent representative work on the mutual trajectory synchronization of multi-link robots is [4]. The paper [4] proposes a nonlinear tracking control law to synchronize multiple robot manipulators in order to track a common desired trajectory. In essence, [4] proves the stability of the control law with estimated velocity and acceleration states from a set of nonlinear observers, which were used for an experimental demonstration of a two-robot system. The following difficulties can be identified. The number of variables to be estimated increases with the number of robots to be synchronized, which imposes a significant communication burden. Additionally, the feedback of estimated acceleration errors requires unnecessary information and complexity. Thus, a method to eliminate both the all-to-all coupling and 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 inputoutput dynamics [6], [7], [3]. Its property of robustness to time-delays is particularly attractive. However,

3 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 enforces them to follow a common reference trajectory, is not addressed. In particular, it shall be shown that our proposed control law is a generalized version of the control law presented in [7]. Another recent work using the passive decomposition [], [4] is interesting in the sense that it shares the same philosophy of mutual 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 the total group maneuver. One drawback of [] is its dependency on a centralized control architecture; the decoupling is not generally ensured under the decentralized control. [4] only considers linear double integrator models, which can degenerate into a simpler problem (see Section IV). We believe our approach using contraction analysis has a clear advantage in its broad applications to a larger class of identical or nonidentical nonlinear systems even with time-delays, non-passive input-output, and complex coupling geometry including concurrent synchronization and partial degrees-of-freedom coupling, while ensuring a simple decentralized coupling control law (see Figure for network structures permitted here). B. Organization The 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 tracking control law, which synchronizes each robot to track the same desired trajectory, is proposed. The proof of exponential synchronization is more involved than that of the tracking stability and treated separately in Section IV. The remainder of the paper further highlights the unique contributions of this work. The main idea of this paper is extended to networks with linear Proportional-Derivative (PD) coupling in Section V-A. A few examples of dynamics networks are given in Section V for validating the effectiveness of the 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 Section VI. An adaptive control version of the proposed synchronization strategy is presented in Section VII while Section VIII offers a potential application of the proposed scheme to the concurrent synchronization and leader-follower problem. II. MODELING OF MULTI-ROBOT NETWORK An exponentially stabilizing nonlinear control law that can synchronize an arbitrary number of robots to track a common reference trajectory is introduced in this paper. The proposed controller is much simpler than earlier work, such as the synchronization law in [4], in terms both of the computational load and the required signals. This paper is devoted to the use of the Lagrangian formulation for its simplicity in dealing with complex systems involving multiple dynamics. The equations of motion for a robot with multiple joints (q i R n ) can be derived by exploiting the Euler-Lagrange equations: L i (q i, q i ) = K i (q i, q i ) V i (q i ) = qt i M i (q i ) q i V i (q i ) d L i (q i, q i ) L i(q i, q i ) = τ i () dt q i q i where i, ( i p) denotes the index of robots or dynamical systems comprising a network, and p is the total number of the individual elements. Equation () can be represented as M i (q i ) q i + C i (q i, q i ) q i + g i (q i ) = τ i () where g i (q i ) = V i q i, and, τ i is a generalized force or torque acting on the i-th robot. Note that we define C i (q i, q i ) such that (Ṁ i C i ) is skew-symmetric [44], and this property plays a central role in our stability analysis using contraction theory [9].

4 (a) identical robots (b) non-identical robots Fig.. 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 () 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 the robot workspace [44]. III. A NEW APPROACH TO SYNCHRONIZATION TRACKING CONTROL We consider the synchronization of multiple robots following a common time-varying trajectory. A tracking controller introduced in this section achieves not only global and exponential synchronization of the configuration variables, but also global exponential convergence to the desired trajectory. A. Proposed Synchronization Control Strategy We introduce a simple alternative control law for the mutual synchronization problem of multiple robots, as seen in [4]. The following tracking control law with two-way-ring symmetry is proposed for the i-th robot in the network comprised of p identical robots (see Figure (a)): τ i = M(q i ) q ir + C(q i, q i ) q ir + g(q i ) (3) K ( q i q ir ) + K ( q i q i,r ) + K ( q i+ q i+,r ) where a positive-definite matrix K R n n is a feedback gain for the i-th robot, and another positivedefinite matrix K R n n is a coupling gain with the adjacent members (i and i + ). The above control law can also be applied to a network consisting of p non-identical robots (Figure (b)), as shall be seen in Section V-C. While the common desired time-varying trajectory (or the virtual leader dynamics) is denoted by q d (t), the reference velocity vector, q ir is given by shifting the common desired velocity q d with the position error: q ir = q d Λ q i = q d Λ(q i q d ) (4) where Λ is a positive diagonal matrix. In contrast with [4], the proposed control law requires only the coupling feedback of the most adjacent robots (i and i + ) for exponential convergence (see Figure ). Note that the last (p-th) robot is connected with the first robot to form a ring network as suggested in [5]. Moreover, estimates of q are no longer required. The closed-loop dynamics using () and (3) become M(q i )ṡ i + C(q i, q i )s i + K s i K s i K s i+ = (5) where s i denotes the composite variable s i = q i q ir. This system is equivalent to M(q i )ṡ i + C(q i, q i )s i + K s i K (s i + s i+ ) + K p s j = K j= p s j (6) j=

