Recently, intensive attention has been paid to distributed control for Euler-Lagrange systems due to its broad applications. Several approaches have been proposed to deal with the distributed tracking problem, see, for example, sliding-mode method [1, 2], disturbance observer , and extended proportional-integral control scheme . Extension to handle unknown parameter uncertainties can be found in , where an adaptive controller is proposed to synchronize nonidentical Euler-Lagrange systems with communication time delays. Later,  solves the synchronization problem of networked robotic systems with both the kinematic and dynamic uncertainties using passivity theory. It has also been shown that, under a jointly connected switching network topology, leader-following consensus can be achieved for multiple Euler-Lagrange systems by employing adaptive control , in which various reference signals, such as sinusoidal and ramp signals, generated by an exosystem are considered.
In order to relax the restrictive requirement for full state measurements in designing the controllers in the existing results, some efforts on partial state feedback control have been made. To estimate the unmeasurable velocity, observers are constructed by invoking the Immersion and Invariance (II) techniques [8, 9, 10]. Through introducing two extra states, some lower dimensional observer is proposed in  in comparison to that in , and moreover, the explicit expressions of the observer have been given. In , dynamic scaling and high-gain terms have been adopted to perform the Lyapunov stability analysis. Note that the dynamics of the observers relying on II techniques are generally high dimensional and complex. In addition, it is required to find a certain attractive and invariant manifold in the extended state space of the plant and the observer, which will likely increase the computational burden.
For multi-agent systems, a consensus algorithm using linear observers is first proposed in . And, in , a distributed control law with time varying control gains is designed to compensate for the lack of neighbors’ velocity measurements. For the distributed tracking problem,  presents a sliding mode observer-based controller to track the leader with constant velocity in finite time. The more challenging problems of tracking a leader with varying velocity have also been investigated in [14, 15, 16]. When only nominal parameters of Euler-Lagrange systems are available, global asymptotic stability can be ensured using continuous control algorithms with adaptive coupling gains . More generally, in cases when we do not have access to any velocity measurement, it is desirable to coordinate the agents using output-feedback strategies. However, some drawbacks of the available results still exist. For example, tracking errors can only be guaranteed to be uniformly ultimately bounded but not converging to zero  and the resulted closed-loop system is only locally but not globally stable under the designed control laws . For some specific class of nonlinear systems, the global output feedback control problem has been investigated recently in  and , where the cyclic-small-gain approach and distributed internal model have been introduced respectively to achieve global convergence. To deal with the leader’s unavailable velocity measurements, distributed observers are designed for second-order agents in .
The goal of this paper is to address the problem of distributed global output-feedback tracking for multiple Euler-Lagrange systems modeling a class of two-link revolute robot manipulators. Up to now, there is no known result for such distributed global tracking algorithms due to several technical challenges. The main difficulties in achieving global stability lie in the quadratic nonlinearities and the cross terms of the unmeasurable velocity states derived from the Coriolis and centrifugal torques. To eliminate such quadratic terms, different state transformation methods have been utilized in [20, 21, 22, 23]. In  and 
, the coordinate transformation strategies are first applied to simplify the nonlinear models, and then controllers are proposed for one-degree-of-freedom Euler-Lagrange systems and underactuated mechanical systems in their Hamiltonian forms, respectively. It should also be noted that both of the techniques in and  pose constraints on the system model, i.e., a class of nonlinear systems that are linear in unmeasured states, globally stabilizable using output feedback 
, and with skew-symmetric Coriolis terms. However, the models of the two-link revolute robot manipulators considered in this paper do not possess any of the properties just mentioned that contribute to the simplification of the system model. So all of these approaches cannot be directly applied to the robot manipulators discussed in this paper. Inspired by  and , we shall focus on how to partially linearize the dynamics of the robot manipulators through coordinate transformation and state reconstruction. With the help of the model transformation, a distributed velocity observer is proposed, which enables us to implement the output-feedback control for multiple robot manipulators such that the tracking errors uniformly globally exponentially converge to zero.
The rest of the paper is organized as follows. Section II reviews the system dynamics and presents the method on how to partially linearize the nonlinear system through coordinate transformation. In Section III, an observer-based control strategy is proposed based on the partially linearized system. Section IV gives the main result of this paper, followed by the numerical simulations in Section V. Finally, conclusions are provided in Section VI.
Notations: denotes the determinant of a real square matrix . is used to denote the
-norm of a vector.
represents the identity matrix with dimension, and denotes the column vector whose components are all 1. We use to denote the th element of matrix . And and
are the largest and smallest eigenvalues of a real symmetric matrix, respectively.
Ii Partial linearization
In this section, we first briefly introduce the general expression of Euler-Lagrange systems, followed by the specific dynamics of two-link revolute robot manipulators. Then, we present the process removing the cross terms of the velocity states via coordinate transformation.
Ii-a Dynamics of Robot Manipulator
We consider here a group of mechanical robots, each of which is described by a Euler-Lagrange equation as follows:
where is the vector of the generalized coordinates, is the symmetric positive-definite inertia matrix, is the Coriolis and centrifugal torque, is the vector of the gravitational torques, and is the control torque on robot .
