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The increasing demands of a modern industry require the usage of industrial machines to perform various tasks with complex profiles. Tasks are usually given by a set of motion trajectory via-points; thus the geometric path representation and the proper reparameterization of the trajectory with respect to time are necessary. This study presents a kinematically constrained reparameterization approach for generating the trajectory by formulating an optimal control problem for time and jerk. Both geometric path generation and trajectory reparameterization are adopted by B-splines to ensure the trajectory smoothness. Inequality constraints are proposed to satisfy the axial kinematic limits, and equality constraints are included to ensure zero velocity and acceleration at the start and end of the trajectory. The trade-off between time and jerk is formulated as a bi-objective optimization problem. Moreover, the normalized normal constraint method with divide and conquer algorithm is applied to generate a Pareto front with significant trade-offs. Subsequently, the best trade-off solution is chosen. The proposed method is investigated by simulation with several geometric profiles, and the effectiveness is validated with an experimental result.

A Kinematically Constrained Reparameterization Approach to Optimal Time and Jerk Motion of Industrial Machines