CN102343985B

CN102343985B - Satellite time optimal posture maneuvering method with reaction flywheel

Info

Publication number: CN102343985B
Application number: CN 201110191564
Authority: CN
Inventors: 周浩; 刘冠南; 陈万春
Original assignee: Beihang University
Current assignee: Beihang University
Priority date: 2011-07-08
Filing date: 2011-07-08
Publication date: 2013-07-24
Anticipated expiration: 2031-07-08
Also published as: CN102343985A

Abstract

The invention discloses a satellite time optimal attitude maneuver method with a reaction flywheel, which comprises the following steps: the first step is to establish a spacecraft attitude motion model considering the reaction wheel dynamics, and on this basis to establish a satellite time optimal attitude movement model. Attitude maneuvering model; the second step, obtaining the open-loop optimal control for the spacecraft attitude motion model considering the reaction wheel dynamics; the third step, obtaining a robust feedback controller to realize the spacecraft redirection attitude maneuvering; the present invention realizes aerospace The fastest attitude maneuver of the actuator, and the maneuver control has high precision and strong robustness, which can meet the torque saturation and momentum saturation constraints of the actuator.

Description

Time Optimal Attitude Maneuvering Method for Satellite with Reaction Flywheel

技术领域 technical field

本发明涉及一种航天器快速，高精度机动的控制方法，具体涉及一种带反作用飞轮的卫星时间最优姿态机动方法，属于航天器控制技术领域。The invention relates to a fast and high-precision maneuvering control method for a spacecraft, in particular to a time-optimized attitude maneuvering method for a satellite with a reaction flywheel, and belongs to the technical field of spacecraft control.

背景技术 Background technique

带反作用轮的航天器时间最短重定向机动问题就是要找到一种控制使得航天器在最短的时间内实现从某一稳定的姿态机动到另一稳定的姿态。一些最优控制律是通过采用直接方法或间接方法得到的。K.D.Bilimoria，and B.Wie，“Time-Optimal Three-AxisReorientation of a Rigid Spacecraft，”Journal of Guidance Control and Dynamics，Vol.16，No.3，1993，pp.446-452.公开了一个针对刚体航天器三轴重定向的bang-bang控制，并且证明了绕特征轴的旋转不是时间最优的。H.Shen，and P.Tsiotras，“Time-Optimal Control of Axisymmetric Rigid Spacecraft Using Two Controls，”Journal of Guidance Control and Dynamics，Vol.22，No.5，1999，pp.682-694.中，仅通过两个控制实现了轴对称刚体航天器的最优机动。L.C.Lai，C.C.Yang，and C.J.Wu，“Time-Optimal Maneuvering Control of a Rigid Spacecraft，”ACTA Astronautica，Vol.60，No.10，2007，pp.791-800.中，把时间最优机动控制问题转换成非线性规划问题，把控制参数作为优化设计变量，通过遗传算法求得了最优解。M.V.Levskii，“The Problemof the Time-Optimal Control of Spacecraft Reorientation，”Journal of AppliedMathematics and Mechanics，Vol.73，No.1，2009，pp.16-25.采用庞德里亚金极大值原理求解了考虑航天器角动量约束的最短时间旋转问题。A.Fleming，P.Sekhavat，and I.M.Ross，“Minimum-Time Reorientation of a Rigid Body，”Journal of GuidanceControl and Dynamics，Vol.33，No.1，2010，pp.160-170.采用了间接方法和伪谱法求得了考虑约束的最优重定向问题。S.Liu，and T.Singh，“Fuel/Time Optimal Control ofSpacecraft Maneuvers，”Journal of Guidance Control and Dynamics，Vol.20，No.2，1996，pp.394-397.开发了STO算法并解决了在三个独立有界的脉冲控制下的燃料最优和时间最优姿态机动问题。X.Bao，and J.L.Junkins，“New Results for Time-OptimalThree-Axis Reorientation of a Rigid Spacecraft”Journal of Guidance Control andDynamics，Vol.32，No.4，2009，pp.1071-1076.的研究显示航天器绕特征轴的机动在控制输入约束的前提下是时间最优的方法。R.G.Melton，“Boundary Points and Arcs inConstrained，Time-Optimal Satellite Reorientation Maneuvers，”AIAA/AASAstrodynamics Specialist Conference，2-5August 2010，Toronto，Ontario，Canada，pp.1-16中，求解卫星时间最优重定向机动问题时考虑边界点和边界弧作为约束。The shortest time redirection maneuver problem of spacecraft with reaction wheels is to find a control that makes the spacecraft maneuver from a stable attitude to another stable attitude in the shortest time. Some optimal control laws are obtained by using direct methods or indirect methods. K.D.Bilimoria, and B.Wie, "Time-Optimal Three-AxisReorientation of a Rigid Spacecraft," Journal of Guidance Control and Dynamics, Vol.16, No.3, 1993, pp.446-452. bang-bang control for three-axis reorientation of a device, and demonstrate that the rotation around the characteristic axis is not time-optimal. H.Shen, and P.Tsiotras, "Time-Optimal Control of Axisymmetric Rigid Spacecraft Using Two Controls," Journal of Guidance Control and Dynamics, Vol.22, No.5, 1999, pp.682-694., only through Two controls achieve optimal maneuvering of an axisymmetric rigid-body spacecraft. L.C.Lai, C.C.Yang, and C.J.Wu, "Time-Optimal Maneuvering Control of a Rigid Spacecraft," ACTA Astronautica, Vol.60, No.10, 2007, pp.791-800. It is converted into a nonlinear programming problem, and the control parameters are used as optimal design variables, and the optimal solution is obtained by genetic algorithm. M.V.Levskii, "The Problem of the Time-Optimal Control of Spacecraft Reorientation," Journal of AppliedMathematics and Mechanics, Vol.73, No.1, 2009, pp.16-25. Considering The shortest-time rotation problem for spacecraft angular momentum constraints. A. Fleming, P. Sekhavat, and I.M. Ross, "Minimum-Time Reorientation of a Rigid Body," Journal of Guidance Control and Dynamics, Vol.33, No.1, 2010, pp.160-170. Using indirect methods and The pseudospectral method solves the optimal reorientation problem considering constraints. S.Liu, and T.Singh, "Fuel/Time Optimal Control of Spacecraft Maneuvers," Journal of Guidance Control and Dynamics, Vol.20, No.2, 1996, pp.394-397. Developed the STO algorithm and solved the problem in Fuel-optimal and time-optimal attitude maneuvers under three independently bounded impulse controls. X.Bao, and J.L.Junkins, "New Results for Time-OptimalThree-Axis Reorientation of a Rigid Spacecraft" Journal of Guidance Control andDynamics, Vol.32, No.4, 2009, pp.1071-1076. Maneuvering around the characteristic axis is the time-optimal method subject to control input constraints. R.G.Melton, "Boundary Points and Arcs in Constrained, Time-Optimal Satellite Reorientation Maneuvers," AIAA/AASAstrodynamics Specialist Conference, 2-5August 2010, Toronto, Ontario, Canada, pp.1-16, Solving Satellite Time Optimal Reorientation Maneuvers The problem considers boundary points and boundary arcs as constraints.

在现有技术中的最优姿态重定向机动没有将执行机构的动力学考虑在卫星的姿态动力学中，精度有待提高。The optimal attitude redirection maneuver in the prior art does not consider the dynamics of the actuator in the attitude dynamics of the satellite, and the accuracy needs to be improved.

发明内容 Contents of the invention

本发明的目的是为了解决上述问题，提出一种带反作用飞轮的卫星时间最优姿态机动方法。The purpose of the present invention is in order to solve the above-mentioned problem, proposes a kind of satellite time optimal attitude maneuvering method with reaction flywheel.

