CN105022402A - Method for determining shortest time of coupled rigid-body spacecraft rapid maneuver - Google Patents

Abstract

The invention relates to a method for determining the shortest time of coupled rigid-body spacecraft rapid maneuver, and aims to solve problems of small effective operating range and long maneuvering time of a non-contact actuator of an existing undisturbed load satellite. The method is implemented through the following technical scheme which comprises the steps of 1, establishing a load platform coordinate system O<s>x<a>y<a>z<a> and a service platform coordinate system O<b>x<b>y<b>z<b>, and marking the load platform coordinate system O<s>x<a>y<a>z<a> and the service platform coordinate system O<b>x<b>y<b>z<b> as an Sa system and an Sb system respectively; 2, writing rotation inertia matrixes of a load platform and a service platform in regards to a system mass center of the coupled rigid-body spacecraft; 3, writing an attitude kinematics equation and an angular momentum conservation equation of the coupled rigid-body spacecraft; 4, calculating an Euler axis e and a rotation angle Phi<f>; 5, writing expressions of the angular acceleration Phi<..>(t), the angular speed Phi<.>(t) and the angle degree Phi(t); 6, writing expressions of an attitude quaternion q<m0> and an attitude quaternion q<m> of the load platform, an attitude matrix C<ao> of the load platform , an attitude matrix C<bo> of the service platform, the attitude angular speed Omega<al><a> of the load platform, the attitude angular speed Omega<bl><b> of the service platform, the attitude angular acceleration Omega<.><al><a> of the load platform and the attitude angular acceleration Omega<.><bl><a> of the service platform; 7, acquiring T<e><a>, T<w><b> and H<w><b> in regards to Phi<..>max, Phi<.>max and t; and 8, solving the constrained shortest maneuvering time by using a Matlab optimization toolbox according to the rotation angle Phi<f>, the T<e><a>, the T<w><b> and the H<w><b>. The method provided by the invention is applied to the field of spacecrafts.

Description

The shortest time defining method of a kind of pair of rigid body spacecraft fast reserve

Technical field

The present invention relates to the shortest time defining method of two rigid body spacecraft fast reserve.

Background technology

The fast reserve that the present invention be directed to " unperturbed load " satellite proposes.The Objective Concept NelsonPedreior of " unperturbed load " satellite proposes, and is applied to the satellite of high precision sensing and high stability demand for control." unperturbed load " satellite is a kind of two rigid body spacecraft, be made up of payload platform and service platform two parts, the attitude of contactless actuator control load platform, extra topworks (as reaction wheel etc.) controls the attitude of service platform, contactless actuator can be isolated extra topworks and be acted on the vibration of external environmental interference of service platform, makes useful load realize high precision and points to and high stability control.But, because the efficient working range of contactless actuator only has several millimeters, therefore the attitude of service platform must follow the tracks of the attitude of payload platform, in large angle maneuver process, in essence, service platform and payload platform carry out motor-driven as a whole, are equivalent to a motor-driven single rigid body satellite, time kept in reserve mainly limits by extra topworks, and the time kept in reserve is long.For " unperturbed load " satellite carrying large-scale useful load, existing configuration cannot meet the demand of fast reserve.

Summary of the invention

The present invention is that the efficient working range of the contactless actuator solving existing unperturbed load satellite and time kept in reserve are long, the problem of the unperturbed load satellite fast reserve of carrying large-scale useful load cannot be met, and propose the shortest time defining method of a kind of pair of rigid body spacecraft fast reserve.

Above-mentioned goal of the invention is achieved through the following technical solutions:

Step one, set up payload platform coordinate system O _sx _ay _az _awith service platform coordinate system O _bx _by _bz _b, payload platform coordinate system is designated as S _asystem, service platform coordinate system is designated as S _bsystem, wherein, O _sbe positioned at ball pivot center, O _bbe positioned at service platform barycenter, O _afor payload platform barycenter, O _cfor two rigid body Space Vehicle System barycenter, the true origin of payload platform coordinate system is positioned at O _s, payload platform and service platform form two rigid body spacecraft, and payload platform is connected by ball pivot with service platform;

Step 2, the S obtained according to step one _asystem and S _bsystem, writes out payload platform and the service platform moment of inertia matrix about two rigid body Space Vehicle System barycenter;

Step 3, the payload platform obtained according to step 2 and service platform, about the moment of inertia matrix of two rigid body Space Vehicle System barycenter, write out the attitude kinematics equations of two rigid body spacecraft and the conservation of angular momentum equation of two rigid body spacecraft;

Step 4, assumed (specified) load platform are by initial attitude motor-driven Euler's axle e to targeted attitude and corner Φ _f;

Step 5, the corner Φ drawn according to step 4 _f, write out the angular acceleration of payload platform pursuit path angular velocity with the expression formula of angle Φ (t);

The angular acceleration of the payload platform pursuit path that step 6, the Euler's axle e drawn according to step 4 and step 5 draw angular velocity with the expression formula of angle Φ (t), write out payload platform attitude quaternion q _m0and q _m, payload platform attitude matrix C _ao, service platform attitude matrix C _bo, payload platform attitude angular velocity the attitude angular velocity of service platform the attitude angle acceleration of payload platform with the attitude angle acceleration of service platform expression formula;

Step 7, by step 5 with Φ (t), and the C in step 6 _ao, c _bo, with substitute into the attitude dynamic equations of the two rigid body spacecrafts in step 3 and the conservation of angular momentum equation of pair rigid body spacecraft, obtain about with t's with

Step 8, the corner Φ drawn according to step 4 _f, in step 7 with use Matlab Optimization Toolbox, solve the shortest time kept in reserve containing constraint.

Invention effect

The gesture stability that the present invention be directed to two rigid body satellites of monitoring for disaster emergency proposes, and two rigid body spacecraft is made up of payload platform and service platform, and payload platform carries useful load, and service platform carries topworks, and the two is connected by ball pivot.The meaning of the two rigid body spacecraft of research is: (1) payload platform can carry more useful load; (2) there are motor and flywheel in the topworks of two rigid body spacecraft, and compared to single rigid body spacecraft of homogenous quantities and moment of inertia, its controllable velocity is faster; (3) payload platform separates with service platform, can reduce the impact of vibration on useful load of topworks.Because one of main task of two rigid body spacecraft is the payload platform fast reserve making to carry useful load, so to the attitude no requirement (NR) of service platform of carrying topworks.The situation that the present invention have chosen makes payload platform fast reserve, service platform keeps absolute orientation.

Achieve the fast reserve of two rigid body spacecraft, motor-driven ball pivot is added between payload platform and service platform, relieve the restriction of contactless actuator efficient working range, the relative attitude making payload platform and service platform can carry out larger angle moves, the moment of inertia matrix become during by introducing two, give the attitude dynamic equations of detailed two rigid body spacecrafts, by introducing reasonable assumption, there are two points of rigid bodies of serious coupling in decoupling zero, give the objective function of time optimal problem and the expression formula of constraint function, use Matlab Optimization Toolbox can obtain the shortest time kept in reserve and corresponding maximum angular acceleration and maximum angular rate easily, carry out in the process of Large Angle Attitude Maneuver at payload platform, because the attitude of service platform is without the need to following the tracks of the attitude of payload platform, namely, without the need to motor-driven whole satellite, only need maneuver load terrace part, greatly can shorten the time kept in reserve, and original technology needs maneuver load platform and service platform simultaneously, assumed load platform and service platform quality close, the fast reserve method that this patent proposes can technically original, time kept in reserve will shorten 50%.The situation that the present invention have chosen makes payload platform fast reserve, service platform keeps absolute orientation, meets the problem of carrying the unperturbed load satellite fast reserve of large-scale useful load.

Accompanying drawing explanation

Fig. 1 is process flow diagram of the present invention;

Fig. 2 is the schematic diagram of coordinate system described in embodiment two, wherein, and O _efor earth centroid, O _cfor two rigid body Space Vehicle System barycenter, O _afor payload platform barycenter, O _bfor service platform barycenter, O _sfor ball pivot barycenter, [i _aj _ak _a] ^tfor S _athe coordinate base of system, i _afor S _athe z-axis of system, j _afor S _athe x-axis of system, k _afor S _athe y-axis of system, T is transposition, [i _bj _bk _b] ^tfor S _bthe coordinate base of system, i _bfor S _bthe x-axis of system, j _bfor S _bthe y-axis of system, k _bfor S _bthe z-axis of system, dm _afor payload platform quality infinitesimal, dm _bfor service platform quality infinitesimal, R _afor by O _epoint to O _avector, R _bfor by O _epoint to O _bvector, R _cfor by O _epoint to O _cvector, r _afor by O _epoint to dm _avector, r _bfor by O _epoint to dm _bvector, d ₁for by O _spoint to O _avector, d ₂for by O _bpoint to O _svector, p ₁for by O _apoint to O _cvector, p ₂for by O _bpoint to O _cvector, q ₁for by O _spoint to dm _avector, q ₂for by O _spoint to dm _bvector;

Fig. 3 is the schematic diagram of payload platform pursuit path described in embodiment six, wherein, and t ₀for motor-driven start time, t ₁for accelerating finish time, t ₂for at the uniform velocity finish time, t _ffor motor-driven finish time, for the angular velocity of payload platform pursuit path, for the maximum angular acceleration of payload platform pursuit path, for the maximum angular rate of payload platform pursuit path, for the angular acceleration of payload platform pursuit path;

Fig. 4 is the attitude angle acceleration plots of figure payload platform in embodiment 1, °/s degree of being/second;

Fig. 5 is figure motor torque modulus value curve map in embodiment 1, and Nm is ox rice;

Fig. 6 is figure flywheel moment modulus value curve map in embodiment 1;

Fig. 7 is figure flywheel angular momentum curve map in embodiment 1.