5 5 Hence, the closed-loop dynamics of each robot have the same excitation input u(t) = K p j= s j, which facilitates the use of contraction analysis. Let us define the following p p block square matrices: A B B A A A B A B [L p A,B ] = , [U p A B A B ] = A A A (7) A A A B B A p p By the definition of the controller in (3), [L p A,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 where p p [M]ẋ + [C]x + ( [L p K, K ] + [U p K ] ) x = [U p K ]x (8) M(q ) C(q, q ) s [M] =....., [C] =....., x =. (9) M(q p ) C(q p, q p ) s p Note that [L p K, K ] can be viewed as the modified Laplacian of the network in the context of graph theory. In other words, [L p K, K ] indicates the connectivity with adjacent systems as well as the strength of the coupling by K. Note that there are only three nonzero elements in each row of the matrix, which implies that there exist diffusive couplings only between adjacent members. The network graphs illustrated in Figure are balanced due to bi-directional coupling [36]. However, it should be noted that the matrix [L p K, K ] 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 a right eigenvector, = (,,, ) T [36]. In contrast, a strictly positive definite [L p K, K ] is required for exponential convergence for the proposed control law in this paper. In other words, unless otherwise noted, [L p K, K ] is assumed to have no zero eigenvalue. For example, the block matrix for p = 4 becomes +K K K [L 4 K, K ] = K +K K K +K K, () K K +K which is positive definite with K K >. For a two-robot network, it is straightforward to verify [ ] [L +K K K, K ] = () K +K requiring the condition K K > for positive definite [L K, K ]. 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 q d. Theorem 3.: Global Exponential Convergence to the Desired Trajectory If [L p K, K ] is positive definite, then every member of the network follows the desired trajectory q d exponentially fast regardless of initial conditions. [L p K, K ] > In other words, if K K >, then q i, (i =,,, p, p 3) converges to q d exponentially fast from any initial conditions. For two-robot systems (p = ), K K > needs to be true instead.

6 6 Proof: We can cancel out the [U p K ] matrix term in (8) to obtain [M]ẋ + [C]x + [L p K, K ]x =. () Equation () corresponds to a conventional tracking problem. We use contraction theory (see the Appendix) to prove that x tends to zero exponentially with [L p K, K ] >. For example, consider the virtual system of y obtained by replacing x with y in (). [M]ẏ + [C]y + [L p K, K ]y = (3) This virtual y system has two particular solutions: x = (s,, s p ) T and. The squared-length analysis with respect to the positive-definite metric [M] yields d ( δy T [M]δy ) = δy T [M]δẏ + δy T [Ṁ]δy (4) dt = δy ( T [C]δy + [L p K, K ]δy ) + δy T [Ṁ]δy = δy T [L p K, K ]δy where we used the skew-symmetric property of [Ṁ] [C]. Accordingly, [L p K, K ] > will make the system contracting (δy ), thus all solutions of y converge to a single trajectory exponentially fast. This in turn indicates that the composite variable of each robot tends to zero exponentially (s ). By the definition of s i = q q d + Λ(q i q d ), the exponential convergence of q i to the common reference trajectory q d is proven (see the hierarchical combination in Theorem A.). The positive-definiteness of [L p K, K ] corresponds to K K > for two-robot systems (p = ). For a network consisting of more than two robots (p 3), it can be shown that K K is a sufficient condition of the positive-definiteness of [L p K, K ] given K >, K >. 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.: Synchronization of Multiple Robots Suppose the conditions in Theorem 3. are true, thus the individual dynamics are exponentially tracking the common desired trajectory. A swarm of p robots synchronize exponentially from any initial conditions if diagonal matrices K >, K > such that [L p K, K ] + [U p K ] > In addition, Λ is a positive diagonal matrix defining a stable composite variable s i = q i + Λ q i. This theorem corresponds to synchronization with stable tracking. The proof is expanded in Section IV by 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 K + K > for two-robot and three-robot networks (p =, 3). A four-robot network (p = 4) would require K + K >. We can also construct a network of multiple robots that can synchronize even without stable tracking. In this case, the follow lemma can be used. Lemma 3.3: Synchronization of Identical Robots Without Trajectory Tracking Suppose the conditions in Theorem 3. 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 matrices K >, K such that [L p K, K ] + [U p K ] > For indifferent tracking, a common desired trajectory q d (t) is no longer required. For unstable tracking, Λ should be sufficiently large such that Λ > K K for p = or Λ > K K for p 3, where σ(m(q)) σ(m(q))