本发明的带反作用飞轮的卫星时间最优姿态机动方法，包括以下几个步骤：The satellite time optimal attitude maneuvering method with reaction flywheel of the present invention comprises the following steps:

第一步、建立考虑反作用轮动力学的航天器姿态运动模型，在此基础上建立卫星时间最优姿态机动模型；The first step is to establish a spacecraft attitude motion model considering the reaction wheel dynamics, and on this basis to establish a satellite time-optimal attitude maneuver model;

第二步、针对考虑了反作用轮动力学的航天器姿态运动模型获取开环最优控制；The second step is to obtain the open-loop optimal control for the spacecraft attitude motion model considering the reaction wheel dynamics;

第三步、获取鲁棒反馈控制器，实现航天器重定向姿态机动；The third step is to obtain a robust feedback controller to realize the spacecraft redirection attitude maneuver;

本发明的优点在于：The advantages of the present invention are:

(1)实现航天器的最快姿态机动；(1) Realize the fastest attitude maneuver of the spacecraft;

(2)机动控制的精度高；(2) High precision of motor control;

(3)鲁棒性强；(3) Strong robustness;

(4)能满足执行机构的力矩饱和和动量饱和约束。(4) It can satisfy the torque saturation and momentum saturation constraints of the actuator.

附图说明 Description of drawings

图1是本发明的方法流程图；Fig. 1 is method flowchart of the present invention;

图2是本发明的有3个反作用轮的卫星布局图；Fig. 2 is the satellite layout diagram that has 3 reaction wheels of the present invention;

图3是本发明的姿态重定向机动；Fig. 3 is the posture redirection maneuver of the present invention;

图4是本发明的实施例中开环最优四元数曲线；Fig. 4 is an open-loop optimal quaternion curve in an embodiment of the present invention;

图5是本发明的实施例中开环最优角速度曲线；Fig. 5 is an open-loop optimal angular velocity curve in an embodiment of the present invention;

图6是本发明的实施例中反作用轮的开环最优角速度曲线；Fig. 6 is the open-loop optimal angular velocity curve of the reaction wheel in an embodiment of the present invention;

图7是本发明的实施例中开环最优控制力矩曲线；Fig. 7 is an open-loop optimal control torque curve in an embodiment of the present invention;

图8是本发明的实施例中在三种控制方案下的姿态重定向机动曲线。Fig. 8 is an attitude redirection maneuver curve under three control schemes in an embodiment of the present invention.

图中：In the picture:

具体实施方式 Detailed ways

下面将结合附图和实施例对本发明作进一步的详细说明。The present invention will be further described in detail with reference to the accompanying drawings and embodiments.

本发明是一种带反作用飞轮的卫星时间最优姿态机动方法，流程如图1所示，针对反作用轮安装在惯性轴上的航天器，从一种稳定的姿态机动到另一种稳定的姿态，包括以下几个步骤：The present invention is a satellite time optimal attitude maneuvering method with a reaction flywheel. The process flow is shown in FIG. 1, aiming at maneuvering from a stable attitude to another stable attitude for a spacecraft with a reaction wheel installed on an inertial axis , including the following steps:

1、建立考虑反作用轮动力学的航天器姿态运动模型；1. Establish a spacecraft attitude motion model considering the reaction wheel dynamics;

姿态运动模型包括姿态动力学模型和姿态运动学模型。The attitude motion model includes attitude dynamics model and attitude kinematics model.

如图2所示，为一个有三个反作用轮安装在惯性轴上的刚体卫星的布局，图中Oxbybzb为飞行器坐标系，O是飞行器的质心，反作用轮主要用于吸收周期扰动力矩，偶尔用于卫星姿态重定向机动。As shown in Figure 2, it is a layout of a rigid body satellite with three reaction wheels installed on the inertial axis. In the figure, Oxbybzb is the coordinate system of the aircraft, and O is the center of mass of the aircraft. The reaction wheels are mainly used to absorb periodic disturbance moments, and occasionally for Satellite attitude redirection maneuver.

姿态是通过四元数来描述的，用四元数描述的姿态运动学模型如下所示。Attitude is described by quaternion, and the attitude kinematics model described by quaternion is as follows.

$\overset{\cdot &Center Dot;}{q q} = = \frac{11}{22} Q Q ((ω ω)) \cdot &Center Dot; q q = = \frac{11}{22} Ξ ξ ((q q)) ω ω - - - - - - ((11))$

其中，q＝[q₁，q₂，q₃，q₄]^T是四元数向量，q₁，q₂，q₃，q₄分别为四元数的四个分量，ω＝[ω₁，ω₂，ω₃]^T是卫星的角速度向量，ω₁、ω₂、ω₃分别为卫星的角速度向量在飞行器坐标系三个轴上的分量，Q(ω)和Ξ(q)是如下矩阵：Among them, q=[q ₁ , q ₂ , q ₃ , q ₄ ] ^T is a quaternion vector, q ₁ , q ₂ , q ₃ , and q ₄ are the four components of the quaternion respectively, ω=[ω ₁ _. _{_} ^_ _{_} _{_} _{_} matrix:

$\begin{matrix} Q Q ((ω ω)) = = [\begin{matrix} - - {ω ω}^{\times \times} & ω ω \\ - - {ω ω}^{T T} & 00 \end{matrix}] & Ξ ξ ((q q)) = = [\begin{matrix} {q q}_{44} {I I}_{33 \times \times 33} + + {q q}_{1313}^{\times \times} \\ - - {q q}_{1313}^{T T} \end{matrix}] \end{matrix} - - - - - - ((22))$

其中，I_3×3表示3×3的单位矩阵，ω^×和是反对称矩阵，如下：where, I _3×3 represents the 3×3 identity matrix, ω ^× and is an antisymmetric matrix, as follows:

$\begin{matrix} {ω ω}^{\times \times} = = [\begin{matrix} 00 & - - {ω ω}_{33} & {ω ω}_{22} \\ {ω ω}_{33} & 00 & - - {ω ω}_{11} \\ - - {ω ω}_{22} & {ω ω}_{11} & 00 \end{matrix}] \end{matrix} {q q}_{1313}^{\times \times} = = [\begin{matrix} 00 & - - {q q}_{33} & {q q}_{22} \\ {q q}_{33} & 00 & - - {q q}_{11} \\ - - {q q}_{22} & {q q}_{11} & 00 \end{matrix}] - - - - - - ((33))$

考虑反作用轮动力学的刚体卫星的姿态动力学模型如下：The attitude dynamics model of the rigid body satellite considering the reaction wheel dynamics is as follows:

$\overset{\cdot \cdot}{ω ω} = = {I I}_{s the s}^{- - 11} ((- - {ω ω}^{\times \times} {I I}_{s the s} ω ω - - {ω ω}^{\times \times} {I I}_{RW RW} Ω Ω - - {T T}_{u u} + + {T T}_{ex ex})) - - - - - - ((44))$

其中，I_s和I_RW分别是卫星和反作用轮的惯性矩矩阵，Ω反作用轮的角速度向量，Ω＝[Ω₁，Ω₂，Ω₃]^T，Ω₁，Ω₂，Ω₃分别为安装在Oxb，Oyb和Ozb轴上的反作用轮的角速度，T_u是反作用轮的力矩向量，T_u＝[T_u1 T_u2 T_u3]^T，T_u1 T_u2 T_u3分别表示安装在Oxb，Oyb和Ozb轴上的反作用轮产生的力矩，T_ex是环境扰动力矩，不考虑此项。忽略扰动力矩的姿态动力学模型(4)为：Among them, I _s and I _RW are the moment of inertia matrix of the satellite and the reaction wheel respectively, the angular velocity vector of the Ω reaction wheel, Ω=[Ω ₁ , Ω ₂ , Ω ₃ ] ^T , Ω ₁ , Ω ₂ , Ω ₃ are the installation The angular velocity of the reaction wheel on the Oxb, Oyb and Ozb axes, T _u is the moment vector of the reaction wheel, T _u = [T _u1 T _u2 T _u3 ] ^T , T _u1 T _u2 T _u3 respectively represent the The torque generated by the reaction wheel on the Ozb axis, T _ex is the environmental disturbance torque, which is not considered. The attitude dynamics model (4) ignoring the disturbance moment is:

$\overset{\cdot &Center Dot;}{ω ω} = = {I I}_{s the s}^{- - 11} ((- - {ω ω}^{\times \times} {I I}_{s the s} ω ω - - {ω ω}^{\times \times} {I I}_{RW RW} Ω Ω - - {T T}_{u u})) - - - - - - ((55))$

反作用轮的姿态动力学模型如下：The attitude dynamic model of the reaction wheel is as follows:

$\overset{\cdot \cdot}{Ω Ω} = = {I I}_{RW RW}^{- - 11} {T T}_{u u} - - - - - - ((66))$

将考虑反作用轮的刚体卫星的姿态运动学、动力学模型(1)(5)(6)进行整合。相应的状态变量和控制变量描述如下：The attitude kinematics and dynamics models (1)(5)(6) of the rigid body satellite considering the reaction wheels are integrated. The corresponding state variables and control variables are described as follows:

x＝[q₁ q₂ q₃ q₄ ω₁ ω₂ ω₃ Ω₁ Ω₂ Ω₃]^T，u＝T_u＝[T_u1 T_u2 T_u3]^T (7)x＝[q ₁ q ₂ q ₃ q ₄ ω ₁ ω ₂ ω ₃ Ω ₁ Ω ₂ Ω ₃ ] ^T , u＝T _u ＝[T _u1 T _u2 T _u3 ] ^T (7)

其中，x是状态变量，u是控制变量。状态变量包括卫星的姿态四元数和角速度以及反作用轮的角速度。控制变量为反作用轮的力矩。姿态运动学、动力学模型(1)，(5)和(6)可以描述成统一的形式即考虑反作用轮动力学的航天器姿态运动模型：Among them, x is the state variable and u is the control variable. The state variables include the attitude quaternion and angular velocity of the satellite and the angular velocity of the reaction wheels. The control variable is the torque of the reaction wheel. Attitude kinematics and dynamics models (1), (5) and (6) can be described in a unified form, that is, the spacecraft attitude motion model considering the reaction wheel dynamics:

$\overset{\cdot &Center Dot;}{x x} = = f f ((x x,, u u)) - - - - - - ((88))$

表示x的导数。

Indicates the derivative of x.

2、建立卫星时间最优姿态机动模型；2. Establish the satellite time optimal attitude maneuver model;

姿态重定向是指将卫星从一个稳定的姿态机动到另一个稳定的姿态。在图3中，姿态角是通过飞行器坐标系Oxbybzb和LVLH坐标系Oxyz的相对旋转定义的，图中A，B，C，D表示卫星飞行过程中经历的四种不同的姿态。最少时间重定向问题就是要设计一组控制力矩以最少的时间实现从两个不同的稳定姿态之间的机动。在最优姿态重定向机动过程中，执行机构的能力必须考虑。在此工作中，反作用轮是控制力矩的提供者，执行机构动力学以及反作用轮的力矩和动量饱和均考虑在卫星最优姿态重定向机动模型中。本发明的控制力矩能够保证在满足最大控制力矩和最大动量约束的前提下实现最短时间的姿态机动。Attitude redirection is the maneuvering of a satellite from one stable attitude to another. In Figure 3, the attitude angle is defined by the relative rotation of the aircraft coordinate system Oxbybzb and the LVLH coordinate system Oxyz, and A, B, C, D in the figure represent four different attitudes experienced by the satellite during flight. The least time reorientation problem is to design a set of control torques to achieve maneuvering between two different stable attitudes in the least amount of time. During optimal attitude redirection maneuvers, the capabilities of the actuators must be considered. In this work, the reaction wheel is the provider of the control torque, and the actuator dynamics as well as the moment and momentum saturation of the reaction wheel are considered in the satellite optimal attitude redirection maneuver model. The control torque of the present invention can ensure the shortest time attitude maneuver under the premise of satisfying the maximum control torque and the maximum momentum constraints.

卫星姿态重定向机动模型的初始稳定状态如下：The initial stable state of the satellite attitude redirection maneuver model is as follows:

$\begin{matrix} {q q}_{11} (({t t}_{00})) = = {q q}_{{11 t t}_{00}} & {q q}_{22} (({t t}_{00})) = = {q q}_{{22 t t}_{00}} & {q q}_{33} (({t t}_{00})) = = {q q}_{{33 t t}_{00}} & {q q}_{44} (({t t}_{00})) = = {q q}_{{44 t t}_{00}} \end{matrix}$

$\begin{matrix} {ω ω}_{11} (({t t}_{00})) = = {ω ω}_{{11 t t}_{00}} & {ω ω}_{22} (({t t}_{00})) = = {ω ω}_{{22 t t}_{00}} & {ω ω}_{33} (({t t}_{00})) = = {ω ω}_{{33 t t}_{00}} & - - - - - - ((99)) \end{matrix}$

$\begin{matrix} {Ω Ω}_{11} (({t t}_{00})) = = {Ω Ω}_{{11 t t}_{00}} & {Ω Ω}_{22} (({t t}_{00})) = = {Ω Ω}_{{22 t t}_{00}} & {Ω Ω}_{33} (({t t}_{00})) = = {Ω Ω}_{{33 t t}_{00}} \end{matrix}$

其中：in:

均为常数值；

are constant values;

q₁(t₀)、q₂(t₀)、q₃(t₀)、q₄(t₀)为初始时刻姿态四元数的值；ω₁(t₀)、ω₂(t₀)、ω₃(t₀)为卫星转动角速度三个分量在初始时刻的值；Ω₁(t₀)、Ω₂(t₀)、Ω₃(t₀)为三个反作用轮在初始时刻的转动角速度的值。q ₁ (t ₀ ), q ₂ (t ₀ ), q ₃ (t ₀ ), q ₄ (t ₀ ) are the values of the attitude quaternion at the initial moment; ω ₁ (t ₀ ), ω ₂ (t ₀ ) , ω ₃ (t ₀ ) are the values of the three components of satellite rotation angular velocity at the initial moment; Ω ₁ (t ₀ ), Ω ₂ (t ₀ ), Ω ₃ (t ₀ ) are the rotations of the three reaction wheels at the initial moment The value of the angular velocity.

卫星姿态重定向机动模型的终端时刻的稳定状态如下：The steady state at the terminal moment of the satellite attitude redirection maneuver model is as follows:

$\begin{matrix} {q q}_{11} (({t t}_{f f})) = = {q q}_{{11 t t}_{f f}} & {q q}_{22} (({t t}_{f f})) = = {q q}_{{22 t t}_{f f}} & {q q}_{33} (({t t}_{f f})) = = {q q}_{{33 t t}_{f f}} & {q q}_{44} (({t t}_{f f})) = = {q q}_{{44 t t}_{f f}} \end{matrix}$

$\begin{matrix} {ω ω}_{11} (({t t}_{f f})) = = {ω ω}_{{11 t t}_{f f}} & {ω ω}_{22} (({t t}_{f f})) = = {ω ω}_{{22 t t}_{f f}} & {ω ω}_{33} (({t t}_{f f})) = = {ω ω}_{{33 t t}_{f f}} \end{matrix} - - - - - - ((1010))$

$\begin{matrix} {Ω Ω}_{11} (({t t}_{f f})) = = {Ω Ω}_{{11 t t}_{f f}} & {Ω Ω}_{22} (({t t}_{f f})) = = {Ω Ω}_{{22 t t}_{f f}} & {Ω Ω}_{33} (({t t}_{f f})) = = {Ω Ω}_{{33 t t}_{f f}} \end{matrix}$

其中：in:

均为常数值；

are constant values;

q₁(t_f)、q₂(t_f)、q₃(t_f)、q₄(t_f)为终端时刻姿态四元数的值；ω₁(t_f)、ω₂(t_f)、ω₃(t_f)为卫星转动角速度三个分量在终端时刻的值；Ω₁(t_f)、Ω₂(t_f)、Ω₃(t_f)为三个反作用轮在终端时刻的转动角速度的值。q ₁ (t _f ), q ₂ (t _f ), q ₃ (t _f ), q ₄ (t _f ) are the values of the attitude quaternion at the terminal moment; ω ₁ (t _f ), ω ₂ (t _f ) , ω ₃ (t _f ) are the values of the three components of satellite rotation angular velocity at the terminal moment; Ω ₁ (t _f ), Ω ₂ (t _f ), Ω ₃ (t _f ) are the rotations of the three reaction wheels at the terminal moment The value of the angular velocity.

