A method for minimizing tracking errors in dynamic systems including obtaining desired trajectory data of the dynamic system, obtaining a set of constraints on at least one of the desired and actual trajectories of the dynamic system, obtaining a set of uniform or non-uniform rational B-splines having known original B-spline basis functions but unknown B-spline coefficients, applying a trajectory optimization process to the desired trajectory data including applying forward filtering to B-spline basis functions and utilizing the original and filtered B-spline basis functions to select optimal coefficients of the B-splines, and outputting an optimal motion command signal in response to the trajectory optimization process to the dynamic system such that a resultant actual trajectory is substantially equal to the desired trajectory while satisfying the set of constraints on the at least one of the desired and actual trajectories.

A control apparatus for a welding tool includes a determination module for determining a normalized displacement signal, in which a deflection of a tong-shaped tool, due to an effect of a mechanical force generated on the tool during a work process using the tong-shaped tool, is compensated for. The control apparatus further includes a force regulation module for regulating a progression of the force which the tong-shaped tool applies to at least one component during the work process on at least one component. The force regulation module is configured to regulate the progression of the force during the work process based on the normalized displacement signal.

A method for minimizing tracking errors in dynamic systems including obtaining desired trajectory data of the dynamic system, obtaining a set of constraints on at least one of the desired and actual trajectories of the dynamic system, obtaining a set of uniform or non-uniform rational B-splines having known original B-spline basis functions but unknown B-spline coefficients, applying a trajectory optimization process to the desired trajectory data including applying forward filtering to B-spline basis functions and utilizing the original and filtered B-spline basis functions to select optimal coefficients of the B-splines, and outputting an optimal motion command signal in response to the trajectory optimization process to the dynamic system such that a resultant actual trajectory is substantially equal to the desired trajectory while satisfying the set of constraints on the at least one of the desired and actual trajectories.