The implicit inversion method for calculating the forward dynamics input Jacobian

This paper presents the implicit inversion method (IIM), a recursive method to evaluate the Jacobian of the forward dynamics w.r.t. the system inputs, using intermediate results obtained from an O(n) forward dynamics algorithm. The resulting coefficient matrix, called the inertia-weighted input matrix (IWIM), can be used to significantly improve the performance of solving optimal control problems that take into account system dynamics for only the current time step. As the relationship between inputs and accelerations appears fixed within a time step, this matrix can be evaluated in the initialization step of the optimization. This means that the forward dynamics only needs to be solved once at the initialization of the optimization, rather than having to solve the equations in every iteration of the optimization. The method presented in this paper especially targets a case where the forward dynamics are calculated using an O(n) method and takes advantage of variables that are already known through the evaluation of that method. These quantities allow us to obtain the inertia-weighted input matrix without having to convert the system to its generalized coordinate form. Exploiting the shape of the resulting equation, it is even possible to avoid an explicit inversion of any matrices in the process. Finally, runtime comparisons between three different methods to calculate the IWIM are made for several examples.