7 7 σ( ) denotes the smallest singular value. In contrast with Theorem 3., the individual dynamics must be identical in the unstable tracking case. This lemma describes synchronization with indifferent dynamics (K K = ) or unstable tracking (K K < ) for p = ; indifferent (K K = ) or unstable (K K < ) for p 3. Section IV provides a unified treatment of both stable tracking (Theorem 3.) and indifferent tracking (Lemma 3.3) while tracking with instability has been proven in [9]. Note that indifferent tracking reduces the main problem into the standard Laplacian problem that does not deal with a common time-varying desired trajectory. Note that we can render the system synchronized first, then follow the common trajectory by tuning the gains properly. For an example of a two-robot network, K > ensures that the two robots synchronize faster than they follow the common desired trajectory, since K + K > K K for K >. This indicates that there exist two different time scales in the closed-loop systems constructed with the proposed controllers. For two-robot systems, the convergence of exponential tracking is proportional to K K whereas the synchronization has a convergence rate of K + K. This multi-time-scale behavior will be exploited in the subsequent sections. IV. PROOF OF EXPONENTIAL SYNCHRONIZATION We prove Theorem 3. and Lemma 3.3 for the exponential synchronization of multiple nonlinear dynamics in this section. First, we describe the difficulties inherent in proving the synchronization of Lagrangian systems that possess nonlinearly coupled inertia matrices in Section IV-A. We then present the synchronization proof Section IV-C. We also show that our method is more general than prior work by reducing our control law to the standard synchronization problem without trajectory tracking in Section IV-D. The main idea about model reduction via synchronization is emphasized in Section IV-F. A. Challenges with Nonlinear Inertia Matrix The 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 which is equivalent to where M(q )ṡ + C(q, q )s + (K + K )s = u(t) M(q )ṡ + C(q, q )s + (K + K )s = u(t) u(t) = K (s + s ) [M ]{ṡ} + [C ]{s} + [K]{s} = [U K ]{s} [ ] [ ] ( ) M(q ) C(q, q [M ] =, [C M(q ) ] = ) s, {s} = C(q, q ) s [ ] [ ] (6) [K] = [L K, K ] + [U K + K K ] =, [U K + K K K K ] = K K Note that s and s are the composite variables defined in (5): s = q q r = q + Λq ( q d + Λq d ) s = q q r = q + Λq ( q d + Λq d ) Direct application of synchronization (Theorem A.4) appears elusive because the original dynamics in () with the control law (3) is in general not contracting. Since () is a second-order differential equation, this can also be viewed as a high-order contraction problem [3]. If we transform the dynamics into a first-order canonical form, we have to prove that they are contracting in the same metric while (5) (7)

8 8 preserving the input symmetry [39]. For example, multiplying (5) by M breaks the input symmetry: i.e., M (q )u(t) M (q )u(t). In essence, M(q ) M(q ) makes this problem intractable in general. Instead, suppose that M(q) remains constant, thereby making C(q, q) zero. Then, we can easily prove s and s tend to each other from Mṡ + (K + K )s = K (s + s ) Mṡ + (K + K )s = K (s + s ) Since the virtual system with the common input u(t) = K (s + s ) (8) Mẏ + (K + K )y = u(t) (9) is contracting with K +K >. Hence, its particular solutions s and s tend to each other exponentially fast according to the synchronization theorem (Theorem A.4). Without loss of generality, this result can easily be extended to arbitrarily large networks. The synchronization of a large network with a constant metric, as seen in (8), is already discussed in [5] using contraction analysis. We now turn to a more difficult problem focused on the synchronization of two robots with non-constant nonlinear metrics (M(q ) M(q )). B. Existence of Flow-Invariant Submanifold We define and prove the existence of a flow-invariant manifold for any system given in (5), regardless of [K], prior to finding the condition of [K] for exponential synchronization. Let us start with the definition of flow-invariant manifold. Consider a flow-invariant differentiable submanifold of the configuration space for a two-robot system, M q = {(q ; q ; q ; q ) q = q, q = q } () Note that M q ia a flow-invariant submanifold for the closed-loop system (5). This follows from the fact that the initial condition, q (t ) = q (t ), q (t ) = q (t ) for an arbitrary t R + yields q ( ) = q ( ), which implies that any trajectory starting in M q remains in M q. The flow-invariant manifold, M q corresponds to the synchronization of the two dynamical systems. Define V sync as a matrix of the orthonormal vectors such that V sync is a projection on M q. Then, it verifies [39] C. Contraction with Multiple Time Scales V sync x = x M q () We present the proof of synchronization for a multi-robot network first, followed by an example of a two-robot network (p = ) in Section IV-E. Recall the closed-loop dynamics given in (8): [M]ẋ + [C]x + [L p K, K ]x = () Since [L p K, K ] 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 flow invariant subspace (the eigenspace). [L p K, K ] = V[D]V T (3) where [D] is a block diagonal matrix, and the square matrix V is composed of the orthonormal eigenvectors such that V T V = VV T = I pn, since the symmetry of [L p K, K ] gives rise to real eigenvalues and orthonormal eigenvectors [48]. Pre-multiplying () by V T and setting x = VV T x result in ( V T [M]V ) V T ẋ + ( V T [C]V ) V T x + ( V T [L p K, K ]V ) V T x = (4)

