An Accurate Numerical Algorithm for Attitude Updating Based on High-Order Polynomial Iteration

Abstract

To effectively improve the accuracy of attitude reconstruction under highly dynamic environments, a new numerical attitude updating algorithm is designed in this paper based on the high-order polynomial iteration according to the differential equation for quaternion. In this algorithm, a high-order polynomial is introduced to fit the angular rate accurately without increasing the number of gyro outputs during per attitude updating interval. This algorithm can provide an exact high-order polynomial solution for quaternion and the process of attitude reconstruction can be implemented efficiently. The algorithm's performance is evaluated as compared with optimal coning algorithm, attitude quaternion updating algorithm based on Picard iteration (QPI), and higher-order rotation vector attitude updating algorithm (Fourth4Rot) under coning motion. The simulation results show that this algorithm can improve the accuracy of attitude computation and clearly outperform the optimal coning algorithm, QPI, and Fourth4Rot in high dynamic environment.INDEX TERMS Attitude updating algorithm, quaternion, high-order polynomial, polynomial iteration.

An attitude updating algorithm using Picard iteration and higher degree polynomial

“…Studies have shown that the last truncation error is small enough and can be neglected when qðNh,0Þ is determined by the sum of the first N þ 3 terms in (8). 20 Consequently, the key to reduce QPI error is to improve the fitting precision of the angular rate, and the ideal case occurs when δω¼ 0.…”

“…Combining ( 2), ( 9), (29), and (30), the Picard series solutions to Qð4hÞ are directly solved. 20 However, there exist potential problems that limit the precision of quaternion reconstruction. To be specific, the process of constructing the three-degree polynomial in (29) involves the computation of the Moore-Penrose pseudoinverse of matrix, which means that c 0 4 and c 0 5 are the minimum-norm solution of and c 0 5 , it is convinced that the substitution of À3=5c 2 × c 3 and 1=5c 2 × c 3 for c 1 × c 0 4 and c 0 × c 0 5 in the process of calculating qð4h,0Þ is beneficial to design high-precision four-sample attitude updating algorithm.…”