Towards a Simulator Tool for Predicting Sprinting and Long Jump Motions with and without Running-Specific Prostheses: An Optimization-Based Approach

The performances of sprinters and long jumpers with below the knee amputation (BKA) have improved continuously since the development of prostheses specifically for athletic movements. In the last years, a number of athletes with BKA have attempted to compete in non-amputee competitions. Due to the specific shape and material properties of the running-specific prosthesis (RSP), concerns exist that it may give athletes an advantage over non-amputee athletes. In this work, we investigate and compare sprinting and long jump movements of athletes with and without unilateral BKA using accurate computer models. In this context, the aim of the work is to describe similarities and differences between the athletes’ movements and to show that the employed model- and optimization-based computations are useful for this purpose. We created subject-specific multi-body models for five different athletes (four non-amputee athletes, one athlete with unilateral BKA) in order to be able to investigate the different movements. Depending on the research question, the models vary in the number of degrees of freedom (DOFs), from 16 DOFs for a two-dimensional model in the sagittal plane to 31 DOFs for a three-dimensional model. For the athlete with BKA, we created a three-segment model of the RSP with one rotational DOF in the sagittal plane. The respective motion is described by a sequence of several phases, which differ by the type of ground contact. Each of these phases is described by its own set of ordinary differential equations (ODEs) or differential algebraic equations (DAEs). We use multi-phase optimal control problems (OCPs) with discontinuities to generate sprint and long jump motions. Three different formulations of OCPs are adopted in this work. (1) We formulate a least squares OCP to reconstruct the dynamics of sprint and long jump motion capture recordings of the individual athletes. (2) For the generation of realistic motions, which can be used for prediction, we formulate a synthesis OCP; this optimizes an objective function consisting of a weighted combination of chosen optimization criteria. (3) Last, in the study of sprint movements, we use an inverse optimal control problem (IOCP): this consists of an inner loop, in which a synthesis OCP is solved, and an outer loop, which adjusts the weights of the individual optimization criteria such that the distance between the inner loop solution and a reference movement becomes minimal. We have successfully applied these three optimization problem formulations to the computation of two sprint steps of three athletes without and one athlete with unilateral transtibial amputation. Here, the movements of the non-amputee athletes differ from that of the amputee athlete in a large number of variables. In particular, the athletes use different actuation strategies for running with and without a RSP. We have observed lower torques in the amputee athlete in the leg affected by the amputation than in the non-amputee control group. In contrast, significantly larger torques occurred in the joints of the upper extremity in the amputee athlete. Furthermore, the comparison has shown that the asymmetry created by the RSP is reflected throughout the body and affects the entire movement. Using the OCPs for motion reconstruction (1) and synthesis (2), we have successfully computed the last three steps of the approach and the jump of a long jump for an athlete without and an athlete with unilateral amputation. In the reconstructed solutions, the amputee athlete achieves a greater jump distance compared to the non-amputee athlete, despite a slower approach velocity, because his take-off is more efficient. In the synthesis solutions, on the other hand, the non-amputee athlete achieves the greater jump distance because he generates a greater vertical force during the take-off and achieves a better ratio of gain of vertical to loss of horizontal velocity. Finally, we have presented our idea of a simulator tool to compare the amputee athlete with himself without amputation and have demonstrated it using the sprint and long jump movements. For this purpose, we have kept the model of the athlete with unilateral transtibial amputation from the previous studies and have created a non-amputee version of the same model by mirroring the biological leg. We have selected one objective function each for sprinting and for long jump and have solved the OCP for motion synthesis (2) for both model versions. Using the differences to the solutions based on the models of two real athletes, we have highlighted the importance of the simulator tool in the evaluation of advantages and disadvantages due to the use of the RSP