CN105739511A - Under-actuated spacecraft hover asymptotic control method for lacking of trace control - Google Patents

Abstract

The invention provides an under-actuated spacecraft hover sliding mode control method suitable for lacking of trace control. A kinetic model is established by aiming at the problem of under-actuated spacecraft hover control. System controllability under the under-actuated condition of lacking of trace control acceleration is analyzed based on the kinetic model, and a hover direction feasible set under the condition is given. The model acts as a controlled object, and a closed-loop control law under the trace under-actuated condition is designed by adopting the sliding mode control method. The under-actuated controller can drive a tracking spacecraft to be asymptotically stabilized to the given feasible hover direction, and the closed-loop system has great robustness and dynamic performance of external perturbation and model error so that the problem of under-actuated spacecraft hover control for lacking of trace control can be solved.

Description

A kind of lack mark to control underactuated spacecraft hover asymptotic control method

Technical field

The present invention relates to a kind of spacecraft flight control method, in particular, provide a kind of asymptotically stable control method for disappearance mark to the underactuated spacecraft hovering controlled particularly to a kind of.

Background technology

Spacecraft hovers, and refers to by pursuit spacecraft is applied lasting control power effect so that it is remain unchanged relative to the relative position of space passive space vehicle.Hovering technology has a extensive future in space tasks, for instance, in asteroid overhead, it can be carried out effective high-resolution scientific observation by hovering.Additionally, earth Orbital Space Vehicle is hovered, make pursuit spacecraft keep the geo-stationary to passive space vehicle, be conducive to carrying out the low coverage operations such as space maintenance, spatial observation, reduce space tasks operation easier and risk.Existing spacecraft Hovering control method all assumes that hovering dynamic system is full driving control system (the control input dimension of system is equal with degree of freedom in system), namely at radial direction, mark to all there is an independent control passage with normal direction.If the controller of a direction breaks down, cause that the direction cannot provide control power effect, then hovering dynamic system becomes under-actuated systems.For this under-actuated systems, existing full driving control method cannot be suitable for, and causes hovering mission failure.Generally, conventional solution is for installing thrust reserve device additional to tackle above-mentioned failure condition, but this certainly will cause the quality of spacecraft to increase with cost.Considering the constraints such as the architecture quality of spacecraft, manufacturing cost and launch cost, more economical and practical method should be design underactuated control device, even if thus under drive lacking case conditions, also can realize spacecraft hovering task.

Existing underactuated spacecraft relative orbit control is many based on space tasks application such as Spacecraft Formation Flyings, not yet has underactuated spacecraft Hovering control technique study.Although formation flight and hovering belong to Spacecraft Relative Motion category, but its track attribute is different, thus corresponding controller design exists difference.Specifically, for formation flight, spacecraft all runs on Keplerian orbit, and the relative orbit between spacecraft is relative orbit free period, namely need not apply control power to maintain relative orbit.But for hovering, it usually needs tracker being continuously applied control masterpiece in order to maintain relative orbit, namely hover configuration, thus tracker runs on non-Keplerian orbit.At present, the underactuated control of non-Kepler's relative orbit is theoretical and method research is also little, and therefore, the present invention is with underactuated spacecraft hovering for application, it is proposed that a kind of underactuated control method of non-Kepler's relative orbit suitable in this application.

Summary of the invention

The present invention solves the problem that underactuated spacecraft hovers, it is proposed that a kind of sliding-mode control.For underactuated spacecraft Hovering control problem, establish its kinetic model.Based on this kinetic model, analyze disappearance mark to the System Controllability controlled in acceleration situation, and give the hovering orientation feasible set in this situation.Additionally, due to lack mark to controlling input channel, the input channel of external perturbation and model error is no longer identical with the control input channel of system, becomes dismatching disturbance.Under there is dismatching disturbance situation, how to realize disappearance mark hover to the spacecraft of control action, be emphasis and the difficult point of the present invention.The present invention, with the drive lacking hovering kinetic model set up for controll plant, ingenious utilizes dynamics of relative motion coupled characteristic in orbital plane, adopts sliding-mode control design closed loop control rule in this drive lacking situation.The advantage of this underactuated control device is in that: (1) can drive pursuit spacecraft Asymptotic Stability to given feasible hovering orientation lacking mark to when controlling acceleration, and hovering position control accuracy is high；(2) closed loop system has good dynamic property, and the external perturbation and model error to dismatching has good robustness and inhibitory action；(3) than installing thrust reserve device to tackle the conventional method of Actuator failure, the underactuated control utensil that the present invention proposes has minimizing spacecraft structure quality, reduces the remarkable advantages such as spacecraft manufacturing cost, launch cost.The present invention creatively solves the underactuated control problem of spacecraft this kind of non-Kepler's relative orbit of hovering, proposed controller can lack radial direction or mark to completing circular orbit spacecraft hovering task when controlling acceleration, Project Realization for underactuated spacecraft hovering provides effective scheme, may be directly applied to the actual hovering tasks such as space asteroid hovering detection and earth orbit space service.

Technical scheme is as follows:

First feasible name hovering orientation is given according to drive lacking situation, corresponding nominal relative motion state is calculated based on this, then calculate the margin of error of actual relative motion state and name relative motion state, finally adopt sliding-mode control design control law, calculate actual controlled quentity controlled variable.In practical application, pursuit spacecraft and the real-time relative motion state of passive space vehicle are obtained by relative navigation system measurement on pursuit spacecraft star, will can be realized underactuated spacecraft Hovering control function by the calculated controlled quentity controlled variable transmission of the method to actuator.

The present invention " a kind of lack mark to control underactuated spacecraft hover asymptotic control method ", it comprises the following steps that, as shown in Figure 1:

Step one: drive lacking situation judges: if disappearance mark is to controlling acceleration, then U_y=0；

Step 2: given name hovering orientation also solves the nominal controlled quentity controlled variable of correspondence: solve disappearance mark to the hovering orientation feasible set in control acceleration situation according to actual drive lacking situationAnd in feasible set, select name hovering orientation ρ_d=[x_dy_dz_d]^T, solve the nominal controlled quentity controlled variable U of correspondence_2d；

Step 3: the margin of error calculates: calculate the margin of error e between actual relative motion state and name relative motion state₂；

Step 4: design of control law: choose sliding-mode surface and Reaching Law, adopts sliding-mode control design underactuated spacecraft Hovering control rule, calculates actual controlled quentity controlled variable U₂；

Wherein, the U described in step one_yFor mark to controlling acceleration；

Wherein, the name hovering orientation described in step 2 is ρ_d=[x_dy_dz_d]^T, x in formula_d、y_dAnd z_dRespectively name radially, mark to normal direction hovering position, subscript T represents the transposition of vector or matrix；For hovering orientation feasible set, its solution procedure is divided into three steps, concrete method for solving to be:

1) mathematical model of underactuated spacecraft hovering is set up

The coordinate system definition describing spacecraft hovering kinetic model is as follows；As in figure 2 it is shown, O_EX_IY_IZ_IFor geocentric inertial coordinate system, wherein O_EFor the earth's core；O_TXyz is that initial point is positioned at passive space vehicle barycenter O_TRelative motion coordinate system, wherein x-axis is along passive space vehicle radially, and z-axis is along passive space vehicle orbital plane normal direction, and y-axis constitutes Descartes's right hand rectangular coordinate system with x, z-axis；O_CFor pursuit spacecraft barycenter；R_CWith R_TRespectively pursuit spacecraft and the earth's core of passive space vehicle are from vector；Make ρ=[xyz]^TWith $v={\left[\begin{array}{ccc}\stackrel{\·}{x}& \stackrel{\·}{y}& \stackrel{\·}{z}\end{array}\right]}^T$ The respectively Relative position vector of pursuit spacecraft and passive space vehicle and relative velocity vector statement in relative motion coordinate system, then underactuated spacecraft hovering kinetic model is

${\stackrel{\·}{X}}_2=F_2\left(X_2\right)+{\mathrm{BU}}_2---\left(1\right)$

Wherein

F₂=[0_1×3f_yf_xf_z]^T(2)

$\left[\begin{array}{c}{f}_x\\ f_y\\ f_z\end{array}\right]=\left[\begin{array}{c}2{\stackrel{\·}{u}}_T\stackrel{\·}{y}+{\stackrel{\·}{u}}_T^{2}x+{\stackrel{\·\·}{u}}_{T}y+n_T^2{R}_T-n_C^2(R_T+x)\\ -2{\stackrel{\·}{u}}_T\stackrel{\·}{x}+{\stackrel{\·}{u}}_T^{2}y-{\stackrel{\·\·}{u}}_{T}x-n_C^{2}y\\ -n_C^{2}z\end{array}\right]---\left(3\right)$

B=[0_2×4I_2×2]^T(4)

U₂=[U_xU_z]^T(5)

