CN102494690A - Any eulerian angle step length orthogonal series approximation output method based on angular speed - Google Patents

Abstract

The invention discloses a eulerian angle any step length orthogonal series approximation output method based on an angular speed, which is used for solving the technical problem that the eulerian angle output precision is poor when the existing aircraft maneuvers. The technical scheme is as follows: a plurality of parameters are introduced, the tumbling, the pitching and the jaw rate are developed and approximated by adopting the improved recurrence mode similar to Chebyshev orthogonal polynomial, according to the pitching angle, the tumbling angle and the yaw angle, the high-order approximate integration is conducted on the eulerian angle expression directly, the solution of the eulerian angle linear approximated linearly according to superlinearity, and the determination of the time update iterative computation accuracy of the eulerian angle can be ensured, so that the accuracy of the flight attitude output by inertia equipment can be improved.

Description

Based on the approximate output intent of any step-length orthogonal series of the Eulerian angle of angular velocity

Technical field

The present invention relates to a kind of aircraft maneuvering flight and confirm method, particularly relate to the approximate output intent of any step-length orthogonal series of a kind of Eulerian angle based on angular velocity.

Background technology

Inertial equipment has vital role in movable body navigation and control; The acceleration of rigid motion, angular velocity and attitude etc. all depend on inertial equipment output usually, and the output accuracy that therefore improves inertial equipment has clear and definite practical significance; In inertial equipment; Acceleration adopts accelerometer, angular velocity to adopt the direct metering system of angular rate gyroscope; The attitude accuracy of rigid body requires when very high to wait like the flight test and adopts the attitude gyro to measure, but all has measurement such as angular velocity directly resolve output in the plurality of applications field; Main cause is because dynamic attitude sensor costs an arm and a leg, volume is big; Cause a lot of aircraft to adopt angular rate gyroscopes etc. to resolve three Eulerian angle; Make the attitude time upgrade output and become core contents such as navigation; Therefore it is become influences one of inertial navigation system accuracy factors, designs and adopts the rational attitude time to upgrade the hot subject that output intent just becomes research; From the document of publishing, attitude output is mainly adopted the direct method of approximation of Eulerian equation based on angular velocity or adopted approximate Long Gekuta method to resolve (Sun Li, Qin Yongyuan; SINS attitude algorithm relatively; China's inertial technology journal, 2006, Vol.14 (3): 6-10; Pu Li; Wang TianMiao, Liang JianHong, Wang Song; An Attitude Estimate Approach using MEMS Sensors for Small UAVs; 2006, IEEE International Conference on Industrial Informatics, 1113-1117); Because three Eulerian angle are coupled mutually in the Eulerian equation, belong to nonlinear differential equation, different in different starting condition with error range under the different flight state, be difficult to guarantee the precision of actual engine request.

Summary of the invention

The problem of Eulerian angle output accuracy difference when overcoming existing aircraft maneuvering flight, the present invention provides any step-length orthogonal series of a kind of Eulerian angle based on angular velocity to be similar to output intent.This method is through introducing a plurality of parameters and adopting the recursive form of improved similar Chebyshev's orthogonal polynomial to launch to approach lift-over, pitching, yaw rate; Through according to finding the solution the angle of pitch, roll angle, crab angle successively; Directly the expression formula of Eulerian angle is carried out high-order approaches integration; Make finding the solution of Eulerian angle approach, thereby can guarantee to confirm the time renewal iterative computation precision of Eulerian angle and the output accuracy of inertance element according to ultralinear.

The technical solution adopted for the present invention to solve the technical problems is: any step-length orthogonal series of a kind of Eulerian angle based on angular velocity is similar to output intent, is characterized in may further comprise the steps:

1, (a) is according to Eulerian equation:

In the formula:

refers to lift-over, pitching, crab angle respectively; P, q, r are respectively lift-over, pitching, yaw rate; Parameter-definition is identical in full; The calculating of these three Eulerian angle is carried out according to the step of finding the solution the angle of pitch, roll angle, crab angle successively; Lift-over, pitching, yaw rate p, q, the n rank expansion of r is respectively

p(t)＝[p ₀?p ₁?L?p _n-1?p _n][ξ ₀(t)?ξ ₁(t)?L?ξ _n-1(t)?ξ _n(t)] ^T

q(t)＝[q ₀?q ₁?L?q _n-1?q _n][ξ ₀(t)?ξ ₁(t)?L?ξ _n-1(t)?ξ _n(t)] ^T

r(t)＝[r ₀?r ₁?L?r _n-1?r _n][ξ ₀(t)?ξ ₁(t)?L?ξ _n-1(t)?ξ _n(t)] ^T

Wherein

$\left.\begin{array}{c}{\mathrm{\ξ}}_0\left(t\right)=1\\ {\mathrm{\ξ}}_1\left(t\right)=\cos \left[a{\cos }^{-1}(1-2t/b)\right]\\ {\mathrm{\ξ}}_2\left(t\right)=2{\mathrm{\ξ}}_1\left(t\right)\·{\mathrm{\ξ}}_1\left(t\right)-1\\ M\\ {\mathrm{\ξ}}_{i+1}\left(t\right)2{\mathrm{\ξ}}_1\left(t\right){\mathrm{\ξ}}_i\left(t\right)-{\mathrm{\ξ}}_{i-1}\left(t\right)\end{array}\right\},i=\mathrm{2,3},L,n-\mathrm{1,0}\≤t\≤\mathrm{NT},b=\mathrm{NT}$

Be the recursive form of Chebyshev (Chebyshev) orthogonal polynomial, a is any real number, and T is the sampling period;

(b) time of the angle of pitch upgrades and to find the solution formula and be:

In the formula:

a ₁＝1+[p ₀?p ₁?L?p _n-1?p _n]Ω[p ₀?p ₁?L?p _n-1?p _n] ^T

+[q ₀?q ₁?L?q _n-1?q _n]Ω[q ₀?q ₁?L?q _n-1?q _n] ^T

+[r ₀?r ₁?L?r _n-1?r _n]Ω[r ₀?r ₁?L?r _n-1?r _n] ^T

+0.25{[p ₀?p ₁?L?p _n-1?p _n]ζ} ²+0.25{[q ₀?q ₁?L?q _n-1?q _n]ζ} ²

-0.25{[r ₀?r ₁?L?r _n-1?r _n]ζ} ²

a ₂＝[q ₀?q ₁?L?q _n-1?q _n]ζ

-0.5[r ₀?r ₁?L?r _n-1?r _n]Ω[p ₀?p ₁?L?p _n-1?p _n] ^T

a ₃＝[r ₀?r ₁?L?r _n-1?r _n]ζ

+0.5[q ₀?q ₁?L?q _n-1?q _n]Ω[p ₀?p ₁?L?p _n-1?p _n] ^T

$\mathrm{\ξ}={\left[\begin{array}{cccc}{\mathrm{\ξ}}_0& {\mathrm{\ξ}}_1& L& {\mathrm{\ξ}}_n\end{array}\right]}^T={\∫}_{\mathrm{kT}}^{(k+1)T}\mathrm{\ξ}\left(t\right)\mathrm{dt}$

${\mathrm{\ξ}}_i={\∫}_{\mathrm{kT}}^{(k+1)T}{\mathrm{\ξ}}_i\left(t\right)\mathrm{dt}={\∫}_{\mathrm{kT}}^{(k+1)T}\cos \left[\mathrm{ai}{\cos }^{-1}(1-2t/b)\right]\mathrm{dt}$

$=\frac{b}{4}\left\{\frac{1}{\mathrm{ai}-1}\cos \right[(\mathrm{ai}-1){\cos }^{-1}(1-2t/b)]$

$-\frac{1}{\mathrm{ai}+1}\cos \left[(\mathrm{ai}+1){\cos }^{-1}(1-2t/b)\right]\}{|}_{\mathrm{kT}}^{(k+1)T}$

$\mathrm{\Ω}={\left\{{\mathrm{\Ω}}_{\mathrm{ji}}\right\}}_{j=\mathrm{0,1},L,n;i=\mathrm{1,2},L,n}={\∫}_{\mathrm{kT}}^{(k+1)T}\mathrm{\ξ}\left(t\right){\∫}_{\mathrm{kT}}^T\mathrm{\ξ}\left(\mathrm{\τ}\right)\mathrm{d\τdt}$

