In this paper, the lowest harmonic trajectory pattern for point-to-point motions with zero end point velocity, acceleration and jerk of fully controlled nonlinear dynamics systems that can be synthesized is proved to be a unique combination of a fundamental frequency harmonic and its second harmonic. For fully controlled dynamic systems such as robot manipulators and other computer controlled machines with rigid links, it is shown that such a trajectory pattern would yield the minimum number of harmonic in the required actuation forces/torques. Such trajectory patterns are therefore suitable for point-to-point motion synthesis of high-precision and high-speed computer controlled machinery and can also be synthesized to minimize excitation of the system natural modes of vibrations.

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