In formula, subscript 2 represents the disappearance mark drive lacking situation to control acceleration； $X_2={\left[\begin{array}{cc}{X}_{2u}^T& X_{2a}^T\end{array}\right]}^T$ For by non-driven state X_2uWith driving condition X_2aThe relative motion state vector of composition；Owing to disappearance mark is to controlling acceleration, $X_{2u}={\left[\begin{array}{cccc}x& y& z& \stackrel{\·}{y}\end{array}\right]}^T$ And $X_{2a}={\left[\begin{array}{cc}\stackrel{\·}{x}& \stackrel{\·}{z}\end{array}\right]}^T;$ U₂=[U_xU_z]^TFor controlling input, wherein U_xAnd U_zRespectively radially control acceleration with normal direction；0_m×nIt is the null matrix of m × n for dimension, I_m×nIt is the unit matrix of m × n for dimension；u_TFor passive space vehicle latitude argument,WithRespectively passive space vehicle orbit angular velocity and angular acceleration；AndWherein R_TWith R_C=[(R_T+x)²+y²+z²]^1/2Respectively passive space vehicle and pursuit spacecraft the earth's core from, μ is Gravitational coefficient of the Earth；

2) drive lacking hovering dynamic system controllability is analyzed

If (namely passive space vehicle is positioned at circular orbitAnd), and pursuit spacecraft and passive space vehicle relative distance much smaller than its earth's core from, then drive lacking hovering kinetic model available linearization is

${\stackrel{\·}{X}}_2=A_2{X}_2+{\mathrm{BU}}_2---\left(6\right)$

Wherein

$A_2=\left[\begin{array}{cccccc}0& 0& 0& 0& 1& 0\\ 0& 0& 0& 1& 0& 0\\ 0& 0& 0& 0& 0& 1\\ 0& 0& 0& 0& -2n_T& 0\\ 3n_T^2& 0& 0& 2n_T& 0& 0\\ 0& 0& -n_T^2& 0& 0& 0\end{array}\right]---\left(7\right)$

Adopt lineary system theory to above-mentioned disappearance mark to control acceleration drive lacking when linearized system formula (6) carry out controllability analysis, analysis result shows, if disappearance mark is to controlling acceleration, under-actuated systems formula (6) is non-fully controlled, can be analyzed to and can control as follows and can not control subspace

$\left[\begin{array}{c}{\stackrel{\·}{\stackrel{\‾}{X}}}_2^c\\ {\stackrel{\·}{\stackrel{\‾}{X}}}_2^u\end{array}\right]=\left[\begin{array}{cc}{\stackrel{\‾}{A}}_2^{cc}& {\stackrel{\‾}{A}}_2^{cu}\\ 0_{1\×5}& {\stackrel{\‾}{A}}_2^{uu}\end{array}\right]\left[\begin{array}{c}{\stackrel{\‾}{X}}_2^c\\ {\stackrel{\‾}{X}}_2^u\end{array}\right]+\left[\begin{array}{c}{\stackrel{\‾}{B}}_2^c\\ 0_{1\×2}\end{array}\right]U_2---\left(8\right)$

Wherein

$\begin{array}{cc}{\stackrel{\‾}{A}}_2^{cc}=\left[\begin{array}{ccccc}0& 0& 0& 1& 0\\ 0& 0& 0& 0& 1\\ 1& 0& 0& 0& 0\\ -n_T^2& 0& 0& 0& 0\\ 0& -n_T^2& 0& 0& 0\end{array}\right],& {\stackrel{\‾}{A}}_2^{cu}=\left[\begin{array}{c}0\\ 0\\ 0\\ 3n_T^2\\ 0\end{array}\right]\\ {\stackrel{\‾}{A}}_2^{uu}=0,& {\stackrel{\‾}{B}}_2^c={\left[\begin{array}{cc}{0}_{2\×3}& I_{2\×2}\end{array}\right]}^T\end{array}---\left(9\right)$

In formula, ${\stackrel{\‾}{X}}_2^c={\left[\begin{array}{ccccc}-\stackrel{\·}{y}/\left(2n_T\right)& z& -y/\left(2n_T\right)& \stackrel{\·}{x}& \stackrel{\·}{z}\end{array}\right]}^T$ With ${\stackrel{\‾}{X}}_2^u=x+\stackrel{\·}{y}/\left(2n_T\right)$ Respectively controlled and uncontrollable state vector；

3) drive lacking hovering orientation feasible set is solved

According to hovering definition, the relative position of pursuit spacecraft and passive space vehicle remains unchanged in relative motion coordinate system；If definition name hovering orientation is ρ_d=[x_dy_dz_d]^T, then $v_d={\stackrel{\·}{\mathrm{\ρ}}}_d={\left[\begin{array}{ccc}{\stackrel{\·}{x}}_d& {\stackrel{\·}{y}}_d& {\stackrel{\·}{z}}_d\end{array}\right]}^T=0_{3\×1},$ And ${\stackrel{\·}{v}}_d={\stackrel{\·\·}{\mathrm{\ρ}}}_d={\left[\begin{array}{ccc}{\stackrel{\·\·}{x}}_d& {\stackrel{\·\·}{y}}_d& {\stackrel{\·\·}{z}}_d\end{array}\right]}^T=0_{3\×1};$ If it is further assumed that (namely passive space vehicle is positioned at circular orbitAnd), then obtained by formula (3),

$\left[\begin{array}{c}{f}_x\\ f_y\\ f_z\end{array}\right]=\left[\begin{array}{c}(n_T^2-n_C^2)(R_T+x_d)\\ (n_T^2-n_C^2)y_d\\ -n_C^2{z}_d\end{array}\right]---\left(10\right)$

The nominal controlled quentity controlled variable U of drive lacking hovering orientation feasible set and correspondence when below solving disappearance mark to Acceleration Control_2d；

When disappearance mark is to control acceleration, i.e. U_yWhen=0, formula (1) obtain,

$\left[\begin{array}{c}{\stackrel{\·\·}{x}}_d\\ {\stackrel{\·\·}{y}}_d\\ {\stackrel{\·\·}{z}}_d\end{array}\right]=\left[\begin{array}{c}{f}_x+U_{xd}\\ f_y\\ f_z+U_{zd}\end{array}\right]=\left[\begin{array}{c}(n_T^2-n_C^2)(R_T+x_d)+U_{xd}\\ (n_T^2-n_C^2)y_d\\ -n_C^2{z}_d+U_{zd}\end{array}\right]=0_{3\×1}---\left(11\right)$

Visible, for realizing hovering, it is desirable to f_y=0, namelySolve hovering orientation feasible set when namely the equation obtains disappearance mark to control acceleration；It is easy to get, Equation f_yThe solution of=0 is n_T=n_COr y_d=0；Solve n_T=n_CThe hovering orientation feasible set obtaining correspondence is Γ₂₁={ ρ_d|2R_Tx_d+||ρ_d||²=0}, in formula,For relative distance, and symbol | | | | represent the norm of vector, solve y_d=0 feasible solution obtaining correspondence is Γ₂₂={ ρ_d|y_d=0}；Thus, disappearance mark to hovering orientation feasible set when controlling acceleration is

Γ₂=Γ₂₁∪Γ₂₂(12)

In formula, symbol ∪ represents union of sets collection；Additionally, by 2) in System Controllability analyze, disappearance mark is non-fully controlled to controlling system in acceleration situation, and uncontrollable state isThen in whole control process, uncontrollable stateKeep its initial value constant, namelyX in formula₀WithRespectively initial time fractional radial position and relative mark are to speed；Consider that hovering definition requires the final momentThen the feasible hovering position of radial direction in final moment isBased on above-mentioned analysis, non-fully controlled due to system, disappearance mark is modified to hovering orientation feasible set when controlling acceleration

${\widetilde{\mathrm{\Γ}}}_2=\left\{{\mathrm{\ρ}}_d\right|{\mathrm{\ρ}}_d\∈{\mathrm{\Γ}}_2,x_d=x_0+{\stackrel{\·}{y}}_0/\left(2n_T\right)\}---\left(13\right)$

Meanwhile, by Equation f_x+U_xd=0 and f_z+U_zd=0, name controlled quentity controlled variable U can be obtained_2dFor

$U_{2d}=\left[\begin{array}{c}{U}_{xd}\\ U_{zd}\end{array}\right]=\left[\begin{array}{c}(n_C^2-n_T^2)(R_T+x_d)\\ n_C^2{z}_d\end{array}\right]---\left(14\right)$

Wherein, the margin of error calculated between actual relative motion state and name relative motion state described in step 3 of the present invention, its computational methods are:

e₂=X₂-X_2d(15)