Embodiment

Embodiment one: composition graphs 1 illustrates present embodiment, the shortest time defining method of a kind of pair of rigid body spacecraft fast reserve specifically carries out according to the following steps:

Step one, set up payload platform coordinate system O _sx _ay _az _awith service platform coordinate system O _bx _by _bz _b, payload platform coordinate system is designated as S _asystem, service platform coordinate system is designated as S _bsystem, wherein, O _sbe positioned at ball pivot center, O _bbe positioned at service platform barycenter, O _afor payload platform barycenter, O _cfor two rigid body Space Vehicle System barycenter, the true origin of payload platform coordinate system is positioned at O _s, payload platform and service platform form two rigid body spacecraft, and payload platform is connected by ball pivot with service platform;

Step 2, the S obtained according to step one _asystem and S _bsystem, writes out payload platform and the service platform moment of inertia matrix about two rigid body Space Vehicle System barycenter;

Step 3, the payload platform obtained according to step 2 and service platform, about the moment of inertia matrix of two rigid body Space Vehicle System barycenter, write out the attitude kinematics equations of two rigid body spacecraft and the conservation of angular momentum equation of two rigid body spacecraft;

Step 4, assumed (specified) load platform are by initial attitude motor-driven Euler's axle e to targeted attitude and corner Φ _f;

Step 5, the corner Φ drawn according to step 4 _f, write out the angular acceleration of payload platform pursuit path angular velocity with the expression formula of angle Φ (t);

The angular acceleration of the payload platform pursuit path that step 6, the Euler's axle e drawn according to step 4 and step 5 draw angular velocity with the expression formula of angle Φ (t), write out payload platform attitude quaternion q _m0and q _m, payload platform attitude matrix C _ao, service platform attitude matrix C _bo, payload platform attitude angular velocity the attitude angular velocity of service platform the attitude angle acceleration of payload platform with the attitude angle acceleration of service platform expression formula;

Step 7, based on a zero momentum system, ignore disturbance torque, by step 5 with Φ (t), and the C in step 6 _ao, c _bo, with substitute into the attitude dynamic equations of the two rigid body spacecrafts in step 3 and the conservation of angular momentum equation of pair rigid body spacecraft, obtain about with t's with

Step 8, the corner Φ drawn according to step 4 _f, in step 7 with use Matlab Optimization Toolbox, solve the shortest time kept in reserve containing constraint.

Embodiment two: composition graphs 2 illustrates present embodiment, present embodiment and embodiment one unlike: set up payload platform coordinate system O in described step one _sx _ay _az _awith service platform coordinate system O _bx _by _bz _b, payload platform coordinate system is designated as S _asystem, service platform coordinate system is designated as S _bsystem, wherein, O _sbe positioned at ball pivot center, O _bbe positioned at service platform barycenter, O _afor payload platform barycenter, O _cfor two rigid body Space Vehicle System barycenter, the true origin of payload platform coordinate system is positioned at O _s, payload platform and service platform form two rigid body spacecraft, and payload platform is connected by ball pivot with service platform; Detailed process is:

Set up payload platform coordinate system O _sx _ay _az _awith service platform coordinate system O _bx _by _bz _b, payload platform coordinate system O _sx _ay _az _abe designated as S _asystem, service platform coordinate system is designated as S _bsystem, wherein, O _sfor ball pivot center, O _bfor service platform barycenter, x _afor payload platform coordinate system x-axis, y _afor payload platform coordinate system y-axis, z _afor payload platform coordinate system z-axis, x _bfor service platform coordinate system x-axis, y _bfor service platform coordinate system y-axis, z _bfor service platform coordinate system z-axis; O _afor payload platform barycenter, O _cfor two rigid body Space Vehicle System barycenter, for convenience of the attitude dynamic equations of the two rigid body spacecraft of follow-up derivation, the true origin of payload platform coordinate system is not positioned at O _a, and be positioned at O _s.

Other step and parameter identical with embodiment one.

Embodiment three: present embodiment and embodiment one or two unlike: according to the S that step one obtains in described step 2 _asystem and S _bsystem, writes out payload platform and the service platform moment of inertia matrix about two rigid body Space Vehicle System barycenter; Detailed process is:

Payload platform is about O _cangular-momentum vector be:

In formula, dm _afor the quality infinitesimal of payload platform, q ₁for by O _spoint to dm _avector, p ₁for by O _apoint to O _cvector, v _aIfor dm _aabsolute velocity vector, for O _babsolute velocity vector, I ₁for payload platform is about O _sinertia dyad, H _afor payload platform is about O _cangular-momentum vector, d ₁for by O _spoint to O _avector, ω _aIfor S _athe absolute angular velocities vector of system, ω _bIfor S _bthe absolute angular velocities vector of system, d ₂for by O _spoint to O _bvector;

Service platform is about O _cangular-momentum vector be:

In formula, dm _bfor the quality infinitesimal of service platform, q ₂for by O _bpoint to dm _bvector, p ₂for by O _bpoint to O _cvector, v _bIfor dm _babsolute velocity vector, I ₂for service platform is about O _binertia dyad, H _bfor service platform is about O _cangular-momentum vector, ω _bIfor S _bthe absolute angular velocities vector of system;

Two rigid body Space Vehicle System is about O _ctotal angular momentum be:

In formula, for O _babsolute velocity vector, H _wfor the angular-momentum vector of flywheel, H is that two rigid body Space Vehicle System is about O _cangular-momentum vector;

Because m _ap ₁+ m _bp ₂=0, and p ₂-p ₁=d ₁+ d ₂, so

$p_1=-\frac{m_b}{m_a+m_b}(d_1+d_2)---\left(29\right)$

$p_2=\frac{m_a}{m_a+m_b}(d_1+d_2)---\left(30\right)$

Formula (29) and formula (30) are substituted into formula (28), obtains

In formula, E is unit inertia dyad;

Therefore payload platform and service platform are about O _cinertia dyad be respectively:

In formula, I _afor payload platform is about O _cinertia dyad, I _bfor service platform is about O _cinertia dyad;

Then formula (31) can be expressed as

H＝I _a·ω _aI+I _b·ω _bI+H _w(34)

I _aat S _acomponent Matrices under system is payload platform about O _cmoment of inertia matrix:

$I_a^a=I_1^a-\frac{m_a{m}_b}{m_a+m_b}\&lsqb;{\left(d_1^a\right)}^{\&times;}{\left(d_1^a\right)}^{\&times;}+{\left(d_1^a\right)}^{\&times;}{C}_{ab}{\left(d_2^b\right)}^{\&times;}{C}_{ba}\&rsqb;---\left(35\right)$

In formula, for payload platform is about O _cmoment of inertia matrix, for payload platform is about O _smoment of inertia matrix, m _afor the quality of payload platform, m _bfor the quality of service platform, for by O _spoint to O _athe component array of vector, for by O _spoint to O _bthe component array of vector, C _abfor S _abe relative S _bthe attitude matrix of system, C _bafor S _bbe relative S _athe attitude matrix of system, for multiplication cross matrix, for multiplication cross matrix;

I _bat S _bcomponent Matrices under system is service platform about O _cmoment of inertia matrix:

$I_b^b=I_2^b-\frac{m_a{m}_b}{m_a+m_b}\&lsqb;{\left(d_2^b\right)}^{\&times;}{\left(d_2^b\right)}^{\&times;}+{\left(d_2^b\right)}^{\&times;}{C}_{ba}{\left(d_1^a\right)}^{\&times;}{C}_{ab}\&rsqb;---\left(36\right)$

In formula, superscript a and b is respectively vector or dyad at S _asystem and S _bcomponent array under system or Component Matrices, for service platform is about O _cmoment of inertia matrix, for service platform is about O _bmoment of inertia matrix;

And with first order derivative be respectively

${\stackrel{\&CenterDot;}{I}}_a^a=\frac{m_a{m}_b}{m_a+m_b}{\left(d_1^a\right)}^{\&times;}{\&lsqb;{({\mathrm{\&omega;}}_{aI}^a-C_{ab}{\mathrm{\&omega;}}_{bI}^b)}^{\&times;}{C}_{ab}{d}_2^b\&rsqb;}^{\&times;}---\left(37\right)$

${\stackrel{\&CenterDot;}{I}}_b^b=\frac{m_a{m}_b}{m_a+m_b}{\left(d_2^b\right)}^{\&times;}{\&lsqb;{({\mathrm{\&omega;}}_{bI}^b-C_{ba}{\mathrm{\&omega;}}_{aI}^a)}^{\&times;}{C}_{ba}{d}_1^a\&rsqb;}^{\&times;}---\left(38\right)$

In formula, for S _athe absolute angular velocities vector of system is at S _acomponent array under system, for S _bthe absolute angular velocities vector of system is at S _bcomponent array under system, for multiplication cross matrix, for multiplication cross matrix.

Other step and parameter identical with embodiment one or two.

