CN105353763A

CN105353763A - Relative orbit attitude finite time control method for non-cooperative target spacecraft

Info

Publication number: CN105353763A
Application number: CN201510869675.4A
Authority: CN
Inventors: 龚有敏; 孙延超; 马广富; 耿远卓; 凌惠祥; 李传江
Original assignee: Harbin Institute of Technology Shenzhen
Current assignee: Harbin Institute of Technology Shenzhen
Priority date: 2015-12-01
Filing date: 2015-12-01
Publication date: 2016-02-24
Anticipated expiration: 2035-12-01
Also published as: CN105353763B

Abstract

The invention relates to a finite-time control method of the relative orbital attitude of a non-cooperative target spacecraft, which relates to the field of aerospace. It solves the existing problems in the joint control of the relative orbital attitude of the non-cooperative target spacecraft. A finite-time control method for the relative orbital attitude of a non-cooperative target spacecraft comprises the following steps: Step 1: project the relative orbital dynamics model represented by the inertial system to the line-of-sight system, and use the line-of-sight system to describe the relative orbital dynamics model of the spacecraft; Step 2: Establish attitude dynamics model and attitude kinematics model; Step 3: Express relative orbit dynamics model, attitude dynamics model and attitude kinematics model in state space to obtain relative orbit attitude dynamics model; Step 4: According to The finite-time continuous controller is obtained relative to the orbit attitude dynamics model and the finite-time control theory. The invention is suitable for joint control of relative orbital attitude of non-cooperative target spacecraft.

Description

A finite-time control method for the relative orbital attitude of a non-cooperative target spacecraft

技术领域technical field

本发明涉及航空航天领域。The invention relates to the field of aerospace.

背景技术Background technique

随着空间资源的开发与利用，各个国家都纷纷发射自己的卫星以及建造自己的空间站，但是随着卫星和空间站数量的不断增加，空间资源越来越缺乏，而且由于卫星的寿命或者失效问题，已经产生了大量的空间垃圾，占用很多轨道资源。为此，对空间碎片、失效卫星等非合作目标的接近和近距离跟踪监控，对失效的航天器进行维护使其恢复工作等空间任务已经成为当今航天领域的一个研究热点与难点。尽管这些空间任务目标各种各样，但都有同样的一个要求，即要求航天器在受控飞行过程中，其轨道与姿态能够同时以高控制精度达到期望的状态，也就是说要求航天器具备高精度的相对位置与姿态联合控制精度。从另一角度来说，航天器与目标之间的相对位置和相对姿态的精确控制对于能否完成空间任务起到了关键性作用。而且，当今世界航空航天技术已经进入了发展的快车道，各国纷纷展开空间活动，空间任务也越来越多样化，对空间系统的功能要求越来越高，对航天器近距离运动的航天器的位姿控制技术也面临着更多的挑战。因此，有必要对非合作目标的姿轨联合控制问题进行研究。With the development and utilization of space resources, various countries have launched their own satellites and built their own space stations, but as the number of satellites and space stations continues to increase, space resources are becoming more and more scarce, and due to the lifespan or failure of satellites, A large amount of space junk has been produced, occupying a lot of orbital resources. For this reason, space tasks such as approaching and close-range tracking and monitoring of non-cooperative targets such as space debris and failed satellites, and maintaining failed spacecraft to restore them to work have become a research hotspot and difficulty in the aerospace field today. Although the goals of these space missions are various, they all have the same requirement, that is, the orbit and attitude of the spacecraft can reach the desired state with high control accuracy at the same time during the controlled flight process. It has high-precision relative position and attitude joint control accuracy. From another point of view, the precise control of the relative position and attitude between the spacecraft and the target plays a key role in the completion of space missions. Moreover, aerospace technology in the world today has entered the fast lane of development. Countries have launched space activities one after another, space missions are becoming more and more diverse, and the requirements for the functions of space systems are getting higher and higher. The pose control technology is also facing more challenges. Therefore, it is necessary to study the problem of joint attitude-orbit control of non-cooperative targets.

航天器的相对轨道机动是研究追踪航天器处于目标航天器或虚拟航天器周围的持续运动规律，是研究轨道机动、编队飞行、在轨维护、交会对接、跟踪监视等空间任务的基础。The relative orbital maneuver of the spacecraft is to study the continuous motion law of the tracking spacecraft around the target spacecraft or virtual spacecraft, and is the basis for the study of orbital maneuvering, formation flight, on-orbit maintenance, rendezvous and docking, tracking and monitoring and other space tasks.

根据控制器输出的形式，航天器相对轨道控制的方法可以分为脉冲控制和连续推力控制。脉冲控制是指控制推进装置提供一个或多个短暂的速度增量进行相对轨道转移的控制方法，由于推进装置的工作时间很小，可以忽略，将产生的推力看成是脉冲。脉冲控制简单，但无法应对转移过程的未知干扰，控制不灵活。连续推力控制是指航天器发生相对转移的过程中始终施加控制作用，随着推进装置的发展和控制理论的完善，连续推力控制在航天器的相对轨道控制中得到了越来越多的应用，而且连续控制由于在转移的过程中一直存在控制作用，应对转移过程的未知干扰能力较强，近些年已经成为相对轨道控制的研究热点。According to the output form of the controller, the methods of spacecraft relative orbit control can be divided into pulse control and continuous thrust control. Pulse control refers to the control method of controlling the propulsion device to provide one or more short-term speed increments for relative orbit transfer. Since the working time of the propulsion device is very small, it can be ignored, and the generated thrust is regarded as a pulse. The pulse control is simple, but it cannot cope with the unknown disturbance in the transfer process, and the control is inflexible. Continuous thrust control means that the control effect is always applied during the relative transfer of the spacecraft. With the development of propulsion devices and the improvement of control theory, continuous thrust control has been more and more used in the relative orbit control of spacecraft. Moreover, continuous control has a strong ability to deal with unknown disturbances in the transfer process because it always has a control effect in the transfer process, and has become a research hotspot in relative orbit control in recent years.

航天器的相对轨道控制中，分为两类，一类是合作目标的航天器相对轨道控制，一类是非合作目标的航天器相对轨道控制。In the relative orbit control of spacecraft, it is divided into two categories, one is the relative orbit control of the spacecraft of the cooperative target, and the other is the relative orbit control of the spacecraft of the non-cooperative target.

所谓的非合作目标航天器是指追踪航天器无法获取目标航天器的相对轨道参数的目标航天器，比如失效的航天器、空间碎片和敌方的航天器等等。针对非合作目标航天器的相对轨道控制在空间非合作目标的跟踪监视、干扰、打击等空间任务中起到举足轻重的作用。The so-called non-cooperative target spacecraft refers to the target spacecraft that the tracking spacecraft cannot obtain the relative orbit parameters of the target spacecraft, such as invalid spacecraft, space debris and enemy spacecraft and so on. Relative orbit control for non-cooperative target spacecraft plays a pivotal role in space missions such as tracking, monitoring, jamming, and strike of non-cooperative targets in space.

在非合作目标的相对轨道控制中，一方面不仅要求追踪航天器能够有效的估计出目标航天器的相对位置信息和相对速度信息，另一方面还要求追踪航天器能够利用估计出来的非合作目标的位置信息和速度信息设计控制律，使得追踪航天器能够转移到目标航天器的周围。In the relative orbit control of non-cooperative targets, on the one hand, it is not only required that the tracking spacecraft can effectively estimate the relative position information and relative velocity information of the target spacecraft, but on the other hand, it is also required that the tracking spacecraft can use the estimated non-cooperative target The control law is designed based on the position information and velocity information, so that the tracking spacecraft can move around the target spacecraft.

航天器姿态控制是指获得并保存航天器在空间定向的技术，其一般是指相对于某参考系，要求航天器姿态以给定的要求或规律变化。Spacecraft attitude control refers to the technology of obtaining and preserving the orientation of the spacecraft in space, which generally refers to requiring the attitude of the spacecraft to change according to a given requirement or law relative to a certain reference system.

航天器的姿态控制可以分为姿态调节控制与姿态跟踪控制。航天器姿态调节控制是设计一个控制器将航天器的姿态镇定到平衡点附近。相比于航天器姿态调节控制，当期望的参考轨迹为时变信号时，航天器姿态控制问题称为姿态跟踪控制。一般来讲，姿态跟踪问题比姿态调节问题更难于处理，因为对于姿态跟踪问题，控制器不仅要镇定整个系统的状态变量，还需使系统的输出跟踪上时变的期望轨迹。Attitude control of spacecraft can be divided into attitude adjustment control and attitude tracking control. The spacecraft attitude adjustment control is to design a controller to stabilize the attitude of the spacecraft near the equilibrium point. Compared with spacecraft attitude regulation control, when the desired reference trajectory is a time-varying signal, the spacecraft attitude control problem is called attitude tracking control. Generally speaking, the attitude tracking problem is more difficult to deal with than the attitude adjustment problem, because for the attitude tracking problem, the controller not only needs to stabilize the state variables of the entire system, but also needs to make the output of the system track a time-varying desired trajectory.

文献“XinM,PanH.Nonlinearoptimalcontrolofspacecraftapproachingatumblingtarget[J].AerospaceScienceandTechnology,2011,15(2):79-89.”研究了近距离非合作目标的相对轨道姿态联合控制问题，采用θ-D方法设计控制器，该控制器具有控制精度高，误差小等优点，但是没能考虑目标轨道机动时动力学的不确定性，因此控制效果不佳。The literature "XinM, PanH.Nonlinearoptimalcontrolofspacecraftapproachingatumblingtarget[J].AerospaceScienceandTechnology,2011,15(2):79-89." studied the relative orbit attitude joint control of non-cooperative targets at close range, and designed the controller using the θ-D method. The controller has the advantages of high control precision and small error, but it fails to consider the uncertainty of the dynamics when maneuvering in the target orbit, so the control effect is not good.

文献“高登巍,罗建军,马卫华,康志宇,陈晓光.接近和跟踪非合作机动目标的非线性最优控制[J].宇航学报,2013,06:773-781.”研究了非合作目标的接近和视线跟踪问题，设计了θ-D控制器。根据Lyapunov最小-最大原理设计了θ-D修正控制器，来减小跟踪同时存在轨道和姿态机动的非合作目标的控制误差，改善跟踪非合作目标的姿轨联合控制的性能。上述方法虽然能获得满意的控制效果，但是其是根据渐近稳定的思想设计的控制器，也就是说理论上系统只有当时间为无穷大时系统才会收敛到平衡点。The literature "Gao Dengwei, Luo Jianjun, Ma Weihua, Kang Zhiyu, Chen Xiaoguang. Non-linear optimal control for approaching and tracking non-cooperative maneuvering targets[J]. Acta Astronautics Sinica, 2013, 06:773-781." studied the approach and control of non-cooperative targets. For the gaze tracking problem, a θ-D controller is designed. Based on Lyapunov's minimum-maximum principle, a θ-D correction controller is designed to reduce the control error of tracking non-cooperative targets with both orbit and attitude maneuvers, and improve the performance of joint attitude-orbit control for tracking non-cooperative targets. Although the above method can obtain a satisfactory control effect, it is a controller designed based on the idea of asymptotic stability, that is to say, the system will converge to the equilibrium point only when the time is infinite in theory.