${\mathrm{\Ω}}_{\mathrm{ji}}={\∫}_{\mathrm{kT}}^{(k+1)T}{\mathrm{\ξ}}_j\left(t\right){\∫}_{\mathrm{kT}}^t{\mathrm{\ξ}}_i\left(\mathrm{\τ}\right)\mathrm{d\τdt}$

$=\frac{b}{8}\left\{\frac{1}{\mathrm{ai}-1}{\∫}_{\mathrm{kT}}^{(k+1)T}\right\{\cos \left[(\mathrm{aj}-\mathrm{ai}+1){\cos }^{-1}(1-2t/b)\right]+\cos \left[(\mathrm{aj}+\mathrm{ai}-1){\cos }^{-1}(1-2t/b)\right]\}\mathrm{dt}$

$-\frac{1}{\mathrm{ai}+1}{\∫}_{\mathrm{kT}}^{(k+1)T}\left\{\cos \right[(\mathrm{aj}-\mathrm{ai}-1){\cos }^{-1}(1-2t/b)]+\cos [(\mathrm{aj}+\mathrm{ai}+1){\cos }^{-1}(1-2t/b)\left]\right\}\mathrm{dt}\}$

$-\frac{b}{4}\left\{\frac{1}{\mathrm{ai}-1}\cos \right[(\mathrm{ai}-1){\cos }^{-1}(1-2\mathrm{kT}/b)]$

$-\frac{1}{\mathrm{ai}+1}\cos \left[(\mathrm{ai}+1){\cos }^{-1}(1-2\mathrm{kT}/b)\right]\left\}{\∫}_{\mathrm{kT}}^{(k+1)T}\cos \right[\mathrm{aj}{\cos }^{-1}(1-2t/b)]\mathrm{dt}$

2, (a) under the situation of the known angle of pitch, the time of roll angle upgrades and to find the solution formula and be:

Wherein

a ₄＝1+[p ₀?p ₁?L?p _n-1?p _n]Ω[p ₀?p ₁?L?p _n-1?p _n] ^T

+[q ₀?q ₁?L?q _n-1?q _n]Ω[q ₀?q ₁?L?q _n-1?q _n] ^T

+[r ₀?r ₁?L?r _n-1?r _n]Ω[r ₀?r ₁?L?r _n-1?r _n] ^T

+0.25{[p ₀?p ₁?L?p _n-1?p _n]ζ(t)} ²-0.25{[r ₀?r ₁?L?r _n-1?r _n]ζ(t)} ²

-0.25{[q ₀?q ₁?L?q _n-1?q _n]ζ(t)} ²

a ₅＝[p ₀?p ₁?L?p _n-1?p _n]ζ

+0.5[r ₀?r ₁?L?r _n-1?r _n]Ω[q ₀?q ₁?L?q _n-1?q _n] ^T

a ₆＝[r ₀?r ₁?L?r _n-1?r _n]ζ

-0.5[p ₀?p ₁?L?p _n-1?p _n]Ω[q ₀?q ₁?L?q _n-1?q _n] ^T

(b) deposit under the angle of pitch, the roll angle known case, the formula of finding the solution of crab angle is:

$\mathrm{\ψ}\left(t\right)=\mathrm{\ψ}\left(\mathrm{kT}\right)+{\∫}_{\mathrm{kT}}^t[b_1\left(t\right)+b_2\left(t\right)]\mathrm{dt}$

In the formula:

The invention has the beneficial effects as follows: because through introducing a plurality of parameters and adopting the recursive form of improved similar Chebyshev's orthogonal polynomial to launch to approach lift-over, pitching, yaw rate; Through according to finding the solution the angle of pitch, roll angle, crab angle successively; Directly the expression formula of Eulerian angle is carried out high-order approaches integration; Make finding the solution of Eulerian angle approach, thereby guaranteed the time renewal iterative computation precision of definite Eulerian angle and the output accuracy of inertance element according to ultralinear.

Below in conjunction with embodiment the present invention is elaborated.