在重定向问题中，卫星在初始时刻和终端时刻的角速度都为0，则In the reorientation problem, the angular velocity of the satellite at the initial moment and the terminal moment are both 0, then

${ω ω}_{11 {t t}_{00}} = = {ω ω}_{22 {t t}_{00}} = = {ω ω}_{33 {t t}_{00}} = = {ω ω}_{11 {t t}_{f f}} = = {ω ω}_{22 {t t}_{f f}} = = {ω ω}_{33 {t t}_{f f}} = = 00 . .$

状态方程考虑了执行机构的动力学。反作用轮的最大力矩和最大动量分别是控制约束和状态约束。反作用轮的角动量的限制可以转换成角速度的限制，如下：The equation of state takes into account the dynamics of the actuator. The maximum moment and maximum momentum of the reaction wheel are control constraints and state constraints, respectively. The limitation of angular momentum of the reaction wheel can be converted into a limitation of angular velocity as follows:

$\begin{matrix} | | {Ω Ω}_{11} | | \leq \leq \overset{&OverBar; &OverBar;}{Ω Ω} & | | {Ω Ω}_{22} | | \leq \leq \overset{&OverBar; &OverBar;}{Ω Ω} & | | {Ω Ω}_{33} | | \leq \leq \overset{&OverBar; &OverBar;}{Ω Ω} \end{matrix} - - - - - - ((1111))$

在此，反作用轮的最大角速度。最大控制力矩约束如下：here, The maximum angular velocity of the reaction wheel. The maximum control torque constraints are as follows:

$\begin{matrix} | | {T T}_{u u 11} | | \leq \leq {\overset{&OverBar; &OverBar;}{T T}}_{u u} & | | {T T}_{u u 22} | | \leq \leq {\overset{&OverBar; &OverBar;}{T T}}_{u u} & | | {T T}_{u u 33} | | \leq \leq {\overset{&OverBar; &OverBar;}{T T}}_{u u} \end{matrix} - - - - - - ((1212))$

在此，

反作用轮的最大力矩。卫星角速度也需要满足一定的约束，描述如下：here,

The maximum torque of the reaction wheel. The angular velocity of the satellite also needs to meet certain constraints, which are described as follows:

$\begin{matrix} | | {ω ω}_{11} | | \leq \leq \overset{&OverBar; &OverBar;}{ω ω} & | | {ω ω}_{22} | | \leq \leq \overset{&OverBar; &OverBar;}{ω ω} & | | {ω ω}_{33} | | \leq \leq \overset{&OverBar; &OverBar;}{ω ω} \end{matrix} - - - - - - ((1313))$

是最大角速度。根据四元数的定义，四元数分量必须满足一下的条件。

is the maximum angular velocity. According to the definition of quaternion, the quaternion components must meet the following conditions.

${q q}_{11}^{22} + + {q q}_{22}^{22} + + {q q}_{33}^{22} + + {q q}_{44}^{22} = = 11 - - - - - - ((1414))$

显然，|q_i|≤1(i＝1，2，3，4)。式(8)在优化问题中也是一项等式约束。为了得到时间最短重定向机动问题的最优控制，给出时间最短问题的最优性能指标如下：Obviously, |q _i |≤1 (i=1, 2, 3, 4). Equation (8) is also an equality constraint in optimization problems. In order to obtain the optimal control of the shortest time redirection maneuver problem, the optimal performance index of the shortest time problem is given as follows:

$\underset{u u}{min min} J J = = \underset{u u}{min min} (({t t}_{f f} - - {t t}_{00})) - - - - - - ((1515))$

t₀是初始时间，t_f是终端时间。t ₀ is the initial time and t _f is the terminal time.

方程(8)-(15)就构成了卫星时间最优姿态机动模型。Equations (8)-(15) constitute the satellite time optimal attitude maneuver model.

(1)对航天器姿态运动模型进行归一化处理(1) Normalize the attitude motion model of the spacecraft

状态变量和控制变量归一化如下：The state variables and control variables are normalized as follows:

$\begin{matrix} \overset{~ ~}{ω ω} = = \frac{ω ω}{\overset{&OverBar; &OverBar;}{ω ω}} & \overset{~ ~}{Ω Ω} = = \frac{Ω Ω}{\overset{&OverBar; &OverBar;}{Ω Ω}} & {\overset{~ ~}{T T}}_{u u} = = \frac{{T T}_{u u}}{{\overset{&OverBar; &OverBar;}{T T}}_{u u}} \end{matrix} - - - - - - ((1616))$

或者or

$\begin{matrix} ω ω = = \overset{&OverBar; &OverBar;}{ω ω} \overset{~ ~}{ω ω} & Ω Ω = = \overset{&OverBar; &OverBar;}{Ω Ω} \overset{~ ~}{Ω Ω} & {T T}_{u u} = = {\overset{&OverBar; &OverBar;}{T T}}_{u u} {\overset{~ ~}{T T}}_{u u} \end{matrix} - - - - - - ((1717))$

用归一化变量描述的航天器姿态运动模型描述如下：The spacecraft attitude motion model described by normalized variables is described as follows:

$\{\begin{matrix} \overset{\cdot \cdot}{q q} = = \frac{11}{22} \overset{&OverBar; &OverBar;}{ω ω} Q Q ((\overset{~ ~}{ω ω})) \cdot &Center Dot; q q \\ \overset{\overset{\cdot &Center Dot;}{~ ~}}{ω ω} = = {I I}_{s the s}^{- - 11} ((- - \overset{&OverBar; &OverBar;}{ω ω} {\overset{~ ~}{ω ω}}^{\times \times} {I I}_{s the s} \overset{~ ~}{ω ω} - - \overset{&OverBar; &OverBar;}{Ω Ω} {\overset{~ ~}{ω ω}}^{\times \times} {I I}_{RW RW} \overset{~ ~}{Ω Ω} - - \frac{{\overset{&OverBar; &OverBar;}{T T}}_{u u}}{\overset{&OverBar; &OverBar;}{ω ω}} {\overset{~ ~}{\overset{&OverBar; &OverBar;}{T T}}}_{u u})) \\ \overset{\overset{\cdot &Center Dot;}{~ ~}}{Ω Ω} = = \frac{{\overset{&OverBar; &OverBar;}{T T}}_{u u}}{\overset{&OverBar; &OverBar;}{Ω Ω}} {I I}_{RW RW}^{- - 11} {\overset{~ ~}{\overset{&OverBar; &OverBar;}{T T}}}_{u u} \end{matrix} - - - - - - ((1818))$

四元数的最大数值为1，同时所有四元数的分量都位于区间[-1，1]。The maximum value of a quaternion is 1, and all components of a quaternion are in the interval [-1, 1].

(2)用归一化参数描述的最优控制问题(2) Optimal control problem described by normalized parameters

对应于最优控制问题，归一化变量描述的动力学方程(18)如下所示。Corresponding to the optimal control problem, the dynamic equation (18) described by the normalized variables is shown below.

$\overset{\overset{\cdot \cdot}{~ ~}}{x x} = = f f ((\overset{~ ~}{x x},, \overset{~ ~}{u u})) - - - - - - ((1919))$

其中in

$\overset{~ ~}{x x} = = {[\begin{matrix} {q q}_{11} & {q q}_{22} & {q q}_{33} & {q q}_{44} & {\overset{~ ~}{ω ω}}_{11} & {\overset{~ ~}{ω ω}}_{22} & {\overset{~ ~}{ω ω}}_{33} & {\overset{~ ~}{Ω Ω}}_{11} & {\overset{~ ~}{Ω Ω}}_{22} & {\overset{~ ~}{Ω Ω}}_{33} \end{matrix}]}^{T T}$

(20)(20)

最优控制的性能指标描述如下：The performance index of optimal control is described as follows:

J＝t_f-t₀ (21)J＝t _f -t ₀ (21)

不等式约束如下：The inequality constraints are as follows:

$\begin{matrix} {\overset{~ ~}{x x}}_{min min} \leq \leq \overset{~ ~}{x x} \leq \leq {\overset{~ ~}{x x}}_{max max} & {\overset{~ ~}{u u}}_{min min} \leq \leq \overset{~ ~}{u u} \leq \leq {\overset{~ ~}{u u}}_{max max} \end{matrix} - - - - - - ((22 twenty two))$

其中， ${\tilde{x}}_{\max} = [\begin{matrix} 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \end{matrix}], {\tilde{x}}_{\min} = - {\tilde{x}}_{\max}, {\tilde{u}}_{\max} = [\begin{matrix} 1 & 1 & 1 \end{matrix}], {\tilde{u}}_{\min} = - {\tilde{u}}_{\max} .$ in, ${\tilde{x}}_{\max} = [\begin{matrix} 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \end{matrix}], {\tilde{x}}_{\min} = - {\tilde{x}}_{\max}, {\tilde{u}}_{\max} = [\begin{matrix} 1 & 1 & 1 \end{matrix}], {\tilde{u}}_{\min} = - {\tilde{u}}_{\max} .$

等式约束见方程(14)。优化的目标是要找到一个归一化的控制

使得卫星姿态重定向的时间最短。See equation (14) for equality constraints. The goal of optimization is to find a normalized control

Make the satellite attitude reorientation time the shortest.

