1 Introduction

In recent years,the coordinated control of multi-agent systems has received significant attention in a range of fields,including physics,biology,computer science,and control engineering^{[1, 2, 3]}. In particular,the coordinated control of networked multibody systems modelled by Lagrangian dynamics (or networked Lagrangian systems) is of considerable interest for two main reasons. The first reason is that the networked Lagrangian systems can represent a class of networked mechanical systems due to their ability in dealing with complex systems involving multiple dynamics,such as multiple robotic manipulators,formation flying spacecrafts,and autonomous vehicles. The other is that the coordinated control of networked multibody systems has a wide range of engineering applications,especially in the complex and integrated production process,where the flexibility,reliability,manipulability,and scalability are highly required^{[4, 5]}. Many researchers have contributed to the understanding of the coordination and synchronization of Lagrangian systems^{[6, 7, 8, 9, 10, 11, 12]}.

Coordination problems encountered in engineering can often be rephrased as the controlled synchronization (or consensus) problems of networked multi-agent systems,which aim to enhance both robustness and performance of networked control systems^[13]. An important task undertaken in the coordinated control is to design an appropriate synchronization protocol (or algorithm) such that a group of interconnected dynamical systems can achieve a certain desired global behaviour of common interest through the information exchange between individual systems. As a result,a large number of synchronization protocols (or algorithms) have recently been proposed from the control theory point of view. For example,a passivity-based control framework was formulated to achieve the synchronization of networked Lagrangian systems^[4]. The cooperative tracking control laws were used for multiple robotic manipulators^[5]. The collision avoidance strategy was developed for the output synchronization of multiple mechanical systems with communication delays^[14]. A passivity-based control law preserving Lagrangian structure was proposed to stabilize the synchronization of networked mechanical systems with unstable dynamics^[15]. A distributed adaptive synchronization scheme was designed for Lagrangian networks with parametric uncertainties^[7]. A distributed finite-time algorithm without velocity measurements was introduced to deal with the tracking synchronization problem of multiple Lagrangian systems^[16]. A constraint control strategy was developed to realize the impulsive synchronization motion of networked open-loop multibody systems^[8].

From the analytical mechanics point of view,the control of complex multibody systems can be formulated as the issue of the constrained motion of the corresponding multibody systems. Thus,the modelling and control of multibody systems can be studied using a unified methodology. The development of the equations of motion of a constrained discrete mechanical system is widely recognized to be one of the key issues in multibody dynamics. Researchers introduced the notion of generalized inverses to obtain simple and general equations of motion for constrained mechanical systems,which provides a different perspective on the constrained motion of multibody systems^{[17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29]}. The proposed approach is simple in implementation because the control forces required are the continuous functions of time and can be obtained explicitly in the closed form^[20] by performing the simple operation of matrix multiplications and additions,thereby making the method attractive for real-time on-line control of nonlinear mechanical systems^[19]. In addition,the modeling approach developed imposes no restrictions on the number and nature of constraints used to construct the equations of motion,and has been successfully applied in the control of complex multibody systems,including synchronization of multiple gyroscopes^{[20, 21]},tracking control of robot in joint space^[25],and tumbling control of dumbbell spacecraft systems^[26].

The main objective of this paper is to investigate the controlled synchronization problem of networked multibody systems based on the framework of analytical mechanics^{[17, 18, 19, 29]}. A novel optimal control scheme,which allows the development of a unified methodology for studying the synchronization of networked mechanical systems formulated by Lagrangian dynamics,will be developed by making use of some recent advances in analytical dynamics^{[19, 20, 21, 27]}. The synchronization problem will be reformulated as a problem of the constrained motion of networked multibody systems. In particular,the network structure will be introduced into the control requirements by using the algebraic graph theory. The explicit closed-form equations of motion in terms of the fundamental equation of mechanics will be used to obtain a set of control forces for the synchronization of networked multibody systems (by imposing the constraints). Accordingly, the minimum forces for the synchronization of nonidentical nonlinear mechanical systems will be obtained,and a novel robust optimal control law for the complete synchronization of networked multibody systems will be derived. Stability analysis for the system over the undirected connected network topology will be carried out (the analysis procedure is also applicable for the directed network topology with a spanning tree). Finally,illustrative examples will be given to demonstrate effectiveness and easy implementation of the proposed control strategy.

