CN102495831B - Quaternion Hermitian approximate output method based on angular velocities for aircraft during extreme flight - Google Patents

Abstract

The invention discloses a quaternion Hermitian approximate output method based on angular velocities for an aircraft during extreme flight, which is used for solving the technical problem of poor precision of quaternion outputted by existing inertial equipment when an aircraft is in extreme flight. The technical scheme includes that approximate prescription for a rolling angular velocity p, a pitching angular velocity q and a yawing angular velocity r is realized by the aid of a Hermitian orthogonal polynomial, a quaternion state transition matrix is directly obtained, and accordingly iterative computation precision of quaternion is guaranteed. The quaternion Hermitian approximate output method has the advantages that orders of the Hermitian orthogonal polynomial for the rolling angular velocity p, the pitching angular velocity q and the yawing angular velocity r can be determined according to the requirement of engineering precision, superlinear approximation for a quaternion state equation transition matrix phie[(k+1)T, kT] is realized, iterative computation precision of specified quaternion is guaranteed, and accordingly output precision of inertial equipment is improved when the aircraft is in extreme flight.

Description

Hypercomplex number Emmett approximation output method when 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 Emmett approximation output method when 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, be that pitching, rolling and crab angle are described with three Eulerian angle conventionally, conventionally all resolve rear output by pitching in Airborne Inertial equipment, rolling and yaw rate.When rigid body is in the time that 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 calculating attitude matrix with direction cosine method 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 and recursive algorithm etc. on this basis.While studying rotating vector in document, it is all the algorithm that is output as angle increment based on rate gyro.But in Practical Project, the output of some gyros is angle rate signals, as optical fibre gyro, dynamic tuned gyroscope etc.In the time that 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, there is obvious minimizing than direction cosine attitude matrix differential equation calculated amount, can meet the requirement to real-time in engineering practice.Its conventional computing method have the card of finishing approximatioss, second order, fourth-order Runge-Kutta method and three rank Taylor expansions etc. (Paul G.Savage.A Unified MathematicalFramework for Strapdown Algorithm Design[J] .Journal of guidance, control, anddynamics, 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.Adopt fourth-order Runge-Kutta method while solving quaternion differential equation, along with the continuous accumulation of integral error, there will be exceed ± 1 phenomenon of trigonometric function value, disperse thereby cause calculating.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 Emmett approximation output method while the invention provides a kind of aircraft extreme flight based on angular velocity, the method adopts Emmett orthogonal polynomial to rolling, pitching, yaw rate p, q, r carries out close approximation description, can ensure to determine the iterative computation precision of hypercomplex number, thus inertial equipment output hypercomplex number precision while improving aircraft extreme flight;

The technical scheme that the present invention solves its technical matters employing is that hypercomplex number Emmett approximation output method when 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)

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

Ф _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

$\left.\begin{array}{c}{\mathrm{\ξ}}_0\left(t\right)=1\\ {\mathrm{\ξ}}_1\left(t\right)=2t\\ {\mathrm{\ξ}}_2\left(t\right)=4t^2-2\\ {\mathrm{\ξ}}_3\left(t\right)=8t^3-12t\\ {\mathrm{\ξ}}_4\left(t\right)=16t^4-48t^2+12\\ {\mathrm{\ξ}}_5\left(t\right)=32t^5-160t^3+120t\\ {\mathrm{\ξ}}_6\left(t\right)=64t^6-480t^4+720t-120\\ .\\ .\\ .\\ {\mathrm{\ξ}}_{i+1}\left(t\right)=2t{\mathrm{\ξ}}_i\left(t\right)-2i{\mathrm{\ξ}}_{i-1}\left(t\right)\end{array}\right\}$ i＝2，3，…，n-1

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

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

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

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

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

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

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

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

h ₁₂＝0.5，h ₂₃＝0.25， $h_{i(i+1)}=\frac{h_{(i-1)i}}{1+2h_{(i-1)i}},$ …

h ₂₁＝0.5，h ₄₁＝-1.5， $h_{\left(2i\right)1}=\frac{4i{h}_{(2i-2)1}}{1+2h_{\left(2i\right)(2i+1)}},$ …

All the other h _ij=0.

The invention has the beneficial effects as follows: due to according to the requirement of engineering precision, determine rolling, pitching, yaw rate p, q, the order of r Emmett orthogonal polynomial, realizes hypercomplex number state equation transition matrix Ф _ethe ultralinear of [(k+1) T, kT] is approached, and has ensured the iterative computation precision of definite hypercomplex number, thus inertial equipment output accuracy while having improved aircraft extreme flight.

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)

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

Ф _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

Emmett orthogonal polynomial is

$\left.\begin{array}{c}{\mathrm{\ξ}}_0\left(t\right)=1\\ {\mathrm{\ξ}}_1\left(t\right)=2t\\ {\mathrm{\ξ}}_2\left(t\right)=4t^2-2\\ {\mathrm{\ξ}}_3\left(t\right)=8t^3-12t\\ {\mathrm{\ξ}}_4\left(t\right)=16t^4-48t^2+12\\ {\mathrm{\ξ}}_5\left(t\right)=32t^5-160t^3+120t\\ {\mathrm{\ξ}}_6\left(t\right)=64t^6-480t^4+720t-120\\ .\\ .\\ .\\ {\mathrm{\ξ}}_{i+1}\left(t\right)=2t{\mathrm{\ξ}}_i\left(t\right)-2i{\mathrm{\ξ}}_{i-1}\left(t\right)\end{array}\right\}$ i＝2，3，…，n-1

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

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

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

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

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

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

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

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

h ₁₂＝0.5，h ₂₃＝0.25， $h_{i(i+1)}=\frac{h_{(i-1)i}}{1+2h_{(i-1)i}},$ …

h ₂₁＝0.5，h ₄₁＝-1.5， $h_{\left(2i\right)1}=\frac{4i{h}_{(2i-2)1}}{1+2h_{\left(2i\right)(2i+1)}},$ …

All the other h _ij=0;

When inertial equipment is directly exported to rolling, pitching, yaw rate p, q, r adopts three rank to approach while description, and acquired results also approaches O (T ³), compare the O (T that finishes the methods such as card approaches ²) precision will height.

Claims (1)

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

Hypercomplex number continuous state equation is

$\stackrel{\·}{e}=A_{e}e,$

Discrete state equations is

e(k+1)＝Φ _e[(k+1)T，kT]e(k)，

Wherein e=[e ₁, e ₂, e ₃, e ₄] ^t

e ₁＝cos(φ/2)cos(θ/2)cos(ψ/2)+sin(φ/2)sin(θ/2)sin(ψ/2)，

e ₂＝sin(φ/2)cos(θ/2)cos(ψ/2)-cos(φ/2)sin(θ/2)sin(ψ/2)，

e ₃＝cos(φ/2)sin(θ/2)cos(ψ/2)+sin(φ/2)cos(θ/2)sin(ψ/2)，

e ₄＝cos(φ/2)cos(θ/2)cos(ψ/2)-sin(φ/2)sin(θ/2)cos(ψ/2)，

$A_e=\frac{1}{2}\left[\begin{array}{cccc}0& -p& -q& -r\\ p& 0& r& -q\\ q& -r& 0& p\\ r& q& -p& 0\end{array}\right]$

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 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 approximate output valve of hypercomplex number Emmett;

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，

$\left.\begin{array}{c}{\mathrm{\ξ}}_0\left(t\right)=1\\ {\mathrm{\ξ}}_1\left(t\right)=2t\\ {\mathrm{\ξ}}_2\left(t\right)={4t}^2-2\\ {\mathrm{\ξ}}_3\left(t\right)={8t}^3-12t\\ {\mathrm{\ξ}}_4\left(t\right)={16t}^4-{48t}^2+12\\ {\mathrm{\ξ}}_5\left(t\right)={32t}^5-{160t}^3+120t\\ {\mathrm{\ξ}}_6\left(t\right)={64t}^6-{480t}^4+720t-120\\ \·\\ \·\\ \·\\ {\mathrm{\ξ}}_{i+1}\left(t\right)={2\mathrm{t\ξ}}_i\left(t\right)-{2\mathrm{i\ξ}}_{i-1}\left(t\right)\end{array}\right\}i=\mathrm{2,3},\·\·\·,n-1$

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

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

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

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

$\begin{array}{c}\mathrm{\Π}=\frac{1}{2}\{\left[\begin{array}{cccc}0& -1& 0& 0\\ 1& 0& 0& 0\\ 0& 0& 0& 1\\ 0& 0& -1& 0\end{array}\right]\left[\begin{array}{ccccc}{p}_0& p_1& \·\·\·& p_{n-1}& p_n\end{array}\right]\\ +\left[\begin{array}{cccc}0& 0& -1& 0\\ 0& 0& 0& -1\\ 1& 0& 0& 0\\ 0& 1& 0& 0\end{array}\right]\left[\begin{array}{ccccc}{q}_0& q_1& \·\·\·& q_{n-1}& q_n\end{array}\right]+\left[\begin{array}{cccc}0& 0& 0& -1\\ 0& 0& 1& 0\\ 0& -1& 0& 0\\ 1& 0& 0& 0\end{array}\right]\left[\begin{array}{ccccc}{r}_0& r_1& \·\·\·& r_{n-1}& r_n\end{array}\right]\}\end{array}$

$\begin{array}{c}{\mathrm{\Π}}_1=\frac{1}{2}\{{\left[\begin{array}{ccccc}{p}_0& p_1& \·\·\·& p_{n-1}& p_n\end{array}\right]}^T\left[\begin{array}{cccc}0& -1& 0& 0\\ 1& 0& 0& 0\\ 0& 0& 0& 1\\ 0& 0& -1& 0\end{array}\right]\\ +{\left[\begin{array}{ccccc}{q}_0& q_1& \·\·\·& q_{n-1}& q_n\end{array}\right]}^T\left[\begin{array}{cccc}0& 0& -1& 0\\ 0& 0& 0& -1\\ 1& 0& 0& 0\\ 0& 1& 0& 0\end{array}\right]+{\left[\begin{array}{ccccc}{r}_0& r_1& \·\·\·& r_{n-1}& r_n\end{array}\right]}^T\left[\begin{array}{cccc}0& 0& 0& -1\\ 0& 0& 1& 0\\ 0& -1& 0& 0\\ 1& 0& 0& 0\end{array}\right]\}\end{array}$

$h_{12}=0.5,h_{23}=0.25,h_{i(i+1)}=\frac{h_{(i-1)i}}{1+{2h}_{(i-1)i}},\·\·\·$

$h_{21}=0.5,h_{41}=-1.5,h_{\left(2i\right)1}=\frac{{4\mathrm{ih}}_{(2i-2)1}}{1+{2h}_{\left(2i\right)(2i+1)}},\·\·\·$

All the other h _ij=0.