Star sensor calibration with separation of intrinsic and extrinsic parameters

Qiaoyun Fan; Qiaoyun Fan; Kangkang He; Kangkang He; Gangyi Wang; Gangyi Wang

doi:10.1364/OE.396996

1. Introduction

As a high-precision attitude measurement device, the star sensor must be calibrated for imaging parameters such as focal length, principal point, and distortion before use [1–14]. Star sensor calibration methods can be generally categorized into three classes according to the way of obtaining calibration data: night sky calibration methods [2], laboratory calibration methods based on a star simulator and a high-precision rotary table [3–15], and calibration methods based on the self-collimation theodolite [16]. Among the calibration methods, laboratory calibration methods based on a star simulator and a high-precision rotary table are widely used for their high accuracy and strong operability. In this calibration method, the ideal situation is that the star sensor is installed on the rotary table, the star sensor coordinate frame is consistent with the rotary table coordinate frame, and the optical axis of the star simulator is perpendicular to the zero position of the rotary table. In real situation, the mounting deviation between the star sensor coordinate frame and the rotary table coordinate frame is inevitable. In addition, there are also inevitable mounting deviations between the optical axis of the star simulator and the zero position of the rotary table. Therefore, if these deviations are not considered, complex installation alignment operations are required. For this reason, Zhang et al. [9] proposed a calibration method based on integrated modeling with intrinsic and extrinsic parameters. In that paper, the focal length, principal point and distortion are defined as intrinsic parameters, the mounting deviations of the above calibration devices are defined as extrinsic parameters, and an integrated calibration model of intrinsic and extrinsic parameters is established according to the coordinate transformation and projection transformation of the starlight vector. The calibration method does not require mounting alignment, which simplifies the operation and improves the accuracy. However, the intrinsic and extrinsic parameters are solved together by the optimization method in this method, so the coupling between intrinsic and extrinsic parameters is inevitable. The consequence of this coupling is that high-precision results can only be obtained when all parameters are used together. However, due to the coupling, it is not guaranteed that each parameter is consistent with the actual value, so the accuracy will decrease when only part of the parameters are used or they are used individually.

For the star sensor applied to the satellite, only intrinsic parameters and part of the extrinsic parameters are used. It is the same for the application of missile. Additionally, the extrinsic parameters need to be used individually for missile. Therefore, the accuracy of this method will decrease in the above applications. In order to ensure the accuracy, it is necessary to separate the parameters according to the actual situation. For applications mentioned above, parameters need to be separated are determined and the corresponding separation methods are proposed in this paper.

To solve the problem of parameter coupling in the calibration method based on integrated modeling with intrinsic and extrinsic parameters, Xiong et al. [10] proposed a calibration method based on the decoupling of intrinsic and extrinsic parameters of the star sensor with distinct extrinsic parameters. It collects calibration data repeatedly with the same trajectory of rotary table under different extrinsic parameters each time on the premise that the intrinsic parameters remain stationary throughout the calibration process, which reduces the coupling effect of extrinsic parameters on intrinsic parameters to some extent. Nevertheless, the intrinsic and extrinsic parameters are not completely decoupled, and the operation process in the method is complex.

This paper is outlined as follows. Section 2 conducts simulation analysis on the coupling form and degree of the parameters based on integrated modeling with intrinsic and extrinsic parameters. Based on the analysis results, Section 3 describes the specific calibration method with separation of intrinsic and extrinsic parameters for the applications of star sensors in satellites and missiles. In Section 4, actual experiments are carried out to verify the calibration method. Finally, conclusions are offered in Section 5.

2. Coupling analysis of intrinsic and extrinsic parameters

2.1 Integrated modeling with intrinsic and extrinsic parameters

The star sensor calibration system is shown in Fig. 1. The parallel incident light similar to real starlight is shed by star simulator. Distinct angles can be generated accurately by the high-precision rotary table. The star sensor is mounted on the internal frame of the rotary table with the boresight pointing to the star simulator. The X-axis (${X_m}$) and Y-axis (${Y_m}$) of star sensor coordinate frame (S) are the row and column of the image sensor plane, respectively. The Z-axis (${Z_m}$) is normal to the ${X_m}\textrm{ - }{Y_m}$ plane.

Fig. 1. Star sensor calibration system (Ref. [10], Fig. 2).

Download Full Size | PDF

The imaging parameters of the star sensor are taken as intrinsic parameters, and the mounting deviations between the star sensor coordinate frame and the rotary table coordinate frame as well as the initial alignment errors of the star simulator are taken as extrinsic parameters. The calibration model of integrated intrinsic and extrinsic parameters are as follows [9,17–19]:

Firstly, the starlight vector ${\textbf{V}_\textbf{s}}$ in frame S which can be expressed as

(1)$${\textbf{V}_\textbf{s}} = {\left[ {\begin{array}{ccc} {{f_1}}&{{f_2}}&{{f_3}} \end{array}} \right]^T} = {\textbf{R}_{\textbf{sr}}}{\textbf{R}_\textbf{r}}\textbf{V},$$

where $\textbf{V}$ is the simulated starlight vector generated by star simulator in the rotary table coordinate frame (R), which can be expressed by the yaw (azimuth) angle ${\beta _1}$ and the pitch (inclination) angle ${\beta _2}$, and the expression is as follows:

(2)$$\textbf{V} = {\left[ {\begin{array}{ccc} {\cos ({\beta_1})\cos ({\beta_2})}&{\sin ({\beta_1})\cos ({\beta_2})}&{\sin ({\beta_2})} \end{array}} \right]^T}.$$

${\textbf{R}_\textbf{r}}$ is the rotation matrix, providing different incident angles for simulated starlight, which can be expressd by the rotation angles ${\theta _1}$ and ${\theta _2}$ of the external and middle frames of the rotary table, and the expression is as follows:

(3)$${\textbf{R}_\textbf{r}} = \left[ {\begin{array}{ccc} {\cos ({\theta_2})}&0&{ - \sin ({\theta_2})}\\ 0&1&0\\ {\sin ({\theta_2})}&0&{\cos ({\theta_2})} \end{array}} \right]\left[ {\begin{array}{ccc} 1&0&0\\ 0&{\cos ({\theta_1})}&{\sin ({\theta_1})}\\ 0&{ - \sin ({\theta_1})}&{\cos ({\theta_1})} \end{array}} \right].$$

${\textbf{R}_{\textbf{sr}}}$ is the transformation matrix from frame R to frame S, which can be expressed by the mounting deviation angles ${\varphi _1}$, ${\varphi _2}$ and ${\varphi _3}$ on the three axes of frame R with respect to the frame S, and the expression is as follows:

(4)$${\textbf{R}_{\textbf{sr}}} = \left[ {\begin{array}{ccc} {\cos ({\varphi_2})}&0&{ - \sin ({\varphi_2})}\\ 0&1&0\\ {\sin ({\varphi_2})}&0&{\cos ({\varphi_2})} \end{array}} \right]\left[ {\begin{array}{ccc} 1&0&0\\ 0&{\cos ({\varphi_1})}&{\sin ({\varphi_1})}\\ 0&{ - \sin ({\varphi_1})}&{\cos ({\varphi_1})} \end{array}} \right]\left[ {\begin{array}{ccc} {\cos ({\varphi_3})}&{\sin ({\varphi_3})}&0\\ { - \sin ({\varphi_3})}&{\cos ({\varphi_3})}&0\\ 0&0&1 \end{array}} \right].$$