Embodiment four: present embodiment and embodiment one, two or three unlike: the payload platform obtained according to step 2 in step 3 described in described step 3 and service platform, about the moment of inertia matrix of two rigid body Space Vehicle System barycenter, write out the attitude kinematics equations of two rigid body spacecraft and the conservation of angular momentum equation of two rigid body spacecraft; Detailed process is:

The payload platform obtained according to step 2 and service platform, about the moment of inertia matrix of two rigid body Space Vehicle System barycenter, write out attitude kinematics equations and the conservation of angular momentum equation of two rigid body spacecraft:

The attitude kinematics equations of two rigid body spacecraft is as formula (3), (4):

$\begin{array}{c}\&lsqb;I_1^a-\frac{m_a{m}_b}{m_a+m_b}{\left(d_1^a\right)}^{\&times;}{\left(d_1^a\right)}^{\&times;}\&rsqb;{\stackrel{\&CenterDot;}{\mathrm{\&omega;}}}_{aI}^a-\frac{m_a{m}_b}{m_a+m_b}{\left(d_1^a\right)}^{\&times;}{C}_{ab}{\left(d_2^b\right)}^{\&times;}{\stackrel{\&CenterDot;}{\mathrm{\&omega;}}}_{bI}^b=\\ T_e^a+T_{da}^a-{\left({\mathrm{\&omega;}}_{aI}^a\right)}^{\&times;}{I}_1^a{\mathrm{\&omega;}}_{aI}^a+\\ {\left(d_1^a\right)}^{\&times;}\{{\mathrm{\&mu;m}}_a{C}_{ao}(\frac{R_c^o}{{R_c}^3}-\frac{R_a^o}{{R_a}^3})-\frac{m_a{m}_b}{m_a+m_b}\left[\begin{array}{c}{\left({\mathrm{\&omega;}}_{aI}^a\right)}^{\&times;}{\left({\mathrm{\&omega;}}_{aI}^a\right)}^{\&times;}{d}_1^a+\\ C_{ab}{\left({\mathrm{\&omega;}}_{bI}^b\right)}^{\&times;}{\left({\mathrm{\&omega;}}_{bI}^b\right)}^{\&times;}{d}_2^b\end{array}\right]\}\end{array}---\left(3\right)$

$\begin{array}{c}{C}_{ba}{I}_a^a{\stackrel{\&CenterDot;}{\mathrm{\&omega;}}}_{aI}^a+I_b^b{\stackrel{\&CenterDot;}{\mathrm{\&omega;}}}_{bI}^b=T_w^b+T_d^b-\\ \{C_{ba}\&lsqb;{\left({\mathrm{\&omega;}}_{aI}^a\right)}^{\&times;}{I}_a^a{\mathrm{\&omega;}}_{aI}^a+{\stackrel{\&CenterDot;}{I}}_a^a{\mathrm{\&omega;}}_{aI}^a\&rsqb;+{\left({\mathrm{\&omega;}}_{bI}^b\right)}^{\&times;}{I}_b^b{\mathrm{\&omega;}}_{bI}^b+{\stackrel{\&CenterDot;}{I}}_b^b{\mathrm{\&omega;}}_{bI}^b+{\left({\mathrm{\&omega;}}_{bI}^b\right)}^{\&times;}{H}_w^b\}\end{array}---\left(4\right)$

The conservation of angular momentum equation of two rigid body spacecraft is as formula (5):

$C_{ba}{I}_a^a{\mathrm{\&omega;}}_{aI}^a+I_b^b{\mathrm{\&omega;}}_{bI}^b+H_w^b=H_0^b---\left(5\right)$

In formula, for derivative, for derivative, for derivative, for derivative, superscript o is component array under the orbital coordinate system of two rigid body spacecraft of vector or dyad or Component Matrices, for S _athe component array of the absolute angular velocities vector of system, for S _bthe component array of the absolute angular velocities vector of system, C _aofor S _abe the attitude matrix of relative orbit system, for the coordinate array of motor torque vector, for the coordinate array of flywheel moment vector, for acting on the component array of the disturbance torque vector of payload platform, for acting on the component array of the disturbance torque vector of two rigid body Space Vehicle System, for the component array of flywheel angular-momentum vector, for the component array of two rigid body Space Vehicle System angular-momentum vector, μ is geocentric gravitational constant, for being pointed to the component array of the vector of payload platform barycenter by the earth's core, for being pointed to the component array of the vector of two rigid body Space Vehicle System barycenter by the earth's core, R _cfor mould, R _afor mould, for multiplication cross matrix, for multiplication cross matrix;

Payload platform is around O _athe attitude kinematics equations rotated is:

In formula, for vector or dyad are at S _aderivative under system, T _efor motor torque vector, T _bafor service platform acts on the moment vector of payload platform, T _dafor acting on the disturbance torque vector of payload platform, ω _aIfor S _athe absolute angular velocities vector of system;

Act on the power F making a concerted effort to be acted on by service platform payload platform of payload platform _baand external environment acts on the perturbed force F of payload platform _dacomposition, according to Newton second law, has

$F_{ba}+F_{da}=m_a\frac{{d_I}^2{R}_a}{{\mathrm{dt}}^2}=m_a\frac{{d_I}^2(R_c-p_1)}{{\mathrm{dt}}^2}---\left(43\right)$

In formula, the derivative under inertial system for vector or dyad, R _afor pointing to O by the earth's core _avector, R _cfor pointing to O by the earth's core _cvector, p ₁for by O _apoint to O _cvector;

Based on the earth-spacecraft two system, have

$\frac{{d_I}^2\left|R_c\right|}{{\mathrm{dt}}^2}=-\frac{\mathrm{\&mu;}}{{\left|R_c\right|}^3}{R}_c---\left(44\right)$

In formula, μ is Gravitational coefficient of the Earth; | R _c| be R _cmodulus value;

Because F _dafor terrestrial attraction, so get

$F_{da}=-\frac{{\mathrm{\&mu;m}}_a}{{\left|R_a\right|}^3}{R}_a---\left(45\right)$

In formula, | R _a| be R _amodulus value;

Formula (29), formula (44) and formula (45) are substituted into formula (43), obtains

$\begin{array}{c}{F}_{ba}={\mathrm{\&mu;m}}_a(\frac{R_a}{{\left|R_a\right|}^3}-\frac{R_c}{{\left|R_c\right|}^3})+\\ \frac{m_a{m}_b}{m_a+m_b}\&lsqb;{\mathrm{\&omega;}}_{aI}\&times;({\mathrm{\&omega;}}_{aI}\&times;d_1)+\frac{d_a{\mathrm{\&omega;}}_{aI}}{dt}\&times;d_1+{\mathrm{\&omega;}}_{bI}\&times;({\mathrm{\&omega;}}_{bI}\&times;d_2)+\frac{d_b{\mathrm{\&omega;}}_{bI}}{dt}\&times;d_2\&rsqb;\end{array}---\left(46\right)$

In formula, for vector or dyad are at S _bderivative under system;

Again by T _ba=-d ₁× F _basubstitution formula (42), obtains payload platform around O _athe attitude kinematics equations rotated:

Above formula is at S _acomponent type under system is

$\begin{array}{c}\&lsqb;I_1^a-\frac{m_a{m}_b}{m_a+m_b}{\left(d_1^a\right)}^{\&times;}{\left(d_1^a\right)}^{\&times;}\&rsqb;{\stackrel{\&CenterDot;}{\mathrm{\&omega;}}}_{aI}^a-\frac{m_a{m}_b}{m_a+m_b}{\left(d_1^a\right)}^{\&times;}{C}_{ab}{\left(d_2^b\right)}^{\&times;}{\stackrel{\&CenterDot;}{\mathrm{\&omega;}}}_{bI}^b=\\ T_e^a+T_{da}^a-{\left({\mathrm{\&omega;}}_{aI}^a\right)}^{\&times;}{I}_1^a{\mathrm{\&omega;}}_{aI}^a+\\ {\left(d_1^a\right)}^{\&times;}\{{\mathrm{\&mu;m}}_a{C}_{ao}(\frac{R_c^o}{{\left|R_c\right|}^3}-\frac{R_a^o}{{\left|R_a\right|}^3})-\frac{m_a{m}_b}{m_a+m_b}\left[\begin{array}{c}{\left({\mathrm{\&omega;}}_{aI}^a\right)}^{\&times;}{\left({\mathrm{\&omega;}}_{aI}^a\right)}^{\&times;}{d}_1^a+\\ C_{ab}{\left({\mathrm{\&omega;}}_{bI}^b\right)}^{\&times;}{\left({\mathrm{\&omega;}}_{bI}^b\right)}^{\&times;}{d}_2^b\end{array}\right]\}\end{array}---\left(48\right)$

In formula, for multiplication cross matrix, for multiplication cross matrix, for S _athe absolute angular velocities vector of system is at S _acomponent array under system, for S _bthe absolute angular velocities vector of system is at S _bcomponent array under system, for multiplication cross matrix, for multiplication cross matrix, for being pointed to the component array of the vector of payload platform barycenter by ball pivot center, for being pointed to the component array of the vector of service platform barycenter by ball pivot center, for derivative, for derivative, for motor torque vector is at S _acomponent array under system, for disturbance torque vector is at S _acomponent array under system;

To formula (34) differentiate, obtain two Rigid-body System around O _cthe attitude kinematics equations rotated:

In formula, T _wfor flywheel moment vector, T _dfor acting on the disturbance torque vector H of two rigid body Space Vehicle System _wfor flywheel angular-momentum vector, ω _aIfor S _athe absolute angular velocities vector of system, ω _bIfor S _bthe absolute angular velocities vector of system, H _wfor the angular-momentum vector of flywheel, I _afor payload platform is about O _cinertia dyad, I _bfor service platform is about O _cinertia dyad;

Formula (49) is at S _bcomponent type under system is:

$\begin{array}{c}{C}_{ba}{I}_a^a{\stackrel{\&CenterDot;}{\mathrm{\&omega;}}}_{aI}^a+I_b^b{\stackrel{\&CenterDot;}{\mathrm{\&omega;}}}_{bI}^b=T_w^b+T_d^b-\\ \{C_{ba}\&lsqb;{\left({\mathrm{\&omega;}}_{aI}^a\right)}^{\&times;}{I}_a^a{\mathrm{\&omega;}}_{aI}^a+{\stackrel{\&CenterDot;}{I}}_a^a{\mathrm{\&omega;}}_{aI}^a\&rsqb;+{\left({\mathrm{\&omega;}}_{bI}^b\right)}^{\&times;}{I}_b^b{\mathrm{\&omega;}}_{bI}^b+{\stackrel{\&CenterDot;}{I}}_b^b{\mathrm{\&omega;}}_{bI}^b+{\left({\mathrm{\&omega;}}_{bI}^b\right)}^{\&times;}{H}_w^b\}\end{array}---\left(50\right)$

In formula, C _bafor S _bbe relative S _athe attitude matrix of system, for I _aat S _acomponent Matrices under system, for I _bat S _bcomponent Matrices under system, for T _wat S _bcomponent array under system, for T _dat S _bcomponent array under system, for derivative, for derivative, for H _wat S _bcomponent array under system;

Formula (34) is at S _bcomponent type under system is

$C_{ba}{I}_a^a{\mathrm{\&omega;}}_{aI}^a+I_b^b{\mathrm{\&omega;}}_{bI}^b+H_w^b=H_0^b---\left(41\right)$

In formula, for the total angular momentum vector of two rigid body spacecraft is at S _bcomponent array under system;

Other step and parameter and embodiment one, two or three identical.