Embodiment

1, (a) is according to rigid body attitude equation (Eulerian equation):

Wherein:

refers to lift-over, pitching, crab angle respectively; P, q, r are respectively lift-over, pitching, yaw rate; Parameter-definition is identical in full; The calculating of these three Eulerian angle is carried out according to the step of finding the solution the angle of pitch, roll angle, crab angle successively; Lift-over, pitching, yaw rate p, q, the n rank expansion of r is respectively

p(t)＝[p ₀?p ₁?L?p _n-1?p _n][ξ ₀(t)?ξ ₁(t)?L?ξ _n-1(t)?ξ _n(t)] ^T

q(t)＝[q ₀?q ₁?L?q _n-1?q _n][ξ ₀(t)?ξ ₁(t)?L?ξ _n-1(t)?ξ _n(t)] ^T

r(t)＝[r ₀?r ₁?L?r _n-1?r _n][ξ ₀(t)?ξ ₁(t)?L?ξ _n-1(t)?ξ _n(t)] ^T

Wherein

$\left.\begin{array}{c}{\mathrm{\ξ}}_0\left(t\right)=1\\ {\mathrm{\ξ}}_1\left(t\right)=\cos \left[a{\cos }^{-1}(1-2t/b)\right]\\ {\mathrm{\ξ}}_2\left(t\right)=2{\mathrm{\ξ}}_1\left(t\right)\·{\mathrm{\ξ}}_1\left(t\right)-1\\ M\\ {\mathrm{\ξ}}_{i+1}\left(t\right)2{\mathrm{\ξ}}_1\left(t\right){\mathrm{\ξ}}_i\left(t\right)-{\mathrm{\ξ}}_{i-1}\left(t\right)\end{array}\right\},i=\mathrm{2,3},L,n-\mathrm{1,0}\≤t\≤\mathrm{NT},b=\mathrm{NT}$

Be the recursive form of Chebyshev (Chebyshev) orthogonal polynomial, a is any real number, and T is the sampling period;

(b) time of the angle of pitch upgrades and to find the solution formula and be:

In the formula:

a ₁＝1+[p ₀?p ₁?L?p _n-1?p _n]Ω[p ₀?p ₁?L?p _n-1?p _n] ^T

+[q ₀?q ₁?L?q _n-1?q _n]Ω[q ₀?q ₁?L?q _n-1?q _n] ^T

+[r ₀?r ₁?L?r _n-1?r _n]Ω[r ₀?r ₁?L?r _n-1?r _n] ^T

+0.25{[p ₀?p ₁?L?p _n-1?p _n]ζ} ²+0.25{[q ₀?q ₁?L?q _n-1?q _n]ζ} ²

-0.25{[r ₀?r ₁?L?r _n-1?r _n]ζ} ²

a ₂＝[q ₀?q ₁?L?q _n-1?q _n]ζ

-0.5[r ₀?r ₁?L?r _n-1?r _n]Ω[p ₀?p ₁?L?p _n-1?p _n] ^T

a ₃＝[r ₀?r ₁?L?r _n-1?r _n]ζ

+0.5[q ₀?q ₁?L?q _n-1?q _n]Ω[p ₀?p ₁?L?p _n-1?p _n] ^T

$\mathrm{\ξ}={\left[\begin{array}{cccc}{\mathrm{\ξ}}_0& {\mathrm{\ξ}}_1& L& {\mathrm{\ξ}}_n\end{array}\right]}^T={\∫}_{\mathrm{kT}}^{(k+1)T}\mathrm{\ξ}\left(t\right)\mathrm{dt}$

${\mathrm{\ξ}}_i={\∫}_{\mathrm{kT}}^{(k+1)T}{\mathrm{\ξ}}_i\left(t\right)\mathrm{dt}={\∫}_{\mathrm{kT}}^{(k+1)T}\cos \left[\mathrm{ai}{\cos }^{-1}(1-2t/b)\right]\mathrm{dt}$

$=\frac{b}{4}\left\{\frac{1}{\mathrm{ai}-1}\cos \right[(\mathrm{ai}-1){\cos }^{-1}(1-2t/b)]$

$-\frac{1}{\mathrm{ai}+1}\cos \left[(\mathrm{ai}+1){\cos }^{-1}(1-2t/b)\right]\}{|}_{\mathrm{kT}}^{(k+1)T}$

$\mathrm{\Ω}={\left\{{\mathrm{\Ω}}_{\mathrm{ji}}\right\}}_{j=\mathrm{0,1},L,n;i=\mathrm{1,2},L,n}={\∫}_{\mathrm{kT}}^{(k+1)T}\mathrm{\ξ}\left(t\right){\∫}_{\mathrm{kT}}^T\mathrm{\ξ}\left(\mathrm{\τ}\right)\mathrm{d\τdt}$

${\mathrm{\Ω}}_{\mathrm{ji}}={\∫}_{\mathrm{kT}}^{(k+1)T}{\mathrm{\ξ}}_j\left(t\right){\∫}_{\mathrm{kT}}^t{\mathrm{\ξ}}_i\left(\mathrm{\τ}\right)\mathrm{d\τdt}$

$=\frac{b}{8}\left\{\frac{1}{\mathrm{ai}-1}{\∫}_{\mathrm{kT}}^{(k+1)T}\right\{\cos \left[(\mathrm{aj}-\mathrm{ai}+1){\cos }^{-1}(1-2t/b)\right]+\cos \left[(\mathrm{aj}+\mathrm{ai}-1){\cos }^{-1}(1-2t/b)\right]\}\mathrm{dt}$

$-\frac{1}{\mathrm{ai}+1}{\∫}_{\mathrm{kT}}^{(k+1)T}\left\{\cos \right[(\mathrm{aj}-\mathrm{ai}-1){\cos }^{-1}(1-2t/b)]+\cos [(\mathrm{aj}+\mathrm{ai}+1){\cos }^{-1}(1-2t/b)\left]\right\}\mathrm{dt}\}$

$-\frac{b}{4}\left\{\frac{1}{\mathrm{ai}-1}\cos \right[(\mathrm{ai}-1){\cos }^{-1}(1-2\mathrm{kT}/b)]$

$-\frac{1}{\mathrm{ai}+1}\cos \left[(\mathrm{ai}+1){\cos }^{-1}(1-2\mathrm{kT}/b)\right]\left\}{\∫}_{\mathrm{kT}}^{(k+1)T}\cos \right[\mathrm{aj}{\cos }^{-1}(1-2t/b)]\mathrm{dt}$

2, (a) under the situation of the known angle of pitch, the time of roll angle upgrades and to find the solution formula and be:

Wherein

a ₄＝1+[p ₀?p ₁?L?p _n-1?p _n]Ω[p ₀?p ₁?L?p _n-1?p _n] ^T

+[q ₀?q ₁?L?q _n-1?q _n]Ω[q ₀?q ₁?L?q _n-1?q _n] ^T

+[r ₀?r ₁?L?r _n-1?r _n]Ω[r ₀?r ₁?L?r _n-1?r _n] ^T

+0.25{[p ₀?p ₁?L?p _n-1?p _n]ζ(t)} ²-0.25{[r ₀?r ₁?L?r _n-1?r _n]ζ(t)} ²

-0.25{[q ₀?q ₁?L?q _n-1?q _n]ζ(t)} ²

a ₅＝[p ₀?p ₁?L?p _n-1?p _n]ζ

+0.5[r ₀?r ₁?L?r _n-1?r _n]Ω[q ₀?q ₁?L?q _n-1?q _n] ^T

a ₆＝[r ₀?r ₁?L?r _n-1?r _n]ζ

-0.5[p ₀?p ₁?L?p _n-1?p _n]Ω[q ₀?q ₁?L?q _n-1?q _n] ^T

(b) under the angle of pitch, roll angle known case, the formula of finding the solution of crab angle is:

$\mathrm{\ψ}\left(t\right)=\mathrm{\ψ}\left(\mathrm{kT}\right)+{\∫}_{\mathrm{kT}}^t[b_1\left(t\right)+b_2\left(t\right)]\mathrm{dt}$

In the formula:

When inertial equipment is directly exported lift-over, pitching, yaw rate p, q, r adopt three rank to approach when describing, and the gained result is also near O (T ³), the O (T of methods such as comparing the direct method of approximation of Eulerian equation or adopt that approximate Long Gekuta method is resolved ²) precision will height.