Finally, the transformation process from starlight vector in frame S to imaging star spot in the focal plane of the star sensor can be expressed as

(5)$$\left\{ {\begin{array}{c} {\widehat x = \frac{F}{{{d_x}}}\frac{{{f_1}}}{{{f_3}}} + {x_0} + {\delta_x}}\\ {\widehat y = \frac{F}{{{d_x}}}\frac{{{f_2}}}{{{f_3}}} + {y_0} + {\delta_y}} \end{array}} \right.,$$

(6)$$\left\{ {\begin{array}{c} {{\delta_x} = \bar{x}({q_1}{r^2} + {q_2}{r^4}) + [{p_1}({r^2} + 2{{\bar{x}}^2}) + 2{p_2}\bar{x}\bar{y}]}\\ {{\delta_y} = \bar{y}({q_1}{r^2} + {q_2}{r^4}) + [{p_2}({r^2} + 2{{\bar{y}}^2}) + 2{p_1}\bar{x}\bar{y}]} \end{array}} \right.,$$

where $(\hat{x},\hat{y})$ is the centroid coordinate of the imaging star spot, F is the focal length of the optical lens, ${d_x}$ and ${d_y}$ are the pixel sizes of the image sensor in the X and Y directions respectively, (${x_0}$, ${y_0}$) is the coordinate of the principal point, ${\delta _x}$ and ${\delta _y}$ are the distortion of the image in the X and Y directions respectively, ${q_1}$ and ${q_2}$ are the radial distortion coefficients, ${p_1}$ and ${p_2}$ are tangential distortion coefficients [20].

In summary, there are 12 parameters in the calibration system, including 7 intrinsic parameters (F, ${x_0}$, ${y_0}$, ${q_1}$, ${q_2}$, ${p_1}$ and ${p_2}$) and 5 extrinsic parameters (${\beta _1}$, ${\beta _2}$, ${\varphi _1}$, ${\varphi _2}$ and ${\varphi _3}$). Obviously, the intrinsic and extrinsic parameters of the calibration model are established and optimized uniformly, so the coupling of parameters is inevitable.

2.2 Coupling form analysis

The error of each parameter in the above model will lead to the change of star spot position. If the change form of star spot position caused by the two parameters errors is the same, the two parameters are coupled, otherwise the two parameters are independent of each other.

For the analysis, a digital star sensor with parameters in Table 1 is established. A total of 121 calibration points, that is, the locus data of the rotary table, used in the simulation are distributed over the range of (-7°, +7°) × (-7°, +7°) with a step size of 1.4°. The parameters in Table 1 and the calibration point data are used to generate star spots based on the integrated modeling with intrinsic and extrinsic parameters. Moreover, the Gaussian noise with a mean of 0 and standard deviation of 0.01 pixels is added to each star spot to make the simulation close to the real situation. The simulation results are presented in turn.

(1) Change of star spot position with respect to the starlight vector error
The changes of star spot position are shown in Figs. 2(a) and (b) when errors of ${\beta _1}$ are +5° and +180° respectively with other parameters in Table 1 remaining unchanged. Similarly, the changes of star spot position are shown in Fig. 2(c) when error of ${\beta _2}$ is +0.5° with other parameters in Table 1 remaining unchanged.
Clearly, the errors of the azimuth angle ${\beta _1}$ and the inclination angle ${\beta _2}$ will cause the overall linear offset of star spots, and the direction of offset will be different with different errors.
(2) Change of star spot position with respect to the errors of mounting deviation angles between the star sensor and the rotary table
The corresponding change of star spot position is shown in Fig. 3 when errors of ${\varphi _1}$, ${\varphi _2}$ and ${\varphi _3}$ are +0.1° respectively with other parameters in Table 1 remaining unchanged.
As can be seen from Figs. 3(a) and (b), the errors of ${\varphi _1}$ and ${\varphi _2}$ will also cause the overall linear offset of star spots, but ${\varphi _3}$ will not.
(3) Change of star spot position with respect to the errors of intrinsic parameters
According to Eq. (5), with other parameters in Table 1 remaining unchanged, the change of star spot position caused by errors of the principal point is similar to that in Fig. 2, that is the overall linear offset of star spots.

Fig. 2. Changes of star spot position under different errors of the starlight vector: (a) The error of ${\beta _1}$ is +5°. (b) The error of ${\beta _1}$ is +180°. (c) The error of ${\beta _2}$ is +0.5°.

Download Full Size | PDF

Fig. 3. Changes of star spot position caused by errors of the mounting deviation angles: (a) The error of ${\varphi _1}$ is +0.1°. (b) The error of ${\varphi _2}$ is +0.1°. (c) The error of ${\varphi _3}$ is +0.1°.

Download Full Size | PDF

Table 1. Parameters of a digital star sensor

View Table | View all tables in this article

Figure 4(a) shows the change of star spot position caused by a +0.1mm error of F. Figures 4(b) and (c) show the changes of star spot position caused by errors of radial and tangential distortion coefficients, respectively.

Fig. 4. Changes of star spot position caused by estimation errors of the focal length and distortion coefficients: (a) The error of F is +0.1mm. (b) ${q_1}$ = ${q_2}$ = 0. (c) ${p_1}$ = ${p_2}$ = 0.

Download Full Size | PDF

To compare and analyze the simulation results from Fig. 2 to Fig. 4, it is obvious that the errors of ${\beta _1}$, ${\beta _2}$, ${\varphi _1}$, ${\varphi _2}$, ${x_0}$ and ${y_0}$ will lead to the overall linear offset of star spots, so there is coupling between these parameters.

2.3 Coupling degree analysis

In this section, the coupling degree of intrinsic (${x_0}$ and ${y_0}$) and extrinsic parameters (${\beta _1}$, ${\beta _2}$, ${\varphi _1}$ and ${\varphi _2}$) are further analyzed. The adopted method is to transform the coupling relationship between parameters into the correlation relationship. In this paper, the estimated values of ${\beta _1}$, ${\beta _2}$, ${\varphi _1}$ and ${\varphi _2}$ are all likely to affect the estimated results of ${x_0}$and ${y_0}$, which belongs to the case with multiple variables. Therefore, the partial correlation analysis is applied, and the coupling degree is measured by the partial correlation coefficient with a significance test, denoted as PCC [21–23].

2.3.1 Partial correlation analysis model

In this work, there are always 3 controlled variables, called the third-order partial correlation. To establish a partial correlation analysis model involving the calculation of PCC and significance test, ${x_0}$, ${y_0}$, ${\beta _1}$, ${\beta _2}$, ${\varphi _1}$ and ${\varphi _2}$ are denoted as 0, 1, 2, 3, 4 and 5, respectively, for simplicity.

(1) The calculation of PCC
Taking the third-order PCC ${r_{02 - 345}}$ between ${x_0}$ and ${\beta _1}$ under controlling the linear effects of ${\beta _2}$, ${\varphi _1}$ and ${\varphi _2}$as an example, the expression is as follows:
$(7)$${r_{02 - 345}} = \frac{{{r_{02 - 34}} - {r_{05 - 34}}{r_{25 - 34}}}}{{\sqrt {(1 - r_{05\textbf{ - }34}^2)(1 - r_{25\textrm{ - }34}^2)} }},$$$ where ${r_{02 - 34}}$is the second-order PCC between ${x_0}$ and ${\beta _1}$ under controlling the linear effects of ${\beta _2}$ and ${\varphi _1}$, ${r_{05 - 34}}$ and ${r_{25 - 34}}$ are both second-order PCCs, similar to ${r_{02 - 34}}$.
By analogy, the third-order PCC ${r_{02 - 345}}$ can be finally expressed by the zero-order PCC, that is, the Pearson coefficient:
$(8)$${r_{ab}} = \frac{{{\mathop{\rm cov}} (a,b)}}{{\sqrt {{\mathop{\rm var}} [a]{\mathop{\rm var}} [b]} }},$$$ where ${\mathop{\rm var}} [a]$ and ${\mathop{\rm var}} [b]$ are the variance of variables a and b, respectively, and ${\mathop{\rm cov}} (a,b)$ is the covariance of variables a and b.
The range of the PCC is from -1 to 1 in a sample. The two variables are positively correlated when PCC > 0. In contrast, the two variables are negatively correlated when PCC < 0. Furthermore, the larger the absolute value of PCC, the stronger the partial correlation between the two variables [22].
(2) The significance test of PCC
The significance test of PCC is conducted in three steps. Propose null and alternative hypotheses, construct and calculate T-statistics, choose the appropriate significance level $\alpha $ and make the statistical decision according to P-value used to judge the result of a significance test [23]. In this paper, $\alpha $ is 0.01. If P < $0.01$, reject the null hypothesis. Otherwise, accept the null hypothesis. The closer the P-value is to zero, the stronger the partial correlation significance.

Equivalently, the process of building the partial correlation analysis model between other parameters is similar to the above process.

2.3.2 Analysis results of coupling degree

To perform the partial correlation analysis effectively, the same digital star sensor as in Section 2.2 is used. Then, 30 groups of simulated calibration results of intrinsic and extrinsic parameters are obtained, including 30 groups of values of ${x_0}$, ${y_0}$, ${\beta _1}$, ${\beta _2}$, ${\varphi _1}$ and ${\varphi _2}$, according to the calibration method based on integrated modeling with intrinsic and extrinsic parameters.

To get more reliable and accurate analysis results, 30 groups of calibration results are further divided 2 groups for the mutual verification of partial correlation analysis. The first 15 groups of data are denoted as Group A, and the last 15 groups of data are denoted as Group B. The PCCs and the 2-tailed significance P-values between the intrinsic parameters and the extrinsic parameters are calculated, and the analysis results are summarized in Table 2.

Table 2. The partial correlation analysis results of Groups A and B

View Table | View all tables in this article

In summary, the following conclusions can be drawn under the above simulation conditions, and the conclusions further verify the analysis results in Section 2.2.

1) ${\beta _1}$ is coupled with ${x_0}$ and ${y_0}$, and the PCCs are -0.988 and 0.993 respectively.
2) ${\beta _2}$ is coupled with ${x_0}$ and ${y_0}$, and the PCCs are -0.992 and -0.994 respectively.
3) ${\varphi _1}$ is only coupled with ${y_0}$, and the PCC is 0.996.
4) ${\varphi _2}$ is only coupled with ${x_0}$, and the PCC is -0.994.

In addition, according to the pinhole imaging theory, the changes of star spot position are closely related to the actual value of ${\beta _1}$ in the case of a certain error of ${\beta _1}$ or ${\beta _2}$, so the partial correlation between ${\beta _1}$ or ${\beta _2}$ and ${x_0}$ or ${y_0}$ is different. However, at least one of ${\beta _1}$ and ${\beta _2}$ is correlated with ${x_0}$ or ${y_0}$ regardless of the value of ${\beta _1}$.

3. Calibration with separation of intrinsic and extrinsic parameters

As can be seen from the analysis in Section 2, there is coupling between intrinsic and extrinsic parameters, especially between ${x_0}$, ${y_0}$ and ${\beta _1}$, ${\beta _2}$, ${\varphi _1}$, ${\varphi _2}$, and the values of PCCs are greater than 0.9. If all the intrinsic and extrinsic parameters are used together, high-precision measurement results can be obtained. However, in typical applications of star sensors, not all parameters are used together, so the parameters need to be separated. In this section, for the applications of star sensors in satellites and missiles, the parameters to be separated are determined, and the corresponding calibration methods are proposed.

3.1 Calibration with partial separation of extrinsic parameters

Generally, an optical cubic prism is installed on the case of star sensor as a reference for its coordinate frame, as shown in Fig. 5. For the star sensor applied to the satellite [24], it is required that the attitude information generated by the star sensor is established in the prism coordinate frame (P). Therefore, the starlight vector in the star sensor coordinate frame (S) must be transformed into the prism coordinate frame (P).

Fig. 5. Structure of the star sensor.

Download Full Size | PDF

3.1.1 Parameters to be separated

The transformation process from frame S to the prism coordinate frame P can be expressed as

(9)$${\textbf{V}_\textbf{p}} = {\textbf{R}_{\textbf{pr}}}\textbf{R}_{\textbf{sr}}^\textbf{T}\textbf{V}_\textbf{s}^{\prime},$$

where ${\textbf{V}_\textbf{p}}$ is the starlight vector in frame P, $\textbf{R}_{\textbf{sr}}^\textbf{T}$ is the transposed matrix of ${\textbf{R}_{\textbf{sr}}}$, which denotes the transformation from frame R to frame S, matrix ${\textbf{R}_{\textbf{pr}}}$ denotes the transformation from frame R to frame P, and $\textbf{V}_\textbf{s}^{\prime}$ is the measured starlight vector in frame S after distortion correction, which is expressed as follows:

(10)$$\textbf{V}_\textbf{s}^{\prime} = \frac{1}{{\sqrt {{{({x_c} - \delta {x_c} - {x_0})}^2} + {{({y_c} - \delta {y_c} - {y_0})}^2} + {F^2}} }}\left[ {\begin{array}{c} { - ({x_c} - \delta {x_c} - {x_0})}\\ { - ({y_c} - \delta {y_c} - {y_0})}\\ F \end{array}} \right],$$

where ${({x_c},{y_c})^T}$ is the coordinate of the measured star spot, $\delta {x_c}$ and $\delta {y_c}$ are the distortion of the imaging spot in the X and Y directions respectively, which can be expressed by distortion coefficients ${q_1}$, ${q_2}$, ${p_1}$ and ${p_2}$.

From Eq. (10), all intrinsic parameters, the extrinsic parameters ${\varphi _1}$, ${\varphi _2}$, ${\varphi _3}$ and the transformation matrix ${\textbf{R}_{\textbf{pr}}}$ are used to calculate the starlight vector $\textbf{V}_\textbf{s}^{\prime}$. The matrix ${\textbf{R}_{\textbf{pr}}}$ can be measured with the assistance of three self-collimation theodolites, so only the remaining parameters need to be determined. To compare with the mathematical model of the star sensor calibration based on the integrated modeling with intrinsic and extrinsic parameters, only ${\beta _1}$ and ${\beta _2}$ are not used among the above parameters in the transformation process of starlight vector from frame R to frame P. Hence, if ${\beta _1}$ and ${\beta _2}$ can be separated accurately in the process of star sensor calibration, that is, the coupling of ${\beta _1}$ and ${\beta _2}$ with other parameters to be eliminated, the accurate starlight vector in frame P can be obtained. Thus, only ${\beta _1}$ and ${\beta _2}$ need to be separated from other parameters in this application.

3.1.2 Separation of${\beta _\textbf{1}}$and${\beta _\textbf{2}}$from other parameters

For the star sensor calibration methods based on the star simulator and high-precision rotary table, a two-axis rotary table is sufficient. To implement the calibration methods with separation of intrinsic and extrinsic parameters, the rotary table used in this paper is a three-axis rotary table, as shown in Fig. 6.

Fig. 6. Sketch of a high-precision three-axis rotary table.

Download Full Size | PDF

Based on the above the high-precision three-axis rotary table and star simulator, when the internal frame is in the zero position, and the rotation angles of the external and middle frames are a1 and b1, respectively, the starlight vector in the star sensor coordinate frame (S) is denoted as ${\textbf{V}_{\textbf{s1}}}$, expressed in Eq. (11), and the star spot position in the focal plane of the star sensor is denoted as ${P_1}$.

(11)$${\textbf{V}_{\textbf{s1}}} = {\textbf{R}_{\textbf{sr}}}{\textbf{R}_{\textbf{r1}}}\textbf{V},$$

where ${\textbf{R}_{\textbf{r1}}}$ is the rotation matrix formed by rotating the external, middle and internal frames by a1, b1 and 0°, $\textbf{V}$ and ${\textbf{R}_{\textbf{sr}}}$, as expressed in Eq. (1), denotes the simulated starlight vector generated by star simulator in the rotary table coordinate frame (R) and the transformation matrix from frame R to frame S, respectively.

Similarly, when the internal frame is in the 90° position, and the rotation angles of the external and middle frames are still a1 and b1, respectively, the starlight vector in frame S is denoted as ${\textbf{V}_{\textbf{s2}}}$, expressed in Eq. (12), and the star spot position is denoted as ${P_2}$.

(12)$${\textbf{V}_{\textbf{s2}}} = {\textbf{R}_{\textbf{sr}}}{\textbf{R}_{\textbf{r2}}}\textbf{V},$$

where ${\textbf{R}_{\textbf{r2}}}$ is the rotation matrix formed by rotating the external, middle and internal frames by a1, b1 and 90°.

If the mounting deviation between frames S and R and the initial alignment error of the star simulator are both zero, that is, ${\textbf{R}_{\textbf{sr}}}$ is an identity matrix and $\textbf{V} = {\left[ {\begin{array}{ccc} 0&0&1 \end{array}} \right]^T}$, ${P_2}$ should coincide with ${P_1}$. In practice, due to the existence of these errors, the two star spot positions do not coincide. However, while keeping the internal frame rotated 90°, the rotation angles of the external and middle frames of the rotary table can be adjusted to make the two star spot positions coincide. Assuming that the rotation angles of the external and middle frames of the rotary table after adjustment are a2 and b2 respectively, the corresponding star spot position is denoted as ${P_3}$, and the starlight vector in frame S is denoted as ${\textbf{V}_{\textbf{s3}}}$, as expressed in Eq. (13).

(13)$${\textbf{V}_{\textbf{s3}}} = {\textbf{R}_{\textbf{sr}}}{\textbf{R}_{\textbf{r3}}}\textbf{V},$$

where ${\textbf{R}_{\textbf{r3}}}$ is the rotation matrix formed by rotating the external, middle and internal frames by a2, b2 and 90°.

Since ${P_3}$ coincides with ${P_1}$, Eq. (14) can be obtained base on the integrated imaging model of the star sensor described in Eq. (1)-(6) on the premise that the intrinsic parameters of the star sensor remain stationary during the rotation of the rotary table.

(14)$${\textbf{V}_{\textbf{s1}}} = {\textbf{V}_{\textbf{s3}}}.$$

Substituting Eq. (11) and (13) into Eq. (14), yields

(15)$${\textbf{R}_{\textbf{sr}}}{\textbf{R}_{\textbf{r1}}}\textbf{V} = {\textbf{R}_{\textbf{sr}}}{\textbf{R}_{\textbf{r3}}}\textbf{V}.$$

Since ${\textbf{R}_{\textbf{sr}}}$ is non-singular, Eq. (16) can be obtained as

(16)$$({\textbf{R}_{\textbf{r1}}} - {\textbf{R}_{\textbf{r3}}})\textbf{V} = {\textbf {0}}.$$

Let $\textbf{A} = {\textbf{R}_{\textbf{r1}}} - {\textbf{R}_{\textbf{r3}}}$, Eq. (16) can be simplified as

(17)$$\textbf{AV} = {\textbf{0}},$$

where the matrix $\textbf{A}$ is non-singular, that is, ${\textbf{R}_{\textbf{r1}}}$ is not equal to ${\textbf{R}_{\textbf{r3}}}$, and $\textbf{V}$ is a unit vector.

Combining Eq. (2), Eq. (17) can be re-expressed as

(18)$$\textbf{A}{\left[ {\begin{array}{ccc} {\cos ({\beta_1})\cos ({\beta_2})}&{\sin ({\beta_1})\cos ({\beta_2})}&{\sin ({\beta_2})} \end{array}} \right]^T} = {\textbf{0}}.$$

Then, ${\beta _1}$ and ${\beta _2}$ can be obtained in MATLAB as

(19)$${\beta _1} = a\tan 2(V(2),V(1)),{\beta _2} = a\sin (V(3)).$$

The rotation data of external, middle and internal frames before and after the rotary table adjustment, that is (a1,b1,0°) and (a2,b2,90°), form a group of separation data of ${\beta _1}$ and ${\beta _2}$ from other parameters. According to the above analysis, the vector $\textbf{V}$ can be calculated only from a group of separation data, but the calculation result is not accurate because of the existence of errors, such as the reading error of the star spot position and so on. To get more accurate and reliable calculation result of vector $\textbf{V}$, more groups of separation data are acquired by varying a1, b1, a2 and b2 in the process of keeping the star point position consistent before and after the rotary table adjustment to optimize the vector $\textbf{V}$. The whole process of separation data acquisition is performed in a dark room to eliminate the interference of stray light.

The optimization process of vector $\textbf{V}$ is presented as follows. Equation (17) can be transformed into the following optimization problem:

(20)$$\min ||{\textbf{AV}} ||,s.t.||\textbf{V} ||= 1.$$

Singular value decomposition (SVD) of A is carried out [26], expressed as follows:

(21)$$\textbf{A} = \textbf{U}\sum {\textbf{Q}^\textbf{T}},$$

where $\textbf{U}$ and $\textbf{Q}$ are unitary matrices, and $\sum$ is a 3×3 diagonal matrix

(22)$$\sum = \left[ {\begin{array}{ccc} {\sqrt {{\lambda_1}} }&0&0\\ 0&{\sqrt {{\lambda_2}} }&0\\ 0&0&{\sqrt {{\lambda_3}} } \end{array}} \right],$$

where $\sqrt {{\lambda _1}}$, $\sqrt {{\lambda _2}}$ and $\sqrt {{\lambda _3}}$ are the singular values of the square matrix $\textbf{A}.$

By substituting Eq. (21) into Eq. (20), it is re-expressed as

(23)$$\min ||{\textbf{AV}} ||= \min \left\|{\textbf{U}\sum {\textbf{Q}^\textbf{T}}\textbf{V}} \right\|= \min \left\|{\sum {\textbf{Q}^\textbf{T}}\textbf{V}} \right\|.$$

Let $\textbf{M} = {\textbf{Q}^\textbf{T}}\textbf{V}$, Eq. (23) can be re-expressed as

(24)$$\min \left\|{\sum \textbf{M}} \right\|,s.t.||\textbf{M} ||= 1.$$

According to Eq. (22), $\sum $ is a 3×3 diagonal matrix, and $\sqrt {{\lambda _1}}$, $\sqrt {{\lambda _2}}$ and $\sqrt {{\lambda _3}}$ are arranged in decreasing order. Thus, the optimal solution of $\textbf{V}$ is obtained at $\textbf{M} = {\left[ {\begin{array}{ccc} 0&0&1 \end{array}} \right]^T}$. Since $\textbf{V} = \textbf{Q} \cdot \textbf{M}$, the optimal solution is the column vector corresponding to the smallest singular value of the unitary matrix $\textbf{Q}$, that is, the last column of the unitary matrix $\textbf{Q}$.

3.1.3 Determination of remaining 10 parameters and ${\textbf{R}_{\textbf{pr}}}$

According to Eq. (9) and (10), after the separation of ${\beta _1}$ and ${\beta _2}$, as along as remaining 10 parameters and ${\textbf{R}_{\textbf{pr}}}$ are determined, the application requirements of the star sensors in satellites can be fully satisfied. Next, the determination methods of them are introduced.

(1) Determination of the remaining 10 parameters
Base on the analysis in Section 3.1.1, the coupling between remaining intrinsic and extrinsic parameters has no effect on the accurate acquisition of the starlight vector in the prism coordinate frame (P), so they can be optimized as a whole with the assistance of calibration data. The calibration data, including calibration point data and star spot data, are acquired by using the rotary table, star simulator and a star sensor to be calibrated in a dark room. The calibration point data is composed of the rotation angles of the external and internal frames of the rotary table. The rotary table is rotated at a certain angle so that the star spots are evenly distributed in the focal plane of the star sensor, and the star spot data can be obtained by averaging the coordinate information collected 11 times at each position.
(2) Determination of ${\textbf{R}_{\textbf{pr}}}$
${\textbf{R}_{\textbf{pr}}}$ denotes the transformation from frame R to frame P. The determination schematic of ${\textbf{R}_{\textbf{pr}}}$ is shown in Fig. 7, with the assistance of three self-collimation theodolites. The rotary table prism is used as a reference for the frame R. ${\textbf{v}_\textbf{1}}$ is the normal vector of the rotary table prism measured by theodolite No.1 after the external frame of the rotary table is rotated by an arbitrary angle from the zero position. The angle is 65° in our experiment. ${\textbf{v}_2}$ is the normal vector of the rotary table prism measured by theodolite No.2 after the external frame of the rotary table is rotated by another angle from the zero position. The angle is 115° in our experiment. The measurement results are unified in the theodolite No. 2 coordinate frame (E) by mutual alignment between theodolites. Then, a mutually orthogonal set of three vectors can be created with the assistance of the direction vectors of ${\textbf{v}_\textbf{1}}$ and ${\textbf{v}_2}$ to obtain the transformation matrix ${\textbf{R}_{\textbf{er}}}$ from frame R to frame E, based on the Triad algorithm [27].

Fig. 7. Sketch of ${\textbf{R}_{\textbf{pr}}}$ determination.

Download Full Size | PDF

Similarly, ${\textbf{v}_3}$ is the normal vector of the star sensor prism measured by theodolite No.3 in the first position when the middle frame of the rotary table is in the zero position. ${\textbf{v}_4}$ is the normal vector of the star sensor prism measured by theodolite No.3 in the second position after the middle frame of the rotary table is rotated 90° from the zero position. Finally, the matrix ${\textbf{R}_{\textbf{pr}}}$ can be obtained with the assistance of the transformation matrix ${\textbf{R}_{\textbf{er}}}$ and the direction vectors of ${\textbf{v}_3}$ and ${\textbf{v}_4}$, based on the Triad algorithm.

3.2 Calibration with full separation of extrinsic parameters

For the star sensor applied to the missile [25], as for the star sensor applied to the satellite, the accurate acquisition of attitude information in frame P is required. The difference is that the deviation between the reference prism and the image sensor also needs to be known exactly. Because the lens hood is installed separately from the star sensor body for some missiles, as shown in Fig. 8. For the small FOV (typically 3°∼ 5°) in this application, the deviation between the lens hood and the image sensor should be as small as possible to guarantee effective shading. Because the optical cubic prism is observable from outside, the lens hood can be installed strictly aligned with the prism coordinate frame (P). Therefore, to minimize the deviation between the image sensor and the prism has the same effect as to minimize the deviation between the image sensor and the lens hood. Consequently, not only the starlight vector in frame P need to be acquired, but also the deviation between the reference prism and the image sensor also needs to be known exactly in this application.

Fig. 8. Assembly sketch of the missile-borne star sensor.

Download Full Size | PDF

3.2.1 Parameters to be separated

As described in section 3.1.1, to determine starlight vector in frame P, the unused extrinsic parameters ${\beta _1}$ and ${\beta _2}$ need to be separated from other parameters. Beyond that, the deviation between the reference prism and the image sensor need to be known exactly in this application, that is, ${\textbf{R}_{\textbf{sr}}}$ needs to be obtained individually and accurately. From Eq. (4), ${\textbf{R}_{\textbf{sr}}}$ is determined by the extrinsic parameters ${\varphi _1}$, ${\varphi _2}$ and ${\varphi _3}$. Therefore, not only the unused parameters ${\beta _1}$ and ${\beta _2}$ need to be separated, but also the individually used parameters ${\varphi _1}$, ${\varphi _2}$ and ${\varphi _3}$need to be separated from other parameters.

Separation of ${\beta _1}$ and ${\beta _2}$ can be done as described in Section 3.1.2. According to the analysis in Sections 2.2 and 2.3, ${\varphi _1}$ is strongly coupled with ${y_0}$ and ${\varphi _2}$ is strongly coupled with ${x_0}$, so ${x_0}$ and ${y_0}$ need to be separated first. Then, ${\varphi _1}$, ${\varphi _2}$ and ${\varphi _3}$ are separated subsequently.

3.2.2 Separation of ${x_\textbf{0}}$ and ${y_\textbf{0}}$ from other parameters

As shown in Fig. 9, the laser of the theodolite passes through the cross reticle and is reflected to the focal plane of the star sensor by a beam splitter at 45° to the boresight. The theodolite is adjusted, and the light reflected from the focal plane is imaged in the field of view of the eyepiece of the theodolite through a beam splitter. When the aiming cross is completely coincident with the reflected cross image, the star sensor acquires multiple pictures. Under the premise that the collimation axis of the theodolite is consistent with the optical axis of the star sensor, the imaging position of the cross in the focal plane is considered as the principal point position [16].

Fig. 9. Schematic diagram of principal point measurement.

Download Full Size | PDF

3.2.3 Separation of ${\varphi _\textbf{1}}$, ${\varphi _\textbf{2}}$ and ${\varphi _\textbf{3}}$ from other parameters

Also based on the analysis conclusions in Sections 2.2 and 2.3, it is verified that the focal length F is not coupled with ${\varphi _1}$, ${\varphi _2}$ and ${\varphi _3}$, so they can be optimized as a whole with the assistance of calibration data. To facilitate the optimization calculation, the star spot data located in the central area of the focal plane and the corresponding calibration point data are used because optical distortion is negligible at the central area, so optimization of the distortion coefficients is not involved in this case. Moreover, the optimized values of ${\varphi _1}$, ${\varphi _2}$ and ${\varphi _3}$ are used as the initial values, which are further optimized by using the calibration points in the full field of view range.

4. Experiments and verification

Experiments are conducted in the laboratory to verify the proposed method in Section 3.1, and the laboratory setup is shown in Fig. 10. The star sensor used to verify the calibration method with partial separation of extrinsic parameters is the one used for satellite platforms, which has an approximate ${15^ \circ } \times {15^ \circ }$ FOV and 42 mm focal length. The star sensor used to verify the calibration method with full separation of extrinsic parameters is the one used for missile platforms, which has an approximate ${5^ \circ } \times {5^ \circ }$ FOV and 90 mm focal length. Both of these star sensors use an image sensor with 2048 pixels × 2048 pixels size and 5.5µm × 5.5µm pixel size.

Fig. 10. Laboratory setup: (a) Star simulator and high-precision three-axis rotary table. (b) Self-collimation theodolite.

Download Full Size | PDF

4.1 Verification of the calibration method with partial separation

The calibration method based on integrated modeling with intrinsic and extrinsic parameters and our proposed method are denoted as method 1 and method 2, respectively, for simplicity.

(1) Determination of ${\beta _1}$ and ${\beta _2}$
The experimental results for separating ${\beta _1}$ and ${\beta _2}$ based on the method described in Section 3.1.2 are shown in Table 3.
(2) Determination of the remaining 10 parameters
Taking the values of ${\beta _1}$ and ${\beta _2}$ determined in step 1 as fixed values, the results obtained by optimizing the remaining 10 parameters, and the final calibration accuracy are shown in Table 4. At the same time, they are compared with those obtained by method 1.
(3) Determination of ${\textbf{R}_{\textbf{pr}}}$

Table 3. Experimental results for separation of ${\beta _\textbf{1}}$ and ${\beta _\textbf{2}}$

View Table | View all tables in this article

Table 4. Comparison of optimization values of remaining parameters and calibration accuracy

View Table | View all tables in this article

According to the measurement results of theodolites and the method described in Section 3.1.3, ${R_{pr}}$ can be obtained as

(25)$${R_{pr}}\textbf{ = }\left[ {\begin{array}{ccc} {0.999999}&{\textbf{ - }0.000723}&{\textbf{ - }0.116020}\\ {0.000725}&{0.999904}&{0.028674}\\ {0.116016}&{\textbf{ - }0.028682}&{0.999916} \end{array}} \right].$$

Finally, the accurate starlight vector in the prism coordinate frame (P) can be obtained based on the above experimental results.

To verify the above conclusion, a self-collimation star simulator with 700" ${\times} $ 700” FOV is used to measure the starlight vector in frame P directly. The self-collimation laser of the star simulator is parallel to the starlight vector generated, and the measurement results are compared with the results obtained by the calibration method with partial separation of extrinsic parameters. At the same time, they are compared with those obtained by method 1, as shown in Table 5.

Table 5. Experimental results of starlight vector determination in frame P

View Table | View all tables in this article

As shown in Table 4, the accuracy of parameters optimization, that is the calibration accuracy, of the proposed method is comparable to that of the integrated model calibration method. However, it can be seen from Table 5 that the accuracy of the starlight vector in frame P obtained by the proposed method is higher than that obtained by the integrated model calibration method. Therefore, our proposed method improves the measurement accuracy of the star sensor when it is used on the satellite platform.

4.2 Verification of the calibration method with full separation

Similarly, the calibration method based on integrated modeling with intrinsic and extrinsic parameters and our proposed method are denoted as method 1 and method 2, respectively, for simplicity.

(1) Determination of ${\beta _1}$ and ${\beta _2}$
Using the same method as in Section 4.1, the values of ${\beta _1}$ and ${\beta _2}$ are determined to be -103.263° and 89.374°, respectively.
(2) Determination of ${\textbf{R}_{\textbf{pr}}}$
Using the same method as in Section 4.1, ${\textbf{R}_{\textbf{pr}}}$ are determined to be
$(26)$${R_{pr}}\textbf{ = }\left[ {\begin{array}{ccc} {0.999999}&{0.001131}&{\textbf{ - }0.000528}\\ {\textbf{ - }0.001138}&{0.999913}&{\textbf{ - }0.013157}\\ {0.000513}&{0.013158}&{0.999913} \end{array}} \right].$$$
(3) Determination of ${x_0}$ and ${y_0}$
The cross image obtained by the star sensor for determining ${x_0}$ and ${y_0}$ is shown in Fig. 11. According to Fig. 11, the determined ${x_0}$ and ${y_0}$ are 995.9 pixels and 1065.5 pixels, respectively.
(4) Determination of ${\varphi _1}$, ${\varphi _2}$ and ${\varphi _3}$

Fig. 11. Cross image.

Download Full Size | PDF

Take ${\beta _1}$, ${\beta _2}$, ${x_0}$ and ${y_0}$ determined in steps 1 and 3 as fixed values, and set distortion coefficients to 0. The optimized values of F, ${\varphi _1}$, ${\varphi _2}$ and ${\varphi _3}$ are obtained by using the calibration points in the ${0.5^ \circ } \times {0.5^ \circ }$ FOV. Moreover, the optimized values are used as the initial values for further optimization. The parameter optimization results and the final calibration accuracy are obtained by using the calibration points in the full field of view range. At the same time, they are compared with those obtained by method 1, as shown in Table 6.

Table 6. Comparison of optimization values of parameters and calibration accuracy

View Table | View all tables in this article

Finally, the accurate starlight vector in frame P can be obtained based on the above experimental results. At the same time, the mounting deviation between the prism and the star sensor can be determined accurately.

To verify the above conclusion, a coordinate measuring machine (CMM) is used to measure the transformation relationship between the star sensor coordinate frame (S) and the prism coordinate frame (P) directly. The measurement results are compared with those obtained by the calibration method with full separation of extrinsic parameters. At the same time, they are compared with those obtained by method 1, as shown in Table 7.

Table 7. Experimental results of the transformation relationship determination between frames S and P

View Table | View all tables in this article

As shown in Table 6, the accuracy of parameters optimization, that is the calibration accuracy, of the proposed method is comparable to that of the integrated model calibration method. At the same time, it can be seen from Table 7 that the mounting deviation between the prism and the star sensor can be determined more accurately by this method, thereby ensuring the control of the mounting deviation when the star sensor is used on the missile platform and the hood is separated from the star sensor. In addition, the experimental results further verify the analysis results in Section 2.

5. Conclusions

This study focuses on the decoupling of intrinsic and extrinsic parameters for the star sensor calibration. First, the coupling form and degree of the parameters based on integrated modeling with intrinsic and extrinsic parameters are acquired effectively. Thereafter, two calibration methods are proposed based on the applications of star sensors in satellites and missiles. For the calibration with partial separation of extrinsic parameters, the transformation of different reference frames is taken full account of and the extrinsic parameters which have a significant impact on the whole transformation are separated with the assistance of a high-precision three-axis rotary table and a star simulator. For the calibration with full separation of extrinsic parameters, the extrinsic parameters coupled with the intrinsic parameters are separated individually with the assistance of self-collimation theodolites, and the intrinsic and extrinsic parameters without coupling are optimized nonlinearly as a whole. Last, the experimental results verify the validity of the proposed method. The calibration method with partial separation of extrinsic parameters applied to the satellite-borne star sensor can improve the measurement accuracy of the starlight vector in its prism coordinate frame. The calibration method with full separation of extrinsic parameters applied to the missile-borne star sensor can not only ensure the measurement accuracy of the starlight vector, but also can determine the transformation matrix between the star sensor coordinate frame and the prism coordinate frame with high precision to ensure the control of the mounting deviation between the hood and the main body of the missile-borne star sensor. The methods proposed in this paper have been successfully applied to practical products.

Funding

National Natural Science Foundation of China (61605007); National Key Research and Development Program of China (2019YFA0706002).

Disclosures

The authors declare that there are no conflicts of interest related to this article.

References

1. C. C. Liebe, “Star trackers for attitude determination,” IEEE Trans. Aerosp. Electron. Syst. 10(6), 10–16 (1995). [CrossRef]

2. C. C. Liebe, “Accuracy performance of star trackers-a tutorial,” IEEE Trans. Aerosp. Electron. Syst. 38(2), 587–599 (2002). [CrossRef]

3. D.S. Anderson, “Autonomous star sensing and pattern recognition for spacecraft attitude determination doctoral thesis,” Ph.D. dissertation (Texas A&M University, 1991).

4. G. Zhang, Star Identification (Springer, 2017), Chap. 3.

5. W. Quan, J. Fang, and W. Zhang, “A method of optimization for the distorted model of star map based on improved genetic algorithm,” Aerosp. Sci. Technol. 15(2), 103–107 (2011). [CrossRef]

6. H. Liu, X. Li, and J. Tan, “Novel approach for laboratory calibration of star tracker,” Opt. Eng. 49(7), 073601 (2010). [CrossRef]

7. Q. Fan, G. Zhang, and X. Wei, “Sun sensor calibration based on exact modeling with intrinsic and extrinsic parameters,” J. Beijing Univ. Aeronaut. Astronaut. 37(10), 1293–1297 (2011).

8. Y. Li, J. Zhang, W. Hu, and J. Tian, “Laboratory calibration of star sensor with installation error using a nonlinear distortion model,” Appl. Phys. B: Lasers Opt. 115(4), 561–570 (2014). [CrossRef]

9. X. Wei, G. Zhang, Q. Fan, J. Jiang, and J. Li, “Star sensor calibration based on integrated modeling with intrinsic and extrinsic parameters,” Measurement 55(9), 117–125 (2014). [CrossRef]

10. K. Xiong, X. Wei, G. Zhang, and J. Jiang, “High-accuracy star sensor calibration based on intrinsic and extrinsic parameter decoupling,” Opt. Eng. 54(3), 034112 (2015). [CrossRef]

11. F. Zhou, T. Ye, X. Chai, X. Wang, and L. Chen, “Novel autonomous on-orbit calibration method for star sensors,” Opt. Laser. Eng. 67, 135–144 (2015). [CrossRef]

12. C. Zhang, Y. Niu, H. Zhang, and J. Lu, “Optimized star sensors laboratory calibration method using a regularization neural network,” Appl. Opt. 57(5), 1067–1074 (2018). [CrossRef]

13. L. Blarre, J. Ouaknine, L. Oddosmarcel, and P. Martinez, “High accuracy sodern star trackers: recent improvements proposed on SED36 and HYDRA star trackers,” in 7th American Institute of Aeronautics and Astronautics Guidance, Navigation, and Control Conference and Exhibit (AIAA GNC) (2006), 6046.

14. U. Schmidt, C. Elstner, and K. Michel, “ASTRO 15 star tracker flight experience and further improvements towards the ASTRO APS star tracker,” in 9th American Institute of Aeronautics and Astronautics Guidance, Navigation and Control Conference and Exhibit (AIAA GNC) (2008), 6649.

15. W. H. Steyn, M. J. Jacobs, and P J Oosthuizen, “A high performance star sensor system for full attitude determination on a microsatellite,” in 20th Annual AAS Guidance and Control Conference (Annual AAS GC) (1997), pp. 1–6.

16. T. Sun, F. Xing, and Z. You, “Optical system error analysis and calibration method of high-accuracy star trackers,” Sensors 13(4), 4598–4623 (2013). [CrossRef]

17. W. Faig, “Calibration of close-range photogrammetry systems: Mathematical formulation,” Photogramm. Eng. Rem. S. 41(12), 1479–1486 (1975).

18. J. Weng, P. Cohen, and M. Herniou, “Camera calibration with distortion models and accuracy evaluation,” IEEE Trans. Pattern Anal. Machine Intell. 14(10), 965–980 (1992). [CrossRef]

19. R. Y. Tsai, “A versatile camera calibration technique for high-accuracy 3D machine vision metrology using off-the-shelf TV cameras and lenses,” IEEE J. Robot. Automat. 3(4), 323–344 (1987). [CrossRef]

20. D. C. Brown, “Close-range camera calibration,” Photogramm. Eng. 37, 855–866 (1971).

21. K. Liu, “Measurement error and its impact on partial correlation and multiple linear regression analyses,” Am. J. Epidemiol. 127(4), 864–874 (1988). [CrossRef]

22. J. Wang, Partial Correlation Coefficient (Springer, 2013), Chap. 2.

23. C. A. Adery, “A simplified monte carlo significance test procedure,” J. R. Stat. Soc. B 30(3), 582–598 (1968). [CrossRef]

24. K. Xiong, C. Wei, and L. Liu, “Autonomous navigation for a group of satellites with star sensors and inter-satellite links,” Acta Astronaut. 86(3), 10–23 (2013). [CrossRef]

25. J. Yang, L. Kong, and K. Xiong, “Study on the fast optimal direction determination for missile-borne star sensor,” Proc. SPIE 9042, 904209 (2013). [CrossRef]

26. A. Hoecker and V. Kartvelishvili, “SVD approach to data unfolding,” Nucl. Instrum. Methods Phys. Res. A. 72(3), 469–481 (1996). [CrossRef]

27. M. D. Shuster and S. D. Oh, “Three-axis attitude determination from vector observations,” J. Guid. Control 4(1), 70–77 (1981). [CrossRef]

Parameter	Value	Parameter	Value	Parameter	Value
Field of view	18° × 18°	F	35.5591 mm	$φ_{3}$	1°
Pixel resolution	2048 pixels × 2048 pixels	$β_{1}$	45°	$q_{1}$	-2.00e-05
Pixel size	0.0055 mm × 0.0055 mm	$β_{2}$	89°	$q_{2}$	-1.00e-06
$x_{0}$	1022 pixels	$φ_{1}$	1°	$p_{1}$	1.00e-05
$y_{0}$	1030 pixels	$φ_{2}$	1°	$p_{2}$	2.00e-05

Group	Initial data (°)			Adjustment data (°)			$β_{1}$ (°)	$β_{2}$ (°)
Group	External	Middle	Internal	External	Middle	Internal	$β_{1}$ (°)	$β_{2}$ (°)
1	0	0	0	1.6433	-1.8220	90	-92.953	88.265
2	0.5	0	0	1.6451	-1.3210	90	-92.906	88.265
3	0	0.5	0	1.1452	-1.8205	90	-92.899	88.265
Result							-92.919	88.265

Parameter	$x_{0}$ (pixels)	$y_{0}$ (pixels)	F (mm)	$φ_{1}$ (°)	$φ_{2}$ (°)	$φ_{3}$ (°)	Accuracy (X/Y) (“)
method 1	1013.1	1033.9	42.73	0.4288	0.3398	0.1518	method 1: (1.35/1.30)
method 2	1011.4	1035.5	42.73	0.4960	0.2717	0.1520
Parameter	$β_{1}$ (°)	$β_{2}$ (°)	$q_{1}$	$q_{2}$	$p_{1}$	$p_{2}$
method 1	86.803	91.734	-1.796e-6	-1.905e-7	1.353e-5	3.604e-6	method 2: (1.33/1.30)
method 2	—	—	-1.896e-6	-1.877e-7	1.604e-5	3.086e-6	method 2: (1.33/1.30)

self-collimation star simulator (Measured value)		Starlight vector in frame P (method 1)		Starlight vector in frame P (method 2)
X (“)	Y (“)	X (“)	Y (“)	X (“)	Y (“)
0.36	0.63	2.29	2.58	0.84	1.16
-171.26	0.56	-175.43	3.82	-173.05	1.67
-345.37	0.47	-349.91	4.63	-347.34	2.29
172.15	-0.71	168.58	-4.05	170.62	-1.84
345.58	-0.54	341.37	-4.89	343.73	-2.38
0.47	172.94	3.64	176.33	1.65	174.28
0.68	344.08	4.65	348.98	2.57	346.02
-0.55	-173.57	-4.26	-170.21	-2.16	-172.29
-0.62	-345.95	-4.91	-341.37	-2.58	-344.16

Parameter	$x_{0}$ (pixels)	$y_{0}$ (pixels)	F (mm)	$φ_{1}$ (°)	$φ_{2}$ (°)	$φ_{3}$ (°)	Accuracy (X/Y) (“)
method 1	1024	1024	90.145	0.5558	-0.7563	-0.0334	method 1: (0.41/0.31)
method 2	—	—	90.142	0.7506	-0.0453	-0.0361
Parameter	$β_{1}$ (°)	$β_{2}$ (°)	$q_{1}$	$q_{2}$	$p_{1}$	$p_{2}$
method 1	36.43	90.94	4.834e-5	-5.492e-7	6.710e-6	4.147e-6	method 2: (0.43/0.39)
method 2	—	—	4.898e-5	-5.274e-7	-4.862e-6	2.686e-5	method 2: (0.43/0.39)

Star sensor calibration with separation of intrinsic and extrinsic parameters

Abstract

1. Introduction

2. Coupling analysis of intrinsic and extrinsic parameters

2.1 Integrated modeling with intrinsic and extrinsic parameters

2.2 Coupling form analysis

2.3 Coupling degree analysis

2.3.1 Partial correlation analysis model

2.3.2 Analysis results of coupling degree

3. Calibration with separation of intrinsic and extrinsic parameters

3.1 Calibration with partial separation of extrinsic parameters

3.1.1 Parameters to be separated

3.1.2 Separation of${\beta _\textbf{1}}$and${\beta _\textbf{2}}$from other parameters

3.1.3 Determination of remaining 10 parameters and ${\textbf{R}_{\textbf{pr}}}$

3.2 Calibration with full separation of extrinsic parameters

3.2.1 Parameters to be separated

3.2.2 Separation of ${x_\textbf{0}}$ and ${y_\textbf{0}}$ from other parameters

3.2.3 Separation of ${\varphi _\textbf{1}}$, ${\varphi _\textbf{2}}$ and ${\varphi _\textbf{3}}$ from other parameters

4. Experiments and verification

4.1 Verification of the calibration method with partial separation

4.2 Verification of the calibration method with full separation

5. Conclusions

Funding

Disclosures

References

Cited By

Figures (11)

Tables (7)

Equations (26)

Optics Express

		$x_{0}$		$y_{0}$
		PCC	P-value	PCC	P-value
$β_{1}$	A	-0.987	0.000	0.993	0.000
	B	-0.989	0.000	0.993	0.000
	Average	-0.988	0.000	0.993	0.000
$β_{2}$	A	-0.988	0.000	-0.992	0.000
	B	-0.995	0.000	-0.995	0.000
	Average	-0.992	0.000	-0.994	0.000
$φ_{1}$	A	-0.155	0.631	0.995	0.000
	B	-0.421	0.173	0.997	0.000
	Average	-0.288	0.402	0.996	0.000
$φ_{2}$	A	-0.992	0.000	0.242	0.449
	B	-0.995	0.000	-0.562	0.057
	Average	-0.994	0.000	-0.160	0.253

	Rotation angle about	Rotation angle about	Rotation angle about
	X axis (arcmins)	Y axis (arcmins)	Z axis (arcmins)
CMM	0.34	0.89	-1.85
method 1	11.75	43.65	-2.04
method 2	0.04	0.93	-1.72