Embodiment five: present embodiment and embodiment one, two, three or four unlike: calculate payload platform in described step 4 by initial attitude motor-driven Euler's axle e to targeted attitude and corner Φ _f; Detailed process is:

Φ _f＝2arccosq _m0(6)

$e=\frac{1}{sin\frac{{\mathrm{\&Phi;}}_f}{2}}{q}_m---\left(7\right)$

In formula, q _m0for payload platform is by the motor-driven hypercomplex number Q to targeted attitude of initial attitude _mmark portion, q _mfor payload platform is by the motor-driven hypercomplex number Q to targeted attitude of initial attitude _marrow portion, Q _m=[q _m0q _m] ^tbe four-vector, expression formula is:

$Q_m={Q_0}^{*}\&CircleTimes;Q_f---\left(8\right)$

In formula, Q ₀for the initial attitude hypercomplex number of payload platform, Q _ffor the targeted attitude hypercomplex number of payload platform, Q _mfor the hypercomplex number of targeted attitude, * is hypercomplex number transposition, for hypercomplex number multiplication, T is the transposition of array;

Other step and parameter and embodiment one, two, three or four identical.

Embodiment six: composition graphs 3 illustrates present embodiment, present embodiment and embodiment one, two, three, four or five unlike: according to the corner Φ that step 4 draws in described step 5 _f, write out the angular acceleration of payload platform pursuit path angular velocity with the expression formula of angle Φ (t); Detailed process is:

If Φ is (t ₀)=0, Φ (t _f)=Φ _f

$\stackrel{\&CenterDot;\&CenterDot;}{\mathrm{\&Phi;}}\left(t\right)=\left\{\begin{array}{cc}{\stackrel{\&CenterDot;\&CenterDot;}{\mathrm{\&Phi;}}}_{max}& t_0\&le;t\&le;t_1\\ 0& t_1<t<t_2\\ -{\stackrel{\&CenterDot;\&CenterDot;}{\mathrm{\&Phi;}}}_{max}& t_2\&le;t\&le;t_f\end{array}\right.---\left(9\right)$

$\stackrel{\&CenterDot;}{\mathrm{\&Phi;}}\left(t\right)=\left\{\begin{array}{cc}{\stackrel{\&CenterDot;\&CenterDot;}{\mathrm{\&Phi;}}}_{max}(t-t_0)& t_0\&le;t\&le;t_1\\ {\stackrel{\&CenterDot;}{\mathrm{\&Phi;}}}_{max}& t_1<t<t_2\\ {\stackrel{\&CenterDot;}{\mathrm{\&Phi;}}}_{\max }-{\stackrel{\&CenterDot;\&CenterDot;}{\mathrm{\&Phi;}}}_{max}(t-t_2)& t_2\&le;t\&le;t_f\end{array}\right.---\left(10\right)$

$\mathrm{\&Phi;}\left(t\right)=\left\{\begin{array}{cc}\frac{1}{2}{\stackrel{\&CenterDot;\&CenterDot;}{\mathrm{\&Phi;}}}_{\max }{(t-t_0)}^2& t_0\&le;t\&le;t_1\\ \frac{1}{2}{\stackrel{\&CenterDot;\&CenterDot;}{\mathrm{\&Phi;}}}_{\max }{(t_1-t_0)}^2+{\stackrel{\&CenterDot;}{\mathrm{\&Phi;}}}_{\max }(t-t_1)& t_1<t<t_2\\ \frac{1}{2}{\stackrel{\&CenterDot;\&CenterDot;}{\mathrm{\&Phi;}}}_{\max }{(t_1-t_0)}^2+{\stackrel{\&CenterDot;}{\mathrm{\&Phi;}}}_{\max }(t-t_1)-\frac{1}{2}{\stackrel{\&CenterDot;\&CenterDot;}{\mathrm{\&Phi;}}}_{\max }{(t-t_2)}^2& t_2\&le;t\&le;t_f\end{array}\right.---\left(11\right)$

In formula, for the angular acceleration of payload platform pursuit path, for the angular velocity of payload platform pursuit path, Φ (t) is the angle of payload platform pursuit path, for the maximum angular acceleration of payload platform pursuit path, for the maximum angular rate of payload platform pursuit path, t ₀for motor-driven start time, t ₁for accelerating finish time, t ₂for at the uniform velocity finish time, t _ffor motor-driven finish time, t is the time, and unit is second.

Other step and parameter and embodiment one, two, three, four or five identical.

Embodiment seven: present embodiment and embodiment one, two, three, four, five or six are unlike the angular acceleration of the payload platform pursuit path that the Euler's axle e drawn according to step 4 in described step 6 and step 5 draw angular velocity with the expression formula of angle Φ (t), write out payload platform attitude quaternion q _m0and q _m, payload platform attitude matrix C _ao, service platform attitude matrix C _bo, payload platform attitude angular velocity the attitude angular velocity of service platform the attitude angle acceleration of payload platform with the attitude angle acceleration of service platform expression formula; Detailed process is:

Payload platform pursuit path followed the tracks of by assumed load platform, and service platform is stablized;

Then payload platform attitude quaternion, as formula (12) (13):

$q_{m0}=cos\frac{\mathrm{\&Phi;}\left(t\right)}{2}---\left(12\right)$

$q_m=esin\frac{\mathrm{\&Phi;}\left(t\right)}{2}---\left(13\right)$

The attitude matrix of payload platform, as formula (14):

C _ao＝q _mq _m ^T+[q _m0E-(q _m) ^×] ²(14)

The attitude angular velocity of payload platform, as formula (15):

${\mathrm{\&omega;}}_{aI}^a=\stackrel{\&CenterDot;}{\mathrm{\&Phi;}}\left(t\right)e+C_{ao}{\mathrm{\&omega;}}_{oI}^o---\left(15\right)$

The attitude angle acceleration of payload platform, as formula (16):

${\stackrel{\&CenterDot;}{\mathrm{\&omega;}}}_{aI}^a=\stackrel{\&CenterDot;\&CenterDot;}{\mathrm{\&Phi;}}\left(t\right)e-\stackrel{\&CenterDot;}{\mathrm{\&Phi;}}\left(t\right)e^{\&times;}{C}_{ao}{\mathrm{\&omega;}}_{oI}^o---\left(16\right)$

The attitude matrix of service platform, as formula (17):

C _bo＝E (17)

The attitude angular velocity of service platform, as formula (18):

${\mathrm{\&omega;}}_{bI}^b={\mathrm{\&omega;}}_{oI}^o---\left(18\right)$

The attitude angle acceleration of service platform, as formula (19):

${\stackrel{\&CenterDot;}{\mathrm{\&omega;}}}_{bI}^b=0---\left(19\right)$

In formula, q _m0for payload platform is by the motor-driven hypercomplex number Q to targeted attitude of initial attitude _mmark portion, q _mfor payload platform is by the motor-driven hypercomplex number Q to targeted attitude of initial attitude _marrow portion, C _bofor S _bbe the attitude matrix of relative orbit system, C _obfor the relative S of track system _bthe attitude matrix of system, E is unit matrix, and e is Euler's axle of payload platform pursuit path, q _m ^tfor q _mtransposition, (q _m) ^×for q _mmultiplication cross matrix, for the coordinate array of orbit angular velocity vector under spacecraft orbit system, e ^×for the multiplication cross matrix of e, Φ (t) is the angle of payload platform pursuit path, for the angular velocity of payload platform pursuit path, for the angular acceleration of payload platform pursuit path, C _bofor S _bbe relative S _othe attitude matrix of system;

Other step and parameter and embodiment one, two, three, four, five or six identical.

Embodiment eight: present embodiment and embodiment one, two, three, four, five, six or seven unlike: based on a zero momentum system in described step 7, ignore disturbance torque, by step 5 with Φ (t), and the C in step 6 _ao, c _bo, with substitute into the attitude dynamic equations of the two rigid body spacecrafts in step 3 and the conservation of angular momentum equation of pair rigid body spacecraft, obtain about with t's with detailed process is:

By the q in step 6 _m0and q _msubstitute into C _ao, obtain the C about Φ (t) _ao, then by C _aosubstitute into with in, obtain about with Φ's (t) and about with Φ's (t) finally by C _ao, c _bo, with substitute into the attitude dynamic equations of the two rigid body spacecrafts in step 3 and the conservation of angular momentum equation of two rigid body spacecraft, available represent with t; available represent with t; available represent with t, obtain about with Φ's (t) with

In formula, for the component array of motor torque vector, for the component array of flywheel moment vector, for the component array of flywheel angular-momentum vector;

Wherein, for the maximum angular acceleration of payload platform pursuit path function, for the maximum angular rate of payload platform pursuit path function, for the function of time t.

