CN102519467A - Approximate output method for eulerian angle Chebyshev index on basis of angular velocity - Google Patents

Abstract

The invention discloses an approximate output method for an eulerian angle Chebyshev index on the basis of angular velocity, which is used for solving the problem of poor eulerian angle output accuracy during maneuver flight of an existing aircraft. The technical scheme is that high order approximation integral is directly performed on express of an eulerian angle by introducing a plurality of parameters and unfolding speed of a roll angle, a pitch angle and a yaw angle according to Chebyshev orthogonal polynomial in a change interval according to the sequence of solving the pitch angle, the roll angle and the yaw angle, the solving of the eulerian angle is approximated in super linear mode, time update iterative computation accuracy for determining the eulerian angle is ensured, and accuracy of output flight posture of an inert device is improved.

Description

Eulerian angle Chebyshev's exponential approximation output intent based on angular velocity

Technical field

The present invention relates to a kind of aircraft maneuvering flight and confirm method, particularly relate to a kind of Eulerian angle Chebyshev's exponential approximation output intent 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 a kind of Eulerian angle Chebyshev's exponential approximation output intent based on angular velocity.This method through introduce a plurality of parameters and with lift-over, pitching, yaw rate according to the Chebyshev's orthogonal polynomial expansion between the fluctuation zone; 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, 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: a kind of Eulerian angle Chebyshev's exponential approximation output intent based on angular velocity 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 expansion of r is respectively

p(t)＝pξ，q(t)＝qξ，r(t)＝rξ

Wherein

p＝[p ₀?p ₁?L?p _n-1?p _n]?q＝[q ₀?q ₁?L?q _n-1?q _n]

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

$\left.\begin{array}{c}{\mathrm{\ξ}}_0\left(t\right)=1\\ {\mathrm{\ξ}}_1\left(t\right)=1-2t/b\\ {\mathrm{\ξ}}_2\left(t\right)=8{(t/b)}^2-8(t/b)+1\\ M\\ {\mathrm{\ξ}}_{i+1}\left(t\right)={2\mathrm{\ξ}}_1\left(t\right){\mathrm{\ξ}}_i\left(t\right)-{\mathrm{\ξ}}_{i-1}\left(t\right)\end{array}\right\}i=\mathrm{2,3},L,n-\mathrm{1,0}\≤t\≤\mathrm{NT},b=\mathrm{NT}$

Be the recursive form of Chebyshev's orthogonal polynomial, T is the sampling period;

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

In the formula:

$a_1={\left(\mathrm{qH}{\mathrm{\ξ}|}_{\mathrm{kT}}^{(k+1)T}\right)}^2+{\left(\mathrm{rH}{\mathrm{\ξ}|}_{\mathrm{kT}}^{(k+1)T}\right)}^2-{\left(\mathrm{pH}{\mathrm{\ξ}|}_{\mathrm{kT}}^{(k+1)T}\right)}^2$

$a_2=p\{\mathrm{\Ω}\⊗[\mathrm{H\ξ}\left(t\right){|}_{\mathrm{kT}}^{(k+1)T}\left]\right\}{H}^T{r}^T-\mathrm{pH}{\mathrm{\ξ}|}_{\mathrm{kT}}^{(k+1)T}{\mathrm{rH\ξ}|}_{\mathrm{kT}}$

$a_3=p\{\mathrm{\Ω}\⊗[\mathrm{H\ξ}\left(t\right){|}_{\mathrm{kT}}^{(k+1)T}\left]\right\}{H}^T{q}^T-\mathrm{pH}{\mathrm{\ξ}|}_{\mathrm{kT}}^{(k+1)T}{\mathrm{qH\ξ}|}_{\mathrm{kT}}$

$\left|\mathrm{\λ}\right|=\left\{p\right\{\mathrm{\Ω}\⊗\left[H{\mathrm{\ξ}\left(t\right)|}_{\mathrm{kT}}^{(k+1)T}\right]\}{H}^T{p}^T-\mathrm{pH}{\mathrm{\ξ}|}_{\mathrm{kT}}^{(k+1)T}{\mathrm{pH\ξ}|}_{\mathrm{kT}}$

$+q\{\mathrm{\Ω}\⊗[\mathrm{H\ξ}{\left(t\right)|}_{\mathrm{kT}}^{(k+1)T}\left]\right\}{H}^T{q}^T-\mathrm{qH}{\mathrm{\ξ}|}_{\mathrm{kT}}^{(k+1)T}{\mathrm{qH\ξ}|}_{\mathrm{kT}}$

$+r\{\mathrm{\Ω}\⊗[\mathrm{H\ξ}{\left(t\right)|}_{\mathrm{kT}}^{(k+1)T}\left]\right\}{H}^T{r}^T-\mathrm{rH}{\mathrm{\ξ}|}_{\mathrm{kT}}^{(k+1)T}{\mathrm{rH\ξ}|}_{\mathrm{kT}}{\}}^{\frac{1}{2}}$

