CN102519467A

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

Info

Publication number: CN102519467A
Application number: CN2011103805672A
Authority: CN
Inventors: 史忠科
Original assignee: Northwestern Polytechnical University
Current assignee: Northwestern Polytechnical University
Priority date: 2011-11-25
Filing date: 2011-11-25
Publication date: 2012-06-27
Anticipated expiration: 2031-11-25
Also published as: CN102519467B

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

Approximate Output Method of Euler Angle Chebyshev Exponent Based on Angular Velocity

技术领域 technical field

本发明涉及一种飞行器机动飞行姿态确定方法，特别是涉及一种基于角速度的欧拉角切比雪夫指数近似输出方法。 The invention relates to a method for determining the maneuvering flight attitude of an aircraft, in particular to an approximate output method of Euler angle Chebyshev exponent based on angular velocity. the

背景技术 Background technique

惯性设备在运动体导航和控制中具有重要作用；刚体运动的加速度、角速度和姿态等通常都依赖于惯性设备输出，因此提高惯性设备的输出精度具有明确的实际意义；在惯性设备中，加速度采用加速度计、角速度采用角速率陀螺直接测量方式，刚体的姿态精度要求很高时如飞行试验等采用姿态陀螺测量，但在很多应用领域都有角速度等测量直接解算输出；主要原因是由于动态姿态传感器价格昂贵、体积大，导致很多飞行器采用角速率陀螺等解算三个欧拉角，使得姿态时间更新输出成为导航等核心内容，也使其成为影响惯导系统精度的主要因素之一，因此设计和采用合理的姿态时间更新输出方法就成为研究的热点课题；从公开发表的文献中对姿态输出主要基于角速度采用欧拉方程直接近似法或采用近似龙格库塔方法解算(孙丽、秦永元，捷联惯导系统姿态算法比较，中国惯性技术学报，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)；由于欧拉方程中三个欧拉角互相耦合，属于非线性微分方程，在不同初始条件和不同飞行状态下的误差范围不同，难以保证实际工程要求的精度。 Inertial equipment plays an important role in the navigation and control of moving bodies; the acceleration, angular velocity and attitude of rigid body motion usually depend on the output of inertial equipment, so improving the output accuracy of inertial equipment has clear practical significance; in inertial equipment, the acceleration adopts The accelerometer and angular velocity are directly measured by the angular rate gyro. When the attitude accuracy of the rigid body is very high, such as the flight test, the attitude gyro is used for measurement. However, in many application fields, the angular velocity and other measurements are directly calculated and output; the main reason is that the dynamic attitude The sensor is expensive and bulky, which causes many aircraft to use angular rate gyroscopes to solve the three Euler angles, making the attitude time update output become the core content of navigation, and also makes it one of the main factors affecting the accuracy of the inertial navigation system. Therefore, Designing and adopting a reasonable attitude time update output method has become a hot topic of research; from the published literature, the attitude output is mainly based on the angular velocity using the direct approximation method of the Euler equation or the approximate Runge-Kutta method (Sun Li, Qin Yongyuan, Comparison of attitude algorithms for strapdown inertial navigation systems, Chinese Journal of Inertial Technology, 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 the three Euler angles in the Euler equation are coupled to each other, it belongs to a nonlinear differential equation, and the error range is different under different initial conditions and different flight states, so it is difficult to Guarantee the accuracy required by actual engineering. the

发明内容 Contents of the invention

为了克服现有的飞行器机动飞行时欧拉角输出精度差的问题，本发明提供一种基于角速度的欧拉角切比雪夫指数近似输出方法。该方法通过引入多个参数并将滚转、俯仰、偏航角速度按照变动区间的切比雪夫正交多项式展开，通过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和惯性单元的输出精度。 In order to overcome the problem of poor output accuracy of Euler angles during maneuvering flight of the existing aircraft, the present invention provides an approximate output method of Chebyshev exponents of Euler angles based on angular velocity. This method introduces multiple parameters and expands the roll, pitch, and yaw angular velocities according to the Chebyshev orthogonal polynomials in the variation interval, and directly expresses the Euler angle by solving the pitch angle, roll angle, and yaw angle in sequence. The high-order approximation integration is carried out according to the formula, so that the solution of the Euler angles is approximated by superlinearity, which can ensure the time update iteration calculation accuracy and the output accuracy of the inertial unit to determine the Euler angles. the

本发明解决其技术问题所采用的技术方案是：一种基于角速度的欧拉角切比雪夫指数近似输出方法，其特点是包括以下步骤： The technical scheme that the present invention solves its technical problem is: a kind of Euler angle Chebyshev exponent approximate output method based on angular velocity, it is characterized in that comprising the following steps:

1、(a)根据欧拉方程： 1. (a) According to the Euler equation:

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；全文参数定义相同；这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的展开式分别为 In the formula:

They refer to roll, pitch, and yaw angles respectively; p, q, and r are roll, pitch, and yaw angular velocities respectively; the definitions of the parameters throughout the text are the same; the calculation of these three Euler angles is to solve the pitch angle, roll angle, and yaw angle in sequence. The steps of the flight angle are carried out; the expansion expressions of the roll, pitch, and yaw angular velocities p, q, and r are respectively

p(t)＝pξ，q(t)＝qξ，r(t)＝rξ p(t)=pξ, q(t)=qξ, r(t)=rξ

其中 in

p＝[p₀ p₁ L p_n-1 p_n] q＝[q₀ q₁ L q_n-1 q_n] p=[p ₀ p ₁ L p _n-1 p _n ] q=[q ₀ q ₁ L q _n-1 q _n ]

r＝[r₀ r₁ L r_n-1 r_n] ξ＝[ξ₀(t) ξ₁(t) L ξ_n-1(t) ξ_n(t)]^T 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

$\begin{matrix} {ξ ξ}_{00} ((t t)) = = 11 \\ {ξ ξ}_{11} ((t t)) = = 11 - - 22 t t / / b b \\ {ξ ξ}_{22} ((t t)) = = 88 {((t t / / b b))}^{22} - - 88 ((t t / / b b)) + + 11 \\ M m \\ {ξ ξ}_{i i + + 11} ((t t)) = = {22 ξ ξ}_{11} ((t t)) {ξ ξ}_{i i} ((t t)) - - {ξ ξ}_{i i - - 11} ((t t)) \end{matrix}\} i i = = 2,3 2,3,, L L,, n no - - 1,0 1,0 \leq \leq t t \leq \leq NT NT,, b b = = NT NT$

为切比雪夫正交多项式的递推形式，T为采样周期； is the recursive form of the Chebyshev orthogonal polynomial, T is the sampling period;

(b)俯仰角的时间更新求解式为： (b) The time update solution of the pitch angle is:

式中： In the formula:

${a a}_{11} = = {((qH wxya {ξ ξ | |}_{kT kT}^{((k k + + 11)) T T}))}^{22} + + {((rH rH {ξ ξ | |}_{kT kT}^{((k k + + 11)) T T}))}^{22} - - {((pH pH {ξ ξ | |}_{kT kT}^{((k k + + 11)) T T}))}^{22}$

${a a}_{22} = = p p {{Ω Ω &CircleTimes; &CircleTimes; [[Hξ Hξ ((t t)) {| |}_{kT kT}^{((k k + + 11)) T T}]]}} {H h}^{T T} {r r}^{T T} - - pH pH {ξ ξ | |}_{kT kT}^{((k k + + 11)) T T} {rHξ rHξ | |}_{kT kT}$

${a a}_{33} = = p p {{Ω Ω &CircleTimes; &CircleTimes; [[Hξ Hξ ((t t)) {| |}_{kT kT}^{((k k + + 11)) T T}]]}} {H h}^{T T} {q q}^{T T} - - pH pH {ξ ξ | |}_{kT kT}^{((k k + + 11)) T T} {qHξ qHξ | |}_{kT kT}$

$| | λ λ | | = = {{p p {{Ω Ω &CircleTimes; &CircleTimes; [[H h {ξ ξ ((t t)) | |}_{kT kT}^{((k k + + 11)) T T}]]}} {H h}^{T T} {p p}^{T T} - - pH pH {ξ ξ | |}_{kT kT}^{((k k + + 11)) T T} {pHξ pHξ | |}_{kT kT}$