文献“王磊,袁建平,罗建军.接近非合作目标的部分状态反馈姿轨联合控制[J].计算机仿真,2013,09:41-45+73.”针对非合作翻滚目标航天器的自主接近跟踪问题，建立了航天器六自由度类拉格朗日动力学方程，利用相对位置和相对姿态反馈信息并针对航天器惯性参数不确定性，采用自适应非线性输出反馈控制和神经网络逼近控制方法设计了姿轨联合控制器。Literature "Wang Lei, Yuan Jianping, Luo Jianjun. Partial state feedback attitude-orbit joint control approaching non-cooperative targets [J]. Computer Simulation, 2013, 09: 41-45+73." Autonomous approaching tracking problem for non-cooperative rolling target spacecraft , the Lagrange-like dynamic equation with six degrees of freedom of the spacecraft is established, using the relative position and attitude feedback information and considering the uncertainty of the inertial parameters of the spacecraft, the adaptive nonlinear output feedback control and the neural network approximation control method are used to design combined attitude-orbit controller.

方案的具体内容如下：The specific content of the plan is as follows:

基于视线坐标系建立轨道动力学方程：The orbital dynamics equation is established based on the line-of-sight coordinate system:

$\begin{matrix} \overset{\cdot\cdot \cdot\cdot}{r r} - - r r {\overset{\cdot &Center Dot;}{ψ ψ}}^{22} - - r r {((\overset{\cdot &Center Dot;}{θ θ} - - ω ω))}^{22} {cos cos}^{22} ψ ψ = = \frac{μ μ}{{r r}_{T T}^{33}} ((- - r r + + 33 r r {sin sin}^{22} {θcos θcos}^{22} ψ ψ)) + + {a a}_{x x 22} \\ r r \overset{\cdot\cdot \cdot\cdot}{ψ ψ} + + 22 \overset{\cdot &Center Dot;}{r r} \overset{\cdot \cdot}{ψ ψ} + + r r {((\overset{\cdot \cdot}{θ θ} - - ω ω))}^{22} sin sin ψ ψ cos cos ψ ψ = = \frac{μ μ}{{r r}_{T T}^{33}} ((- - 33 r r {sin sin}^{22} θ θ cos cos ψ ψ sin sin ψ ψ)) + + {a a}_{y the y 22} \\ r r ((\overset{\cdot\cdot \cdot\cdot}{θ θ} - - \overset{\cdot \cdot}{ω ω})) + + 22 \overset{\cdot \cdot}{r r} ((\overset{\cdot \cdot}{θ θ} - - ω ω)) - - 22 r r \overset{\cdot \cdot}{ψ ψ} ((\overset{\cdot \cdot}{θ θ} - - ω ω)) tan the tan ψ ψ = = \frac{μ μ}{{r r}_{T T}^{33}} ((33 r r cos cos θ θ sin sin θ θ)) + + {a a}_{z z 22} \end{matrix},, - - - - - - ((11))$

其中μ为地球引力常数，a_x2、a_y2、a_z2为沿视线坐标系三轴的控制加速度，r为两航天器之间的视线距离，r_T为目标与地心之间的距离，ψ为视线与其在目标轨道面上的投影之间的夹角，这里称为视线倾角，θ为视线偏角。ω为航天器姿态角速度各式两端同时乘以追踪航天器质量m_c得到类拉格朗日形式方程：Among them, μ is the gravitational constant of the earth, a _x2 , a _y2 , and a _z2 are the control acceleration along the three axes of the line-of-sight coordinate system, r is the line-of-sight distance between two spacecraft, r _T is the distance between the target and the center of the earth, ψ is the angle between the line of sight and its projection on the target orbital surface, which is called the line of sight inclination here, and θ is the line of sight declination. ω is the spacecraft attitude angular velocity multiplied by the mass m _c of the tracking spacecraft at the same time to obtain the Lagrange-like equation:

${J J}_{o o} (({q q}_{s the s})) {\overset{\cdot\cdot \cdot\cdot}{q q}}_{s the s} + + {C C}_{o o} (({q q}_{s the s},, {\overset{\cdot \cdot}{q q}}_{s the s})) {\overset{\cdot &Center Dot;}{q q}}_{s the s} + + {L L}_{o o} (({q q}_{s the s})) = = {u u}_{o o},, - - - - - - ((22))$

其中：in:

q_s＝[rψθ]^T q _s =[rψθ] ^T

u_o＝[a_x2ra_y2ra_z2cos²ψ]^T u _o ＝[a _x2 ra _y2 ra _z2 cos ² ψ] ^T

${J J}_{o o} (({q q}_{s the s})) = = {m m}_{c c} [\begin{matrix} 11 & 00 & 00 \\ 00 & {r r}^{22} & 00 \\ 00 & 00 & {r r}^{22} {cos cos}^{22} ψ ψ \end{matrix}]$

${C C}_{o o} (({q q}_{s the s},, {\overset{\cdot &Center Dot;}{q q}}_{s the s})) = = {m m}_{c c} [\begin{matrix} 00 & - - r r \overset{\cdot &Center Dot;}{ψ ψ} & - - r r \overset{\cdot &Center Dot;}{θ θ} {cos cos}^{22} ψ ψ + + 22 {rωcos rωcos}^{22} ψ ψ \\ r r \overset{\cdot &Center Dot;}{ψ ψ} & r r \overset{\cdot &Center Dot;}{r r} & {r r}^{22} θ θ sin sin ψ ψ cos cos ψ ψ - - 22 {r r}^{22} ω ω sin sin ψ ψ cos cos ψ ψ \\ r r \overset{\cdot &Center Dot;}{θ θ} {cos cos}^{22} ψ ψ - - 22 {rωcos rωcos}^{22} ψ ψ & - - {r r}^{22} \overset{\cdot \cdot}{θ θ} sin sin ψ ψ cos cos ψ ψ + + 22 {r r}^{22} ω ω sin sin ψ ψ cos cos ψ ψ & r r \overset{\cdot &Center Dot;}{r r} {cos cos}^{22} ψ ψ - - {r r}^{22} \overset{\cdot &Center Dot;}{ψ ψ} cos cos ψ ψ sin sin ψ ψ \end{matrix}]$

追踪航天器的姿态动力学方程为：The attitude dynamic equation of the tracking spacecraft is:

${J J}_{c c} \overset{\cdot \cdot}{ω ω} + + ω ω \times \times {J J}_{c c} ω ω = = {t t}_{c c},, - - - - - - ((33))$

式中J_c为追踪航天器转动惯量，t_c为航天器所受的控制力矩，ω为航天器姿态角速度。In the formula, J _c is the moment of inertia of the tracking spacecraft, t _c is the control torque on the spacecraft, and ω is the attitude angular velocity of the spacecraft.

追踪航天器姿态运动学方程为：The attitude kinematics equation of the tracking spacecraft is:

$\overset{\cdot &Center Dot;}{σ σ} = = B B ((σ σ)) ω ω,, - - - - - - ((44))$

其中：in:

$\begin{matrix} B B ((σ σ)) = = \frac{11}{44} ((((11 - - {σ σ}^{T T} σ σ)) [[{I I}_{33 \times \times 33}]] + + 22 {σσ σσ}^{T T} + + 22 [[\overset{~ ~}{σ σ}]])) \\ = = \frac{11}{44} * * [\begin{matrix} ((11 - - {σ σ}^{22} + + 22 {σ σ}_{i i}^{22})) & 22 (({σ σ}_{i i} {σ σ}_{j j} - - {σ σ}_{k k})) & 22 (({σ σ}_{i i} {σ σ}_{k k} + + {σ σ}_{j j})) \\ 22 (({σ σ}_{i i} {σ σ}_{j j} + + {σ σ}_{k k})) & ((11 - - {σ σ}^{22} + + 22 {σ σ}_{j j}^{22})) & 22 (({σ σ}_{j j} {σ σ}_{k k} - - {σ σ}_{j j})) \\ 22 (({σ σ}_{i i} {σ σ}_{k k} - - {σ σ}_{j j})) & 22 (({σ σ}_{j j} {σ σ}_{k k} + + {σ σ}_{j j})) & ((11 - - {σ σ}^{22} + + 22 {σ σ}_{k k}^{22})) \end{matrix}] \end{matrix}$

$[[\overset{~ ~}{σ σ}]] = = [\begin{matrix} 00 & - - {σ σ}_{k k} & {σ σ}_{j j} \\ {σ σ}_{k k} & 00 & - - {σ σ}_{i i} \\ - - {σ σ}_{j j} & {σ σ}_{i i} & 00 \end{matrix}]$

根据式(3)和式(4)整理可得类拉格朗日形式的姿态动力学方程为(这里下标a表示姿态)：According to formula (3) and formula (4), the attitude dynamics equation of Lagrange-like form can be obtained (here subscript a represents attitude):

${J J}_{a a} ((σ σ)) \overset{\cdot\cdot \cdot\cdot}{σ σ} + + {C C}_{a a} ((σ σ,, \overset{\cdot \cdot}{σ σ})) \overset{\cdot \cdot}{σ σ} = = {u u}_{a a},, - - - - - - ((55))$

其中：in:

J_a(σ)＝B^-T(σ)J_cB^-1(σ)，J _a (σ) = B ^-T (σ) J _c B ^-1 (σ),

$S S (({J J}_{c c} {B B}^{- - 11} \overset{\cdot \cdot}{σ σ})) = = S S (({J J}_{c c} ω ω)),,$

u_a＝B^-T(σ)t_c，u _a =B ^-T (σ)t _c ,

${C C}_{a a} ((σ σ,, \overset{\cdot \cdot}{σ σ})) = = - - {B B}^{- - T T} {J J}_{c c} {B B}^{- - 11} \overset{\cdot \cdot}{B B} {B B}^{- - 11} - - {B B}^{- - T T} S S (({J J}_{c c} {B B}^{- - 11} \overset{\cdot \cdot}{σ σ})) {B B}^{- - 11} . .$

联合式(2)和式(5)可得追踪航天器六自由度类拉格朗日动力学方程：Combining Equation (2) and Equation (5), the six-DOF Lagrange-like dynamic equation of the tracking spacecraft can be obtained:

${J J}^{* *} \overset{\cdot\cdot \cdot\cdot}{x x} + + C C \overset{\cdot \cdot}{x x} + + L L = = U u,, - - - - - - ((66))$

其中： $J^{*} = [\begin{matrix} J_{o} \\ J_{a} \end{matrix}], C = [\begin{matrix} C_{o} \\ C_{a} \end{matrix}], L = [\begin{matrix} L_{o} \\ 0_{3 \times 1} \end{matrix}],$ U＝[u_ou_a]^T，x＝[q_sσ]^T。in: $J^{*} = [\begin{matrix} J_{o} \\ J_{a} \end{matrix}], C = [\begin{matrix} C_{o} \\ C_{a} \end{matrix}], L = [\begin{matrix} L_{o} \\ 0_{3 \times 1} \end{matrix}],$ U=[u _o u _a ] ^T , x=[q _s σ] ^T .

首先定义参数估计误差其中表示质量估计误差，表示转动惯量估计误差并假设:定义为常值、未知参数矢量，即：其中为的估计值。定义相对跟踪误差e＝[e_oe_a]^T∈R⁶，伪速率滤波器输出e_f＝[e_foe_fa]^T∈R⁶，辅助滤波器变量p(t)∈R⁶，则伪速率滤波器动态方程：First define the parameter estimation error in represents the mass estimation error, represents the moment of inertia estimation error and assumes: definition is a constant, unknown parameter vector, that is: in for estimated value. Define the relative tracking error e＝[e _o e _a ] ^T ∈R ⁶ , the pseudo-rate filter output e _f ＝[e _fo e _fa ] ^T ∈R ⁶ , the auxiliary filter variable p(t)∈R ⁶ , then the pseudo-rate filter Rate filter dynamic equation:

$\begin{matrix} {e e}_{f f} = = - - K K e e + + p p \\ \overset{\cdot \cdot}{p p} = = - - ((K K + + I I)) p p + + (({K K}^{22} + + I I)) e e \\ p p ((00)) = = K K e e ((00)) \end{matrix},, - - - - - - ((77))$

其中K∈R^6×6是常值矩阵。引入辅助跟踪误差变量η＝[η₁η₂]∈R⁶，其形式为假定x_d是式(6)中x的期望值，L_d为式(6)中L矩阵的期望值。从而得到闭环误差动力学方程：where K ∈ R ^6×6 is a constant matrix. Introduce the auxiliary tracking error variable η=[η ₁ η ₂ ]∈R ⁶ , whose form is Suppose x _d is the expected value of x in formula (6), and L _d is the expected value of L matrix in formula (6). Thus, the closed-loop error dynamic equation is obtained:

其中 $\tilde{L} = L - L_{d},$ in $\tilde{L} = L - L_{d},$

$χ = J^{*} (x) {\overset{\cdot\cdot}{x}}_{d} + C (x, \overset{\cdot}{x}) {\overset{\cdot\cdot}{x}}_{d} - J^{*} (x_{d}) {\overset{\cdot\cdot}{x}}_{d} - C (x_{d}, {\overset{\cdot}{x}}_{d}) {\overset{\cdot\cdot}{x}}_{d} + C (x, \overset{\cdot}{x}) (e_{f} + e) + J^{*} (x) η - 2 J^{*} (x) e_{f} - \tilde{L},$ 神经网络的激活函数选为高斯型函数则神经网络控制器为 $χ = J^{*} (x) {\overset{\cdot\cdot}{x}}_{d} + C (x, \overset{&Center Dot;}{x}) {\overset{\cdot\cdot}{x}}_{d} - J^{*} (x_{d}) {\overset{\cdot\cdot}{x}}_{d} - C (x_{d}, {\overset{&Center Dot;}{x}}_{d}) {\overset{\cdot\cdot}{x}}_{d} + C (x, \overset{&Center Dot;}{x}) (e_{f} + e) + J^{*} (x) η - 2 J^{*} (x) e_{f} - \tilde{L},$ The activation function of the neural network is selected as a Gaussian function Then the neural network controller is

${U u}_{22} = = - - K K {[\begin{matrix} tanh tanh (({δe δe}_{f f 11})) & ... ... & tanh tanh (({δe δ e}_{f f n no})) \end{matrix}]}^{T T} + + \overset{^^}{χ χ},, - - - - - - ((99))$

自适应非线性输出反馈控制器设计为：The adaptive nonlinear output feedback controller is designed as:

可得如下的姿轨联合控制器U＝U₁+U₂，令估计值为基函数，是权重矩阵的估计值，ε是逼近误差，δ为正的控制参数。The following combined attitude-orbit controller U＝U ₁ +U ₂ can be obtained, let estimated value is the basis function, is the estimated value of the weight matrix, ε is the approximation error, and δ is a positive control parameter.

则不包含速度误差项的自适应参数估计变化律为：Then the change law of adaptive parameter estimation that does not include the velocity error term is:

其中Γ、Γ_m、k_w为正的控制参数。Among them, Γ, Γ _m and k _w are positive control parameters.

该方法中控制器的形式非常复杂，这限制了其在实际工程应用中的使用，而且，该控制是基于渐近稳定的思想设计的，理论上只有当时间为无穷大时，系统才会达到稳定，当要求系统在有限时间内达到稳定状态时就无法应用。The form of the controller in this method is very complicated, which limits its use in practical engineering applications. Moreover, the control is designed based on the idea of asymptotic stability. In theory, the system will reach stability only when the time is infinite. , it cannot be applied when the system is required to reach a steady state within a finite time.

文献“苏晏,黎康.近距离星间相对姿轨耦合动力学建模与控制[J].空间控制技术与应用,2014,04:20-25.”针对在轨服务过程中近距离星间相对位置以及相对姿态精确控制的问题，设计了相对姿态轨道联合控制算法，实现了对目标星的精确指向绕飞运动。Document "Su Yan, Li Kang. Dynamic modeling and control of relative attitude-orbit coupling between short-distance satellites [J]. Space Control Technology and Application, 2014, 04:20-25." To solve the problem of precise control of relative position and relative attitude, a relative attitude-orbit joint control algorithm is designed to realize the precise pointing and orbiting motion of the target star.

方案的具体内容如下：The specific content of the plan is as follows:

建立相对姿轨耦合动力学模型：Establish relative attitude-orbit coupling dynamics model:

$\{\begin{matrix} \overset{\cdot &Center Dot;}{ρ ρ} = = ν ν \\ \overset{\cdot &Center Dot;}{ν ν} = = f f ((ω ω,, \overset{\cdot &Center Dot;}{ω ω},, ρ ρ,, \overset{\cdot \cdot}{ρ ρ},, {r r}_{t t})) + + {C C}_{t t o o} g g ((ω ω,, \overset{\cdot \cdot}{ω ω},, ρ ρ,, \overset{\cdot \cdot}{ρ ρ},, {J J}_{22})) + + {C C}_{t t c c} a a \\ {\overset{\cdot \cdot}{q q}}_{e e} = = \frac{11}{22} Ω Ω (({q q}_{e e})) {C C}_{c c o o}^{T T} (({ω ω}_{d d b b} - - {ω ω}_{c c b b})) \\ {\overset{\cdot \cdot}{ω ω}}_{c c b b} = = {J J}_{c c}^{- - 11} ((- - {ω ω}_{c c b b}^{\times \times} {J J}_{c c} {ω ω}_{c c b b} + + T T + + d d)) \end{matrix},, - - - - - - ((1313))$

式中：In the formula:

$f f ((ω ω,, \overset{\cdot &Center Dot;}{ω ω},, ρ ρ,, \overset{\cdot \cdot}{ρ ρ},, {r r}_{t t})) = = - - 22 ω ω \times \times \overset{\cdot &Center Dot;}{ρ ρ} - - ω ω \times \times ((ω ω \times \times ρ ρ)) - - \overset{\cdot \cdot}{ω ω} \times \times ρ ρ + + {C C}_{t t o o} \frac{μ μ}{{r r}_{t t}^{33}} ((\frac{33 {xr xr}_{t t}}{{r r}_{t t}} - - ρ ρ)),,$

$g g ((ω ω,, \overset{\cdot &Center Dot;}{ω ω},, ρ ρ,, \overset{\cdot \cdot}{ρ ρ},, {J J}_{22})) = = \frac{{μJ μJ}_{22} {R R}_{E E.}^{22} ((11 - - 55 {sin sin}^{22} i i {sin sin}^{22} u u))}{{r r}_{t t}^{44} (({r r}_{t t} + + 22 x x))} {C C}_{t t o o} \cdot \cdot {[\begin{matrix} 66 x x & \frac{33 y the y}{22} & \frac{33 z z}{22} \end{matrix}]}^{T T},,$

$Ω (q_{e}) = [\begin{matrix} q_{e 4} I_{3} + q_{e v}^{\times} \\ - q_{e v}^{T} \end{matrix}],$ C_tc＝C_toC_oc，为基准坐标系向基准星轨道坐标系的转换矩阵，μ为引力常数，b_t和b_c分别为各自摄动力引起的加速度，ω和分别为目标轨道坐标系角速度矢量和角加速度矢量，r_t、r_c分别为两颗卫星的地心距，表示惯性系到基准星轨道坐标系的转换矩阵，x、y、z表示位置矢量在基准星轨道坐标系的分量大小，J_c为编队星的转动惯量在编队星本体坐标系下的表示。 $Ω (q_{e}) = [\begin{matrix} q_{e 4} I_{3} + q_{e v}^{\times} \\ - q_{e v}^{T} \end{matrix}],$ C _tc = C _to C _oc , is the transformation matrix from the reference coordinate system to the reference star orbit coordinate system, μ is the gravitational constant, b _t and b _c are the accelerations caused by their respective perturbations, ω and are the angular velocity vector and angular acceleration vector of the target orbital coordinate system respectively, r _t and r _c are the earth center distances of the two satellites respectively, which represent the conversion matrix from the inertial system to the reference satellite orbital coordinate system, and x, y, z represent the position vector at The component size of the reference star orbit coordinate system, _Jc is the expression of the formation star's moment of inertia in the formation star body coordinate system.

基准坐标系的x_d轴保持对基准星指向；y_d轴与x_d垂直且与矢量m夹角最小，其中m为对航天器与地球质心的指向矢量；z_d轴的选取遵循右手定则。基准坐标系如下:x_d＝-ρ₀/||ρ₀||，y_d＝(x_d×m)×x_d/||(x_d×m)×x_d||，z_d＝(x_d×y_d)×x_d，C_do＝[x_dy_dz_d]^T。期望获得的角速度和角加速度在基准坐标系下的表示为： $ω_{d b}^{\times} = - {\overset{\cdot}{C}}_{d o} C_{d o}^{T}, {\overset{\cdot}{ω}}_{d b}^{\times} = - {\overset{\cdot}{C}}_{d o} {\overset{\cdot}{C}}_{d o}^{T} - {\overset{\cdot\cdot}{C}}_{d o} {\overset{\cdot}{C}}_{d o}^{T} .$ The x _d axis of the reference coordinate system keeps pointing to the reference star; the y _d axis is perpendicular to x _d and has the smallest angle with the vector m, where m is the pointing vector to the spacecraft and the center of mass of the earth; the selection of the z _d axis follows the right-hand rule . The reference coordinate system is as follows: x _d ＝-ρ ₀ /||ρ ₀ ||, y _d ＝(x _d ×m)×x _d /||(x _d ×m)×x _d ||, z _d ＝( x _d ×y _d )×x _d , C _do =[x _d y _d z _d ] ^T . The desired angular velocity and angular acceleration are expressed in the reference coordinate system as: $ω_{d b}^{\times} = - {\overset{&Center Dot;}{C}}_{d o} C_{d o}^{T}, {\overset{&Center Dot;}{ω}}_{d b}^{\times} = - {\overset{&Center Dot;}{C}}_{d o} {\overset{&Center Dot;}{C}}_{d o}^{T} - {\overset{\cdot\cdot}{C}}_{d o} {\overset{&Center Dot;}{C}}_{d o}^{T} .$

定义跟踪误差Define Tracking Error

${{\begin{matrix} {e e}_{ρ ρ} = = ρ ρ - - {ρ ρ}_{d d} \\ {e e}_{v v} = = v v - - {v v}_{d d} \\ {e e}_{ω ω} = = {ω ω}_{c c b b} - - {ω ω}_{d d b b} - - {ω ω}_{e e d d} \end{matrix},, - - - - - - ((1414))$

其中ρ_d为期望相对位置，v_d为期望相对速度，ω_db为期望角加速度，ω_ed为期望角加速度差。Where ρ _d is the expected relative position, v _d is the expected relative velocity, ω _db is the expected angular acceleration, and ω _ed is the expected angular acceleration difference.

设计轨道控制加速度为：The design trajectory control acceleration is:

$a a = = {C C}_{t t c c}^{- - 11} [[- - f f ((ω ω,, \overset{\cdot &Center Dot;}{ω ω},, ρ ρ,, \overset{\cdot &Center Dot;}{ρ ρ},, {r r}_{t t})) - - {C C}_{t t o o} g g ((ω ω,, \overset{\cdot &Center Dot;}{ω ω},, ρ ρ,, \overset{\cdot &Center Dot;}{ρ ρ},, {J J}_{22})) + + {\overset{\cdot &Center Dot;}{v v}}_{d d} - - {Re Re}_{ρ ρ} - - {Qe Q}_{v v}]],, - - - - - - ((1515))$

取姿态控制力矩take attitude control torque

T＝T_c1+T_c2，(16)T=T _c1 +T _c2 , (16)

其中，T_c2用来消除编队星由于质量偏心等因素引入的轨道控制对姿态的干扰力矩。Among them, T _c2 is used to eliminate the interference moment of the orbit control on the attitude of the formation star due to factors such as mass eccentricity.

设计控制力矩Design control torque

${T T}_{c c 11} = = {ω ω}_{c c b b}^{\times \times} {J J}_{c c} {ω ω}_{c c b b} + + {J J}_{c c} ((- - {ω ω}_{c c b b} + + {ω ω}_{c c b b d d} + + {\overset{\cdot &Center Dot;}{ω ω}}_{c c b b d d})),, - - - - - - ((1717))$

${T T}_{c c 22} = = - - (({d d}_{m m a a x x} + + ξ ξ)) \frac{s the s}{| | | | s the s | | | |},, - - - - - - ((1818))$

式中d_max为轨控推力引入的最大姿态干扰力矩，ξ为很小的正数，d_max+ξ>||d||，d∈M＝{d:||d||≤d_max＝l_max×F}，l_max为最大干扰力臂，F为轨控推力，P为正定矩阵。In the formula, d _max is the maximum attitude disturbance moment introduced by the orbit control thrust, ξ is a small positive number, d _max +ξ>||d||, d∈M={d:||d||≤d _max = l _max ×F}, l _max is the maximum disturbance arm, F is the orbit control thrust, P is a positive definite matrix.

在该方案中，所用的动力学模型比较复杂，控制器的形式相对复杂，其仍然是一个渐近稳定的系统，理论上在该控制器的作用下，追踪航天器未能在有限时间内到达系统的平衡点。In this scheme, the dynamics model used is relatively complicated, and the form of the controller is relatively complicated. It is still an asymptotically stable system. Theoretically, under the action of this controller, the tracking spacecraft fails to reach The balance point of the system.

发明内容Contents of the invention

本发明为了解决目前非合作目标的航天器相对轨道姿态联合控制中所存在的问题，提出了一种非合作目标航天器相对轨道姿态有限时间控制方法，In order to solve the existing problems in the joint control of the relative orbital attitude of the non-cooperative target spacecraft, the present invention proposes a limited time control method for the relative orbital attitude of the non-cooperative target spacecraft.

一种非合作目标航天器相对轨道姿态有限时间控制方法包括以下步骤：A finite-time control method for the relative orbital attitude of a non-cooperative target spacecraft comprises the following steps:

步骤一、将用惯性系表示的相对轨道动力学模型投影到视线系，采用视线系描述航天器的相对轨道动力学模型；Step 1. Project the relative orbital dynamics model represented by the inertial system to the line of sight system, and use the line of sight system to describe the relative orbital dynamics model of the spacecraft;

步骤二、建立姿态动力学模型和姿态运动学模型；Step 2, establishing attitude dynamics model and attitude kinematics model;

步骤三、将相对轨道动力学模型、姿态动力学模型和姿态运动学模型进行状态空间表示，获得相对轨道姿态动力学模型；Step 3, performing state space representation on the relative orbital dynamics model, attitude dynamics model and attitude kinematics model to obtain the relative orbital attitude dynamics model;

步骤四、根据相对轨道姿态动力学模型和有限时间控制理论获得有限时间连续控制器。Step 4: Obtain a finite-time continuous controller based on the relative orbital attitude dynamics model and finite-time control theory.

有益效果：本发明专利研究了非合作目标航天器的相对轨道姿态联合控制问题，考虑了未知干扰和不确定性，结合有限时间控制理论，提出了姿轨联合控制的有限时间控制器。Beneficial effects: the patent of the present invention studies the problem of joint control of relative orbital attitude of non-cooperative target spacecraft, considers unknown interference and uncertainty, and combines finite time control theory to propose a finite time controller for joint control of attitude and orbit.

本发明针对的是非合作航天器的相对轨道、姿态控制，只需要知道目标航天器的逃逸上界就能够获得非常好的控制效果，而无需对目标的逃逸进行估计；由于考虑了系统的不确定性与外界未知干扰，即便存在不确定性的外界干扰时，追踪航天器仍能很好的跟踪上逃逸的非合作目标航天器；利用有限时间控制原理获得控制器，能够使航天器能够在有限的时间内跟踪上非合作目标的航天器并对其进行跟踪；同时，控制器的形式简单、计算量小，适合在星载计算机上使用，易于工程实现，具有工程实际意义。The present invention is aimed at the relative orbit and attitude control of the non-cooperative spacecraft, and only needs to know the escape upper bound of the target spacecraft to obtain a very good control effect without estimating the escape of the target; due to the consideration of the uncertainty of the system Even when there is uncertain external interference, the tracking spacecraft can still track the escaped non-cooperative target spacecraft well; using the finite time control principle to obtain the controller can make the spacecraft be able to Track the spacecraft on the non-cooperative target and track it within a certain period of time; at the same time, the controller is simple in form and has a small amount of calculation, which is suitable for use on the on-board computer, easy to implement, and has engineering practical significance.

附图说明Description of drawings

图1为具体实施方式二所述的惯性系与视线系以及它们之间的关系示意图；Fig. 1 is a schematic diagram of the inertial system and line-of-sight system and the relationship between them described in the second embodiment;

图2为在接近和跟踪非合作目标时，轨道相关参数随时间变化曲线图，包括相对距离、视线倾角和视线偏角；Figure 2 is a graph of orbit-related parameters changing over time when approaching and tracking a non-cooperative target, including relative distance, line-of-sight inclination and line-of-sight deviation;

图3为在接近和跟踪非合作目标时，姿态角随时间变化曲线图；Fig. 3 is when approaching and tracking the non-cooperative target, the curve diagram of attitude angle changing with time;

图4为在接近和跟踪非合作目标时，控制加速度随时间变化曲线图；Figure 4 is a curve diagram of control acceleration versus time when approaching and tracking a non-cooperative target;

图5为在接近和跟踪非合作目标时，控制力矩随时间变化曲线图；Figure 5 is a curve diagram of the control torque over time when approaching and tracking a non-cooperative target;

图6为轨道姿态参数与相应的期望参数之间的偏差随时间变化曲线图；Fig. 6 is a graph showing the deviation over time between the orbital attitude parameter and the corresponding desired parameter;

图7为y＝tanh(x)曲线与y＝x曲线关系示意图。Fig. 7 is a schematic diagram of the relationship between the y=tanh(x) curve and the y=x curve.

具体实施方式detailed description

具体实施方式一、本具体实施方式所述的一种非合作目标航天器相对轨道姿态有限时间控制方法包括以下步骤：DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENT 1. A non-cooperative target spacecraft relative orbital attitude finite time control method described in this specific embodiment includes the following steps:

本实施方式通过有限时间控制理论，采用连续控制的方式，获得有限时间连续控制器，进而实现非合作目标的航天器相对轨道姿态联合控制。This embodiment adopts the finite-time control theory and adopts the continuous control method to obtain the finite-time continuous controller, and then realize the joint control of the relative orbital attitude of the non-cooperative target spacecraft.

具体实施方式二、结合图1说明本具体实施方式，本具体实施方式与具体实施方式一所述的一种非合作目标航天器相对轨道姿态有限时间控制方法的区别在于，步骤一所述的将用惯性系表示的相对轨道动力学模型投影到视线系，采用视线系描述航天器的相对轨道动力学模型的具体过程为：Specific Embodiment 2. This specific embodiment is described in conjunction with FIG. 1. The difference between this specific embodiment and the method for controlling the relative orbital attitude of a non-cooperative target spacecraft for a limited time in the specific embodiment 1 is that the method described in step 1 will The relative orbital dynamics model represented by the inertial system is projected to the line of sight system, and the specific process of using the line of sight system to describe the relative orbital dynamics model of the spacecraft is as follows:

将用惯性系O_ix_iy_iz_i表示的相对轨道动力学模型投射到视线系O_lx_ly_lz_l：Project the relative orbital dynamics model represented by the inertial system O _i x _i y _i z _i to the line of sight system O _l x _ly _l z _l :

$\begin{matrix} {((\frac{{d d}^{22} ρ ρ}{{dt dt}^{22}}))}_{l l} = = \frac{{d d}^{22} {((ρ ρ))}_{l l}}{{dt dt}^{22}} + + {(({\overset{\cdot \cdot}{ω ω}}_{l l}))}_{l l}^{\times \times} {((ρ ρ))}_{l l} + + 22 {(({ω ω}_{l l}))}_{l l}^{\times \times} \frac{d d {((ρ ρ))}_{l l}}{d d t t} + + {(({ω ω}_{l l}))}_{l l}^{\times \times} {(({ω ω}_{l l}))}_{l l}^{\times \times} {((ρ ρ))}_{l l} \\ = = {((Δ Δ g g))}_{l l} + + {((f f))}_{l l} - - (({u u}_{c c})) l l \end{matrix},, - - - - - - ((11 - - 11))$

其中，ρ为目标航天器相对于追踪航天器的位置矢量，上标×表示向量的反对称矩阵，△g＝[△g_x△g_y△g_z]^T为两空间飞行器的引力差项，当进行近距离相对转移和跟踪时，△g等于0。f＝[f_xf_yf_z]^T为目标空间飞行器的逃逸，u_c＝[u_cxu_cyu_cz]^T为输入的控制矢量；Among them, ρ is the position vector of the target spacecraft relative to the tracking spacecraft, the superscript × indicates the antisymmetric matrix of the vector, △g=[△g _x △g _y △g _z ] ^T is the gravitational difference term of the two spacecraft, When carrying out short distance relative transfer and tracking, △g is equal to 0. f=[f _x f _y f _z ] ^T is the escape of the target spacecraft, u _c =[u _cx u _cy u _cz ] ^T is the input control vector;

将式(1-1)按照分量展开，即可获得采用视线系描述航天器的相对轨道动力学模型：By expanding the formula (1-1) according to the components, the relative orbital dynamics model of the spacecraft described by the line of sight can be obtained:

$\{\begin{matrix} \overset{\cdot\cdot \cdot\cdot}{ρ ρ} - - ρ ρ (({\overset{\cdot \cdot}{q q}}_{ϵ ϵ}^{22} + + {\overset{\cdot \cdot}{q q}}_{β β}^{22} {cos cos}^{22} {q q}_{ϵ ϵ})) = = {Δg Δ g}_{x x} + + {f f}_{x x} - - {u u}_{c c x x} \\ ρ ρ {\overset{\cdot\cdot \cdot\cdot}{q q}}_{ϵ ϵ} + + 22 \overset{\cdot \cdot}{ρ ρ} {\overset{\cdot \cdot}{q q}}_{ϵ ϵ} + + ρ ρ {\overset{\cdot \cdot}{q q}}_{β β}^{22} {sinq sinq}_{ϵ ϵ} {cosq cosq}_{ϵ ϵ} = = {Δg Δg}_{y the y} + + {f f}_{y the y} - - {u u}_{c c y the y} \\ - - ρ ρ {\overset{\cdot\cdot \cdot\cdot}{q q}}_{β β} {cosq cosq}_{ϵ ϵ} + + 22 ρ ρ {\overset{\cdot &Center Dot;}{q q}}_{β β} {\overset{\cdot &Center Dot;}{q q}}_{ϵ ϵ} {sinq sinq}_{ϵ ϵ} - - 22 \overset{\cdot &Center Dot;}{ρ ρ} {\overset{\cdot \cdot}{q q}}_{β β} {cosq cosq}_{ϵ ϵ} = = {Δg Δg}_{z z} + + {f f}_{z z} - - {u u}_{c c z z} \end{matrix},, - - - - - - ((11 - - 22))$

其中q_ε为视线倾角，q_β为视线偏角。Where q _ε is the inclination angle of the line of sight, and q _β is the declination angle of the line of sight.

具体实施方式三、本具体实施方式与具体实施方式二所述的一种非合作目标航天器相对轨道姿态有限时间控制方法的区别在于，步骤二所述的建立姿态动力学模型和姿态运动学模型的具体过程为：Specific Embodiment Three. The difference between this specific embodiment and the finite-time control method for the relative orbital attitude of a non-cooperative target spacecraft described in the second specific embodiment is that the establishment of the attitude dynamics model and the attitude kinematics model described in step two The specific process is:

用下标b表示体坐标系，t表示目标飞行器，c表示追踪飞行器，则追踪航天器姿态动力学方程，即姿态动力学模型为：Use the subscript b to represent the body coordinate system, t to represent the target aircraft, and c to represent the tracking aircraft, then the attitude dynamics equation of the tracking spacecraft, that is, the attitude dynamics model is:

${J J}_{c c} {\overset{\cdot \cdot}{ω ω}}_{b b c c} + + {ω ω}_{b b c c}^{\times \times} {J J}_{c c} {ω ω}_{b b c c} = = {T T}_{c c},, - - - - - - ((11 - - 33))$

其中，J_c＝[J_cxJ_cyJ_cz]^T是转动惯量，ω_bc＝[ω_xω_yω_z]^T是姿态角速度，T_c＝[T_cxT_cyT_cz]^T是控制力矩，Among them, J _c ＝[J _cx J _cy J _cz ] ^T is moment of inertia, ω _bc ＝[ω _x ω _y ω _z ] ^T is attitude angular velocity, T _c ＝[T _cx T _cy T _cz ] ^T is control torque,

定义θ、ψ分别是追踪飞行器绕本体x、y、z轴的转角，则用欧拉角表示的姿态为：definition θ and ψ are the rotation angles of the tracking aircraft around the x, y, and z axes of the body, respectively, and the attitude represented by Euler angles is:

从而获得姿态角速度为：Thus, the attitude angular velocity is obtained as:

定义矩阵R为：Define the matrix R as:

则姿态运动学模型为：Then the posture kinematics model is:

具体实施方式四、本具体实施方式与具体实施方式三所述的一种非合作目标航天器相对轨道姿态有限时间控制方法的区别在于，步骤三所述的将相对轨道动力学模型、姿态动力学模型和姿态运动学模型进行状态空间表示，获得相对轨道姿态动力学模型的具体过程为：Embodiment 4. The difference between this embodiment and the non-cooperative target spacecraft relative orbit attitude finite time control method described in Embodiment 3 is that the relative orbit dynamics model, attitude dynamics described in step 3 The state space representation of the model and the attitude kinematics model, and the specific process of obtaining the relative orbital attitude dynamics model are as follows:

将式(1-2)、式(1-3)和式(1-7)表示为状态空间的形式：Express the formula (1-2), formula (1-3) and formula (1-7) as the form of the state space:

$[\begin{matrix} \overset{\cdot\cdot \cdot\cdot}{ρ ρ} \\ {\overset{\cdot\cdot \cdot\cdot}{q q}}_{ϵ ϵ} \\ {\overset{\cdot\cdot \cdot\cdot}{q q}}_{β β} \\ {\overset{\cdot &Center Dot;}{ω ω}}_{x x} \\ {\overset{\cdot &Center Dot;}{ω ω}}_{y the y} \\ {\overset{\cdot \cdot}{ω ω}}_{z z} \end{matrix}] = = [\begin{matrix} ρ ρ {\overset{\cdot &Center Dot;}{q q}}_{ϵ ϵ}^{22} + + ρ ρ {\overset{\cdot \cdot}{q q}}_{β β}^{22} {cos cos}^{22} {q q}_{ϵ ϵ} \\ - - \frac{22 \overset{\cdot \cdot}{ρ ρ}}{ρ ρ} {\overset{\cdot \cdot}{q q}}_{ϵ ϵ} - - {\overset{\cdot &Center Dot;}{q q}}_{β β}^{22} {sinq sinq}_{ϵ ϵ} {cosq cosq}_{ϵ ϵ} \\ \frac{22 {\overset{\cdot &Center Dot;}{q q}}_{β β} {\overset{\cdot \cdot}{q q}}_{ϵ ϵ} {sinq sinq}_{ϵ ϵ}}{{cosq cosq}_{ϵ ϵ}} - - \frac{22 \overset{\cdot &Center Dot;}{ρ ρ} {\overset{\cdot &Center Dot;}{q q}}_{β β}}{ρ ρ} \\ \frac{(({J J}_{c c y the y} - - {J J}_{c c z z}))}{{J J}_{c c x x}} {ω ω}_{y the y} {ω ω}_{z z} \\ \frac{(({J J}_{c c z z} - - {J J}_{c c x x}))}{{J J}_{c c y the y}} {ω ω}_{x x} {ω ω}_{z z} \\ \frac{(({J J}_{c c x x} - - {J J}_{c c y the y}))}{{J J}_{c c z z}} {ω ω}_{y the y} {ω ω}_{x x} \end{matrix}] + + [\begin{matrix} {f f}_{x x} \\ \frac{{f f}_{y the y}}{ρ ρ} \\ \frac{- - {f f}_{z z}}{{ρcosq ρ cosq}_{ϵ ϵ}} \\ 00 \\ 00 \\ 00 \end{matrix}] + + [\begin{matrix} - - {u u}_{c c x x} \\ \frac{- - {u u}_{c c y the y}}{ρ ρ} \\ \frac{{u u}_{c c z z}}{{ρcosq ρ cosq}_{ϵ ϵ}} \\ \frac{{T T}_{c c x x}}{{J J}_{c c x x}} \\ \frac{{T T}_{c c y the y}}{{J J}_{c c y the y}} \\ \frac{{T T}_{c c z z}}{{J J}_{c c z z}} \end{matrix}],, - - - - - - ((11 - - 99))$

设Assume

则式(1-8)和式(1-9)成为如下状态空间形式，即相对轨道姿态动力学模型为：Then Equation (1-8) and Equation (1-9) become the following state space form, that is, the relative orbit attitude dynamic model is:

$\{\begin{matrix} {\overset{\cdot &Center Dot;}{x x}}_{11} = = A A (({x x}_{11})) {x x}_{22} \\ {\overset{\cdot &Center Dot;}{x x}}_{22} = = f f ((x x)) + + w w ((x x)) + + g g ((x x)) u u \end{matrix},, - - - - - - ((11 - - 1010))$

其中，in,

w(x)是系统的不确定性和外部干扰的总和，满足||w(x)||≤w_d，其中w_d>0，u∈Rⁿ是系统的输入。w(x) is the sum of system uncertainty and external disturbance, satisfying ||w(x)||≤w _d , where w _d >0, u∈R ⁿ is the input of the system.

具体实施方式五、本具体实施方式与具体实施方式四所述的一种非合作目标航天器相对轨道姿态有限时间控制方法的区别在于，步骤四所述的根据相对轨道姿态动力学模型和有限时间控制理论获得有限时间连续控制器的具体过程为：Embodiment 5. The difference between this embodiment and the finite-time control method for the relative orbital attitude of a non-cooperative target spacecraft described in Embodiment 4 is that the method described in step 4 is based on the relative orbital attitude dynamics model and the finite time The specific process of obtaining a finite-time continuous controller in control theory is as follows:

记辅助控制器：Note the auxiliary controller:

v(x₁)＝-A^-1(x₁)K₁sign(x₁)tanh^α(|x₁|)，(1-11)v(x ₁ )＝-A ^-1 (x ₁ )K ₁ sign(x ₁ )tanh ^α (|x ₁ |), (1-11)

其中，K₁＝diag(k₁₁,...,k_1n)，k_1i>0，0<α<1，记sign(x)tanh^α(|x|)＝th^α(x)，则：Among them, K ₁ = diag(k ₁₁ ,...,k _1n ), k _1i >0, 0<α<1, write sign(x)tanh ^α (|x|)=th ^α (x), then:

v(x₁)＝-A^-1(x₁)K₁th^α(x₁)，(1-12)v(x ₁ )＝-A ^-1 (x ₁ )K ₁ th ^α (x ₁ ), (1-12)

定义误差变量：Define the error variable:

z＝x₂-v(x₁)，(1-13)z=x ₂ −v(x ₁ ), (1-13)

将式(1-12)和式(1-13)代入式(1-10)中，得：Substituting formula (1-12) and formula (1-13) into formula (1-10), we get:

$\begin{matrix} {\overset{\cdot &Center Dot;}{x x}}_{11} = = {A A}^{- - 11} (({x x}_{11})) ((z z + + v v (({x x}_{11})))) = = - - {K K}_{11} {th the th}^{α α} (({x x}_{11})) + + A A (({x x}_{11})) z z \\ \overset{\cdot &Center Dot;}{z z} = = f f ((x x)) + + g g ((x x)) u u + + w w ((x x)) - - \overset{\cdot &Center Dot;}{v v} (({x x}_{11})) \end{matrix},, - - - - - - ((11 - - 1414))$

则有限时间连续控制器为：Then the finite-time continuous controller is:

$u u = = {g g}^{- - 11} ((x x)) ((\overset{\cdot &Center Dot;}{v v} (({x x}_{11})) - - f f ((x x)) - - {w w}_{d d} s the s i i g g n no ((z z)) - - {A A}^{T T} (({x x}_{11})) {x x}_{11} - - {K K}_{22} {th the th}^{α α} ((z z)))),, - - - - - - ((11 - - 1515))$

其中，K₂＝diag(k₂₁...k_2n)>0，w_d为系统不确定性与外界未知干扰的总和的上界。Wherein, K ₂ =diag(k ₂₁ ...k _2n )>0, w _d is the upper bound of the sum of system uncertainty and external unknown interference.

本实施方式中涉及到有限时间稳定理论，具体如下：The finite time stability theory is involved in this embodiment, as follows:

考虑一个形式如下的非线性时变系统Consider a nonlinear time-varying system of the form

$\overset{\cdot \cdot}{x x} = = f f ((x x)),, f f ((00)) = = 00,, x x &Element; &Element; {R R}^{n no},, - - - - - - ((22 - - 11))$

其中f:U→Rⁿ为开区域U上对x连续的函数，且开区域U包含原点。where f:U→R ⁿ is a function continuous to x on the open region U, and the open region U contains the origin.

引理1：对于非线性系统(1-1)，假设存在定义在Rⁿ原点邻域内的连续函数V(x)，且实数c>0,0<α<1，满足：Lemma 1: For the nonlinear system (1-1), suppose there exists a neighborhood defined in the origin of R ⁿ The continuous function V(x) inside, and the real number c>0,0<α<1, satisfy:

(1)V(x)在中正定(1) V(x) at Zhongzhengding

(2) $\overset{\cdot}{V} (x) + {cV}^{α} (x) \leq 0, &ForAll; x &Element; \hat{U}$ (2) $\overset{&Center Dot;}{V} (x) + {cV}^{α} (x) \leq 0, &ForAll; x &Element; \hat{u}$

则系统(1-1)的原点是局部有限时间稳定。稳定时间取决于初始状态x(0)＝x₀，满足：Then the origin of system (1-1) is locally finite-time stable. The settling time depends on the initial state x(0)=x ₀ , satisfying:

${T T}_{x x} (({x x}_{00})) \leq \leq \frac{V V {(({x x}_{00}))}^{11 - - α α}}{c c ((11 - - α α))},, - - - - - - 22 - - 22))$

对于原点一些开邻域中的所有x₀成立。若且V(x)径向无界(V(x)→+∞时，||×||→+∞)，则系统(1-1)的原点是全局有限时间稳定。holds for all x ₀ in some open neighborhood of the origin. like And V(x) is radially unbounded (when V(x)→+∞, ||×||→+∞), then the origin of system (1-1) is globally finite-time stable.