The neighbor relationships between the robots are described by a directed graph with the vertex set and the edge set . We use to denote the adjacency matrix, where means there is an edge between robots and , and robot can obtain information from robot , but not vice versa, and otherwise. There is one leader robot and the rest are followers. The interaction relationships among the followers and the leader is denoted by the matrix , where if the leader is a neighbor of robot , and otherwise. The Laplacian matrix is defined by and , where denotes the set of neighbors of robot .
It is well known that a wide range of mechanical systems can be represented by Euler-Lagrange equations, such as robot manipulators, mobile robots and rigid bodies. Here, we focus on a class of two-link revolute robot manipulators, whose dynamics are given by (see  for more details)
where is the acceleration of gravity, represents the joint angles of the two links and , in which the variables and
are, respectively, used to denote the masses, the lengths and the moments of inertia of link, and represents the distance from the previous joint to the center of mass of link , . The inertia matrix satisfies the following property: for all , there exist positive constants and such that .
Ii-B Coordinate Transformation
In order to linearize the quadratic velocity terms in and to simplify the dynamics model, motivated by , we introduce the following coordinate transformation
where , a nonsingular matrix with bounded elements to be determined, is constructed as follows
where needs to be determined. Here, instead of fully linearizing system (1), we aim at partially linearizing the nonlinear mechanical system. Hence, the transformation matrix is chosen to be in its upper triangular form (3
), which not only simplifies the system model, but also reduces the computational complexity greatly when solving a set of partial differential equations (PDEs). Considering the system model (1), the dynamics of the new state can be described by
Here, for the purpose of removing the cross coupling term in , we let
With (6), the dynamics of reduce to
One can check that one solution to (6) is
So, the globally nonsingular transformation matrix is obtained as follows
Consequently, the coordinate transformation (2) results in the partially linearized system with the state , output and input
It can be seen that the quadratic cross terms of the unmeasurable velocities have been removed from the system dynamics (10). Moreover, the matrices and are both independent of the velocity states and bounded. Both of the above properties will facilitate the design of globally stable observers and controllers.
For future reference, denote
It follows from the positive definiteness of the inertia matrix that and . Since is bounded, we have
The simplification of Euler-Lagrange systems was previously studied in , where the conditions for the existence of the transformation matrix were presented based on the equation . However, for a class of Euler-Lagrange systems, such as the robot manipulators we discussed here and unicycle-type mobile robots , such a nonsingular matrix solution does not exist. So, in this paper, a wide class of transformation matrices is derived from the relaxed equation, i.e., resulted from (II-B).
It can be seen that the computation of the nonsingular coordinate transformation matrix (9) relies on the exact knowledge of the inertia parameters. When the parameter uncertainties are taken into account, the construction of robust adaptive controllers needs to be considered based on the parameter linearizability property of Euler-Lagrange systems.
Ii-C Problem Formulation
Consider a group of followers modeled by (1), and the leader labeled by with the same dynamics as the followers. Hence, by employing (2), the leader’s dynamics can also be transformed to (10) with the states . The distributed global output-feedback tracking problem is to design local control protocols using only output information for all the followers, such that all the followers’ states synchronize to the leader’s state globally, i.e.,
Iii Output-feedback tracking control
The purpose of this section is to present an observer-based control law to solve the distributed output-feedback tracking problem. Toward this end, we first design the observers to estimate the unmeasurable velocities.
Iii-a Observer Design
Note that system (10) can be rewritten into the following two sub-systems
It can be observed from (13) that the dynamics of incorporate the state of the second sub-system. Similarly, the dynamics of also depend on the state of the first sub-system in (14). This implies that when we design the velocity observers for both of the sub-systems, the convergence analysis of the observation errors for each sub-system is still related to each other, which makes it challenging to design globally stable observers. To handle this problem, motivated by , we aim at fully decoupling the sub-systems by constructing the new sates as follows:
in which the dynamics of the first sub-system are independent of the second one . Consequently, it is relatively straightforward to design the observers for the two sub-systems in (16). For the first sub-system, the observer is designed as
where and are the observations of and , respectively. Here, and are positive observer gains. Correspondingly, the observation errors are defined as and , whose dynamics are of the form
It can be easily checked that matrix is Hurwitz, and therefore system (18) is exponentially stable at the origin. So
For the second sub-system , the observer is constructed as
where is continuous in and , and locally Lipschitz in . Note that both (18) and the nominal part of (21) are uniformly globally exponentially stable (UGES). Then, the origin of the cascaded system (18) and (21) is UGES , namely, and uniformly globally exponentially converge to zero.
Iii-B Observer-Based Control Law Design
The following assumptions are made throughout this paper.
The leader’s state information satisfies .
The communication relationships among the robots form a directed graph that contains a spanning tree rooted at the leader.
In order to keep this paper self-contained, two lemmas are presented.
 Let . Then
And, for all and ,
To come up with the observer-based distributed control laws, an auxiliary variable is introduced as follows:
where is a constant. The local differences are defined as
The auxiliary variable can be written into a compact form
The distributed control law for robot is proposed as follows
where . Here, can be any positive number, and and are positive numbers satisfying
Iv Main results
The main result of this paper is given below.
Proof of Theorem 1.
The Lyapunov function candidate is chosen as
where , and is a positive scalar satisfying . It is straightforward to check that matrix is positive definite and
The generalized derivative of (see [31, Remark 3.7]) is given by
Also, the control input (30) can be written in its stacked form as
where Lemma 1 and the equality that have been used.