$+ + q q {{Ω Ω &CircleTimes; &CircleTimes; [[Hξ Hξ {((t t)) | |}_{kT kT}^{((k k + + 11)) T T}]]}} {H h}^{T T} {q q}^{T T} - - qH wxya {ξ ξ | |}_{kT kT}^{((k k + + 11)) T T} {qHξ qHξ | |}_{kT kT}$

$+ + r r {{Ω Ω &CircleTimes; &CircleTimes; [[Hξ Hξ {((t t)) | |}_{kT kT}^{((k k + + 11)) T T}]]}} {H h}^{T T} {r r}^{T T} - - rH rH {ξ ξ | |}_{kT kT}^{((k k + + 11)) T T} {rHξ rHξ | |}_{kT kT} {}}}^{\frac{11}{22}}$

$H h = = [\begin{matrix} {H h}_{00} \\ {H h}_{11} \\ M m \\ {H h}_{n no} \end{matrix}] = = b b \cdot \cdot [\begin{matrix} \frac{11}{22} & - - \frac{11}{22} & 00 & L L & 00 & 00 & 00 \\ \frac{11}{88} & 00 & - - \frac{11}{88} & L L & 00 & 00 & 00 \\ - - \frac{11}{66} & \frac{11}{44} & 00 & L L & 00 & 00 & 00 \\ - - \frac{11}{1616} & 00 & \frac{11}{88} & L L & 00 & 00 & 00 \\ M m & M m & M m & O o & M m & M m & M m \\ \frac{- - 11}{22 ((m m - - 11)) ((m m - - 33))} & 00 & 00 & L L & \frac{11}{44 ((m m - - 33))} & 00 & \frac{- - 11}{44 ((m m - - 11))} \\ \frac{- - 11}{22 m m ((m m - - 22))} & 00 & 00 & L L & 00 & \frac{11}{44 ((m m - - 22))} & 00 \end{matrix}]$

当p，q，r的展开式最高次项n为奇数时，m＝4，6，K，n+1，高次项n为偶数时m＝5，7，K，n+1，Hi(i＝1，2，L，n)为H相应的行向量； When p, q, when the highest order item n of the expanded formula of r is an odd number, m=4,6, K, n+1, m=5,7, K, n+1 when the high order item n is an even number, Hi( i=1, 2, L, n) is the corresponding row vector of H;

$Ω Ω = = \frac{11}{22} [\begin{matrix} {H h}_{22} + + {H h}_{00} & {H h}_{33} + + {H h}_{11} & L L & {H h}_{n no + + 11} + + {H h}_{n no - - 11} \\ {H h}_{33} + + {H h}_{11} & {H h}_{44} + + {H h}_{00} & L L & {H h}_{n no + + 22} + + {H h}_{n no - - 22} \\ M m & M m & O o & M m \\ {H h}_{n no + + 11} + + {H h}_{n no - - 11} & {H h}_{n no + + 22} + + {H h}_{n no - - 22} & L L & {H h}_{22 n no} + + {H h}_{00} \end{matrix}]$

2、在已知俯仰角的情况下，滚转角的时间更新求解式为： 2. In the case of known pitch angle, the time update solution formula of roll angle is:

其中 in

${a a}_{44} = = {(({pHξ pHξ | |}_{kT kT}^{((k k + + 11)) T T}))}^{22} + + {(({rHξ rHξ | |}_{kT kT}^{((k k + + 11)) T T}))}^{22} - - {(({qHξ qHξ | |}_{kT kT}^{((k k + + 11)) T T}))}^{22}$

${a a}_{55} = = q q {{Ω Ω &CircleTimes; &CircleTimes; [[Hξ Hξ ((t t)) {| |}_{kT kT}^{((k k + + 11)) T T}]]}} {H h}^{T T} {p p}^{T T} - - qH wxya {ξ ξ | |}_{kT kT}^{((k k + + 11)) T T} {qHξ qHξ | |}_{kT kT}$

${a a}_{66} = = q q {{Ω Ω &CircleTimes; &CircleTimes; [[Hξ Hξ ((t t)) {| |}_{kT kT}^{((k k + + 11)) T T}]]}} {H h}^{T T} {r r}^{T T} - - qH wxya {ξ ξ | |}_{kT kT}^{((k k + + 11)) T T} {rHξ rHξ | |}_{kT kT}$

3、在俯仰角、滚转角已知情况下，偏航角的求解式为： 3. When the pitch angle and roll angle are known, the formula for solving the yaw angle is:

$ψ ψ ((t t)) = = ψ ψ ((kT kT)) + + {&Integral; &Integral;}_{kT kT}^{t t} [[{b b}_{11} ((t t)) + + {b b}_{22} ((t t))]] dt dt$

式中： In the formula:

本发明的有益效果是：由于引入多个参数并将滚转、俯仰、偏航角速度按照变动区间的勒让德正交多项式展开，通过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，从而保证了确定欧拉角的时间更新迭代计算精度和惯性单元的输出精度。 The beneficial effects of the present invention are: due to the introduction of multiple parameters and the roll, pitch, and yaw angular velocities are expanded according to the Legendre orthogonal polynomials in the variation interval, by solving the pitch angle, roll angle, and yaw angle in sequence, directly The expression of Euler angles is integrated by high-order approximation, so that the solution of Euler angles is approximated by super-linearity, thus ensuring the accuracy of the time update iteration calculation and the output accuracy of the inertial unit for determining the Euler angles. the

下面结合具体实施方式对本发明作详细说明。 The present invention will be described in detail below in combination with specific embodiments. the

具体实施方式 Detailed ways

1、(a)根据刚体姿态方程(欧拉方程)： 1. (a) According to the rigid body attitude equation (Euler equation):

式中：

p(t)＝pξ，q(t)＝qξ，r(t)＝rξ p(t)=pξ, q(t)=qξ, r(t)=rξ

其中 in

$\begin{matrix} {ξ ξ}_{00} ((t t)) = = 11 \\ {ξ ξ}_{11} ((t t)) = = 22 t t / / b b - - 11 \\ {ξ ξ}_{22} ((t t)) = = 66 {((t t / / b b))}^{22} - - 66 ((t t / / b b)) + + 11 \\ M m \\ ((i i + + 11)) {ξ ξ}_{i i + + 11} ((t t)) = = {((22 i i + + 11)) ξ ξ}_{11} ((t t)) {ξ ξ}_{i i} ((t t)) - - {iξ iξ}_{i i - - 11} ((t t)) \end{matrix}\} i i = = 2,3 2,3,, L L,, n no - - 11;; 00 \leq \leq t t \leq \leq NT NT;; b b = = NT NT$

为勒让德正交多项式的递推形式，T为采样周期； is the recursive form of the Legendre orthogonal polynomial, T is the sampling period;

式中： In the formula:

${a a}_{22} = = p p {&Integral; &Integral;}_{kT kT}^{((k k + + 11)) T T} [[ξ ξ ((t t)) {ξ ξ}^{T T} ((t t))]] dt dt {H h}^{T T} {r r}^{T T} - - pH pH {ξ ξ | |}_{kT kT}^{((k k + + 11)) T T} {rHξ rHξ | |}_{kT kT}$

${a a}_{33} = = p p {&Integral; &Integral;}_{kT kT}^{((k k + + 11)) T T} [[ξ ξ ((t t)) {ξ ξ}^{T T} ((t t))]] dt dt {H h}^{T T} {q q}^{T T} - - pH pH {ξ ξ | |}_{kT kT}^{((k k + + 11)) T T} {qHξ qHξ | |}_{kT kT}$

$| | λ λ | | = = {{p p {&Integral; &Integral;}_{kT kT}^{((k k + + 11)) T T} [[ξ ξ ((t t)) {ξ ξ}^{T T} ((t t))]] dt dt {H h}^{T T} {p p}^{T T} - - pH pH {ξ ξ | |}_{kT kT}^{((k k + + 11)) T T} {pHξ pHξ | |}_{kT kT}$

