Two efficient static optimization algorithms that account for muscle-tendon equilibrium: approaching the constraint Jacobian via a constant or a cubic spline function

Despite recent advances in algorithms for estimating muscle activities, static optimization remains the most used. Static optimization estimates muscle activations required to obtain a particular set of estimated kinematics. Although fast, static optimization may require considerable time for long trials. Improvements have been proposed in the past, but the current implementations are either accurate and slow (such as the most traditional implementation) or fast but less accurate (such as the linearized at maximal activations method used by OpenSim). Two innovative algorithms are proposed to improve both optimization time and accuracy of the static optimization. The first, designed to be fast, linearizes the constraint—i.e., constructing a constant constraint Jacobian—at the activations of the previous frame. The second, designed to be as accurate as possible, approaches the constraint Jacobian by cubic splines. Their performance and accuracy are compared to the traditional and OpenSim implementations. The linearized method performed as fast as the OpenSim implementation and was more accurate (0.3% of RMSE versus 5.9%). The spline method had excellent accuracy (0.1% of RMSE), but was 2X slower than the linearized approaches. Nevertheless, it was 100X faster than the traditional implementation. Our linearized method is therefore recommended when fast computation is needed, such as real-time applications, while the spline method is recommended otherwise.