In formula, $X_2={\left[\begin{array}{cccccc}x& y& z& \stackrel{\·}{y}& \stackrel{\·}{x}& \stackrel{\·}{z}\end{array}\right]}^T$ For disappearance mark to control acceleration when actual relative motion state, wherein x, y, z, WithRespectively actual diametrically opposite position, mark are to relative position, normal direction relative position, diametrically speed, mark to relative velocity and normal direction relative velocity； $X_{2d}={\left[\begin{array}{cc}{\mathrm{\ρ}}_d^T& 0_{1\×3}\end{array}\right]}^T$ For name relative motion state；

Wherein, the design sliding formwork control law described in step 4 of the present invention, calculate actual controlled quentity controlled variable U₂, its method is:

Consider the external perturbation power effect in real space environment, then by the drive lacking hovering kinetic model taken the photograph under condition be

${\stackrel{\·}{X}}_2=A_2{X}_2+{\mathrm{BU}}_2+{\stackrel{\‾}{D}}_2+{\mathrm{\ΔF}}_2\left(X_2\right)---\left(16\right)$

In formula,For external perturbation force vector, Δ F₂(X₂)=F₂(X₂)-A₂X₂For linearized stability vector；

By 2) in analyze, name hovering kinetics equation be

${\stackrel{\·}{X}}_{2d}=F_2\left(X_{2d}\right)+{\mathrm{BU}}_{2d}=0_{6\×1}---\left(17\right)$

Definition error relative motion state is $e_2=X_2-X_{2d}={\left[\begin{array}{cccccc}{e}_x& e_y& e_z& {\stackrel{\·}{e}}_y& {\stackrel{\·}{e}}_x& {\stackrel{\·}{e}}_z\end{array}\right]}^T,$ Wherein e_x、e_yAnd e_zRespectively radially, mark to normal direction the relative position error,WithRespectively radially, mark to normal direction relative velocity error；Being obtained error dynamics model by formula (16) with (17) work difference is

${\stackrel{\·}{e}}_2=A_2{e}_2+{\mathrm{Bu}}_2+D_2---\left(18\right)$

Wherein

$D_2={\stackrel{\‾}{D}}_2+{\mathrm{\ΔF}}_2\left(X_2\right)-{\mathrm{\ΔF}}_2\left(X_{2d}\right)---\left(19\right)$

In formula, u₂=U₂-U_2dFor error control amount； $D_2={\left[\begin{array}{cc}{0}_{1\×3}& d_2^T\end{array}\right]}^T$ For total perturbing vector of external perturbation and linearized stability composition, wherein, d₂=[d_yd_xd_z]^T, d_x、d_yAnd d_zRespectively radially, mark to and normal disturbance；

Hereinafter design sliding mode controller:

The error dynamics equation (18) method shown in formula (8) is carried out controllability STRUCTURE DECOMPOSITION, and can control part rewrite as follows by therein

$\left\{\begin{array}{c}{\stackrel{\·}{\stackrel{\‾}{e}}}_{2u}=A_{21}{\stackrel{\‾}{e}}_{2u}+A_{22}{e}_{2a}+d_{2u}\\ {\stackrel{\·}{e}}_{2a}=A_{23}{e}_{2u}+A_{24}{e}_{2a}+B_{2a}{U}_2+d_{2a}\end{array}\right.---\left(20\right)$

Wherein

$\begin{array}{cc}{A}_{21}=\left[\begin{array}{ccc}0& 0& 0\\ 0& 0& 0\\ 1& 0& 0\end{array}\right],& A_{22}=\left[\begin{array}{cc}1& 0\\ 0& 1\\ 0& 0\end{array}\right]\\ A_{23}=\left[\begin{array}{cccc}3n_T^2& 0& 0& 2n_T\\ 0& 0& -n_T^2& 0\end{array}\right],& A_{24}=\left[\begin{array}{cc}0& 0\\ 0& 0\end{array}\right]\end{array}---\left(21\right)$

In formula, ${\stackrel{\‾}{e}}_{2u}={\left[\begin{array}{ccc}-{\stackrel{\·}{e}}_y/\left(2n_T\right)& e_z& -e_y/\left(2n_T\right)\end{array}\right]}^T,e_{2a}={\left[\begin{array}{cc}{\stackrel{\·}{e}}_x& {\stackrel{\·}{e}}_z\end{array}\right]}^T$ And $e_{2u}={\left[\begin{array}{cccc}{e}_x& e_y& e_z& {\stackrel{\·}{e}}_y\end{array}\right]}^T;$ Additionally, d_2u=[-d_y/(2n_T)00]^TAnd d_2a=[d_xd_z]^T；

In like manner, it is contemplated thatButRightDo linear transformation ${\tilde{e}}_{2u}=P_{21}{\stackrel{\‾}{e}}_{2u}$ MakeWherein matrixIt is defined as

$P_{21}=\left[\begin{array}{cc}{k}_{21}^{-1}& 0\\ 0& 1\end{array}\right]\left[\begin{array}{ccc}{k}_{21}& 0& k_{22}\\ 0& 1& 0\end{array}\right]---\left(22\right)$

In formula, k₂₁And k₂₂For design parameter, meet k₂₁k₂₂> 0；

Notice P₂₁A₂₂=I_2×2, thenKinetics equation be

${\stackrel{\·}{\tilde{e}}}_{2u}=P_{22}{\stackrel{\‾}{e}}_{2u}+e_{2a}+P_{21}{d}_{2u}---\left(23\right)$

In formula, P₂₂=P₂₁A₂₁；

Definition sliding-mode surface is

$s_2=e_{2a}+P_{22}{\stackrel{\‾}{e}}_{2u}+{\mathrm{\α}}_2{\tilde{e}}_{2u}+{\mathrm{\β}}_{2}g\left({\tilde{e}}_{2u}\right)---\left(24\right)$

Wherein

$g_i\left({\tilde{e}}_{2ui}\right)=\left\{\begin{array}{cc}{\tilde{e}}_{2ui}^{p/q},& \begin{array}{ccc}{\tilde{s}}_{2i}=0& or& {\tilde{s}}_{2i}\&NotEqual;0,\left|{\tilde{e}}_{2ui}\right|\&GreaterEqual;\mathrm{\&delta;}\end{array}\\ {\mathrm{\&nu;}}_{1i}{\tilde{e}}_{2ui}+{\mathrm{\&nu;}}_{2i}{\tilde{e}}_{2ui}^2\mathrm{sgn}\left({\tilde{e}}_{1ui}\right),& {\tilde{s}}_{2i}\&NotEqual;0,\left|{\tilde{e}}_{2ui}\right|<\mathrm{\&delta;}\end{array}\right.,(i=1,2)---\left(25\right)$

And

${\tilde{s}}_2={\left[\begin{array}{cc}{\tilde{s}}_{21}& {\tilde{s}}_{22}\end{array}\right]}^T=e_{2a}+P_{22}{\stackrel{\&OverBar;}{e}}_{2u}+{\mathrm{\&alpha;}}_2{\tilde{e}}_{2u}+{\mathrm{\&beta;}}_2{\tilde{e}}_{2u}^{p/q}---\left(26\right)$

In formula, α₂> 0 and β₂> 0 is design parameter；VectorFor $g\left({\tilde{e}}_{2u}\right)={\left[\begin{array}{cc}{g}_1\left({\tilde{e}}_{2u1}\right)& g_2\left({\tilde{e}}_{2u2}\right)\end{array}\right]}^T;$ P and q is positive odd number, and p < q；Coefficient ν_1iAnd ν_2iFor ν_1i=(2-p/q) δ^p/q-1And v_2i=(p/q-1) δ^p/q-2, δ > 0 is design parameter；

The Reaching Law chosen is

${\stackrel{\&CenterDot;}{s}}_2=u_{2s}=-K_{21}{s}_2-K_{22}{\mathrm{sig}}^{{\mathrm{\&gamma;}}_2}\left(s_2\right)---\left(27\right)$

In formula,WithFor positive definite diagonal matrix；VectorFor ${\mathrm{sig}}^{{\mathrm{\&gamma;}}_2}\left(s_2\right)={\left[\begin{array}{cc}{\left|s_{21}\right|}^{{\mathrm{\&gamma;}}_2}\mathrm{sgn}\left(s_{21}\right)& {\left|s_{22}\right|}^{{\mathrm{\&gamma;}}_2}\mathrm{sgn}\left(s_{22}\right)\end{array}\right]}^T,$ Wherein 0 < γ₂< 1 is design parameter；

The error control rule obtained based on above-mentioned sliding-mode surface and Reaching Law is

u₂=u_2eq+u_2s(28)

Wherein

$u_{2eq}=-(A_{23}{e}_{2u}+A_{24}{e}_{2a})-P_{22}(A_{21}{\stackrel{\&OverBar;}{e}}_{2u}+A_{22}{e}_{2a})-{\mathrm{\&alpha;}}_2(P_{22}{\stackrel{\&OverBar;}{e}}_{2u}+e_{2a})-{\mathrm{\&beta;}}_2\stackrel{\&CenterDot;}{g}\left({\tilde{e}}_{2u}\stackrel{\&CenterDot;}{,}{\stackrel{\&CenterDot;}{\tilde{e}}}_{2u}\right)---\left(29\right)$

