3.1.

Focal Plane

The LAMOST focal plane is actually a spherical crown with a diameter of 1.75 m and a curvature radius of $\sim 19.88\mathrm{m}$; a total of 4000 fiber positioners distributed on the focal plane are shown in Fig. 2. Each fiber positioner has a double gyration mechanism, as shown in Fig. 3. The position range of the fiber is 33 mm, allowing the telescope to observe and record the spectra of 4000 celestial targets simultaneously. The fiber positioner requires several steps to locate the target position in the corresponding detection area because of the mechanical drive error. Position feedback and open-loop compensation are used to compensate for the positioning error. The accuracy of compensation value is determined by the accuracy of the fiber position measurement.

Fig. 2

(a) LAMOST focal surface and (b) fiber positioner.

Fig. 3

Double gyration mechanism.

The $Z$ axis of the focal plane’s world coordinate frame is the normal direction at point $O$, and the $X\text{-}Y$ plane coincides with the tangent plane at the $O$ point. Let ${\delta }_Z$ represent the error of the fiber in the $Z$ direction, ${\delta }_{XY}$ represent the error in the $X\text{-}Y$ plane, and ${\delta }_S$ represent the error on the spherical surface of the focal plane. According to the geometric relationship shown in Fig. 4, the following equation is obtained. Given that angle $\varphi \approx 0$, the positioning error of fiber ${\delta }_S$ is mainly reflected in ${\delta }_{XY}$, and $Z$ direction error ${\delta }_Z$ is negligible:²⁰

Fig. 4

Focal plane error model.

Due to the limit of the hardware system, it is difficult for a camera to cover the whole field of view. At present, the focal plane is usually divided into many smaller subfocal planes (SFP), each equipped with its own measuring camera.²¹ According to the shape of the focal surface, the relation between the flatness error and the SFP’s radius is shown in Fig. 5. The geometrical shape of a SFP with a smaller area can be approximated as a plane, that is to say, multiple fibers are distributed in the SFP discretely.

Fig. 5

Focal plane flatness error.

3.2.

Construction Model of EPT

The process of constructing the EPT is revealed in Fig. 6. The EPT is installed on the SFP in parallel, and the $Z$ axis of the EPT is defined to coincide with the $Z$ axis of the SFP. Let $p_i$ and $q_i$ represent the $i$’th feature point on the EPT and its corresponding image point, respectively, and $d_j$ and $b_j$ represent the $j$’th feature point on the SFP and its corresponding image point, respectively. $p_i(x_{wTi},y_{wTi},z_{wTi},1)$ represents the homogeneous coordinates of the $i$’th feature points in the EPT’s world coordinate frame (TWCF), and $d_j(x_{wFj},y_{wFj},z_{wFj},1)$ represents the homogeneous coordinates of the $j$’th feature points in the SFP’s world coordinate frame (FWCF). $q_i(u_i,v_i,1)$ and $b_j(u_j,v_j,1)$ are homogeneous coordinates of image points in the pixel coordinate frame corresponding to $p_i$ and $d_j$, respectively. ${\tilde{d}}_j({\tilde{x}}_{wTj},{\tilde{y}}_{wTj},{\tilde{z}}_{wTj},1)$ denotes the homogeneous coordinates of the $j$’th feature points on the SFP in the TWCF.

Fig. 6

Process of constructing the EPT.

According to Eq. (1), we obtain

As mentioned in Sec. 1, the focal plane lacks high-precision internal control points, which means that the parameters $[\begin{array}{cc}{R}_F& t_F\end{array}]$ are inaccurate. So, it is difficult to get the exact fiber world coordinates from the pixel coordinates from Eq. (5). Because the position of each feature point on the EPT plane was accurate and known, the parameters $[\begin{array}{cc}{R}_T& t_T\end{array}]$ of the camera coordinate frame to TWCF can be calculated by the space resection method. FWCF and TWCF are parallel to each other in the $X\text{-}Y$ direction, and their $Z$ axis coincides. So, ${\tilde{d}}_j$ coordinates are calculated from the following equation:

Here, $\mathrm{\Delta }z$represents the theoretical distance between WFCF and TWCF in the $Z$ direction. According to Eqs. (6) and (4), precise coordinates of the feature point of the SFP are obtained. However, the EPT hardly parallels with SPF completely. That is to say, the feature points on the EPT plane act as constraints only in the $X\text{-}Y$ direction, so the effect of $\mathrm{\Delta }z$’s error on measurement accuracy must be evaluated. Let $d_{cj}(x_{cj},y_{cj},z_{cj})$ represent the $j$’th points in the camera coordinate system, with coordinates that are calculated as

Here, $\mathrm{\Delta }{z}_c$ represents $\mathrm{\Delta }z$ in the camera coordinates system, and as the collinear equation of the projection, the relation between $b_j(u_j,v_j,1)$ and $d_{cj}(x_{cj},y_{cj},z_{cj})$ is as follows:

Here, $z_{cj}$ denotes that the distance between the camera and the SFP is more than 500 cm. We noticed that $\mathrm{\Delta }{z}_c=\mathrm{\Delta }{z}^{\prime }+\delta z_1+\delta z_2$. $\mathrm{\Delta }{z}^{\prime }$ is the theoretical design value. $\delta z_1$ represents the mechanical assembly error of the EPT and focal plane in the $Z$ axis, which is usually <0.1 mm. $\delta z_2$ denotes the flatness error of the SFP, which is at most about 2 mm. Therefore, the pixel error due to $\mathrm{\Delta }{z}_c$ in Eq. (8) is almost 0, which means it only has a tiny effect on the measurement accuracy in the $X\text{-}Y$ direction of the SFP.

4.1.

Bundle Adjustment

In computer vision, bundle adjustment is a key part of the feature-based 3D reconstruction algorithm, which comes down to minimizing the reprojection error between observation and prediction image points to optimize the camera parameters and 3D coordinates of the feature points. So multiview reconstruction and bundle adjustment are adopted to improve the measurement accuracy.²² In the $i$'th image, $d_{ij}$ denotes the homogeneous coordinate of the $j$’th feature point on the SFP in the FWCF, and ${\tilde{d}}_{ij}$ denotes the homogeneous coordinate of the same feature point in the TWCF. The image coordinate given by the camera, $b_{ij}$, is different from the image coordinate obtained from the perspective projection model, ${\tilde{b}}_{ij}$, which is given as

Here, $\mathrm{\Delta }T=[\begin{array}{ccc}0& 0& \mathrm{\Delta }z\end{array}{]}^{\prime }$. The LAMOST pays more attention to the positioning accuracy of the $X\text{-}Y$ direction on the focal plane, so we add external control points as $X\text{-}Y$ position constraints on the EPT plane. $p_{ij}$ denotes the coordinate of the $j$’th feature point on the EPT plane in the TWCF. The image coordinate given by the camera, $q_{ij}$, is different from the image coordinate obtained from the perspective projection model, ${\tilde{q}}_{ij}$, which is given as

The objective function of the nonlinear optimization is established as follows:

4.2.

Initial Value Estimation

The bundle adjustment is a nonlinear optimization iterative process, and a better initial value can avoid the solution falling into local optimization. The initial value of the intrinsic camera parameters is obtained using Zhang’s algorithm²³ and a precision calibration template. The relatively accurate extrinsic camera parameters can be determined by space resection using the internal control points in the field of view. The rotation angles of the center rotation axis and the eccentric rotation axis of the fiber positioner are calculated in advance. Considering the mechanical transmission error, the distance between the fiber’s actual and theoretical positions is usually $<200\mu \mathrm{m}$. Therefore, the theoretical position of the fiber can be considered to be the initial position of the fiber tip on the focal plane. The fiber’s initial $X\text{-}Y$ coordinate in the FWCF can be taken as the design value, and the initial coordinate of $Z$ requires consideration of the vertical distance between the EPT and SFP after assembly.

4.3.

Measurement Process

The overall process for the fiber position measurement method based on the EPT proposed in this paper is shown in Fig. 7. First, the camera is calibrated to get the initial value of the intrinsic camera parameters; second, the EPT is assembled on the SFP and the camera captures $m$ pieces of images from different positions and directions; third, the initial values of the extrinsic camera parameters are calculated through the pyramid method²⁴ and then optimized through the space resection method; and finally, all parameters including the initial value of the intrinsic parameters ($A$), extrinsic camera parameters $[\begin{array}{cc}{R}_T& t_T\end{array}]$, lens distortion parameters ($K$), and fibers position$(d_{ij})$ are calculated using the bundle adjustment.

Fig. 7

Flowchart of the measurement process.

5.1.

Simulation Experiments

To test the reliability and accuracy of the proposed measurement method, simulation experiments are carried out with the following settings. The intrinsic parameter matrix is $A=\left[\begin{array}{cccc}2968.37& 0& 539.37& 0\\ 0& 2968.12& 490.96& 0\\ 0& 0& 1& 0\end{array}\right]$; furthermore, the distortion coefficients are $k_1=0.00261$, $k_2=-0.33046$, $k_3=-0.00096$, $k_4=-0.00068$, and $k_5=0$. These parameters are used as initial values for optimization.

As shown in Fig. 8, four identical small calibration templates ($A,B,C,D$) are assembled on top of the large board. The feature points of the small calibration templates are set to be measuring points, and the feature points on the large board are measured by a universal toolmaker’s microscope as the external control points. The measurement error of the universal toolmaker’s microscope is $1\mu \mathrm{m}$. Gaussian noise with mean and standard deviation of $200\mu \mathrm{m}$ was added to the initial position of the measuring points, as shown in Fig. 9. The bundle adjustment was used to optimize the coordinates of the measuring points.

Fig. 8

EPT in simulation experiment.

Fig. 9

Initial position error.

The real distance between feature points of the small calibration template is 5.000 mm. So, the distance of the measuring points in the $X\text{-}Y$ plane is selected as the measured value to verify the accuracy of the result. To avoid repetition, only the distance between a point and its neighbors on the left and below are calculated, amounting to 336 data points. Then, an error statistical histogram of all of the measured data points is established, as shown in Fig. 10. The mean error was 0.001 mm, and the standard deviation was 0.0063 mm, with most of the position errors conforming to the accuracy requirement ($40\mu \mathrm{m}$) and 96% of measuring points having a position error of $<0.01\mathrm{mm}$. Obviously, the $Z$-direction’s assembly error has little effect on the measurement accuracy in the $X\text{-}Y$ direction. Another similarly sized calibration template is used for comparative experiments with internal control points, and the real distance between feature points is 6.250 mm. Four feature points at the four corners are used as internal control points. The same initial parameters are used for bundle adjustment. Figure 11 indicates that measurement accuracy improves very little using the internal control points. However, the internal control points are difficult to obtain in practice; thus the measurement method based on the EPT is flexible and has high precision.

Fig. 10

Results with external control points.

Fig. 11

Results with internal control points.

At least three control points are required to calculate the initial value of the extrinsic camera parameters by the pyramid method, and we range the number of control points from 3 to 9. We conclude from the results shown in Fig. 12 that the mean value has a slightly decreasing trend with the increase of the number of control points. However, increasing the number of control points will affect the spectroscopic observation efficiency. In the real data experiments with fibers, to distribute the control points evenly on the focal plane, five control points were placed on the EPT plane: four at corners and one at the center.

Fig. 12

Results of the number of control points.

5.2.

Real Data Experiments

The small size focal plane processed by CNC was applied to the real data experiments, and the size was also $120\times 120\mathrm{mm}$. All measuring fibers were installed on the focal plane, and the position of each fiber on the focal plane was measured by a universal toolmaker’s microscope, which was used as the real position. The initial position errors of all measuring fibers are a Gaussian distribution with the mean and standard deviation of $200\mu \mathrm{m}$. Five fibers were installed on the EPT as external control points, and their position coordinates were measured by a universal toolmaker’s microscope. The experimental platform is illustrated in Fig. 13. Because the diameter of the fiber is micron in scale, extracting the image feature of fiber under natural light will add significant error to the measurement result. Taking advantage of the ability of fibers to transmit light, the fiber position measurement with the help of backward lighting is feasible in the dark environment, in which the fibers transfer the backward lighting to the measured fiber tips to form a light spot and the camera takes the light spot images from different directions. The experiment results are shown in Fig. 14. Most of the fiber position errors are $<20\mu \mathrm{m}$. Because of the damage of the fiber end surface, the quality of the spot is affected, and the position error of a few fibers is nearly $40\mu \mathrm{m}$.

Fig. 13

(a) Fiber real position detection and (b) EPT in real data experiment.

Fig. 14

Results with real data.