$H=\left[\begin{array}{c}{H}_0\\ H_1\\ M\\ H_n\end{array}\right]=b\·\left[\begin{array}{ccccccc}\frac{1}{2}& -\frac{1}{2}& 0& L& 0& 0& 0\\ \frac{1}{8}& 0& -\frac{1}{8}& L& 0& 0& 0\\ -\frac{1}{6}& \frac{1}{4}& 0& L& 0& 0& 0\\ -\frac{1}{16}& 0& \frac{1}{8}& L& 0& 0& 0\\ M& M& M& O& M& M& M\\ \frac{-1}{2(m-1)(m-3)}& 0& 0& L& \frac{1}{4(m-3)}& 0& \frac{-1}{4(m-1)}\\ \frac{-1}{2m(m-2)}& 0& 0& L& 0& \frac{1}{4(m-2)}& 0\end{array}\right]$

Work as p, q, when the high-order term n of the expansion of r is odd number, m=4,6, K, n+1, m=5 when high-order term n is even number, 7, K, n+1, Hi (i=1,2, L n) is H row vector accordingly;

$\mathrm{\Ω}=\frac{1}{2}\left[\begin{array}{cccc}{H}_2+H_0& H_3+H_1& L& H_{n+1}+H_{n-1}\\ H_3+H_1& H_4+H_0& L& H_{n+2}+H_{n-2}\\ M& M& O& M\\ H_{n+1}+H_{n-1}& H_{n+2}+H_{n-2}& L& H_{2n}+H_0\end{array}\right]$

2, under the situation of the known angle of pitch, the renewal of the time of roll angle is found the solution formula and is:

Wherein

$a_4={\left({\mathrm{pH\ξ}|}_{\mathrm{kT}}^{(k+1)T}\right)}^2+{\left({\mathrm{rH\ξ}|}_{\mathrm{kT}}^{(k+1)T}\right)}^2-{\left({\mathrm{qH\ξ}|}_{\mathrm{kT}}^{(k+1)T}\right)}^2$

$a_5=q\{\mathrm{\Ω}\⊗[\mathrm{H\ξ}\left(t\right){|}_{\mathrm{kT}}^{(k+1)T}\left]\right\}{H}^T{p}^T-\mathrm{qH}{\mathrm{\ξ}|}_{\mathrm{kT}}^{(k+1)T}{\mathrm{qH\ξ}|}_{\mathrm{kT}}$

$a_6=q\{\mathrm{\Ω}\⊗[\mathrm{H\ξ}\left(t\right){|}_{\mathrm{kT}}^{(k+1)T}\left]\right\}{H}^T{r}^T-\mathrm{qH}{\mathrm{\ξ}|}_{\mathrm{kT}}^{(k+1)T}{\mathrm{rH\ξ}|}_{\mathrm{kT}}$

3, 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:

The invention has the beneficial effects as follows: since introduce a plurality of parameters and with lift-over, pitching, yaw rate according to the Legendre's orthogonal polynomial expansion between the fluctuation zone; 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):

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 expansion of r is respectively

p(t)＝pξ，q(t)＝qξ，r(t)＝rξ

Wherein

p＝[p ₀?p ₁?L?p _n-1?p _n]?q＝[q ₀?q ₁?L?q _n-1?q _n]

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

$\left.\begin{array}{c}{\mathrm{\ξ}}_0\left(t\right)=1\\ {\mathrm{\ξ}}_1\left(t\right)=2t/b-1\\ {\mathrm{\ξ}}_2\left(t\right)=6{(t/b)}^2-6(t/b)+1\\ M\\ (i+1){\mathrm{\ξ}}_{i+1}\left(t\right)={(2i+1)\mathrm{\ξ}}_1\left(t\right){\mathrm{\ξ}}_i\left(t\right)-{\mathrm{i\ξ}}_{i-1}\left(t\right)\end{array}\right\}i=\mathrm{2,3},L,n-1;0\≤t\≤\mathrm{NT};b=\mathrm{NT}$

Be the recursive form of Legendre's orthogonal polynomial, T is the sampling period;

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

In the formula:

$a_1={\left(\mathrm{qH}{\mathrm{\ξ}|}_{\mathrm{kT}}^{(k+1)T}\right)}^2+{\left(\mathrm{rH}{\mathrm{\ξ}|}_{\mathrm{kT}}^{(k+1)T}\right)}^2-{\left(\mathrm{pH}{\mathrm{\ξ}|}_{\mathrm{kT}}^{(k+1)T}\right)}^2$

$a_2=p{\∫}_{\mathrm{kT}}^{(k+1)T}\left[\mathrm{\ξ}\left(t\right){\mathrm{\ξ}}^T\left(t\right)\right]\mathrm{dt}{H}^T{r}^T-\mathrm{pH}{\mathrm{\ξ}|}_{\mathrm{kT}}^{(k+1)T}{\mathrm{rH\ξ}|}_{\mathrm{kT}}$

