CN102494688B - Quaternion Laguerre approximate output method based on angular speed used during extreme flight of flying vehicle - Google Patents

Abstract

The invention discloses a quaternion Laguerre approximate output method based on an angular speed used during extreme flight of a flying vehicle, which is used for solving the technical problem of poor quaternion output accuracy of inertia equipment during extreme flight of the conventional flying vehicle. In a technical scheme, approximate approach description is performed on a rolling angular speed, a pitching angular speed and a yawing angular speed p, q and r by adopting a Shifted Lagnuerre orthogonal polynomial to directly obtain a quaternion state transfer matrix, so that the iterative computation accuracy of a determined quaternion is ensured. Shifted Lagnuerre orthogonal polynomial orders for the rolling angular speed, the pitching angular speed and the yawing angular speed p, q and r can be determined according to engineering accuracy, so that ultra-linear approximation for a quaternion state equation transfer matrix phie[(k+1)T, kT] is realized, the iterative computation accuracy of the determined quaternion is ensured, and the output accuracy of the inertia equipment during extreme flight of the flying vehicle is increased.

Description

Hypercomplex number Laguerre approximation output method during aircraft extreme flight based on angular velocity

Technical field

The present invention relates to a kind of attitude output intent of air craft carried inertial equipment, particularly hypercomplex number Laguerre approximation output method during a kind of aircraft extreme flight based on angular velocity.

Background technology

Conventionally, the acceleration of rigid motion, angular velocity and attitude etc. all depend on inertial equipment output, and the output accuracy that therefore improves inertial equipment has clear and definite practical significance.The spatial movements such as aircraft, torpedo, spacecraft in most of the cases all adopt the rigid motion differential equation; And the differential equation of portraying rigid body attitude is core wherein, with three Eulerian angle, be that pitching, rolling and crab angle are described conventionally, all pitching in Airborne Inertial equipment, rolling and yaw rate are resolved rear output conventionally.When rigid body is when the angle of pitch is ± 90 °, roll angle and crab angle cannot definite values, and it is excessive that the region of simultaneously closing on this singular point solves error, causes intolerable error in engineering and can not use; For fear of this problem, people adopt the method for restriction angle of pitch span, and this degenerates equation, attitude work entirely, thereby be difficult to be widely used in engineering practice.For this reason, pitching, rolling and the yaw rate direct measured value of people based in Airborne Inertial equipment, and adopted the output flight attitudes such as direction cosine method, Rotation Vector, Quaternion Method.

Direction cosine method has been avoided Euler method " unusual " phenomenon, and with direction cosine method, calculating attitude matrix does not have equation degenerate problem, attitude work entirely, but need to solve nine differential equations, calculated amount is larger, and real-time is poor, cannot meet engineering practice requirement.Rotation Vector is as list sample recursion, Shuangzi sample gyration vector, three increment gyration vectors and four increment rotating vector methods and various correction algorithms on this basis and recursive algorithm etc.While studying rotating vector in document, be all based on rate gyro, to be output as the algorithm of angle increment.Yet in Practical Project, the output of some gyros is angle rate signals, as optical fibre gyro, dynamic tuned gyroscope etc.When rate gyro is output as angle rate signal, the Algorithm Error of rotating vector method obviously increases.Hypercomplex number method is the most widely used method, the method is that the function of four Eulerian angle of definition calculates boat appearance, can effectively make up the singularity of Euler method, as long as separate four differential equation of first order formula groups, than direction cosine attitude matrix differential equation calculated amount, there is obvious minimizing, can meet the requirement to real-time in engineering practice.Its conventional computing method have finish card approximatioss, second order, fourth-order Runge-Kutta method and three rank Taylor expansions etc. (Paul G.Savage.A Unified Mathematical Framework for Strapdown Algorithm Design[J] .Journal of guidance, control, and dynamics, 2006,29 (2): 237-248).Finishing card approximatioss essence is list sample algorithm, and what limited rotation was caused can not compensate by exchange error, and the algorithm drift under high current intelligence in attitude algorithm can be very serious.While adopting fourth-order Runge-Kutta method to solve quaternion differential equation, along with the continuous accumulation of integral error, there will be exceed ± 1 phenomenon of trigonometric function value, thereby cause calculating, disperse.Taylor expansion is also because the deficiency of computational accuracy is restricted, particularly for aircraft maneuvering flight, attitude orientation angular speed is conventionally all larger, and the estimated accuracy of attitude has been proposed to requirements at the higher level, and the parameters such as hypercomplex number determine that the error of bringing makes said method in most cases can not meet engineering precision.

