To be successful at manipulating objects one needs to apply simultaneously well controlled movements and contact forces. We present a computational theory of how the brain may successfully generate a vast spectrum of interactive behaviors by combining two independent processes. One process is competent to control movements in free space and the other is competent to control contact forces against rigid constraints. Free space and rigid constraints are singularities at the boundaries of a continuum of mechanical impedance. Within this continuum, forces and motions occur in “compatible pairs” connected by the equations of Newtonian dynamics. The force applied to an object determines its motion. Conversely, inverse dynamics determine a unique force trajectory from a movement trajectory. In this perspective, we describe motor learning as a process leading to the discovery of compatible force/motion pairs. The learned compatible pairs constitute a local representation of the environment's mechanics. Experiments on force field adaptation have already provided us with evidence that the brain is able to predict and compensate the forces encountered when one is attempting to generate a motion. Here, we tested the theory in the dual case, i.e., when one attempts at applying a desired contact force against a simulated rigid surface. If the surface becomes unexpectedly compliant, the contact point moves as a function of the applied force and this causes the applied force to deviate from its desired value. We found that, through repeated attempts at generating the desired contact force, subjects discovered the unique compatible hand motion. When, after learning, the rigid contact was unexpectedly restored, subjects displayed after effects of learning, consistent with the concurrent operation of a motion control system and a force control system. Together, theory and experiment support a new and broader view of modularity in the coordinated control of forces and motions.