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<p>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.</p>

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Rapid search for the parameterized equilibrium of the state-dependent delay differential equation in low-stiffness milling systems: towards robotic milling stability

  • 期刊论文

  • 发表日期：

    2025年8月21日

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  • SiHao Mao,
  • YanRu Jiang,
  • SongTao Ye,
  • XiaoJian Zhang,
  • SiJie Yan,
  • Han Ding

摘要

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.