A Fast Chebyshev Collocation Method for Stability Analysis of a Robotic Machining System with Time Delay

In the machining process that material removal involves, the time-delay effect due to the regenerative cutting is the root cause of tool chatter, which is a severe nonlinear vibration that leads to system failures. The Chebyshev collocation method (CCM) can be applied to the stability analysis of the delay-affected machining systems. However, when the degree of freedom (DOF) of the system is high, the computational efficiency of the CCM is far lower than the commonly used semi-discretization method and full-discretization method (FDM). In this article, a robotic milling model allowing arbitrarily high DOF is proposed as a benchmark to evaluate the computation performance for different stability analysis algorithms. Then, an improved algorithm named the Fast Chebyshev Collocation Method (FCCM) is proposed to handle the delay differential equation with high DOFs. The proposed FCCM accelerates the traditional CCM in two approaches: one is the inversion of the matrix when constructing the transition matrix, and the other is the reduction of the dimension of the transition matrix by applying the Sherman-Morrision-Woodbury formula. Subsequently, both FDM and the proposed FCCM are applied to the robotic milling system to show their convergence rate, computation efficiency, and accuracy. The results demonstrate that in most cases, the proposed method is overall advantageous to FDM in convergence, computational efficiency, and accuracy even when the DOF is high, implying that the proposed FCCM can be a potential alternative tool to deal with the stability analysis for complex time-delay systems.