Methods for computing the inverse dynamics of multibody systems with contacts are disclosed. Inverse dynamics means computing external forces that cause a system to move along a given trajectory. Such computations have been used routinely for data analysis and control synthesis in the absence of contacts between rigid bodies. The disclosed inverse dynamics methods include the ability to handle contacts. The disclosed methods include the following steps: projecting the discrete-time equations of motion from joint space to contact space; defining the forward dynamics in contact space as the solution to an optimization problem; using the features of this optimization problem to obtain a unique inverse—which turns out to correspond to the solution to a dual optimization problem; solving the latter using standard methods for numerical optimization; projecting the solution from contact space back to joint space and finding the external forces. The corresponding methods for computing forward dynamics are also disclosed.