$a_3=p{\∫}_{\mathrm{kT}}^{(k+1)T}\left[\mathrm{\ξ}\left(t\right){\mathrm{\ξ}}^T\left(t\right)\right]\mathrm{dt}{H}^T{q}^T-\mathrm{pH}{\mathrm{\ξ}|}_{\mathrm{kT}}^{(k+1)T}{\mathrm{qH\ξ}|}_{\mathrm{kT}}$

$\left|\mathrm{\λ}\right|=\left\{p{\∫}_{\mathrm{kT}}^{(k+1)T}\right[\mathrm{\ξ}\left(t\right){\mathrm{\ξ}}^T\left(t\right)]\mathrm{dt}{H}^T{p}^T-\mathrm{pH}{\mathrm{\ξ}|}_{\mathrm{kT}}^{(k+1)T}{\mathrm{pH\ξ}|}_{\mathrm{kT}}$

$+q{\∫}_{\mathrm{kT}}^{(k+1)T}\left[\mathrm{\ξ}\left(t\right){\mathrm{\ξ}}^T\left(t\right)\right]\mathrm{dt}{H}^T{q}^T-\mathrm{qH}{\mathrm{\ξ}|}_{\mathrm{kT}}^{(k+1)T}{\mathrm{qH\ξ}|}_{\mathrm{kT}}$

$+r{\∫}_{\mathrm{kT}}^{(k+1)T}\left[\mathrm{\ξ}\left(t\right){\mathrm{\ξ}}^T\left(t\right)\right]\mathrm{dt}{H}^T{r}^T-\mathrm{rH}{\mathrm{\ξ}|}_{\mathrm{kT}}^{(k+1)T}{\mathrm{rH\ξ}|}_{\mathrm{kT}}{\}}^{\frac{1}{2}}$

$H=b\·\left[\begin{array}{cccccccc}\frac{1}{2}& \frac{1}{6}& 0& 0& L& 0& 0& 0\\ -\frac{1}{2}& 0& \frac{1}{10}& 0& L& 0& 0& 0\\ 0& -\frac{1}{6}& 0& \frac{1}{14}& L& 0& 0& 0\\ M& M& M& M& O& M& M& M\\ 0& 0& 0& 0& L& \frac{-1}{2(2m-5)}& 0& \frac{1}{2(2m-1)}\\ 0& 0& 0& 0& L& 0& \frac{-1}{2(2m-3)}& 0\end{array}\right]$

Work as p, q, when the high-order term n of the expansion of r is odd number, m=2,4, K, n+1, m=3 when high-order term n is even number, 5, K, n+1;

2, under the situation of the known angle of pitch, the renewal of the time of roll angle is found the solution formula and is:

Wherein

$a_4={\left({\mathrm{pH\ξ}|}_{\mathrm{kT}}^{(k+1)T}\right)}^2+{\left({\mathrm{rH\ξ}|}_{\mathrm{kT}}^{(k+1)T}\right)}^2-{\left({\mathrm{qH\ξ}|}_{\mathrm{kT}}^{(k+1)T}\right)}^2$

$a_5=q{\∫}_{\mathrm{kT}}^{(k+1)T}\left[\mathrm{\ξ}\left(t\right){\mathrm{\ξ}}^T\left(t\right)\right]\mathrm{dt}{H}^T{p}^T-\mathrm{qH}{\mathrm{\ξ}|}_{\mathrm{kT}}^{(k+1)T}{\mathrm{qH\ξ}|}_{\mathrm{kT}}$

$a_6=q{\∫}_{\mathrm{kT}}^{(k+1)T}\left[\mathrm{\ξ}\left(t\right){\mathrm{\ξ}}^T\left(t\right)\right]\mathrm{dt}{H}^T{r}^T-\mathrm{qH}{\mathrm{\ξ}|}_{\mathrm{kT}}^{(k+1)T}{\mathrm{rH\ξ}|}_{\mathrm{kT}}$

3, 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 (T3), and O (T2) precision of methods such as comparing the direct method of approximation of Eulerian equation or adopt that approximate Long Gekuta method is resolved wants high.

Claims (1)

1. Eulerian angle Chebyshev's exponential approximation output intent based on angular velocity 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 expansion of r is respectively

p(t)＝pξ，q(t)＝qξ，r(t)＝rξ

Wherein

p＝[p ₀?p ₁?L?p _n-1?p _n]?q＝[q ₀?q ₁?L?q _n-1?q _n]

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

Be the recursive form of Chebyshev's orthogonal polynomial, T is the sampling period;

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

In the formula:

Work as p, q, when the high-order term n of the expansion of r is odd number, m=4,6, K, n+1, m=5 when high-order term n is even number, 7, K, n+1, H _i(i=1,2, L n) is H row vector accordingly;

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

Wherein

Step 3, under the angle of pitch, roll angle known case, the formula of finding the solution of crab angle is:

In the formula: