In this work, we present a constructive method to design a family of virtual contraction based controllers that solve the standard trajectory tracking problem of flexible-joint robots in the port-Hamiltonian framework. The proposed design method, called virtual contraction based control, combines the concepts of virtual control systems and contraction analysis. It is shown that under potential energy matching conditions, the closed-loop virtual system is contractive and exponential convergence to a predefined trajectory is guaranteed. Moreover, the closed-loop virtual system exhibits properties such as structure preservation, differential passivity, and the existence of (incrementally) passive maps. The method is later applied to a planar RR robot, and two nonlinear tracking control schemes in the developed controllers family are designed using different contraction analysis approaches. Experiments confirm the theoretical results for each controller. K E Y W O R D Scontraction, flexible-joints robots, port-Hamiltonian systems, tracking control, virtual control systems INTRODUCTIONControl problems in rigid robots have been widely studied in the literature due to they are instrumental in modern manufacturing systems. However, as pointed out in Nicosia and Tomei 1 the elasticity in the joints often cannot be neglected for accurate position tracking. For every joint, that is, actuated by a motor, we basically need two degrees of freedom (dof) instead of one. Such flexible-joint robots (FJRs) are therefore underactuated mechanical systems. In the work of Spong 2 two state feedback control laws based, respectively, on feedback linearization and singular perturbation theory are presented for a simplified FJRs model. Similarly, in de Wit et al 3 a dynamic feedback controller for a more detailed model is presented. In Loría and Ortega 4 a computed-torque controller for FJRs is designed, which does not need jerk measurements. In Ailon and Ortega 5 and Brogliato et al 6 passivity-based control (PBC) schemes are proposed. The first one is an observer-based controller, which requires only motor position measurements. In the latter one, a PBC controller is designed and compared with backstepping and decoupling techniques. For further details on PBC of FJRs, we refer Partial results were presented in the IFAC Workshop on Lagrangian and Hamiltonian Methods in Nonlinear Control 2018. Int J Robust Nonlinear Control. 2020;30:3269-3295. wileyonlinelibrary.com/journal/rnc

In this paper, we propose a trajectory tracking controller for fully-actuated portHamiltonian (pH) mechanical systems, which is based on recent advances in contraction analysis and differential Lyapunov theory. The tracking problem is solved by defining a suitable invariant sliding manifold which provides a desired steady state behavior. The manifold is then made attractive via contraction techniques. Finally, we present numerical simulation results where a SCARA robot is commanded by the proposed tracking control law.

Based on recent advances in contraction methods in systems and control, in this paper we present the virtual differential passivity based control (v-dPBC) technique. This is a constructive design method that combines the concept of virtual systems and of differential passivity. We apply the method to the tracking control problem of flexible joints robots (FJRs) which are formulated in the port-Hamiltonian (pH) framework. Simulations on a single flexible joint link are presented for showing the performance of a controller obtained with this approach.

who always challenged me to go further. I also would like to thank to the reading committee Prof.dr.ir. Jacquelien Scherpen, Prof.dr. Rafal Wisniewski and Prof.dr. Ian Manchester, for their nice comments and feedback of my thesis document. I am also grateful to Prof.dr. Henk Broer, Prof.dr.ir. Nathan van de Wouw, Prof.dr. Claudio de Persis, Dr. Hildeberto Jardón-Kojakhmetov and Dr.ir. Bart Besselink for accepting being part of the PhD committee. 2 Er gaat niets boven Groningen! To my paranymphs and beloved friends Pablo Borja and Alain Govaert, for their unconditional friendship and companionship during the different stages of my PhD time. The friendship with Pablo goes back to October 2013, when we meet during the Mexican Congress of Automatic Control in Ensenada Baja California, which was the first control conference for the both of us; remarkably, we did not meet at the conference itself but at the Mexican control scientists favorite networking spot, the Hussongs Cantina. On the other hand, I had the pleasure of meeting Alain when he started his PhD studies, first as classmate during the Mondays of DISC courses in Utrecht, and later as friend. I keep a lots of good memories with him exploring the different social spots in Groningen, networking sessions at the Benelux meetings on systems and control, and the recent trip to the ECC 2019 in Naples Italy. I also thank Alain's fellowship and effort for motivating me to do sports; it did not work though hehe. He is my best Dutch friend. Since probably it will take too long if I thank person per person, I want to thank in general to all the members and former members of the Jan C. Willems for Systems

In this work we propose a family of trajectory tracking controllers for marine craft in the port-Hamiltonian (pH) framework using virtual differential passivity based control (v-dPBC). Two pH models of marine craft are considered, one in a body frame and another in an inertial frame. The structure and workless forces of pH models are exploited to design two virtual control systems which are related to the original marine craft's pH models. These virtual systems are rendered differentially passive with an imposed steady-state trajectory, both by means of a control scheme. Finally, the original marine craft pH models in closed-loop with above controllers solve the trajectory tracking problem. The performance of the closedloop system is evaluated on numerical simulations.R. Reyes-Báez, B. Jayawardhana and A.J. van der Schaft are with the Jan C.