Other step and parameter and embodiment one, two, three, four, five, six or seven identical.

Embodiment nine: present embodiment and embodiment one, two, three, four, five, six, seven or eight unlike: according to the corner Φ that step 4 draws in described step 8 _f, in step 7 with use Matlab Optimization Toolbox, solve the shortest time kept in reserve containing constraint; Detailed process is:

Use Matlab Optimization Toolbox, separate the minimum value containing constraint, wherein, variable is with objective function is the time kept in reserve:

$\mathrm{\&Delta;}t=\frac{{\mathrm{\&Phi;}}_f}{{\stackrel{\&CenterDot;}{\mathrm{\&Phi;}}}_{max}}+\frac{{\stackrel{\&CenterDot;}{\mathrm{\&Phi;}}}_{max}}{{\stackrel{\&CenterDot;\&CenterDot;}{\mathrm{\&Phi;}}}_{max}}---\left(20\right)$

Constraint function is:

$\left|T_e^a\right|\&le;T_{emax}---\left(21\right)$

$\left|T_w^b\right|\&le;T_{wmax}---\left(22\right)$

$\left|H_w^b\right|\&le;H_{wmax}---\left(23\right)$

In formula, T _emaxfor the minimum envelop radius of motor torque, T _wmaxfor the minimum envelop radius of flywheel moment, H _wmaxfor the minimum envelop radius of flywheel angular momentum, Φ _ffor corner, Δ t is the time kept in reserve, for the component array of motor torque vector, for the component array of flywheel moment vector, for the component array of flywheel angular-momentum vector;

The shortest time kept in reserve is gone out according to objective function and constraint function call.

Other step and parameter and embodiment one, two, three, four, five, six, seven or eight identical.

Embodiment

Adopt following experimental verification beneficial effect of the present invention:

Embodiment 1

The shortest time defining method of a kind of pair of rigid body spacecraft fast reserve specifically carries out according to the following steps:

Step one, set up payload platform coordinate system O _sx _ay _az _awith service platform coordinate system O _bx _by _bz _b, payload platform coordinate system is designated as S _asystem, service platform coordinate system is designated as S _bsystem, wherein, O _sbe positioned at ball pivot center, O _bbe positioned at service platform barycenter, O _afor payload platform barycenter, O _cfor two rigid body Space Vehicle System barycenter, the true origin of payload platform coordinate system is positioned at O _s, payload platform and service platform form two rigid body spacecraft, and payload platform is connected by ball pivot with service platform;

Step 2, the S obtained according to step one _asystem and S _bsystem, writes out payload platform and the service platform moment of inertia matrix about two rigid body Space Vehicle System barycenter;

Step 3, the payload platform obtained according to step 2 and service platform, about the moment of inertia matrix of two rigid body Space Vehicle System barycenter, write out the attitude kinematics equations of two rigid body spacecraft and the conservation of angular momentum equation of two rigid body spacecraft;

Step 4, assumed (specified) load platform are by initial attitude motor-driven Euler's axle e to targeted attitude and corner Φ _f;

Step 5, the corner Φ drawn according to step 4 _f, write out the angular acceleration of payload platform pursuit path angular velocity with the expression formula of angle Φ (t);

The angular acceleration of the payload platform pursuit path that step 6, the Euler's axle e drawn according to step 4 and step 5 draw angular velocity with the expression formula of angle Φ (t), write out payload platform attitude quaternion q _m0and q _m, payload platform attitude matrix C _ao, service platform attitude matrix C _bo, payload platform attitude angular velocity the attitude angular velocity of service platform the attitude angle acceleration of payload platform with the attitude angle acceleration of service platform expression formula;

Step 7, based on a zero momentum system, ignore disturbance torque, by step 5 with Φ (t), and the C in step 6 _ao, c _bo, with substitute into the attitude dynamic equations of the two rigid body spacecrafts in step 3 and the conservation of angular momentum equation of pair rigid body spacecraft, obtain about with t's with

Step 8, the corner Φ drawn according to step 4 _f, in step 7 with use Matlab Optimization Toolbox, solve the shortest time kept in reserve containing constraint;

The moment of inertia matrix of payload platform and service platform is respectively

$I_1=\left[\begin{array}{ccc}2000& -10& 15\\ -10& 1000& -20\\ 15& -20& 3000\end{array}\right]kg\&CenterDot;m^2,$

$I_2=\left[\begin{array}{ccc}2500& 12& -5\\ 12& 2000& -10\\ -5& -10& 3200\end{array}\right]kg\&CenterDot;m^2.$

The quality of payload platform is m _a=500kg, the quality of service platform is m _b=600kg. $d_1^a={\left[\begin{array}{ccc}0& 0& 1\end{array}\right]}^{T}m,d_2^b={\left[\begin{array}{ccc}0& 0& 1\end{array}\right]}^{T}m.$ Orbit angular velocity is ω _o=0.0011rad/s, the position in the relative the earth's core of payload platform barycenter is $R_a^o={\left[\begin{array}{ccc}0& 0& -7\&times;{10}^6+0.5\end{array}\right]}^{T}m,$ The position in the relative the earth's core of two rigid body Space Vehicle System barycenter is $R_c^o={\left[\begin{array}{ccc}0& 0& -1\&times;{10}^6\end{array}\right]}^{T}m.$ Motor-driven start time is t ₀=100s, payload platform initial angular velocity is ${\mathrm{\&omega;}}_{\mathrm{ao}}^a\left(t_0\right)={\left[\begin{array}{ccc}0& 0& 0\end{array}\right]}^{\mathrm{oT}}/s,$ Target angular velocity is ${\mathrm{\&omega;}}_{\mathrm{ao}}^a\left(t_f\right)={\left[\begin{array}{ccc}0& 0& 0\end{array}\right]}^{\mathrm{oT}}/s.$ The initial attitude of payload platform is over the ground [0 0 0] ^oT, targeted attitude is over the ground [50 7 12] ^oTthen Euler's axle is e=[0.9600 0.2300 0.1599] ^t, target rotation angle is Φ _f=51.0603 °.Service platform keeps three-axis stabilization over the ground in whole mobile process.Attitude angle error is three axles 5 × 10 ^-4o(3 σ), attitude angular velocity measuring error is three axles 10 ^-4o/ s (3 σ).The disturbance torque acting on payload platform and two rigid body Space Vehicle System is respectively

$T_{da}^a=\left(\left[\begin{array}{c}1\&times;{10}^{-4}\sin \left({\mathrm{\&omega;}}_{o}t\right)\\ 0.75\&times;{10}^{-4}\cos \left({\mathrm{\&omega;}}_{o}t\right)\\ 1\&times;{10}^{-4}\sin \left({\mathrm{\&omega;}}_{o}t\right)\end{array}\right]+\left[\begin{array}{c}0.5\&times;{10}^{-4}\\ 0.5\&times;{10}^{-4}\\ 0.5\&times;{10}^{-4}\end{array}\right]\right)Nm,$

$T_d^b=\left(\left[\begin{array}{c}2\&times;{10}^{-4}\sin \left({\mathrm{\&omega;}}_{o}t\right)\\ 1.5\&times;{10}^{-4}\cos \left({\mathrm{\&omega;}}_{o}t\right)\\ 2\&times;{10}^{-4}\sin \left({\mathrm{\&omega;}}_{o}t\right)\end{array}\right]+\left[\begin{array}{c}1\&times;{10}^{-4}\\ 1\&times;{10}^{-4}\\ 1\&times;{10}^{-4}\end{array}\right]\right)Nm.$

In controller, take from angular frequency of shaking _n=0.2rad/s, damping ratio ξ=0.9, then K _p=ω _n ²=0.04, K _d=2 ξ ω _n=0.36.Flywheel adopts four angle mount configurations, and the maximum moment that flywheel can provide is 0.4Nm, and maximum angular momentum is 32Nm × s, and the maximum moment that motor can provide is 0.5Nm.

According to the method for the invention, try to achieve maximum angular acceleration maximum angular rate time kept in reserve Δ t=151.8272s.Analogous diagram is as Fig. 4, Fig. 5, Fig. 6 and Fig. 7;

From analogous diagram Fig. 4, Fig. 5, Fig. 6 and Fig. 7, flywheel moment reaches the upper limit, and motor torque and flywheel angular momentum all do not reach the upper limit, explanation with value meet the constraint of motor torque, flywheel moment and flywheel angular momentum, time kept in reserve optimization method is feasible.Why time kept in reserve cannot be faster, is because the flywheel moment upper limit is relatively little.