9 9 By setting V T x = z, (4) becomes ( V T [M]V ) ż + ( V T [C]V ) z + [D]z = (5) Then, we can develop the squared-length analysis. Notice that ( V T [M]V ) is always symmetric positive definite since [M] is symmetric positive definite. Also, note that d ( V T [M]V ) ( V T [C]V ) (6) dt is skew-symmetric, which facilitates our stability analysis. Since the modified Laplacian [L p K, K ] represents a regular graph, where each member has the same number of neighbors (= for p 3), [] = p [I n, I n,, I n ] T (7) is the pn n block column matrix of eigenvectors associated with the eigenvalues λ(k K ) for p = or λ(k K ) for p 3. Note that I n denotes the n n identity matrix, and the [] matrix consists of p matrices of I n. The eigenvector matrix [] represents the common trajectory tracking state (e.g., s + s + + s p ). Similar to (), we can define a pn (p )n matrix V sync that consists of the orthonormal eigenvectors other than [] such that ( ) [ ] [ ] [] V T T ([] ) [] V = T [] [] T V Vsync = sync I Vsync[] T VsyncV T = n n (p )n (8) sync (p )n n I p(n ) V T sync where we used the orthogonality between [] and V sync. Accordingly, the block diagonal matrix [D], which represents the eigenvalues of [L p K, K ], can be partitioned from (3) [ [] V T [L p T [L p K K, K ]V =, K ][] [] T [L p ] [ ] K, K ]V sync D Vsync[L T p K, K ][] Vsync[L T p = n (p )n (9) K, K ]V sync (p )n n D It should be emphasized that D = K K for p 3 (or D = K K for p = ) represents the tracking gain while D corresponds to the synchronization gain. We can choose the diagonal control gain matrices K and K such that D = V T sync[l p K, K ]V sync > D = [] T [L p K, K ][], (3) thereby ensuring that the robots synchronize faster than they follow the common desired trajectory. The condition in (3) can be written as D = K + K > D = K K for two robots (p = ). This multi timescale behavior is graphically illustrated in Figure. The figure depicts that s and s synchronize first, then they converge to the desired trajectory while staying together. This observation motivates separation of the two different time scales, namely K + K and K K. Consider the virtual system of y = (y t, y s ) T which has the following two particular solutions: ( yt = [] T x y s = Vsyncx ) T T ( and yt = y s = ) T (3) ( V T [M]V ) ẏ + ( V T [C]V ) y + [D]y = (3) For [D] >, we can show that the above virtual system is contracting (i.e., δy globally and exponentially). We take the symmetric positive definite block matrix V T [M]V as our contraction metric.

10 S =S synchronization tracking S S S =S = Fig.. Multiple timescales of synchronization (faster) and tracking (slower). The dashed line indicates the desired trajectory, and arrows indicate increasing time. The drawing is conceptual, since strictly speaking s and s synchronize exponentially. Performing the squared-length analysis with respect to this metric yields d ( dt δyt V T [M]V ) δy = δy ( T V T [M]V ) ( ) δẏ + δy T V T [Ṁ]V δy (33) ( (V = δy T T [C]V ) ) ( ) δy + [D]δy + δy T V T [Ṁ]V δy = δy T [D]δy ( ) where we used the skew-symmetric property of V T [Ṁ]V ( V T [C]V ). The above equation can be rewritten in terms of two different time scales ( ) T [ ] ( ) ( ) d δyt [] T [M][] [] T T [ ] ( ) [M]V sync δyt δyt D δyt dt δy s Vsync[M][] T Vsync[M]V T = sync δy s δy s D δy s If [D] >, which is in fact equivalent to [L p K, K ] >, the combined virtual system in (3) is contracting. In other words, δy exponentially fast. This in turn implies all solutions of y tend to the single trajectory. As a result, [] T x = / p(s + + s p ) and V T syncx tend to zero exponentially. Note that s,, s p has already been proven for Theorem 3., which is a sufficient condition to make the sum of the composite variables also tend to zero (i.e., [] T x ). The new result in this section is the synchronization V T syncx. It is straightforward to show that V T syncx and Λ > also hierarchically make q,, q p synchronize exponentially (see Theorem A.). This can be verified by the following contracting dynamics constructed from (7) V T sync{ q} + ( V T sync[λ]v sync ) V T sync {q} = V T syncx (35) where {q} = (q,, q p ) T. Note that the orthonormal vectors V sync cancel the common input term q d + Λq d. Also, [Λ] is a block diagonal matrix of Λ >, thereby yielding V T sync[λ][] = from (34) V sync V T sync{q} + [][] T {q} = {q} (36) Consequently, Vsync{q} T implies the synchronization of the original state variable q i, i =,, p. This also implies that the diagonal terms of the metric, Vsync[M][] T tend to zero exponentially, thereby eliminating the coupling of the inertia term V T [M]V in (33,34). Note that we assume here that q d (t) is identical for each member. If q d (t) were different for each dynamics, s i s j would imply the synchronization of q i q j to the difference of the desired trajectories, which is useful to construct phase synchronization.

