求解捷联式方向余弦矩阵的一种新方法

本文提出求解捷联式方向余弦矩阵的一种新方法——三角函数法。建立了三角函数的微分方程;讨论了求解位置矩阵、姿态矩阵正交化等有关问题;对三角函数法的计算误差进行了分析。通过大量的数学模拟,对四元数法、方向余弦法及三角函数法的计算精度作了比较。结果表明,三角函数法运算量较小,精度高,是求解方向余弦矩阵的可行方法。

A NEW METHOD FOR CALCULATING STRAPDOWN DIRECTION COSINE MATRIX

A new method,"Circular Function" method,is presented for calculating the strapdown direction cosine matrix.The differential equation of Circular Function（6）and equation（9）for position matrix are derived.The errors of calculation with Circular Function are analyzed.A step by step method is employed in the Circular Function method.It consists of determining the matrix of Cn+1n,by equation（6）first and obtaining the matrix of Cn+1 from equation（8）then.As there is almost no non-orthogonal error in the matrix of Cn+1n,the matrix of Cn+1,possesses good orthogonality and higher accuracy.Because the Circular Function method needs solving only three equations with specific preliminary values,the workload of calculation is reduced. When the angle of 9 changes in a small range,it is possible to calculate directly the attitude matrix by equation（5）and orthogonalize the matrix by normalizing circular functions.Through a great deal of mathematic simulation and comparison of its accuracy with those of the Quaternion and Direction Cosine methods,relationship of errors in calculation with input angular velocity,time and period of calculation is clarified.The conclusions have been drawn as follows:1 The accuracy of the Circular Function method is a bit higher than those of the Quaternion and Direction Cosine methods.2 The error in calculating increases more or less linearly with calculation time.3 It is possible to enhance accuracy by choosing the calculation period reasonably.