Claims (9)

1. a shortest time defining method for two rigid body spacecraft fast reserve, is characterized in that, the shortest time defining method of a kind of pair of rigid body spacecraft fast reserve specifically carries out according to the following steps:

Step one, set up payload platform coordinate system O _sx _ay _az _awith service platform coordinate system O _bx _by _bz _b, payload platform coordinate system is designated as S _asystem, service platform coordinate system is designated as S _bsystem, wherein, O _sbe positioned at ball pivot center, O _bbe positioned at service platform barycenter, O _afor payload platform barycenter, O _cfor two rigid body Space Vehicle System barycenter, the true origin of payload platform coordinate system is positioned at O _s, payload platform and service platform form two rigid body spacecraft, and payload platform is connected by ball pivot with service platform;

Step 2, the S obtained according to step one _asystem and S _bsystem, writes out payload platform and the service platform moment of inertia matrix about two rigid body Space Vehicle System barycenter;

Step 3, the payload platform obtained according to step 2 and service platform, about the moment of inertia matrix of two rigid body Space Vehicle System barycenter, write out the attitude kinematics equations of two rigid body spacecraft and the conservation of angular momentum equation of two rigid body spacecraft;

Step 4, assumed (specified) load platform are by initial attitude motor-driven Euler's axle e to targeted attitude and corner Φ _f;

Step 5, the corner Φ drawn according to step 4 _f, write out the angular acceleration of payload platform pursuit path angular velocity with the expression formula of angle Φ (t);

The angular acceleration of the payload platform pursuit path that step 6, the Euler's axle e drawn according to step 4 and step 5 draw angular velocity with the expression formula of angle Φ (t), write out payload platform attitude quaternion q _m0and q _m, payload platform attitude matrix C _ao, service platform attitude matrix C _bo, payload platform attitude angular velocity the attitude angular velocity of service platform the attitude angle acceleration of payload platform with the attitude angle acceleration of service platform expression formula;

Step 7, by step 5 with Φ (t), and the C in step 6 _ao, c _bo, with substitute into the attitude dynamic equations of the two rigid body spacecrafts in step 3 and the conservation of angular momentum equation of pair rigid body spacecraft, obtain about with t's with

Step 8, the corner Φ drawn according to step 4 _f, in step 7 with use Matlab Optimization Toolbox, solve the shortest time kept in reserve containing constraint.

2. the shortest time defining method of a kind of pair of rigid body spacecraft fast reserve according to claim 1, is characterized in that, sets up payload platform coordinate system O in described step one _sx _ay _az _awith service platform coordinate system O _bx _by _bz _b, payload platform coordinate system is designated as S _asystem, service platform coordinate system is designated as S _bsystem, wherein, O _sbe positioned at ball pivot center, O _bbe positioned at service platform barycenter, O _afor payload platform barycenter, O _cfor two rigid body Space Vehicle System barycenter, the true origin of payload platform coordinate system is positioned at O _s, payload platform and service platform form two rigid body spacecraft, and payload platform is connected by ball pivot with service platform;

Detailed process is:

Set up payload platform coordinate system O _sx _ay _az _awith service platform coordinate system O _bx _by _bz _b, payload platform coordinate system O _sx _ay _az _abe designated as S _asystem, service platform coordinate system is designated as S _bsystem, wherein, O _sfor ball pivot center, O _bfor service platform barycenter, x _afor payload platform coordinate system x-axis, y _afor payload platform coordinate system y-axis, z _afor payload platform coordinate system z-axis, x _bfor service platform coordinate system x-axis, y _bfor service platform coordinate system y-axis, z _bfor service platform coordinate system z-axis; O _afor payload platform barycenter, O _cfor two rigid body Space Vehicle System barycenter, the true origin of payload platform coordinate system is positioned at O _s.

3. the shortest time defining method of a kind of pair of rigid body spacecraft fast reserve according to claim 2, is characterized in that, according to the S that step one obtains in described step 2 _asystem and S _bsystem, writes out payload platform and the service platform moment of inertia matrix about two rigid body Space Vehicle System barycenter; Detailed process is:

Payload platform is about O _cangular-momentum vector be:

In formula, dm _afor the quality infinitesimal of payload platform, q ₁for by O _spoint to dm _avector, p ₁for by O _apoint to O _cvector, v _aIfor dm _aabsolute velocity vector, for O _babsolute velocity vector, for payload platform is about O _sinertia dyad, H _afor payload platform is about O _cangular-momentum vector, d ₁for by O _spoint to O _avector, ω _aIfor S _athe absolute angular velocities vector of system, ω _bIfor S _bthe absolute angular velocities vector of system, d ₂for by O _spoint to O _bvector;

Service platform is about O _cangular-momentum vector be:

In formula, dm _bfor the quality infinitesimal of service platform, q ₂for by O _bpoint to dm _bvector, p ₂for by O _bpoint to O _cvector, v _bIfor dm _babsolute velocity vector, for service platform is about O _binertia dyad, H _bfor service platform is about O _cangular-momentum vector, ω _bIfor S _bthe absolute angular velocities vector of system;

Two rigid body Space Vehicle System is about O _ctotal angular momentum be:

In formula, for O _babsolute velocity vector, H _wfor the angular-momentum vector of flywheel, H is that two rigid body Space Vehicle System is about O _cangular-momentum vector;

Because m _ap ₁+ m _bp ₂=0, and p ₂-p ₁=d ₁+ d ₂, so

$p_1=-\frac{m_b}{m_a+m_b}(d_1+d_2)---\left(29\right)$

$p_2=\frac{m_a}{m_a+m_b}(d_1+d_2)---\left(30\right)$

Formula (29) and formula (30) are substituted into formula (28), obtains

In formula, for unit inertia dyad;

Therefore payload platform and service platform are about O _cinertia dyad be respectively:

In formula, for payload platform is about O _cinertia dyad, for service platform is about O _cinertia dyad;

Then formula (31) can be expressed as

at S _acomponent Matrices under system is payload platform about O _cmoment of inertia matrix:

$I_a^a=I_1^a-\frac{m_a{m}_b}{m_a+m_b}\&lsqb;{\left(d_1^a\right)}^{\&times;}{\left(d_1^a\right)}^{\&times;}+{\left(d_1^a\right)}^{\&times;}{C}_{ab}{\left(d_2^b\right)}^{\&times;}{C}_{ba}\&rsqb;---\left(35\right)$

In formula, for payload platform is about O _cmoment of inertia matrix, for payload platform is about O _smoment of inertia matrix, m _afor the quality of payload platform, m _bfor the quality of service platform, for by O _spoint to O _athe component array of vector, for by O _spoint to O _bthe component array of vector, C _abfor S _abe relative S _bthe attitude matrix of system, C _bafor S _bbe relative S _athe attitude matrix of system, for multiplication cross matrix, for multiplication cross matrix;

at S _bcomponent Matrices under system is service platform about O _cmoment of inertia matrix:

$I_b^b=I_2^b-\frac{m_a{m}_b}{m_a+m_b}\&lsqb;{\left(d_2^b\right)}^{\&times;}{\left(d_2^b\right)}^{\&times;}+{\left(d_2^b\right)}^{\&times;}{C}_{ba}{\left(d_1^a\right)}^{\&times;}{C}_{ab}\&rsqb;---\left(36\right)$

In formula, superscript a and b is respectively vector or dyad at S _asystem and S _bcomponent array under system or Component Matrices, for service platform is about O _cmoment of inertia matrix, for service platform is about O _bmoment of inertia matrix;

And with first order derivative be respectively

${\stackrel{\&CenterDot;}{I}}_a^a=\frac{m_a{m}_b}{m_a+m_b}{\left(d_1^a\right)}^{\&times;}{\&lsqb;{({\mathrm{\&omega;}}_{aI}^a-C_{ab}{\mathrm{\&omega;}}_{bI}^b)}^{\&times;}{C}_{ab}{d}_2^b\&rsqb;}^{\&times;}---\left(37\right)$

${\stackrel{\&CenterDot;}{I}}_b^b=\frac{m_a{m}_b}{m_a+m_b}{\left(d_2^b\right)}^{\&times;}{\&lsqb;{({\mathrm{\&omega;}}_{bI}^b-C_{ba}{\mathrm{\&omega;}}_{aI}^a)}^{\&times;}{C}_{ba}{d}_1^a\&rsqb;}^{\&times;}---\left(38\right)$

In formula, for S _athe absolute angular velocities vector of system is at S _acomponent array under system, for S _bthe absolute angular velocities vector of system is at S _bcomponent array under system, for multiplication cross matrix, for multiplication cross matrix.

4. the shortest time defining method of a kind of pair of rigid body spacecraft fast reserve according to claim 3, it is characterized in that, the payload platform obtained according to step 2 in described step 3 and service platform, about the moment of inertia matrix of two rigid body Space Vehicle System barycenter, write out the attitude kinematics equations of two rigid body spacecraft and the conservation of angular momentum equation of two rigid body spacecraft; Detailed process is:

The payload platform obtained according to step 2 and service platform, about the moment of inertia matrix of two rigid body Space Vehicle System barycenter, write out attitude kinematics equations and the conservation of angular momentum equation of two rigid body spacecraft:

The attitude kinematics equations of two rigid body spacecraft is as formula (3), (4):

$\begin{array}{c}\&lsqb;I_1^a-\frac{m_a{m}_b}{m_a+m_b}{\left(d_1^a\right)}^{\&times;}{\left(d_1^a\right)}^{\&times;}\&rsqb;{\stackrel{\&CenterDot;}{\mathrm{\&omega;}}}_{aI}^a-\frac{m_a{m}_b}{m_a+m_b}{\left(d_1^a\right)}^{\&times;}{C}_{ab}{\left(d_2^a\right)}^{\&times;}{\stackrel{\&CenterDot;}{\mathrm{\&omega;}}}_{bI}^b=\\ T_e^a+T_{da}^a-{\left({\mathrm{\&omega;}}_{aI}^a\right)}^{\&times;}{I}_1^a{\mathrm{\&omega;}}_{aI}^a+\\ {\left(d_1^a\right)}^{\&times;}\{{\mathrm{\&mu;m}}_a{C}_{ao}(\frac{R_c^o}{{R_c}^3}-\frac{R_a^o}{{R_a}^3})-\frac{m_a{m}_b}{m_a+m_b}\left[\begin{array}{c}{\left({\mathrm{\&omega;}}_{aI}^a\right)}^{\&times;}{\left({\mathrm{\&omega;}}_{aI}^a\right)}^{\&times;}{d}_1^a+\\ C_{ab}{\left({\mathrm{\&omega;}}_{bI}^b\right)}^{\&times;}{\left({\mathrm{\&omega;}}_{bI}^b\right)}^{\&times;}{d}_2^b\end{array}\right]\}\end{array}---\left(3\right)$

$\begin{array}{c}{C}_{ba}{I}_a^a{\stackrel{\&CenterDot;}{\mathrm{\&omega;}}}_{aI}^a+I_b^b{\stackrel{\&CenterDot;}{\mathrm{\&omega;}}}_{bI}^b=T_w^b+T_d^b-\\ \{C_{ba}\&lsqb;{\left({\mathrm{\&omega;}}_{aI}^a\right)}^{\&times;}{I}_a^a{\mathrm{\&omega;}}_{aI}^a+{\stackrel{\&CenterDot;}{I}}_a^a{\mathrm{\&omega;}}_{aI}^a\&rsqb;+{\left({\mathrm{\&omega;}}_{bI}^b\right)}^{\&times;}{I}_b^b{\mathrm{\&omega;}}_{bI}^b+{\stackrel{\&CenterDot;}{I}}_b^b{\mathrm{\&omega;}}_{bI}^b+{\left({\mathrm{\&omega;}}_{bI}^b\right)}^{\&times;}{H}_w^b\}\end{array}---\left(4\right)$

The conservation of angular momentum equation of two rigid body spacecraft is as formula (5):

$C_{ba}{I}_a^a{\mathrm{\&omega;}}_{aI}^a+I_b^b{\mathrm{\&omega;}}_{bI}^b+H_w^b=H_0^b---\left(5\right)$

In formula, for derivative, for derivative, for derivative, for derivative, superscript o is component array under the orbital coordinate system of two rigid body spacecraft of vector or dyad or Component Matrices, for S _athe component array of the absolute angular velocities vector of system, for S _bthe component array of the absolute angular velocities vector of system, C _aofor S _abe the attitude matrix of relative orbit system, for the coordinate array of motor torque vector, for the coordinate array of flywheel moment vector, for acting on the component array of the disturbance torque vector of payload platform, for acting on the component array of the disturbance torque vector of two rigid body Space Vehicle System, for the component array of flywheel angular-momentum vector, for the component array of two rigid body Space Vehicle System angular-momentum vector, μ is geocentric gravitational constant, for being pointed to the component array of the vector of payload platform barycenter by the earth's core, for being pointed to the component array of the vector of two rigid body Space Vehicle System barycenter by the earth's core, R _cfor mould, R _afor mould, for multiplication cross matrix, for multiplication cross matrix;

Payload platform is around O _athe attitude kinematics equations rotated is:

In formula, for vector or dyad are at S _aderivative under system, T _efor motor torque vector, T _bafor service platform acts on the moment vector of payload platform, T _dafor acting on the disturbance torque vector of payload platform, ω _aIfor S _athe absolute angular velocities vector of system;

Act on the power F making a concerted effort to be acted on by service platform payload platform of payload platform _baand external environment acts on the perturbed force F of payload platform _dacomposition, according to Newton second law, has

$F_{ba}+F_{da}=m_a\frac{{d_I}^2{R}_a}{{\mathrm{dt}}^2}=m_a\frac{{d_I}^2(R_c-p_1)}{{\mathrm{dt}}^2}---\left(43\right)$

In formula, the derivative under inertial system for vector or dyad, R _afor pointing to O by the earth's core _avector, R _cfor pointing to O by the earth's core _cvector, p ₁for by O _apoint to O _cvector;

Based on the earth-spacecraft two system, have

$\frac{{d_I}^2\left|R_c\right|}{{\mathrm{dt}}^2}=-\frac{\mathrm{\&mu;}}{{\left|R_c\right|}^3}{R}_c---\left(44\right)$

In formula, μ is Gravitational coefficient of the Earth; | R _c| be R _cmodulus value;

Because F _dafor terrestrial attraction, so get

$F_{da}=-\frac{{\mathrm{\&mu;m}}_a}{{\left|R_a\right|}^3}{R}_a---\left(45\right)$

In formula, | R _a| be R _amodulus value;

Formula (29), formula (44) and formula (45) are substituted into formula (43), obtains

$\begin{array}{c}{F}_{ba}={\mathrm{\&mu;m}}_a(\frac{R_a}{{\left|R_a\right|}^3}-\frac{R_c}{{\left|R_c\right|}^3})+\\ \frac{m_a{m}_b}{m_a+m_b}\&lsqb;{\mathrm{\&omega;}}_{aI}\&times;({\mathrm{\&omega;}}_{aI}\&times;d_1)+\frac{d_a{\mathrm{\&omega;}}_{aI}}{dt}\&times;d_1+{\mathrm{\&omega;}}_{bI}\&times;({\mathrm{\&omega;}}_{bI}\&times;d_2)+\frac{d_b{\mathrm{\&omega;}}_{bI}}{dt}\&times;d_2\&rsqb;\end{array}---\left(46\right)$

In formula, for vector or dyad are at S _bderivative under system;

Again by T _ba=-d ₁× F _basubstitution formula (42), obtains payload platform around O _athe attitude kinematics equations rotated:

Above formula is at S _acomponent type under system is

$\begin{array}{c}\&lsqb;I_1^a-\frac{m_a{m}_b}{m_a+m_b}{\left(d_1^a\right)}^{\&times;}{\left(d_1^a\right)}^{\&times;}\&rsqb;{\stackrel{\&CenterDot;}{\mathrm{\&omega;}}}_{aI}^a-\frac{m_a{m}_b}{m_a+m_b}{\left(d_1^a\right)}^{\&times;}{C}_{ab}{\left(d_1^a\right)}^{\&times;}{\stackrel{\&CenterDot;}{\mathrm{\&omega;}}}_{bI}^b=\\ T_e^a+T_{da}^a-{\left({\mathrm{\&omega;}}_{aI}^a\right)}^{\&times;}{I}_1^a{\mathrm{\&omega;}}_{aI}^a+\\ {\left(d_1^a\right)}^{\&times;}\{{\mathrm{\&mu;m}}_a{C}_{ao}(\frac{R_c^o}{{\left|R_c\right|}^3}-\frac{R_a^o}{{\left|R_a\right|}^3})-\frac{m_a{m}_b}{m_a+m_b}\left[\begin{array}{c}{\left({\mathrm{\&omega;}}_{aI}^a\right)}^{\&times;}{\left({\mathrm{\&omega;}}_{aI}^a\right)}^{\&times;}{d}_1^a+\\ C_{ab}{\left({\mathrm{\&omega;}}_{bI}^b\right)}^{\&times;}{\left({\mathrm{\&omega;}}_{bI}^b\right)}^{\&times;}{d}_2^b\end{array}\right]\}\end{array}---\left(48\right)$

In formula, for multiplication cross matrix, for multiplication cross matrix, for S _athe absolute angular velocities vector of system is at S _acomponent array under system, for S _bthe absolute angular velocities vector of system is at S _bcomponent array under system, for multiplication cross matrix, for multiplication cross matrix, for being pointed to the component array of the vector of payload platform barycenter by ball pivot center, for being pointed to the component array of the vector of service platform barycenter by ball pivot center, for derivative, for derivative, for motor torque vector is at S _acomponent array under system, for disturbance torque vector is at S _acomponent array under system;

To formula (34) differentiate, obtain two Rigid-body System around O _cthe attitude kinematics equations rotated:

In formula, T _wfor flywheel moment vector, T _dfor acting on the disturbance torque vector H of two rigid body Space Vehicle System _wfor flywheel angular-momentum vector, ω _aIfor S _athe absolute angular velocities vector of system, ω _bIfor S _bthe absolute angular velocities vector of system, H _wfor the angular-momentum vector of flywheel, for payload platform is about O _cinertia dyad, for service platform is about O _cinertia dyad;

Formula (49) is at S _bcomponent type under system is:

$\begin{array}{c}{C}_{ba}{I}_a^a{\stackrel{\&CenterDot;}{\mathrm{\&omega;}}}_{aI}^a+I_b^b{\stackrel{\&CenterDot;}{\mathrm{\&omega;}}}_{bI}^b=T_w^b+T_d^b-\\ \{C_{ba}\&lsqb;{\left({\mathrm{\&omega;}}_{aI}^a\right)}^{\&times;}{I}_a^a{\mathrm{\&omega;}}_{aI}^a+{\stackrel{\&CenterDot;}{I}}_a^a{\mathrm{\&omega;}}_{aI}^a\&rsqb;+{\left({\mathrm{\&omega;}}_{bI}^b\right)}^{\&times;}{I}_b^b{\mathrm{\&omega;}}_{bI}^b+{\stackrel{\&CenterDot;}{I}}_b^b{\mathrm{\&omega;}}_{bI}^b+{\left({\mathrm{\&omega;}}_{bI}^b\right)}^{\&times;}{H}_w^b\}\end{array}---\left(50\right)$

In formula, C _bafor S _bbe relative S _athe attitude matrix of system, for at S _acomponent Matrices under system, for at S _bcomponent Matrices under system, for T _wat S _bcomponent array under system, for T _dat S _bcomponent array under system, for derivative, for derivative, for H _wat S _bcomponent array under system;

Formula (34) is at S _bcomponent type under system is

$C_{ba}{I}_a^a{\mathrm{\&omega;}}_{aI}^a+I_b^b{\mathrm{\&omega;}}_{bI}^b+H_w^b=H_0^b---\left(41\right)$

In formula, for the total angular momentum vector of two rigid body spacecraft is at S _bcomponent array under system.