引理2：假定x:[0,∞)→R一阶连续，且可微，而且t→∞时存在极限，那么若存在且有界，那么 Lemma 2: Suppose x:[0,∞)→R is first-order continuous and differentiable, and there is a limit when t→∞, then if exists and is bounded, then

本实施方式中，由于th^α(x₁)在x_1i＝0且时的微分为无穷大，为了避免这种奇异问题，设置一个阈值λ来判断奇异，因此定义如下In this embodiment, since th ^α (x ₁ ) is at x _1i =0 and When the differential is infinite, in order to avoid this singularity problem, a threshold λ is set to judge the singularity, so the definition as follows

$\overset{\cdot \cdot}{v v} (({x x}_{11})) = = \{\begin{matrix} - - {\overset{\cdot \cdot}{A A}}^{- - 11} (({x x}_{11})) {K K}_{11} {th the th}^{α α} (({x x}_{11})) - - {A A}^{- - 11} (({x x}_{11})) η η (({x x}_{11})),, & {\overset{\cdot &Center Dot;}{x x}}_{11} &NotEqual; &NotEqual; 00 \\ 00,, & {\overset{\cdot \cdot}{x x}}_{11} = = 00 \end{matrix},, - - - - - - ((11 - - 1616))$

${η η}_{i i} (({x x}_{11 i i})) = = \{\begin{matrix} {k k}_{11 i i} α α {| | {x x}_{11 i i} | |}^{α α - - 11} {\overset{\cdot &Center Dot;}{x x}}_{11 i i},, | | {x x}_{11 i i} | | &GreaterEqual; &Greater Equal; λ λ \\ {k k}_{11 i i} α α {| | {Δ Δ}_{i i} | |}^{α α - - 11} {\overset{\cdot &Center Dot;}{x x}}_{11 i i},, | | {x x}_{11 i i} | | < < λ λ \end{matrix}$

其中λ和△_i都是小的正常数，x_1i为x₁的第i个元素，η_i(x_1i)为η(x₁)中的第i个元素。Where λ and △ _i are small normal constants, x _1i is the i-th element of x ₁ , and η _i (x _1i ) is the i-th element in η(x ₁ ).

因为f(x)、g(x)和tanh(x)都是连续函数，因此该控制器还是连续的。Since f(x), g(x), and tanh(x) are all continuous functions, the controller is also continuous.

将控制器代入式(1-14)得Substituting the controller into formula (1-14) to get

$\begin{matrix} {\overset{\cdot &Center Dot;}{x x}}_{11} = = - - {K K}_{11} {th the th}^{α α} (({x x}_{11})) + + A A (({x x}_{11})) z z \\ \overset{\cdot &Center Dot;}{z z} = = - - {K K}_{22} {th the th}^{α α} ((z z)) - - {A A}^{T T} (({x x}_{11})) {x x}_{11} + + w w ((x x)) - - {w w}_{d d} s the s i i g g n no ((z z)) \end{matrix},, - - - - - - ((11 - - 1717))$

证明：prove:

第一步：证明全局渐近稳定Step 1: Prove that the global asymptotically stable

选取Lyapunov函数Choose the Lyapunov function

$V V = = \frac{11}{22} {x x}_{11}^{T T} {x x}_{11} + + \frac{11}{22} {z z}^{T T} z z$

则but

$\begin{matrix} \overset{\cdot &Center Dot;}{V V} = = {x x}_{11}^{T T} {\overset{\cdot \cdot}{x x}}_{11} + + {z z}^{T T} \overset{\cdot &Center Dot;}{z z} \\ = = {x x}_{11}^{T T} ((- - {K K}_{11} {th the th}^{α α} (({x x}_{11})) + + A A (({x x}_{11})) z z)) \\ + + {z z}^{T T} ((- - {K K}_{22} {th the th}^{α α} ((z z)) - - {A A}^{T T} (({x x}_{11})) {x x}_{11} + + w w ((x x)) - - {w w}_{d d} s the s i i g g n no ((z z)))) \\ = = - - {x x}_{11}^{T T} {K K}_{11} {th the th}^{α α} (({x x}_{11})) + + {x x}_{11}^{T T} A A (({x x}_{11})) z z - - {z z}^{T T} {K K}_{22} {th the th}^{α α} ((z z)) - - {z z}^{T T} {A A}^{T T} (({x x}_{11})) {x x}_{11} \\ + + {z z}^{T T} w w ((x x)) - - {z z}^{T T} {w w}_{d d} s the s i i g g n no ((z z)) \\ = = - - {x x}_{11}^{T T} {K K}_{11} {th the th}^{α α} (({x x}_{11})) - - {z z}^{T T} {K K}_{22} {th the th}^{α α} ((z z)) + + {z z}^{T T} w w ((x x)) - - {z z}^{T T} {w w}_{d d} s the s i i g g n no ((z z)) \end{matrix}$

由于||w(x)||≤w_d，所以z^Tw(x)≤z^Tw_dsign(z)，因此Since ||w(x)||≤w _d , so z ^T w(x)≤z ^T w _d sign(z), so

$\overset{\cdot &Center Dot;}{V V} \leq \leq - - {x x}_{11}^{T T} {K K}_{11} {th the th}^{α α} (({x x}_{11})) - - {z z}^{T T} {K K}_{22} {th the th}^{α α} ((z z)) \leq \leq 00$

x₁和z的L₂范数有界，由v(x₁)和z的定义知x₂有界，对于绝大多数系统有界，由引理2可知，当t→∞时，x₁→0，z→0，x₂→0，故系统(1-17)全局渐近稳定。The L ₂ norm of x ₁ and z is bounded. From the definition of v(x ₁ ) and z, we know that x ₂ is bounded. For most systems Bounded. From Lemma 2, we know that when t→∞, x ₁ →0, z→0, x ₂ →0, so the system (1-17) is globally asymptotically stable.

第二步：证明全局有限时间稳定Step 2: Prove that the global finite-time stability

y＝tanh(x)曲线与y＝x曲线如图7所示，当x₁,z在x₁＝0,z＝0的邻域内时，即||x₁||≤δ,||z||≤δ,δ≤0.5，th^α(x₁)≈sig^α(x₁)，th^α(z)≈sig^α(z)，此时，辅助控制器为The y=tanh(x) curve and the y=x curve are shown in Figure 7, when x ₁ , z is in the neighborhood of x ₁ =0, z=0, that is ||x ₁ ||≤δ,||z ||≤δ,δ≤0.5, th ^α (x ₁ )≈sig ^α (x ₁ ), th ^α (z)≈sig ^α (z), at this time, the auxiliary controller is

v(x₁)＝-A^-1(x₁)K₁sig^α(x₁)，(1-18)v(x ₁ )＝-A ^-1 (x ₁ )K ₁ sig ^α (x ₁ ), (1-18)

控制器为The controller is

$u u = = {g g}^{- - 11} ((x x)) ((\overset{\cdot &Center Dot;}{v v} (({x x}_{11})) - - f f ((x x)) - - {w w}_{d d} s the s i i g g n no ((z z)) - - {K K}_{22} {sig sig}^{α α} ((z z)) - - {A A}^{T T} (({x x}_{11})) {x x}_{11})),, - - - - - - ((11 - - 1919))$

将控制器代入式(1-15)得Substituting the controller into formula (1-15) to get

$\begin{matrix} {\overset{\cdot \cdot}{x x}}_{11} = = - - {K K}_{11} {sig sig}^{α α} (({x x}_{11})) + + A A (({x x}_{11})) z z \\ \overset{\cdot \cdot}{z z} = = - - {K K}_{22} {sig sig}^{α α} ((z z)) - - {A A}^{T T} (({x x}_{11})) {x x}_{11} - - {w w}_{d d} s the s i i g g n no ((z z)) + + w w ((x x)) \end{matrix},, - - - - - - ((11 - - 2020))$

只需要证明系统(1-20)在||x₁||≤δ,||z||≤δ,δ≤0.5内全局有限时间稳定就说明系统(1-17)是全局有限时间稳定。It is only necessary to prove that the system (1-20) is globally finite-time stable within ||x ₁ ||≤δ, ||z||≤δ, δ≤0.5, which means that the system (1-17) is globally finite-time stable.

选取Lyapunov函数Choose the Lyapunov function

${V V}_{11} = = \frac{11}{22} {x x}_{11}^{T T} {x x}_{11} + + \frac{11}{22} {z z}^{T T} z z$

对Lyapunov函数进行求导，可以得到Deriving the Lyapunov function, we can get

$\begin{matrix} {\overset{\cdot &Center Dot;}{V V}}_{11} = = {x x}_{11}^{T T} {\overset{\cdot &Center Dot;}{x x}}_{11} + + {z z}^{T T} \overset{\cdot &Center Dot;}{z z} \\ = = {x x}_{11}^{T T} ((- - {K K}_{11} {sig sig}^{α α} (({x x}_{11})) + + A A (({x x}_{11})) z z)) + + {z z}^{T T} ((- - {K K}_{22} {sig sig}^{α α} ((z z)) - - {A A}^{T T} (({x x}_{11})) {x x}_{11} - - {w w}_{d d} s the s i i g g n no ((z z)) + + w w ((x x)))) \\ = = - - {x x}_{11}^{T T} {K K}_{11} {sig sig}^{α α} (({x x}_{11})) + + {x x}_{11}^{T T} A A (({x x}_{11})) z z - - {z z}^{T T} {K K}_{22} {sig sig}^{α α} ((z z)) - - {z z}^{T T} {A A}^{T T} (({x x}_{11})) {x x}_{11} + + {z z}^{T T} ((w w ((x x)) - - {w w}_{d d} s the s i i g g n no ((z z)))) \\ = = - - {x x}_{11}^{T T} {K K}_{11} {sig sig}^{α α} (({x x}_{11})) - - {z z}^{T T} {K K}_{22} {sig sig}^{α α} ((z z)) + + {z z}^{T T} ((w w ((x x)) - - {w w}_{d d} s the s i i g g n no ((z z)))) \end{matrix},,$

$\begin{matrix} {\overset{\cdot &Center Dot;}{V V}}_{11} \leq \leq - - {x x}_{11}^{T T} {K K}_{11} {sig sig}^{α α} (({x x}_{11})) - - {z z}^{T T} {K K}_{22} {sig sig}^{α α} ((z z)) \\ = = - - {Σ Σ}_{i i = = 11}^{n no} {k k}_{11 i i} {| | {x x}_{11 i i} | |}^{11 + + α α} - - {Σ Σ}_{i i = = 11}^{n no} {k k}_{22 i i} {| | {z z}_{i i} | |}^{11 + + α α} \\ \leq \leq - - {\overset{&OverBar; &OverBar;}{k k}}_{11} {((\frac{11}{22} {Σ Σ}_{i i = = 11}^{n no} {x x}_{11 i i}^{22}))}^{μ μ} - - {\overset{&OverBar; &OverBar;}{k k}}_{2211} {((\frac{11}{22} {Σ Σ}_{i i = = 11}^{n no} {z z}_{i i}^{22}))}^{μ μ} \\ \leq \leq - - \overset{&OverBar; &OverBar;}{k k} {V V}_{11}^{μ μ} \end{matrix}$

其中 $μ = \frac{(1 + α)}{2}, \frac{1}{2} < μ < 1,$ k_1min＝min{k_1i},k_2min＝min{k_2i}， ${\overset{&OverBar;}{k}}_{1} = 2^{μ} k_{1 \min}, {\overset{&OverBar;}{k}}_{2} = 2^{μ} k_{2 \min},$ $\overset{&OverBar;}{k} = m i n {{\overset{&OverBar;}{k}}_{1}, {\overset{&OverBar;}{k}}_{2}} .$ in $μ = \frac{(1 + α)}{2}, \frac{1}{2} < μ < 1,$ k _1min = min{k _1i }, k _2min = min{k _2i }, ${\overset{&OverBar;}{k}}_{1} = 2^{μ} k_{1 \min}, {\overset{&OverBar;}{k}}_{2} = 2^{μ} k_{2 \min},$ $\overset{&OverBar;}{k} = m i no {{\overset{&OverBar;}{k}}_{1}, {\overset{&OverBar;}{k}}_{2}} .$

所以，根据引理1可以知道，对于任意给定的初始值x(0)＝x₀，x₁和z将在时间T之内稳定到原点，T为稳定时间。由v(x₁)和z的定义可知，当x₁＝0,z＝0，可以得到x₂＝0，从而系统(1-20)全局有限时间稳定。Therefore, according to Lemma 1, it can be known that for any given initial value x(0)=x ₀ , x ₁ and z will stabilize to the origin within time T, where T is the stabilization time. From the definitions of v(x ₁ ) and z, it can be seen that when x ₁ =0, z=0, x ₂ =0 can be obtained, so that the system (1-20) is globally stable in finite time.