Summary of the invention

In order to overcome the large problem of existing hypercomplex number output error, hypercomplex number Laguerre approximation output method while the invention provides a kind of aircraft extreme flight based on angular velocity, the method adopts change Laguerre (Shifted Laguerre) orthogonal polynomial to rolling, pitching, yaw rate p, q, r carries out close approximation description, directly obtain hypercomplex number state-transition matrix, thereby can guarantee to determine the iterative computation precision of hypercomplex number.

The technical scheme that the present invention solves its technical matters employing is that hypercomplex number Laguerre approximation output method during a kind of aircraft extreme flight based on angular velocity, is characterized in comprising the following steps:

According to hypercomplex number continuous state equation

$\stackrel{\·}{e}=A_{e}e$

And discrete state equations

e(k+1)＝Φ _e[(k+1)T，kT]e(k)

E=[e wherein ₁, e ₂, e ₃, e ₄] ^t $A_e=\frac{1}{2}\left[\begin{array}{cccc}0& -p& -q& -r\\ p& 0& r& -q\\ q& -r& 0& p\\ r& q& -p& 0\end{array}\right]$

Φ _e[(k+1) T, kT] is A _estate-transition matrix, T is the sampling period, in full symbol definition is identical;

P, q, r is respectively rolling, pitching, yaw rate; Eulerian angle

θ, ψ refers to respectively rolling, pitching, crab angle;

State-transition matrix is according to approximant

${\mathrm{\Φ}}_e[(k+1)T,\mathrm{kT}]\≈I+\mathrm{\ΠH\ξ}\left(t\right){|}_{\mathrm{kT}}^{(k+1)T}+\mathrm{\Π}{\∫}_{\mathrm{kT}}^{(k+1)T}\left[\mathrm{\ξ}\left(t\right){\mathrm{\ξ}}^T\left(t\right)\right]\mathrm{dt}{H}^T{\mathrm{\Π}}_1-\mathrm{\ΠH\ξ}\left(t\right){|}_{\mathrm{kT}}^{(k+1)T}\mathrm{\ΠH\ξ}\left(\mathrm{kT}\right)$

And e (k+1)=Φ _e[(k+1) T, kT] e (k) obtains the time renewal value of hypercomplex number;

Wherein $I=\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right],$ ξ(t)＝[ξ ₀(t)ξ ₁(t)…ξ _n-1(t)ξ _n(t)] ^T

$\left.\begin{array}{c}{\mathrm{\ξ}}_0\left(t\right)=1\\ {\mathrm{\ξ}}_1\left(t\right)=1-t\\ {\mathrm{\ξ}}_2\left(t\right)=1-2t+0.5t^2\\ \·\\ \·\\ \·\\ (i+1){\mathrm{\ξ}}_{i+1}\left(t\right)=(i+2i-t){\mathrm{\ξ}}_i\left(t\right)-i{\mathrm{\ξ}}_{i-1}\left(t\right)\end{array}\right\}$ i＝2，3，…，n-1

For the recursive form of Laguerre orthogonal polynomial, rolling, pitching, yaw rate p, q, the expansion of r is respectively

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

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

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

$\mathrm{\Π}=\frac{1}{2}\{\left[\begin{array}{cccc}0& -1& 0& 0\\ 1& 0& 0& 0\\ 0& 0& 0& 1\\ 0& 0& -1& 0\end{array}\right]\left[\begin{array}{ccccccc}{p}_0& p_1& \·& \·& \·& p_{n-1}& p_n\end{array}\right]$

