Rapid search for the parameterized equilibrium of the state-dependent delay differential equation in robotic milling

The large forced vibration significantly impacts the tool-workpiece interaction in robotic milling, resulting in the state-dependent delay stability problem (SDD-SP). To solve SDD-SP, it is necessary to search the equilibrium by minimizing the residual of the state-dependent delay differential equation (SDD-DE) with periodic boundary conditions.Previous research has shown that it is complicated and time-consuming because of the nonanalytic iteration process and reduplicative root-finding for SDD.This paper introduces a rapid algorithm for searching for equilibrium using time-domain harmonic-balance-like methods (TDHBM). To solve the time-consuming challenge, the parameterized form of the equilibrium is initially obtained based on Fourier expansion, which spontaneously meets the periodic boundary condition.Then, by substituting the parameterized form into the SDD-DE, the semi-analytic Jacobian matrix for optimization is derived to improve iteration efficiency. Meanwhile, the iteration count for root-finding is significantly reduced by substituting numerical integration of the first-order differential SDD for root-finding.As a result of the simulation, the TDHBM reduces the computation time of searching for the equilibrium by 80% compared with the shooting method (SM), and the equilibrium maintains higher accuracy with the same collocation number.