Trajectory Tracking of Mechanical Systems with Unilateral Constraints: Experimental Results of a Recently Introduced Hybrid PD Feedback Controller Gian Paolo Incremona, Alessandro Saccon, Antonella Ferrara and Henk Nijmeijer Abstract— The problem of tracking a time-varying reference trajectory of a mechanical system with unilateral position constraints is addressed in this paper. We present for the first time simulation and experimental results of a recently introduced trajectory tracking controller for hybrid systems with state jumps. The controller is applied to (locally) stabilize a time-varying trajectory of a 1-DOF robotic arm impacting and bouncing off an aluminum rod. The arm is modeled as a rigid link with viscous and Coulomb friction. The impact phenomenon is assumed to instantaneously reset the velocity in accordance with the classical Newton’s law of restitution. Kine- matic and dynamic identified parameter values are reported. The employed controller, hereafter called “hybrid PD controller with acceleration feedforward”, requires the real time detection on each impact in order to properly define the error signal. To this end, a force sensor and a triggering logic based on a force threshold are employed. I. I NTRODUCTION Hybrid systems exhibit both continuous (“flow”) and discrete (“jump”) behaviors [1], [2]. Under suitable conditions, they may be used to describe the dynamics of mechanical systems with unilateral constraints, such as bipedal walkers [3], [4], juggling robots [5], or constrained mechanisms [6]. In this way, a trajectory tracking control problem for a mechanical system with unilateral constraint can be posed as a trajectory tracking control problem for a hybrid system with state-triggered jumps. A state-triggered jump corresponds, when considering the original mechanical formulation, to the occurrence of an impact between the mechanical system and the rigid obstacle defining the unilateral constraint [7]. Trajectory tracking of discontinuous state trajectories of a hybrid system is an active field of research [8]–[10]. We are interested in the specific situation when the jump times of the reference and the plant are different, similarly to what presented in [11] and [12]. A distinctive feature of [11] and [12] is the introduction of a new notion of tracking error, based on the idea of mirror reference trajectory. Further results in this direction include [12] and [13]. In [14], while addressing the same tracking problem studied in [11] and [12], the authors suggest the use of a new notion of tracking error, moving away from the concept of mirror This is the final version of the accepted paper submitted for inclusion in the Proceedings of the IEEE 54th Conference on Decision and Control, Osaka, Japan, Dec. 2015. Gian Paolo In- cremona and Antonella Ferrara are with the Dipartimento di In- gegneria Industriale e dell’Informazione, University of Pavia, via Ferrata 1, 27100, Pavia, Italy. gp.incremona@gmail.com, antonella.ferrara@unipv.it Alessandro Saccon and Henk Nijmeijer are with the Department of Me- chanical Engineering, University of Technology, Eindhoven, the Netherlands. {a.saccon,h.nijmeijer}@tue.nl Motor, encoder Outer segment Torque sensor Inner segment End-effector Rod Fig. 1. The considered 1-DOF robotic setup and the aluminum rod: the torque sensor is made of two inductive sensors and two leaf springs that connect the outer with the inner segment. trajectory. This work is a continuation of that line of thought. The new notion of error requires the (local) extension of the reference trajectory in each neighborhood of the nominal jump times. These extensions are referred to as extented ante- and post-event trajectories [14]. Further details about this control strategy and the synthesis of optimal gains can be found in [15]. Starting from the hybrid control law introduced in [14], this paper presents the first simulation and experimental results obtained by employing this hybrid controller for tracking a reference trajectory of a 1-DOF robot arm bouncing against an aluminum rod. Moreover, estimates of the coefficient of restitution obtained via experimental tests are also reported. After a standard least-square identification of the friction and inertia parameters, a reference trajectory is generated using Bézier curves to include the robot workspace constraints and the impact law. A suitable detection mechanism has been introduced by considering the comparison between contact force, exerted by the bouncing rod on an aluminum cylinder, and a pre-specified threshold, selected on the basis of experimental data. Numerical simulations show that the hybrid trajectory tracking controller employed in this work behaves effectively even when a compliant contact model (the Hunt and Crossley model [16], [17]), rather than a nonsmooth contact model, is employed. II. THE 1-DOF SETUP:MODELING AND I DENTIFICATION The experimental setup used in this work is shown in Fig. 1. The setup consists of one rotational joint actuated by an electrical motor (design and construction details can be found in [18, Chapter 4]). The device presents two main segments,