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

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 10 000 times, which further verified the superiority of the new numerical difference formula.