Minimum-jerk trajectory planning pertaining to a translational 3-degree-of-freedom parallel manipulator through piecewise quintic polynomials interpolation

Song Lu,Bingxiao Ding,Yangmin Li

doi:10.1177/1687814020913667

Abstract

This article aims to present a minimum-jerk trajectory planning approach to address the smooth trajectory generation problem of 3-prismatic-universal-universal translational parallel kinematic manipulator. First, comprehensive kinematics and dynamics characteristics of this 3-prismatic-universal-universal parallel kinematic manipulator are analyzed by virtue of the accepted link Jacobian matrices and proverbial virtual work principle. To satisfy indispensable continuity and smoothness requirements, the discretized piecewise quintic polynomials are employed to interpolate the sequence of joints’ angular position knots which are transformed from these predefined via-points in Cartesian space. Furthermore, the trajectory planning problem is directly converted into a constrained nonlinear multi-variables optimization problem of which objective function is to minimize the maximum of the joints’ angular jerk throughout the whole trajectory. Finally, two typical application simulations using the reliable sequential quadratic programming algorithm demonstrate that this proposed minimum-jerk trajectory planning approach is of explicit feasibility and appreciable effectiveness.

Highlights

The last three decades have witnessed a considerable development of parallel kinematic manipulators (PKMs) which can be treated as positive encouraging candidates for many advanced automation processes and digitalization applications.[1,2,3,4,5]
Arising from the proposed minimum-jerk trajectory planning algorithm imposed upon the 3-PUU PKM, all strict and necessary constraints consist of (1) meeting the predefined state conditions at the initial and ultimate time; (2) fulfilling the definite continuity and smoothness requirements with regard to position, velocity, acceleration, and jerk for specific relevant via-points; (3) satisfying the explicit kinematics restraints; (4) abiding by the specific dynamics bounds; and (5) subjecting to the limitation of total execution time
This article focuses on taking jerk minimization as the most important consideration for trajectory planning issue of parallel manipulators

Summary

Introduction

The last three decades have witnessed a considerable development of parallel kinematic manipulators (PKMs) which can be treated as positive encouraging candidates for many advanced automation processes and digitalization applications.[1,2,3,4,5] Undoubtedly, the apparent advantages of PKMs are low moving inertia, high stiffness, high dexterity, and high payload-toweight ratio. Except the exhaustive kinematics equations analyzed above, the investigation of dynamics features is essential to the smooth trajectory planning of 3-PUU PKM.[37] First and foremost, we propose two prevalent hypotheses: (1) even though tiny elastic deformations of lightweight struts are inevitable in practice process, all components existing in the whole mechanical system are deemed ideal rigid bodies, which means that they can withstand the applied forces and do not deform; (2) the effect of frictional force among the joints and active components is negligible, and there is no energy loss caused by friction, the exclusive variation of virtual work is affected directly by the movement of the input and output forces/torques. Arising from the proposed minimum-jerk trajectory planning algorithm imposed upon the 3-PUU PKM, all strict and necessary constraints consist of (1) meeting the predefined state conditions at the initial and ultimate time; (2) fulfilling the definite continuity and smoothness requirements with regard to position, velocity, acceleration, and jerk for specific relevant via-points; (3) satisfying the explicit kinematics restraints; (4) abiding by the specific dynamics bounds; and (5) subjecting to the limitation of total execution time.

Objective function

Conclusion