CN106649947A - Satellite attitude numerical simulation method based on Lie group spectral algorithm - Google Patents

Abstract

The invention discloses a satellite attitude numerical simulation method based on the Lie group spectral algorithm. The method includes the following steps that 1, a satellite attitude kinematics and dynamics Lie group model is established based on the SO(3) group; 2, canonical coordinates are selected to convert a satellite attitude Lie group equation into an equivalent Lie algebra equation and an equivalent Lie group reconstruction equation; 3, the Lie algebra equation is solved with the spectral method to obtain the angular speed of satellite attitude rotation, and a satellite attitude matrix is solved with the Lie group reconstruction equation. According to the method, the satellite attitude kinematics and dynamics Lie group model is established based on the SO(3) group, satellite attitude expression is simple without singularity, and the unwinding phenomenon is avoided; as the Lie group spectral method is adopted to simulate the attitude dynamics model, the geometric structure and physical property of a satellite system can be kept for a long time, precision is high, and stability is high.

Description

The attitude of satellite numerical value emulation method of algorithm is composed based on Lie group

Technical field

The present invention relates to satellite gravity anomaly technical field, and in particular to a kind of attitude of satellite number that algorithm is composed based on Lie group Value emulation mode.

Background technology

The background of related of the present invention is illustrated below, but these explanations might not constitute the existing of the present invention Technology.

Mainly the foundation comprising Satellite Attitude Dynamics model and numerical value are imitative for existing Satellite Attitude Dynamics emulation technology True algorithm.

The existing attitude of satellite describes method mainly to be included, Eulerian angles, the method such as direction cosines and the element of unit four.

(1) Eulerian angles description

It is Eulerian angles that the attitude of satellite describes most common method.It is general with three Eulerian angles-(azimuth ψ, pitching angle theta, incline Oblique angle) represent attitude of the noninertial system of coordinates relative to inertial coodinate system.Inertial coodinate system is rotated around corresponding reference axis Angle, can be completely superposed with noninertial system of coordinates.Assume coordinate difference of the vector v under inertial coodinate system and noninertial system of coordinates For (v_x, v_y, v_z)^T(v_bx, v_by, v_bz)^T.There is following relation between them：

Relation under noninertial system of coordinates between angular speed and the time-derivative of Eulerian angles is represented by：

Wherein, p, q, r are the angular speed under body coordinate system.

(2) unit quaternion description

Quaternary number and the number being made up of four units：Q(q₀,q₁,q₂,q₃)=q₀+q₁×i+q₂×j+q₃×k.Wherein q₀,q₁, q₂,q₃It is real number, i, j, k are both synergistic orthogonal variable and empty unit.It is by the Coordinate Conversion under noninertial system of coordinates Coordinate under inertial coodinate system, its direction cosine matrix is represented by：

And the first derivative of unit quaternion meets：

The conventional numerical simulation algorithm of Satellite Attitude Dynamics is mainly explicit, including Euler method, Runge-Kutta methods Deng.

(1) Euler method

For ODE

Independent variable t is divided into into equidistant segment, i.e. t_n=t₀+nh.The numeric format of Euler method is：

y_n+1=y_n+hf′(t,y_n), n=0,1,2...

(2) Runge-Kutta methods

Runge-Kutta methods have various ways, including explicit form and implied format.Below we are given the most frequently used The explicit Runge-Kutta methods of quadravalence,

The shortcoming of prior art mainly includes the shortcoming of model and the shortcoming of numerical algorithm.Satellite wherein based on Eulerian angles Attitude description is local, and with singularity, and the attitude of satellite description for being based on unit quaternion will cause unwinding (unwinding) phenomenon, so as to cause satellite control in big energy waste.Euler method in numerical algorithm is that a class is explicit Single step algorithm, although fast operation, but only first derivation precision.And conventional explicit Runge-Kutta methods can have There is a higher order accuracy, but the less stable of algorithm.The Runge-Kutta methods computation complexity of implicit expression is again higher.And with On algorithm tend not to keep the geometry and physics conserved quantity, such as SO (3) group structure of rigid body that emulates dynamic system, it is dynamic Amount, energy etc..

The content of the invention

In order to solve problems of the prior art, the present invention proposes a kind of attitude of satellite number that algorithm is composed based on Lie group Value emulation mode, comprises the steps：

S1, attitude kinematics and the dynamic (dynamical) Lie group model that satellite is set up based on SO (3) group；

S2, selection canoncial coordinates, the attitude Lie group of satellite is equations turned for Lie algebra equation and Lie group reconstruct of equal value Equation；

S3, the angular speed that attitude of satellite rotation is obtained with spectral method solution Lie algebra equation are simultaneously asked using Lie group reconstruct equation The attitude matrix of solution satellite.

Preferably, Lie group model is：

In formula, g (t) is the transition matrix that t body coordinate is tied to inertial coodinate system, and g (t) is Lie group,For g's (t) Derivative；V (t, g (t)) is speed of the t satellite in inertial coodinate system, and V (t, g (t)) is Lie algebra (3 × 3 antisymmetry squares Battle array)；F (g (t), V (t, g (t))) is with regard to g (t) and the function of V (t, g (t))；T is the time；U for satellite control moment and The summation of disturbance torque, unit is Nm.

Preferably,

In formula, ω₁For the attitude angular velocity that satellite rotates around inertial coodinate system x-axis；ω₂It is satellite around inertial coodinate system y-axis The attitude angular velocity of rotation；ω₃For the attitude angular velocity that satellite rotates around inertial coodinate system z-axis.

Preferably, Lie group equation is：

Lie algebra equation is：

Lie group reconstructs equation：

G (t)=τ (θ (t)) g₀Equation 3；

Wherein, θ (t) is intermediate variable, Lie algebra；For the derivative of θ (t)；τ (θ (t)) is the function of θ (t)；g₀To defend Star initial attitude lower body coordinate is tied to the transition matrix of inertial coodinate system.

Preferably, canoncial coordinates are index coordinates.

Preferably, adopt Rodrigues formula to represent index coordinates exp (θ) for：

MappingIt is expressed as：

In formula, I is 3 × 3 unit matrix.

Preferably, canoncial coordinates are Cayley coordinates.

Preferably, Cayley coordinates cay (θ) is：

MappingIt is expressed as：

In formula, I is 3 × 3 unit matrix.

Advantage of the invention is mainly manifested in：The attitude dynamics model of satellite, Satellite Attitude are set up based on SO (3) group State represents succinct, without singularity and avoids the unwinding phenomenon of unit quaternion method；Appearance is solved using Lie group spectral method State kinetic model, can for a long time keep the geometry and physical characteristic of satellite, high precision and stability is strong.

Description of the drawings

By the specific embodiment part of offer referring to the drawings, the features and advantages of the present invention will become more It is easy to understand, in the accompanying drawings：

Fig. 1 is the schematic flow sheet of the attitude of satellite numerical value emulation method that algorithm is composed based on Lie group of the invention；

Fig. 2 is the distance between inertial coodinate system Satellite is pointed to and target is pointed in preferred embodiment of the present invention curve；

Fig. 3 is that preferred embodiment of the present invention Satellite points to the coordinate curve in inertial coodinate system；

Fig. 4 is angular speed curve of the preferred embodiment of the present invention Satellite in inertial coodinate system；

Fig. 5 is that preferred embodiment of the present invention Satellite is pointed in two-dimensional sphere S²On track schematic diagram.

Specific embodiment

The illustrative embodiments of the present invention are described in detail with reference to the accompanying drawings.Illustrative embodiments are retouched State merely for the sake of demonstration purpose, and be definitely not to the present invention and its application or the restriction of usage.

SO (3) group is three-dimensional rotation Lie group.It is isotropic in common coordinate space that it is not only description physical system Symmetric group, is also to process the symmetric useful tool of physics internal system, is occupied in physical application highly importantly Position.The present invention sets up the attitude dynamics model of satellite based on SO (3) group, and using spectral method attitude dynamics model is solved, and defends Star attitude represents succinct, without singularity, is avoided that unwinding phenomenon, can for a long time keep the geometry of satellite special with physics Property, high precision, stability are strong.

Referring to Fig. 1, the attitude of satellite numerical value emulation method that algorithm is composed based on Lie group of the present invention, comprise the steps：

S1, attitude kinematics and the dynamic (dynamical) Lie group model that satellite is set up based on SO (3) group；

S2, selection canoncial coordinates, the attitude Lie group of satellite is equations turned for Lie algebra equation and Lie group reconstruct of equal value Equation；

S3, the angular speed that attitude of satellite rotation is obtained with spectral method solution Lie algebra equation are simultaneously asked using Lie group reconstruct equation The attitude matrix of solution satellite.

Attitude of satellite description in prior art based on Eulerian angles is local, and with singularity；Based on unit four The attitude of satellite description of first number has ambiguity, can cause to unwind (unwinding) phenomenon, so as to cause satellite control in it is a large amount of The waste of energy.The present invention sets up the attitude dynamics model of satellite based on SO (3) group, due to element and the boat of each Lie group The attitude of its device is one-to-one, therefore the attitude dynamics model of the present invention does not have singularity, can be prevented effectively from unwinding existing As.

In some embodiments of the invention, the Lie group model for setting up satellite based on SO (3) group is：

In formula, g (t) is the transition matrix that t body coordinate is tied to inertial coodinate system, and g (t) is Lie group,For g's (t) Derivative；V (t, g (t)) is angular speed of the t satellite in inertial coodinate system, and V (t, g (t)) is Lie algebra；f(g(t),V(t,g (t))) it is with regard to g (t) and the function of V (t, g (t))；T is the time；U is the control moment of satellite and the summation of disturbance torque, single Position is Nm.

V (t, g (t)) is 3 × 3 matrixes, it is preferable that

In formula, ω₁For the attitude angular velocity that satellite rotates around inertial coodinate system x-axis；ω₂It is satellite around inertial coodinate system y-axis The attitude angular velocity of rotation；ω₃For the attitude angular velocity that satellite rotates around inertial coodinate system z-axis.

After setting up the attitude dynamics model of satellite, the present invention solves attitude dynamics model using Lie group spectral method, Determine that satelloid coordinate is tied to the transition matrix of inertial coodinate system.Attitude dynamics model is solved using Lie group spectral method, can Simulation algorithm is set to keep the geometry and physical characteristic of satellite for a long time, high precision and stability is strong.

In some embodiments of the invention, Lie group model is solved using Lie group spectral method, wherein, Lie group equation is：

Lie algebra equation is：

Lie group reconstructs equation：

G (t)=τ (θ (t)) g₀Equation 3；

Wherein, θ (t) is intermediate variable, Lie algebra；For the derivative of θ (t)；τ (θ (t)) is the function of θ (t)；g₀To defend Star initial attitude lower body coordinate is tied to the transition matrix of inertial coodinate system.

Those skilled in the art can select the numerical value of suitable spectrum point collocation and Nonlinear System of Equations according to actual conditions Solution solves transition matrix g (t) that satelloid coordinate is tied to inertial coodinate system.The spectrum point collocation that for example can be selected includes The numerical solution of Chebyshev methods, Legendre methods etc. and Nonlinear System of Equations includes Newton-Raphson methods, quasi-Newton method Deng.In some embodiments of the invention, above-mentioned canoncial coordinates be index coordinates, the applied range of index coordinates method, substantially It is upper to be applied to all of Lie algebra.Preferably, adopt Rodrigues formula to represent index coordinates exp (θ) for：

MappingIt is expressed as：

In formula, I is 3 × 3 unit matrix.

The present invention other embodiments in, above-mentioned canoncial coordinates be Cayley coordinates, the calculating of Cayley coordinate methods Speed is fast, it is adaptable to second order Lie algebra.Preferably, Cayley coordinates cay (θ) is：

MappingIt is expressed as：

In formula, I is 3 × 3 unit matrix.

Embodiment

It is described in detail by taking the spin clusters of satellite as an example below, the parameter value of satellite simulation system is referring to following table 1。

The attitude dynamics model for setting up satellite based on SO (3) group is：

Wherein e₁=[1,0,0] T, e₂=[0,1,0]^T。

Control rate is taken as：

WhereinQ=Re0 represents the sensing of satellite.

It is satellite initial angular momentum M₀Estimation, it then follows rate of change below：

Wherein, dist (q₁, q₂) represent direction vector q₁, q₂Between angle, i.e. dist (q₁, q₂)=arccos (<q₁,q₂ >), e₀=[0,0,1]^T。

The parameter value of the satellite simulation system of table 1

Above-mentioned simulation result is referring to Fig. 2-5.Fig. 2 is the distance between inertial coodinate system Satellite is pointed to and target is pointed to song Line, figure it is seen that satellite is pointed to gradually go to zero with the distance between target sensing, it is and basicly stable after 19s. Fig. 3 is that satellite points to the coordinate curve in inertial coodinate system, and Fig. 4 is angular speed curve of the satellite in inertial coodinate system, from As can be seen that after 19s, satellite is pointed to and angular speed maintains essentially in same level in figure.Fig. 5 is that satellite is pointed in two dimension Spherical Surface S²On track schematic diagram.From Fig. 2-5 it is found that the attitude simulation method of the present invention can effectively keep satellite The Lie group structure of attitude, confidence level is higher.

Although with reference to illustrative embodiments, invention has been described, but it is to be understood that the present invention does not limit to The specific embodiment that Yu Wenzhong is described in detail and illustrated, in the case of without departing from claims limited range, this Art personnel can make various changes to the illustrative embodiments.

Claims (8)

1. the attitude of satellite numerical value emulation method of algorithm is composed based on Lie group, it is characterised in that comprised the steps：

S1, attitude kinematics and the dynamic (dynamical) Lie group model that satellite is set up based on SO (3) group；

S2, selection canoncial coordinates, the attitude Lie group of satellite is equations turned for Lie algebra equation and Lie group reconstruct equation of equal value；

S3, the angular speed that attitude of satellite rotation is obtained with spectral method solution Lie algebra equation are simultaneously defended using Lie group reconstruct equation solution The attitude matrix of star.

2. attitude of satellite numerical value emulation method as claimed in claim 1, wherein, the Lie group model is：

$\left\{\begin{array}{c}\stackrel{\&CenterDot;}{g}\left(t\right)=V(t,g(t\left)\right)g\left(t\right)\\ \stackrel{\&CenterDot;}{V}(t,g(t\left)\right)=f\left(g\right(t),V(t,g\left(t\right)\left)\right)+U\end{array}\right.$

In formula, g (t) is the transition matrix that t body coordinate is tied to inertial coodinate system, and g (t) is Lie group,Leading for g (t) Number；V (t, g (t)) is speed of the t satellite in inertial coodinate system, and V (t, g (t)) is Lie algebra (3 × 3 antisymmetric matrix)；f (g (t), V (t, g (t))) is with regard to g (t) and the function of V (t, g (t))；T is the time；U is the control moment and perturbed force of satellite The summation of square, unit is Nm.

3. attitude of satellite numerical value emulation method as claimed in claim 2, wherein,

$V=\left[\begin{array}{ccc}0& -{\mathrm{\&omega;}}_3& {\mathrm{\&omega;}}_2\\ {\mathrm{\&omega;}}_3& 0& -{\mathrm{\&omega;}}_1\\ -{\mathrm{\&omega;}}_2& {\mathrm{\&omega;}}_1& 0\end{array}\right]$

In formula, ω₁For the attitude angular velocity that satellite rotates around inertial coodinate system x-axis；ω₂Rotate around inertial coodinate system y-axis for satellite Attitude angular velocity；ω₃For the attitude angular velocity that satellite rotates around inertial coodinate system z-axis.

4. attitude of satellite numerical value emulation method as claimed in claim 3, wherein, Lie group equation is：

Lie algebra equation is：

Lie group reconstructs equation：

G (t)=τ (θ (t)) g₀Equation 3；

Wherein, θ (t) is intermediate variable, Lie algebra；For the derivative of θ (t)；τ (θ (t)) is the function of θ (t)；g₀At the beginning of satellite Beginning attitude lower body coordinate is tied to the transition matrix of inertial coodinate system.

5. attitude of satellite numerical value emulation method as claimed in claim 4, wherein, the canoncial coordinates are index coordinates.

6. attitude of satellite numerical value emulation method as claimed in claim 5, wherein, represent that index is sat using Rodrigues formula Marking exp (θ) is：

$\exp \left(\mathrm{\&theta;}\right)=I+\frac{\sin \left|\right|\mathrm{\&theta;}\left|\right|}{\left|\right|\mathrm{\&theta;}\left|\right|}\mathrm{\&theta;}+\frac{1-\cos {\left|\right|\mathrm{\&theta;}\left|\right|}^2}{\left|\right|\mathrm{\&theta;}\left|\right|}{\mathrm{\&theta;}}^2$

MappingIt is expressed as：

$d{\exp }_{\mathrm{\&theta;}}^{-1}=I-\frac{1}{2}\mathrm{\&theta;}+\frac{1-\frac{\left|\right|\mathrm{\&theta;}\left|\right|}{2}\cot \frac{\left|\right|\mathrm{\&theta;}\left|\right|}{2}}{{\left|\right|\mathrm{\&theta;}\left|\right|}^2}{\mathrm{\&theta;}}^2$

In formula, I is 3 × 3 unit matrix.

7. attitude of satellite numerical value emulation method as claimed in claim 4, wherein, the canoncial coordinates are Cayley coordinates.

8. attitude of satellite numerical value emulation method as claimed in claim 7, wherein, Cayley coordinate cay (θ) is：

$\mathrm{cay}\left(\mathrm{\&theta;}\right)=I+\frac{1}{1+0.25{\left|\right|\mathrm{\&theta;}\left|\right|}^2}(\mathrm{\&theta;}+0.5{\mathrm{\&theta;}}^2)$

MappingIt is expressed as：

${\mathrm{dcay}}_{\mathrm{\&theta;}}^{-1}=(1+\frac{1}{4}{\left|\right|\mathrm{\&theta;}\left|\right|}^2)I-\frac{1}{2}\mathrm{\&theta;}+\frac{1}{4}{\mathrm{\&theta;}}^2$

In formula, I is 3 × 3 unit matrix.