$+\left[\begin{array}{cccc}0& 0& -1& 0\\ 0& 0& 0& -1\\ 1& 0& 0& 0\\ 0& 1& 0& 0\end{array}\right]\left[\begin{array}{ccccccc}{q}_0& q_1& \·& \·& \·& q_{n-1}& q_n\end{array}\right]+\left[\begin{array}{cccc}0& 0& 0& -1\\ 0& 0& 1& 0\\ 0& -1& 0& 0\\ 1& 0& 0& 0\end{array}\right]\left[\begin{array}{ccccccc}{r}_0& r_1& \·& \·& \·& r_{n-1}& r_n\end{array}\right]\}$

${\mathrm{\Π}}_1=\frac{1}{2}\{{\left[\begin{array}{ccccccc}{p}_0& p_1& \·& \·& \·& p_{n-1}& p_n\end{array}\right]}^T\left[\begin{array}{cccc}0& -1& 0& 0\\ 1& 0& 0& 0\\ 0& 0& 0& 1\\ 0& 0& -1& 0\end{array}\right]$

$+{\left[\begin{array}{ccccccc}{q}_0& q_1& \·& \·& \·& q_{n-1}& q_n\end{array}\right]}^T\left[\begin{array}{cccc}0& 0& -1& 0\\ 0& 0& 0& -1\\ 1& 0& 0& 0\\ 0& 1& 0& 0\end{array}\right]+{\left[\begin{array}{ccccccc}{r}_0& r_1& \·& \·& \·& r_{n-1}& r_n\end{array}\right]}^T\left[\begin{array}{cccc}0& 0& 0& -1\\ 0& 0& 1& 0\\ 0& -1& 0& 0\\ 1& 0& 0& 0\end{array}\right]\}$

Work as p, q, when the high-order term n of expansion of r is odd number, m=4,6 ..., n+1, m=5 when high-order term n is even number, 7 ..., n+1.

The invention has the beneficial effects as follows: owing to adopting change Laguerre (Shifted Laguerre) orthogonal polynomial to rolling, pitching, yaw rate p, q, r carries out close approximation description, directly obtain hypercomplex number state-transition matrix, thereby guaranteed the iterative computation precision of definite hypercomplex number.

Below in conjunction with embodiment, the present invention is elaborated.

Embodiment

According to hypercomplex number continuous state equation

$\stackrel{\·}{e}=A_{e}e$

And discrete state equations

e(k+1)＝Φ _e[(k+1)T，kT]e(k)

E=[e wherein ₁, e ₂, e ₃, e ₄] ^t $A_e=\frac{1}{2}\left[\begin{array}{cccc}0& -p& -q& -r\\ p& 0& r& -q\\ q& -r& 0& p\\ r& q& -p& 0\end{array}\right]$

Φ _e[(k+1) T, kT] is A _estate-transition matrix, T is the sampling period,

P, q, r is respectively rolling, pitching, yaw rate; Eulerian angle

θ, ψ refers to respectively rolling, pitching, crab angle;

State-transition matrix is according to approximant

${\mathrm{\Φ}}_e[(k+1)T,\mathrm{kT}]\≈I+\mathrm{\ΠH\ξ}\left(t\right){|}_{\mathrm{kT}}^{(k+1)T}+\mathrm{\Π}{\∫}_{\mathrm{kT}}^{(k+1)T}\left[\mathrm{\ξ}\left(t\right){\mathrm{\ξ}}^T\left(t\right)\right]\mathrm{dt}{H}^T{\mathrm{\Π}}_1-\mathrm{\ΠH\ξ}\left(t\right){|}_{\mathrm{kT}}^{(k+1)T}\mathrm{\ΠH\ξ}\left(\mathrm{kT}\right)$

And e (k+1)=Φ _e[(k+1) T, kT] e (k) obtains the time renewal value of hypercomplex number;

Wherein $I=\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right],$ ξ(t)＝[ξ ₀(t)ξ ₁(t)…ξ _n-1(t)ξ _n(t)] ^T

$\left.\begin{array}{c}{\mathrm{\ξ}}_0\left(t\right)=1\\ {\mathrm{\ξ}}_1\left(t\right)=1-t\\ {\mathrm{\ξ}}_2\left(t\right)=1-2t+0.5t^2\\ \·\\ \·\\ \·\\ (i+1){\mathrm{\ξ}}_{i+1}\left(t\right)=(i+2i-t){\mathrm{\ξ}}_i\left(t\right)-i{\mathrm{\ξ}}_{i-1}\left(t\right)\end{array}\right\}$ i＝2，3，…，n-1

For the recursive form of Laguerre (Laguerre) orthogonal polynomial, rolling, pitching, yaw rate p, q, the expansion of r is respectively

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

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

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

$\mathrm{\Π}=\frac{1}{2}\{\left[\begin{array}{cccc}0& -1& 0& 0\\ 1& 0& 0& 0\\ 0& 0& 0& 1\\ 0& 0& -1& 0\end{array}\right]\left[\begin{array}{ccccccc}{p}_0& p_1& \·& \·& \·& p_{n-1}& p_n\end{array}\right]$

$+\left[\begin{array}{cccc}0& 0& -1& 0\\ 0& 0& 0& -1\\ 1& 0& 0& 0\\ 0& 1& 0& 0\end{array}\right]\left[\begin{array}{ccccccc}{q}_0& q_1& \·& \·& \·& q_{n-1}& q_n\end{array}\right]+\left[\begin{array}{cccc}0& 0& 0& -1\\ 0& 0& 1& 0\\ 0& -1& 0& 0\\ 1& 0& 0& 0\end{array}\right]\left[\begin{array}{ccccccc}{r}_0& r_1& \·& \·& \·& r_{n-1}& r_n\end{array}\right]\}$

${\mathrm{\Π}}_1=\frac{1}{2}\{{\left[\begin{array}{ccccccc}{p}_0& p_1& \·& \·& \·& p_{n-1}& p_n\end{array}\right]}^T\left[\begin{array}{cccc}0& -1& 0& 0\\ 1& 0& 0& 0\\ 0& 0& 0& 1\\ 0& 0& -1& 0\end{array}\right]$

$+{\left[\begin{array}{ccccccc}{q}_0& q_1& \·& \·& \·& q_{n-1}& q_n\end{array}\right]}^T\left[\begin{array}{cccc}0& 0& -1& 0\\ 0& 0& 0& -1\\ 1& 0& 0& 0\\ 0& 1& 0& 0\end{array}\right]+{\left[\begin{array}{ccccccc}{r}_0& r_1& \·& \·& \·& r_{n-1}& r_n\end{array}\right]}^T\left[\begin{array}{cccc}0& 0& 0& -1\\ 0& 0& 1& 0\\ 0& -1& 0& 0\\ 1& 0& 0& 0\end{array}\right]\}$

Work as p, q, when the high-order term n of expansion of r is odd number, m=4,6 ..., n+1, m=5 when high-order term n is even number, 7 ..., n+1.

Claims (1)

1. the hypercomplex number Laguerre approximation output method during aircraft extreme flight based on angular velocity, is characterized in that comprising the following steps:

According to hypercomplex number continuous state equation

$\stackrel{\·}{e}=A_{e}e$

And discrete state equations

e(k+1)＝Φ _e[(k+1)T，kT]e(k)

E=[e wherein ₁, e ₂, e ₃, e ₄] ^t $A_e=\frac{1}{2}\left[\begin{array}{cccc}0& -p& -q& -r\\ p& 0& r& -q\\ q& -r& 0& p\\ r& q& -p& 0\end{array}\right]$

Φ _e[(k+1) T, kT] is A _estate-transition matrix, T is the sampling period;

P, q, r is respectively rolling, pitching, yaw rate; Eulerian angle

θ, Ψ refers to respectively rolling, pitching, crab angle;

State-transition matrix is according to approximant

${\mathrm{\Φ}}_e[(k+1)T,\mathrm{kT}]\≈I+\mathrm{\ΠH\ξ}\left(t\right){|}_{\mathrm{kT}}^{(k+1)T}+\mathrm{\Π}{\∫}_{\mathrm{kT}}^{(k+1)T}\left[\mathrm{\ξ}\left(t\right){\mathrm{\ξ}}^T\left(t\right)\right]\mathrm{dt}{H}^T{\mathrm{\Π}}_1-\mathrm{\ΠH\ξ}\left(t\right){|}_{\mathrm{kT}}^{(k+1)T}\mathrm{\ΠH\ξ}\left(\mathrm{kT}\right)$

And e (k+1)=Φ _e[(k+1) T, kT] e (k) obtains the time renewal value of hypercomplex number;

Wherein $I=\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right],$ ξ(t)＝[ξ ₀(t)?ξ ₁(t)?…?ξ _n-1(t)?ξ _n(t)] ^T

$\left.\begin{array}{c}{\mathrm{\ξ}}_0\left(t\right)=1\\ {\mathrm{\ξ}}_1\left(t\right)=1-t\\ {\mathrm{\ξ}}_2\left(t\right)=1-2t+0.5t^2\\ \·\\ \·\\ \·\\ (i+1){\mathrm{\ξ}}_{i+1}\left(t\right)=(1+2i-t){\mathrm{\ξ}}_i\left(t\right)-i{\mathrm{\ξ}}_{i-1}\left(t\right)\end{array}\right\}i=\mathrm{2,3},\·\·\·,n-1$

For the recursive form of Laguerre orthogonal polynomial, rolling, pitching, yaw rate p, q, the expansion of r is respectively

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

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

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

$\mathrm{\Π}=\frac{1}{2}\{\left[\begin{array}{cccc}0& -1& 0& 0\\ 1& 0& 0& 0\\ 0& 0& 0& 1\\ 0& 0& -1& 0\end{array}\right]\left[\begin{array}{ccccc}{p}_0& p_1& \·\·\·& p_{n-1}& p_n\end{array}\right]$

$+\left[\begin{array}{cccc}0& 0& -1& 0\\ 0& 0& 0& -1\\ 1& 0& 0& 0\\ 0& 1& 0& 0\end{array}\right]\left[\begin{array}{ccccc}{q}_0& q_1& \·\·\·& q_{n-1}& q_n\end{array}\right]+\left[\begin{array}{cccc}0& 0& 0& -1\\ 0& 0& 1& 0\\ 0& -1& 0& 0\\ 1& 0& 0& 0\end{array}\right]\left[\begin{array}{ccccc}{r}_0& r_1& \·\·\·& r_{n-1}& r_n\end{array}\right]\}$

${\mathrm{\Π}}_1=\frac{1}{2}\{{\left[\begin{array}{ccccc}{p}_0& p_1& \·\·\·& p_{n-1}& p_n\end{array}\right]}^T\left[\begin{array}{cccc}0& -1& 0& 0\\ 1& 0& 0& 0\\ 0& 0& 0& 1\\ 0& 0& -1& 0\end{array}\right]$

$+{\left[\begin{array}{ccccc}{q}_0& q_1& \·\·\·& q_{n-1}& q_n\end{array}\right]}^T\left[\begin{array}{cccc}0& 0& -1& 0\\ 0& 0& 0& -1\\ 1& 0& 0& 0\\ 0& 1& 0& 0\end{array}\right]+{\left[\begin{array}{ccccc}{r}_0& r_1& \·\·\·& r_{n-1}& r_n\end{array}\right]}^T\left[\begin{array}{cccc}0& 0& 0& -1\\ 0& 0& 1& 0\\ 0& -1& 0& 0\\ 1& 0& 0& 0\end{array}\right]\}$