Smooth trajectory generation based on contour error constraint and parameter correction b-spline

In terms of CNC machining task, high feedrate and high accuracy are of great significance. In this paper, an integrated jerk-limited method of minimal time trajectory planning with confined contour error (IJLMTPC) is proposed. Firstly, an optimal control method is utilized to obtain the minimal time feedrate profile. To compensate the contour error, it is added into the optimal control framework as a constraint. Based on the high-order transfer functions of every axis servo model, the contour errors constraint is formulated as the function of tracking errors. However, the optimal control problem (OCP) is difficult to be solved. Hence, the OCP is transformed into two convex subproblems. Since the axial jerk elements are included in the contour error constraint, they are no longer considered as dependent constraints in the two subproblems specially. Afterwards, two subproblems are discretized with control vector parameterization. Consequently, two subproblems are more efficient to be solved with nonlinear programming, and the minimal time feedrate is obtained. Secondly, considering that the OCP is solved on the domain of curve parameter, it is inconvenient for real-time interpolation. The resulting feedrate from solving the OCP is transformed into the corresponding profile on time domain, while the limited tangent jerk is imposed on. Hence, the feedrate is smoothed. Finally, to alleviate the fluctuation of feedrate, the real-time interpolation is performed based on the parameter correction b-spline. Two benchmark tool paths are adopted to test the proposed scheme, and the effectiveness is verified.