基于泰勒展开的新型差分公式设计与应用研究  

Design and application of a new difference formula based on Taylor’s Expansion

作　　者：余怡霖 仓乃梦 郭东生 李煊鲜 邵辉[2] YU Yilin;CANG Naimeng;GUO Dongsheng;LI Xuanxian;SHAO Hui(School of Information and Communication Engineering,Hainan University,Haikou 570228,China;School of Information Science and Engineering,Huaqiao University,Xiamen 361021,China)

机构地区：[1]海南大学信息与通信工程学院,海南海口570228 [2]华侨大学信息科学与工程学院,福建厦门361021

出　　处：《海南大学学报(自然科学版中英文)》2025年第1期73-80,共8页Natural Science of Hainan University

基　　金：国家自然科学基金项目(U2141234);海南省重点研发项目(ZDYF2024GXJS003);海南大学科研启动基金项目(KYQD(ZR)23025)。

摘　　要：针对欧拉差分公式精度不高的问题,提出一种新型数值差分公式用以实现对目标函数一阶导数的估算。基于泰勒级数展开原理,研究目标函数在不同数据点下的展开式,通过移项变换消除展开式的高阶项,从而推导得到具有较高计算精度的新型数值差分公式,并通过理论分析给出新型数值差分公式的最优步长,数值实验结果验证了所提出的新型数值差分公式的有效性。UR5机械臂的仿真实验证明了采样时间为0.01 s时,机械臂运动精度提高10 000倍,进一步验证了新型数值差分公式的优越性。Aimed at the problem that the accuracy of Euler's difference formula is low,in the report,a new numerical difference formula for estimating the first-order derivative of the objective function was proposed.Based on Taylor Series Expansions,the expanded forms of the objective function at difference data points were presented.Aided with the transposition and conversion,the higher order terms in the expanded forms were eliminated,and the new numerical difference formula with high computational precision was thus derived.The theoretical analysis was performed to obtain the optimal step size of the difference formula.The effectiveness of the proposed new numerical difference formula was verified by the numerical experimental results.The simulation experiment of UR5 manipulator proved that when the sampling time was 0.01 s,the motion accuracy of the manipulator was increased by 10000 times,which further verified the superiority of the new numerical difference formula.

关 键 词：泰勒级数展开 数值差分公式 一阶导数 最优步长 

分 类 号：O174.42[理学—数学]