Output synchronization control of Euler-Lagrange systems with nonlinear damping terms Erik Kyrkjebø and Kristin Y. Pettersen Abstract— A coordinated synchronization control scheme to synchronize two or more Euler-Lagrange systems with nonlin- ear damping in a leader-follower configuration is presented. The scheme is based on position measurements only, and no mathematical model of the leader is required. Observers are designed to estimate the velocity and acceleration of the systems, and the scheme yields semi-global ultimately bounded closed- loop errors for the output synchronization problem. The control scheme is valid for systems with nonlinear damping. I. INTRODUCTION Synchronization of two systems in a leader-follower con- figuration can be considered as a tracking control problem where the reference is a physical object with dynamics that is subject to disturbances and actuator limitations (e.g. robot arm, ship, satellite, underwater vehicle). As opposed to tracking a theoretical and ideal reference path, the actual states of the reference object can diverge from its ideal path due to disturbances, unmodeled dynamics, actuator limitations, poor control design or actuator failure. Under these constraints we cannot guarantee that the reference object tracks its desired path perfectly, and knowledge of the desired path of the reference may thus not be enough to assure synchronization in the leader-follower system. In particular, any two physical systems that is not identical in their design will experience different impacts from en- vironmental forces such as wind, drag, current, terrain or waves. This difference may possibly lead to critical situations when employing simple tracking controllers to predefined reference paths where the coordination of the two systems is only done at the path planning stage, and not through active control. The output synchronization control scheme with only position measurements of the physical reference is also different from the output tracking problem in that the velocity and acceleration of the reference is unknown, and must be estimated based only on the position measurements. Output synchronization control is an important aspect in applications such as formation control of vehicles and teleoperation. Synchronization is found both as a natural phenomenon in nature like in the flashing of fireflies, choruses of crickets and musical dancing, as well as the controlled synchronization of a pacemaker or a transmitter-receiver system. Synchro- nization has recently attracted an increasing interest from This work was partially supported by the Norwegian Research Council under grant 159556/130 E. Kyrkjebø is with the Department of Engineering Cybernetics, Norwe- gian University of Science and Technology, NO-7491 Trondheim, Norway Erik.Kyrkjebo@itk.ntnu.no K. Y. Pettersen is with the Department of Engineering Cybernetics, Norwegian University of Science and Technology, NO-7491 Trondheim, Norway Kristin.Y.Pettersen@itk.ntnu.no researchers within physics, dynamical systems, circuit theory, and more lately control theory through [1]. The synchroniza- tion control problem can be seen as making a set of physical objects cooperate in their states. [2] and [3] have expanded on traditional tracking methods with predefined paths, and introduced a feedback from the actual position of a object (subject to disturbances) to the other objects through a path parametrization variable. All objects have predefined paths with individual tracking controllers requiring mathematical models and control availability, and the objects synchronize in terms of progression along the path. Thus, disturbances affecting tracking performance along the path is canceled, but cross-track errors due to any difference in disturbances are not. [2] used a coordinated approach with a leader and a follower, while [3] allowed for a cooperative approach where all objects mutually coordinate to the reference. Both a coordinated and a cooperative synchronization control scheme where the objects are synchronized in their states were presented in [4] and [5], and applied to robot control. Based on these results, a synchronization scheme for ship rendezvous control at sea for underway replenishment was presented in [6] with experimental results in [7]. There is no need for a predefined path or a dynamic model for the leader in the coordinated schemes of [5] and [6], and the coordination of the objects is achieved using a controller that synchronizes the position and velocity of each follower system to the leader based on position measurements only. This places all the control responsibility on the followers, and permits coordinated motion between a leader and a follower in situations where the control design and math- ematical model of the leader is unknown or unavailable. Disturbances affecting the objects differently are inherently canceled through the synchronization. For a view on the output tracking problem from a synchronization perspective, see [8]. Passivity-based tracking control of Euler-Lagrange sys- tems through energy-shaping and damping injection has been thoroughly elaborated in [9] for state-feedback systems, while a nonlinear dynamic output feedback control approach for a class of Euler-Lagrange systems was suggested in [10]. The nonlinear damping was injected without velocity measurements using a dynamic extension technique and a dissipation propagation condition. The results were extended in [11] and [12] for systems with input constraints. [13] suggested a tracking controller with a velocity observer for a class of mechanical systems with nonlinear damping terms, and [14] proposed an output tracking observer-controller scheme to estimate the velocity by imposing a monotone Proceedings of the 44th IEEE Conference on Decision and Control, and the European Control Conference 2005 Seville, Spain, December 12-15, 2005 WeB03.4 0-7803-9568-9/05/$20.00 ©2005 IEEE 4951