A close-form command shaping control for point-to-point maneuver with nonzero initial and final conditions

Command shaping algorithms are generally associated with nonzero initial and final conditions. However, initial disturbances may be inevitable, which compromises the effectiveness of shaped commands. A common example is when the crane’s jib is not or cannot be initially positioned directly above the payload, which forces the transfer operation to start with an initial angle. Shaped commands based on zero initial conditions would most probably result in unwanted oscillations at the target location of the transfer maneuver. Other applications may, by design, require nonzero final conditions such as the case of demolition cranes. In this work, a closed-form command shaping algorithm is developed considering nonzero initial angle and angular velocity disturbances. The algorithm is capable of setting the final angle and angular velocity to pre-determined design values. The equation of motion is derived and then solved to eliminate the associated residual vibrations using our proposed shaper. Using complex mathematical manipulations, the shapers parameters are derived analytically for the two cases; initial and final conditions. To test the proposed technique, several scenarios are simulated and tested experimentally. Simulation and experimental results illustrate the ability of the shaped command to either eliminate residual oscillations for nonzero initial conditions or conclude a maneuver at a pre-determined final angle value.