# Minimising the Userâ€™s Effort during Wheelchair Propulsion using an Optimal Control Problem

### Ouazna Oukacha, Chouki Sentouh, Philippe Pudlo

#### Abstract

This paper proposes a study of the optimal control problem with state constraint, using two types of a power-assist wheelchair propulsion. The cost function is given by the metabolic function, which represented by a compromise between the work exerted by the joints muscles (mechanical effect) and an efficiency function that converts chemical into mechanical energy (biomechanical effect). The dynamic wheelchair is given by a simple model, which connects the push force to the wheelchair speed. An upper bound constraint is considered in order to limit the energy consumed by the motor. This study used an approach that calls the Pontryaginâ€™s maximum principle, the optimal solution varies with the parameters of the problem. Finally, a numerical comparison is enabled using two types of assistance: constant and proportional. This numerical comparison is based on the framework of the optimal control theory with two different costs. The first cost is given by the integral of the squared userâ€™s force and the second by the integral of the metabolic function. This Numerical results show that the user provides less effort with metabolic cost than with the energy userâ€™s force.

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