2 Preliminary and fundamental equations 2.1 Notation and definitions

Throughout this paper,the following notations and definitions will be used. Let R = (-∞,+∞) be the set of real numbers. For the vector u ∈Rⁿ,u^T denotes its transpose,the norm of the vector u is defined as . R^n×n stands for the set of n × n real matrices. For the matrix A ∈ R^n×n,λ_min(A) and λ_max(A) denote its minimum and maximum eigenvalues,respectively. A^-1 denotes the inverse of matrix A. Assume that A is a symmetric matrix,A > 0 (A ≥ 0) denotes the positive definite (semi-positive definite) matrix. For any matrix B ∈ R^m×n,B⁺ denotes the Moore-Penrose generalized inverse of matrix B. 0_n ∈ R_n is the vector with all zeros,1_n ∈ Rⁿ is the vector with all ones,and I_n ∈ R^n×n is the identity matrix of order n. A⊗B denotes the Kronecker product of matrices A and B. Matrix dimensions,if not explicitly stated,are assumed to be compatible for algebraic operations. For a complex number z,Re(z) and Im(z) represent its real and imaginary parts,respectively.

2.2 Graph theory

In this paper,the graph theory will be used to model the interactions or communication topology among the agents of networked multibody systems. Some basic concepts of the graph theory are summarized as follows^{[8, 30, 31, 32, 33]}.

Consider a weighted directed graph G=(V ,E,A) of order N (N ≥ 2) with a set of nodes V = (v₁,v₂,· · · ,v_N),a set of edges E ⊂ V ×V ,and a weighted adjacency matrix A = (a_ij) ∈ RN×N having nonnegative adjacency elements aij . A directed edge in a directed graph is denoted by (v_i,v_j ). (v_i,v_j) ∈ E means that the node v_j can receive information from the node vⁱ,where vⁱ is called as the parent node of v^j,and v^j is called as the child node of vⁱ. The element a_ij of the weighted adjacency matrix A is positive if and only if there is a directed edge (v_j,v_i) in G,otherwise a_ij = 0. A directed tree is a directed graph,where every node has exactly one parent except for one node,called the root,which has no parent but a directed path to any other node. A directed spanning tree of a directed graph is a directed tree consisting of all the nodes and some edges in G. On the contrary,an undirected edge in an undirected graph is unordered,where an edge (v_i,v_j) denotes that vi and v_j can obtain information from one another. The adjacency matrix of an undirected graph is defined analogously except that a_ij = a_ji for i ≠ j. The undirected graph G is said connected if any pair of distinct nodes can be connected by an undirected path. All of the graphs are assumed to have no self-loop. The elements of the Laplacian matrix L = (l_ij) ∈ R^N×N associated with the graph G are defined asfollows: a_ij and l_ij = -a_ij,where i ≠ j. For a directed graph,the Laplacian matrix L is generally asymmetric,whereas for an undirected graph,the Laplacian matrix L = (l_ij) is symmetric positive semidefinite.

2.3 Fundamental equation of mechanics

The explicit equations of motion obtained in Refs. ^[17] and ^[19] provide a different perspective on the constrained motion of complex multibody systems,when the systems are forced to satisfy a set of consistent constraints,which will be introduced as follows.

An unconstrained mechanical system with independent coordinates is first considered here. The equation of motion of the system can be obtained using Lagrange’s equation or Newtonian mechanics as

where q ∈ Rⁿ is the vector of generalized coordinates used to describe the configuration of the system,M(q,t) ∈ Rn×n is the symmetric positive definite inertia matrix (or mass matrix), and F(q,,t) ∈ Rⁿ is a known function (or force vector) of q,,and t. Here,the dots denote the differentiation with respect to the time.

The unconstrained system is assumed to be subjected to a set of m sufficiently smooth control requirements given by

where H is an m-vector. The control constraint described by (2) includes all the general types of holonomic and/or nonholonomic constraints. The initial conditions are assumed to satisfy the control requirements (2),i.e.,

Under the assumption of sufficient smoothness,differentiating the control requirements (2) twice with respect to the time t gives rise to

where A is an m×n matrix. It is noted that each row of A arises by appropriately differentiating one of the m control requirements given in (2).