故系统(1-17)是全局有限时间稳定。So the system (1-17) is globally finite-time stable.

下面将对本发明所提出的方法进行验证，首先对期望轨道和期望姿态进行计算。The method proposed by the present invention will be verified below. First, the expected orbit and expected attitude are calculated.

(1)期望轨道(1) Desired track

追踪航天器轨道期望之所以发生变化是因为目标的姿态运动导致了特征点在视线坐标系的相对位置变化。记目标空间飞行器特征点在其体坐标系下的单位矢量方向为n_b，则-n_b就是进行视线跟踪的期望轨道方向，所以，将追踪航天器的期望方向投影到惯性系得：The reason why the trajectory expectation of the tracking spacecraft changes is that the relative position of the feature points in the line-of-sight coordinate system changes due to the attitude movement of the target. Note that the unit vector direction of the feature point of the target spacecraft in its body coordinate system is n _b , then -n _b is the desired orbital direction for line-of-sight tracking, so project the desired direction of the tracking spacecraft into the inertial system:

${ρ ρ}_{i i} = = {C C}_{i i}^{b b t t} ((- - {n no}_{b b} {ρ ρ}_{f f})) = = {[\begin{matrix} {x x}_{i i} & {y the y}_{i i} & {z z}_{i i} \end{matrix}]}^{T T},, - - - - - - ((11 - - 21 twenty one))$

其中就是将目标体坐标系变换到惯性系的矩阵。在追踪航天器视线坐标系下表示的最终期望跟踪方向为ρ_l＝[ρ_f00]^T，ρ_f为期望的相对距离，从而得到在惯性坐标系下的表达式：in It is the matrix that transforms the target body coordinate system to the inertial system. The final expected tracking direction expressed in the line-of-sight coordinate system of the tracking spacecraft is ρ _l =[ρ _f 00] ^T , and ρ _f is the expected relative distance, so the expression in the inertial coordinate system is obtained:

${ρ ρ}_{i i} = = {C C}_{i i}^{l l} {ρ ρ}_{l l},, - - - - - - ((11 - - 22 twenty two))$

其中就是将视线坐标系变换到惯性坐标系的矩阵。in It is the matrix that transforms the line-of-sight coordinate system to the inertial coordinate system.

根据式(1-21)和式(1-22)容易得到期望视线倾角q_εf和期望的视线偏角q_βf。According to formula (1-21) and formula (1-22), it is easy to obtain the expected line-of-sight inclination q _εf and the expected line-of-sight declination q _βf .

对惯性坐标系而言，目标体坐标系是转动的，将该角速度投影到惯性坐标系下，记为ω_bt,i，可由下式计算：For the inertial coordinate system, the target body coordinate system is rotating, and the angular velocity is projected onto the inertial coordinate system, denoted as ω _bt,i , which can be calculated by the following formula:

${ω ω}_{b b t t,, i i} = = {C C}_{i i}^{b b t t} {ω ω}_{b b t t},, - - - - - - ((11 - - 23 twenty three))$

其中ω_bt是姿态角速度，其是目标相对于惯性坐标系的，则两航天器之间距离的变化率为：Where _ωbt is the attitude angular velocity, which is the target relative to the inertial coordinate system, then the rate of change of the distance between the two spacecraft is:

${\overset{\cdot &Center Dot;}{ρ ρ}}_{i i} = = {[\begin{matrix} {\overset{\cdot &Center Dot;}{x x}}_{i i} & {\overset{\cdot &Center Dot;}{y the y}}_{i i} & {\overset{\cdot &Center Dot;}{z z}}_{i i} \end{matrix}]}^{T T} = = {(({ω ω}_{b b t t,, i i}))}^{\times \times} {[\begin{matrix} {x x}_{i i} & {y the y}_{i i} & {z z}_{i i} \end{matrix}]}^{T T},, - - - - - - ((11 - - 24 twenty four))$

这样，根据式(1-22)和式(1-24)容易得到期望的视线倾角角速率和期望的视线偏角速率 In this way, according to formula (1-22) and formula (1-24), it is easy to get the desired line-of-sight inclination angle rate and the desired line-of-sight declination rate

(2)期望姿态(2) Expectation attitude

在进行近距离的非合作目标空间飞行器相对轨道的视线跟踪时，要求追踪飞行器能够实时监视目标。假设对测量设备进行安装时，其中心轴线与x_bcf轴方向相同，x_bc的方向与视线轴方向相同，则可以得到：When performing line-of-sight tracking of non-cooperative target space vehicles relative to the orbit at close range, it is required that the tracking vehicle can monitor the target in real time. Assuming that when the measuring equipment is installed, its central axis is in the same direction as the x _bcf axis, and the direction of x _bc is the same as the line of sight axis, then you can get:

${x x}_{b b c c f f} = = \frac{{ρ ρ}_{i i}}{{ρ ρ}_{f f}},, {y the y}_{b b c c f f} = = \frac{{ρ ρ}_{i i}^{\times \times} \overset{^^}{s the s}}{| | | | {ρ ρ}_{i i}^{\times \times} \overset{^^}{s the s} | | {| |}_{22}},, {z z}_{b b c c f f} = = {x x}_{b b c c f f}^{\times \times} {y the y}_{b b c c f f},, - - - - - - ((11 - - 2525))$

式中是惯性坐标系下太阳光线的方向，在进行跟踪时，要求入射光线垂直太阳能帆板，故安装太阳能帆板时，要求沿着追踪航天器体坐标系y轴。In the formula It is the direction of the sun's rays in the inertial coordinate system. When tracking, the incident light is required to be perpendicular to the solar panel. Therefore, when installing the solar panel, it is required to follow the y-axis of the spacecraft body coordinate system.

再由下式Then by the following formula

${I I}_{33} = = {C C}_{b b c c}^{i i} [\begin{matrix} {x x}_{b b c c f f} & {y the y}_{b b c c f f} & {z z}_{b b c c f f} \end{matrix}],, - - - - - - ((11 - - 2626))$

即可求解追踪航天器的期望姿态角θ_f、ψ_f，再对式(1-25)进行求导再联立式(1-26)即可求解出追踪航天器的期望姿态角速度。The expected attitude angle of the tracking spacecraft can be solved θ _f , ψ _f , then deriving Equation (1-25) and then combining Equation (1-26) to obtain the expected attitude angular velocity of the tracking spacecraft.

选取实际值与期望值之差作为状态变量，设开始时两航天器的相对距离为260m，首先转移到距离目标100m处，再进行视线跟踪。The difference between the actual value and the expected value is selected as the state variable, and the relative distance between the two spacecraft is set to be 260m at the beginning, first shift to a distance of 100m from the target, and then perform line-of-sight tracking.

目标在惯性系的初始位置是[2000，0，0]m，初始本体系与惯性系对齐，运行过程中角速度在本体系中是[-0.00250.002-0.002]rad/s，特征点在本体系中的单位方向矢量是轨道机动在本体系每个轴上为[-1,1]m/s²上服从均匀分布。The initial position of the target in the inertial system is [2000, 0, 0]m, the initial system is aligned with the inertial system, the angular velocity in the system is [-0.00250.002-0.002]rad/s during operation, and the feature points are in this system The unit direction vector in the system is Orbital maneuvers follow a uniform distribution on each axis of the system at [-1,1]m/s ² .

追踪航天器初始视线倾角为0.9rad，初始视线偏角为-3.1rad，初始姿态角为[0.2,0,3]rad，设太阳光照方向为转动惯量J_c＝[30,25,20]，在实际系统中，控制器的输出总是会饱和的，故在仿真中限制每轴所能提供的最大控制加速度为5m/s²，最大控制力矩为1Nm。按照式(1-15)设计控制器，w_d＝diag(1,1,1)，K₁＝diag(1.5,0.1,0.1,0.2,0.1,0.1)，K₂＝diag(75,2.5,4,2,3,3.5)，α＝0.8，λ＝0.01，仿真时间1000s，定步长为0.01s。The initial line-of-sight inclination angle of the tracking spacecraft is 0.9rad, the initial line-of-sight declination angle is -3.1rad, and the initial attitude angle is [0.2,0,3]rad. Let the sun illumination direction be Moment of inertia J _c =[30,25,20], in the actual system, the output of the controller will always be saturated, so in the simulation, the maximum control acceleration that can be provided by each axis is limited to 5m/s ² , the maximum control The torque is 1Nm. Design the controller according to formula (1-15), w _d ＝diag(1,1,1), K ₁ ＝diag(1.5,0.1,0.1,0.2,0.1,0.1), K ₂ ＝diag(75,2.5, 4,2,3,3.5), α=0.8, λ=0.01, the simulation time is 1000s, and the fixed step is 0.01s.

图2为在接近和跟踪非合作目标时，轨道相关参数随时间变化曲线，包括相对距离、视线倾角和视线偏角，从图中可以看出，在大约40s的时候，从相距目标转移到了，并保持跟踪上期望轨道。Figure 2 is the time-varying curve of orbit-related parameters when approaching and tracking a non-cooperative target, including relative distance, line-of-sight inclination and line-of-sight declination. And keep track of the desired track.

图3为在接近和跟踪非合作目标时，姿态角随时间变化曲线，根据图可以得到，大约40s后，姿态角快速趋向于期望值，能够在期望值附近运动，实现了对非合作目标航天器特定方向指向。Figure 3 is the time-varying curve of the attitude angle when approaching and tracking a non-cooperative target. According to the figure, it can be obtained that after about 40s, the attitude angle quickly tends to the expected value, and can move around the expected value, realizing the specificity of the non-cooperative target spacecraft. direction pointing.

图4为相对轨道的控制加速度的变化曲线，根据图可以看出，由于目标航天器是处于逃逸状态，追踪航天器为了能够跟踪上目标航天器，其控制加速度一直保持输出。Figure 4 is the change curve of the control acceleration relative to the orbit. According to the figure, it can be seen that since the target spacecraft is in an escape state, the tracking spacecraft keeps outputting its control acceleration in order to be able to track the target spacecraft.