${\stackrel{\&CenterDot;}{g}}_i({\tilde{e}}_{2ui},{\stackrel{\&CenterDot;}{\tilde{e}}}_{2ui})=\left\{\begin{array}{cc}(p/q){\tilde{e}}_{2ui}^{p/q-1}{\stackrel{\&CenterDot;}{\tilde{e}}}_{2ui},& \begin{array}{ccc}{\tilde{s}}_{2i}=0& or& {\tilde{s}}_{2i}\&NotEqual;0,\left|{\tilde{e}}_{2ui}\right|\&GreaterEqual;\mathrm{\&delta;}\end{array}\\ {\mathrm{\&nu;}}_{1i}{\stackrel{\&CenterDot;}{\tilde{e}}}_{2ui}+2{\mathrm{\&nu;}}_{2i}{\tilde{e}}_{2ui}{\stackrel{\&CenterDot;}{\tilde{e}}}_{2ui}\mathrm{sgn}\left({\tilde{e}}_{2ui}\right),& {\tilde{s}}_{2i}\&NotEqual;0,\left|{\tilde{e}}_{2ui}\right|<\mathrm{\&delta;}\end{array}\right.,(i=1,2)---\left(30\right)$

In formula, u_2eqFor equivalent control, vectorFor $\stackrel{\&CenterDot;}{g}({\tilde{e}}_{2u},{\stackrel{\&CenterDot;}{\tilde{e}}}_{2u})={\left[\begin{array}{cc}{\stackrel{\&CenterDot;}{g}}_1({\tilde{e}}_{2u1},{\stackrel{\&CenterDot;}{\tilde{e}}}_{2u1})& {\stackrel{\&CenterDot;}{g}}_2({\tilde{e}}_{2u2},{\stackrel{\&CenterDot;}{\tilde{e}}}_{2u2})\end{array}\right]}^T;$

To sum up, actual controlled quentity controlled variable is

U₂=U_2d+u₂=U_2d+u_2eq+u_2s(31)

In formula, U_2d、u_2eqAnd u_2sExpression formula respectively as shown in formula (14), (29) and (27).

The invention has the beneficial effects as follows, the present invention " a kind of lack mark to control underactuated spacecraft hover asymptotic control method ", it compared with prior art has the advantage that

(1). this method give disappearance mark to the name hovering orientation feasible set in the drive lacking situation controlled；

(2). the method can set up arbitrary feasible name hovering configuration at disappearance mark in the drive lacking situation controlled, and can guarantee that the asymptotic stability of closed-loop control system；

(3). the method is by choosing suitable sliding-mode surface and Reaching Law design sliding formwork control law so that model linearization error and external disturbance are had good robustness by system；

(4). than installing thrust reserve device to tackle the conventional methods of Actuator failure, the method can reduce spacecraft structure quality, reduce spacecraft manufacture and launch cost.

Control engineer to hover for reality the feature of application task (such as ASTEREX and On-orbit servicing etc.), arbitrary feasible hovering orientation is given according to actual drive lacking situation, and the controlled quentity controlled variable obtained by the method is transmitted to actuator (such as star lifting force device etc.), disappearance mark can be realized and control function to underactuated spacecraft hovering Asymptotic Stability when controlling acceleration.Thus, theoretical mechanism of the present invention is distinct, solve the hovering task Problem of Failure that thrust disappearance (such as Actuator failure etc.) causes innovatively, than the conventional method installing thrust reserve device, the inventive method is effectively reduced architecture quality, reduces manufacture and production cost, thus practicality is higher and engineer applied is worth higher.

Accompanying drawing explanation

In order to be illustrated more clearly that the embodiment of the present invention or technical scheme of the prior art, the accompanying drawing used required in embodiment or description of the prior art will be briefly described below, apparently, accompanying drawing in the following describes is only embodiments of the invention, for those of ordinary skill in the art, under the premise not paying creative work, it is also possible to obtain other accompanying drawing according to the accompanying drawing provided.

Fig. 1 is the underactuated spacecraft of the present invention asymptotic control method flow chart of steps of hovering

Fig. 2 is drive lacking of the present invention hovering kinetic model coordinate system definition figure

Fig. 3 be disappearance mark of the present invention to acceleration when relative position track

Fig. 4 be disappearance mark of the present invention to acceleration when the relative position error change curve

Fig. 5 be disappearance mark of the present invention to acceleration when relative velocity error change curve

Fig. 6 be disappearance mark of the present invention to acceleration when controlled quentity controlled variable change curve

In figure, symbol description is as follows:

O_CPursuit spacecraft barycenter

O_EX_IY_IZ_IGeocentric inertial coordinate system (O_EFor the earth's core)

O_TXyz relative motion coordinate system (O_TFor passive space vehicle barycenter)

R_CPursuit spacecraft the earth's core is from vector

R_TPassive space vehicle the earth's core is from vector

U_xRadially control acceleration

U_yMark is to controlling acceleration

U_zNormal direction controls acceleration

u_TPassive space vehicle latitude argument

X is radially

Y mark to

Z normal direction

ρ pursuit spacecraft and passive space vehicle Relative position vector

Specific embodiments

In order to make those skilled in the art be more fully understood that the technical scheme in the application, below in conjunction with the accompanying drawing in the embodiment of the present application, technical scheme in the embodiment of the present application is clearly and completely described, obviously, described embodiment is only some embodiments of the present application, rather than whole embodiments.

Based on the embodiment in the application, the every other embodiment that those of ordinary skill in the art obtain under not making creative work premise, all should belong to the scope of the application protection.

Below in conjunction with accompanying drawing, the method for designing in the present invention is further described:

The present invention " a kind of lack mark to control underactuated spacecraft hover asymptotic control method ", it specifically comprises the following steps that

Step one: drive lacking situation judges

If disappearance mark is to controlling acceleration, i.e. U_y=0.

Step 2: given name hovering orientation also solves corresponding nominal controlled quentity controlled variable

Disappearance mark is to when controlling acceleration, and hovering orientation feasible set is

${\widetilde{\mathrm{\&Gamma;}}}_2=\left\{{\mathrm{\&rho;}}_d\right|{\mathrm{\&rho;}}_d\&Element;{\mathrm{\&Gamma;}}_2,x_d=x_0+{\stackrel{\&CenterDot;}{y}}_0/\left(2n_T\right)\}---\left(32\right)$

Wherein

Γ₂={ ρ_d|2R_Tx_d+||ρ_d||²=0} ∪ { ρ_d|y_d=0} (33)

In formula, x₀WithRespectively initial time fractional radial position and relative mark are to speed.ρ_d=[x_dy_dz_d]^TFor hovering orientation, wherein x_d、y_dWith z_dRespectively radially, mark to normal direction hovering position.R_TFor passive space vehicle the earth's core from.For the relative distance of pursuit spacecraft Yu passive space vehicle, distance of namely hovering, wherein | | | | represent the norm of vector.

Table 1 initial time passive space vehicle orbital tracking

Assume in this example that initial time passive space vehicle orbital tracking is as shown in table 1, and assume that initial time relative motion state is

$\begin{array}{c}{X}_2\left(0\right)={\left[\begin{array}{cccccc}{x}_0& y_0& z_0& {\stackrel{\&CenterDot;}{y}}_0& {\stackrel{\&CenterDot;}{x}}_0& {\stackrel{\&CenterDot;}{z}}_0\end{array}\right]}^T\\ ={\left[\begin{array}{cccccc}900m& 500m& -500m& 0.22m/s& 1m/s& 1m/s\end{array}\right]}^T\end{array}---\left(34\right)$

Being obtained by above formula, feasible radial direction hovering position isThis feasible hovering orientation chosen of base is ρ_d=[100000]^TM, it is possible to checking ${\mathrm{\&rho;}}_d\&Element;{\widetilde{\mathrm{\&Gamma;}}}_2.$

Disappearance mark is to when controlling acceleration, it is achieved the nominal controlled quentity controlled variable in feasible hovering orientation is

$U_{2d}=\left[\begin{array}{c}{U}_{xd}\\ U_{zd}\end{array}\right]=\left[\begin{array}{c}(n_C^2-n_T^2)(R_T+x_d)\\ n_C^2{z}_d\end{array}\right]---\left(35\right)$

The name hovering orientation ρ that will choose in this example_dObtaining name controlled quentity controlled variable in substitution formula (35) is

U_2d=[-3.64 × 10^-30]^Tm/s²(36)

Step 3: the margin of error calculates

Calculate the margin of error e of actual relative motion state and name relative motion state₂, namely