5. the shortest time defining method of a kind of pair of rigid body spacecraft fast reserve according to claim 4, is characterized in that, calculates payload platform by initial attitude motor-driven Euler's axle e to targeted attitude and corner Φ in described step 4 _f; Detailed process is:

Φ _f＝2arccosq _m0(6)

$e=\frac{1}{sin\frac{{\mathrm{\&Phi;}}_f}{2}}{q}_m---\left(7\right)$

In formula, q _m0for payload platform is by the motor-driven hypercomplex number Q to targeted attitude of initial attitude _mmark portion, q _mfor payload platform is by the motor-driven hypercomplex number Q to targeted attitude of initial attitude _marrow portion, Q _m=[q _m0q _m] ^tbe four-vector, expression formula is:

$Q_m={Q_0}^{*}\&CircleTimes;Q_f---\left(8\right)$

In formula, Q ₀for the initial attitude hypercomplex number of payload platform, Q _ffor the targeted attitude hypercomplex number of payload platform, Q _mfor the hypercomplex number of targeted attitude, * is hypercomplex number transposition, for hypercomplex number multiplication, T is the transposition of array.

6. the shortest time defining method of a kind of pair of rigid body spacecraft fast reserve according to claim 5, is characterized in that, according to the corner Φ that step 4 draws in described step 5 _f, write out the angular acceleration of payload platform pursuit path angular velocity with the expression formula of angle Φ (t); Detailed process is:

If Φ is (t ₀)=0, Φ (t _f)=Φ _f

$\stackrel{\&CenterDot;\&CenterDot;}{\mathrm{\&Phi;}}\left(t\right)=\left\{\begin{array}{cc}{\stackrel{\&CenterDot;\&CenterDot;}{\mathrm{\&Phi;}}}_{max}& t_0\&le;t\&le;t_1\\ 0& t_1<t<t_2\\ -{\stackrel{\&CenterDot;\&CenterDot;}{\mathrm{\&Phi;}}}_{max}& t_2\&le;t\&le;t_f\end{array}\right.---\left(9\right)$

$\stackrel{\&CenterDot;}{\mathrm{\&Phi;}}\left(t\right)=\left\{\begin{array}{cc}{\stackrel{\&CenterDot;\&CenterDot;}{\mathrm{\&Phi;}}}_{max}(t-t_0)& t_0\&le;t\&le;t_1\\ {\stackrel{\&CenterDot;}{\mathrm{\&Phi;}}}_{max}& t_1<t<t_2\\ {\stackrel{\&CenterDot;}{\mathrm{\&Phi;}}}_{max}-{\stackrel{\&CenterDot;\&CenterDot;}{\mathrm{\&Phi;}}}_{max}(t-t_2)& t_2\&le;t\&le;t_f\end{array}\right.---\left(10\right)$

$\mathrm{\&Phi;}\left(t\right)=\left\{\begin{array}{cc}\frac{1}{2}{\stackrel{\&CenterDot;\&CenterDot;}{\mathrm{\&Phi;}}}_{\max }{(t-t_0)}^2& t_0\&le;t\&le;t_1\\ \frac{1}{2}{\stackrel{\&CenterDot;\&CenterDot;}{\mathrm{\&Phi;}}}_{\max }{(t_1-t_0)}^2+{\stackrel{\&CenterDot;}{\mathrm{\&Phi;}}}_{\max }(t-t_1)& t_1<t<t_2\\ \frac{1}{2}{\stackrel{\&CenterDot;\&CenterDot;}{\mathrm{\&Phi;}}}_{\max }{(t_1-t_0)}^2+{\stackrel{\&CenterDot;}{\mathrm{\&Phi;}}}_{\max }(t-t_1)-\frac{1}{2}{\stackrel{\&CenterDot;\&CenterDot;}{\mathrm{\&Phi;}}}_{\max }{(t-t_2)}^2& t_2\&le;t\&le;t_f\end{array}\right.---\left(11\right)$

In formula, for the angular acceleration of payload platform pursuit path, for the angular velocity of payload platform pursuit path, Φ (t) is the angle of payload platform pursuit path, for the maximum angular acceleration of payload platform pursuit path, for the maximum angular rate of payload platform pursuit path, t ₀for motor-driven start time, t ₁for accelerating finish time, t ₂for at the uniform velocity finish time, t _ffor motor-driven finish time, t is the time, and unit is second.

7. the shortest time defining method of a kind of pair of rigid body spacecraft fast reserve according to claim 6, is characterized in that, the angular acceleration of the payload platform pursuit path that the Euler's axle e drawn according to step 4 in described step 6 and step 5 draw angular velocity with the expression formula of angle Φ (t), write out payload platform attitude quaternion q _m0and q _m, payload platform attitude matrix C _ao, service platform attitude matrix C _bo, payload platform attitude angular velocity the attitude angular velocity of service platform the attitude angle acceleration of payload platform with the attitude angle acceleration of service platform expression formula; Detailed process is:

Payload platform pursuit path followed the tracks of by assumed load platform, and service platform is stablized;

Then payload platform attitude quaternion, as formula (12) (13):

$q_{m0}=cos\frac{\mathrm{\&Phi;}\left(t\right)}{2}---\left(12\right)$

$q_m=esin\frac{\mathrm{\&Phi;}\left(t\right)}{2}---\left(13\right)$

The attitude matrix of payload platform, as formula (14):

C _ao＝q _mq _m ^T+[q _m0E-(q _m) ^×] ²(14)

The attitude angular velocity of payload platform, as formula (15):

${\mathrm{\&omega;}}_{aI}^a=\stackrel{\&CenterDot;}{\mathrm{\&Phi;}}\left(t\right)e+C_{ao}{\mathrm{\&omega;}}_{oI}^o---\left(15\right)$

The attitude angle acceleration of payload platform, as formula (16):

${\stackrel{\&CenterDot;}{\mathrm{\&omega;}}}_{aI}^a=\stackrel{\&CenterDot;\&CenterDot;}{\mathrm{\&Phi;}}\left(t\right)e-\stackrel{\&CenterDot;}{\mathrm{\&Phi;}}\left(t\right)e^{\&times;}{C}_{ao}{\mathrm{\&omega;}}_{oI}^o---\left(16\right)$

The attitude matrix of service platform, as formula (17):

C _bo＝E (17)

The attitude angular velocity of service platform, as formula (18):

${\mathrm{\&omega;}}_{bI}^b={\mathrm{\&omega;}}_{oI}^o---\left(18\right)$

The attitude angle acceleration of service platform, as formula (19):

${\stackrel{\&CenterDot;}{\mathrm{\&omega;}}}_{bI}^b=0---\left(19\right)$

In formula, q _m0for payload platform is by the motor-driven hypercomplex number Q to targeted attitude of initial attitude _mmark portion, q _mfor payload platform is by the motor-driven hypercomplex number Q to targeted attitude of initial attitude _marrow portion, C _bofor S _bbe the attitude matrix of relative orbit system, C _obfor the relative S of track system _bthe attitude matrix of system, E is unit matrix, and e is Euler's axle of payload platform pursuit path, q _m ^tfor q _mtransposition, (q _m) ^×for q _mmultiplication cross matrix, for the coordinate array of orbit angular velocity vector under spacecraft orbit system, e ^×for the multiplication cross matrix of e, Φ (t) is the angle of payload platform pursuit path, for the angular velocity of payload platform pursuit path, for the angular acceleration of payload platform pursuit path, C _bofor S _bbe relative S _othe attitude matrix of system.

8. the shortest time defining method of a kind of pair of rigid body spacecraft fast reserve according to claim 7, is characterized in that, by step 5 in described step 7 with Φ (t), and the C in step 6 _ao, c _bo, with substitute into the attitude dynamic equations of the two rigid body spacecrafts in step 3 and the conservation of angular momentum equation of pair rigid body spacecraft, obtain about with t's with detailed process is:

By the q in step 6 _m0and q _msubstitute into C _ao, obtain the C about Φ (t) _ao, then by C _aosubstitute into with in, obtain about with Φ's (t) and about with Φ's (t) finally by C _ao, c _bo, with substitute into the attitude dynamic equations of the two rigid body spacecrafts in step 3 and the conservation of angular momentum equation of pair rigid body spacecraft, obtain about with Φ's (t) with

In formula, for the component array of motor torque vector, for the component array of flywheel moment vector, for the component array of flywheel angular-momentum vector;

Wherein, for the maximum angular acceleration of payload platform pursuit path function, for the maximum angular rate of payload platform pursuit path function, for the function of time t.

9. the shortest time defining method of a kind of pair of rigid body spacecraft fast reserve according to claim 8, is characterized in that, according to the corner Φ that step 4 draws in described step 8 _f, in step 7 with use Matlab Optimization Toolbox, solve the shortest time kept in reserve containing constraint; Detailed process is:

Use Matlab Optimization Toolbox, separate the minimum value containing constraint, wherein, variable is with objective function is the time kept in reserve:

$\mathrm{\&Delta;}t=\frac{{\mathrm{\&Phi;}}_f}{{\stackrel{\&CenterDot;}{\mathrm{\&Phi;}}}_{max}}+\frac{{\stackrel{\&CenterDot;}{\mathrm{\&Phi;}}}_{max}}{{\stackrel{\&CenterDot;\&CenterDot;}{\mathrm{\&Phi;}}}_{max}}---\left(20\right)$

Constraint function is:

$\left|T_e^a\right|\&le;T_{emax}---\left(21\right)$

$\left|T_w^b\right|\&le;T_{wmax}---\left(22\right)$

$\left|H_w^b\right|\&le;H_{wmax}---\left(23\right)$

In formula, T _emaxfor the minimum envelop radius of motor torque, T _wmaxfor the minimum envelop radius of flywheel moment, H _wmaxfor the minimum envelop radius of flywheel angular momentum, Φ _ffor corner, △ t is the time kept in reserve, for the component array of motor torque vector, for the component array of flywheel moment vector, for the component array of flywheel angular-momentum vector;

The shortest time kept in reserve is gone out according to objective function and constraint function call.