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Ankle injuries are among the most common injuries in sport and daily life. However, for their recovery, it is important for patients to perform rehabilitation exercises. These exercises are usually done with a therapist’s guidance to help strengthen the patient’s ankle joint and restore its range of motion. However, in order to share the load with therapists so that they can offer assistance to more patients, and to provide an efficient and safe way for patients to perform ankle rehabilitation exercises, we propose a framework that integrates learning techniques with a 3-PRS parallel robot, acting together as an ankle rehabilitation device. In this paper, we propose to use passive rehabilitation exercises for dorsiflexion/plantar flexion and inversion/eversion ankle movements. The therapist is needed in the first stage to design the exercise with the patient by teaching the robot intuitively through learning from demonstration. We then propose a learning control scheme based on dynamic movement primitives and iterative learning control, which takes the designed exercise trajectory as a demonstration (an input) together with the recorded forces in order to reproduce the exercise with the patient for a number of repetitions defined by the therapist. During the execution, our approach monitors the sensed forces and adapts the trajectory by adding the necessary offsets to the original trajectory to reduce its range without modifying the original trajectory and subsequently reducing the measured forces. After a predefined number of repetitions, the algorithm restores the range gradually, until the patient is able to perform the originally designed exercise. We validate the proposed framework with both real experiments and simulation using a Simulink model of the rehabilitation parallel robot that has been developed in our lab.

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Abstract. Modular robots are systems that can change its geometry or configuration when connecting more modules or when rearranging them in a different manner to perform a variety of tasks. Graph theory can be used to describe modular robots configurations, hence the possibility to determine the flexibility of the robot to move from one point to another. When the robot's configurations are represented in a mathematical way, forward kinematics can be obtained.

The present work describes a symbolic formulation based on Lie groups and graph theory to obtain the dynamic equations of tree-structure robotic mechanisms (TRM)s. The resulting equation is the equivalent geometric form of the classical Newton-Euler's equation of motion. Such formulation is valid for any TRM without closed kinematic chains and whose joints have one degree of freedom (revolute and/or prismatic). For instance, open chain manipulator, robotic hands or humanoids are included within TRM. The structure of the proposed formulation allows analytic differentiation, which is required for techniques such as optimal control or robust control. Moreover, the assembly parameters are explicitly held in matrices, so that they can be mathematically manipulated in order to be adapted for simulation, analysis or identification algorithms. Finally, the formulation is applied to a 7DOF manipulator and to a two arm torso of 16 degrees of freedom. The results are compared with the multi-body simulation software package MSC/ADAMS© proving the correctness of the formulation.