Jerk-Optimal Piecewise Planning of Tool Orientation for 5-Axis Ball-End Machining With Linearized Kinematic Constraints

When planning tool orientations for 5-axis machining from the perspective of kinematics, existing works often tend to optimize the angular velocity or acceleration of the rotary axes rather than the angular jerk due to its calculation sensitivity to discretization variations, though the jerk is fundamentally tied to the tracking errors and residual vibrations of the actuators. In this paper, a study is reported on how to optimize the angular jerk of the rotary axes while comprehensively considering the kinematic constraints, thus achieving a jerk-optimal tool orientation along the tool path, with driving capacity of the rotary axes respected. In this method, the displacements of the rotary axes are continuously represented by two quintic B-spline curves, and then the angular velocity, acceleration, and jerk of the rotary axes, which are the derivatives of the displacements, can be succinctly represented as a B-spline curve. Taking advantage of the convex hull property of B-spline curve, the linear analytical representations of the kinematic constraints of the rotary axes can be successfully derived in form of control coefficient combinations. To prevent the machining interference at the same time, a greedy strategy that incorporates a process of alternately smoothing tool orientation and checking machining interference is employed. Then, the smooth displacement splines of the rotary axes can be obtained by solving a constructed quadratic programming (QP) model that minimizes the angular jerk along the tool path, while satisfying kinematic constraints and without machining interference. Moreover, to generate efficiently tool orientations for long tool paths, a piecewise planning strategy that optimizes the tool orientation from coarse to fine is developed. Finally, the conducted experiments validate the proposed method.