基于角速度的欧拉角傅里埃指数近似输出方法

摘要

本发明公开了一种基于角速度的欧拉角傅里埃指数近似输出方法，用于解决现有的飞行器机动飞行时欧拉角输出精度差的技术问题。技术方案是通过引入多个参数并采用傅里埃级数对滚转、俯仰、偏航角速度p，q，r进行近似逼近描述，可以对滚转、俯仰、偏航角速度p，q，r傅里埃级数描述的阶次，按照依次求解俯仰角、滚转角、偏航角，直接对欧拉角的表达式进行高阶逼近积分，使得欧拉角的求解按照超线性逼近，可以保证确定欧拉角的时间更新迭代计算精度，从而提高了惯性设备输出飞行姿态的准确性。

主权项

一种基于角速度的欧拉角傅里埃指数近似输出模型建模方法，其特征在于包括以下步骤：步骤1、（a）根据欧拉方程:式中：分别指滚转、俯仰、偏航角；p,q,r分别为滚转、俯仰、偏航角速度；三个欧拉角的计算按照依次求解俯仰角、滚转角、偏航角的步骤进行；滚转、俯仰、偏航角速度p,q,r的n阶展开式分别为p(t)=p_A[1 cos(ωt)…cos[(n‑1)ωt]cos(nωt)]^T+p_B[sin(ωt)sin(2ωt)…sin[(n‑1)ωt]sin(nωt)]^Tq(t)=q_A[1 cos(ωt)…cos[(n‑1)ωt]cos(nωt)]^T+q_B[sin(ωt)sin(2ωt)…sin[(n‑1)ωt]sin(nωt)]^Tr(t)=r_A[cos(ωt)cos(2ωt)…cos[(n‑1)ωt]cos(nωt)]^T+r_B[sin(ωt)sin(2ωt)…sin[(n‑1)ωt]sin(nωt)]^T其中，ω为角频率，p_A=[p_a0 p_a1…p_a(n‑1) p_an],p_B=[p_b1 p_b2… p_b(n‑1)p_bn]q_A=[q_a0 q_a1…q_a(n‑1) q_an],q_B=[q_b1 q_b2…q_b(n‑1) q_bn]r_A=[r_a0 r_a1…r_a(n‑1) r_an],r_B=[r_b1 r_b2…r_b(n‑1) r_bn]（b）俯仰角的时间更新求解式为：式中：T为采样周期，$\begin{array}{c}{a}_1={[q_A{H}_A{\xi }_{\mathrm{AI}}\left(t\right){|}_{\mathrm{kT}}^{(k+1)T}+q_B{H}_B{\xi }_{\mathrm{BI}}\left(t\right){|}_{\mathrm{kT}}^{(k+1)T}]}^2\\ +{[r_A{H}_A{\xi }_{\mathrm{AI}}\left(t\right){|}_{\mathrm{kT}}^{(k+1)T}+r_B{H}_B{\xi }_{\mathrm{BI}}\left(t\right){|}_{\mathrm{kT}}^{(k+1)T}]}^2\\ -{[p_A{H}_A{\xi }_{\mathrm{AI}}\left(t\right){|}_{\mathrm{kT}}^{(k+1)T}+p_B{H}_B{\xi }_{\mathrm{BI}}\left(t\right){|}_{\mathrm{kT}}^{(k+1)T}]}^2\end{array}$$\begin{array}{c}{a}_2=p_A{\Omega }_{\mathrm{AAI}}{|}_{\mathrm{kT}}^{(k+1)T}{H}_A^T{r}_A^T+p_B{\Omega }_{\mathrm{BAI}}{|}_{\mathrm{kT}}^{(k+1)T}{H}_A^T{r}_A^T\\ +p_A{\Omega }_{\mathrm{ABI}}{|}_{\mathrm{kT}}^{(k+1)T}{H}_B^T{r}_B^T+p_B{\Omega }_{\mathrm{BBI}}{|}_{\mathrm{kT}}^{(k+1)T}{H}_B^T{r}_B^T\\ -[p_A{H}_A{\xi }_{\mathrm{AI}}\left(t\right)+p_B{H}_B{\xi }_{\mathrm{BI}}\left(t\right)]{|}_{\mathrm{kT}}^{(k+1)T}[{\xi }_{\mathrm{AI}}^T\left(t\right)H_A^T{r}_A^T+{\xi }_{\mathrm{BI}}^T\left(t\right)H_B^T{r}_B^T]{|}_{\mathrm{kT}}\end{array}$$\begin{array}{c}{a}_3=p_A{\Omega }_{\mathrm{AAI}}{|}_{\mathrm{kT}}^{(k+1)T}{H}_A^T{q}_A^T+p_B{\Omega }_{\mathrm{BAI}}{|}_{\mathrm{kT}}^{(k+1)T}{H}_A^T{q}_A^T\\ +p_A{\Omega }_{\mathrm{ABI}}{|}_{\mathrm{kT}}^{(k+1)T}{H}_B^T{q}_B^T+p_B{\Omega }_{\mathrm{BBI}}{|}_{\mathrm{kT}}^{(k+1)T}{H}_B^T{q}_B^T\\ -[p_A{H}_A{\xi }_{\mathrm{AI}}\left(t\right)+p_B{H}_B{\xi }_{\mathrm{BI}}\left(t\right)]{|}_{\mathrm{kT}}^{(k+1)T}[{\xi }_{\mathrm{AI}}^T\left(t\right)H_A^T{q}_A^T+{\xi }_{\mathrm{BI}}^T\left(t\right)H_B^T{q}_B^T]{|}_{\mathrm{kT}}\end{array}$$\begin{array}{c}\left|\lambda \right|=\{p_A{\Omega }_{\mathrm{AAI}}{|}_{\mathrm{kT}}^{(k+1)T}{H}_A^T{p}_A^T+p_B{\Omega }_{\mathrm{BAI}}{|}_{\mathrm{kT}}^{(k+1)T}{H}_A^T{p}_A^T\\ +p_A{\Omega }_{\mathrm{ABI}}{|}_{\mathrm{kT}}^{(k+1)T}{H}_B^T{p}_B^T+p_B{\Omega }_{\mathrm{BBI}}{|}_{\mathrm{kT}}^{(k+1)T}{H}_B^T{p}_B^T\\ -[p_A{H}_A{\xi }_{\mathrm{AI}}\left(t\right)+p_B{H}_B{\xi }_{\mathrm{BI}}\left(t\right)]{|}_{\mathrm{kT}}^{(k+1)T}[{\xi }_{\mathrm{AI}}^T\left(t\right)H_A^T{p}_A^T+{\xi }_{\mathrm{BI}}^T\left(t\right)H_B^T{p}_B^T]{|}_{\mathrm{kT}}\\ +q_A{\Omega }_{\mathrm{AAI}}{|}_{\mathrm{kT}}^{(k+1)T}{H}_A^T{q}_A^T+q_B{\Omega }_{\mathrm{BAT}}{|}_{\mathrm{kT}}^{(k+1)T}{H}_A^T{q}_A^T\\ +q_A{\Omega }_{\mathrm{ABI}}{|}_{\mathrm{kT}}^{(k+1)T}{H}_B^T{q}_B^T+q_B{\Omega }_{\mathrm{BBI}}{|}_{\mathrm{kT}}^{(k+1)T}{H}_B^T{q}_B^T\\ -[q_A{H}_A{\xi }_{\mathrm{AI}}\left(t\right)+q_B{H}_B{\xi }_{\mathrm{BI}}\left(t\right)]{|}_{\mathrm{kT}}^{(k+1)T}[{\xi }_{\mathrm{AI}}^T\left(t\right)H_A^T{q}_A^T+{\xi }_{\mathrm{BI}}^T\left(t\right)H_B^T{q}_B^T]{|}_{\mathrm{kT}}\\ +r_A{\Omega }_{\mathrm{AAI}}{|}_{\mathrm{kT}}^{(k+1)T}{H}_A^T{r}_A^T+r_B{\Omega }_{\mathrm{BAI}}{|}_{\mathrm{kT}}^{(k+1)T}{H}_A^T{r}_A^T\\ +r_A{\Omega }_{\mathrm{ABI}}{|}_{\mathrm{kT}}^{(k+1)T}{H}_B^T{r}_B^T+r_B{\Omega }_{\mathrm{BBI}}{|}_{\mathrm{kT}}^{(k+1)T}{H}_B^T{r}_B^T\\ -[r_A{H}_A{\xi }_{\mathrm{AI}}\left(t\right)+r_B{H}_B{\xi }_{\mathrm{BI}}\left(t\right)]{|}_{\mathrm{kT}}^{(k+1)T}\left[{\xi }_{\mathrm{AI}}^T\left(t\right)H_A^T{r}_A^T\left(t\right)H_B^T{r}_B^T\right]{|}_{\mathrm{kT}}{\}}^{\frac{1}{2}}\end{array}$ξ_AI(t)=[t sin(ωt)…sin[(n‑1)ωt]sin(nωt)]^Tξ_BI(t)=[cos(ωt)cos(2ωt)…cos[(n‑1)ωt]cos(nωt)]^T$H_A={\left[\begin{array}{cccc}{H}_0^T& H_1^T& ...