Claims (1)

1. any step-length orthogonal series of the Eulerian angle based on angular velocity is similar to output intent, it is characterized in that may further comprise the steps:

Step 1, (a) are according to Eulerian equation:

In the formula:

refers to lift-over, pitching, crab angle respectively; P, q, r are respectively lift-over, pitching, yaw rate; Parameter-definition is identical in full; The calculating of these three Eulerian angle is carried out according to the step of finding the solution the angle of pitch, roll angle, crab angle successively; Lift-over, pitching, yaw rate p, q, the n rank expansion of r is respectively

p(t)＝[p ₀?p ₁?L?p _n-1?p _n][ξ ₀(t)?ξ ₁(t)?L?ξ _n-1(t)?ξ _n(t)] ^T

q(t)＝[q ₀?q ₁?L?q _n-1?q _n][ξ ₀(t)?ξ ₁(t)?L?ξ _n-1(t)?ξ _n(t)] ^T

r(t)＝[r ₀?r ₁?L?r _n-1?r _n][ξ ₀(t)?ξ ₁(t)?L?ξ _n-1(t)?ξ _n(t)] ^T

Wherein

$\left.\begin{array}{c}{\mathrm{\ξ}}_0\left(t\right)=1\\ {\mathrm{\ξ}}_1\left(t\right)=\cos \left[a{\cos }^{-1}(1-2t/b)\right]\\ {\mathrm{\ξ}}_2\left(t\right)=2{\mathrm{\ξ}}_1\left(t\right)\·{\mathrm{\ξ}}_1\left(t\right)-1\\ M\\ {\mathrm{\ξ}}_{i+1}\left(t\right)2{\mathrm{\ξ}}_1\left(t\right){\mathrm{\ξ}}_i\left(t\right)-{\mathrm{\ξ}}_{i-1}\left(t\right)\end{array}\right\},i=\mathrm{2,3},L,n-\mathrm{1,0}\≤t\≤\mathrm{NT},b=\mathrm{NT}$

Be the recursive form of Chebyshev (Chebyshev) orthogonal polynomial, a is any real number, and T is the sampling period;

(b) time of the angle of pitch upgrades and to find the solution formula and be:

In the formula:

a ₁＝1+[p ₀?p ₁?L?p _n-1?p _n]Ω[p ₀?p ₁?L?p _n-1?p _n] ^T

+[q ₀?q ₁?L?q _n-1?q _n]Ω[q ₀?q ₁?L?q _n-1?q _n] ^T

+[r ₀?r ₁?L?r _n-1?r _n]Ω[r ₀?r ₁?L?r _n-1?r _n] ^T

+0.25{[p ₀?p ₁?L?p _n-1?p _n]ζ} ²+0.25{[q ₀?q ₁?L?q _n-1?q _n]ζ} ²

-0.25{[r ₀?r ₁?L?r _n-1?r _n]ζ} ²

a ₂＝[q ₀?q ₁?L?q _n-1?q _n]ζ

-0.5[r ₀?r ₁?L?r _n-1?r _n]Ω[p ₀?p ₁?L?p _n-1?p _n] ^T

a ₃＝[r ₀?r ₁?L?r _n-1?r _n]ζ

+0.5[q ₀?q ₁?L?q _n-1?q _n]Ω[p ₀?p ₁?L?p _n-1?p _n] ^T

$\mathrm{\ξ}={\left[\begin{array}{cccc}{\mathrm{\ξ}}_0& {\mathrm{\ξ}}_1& L& {\mathrm{\ξ}}_n\end{array}\right]}^T={\∫}_{\mathrm{kT}}^{(k+1)T}\mathrm{\ξ}\left(t\right)\mathrm{dt}$

${\mathrm{\ξ}}_i={\∫}_{\mathrm{kT}}^{(k+1)T}{\mathrm{\ξ}}_i\left(t\right)\mathrm{dt}={\∫}_{\mathrm{kT}}^{(k+1)T}\cos \left[\mathrm{ai}{\cos }^{-1}(1-2t/b)\right]\mathrm{dt}$

$=\frac{b}{4}\left\{\frac{1}{\mathrm{ai}-1}\cos \right[(\mathrm{ai}-1){\cos }^{-1}(1-2t/b)]$