$\begin{array}{c}{e}_2=X_2-X_{2d}={\left[\begin{array}{cccccc}{e}_x& e_y& e_z& {\stackrel{\&CenterDot;}{e}}_y& {\stackrel{\&CenterDot;}{e}}_x& {\stackrel{\&CenterDot;}{e}}_z\end{array}\right]}^T\\ ={\left[\begin{array}{cccccc}x-x_d& y-y_d& z-z_d& \stackrel{\&CenterDot;}{y}& \stackrel{\&CenterDot;}{x}& \stackrel{\&CenterDot;}{z}\end{array}\right]}^T\end{array}---\left(37\right)$ The actual relative motion state of initial time provided by formula (34), it is possible to the relative motion state error amount calculating initial time is

e₂(0)=[-100m500m-500m0.22m/s1m/s1m/s]^T(38)

Step 4: design of control law

Disappearance mark is to when controlling acceleration, it is considered to external perturbation with the energy control part in the drive lacking hovering kinetic model of linearized stability be

$\left\{\begin{array}{c}{\stackrel{\&CenterDot;}{\stackrel{\&OverBar;}{e}}}_{2u}=A_{21}{\stackrel{\&OverBar;}{e}}_{2u}+A_{22}{e}_{2a}+d_{2u}\\ {\stackrel{\&CenterDot;}{e}}_{2a}=A_{23}{e}_{2u}+A_{24}{e}_{2a}+B_{2a}{U}_2+d_{2a}\end{array}\right.---\left(39\right)$

Wherein

$\begin{array}{cc}{A}_{21}=\left[\begin{array}{ccc}0& 0& 0\\ 0& 0& 0\\ 1& 0& 0\end{array}\right],& A_{22}=\left[\begin{array}{cc}1& 0\\ 0& 1\\ 0& 0\end{array}\right]\\ A_{23}=\left[\begin{array}{cccc}3n_T^2& 0& 0& 2n_T\\ 0& 0& -n_T^2& 0\end{array}\right],& A_{24}=\left[\begin{array}{cc}0& 0\\ 0& 0\end{array}\right]\end{array}---\left(40\right)$

In formula, ${\stackrel{\&OverBar;}{e}}_{2u}={\left[\begin{array}{ccc}-{\stackrel{\&CenterDot;}{e}}_y/\left(2n_T\right)& e_z& -e_y/\left(2n_T\right)\end{array}\right]}^T,e_{2a}={\left[\begin{array}{cc}{\stackrel{\&CenterDot;}{e}}_x& {\stackrel{\&CenterDot;}{e}}_z\end{array}\right]}^T$ And $e_{2u}={\left[\begin{array}{cccc}{e}_x& e_y& e_z& {\stackrel{\&CenterDot;}{e}}_y\end{array}\right]}^T.$ d_2u=[-d_y/(2n_T)00]^TWith d_2a=[d_xd_z]^TFor uncertain disturbance vector.

Choosing sliding-mode surface is

$s_2=e_{2a}+P_{22}{\stackrel{\&OverBar;}{e}}_{2u}+{\mathrm{\&alpha;}}_2{\tilde{e}}_{2u}+{\mathrm{\&beta;}}_{2}g\left({\tilde{e}}_{2u}\right)---\left(41\right)$

In formula, α₂> 0 and β₂> 0 is design parameter.P₂₂=P₂₁A₂₁, wherein matrix P₂₁For

$P_{21}=\left[\begin{array}{cc}{k}_{21}^{-1}& 0\\ 0& 1\end{array}\right]\left[\begin{array}{ccc}{k}_{21}& 0& k_{22}\\ 0& 1& 0\end{array}\right]---\left(42\right)$

In formula, k₂₁And k₂₂For controller parameter, meet k₂₁k₂₂> 0.

VectorFor $g\left({\tilde{e}}_{2u}\right)={\left[\begin{array}{cc}{g}_1\left({\tilde{e}}_{2u1}\right)& g_2\left({\tilde{e}}_{2u2}\right)\end{array}\right]}^T,$ Wherein

$g_i\left({\tilde{e}}_{2ui}\right)=\left\{\begin{array}{cc}{\tilde{e}}_{2ui}^{p/q},& \begin{array}{ccc}{\tilde{s}}_{2i}=0& or& {\tilde{s}}_{2i}\&NotEqual;0,\left|{\tilde{e}}_{2ui}\right|\&GreaterEqual;\mathrm{\&delta;}\end{array}\\ {\mathrm{\&nu;}}_{1i}{\tilde{e}}_{2ui}+{\mathrm{\&nu;}}_{2i}{\tilde{e}}_{2ui}^2\mathrm{sgn}\left({\tilde{e}}_{1ui}\right),& {\tilde{s}}_{2i}\&NotEqual;0,\left|{\tilde{e}}_{2ui}\right|<\mathrm{\&delta;}\end{array}\right.,(i=1,2)---\left(43\right)$

And

${\tilde{s}}_2={\left[\begin{array}{cc}{\tilde{s}}_{21}& {\tilde{s}}_{22}\end{array}\right]}^T=e_{2a}+P_{22}{\stackrel{\&OverBar;}{e}}_{2u}+{\mathrm{\&alpha;}}_2{\tilde{e}}_{2u}+{\mathrm{\&beta;}}_2{\tilde{e}}_{2u}^{p/q}---\left(44\right)$

In formula, p and q is positive odd number, and p < q.Coefficient ν_1iAnd ν_2iFor ν_1i=(2-p/q) δ^p/q-1And ν_2i=(p/q-1) δ^p/q-2, δ > 0 is design parameter.

The Reaching Law chosen is

${\stackrel{\&CenterDot;}{s}}_2=u_{2s}=-K_{21}{s}_2-K_{22}{\mathrm{sig}}^{{\mathrm{\&gamma;}}_2}\left(s_2\right)---\left(45\right)$

In formula,WithFor positive definite diagonal matrix.VectorFor ${\mathrm{sig}}^{{\mathrm{\&gamma;}}_2}\left(s_2\right)={\left[\begin{array}{cc}{\left|s_{21}\right|}^{{\mathrm{\&gamma;}}_2}\mathrm{sgn}\left(s_{21}\right)& {\left|s_{22}\right|}^{{\mathrm{\&gamma;}}_2}\mathrm{sgn}\left(s_{22}\right)\end{array}\right]}^T,$ Wherein 0 < γ₂< 1 is design parameter, and sgn is sign function, namely

$\mathrm{sgn}\left(x\right)=\left\{\begin{array}{cc}1,& x>0\\ 0,& x=0\\ -1,& x<0\end{array}\right.---\left(46\right)$

The control parameter chosen in this example is listed in table 2.

Table 2 controller design parameter (disappearance mark is to controlling acceleration situation)

The error control rule obtained based on above-mentioned sliding-mode surface and Reaching Law is

u₂=u_2eq+u_2s(47)

Wherein

$u_{2eq}=-(A_{23}{e}_{2u}+A_{24}{e}_{2a})-P_{22}(A_{21}{\stackrel{\&OverBar;}{e}}_{2u}+A_{22}{e}_{2a})-{\mathrm{\&alpha;}}_2(P_{22}{\stackrel{\&OverBar;}{e}}_{2u}+e_{2a})-{\mathrm{\&beta;}}_2\stackrel{\&CenterDot;}{g}({\tilde{e}}_{2u},{\stackrel{\&CenterDot;}{\tilde{e}}}_{2u})---\left(48\right)$

${\stackrel{\&CenterDot;}{g}}_i({\tilde{e}}_{2ui},{\stackrel{\&CenterDot;}{\tilde{e}}}_{2ui})=\left\{\begin{array}{cc}(p/q){\tilde{e}}_{2ui}^{p/q-1}{\stackrel{\&CenterDot;}{\tilde{e}}}_{2ui},& \begin{array}{ccc}{\tilde{s}}_{2i}=0& or& {\tilde{s}}_{2i}\&NotEqual;0,\left|{\tilde{e}}_{2ui}\right|\&GreaterEqual;\mathrm{\&delta;}\end{array}\\ {\mathrm{\&nu;}}_{1i}{\stackrel{\&CenterDot;}{\tilde{e}}}_{2ui}+2{\mathrm{\&nu;}}_{2i}{\tilde{e}}_{2ui}{\stackrel{\&CenterDot;}{\tilde{e}}}_{2ui}\mathrm{sgn}\left({\tilde{e}}_{2ui}\right),& {\tilde{s}}_{2i}\&NotEqual;0,\left|{\tilde{e}}_{2ui}\right|<\mathrm{\&delta;}\end{array}\right.,(i=1,2)---\left(49\right)$

In formula, u_2eqFor equivalent control, vectorFor $\stackrel{\&CenterDot;}{g}({\tilde{e}}_{2u},{\stackrel{\&CenterDot;}{\tilde{e}}}_{2u})={\left[\begin{array}{cc}{\stackrel{\&CenterDot;}{g}}_1({\tilde{e}}_{2u1},{\stackrel{\&CenterDot;}{\tilde{e}}}_{2u1})& {\stackrel{\&CenterDot;}{g}}_2({\tilde{e}}_{2u2},{\stackrel{\&CenterDot;}{\tilde{e}}}_{2u2})\end{array}\right]}^T.$

Therefore, actual controlled quentity controlled variable is

U₂=U_2d+u₂=U_2d+u_2eq+u_2s(50)

In formula, U_2d、u_2eqWith u_2sRespectively as shown in formula (36), (48) and formula (45).Controller parameter in table 2 is substituted into control law and can calculate actual controlled quentity controlled variable.

Choose J equally₂Perturbative force as external perturbation, then lacks mark to the drive lacking Hovering control result of control acceleration as shown in Figures 3 to 6 in this example.Fig. 3 gives pursuit spacecraft and passive space vehicle relative position variation track, it is seen then that pursuit spacecraft, from initial relative position, arrives and maintain name hovering orientation, it was demonstrated that the effectiveness of the control method that the present invention proposes and correctness.Fig. 4 and Fig. 5 sets forth relative position and relative velocity error change curve, and wherein, stable state the relative position error is 10⁰The m order of magnitude, and maximum the relative position error be about hovering distance 0.2%, velocity error is 10 by equilibrium transport^-3The m/s order of magnitude.Consider that this control method is underactuated control, steady-state error has met required precision, it was demonstrated that the control method that the present invention proposes has higher control accuracy.Fig. 6 gives and realizes the control acceleration change track that hovering is required, it is seen then that actual controlled quentity controlled variable converges near name controlled quentity controlled variable after half orbital period, and its order of magnitude is each about 10^-3m/s², meet engineering reality, can realize in actual hovering task.

Claims (4)

1. lack mark to the underactuated spacecraft asymptotic control method of hovering controlled, sequentially include the following steps:

Step one: drive lacking situation judges: if disappearance mark is to controlling acceleration, then U_y=0, U_yFor mark to controlling acceleration；

Step 2: given name hovering orientation also solves the nominal controlled quentity controlled variable of correspondence: solve disappearance mark to the hovering orientation feasible set in control acceleration situation according to actual drive lacking situationAnd in feasible set, select name hovering orientation ρ_d=[x_dy_dz_d]^T, solve the nominal controlled quentity controlled variable U of correspondence_2d；

Step 3: the margin of error calculates: calculate the margin of error e between actual relative motion state and name relative motion state₂；

Step 4: design of control law: choose sliding-mode surface and Reaching Law, adopts sliding-mode control design underactuated spacecraft Hovering control rule, calculates actual controlled quentity controlled variable U₂。

2. a kind of mark that lacks as claimed in claim 1 hovers asymptotic control method to the underactuated spacecraft controlled, it is characterised in that: name described in step 2 hovering orientation is ρ_d=[x_dy_dz_d]^T, x in formula_d、y_dAnd z_dRespectively name radially, mark to normal direction hovering position, subscript T represents the transposition of vector or matrix；For hovering orientation feasible set, its solution procedure is divided into three steps, concrete method for solving to be:

1) mathematical model of underactuated spacecraft hovering

The coordinate system definition describing spacecraft hovering kinetic model is as follows, as shown in Figure 2；O_EX_IY_IZ_IFor geocentric inertial coordinate system, wherein O_EFor the earth's core, O_TXyz is that initial point is positioned at passive space vehicle barycenter O_TRelative motion coordinate system, wherein x-axis is along passive space vehicle radially, and z-axis is along passive space vehicle orbital plane normal direction, and y-axis constitutes Descartes's right hand rectangular coordinate system, O with x, z-axis_CFor pursuit spacecraft barycenter, R_CWith R_TRespectively pursuit spacecraft and the earth's core of passive space vehicle are from vector；Make ρ=[xyz]^TWith $v={\left[\begin{array}{ccc}\stackrel{\&CenterDot;}{x}& \stackrel{\&CenterDot;}{y}& \stackrel{\&CenterDot;}{z}\end{array}\right]}^T$ The respectively Relative position vector of pursuit spacecraft and passive space vehicle and relative velocity vector statement in relative motion coordinate system, then underactuated spacecraft hovering kinetic model is

${\stackrel{\&CenterDot;}{X}}_2=F_2\left(X_2\right)+{\mathrm{BU}}_2---\left(1\right)$

Wherein

F₂=[0_1×3f_yf_xf_z]^T(2)

$\left[\begin{array}{c}{f}_x\\ f_y\\ f_z\end{array}\right]=\left[\begin{array}{c}2{\stackrel{\&CenterDot;}{u}}_T\stackrel{\&CenterDot;}{y}+{\stackrel{\&CenterDot;}{u}}_T^{2}x+{\stackrel{\&CenterDot;\&CenterDot;}{u}}_{T}y+n_T^2{R}_T-n_c^2(R_T+x)\\ -2{\stackrel{\&CenterDot;}{u}}_T\stackrel{\&CenterDot;}{x}+{\stackrel{\&CenterDot;}{u}}_T^{2}y-{\stackrel{\&CenterDot;\&CenterDot;}{u}}_{T}x-n_C^{2}y\\ -n_C^{2}z\end{array}\right]---\left(3\right)$

B=[0_2×4I_2×2]^T(4)

U₂=[U_xU_z]^T(5)

In formula, subscript 2 represents disappearance mark to the drive lacking situation controlling acceleration, $X_2={\left[\begin{array}{cc}{X}_{2u}^T& X_{2a}^T\end{array}\right]}^T$ For by non-driven state X_2uWith driving condition X_2aThe relative motion state vector of composition, owing to disappearance mark is to controlling acceleration, $X_{2u}={\left[\begin{array}{cccc}x& y& z& \stackrel{\&CenterDot;}{y}\end{array}\right]}^T$ And $X_{2a}={\left[\begin{array}{cc}\stackrel{\&CenterDot;}{x}& \stackrel{\&CenterDot;}{z}\end{array}\right]}^T;$ U₂=[U_xU_z]^TFor controlling input, wherein U_xAnd U_zRespectively radially control acceleration with normal direction；0_m×nIt is the null matrix of m × n for dimension, I_m×nIt is the unit matrix of m × n for dimension；u_TFor passive space vehicle latitude argument,WithRespectively passive space vehicle orbit angular velocity and angular acceleration,AndWherein R_TWith R_C=[(R_T+x)²+y²+z²]^1/2Respectively passive space vehicle and pursuit spacecraft the earth's core from, μ is Gravitational coefficient of the Earth；

2) drive lacking hovering dynamic system controllability is analyzed

If (namely passive space vehicle is positioned at circular orbitAnd), and pursuit spacecraft and passive space vehicle relative distance much smaller than its earth's core from, then drive lacking hovering kinetic model available linearization is

${\stackrel{\&CenterDot;}{X}}_2=A_2{X}_2+{\mathrm{BU}}_2---\left(6\right)$

Wherein

$A_2=\left[\begin{array}{cccccc}0& 0& 0& 0& 1& 0\\ 0& 0& 0& 1& 0& 0\\ 0& 0& 0& 0& 0& 1\\ 0& 0& 0& 0& -2n_T& 0\\ 3n_T^2& 0& 0& 2n_T& 0& 0\\ 0& 0& -n_T^2& 0& 0& 0\end{array}\right]---\left(7\right)$

Adopt lineary system theory to above-mentioned disappearance mark to control acceleration drive lacking when linearized system formula (6) carry out controllability analysis, analysis result shows, if disappearance mark is to controlling acceleration, under-actuated systems formula (6) is non-fully controlled, can be analyzed to and can control as follows and can not control subspace

$\left[\begin{array}{c}{\stackrel{\&CenterDot;}{\stackrel{\&OverBar;}{X}}}_2^c\\ {\stackrel{\&CenterDot;}{\stackrel{\&OverBar;}{X}}}_2^u\end{array}\right]=\left[\begin{array}{cc}{\stackrel{\&OverBar;}{A}}_2^{cc}& {\stackrel{\&OverBar;}{A}}_2^{cu}\\ 0_{1\&times;5}& {\stackrel{\&OverBar;}{A}}_2^{uu}\end{array}\right]\left[\begin{array}{c}{\stackrel{\&OverBar;}{X}}_2^c\\ {\stackrel{\&OverBar;}{X}}_2^u\end{array}\right]+\left[\begin{array}{c}{\stackrel{\&OverBar;}{B}}_2^c\\ 0_{1\&times;2}\end{array}\right]U_2---\left(8\right)$

Wherein

${\stackrel{\&OverBar;}{A}}_2^{cc}=\left[\begin{array}{ccccc}0& 0& 0& 1& 0\\ 0& 0& 0& 0& 1\\ 1& 0& 0& 0& 0\\ -n_T^2& 0& 0& 0& 0\\ 0& -n_T^2& 0& 0& 0\end{array}\right],$ ${\stackrel{\&OverBar;}{A}}_2^{cu}=\left[\begin{array}{c}0\\ 0\\ 0\\ 3n_T^2\\ 0\end{array}\right]---\left(9\right)$

${\stackrel{\&OverBar;}{A}}_2^{uu}=0,$ ${\stackrel{\&OverBar;}{B}}_2^c={\left[\begin{array}{cc}{0}_{2\&times;3}& I_{2\&times;2}\end{array}\right]}^T$

In formula, ${\stackrel{\&OverBar;}{X}}_2^c={\left[\begin{array}{ccccc}-\stackrel{\&CenterDot;}{y}/\left(2n_T\right)& z& -y/\left(2n_T\right)& \stackrel{\&CenterDot;}{x}& \stackrel{\&CenterDot;}{z}\end{array}\right]}^T$ With ${\stackrel{\&OverBar;}{X}}_2^u=x+\stackrel{\&CenterDot;}{y}/\left(2n_T\right)$ Respectively controlled and uncontrollable state vector；

3) drive lacking hovering orientation feasible set is solved

According to hovering definition, the relative position of pursuit spacecraft and passive space vehicle remains unchanged in relative motion coordinate system, if definition name hovering orientation is ρ_d=[x_dy_dz_d]^T, then $v_d={\stackrel{\&CenterDot;}{\mathrm{\&rho;}}}_d={\left[\begin{array}{ccc}{\stackrel{\&CenterDot;}{x}}_d& {\stackrel{\&CenterDot;}{y}}_d& {\stackrel{\&CenterDot;}{z}}_d\end{array}\right]}^T=0_{3\&times;1},$ And ${\stackrel{\&CenterDot;}{v}}_d={\stackrel{\&CenterDot;\&CenterDot;}{\mathrm{\&rho;}}}_d={\left[\begin{array}{ccc}{\stackrel{\&CenterDot;\&CenterDot;}{x}}_d& {\stackrel{\&CenterDot;\&CenterDot;}{y}}_d& {\stackrel{\&CenterDot;\&CenterDot;}{z}}_d\end{array}\right]}^T=0_{3\&times;1},$ If it is further assumed that (namely passive space vehicle is positioned at circular orbitAnd), then obtained by formula (3),

$\left[\begin{array}{c}{f}_x\\ f_y\\ f_z\end{array}\right]=\left[\begin{array}{c}(n_T^2-n_C^2)(R_T+x_d)\\ (n_T^2-n_C^2)y_d\\ -n_C^2{z}_d\end{array}\right]---\left(10\right)$

Hereinafter solve the nominal controlled quentity controlled variable U of drive lacking hovering orientation feasible set and correspondence_2d:

When disappearance mark is to control acceleration, i.e. U_yWhen=0, formula (1) obtain,

$\left[\begin{array}{c}{\stackrel{\&CenterDot;\&CenterDot;}{x}}_d\\ {\stackrel{\&CenterDot;\&CenterDot;}{y}}_d\\ {\stackrel{\&CenterDot;\&CenterDot;}{z}}_d\end{array}\right]=\left[\begin{array}{c}{f}_x+U_{xd}\\ f_y\\ f_z+U_{zd}\end{array}\right]=\left[\begin{array}{c}(n_T^2-n_C^2)(R_T+x_d)+U_{xd}\\ (n_T^2-n_C^2)y_d\\ -n_c^2{z}_d+U_{zd}\end{array}\right]=0_{3\&times;1}---\left(11\right)$

Visible, for realizing hovering, it is desirable to f_y=0, namelySolve hovering orientation feasible set when namely the equation obtains disappearance mark to control acceleration；It is easy to get, Equation f_yThe solution of=0 is n_T=n_COr y_d=0, solve n_T=n_CThe hovering orientation feasible set obtaining correspondence is Γ₂₁={ ρ_d|2R_Tx_d+||ρ_d||²=0}, in formula,For relative distance, and symbol | | | | represent the norm of vector, solve y_d=0 feasible solution obtaining correspondence is Γ₂₂={ ρ_d|y_d=0}, thus, disappearance mark to hovering orientation feasible set when controlling acceleration is

Γ₂=Γ₂₁∪Γ₂₂(12)

In formula, symbol ∪ represents union of sets collection；Additionally, by 2) in System Controllability analyze, disappearance mark is non-fully controlled to controlling system in acceleration situation, and uncontrollable state isThen in whole control process, uncontrollable stateKeep its initial value constant, namelyX in formula₀WithRespectively initial time fractional radial position and relative mark are to speed；Consider that hovering definition requires the final momentThen the feasible hovering position of radial direction in final moment isBased on above-mentioned analysis, non-fully controlled due to system, disappearance mark is modified to hovering orientation feasible set when controlling acceleration

${\widetilde{\mathrm{\&Gamma;}}}_2=\left\{{\mathrm{\&rho;}}_d\right|{\mathrm{\&rho;}}_d\&Element;{\mathrm{\&Gamma;}}_2,x_d=x_0+{\stackrel{\&CenterDot;}{y}}_0/\left(2n_T\right)\}---\left(13\right)$

Meanwhile, by Equation f_x+U_xd=0 and f_z+U_zd=0, name controlled quentity controlled variable U can be obtained_2dFor

$U_{2d}=\left[\begin{array}{c}{U}_{xd}\\ U_{zd}\end{array}\right]=\left[\begin{array}{c}(n_C^2-n_T^2)(R_T+x_d)\\ n_C^2{z}_d\end{array}\right]---\left(14\right).$

3. as claimed in claim 1 a kind of lack mark to control underactuated spacecraft hover asymptotic control method, it is characterized in that: the margin of error calculated between actual relative motion state and name relative motion state described in step 3 of the present invention, its computational methods are:

e₂=X₂-X_2d(15)

In formula, $X_2={\left[\begin{array}{cccccc}x& y& z& \stackrel{\&CenterDot;}{y}& \stackrel{\&CenterDot;}{x}& \stackrel{\&CenterDot;}{z}\end{array}\right]}^T$ For disappearance mark to control acceleration when actual relative motion state, wherein x, y, z, WithRespectively actual diametrically opposite position, mark to relative position, normal direction relative position, diametrically speed, mark to relative velocity and normal direction relative velocity, $X_{2d}={\left[\begin{array}{cc}{\mathrm{\&rho;}}_d^T& 0_{1\&times;3}\end{array}\right]}^T$ For name relative motion state.

4. as claimed in claim 1 a kind of lack mark to control underactuated spacecraft hover asymptotic control method, it is characterised in that: the design sliding formwork control law described in step 4 of the present invention, calculate actual controlled quentity controlled variable U₂, its method is:

Consider the external perturbation power effect in real space environment, then by the drive lacking hovering kinetic model taken the photograph under condition be

${\stackrel{\&CenterDot;}{X}}_2=A_2{X}_2+{\mathrm{BU}}_2+{\stackrel{\&OverBar;}{D}}_2+{\mathrm{\&Delta;F}}_2\left(X_2\right)---\left(16\right)$

In formula,For external perturbation force vector, Δ F₂(X₂)=F₂(X₂)-A₂X₂For linearized stability vector；

Name hovering kinetics equation is

${\stackrel{\&CenterDot;}{X}}_{2d}=F_2\left(X_{2d}\right)+{\mathrm{BU}}_{2d}=0_{6\&times;1}---\left(17\right)$

Definition error relative motion state is $e_2=X_2-X_{2d}={\left[\begin{array}{cccccc}{e}_z& e_y& e_z& {\stackrel{\&CenterDot;}{e}}_y& {\stackrel{\&CenterDot;}{e}}_x& {\stackrel{\&CenterDot;}{e}}_z\end{array}\right]}^T,$ Wherein e_x、e_yAnd e_zRespectively radially, mark to normal direction the relative position error,WithRespectively radially, mark to normal direction relative velocity error, formula (16) and formula (17) make difference and obtain error dynamics model and be

${\stackrel{\&CenterDot;}{e}}_2=A_2{e}_2+{\mathrm{Bu}}_2+D_2---\left(18\right)$

Wherein

$D_2={\stackrel{\&OverBar;}{D}}_2+{\mathrm{\&Delta;F}}_2\left(X_2\right)-{\mathrm{\&Delta;F}}_2\left(X_{2d}\right)---\left(19\right)$

In formula, u₂=U₂-U_2dFor error control amount, $D_2={\left[\begin{array}{cc}{0}_{1\&times;3}& d_2^T\end{array}\right]}^T$ For total perturbing vector of external perturbation and linearized stability composition, wherein, d₂=[d_yd_xd_z]^T, d_x、d_yAnd d_zRespectively radially, mark to and normal disturbance；

Hereinafter design sliding mode controller:

The error dynamics equation (18) method shown in formula (8) is carried out controllability STRUCTURE DECOMPOSITION, and can control part rewrite as follows by therein

$\{\begin{array}{c}{\stackrel{\&CenterDot;}{\stackrel{\&OverBar;}{e}}}_{2u}=A_{21}{\stackrel{\&OverBar;}{e}}_{2u}+A_{22}{e}_{2a}+d_{2u}\\ {\stackrel{\&CenterDot;}{e}}_{2a}=A_{23}{e}_{2u}+A_{24}{e}_{2a}+B_{2a}{U}_2+d_{2a}\end{array}---\left(20\right)$

Wherein

$A_{21}=\left[\begin{array}{ccc}0& 0& 0\\ 0& 0& 0\\ 1& 0& 0\end{array}\right],$ $A_{22}=\left[\begin{array}{cc}1& 0\\ 0& 1\\ 0& 0\end{array}\right]---\left(21\right)$

$A_{23}=\left[\begin{array}{cccc}3n_T^2& 0& 0& 2n_T\\ 0& 0& -n_T^2& 0\end{array}\right],$ $A_{24}=\left[\begin{array}{cc}0& 0\\ 0& 0\end{array}\right]$

In formula, ${\stackrel{\&OverBar;}{e}}_{2u}={\left[\begin{array}{ccc}-{\stackrel{\&CenterDot;}{e}}_y/\left(2n_T\right)& e_z& -e_y/\left(2n_T\right)\end{array}\right]}^T,$ $e_{2a}={\left[\begin{array}{cc}{\stackrel{\&CenterDot;}{e}}_K& {\stackrel{\&CenterDot;}{e}}_z\end{array}\right]}^T$ And $e_{2u}={\left[\begin{array}{cccc}{e}_x& e_y& e_z& {\stackrel{\&CenterDot;}{e}}_y\end{array}\right]}^T;$ Additionally, d_2u=[-d_y/(2n_T)00]^TAnd d_2a=[d_xd_z]^T；

In like manner, it is contemplated thatButRightDo linear transformation ${\tilde{e}}_{2u}=P_{21}{\stackrel{\&OverBar;}{e}}_{2u}$ MakeWherein matrixIt is defined as

$P_{21}=\left[\begin{array}{cc}{k}_{21}^{-1}& 0\\ 0& 1\end{array}\right]\left[\begin{array}{ccc}{k}_{21}& 0& k_{22}\\ 0& 1& 0\end{array}\right]---\left(22\right)$

In formula, k₂₁And k₂₂For design parameter, meet k₂₁k₂₂> 0；

Notice P₂₁A₂₂=I_2×2, thenKinetics equation be

${\stackrel{\&CenterDot;}{\tilde{e}}}_{2u}=P_{22}{\stackrel{\&OverBar;}{e}}_{2u}+e_{2a}+P_{21}{d}_{2u}---\left(23\right)$

In formula, P₂₂=P₂₁A₂₁；

Definition sliding-mode surface is

$s_2=e_{2a}+P_{22}{\stackrel{\&OverBar;}{e}}_{2u}+{\mathrm{\&alpha;}}_2{\tilde{e}}_{2u}+{\mathrm{\&beta;}}_{2}g\left({\tilde{e}}_{2u}\right)---\left(24\right)$

Wherein

$g_i\left({\tilde{e}}_{2ui}\right)=\left\{\begin{array}{cc}{\tilde{e}}_{2ui}^{p/q},& \begin{array}{ccc}{\tilde{s}}_{2i}=0& or& {\tilde{s}}_{2i}\&NotEqual;0,\left|{\tilde{e}}_{2ui}\right|\end{array}\&GreaterEqual;\mathrm{\&delta;}\\ v_{1i}{\tilde{e}}_{2ui}+v_{2i}{\tilde{e}}_{2ui}^2\mathrm{sgn}\left({\tilde{e}}_{1ui}\right),& {\tilde{s}}_{2i}\&NotEqual;0,\left|{\tilde{e}}_{2ui}\right|<\mathrm{\&delta;}\end{array}\right.,(i=1,2)---\left(25\right)$

And

${\tilde{s}}_2={\left[\begin{array}{cc}{\tilde{s}}_{21}& {\tilde{s}}_{22}\end{array}\right]}^T=e_{2a}+P_{22}{\stackrel{\&OverBar;}{e}}_{2u}+{\mathrm{\&alpha;}}_2{\tilde{e}}_{2u}+{\mathrm{\&beta;}}_2{\tilde{e}}_{2u}^{p/q}---\left(26\right)$

In formula, α₂> 0 and β₂> 0 is design parameter, vectorFor $g\left({\tilde{e}}_{2u}\right)={\left[\begin{array}{cc}{g}_1\left({\tilde{e}}_{2u1}\right)& g_2\left({\tilde{e}}_{2u2}\right)\end{array}\right]}^T,$ P and q is positive odd number, and p < q, coefficient ν_1iAnd ν_2iFor ν_1i=(2-p/q) δ^p/q-1And v_2i=(p/q-1) δ^p/q-2, δ > 0 is design parameter；

The Reaching Law chosen is

${\stackrel{\&CenterDot;}{s}}_2=u_{2s}=-K_{21}{s}_2-K_{22}{\mathrm{Ng}}^{{\mathrm{\&gamma;}}_2}\left(s_2\right)---\left(27\right)$

In formula,WithFor positive definite diagonal matrix；VectorFor ${\mathrm{sig}}^{{\mathrm{\&gamma;}}_2}\left(s_2\right)={\left[\begin{array}{cccc}{\left|s_{21}\right|}^{{\mathrm{\&gamma;}}_2}& \mathrm{sgn}\left(s_{21}\right)& {\left|s_{22}\right|}^{{\mathrm{\&gamma;}}_2}& \mathrm{sgn}\left(s_{22}\right)\end{array}\right]}^T,$ Wherein 0 < γ₂< 1 is design parameter；

The error control rule obtained based on above-mentioned sliding-mode surface and Reaching Law is

u₂=u_2eq+u_2s(28)

Wherein

$u_{2eq}=-(A_{23}{e}_{2u}+A_{24}{e}_{2a})-P_{22}(A_{21}{\stackrel{\&OverBar;}{e}}_{2u}+A_{22}{e}_{2a})-{\mathrm{\&alpha;}}_2(P_{22}{\stackrel{\&OverBar;}{e}}_{2u}+e_{2a})-{\mathrm{\&beta;}}_2\stackrel{\&CenterDot;}{g}({\tilde{e}}_{2u},{\stackrel{\&CenterDot;}{\tilde{e}}}_{2u})---\left(29\right)$

${\stackrel{\&CenterDot;}{g}}_i({\tilde{e}}_{2ui},{\stackrel{\&CenterDot;}{\tilde{e}}}_{2ui})=\left\{\begin{array}{cc}(p/q){\tilde{e}}_{2ui}^{p/q-1}{\stackrel{\&CenterDot;}{\tilde{e}}}_{2ui},& \begin{array}{ccc}{\tilde{s}}_{2i}=0& or& {\tilde{s}}_{2i}\&NotEqual;0,\left|{\tilde{e}}_{2ui}\right|\&GreaterEqual;\mathrm{\&delta;}\end{array}\\ v_{1i}{\stackrel{\&CenterDot;}{\tilde{e}}}_{2ui}+2v_{2i}{\tilde{e}}_{2ui}{\stackrel{\&CenterDot;}{\tilde{e}}}_{2ui}\mathrm{sgn}\left({\tilde{e}}_{2ui}\right),& {\tilde{s}}_{2i}\&NotEqual;0,\left|{\tilde{e}}_{2ui}\right|<\mathrm{\&delta;}\end{array}\right.,(i=1,2)---\left(30\right)$

In formula, u_2eqFor equivalent control, vectorFor $\stackrel{\&CenterDot;}{g}({\tilde{e}}_{2u},{\stackrel{\&CenterDot;}{\tilde{e}}}_{2u})={\left[\begin{array}{cc}{\stackrel{\&CenterDot;}{g}}_1({\tilde{e}}_{2u1},{\stackrel{\&CenterDot;}{\tilde{e}}}_{2u1})& {\stackrel{\&CenterDot;}{g}}_2({\tilde{e}}_{2u2},{\stackrel{\&CenterDot;}{\tilde{e}}}_{2u2})\end{array}\right]}^T;$

To sum up, actual controlled quentity controlled variable is

U₂=U_2d+u₂=U_2d+u_2eq+u_2s(31)

In formula, U_2d、u_2eqAnd u_2sExpression formula respectively as shown in formula (14), (29) and (27).