METHOD FOR CORRECTING POINTING ERRORS OF BIAXIAL ROTATION SYSTEM BASED ON SPHERICAL CAP FUNCTION

Abstract

The invention discloses a method for correcting the pointing errors of a biaxial rotation system based on the spherical cap function, comprising: error collection: selecting stars or radio sources distributed evenly in a star catalogue for tracking and observation to obtain the theoretical position and measurement position of the stars, and subtracting the measurement positions and the theoretical positions to obtain the error distribution; error model fitting: selecting a suitable orthogonal spherical cap function for the obtained error distribution and performing fitting to calculate an error fitting coefficient, the orthogonal spherical cap function model comprising a hemispheric harmonic function HSH, a Zernike spherical cap function ZSF, and a longitudinal spherical cap function LSF; and error control and compensation: putting the error model and the related fitting coefficient into a pointing control system for compensation. In the present method for correcting the pointing errors of a biaxial rotation system based on a spherical cap function, the model has strong stability and is not easily affected by measurement noise; there is no need to determine the form of the model on the bases of the frame form of the telescope, and the correction accuracy is high.

Claims

1. Method for correcting the pointing errors of a biaxial rotation system based on the spherical cap function, characterized in that it comprising: Error collection: selecting stars or radio distributed evenly in a star catalogue for tracking and observation to obtain the theoretical position and measurement position of the stars, and subtracting the measurement positions and the theoretical positions to obtain the error distribution; for the small twin-axis rotating system such as double swing milling head, the error distribution on the spherical cap is measured by laser tracker; Error model fitting: selecting a suitable orthogonal spherical cap function for the obtained error distribution and performing fitting to calculate an error fitting coefficient; the orthogonal spherical cap function model comprising a hemispheric harmonic function HSH, a Zernike spherical cap function ZSF and a longitudinal spherical cap function LSF; Error control and compensation: putting the error model and the related fitting coefficient into a pointing control system for compensation.

2. Method for correcting the pointing errors of a biaxial rotation system based on the spherical cap function according to claim 1, wherein the specific process of the method is: S1, Error collecting; S2, Determining the maximum zenith angle and the fitting function; Determining the maximum zenith angle, and the error obtained in step S1 was fitted using the orthogonal spherical cap function model to calculate the error fitting coefficient; S3, Calculating the error fitting and the fitting coefficient was obtained; S4, Obtaining the pointing error model; S5, Putting it into the control system for error correction.

3. Method for correcting the pointing errors of a biaxial rotation system based on the spherical cap function according to claim 2, wherein the specific process of step S1 is: S1-1, Selecting theoretical value Spherical crown is divided into s bins (optimizing evenly divided), measured in each bin, Zenith direction θ and azimuth direction ϕ and the radial direction of the theoretical value r can be got; S1-2, The experimental equipment is used for testing and the measured value is obtained; S1-3, The theoretical and experimental values are subtracted to obtain the pointing error.

4. Method for correcting the pointing errors of a biaxial rotation system based on the spherical cap function according to claim 2, wherein the specific process of step S2 is: S2-1, Selecting the ith orthogonal spherical cap function model function; S2-2, Determining the corresponding angle to the jth direction (1≤j≤s); S2-3, Calculating the ith function valueh.sub.i.sup.j.

5. Method for correcting the pointing errors of a biaxial rotation system based on the spherical cap function according to claim 2, wherein step S2: (1) Hemispheric harmonic function: ${\mathrm{HSH}}_n^m\left(s,\phi \right)={\left(-1\right)}^{\left|m\right|}\sqrt{\frac{\left(2n+1\right)\left(n-.Math.m.Math.\right)!}{2\left(n+.Math.m.Math.\right)!}}.Math.P_n^{.Math.m.Math.}\left\{\cos \left(\theta \right)\right\}.Math.{\Phi }_m\left(\phi \right),\left(\begin{array}{c}n=0,2,4,.Math.;m=0,\begin{array}{cc}\pm 2,.Math.,& \pm n\end{array}\\ n=1,3,5,.Math.;m=\pm 1,\begin{array}{cc}\pm 3,.Math.,& \pm n\end{array}\end{array}\right)$ Where $P_n^{.Math.m.Math.}\left(x\right)=\frac{{\left(-1\right)}^{.Math.m.Math.}}{2^{n}n!}{\left(1-x^2\right)}^{\frac{.Math.m.Math.}{2}}{\left(\frac{d}{dx}\right)}^{n+.Math.m.Math.}{\left(x^2-1\right)}^n,$  HSH is a complete set of orthogonal functions on a hemispheric plane; (2) Zernike Spherical cap function ZSF: $ZS{F}_n^m\left(t,\phi \right)=\sqrt{2\left(n+1\right)}.Math.R_n^m\left(t\right).Math.{\Phi }_m\left(\phi \right),\left(\begin{array}{ccc}n=0,2,4.Math.;m=0,& \pm 2,.Math.,& \pm n\\ n=1,3,5.Math.;m=\pm 1,& \pm 3,.Math.,& \pm n\end{array}\right)$ Among, $t=\sin \left(\theta /2\right)/\sin \left({\theta }_0/2\right),R_n^m\left(t\right)=\underset{k=0}{\overset{\left(n-\left|m\right|\right)/2}{.Math.}}\frac{{\left(-1\right)}^k\left(n-k\right)!}{k!\left(\frac{n+.Math.m.Math.}{2}-k\right)!\left(\frac{n-.Math.m.Math.}{2}-k\right)!}{t}^{n-2k},$  ZSF is a complete set of orthogonal functions on the spherical cap, θ.sub.0 is the maximum Zenith angle of the spherical crown (0<θ.sub.0<π); (3) Longitudinal spherical cap function LSF: $LS{F}_n^m\left(w,\phi \right)=\sqrt{\frac{2^{\left|m\right|+2.5}}{{\gamma }_v^{\left(a,b\right)}}}{w}^{\left|m\right|}J\left(\frac{n-\left|m\right|}{2},0,\frac{2\left|m\right|-1}{2},2w^2-1\right).Math.{{\Phi }_m\left(\phi \right)}_{,}\left(\begin{array}{ccc}n=0,2,.Math.;m=0,& \pm 2,.Math.,& \pm n\\ n=1,3,.Math.;m=\pm 1,& \pm 3,.Math.,& \pm n\end{array}\right)$ Among $\begin{array}{cc}w=\left[1-\cos \left(\theta \right)\right]/\left[1-\cos \left({\theta }_0\right)\right]& J\left(v,a,b,x\right)=\end{array}\frac{1}{v!}\underset{k=0}{\overset{v}{.Math.}}\frac{{\left(-v\right)}_k{\left(a+b+v+1\right)}_k{\left(a+k+1\right)}_{v-k}}{k!}{\left(\frac{1-x}{2}\right)}^{k}ZSF$  is a complete set of orthogonal functions on the spherical cap, θ.sub.0 is the maximum Zenith angle of the spherical crown (0<θ.sub.0<π);

6. Method for correcting the pointing errors of a biaxial rotation system based on the spherical cap function according to claim 2, wherein step S2: These lection of maximum Zenith angle θ.sub.0 is the maximum use range of the height axis in the two-axis rotation system. When θ.sub.0<90°, the fitting function can choose the hemispheric harmonic function or Zernike spherical cap function ZSF or Longitudinal spherical cap function LSF; when θ.sub.0>90°, the fitting function can choose Zernike spherical cap function ZSF, Longitudinal spherical cap function LSF.

7. Method for correcting the pointing errors of a biaxial rotation system based on the spherical cap function according to claim 2, wherein step S3 includes: The Zenith direction θ and azimuth direction ϕ and the radial direction of the theoretical value r can be got.

8. Method for correcting the pointing errors of a biaxial rotation system based on the spherical cap function according to claim 2, wherein step S4 includes: The Zenith direction θ and azimuth direction ϕ and the radial direction of the theoretical value rcan be got.

9. Method for correcting the pointing errors of a biaxial rotation system based on the spherical cap function according to claim 2, wherein the specific process of step S5 is: S5-1, Getting pointing command; S5-2, Substituting the pointing command into the pointing error model of Zenith angle to obtain the pointing error of Zenith angle; S5-3, Fixing zenith pointing command; S5-4, Putting the modified command into the control system for execution.

Description

DESCRIPTION OF THE FIGURES

[0047] FIG. 1 is the control flow chart of the method of the invention;

[0048] FIG. 2 is a schematic diagram of the correction model of spherical cap function;

[0049] FIG. 3 is a schematic diagram of the orthogonal biaxial rotation system;

[0050] FIG. 4 is the schematic diagram of correction of pointing through pointing error model.

DETAILED DESCRIPTION

[0051] The following is a further detailed description of the invention in combination with the attached drawings.

[0052] The method of the invention is applicable in two situations:

[0053] (1) If the area covered by the error model is smaller than the hemisphere, HSH, ZSF and LSF can be used.

[0054] (2) If the error model covers more than one hemisphere, both ZSF and LSF can be used, and the maximum Zenith angle θ.sub.0 can be determined according to the distribution of error data.

[0055] For convenience, the following discussion takes HSH as an example (ZSF and LSF are used in the same way as HSH)

[0056] The method and steps of the invention are shown in FIG. 1, including:

Step 1: error collection
Step 1-1 Selecting the theoretical value

[0057] Spherical crown is divided into s bins (optimizing evenly divided), measured in each bin, Zenith direction Band azimuth direction ϕ and the radial direction of the theoretical value rcan be got:

[00007]$\{\begin{array}{c}\Theta ={\left[\begin{array}{cccc}{\theta }_1& {\theta }_2& .Math.& {\theta }_s\end{array}\right]}^T\\ \Phi ={\left[\begin{array}{cccc}{\phi }_1& {\phi }_2& .Math.& {\phi }_s\end{array}\right]}^T\\ R={\left[\begin{array}{cccc}{r}_1& r_2& .Math.& r_s\end{array}\right]}^T\end{array}$

Step 1-2 Testing with the experimental equipment to obtain the measured value

[00008]$\{\begin{array}{c}\tilde{\Theta }=[\begin{array}{cccc}{\tilde{\theta }}_1& {\tilde{\theta }}_2& .Math.& {{\tilde{\theta }}_s]}^T\end{array}\\ \tilde{\Phi }={\left[\begin{array}{cccc}{\tilde{\phi }}_1& {\tilde{\phi }}_2& .Math.& {\tilde{\phi }}_s\end{array}\right]}^T\\ \tilde{R}={\left[\begin{array}{cccc}{\tilde{r}}_1& {\tilde{r}}_2& .Math.& {\tilde{r}}_s\end{array}\right]}^T\end{array}$

Step 1-3 Subtracting the theoretical value from the experimental value to obtain the pointing error:

[00009]$\{\begin{array}{c}d\Theta =\Theta -\tilde{\Theta }\\ d\Phi =\Phi -\tilde{\Theta }\\ \mathrm{dR}=R-\tilde{R}\end{array}$

Step 2: error model fitting

[0058] The error distribution obtained above is selected to fit the orthogonal spherical cap function model, and the error fitting coefficient was calculated. The orthogonal spherical cap function model includes HSH (Hemi-Spherical Harmonics), ZSF (Zernike Spherical Function) and LSF (Longitudinal Spherical Function).

(1) the analytical expression formula of HSH is as follows:

[00010]${\mathrm{HSH}}_n^m\left(s,\phi \right)={\left(-1\right)}^{.Math.m.Math.}\sqrt{\frac{\left(2n+1\right)}{2}\frac{\left(n-.Math.m.Math.\right)!}{\left(n+.Math.m.Math.\right)!}}.Math.P_n^{.Math.m.Math.}\left\{\cos \left(\theta \right)\right\}.Math.{\Phi }_m\left(\phi \right),\left(\begin{array}{c}n=0,2,4,.Math.;m=0,\pm 2,.Math.,\pm n\\ n=1,3,5,.Math.;m=\pm 1,\pm 3,.Math.,\pm n\end{array}\right)$

Among, P.sub.n.sup.|m| is the Associated Legendre Polynomial (Associated Legendre Polynomials) Here are the expressions:

[00011]$P_n^{.Math.m.Math.}\left(x\right)=\frac{{\left(-1\right)}^{.Math.m.Math.}}{2^{n}n!}{\left(1-x^2\right)}^{\frac{.Math.m.Math.}{2}}{\left(\frac{d}{\mathrm{dx}}\right)}^{n+.Math.m.Math.}{\left(x^2-1\right)}^n$

HSH is a complete set of orthogonal functions on a hemispheric plane.
(2) The analytic expression of Zernike spherical cap function ZSF:

[00012]${\mathrm{ZSF}}_n^m\left(t,\phi \right)=\sqrt{2\left(n+1\right)}.Math.R_n^m\left(t\right).Math.{\Phi }_m\left(\phi \right),\left(\begin{array}{c}n=0,2,4,.Math.;m=0,\pm 2,.Math.,\pm n\\ n=1,3,5,.Math.;m=\pm 1,\pm 3,.Math.,\pm n\end{array}\right)$

Among, t=sin(θ/2)/sin(θ.sub.0/2). R.sub.n.sup.m(t) is the Zernike polynomial, here is the expression:

[00013]$R_n^m\left(t\right)={.Math.}_{k=0}^{\left(n-.Math.m.Math.\right)/2}\frac{{\left(-1\right)}^k\left(n-k\right)!}{k!\left(\frac{n+.Math.m.Math.}{2}-k\right)!\left(\frac{n-.Math.m.Math.}{2}-k\right)!}{t}^{n-2k},$

ZSF is a complete set of orthogonal functions on a spherical crown with a maximum Zenith angle θ.sub.0.
(3) Longitudinal spherical cap function LSF:

[00014]${\mathrm{LSF}}_n^m\left(w,\phi \right)=\sqrt{\frac{2^{\left|m\right|+2.5}}{{\gamma }_v^{\left(a,b\right)}}}{w}^{\left|m\right|}J\left(\frac{n-\left|m\right|}{2},0,\frac{2\left|m\right|-1}{2},2w^2-1\right).Math.{{\Phi }_m\left(\phi \right)}_{,}\left(\begin{array}{cc}n=0,2,.Math.;m=0,& \pm 2,.Math.,\pm n\\ n=1,3,.Math.;m=\pm 1,& \pm 3,.Math.,\pm n\end{array}\right)$

[0059] Among,

[00015]$w=\left[1-\cos \left(\theta \right)\right]/\left[1-\cos \left({\theta }_0\right)\right]J\left(v,a,b,x\right)=\frac{1}{v!}\underset{k=0}{\overset{v}{.Math.}}\frac{{\left(-v\right)}_k{\left(a+b+v+1\right)}_k{\left(a+k+1\right)}_{v-k}}{k!}{\left(\frac{1-x}{2}\right)}^k,$

LSF is a complete set of orthogonal functions on a spherical crown with a maximum Zenith angle θ.sub.0.

[0060] Selection of maximum Zenith angle θ.sub.0: the maximum range of use for the height axis in a two-axis rotation system should be selected. When θ.sub.0 less than 90 degrees, the fitting function can choose the hemispheric harmonic function, or Zernike spherical cap function ZSF, or longitudinal spherical cap function LSF.

S2-1, Selecting the ith HSH: HSH.sub.i (1≤i≤N);
S2-2, Determining the corresponding angle (θ.sub.j, ˜.sub.j) to the jth direction (1≤j≤s);
S2-3, Calculating the function valueh.sub.i.sup.j to the jth direction to form fitting matrix H.

[00016]$H={\left[\begin{array}{cccc}{h}_1^1& h_2^1& .Math.& h_N^1\\ h_1^2& h_2^2& .Math.& h_N^2\\ .Math.& .Math.& \ddots & .Math.\\ h_1^s& h_2^s& .Math.& h_N^s\end{array}\right]}_{s\times N}$

Step 3: Calculating the coefficients in the fitting model
Step 3-1 Calculating the coefficient of Zenith angle θ

{right arrow over (A)}=[a.sub.1 a.sub.1 . . . a.sub.N].sup.T=(H.sup.TH).sup.−1H.sup.T(dΘ)

Step 3-2 Calculating the coefficient of azimuth angle ϕ

{right arrow over (B)}=[b.sub.1 b.sub.2 . . . b.sub.N].sup.T=(H.sup.TH).sup.−1H.sup.T(dΦ)

Step 3-3 Calculating the coefficient of radial direction r

{right arrow over (C)}=[c.sub.1 c.sub.2 . . . c.sub.N].sup.T=(H.sup.TH).sup.−1H.sup.T(dR)

Step 4: Obtaining the pointing error model
Step 4-1 The error model to direction θ of Zenith angle is as follows:

[00017]$E_{\theta }\left(\theta ,\phi \right)=\underset{i=1}{\overset{N}{.Math.}}a,.Math.HS{H}_i\left(\theta ,\phi \right)$

Step 4-2 The error model to direction ϕ of Azimuth angle is as follows:

[00018]$E_{\phi }\left(\theta ,\phi \right)=\underset{i=1}{\overset{N}{.Math.}}{b}_i.Math.HS{H}_i\left(\theta ,\phi \right)$

Step 4-3 The error model to radical direction r is as follows:

[00019]$E_r\left(\theta ,\phi \right)=\underset{i=1}{\overset{N}{.Math.}}{c}_i.Math.HS{H}_i\left(\theta ,\phi \right)$

Step 5: Error correction by putting in the pointing control system, as shown in FIG. 2-3.

[0061] Taking Zenith angle θ as an example, correction is made through pointing error model.

Step 5-1 Getting pointing command (θ.sub.m, φ.sub.m);
Step 5-2 The pointing command is substituted into the pointing error model of Zenith angle, and the pointing error of Zenith angle is calculated:

[00020]$e_{\theta }=\underset{i=1}{\overset{N}{.Math.}}{a}_i.Math.HS{H}_i\left({\theta }_m,{\phi }_m\right)$

Step 5-3 Fixed zenith pointing command:

{circumflex over (θ)}.sub.m=θ.sub.m−e.sub.θ

Step 5-4 Put the modified command into the control system for execution. See FIG. 5 for the specific block diagram.
Step 6: Put in the pointing control system for error correction

[0062] The azimuth flow is the same.

[0063] The correction of radial r is more complicated than Azimuth angle and Zenith angle, so it cannot be directly corrected. In five-axis numerical control equipment, its correction should be compensated by linear motion in three directions (X, Y, Z), and the formula is as follows:

[00021]$e_r=\underset{i=1}{\overset{N}{.Math.}}{c}_i.Math.HS{H}_i\left({\theta }_m,{\phi }_m\right)$

[00022]$\{\begin{array}{c}{e}_x=e_r.Math.\sin \left({\theta }_m\right)\cos \left({\phi }_m\right)\\ e_y=e_r.Math.\sin \left({\theta }_m\right)\sin \left({\phi }_m\right)\\ e_z=e_r.Math.\cos \left({\theta }_m\right)\end{array}$

Then put in the control system of X, Y and Z for correction;

[0064] As mentioned above is only a relatively good embodiment of the invention and is not used to restrict the invention. Any modification, equivalent replacement and improvement etc. made within the spirit and principles of the invention shall be included in the protection scope of the invention.