11 Now let us relate the condition in Theorem 3. with [D] >. The block matrix [U p K ] also has [] as its eigenvector. We multiply [L p K, K ] + [U p K ] by its orthonormal eigenvectors other than []: [L p K, K ]V sync + [U p K ]V sync = [L p K, K ]V sync = V sync D (37) which shows that V sync also represents the orthonormal eigenvectors of [K] = [L p K, K ] + [U p K ]. In other words, D corresponds to the eigenvalues of [K]. The remaining eigenvalue of [K], associated with [], is K + (p )K, which is greater than the tracking eigenvalue D = K K for p 3. Hence, the synchronization occurs if [L p K, K ]+[U K ] > : This completes the proof of Theorem 3.. [D] > D > and [L p K, K ] + [U K ] >. (38) D. Synchronization Without a Common Reference Trajectory We now turn into the more standard synchronization problem where the tracking gain is zero D = K K = for p = or D = K K = for p 3, which fails the exponential stability condition in Theorem 3.. In this case, our modified Laplacian [L p K, K ] reduces to the standard weighted Laplacian whose row sum is zero. For synchronization to the weighted average of the initial conditions, we do not require the common desired trajectory q d, and q d can simply be set to zero as follows: q ir = Λq i, s i = q i + Λq i (39) In other words, our control strategy represents a more generalized framework for the synchronization of multiple Lagrangian systems. The combined virtual system per se is then semi-contracting ([8], [9]) since the squared-length analysis in (33) yields the negative semi-definite matrix: d dt ( ) T [ δyt [] T [M][] δy s Vsync[M][] T ] ( ) [] T [M]V sync δyt Vsync[M]V T = sync δy s ( ) T [ ] ( ) δyt δyt (4) δy s D δy s While δy t, representing the tracking dynamics, remains in a finite ball due to D =, δy s tends to zero asymptotically due to D >. This result can be proven as follows. V is uniformly continuous since a bounded δẏ s from (3) leads to a bounded V since V = 4δy T s D δẏ s (4) Due to V, the use of Barbalat s lemma [44] verifies that V as t. This implies that δys tends to zero asymptotically fast. As a result, V T syncx tends to zero asymptotically. From the hierarchical combination discussed in (35), this also implies V T sync{q} =. This completes the proof of Lemma 3.3 with indifferent tracking. This will eventually decouple the metric matrix with Λ >, as seen in (34), since V T sync[m][] tends to zero simultaneously as V T sync{q} =. As a result, when M(q i ) M(q j ) is sufficiently close to zero, the convergence of δy s turns exponential. We can also prove synchronization in the presence of tracking instability (D < ) with sufficiently small D by decoupling the unstable dynamics from the stable synchronization dynamics. We refer the readers to [9] for details. In essence, we can show the synchronization can occur fast enough to overcome the tracking instability (see the simulation results in Section V-D).

12 Fast Inhibition: The dynamics of a large network with semi-contracting stability as in (4) can be instantaneously transformed to contracting dynamics by the addition of a single inhibitory coupling link. In other words, a single inhibitory link will also make δy t. For instance, we can add a single inhibitory link between two arbitrary elements a and b while we keep the rest of the elements the same: [5] τ a = M(q a ) q ar + C(q a, q a ) q ar + g(q a ) K s a + K s a + K s a+ K(s a + s b ) (4) τ b = M(q b ) q br + C(q b, q b ) q br + g(q b ) K s b + K s b + K s b+ K(s a + s b ) where K is substituted for K, and K >. Hence, we can easily show that [L p K, K ] is now strictly positive definite, in contrast with the original semi-contracting system. As a result, the closed-loop system is contracting, resulting in s i and q i from (39). This fast inhibition is useful to rapidly destroy unwanted synchronized oscillation of robots. E. Examples For the case of a two-agent network, we can easily verify that [ M(q V T )+M(q ) M(q ) M(q ] ( ) ) [M]V = [], z = V T T x = x = ( V T [C]V ) = M(q ) M(q ) M(q )+M(q ) [ C(q, q )+C(q, q ) C(q, q ) C(q, q ) C(q, q ) C(q, q ) C(q, q )+C(q, q ) ], [D] = V T sync [ I I I I [ ] K K K + K ] ( s If K + K > K K >, the rate of the virtual length in (33) is uniformly negative definite for nonzero δy t and δy s : ( ) T [ d M(q)+M(q) M(q ) M(q ] ( ) ) δyt δyt dt δy M(q ) M(q ) M(q )+M(q ) (44) s δy s ( ) T [ ] ( ) δyt K K = δyt <. δy s K + K δy s s ), Consequently, the combined virtual system in (3) is contracting. This in turn implies all solutions of y t and y s tend to the single trajectory. As a result, s p = s + s and s m = s s tend to zero exponentially. It is straightforward to show that s m also hierarchically makes q tend to q exponentially (see Theorem A.). From the definition of the composite variables in (7), we can find the following contracting dynamics, M(q ) M(q ) (43) ( q q ) + Λ(q q ) = s m (45) Note that ẏ + Λy = is contracting with Λ >. Consequently, Λ > and s m make q q M(q exponentially fast. This also implies that the diagonal terms of the metric, ) M(q ) tend to zero exponentially, thereby eliminating the coupling of the inertia term. [ M(q )+M(q ) M(q ) M(q ] [ ] ) M(q ) δm (46) δm M(q ) M(q )+M(q ) where the perturbation term δm = M(q ) M(q ) tends exponentially to zero, since δm m q q. We can extend the method in this section to arbitrarily large networks. For example, a network of three robots has the following V whose columns are orthonormal eigenvectors of [L p=3 V = [ ] 3 I 6 I [] V sync = 3 I 6 I 3 I 6 I I I K, K ]:. (47)