图5为追踪航天器的姿态控制力矩变化曲线，根据图可以看出，大约在40s后，追踪航天器完成姿态指向后，追踪航天器姿态控制力矩的输出为0。Figure 5 is the attitude control torque change curve of the tracking spacecraft. According to the figure, it can be seen that after about 40s, after the tracking spacecraft completes the attitude pointing, the output of the attitude control torque of the tracking spacecraft is 0.

图6为轨道姿态参数与相应的期望参数之间的偏差随时间变化曲线，e1、e2、e3、e4、e5、e6分别为相对距离、视线倾角、视线偏角、的偏差，可以看出，当跟踪上期望信号之后，能够保持实际的轨道姿态参数偏离期望值非常小。Figure 6 is the time-varying curve of the deviation between the orbital attitude parameters and the corresponding expected parameters. e1, e2, e3, e4, e5, and e6 are the deviations of the relative distance, line-of-sight inclination, and line-of-sight declination, respectively. It can be seen that, After tracking the expected signal, the deviation of the actual orbital attitude parameters from the expected value can be kept very small.

Claims

1. A non-cooperative target spacecraft relative orbit attitude finite time control method is characterized by comprising the following steps:

projecting a relative orbit dynamic model represented by an inertial system to a sight line, and describing the relative orbit dynamic model of the spacecraft by using the sight line;

establishing a posture dynamics model and a posture kinematics model;

performing state space representation on the relative orbit dynamics model, the attitude dynamics model and the attitude kinematics model to obtain a relative orbit attitude dynamics model;

and step four, obtaining the finite time continuous controller according to the relative orbit attitude dynamics model and the finite time control theory.

2. The method for the finite-time control of the relative orbital attitude of a non-cooperative target spacecraft according to claim 1, wherein the step one of projecting the relative orbital dynamics model represented by the inertial system to the line-of-sight system, and the specific process of describing the relative orbital dynamics model of the spacecraft by the line-of-sight system comprises:

will use the inertia system O_ix_iy_iz_iProjection of the relative orbital dynamics model represented onto the visual system O_lx_ly_lz_l：

\begin{matrix} {(\frac{d^{2} ρ}{{dt}^{2}})}_{l} = \frac{d^{2} {(ρ)}_{l}}{{dt}^{2}} + {({\overset{\cdot}{ω}}_{l})}_{l}^{\times} {(ρ)}_{l} + 2 {(ω_{l})}_{l}^{\times} \frac{d {(ρ)}_{l}}{d t} + {(ω_{l})}_{l}^{\times} {(ω_{l})}_{l}^{\times} {(ρ)}_{l} \\ = {(Δ g)}_{l} + {(f)}_{l} - (u_{c}) l \end{matrix}, - - - (1 - 1)

Where ρ is the position vector of the target spacecraft relative to the tracking spacecraft, the superscript × represents the antisymmetric matrix of the vectors, △ g ═ △ g_x△g_y△g_z]^TFor gravity difference terms of two spacecraft, when performing close-range relative transfer and tracking, △ g equals 0 ═ f_xf_yf_z]^TFor the escape of the target spacecraft u_c＝[u_cxu_cyu_cz]^TIs an input control vector;

and (3) developing the formula (1-1) according to components to obtain a relative orbit dynamics model for describing the spacecraft by adopting a line-of-sight system:

\{\begin{matrix} \overset{\cdot\cdot}{ρ} - ρ ({\overset{\cdot}{q}}_{ϵ}^{2} + {\overset{\cdot}{q}}_{β}^{2} \cos^{2} q_{ϵ}) = {Δg}_{x} + f_{x} - u_{c x} \\ ρ {\overset{\cdot\cdot}{q}}_{ϵ} + 2 \overset{\cdot}{ρ} {\overset{\cdot}{q}}_{ϵ} + ρ {\overset{\cdot}{q}}_{β}^{2} \sin q_{ϵ} \cos q_{ϵ} = {Δg}_{y} + f_{y} - u_{c y} \\ - ρ {\overset{\cdot\cdot}{q}}_{β} \cos q_{ϵ} + 2 ρ {\overset{\cdot}{q}}_{β} {\overset{\cdot}{q}}_{ϵ} \sin q_{ϵ} - 2 \overset{\cdot}{ρ} {\overset{\cdot}{q}}_{β} \cos q_{ϵ} = {Δg}_{z} + f_{z} - u_{c z} \end{matrix}, - - - (1 - 2)

wherein q isIs the inclination angle of the line of sight, q_βIs the declination of the line of sight.

3. The method for controlling the relative orbit attitude of the non-cooperative target spacecraft according to claim 2, wherein the specific process of establishing the attitude kinetic model and the attitude kinematic model in the second step is as follows:

and b, a body coordinate system is represented by subscript b, t represents a target aircraft, c represents a tracking aircraft, and then the attitude dynamics equation of the tracking spacecraft, namely an attitude dynamics model, is as follows:

J_{c} {\overset{\cdot}{ω}}_{b c} + ω_{b c}^{\times} J_{c} ω_{b c} = T_{c}, - - - (1 - 3)

wherein, J_c＝[J_cxJ_cyJ_cz]^TIs moment of inertia, ω_bc＝[ω_xω_yω_z]^TIs attitude angular velocity, T_c＝[T_cxT_cyT_cz]^TIs the control torque of the motor to be controlled,

definition ofTheta and psi are angles of rotation of the tracked aircraft about the x, y and z axes of the body, respectively, and the attitude expressed by the Euler angle is as follows:

thus obtaining an attitude angular velocity of:

the definition matrix R is:

the pose kinematics model is then:

4. the method for the finite time control of the relative orbit attitude of the non-cooperative target spacecraft according to claim 3, wherein the state space representation of the relative orbit dynamics model, the attitude dynamics model and the attitude kinematics model in the third step is performed, and the specific process for obtaining the relative orbit attitude dynamics model is as follows:

expressions (1-2), (1-3) and (1-7) are expressed in the form of a state space:

[\begin{matrix} \overset{\cdot\cdot}{ρ} \\ {\overset{\cdot\cdot}{q}}_{ϵ} \\ {\overset{\cdot\cdot}{q}}_{β} \\ {\overset{\cdot}{ω}}_{x} \\ {\overset{\cdot}{ω}}_{y} \\ {\overset{\cdot}{ω}}_{z} \end{matrix}] = [\begin{matrix} ρ {\overset{\cdot}{q}}_{ϵ}^{2} + ρ {\overset{\cdot}{q}}_{β}^{2} \cos^{2} q_{ϵ} \\ - \frac{2 \overset{\cdot}{ρ}}{ρ} {\overset{\cdot}{q}}_{ϵ} - {\overset{\cdot}{q}}_{β}^{2} \sin q_{ϵ} \cos q_{ϵ} \\ \frac{2 {\overset{\cdot}{q}}_{β} {\overset{\cdot}{q}}_{ϵ} \sin q_{ϵ}}{\cos q_{ϵ}} - \frac{2 \overset{\cdot}{ρ} {\overset{\cdot}{q}}_{β}}{ρ} \\ \frac{(J_{c y} - J_{c z})}{J_{c x}} ω_{y} ω_{z} \\ \frac{(J_{c z} - J_{c x})}{J_{c y}} ω_{x} ω_{z} \\ \frac{(J_{c x} - J_{c y})}{J_{c z}} ω_{y} ω_{x} \end{matrix}] + [\begin{matrix} f_{x} \\ \frac{f_{y}}{ρ} \\ \frac{- f_{z}}{ρ \cos q_{ϵ}} \\ 0 \\ 0 \\ 0 \end{matrix}] + [\begin{matrix} - u_{c x} \\ \frac{- u_{c y}}{ρ} \\ \frac{u_{c z}}{ρ \cos q_{ϵ}} \\ \frac{T_{c x}}{J_{c x}} \\ \frac{T_{c y}}{J_{c y}} \\ \frac{T_{c z}}{J_{c z}} \end{matrix}], - - - (1 - 9)

is provided with

Then the equations (1-8) and (1-9) are in the form of the following state space, i.e. the dynamic model of the relative orbit attitude is:

\{\begin{matrix} {\overset{\cdot}{x}}_{1} = A (x_{1}) x_{2} \\ {\overset{\cdot}{x}}_{2} = f (x) + w (x) + g (x) u \end{matrix}, - - - (1 - 10)

wherein,

f (x) = [\begin{matrix} ρ {\overset{\cdot}{q}}_{ϵ}^{2} + ρ {\overset{\cdot}{q}}_{β}^{2} \cos^{2} q_{ϵ} \\ - \frac{2 \overset{\cdot}{ρ}}{ρ} {\overset{\cdot}{q}}_{ϵ} - {\overset{\cdot}{q}}_{β}^{2} \sin q_{ϵ} \cos q_{ϵ} \\ \frac{2 {\overset{\cdot}{q}}_{β} {\overset{\cdot}{q}}_{ϵ} \sin q_{ϵ}}{\cos q_{ϵ}} - \frac{2 \overset{\cdot}{ρ} {\overset{\cdot}{q}}_{β}}{ρ} \\ \frac{(J_{c y} - J_{c z})}{J_{c x}} ω_{y} ω_{z} \\ \frac{(J_{c z} - J_{c x})}{J_{c y}} ω_{x} ω_{z} \\ \frac{(J_{c x} - J_{c y})}{J_{c z}} ω_{y} ω_{x} \end{matrix}], w (x) = [\begin{matrix} f_{x} \\ \frac{f_{y}}{ρ} \\ \frac{- f_{z}}{ρ \cos q_{ϵ}} \\ 0 \\ 0 \\ 0 \end{matrix}], g (x) = [\begin{matrix} - 1 \\ \frac{- 1}{ρ} \\ \frac{1}{ρ \cos q_{ϵ}} \\ \frac{1}{J_{c x}} \\ \frac{1}{J_{c y}} \\ \frac{1}{J_{c z}} \end{matrix}],

w (x) is the sum of uncertainty and external interference of the system, and satisfies | | | w (x) | ≦ w_dWherein w is_d>0，u∈RⁿIs an input to the system.

5. The method for the finite time control of the relative orbit attitude of the non-cooperative target spacecraft according to claim 4, wherein the specific process of obtaining the finite time continuous controller according to the relative orbit attitude dynamics model and the finite time control theory in the step four is as follows:

designing an auxiliary controller:

v(x₁)＝-A^-1(x₁)K₁sign(x₁)tanh^α(|x₁|)，(1-11)

wherein, K₁＝diag(k₁₁,…,k_1n)，k_1i>0，0<α<1, sign (x) tanh^α(|x|)＝th^α(x) And then:

v(x₁)＝-A^-1(x₁)K₁th^α(x₁)，(1-12)

defining an error variable:

z＝x₂-v(x₁)，(1-13)

substituting the formulae (1-12) and (1-13) into the formulae (1-10) to obtain:

\begin{matrix} {\overset{\cdot}{x}}_{1} = A^{- 1} (x_{1}) (z + v (x_{1})) = - K_{1} {th}^{α} (x_{1}) + A (x_{1}) z \\ \overset{\cdot}{z} = f (x) + g (x) u + w (x) - \overset{\cdot}{v} (x_{1}) \end{matrix}, - - - (1 - 14)

the finite time continuous controller is:

u = g^{- 1} (x) (\overset{\cdot}{v} (x_{1}) - f (x) - w_{d} s i g n (z) - A^{T} (x_{1}) x_{1} - K_{2} {th}^{α} (z)), - - - (1 - 15)

wherein, K₂＝diag(k₂₁...k_2n)>0，w_dIs an upper bound on the sum of the system uncertainty and the external unknown interference.