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

Abstract

The invention discloses a relative orbit attitude finite time control method for a non-cooperative target spacecraft, and relates to the field of aerospace. The method solves the problems in present relative orbit attitude joint control of non-cooperative target spacecrafts. The relative orbit attitude finite time control method for the non-cooperative target spacecraft comprises the following steps: 1, projecting a relative orbit dynamics model expressed by an inertial system to a sight system, and describing the relative orbit dynamics model of the spacecraft by adopting the sight system; 2, establishing an attitude dynamics model and an attitude kinematics model; 3, 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 4, obtaining a finite time continuous controller according to the relative orbit attitude dynamics model and the finite time control theory. The method is suitable for relative orbit attitude joint control of non-cooperative target spacecrafts.

Description

Non-cooperative target spacecraft relative orbit attitude finite time control method

Technical Field

The invention relates to the field of aerospace.

Background

With the development and utilization of space resources, each country has to launch its own satellite and build its own space station, but with the increasing number of satellites and space stations, space resources are increasingly lacking, and due to the problem of the life or failure of satellites, a large amount of space waste is generated, occupying a lot of orbit resources. Therefore, the space tasks of approaching and closely tracking and monitoring non-cooperative targets such as space debris, invalid satellites and the like, maintaining invalid spacecrafts to enable the invalid spacecrafts to recover work and the like become research hotspots and difficulties in the field of current spaceflight. Although these space mission targets are various, there is a requirement that the orbit and attitude of the spacecraft can reach a desired state with high control accuracy during controlled flight, that is, the spacecraft is required to have high-accuracy relative position and attitude combined control accuracy. From another perspective, precise control of the relative position and relative attitude between the spacecraft and the target plays a critical role in whether a space mission can be completed. Moreover, the world aerospace technology has entered into the developed express way, and various countries develop space activities, so that space tasks are more and more diversified, the requirements on the functions of space systems are higher and higher, and the pose control technology of the spacecraft which moves in a short distance faces more challenges. Therefore, it is necessary to study the attitude and orbit joint control problem of non-cooperative targets.

The relative orbit maneuver of the spacecraft is a continuous motion rule for researching and tracking the spacecraft around a target spacecraft or a virtual spacecraft, and is a basis for researching space tasks such as orbit maneuver, formation flight, on-orbit maintenance, rendezvous and docking, tracking and monitoring and the like.

The methods of spacecraft relative orbit control can be divided into pulsed control and continuous thrust control, depending on the form of the controller output. Pulse control refers to a control method for controlling a propulsion device to provide one or more short speed increments for relative orbit transfer, and the generated thrust is regarded as a pulse because the working time of the propulsion device is very small and can be ignored. The pulse control is simple, but the unknown interference of the transfer process cannot be dealt with, and the control is not flexible. Continuous thrust control means that a control action is always exerted in the relative transfer process of a spacecraft, the continuous thrust control is increasingly applied to the relative orbit control of the spacecraft along with the development of a propulsion device and the improvement of a control theory, and the continuous control has stronger unknown interference capacity to the transfer process due to the control action in the transfer process, so that the continuous thrust control becomes a research hotspot of the relative orbit control in recent years.

The relative orbit control of the spacecraft is divided into two types, one is the relative orbit control of the spacecraft with a cooperative target, and the other is the relative orbit control of the spacecraft with a non-cooperative target.

By non-cooperative target spacecraft is meant a target spacecraft, such as a failed spacecraft, space debris, and hostile spacecraft, that the tracking spacecraft is unable to acquire the relative orbital parameters of the target spacecraft. The relative orbit control aiming at the non-cooperative target spacecraft plays a significant role in space tasks such as tracking, monitoring, interference, striking and the like of the space non-cooperative target.

In the relative orbit control of the non-cooperative target, on one hand, the tracking spacecraft is required to be capable of effectively estimating the relative position information and the relative speed information of the target spacecraft, and on the other hand, the tracking spacecraft is required to be capable of designing a control law by utilizing the estimated position information and the estimated speed information of the non-cooperative target, so that the tracking spacecraft can be transferred to the periphery of the target spacecraft.

Spacecraft attitude control refers to a technique for obtaining and preserving the orientation of a spacecraft in space, and generally refers to requiring the attitude of the spacecraft to change with a given requirement or rule relative to some reference frame.

The attitude control of the spacecraft can be divided into attitude regulation control and attitude tracking control. The spacecraft attitude adjustment control is to design a controller to stabilize the attitude of the spacecraft to be near a balance point. Compared with spacecraft attitude adjustment control, when a desired reference trajectory is a time-varying signal, the spacecraft attitude control problem is called attitude tracking control. Generally speaking, the gesture tracking problem is more difficult to deal with than the gesture adjustment problem because for the gesture tracking problem, the controller needs to not only stabilize the state variables of the whole system, but also make the output of the system track the desired trajectory in time-varying.

The document "XinM, panh, nonlinear control of a tracking and tracking target [ J ]. aerotopace science and technology,2011,15(2): 79-89" researches the relative orbit attitude joint control problem of a close-range non-cooperative target, and adopts a theta-D method to design a controller, wherein the controller has the advantages of high control precision, small error and the like, but the uncertainty of dynamics during target orbit maneuvering cannot be considered, so the control effect is poor.

The literature' Gaodeng Wei, Luojian Jun, Mawei, Kangshiyu, Chengxing Guang, nonlinear optimal control [ J ] astronavigation newspaper for approaching and tracking non-cooperative maneuvering targets, 2013,06: 773-. A theta-D correction controller is designed according to the Lyapunov minimum-maximum principle to reduce the control error of a non-cooperative target with a track and a posture maneuver existing simultaneously in tracking and improve the posture and orbit joint control performance of tracking the non-cooperative target. Although the method can obtain satisfactory control effect, the method is a controller designed according to the idea of asymptotic stability, that is, theoretically, the system can only converge to the equilibrium point when the time is infinite.

The method is characterized by comprising the following steps of establishing a six-degree-of-freedom Lagrange dynamical equation of a spacecraft aiming at the problem of autonomous approaching tracking of a non-cooperative rolling target spacecraft by the aid of partial state feedback attitude and orbit joint control [ J ] of a non-cooperative target, computer simulation, 2013,09:41-45+73 ], and designing an attitude and orbit joint controller by means of self-adaptive nonlinear output feedback control and neural network approximation control according to feedback information of relative positions and relative postures and uncertainty of inertial parameters of the spacecraft.

The specific contents of the scheme are as follows:

establishing a track dynamics equation based on the sight line coordinate system:

$\begin{array}{c}\stackrel{\·\·}{r}-r{\stackrel{\·}{\mathrm{\ψ}}}^2-r{(\stackrel{\·}{\mathrm{\θ}}-\mathrm{\ω})}^2{\cos }^2\mathrm{\ψ}=\frac{\mathrm{\μ}}{r_T^3}(-r+3r{\sin }^2{\mathrm{\θcos}}^2\mathrm{\ψ})+a_{x2}\\ r\stackrel{\·\·}{\mathrm{\ψ}}+2\stackrel{\·}{r}\stackrel{\·}{\mathrm{\ψ}}+r{(\stackrel{\·}{\mathrm{\θ}}-\mathrm{\ω})}^2\sin \mathrm{\ψ}\cos \mathrm{\ψ}=\frac{\mathrm{\μ}}{r_T^3}(-3r{\sin }^2\mathrm{\θ}\cos \mathrm{\ψ}\sin \mathrm{\ψ})+a_{y2}\\ r(\stackrel{\·\·}{\mathrm{\θ}}-\stackrel{\·}{\mathrm{\ω}})+2\stackrel{\·}{r}(\stackrel{\·}{\mathrm{\θ}}-\mathrm{\ω})-2r\stackrel{\·}{\mathrm{\ψ}}(\stackrel{\·}{\mathrm{\θ}}-\mathrm{\ω})\tan \mathrm{\ψ}=\frac{\mathrm{\μ}}{r_T^3}\left(3r\cos \mathrm{\θ}\sin \mathrm{\θ}\right)+a_{z2}\end{array},---\left(1\right)$

where μ is the gravitational constant, a_x2、a_y2、a_z2For controlling acceleration along three axes of a line-of-sight coordinate system, r being that of two spacecraftsThe line-of-sight distance between r_TPsi is the angle between the line of sight and its projection on the orbital plane of the target, referred to herein as the line of sight tilt angle, and theta is the line of sight declination. Omega is the simultaneous multiplication of the tracking spacecraft mass m by the two ends of various attitude angular velocities of the spacecraft_cA lagrange-like formal equation is obtained:

$J_o\left(q_s\right){\stackrel{\·\·}{q}}_s+C_o(q_s,{\stackrel{\·}{q}}_s){\stackrel{\·}{q}}_s+L_o\left(q_s\right)=u_o,---\left(2\right)$

wherein:

q_s＝[rψθ]^T

u_o＝[a_x2ra_y2ra_z2cos²ψ]^T

$J_o\left(q_s\right)=m_c\left[\begin{array}{ccc}1& 0& 0\\ 0& r^2& 0\\ 0& 0& r^2{\cos }^2\mathrm{\ψ}\end{array}\right]$

$C_o(q_s,{\stackrel{\·}{q}}_s)=m_c\left[\begin{array}{ccc}0& -r\stackrel{\·}{\mathrm{\ψ}}& -r\stackrel{\·}{\mathrm{\θ}}{\cos }^2\mathrm{\ψ}+2{\mathrm{r\ωcos}}^2\mathrm{\ψ}\\ r\stackrel{\·}{\mathrm{\ψ}}& r\stackrel{\·}{r}& r^2\mathrm{\θ}\sin \mathrm{\ψ}\cos \mathrm{\ψ}-2r^2\mathrm{\ω}\sin \mathrm{\ψ}\cos \mathrm{\ψ}\\ r\stackrel{\·}{\mathrm{\θ}}{\cos }^2\mathrm{\ψ}-2{\mathrm{r\ωcos}}^2\mathrm{\ψ}& -r^2\stackrel{\·}{\mathrm{\θ}}\sin \mathrm{\ψ}\cos \mathrm{\ψ}+2r^2\mathrm{\ω}\sin \mathrm{\ψ}\cos \mathrm{\ψ}& r\stackrel{\·}{r}{\cos }^2\mathrm{\ψ}-r^2\stackrel{\·}{\mathrm{\ψ}}\cos \mathrm{\ψ}\sin \mathrm{\ψ}\end{array}\right]$

the attitude dynamics equation of the tracking spacecraft is as follows:

$J_c\stackrel{\·}{\mathrm{\ω}}+\mathrm{\ω}\×J_c\mathrm{\ω}=t_c,---\left(3\right)$

in the formula J_cTo track the moment of inertia of the spacecraft, t_cThe control moment borne by the spacecraft is omega, and the attitude angular velocity of the spacecraft is omega.

The kinematic equation for tracking the attitude of the spacecraft is as follows:

$\stackrel{\·}{\mathrm{\σ}}=B\left(\mathrm{\σ}\right)\mathrm{\ω},---\left(4\right)$

wherein:

$\begin{array}{c}B\left(\mathrm{\σ}\right)=\frac{1}{4}\left(\right(1-{\mathrm{\σ}}^T\mathrm{\σ})\[I_{3\×3}\]+2{\mathrm{\σ\σ}}^T+2\[\widetilde{\mathrm{\σ}}\])\\ =\frac{1}{4}*\left[\begin{array}{ccc}(1-{\mathrm{\σ}}^2+2{\mathrm{\σ}}_i^2)& 2({\mathrm{\σ}}_i{\mathrm{\σ}}_j-{\mathrm{\σ}}_k)& 2({\mathrm{\σ}}_i{\mathrm{\σ}}_k+{\mathrm{\σ}}_j)\\ 2({\mathrm{\σ}}_i{\mathrm{\σ}}_j+{\mathrm{\σ}}_k)& (1-{\mathrm{\σ}}^2+2{\mathrm{\σ}}_j^2)& 2({\mathrm{\σ}}_j{\mathrm{\σ}}_k-{\mathrm{\σ}}_j)\\ 2({\mathrm{\σ}}_i{\mathrm{\σ}}_k-{\mathrm{\σ}}_j)& 2({\mathrm{\σ}}_j{\mathrm{\σ}}_k+{\mathrm{\σ}}_j)& (1-{\mathrm{\σ}}^2+2{\mathrm{\σ}}_k^2)\end{array}\right]\end{array}$

$\[\widetilde{\mathrm{\σ}}\]=\left[\begin{array}{ccc}0& -{\mathrm{\σ}}_k& {\mathrm{\σ}}_j\\ {\mathrm{\σ}}_k& 0& -{\mathrm{\σ}}_i\\ -{\mathrm{\σ}}_j& {\mathrm{\σ}}_i& 0\end{array}\right]$

the pose dynamics equation for the lagrange-like form, which can be formulated according to equations (3) and (4), is (where subscript a denotes the pose):

$J_a\left(\mathrm{\σ}\right)\stackrel{\·\·}{\mathrm{\σ}}+C_a(\mathrm{\σ},\stackrel{\·}{\mathrm{\σ}})\stackrel{\·}{\mathrm{\σ}}=u_a,---\left(5\right)$

wherein:

J_a(σ)＝B^-T(σ)J_cB^-1(σ)，

$S\left(J_c{B}^{-1}\stackrel{\·}{\mathrm{\σ}}\right)=S\left(J_c\mathrm{\ω}\right),$

u_a＝B^-T(σ)t_c，

$C_a(\mathrm{\σ},\stackrel{\·}{\mathrm{\σ}})=-B^{-T}{J}_c{B}^{-1}\stackrel{\·}{B}{B}^{-1}-B^{-T}S\left(J_c{B}^{-1}\stackrel{\·}{\mathrm{\σ}}\right)B^{-1}.$

the six-degree-of-freedom Lagrange dynamics equation of the tracked spacecraft can be obtained by combining the formula (2) and the formula (5):

$J^{*}\stackrel{\·\·}{x}+C\stackrel{\·}{x}+L=U,---\left(6\right)$

wherein: $J^{*}=\left[\begin{array}{cc}{J}_o& \\ & J_a\end{array}\right],C=\left[\begin{array}{cc}{C}_o& \\ & C_a\end{array}\right],L=\left[\begin{array}{c}{L}_o\\ 0_{3\×1}\end{array}\right],$ U＝[u_ou_a]^T，x＝[q_sσ]^T。

firstly, defining parameter estimation errorWhereinWhich is indicative of the error in the quality estimation,represents the moment of inertia estimation error and assumes:definition ofIs a constant, unknown parameter vector, namely:whereinIs composed ofAn estimate of (d). Defining a relative tracking error e ═ e_oe_a]^T∈R⁶Output e of pseudo rate filter_f＝[e_foe_fa]^T∈R⁶Auxiliary filter variable p (t) ∈ R⁶Then pseudo-rate filter dynamic equation:

$\begin{array}{c}{e}_f=-Ke+p\\ \stackrel{\·}{p}=-(K+I)p+(K^2+I)e\\ p\left(0\right)=Ke\left(0\right)\end{array},---\left(7\right)$

wherein K ∈ R^6×6Is a constant matrix, the introduced auxiliary tracking error variable η is η₁η₂]∈R⁶In the form ofSuppose x_dIs the expected value of x in formula (6), L_dIs the expected value of the L matrix in equation (6). Thereby obtaining a closed-loop error kinetic equation:

wherein $\tilde{L}=L-L_d,$

$\mathrm{\χ}=J^{*}\left(x\right){\stackrel{\·\·}{x}}_d+C(x,\stackrel{\·}{x}){\stackrel{\·\·}{x}}_d-J^{*}\left(x_d\right){\stackrel{\·\·}{x}}_d-C(x_d,{\stackrel{\·}{x}}_d){\stackrel{\·\·}{x}}_d+C(x,\stackrel{\·}{x})(e_f+e)+J^{*}\left(x\right)\mathrm{\η}-2J^{*}\left(x\right)e_f-\tilde{L},$ The activation function of the neural network is selected as a Gaussian functionThe neural network controller is

$U_2=-K{\left[\begin{array}{ccc}\tanh \left({\mathrm{\δe}}_{f1}\right)& \mathrm{...}& \tanh \left({\mathrm{\δe}}_{fn}\right)\end{array}\right]}^T+\widehat{\mathrm{\χ}},---\left(9\right)$

The self-adaptive nonlinear output feedback controller is designed as follows:

the following attitude and orbit combined controller U can be obtained₁+U₂Let us orderEstimated value In order to be the basis function(s),is an estimated value of the weight matrix, is an approximation error, and is a positive control parameter.

Then the adaptive parameter estimation change law without the velocity error term is:

wherein,_m、k_wIs a positive control parameter.

The form of the controller in this method is very complicated, which limits its use in practical engineering applications, and the control is designed based on the idea of asymptotic stability, which theoretically is only achieved when the time is infinite, and is not applicable when the system is required to reach a stable state within a limited time.

The document ' suyan, likang ' short-distance inter-satellite relative attitude and orbit coupling dynamics modeling and control [ J ]. space control technology and application 2014,04:20-25 ' designs a relative attitude and orbit combined control algorithm aiming at the problem of accurately controlling the short-distance inter-satellite relative position and relative attitude in the in-orbit service process, and realizes the accurate pointing fly-around motion of a target satellite.

The specific contents of the scheme are as follows:

establishing a relative attitude and orbit coupling dynamic model:

$\left\{\begin{array}{c}\stackrel{\·}{\mathrm{\ρ}}=\mathrm{\ν}\\ \stackrel{\·}{\mathrm{\ν}}=f(\mathrm{\ω},\stackrel{\·}{\mathrm{\ω}},\mathrm{\ρ},\stackrel{\·}{\mathrm{\ρ}},r_t)+C_{to}g(\mathrm{\ω},\stackrel{\·}{\mathrm{\ω}},\mathrm{\ρ},\stackrel{\·}{\mathrm{\ρ}},J_2)+C_{tc}a\\ {\stackrel{\·}{q}}_e=\frac{1}{2}\mathrm{\Ω}\left(q_e\right)C_{co}^T({\mathrm{\ω}}_{db}-{\mathrm{\ω}}_{cb})\\ {\stackrel{\·}{\mathrm{\ω}}}_{cb}=J_c^{-1}(-{\mathrm{\ω}}_{cb}^{\×}{J}_c{\mathrm{\ω}}_{cb}+T+d)\end{array}\right.,---\left(13\right)$

in the formula:

$f(\mathrm{\ω},\stackrel{\·}{\mathrm{\ω}},\mathrm{\ρ},\stackrel{\·}{\mathrm{\ρ}},r_t)=-2\mathrm{\ω}\×\stackrel{\·}{\mathrm{\ρ}}-\mathrm{\ω}\×(\mathrm{\ω}\×\mathrm{\ρ})-\stackrel{\·}{\mathrm{\ω}}\×\mathrm{\ρ}+C_{to}\frac{\mathrm{\μ}}{r_t^3}(\frac{3{\mathrm{xr}}_t}{r_t}-\mathrm{\ρ}),$

$g(\mathrm{\ω},\stackrel{\·}{\mathrm{\ω}},\mathrm{\ρ},\stackrel{\·}{\mathrm{\ρ}},J_2)=\frac{{\mathrm{\μJ}}_2{R}_E^2(1-5{\sin }^{2}i{\sin }^{2}u)}{r_t^4(r_t+2x)}{C}_{to}\·{\left[\begin{array}{ccc}6x& \frac{3y}{2}& \frac{3z}{2}\end{array}\right]}^T,$

$\mathrm{\Ω}\left(q_e\right)=\left[\begin{array}{c}{q}_{e4}{I}_3+q_{ev}^{\×}\\ -q_{ev}^T\end{array}\right],$ C_tc＝C_toC_oc，is a conversion matrix from a reference coordinate system to a reference star orbit coordinate system, mu is a gravitational constant, b_tAnd b_cAcceleration, ω andrespectively an angular velocity vector and an angular acceleration vector, r, of the target orbital coordinate system_t、r_cThe earth center distances of two satellites respectively represent transformation matrixes from an inertia system to a reference satellite orbit coordinate system, x, y and z represent the component sizes of position vectors in the reference satellite orbit coordinate system, J_cThe method is a representation of the moment of inertia of the formation stars under the body coordinate system of the formation stars.

X of the reference coordinate system_dThe shaft remains pointing to the reference star; y is_dAxis and x_dPerpendicular to the direction of the spacecraft and the earth mass center, and has the minimum included angle with a vector m, wherein m is a pointing vector for the spacecraft and the earth mass center; z is a radical of_dThe selection of axes follows the right hand rule. The reference coordinate system is as follows 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. The angular velocity and angular acceleration desired to be obtained are expressed in a reference coordinate system as: ${\mathrm{\ω}}_{db}^{\×}=-{\stackrel{\·}{C}}_{do}{C}_{do}^T,{\stackrel{\·}{\mathrm{\ω}}}_{db}^{\×}=-{\stackrel{\·}{C}}_{do}{\stackrel{\·}{C}}_{do}^T-{\stackrel{\·\·}{C}}_{do}{\stackrel{\·}{C}}_{do}^T.$

defining tracking error

$\{\begin{array}{c}{e}_{\mathrm{\ρ}}=\mathrm{\ρ}-{\mathrm{\ρ}}_d\\ e_v=v-v_d\\ e_{\mathrm{\ω}}={\mathrm{\ω}}_{cb}-{\mathrm{\ω}}_{db}-{\mathrm{\ω}}_{ed}\end{array},---\left(14\right)$

Where ρ is_dTo a desired relative position, v_dTo the desired relative velocity, ω_dbTo expect angular acceleration, ω_edIs the desired angular acceleration difference.

The design track control acceleration is as follows:

$a=C_{tc}^{-1}\[-f(\mathrm{\ω},\stackrel{\·}{\mathrm{\ω}},\mathrm{\ρ},\stackrel{\·}{\mathrm{\ρ}},r_t)-C_{to}g(\mathrm{\ω},\stackrel{\·}{\mathrm{\ω}},\mathrm{\ρ},\stackrel{\·}{\mathrm{\ρ}},J_2)+{\stackrel{\·}{v}}_d-{\mathrm{Re}}_{\mathrm{\ρ}}-{\mathrm{Qe}}_v\],---\left(15\right)$

control moment for taking attitude

T＝T_c1+T_c2，(16)

Wherein, T_c2The method is used for eliminating disturbance moment of attitude caused by track control introduced by the formation stars due to mass eccentricity and the like.

Design control moment

$T_{c1}={\mathrm{\ω}}_{cb}^{\×}{J}_c{\mathrm{\ω}}_{cb}+J_c(-{\mathrm{\ω}}_{cb}+{\mathrm{\ω}}_{cbd}+{\stackrel{\·}{\mathrm{\ω}}}_{cbd}),---\left(17\right)$

$T_{c2}=-(d_{max}+\mathrm{\ξ})\frac{s}{\left|\right|s\left|\right|},---\left(18\right)$

In the formula d_maxMaximum attitude disturbance moment introduced for rail-controlled thrust, ξ being a very small positive number, d_max+ξ>||d||，d∈M＝{d:||d||≤d_max＝l_max×F}，l_maxIs the maximum interference force arm, F is the rail control thrust,p is a positive definite matrix.

In this solution, the dynamic model used is complex, the form of the controller is relatively complex, it is still an asymptotically stable system, and theoretically, under the action of the controller, the tracking spacecraft can not reach the equilibrium point of the system in a limited time.

Disclosure of Invention

The invention provides a relative orbit attitude limited time control method of a non-cooperative target spacecraft, aiming at solving the problems in the prior relative orbit attitude joint control of the spacecraft of the non-cooperative target,

a non-cooperative target spacecraft relative orbit attitude finite time control method comprises 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.

Has the advantages that: the invention researches the relative orbit attitude joint control problem of the non-cooperative target spacecraft, considers unknown interference and uncertainty, combines a finite time control theory and provides an attitude and orbit joint control finite time controller.

Aiming at the relative orbit and attitude control of the non-cooperative spacecraft, the invention can obtain good control effect only by knowing the escape upper bound of the target spacecraft without estimating the escape of the target; due to the fact that uncertainty of the system and unknown external interference are considered, even if the uncertain external interference exists, the tracking spacecraft can still well track the non-cooperative target spacecraft which escapes; the controller is obtained by utilizing a finite time control principle, so that the spacecraft can track the spacecraft with a non-cooperative target in a finite time and track the spacecraft; meanwhile, the controller is simple in form and small in calculation amount, is suitable for being used on a satellite borne computer, is easy to realize engineering, and has practical engineering significance.

Drawings

FIG. 1 is a schematic diagram of an inertial system and a line of sight system and their relationship according to a second embodiment;

FIG. 2 is a graph of a time-varying trajectory-related parameter including relative distance, gaze inclination, and gaze declination as a non-cooperative target is approached and tracked;

FIG. 3 is a graph of attitude angle versus time as non-cooperative targets are approached and tracked;

FIG. 4 is a graph of control acceleration versus time as a non-cooperative target is approached and tracked;

FIG. 5 is a graph of control torque versus time as a non-cooperative target is approached and tracked;

FIG. 6 is a graph of deviation between an orbital attitude parameter and a corresponding desired parameter over time;

fig. 7 is a graph showing the relationship between the y-tanh (x) curve and the y-x curve.

Detailed Description

In a first specific embodiment, a method for controlling a relative orbit attitude of a non-cooperative target spacecraft in a limited time according to the first specific embodiment includes 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.

According to the method, the finite time continuous controller is obtained by adopting a continuous control mode through a finite time control theory, and then the spacecraft relative orbit attitude joint control of a non-cooperative target is realized.

In a second embodiment, the present embodiment is described with reference to fig. 1, and the difference between the present embodiment and the method for controlling the relative orbit attitude finite time of the non-cooperative target spacecraft in the first embodiment is that, in the first step, the relative orbit dynamical model represented by the inertial system is projected to the line-of-sight system, and a specific process for describing the relative orbit dynamical model of the spacecraft by using the line-of-sight system includes:

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

$\begin{array}{c}{\left(\frac{d^2\mathrm{\ρ}}{{\mathrm{dt}}^2}\right)}_l=\frac{d^2{\left(\mathrm{\ρ}\right)}_l}{{\mathrm{dt}}^2}+{\left({\stackrel{\·}{\mathrm{\ω}}}_l\right)}_l^{\×}{\left(\mathrm{\ρ}\right)}_l+2{\left({\mathrm{\ω}}_l\right)}_l^{\×}\frac{d{\left(\mathrm{\ρ}\right)}_l}{dt}+{\left({\mathrm{\ω}}_l\right)}_l^{\×}{\left({\mathrm{\ω}}_l\right)}_l^{\×}{\left(\mathrm{\ρ}\right)}_l\\ ={\left(\mathrm{\Δ}g\right)}_l+{\left(f\right)}_l-\left(u_c\right)l\end{array},---(1-1)$

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:

$\left\{\begin{array}{c}\stackrel{\·\·}{\mathrm{\ρ}}-\mathrm{\ρ}({\stackrel{\·}{q}}_{\mathrm{\ϵ}}^2+{\stackrel{\·}{q}}_{\mathrm{\β}}^2{\cos }^2{q}_{\mathrm{\ϵ}})={\mathrm{\Δg}}_x+f_x-u_{cx}\\ \mathrm{\ρ}{\stackrel{\·\·}{q}}_{\mathrm{\ϵ}}+2\stackrel{\·}{\mathrm{\ρ}}{\stackrel{\·}{q}}_{\mathrm{\ϵ}}+\mathrm{\ρ}{\stackrel{\·}{q}}_{\mathrm{\β}}^2{\mathrm{sinq}}_{\mathrm{\ϵ}}{\mathrm{cosq}}_{\mathrm{\ϵ}}={\mathrm{\Δg}}_y+f_y-u_{cy}\\ -\mathrm{\ρ}{\stackrel{\·\·}{q}}_{\mathrm{\β}}{\mathrm{cosq}}_{\mathrm{\ϵ}}+2\mathrm{\ρ}{\stackrel{\·}{q}}_{\mathrm{\β}}{\stackrel{\·}{q}}_{\mathrm{\ϵ}}{\mathrm{sinq}}_{\mathrm{\ϵ}}-2\stackrel{\·}{\mathrm{\ρ}}{\stackrel{\·}{q}}_{\mathrm{\β}}{\mathrm{cosq}}_{\mathrm{\ϵ}}={\mathrm{\Δg}}_z+f_z-u_{cz}\end{array}\right.,---(1-2)$

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

The difference between the third specific embodiment and the second specific embodiment in the method for controlling the relative orbit attitude finite time of the non-cooperative target spacecraft is that 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:

$J_c{\stackrel{\·}{\mathrm{\ω}}}_{bc}+{\mathrm{\ω}}_{bc}^{\×}{J}_c{\mathrm{\ω}}_{bc}=T_c,---(1-3)$

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:

a difference between the fourth specific embodiment and the third specific embodiment in the method for controlling the relative orbit attitude finite time of the non-cooperative target spacecraft is that the third step represents the relative orbit dynamics model, the attitude dynamics model and the attitude kinematics model in a state space, and a 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:

$\left[\begin{array}{c}\stackrel{\·\·}{\mathrm{\ρ}}\\ {\stackrel{\·\·}{q}}_{\mathrm{\ϵ}}\\ {\stackrel{\·\·}{q}}_{\mathrm{\β}}\\ {\stackrel{\·}{\mathrm{\ω}}}_x\\ {\stackrel{\·}{\mathrm{\ω}}}_y\\ {\stackrel{\·}{\mathrm{\ω}}}_z\end{array}\right]=\left[\begin{array}{c}\mathrm{\ρ}{\stackrel{\·}{q}}_{\mathrm{\ϵ}}^2+\mathrm{\ρ}{\stackrel{\·}{q}}_{\mathrm{\β}}^2{\cos }^2{q}_{\mathrm{\ϵ}}\\ -\frac{2\stackrel{\·}{\mathrm{\ρ}}}{\mathrm{\ρ}}{\stackrel{\·}{q}}_{\mathrm{\ϵ}}-{\stackrel{\·}{q}}_{\mathrm{\β}}^2{\mathrm{sinq}}_{\mathrm{\ϵ}}{\mathrm{cosq}}_{\mathrm{\ϵ}}\\ \frac{2{\stackrel{\·}{q}}_{\mathrm{\β}}{\stackrel{\·}{q}}_{\mathrm{\ϵ}}{\mathrm{sinq}}_{\mathrm{\ϵ}}}{{\mathrm{cosq}}_{\mathrm{\ϵ}}}-\frac{2\stackrel{\·}{\mathrm{\ρ}}{\stackrel{\·}{q}}_{\mathrm{\β}}}{\mathrm{\ρ}}\\ \frac{(J_{cy}-J_{cz})}{J_{cx}}{\mathrm{\ω}}_y{\mathrm{\ω}}_z\\ \frac{(J_{cz}-J_{cx})}{J_{cy}}{\mathrm{\ω}}_x{\mathrm{\ω}}_z\\ \frac{(J_{cx}-J_{cy})}{J_{cz}}{\mathrm{\ω}}_y{\mathrm{\ω}}_x\end{array}\right]+\left[\begin{array}{c}{f}_x\\ \frac{f_y}{\mathrm{\ρ}}\\ \frac{-f_z}{{\mathrm{\ρcosq}}_{\mathrm{\ϵ}}}\\ 0\\ 0\\ 0\end{array}\right]+\left[\begin{array}{c}-u_{cx}\\ \frac{-u_{cy}}{\mathrm{\ρ}}\\ \frac{u_{cz}}{{\mathrm{\ρcosq}}_{\mathrm{\ϵ}}}\\ \frac{T_{cx}}{J_{cx}}\\ \frac{T_{cy}}{J_{cy}}\\ \frac{T_{cz}}{J_{cz}}\end{array}\right],---(1-9)$

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:

$\left\{\begin{array}{c}{\stackrel{\·}{x}}_1=A\left(x_1\right)x_2\\ {\stackrel{\·}{x}}_2=f\left(x\right)+w\left(x\right)+g\left(x\right)u\end{array}\right.,---(1-10)$

wherein,

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.

A fifth specific embodiment, the difference between this specific embodiment and the fourth specific embodiment, is that 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 fourth step is as follows:

recording 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:

$\begin{array}{c}{\stackrel{\&CenterDot;}{x}}_1=A^{-1}\left(x_1\right)(z+v(x_1\left)\right)=-K_1{\mathrm{th}}^{\mathrm{\&alpha;}}\left(x_1\right)+A\left(x_1\right)z\\ \stackrel{\&CenterDot;}{z}=f\left(x\right)+g\left(x\right)u+w\left(x\right)-\stackrel{\&CenterDot;}{v}\left(x_1\right)\end{array},---(1-14)$

the finite time continuous controller is:

$u=g^{-1}\left(x\right)\left(\stackrel{\&CenterDot;}{v}\right(x_1)-f(x)-w_{d}sign(z)-A^T(x_1)x_1-K_2{\mathrm{th}}^{\mathrm{\&alpha;}}(z\left)\right),---(1-15)$

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

The present embodiment relates to the finite time stability theory, which is specifically as follows:

consider a nonlinear time-varying system of the form

$\stackrel{\&CenterDot;}{x}=f\left(x\right),f\left(0\right)=0,x\&Element;R^n,---(2-1)$

Wherein f is U → RⁿIs a continuous function of x over the open area U, and the open area U contains the origin.

Introduction 1: for the nonlinear system (1-1), it is assumed that there is a definition at RⁿNeighborhood of originInner continuous function V (x), and real number c>0,0<α<1, satisfying:

(1) v (x) is inMiddle school positive definition

(2) $\stackrel{\&CenterDot;}{V}\left(x\right)+{\mathrm{cV}}^{\mathrm{\&alpha;}}\left(x\right)\&le;0,\&ForAll;x\&Element;\hat{U}$

The origin of the system (1-1) is locally time-limited stable. The settling time depends on the initial state x (0) ═ x₀And satisfies the following conditions:

$T_x\left(x_0\right)\&le;\frac{V{\left(x_0\right)}^{1-\mathrm{\&alpha;}}}{c(1-\mathrm{\&alpha;})},---2-2)$

some all x in open neighborhood for origin₀This is true. If it isAnd v (x) radial unbounded (v (x) → + ∞, | × | → + ∞), then the origin of system (1-1) is globally time-limited stable.

2, leading: assuming that x: [0, ∞) → R is first order continuous and differentiable, and that there is a limit t → ∞ then ifIs present and bounded, then

In this embodiment, th^α(x₁) At x_1iIs equal to 0 andthe differential is infinite, and in order to avoid the singularity problem, a threshold lambda is set to determine the singularity, thus definingAs follows

$\stackrel{\&CenterDot;}{v}\left(x_1\right)=\left\{\begin{array}{cc}-{\stackrel{\&CenterDot;}{A}}^{-1}\left(x_1\right)K_1{\mathrm{th}}^{\mathrm{\&alpha;}}\left(x_1\right)-A^{-1}\left(x_1\right)\mathrm{\&eta;}\left(x_1\right),& {\stackrel{\&CenterDot;}{x}}_1\&NotEqual;0\\ 0,& {\stackrel{\&CenterDot;}{x}}_1=0\end{array}\right.,---(1-16)$

${\mathrm{\&eta;}}_i\left(x_{1i}\right)=\left\{\begin{array}{c}{k}_{1i}\mathrm{\&alpha;}{\left|x_{1i}\right|}^{\mathrm{\&alpha;}-1}{\stackrel{\&CenterDot;}{x}}_{1i},\left|x_{1i}\right|\&GreaterEqual;\mathrm{\&lambda;}\\ k_{1i}\mathrm{\&alpha;}{\left|{\mathrm{\&Delta;}}_i\right|}^{\mathrm{\&alpha;}-1}{\stackrel{\&CenterDot;}{x}}_{1i},\left|x_{1i}\right|<\mathrm{\&lambda;}\end{array}\right.$

Where λ and △_iAre all small normal numbers, x_1iIs x₁η_i(x_1i) Is η (x)₁) The ith element in (1).

Because f (x), g (x), and tanh (x) are continuous functions, the controller is also continuous.

The controller is substituted into (1-14)

$\begin{array}{c}{\stackrel{\&CenterDot;}{x}}_1=-K_1{\mathrm{th}}^{\mathrm{\&alpha;}}\left(x_1\right)+A\left(x_1\right)z\\ \stackrel{\&CenterDot;}{z}=-K_2{\mathrm{th}}^{\mathrm{\&alpha;}}\left(z\right)-A^T\left(x_1\right)x_1+w\left(x\right)-w_{d}sign\left(z\right)\end{array},---(1-17)$

And (3) proving that:

the first step is as follows: prove global asymptotic stabilization

Selecting Lyapunov function

$V=\frac{1}{2}{x}_1^T{x}_1+\frac{1}{2}{z}^{T}z$

Then

$\begin{array}{c}\stackrel{\&CenterDot;}{V}=x_1^T{\stackrel{\&CenterDot;}{x}}_1+z^T\stackrel{\&CenterDot;}{z}\\ =x_1^T(-K_1{\mathrm{th}}^{\mathrm{\&alpha;}}(x_1)+A(x_1\left)z\right)\\ +z^T(-K_2{\mathrm{th}}^{\mathrm{\&alpha;}}(z)-A^T(x_1)x_1+w(x)-w_{d}sign(z\left)\right)\\ =-x_1^T{K}_1{\mathrm{th}}^{\mathrm{\&alpha;}}\left(x_1\right)+x_1^{T}A\left(x_1\right)z-z^T{K}_2{\mathrm{th}}^{\mathrm{\&alpha;}}\left(z\right)-z^T{A}^T\left(x_1\right)x_1\\ +z^{T}w\left(x\right)-z^T{w}_{d}sign\left(z\right)\\ =-x_1^T{K}_1{\mathrm{th}}^{\mathrm{\&alpha;}}\left(x_1\right)-z^T{K}_2{\mathrm{th}}^{\mathrm{\&alpha;}}\left(z\right)+z^{T}w\left(x\right)-z^T{w}_{d}sign\left(z\right)\end{array}$

W is less than or equal to | w (x) | | w_dSo that z is^Tw(x)≤z^Tw_dsign (z) therefore

$\stackrel{\&CenterDot;}{V}\&le;-x_1^T{K}_1{\mathrm{th}}^{\mathrm{\&alpha;}}\left(x_1\right)-z^T{K}_2{\mathrm{th}}^{\mathrm{\&alpha;}}\left(z\right)\&le;0$

x₁And L of z₂Norm is bounded by v (x)₁) And z is defined as x₂Bounded, for most systemsBounded, as can be seen from the lemma 2, when t → ∞ x₁→0，z→0，x₂→ 0, so the system (1-17) is globally asymptotically stable.

The second step is that: proving global finite time stability

The y-tanh (x) curve and the y-x curve are shown in fig. 7, when x is₁Z is at x₁0, z is in the neighborhood of 0, i.e. | | x₁||≤,||z||≤,≤0.5，th^α(x₁)≈sig^α(x₁)，th^α(z)≈sig^α(z) at this time, the auxiliary controller is

v(x₁)＝-A^-1(x₁)K₁sig^α(x₁)，(1-18)

The controller is

$u=g^{-1}\left(x\right)\left(\stackrel{\&CenterDot;}{v}\right(x_1)-f(x)-w_{d}sign(z)-K_2{\mathrm{sig}}^{\mathrm{\&alpha;}}(z)-A^T(x_1\left)x_1\right),---(1-19)$

The controller is substituted into (1-15)

$\begin{array}{c}{\stackrel{\&CenterDot;}{x}}_1=-K_1{\mathrm{sig}}^{\mathrm{\&alpha;}}\left(x_1\right)+A\left(x_1\right)z\\ \stackrel{\&CenterDot;}{z}=-K_2{\mathrm{sig}}^{\mathrm{\&alpha;}}\left(z\right)-A^T\left(x_1\right)x_1-w_{d}sign\left(z\right)+w\left(x\right)\end{array},---(1-20)$

It is only necessary to prove that the system (1-20) is in | | | x₁The system (1-17) is globally limited in time stable if the globally limited time is less than or equal to 0.5.

Selecting Lyapunov function

$V_1=\frac{1}{2}{x}_1^T{x}_1+\frac{1}{2}{z}^{T}z$

The Lyapunov function is subjected to derivation to obtain

$\begin{array}{c}{\stackrel{\&CenterDot;}{V}}_1=x_1^T{\stackrel{\&CenterDot;}{x}}_1+z^T\stackrel{\&CenterDot;}{z}\\ =x_1^T(-K_1{\mathrm{sig}}^{\mathrm{\&alpha;}}(x_1)+A(x_1\left)z\right)+z^T(-K_2{\mathrm{sig}}^{\mathrm{\&alpha;}}(z)-A^T(x_1)x_1-w_{d}sign(z)+w(x\left)\right)\\ =-x_1^T{K}_1{\mathrm{sig}}^{\mathrm{\&alpha;}}\left(x_1\right)+x_1^{T}A\left(x_1\right)z-z^T{K}_2{\mathrm{sig}}^{\mathrm{\&alpha;}}\left(z\right)-z^T{A}^T\left(x_1\right)x_1+z^T\left(w\right(x)-w_{d}sign(z\left)\right)\\ =-x_1^T{K}_1{\mathrm{sig}}^{\mathrm{\&alpha;}}\left(x_1\right)-z^T{K}_2{\mathrm{sig}}^{\mathrm{\&alpha;}}\left(z\right)+z^T\left(w\right(x)-w_{d}sign(z\left)\right)\end{array},$

W is less than or equal to | w (x) | | w_dSo that z is^Tw(x)≤z^Tw_dsign (z) therefore

$\begin{array}{c}{\stackrel{\&CenterDot;}{V}}_1\&le;-x_1^T{K}_1{\mathrm{sig}}^{\mathrm{\&alpha;}}\left(x_1\right)-z^T{K}_2{\mathrm{sig}}^{\mathrm{\&alpha;}}\left(z\right)\\ =-\underset{i=1}{\overset{n}{\&Sigma;}}{k}_{1i}{\left|x_{1i}\right|}^{1+\mathrm{\&alpha;}}-\underset{i=1}{\overset{n}{\&Sigma;}}{k}_{2i}{\left|z_i\right|}^{1+\mathrm{\&alpha;}}\\ \&le;-{\stackrel{\&OverBar;}{k}}_1{\left(\frac{1}{2}\underset{i=1}{\overset{n}{\&Sigma;}}{x}_{1i}^2\right)}^{\mathrm{\&mu;}}-{\stackrel{\&OverBar;}{k}}_{21}{\left(\frac{1}{2}\underset{i=1}{\overset{n}{\&Sigma;}}{z}_i^2\right)}^{\mathrm{\&mu;}}\\ \&le;-\stackrel{\&OverBar;}{k}{V}_1^{\mathrm{\&mu;}}\end{array}$

Wherein $\mathrm{\&mu;}=\frac{(1+\mathrm{\&alpha;})}{2},\frac{1}{2}<\mathrm{\&mu;}<1,$ k_1min＝min{k_1i},k_2min＝min{k_2i}， ${\stackrel{\&OverBar;}{k}}_1=2^{\mathrm{\&mu;}}{k}_{1\min },{\stackrel{\&OverBar;}{k}}_2=2^{\mathrm{\&mu;}}{k}_{2\min },$ $\stackrel{\&OverBar;}{k}=min\{{\stackrel{\&OverBar;}{k}}_1,{\stackrel{\&OverBar;}{k}}_2\}.$

Therefore, as can be seen from lemma 1, for any given initial value x (0) ═ x₀，x₁And z will settle to the origin within time T, which is the settling time. From v (x)₁) And z, when x is defined₁X can be obtained when z is 0₂0, so that the system (1-20) is globally time-limited stable.

The system (1-17) is globally time-limited stable.

The method proposed by the invention will be verified below, first calculating the desired trajectory and the desired pose.

(1) Desired track

The change in the tracked spacecraft orbit is expected because the attitude motion of the target causes the relative position of the feature point in the line-of-sight coordinate system to change. Recording the unit vector direction of the target spacecraft feature point under the body coordinate system as n_bThen-n_bThat is, the desired orbital direction for line-of-sight tracking, so projecting the desired direction for tracking the spacecraft into the inertial system yields:

${\mathrm{\&rho;}}_i=C_i^{bt}(-n_b{\mathrm{\&rho;}}_f)={\left[\begin{array}{ccc}{x}_i& y_i& z_i\end{array}\right]}^T,---(1-21)$

whereinThe target body coordinate system is transformed to a matrix of the inertial system. The final desired tracking direction represented in the tracked spacecraft line-of-sight coordinate system is p_l＝[ρ_f00]^T，ρ_fThe desired relative distance is obtained, thereby obtaining the expression under the inertial coordinate system:

${\mathrm{\&rho;}}_i=C_i^l{\mathrm{\&rho;}}_l,---(1-22)$

whereinIs a matrix that transforms the line-of-sight coordinate system to the inertial coordinate system.