The fundamental equation of motion of the constrained system that satisfies these constraints (or the controlled nominal system) can then be explicitly written as

where Here,N > 0 is a given positive definite matrix,the superscript ‘+’ denotes the Moore-Penrose generalized inverse of the matrix,and the control force F^c is employed to optimally minimize the quantity (F^c)^TNF^c (i.e.,the weighted norm of the active control force F^c) at each time instant^[19].

When the initial values do not satisfy the constraints,the control constraints can be modified as Ḣ (q, , t) = f(H,t; α),where f(H,t; α) is an m-vector,which may contain a parameter p-vector α. This m-vector f(H,t; α) is usually chosen in a way such that the system (2) has the following two properties^[19]: (i) H = 0 is an equilibrium point of the system; (ii) The equilibrium point is globally asymptotically stable.

3 Synchronization of networked multibody systems 3.1 Problem formulations

Consider a network consisting of N agents described by Lagrangian dynamics as

where q_i = (q_i1,q_i2,· · · ,q_in)T ∈ Rⁿ is the vector of the generalized coordinates for the agent i, ∈ Rⁿ are the vectors of the corresponding generalized velocities and accelerations,respectively,and F_i(q_i,_i,t) ∈ R^N is a known nonlinear function of q_i,_i,and t. In the context of mechanical vibrations,these agents are n degrees of freedom mechanical systems whose individual systems need not to be identical. The model (7) has also been referred to as a networked multibody system in the literature.

Introducing a control input U_i into the right hand side of (7) gives the following controlled networked multibody system:

The main objective here is to implement a suitable control strategy of preliminary state feedback such that the motion of all the agents in the network will enter into the synchronization manifold of the controlled network (8)^[25],

Then,a complete synchronization of the networked multibody system with respect to the generalized coordinates and their velocities will be realized,i.e.,for i =1,2,···,N,

It is worth mentioning that in order to generate interactions between the agents and guarantee the synchronization behavior of networked multibody systems,the Laplacian matrix L with respect to the graph G is usually used to design an appropriate synchronization control law, which plays an important role in determining the synchronization dynamics of the controlled networks^{[3, 33, 34]}.

3.2 Synchronization framework

In this subsection,the fundamental equations of motion in analytical dynamics developed by Udwadia and Kalaba^[17],Udwadia^[19],and Peters et al.^[25] will be used to formulate the synchronization framework. The synchronization problem of networked multibody systems under an undirected connected graph will be firstly addressed.

With the definition of q=(q₁^T,q₂T ,···,q_N^T )^T,M=diag{M₁,M₂,···,M_N},F =(F₁^T ,F₂^T ,··· ,F_N^T)^T,and U =(U^1T,U₂^T ,··· ,U_N^T)^T,Eq. (8) can be rewritten in a compact vector form as

In order to demonstrate the effect of network topology on synchronization dynamics of the network,the problem of complete synchronization can be interpretedin terms of a constraint,which is explicitly given by

According to the property of the Laplacianmatrix L,it is easy to verify that the enforcing constraint (12) will make the motion of all the agents approach the synchronization manifold Λ.

In general,the initial conditions of Eq. (8) may not satisfy the constraint (12). Then,the constraint can be further modiffed by differentiating the constraint (12) twice with respect to the time t,

where c > 0and α > 0are positive constants. It is easy to note that (L ⊗ In)q =0 is an equi-librium point of Eq. (13),which is globally (exponentially) asymptotically stable (see the next subsection). The complete synchronization of networked multibody systems is then achieved.

To facilitate optimal control framework for the controlled networked multibody system (11),the control force will be minimized with respect to the certain given positive definite metric N > 0,i.e.,J (t) = U TN (t)U at each time instant. By defining

the generalized accelerations can be written as

Then,substituting (15) into the synchronization constraint (13) gives rise to

By utilizing fundamental equations (5) and (6),the explicit generalized control force U ,which minimizes J (t) and satisfies Eq. (13),can be obtained as

Just as stated in Ref.[20],the selection of the metric N plays a key role in determining the type of solution. For instance,in some engineering applications such as multibody dynamics and robotic control,N is often chosen to be M ^-2[8]. In other cases,the control force U may be required to comply with the principle of virtual displacements governedby d’Alembert’s principle for which the metric N = M ^-1 is usually more appropriate^{[19, 25]}.