13 3 Note that V T V = VV T = I 3n. The new transformed inertia matrix, V T [M]V is written as M 3 + M 3 + M 3 3 M M + 3 M 3 6 M + 6 M 3 M M + 3 M 3 M 3 + M 6 + M 6 3 M M + 6 M 3 M 3 + M 3 3 M + M 3 3 M 3 (48) where M = M(q ), M = M(q ), and M 3 = M(q 3 ) for notational simplicity. This matrix is symmetric and positive definite. Also notice that its off-diagonal terms vanish as q q, q q 3. The diagonal matrix [D] is also computed as K K K + K (49) K + K For a four-robot network p = 4 (see Figure ), [D] is given as K K K + K K (5) K where K represents the synchronization gain associated with the synchronization of diagonal members (q = q 3, q = q 4 ), while K + K represents the synchronization gain of direct couplings (e.g., q = q, q 3 = q 4 ). This is a percolation effect discussed in [39]. The percolation effect can be exploited in order to prove the synchronization to the large unknown invariant set (e.g., V T syncx = from a large network) by verifying the synchronization to a known, not necessarily invariant, subset of the global flow-invariant set [4]. F. A Perspective on Model Reduction by Synchronization It 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 (D ) is faster than the tracking rate (D ). Assuming that the dynamics are synchronized, the augmented dynamics in () reduces to M(q)ṡ + C(q, q)s + (K K )s = (5) where q = q = = q p, and D = K K is replaced by K K for p =. This implies that once the network is proven to synchronize, we can regard a network as a single set of synchronized 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 Robots One may consider the following Proportional and Derivative (PD) coupling control law for two identical robots from () with p =, τ = K ( q + Λ q ) + K ( q + Λ q ) τ = K ( q + Λ q ) + K ( q + Λ q ) where q i = q i q d and the bounded reference position q d has zero velocity such that q i = q i. (5)

14 4 For simplicity, the gravity term in () is assumed to be zero or canceled by a feed-forward control law. Then, the closed-loop dynamics satisfy M(q ) q + C(q, q ) q + K q + KΛ q = u(t) M(q ) q + C(q, q ) q + K q + KΛ q = u(t) where K = K + K and u(t) = K ( q + q ) + K Λ( q + q ). Similar to Section IV-C, we can perform a spectral decomposition: ( V T [M]V ) V T ẍ + ( V T [C]V ) V T ẋ + ( V T [L p K, K ]V ) V T ẋ + ( V T [L p K Λ, K Λ ]V) V T x = Using the following Lyapunov function, it is straightforward to show that this PD coupling control law drives the system to the desired rest state q d globally and asymptotically while tending to the synchronized flow-invariant manifold, M q in (): V = ( ) T [ M+M qp M q M m where M M M +M ] ( qp q m ) + M = M(q ), M = M(q ), V T x = ( qp (53) (54) ( ) T [ ] ( ) qp (K K )Λ qp. (55) q m (K + K )Λ q m q m ) = ( ) q + q, q q q p = ( q + q ). (56) The rate of V can be computed as ( ) T [ dv dt = qp K K q m ] ( ) qp K + K q m (57) which implies that V is negative semi-definite with K + K > and K K >. By invoking LaSalle s invariant set theorem [44], we can conclude that q p, q p, q m, and q m tend to zero with global and asymptotic convergence. This implies that q and q will follow q d while q and q synchronize asymptotically. Theorem 5.: PD Synchronization of Two Identical Robots Two identical robots synchronize asymptotically and globally with the PD coupling law in (5) if K > such that K + K > Λ > In addition, K K > is required for asymptotic stability of the PD control. B. Velocity Coupling, Λ = Note that if Λ =, the PD coupling control law in (5) reduces to τ = K q + K q τ = K q + K q (58) This velocity coupling control can also be derived from the exponential tracking control law in (3) by setting q d =, q d = and Λ =. Therefore, the proof of Theorem 5. with Λ = is the same as Section IV-C whereas the convergence rate is now exponential compared to the asymptotic convergence of the PD control. On the other hand, we can find that positions do not synchronize in the absence of the gravity term even though the velocities synchronize exponentially fast.