The desired line-of-sight inclination angle q can be easily obtained from the expressions (1-21) and (1-22)_fAnd a desired gaze declination q_βf。

For the inertial coordinate system, the target coordinate system is rotated, and the angular velocity is projected under the inertial coordinate system and is denoted by ω_bt,iIt can be calculated by the following formula:

${\mathrm{\&omega;}}_{bt,i}=C_i^{bt}{\mathrm{\&omega;}}_{bt},---(1-23)$

wherein ω is_btIs the attitude angular velocity, which is the target relative to the inertial coordinate system, the rate of change of the distance between the two spacecraft is:

${\stackrel{\&CenterDot;}{\mathrm{\&rho;}}}_i={\left[\begin{array}{ccc}{\stackrel{\&CenterDot;}{x}}_i& {\stackrel{\&CenterDot;}{y}}_i& {\stackrel{\&CenterDot;}{z}}_i\end{array}\right]}^T={\left({\mathrm{\&omega;}}_{bt,i}\right)}^{\&times;}{\left[\begin{array}{ccc}{x}_i& y_i& z_i\end{array}\right]}^T,---(1-24)$

thus, the desired line-of-sight inclination angle rate can be easily obtained from the expressions (1-22) and (1-24)And desired gaze declination rate

(2) Desired attitude

When the sight tracking of the close-distance non-cooperative target space vehicle relative to the orbit is carried out, the tracking vehicle is required to be capable of monitoring the target in real time. Assuming that the measuring device is installed, its central axis and x_bcfSame axial direction, x_bcIs the same as the direction of the sight axis, then:

$x_{bcf}=\frac{{\mathrm{\&rho;}}_i}{{\mathrm{\&rho;}}_f},y_{bcf}=\frac{{\mathrm{\&rho;}}_i^{\&times;}\hat{s}}{\left|\right|{\mathrm{\&rho;}}_i^{\&times;}\hat{s}|{|}_2},z_{bcf}=x_{bcf}^{\&times;}{y}_{bcf},---(1-25)$

in the formulaThe direction of the solar ray under the inertial coordinate system requires that the incident ray is vertical to the solar sailboard when tracking is carried out, so that the y axis of the coordinate system of the spacecraft is required to be tracked when the solar sailboard is installed.

Then the following formula

$I_3=C_{bc}^i\left[\begin{array}{ccc}{x}_{bcf}& y_{bcf}& z_{bcf}\end{array}\right],---(1-26)$

Namely, the expected attitude angle of the tracked spacecraft can be solvedθ_f、ψ_fAnd then the expected attitude angular velocity of the tracking spacecraft can be solved by carrying out derivation on the formula (1-25) and then carrying out the connection and the disconnection (1-26).

And selecting the difference between the actual value and the expected value as a state variable, setting the relative distance between the two spacecrafts to be 260m at the beginning, firstly transferring to a position 100m away from the target, and then carrying out sight tracking.

The initial position of the target in the inertial system is [2000, 0 ]]m, the initial body system is aligned with the inertial system, and the angular velocity is in the original position during operationIn the system is [ -0.00250.002-0.002]rad/s, unit direction vector of feature points in the system isThe rail motor is [ -1,1 ] on each axle of the main system]m/s²And uniformly distributed.

The initial line-of-sight inclination angle of the tracking spacecraft is 0.9rad, the initial line-of-sight declination angle is-3.1 rad, and the initial attitude angle is [0.2,0,3 ]]rad, set the sun illumination direction asMoment of inertia J_c＝[30,25,20]In a real system, the output of the controller is always saturated, so that the maximum control acceleration provided by each axis is limited to 5m/s in simulation²The maximum control torque is 1 Nm. The controller, w, is designed according to the formula (1-15)_d＝diag(1,1,1)，K₁＝diag(1.5,0.1,0.1,0.2,0.1,0.1)，K₂Biag (75,2.5,4,2,3,3.5), α 0.8, λ 0.01, simulation time 1000s, fixed step size 0.01 s.

Fig. 2 is a graph of the time-dependent parameters of the trajectory, including relative distance, line-of-sight inclination and line-of-sight declination, as it appears, at approximately 40s, shifted from a distant target and maintained tracking on a desired trajectory, as it approaches and tracks a non-cooperative target.

Fig. 3 is a time-varying curve of the attitude angle when approaching and tracking a non-cooperative target, and it can be obtained from the graph that after about 40s, the attitude angle rapidly approaches to the expected value, and can move around the expected value, so that the non-cooperative target spacecraft is pointed in a specific direction.

Fig. 4 is a variation curve of the control acceleration with respect to the orbit, and it can be seen from the graph that the control acceleration of the tracking spacecraft is always kept output in order to track the target spacecraft because the target spacecraft is in an escape state.

Fig. 5 is a variation curve of the attitude control moment of the tracking spacecraft, and it can be seen from the graph that after about 40s, the output of the attitude control moment of the tracking spacecraft is 0 after the tracking spacecraft completes attitude pointing.

Fig. 6 is a time-varying curve of the deviation between the orbit attitude parameter and the corresponding desired parameter, and e1, e2, e3, e4, e5, e6 are the deviations of the relative distance, the inclination angle of the line of sight, the declination angle of the line of sight, respectively.

Claims (5)

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：

$\begin{array}{c}{\left(\frac{d^2\mathrm{\&rho;}}{{\mathrm{dt}}^2}\right)}_l=\frac{d^2{\left(\mathrm{\&rho;}\right)}_l}{{\mathrm{dt}}^2}+{\left({\stackrel{\&CenterDot;}{\mathrm{\&omega;}}}_l\right)}_l^{\&times;}{\left(\mathrm{\&rho;}\right)}_l+2{\left({\mathrm{\&omega;}}_l\right)}_l^{\&times;}\frac{d{\left(\mathrm{\&rho;}\right)}_l}{dt}+{\left({\mathrm{\&omega;}}_l\right)}_l^{\&times;}{\left({\mathrm{\&omega;}}_l\right)}_l^{\&times;}{\left(\mathrm{\&rho;}\right)}_l\\ ={\left(\mathrm{\&Delta;}g\right)}_l+{\left(f\right)}_l-\left(u_c\right)l\end{array},---\left(1-1\right)$

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:

$\left\{\begin{array}{c}\stackrel{\&CenterDot;\&CenterDot;}{\mathrm{\&rho;}}-\mathrm{\&rho;}({\stackrel{\&CenterDot;}{q}}_{\mathrm{\&epsiv;}}^2+{\stackrel{\&CenterDot;}{q}}_{\mathrm{\&beta;}}^2{\cos }^2{q}_{\mathrm{\&epsiv;}})={\mathrm{\&Delta;g}}_x+f_x-u_{cx}\\ \mathrm{\&rho;}{\stackrel{\&CenterDot;\&CenterDot;}{q}}_{\mathrm{\&epsiv;}}+2\stackrel{\&CenterDot;}{\mathrm{\&rho;}}{\stackrel{\&CenterDot;}{q}}_{\mathrm{\&epsiv;}}+\mathrm{\&rho;}{\stackrel{\&CenterDot;}{q}}_{\mathrm{\&beta;}}^2\sin q_{\mathrm{\&epsiv;}}\cos q_{\mathrm{\&epsiv;}}={\mathrm{\&Delta;g}}_y+f_y-u_{cy}\\ -\mathrm{\&rho;}{\stackrel{\&CenterDot;\&CenterDot;}{q}}_{\mathrm{\&beta;}}\cos q_{\mathrm{\&epsiv;}}+2\mathrm{\&rho;}{\stackrel{\&CenterDot;}{q}}_{\mathrm{\&beta;}}{\stackrel{\&CenterDot;}{q}}_{\mathrm{\&epsiv;}}\sin q_{\mathrm{\&epsiv;}}-2\stackrel{\&CenterDot;}{\mathrm{\&rho;}}{\stackrel{\&CenterDot;}{q}}_{\mathrm{\&beta;}}\cos q_{\mathrm{\&epsiv;}}={\mathrm{\&Delta;g}}_z+f_z-u_{cz}\end{array}\right.,---\left(1-2\right)$

wherein q is Is 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:

$J_c{\stackrel{\&CenterDot;}{\mathrm{\&omega;}}}_{bc}+{\mathrm{\&omega;}}_{bc}^{\&times;}{J}_c{\mathrm{\&omega;}}_{bc}=T_c,---(1-3)$

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:

$\left[\begin{array}{c}\stackrel{\&CenterDot;\&CenterDot;}{\mathrm{\&rho;}}\\ {\stackrel{\&CenterDot;\&CenterDot;}{q}}_{\mathrm{\&epsiv;}}\\ {\stackrel{\&CenterDot;\&CenterDot;}{q}}_{\mathrm{\&beta;}}\\ {\stackrel{\&CenterDot;}{\mathrm{\&omega;}}}_x\\ {\stackrel{\&CenterDot;}{\mathrm{\&omega;}}}_y\\ {\stackrel{\&CenterDot;}{\mathrm{\&omega;}}}_z\end{array}\right]=\left[\begin{array}{c}\mathrm{\&rho;}{\stackrel{\&CenterDot;}{q}}_{\mathrm{\&epsiv;}}^2+\mathrm{\&rho;}{\stackrel{\&CenterDot;}{q}}_{\mathrm{\&beta;}}^2{\cos }^2{q}_{\mathrm{\&epsiv;}}\\ -\frac{2\stackrel{\&CenterDot;}{\mathrm{\&rho;}}}{\mathrm{\&rho;}}{\stackrel{\&CenterDot;}{q}}_{\mathrm{\&epsiv;}}-{\stackrel{\&CenterDot;}{q}}_{\mathrm{\&beta;}}^2\sin q_{\mathrm{\&epsiv;}}\cos q_{\mathrm{\&epsiv;}}\\ \frac{2{\stackrel{\&CenterDot;}{q}}_{\mathrm{\&beta;}}{\stackrel{\&CenterDot;}{q}}_{\mathrm{\&epsiv;}}\sin q_{\mathrm{\&epsiv;}}}{\cos q_{\mathrm{\&epsiv;}}}-\frac{2\stackrel{\&CenterDot;}{\mathrm{\&rho;}}{\stackrel{\&CenterDot;}{q}}_{\mathrm{\&beta;}}}{\mathrm{\&rho;}}\\ \frac{\left(J_{cy}-J_{cz}\right)}{J_{cx}}{\mathrm{\&omega;}}_y{\mathrm{\&omega;}}_z\\ \frac{\left(J_{cz}-J_{cx}\right)}{J_{cy}}{\mathrm{\&omega;}}_x{\mathrm{\&omega;}}_z\\ \frac{\left(J_{cx}-J_{cy}\right)}{J_{cz}}{\mathrm{\&omega;}}_y{\mathrm{\&omega;}}_x\end{array}\right]+\left[\begin{array}{c}{f}_x\\ \frac{f_y}{\mathrm{\&rho;}}\\ \frac{-f_z}{\mathrm{\&rho;}\cos q_{\mathrm{\&epsiv;}}}\\ 0\\ 0\\ 0\end{array}\right]+\left[\begin{array}{c}-u_{cx}\\ \frac{-u_{cy}}{\mathrm{\&rho;}}\\ \frac{u_{cz}}{\mathrm{\&rho;}\cos q_{\mathrm{\&epsiv;}}}\\ \frac{T_{cx}}{J_{cx}}\\ \frac{T_{cy}}{J_{cy}}\\ \frac{T_{cz}}{J_{cz}}\end{array}\right],---\left(1-9\right)$

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:

$\left\{\begin{array}{c}{\stackrel{\&CenterDot;}{x}}_1=A\left(x_1\right)x_2\\ {\stackrel{\&CenterDot;}{x}}_2=f\left(x\right)+w\left(x\right)+g\left(x\right)u\end{array}\right.,---\left(1-10\right)$

wherein,

$f\left(x\right)=\left[\begin{array}{c}\mathrm{\&rho;}{\stackrel{\&CenterDot;}{q}}_{\mathrm{\&epsiv;}}^2+\mathrm{\&rho;}{\stackrel{\&CenterDot;}{q}}_{\mathrm{\&beta;}}^2{\cos }^2{q}_{\mathrm{\&epsiv;}}\\ -\frac{2\stackrel{\&CenterDot;}{\mathrm{\&rho;}}}{\mathrm{\&rho;}}{\stackrel{\&CenterDot;}{q}}_{\mathrm{\&epsiv;}}-{\stackrel{\&CenterDot;}{q}}_{\mathrm{\&beta;}}^2\sin q_{\mathrm{\&epsiv;}}\cos q_{\mathrm{\&epsiv;}}\\ \frac{2{\stackrel{\&CenterDot;}{q}}_{\mathrm{\&beta;}}{\stackrel{\&CenterDot;}{q}}_{\mathrm{\&epsiv;}}\sin q_{\mathrm{\&epsiv;}}}{\cos q_{\mathrm{\&epsiv;}}}-\frac{2\stackrel{\&CenterDot;}{\mathrm{\&rho;}}{\stackrel{\&CenterDot;}{q}}_{\mathrm{\&beta;}}}{\mathrm{\&rho;}}\\ \frac{\left(J_{cy}-J_{cz}\right)}{J_{cx}}{\mathrm{\&omega;}}_y{\mathrm{\&omega;}}_z\\ \frac{\left(J_{cz}-J_{cx}\right)}{J_{cy}}{\mathrm{\&omega;}}_x{\mathrm{\&omega;}}_z\\ \frac{\left(J_{cx}-J_{cy}\right)}{J_{cz}}{\mathrm{\&omega;}}_y{\mathrm{\&omega;}}_x\end{array}\right],w\left(x\right)=\left[\begin{array}{c}{f}_x\\ \frac{f_y}{\mathrm{\&rho;}}\\ \frac{-f_z}{\mathrm{\&rho;}\cos q_{\mathrm{\&epsiv;}}}\\ 0\\ 0\\ 0\end{array}\right],g\left(x\right)=\left[\begin{array}{c}-1\\ \frac{-1}{\mathrm{\&rho;}}\\ \frac{1}{\mathrm{\&rho;}\cos q_{\mathrm{\&epsiv;}}}\\ \frac{1}{J_{cx}}\\ \frac{1}{J_{cy}}\\ \frac{1}{J_{cz}}\end{array}\right],$

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:

$\begin{array}{c}{\stackrel{\&CenterDot;}{x}}_1=A^{-1}\left(x_1\right)\left(z+v\left(x_1\right)\right)=-K_1{\mathrm{th}}^{\mathrm{\&alpha;}}\left(x_1\right)+A\left(x_1\right)z\\ \stackrel{\&CenterDot;}{z}=f\left(x\right)+g\left(x\right)u+w\left(x\right)-\stackrel{\&CenterDot;}{v}\left(x_1\right)\end{array},---\left(1-14\right)$

the finite time continuous controller is:

$u=g^{-1}\left(x\right)\left(\stackrel{\&CenterDot;}{v}\right(x_1)-f(x)-w_{d}sign(z)-A^T(x_1)x_1-K_2{\mathrm{th}}^{\mathrm{\&alpha;}}(z\left)\right),---(1-15)$

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