(3)采用勒让德伪谱法将最优控制问题转化成非线性规划问题(3) Transform the optimal control problem into a nonlinear programming problem by using the Legendre pseudospectral method

采用勒让德伪谱法离散前面描述的最优控制问题，此方法是建立在用拉格朗日插值多项式估计状态变量和控制变量的基础上。未知的系数是插值节点变量的值，叫做求积点，或叫勒让德高斯兰伯特(LGL)点。因为LGL点位于区间[-1，1]，前面描述的最优控制问题描述在时间区间[t₀，t_f]，所以我们采用如下形式对LGL区间和物理时间区间进行转换：τ∈[τ₀，τ_N]＝[-1，1]Using the Legendre pseudospectral method to discretize the optimal control problem described above, this method is based on estimating the state variables and control variables with Lagrangian interpolation polynomials. The unknown coefficients are the values of the interpolation node variables, called product points, or Legendre Gauss-Lambert (LGL) points. Because the LGL point is located in the interval [-1, 1], the optimal control problem described above is described in the time interval [t ₀ , t _f ], so we use the following form to convert the LGL interval and the physical time interval: τ∈[τ ₀ , τ _N ]=[-1, 1]

$t t = = \frac{(({t t}_{f f} - - {t t}_{00})) τ τ + + (({t t}_{f f} + + {t t}_{00}))}{22} - - - - - - ((23 twenty three))$

归一化的姿态运动模型如下：The normalized attitude motion model is as follows:

$\overset{\overset{\cdot &Center Dot;}{~ ~}}{x x} ((τ τ)) = = \frac{{t t}_{f f} - - {t t}_{00}}{22} f f ((\overset{~ ~}{x x} ((τ τ)),, \overset{~ ~}{u u ((τ τ))})) - - - - - - ((24 twenty four))$

$\begin{matrix} \overset{~ ~}{x x} ((- - 11)) = = {\overset{~ ~}{x x}}_{{t t}_{00}} & \overset{~ ~}{x x} ((11)) = = {\overset{~ ~}{x x}}_{{t t}_{f f}} \end{matrix}$

其中：

表示τ对应时刻的归一化的状态变量的导数、归一化的状态变量和控制变量的值，

表示初始时刻和终端时刻归一化的状态变量的值，均为常值。经过转换后，通过N阶多项式的形式来估计连续状态变量和控制变量，如下所示。in:

Indicates the derivative of the normalized state variable, the value of the normalized state variable and the control variable at the time corresponding to τ,

Represents the value of the state variable normalized at the initial time and terminal time, are constant values. After conversion, the continuous state variables and control variables are estimated in the form of an N-order polynomial, as shown below.

$\begin{matrix} \overset{~ ~}{x x} \approx \approx {\overset{~ ~}{x x}}^{N N} ((τ τ)) = = {Σ Σ}_{t t = = 00}^{N N} {\overset{~ ~}{x x}}_{l l} {φ φ}_{11} ((τ τ)) & \overset{~ ~}{u u} \approx \approx {\overset{~ ~}{u u}}^{N N} ((τ τ)) = = {Σ Σ}_{t t = = 00}^{N N} {\overset{~ ~}{u u}}_{l l} {φ φ}_{l l} ((τ τ)) \end{matrix} - - - - - - ((2525))$

其中，l＝0，1，…，N，N表示一选定的正整数，

表示N个点拟合的状态变量和控制变量在τ对应时刻的值。Wherein, l=0,1,...,N, N represents a selected positive integer,

Indicates the value of the state variable and control variable fitted by N points at the time corresponding to τ.

${φ φ}_{l l} ((τ τ)) = = \frac{11}{N N ((N N + + 11)) {L L}_{N N} (({τ τ}_{l l}))} \frac{(({τ τ}^{22} - - 11)) {\overset{\cdot \cdot}{L L}}_{N N} ((τ τ))}{τ τ - - {τ τ}_{l l}} = = \{\begin{matrix} 11 & if if & l l = = j j \\ 00 & if if & l l &NotEqual; &NotEqual; j j \end{matrix} - - - - - - ((2626))$

上式是N阶的拉格朗日插值多项式，L_N(τ_l)勒让德多项式。The above formula is a Lagrangian interpolation polynomial of order N, L _N (τ _l ) Legendre polynomial.

${\overset{~ ~}{x x}}_{l l} = = {\overset{~ ~}{x x}}^{N N} (({τ τ}_{l l})),, {\overset{~ ~}{u u}}_{l l} = = {\overset{~ ~}{u u}}^{N N} (({τ τ}_{l l})),, {τ τ}_{l l} = = τ τ (({t t}_{l l})) - - - - - - ((2727))$

其中：t_l表示第l个节点对应的时刻，τ_l表示第l个节点对应的τ的值。表示第l个节点对应的归一化的状态变量和控制变量的值。为了根据在节点τ_l的值

来表达状态变量的导数

相应的状态方程(19)可描述成以下形式：Where: t _l represents the moment corresponding to the lth node, and τ _l represents the value of τ corresponding to the lth node. Indicates the values of the normalized state variables and control variables corresponding to the lth node. In order according to the value at the node τ _l

to express the derivative of the state variable

The corresponding state equation (19) can be described as the following form:

${\overset{\overset{\cdot &Center Dot;}{~ ~}}{x x}}^{N N} (({τ τ}_{k k})) = = {Σ Σ}_{t t = = 00}^{N N} {D D.}_{kl kl} \overset{~ ~}{x x} (({τ τ}_{l l})) - - - - - - ((2828))$

其中D_kl是(N+1)×(N+1)的差分矩阵D的分量：where D _kl is the component of the difference matrix D of (N+1)×(N+1):

$D D. = = [[{D D.}_{kl kl}]] = = \{\begin{matrix} \frac{{L L}_{N N} (({τ τ}_{k k}))}{{L L}_{N N} (({τ τ}_{l l}))} \frac{11}{(({τ τ}_{k k} - - {τ τ}_{l l}))} & k k &NotEqual; &NotEqual; l l \\ - - \frac{N N ((N N + + 11))}{44} & k k = = l l = = 00 \\ \frac{N N ((N N + + 11))}{44} & k k = = l l = = N N \\ 00 & otherwise otherwise \end{matrix} - - - - - - ((2929))$

因而，在最优控制问题中的状态方程的等式约束可以通过离散状态描述为：Thus, the equality constraints of the state equation in the optimal control problem can be described by discrete states as:

${Σ Σ}_{l l = = 00}^{N N} {D D.}_{kl kl} \overset{~ ~}{x x} (({τ τ}_{l l})) - - \frac{{t t}_{f f} - - {t t}_{00}}{22} f f (({Σ Σ}_{l l = = 00}^{N N} \overset{~ ~}{x x} (({τ τ}_{l l})) {φ φ}_{l l} (({τ τ}_{k k})),, {Σ Σ}_{l l = = 00}^{N N} \overset{~ ~}{u u} (({τ τ}_{l l})) {φ φ}_{l l} (({τ τ}_{k k})))) = = 00 - - - - - - ((3030))$

其它的等式约束见方程(14)，不等式约束如下所示。See equation (14) for other equality constraints, and inequality constraints are shown below.

$\begin{matrix} {\overset{~ ~}{x x}}_{min min} \leq \leq \overset{~ ~}{x x} (({τ τ}_{l l})) \leq \leq {\overset{~ ~}{x x}}_{max max} & {\overset{~ ~}{u u}}_{min min} \leq \leq \overset{~ ~}{u u} (({τ τ}_{l l})) \leq \leq {\overset{~ ~}{u u}}_{max max} \end{matrix} - - - - - - ((3131))$

最优控制问题的性能指标见方程(21)，优化变量是和

这样最优控制问题就转化成了一个非线性规划问题。The performance index for the optimal control problem is given in equation (21), and the optimization variable is and

In this way, the optimal control problem is transformed into a nonlinear programming problem.