15 5 C. Synchronization of Heterogeneous Robots Notice that the proposed tracking and synchronization control law in (3) can easily be applied to a network consisting of heterogenous robots in () if the stable tracking condition in Theorem 3. is true. τ i = M i (q i ) q ir + C i (q i, q i ) q ir + g i (q i ) K ( q i q ir ) + K ( q i q i,r ) + K ( q i+ q i+,r ) (59) where M i, C i, and g i (q i ) represent the i-th robot dynamics, which can be different from robot to robot. Each robot has the same number of configuration variables (q i R n ). For example, the M matrix and C matrix notations used in the previous sections can be interpreted as M(q ) M (q ) and M(q ) M (q ) with M ( ) M ( ) (the same for the C matrices). Hence, the assumption of non-identical dynamics does not alter the proof of exponential tracking in Theorem 3. and stable synchronization in Section IV. However, the synchronization with indifferent or unstable tracking (Lemma 3.3) is no longer true for the case of non-identical robots since Vsyncx T = is not a flow-invariant manifold, and q = q does not cancel the off-diagonal terms M (q ) M (q ) in the metric matrix. The above discussion is summarized in the following lemma. Lemma 5.: Synchronization of Multiple Non-Identical Robots If [L p K, K ] is positive definite, then every member of the network follows the desired trajectory q d exponentially fast. [L p K, K ] > (6) In addition, a swarm of p non-identical robots synchronize exponentially from any initial conditions if diagonal matrices K >, K > such that [L p K, K ] + [U p K ] > where the closed-loop dynamics in (8) can be rewritten to account for the non-identical dynamics of each robot: [M]ẋ + [C]x + ( [L p K, K ] + [U p K ] ) x = [U p K ]x (6) where M (q ) C (q, q ) s [M] =....., [C] =....., x =.. (6) M p (q p ) C p (q p, q p ) s p D. Cooperative Control of Robot Manipulators The 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 compare with the performance of the controller for a network consisting of non-identical robots, consider the two different robots in Figure 3(b). The first robot has the same physical parameters as in Figure 3(a), and the second robot s parameters are given in Table I. TABLE I PHYSICAL PARAMETERS OF THE ROBOTS SHOWN IN FIGURES 3. Physical Parameters m l m e I l c I e l ce Robots in Fig. 3(a) kg m kg. kgm.5 m.5 kgm.6 m Left robot in Fig. 3(b) kg m kg. kgm.5 m.5 kgm.6 m Right robot in Fig. 3(b) kg. m.5 kg.5 kgm.6 m.4 kgm.7 m

16 6 (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 Identical Robot, Stable Tracking Non Identical Robot, Stable Tracking q d q d Joint [rad].5 q q Joint [rad].5 q q time[s] time[s] q d q d Joint [rad] q q Joint [rad] q q time[s] time[s] (a) Identical robots (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 K = 5I, K =.5I, and Λ = 5I. The desired trajectory is q d = (.3( cos πt),.5( e t ) ) T. The lower joints are initially at.3 rad with rad/s (first robot) and. rad at rest (second robot). The upper joint of the first robot is initially at.4 rad. The rest of the initial conditions are zero. The same initial conditions are used for both the robots in Figures 3 (a) and (b). The synchronization gain of s and s corresponds to K + K = 6.5I, which is larger than the tracking convergence gain K K = 3.5I. As a result, we can see in Figure 4 that the first and second robots exponentially synchronize first. Then, they exponentially converge together to the desired trajectory. Let us now investigate the synchronization under the instability condition. For example, the control gains are re-defined as K = 5I, K = 5.3I, and Λ = I. Now the tracking gain, K K =.3I is negative, thus instability occurs. Figures 5 shows the simulation result with the instability. For the network with two identical robots, both robots synchronize exponentially fast even though the synchronized trajectory becomes unstable. In contrast, the exponential synchronization of the two different robots is no longer guaranteed in the presence of tracking instability, as seen in Figure 5(b).