$+ + q q {&Integral; &Integral;}_{kT kT}^{((k k + + 11)) T T} [[ξ ξ ((t t)) {ξ ξ}^{T T} ((t t))]] dt dt {H h}^{T T} {q q}^{T T} - - qH wxya {ξ ξ | |}_{kT kT}^{((k k + + 11)) T T} {qHξ qHξ | |}_{kT kT}$

$+ + r r {&Integral; &Integral;}_{kT kT}^{((k k + + 11)) T T} [[ξ ξ ((t t)) {ξ ξ}^{T T} ((t t))]] dt dt {H h}^{T T} {r r}^{T T} - - rH rH {ξ ξ | |}_{kT kT}^{((k k + + 11)) T T} {rHξ rHξ | |}_{kT kT} {}}}^{\frac{11}{22}}$

$H h = = b b \cdot \cdot [\begin{matrix} \frac{11}{22} & \frac{11}{66} & 00 & 00 & L L & 00 & 00 & 00 \\ - - \frac{11}{22} & 00 & \frac{11}{1010} & 00 & L L & 00 & 00 & 00 \\ 00 & - - \frac{11}{66} & 00 & \frac{11}{1414} & L L & 00 & 00 & 00 \\ M m & M m & M m & M m & O o & M m & M m & M m \\ 00 & 00 & 00 & 00 & L L & \frac{- - 11}{22 ((22 m m - - 55))} & 00 & \frac{11}{22 ((22 m m - - 11))} \\ 00 & 00 & 00 & 00 & L L & 00 & \frac{- - 11}{22 ((22 m m - - 33))} & 00 \end{matrix}]$

当p，q，r的展开式最高次项n为奇数时，m＝2，4，K，n+1，高次项n为偶数时m＝3，5，K，n+1； When p, q, the highest order item n of the expanded formula of r is an odd number, m=2, 4, K, n+1, and when the high order item n is an even number, m=3, 5, K, n+1;

其中 in

${a a}_{55} = = q q {&Integral; &Integral;}_{kT kT}^{((k k + + 11)) T T} [[ξ ξ ((t t)) {ξ ξ}^{T T} ((t t))]] dt dt {H h}^{T T} {p p}^{T T} - - qH wxya {ξ ξ | |}_{kT kT}^{((k k + + 11)) T T} {qHξ qHξ | |}_{kT kT}$

${a a}_{66} = = q q {&Integral; &Integral;}_{kT kT}^{((k k + + 11)) T T} [[ξ ξ ((t t)) {ξ ξ}^{T T} ((t t))]] dt dt {H h}^{T T} {r r}^{T T} - - qH wxya {ξ ξ | |}_{kT kT}^{((k k + + 11)) T T} {rHξ rHξ | |}_{kT kT}$

式中： In the formula:

当对惯性设备直接输出滚转、俯仰、偏航角速度p，q，r采用三阶逼近描述时，所得结果也接近O(T3)，相比欧拉方程直接近似法或采用近似龙格库塔方法解算等方法的O(T2)精度要高。 When the inertial equipment directly outputs the roll, pitch, and yaw angular velocities p, q, and r using the third-order approximation description, the obtained result is also close to O(T3), compared with the direct approximation method of the Euler equation or the approximate Runge-Kutta The O(T2) precision of methods such as method solve should be high. the

Claims

1. a kind of Euler angle Chebyshev exponent approximate output method based on angular velocity, it is characterized in that comprising the following steps:

Step 1, (a) according to the Euler equation:

In the formula:

p(t)=pξ, q(t)=qξ, r(t)=rξ

in

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

is the recursive form of the Chebyshev orthogonal polynomial, T is the sampling period;

(b) The time update solution of the pitch angle is:

In the formula:

When p, q, and the highest order n of the expanded formula of r are odd numbers, m=4, 6, K, n+1, and when the higher order n is an even number, m=5, 7, K, n+1, H _i (i=1, 2, L, n) is the corresponding row vector of H;

Step 2. In the case of known pitch angle, the time update solution formula of roll angle is:

in

Step 3. When the pitch angle and roll angle are known, the formula for solving the yaw angle is:

In the formula: