CN102343985A - Satellite time optimal posture maneuvering method with reaction flywheel - Google Patents

Abstract

The invention discloses a satellite time optimal posture maneuvering method with a reaction flywheel. The method comprises the following steps of: (1) establishing a spacecraft posture motion model considering reaction wheel dynamics, and establishing a satellite time optimal posture maneuvering model on this basis; (2) obtaining an open-loop optimal control for the spacecraft posture motion model considering reaction wheel dynamics; and (3) obtaining a robust feedback controller to realize spacecraft redirection posture maneuvering. According to the invention, fastest posture maneuvering of the spacecraft is realized; moreover, high accuracy and strong robustness of the maneuvering control are obtained; and moment saturation and momentum saturation restriction of an executing mechanism can be satisfied.

Description

The motor-driven method of satellite time optimum attitude of band counteraction flyback

Technical field

It is quick to the present invention relates to a kind of spacecraft, and the power operated control method of high precision is specifically related to a kind of motor-driven method of satellite time optimum attitude with counteraction flyback, belongs to the Spacecraft Control technical field.

Background technology

The short weight orientation maneuver problem of spacecraft time of band reaction wheel is exactly to find a kind of control to make spacecraft in the shortest time, realize from a certain stable attitude maneuver to another stable attitude.Some optimal control laws obtain through adopting direct method or indirect method.K.D.Bilimoria; And B.Wie; " Time-Optimal Three-Axis Reorientation of a Rigid Spacecraft; " Journal of Guidance Control and Dynamics; Vol.16, No.3,1993; Pp.446-452. disclose one to three redirected bang-bang controls of rigid body spacecraft, and proved that the rotation around feature axis is not time optimal.H.Shen; And P.Tsiotras; " Time-Optimal Control of Axisymmetric Rigid Spacecraft Using Two Controls; " Journal of Guidance Control and Dynamics; Vol.22, No.5,1999; Pp.682-694. in, only realized the optimum motor-driven of rotational symmetry rigid body spacecraft through two controls.L.C.Lai; C.C.Yang, and C.J.Wu, " Time-Optimal Maneuvering Control of a Rigid Spacecraft; " ACTA Astronautica; Vol.60, No.10,2007; Pp.791-800. in; The motor-driven control problem of time optimal is converted to nonlinear programming problem, controlled variable as the optimal design variable, has been tried to achieve optimal solution through genetic algorithm.M.V.Levskii; " The Problem of the Time-Optimal Control of Spacecraft Reorientation; " Journal of Applied Mathematics and Mechanics; Vol.73; No.1; 2009, pp.16-25. adopts Pang De lia king maximum principle to find the solution the shortest time rotation problem of consideration spacecraft moment of momentum constraint.A.Fleming; P.Sekhavat; And I.M.Ross; " Minimum-Time Reorientation of a Rigid Body; " Journal of Guidance Control and Dynamics, Vol.33, No.1; 2010, pp.160-170. has adopted indirect method and pseudo-spectrometry to try to achieve the optimum of considering constraint and has been redirected problem.S.Liu; And T.Singh; " Fuel/Time Optimal Control of Spacecraft Maneuvers; " Journal of Guidance Control and Dynamics; Vol.20; No.2,1996, pp.394-397. has developed the STO algorithm and has solved the optimum and time optimal attitude maneuver problem of fuel under the control by pulses of three independent boundeds.X.Bao; And J.L.Junkins; " New Results for Time-Optimal Three-Axis Reorientation of a Rigid Spacecraft " Journal of Guidance Control and Dynamics; Vol.32; No.4; 2009, the research of pp.1071-1076. shows that spacecraft is time optimal method around the motor-driven of feature axis under the prerequisite of control input constraint.R.G.Melton; " Boundary Points and Arcs in Constrained; Time-Optimal Satellite Reorientation Maneuvers; " AIAA/AASAstrodynamics Specialist Conference, 2-5August 2010, Toronto; Ontario; Canada among the pp.1-16, finds the solution when satellite time is optimum to be redirected motor-driven problem and considers that boundary point and border arc are as constraint.

Optimum attitude in the prior art is redirected motor-driven considers that with the dynam of actuating unit precision has much room for improvement in the attitude dynamics of satellite.

Summary of the invention

To the objective of the invention is in order addressing the above problem, to propose a kind of motor-driven method of satellite time optimum attitude with counteraction flyback.

The motor-driven method of satellite time optimum attitude of band counteraction flyback of the present invention comprises following step:

The dynamic (dynamical) spacecraft attitude kinematic model of reaction wheel is considered in the first step, foundation, sets up the motor-driven model of satellite time optimum attitude on this basis;

Second goes on foot, is directed against and has considered that the dynamic (dynamical) spacecraft attitude kinematic model of reaction wheel obtains the open loop optimal control;

The 3rd goes on foot, obtains the robust feedback controller, realizes that spacecraft is redirected attitude maneuver;

The invention has the advantages that:

(1) the fastest attitude maneuver of realization spacecraft;

(2) precision of motor-driven control is high;

(3) strong robustness;

(4) can satisfy the moment of actuating unit saturated with the momentum constraint of saturation.

Description of drawings

Fig. 1 is a method flow diagram of the present invention;

Fig. 2 is the satellite layout that 3 reaction wheels are arranged of the present invention;

Fig. 3 is that attitude of the present invention is redirected motor-driven;

Fig. 4 is the optimum quaternion curve of open loop in the embodiments of the invention;

Fig. 5 is an open loop optimal corner velocity curve in the embodiments of the invention;

Fig. 6 is the open loop optimal corner velocity curve of reaction wheel in the embodiments of the invention;

Fig. 7 is an open loop optimal control M curve in the embodiments of the invention;

Fig. 8 is that the attitude under three kinds of controlling schemes is redirected motor-driven curve in the embodiments of the invention.

Among the figure:

The specific embodiment

To combine accompanying drawing and embodiment that the present invention is done further detailed description below.

The present invention is a kind of motor-driven method of satellite time optimum attitude with counteraction flyback, and flow process is installed in the spacecraft on the axes of inertia to reaction wheel as shown in Figure 1, to the stable attitude of another kind, comprises following step from a kind of stable attitude maneuver:

The dynamic (dynamical) spacecraft attitude kinematic model of reaction wheel is considered in the first step, foundation, sets up the motor-driven model of satellite time optimum attitude on this basis;

1, sets up the dynamic (dynamical) spacecraft attitude kinematic model of consideration reaction wheel;

The attitude motion model comprises attitude dynamics model and attitude motion model.

As shown in Figure 2; Be a layout that has three reaction wheels to be installed in the rigid body satellite on the axes of inertia, Oxbybzb is the aircraft system of axes among the figure, and O is the barycenter of aircraft; Reaction wheel is mainly used in and absorbs periodic perturbation moment, is used for satellite attitude once in a while and is redirected motor-driven.

Attitude is described through quaternion, and is as follows with the attitude motion model that quaternion is described.

$\stackrel{\·}{q}=\frac{1}{2}Q\left(\mathrm{\ω}\right)\·q=\frac{1}{2}\mathrm{\Ξ}\left(q\right)\mathrm{\ω}---\left(1\right)$

Wherein, q=[q ₁, q ₂, q ₃, q ₄] ^TBe quaternionic vector, q ₁, q ₂, q ₃, q ₄Be respectively four components of quaternion, ω=[ω ₁, ω ₂, ω ₃] ^TBe the angular velocity vector of satellite, ω ₁, ω ₂, ω ₃Be respectively the component of angular velocity vector on three axles of aircraft system of axes of satellite, Q (ω) and Ξ (q) are following matrixes:

$\begin{array}{cc}Q\left(\mathrm{\&omega;}\right)=\left[\begin{array}{cc}-{\mathrm{\&omega;}}^{\&times;}& \mathrm{\&omega;}\\ -{\mathrm{\&omega;}}^{\mathrm{<figure-callout\; id="0"\; label="T"\; filenames="DEST\_PATH\_HDA0000083179400000041.png"\; state="\{\{state\}\}">T</figure-callout>}}& 0\end{array}\right]& \mathrm{\&Xi;}\left(q\right)=\left[\begin{array}{c}{q}_4{\mathrm{<figure-callout\; id="3"\; label="I"\; filenames="DEST\_PATH\_HDA0000083179400000031.png"\; state="\{\{state\}\}">I</figure-callout>}}_{3\&times;3}+{\mathrm{<figure-callout\; id="13"\; label="q"\; filenames="DEST\_PATH\_HDA0000083179400000041.png"\; state="\{\{state\}\}">q</figure-callout>}}_{13}^{\&times;}\\ -q_{13}^T\end{array}\right]\end{array}---\left(2\right)$

Wherein, I _{3 * 3}The identity matrix of expression 3 * 3, ω ^*With

Be antisymmetric matrix, as follows:

$\begin{array}{cc}{\mathrm{\&omega;}}^{\&times;}=\left[\begin{array}{ccc}0& -{\mathrm{\&omega;}}_3& {\mathrm{\&omega;}}_2\\ {\mathrm{\&omega;}}_3& 0& -{\mathrm{\&omega;}}_1\\ -{\mathrm{\&omega;}}_2& {\mathrm{\&omega;}}_1& 0\end{array}\right]& \end{array}{\mathrm{<figure-callout\; id="13"\; label="q"\; filenames="DEST\_PATH\_HDA0000083179400000041.png"\; state="\{\{state\}\}">q</figure-callout>}}_{13}^{\&times;}=\left[\begin{array}{ccc}0& -q_3& q_2\\ {\mathrm{<figure-callout\; id="3"\; label="q"\; filenames="DEST\_PATH\_HDA0000083179400000031.png"\; state="\{\{state\}\}">q</figure-callout>}}_3& 0& -q_1\\ -q_2& {\mathrm{<figure-callout\; id="1"\; label="q"\; filenames="DEST\_PATH\_HDA0000083179400000022.png,DEST\_PATH\_HDA0000083179400000041.png"\; state="\{\{state\}\}">q</figure-callout>}}_1& 0\end{array}\right]---\left(3\right)$

The attitude dynamics model of considering the dynamic (dynamical) rigid body satellite of reaction wheel is following:

$\stackrel{\&CenterDot;}{\mathrm{\&omega;}}=I_s^{-1}(-{\mathrm{\&omega;}}^{\&times;}{I}_s\mathrm{\&omega;}-{\mathrm{\&omega;}}^{\&times;}{I}_{\mathrm{RW}}\mathrm{\&Omega;}-T_u+T_{\mathrm{ex}})---\left(4\right)$

Wherein, I _sAnd I _RWBe respectively the moment of inertia matrix of satellite and reaction wheel, the angular velocity vector of Ω reaction wheel, Ω=[Ω ₁, Ω ₂, Ω ₃] ^T, Ω ₁, Ω ₂, Ω ₃Be respectively and be installed in Oxb, the cireular frequency of the reaction wheel on Oyb and the Ozb axle, T _uBe the moment vector of reaction wheel, T _u=[T _U1T _U2T _U3] ^T, T _U1T _U2T _U3Expression is installed in Oxb respectively, the moment that the reaction wheel on Oyb and the Ozb axle produces, T _ExBe environmental perturbation moment, do not consider this.The attitude dynamics model (4) of ignoring distrubing moment is:

$\stackrel{\&CenterDot;}{\mathrm{\&omega;}}=I_s^{-1}(-{\mathrm{\&omega;}}^{\&times;}{I}_s\mathrm{\&omega;}-{\mathrm{\&omega;}}^{\&times;}{I}_{\mathrm{RW}}\mathrm{\&Omega;}-T_u)---\left(5\right)$

The attitude dynamics model of reaction wheel is following:

$\stackrel{\&CenterDot;}{\mathrm{\&Omega;}}=I_{\mathrm{RW}}^{-1}{T}_u---\left(6\right)$

Attitude motion, the kinetic model (1) (5) (6) of considering the rigid body satellite of reaction wheel are integrated.Corresponding state variable and control variable are described below:

x＝[q ₁ q ₂ q ₃ q ₄ ω ₁ ω ₂ ω ₃ Ω ₁ Ω ₂ Ω ₃] ^T，u＝T _u＝[T _u1 T _u2 T _u3] ^T (7)

Wherein, x is a state variable, and u is a control variable.State variable comprises the attitude quaternion of satellite and the cireular frequency of cireular frequency and reaction wheel.Control variable is the moment of reaction wheel.Attitude motion is learned, kinetic model (1), and (5) and (6) can be described as unified form and promptly consider the dynamic (dynamical) spacecraft attitude kinematic model of reaction wheel:

$\stackrel{\&CenterDot;}{x}=f(x,u)---\left(8\right)$

the derivative of x.

2, set up the motor-driven model of satellite time optimum attitude;

Attitude is redirected and is meant satellite from a stable attitude maneuver to another stable attitude.In Fig. 3, attitude angle is through the relative rotation definition of aircraft system of axes Oxbybzb and LVLH system of axes Oxyz, and A among the figure, B, C, D represent four kinds of different attitudes experiencing in the satellite flight process.It is exactly will design one group of control torque to realize motor-driven between two different holding position with the minimum time that the minimum time is redirected problem.Be redirected in the mobile process in optimum attitude, the ability of actuating unit must be considered.In this work, reaction wheel is the supplier of control torque, and the moment of actuating unit dynam and reaction wheel and momentum are saturated all to be considered to be redirected in the motor-driven model in the satellite optimum attitude.Control torque of the present invention can guarantee under the prerequisite that satisfies the constraint of maximum control moment and maximum momentum, to realize the attitude maneuver of shortest time.

The incipient stability state of the redirected motor-driven model of satellite attitude is following:

$\begin{array}{cccc}{q}_1\left(t_0\right)=q_{{1t}_0}& q_2\left(t_0\right)=q_{{2t}_0}& q_3\left(t_0\right)=q_{{3t}_0}& q_4\left(t_0\right)=q_{{4\mathrm{<figure-callout\; id="0"\; label="t"\; filenames="DEST\_PATH\_HDA0000083179400000041.png"\; state="\{\{state\}\}">t</figure-callout>}}_0}\end{array}$

$\begin{array}{cccc}{\mathrm{\&omega;}}_1\left(t_0\right)={\mathrm{\&omega;}}_{{1\mathrm{<figure-callout\; id="0"\; label="t"\; filenames="DEST\_PATH\_HDA0000083179400000041.png"\; state="\{\{state\}\}">t</figure-callout>}}_0}& {\mathrm{\&omega;}}_2\left(t_0\right)={\mathrm{\&omega;}}_{{2\mathrm{<figure-callout\; id="0"\; label="t"\; filenames="DEST\_PATH\_HDA0000083179400000041.png"\; state="\{\{state\}\}">t</figure-callout>}}_0}& {\mathrm{\&omega;}}_3\left(t_0\right)={\mathrm{\&omega;}}_{{3t}_0}& ---\left(9\right)\end{array}$

$\begin{array}{ccc}{\mathrm{\&Omega;}\mathrm{}}_1\left(t_0\right)={\mathrm{\&Omega;}}_{{1\mathrm{<figure-callout\; id="0"\; label="t"\; filenames="DEST\_PATH\_HDA0000083179400000041.png"\; state="\{\{state\}\}">t</figure-callout>}}_0}& {\mathrm{\&Omega;}}_2\left(t_0\right)={\mathrm{\&Omega;}}_{{2\mathrm{<figure-callout\; id="0"\; label="t"\; filenames="DEST\_PATH\_HDA0000083179400000041.png"\; state="\{\{state\}\}">t</figure-callout>}}_0}& {\mathrm{\&Omega;}}_3\left(t_0\right)={\mathrm{\&Omega;}}_{{3\mathrm{<figure-callout\; id="0"\; label="t"\; filenames="DEST\_PATH\_HDA0000083179400000041.png"\; state="\{\{state\}\}">t</figure-callout>}}_0}\end{array}$

Wherein:

are constant values;

q ₁(t ₀), q ₂(t ₀), q ₃(t ₀), q ₄(t ₀) be the value of initial time attitude quaternion; ω ₁(t ₀), ω ₂(t ₀), ω ₃(t ₀) be the values of three components of satellite rotational angular velocity at initial time; Ω ₁(t ₀), Ω ₂(t ₀), Ω ₃(t ₀) be the values of three reaction wheels at the rotational angular velocity of initial time.

The stabilized conditions of the terminal juncture of the redirected motor-driven model of satellite attitude is following:

$\begin{array}{cccc}{q}_1\left(t_f\right)=q_{{1t}_f}& q_2\left(t_f\right)=q_{{2t}_f}& q_3\left(t_f\right)=q_{{3t}_f}& q_4\left(t_f\right)=q_{{4t}_f}\end{array}$

$\begin{array}{cccc}{\mathrm{\&omega;}}_1\left(t_f\right)={\mathrm{\&omega;}}_{{1t}_f}& {\mathrm{\&omega;}}_2\left(t_f\right)={\mathrm{\&omega;}}_{{2t}_f}& {\mathrm{\&omega;}}_3\left(t_f\right)={\mathrm{\&omega;}}_{{3t}_f}& \end{array}---\left(10\right)$

$\begin{array}{ccc}{\mathrm{\&Omega;}}_1\left(t_f\right)={\mathrm{\&Omega;}}_{{1t}_f}& {\mathrm{\&Omega;}}_2\left(t_f\right)={\mathrm{\&Omega;}}_{{2t}_f}& {\mathrm{\&Omega;}}_3\left(t_f\right)={\mathrm{\&Omega;}}_{{3t}_f}\end{array}$

Wherein:

are constant values;

q ₁(t _f), q ₂(t _f), q ₃(t _f), q ₄(t _f) be the value of terminal juncture attitude quaternion; ω ₁(t _f), ω ₂(t _f), ω ₃(t _f) be the values of three components of satellite rotational angular velocity at terminal juncture; Ω ₁(t _f), Ω ₂(t _f), Ω ₃(t _f) be the values of three reaction wheels at the rotational angular velocity of terminal juncture.

In the problem of being redirected, satellite all is 0 at the cireular frequency of initial time and terminal juncture, then

${\mathrm{\&omega;}}_{1{\mathrm{<figure-callout\; id="0"\; label="t"\; filenames="DEST\_PATH\_HDA0000083179400000041.png"\; state="\{\{state\}\}">t</figure-callout>}}_0}={\mathrm{\&omega;}}_{2{\mathrm{<figure-callout\; id="0"\; label="t"\; filenames="DEST\_PATH\_HDA0000083179400000041.png"\; state="\{\{state\}\}">t</figure-callout>}}_0}={\mathrm{\&omega;}}_{3{\mathrm{<figure-callout\; id="0"\; label="t"\; filenames="DEST\_PATH\_HDA0000083179400000041.png"\; state="\{\{state\}\}">t</figure-callout>}}_0}={\mathrm{\&omega;}}_{1t_f}={\mathrm{\&omega;}}_{2t_f}={\mathrm{\&omega;}}_{3t_f}=0.$

Equation of state has been considered the dynam of actuating unit.The maximum torque of reaction wheel and maximum momentum are respectively control constraint and state constraint.The angular motion quantitative limitation of reaction wheel can convert the restriction of cireular frequency to, as follows:

$\begin{array}{ccc}\left|{\mathrm{\&Omega;}}_1\right|\&le;\stackrel{\&OverBar;}{\mathrm{\&Omega;}}& \left|{\mathrm{\&Omega;}}_2\right|\&le;\stackrel{\&OverBar;}{\mathrm{\&Omega;}}& \left|{\mathrm{\&Omega;}}_3\right|\&le;\stackrel{\&OverBar;}{\mathrm{\&Omega;}}\end{array}---\left(11\right)$

At this, the maximum angular rate of

reaction wheel.Maximum control moment retrains as follows:

$\begin{array}{ccc}\left|{\mathrm{<figure-callout\; id="1"\; label="T\; u"\; filenames="DEST\_PATH\_HDA0000083179400000022.png,DEST\_PATH\_HDA0000083179400000041.png"\; state="\{\{state\}\}">T</figure-callout>}}_u\right|\&le;{\stackrel{\&OverBar;}{T}}_u& \left|T_{u2}\right|\&le;{\stackrel{\&OverBar;}{T}}_u& \left|{\mathrm{<figure-callout\; id="3"\; label="T\; u"\; filenames="DEST\_PATH\_HDA0000083179400000031.png"\; state="\{\{state\}\}">T</figure-callout>}}_u\right|\&le;{\stackrel{\&OverBar;}{T}}_u\end{array}---\left(12\right)$

In this,

Maximum torque reaction wheels.Satellite Angle speed also need satisfy certain constraint, is described below:

$\begin{array}{ccc}\left|{\mathrm{\&omega;}}_1\right|\&le;\stackrel{\&OverBar;}{\mathrm{\&omega;}}& \left|{\mathrm{\&omega;}}_2\right|\&le;\stackrel{\&OverBar;}{\mathrm{\&omega;}}& \left|{\mathrm{\&omega;}}_3\right|\&le;\stackrel{\&OverBar;}{\mathrm{\&omega;}}\end{array}---\left(13\right)$

is maximum angular rate.According to the definition of quaternion, quaternion components must satisfy condition.

${\mathrm{<figure-callout\; id="1"\; label="q"\; filenames="DEST\_PATH\_HDA0000083179400000022.png,DEST\_PATH\_HDA0000083179400000041.png"\; state="\{\{state\}\}">q</figure-callout>}}_1^2+q_2^2+{\mathrm{<figure-callout\; id="3"\; label="q"\; filenames="DEST\_PATH\_HDA0000083179400000031.png"\; state="\{\{state\}\}">q</figure-callout>}}_3^2+q_4^2=1---\left(14\right)$

Obviously, | q _i|≤1 (i=1,2,3,4).Formula (8) also is an equality constraint in optimization problem.In order to obtain the optimal control of short weight orientation maneuver problem of time, the optimal performance index of short problem is following to provide the time:

$\underset{u}{\min }J=\underset{u}{\min }(t_f-t_0)---\left(15\right)$

t ₀Be initial time, t _fIt is terminal time.

Equation (8)-(15) have just constituted the motor-driven model of satellite time optimum attitude.

Second goes on foot, is directed against and has considered that the dynamic (dynamical) spacecraft attitude kinematic model of reaction wheel obtains the open loop optimal control;

(1) the spacecraft attitude kinematic model being carried out normalization method handles

State variable and control variable normalization method are following:

$\begin{array}{ccc}\widetilde{\mathrm{\&omega;}}=\frac{\mathrm{\&omega;}}{\stackrel{\&OverBar;}{\mathrm{\&omega;}}}& \widetilde{\mathrm{\&Omega;}}=\frac{\mathrm{\&Omega;}}{\stackrel{\&OverBar;}{\mathrm{\&Omega;}}}& {\tilde{T}}_u=\frac{T_u}{{\stackrel{\&OverBar;}{T}}_u}\end{array}---\left(16\right)$

Perhaps

$\begin{array}{ccc}\mathrm{\&omega;}=\stackrel{\&OverBar;}{\mathrm{\&omega;}}\widetilde{\mathrm{\&omega;}}& \mathrm{\&Omega;}=\stackrel{\&OverBar;}{\mathrm{\&Omega;}}\widetilde{\mathrm{\&Omega;}}& T_u={\stackrel{\&OverBar;}{T}}_u{\tilde{T}}_u\end{array}---\left(17\right)$

Spacecraft attitude kinematic model with the normalization method variable description is described below:

$\left\{\begin{array}{c}\stackrel{\&CenterDot;}{q}=\frac{1}{2}\stackrel{\&OverBar;}{\mathrm{\&omega;}}Q\left(\widetilde{\mathrm{\&omega;}}\right)\&CenterDot;q\\ \stackrel{\stackrel{\&CenterDot;}{~}}{\mathrm{\&omega;}}=I_s^{-1}\left(-\stackrel{\&OverBar;}{\mathrm{\&omega;}}{\widetilde{\mathrm{\&omega;}}}^{\&times;}{I}_s\widetilde{\mathrm{\&omega;}}-\stackrel{\&OverBar;}{\mathrm{\&Omega;}}{\widetilde{\mathrm{\&omega;}}}^{\&times;}{I}_{\mathrm{RW}}\widetilde{\mathrm{\&Omega;}}-\frac{{\stackrel{\&OverBar;}{T}}_u}{\stackrel{\&OverBar;}{\mathrm{\&omega;}}}{\widetilde{\stackrel{\&OverBar;}{T}}}_u\right)\\ \stackrel{\stackrel{\&CenterDot;}{~}}{\mathrm{\&Omega;}}=\frac{{\stackrel{\&OverBar;}{T}}_u}{\stackrel{\&OverBar;}{\mathrm{\&Omega;}}}{I}_{\mathrm{RW}}^{-1}{\widetilde{\stackrel{\&OverBar;}{T}}}_u\end{array}\right.---\left(18\right)$

The greatest measure of quaternion is 1, and the component of all quaternions all is positioned at interval [1,1] simultaneously.

(2) optimal control problem of describing with normalized parameter

Corresponding to optimal control problem, the kinetics equation of normalization method variable description (18) is as follows.

$\stackrel{\stackrel{\&CenterDot;}{~}}{x}=f(\tilde{x},\tilde{u})---\left(19\right)$

Wherein

$\tilde{x}={\left[\begin{array}{cccccccccc}{q}_1& q_2& q_3& q_4& {\widetilde{\mathrm{\&omega;}}}_1& {\widetilde{\mathrm{\&omega;}}}_2& {\widetilde{\mathrm{\&omega;}}}_3& {\widetilde{\mathrm{\&Omega;}}}_1& {\widetilde{\mathrm{\&Omega;}}}_2& {\widetilde{\mathrm{\&Omega;}}}_3\end{array}\right]}^T$

(20)

$\tilde{u}={\left[\begin{array}{ccc}{\tilde{T}}_{u1}& {\tilde{T}}_{u2}& {\tilde{T}}_{u3}\end{array}\right]}^T$

The performance figure of optimal control are described below:

J＝t _f-t ₀ (21)

Inequality constrain is following:

$\begin{array}{cc}{\tilde{x}}_{\min }\&le;\tilde{x}\&le;{\tilde{x}}_{\max }& {\tilde{u}}_{\min }\&le;\stackrel{\text{~}}{u}\&le;{\tilde{u}}_{\max }\end{array}---\left(22\right)$

Wherein, ${\tilde{x}}_{\max }=\left[\begin{array}{cccccccccc}1& 1& 1& 1& 1& 1& 1& 1& 1& 1\end{array}\right],{\tilde{x}}_{\min }=-{\tilde{x}}_{\max },{\tilde{u}}_{\max }=\left[\begin{array}{ccc}1& 1& 1\end{array}\right],{\tilde{u}}_{\min }=-{\tilde{u}}_{\max }.$

The square journey of equality constraint (14).Optimization goal is to find a normalization control

making satellite attitude redirect shortest.

(3) adopt the pseudo-spectrometry of Legendre that optimal control problem is changed into nonlinear programming problem

Adopt the discrete previously described optimal control problem of the pseudo-spectrometry of Legendre, the method is on the basis that is based upon with Lagrange interpolation polynomial estimated state variable and control variable.Unknown coefficient is the value of interpolation knot variable, is called the quadrature point, or is Legendre Gauss Lambert (LGL) point.Because the LGL point is positioned at interval [1,1], previously described optimal control problem is described in time interval [t ₀, t _f], so we adopt following form that LGL interval and physical time interval are changed: τ ∈ [τ ₀, τ _N]=[-1,1]

$t=\frac{(t_f-t_0)\mathrm{\&tau;}+(t_f+t_0)}{2}---\left(23\right)$

Normalized attitude motion model is following:

$\stackrel{\stackrel{\&CenterDot;}{~}}{x}\left(\mathrm{\&tau;}\right)=\frac{t_f-{\mathrm{<figure-callout\; id="0"\; label="t"\; filenames="DEST\_PATH\_HDA0000083179400000041.png"\; state="\{\{state\}\}">t</figure-callout>}}_0}{2}f(\tilde{x}\left(\mathrm{\&tau;}\right),\widetilde{u\left(\mathrm{\&tau;}\right)})---\left(24\right)$

$\begin{array}{cc}\tilde{x}(-1)={\tilde{x}}_{t_0}& \tilde{x}\left(1\right)={\tilde{x}}_{t_f}\end{array}$

Where:

indicates time τ corresponding to the normalized derivative of the state variable, the normalized state variable and the control variable,

represents the initial time and end time of the normalized values of the state variables,

are constant values.After conversion, estimate continuous state variable and control variable through the polynomial form in N rank, as follows.

$\begin{array}{cc}\tilde{x}\&ap;{\tilde{x}}^N\left(\mathrm{\&tau;}\right)=\underset{t=0}{\overset{N}{\mathrm{\&Sigma;}}}{\tilde{x}}_l{\mathrm{\&phi;}}_1\left(\mathrm{\&tau;}\right)& \tilde{u}\&ap;{\tilde{u}}^N\left(\mathrm{\&tau;}\right)=\underset{t=0}{\overset{N}{\mathrm{\&Sigma;}}}{\tilde{u}}_l{\mathrm{\&phi;}}_l\left(\mathrm{\&tau;}\right)\end{array}---\left(25\right)$

Where, l = 0,1, ..., N, N represents a selected positive integer, represents N points fit state variables and control variables in τ corresponding time values.

${\mathrm{\&phi;}}_l\left(\mathrm{\&tau;}\right)=\frac{1}{N(N+1)L_N\left({\mathrm{\&tau;}}_l\right)}\frac{({\mathrm{\&tau;}}^2-1){\stackrel{\&CenterDot;}{L}}_N\left(\mathrm{\&tau;}\right)}{\mathrm{\&tau;}-{\mathrm{\&tau;}}_l}=\left\{\begin{array}{ccc}1& \mathrm{if}& l=\mathrm{<figure-callout\; id="0"\; label="j"\; filenames="DEST\_PATH\_HDA0000083179400000041.png"\; state="\{\{state\}\}">j</figure-callout>}\\ 0& \mathrm{if}& l\&NotEqual;j\end{array}\right.---\left(26\right)$

Following formula is the Lagrange interpolation polynomial on N rank, L _N(τ _l) Legendre polynomial.

${\tilde{x}}_l={\tilde{x}}^N\left({\mathrm{\&tau;}}_l\right),{\tilde{u}}_l={\tilde{u}}^N\left({\mathrm{\&tau;}}_l\right),{\mathrm{\&tau;}}_l=\mathrm{\&tau;}\left(t_l\right)---\left(27\right)$

Wherein: t _lRepresent l node moment corresponding, τ _lThe value of representing l the cooresponding τ of node.

represents the l-th node corresponds to a normalized state variables and control variables.For according at node τ _lValue

Come the derivative of expression status variable

Corresponding state equation (19) can be described as following form:

${\stackrel{\stackrel{\&CenterDot;}{~}}{x}}^N\left({\mathrm{\&tau;}}_k\right)=\underset{t=0}{\overset{N}{\mathrm{\&Sigma;}}}{D}_{\mathrm{kl}}\tilde{x}\left({\mathrm{\&tau;}}_l\right)---\left(28\right)$

D wherein _KlBe the component of the difference matrix D of (N+1) * (N+1):

$D=\left[D_{\mathrm{kl}}\right]=\left\{\begin{array}{cc}\frac{L_N\left({\mathrm{\&tau;}}_k\right)}{L_N\left({\mathrm{\&tau;}}_l\right)}\frac{1}{({\mathrm{\&tau;}}_k-{\mathrm{\&tau;}}_l)}& k\&NotEqual;l\\ -\frac{N(N+1)}{4}& k=l=0\\ \frac{N(N+1)}{4}& k=l=\mathrm{<figure-callout\; id="0"\; label="N"\; filenames="DEST\_PATH\_HDA0000083179400000041.png"\; state="\{\{state\}\}">N</figure-callout>}\\ 0& \mathrm{otherwise}\end{array}\right.---\left(29\right)$

Thereby the equality constraint of the equation of state in optimal control problem can be described as through discrete state:

$\underset{l=0}{\overset{N}{\mathrm{\&Sigma;}}}{D}_{\mathrm{kl}}\tilde{x}\left({\mathrm{\&tau;}}_l\right)-\frac{t_f-{\mathrm{<figure-callout\; id="0"\; label="t"\; filenames="DEST\_PATH\_HDA0000083179400000041.png"\; state="\{\{state\}\}">t</figure-callout>}}_0}{2}f\left(\underset{l=0}{\overset{N}{\mathrm{\&Sigma;}}}\tilde{x}\left({\mathrm{\&tau;}}_l\right){\mathrm{\&phi;}}_l\left({\mathrm{\&tau;}}_k\right),\underset{l=0}{\overset{N}{\mathrm{\&Sigma;}}}\tilde{u}\left({\mathrm{\&tau;}}_l\right){\mathrm{\&phi;}}_l\left({\mathrm{\&tau;}}_k\right)\right)=0---\left(30\right)$

The square journey of other equality constraint (14), inequality constrain is as follows.

$\begin{array}{cc}{\tilde{x}}_{\min }\&le;\tilde{x}\left({\mathrm{\&tau;}}_l\right)\&le;{\tilde{x}}_{\max }& {\tilde{u}}_{\min }\&le;\tilde{u}\left({\mathrm{\&tau;}}_l\right)\&le;{\tilde{u}}_{\max }\end{array}---\left(31\right)$

Performance optimal control problems see equation (21), the optimization variables is

and

this optimal control problem is transformed into a nonlinear programming problem.

(4) computation optimization is found the solution nonlinear programming problem, obtains the motor-driven parameter of open loop optimum attitude

Through utilize some numerical optimization softwares for example SNOPT or matlab just can find the solution this optimization problem, thereby obtained optimum attitude.

In the 3rd step, design robust feedback controller realizes that spacecraft is redirected attitude maneuver;

Above said step obtained the power operated open loop of optimum attitude control through finding the solution optimal control problem.In practical application, owing to have uncertainty and an environmental perturbation, the uncertainty of Dynamic Modeling for example, air resistance disturbance force.So needing a controlled reset with better robustness to restrain follows the tracks of reference locus.According to the attitude error equations robust controller of deriving.

1, sets up attitude error equations

Concerning the rigid body satellite, attitude dynamic equations provides in equation (4), and attitude motion is learned equation and in equation (1), provided.And the attitude motion track of expectation also obtains through separating fwd open loop optimal control problem.q _dAnd ω _dBe defined as expectation attitude quaternary element and rotational angular velocity, q _eBe system of axes F _bExpect system of axes F relatively _dAttitude quaternion.Accordingly from F _dTo F _bTransition matrix C (q _e) as follows.

$C=(q_{e4}^2-q_{e13}^T{q}_{e13})I_{3\&times;3}+2q_{e13}{q}_{e13}^T-{2q}_{e4}{\mathrm{<figure-callout\; id="13"\; label="q\; e"\; filenames="DEST\_PATH\_HDA0000083179400000041.png"\; state="\{\{state\}\}">q</figure-callout>}}_e^{13}---\left(32\right)$

Wherein: q _E1, q _E2, q _E3, q _E4Be quaternion q _eFour components, vectorial q _E13=[q _E1, q _E2, q _E3] ^T,

Be q _E13Transposed vector, Be q _E13Antisymmetric matrix, q and q _d, q _eRelationship description following:

q＝mat(q _d)q _e (33)

Wherein

$\mathrm{mat}\left(q_d\right)=\left[\begin{array}{cccc}{q}_{d4}& q_{d3}& -q_{d2}& q_{d1}\\ -q_{d3}& q_{d4}& -q_{d1}& q_{d2}\\ q_{d2}& q_{d1}& q_{d4}& q_{d3}\\ -q_{d1}& -q_{d2}& -q_{d3}& q_{d4}\end{array}\right]---\left(34\right)$

Wherein: q _D1, q _D2, q _D3, q _D4Be q _dFour components, F _bRelative F _dAngular velocity omega _eAt system of axes F _bIn be described below:

ω _e＝ω-Cω _d (35)

According to ω _eAnd q _eRewrite eq (4) and equation (1), the error equation that has just obtained.

$I_s{\stackrel{\&CenterDot;}{\mathrm{\&omega;}}}_e=-{({\mathrm{\&omega;}}_e+{\mathrm{C\&omega;}}_d)}^{\&times;}{I}_s({\mathrm{\&omega;}}_e+{\mathrm{C\&omega;}}_d)-{({\mathrm{\&omega;}}_e+{\mathrm{C\&omega;}}_d)}^{\&times;}{I}_{\mathrm{RW}}\mathrm{\&Omega;}+I_s({\mathrm{\&omega;}}_e^{\&times;}{\mathrm{C\&omega;}}_d-{C\stackrel{\&CenterDot;}{\mathrm{\&omega;}}}_d)-T_u$

(36)

$\stackrel{\&CenterDot;}{q}=\frac{1}{2}\mathrm{\&Xi;}\left(q_e\right){\mathrm{\&omega;}}_e$

In equation (36), considered the kinetic effect of reaction wheel.

2, obtain the robust feedback controller

We select the Li Yapuluofu function following:

$V=\frac{1}{2}{{\mathrm{\&omega;}}_e}^T{I}_s{\mathrm{\&omega;}}_e+k_1({\mathrm{<figure-callout\; id="1"\; label="q\; e"\; filenames="DEST\_PATH\_HDA0000083179400000022.png,DEST\_PATH\_HDA0000083179400000041.png"\; state="\{\{state\}\}">q</figure-callout>}}_e^1+q_{e2}^2+{\mathrm{<figure-callout\; id="3"\; label="q\; e"\; filenames="DEST\_PATH\_HDA0000083179400000031.png"\; state="\{\{state\}\}">q</figure-callout>}}_e^3+{(1-q_{e4})}^2)---\left(37\right)$

K wherein ₁＞0.Can derive the derivative of speed V:

$\stackrel{\&CenterDot;}{V}={\mathrm{\&omega;}}_e^T\{-{({\mathrm{\&omega;}}_e+{\mathrm{C\&omega;}}_d)}^{\&times;}{I}_s({\mathrm{\&omega;}}_e+{\mathrm{C\&omega;}}_d)-{({\mathrm{\&omega;}}_e+{\mathrm{C\&omega;}}_d)}^{\&times;}{I}_{\mathrm{RW}}\mathrm{\&Omega;}+I_s({\mathrm{\&omega;}}_e^{\&times;}{\mathrm{C\&omega;}}_d-{C\stackrel{\&CenterDot;}{\mathrm{\&omega;}}}_d)-T_u\}+{\mathrm{<figure-callout\; id="1"\; label="k"\; filenames="DEST\_PATH\_HDA0000083179400000022.png,DEST\_PATH\_HDA0000083179400000041.png"\; state="\{\{state\}\}">k</figure-callout>}}_1{\mathrm{\&omega;}}_e^T{q}_{e13}^T$

$={\mathrm{\&omega;}}_e^T\{-{\mathrm{\&omega;}}_e^{\&times;}(I_s({\mathrm{\&omega;}}_e+{\mathrm{C\&omega;}}_d)+I_{\mathrm{RW}}\mathrm{\&Omega;})-({\left(C{\mathrm{\&omega;}}_d\right)}^{\&times;}{I}_s+I_s{\left({\mathrm{C\&omega;}}_d\right)}^{\&times;}){\mathrm{\&omega;}}_e-{\left({\mathrm{C\&omega;}}_d\right)}^{\&times;}{I}_s{\mathrm{C\&omega;}}_d-I_s{C\stackrel{\&CenterDot;}{\mathrm{\&omega;}}}_d$

(38)

$-{\left({\mathrm{C\&omega;}}_d\right)}^{\&times;}{I}_{\mathrm{RW}}\mathrm{\&Omega;}-T_u\}+{\mathrm{<figure-callout\; id="1"\; label="k"\; filenames="DEST\_PATH\_HDA0000083179400000022.png,DEST\_PATH\_HDA0000083179400000041.png"\; state="\{\{state\}\}">k</figure-callout>}}_1{\mathrm{\&omega;}}_e^T{q}_{e13}^T$

$={\mathrm{\&omega;}}_e^T(-{\left({\mathrm{C\&omega;}}_d\right)}^{\&times;}{I}_s{\mathrm{C\&omega;}}_d-I_s{C\stackrel{\&CenterDot;}{\mathrm{\&omega;}}}_d-{\left({\mathrm{C\&omega;}}_d\right)}^{\&times;}{I}_{\mathrm{RW}}\mathrm{\&Omega;}-T_u)+{\mathrm{<figure-callout\; id="1"\; label="k"\; filenames="DEST\_PATH\_HDA0000083179400000022.png,DEST\_PATH\_HDA0000083179400000041.png"\; state="\{\{state\}\}">k</figure-callout>}}_1{\mathrm{\&omega;}}_e^T{q}_{e13}^T$

Proposing the robust feedback controller according to following formula is:

$u=T_u{=(-{\mathrm{C\&omega;}}_d)}^{\&times;}{I}_s{\mathrm{C\&omega;}}_d-I_s{C\stackrel{\&CenterDot;}{\mathrm{\&omega;}}}_d-{\left({\mathrm{C\&omega;}}_d\right)}^{\&times;}{I}_{\mathrm{RW}}\mathrm{\&Omega;}+k_1{\mathrm{<figure-callout\; id="13"\; label="q\; e"\; filenames="DEST\_PATH\_HDA0000083179400000041.png"\; state="\{\{state\}\}">q</figure-callout>}}_e+k_2{\mathrm{\&omega;}}_e---\left(39\right)$

K wherein ₂＞0.Equation substitution equation (38) with top can obtain

$\stackrel{\&CenterDot;}{V}=-k_2{\mathrm{\&omega;}}_e^T{\mathrm{\&omega;}}_e\&le;0---\left(40\right)$

Thereby system is stable under the control law shown in the equation (39).The main task that closed loop is followed the tracks of is the influence of offsetting the distrubing moment that in the open loop optimal trajectory, does not have consideration.First three items is the error that is used to compensate kinetic model in the control law of equation (39), is a feedforward compensation, and last two essence are a PD feedback term of offsetting uncertain disturbance.

Go to follow the tracks of front optimization through the controller of telling a story and find the solution the open loop optimum attitude that obtains, just can realize that quick, high-precision spacecraft is redirected attitude maneuver.And the method for the present invention motor-driven very strong robustness that has that gestures.

Embodiment:

Among the embodiment, be example with the rigid body satellite that has three reaction wheels, three reaction wheels are installed on the axes of inertia of satellite.Satellite dynamics has comprised the dynam of reaction wheel.The parameter of rigid body satellite is as shown in table 1 below, and initial sum SOT state of termination value is seen table 2.

The correlation parameter of table 1 rigid body satellite

Wherein, I _Xx, I _Yy, I _ZzBe respectively the rotor inertia of satellite on three principal axis of inertia, I _wRotor inertia for reaction wheel.

The initial sum terminal condition of this example of table 2

Q (t ₀), ω (t ₀), Ω (t ₀), q (t _f), ω (t _f), Ω (t _f) be illustrated respectively in the value of initial time and terminal juncture attitude quaternion, attitude angular velocity and reaction wheel rotational angular velocity.

Through the pseudo-spectrometry of Legendre with utilize software SNOPT optimize and obtained the open loop optimal control.The short weight orientation maneuver time is 99s.The optimum quaternion curve of corresponding open loop, the Satellite Angle velocity curve, the cireular frequency curve of reaction wheel and control torque curve such as Fig. 4-shown in Figure 7.

In the prior art, optimum redirected problem is three virtual control torques finding on axon, and the actuating unit dynam is not considered in kinetics equation.Its kinetic model is following:

$\stackrel{\&CenterDot;}{\mathrm{\&omega;}}=I_s^{-1}(-{\mathrm{\&omega;}}^{\&times;}{I}_s\mathrm{\&omega;}+T_u)---\left(41\right)$

And equation (5) is relatively, in the formula-ω ^*I _RWΩ does not consider in equation (41).

For the result of distinct methods relatively, with equation (48) as the motor-driven problem of open loop shortest time of kinetics equation also through the pseudo-spectrometry of Legendre with utilize software SNOPT to optimize to find the solution.Under other parameters situation consistent with the front, its corresponding simulation result result shows that the shortest time kept in reserve is 88 seconds.Time kept in reserve is shorter than the time kept in reserve of the present invention, mainly is because ignored the influence of reaction wheel in the model, and does not consider the restriction that the reaction wheel momentum is saturated, so because model not accurate enough, this result is not an optimum, can makes suboptimum and follow the tracks of and use.Then adopt non-linear predication control method (NPC) that this suboptimum track is followed the tracks of.When follow-up control, considered the constraint of moment and momentum; Relevant parameters is shown in Fig. 8 middle column; It is 130 seconds that attitude quaternion reaches the stable time; It is 145 seconds that Satellite Angle speed reaches the stable time; Three wheels are normal operation all, and only wheel moment and momentum in the attitude rotary course all reached saturated.

In order to contrast, adopt the method for quaternion feedback (QFC) to find the solution the redirected problem of same attitude equally.The moment and the moment of momentum constraint of saturation of reaction wheel have been considered in the model.Corresponding results is shown in hurdle, Fig. 8 left side; The method is that the feature axis rotation is motor-driven; The time that attitude quaternion reaches stabilized conditions is 132 seconds; The time that Satellite Angle speed reaches stabilized conditions is 146 seconds; In the attitude maneuver process; Only reaction wheel work, other two wheels remain static, so satellite is closely around an axes of inertia rotation.

The shortest time that the present invention proposes is redirected shown in the right hurdle of power operated Fig. 8 as a result, and the follow-up control rule of employing is seen formula (39).In the attitude maneuver process, three reaction wheels are all worked, and have a moment to reach oversaturation, also have a momentum to reach oversaturation.The time that attitude quaternion reaches stabilized conditions is 99 seconds; The time that cireular frequency reaches stabilized conditions is 100 seconds; So it is 100 seconds that attitude is redirected the time kept in reserve, has shortened 31.5% than quaternion feedback method, has shortened 31% than suboptimization and nonlinear prediction controlling schemes.Corresponding correlation data is as shown in table 3.

The performance ratio of three kinds of methods of table 3

Claims (1)

1. the motor-driven method of satellite time optimum attitude with counteraction flyback is installed in the spacecraft on the axes of inertia to reaction wheel,, it is characterized in that to the stable attitude of another kind from a kind of stable attitude maneuver, comprises following step:

The dynamic (dynamical) spacecraft attitude kinematic model of reaction wheel is considered in the first step, foundation, sets up the motor-driven model of satellite time optimum attitude on this basis;

(1) sets up the dynamic (dynamical) spacecraft attitude kinematic model of consideration reaction wheel;

The attitude motion model comprises attitude dynamics model and attitude motion model;

Attitude motion is learned model:

$\stackrel{\&CenterDot;}{q}=\frac{1}{2}Q\left(\mathrm{\&omega;}\right)\&CenterDot;q=\frac{1}{2}\mathrm{\&Xi;}\left(q\right)\mathrm{\&omega;}---\left(1\right)$

Wherein, q=[q ₁, q ₂, q ₃, q ₄] ^TBe quaternionic vector, q ₁, q ₂, q ₃, q ₄Be respectively four components of quaternion, ω=[ω ₁, ω ₂, ω ₃] ^TBe the angular velocity vector of satellite, ω ₁, ω ₂, ω ₃Be respectively the component of angular velocity vector on three axles of aircraft system of axes of satellite, Q (ω) and Ξ (q) are:

$\begin{array}{cc}Q\left(\mathrm{\&omega;}\right)=\left[\begin{array}{cc}-{\mathrm{\&omega;}}^{\&times;}& \mathrm{\&omega;}\\ -{\mathrm{\&omega;}}^T& 0\end{array}\right]& \mathrm{\&Xi;}\left(q\right)=\left[\begin{array}{c}{q}_4{I}_{3\&times;3}+q_{13}^{\&times;}\\ -q_{13}^T\end{array}\right]\end{array}---\left(2\right)$

Wherein, I _{3 * 3}The identity matrix of expression 3 * 3, ω ^*With

Be antisymmetric matrix, as follows:

$\begin{array}{cc}{\mathrm{\&omega;}}^{\&times;}=\left[\begin{array}{ccc}0& -{\mathrm{\&omega;}}_3& {\mathrm{\&omega;}}_2\\ {\mathrm{\&omega;}}_3& 0& -{\mathrm{\&omega;}}_1\\ -{\mathrm{\&omega;}}_2& {\mathrm{\&omega;}}_1& 0\end{array}\right]& \end{array}{q}_{13}^{\&times;}=\left[\begin{array}{ccc}0& -q_3& q_2\\ q_3& 0& -q_1\\ -q_2& q_1& 0\end{array}\right]---\left(3\right)$

The attitude dynamics model is:

$\stackrel{\&CenterDot;}{\mathrm{\&omega;}}=I_s^{-1}(-{\mathrm{\&omega;}}^{\&times;}{I}_s\mathrm{\&omega;}-{\mathrm{\&omega;}}^{\&times;}{I}_{\mathrm{RW}}\mathrm{\&Omega;}-T_u+T_{\mathrm{ex}})---\left(4\right)$

Wherein, I _sAnd I _RWBe respectively the moment of inertia matrix of satellite and reaction wheel, the angular velocity vector of Ω reaction wheel, Ω=[Ω ₁, Ω ₂, Ω ₃] ^T, Ω ₁, Ω ₂, Ω ₃Be respectively and be installed in Oxb, the cireular frequency of the reaction wheel on Oyb and the Ozb axle, T _uBe the moment vector of reaction wheel, T _u=[T _U1T _U2T _U3] ^T, T _U1T _U2T _U3Expression is installed in Oxb respectively, the moment that the reaction wheel on Oyb and the Ozb axle produces, T _ExBe environmental perturbation moment, do not consider this; The attitude dynamics model (4) of ignoring distrubing moment is:

$\stackrel{\&CenterDot;}{\mathrm{\&omega;}}=I_s^{-1}(-{\mathrm{\&omega;}}^{\&times;}{I}_s\mathrm{\&omega;}-{\mathrm{\&omega;}}^{\&times;}{I}_{\mathrm{RW}}\mathrm{\&Omega;}-T_u)---\left(5\right)$

The attitude dynamics model of reaction wheel is:

$\stackrel{\&CenterDot;}{\mathrm{\&Omega;}}=I_{\mathrm{RW}}^{-1}{T}_u---\left(6\right)$

Through type (1), (5), (6) obtain considering that the dynamic (dynamical) spacecraft attitude kinematic model of reaction wheel is:

$\stackrel{\&CenterDot;}{x}=f(x,u)---\left(7\right)$

Wherein: x is a state variable, and u is a control variable; State variable comprises the attitude quaternion of satellite and the cireular frequency of cireular frequency and reaction wheel; Control variable is the moment of reaction wheel; State variable and control variable are:

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

(2) set up the motor-driven model of satellite time optimum attitude;

The incipient stability state that satellite attitude is redirected motor-driven model is:

$\begin{array}{cccc}{q}_1\left(t_0\right)=q_{{1t}_0}& q_2\left(t_0\right)=q_{{2t}_0}& q_3\left(t_0\right)=q_{{3t}_0}& q_4\left(t_0\right)=q_{{4t}_0}\end{array}$

$\begin{array}{cccc}{\mathrm{\&omega;}}_1\left(t_0\right)={\mathrm{\&omega;}}_{{1t}_0}& {\mathrm{\&omega;}}_2\left(t_0\right)={\mathrm{\&omega;}}_{{2t}_0}& {\mathrm{\&omega;}}_3\left(t_0\right)={\mathrm{\&omega;}}_{{3t}_0}& \end{array}---\left(9\right)$

$\begin{array}{ccc}{\mathrm{\&Omega;}\mathrm{}}_1\left(t_0\right)={\mathrm{\&Omega;}}_{{1t}_0}& {\mathrm{\&Omega;}}_2\left(t_0\right)={\mathrm{\&Omega;}}_{{2t}_0}& {\mathrm{\&Omega;}}_3\left(t_0\right)={\mathrm{\&Omega;}}_{{3t}_0}\end{array}$

Where,

is the initial moment quaternion values ;

for the satellite angular velocity of the three components at the initial time value;

for the three reaction wheels rotating at the initial time angular velocity values ;

The stabilized conditions that satellite attitude is redirected the terminal juncture of motor-driven model is:

$\begin{array}{cccc}{q}_1\left(t_f\right)=q_{{1t}_f}& q_2\left(t_f\right)=q_{{2t}_f}& q_3\left(t_f\right)=q_{{3t}_f}& q_4\left(t_f\right)=q_{{4t}_f}\end{array}$

$\begin{array}{cccc}{\mathrm{\&omega;}}_1\left(t_f\right)={\mathrm{\&omega;}}_{{1t}_f}& {\mathrm{\&omega;}}_2\left(t_f\right)={\mathrm{\&omega;}}_{{2t}_f}& {\mathrm{\&omega;}}_3\left(t_f\right)={\mathrm{\&omega;}}_{{3t}_f}& ---\left(10\right)\end{array}$

$\begin{array}{ccc}{\mathrm{\&Omega;}}_1\left(t_f\right)={\mathrm{\&Omega;}}_{{1t}_f}& {\mathrm{\&Omega;}}_2\left(t_f\right)={\mathrm{\&Omega;}}_{{2t}_f}& {\mathrm{\&Omega;}}_3\left(t_f\right)={\mathrm{\&Omega;}}_{{3t}_f}\end{array}$

Where, for the terminal moment quaternion values;

for the satellite angular velocity three times the value of the components in the terminal;

for the three reaction wheels turning moment in the terminal angular velocity values;

Satellite all is O at the cireular frequency of initial time and terminal juncture, then

${\mathrm{\&omega;}}_{{1t}_0}={\mathrm{\&omega;}}_{{2t}_0}={\mathrm{\&omega;}}_{{3t}_0}={\mathrm{\&omega;}}_{{1t}_f}={\mathrm{\&omega;}}_{{2t}_f}={\mathrm{\&omega;}}_{{3t}_f}=0;$

The angular motion quantitative limitation of reaction wheel converts the restriction of cireular frequency to, as follows:

$\begin{array}{ccc}\left|{\mathrm{\&Omega;}}_1\right|\&le;\stackrel{\&OverBar;}{\mathrm{\&Omega;}}& \left|{\mathrm{\&Omega;}}_2\right|\&le;\stackrel{\&OverBar;}{\mathrm{\&Omega;}}& \left|{\mathrm{\&Omega;}}_3\right|\&le;\stackrel{\&OverBar;}{\mathrm{\&Omega;}}\end{array}---\left(11\right)$

Where, is the maximum reaction wheel angular velocity; maximum control torque constraints are as follows:

$\begin{array}{ccc}\left|T_{u1}\right|\&le;{\stackrel{\&OverBar;}{T}}_u& \left|T_{u2}\right|\&le;{\stackrel{\&OverBar;}{T}}_u& \left|T_{u3}\right|\&le;{\stackrel{\&OverBar;}{T}}_u\end{array}---\left(12\right)$

Where,

is the maximum torque reaction wheel; satellite angular constraint as follows:

$\begin{array}{ccc}\left|{\mathrm{\&omega;}}_1\right|\&le;\stackrel{\&OverBar;}{\mathrm{\&omega;}}& \left|{\mathrm{\&omega;}}_2\right|\&le;\stackrel{\&OverBar;}{\mathrm{\&omega;}}& \left|{\mathrm{\&omega;}}_3\right|\&le;\stackrel{\&OverBar;}{\mathrm{\&omega;}}\end{array}---\left(13\right)$

Where,

is the maximum angular velocity; quaternion component satisfies:

$q_1^2+q_2^2+q_3^2+q_4^2=1---\left(14\right)$

| q _i|≤1, i=1,2,3,4; Formula (7) also is an equality constraint; The optimal performance index of the motor-driven model of satellite time optimum attitude is:

$\underset{u}{\min }J=\underset{u}{\min }(t_f-t_0)---\left(15\right)$

Wherein: t ₀Be initial time, t _fIt is terminal time; Formula (7)-(15) constitute the motor-driven model of satellite time optimum attitude;

Second goes on foot, is directed against and has considered that the dynamic (dynamical) spacecraft attitude kinematic model of reaction wheel obtains the open loop optimal control;

(1) the spacecraft attitude kinematic model being carried out normalization method handles

State variable and control variable are normalized to:

$\begin{array}{ccc}\widetilde{\mathrm{\&omega;}}=\frac{\mathrm{\&omega;}}{\stackrel{\&OverBar;}{\mathrm{\&omega;}}}& \widetilde{\mathrm{\&Omega;}}=\frac{\mathrm{\&Omega;}}{\stackrel{\&OverBar;}{\mathrm{\&Omega;}}}& {\tilde{T}}_u=\frac{T_u}{{\stackrel{\&OverBar;}{T}}_u}\end{array}---\left(16\right)$

Perhaps

$\begin{array}{ccc}\mathrm{\&omega;}=\stackrel{\&OverBar;}{\mathrm{\&omega;}}\widetilde{\mathrm{\&omega;}}& \mathrm{\&Omega;}=\stackrel{\&OverBar;}{\mathrm{\&Omega;}}\widetilde{\mathrm{\&Omega;}}& T_u={\stackrel{\&OverBar;}{T}}_u{\tilde{T}}_u\end{array}---\left(17\right)$

Spacecraft attitude kinematic model after normalization method is handled is:

$\left\{\begin{array}{c}\stackrel{\&CenterDot;}{q}=\frac{1}{2}\stackrel{\&OverBar;}{\mathrm{\&omega;}}Q\left(\widetilde{\mathrm{\&omega;}}\right)\&CenterDot;q\\ \stackrel{\stackrel{\&CenterDot;}{~}}{\mathrm{\&omega;}}=I_s^{-1}\left(-\stackrel{\&OverBar;}{\mathrm{\&omega;}}{\widetilde{\mathrm{\&omega;}}}^{\&times;}{I}_s\widetilde{\mathrm{\&omega;}}-\stackrel{\&OverBar;}{\mathrm{\&Omega;}}{\widetilde{\mathrm{\&omega;}}}^{\&times;}{I}_{\mathrm{RW}}\widetilde{\mathrm{\&Omega;}}-\frac{{\stackrel{\&OverBar;}{T}}_u}{\stackrel{\&OverBar;}{\mathrm{\&omega;}}}{\widetilde{\stackrel{\&OverBar;}{T}}}_u\right)\\ \stackrel{\stackrel{\&CenterDot;}{~}}{\mathrm{\&Omega;}}=\frac{{\stackrel{\&OverBar;}{T}}_u}{\stackrel{\&OverBar;}{\mathrm{\&Omega;}}}{I}_{\mathrm{RW}}^{-1}{\widetilde{\stackrel{\&OverBar;}{T}}}_u\end{array}\right.---\left(18\right)$

The greatest measure of quaternion is 1, and the component of all quaternions all is positioned at interval [1,1] simultaneously;

(2) optimal control problem of describing with normalized parameter;

The spacecraft attitude kinematic model of normalization method variable description does;

$\stackrel{\stackrel{\&CenterDot;}{~}}{x}=f(\tilde{x},\tilde{u})---\left(19\right)$

Wherein

$\tilde{x}={\left[\begin{array}{cccccccccc}{q}_1& q_2& q_3& q_4& {\widetilde{\mathrm{\&omega;}}}_1& {\widetilde{\mathrm{\&omega;}}}_2& {\widetilde{\mathrm{\&omega;}}}_3& {\widetilde{\mathrm{\&Omega;}}}_1& {\widetilde{\mathrm{\&Omega;}}}_2& {\widetilde{\mathrm{\&Omega;}}}_3\end{array}\right]}^T$

(20)

$\tilde{u}={\left[\begin{array}{ccc}{\tilde{T}}_{u1}& {\tilde{T}}_{u2}& {\tilde{T}}_{u3}\end{array}\right]}^T$

The performance figure of optimal control are:

J＝t _f-t ₀ (21)

Inequality constrain is:

$\begin{array}{cc}{\tilde{x}}_{\min }\&le;\tilde{x}\&le;{\tilde{x}}_{\max }& {\tilde{u}}_{\min }\&le;\stackrel{\text{~}}{u}\&le;{\tilde{u}}_{\max }\end{array}---\left(22\right)$

Wherein, ${\tilde{x}}_{\max }=\left[\begin{array}{cccccccccc}1& 1& 1& 1& 1& 1& 1& 1& 1& 1\end{array}\right],{\tilde{x}}_{\min }=-{\tilde{x}}_{\max },{\tilde{u}}_{\max }=\left[\begin{array}{ccc}1& 1& 1\end{array}\right],{\tilde{u}}_{\min }=-{\tilde{u}}_{\max };$

Equality constraint is

optimization objectives: to find a normalized control

making satellite attitude redirect shortest;

(3) adopt the pseudo-spectrometry of Legendre that optimal control problem is changed into nonlinear programming problem

Optimal control problem is described in time interval [t ₀, t _f], adopt following form that interval, Legendre Gauss Lambert and physical time interval are changed: τ ∈ [τ ₀, τ _N]=[-1,1]

$t=\frac{(t_f-t_0)\mathrm{\&tau;}+(t_f+t_0)}{2}---\left(23\right)$

The normalized attitude motion model of spacecraft is following:

$\stackrel{\stackrel{\&CenterDot;}{~}}{x}\left(\mathrm{\&tau;}\right)=\frac{t_f-t_0}{2}f(\tilde{x}\left(\mathrm{\&tau;}\right),\widetilde{u\left(\mathrm{\&tau;}\right)})---\left(24\right)$

$\begin{array}{cc}\tilde{x}(-1)={\tilde{x}}_{t_0}& \tilde{x}\left(1\right)={\tilde{x}}_{t_f}\end{array}$

Where: indicates time τ corresponding to the normalized derivative of the state variable, the normalized state variable and the control variable,

represents the initial time and end time normalized state variable value,

are constant value after conversion by N-order polynomial to estimate the form of a continuous state and control variables, as follows;

$\begin{array}{cc}\tilde{x}\&ap;{\tilde{x}}^N\left(\mathrm{\&tau;}\right)=\underset{t=0}{\overset{N}{\mathrm{\&Sigma;}}}{\tilde{x}}_l{\mathrm{\&phi;}}_1\left(\mathrm{\&tau;}\right)& \tilde{u}\&ap;{\tilde{u}}^N\left(\mathrm{\&tau;}\right)=\underset{t=0}{\overset{N}{\mathrm{\&Sigma;}}}{\tilde{u}}_l{\mathrm{\&phi;}}_l\left(\mathrm{\&tau;}\right)\end{array}---\left(25\right)$

Where, l = 0,1, ..., N, N represents a selected positive integer,

represents N points fit state variables and control variables in τ corresponding time values.

${\mathrm{\&phi;}}_l\left(\mathrm{\&tau;}\right)=\frac{1}{N(N+1)L_N\left({\mathrm{\&tau;}}_l\right)}\frac{({\mathrm{\&tau;}}^2-1){\stackrel{\&CenterDot;}{L}}_N\left(\mathrm{\&tau;}\right)}{\mathrm{\&tau;}-{\mathrm{\&tau;}}_l}=\left\{\begin{array}{ccc}1& \mathrm{if}& l=j\\ 0& \mathrm{if}& l\&NotEqual;j\end{array}\right.---\left(26\right)$

Following formula is the Lagrange interpolation polynomial on N rank, L _N(τ _l) Legendre polynomial;

${\tilde{x}}_l={\tilde{x}}^N\left({\mathrm{\&tau;}}_l\right),{\tilde{u}}_l={\tilde{u}}^N\left({\mathrm{\&tau;}}_l\right),{\mathrm{\&tau;}}_l=\mathrm{\&tau;}\left(t_l\right)---\left(27\right)$

Wherein, t _lRepresent l node moment corresponding, τ _lThe value of representing l the cooresponding τ of node.

indicates the l-th node corresponds to a normalized state variables and control variables.For according at node τ _lValue

Come the derivative of expression status variable Corresponding spacecraft attitude kinematic model formula (19) is:

${\stackrel{\stackrel{\&CenterDot;}{~}}{x}}^N\left({\mathrm{\&tau;}}_k\right)=\underset{t=0}{\overset{N}{\mathrm{\&Sigma;}}}{D}_{\mathrm{kl}}\tilde{x}\left({\mathrm{\&tau;}}_l\right)---\left(28\right)$

Wherein: D _KlBe the component of the difference matrix D of (N+1) * (N+1):

$D=\left[D_{\mathrm{kl}}\right]=\left\{\begin{array}{cc}\frac{L_N\left({\mathrm{\&tau;}}_k\right)}{L_N\left({\mathrm{\&tau;}}_l\right)}\frac{1}{({\mathrm{\&tau;}}_k-{\mathrm{\&tau;}}_l)}& k\&NotEqual;l\\ -\frac{N(N+1)}{4}& k=l=0\\ \frac{N(N+1)}{4}& k=l=N\\ 0& \mathrm{otherwise}\end{array}\right.---\left(29\right)$

Thereby the equality constraint of the equation of state in optimal control problem is represented through discrete state:

$\underset{l=0}{\overset{N}{\mathrm{\&Sigma;}}}{D}_{\mathrm{kl}}\tilde{x}\left({\mathrm{\&tau;}}_l\right)-\frac{t_f-t_0}{2}f\left(\underset{l=0}{\overset{N}{\mathrm{\&Sigma;}}}\tilde{x}\left({\mathrm{\&tau;}}_l\right){\mathrm{\&phi;}}_l\left({\mathrm{\&tau;}}_k\right),\underset{l=0}{\overset{N}{\mathrm{\&Sigma;}}}\tilde{u}\left({\mathrm{\&tau;}}_l\right){\mathrm{\&phi;}}_l\left({\mathrm{\&tau;}}_k\right)\right)=0---\left(30\right)$

Other equality constraint for

inequality constraints is:

$\begin{array}{cc}{\tilde{x}}_{\min }\&le;\tilde{x}\left({\mathrm{\&tau;}}_l\right)\&le;{\tilde{x}}_{\max }& {\tilde{u}}_{\min }\&le;\tilde{u}\left({\mathrm{\&tau;}}_l\right)\&le;{\tilde{u}}_{\max }\end{array}---\left(31\right)$

The performance indications of optimal control problem are formula (21), and optimization variable is that

and

optimal control problem is converted into nonlinear programming problem;

(4) computation optimization is found the solution nonlinear programming problem, obtains the motor-driven parameter of open loop optimum attitude;

The 3rd goes on foot, obtains the robust feedback controller, realizes that spacecraft is redirected attitude maneuver;

Above said step obtained the open loop control of the motor-driven model of optimum attitude through finding the solution optimal control problem, obtain the robust feedback controller according to attitude error equations;

(1) sets up attitude error equations;

q _d, ω _dBe expectation attitude quaternary element and rotational angular velocity, q _eBe system of axes F _bExpect system of axes F relatively _dAttitude quaternion; Accordingly from F _dTo F _bTransition matrix C (q _e) do;

$C=(q_{e4}^2-q_{e13}^T{q}_{e13})I_{3\&times;3}+2q_{e13}{q}_{e13}^T-{2q}_{e4}{q}_{e13}^{\&times;}---\left(32\right)$

Wherein: q _E1, q _E2, q _E3, q _E4Be quaternion q _eFour components, vectorial q _E13=[q _E1, q _E2, q _E3] ^T,

Be q _E13Transposed vector,

Be q _E13Antisymmetric matrix, q and q _d, q _eRelation be:

q＝mat(q _d)q _e (33)

Wherein

$\mathrm{mat}\left(q_d\right)=\left[\begin{array}{cccc}{q}_{d4}& q_{d3}& -q_{d2}& q_{d1}\\ -q_{d3}& q_{d4}& -q_{d1}& q_{d2}\\ q_{d2}& q_{d1}& q_{d4}& q_{d3}\\ -q_{d1}& -q_{d2}& -q_{d3}& q_{d4}\end{array}\right]---\left(34\right)$

Wherein: q _D1, q _D2, q _D3, q _D4Be q _dFour components, F _bRelative F _dAngular velocity omega _eAt system of axes F _bIn be:

ω _e＝ω-Cω _d (35)

According to ω _eAnd q _eRewriting formula (4) and formula (1), then attitude error equations is:

$I_s{\stackrel{\&CenterDot;}{\mathrm{\&omega;}}}_e=-{({\mathrm{\&omega;}}_e+{\mathrm{C\&omega;}}_d)}^{\&times;}{I}_s({\mathrm{\&omega;}}_e+{\mathrm{C\&omega;}}_d)-{({\mathrm{\&omega;}}_e+{\mathrm{C\&omega;}}_d)}^{\&times;}{I}_{\mathrm{RW}}\mathrm{\&Omega;}+I_s({\mathrm{\&omega;}}_e^{\&times;}{\mathrm{C\&omega;}}_d-{C\stackrel{\&CenterDot;}{\mathrm{\&omega;}}}_d)-T_u$

(36)

${\stackrel{\&CenterDot;}{q}}_e=\frac{1}{2}\mathrm{\&Xi;}\left(q_e\right){\mathrm{\&omega;}}_e$

(2) obtain the robust feedback controller

The Li Yapuluofu function is:

$V=\frac{1}{2}{{\mathrm{\&omega;}}_e}^T{I}_s{\mathrm{\&omega;}}_e+k_1(q_{e1}^2+q_{e2}^2+q_{e3}^2+{(1-q_{e4})}^2)---\left(37\right)$

Wherein: k ₁＞0; The derivative that can obtain speed V is:

$\stackrel{\&CenterDot;}{V}={\mathrm{\&omega;}}_e^T\{-{({\mathrm{\&omega;}}_e+{\mathrm{C\&omega;}}_d)}^{\&times;}{I}_s({\mathrm{\&omega;}}_e+{\mathrm{C\&omega;}}_d)-{({\mathrm{\&omega;}}_e+{\mathrm{C\&omega;}}_d)}^{\&times;}{I}_{\mathrm{RW}}\mathrm{\&Omega;}+I_s({\mathrm{\&omega;}}_e^{\&times;}{\mathrm{C\&omega;}}_d-{C\stackrel{\&CenterDot;}{\mathrm{\&omega;}}}_d)-T_u\}+k_1{\mathrm{\&omega;}}_e^T{q}_{e13}^T$

$={\mathrm{\&omega;}}_e^T\{-{\mathrm{\&omega;}}_e^{\&times;}(I_s({\mathrm{\&omega;}}_e+{\mathrm{C\&omega;}}_d)+I_{\mathrm{RW}}\mathrm{\&Omega;})-({\left(C{\mathrm{\&omega;}}_d\right)}^{\&times;}{I}_s+I_s{\left({\mathrm{C\&omega;}}_d\right)}^{\&times;}){\mathrm{\&omega;}}_e-{\left({\mathrm{C\&omega;}}_d\right)}^{\&times;}{I}_s{\mathrm{C\&omega;}}_d-I_s{C\stackrel{\&CenterDot;}{\mathrm{\&omega;}}}_d$

(38)

$-{\left({\mathrm{C\&omega;}}_d\right)}^{\&times;}{I}_{\mathrm{RW}}\mathrm{\&Omega;}-T_u\}+k_1{\mathrm{\&omega;}}_e^T{q}_{e13}^T$

$={\mathrm{\&omega;}}_e^T(-{\left({\mathrm{C\&omega;}}_d\right)}^{\&times;}{I}_s{\mathrm{C\&omega;}}_d-I_s{C\stackrel{\&CenterDot;}{\mathrm{\&omega;}}}_d-{\left({\mathrm{C\&omega;}}_d\right)}^{\&times;}{I}_{\mathrm{RW}}\mathrm{\&Omega;}-T_u)+k_1{\mathrm{\&omega;}}_e^T{q}_{e13}^T$

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

$u=T_u{=(-{\mathrm{C\&omega;}}_d)}^{\&times;}{I}_s{\mathrm{C\&omega;}}_d-I_s{C\stackrel{\&CenterDot;}{\mathrm{\&omega;}}}_d-{\left({\mathrm{C\&omega;}}_d\right)}^{\&times;}{I}_{\mathrm{RW}}\mathrm{\&Omega;}+k_1{q}_{e13}+k_2{\mathrm{\&omega;}}_e---\left(39\right)$

Go on foot (4) through robust feedback controller tracking second and obtain the motor-driven parameter of open loop optimum attitude, realize that spacecraft is redirected attitude maneuver.

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