A Mathematical Study of Sprinting on Artificial Legs

In 2008, the remarkable performance of the double amputee sprinter Oscar Pistorius initiated a discussion of whether his running prostheses might give him an advantage over able-bodied sprinters. He uses carbon fiber Cheetah devices by Össur that have spring-like properties; and the assumption was that the high passive torques and the lower moments of inertia of the prosthetic lower legs more than compensate for the absence of active ankle torques. The purpose of our research is to use mathematical models and optimal control techniques to better understand the underlying mechanics and control of sprinting on prostheses and to bring new insights into the continuing discussion. We established rigid multibody system models for the hybrid dynamics of able-bodied as well as double amputee sprinters. In the present study, we use models in the sagittal plane with 9 bodies and 11 degrees of freedom. In the able-bodied case, there are torque actuators at all eight internal joints; in the double amputee case, the actuators at the ankles are replaced by linear spring damper elements, but the other six actuators remain. Running motions for this model are generated by solving a multiphase optimal control problem with discontinuities and periodicity constraints, using an efficient direct multiple shooting approach.