IFAC PapersOnLine 52-11 (2019) 1–6 ScienceDirect ScienceDirect Available online at www.sciencedirect.com 2405-8963 © 2019, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Peer review under responsibility of International Federation of Automatic Control. 10.1016/j.ifacol.2019.09.109 © 2019, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. 1. INTRODUCTION Nowadays, stroke rehabilitation is based on the evidence of neuroplasticity (Arya et al., 2011), which stands for the ability of the brain (or any other nervous tissue) to change its structure or reorganise as a response to sensory inputs, learning, and experience (Chan et al., 2006). Thus, stroke rehabilitation focuses on therapies that induce these changes in the brain by appropriate training routines of the affected limb, an activity usually referred to as motor reeducation (Stauffer et al., 2009); such reeducation phenomena occurs not only in injured brains, but in healthy subjects too (Hubbard et al., 2009). One of the many advantages of using a device for robotic- assisted therapy is that it can be used for long periods of training while recording the progress of the patient; also, a single device can be programmed to perform different tasks with the patient such as assistive or resistive exercises (Arya et al., 2011). From the point of view of robotics, a device for motor reeducation usually belongs to the class of rigid body systems; when they consist in serial arms, potential and kinetic energies are employed to obtain their dynamics via the Euler-Lagrange equations (Lewis et al., 2003). With the model at hand and based on feedback linearisation (Isidori, 1995), the so-called computed-torque control suc- cessfully drives the manipulator to a desired trajectory (Gilbert and Ha, 1984), provided appropriate issues of path generation (Shin and McKay, 1985), memory (Lewis ⋆ This work has been supported by the ECOS Nord SEP- CONACYT-ANUIES Project (Mexico 291309 / France M17M08). This research is sponsored by ELSAT 2020 of the Haut de France Region, the European Community, the Regional Delegation for Re- search and Technology, the French Ministry of Higher Education and Research, and the French National Center for Scientific Research. and Kamen, 1994), and possibly discretisation ( ˚ Astr¨ om and Wittenmark, 2013) are considered. Nevertheless, if the employed device is a parallel robot, i.e., a closed kinematic chain mechanism whose end effector is linked to the base (Merlet, 2006), Euler-Lagrange method- ologies cannot longer be directly applied (Codourey, 1998). Up to our knowledge, most of the solutions available (Cheng et al., 2003; Shang and Cong, 2009) ignore the fact that the resulting dynamics can be seen as differential algebraic equations (DAEs), i.e., system dynamics (serial arms) subject to algebraic restrictions (parallel character- istics) (Arceo et al., 2018b). As expected, both physically and mathematically, DAEs require proper initialisation as the dynamics are restricted to a subset of the free ones (Rabier and Rheinboldt, 2002); moreover, the Pan- telides algorithm is required to find such conditions as well as recover the missing dynamics for simulation purposes (Pantelides, 1988).  Fig. 1. Photography of the system. Keywords: Computed-Torque Control, Differential Algebraic Equation, Ankle Reeducation Therapy. Abstract: This paper extends the well-known computed-torque control technique for trajectory tracking in parallel robotic manipulators, which belong to the class of nonlinear systems under algebraic restrictions. The proposed methodology is based on Euler-Lagrange modelling which, once the closed-loop kinematic chain is taken into account, leads to a system of differential algebraic equations requiring proper initialisation while controlled. The control scheme is illustrated for an ankle therapy device known as the motoBOTTE, where post-stroke patients are subject to reeducation routines corresponding to different trajectories of the parallel manipulator. * Sonora Institute of Technology, 5 de Febrero 818 Sur, Ciudad Obregon, Sonora, Mexico ** Universit´ e Polytechnique Hauts-de-France, LAMIH UMR CNRS 8201, F-59313 Valenciennes, France. Jorge Alvarez * Juan Carlos Arceo ** Carlos Armenta * Jimmy Lauber ** Miguel Bernal * An Extension of Computed-Torque Control for Parallel Robots in Ankle Reeducation ⋆