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

Abstract

Abstract 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 Chebyshev collocation method is far lower than the commonly used semidiscretization and full-discretization methods (FDMs). In this article, a robotic milling model allowing an arbitrarily high degree-of-freedom is proposed as a benchmark to evaluate the computation performance for different stability analysis algorithms. An improved algorithm named the “fast Chebyshev collocation method” (FCCM) is proposed to handle the delay differential equation (DDE) with a high degree-of-freedom. The proposed fast Chebyshev collocation method accelerates the traditional Chebyshev collocation method 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 the full-discretization method and the proposed method are applied to the robotic milling system to show their convergence rate, computation efficiency, and accuracy. The results demonstrate that the proposed method is overall advantageous to the full-discretization methods in convergence, efficiency, and accuracy even when the degree-of-freedom is sufficiently high, implying that the proposed fast Chebyshev collocation method can be a potential alternative tool to deal with the stability analysis for time-delay systems.