17 7 Identical Robot, Unstable Tracking Non Identical Robot, Unstable Tracking Joint [rad] 5 5 q d q q Joint [rad] q d q q Joint [rad] time[s] time[s] q d q q Joint [rad] time[s] 6 4 q d q q 8 time[s] (a) Identical robots (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 Robots Even 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 of highly nonlinear systems with p 4, a network of four identical 3-DOF robots is considered here (see Figure 6). Each joint is assumed to be frictionless, and the dynamics modeling of the 3-DOF robot is based upon the double inverted pendulum robot on a cart given in [9]. The physical parameters of each robot 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 8 TABLE II PHYSICAL PARAMETERS OF THE ROBOTS SHOWN IN FIGURES 6. Physical Parameters M m l I l c m I l c Values 4 kg kg m. kgm.5 m kg.5 kgm.6 m TABLE III PHYSICAL PARAMETERS OF THE ROBOTS SHOWN IN FIGURES 6. Variable [unit] s [m] ṡ [m/s] θ [rad] θ [rad/s] θ [rad] θ [rad/s] Robot Robot Robot Robot Table III, are driven to synchronize as well as to track the following desired trajectory: s d =.t, ṡ d =., s d = θ d = cos(.πt), θd = sin(.πt)(.π), θd = cos(.πt)(.π) (63) θ d = π 4 ( cos(.8πt)), θd = π 4 sin(.8πt)(.8π), θd = π 4 cos(.8πt)(.8π) Note that this simulation fully considers the nonlinear rotational dynamics of the robots, as opposed to some earlier work on multi-agent coordination [36], [37]. For the control gains in the control law (3), we used K = I, K =.4I, and Λ = I. According to Theorems 3. and 3., the corresponding tracking gain K K =. is smaller than the corresponding synchronization gain K + K. Figure 7 shows that the system states synchronize exponentially fast and the synchronized states converge to the desired trajectory together. F. Synchronization with Partial Degrees-of-Freedom Coupling We 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 having two joint variables (q = (x, x ) T ) such that τ = M(q ) q r + C(q, q ) q r + g(q ) K s + K ( x τ = M(q ) q r + C(q, q ) q r + g(q ) K s + K ( x ) ) ( x + K Λ q ) ) ( x + K Λ q Nevertheless, Theorems 3. and 3. are true with diagonal matrices, K, K and Λ, which can be verified by writing the closed-loop system as in (5): [ ] M(q )ṡ + C(q, q )s + (K + K )s = u(t) [ ] M(q )ṡ + C(q, q )s + (K + K )s = u(t) (65) [ ] u(t) = K (s + s ) It is straightforward to prove that Theorems 3. and 3. still hold. This is because [ ] [ ] (K + K ) and (K K ) (66) q q (64)

19 9 ROBOT.5 ROBOT.5 ROBOT 3.5 ROBOT s[m] (a) Simulation result. 8 Synchronization of four robot network s [m] 6 4 Desired Robot Robot Robot3 Robot Joint [rad].5.5 Desired Robot Robot Robot3 Robot Joint [rad].5.5 Desired Robot Robot Robot3 Robot time [s] (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 convergence to the desired trajectory, respectively. Hence, we did not break any assumption in the proof of Theorem 3.. Figure 8(b) shows exponential synchronization of two identical robots with the partial coupling law above. All simulation parameters are the same as in Figure 4. This partial-state coupling also works for the semi-contracting case with (39), as presented in Section IV-D. While the coupled variables synchronize to 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 Delays Wang and Slotine [5] 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 multiple Lagrangian systems discussed in this paper. In particular, the proposed synchronization coupling control law in (3) is proven to synchronize multiple robots as well as to track the common trajectory, regardless of time delays in the communication. Figure 9 shows two two-link manipulators transmitting their state information to each other via timedelayed transmission channels. While T is a positive constant denoting the time delay in the communication from the first robot to the second robot, T denotes the delay from the second robot to the first robot. Similar to [5], we can modify our original Lagrangian systems consisting of two identical robots in (5) as follows M(q )ṡ + C(q, q )s + (K K )s + G τ = M(q )ṡ + C(q, q )s + (K K )s + G τ = where G and G are constant matrices (R n n ). The communication between the two dynamics occurs by transmitting intermediate wave variables, defined as [5] (67) u = G T s + k τ v = G T s u = G T s + k τ v = G T s (68)

21 Fig. 9. Synchronization of two identical robots with transmission delays where k and k are strictly positive constants. Time delays of T and T result in u (t) = v (t T ) u (t) = v (t T ) (69) Notice that the original dynamics without any communication delays are contracting since K K >. Expanding (67), using the above relationships on the wave variables, yields M(q )ṡ + C(q, q )s + (K K )s k G ( G T s (t T ) + G T s (t) ) = M(q )ṡ + C(q, q )s + (K K )s k G ( G T s (t T ) + G T s (t) ) = We can verify that (7) becomes equivalent to the original two-robot dynamics in (5) by setting k = k = and G = G = K. Note that K is the Cholesky decomposition of the positive-definite symmetric matrix, K. The resultant equations reflecting the time-delayed transmissions become M(q )ṡ + C(q, q )s + K s K s (t T ) = M(q )ṡ + C(q, q )s + K s K s (t T ) = which can be shown to be asymptotically contracting using the following differential length similar to [46], [5]: where V, = t t T δv T δv dɛ + (7) (7) V = δst M(q )δs + δst M(q )δs + V, (7) t t T δv T δv dɛ T δv T δv dɛ T δv T δv dɛ (73) In conclusion, the robot network systems, individually contracting (exponentially converging) and interacting through time-delayed diffusion-like coupling in (7) are asymptotically contracting regardless of the values of the time delays. This in turn implies that individual robots still synchronize and follow the common trajectory asymptotically regardless of the time delays. We have exactly matched the analysis in [5] 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 robustness properties with respect to time delays.