Without loss of generality,by specifying N = M ^-2 in this paper,the complete control law is given by

By considering the structure of Laplacian matrix L with respect to the undirected connected graphs,the following relation can be obtained: L⁺L = I_N -P ,where ,and L⁺L² = L. Hence,by using the property of pseudo-inverse matrix,the generalized control force can be formulated as follows:

or the ith component of U is given by where and

It can be seen that the synchronization of N agents of the networked multibody system will be achieved under the generalized control force U_i acting on the agent i given by (20). The control force U_i can be formulated as two components. The first component U_i1 describes the architecture of the network,which is determined by several elementary factors,such as the direct or indirectinteractions of the forces among the agents,various types of physical constraints,different configurations of communication links among agents,and a variety of network environmental factors^[34]. In contrast,the secondcomponent U_i2 denotes the active control force applying on the agent i,which compels all the agents to reach the synchronized motion of the networked multibody system.

From the control theory point ofview^[3],the control law U_i is a cooperative control algorithm for N agents to follow a synchronization trajectory imposed by the constraint (13). In fact,the first component U_i1 can be rewrittenas

It is easy to note that the first component U_i1 is a distributed coordinated algorithm,while the second component U_i2 is an active feedback control strategy.

It is important to point out that various choices of parameters c and α,as well as the Laplacian matrix L can lead to different “paths” taken by the networked multibody systemstowards their eventual synchronization. In particular,the Laplacian matrix L plays a key role,because it determines the convergenceperformance of the trajectorygoverned by the different “paths”. Accordingly,the developed framework provides a unified and systematic analysis procedure for studying the synchronization motion of networked multi-agent systems by using analytical dynamics instead of the control theory. Thecorresponding global stability of synchronization manifold is also fully guaranteed based on the stability theory of dynamical systems,which will be discussed in the next subsection.

When the Laplacian matrix L is taken as the global coupling configuration,i.e.,

and n =1,M = I_nN ,it is interesting to find that the complete control law U_i given by Eq. (20) is just the same as Eq. (19) obtained in Ref.[20]. Therefore,the synchronization of multiple gyroscopes systems addressed by Udwadia and Han^[20] can be treated as a special case of the present work. 3.3 Stability analysis

The stability of synchronization manifold in the controlled networked multibody system (11) is examined in this subsection. By using the complete control law (19),Eq. (11) can be rewritten as

The governedequation of synchronizedmotion for the agent i is then explicitly given by where ri = _i + αq_i denotes the output information of the systems,which is linear combinations of both generalized coordinate q_i and velocity i for the agent i,and α is a positive constant scalar. c > 0 is a control parameter that represents the couplingstrength between agents,a_ij characterizes the interactionbetween agents iand j (i.e.,a_ij> 0if the agent i can receive the output information r_j of agent j and a_ij= 0 otherwise)^[20].

From the viewpoint of complex networks,the system (26) (or the system (11) with the control law (19)) can be regarded as an undirected dynamical network consistingof N linearly and diffusively coupled nonidentical agents under an additive active controller given by Eq. (22).

According to the geometrical analytic approach of synchronization manifold for coupled dynamical systems^[20],the synchronization state can be defined as

and its evolution equation is given by

It will be convenient to introduce the synchronization error e_i(t) = q_i(t)-q_syn(t) (i = 1,2,···,N ) to examine the stability of synchronization manifold. Then,the corresponding synchronization error system can be written as

where e = (e₁^T,e₂^T,···,e_N^T )^T.

Since the Laplacian matrspan>ix L associated with an undirected connected graph G is symmetric and irreducible,it is easy to notice that L ≥0 and the rank of L is N ^-1. Furthermore,the matrix L has an algebraically simple eigenvalue 0,and all the other eigenvalues are positive. For brevity,these eigenvalues can be ordered as 0 = λ₁ < λ₂ ≤ ···≤ λ_N .

Let L = UJU^T be the eigenvector decomposition of L,where U ∈ R^N×N is a normal orthogonal matrix,and J = diag(λ₁,λ₂,··· ,λ_n). By defining v(t) = (U^T ⊗ I_n)e(t),where v = (v₁^T (t),v₂^T (t),··· ,v_N^T (t))T ∈ R^nN,it then follows from (29) that

It is easy to verify that all the states v_i(t) (i = 2,3,· · · ,N) of the system (30) will globally, exponentially,and asymptotically converge to zero.