& H_n^T\end{array}\right]}^T=\mathrm{diag}\{1,\frac{1}{\omega },\frac{1}{2\omega },...,\frac{1}{\mathrm{n\omega }}\},H_B=-H_A$H_i为H_A阵的行向量，步骤2、在已知俯仰角的情况下，滚转角的时间更新求解式为：其中$\begin{array}{c}{a}_4=a_1+2\left\{\right[p_A{H}_A{\xi }_{\mathrm{AI}}\left(t\right){|}_{\mathrm{kT}}^{(k+1)T}+p_B{H}_B{\xi }_{\mathrm{BI}}\left(t\right){|}_{\mathrm{kT}}^{(k+1)T}{]}^2\\ -[q_A{H}_A{\xi }_{\mathrm{AI}}\left(t\right){|}_{\mathrm{kT}}^{(k+1)T}+q_B{H}_B{\xi }_{\mathrm{BI}}\left(t\right){|}_{\mathrm{kT}}^{(k+1)T}{]}^2\}\end{array}$$\begin{array}{c}{a}_5=q_A{\Omega }_{\mathrm{AAI}}{|}_{\mathrm{kT}}^{(k+1)T}{H}_A^T{p}_A^T+q_B{\Omega }_{\mathrm{BAI}}{|}_{\mathrm{kT}}^{(k+1)T}{H}_A^T{p}_A^T\\ +q_A{\Omega }_{\mathrm{ABI}}{|}_{\mathrm{kT}}^{(k+1)T}{H}_B^T{p}_B^T+q_B{\Omega }_{\mathrm{BBI}}{|}_{\mathrm{kT}}^{(k+1)T}{H}_B^T{p}_B^T\\ -[q_A{H}_A{\xi }_{\mathrm{AI}}\left(t\right)+q_B{H}_B{\xi }_{\mathrm{BI}}\left(t\right)]{|}_{\mathrm{kT}}^{(k+1)T}[{\xi }_{\mathrm{AI}}^T\left(t\right)H_A^T{p}_A^T+{\xi }_{\mathrm{BI}}^T\left(t\right)H_B^T{p}_B^T]{|}_{\mathrm{kT}}\end{array}$$\begin{array}{c}{a}_6=q_A{\Omega }_{\mathrm{AAI}}{|}_{\mathrm{kT}}^{(k+1)T}{H}_A^T{r}_A^T+q_B{\Omega }_{\mathrm{BAI}}{|}_{\mathrm{kT}}^{(k+1)T}{H}_A^T{r}_A^T\\ +q_A{\Omega }_{\mathrm{ABI}}{|}_{\mathrm{kT}}^{(k+1)T}{H}_B^T{r}_B^T+q_B{\Omega }_{\mathrm{BBI}}{|}_{\mathrm{kT}}^{(k+1)T}{H}_B^T{r}_B^T\\ -[q_A{H}_A{\xi }_{\mathrm{AI}}\left(t\right)+q_B{H}_B{\xi }_{\mathrm{BI}}\left(t\right)]{|}_{\mathrm{kT}}^{(k+1)T}[{\xi }_{\mathrm{AI}}^T\left(t\right)H_A^T{r}_A^T+{\xi }_{\mathrm{BI}}^T\left(t\right)H_B^T{r}_B^T]{|}_{\mathrm{kT}}\end{array}$步骤3、在俯仰角、滚转角已知情况下，偏航角的求解为：$\psi \left(t\right)=\psi \left(\mathrm{kT}\right)+{\int }_{\mathrm{kT}}^t[b_1\left(t\right)+b_2\left(t\right)]\mathrm{dt}$式中：