Abstract: A control system for operating any mechanical system, such as a non-linear robotic system, obtains and applies exact control forces so that the non-linear mechanical system satisfies desired trajectory requirements. The system enables a user to set trajectory requirements and allocate weighting factors to the control forces, the weighting factors being set at any values deemed appropriate to optimize the control forces. The trajectory requirements have a functional dependence on a physical measurement vector and time, a velocity of the physical measurement vector and time; or a combination of the two functional dependencies, the physical measurement vector being related to the displacement of one or more locations within the mechanical system.