By noting that u₁ (1,1,· · · ,1)^T∈R^N is an eigenvector with respect to the eigenvalue λ₁ = 0,it can be obtained that v₁(t) = = 0_n. Consequently,the global stability of synchronization manifold in the controlled networked multibody system (26) is guaranteed.

The above analysis procedure indicates that the networked multibody system (7) with the control constraint (18) can always achieve complete synchronization with respect to the generalized coordinates and their velocities. As observed from Eq. (30),the complete synchronization of networked multibody systems depends on not only the coupling strength c,but also the algebraic connectivity λ₂ of the controlled networks,which can well characterize the synchronization performance of the networks^{[33, 34]}. In general,it is desirable to make the coupling strength as small as possible in practice. For a fixed coupling strength c,the larger the number of the algebraic connectivity λ₂ is,the easier the synchronization of a networked multibody system will be. This conclusion is in excellent agreement with some empirical evidences from complex networks^[3].

It should be mentioned that the above analysis procedure is also applicable to the synchronization problem of networked multibody systems with a directed spanning tree. In this case,the Laplacian matrix L associated with a directed graph G has an eigenvalue 0 with multiplicity 1,and all the other eigenvalues have positive real parts.

With the property of the Laplacian matrix L,it can be obtained that L⁺L² = L − PL = where η = (η₁,η₂,· · · ,η_N)^T with η_i =

As expected,similar to the derivation of Eq. (19),the generalized control force with respect to the directed network topology can be rewritten as

Then,the ith component of U acting on the agent i can be written as

where the first two components U_i1 and U_i2 are the same as stated in Eqs. (22) and (23),while the third component U_i3 is explicitly given by Notably,U_i3 is actually a state feedback controller based on the directed topology structure of networked multibody systems.

It is assumed that ξ = (ξ₁,ξ₂,· · · ,ξ_N)^T is an eigenvector with respect to the eigenvalue λ₁ = 0 such that ξ^TL = 0_N^T and By following the geometrical analytic procedure of synchronization stability for directed coupled dynamical systems,the controlled synchronization state can be obtained as q_syn(t) = and the corresponding synchronization error dynamics is given by e_i(t) = q_i(t) − q_syn(t)(i = 1,2,· · · ,N).

Then,by following the same arguments as the undirected topology case,it is easy to verify that the global stability with respect to synchronization manifold is fully guaranteed if and only if the following condition holds:

where λ_j (j = 2,3,· · · ,N) are all the nonzero eigenvalues of the Laplacian matrix L.

Based on the above discussion,it can be concluded that the synchronization motion of networked multibody systems can always be achieved by introducing the generalized control law U_i to the agent i when the directed graph has a spanning tree. The proposed control strategy given by Eqs. (31)-(33) is especially interesting as it has a clear physical interpretation. The generalized control law U_i is a combination of three control components U_i1,U_i2,and U_i3,where U_i1 is a preliminary feedback control input based on the architecture of network topology,U_i2 is an active control law imposed by Lagrangian dynamics of all the agents,and U_i3 is an additional state feedback controller exerted by the topology characteristic of directed network. The control law U_i can be considered to have two main parts. The first part U_i1 + U_i3 is a preliminary feedback control law from the control theory point of view,which describes the interactions between the agents. While the second part U_i2 denotes an active control law that needs to be applied to synchronize these agents. If specify η = 0_n in Eq. (33),the third component U_i3 will vanish. Accordingly,the generalized control law (31) is a natural extension and generalization of (19) from the undirected connected graph to the directed network topology.

4 Illustrative examples and numerical simulation

As an application of the above derived theoretical results,the synchronization problem of a representative network composed of ten identical or nonidentical gyroscopes is discussed in this section,where numerical simulation with different kinds of network structures,including global coupling,nearest-neighbor,and small-world networks,is given to validate the theoretical results.

As stated in Ref.[20],the gyrodynamics has been an area of mechanics of considerable interest for more than a century in various fields of science and engineering. Due to its intrinsic and complex nonlinear dynamics,it has bee regarded as a typical model for analysis of nonlinear phenomena such as fixed points,periodic behavior,period-doubling behavior,quasi-periodic behavior,and chaotic motions. Moreover,it has a wide range of engineering applications including navigation of aircraft,rocket,and spacecraft and the control of complex mechanical systems^[20].