(4)优化计算求解非线性规划问题，得到开环最优姿态机动参数(4) Optimal calculation to solve the nonlinear programming problem, and obtain the optimal attitude maneuver parameters of the open loop

通过利用一些数值优化软件例如SNOPT或matlab就可以求解此优化问题了，从而得到了最优姿态。By using some numerical optimization software such as SNOPT or matlab, this optimization problem can be solved, and the optimal attitude can be obtained.

第三步，设计鲁棒反馈控制器，实现航天器重定向姿态机动；The third step is to design a robust feedback controller to realize the spacecraft redirection attitude maneuver;

上面所述步骤通过求解最优控制问题得到了最优姿态机动的开环控制。在实际应用中，由于存在不确定性和环境扰动，例如动力学建模的不确定性，空气阻力扰动力。所以需要一个具有较好鲁棒性的反馈控制律来对参考轨迹进行跟踪。根据姿态误差方程来推导鲁棒控制器。The above steps obtain the open-loop control of the optimal attitude maneuver by solving the optimal control problem. In practical applications, due to uncertainties and environmental disturbances, such as uncertainties in dynamic modeling, air resistance disturbs the force. Therefore, a robust feedback control law is needed to track the reference trajectory. A robust controller is derived from the attitude error equation.

1、建立姿态误差方程1. Establish attitude error equation

对刚体卫星来说，姿态动力学方程在方程(4)中已给出，姿态运动学方程在方程(1)中给出。而期望的姿态运动轨迹也通过解前面的开环最优控制问题得到。q_d和ω_d定义为期望姿态四元素和转动角速度，q_e为坐标系F_b相对期望坐标系F_d的姿态四元数。相应的从F_d到F_b的转换矩阵C(q_e)如下所示。For rigid satellites, the attitude dynamics equation is given in Equation (4), and the attitude kinematics equation is given in Equation (1). And the desired attitude trajectory is also obtained by solving the previous open-loop optimal control problem. q _d and ω _d are defined as the four elements of the desired attitude and the rotational angular velocity, and q _e is the attitude quaternion of the coordinate system F _b relative to the desired coordinate system F _d . The corresponding transformation matrix C(q _e ) from F _d to F _b is shown below.

$C C = = (({q q}_{e e 44}^{22} - - {q q}_{e e 1313}^{T T} {q q}_{e e 1313})) {I I}_{33 \times \times 33} + + 22 {q q}_{e e 1313} {q q}_{e e 1313}^{T T} - - {22 q q}_{e e 44} {q q}_{e e 1313}^{\times \times} - - - - - - ((3232))$

其中：q_e1、q_e2、q_e3、q_e4为四元数q_e的四个分量，向量q_e13＝[q_e1、q_e2、q_e3]^T，

为q_e13的转置向量，

是q_e13的反对称矩阵，q与q_d，q_e的关系描述如下：Among them: q _e1 , q _e2 , q _e3 , q _e4 are the four components of quaternion q _e , vector q _e13 =[q _e1 , q _e2 , q _e3 ] ^T ,

is the transpose vector of q _e13 ,

is the antisymmetric matrix of q _e13 , the relationship between q and q _d , q _e is described as follows:

q＝mat(q_d)q_e (33)q＝mat(q _d )q _e (33)

其中in

$mat mat (({q q}_{d d})) = = [\begin{matrix} {q q}_{d d 44} & {q q}_{d d 33} & - - {q q}_{d d 22} & {q q}_{d d 11} \\ - - {q q}_{d d 33} & {q q}_{d d 44} & - - {q q}_{d d 11} & {q q}_{d d 22} \\ {q q}_{d d 22} & {q q}_{d d 11} & {q q}_{d d 44} & {q q}_{d d 33} \\ - - {q q}_{d d 11} & - - {q q}_{d d 22} & - - {q q}_{d d 33} & {q q}_{d d 44} \end{matrix}] - - - - - - ((3434))$

其中：q_d1、q_d2、q_d3、q_d4为q_d的四个分量，F_b相对F_d的角速度ω_e在坐标系F_b中描述如下：Among them: q _d1 , q _d2 , q _d3 , and q _d4 are the four components of q _d , and the angular velocity ω _e of F _b relative to F _d is described in the coordinate system F _b as follows:

ω_e＝ω-Cω_d (35)ω _e = ω - Cω _d (35)

根据ω_e和q_e重写方程(4)和方程(1)，就得到了的误差方程。Rewrite equation (4) and equation (1) according to ω _e and q _e , and the error equation of is obtained.

${I I}_{s the s} {\overset{\cdot &Center Dot;}{ω ω}}_{e e} = = - - {(({ω ω}_{e e} + + {Cω Cω}_{d d}))}^{\times \times} {I I}_{s the s} (({ω ω}_{e e} + + {Cω Cω}_{d d})) - - {(({ω ω}_{e e} + + {Cω Cω}_{d d}))}^{\times \times} {I I}_{RW RW} Ω Ω + + {I I}_{s the s} (({ω ω}_{e e}^{\times \times} {Cω Cω}_{d d} - - {C C \overset{\cdot \cdot}{ω ω}}_{d d})) - - {T T}_{u u}$

(36)(36)

$\overset{\cdot &Center Dot;}{q q} = = \frac{11}{22} Ξ ξ (({q q}_{e e})) {ω ω}_{e e}$

在方程(36)中考虑了反作用轮的动力学影响。The dynamic influence of the reaction wheel is considered in equation (36).

2、获取鲁棒反馈控制器2. Obtain a robust feedback controller

我们选择李亚普洛夫函数如下：We choose the Lyapunov function as follows:

$V V = = \frac{11}{22} {ω ω}_{e e}^{T T} {I I}_{s the s} {ω ω}_{e e} + + {k k}_{11} (({q q}_{}^{e e 11} + + {q q}_{}^{e e 22} + + {q q}_{}^{e e 33} + + {((11 - - {q q}_{e e 44}))}^{22})) - - - - - - ((3737))$

其中k₁＞0。可以推导出速度V的导数：where k ₁ >0. The derivative of the velocity V can be derived:

$\overset{\cdot \cdot}{V V} = = {ω ω}_{e e}^{T T} {{- - {(({ω ω}_{e e} + + {Cω Cω}_{d d}))}^{\times \times} {I I}_{s the s} (({ω ω}_{e e} + + {Cω Cω}_{d d})) - - {(({ω ω}_{e e} + + {Cω Cω}_{d d}))}^{\times \times} {I I}_{RW RW} Ω Ω + + {I I}_{s the s} (({ω ω}_{e e}^{\times \times} {Cω Cω}_{d d} - - {C C \overset{\cdot &Center Dot;}{ω ω}}_{d d})) - - {T T}_{u u}}} + + {k k}_{11} {ω ω}_{e e}^{T T} {q q}_{e e 1313}^{T T}$

$= = {ω ω}_{e e}^{T T} {{- - {ω ω}_{e e}^{\times \times} (({I I}_{s the s} (({ω ω}_{e e} + + {Cω Cω}_{d d})) + + {I I}_{RW RW} Ω Ω)) - - (({((C C {ω ω}_{d d}))}^{\times \times} {I I}_{s the s} + + {I I}_{s the s} {(({Cω Cω}_{d d}))}^{\times \times})) {ω ω}_{e e} - - {(({Cω Cω}_{d d}))}^{\times \times} {I I}_{s the s} {Cω Cω}_{d d} - - {I I}_{s the s} {C C \overset{\cdot \cdot}{ω ω}}_{d d}$

(38)(38)

$- - {(({Cω Cω}_{d d}))}^{\times \times} {I I}_{RW RW} Ω Ω - - {T T}_{u u}}} + + {k k}_{11} {ω ω}_{e e}^{T T} {q q}_{e e 1313}^{T T}$

$= = {ω ω}_{e e}^{T T} ((- - {(({Cω Cω}_{d d}))}^{\times \times} {I I}_{s the s} {Cω Cω}_{d d} - - {I I}_{s the s} {C C \overset{\cdot \cdot}{ω ω}}_{d d} - - {(({Cω Cω}_{d d}))}^{\times \times} {I I}_{RW RW} Ω Ω - - {T T}_{u u})) + + {k k}_{11} {ω ω}_{e e}^{T T} {q q}_{e e 1313}^{T T}$

根据上式提出鲁棒反馈控制器为：According to the above formula, the robust feedback controller is proposed as:

$u u = = {T T}_{u u} {= = ((- - {Cω Cω}_{d d}))}^{\times \times} {I I}_{s the s} {Cω Cω}_{d d} - - {I I}_{s the s} {C C \overset{\cdot &Center Dot;}{ω ω}}_{d d} - - {(({Cω Cω}_{d d}))}^{\times \times} {I I}_{RW RW} Ω Ω + + {k k}_{11} {q q}_{e e 1313} + + {k k}_{22} {ω ω}_{e e} - - - - - - ((3939))$

其中k₂＞0。将上面的方程代入方程(38)，可得到where k ₂ >0. Substituting the above equation into equation (38), we can get

$\overset{\cdot &Center Dot;}{V V} = = - - {k k}_{22} {ω ω}_{e e}^{T T} {ω ω}_{e e} \leq \leq 00 - - - - - - ((4040))$

因而，系统在方程(39)所示的控制律下是稳定的。闭环跟踪的主要任务是抵消在开环最优轨迹中没有考虑的扰动力矩的影响。方程(39)的控制律中前三项是用于补偿动力学模型的误差，为一个前馈补偿，最后两项实质为一抵消不确定扰动的PD反馈项。Thus, the system is stable under the control law shown in equation (39). The main task of closed-loop tracking is to counteract the effects of disturbance moments that are not considered in the open-loop optimal trajectory. The first three items in the control law of equation (39) are used to compensate the error of the dynamic model, which is a feed-forward compensation, and the last two items are essentially a PD feedback item to offset the uncertain disturbance.

通过说书的控制器去跟踪前面优化求解得到的开环最优姿态，就可以实现快速、高精度的航天器重定向姿态机动。并且本发明的方法作姿态机动具有很强的鲁棒性。By using the storytelling controller to track the open-loop optimal attitude obtained by the previous optimization solution, fast and high-precision spacecraft redirection attitude maneuvering can be realized. And the method of the present invention has very strong robustness for attitude maneuvering.

实施例：Example:

实施例中，以拥有三个反作用轮的刚体卫星为例，三个反作用轮安装在卫星的惯性轴上。卫星动力学包含了反作用轮的动力学。刚体卫星的参数如下表1所示，初始和终端状态值见表2。In the embodiment, a rigid satellite with three reaction wheels is taken as an example, and the three reaction wheels are installed on the inertial axis of the satellite. Satellite dynamics include the dynamics of the reaction wheels. The parameters of the rigid body satellite are shown in Table 1 below, and the initial and terminal state values are shown in Table 2.

表1刚体卫星的相关参数Table 1 Relevant parameters of rigid body satellites

其中，I_xx、I_yy、I_zz分别为卫星在三个惯性主轴上的转动惯量，I_w为反作用轮的转动惯量。Among them, I _xx , I _yy , and I _zz are the moment of inertia of the satellite on the three inertial axes respectively, and I _w is the moment of inertia of the reaction wheel.

表2此算例的初始和终端条件Table 2 Initial and terminal conditions of this example

q(t₀)、ω(t₀)、Ω(t₀)、q(t_f)、ω(t_f)、Ω(t_f)分别表示在初始时刻和终端时刻姿态四元数、姿态角速度和反作用轮转动角速度的值。q(t ₀ ), ω(t ₀ ), Ω(t ₀ ), q(t _f ), ω(t _f ), Ω(t _f ) represent the attitude quaternion and attitude angular velocity at the initial and terminal moments, respectively and the value of the rotational angular velocity of the reaction wheel.

通过勒让德伪谱法和利用软件TOMLAB\SNOPT优化得到了开环最优控制。最短重定向机动时间为99s。相应的开环最优四元数曲线，卫星角速度曲线，反作用轮的角速度曲线和控制力矩曲线如图4-图7所示。The open-loop optimal control is obtained by Legendre pseudospectral method and software TOMLAB\SNOPT optimization. The shortest redirection maneuver time is 99s. The corresponding open-loop optimal quaternion curve, satellite angular velocity curve, angular velocity curve and control torque curve of the reaction wheel are shown in Fig. 4-Fig. 7.

在现有技术中，最优重定向问题是去发现在体轴上的三个虚拟的控制力矩，执行机构动力学没有考虑在动力学方程中。其动力学模型如下：In the prior art, the optimal reorientation problem is to find three virtual control moments on the body axis, and the actuator dynamics are not considered in the dynamic equations. Its dynamic model is as follows:

$\overset{\cdot &Center Dot;}{ω ω} = = {I I}_{s the s}^{- - 11} ((- - {ω ω}^{\times \times} {I I}_{s the s} ω ω + + {T T}_{u u})) - - - - - - ((4141))$

和方程(5)比较，公式中的-ω^×I_RWΩ在方程(41)中没有考虑。Compared with Equation (5), -ω ^× I _RW Ω in the formula is not considered in Equation (41).

为了比较不同方法的结果，以方程(48)作为动力学方程的开环最短时间机动问题也通过勒让德伪谱法和利用软件SNOPT优化求解。在其他参数与前面一致的情况下，其相应的仿真结果结果显示最短机动时间为88秒。机动时间比本发明机动时间更短，主要是因为模型中忽略了反作用轮的影响，并且没有考虑反作用轮动量饱和的限制，所以由于模型的不够准确，此结果并非最优，可作次优跟踪用。接着采用非线性预测控制方法(NPC)对此次优轨迹进行跟踪。在跟踪控制时考虑了力矩和动量约束，相应的参数如图8中间栏所示，姿态四元数达到稳定的时间为130秒，卫星角速度达到稳定的时间为145秒，三个轮子都正常工作，仅一个轮子在姿态旋转过程中力矩和动量都达到了饱和。In order to compare the results of different methods, the open-loop minimum time maneuver problem with equation (48) as the dynamic equation is also solved by the Legendre pseudospectral method and optimized using the software SNOPT. In the case that other parameters are consistent with the previous ones, the corresponding simulation results show that the shortest maneuvering time is 88 seconds. The maneuvering time is shorter than the maneuvering time of the present invention, mainly because the influence of the reaction wheel is ignored in the model, and the limitation of the momentum saturation of the reaction wheel is not considered, so due to the inaccuracy of the model, this result is not optimal and can be used for suboptimal tracking use. Then, nonlinear predictive control method (NPC) is used to track the optimal trajectory. The torque and momentum constraints are considered in the tracking control, and the corresponding parameters are shown in the middle column of Figure 8. The time for the attitude quaternion to stabilize is 130 seconds, and the time for the satellite angular velocity to stabilize is 145 seconds. All three wheels are working normally. , the torque and momentum of only one wheel are saturated during the attitude rotation process.

同样为了对比，采用四元数反馈(QFC)的方法来求解同样的姿态重定向问题。模型中考虑了反作用轮的力矩和角动量饱和约束。相应的结果如图8左栏所示，此方法为特征轴旋转机动，姿态四元数达到稳定状态的时间为132秒，卫星角速度达到稳定状态的时间为146秒，在姿态机动过程中，仅一个反作用轮工作，其它两个轮子处于静止状态，所以卫星近绕一根惯性轴旋转。Also for comparison, the quaternion feedback (QFC) method is used to solve the same attitude reorientation problem. The torque and angular momentum saturation constraints of the reaction wheel are considered in the model. The corresponding results are shown in the left column of Fig. 8. This method is a characteristic axis rotation maneuver. The time for the attitude quaternion to reach a stable state is 132 seconds, and the time for the satellite angular velocity to reach a stable state is 146 seconds. During the attitude maneuver process, only One reaction wheel works, and the other two wheels are at rest, so the satellite rotates around an inertial axis.

本发明提出的最短时间重定向机动的结果图8右栏所示，采用的跟踪控制律见公式(39)。在姿态机动过程中，三个反作用轮均工作，有一个力矩达到过饱和，还有一个动量达到过饱和。姿态四元数达到稳定状态的时间为99秒，角速度达到稳定状态的时间为100秒，所以姿态重定向机动时间为100秒，比四元数反馈方法缩短了31.5％，比次优化和非线性预测控制方案缩短了31％。相应的对比数据如表3所示。The result of the shortest time redirection maneuver proposed by the present invention is shown in the right column of Fig. 8, and the tracking control law adopted is shown in formula (39). During attitude maneuvering, all three reaction wheels are working, one moment reaches supersaturation, and the other momentum reaches supersaturation. The time for the attitude quaternion to reach a stable state is 99 seconds, and the time for the angular velocity to reach a stable state is 100 seconds, so the attitude redirection maneuver time is 100 seconds, which is 31.5% shorter than the quaternion feedback method, compared with sub-optimization and nonlinear The predictive control scenario was shortened by 31%. The corresponding comparative data are shown in Table 3.

