Calibration method for rotating accelerometer gravity gradiometer

Abstract

Disclosed is a calibration method for a rotating accelerometer gravity gradiometer, wherein linear motion error coefficients, angular motion error coefficients, self-gradient model parameters and scale factors of the rotating accelerometer gravity gradiometer are calibrated once by changing linear motion, angular motion, and self-gradient excitations of the rotating accelerometer gravity gradiometer. The calibrated linear and angular motion error coefficients are used for compensating for motion errors of the gravity gradiometer online, and the calibrated self-gradient model parameters are used for self-gradient compensation. The calibration method provided by the present invention is easy to operate and not limited by any calibration site, thereby being suitable for programmed self-calibration and realizing an important engineering value.

Claims

1. A calibration method for a rotating accelerometer gravity gradiometer, the rotating accelerometer gravity gradiometer undergoing calibration comprises four accelerometers, an origin of a measurement coordinate system is located in a center of a disc, and the three coordinate axes of the measurement coordinate system of the gravity gradiometer are each provided with a gyroscope and an axial accelerometer configured to record the angular velocity and the linear motion along each respective axis, the calibration method comprising the following steps: 1) changing the linear motions and angular motions to the rotating accelerometer gravity gradiometer undergoing calibration, and meanwhile continuously changing attitudes of the rotating accelerometer gravity gradiometer undergoing calibration, and recording linear motions (a.sub.x,a.sub.y,a.sub.z) with the accelerometers, angular motions (ω.sub.x,ω.sub.y,ω.sub.z,ω.sub.ax,ω.sub.ay,ω.sub.az) with the gyroscopes, the attitudes (θ.sub.x,θ.sub.y,θ.sub.z) and an output (G.sub.out) of the rotating accelerometer gravity gradiometer as calibrated data, wherein a start time for the calibrated data is t.sub.1, an end time is t.sub.p, and linear motion data of the rotating accelerometer gravity gradiometer are as follows: $\{\begin{array}{c}{a}_x={\left[a_x\left(t_1\right),\mathrm{.Math.},a_x\left(t\right),\mathrm{.Math.},a_x\left(t_p\right)\right]}^T\\ a_y={\left[a_y\left(t_1\right),\mathrm{.Math.},a_y\left(t\right),\mathrm{.Math.},a_y\left(t_p\right)\right]}^T\\ a_z={\left[a_z\left(t_1\right),\mathrm{.Math.},a_z\left(t\right),\mathrm{.Math.},a_z\left(t_p\right)\right]}^T\end{array}\hspace{1em}$ wherein, a.sub.x represents acceleration data in an X direction of a measurement coordinate system of the rotating accelerometer gravity gradiometer, and a.sub.x(t.sub.1) represents an acceleration in the X direction of the measurement coordinate system of the rotating accelerometer gravity gradiometer at the time t.sub.1; a.sub.x(t) represents an acceleration in the X direction of the measurement coordinate system of the rotating accelerometer gravity gradiometer at time t; a.sub.x(t.sub.p) represents an acceleration in the X direction of the measurement coordinate system of the rotating accelerometer gravity gradiometer at the time t.sub.p; a.sub.y represents acceleration data in a Y direction of the measurement coordinate system of the rotating accelerometer gravity gradiometer, and a.sub.y(t.sub.1) represents an acceleration in the Y direction of the measurement coordinate system of the rotating accelerometer gravity gradiometer at the time t.sub.1; a.sub.y(t) represents an acceleration in the Y direction of the measurement coordinate system of the rotating accelerometer gravity gradiometer at the time t; a.sub.y(t.sub.p) represents an acceleration in the Y direction of the measurement coordinate system of the rotating accelerometer gravity gradiometer at the time t.sub.p; a.sub.z represents acceleration data in a Z direction of the measurement coordinate system of the rotating accelerometer gravity gradiometer, and a.sub.z(t.sub.1) represents an acceleration in the Z direction of the measurement coordinate system of the rotating accelerometer gravity gradiometer at the time t.sub.1; a.sub.z(t) represents an acceleration in the Z direction of the measurement coordinate system of the rotating accelerometer gravity gradiometer at the time t; a.sub.z(t.sub.p) represents an acceleration in the Z direction of the measurement coordinate system of the rotating accelerometer gravity gradiometer at the time t.sub.p; where, [⋅].sup.T represents transposed operation; angular motion data of the rotating accelerometer gravity gradiometer are as follows: $\{\begin{array}{c}{\omega }_x={\left[{\omega }_x\left(t_1\right),\mathrm{.Math.},{\omega }_x\left(t\right),\mathrm{.Math.},{\omega }_x\left(t_p\right)\right]}^T\\ {\omega }_y={\left[{\omega }_y\left(t_1\right),\mathrm{.Math.},{\omega }_y\left(t\right),\mathrm{.Math.},{\omega }_y\left(t_p\right)\right]}^T\\ {\omega }_z={\left[{\omega }_z\left(t_1\right),\mathrm{.Math.},{\omega }_z\left(t\right),\mathrm{.Math.},{\omega }_z\left(t_p\right)\right]}^T\end{array}\hspace{1em}\{\begin{array}{c}{\omega }_{\mathrm{ax}}={\left[{\omega }_{\mathrm{ax}}\left(t_1\right),\mathrm{.Math.},{\omega }_{\mathrm{ax}}\left(t\right),\mathrm{.Math.},{\omega }_{\mathrm{ax}}\left(t_p\right)\right]}^T\\ {\omega }_{\mathrm{ay}}={\left[{\omega }_{\mathrm{ay}}\left(t_1\right),\mathrm{.Math.},{\omega }_{\mathrm{ay}}\left(t\right),\mathrm{.Math.},{\omega }_{\mathrm{ay}}\left(t_p\right)\right]}^T\\ {\omega }_{\mathrm{az}}={\left[{\omega }_{\mathrm{az}}\left(t_1\right),\mathrm{.Math.},{\omega }_{\mathrm{az}}\left(t\right),\mathrm{.Math.},{\omega }_{\mathrm{az}}\left(t_p\right)\right]}^T\end{array}$ wherein, ω.sub.x represents angular velocity data in the X direction of the measurement coordinate system of the rotating accelerometer gravity gradiometer, ω.sub.x(t.sub.1) represents an angular velocity in the X direction of the measurement coordinate system of the rotating accelerometer gravity gradiometer at the start time t.sub.1, ω.sub.x(t) represents an angular velocity in the X direction of the measurement coordinate system of the rotating accelerometer gravity gradiometer at the time t; ω.sub.x(t.sub.p) represents an angular velocity in the X direction of the measurement coordinate system of the rotating accelerometer gravity gradiometer at the end time t.sub.p; ω.sub.y represents angular velocity data in the Y direction of the measurement coordinate system of the rotating accelerometer gravity gradiometer, ω.sub.y(t.sub.1) represents an angular velocity in the Y direction of the measurement coordinate system of the rotating accelerometer gravity gradiometer at the start time t.sub.1, ω.sub.y(t) represents an angular velocity in the Y direction of the measurement coordinate system of the rotating accelerometer gravity gradiometer at the time t, and ω.sub.y(t.sub.p) represents an angular velocity in the Y direction of the measurement coordinate system of the rotating accelerometer gravity gradiometer at the end time t.sub.p; ω.sub.z represents angular velocity data in the Z direction of the measurement coordinate system of the rotating accelerometer gravity gradiometer, ω.sub.z(t.sub.1) represents an angular velocity in the Z direction of the measurement coordinate system of the rotating accelerometer gravity gradiometer at the start time t.sub.1, ω.sub.z(t) represents an angular velocity in the Z direction of the measurement coordinate system of the rotating accelerometer gravity gradiometer at the time t, and ω.sub.z(t.sub.p) represents an angular velocity in the Z direction of the measurement coordinate system of the rotating accelerometer gravity gradiometer at the end time t.sub.p; ω.sub.ax represents angular acceleration data in the X direction of the measurement coordinate system of the rotating accelerometer gravity gradiometer, ω.sub.ax(t.sub.1) represents an angular acceleration in the X direction of the measurement coordinate system of the rotating accelerometer gravity gradiometer at the start time t.sub.1, ω.sub.ax(t) represents an angular acceleration in the X direction of the measurement coordinate system of the rotating accelerometer gravity gradiometer at the time t, and ω.sub.ax(t.sub.p) represents an angular acceleration in the X direction of the measurement coordinate system of the rotating accelerometer gravity gradiometer at the end time t.sub.p; ω.sub.ay represents angular acceleration data in the Y direction of the measurement coordinate system of the rotating accelerometer gravity gradiometer, ω.sub.ay(t.sub.1) represents an angular acceleration in the Y direction of the measurement coordinate system of the rotating accelerometer gravity gradiometer at the start time t.sub.1, ω.sub.ay(t) represents an angular acceleration in the Y direction of the measurement coordinate system of the rotating accelerometer gravity gradiometer at the time t, and ω.sub.ay(t.sub.p) represents an angular acceleration in the Y direction of the measurement coordinate system of the rotating accelerometer gravity gradiometer at the end time t.sub.p; ω.sub.az represents angular acceleration data in the Z direction of the measurement coordinate system of the rotating accelerometer gravity gradiometer, ω.sub.az(t.sub.1) represents an angular acceleration in the Z direction of the measurement coordinate system of the rotating accelerometer gravity gradiometer at the start time t.sub.1, ω.sub.az(t) represents an angular acceleration in the Z direction of the measurement coordinate system of the rotating accelerometer gravity gradiometer at the time t, and ω.sub.az(t.sub.p) represents an angular acceleration in the Z direction of the measurement coordinate system of the rotating accelerometer gravity gradiometer at the end time t.sub.p; the attitude data of the gravity gradiometer are as follows: $\{\begin{array}{c}{\theta }_x={\left[{\theta }_x\left(t_1\right),\mathrm{.Math.},{\theta }_x\left(t\right),\mathrm{.Math.},{\theta }_x\left(t_p\right)\right]}^T\\ {\theta }_y={\left[{\theta }_y\left(t_1\right),\mathrm{.Math.},{\theta }_y\left(t\right),\mathrm{.Math.},{\theta }_y\left(t_p\right)\right]}^T\\ {\theta }_z={\left[{\theta }_z\left(t_1\right),\mathrm{.Math.},{\theta }_z\left(t\right),\mathrm{.Math.},{\theta }_z\left(t_p\right)\right]}^T\end{array}\hspace{1em}$ wherein, θ.sub.x represents attitude angle data of a rotation around the X axis, θ.sub.x(t.sub.1) represents an attitude angle of the rotation around the X axis at the start time t.sub.1, θ.sub.x(t) represents an attitude angle of the rotation around the X axis at the time t, and θ.sub.x(t.sub.p) represents an attitude angle of the rotation around the X axis at the end time t.sub.p; θ.sub.y represents attitude angle data of a rotation around the Y axis, θ.sub.y(t) represents an attitude angle of the rotation around the Y axis at the start time t.sub.1, θ.sub.y(t) represents an attitude angle of the rotation around the X axis at the time t, and θ.sub.y(t.sub.p) represents an attitude angle of the rotation around the X axis at the end time t.sub.p; θ.sub.z represents attitude angle data of a rotation around the Z axis, θ.sub.z(t.sub.1) represents an attitude angle of the rotation around the Z axis at the start time t.sub.1, θ.sub.z(t) represents an attitude angle of the rotation around the Z axis at the time t, and θ.sub.z(t.sub.p) represents an attitude angle of the rotation around the Z axis at the end time t.sub.p; output data of the rotating accelerometer gravity gradiometer is as follows:
G.sub.out=[G.sub.out(t.sub.1), . . . ,G.sub.out(t), . . . ,G.sub.out(t.sub.p)] wherein, G.sub.out is the output data of the rotating accelerometer gravity gradiometer, G.sub.out(t.sub.1) represents an output of the rotating accelerometer gravity gradiometer at the start time t.sub.1, G.sub.out(t) represents an output of the rotating accelerometer gravity gradiometer at the time t, and G.sub.out(t.sub.p) represents an output of the rotating accelerometer gravity gradiometer at the time t.sub.p; 2) calibrating a linear motion error coefficient vector C.sub.m, an angular motion error coefficient vector C.sub.A and a scale factor k.sub.ggi of the rotating accelerometer gravity gradiometer based on the following steps; 2-1) calculating linear motion vectors and angular motion vectors at all times based on the following formula: $L_m\left(t\right)=\left[\begin{array}{c}{a}_x\left(t\right)a_y\left(t\right)\sin 2\mathrm{\Omega t}+0.5\left(a_x^2\left(t\right)-a_y^2\left(t\right)\right)\cos 2\mathrm{\Omega t}\\ a_x\left(t\right)a_y\left(t\right)\cos 2\mathrm{\Omega t}-0.5\left(a_x^2\left(t\right)-a_y^2\left(t\right)\right)\sin 2\mathrm{\Omega t}\\ a_x\left(t\right)\sin \mathrm{\Omega t}-a_y\left(t\right)\cos \mathrm{\Omega t}\\ a_y\left(t\right)\sin \mathrm{\Omega t}+a_x\left(t\right)\cos \mathrm{\Omega t}\\ a_x\left(t\right)a_z\left(t\right)\sin \mathrm{\Omega t}-a_y\left(t\right)a_z\left(t\right)\cos \mathrm{\Omega t}\\ a_y\left(t\right)a_z\left(t\right)\sin \mathrm{\Omega t}+a_x\left(t\right)a_z\left(t\right)\cos \mathrm{\Omega t}\\ 0.5\left(a_x^2\left(t\right)+a_y^2\left(t\right)\right)\\ a_z^2\left(t\right)\\ a_z\left(t\right)\\ 1\end{array}\right],L_a\left(t\right)=\left[\begin{array}{c}-0.5\left({\omega }_x^2\left(t\right)-{\omega }_y^2\left(t\right)\right)\sin 2\mathrm{\Omega t}+{\omega }_x\left(t\right){\omega }_y\left(t\right)\cos 2\mathrm{\Omega t}\\ {\omega }_x\left(t\right){\omega }_y\left(t\right)\sin 2\mathrm{\Omega t}+0\mathrm{.5}\left({\omega }_x^2\left(t\right)-{\omega }_y^2\left(t\right)\right)\cos 2\mathrm{\Omega t}\\ {\omega }_x\left(t\right){\omega }_z\left(t\right)\sin \mathrm{\Omega t}-{\omega }_y\left(t\right){\omega }_z\left(t\right)\cos \mathrm{\Omega t}\\ {\omega }_y\left(t\right){\omega }_z\left(t\right)\sin \mathrm{\Omega t}+{\omega }_x\left(t\right){\omega }_z\left(t\right)\cos \mathrm{\Omega t}\\ {\omega }_{\mathrm{ax}}\left(t\right)\sin \mathrm{\Omega t}-{\omega }_{\mathrm{ay}}\left(t\right)\cos \mathrm{\Omega t}\\ {\omega }_{\mathrm{ay}}\left(t\right)\sin \mathrm{\Omega t}+{\omega }_{\mathrm{ax}}\left(t\right)\cos \mathrm{\Omega t}\\ {\omega }_{\mathrm{az}}\left(t\right)\\ 0.5\left({\omega }_x^2\left(t\right)+{\omega }_y^2\left(t\right)+2{\omega }_z^2\left(t\right)\right)\end{array}\right]$ wherein, L.sub.m(t) represents a linear motion vector at the time t, L.sub.a(t) represents an angular motion vector at the time t, and a.sub.x(t),a.sub.y(t),a.sub.z(t) represents a linear motion data at the time t; ω.sub.x(t), ω.sub.y(t), ω.sub.z(t), ω.sub.ax(t), ω.sub.ay(t), and ω.sub.az(t) represent angular motion data at the time t, and Ω represents an angular frequency of a rotating disc of the rotating accelerometer gravity gradiometer; 2-2) substituting the linear motion vectors and the angular motion vectors at all times into the following formula to calculate a motion matrix L; $L=\left[\begin{array}{c}{L}_m\left(t_1\right),\mathrm{.Math.},L_m\left(t\right),\mathrm{.Math.}{L}_m\left(t_p\right)\\ L_a\left(t_1\right),\mathrm{.Math.},L_a\left(t\right),\mathrm{.Math.}{L}_a\left(t_p\right)\end{array}\right]$ where, L.sub.m(t.sub.1) represents a linear motion vector at the start time t.sub.1, L.sub.m(t) represents the linear motion vector at the time t, and L.sub.m(t.sub.p) represents a linear motion vector at the end time t.sub.p; L.sub.a(t.sub.1) represents an angular motion vector at the start time t.sub.1, L.sub.a(t) represents the angular motion vector at the time t, and L.sub.a(t.sub.p) represents an angular motion vector at the end time t.sub.p; 2-3) calibrating the linear motion error coefficient vector C.sub.m and the angular motion error coefficient vector C.sub.A based on the following formula, wherein C.sub.m is 1×10 vectors, and C.sub.A is 1×8 vectors:
[C.sub.m,C.sub.A]=G.sub.out.Math.L.sup.+ wherein, G.sub.out is the output data of the rotating accelerometer gravity gradiometer, and L.sup.+ represents a generalized inverse of the motion matrix L; the scale factor k.sub.ggi of the gravity gradiometer is equal to a first element of the angular motion error coefficient vector C.sub.A, namely k.sub.ggi=C.sub.A(1); 3) calibrating a self-gradient model parameter based on the following steps: 3-1) calculating modulation vectors at all times based on the following formula:
C.sub.ref(t)=[sin 2Ωt,cos 2Ωt] wherein, C.sub.ref(t) represents a modulation vector at the time t; 3-2) substituting the attitude data into the following formula to calculate attitude characteristic parameters at all times; $\{\begin{array}{c}{a}_{1,1}\left(t\right)=c^2{\theta }_y\left(t\right),a_{1,2}\left(t\right)=0,a_{1,3}\left(t\right)=-2c{\theta }_y\left(t\right)s{\theta }_y\left(t\right),a_{1,4}\left(t\right)=\\ -1,a_{1,5}\left(t\right)=0,\\ a_{1,6}\left(t\right)=s^2{\theta }_y\left(t\right),a_{1,7}\left(t\right)=c^2{\theta }_y\left(t\right),a_{1,8}\left(t\right)=2c{\theta }_y\left(t\right)s{\theta }_y\left(t\right)s{\theta }_x\left(t\right),\\ a_{1,9}\left(t\right)=-2c{\theta }_x\left(t\right)c{\theta }_y\left(t\right)s{\theta }_y\left(t\right),a_{1,10}=-c^2{\theta }_x\left(t\right)+s^2{\theta }_x\left(t\right)s^2{\theta }_y\left(t\right),\\ a_{1,11}=-2c{\theta }_{x}s{\theta }_x{s}^2{\theta }_y\left(t\right)-2c{\theta }_x\left(t\right)s{\theta }_x\left(t\right),a_{1,12}=\\ c^2{\theta }_x\left(t\right)s^2{\theta }_y\left(t\right)-s^2{\theta }_x\left(t\right)\\ a_{1,13}\left(t\right)=-c{\theta }_y\left(t\right)s{\theta }_x\left(t\right)s{\theta }_y\left(t\right)s2{\theta }_z\left(t\right)+0.5[c^2{\theta }_x\left(t\right)+c^2{\theta }_y\left(t\right)-\\ s^2{\theta }_x\left(t\right)s^2{\theta }_y\left(t\right)]c2{\theta }_z\left(t\right)-0.5\left[c^2{\theta }_x\left(t\right)s^2{\theta }_y\left(t\right)-s^2{\theta }_x\left(t\right)\right]\\ a_{1,14}\left(t\right)=\left[c^2{\theta }_x\left(t\right)c^2{\theta }_y\left(t\right)-s^2{\theta }_X\left(t\right)s^2{\theta }_y\left(t\right)\right]s2{\theta }_z\left(t\right)+\\ 2c{\theta }_y\left(t\right)s{\theta }_x\left(t\right)s{\theta }_y\left(t\right)c2{\theta }_z\left(t\right)\\ a_{1,15}\left(t\right)=2\left[c{\theta }_x\left(t\right)s{\theta }_x\left(t\right)s^2{\theta }_y\left(t\right)+c{\theta }_x\left(t\right)s{\theta }_x\left(t\right)\right]s{\theta }_z\left(t\right)-\\ 2c{\theta }_x\left(t\right)c{\theta }_y\left(t\right)s{\theta }_y\left(t\right)c{\theta }_z\left(t\right)\\ a_{1,16}\left(t\right)=c{\theta }_y\left(t\right)s{\theta }_x\left(t\right)s{\theta }_y\left(t\right)s2{\theta }_z\left(t\right)+0.5[s^2{\theta }_x\left(t\right)s^2{\theta }_y\left(t\right)-\\ c^2{\theta }_x\left(t\right)-c^2{\theta }_y\left(t\right)]c2{\theta }_z\left(t\right)-0.5\left[c^2{\theta }_x\left(t\right)s^2{\theta }_y\left(t\right)-s^2{\theta }_x\left(t\right)\right]\\ a_{1,17}\left(t\right)=-2c{\theta }_x\left(t\right)c{\theta }_y\left(t\right)s{\theta }_y\left(t\right)s{\theta }_z\left(t\right)-2[c{\theta }_x\left(t\right)s{\theta }_x\left(t\right)s^2{\theta }_y\left(t\right)+\\ {\mathrm{c\theta }}_x\left(t\right)s{\theta }_x\left(t\right)]c{\theta }_z\left(t\right)\\ a_{1,18}\left(t\right)=c^2{\theta }_x\left(t\right)s^2{\theta }_y\left(t\right)-s^2{\theta }_x\left(t\right)\end{array}\{\begin{array}{c}{a}_{2,1}\left(t\right)=0,a_{2,2}\left(t\right)=c{\theta }_y\left(t\right),a_{2,3}\left(t\right)=0,a_{2,4}\left(t\right)=0,a_{2,5}\left(t\right)=-s{\theta }_y\left(t\right),\\ a_{2,6}\left(t\right)=0,a_{2,7}\left(t\right)=0,\\ a_{2,8}\left(t\right)=c{\theta }_x\left(t\right)c{\theta }_y\left(t\right),a_{2,9}\left(t\right)=c{\theta }_y\left(t\right)s{\theta }_x\left(t\right),a_{2,10}\left(t\right)=c{\theta }_x\left(t\right)s{\theta }_x\left(t\right)s{\theta }_y\left(t\right),\\ a_{2,11}\left(t\right)=s{\theta }_y\left(t\right)\left[s^2{\theta }_x\left(t\right)-c^2{\theta }_x\left(t\right)\right],a_{2,12}\left(t\right)=-c{\theta }_x\left(t\right)s{\theta }_x\left(t\right)s{\theta }_y\left(t\right),\\ a_{2,13}\left(t\right)=-0.5c{\theta }_x\left(t\right)c{\theta }_y\left(t\right)s2{\theta }_z\left(t\right)-0.5c{\theta }_x\left(t\right)s{\theta }_x\left(t\right)s{\theta }_y\left(t\right)c2{\theta }_z\left(t\right)+\\ 0.5c{\theta }_x\left(t\right)s{\theta }_x\left(t\right)s{\theta }_y\left(t\right)\\ a_{2,14}\left(t\right)=-c{\theta }_x\left(t\right)s{\theta }_x\left(t\right)s{\theta }_y\left(t\right)s2{\theta }_z\left(t\right)+c{\theta }_x\left(t\right)c{\theta }_y\left(t\right)c2{\theta }_z\left(t\right)\\ a_{2,15}\left(t\right)=\left[s{\theta }_y\left(t\right)c^2{\theta }_x\left(t\right)-s{\theta }_y\left(t\right)s^2{\theta }_x\left(t\right)\right]s{\theta }_z\left(t\right)+c{\theta }_y\left(t\right)s{\theta }_x\left(t\right)c{\theta }_z\left(t\right)\\ a_{2,16}\left(t\right)=0.5c{\theta }_x\left(t\right)c{\theta }_y\left(t\right)s2{\theta }_z\left(t\right)+0.5c{\theta }_x\left(t\right)s{\theta }_x\left(t\right)s{\theta }_y\left(t\right)c2{\theta }_z\left(t\right)+\\ 0.5c{\theta }_x\left(t\right)s{\theta }_x\left(t\right)s{\theta }_y\left(t\right)\\ a_{2,17}\left(t\right)=c{\theta }_y\left(t\right)s{\theta }_x\left(t\right)s{\theta }_z\left(t\right)+\left[s{\theta }_y\left(t\right)s^2{\theta }_x\left(t\right)-s{\theta }_y\left(t\right)c^2{\theta }_x\left(t\right)\right]c{\theta }_z\left(t\right)\\ a_{2,18}\left(t\right)=-c{\theta }_x\left(t\right)s{\theta }_x\left(t\right)s{\theta }_y\left(t\right)\end{array}$ where, c,c.sup.2 represent cos( ),cos.sup.2( ) respectively, and s,s.sup.2 represent sin( ),sin.sup.2( ) respectively; θ.sub.x(t),θ.sub.y(t),θ.sub.z(t) represents an attitude at the time t, and a.sub.1,1(t), . . . , a.sub.1,8(t), a.sub.21(t), . . . , a.sub.2,18(t) are attitude characteristic parameters at the time t; 3-3) substituting the calculated attitude characteristic parameters into the following formula to calculate attitude characteristic matrices at all times; $\begin{array}{cc}{A}_{\mathrm{attu}}\left(t\right)=\left[\begin{array}{ccc}{a}_{1,1}\left(t\right)& \mathrm{.Math.}& a_{1,18}\left(t\right)\\ a_{2,1}\left(t\right)& \mathrm{.Math.}& a_{2,18}\left(t\right)\end{array}\right]& \end{array}$ where, A.sub.attu(t) represents an attitude characteristic matrix at the time t, and A.sub.attu(t) is 2×18 matrices; 3-4) calibrating the self-gradient model parameter of the rotating accelerometer gravity gradiometer based on the following formula: $\begin{array}{cc}P={\frac{1}{C_A\left(1\right)}\left[\begin{array}{c}{C}_{\mathrm{ref}}\left(t_1\right)A_{\mathrm{attu}}\left(t_1\right)\\ \mathrm{.Math.}\\ C_{\mathrm{ref}}\left(t\right)A_{\mathrm{attu}}\left(t\right)\\ \mathrm{.Math.}\\ C_{\mathrm{ref}}\left(t_p\right)A_{\mathrm{attu}}\left(t_p\right)\\ 1,0,0,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0\\ 0,0,0,0,0,0,1,0,0,1,0,1,0,0,0,0,0,0,0,0\\ 0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,1,0,0\end{array}\right]}^{+}[\begin{array}{c}{G}_{\mathrm{out}}\left(t_1\right)-\left[C_m,C_A\right]\left[\begin{array}{c}{L}_m\left(t_1\right)\\ L_a\left(t_1\right)\end{array}\right]\\ \mathrm{.Math.}\\ G_{out}\left(t\right)-\left[C_m,C_A\right]\left[\begin{array}{c}{L}_m\left(t\right)\\ L_a\left(t\right)\end{array}\right]\\ \mathrm{.Math.}\\ G_{out}\left(t_p\right)-\left[C_m,C_A\right]\left[\begin{array}{c}{L}_m\left(t_p\right)\\ L_a\left(t_p\right)\end{array}\right]\\ 0\\ 0\\ 0\end{array}]& \end{array}$ wherein, P represents a calibrated self-gradient model parameter, C.sub.m represents the linear motion error coefficient vector calibrated in the step 2, C.sub.A is the angular motion error coefficient vector calculated in the step 2, and C.sub.A(1) is the first element of the angular motion error coefficient vector; L.sub.m(t.sub.1) is the linear motion vector at the start time t.sub.1, L.sub.m(t) is the linear motion vector at the time t, and L.sub.m(t.sub.p) is the linear motion vector at the end time t.sub.p; L.sub.a(t) is the angular motion vector at the time t, L.sub.a(t.sub.1) is the angular motion vector at the start time t.sub.1, and L.sub.a(t.sub.p) is the angular motion vector at the end time t.sub.p; C.sub.ref(t) is the modulation vector at the time t, C.sub.ref(t.sub.1) is a modulation vector at the start time t.sub.1, and C.sub.ref(t.sub.p) is a modulation vector at the end time t.sub.p; and A.sub.attu(t) is the attitude characteristic matrix at the time t, A.sub.attu(t.sub.1) is an attitude characteristic matrix at the start time t.sub.1, and A.sub.attu(t.sub.p) is an attitude characteristic matrix at the end time t.sub.p, and calibrating the rotating accelerometer gravity gradiometer before a gravity field exploration by using the self-gradient model and the calculated calibration parameters thereby preventing saturation or damage to a circuit of the rotating accelerometer gravity gradiometer.

Description

BRIEF DESCRIPTION OF THE DRAWINGS

(1) FIG. 1 is an installation diagram of a sensor for angular and linear motions of a gravity gradiometer.

(2) FIG. 2 is the calibration process of a rotating accelerometer gravity gradiometer.

DETAILED DESCRIPTION OF THE EMBODIMENTS

(3) The present invention will be further depicted in combination with the embodiments and the drawings.

(4) As shown in FIG. 1, A1, A2, A3 and A4 are four accelerometers disposed on a rotating accelerometer gravity gradiometer and sensitive elements for a gravitational gradient; the origin of a measurement coordinate system of the gravity gradiometer is located in the center of a disc, x.sub.m is an X axis of the measurement coordinate system of the gravity gradiometer, y.sub.m is a Y axis of the measurement coordinate system of the gravity gradiometer, and z.sub.m is a Z axis of the measurement coordinate system of the gravity gradiometer; a triaxial accelerometer is disposed at a center point of a rotating disc of the gravity gradiometer to record the linear motion of the gravity gradiometer; and three coordinate axes of the measurement coordinate system of the gravity gradiometer are respectively provided with a gyroscope to record angular motions (angular velocity and angular acceleration) of the gravity gradiometer.

(5) As shown in FIG. 2, a calibration method for a rotating accelerometer gravity gradiometer, the method comprising the following steps:

(6) (1) applying linear and angular motions to the rotating accelerometer gravity gradiometer, and meanwhile continuously changing the attitudes of the gravity gradiometer, and recording linear motions (a.sub.x,a.sub.y,a.sub.z), angular motions (ω.sub.x,ω.sub.y,ω.sub.z,ω.sub.ax,ω.sub.ay,ω.sub.az), the attitudes (θ.sub.x,θ.sub.y,θ.sub.z) and an output (G.sub.out) of the gravity gradiometer as calibrated data, wherein start time for the calibrated data is t.sub.1, end time is t.sub.p, and the linear motion data of the gravity gradiometer is as follows:

(7) $\hspace{1em}\{\begin{array}{c}{a}_x={\left[a_x\left(t_1\right),\mathrm{.Math.},a_x\left(t\right),\mathrm{.Math.},a_x\left(t_p\right)\right]}^T\\ a_y={\left[a_y\left(t_1\right),\mathrm{.Math.},a_y\left(t\right),\mathrm{.Math.},a_y\left(t_p\right)\right]}^T\\ a_z={\left[a_z\left(t_1\right),\mathrm{.Math.},a_z\left(t\right),\mathrm{.Math.},a_z\left(t_p\right)\right]}^T\end{array}$
where, a.sub.x represents acceleration data in an X direction of a measurement coordinate system of the gravity gradiometer, and a.sub.x(t.sub.1) represents an acceleration in the X direction of the measurement coordinate system of the gravity gradiometer at the time t.sub.1; a.sub.x(t) represents an acceleration in the X direction of the measurement coordinate system of the gravity gradiometer at time t; a.sub.x(t.sub.2) represents an acceleration in the X direction of the measurement coordinate system of the gravity gradiometer at the time t.sub.2; a.sub.y represents acceleration data in a Y direction of the measurement coordinate system of the gravity gradiometer, and a.sub.y(t.sub.1) represents an acceleration in the Y direction of the measurement coordinate system of the gravity gradiometer at the time t.sub.1; a.sub.y(t) represents an acceleration in the Y direction of the measurement coordinate system of the gravity gradiometer at the time t; a.sub.y(t.sub.2) represents an acceleration in the Y direction of the measurement coordinate system of the gravity gradiometer at the time t.sub.2; a.sub.z represents acceleration data in a Z direction of the measurement coordinate system of the gravity gradiometer, and a.sub.z(t.sub.1) represents an acceleration in the Z direction of the measurement coordinate system of the gravity gradiometer at the time t.sub.1; a.sub.z(t) represents an acceleration in the Z direction of the measurement coordinate system of the gravity gradiometer at the time t; a.sub.z(t.sub.2) represents an acceleration in the Z direction of the measurement coordinate system of the gravity gradiometer at the time t.sub.2; where, [⋅].sup.T represents transposed operation; the angular motion data of the gravity gradiometer is as follows:

(8) 0$\{\begin{array}{c}{\omega }_x={\left[{\omega }_x\left(t_1\right),\mathrm{.Math.},{\omega }_x\left(t\right),\mathrm{.Math.},{\omega }_x\left(t_p\right)\right]}^T\\ {\omega }_y={\left[{\omega }_y\left(t_1\right),\mathrm{.Math.},{\omega }_y\left(t\right),\mathrm{.Math.},{\omega }_y\left(t_p\right)\right]}^T\\ {\omega }_z={\left[{\omega }_z\left(t_1\right),\mathrm{.Math.},{\omega }_z\left(t\right),\mathrm{.Math.},{\omega }_z\left(t_p\right)\right]}^T\end{array}\hspace{1em}\{\begin{array}{c}{\omega }_{\mathrm{ax}}={\left[{\omega }_{\mathrm{ax}}\left(t_1\right),\mathrm{.Math.},{\omega }_{\mathrm{ax}}\left(t\right),\mathrm{.Math.},{\omega }_{\mathrm{ax}}\left(t_p\right)\right]}^T\\ {\omega }_{\mathrm{ay}}={\left[{\omega }_{\mathrm{ay}}\left(t_1\right),\mathrm{.Math.},{\omega }_{\mathrm{ay}}\left(t\right),\mathrm{.Math.},{\omega }_{\mathrm{ay}}\left(t_p\right)\right]}^T\\ {\omega }_{\mathrm{az}}={\left[{\omega }_{\mathrm{az}}\left(t_1\right),\mathrm{.Math.},{\omega }_{\mathrm{az}}\left(t\right),\mathrm{.Math.},{\omega }_{\mathrm{az}}\left(t_p\right)\right]}^T\end{array}$ where, ω.sub.x represents angular velocity data in the X direction of the measurement coordinate system of the gravity gradiometer, ω.sub.x(t.sub.1) represents an angular velocity in the X direction of the measurement coordinate system of the gravity gradiometer at the start time t.sub.1, ω.sub.x(t) represents an angular velocity in the X direction of the measurement coordinate system of the gravity gradiometer at the time t; ω.sub.x(t.sub.p) represents an angular velocity in the X direction of the measurement coordinate system of the gravity gradiometer at the end time t.sub.p; ω.sub.y represents angular velocity data in the Y direction of the measurement coordinate system of the gravity gradiometer, ω.sub.y(t.sub.1) represents an angular velocity in the Y direction of the measurement coordinate system of the gravity gradiometer at the start time t.sub.1, ω.sub.y(t) represents an angular velocity in the Y direction of the measurement coordinate system of the gravity gradiometer at the time t, and ω.sub.y(t.sub.p) represents an angular velocity in the Y direction of the measurement coordinate system of the gravity gradiometer at the end time t.sub.p; ω.sub.z represents angular velocity data in the Z direction of the measurement coordinate system of the gravity gradiometer, ω.sub.z(t.sub.1) represents an angular velocity in the Z direction of the measurement coordinate system of the gravity gradiometer at the start time t.sub.1, ω.sub.z(t) represents an angular velocity in the Z direction of the measurement coordinate system of the gravity gradiometer at the time t, and ω.sub.z(t.sub.p) represents an angular velocity in the Z direction of the measurement coordinate system of the gravity gradiometer at the end time t.sub.p; ω.sub.ax represents angular acceleration data in the X direction of the measurement coordinate system of the gravity gradiometer, ω.sub.ax(t.sub.1) represents an angular acceleration in the X direction of the measurement coordinate system of the gravity gradiometer at the start time t.sub.1, ω.sub.ax(t) represents an angular acceleration in the X direction of the measurement coordinate system of the gravity gradiometer at the time t, and ω.sub.ax(t.sub.p) represents an angular acceleration in the X direction of the measurement coordinate system of the gravity gradiometer at the end time t.sub.p; ω.sub.ay represents angular acceleration data in the Y direction of the measurement coordinate system of the gravity gradiometer, ω.sub.ay(t.sub.1) represents an angular acceleration in the Y direction of the measurement coordinate system of the gravity gradiometer at the start time t.sub.1, ω.sub.ay(t) represents an angular acceleration in the Y direction of the measurement coordinate system of the gravity gradiometer at the time t, and ω.sub.ay(t.sub.p) represents an angular acceleration in the Y direction of the measurement coordinate system of the gravity gradiometer at the end time t.sub.p; ω.sub.az represents angular acceleration data in the Z direction of the measurement coordinate system of the gravity gradiometer, ω.sub.az(t.sub.1) represents an angular acceleration in the Z direction of the measurement coordinate system of the gravity gradiometer at the start time t.sub.1, ω.sub.az(t) represents an angular acceleration in the Z direction of the measurement coordinate system of the gravity gradiometer at the time t, and ω.sub.az(t.sub.p) represents an angular acceleration in the Z direction of the measurement coordinate system of the gravity gradiometer at the end time t.sub.p; the attitude data of the gravity gradiometer is as follows:

(9) $\{\begin{array}{c}{\theta }_x={\left[{\theta }_x\left(t_1\right),\mathrm{.Math.},{\theta }_x\left(t\right),\mathrm{.Math.},{\theta }_x\left(t_p\right)\right]}^T\\ {\theta }_y={\left[{\theta }_y\left(t_1\right),\mathrm{.Math.},{\theta }_y\left(t\right),\mathrm{.Math.},{\theta }_y\left(t_p\right)\right]}^T\\ {\theta }_z={\left[{\theta }_z\left(t_1\right),\mathrm{.Math.},{\theta }_z\left(t\right),\mathrm{.Math.},{\theta }_z\left(t_p\right)\right]}^T\end{array}\hspace{1em}$

(10) where, σ.sub.x represents attitude angle data of a rotation around the X axis, θ.sub.a(t.sub.1) represents an attitude angle of the rotation around the X axis at the start time t.sub.1, θ.sub.x(t) represents an attitude angle of the rotation around the X axis at the time t, and θ.sub.x(t.sub.p) represents an attitude angle of the rotation around the X axis at the end time t.sub.p; θ.sub.y represents attitude angle data of a rotation around the Y axis, θ.sub.y(t.sub.1) represents an attitude angle of the rotation around the Y axis at the start time t.sub.1, θ.sub.y(t) represents an attitude angle of the rotation around the X axis at the time t, and θ.sub.y(t.sub.p) represents an attitude angle of the rotation around the X axis at the end time t.sub.p; θ.sub.z represents attitude angle data of a rotation around the Z axis, θ.sub.z(t.sub.1) represents an attitude angle of the rotation around the Z axis at the start time t.sub.1, θ.sub.z(t) represents an attitude angle of the rotation around the Z axis at the time t, and θ.sub.z(t.sub.p) represents an attitude angle of the rotation around the Z axis at the end time t.sub.p;

(11) output data of the gravity gradiometer is as follows:
G.sub.out=[G.sub.out(t.sub.1), . . . ,G.sub.out(t), . . . ,G.sub.out(t.sub.p)]

(12) where, G.sub.out is the output data of the gravity gradiometer, G.sub.out(t.sub.1) represents an output of the gravity gradiometer at the start time t.sub.1, G.sub.out(t) represents an output of the gravity gradiometer at the time t, and G.sub.out(t.sub.p) represents an output of the gravity gradiometer at the time t.sub.p;

(13) (2) calibrating a linear motion error coefficient vector C.sub.m, an angular motion error coefficient vector C.sub.A and a scale factor k.sub.ggi of the rotating accelerometer gravity gradiometer based on the following formula;

(14) (2-1). calculating linear and angular motion vectors at all times based on the following formula:

(15) $L_m\left(t\right)=\left[\begin{array}{c}{a}_x\left(t\right)a_y\left(t\right)\sin 2\mathrm{\Omega t}+0.5\left(a_x^2\left(t\right)-a_y^2\left(t\right)\right)\cos 2\mathrm{\Omega t}\\ a_x\left(t\right)a_y\left(t\right)\cos 2\mathrm{\Omega t}-0.5\left(a_x^2\left(t\right)-a_y^2\left(t\right)\right)\sin 2\mathrm{\Omega t}\\ a_x\left(t\right)\sin \mathrm{\Omega t}-a_y\left(t\right)\cos \mathrm{\Omega t}\\ a_y\left(t\right)\sin \mathrm{\Omega t}+a_x\left(t\right)\cos \mathrm{\Omega t}\\ a_x\left(t\right)a_z\left(t\right)\sin \mathrm{\Omega t}-a_y\left(t\right)a_z\left(t\right)\cos \mathrm{\Omega t}\\ a_y\left(t\right)a_z\left(t\right)\sin \mathrm{\Omega t}+a_x\left(t\right)a_z\left(t\right)\cos \mathrm{\Omega t}\\ 0.5\left(a_x^2\left(t\right)+a_y^2\left(t\right)\right)\\ a_z^2\left(t\right)\\ a_z\left(t\right)\\ 1\end{array}\right],L_a\left(t\right)=\left[\begin{array}{c}-0.5\left({\omega }_x^2\left(t\right)-{\omega }_y^2\left(t\right)\right)\sin 2\mathrm{\Omega t}+{\omega }_x\left(t\right){\omega }_y\left(t\right)\cos 2\mathrm{\Omega t}\\ {\omega }_x\left(t\right){\omega }_y\left(t\right)\sin 2\mathrm{\Omega t}+0\mathrm{.5}\left({\omega }_x^2\left(t\right)-{\omega }_y^2\left(t\right)\right)\cos 2\mathrm{\Omega t}\\ {\omega }_x\left(t\right){\omega }_z\left(t\right)\sin \mathrm{\Omega t}-{\omega }_y\left(t\right){\omega }_z\left(t\right)\cos \mathrm{\Omega t}\\ {\omega }_y\left(t\right){\omega }_z\left(t\right)\sin \mathrm{\Omega t}+{\omega }_x\left(t\right){\omega }_z\left(t\right)\cos \mathrm{\Omega t}\\ {\omega }_{\mathrm{ax}}\left(t\right)\sin \mathrm{\Omega t}-{\omega }_{\mathrm{ay}}\left(t\right)\cos \mathrm{\Omega t}\\ {\omega }_{\mathrm{ay}}\left(t\right)\sin \mathrm{\Omega t}+{\omega }_{\mathrm{ax}}\left(t\right)\cos \mathrm{\Omega t}\\ {\omega }_{\mathrm{az}}\left(t\right)\\ 0.5\left({\omega }_x^2\left(t\right)+{\omega }_y^2\left(t\right)+2{\omega }_z^2\left(t\right)\right)\end{array}\right]$
where, L.sub.m(t) represents a linear motion vector at the time t, L.sub.a(t) represents an angular motion vector at the time t, and a.sub.x(t),a.sub.y(t),a.sub.z(t) represents a linear motion data at the time t; ω.sub.x(t), ω.sub.y(t), ω.sub.z(t), ω.sub.ax(t), ω.sub.ay(t), and ω.sub.az(t) represent angular motion data at the time t, and Ω represents an angular frequency of a rotating disc of the rotating accelerometer gravity gradiometer;

(16) (2-2). substituting the linear motion vectors and the angular motion vectors at all times into the following formula to calculate a motion matrix L;

(17) $L=\left[\begin{array}{c}{L}_m\left(t_1\right),\mathrm{.Math.},L_m\left(t\right),\mathrm{.Math.}{L}_m\left(t_p\right)\\ L_a\left(t_1\right),\mathrm{.Math.},L_a\left(t\right),\mathrm{.Math.}{L}_a\left(t_p\right)\end{array}\right]$
where, L.sub.m(t.sub.1) represents a linear motion vector at the start time t.sub.1, L.sub.m(t) represents a linear motion vector at the time t, and L.sub.m(t.sub.p) represents a linear motion vector at the end time t.sub.p; L.sub.a(t.sub.1) represents an angular motion vector at the start time t.sub.1, L.sub.a(t) represents an angular motion vector at the time t, and L.sub.a(t.sub.p) represents an angular motion vector at the end time t.sub.p;

(18) (2-3). Calibrating the linear motion error coefficient vector C.sub.m and the angular motion error coefficient vector C.sub.A based on the following formula, wherein C.sub.m is 1×10 vectors, and C.sub.A is 1×8 vectors:
[C.sub.m,C.sub.A]=G.sub.out.Math.L.sup.+
where, G.sub.out is an output of the rotating accelerometer gravity gradiometer, and L.sup.+ represents a generalized inverse of the matrix L; the scale factor k.sub.ggi of the gravity gradiometer is equal to a first element of the angular motion error coefficient vector C.sub.A, namely k.sub.ggi=C.sub.A(1);

(19) (3) Calibrating a self-gradient model parameter based on the following formula:

(20) (3-1). Calculating modulation vectors at all times based on the following formula
C.sub.ref(t)=[sin 2Ωt,cos 2Ωt]
C.sub.ref(t) represents a modulation vector at the time t;

(21) (3-2). Substituting the attitude data into the following formula to calculate attitude characteristic parameters at all times;

(22) $\{\begin{array}{c}{a}_{1,1}\left(t\right)=c^2{\theta }_y\left(t\right),a_{1,2}\left(t\right)=0,a_{1,3}\left(t\right)=-2c{\theta }_y\left(t\right)s{\theta }_y\left(t\right),a_{1,4}\left(t\right)=-1,a_{1,5}\left(t\right)=0,\\ a_{1,6}\left(t\right)=s^2{\theta }_y\left(t\right),a_{1,7}\left(t\right)=c^2{\theta }_y\left(t\right),a_{1,8}\left(t\right)=2c{\theta }_y\left(t\right)s{\theta }_y\left(t\right)s{\theta }_x\left(t\right),\\ a_{1,9}\left(t\right)=-2c{\theta }_x\left(t\right)c{\theta }_y\left(t\right)s{\theta }_y\left(t\right),a_{1,10}=-c^2{\theta }_x\left(t\right)+s^2{\theta }_x\left(t\right)s^2{\theta }_y\left(t\right),\\ a_{1,11}=-2c{\theta }_{x}s{\theta }_x{s}^2{\theta }_y\left(t\right)-2c{\theta }_x\left(t\right)s{\theta }_x\left(t\right),a_{1,12}=c^2{\theta }_x\left(t\right)s^2{\theta }_y\left(t\right)-s^2{\theta }_x\left(t\right)\\ a_{1,13}\left(t\right)=-c{\theta }_y\left(t\right)s{\theta }_x\left(t\right)s{\theta }_y\left(t\right)s2{\theta }_z\left(t\right)+0.5\left[c^2{\theta }_x\left(t\right)+c^2{\theta }_y\left(t\right)-s^2{\theta }_x\left(t\right)s^2{\theta }_y\left(t\right)\right]c2{\theta }_z\left(t\right)\\ -0.5\left[c^2{\theta }_x\left(t\right)s^2{\theta }_y\left(t\right)-s^2{\theta }_x\left(t\right)\right]\\ a_{1,14}\left(t\right)=\left[c^2{\theta }_x\left(t\right)c^2{\theta }_y\left(t\right)-s^2{\theta }_X\left(t\right)s^2{\theta }_y\left(t\right)\right]s2{\theta }_z\left(t\right)+2c{\theta }_y\left(t\right)s{\theta }_x\left(t\right)s{\theta }_y\left(t\right)c2{\theta }_z\left(t\right)\\ a_{1,15}\left(t\right)=2\left[c{\theta }_x\left(t\right)s{\theta }_x\left(t\right)s^2{\theta }_y\left(t\right)+c{\theta }_x\left(t\right)s{\theta }_x\left(t\right)\right]s{\theta }_z\left(t\right)-2c{\theta }_x\left(t\right)c{\theta }_y\left(t\right)s{\theta }_y\left(t\right)c{\theta }_z\left(t\right)\\ a_{1,16}\left(t\right)=c{\theta }_y\left(t\right)s{\theta }_x\left(t\right)s{\theta }_y\left(t\right)s2{\theta }_z\left(t\right)+0.5\left[s^2{\theta }_x\left(t\right)s^2{\theta }_y\left(t\right)-c^2{\theta }_x\left(t\right)-c^2{\theta }_y\left(t\right)\right]c2{\theta }_z\left(t\right)\\ -0.5\left[c^2{\theta }_x\left(t\right)s^2{\theta }_y\left(t\right)-s^2{\theta }_x\left(t\right)\right]\\ a_{1,17}\left(t\right)=-2c{\theta }_x\left(t\right)c{\theta }_y\left(t\right)s{\theta }_y\left(t\right)s{\theta }_z\left(t\right)-2\left[c{\theta }_x\left(t\right)s{\theta }_x\left(t\right)s^2{\theta }_y\left(t\right)+c{\theta }_x\left(t\right)s{\theta }_x\left(t\right)\right]c{\theta }_z\left(t\right)\\ a_{1,18}\left(t\right)=c^2{\theta }_x\left(t\right)s^2{\theta }_y\left(t\right)-s^2{\theta }_x\left(t\right)\end{array}\{\begin{array}{c}{a}_{2,1}\left(t\right)=0,a_{2,2}\left(t\right)=c{\theta }_y\left(t\right),a_{2,3}\left(t\right)=0,a_{2,4}\left(t\right)=0,a_{2,5}\left(t\right)=-s{\theta }_y\left(t\right),a_{2,6}\left(t\right)=0,a_{2,7}\left(t\right)=0,\\ a_{2,8}\left(t\right)=c{\theta }_x\left(t\right)c{\theta }_y\left(t\right),a_{2,9}\left(t\right)=c{\theta }_y\left(t\right)s{\theta }_x\left(t\right),a_{2,10}\left(t\right)=c{\theta }_x\left(t\right)s{\theta }_x\left(t\right)s{\theta }_y\left(t\right),\\ a_{2,11}\left(t\right)=s{\theta }_y\left(t\right)\left[s^2{\theta }_x\left(t\right)-c^2{\theta }_x\left(t\right)\right],a_{2,12}\left(t\right)=-c{\theta }_x\left(t\right)s{\theta }_x\left(t\right)s{\theta }_y\left(t\right),\\ a_{2,13}\left(t\right)=-0.5c{\theta }_x\left(t\right)c{\theta }_y\left(t\right)s2{\theta }_z\left(t\right)-0.5c{\theta }_x\left(t\right)s{\theta }_x\left(t\right)s{\theta }_y\left(t\right)c2{\theta }_z\left(t\right)+0.5c{\theta }_x\left(t\right)s{\theta }_x\left(t\right)s{\theta }_y\left(t\right)\\ a_{2,14}\left(t\right)=-c{\theta }_x\left(t\right)s{\theta }_x\left(t\right)s{\theta }_y\left(t\right)s2{\theta }_z\left(t\right)+c{\theta }_x\left(t\right)c{\theta }_y\left(t\right)c2{\theta }_z\left(t\right)\\ a_{2,15}\left(t\right)=\left[s{\theta }_y\left(t\right)c^2{\theta }_x\left(t\right)-s{\theta }_y\left(t\right)s^2{\theta }_x\left(t\right)\right]s{\theta }_z\left(t\right)+c{\theta }_y\left(t\right)s{\theta }_x\left(t\right)c{\theta }_z\left(t\right)\\ a_{2,16}\left(t\right)=0.5c{\theta }_x\left(t\right)c{\theta }_y\left(t\right)s2{\theta }_z\left(t\right)+0.5c{\theta }_x\left(t\right)s{\theta }_x\left(t\right)s{\theta }_y\left(t\right)c2{\theta }_z\left(t\right)+0.5c{\theta }_x\left(t\right)s{\theta }_x\left(t\right)s{\theta }_y\left(t\right)\\ a_{2,17}\left(t\right)=c{\theta }_y\left(t\right)s{\theta }_x\left(t\right)s{\theta }_z\left(t\right)+\left[s{\theta }_y\left(t\right)s^2{\theta }_x\left(t\right)-s{\theta }_y\left(t\right)c^2{\theta }_x\left(t\right)\right]c{\theta }_z\left(t\right)\\ a_{2,18}\left(t\right)=-c{\theta }_x\left(t\right)s{\theta }_x\left(t\right)s{\theta }_y\left(t\right)\end{array}$
where, c,c.sup.2 represent cos( ),cos.sup.2( ) respectively, and s,s.sup.2 represent sin( ),sin.sup.2( ) respectively; θ.sub.x(t),θ.sub.y(t),θ.sub.z(t) represents an attitude at the time t, and a.sub.1,1(t), . . . , a.sub.1,18(t), a.sub.21(t), . . . , a.sub.2,18 (t) are attitude characteristic parameters at the time t;