$-\frac{1}{\mathrm{ai}+1}\cos \left[(\mathrm{ai}+1){\cos }^{-1}(1-2t/b)\right]\}{|}_{\mathrm{kT}}^{(k+1)T}$

$\mathrm{\Ω}={\left\{{\mathrm{\Ω}}_{\mathrm{ji}}\right\}}_{j=\mathrm{0,1},L,n;i=\mathrm{1,2},L,n}={\∫}_{\mathrm{kT}}^{(k+1)T}\mathrm{\ξ}\left(t\right){\∫}_{\mathrm{kT}}^T\mathrm{\ξ}\left(\mathrm{\τ}\right)\mathrm{d\τdt}$

${\mathrm{\Ω}}_{\mathrm{ji}}={\∫}_{\mathrm{kT}}^{(k+1)T}{\mathrm{\ξ}}_j\left(t\right){\∫}_{\mathrm{kT}}^t{\mathrm{\ξ}}_i\left(\mathrm{\τ}\right)\mathrm{d\τdt}$

$=\frac{b}{8}\left\{\frac{1}{\mathrm{ai}-1}{\∫}_{\mathrm{kT}}^{(k+1)T}\right\{\cos \left[(\mathrm{aj}-\mathrm{ai}+1){\cos }^{-1}(1-2t/b)\right]+\cos \left[(\mathrm{aj}+\mathrm{ai}-1){\cos }^{-1}(1-2t/b)\right]\}\mathrm{dt}$

$-\frac{1}{\mathrm{ai}+1}{\∫}_{\mathrm{kT}}^{(k+1)T}\left\{\cos \right[(\mathrm{aj}-\mathrm{ai}-1){\cos }^{-1}(1-2t/b)]+\cos [(\mathrm{aj}+\mathrm{ai}+1){\cos }^{-1}(1-2t/b)\left]\right\}\mathrm{dt}\}$

$-\frac{b}{4}\left\{\frac{1}{\mathrm{ai}-1}\cos \right[(\mathrm{ai}-1){\cos }^{-1}(1-2\mathrm{kT}/b)]$

$-\frac{1}{\mathrm{ai}+1}\cos \left[(\mathrm{ai}+1){\cos }^{-1}(1-2\mathrm{kT}/b)\right]\left\}{\∫}_{\mathrm{kT}}^{(k+1)T}\cos \right[\mathrm{aj}{\cos }^{-1}(1-2t/b)]\mathrm{dt}$

Step 2, (a) are under the situation of the known angle of pitch, and the renewal of the time of roll angle is found the solution formula and is:

Wherein

a ₄＝1+[p ₀?p ₁?L?p _n-1?p _n]Ω[p ₀?p ₁?L?p _n-1?p _n] ^T

+[q ₀?q ₁?L?q _n-1?q _n]Ω[q ₀?q ₁?L?q _n-1?q _n] ^T

+[r ₀?r ₁?L?r _n-1?r _n]Ω[r ₀?r ₁?L?r _n-1?r _n] ^T

+0.25{[p ₀?p ₁?L?p _n-1?p _n]ζ(t)} ²-0.25{[r ₀?r ₁?L?r _n-1?r _n]ζ(t)} ²

-0.25{[q ₀?q ₁?L?q _n-1?q _n]ζ(t)} ²

a ₅＝[p ₀?p ₁?L?p _n-1?p _n]ζ

+0.5[r ₀?r ₁?L?r _n-1?r _n]Ω[q ₀?q ₁?L?q _n-1?q _n] ^T

a ₆＝[r ₀?r ₁?L?r _n-1?r _n]ζ

-0.5[p ₀?p ₁?L?p _n-1?p _n]Ω[q ₀?q ₁?L?q _n-1?q _n] ^T

(b) under the angle of pitch, roll angle known case, the formula of finding the solution of crab angle is:

$\mathrm{\ψ}\left(t\right)=\mathrm{\ψ}\left(\mathrm{kT}\right)+{\∫}_{\mathrm{kT}}^t[b_1\left(t\right)+b_2\left(t\right)]\mathrm{dt}$

In the formula: