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Discrete gradient-zeroing neural network algorithm for solving future Sylvester equation aided with left–right four-step rule as well as robot arm inverse kinematics

Author

Listed:
  • Guo, Pengfei
  • Zhang, Yunong
  • Yao, Zheng-an

Abstract

The temporal-variant Sylvester equation (TVSE) occupies a significant position in applied mathematics, particularly in the realms of optimal control theory and matrix optimization engineering applications. Within the framework of prediction modeling systems, the future Sylvester equation (FSE) emerges as the discrete manifestation of TVSE, characterized by unknown future information. Leveraging a novel left and right four-step (LRFS) rule, we propose a novel discrete gradient-zeroing neural network (DGZNN) algorithm with order-5 precision, which is developed from the continuous gradient-zeroing neural network (GZNN) model, for solving the FSE problem. The proposed algorithm is named as LRFS-DGZNN algorithm, which stands out as an inverse-less neurodynamic algorithm. Additionally, the convergence properties of the GZNN model in solving the TVSE problem are elucidated through Lyapunov stability theory and matrix spectral theory. Furthermore, the LRFS-DGZNN algorithm’s error pattern property in solving the FSE problem is established and proven using stability theory of linear multi-step methods and ordinary differential equation numerical approximation theory. Three numerical experiments are conducted to evaluate the performance of the proposed GZNN model for solving the TVSE problem and the LRFS-DGZNN algorithm for solving the FSE problem. Moreover, the study showcases the inverse-kinematics solutions and simulations involving planar robot arm with 2 degrees of freedom (DOFs), the Kinova Jaco2 robot arm with 6 DOFs, and the Franka Emika Panda robot arm with 7 DOFs, illustrating the high efficiency of the LRFS-DGZNN algorithm.

Suggested Citation

  • Guo, Pengfei & Zhang, Yunong & Yao, Zheng-an, 2025. "Discrete gradient-zeroing neural network algorithm for solving future Sylvester equation aided with left–right four-step rule as well as robot arm inverse kinematics," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 233(C), pages 475-501.
    • Handle: RePEc:eee:matcom:v:233:y:2025:i:c:p:475-501
    DOI: 10.1016/j.matcom.2025.02.009
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    References listed on IDEAS

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    1. Zhao, Lv & Jin, Jie & Gong, Jianqiang, 2021. "Robust zeroing neural network for fixed-time kinematic control of wheeled mobile robot in noise-polluted environment," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 185(C), pages 289-307.
    2. Xiao, Lin & Yi, Qian & Zuo, Qiuyue & He, Yongjun, 2020. "Improved finite-time zeroing neural networks for time-varying complex Sylvester equation solving," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 178(C), pages 246-258.
    3. Gao, Yuefeng & Tang, Zhichao & Ke, Yuanyuan & Stanimirović, Predrag S., 2024. "New activation functions and Zhangians in zeroing neural network and applications to time-varying matrix pseudoinversion," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 225(C), pages 1-12.
    4. Yi, Chenfu & Zhang, Yunong & Guo, Dongsheng, 2013. "A new type of recurrent neural networks for real-time solution of Lyapunov equation with time-varying coefficient matrices," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 92(C), pages 40-52.
    5. Jin, Jie & Chen, Weijie & Qiu, Lixin & Zhu, Jingcan & Liu, Haiyan, 2023. "A noise tolerant parameter-variable zeroing neural network and its applications," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 207(C), pages 482-498.
    6. Mandal, Dibyendu Sekhar & Chekroun, Abdennasser & Samanta, Sudip & Chattopadhyay, Joydev, 2021. "A mathematical study of a crop-pest–natural enemy model with Z-type control," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 187(C), pages 468-488.
    7. Zhang, Yunong & Zhai, Keke & Chen, Dechao & Jin, Long & Hu, Chaowei, 2016. "Challenging simulation practice (failure and success) on implicit tracking control of double-integrator system via Zhang-gradient method," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 120(C), pages 104-119.
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