Consider a representative network system consisting of ten identical or nonidentical gyroscopes by Lagrangian dynamics as^[20]

where P_i = (α_i,β_i,c_i,e_i,γ_i,ω_i) are the corresponding parameters. It is known from Ref.[20] that with different parameters,the solution trajectory of (31) can exhibit fixed points,periodic behavior and chaotic motions. Three different gyroscopic systems are chosen and illustrated in the (q_i,_i) phase plane for 100 ≤ t ≤ 200,as shown in Fig. 1. The parameter sets used in calculations are: (a) P_chaos1 = (10,1,0.5,0.05,35.5,2)(for chaotic motion),(b) P_chaos2 = (10.5,1,0.5,0.04,38.5,2.1)(for chaotic motion),(c) P_per=(10.5,1,0.45,0.045,36,2.05)(for periodic motion),and the initial values are randomly chosen in the interval [−1,1].

Three kinds of network structures including global coupling,nearest-neighbor,and smallworld networks are considered,as shown in Fig. 2,and the corresponding Laplacian matrices are chosen as

respectively. It is easy to find that their algebraic connectivities λ₂(L_gc) = 10,λ₂(L_nc) = 0.382,and λ₂(L_sw) = 2.290 7,respectively. In general,the global network is easy to achieve synchronization,while the nearest-neighbor network is difficult to realize synchronization^{[33, 34]}.

In order to demonstrate the influence of both the individual dynamics and network topology on synchronization behaviors,three cases will be considered in the following.

4.1 Synchronization state of identical or nonidentical gyroscopes

In this subsection,numerical simulation with the same or different parameter values of gyroscopes will be obtained to show their effects on the synchronization state of the network composed of identical or nonidentical gyroscopes. Let the coupling strength c = 2 and the coefficient α = 1. Then,the complete control law U can be explicitly obtained from Eq. (18) if the parameter values of each gyroscope are fixed.

Two cases of identical gyroscopes are considered with the parameter sets of each gyroscope being chosen as P_i = P_chaos2 and P_i = P_per (i = 1,2,· · · ,10),respectively. Figures 3 and 4 show the phase portraits of the ten synchronized gyroscopes in the (q_i,_i) plane,i = 1,2,· · · ,10,for 100 ≤ t ≤ 150 for the networks with respect to L_gc,L_sw,and L_nc,respectively. It can be observed that the synchronization states display the corresponding chaotic attractors and periodic orbits.

For the nonidentical gyroscope case,the parameter sets of ten different gyroscopes are chosen as P₁ = P₂ = P₃ = Pchaos1,P₄ = P₅ = P₆ = P₇ = P_chaos2,and P₈ = P₉ = P₁₀ = P_per. Figure 5 demonstrates the time response of generalized coordinates q_i and velocities _i,i = 1,2,· · · ,10 for the ten different synchronized gyroscopes with respect to L_gc,where the state variables of all the gyroscopes gradually approach to zero,indicating that the synchronization state is zero.

It is observed from numerical simulation that no matter whether the parameter sets of gyroscopes are the same or not,the network composed of identical or nonidentical gyroscopes can always achieve complete synchronization with respect to the generalized coordinates and velocities. However,their synchronization states are entirely different because they are related to the complex dynamics of the given gyroscope systems.

4.2 Synchronization performance of different network structures

This subsection considers the effects of different network structures on the synchronization performance of the network of nonidentical gyroscopes. For simplicity,the parameter sets of the ten different gyroscopes are chosen the same as the case in the last subsection.

The synchronization error with respect to the generalized coordinates and velocities is introduced as

It is well known that both algebraic connectivity and coupling strength of the network can well formulate practical architecture of network topology^[33]. These two parameters are thus taken as the key parameters for analyzing the synchronization performance of the network.

Figures 6 and 7 display the time response of synchronization process of generalized coordinates q_i and velocities _i (i = 1,2,· · · ,10) for ten nonidentical gyroscopes with respect to Lgc and L_nc.