表3三种方法的性能比较Table 3 Performance comparison of the three methods

Claims

1. a satellite time optimal attitude maneuvering method with reaction wheels, at the satellite that reaction wheels are installed on the inertial axis, maneuvers to another kind of stable attitude from a kind of stable attitude, is characterized in that, comprises following several steps:

The first step is to establish a satellite attitude motion model considering the reaction wheel dynamics, and on this basis to establish a satellite time-optimal attitude maneuver model;

(1) Establish a satellite attitude motion model considering the reaction wheel dynamics;

The satellite attitude motion model includes the satellite attitude dynamic model and the satellite attitude kinematics model;

The satellite attitude kinematics model is:

Among them, q=[q ₁ , q ₂ , q ₃ , q ₄ ] ^T is a quaternion vector, q ₁ , q ₂ , q ₃ , and q ₄ are the four components of the quaternion vector respectively, ω=[ω ₁ , ω ₂ , ω ₃ ] ^T is the satellite's angular velocity vector, ω ₁ , ω ₂ , ω ₃ are the components of the satellite's angular velocity vector on the three axes of the aircraft coordinate system, Q(ω) and Ξ(q) are :

where, q ₁₃ =[q ₁ ,q ₂ ,q ₃ ] ^T ,

is the transpose vector of q ₁₃ ,

is the antisymmetric matrix of q ₁₃ , I _3×3 represents the identity matrix of 3×3, and ω ^× is the antisymmetric matrix of ω, as follows:

The satellite attitude dynamic model is:

Among them, I _s and I _RW are the moment of inertia matrix of the satellite and the reaction wheel respectively, the angular velocity vector of the Ω reaction wheel, Ω=[Ω ₁ ,Ω ₂ ,Ω ₃ ] ^T , Ω ₁ ,Ω ₂ ,Ω ₃ are three The angular velocity of the reaction wheels installed on the Oxb, Oyb and Ozb shafts respectively, T _u is the moment vector of the reaction wheels, T _u = [T _u1 T _u2 T _u3 ] ^T , T _u1 T _u2 T _u3 means three The torque generated by the reaction wheels on the Oxb, Oyb and Ozb axes, T _ex is the environmental disturbance moment, which is not considered; the satellite attitude dynamic model (4) ignoring the disturbance moment is:

The attitude dynamic model of the reaction wheel is:

Through equations (1), (5), and (6), the satellite attitude motion model considering the dynamics of the reaction wheel is obtained as:

Among them: x is a state variable, u is a control variable; the state variable includes the attitude quaternion and angular velocity of the satellite and the angular velocity of the reaction wheel; the control variable is the torque of the reaction wheel; the state variable and the control variable are:

x＝[q ₁ q ₂ q ₃ q ₄ ω ₁ ω ₂ ω ₃ Ω ₁ Ω ₂ Ω ₃ ] ^T , u＝T _u ＝[T _u1 T _u2 T _u3 ] ^T (8)

(2) Establish the satellite time optimal attitude maneuver model;

The initial stable state of the satellite attitude redirection maneuver model is:

in,

is the value of the attitude quaternion at the initial moment;

are the values of the three components of satellite angular velocity at the initial moment;

is the value of the angular velocity of the three reaction wheels at the initial moment;

The steady state at the terminal moment of the satellite attitude redirection maneuver model is:

in,

is the value of the attitude quaternion at the terminal moment;

is the value of the three components of satellite angular velocity at the terminal moment;

is the value of the angular velocity of the three reaction wheels at the terminal moment;

The angular velocity of the satellite at the initial moment and the terminal moment are both 0, then

The limitation of the angular momentum of the reaction wheel is converted into a limitation of the angular velocity as follows:

in,

is the maximum angular velocity of the reaction wheel; the maximum control torque is constrained as follows:

in,

is the maximum moment of the reaction wheel; the constraints on the satellite angular velocity are as follows:

in, is the maximum angular velocity; the quaternion satisfies:

|q _i |≤1, i=1,2,3,4; equation (7) is also an equality constraint; the optimal performance index of the satellite time optimal attitude maneuver model is:

Among them: t ₀ is the initial time, t _f is the terminal time; Equations (7)-(15) constitute the satellite time optimal attitude maneuver model;

The second step is to obtain the open-loop optimal control for the satellite attitude motion model considering the reaction wheel dynamics;

(1) Normalize the satellite attitude motion model

The state variables and control variables are normalized as:

or

express

The anti-symmetric matrix of the satellite attitude motion model after normalization is:

The maximum value of a quaternion is 1, and all quaternions are in the interval [-1,1];

(2) Optimal control problems described by normalized parameters;

The satellite attitude motion model described by normalized variables is;

in

(20)

The performance index of optimal control is:

J = t _f -t ₀ (21)

The inequality constraints are:

in,

The equality constraints are

The goal of optimization is to find a normalized control

Make the satellite attitude redirection time the shortest;

(3) Transform the optimal control problem into a nonlinear programming problem by using the Legendre pseudospectral method

The optimal control problem is described in the time interval [t ₀ ,t _f ], and the Legendre Gauss-Lambert interval and the physical time interval are converted in the following form: τ∈[τ ₀ ,τ _N ]=[-1,1 ]

The normalized satellite attitude motion model is as follows:

(twenty four)

in:

Represents the derivative of the normalized state variable, the value of the normalized state variable and the control variable at the time corresponding to τ,

Represents the value of the state variable normalized at the initial time and terminal time,

Both are constant values. After conversion, the continuous state variables and control variables are estimated in the form of N-order polynomials, as shown below;

Wherein, l=0,1,...,N, N represents a selected positive integer, Indicates the value of the state variable and control variable fitted by N points at the time corresponding to τ,

The above formula is a Lagrange interpolation polynomial of order N, L _N (τ _l ) Legendre polynomial;

Among them, t _l represents the moment corresponding to the l-th node, τ _l represents the value of τ corresponding to the l-th node, Indicates the values of the normalized state variables and control variables corresponding to the lth node, in order to base on the value of the node τ _l to express the derivative of the state variable

The corresponding satellite attitude motion model formula (19) is:

Where: D _kl is the component of the difference matrix D of (N+1)×(N+1):

Thus, the equality constraints of the state equation in the optimal control problem are represented by discrete states:

The other equality constraints are

The inequality constraints are:

The performance index of the optimal control problem is formula (21), and the optimization variable is

and

The optimal control problem is transformed into a nonlinear programming problem;

(4) Optimize the calculation to solve the nonlinear programming problem, and obtain the optimal attitude maneuver parameters of the open loop;

The third step is to obtain a robust feedback controller to realize satellite redirection attitude maneuvering;

The above steps obtain the open-loop control of the optimal attitude maneuver model by solving the optimal control problem, and obtain the robust feedback controller according to the attitude error equation;

(1) Establish attitude error equation;

q _d , ω _d are the desired attitude quaternion vector _and _angular velocity vector, q _e is the attitude quaternion vector of the coordinate system F _b relative to the desired coordinate system F _d ; the corresponding transformation matrix C(q _e ) is;

Among them: q _e1 , q _e2 , q _e3 , q _e4 are four components of quaternion vector q _e , vector q _e13 =[q _e1 , q _e2 , q _e3 ] ^T ,

is the transpose vector of q _e13 ,

is the antisymmetric matrix of q _e13 , the relationship between q and q _d , q _e is:

q=mat(q _d )q _e (33)

in

Among them: q _d1 , q _d2 , q _d3 , q _d4 are the four components of q _d , and the angular velocity ω _e of F _b relative to F _d in the coordinate system F _b is:

ω _e = ω - Cω _d (35)

Rewrite formula (4) and formula (1) according to ω _e and q _e , then the attitude error equation is:

(2) Get a robust feedback controller

The Lyapunov function is:

Among them: k ₁ >0; the derivative of velocity V can be obtained as:

(38)

According to formula (38), the robust feedback controller is:

The second step (4) is tracked by the robust feedback controller to obtain the open-loop optimal attitude maneuver parameters to realize the satellite redirection attitude maneuver. the