For a new class of tendon-driven robotic systems that is generalized to include tensegrity structures, this paper focuses on a method to determine the tendon force inputs from a set of admissible, non-saturating inputs, that will move the rigidbody system from point A to point B along a prescribed path in minimum time. The approach utilizes the existence conditions and solution of a linear algebra problem that describes how the set of admissible tendon forces is mapped onto the set of path-dependent torques. Since this mapping is not one-to-one, free parameters in the control law always exist. An infinity-norm minimization with respect to these free parameters is responsible for saturation avoidance. Characterizing and optimizing these free parameters is the new contribution. Feedback is introduced to attenuate disturbances arising from the tensegrity paradigm. Examples illustrate methods and validate tensegrity's superior saturation avoidance capability.