Figure 8 is the simulation result of the variation process of e(t) for two different connectivities. It can be seen from this figure that the global network has much stronger synchronizability than the nearest-neighbor network since the algebraic connectivity λ₂(L_gc) = 10 > λ₂(L_nc) = 0.382.

Figure 9 exhibits the time evolution of synchronization process of generalized coordinates q_i and its velocities _i (i = 1,2,· · · ,10) for the ten nonidentical gyroscopes under Lsw with the coupling strength c = 20 and c = 0.2,respectively. Figure 10 shows the variation of e(t) for two different coupling strengths. The larger the coupling strength is,the easier the synchronization of the network will be.

The above numerical simulation indicates that both the algebraic connectivity and coupling strength in networks have significant effects on the synchronization performance of the networks. Therefore,the network structure can be used to improve the synchronization performance of networked multibody systems for the purpose of practical control strategy.

4.3 Effects of active control strategy on synchronization

As mentioned before,the generalized control law is composed of two main parts: a preliminary feedback control law and an additional active control law. The effect of the active control law on the synchronization performance will be discussed in order to demonstrate the distinguished features of the proposed control methodology.

A small-world network consisting of ten nonidentical gyroscopes,where the parameter sets of ten different gyroscopes are chosen as P₁ = P₂ = P₃ = P_chaos1,P₄ = P₅ = P₆ = P₇ = P_chaos2 and P₈ = P₉ = P₁₀ = P_per,respectively. For the sake of comparison,the synchronization problem of the small-world network will be addressed under an undirect connected graph and a directed spanning tree,respectively. For the directed topology case,the corresponding Laplacian matrix is taken as

It is easy to notice that both the undirected small-world network and the directed small-world network have the same structure,and the graph with respect to has a directed spanning tree. By following the analysis procedure of synchronization stability,it can be concluded that under the generalized control law U_i,the undirected connected small-world network can always achieve complete synchronization for any coupling strength c > 0,while the directed connected small-world network will achieve synchronization if and only if the coupling strength satisfies c > 0.042 91α. Figures 11 and 12 present the time response of q_i (i = 1,2,· · · ,10) with respect to Lsw. In particular,Figure 11 shows the complete synchronization of the undirected smallworld network with the active control law U_i2,whereas Fig. 12 displays the desynchronization of such a small-world network without the active control law U_i2. Figures 13 and 14 are the simulation results of the directed small-world network associated with ,which yields the same dynamic behavior as the undirected case.

When a preliminary feedback control law U_i1+U_i3 is unable to guarantee the network synchronization of general nonidentical gyroscopes systems,an additive active control law U_i2 is required for such a purpose of control strategy. Then,the complete (global) synchronization of networked gyroscopic systems can be well guaranteed no matter whether the network topology is undirected or directed. Therefore,this paper develops a simple yet efficient control methodology to deal with the synchronization problem of networked nonidentical,highly nonlinear mechanical systems,where the developed active control strategy plays a crucial role in yielding complete synchronization of the network.

5 Conclusions

The issue of synchronization motion of networked multibody systems is investigated in this paper with the analytical dynamics approach rather than the control theory. A novel control framework which allows the development of a unified methodology for studying the synchronization motion of networked multibody systems has been developed based on the fundamental equation of mechanics. A novel optimal control law derived from this generic framework has been applied to achieve the complete synchronization of networked multibody systems,and then the corresponding stability analysis has been fully addressed in detail. A typical network composed of ten identical or nonidentical gyroscopes with three kinds of network structures has been used to validate and visualize the theoretical results.

This paper has developed a systematic and unified methodology for studying the synchronization of networked multibody systems. Of particular note,the network structure is introduced into the control requirements with the algebraic graph theory. The proposed control strategy has a clear physical interpretation and can be divided into two main parts. The first part describes the interaction between the agents,while the second part denotes the active control force that is needed to synchronize these agents modeled by Lagrangian dynamics. It is found that,the network topology plays an important role in generating synchronization dynamics of networked multibody systems.

It should be noted that the present research is only the first step for the cooperative control of networked multibody systems in this direction. It is hoped that the approach and procedure developed in this paper could be extended to deal with the synchronization problems for more general networked multibody systems with various network topologies and coupling characteristics, such as random graph,cluster group,time varying,and delay coupling,as well as the different kinds of physical constraints and impact dynamics. This will be a more interesting and challenging topic for the future research in this direction.