Two-step calibration method of the extrinsic parameters with high accuracy for a bistatic non-orthogonal shafting laser theodolite system

Zefeng Sun; Jiehu Kang; Jiehu Kang; Zhen Zhang; Luyuan Feng; And Bin Wu

doi:10.1364/OE.481204

1. Introduction

The theodolite is a kind of no-contact optical measurement instrument widely used in engineering [1–3], and its internal shafting is required strictly orthogonal to each other. To achieve the cost-effective manufacturing of precision instrument, the non-orthogonal shafting laser theodolite (N-theodolite) was proposed in recent years [4,5]. Different from the traditional theodolite, there is no orthogonality requirement for the internal shafting of the N-theodolite. Wu et al. have raised the working model of the N-theodolite and constructed the bistatic N-theodolite measurement system, the feasibility of non-orthogonal shafting type instrument has been verified [6].

For the bistatic N-theodolite system, the high-accuracy calibration of intrinsic and extrinsic parameters is the premise of 3D large-scale measurement. The intrinsic parameters of N-theodolite include the parameters describing the spatial poses of the rotation axes and laser axis. With a series of methods for calibrating the internal shafting with high accuracy [7–11], the intrinsic parameters calibration is not the main problem to be solved. However, compared with the intrinsic parameters, there are few studies on the extrinsic parameters calibration of the bistatic N-theodolite system. The extrinsic parameters are the coordinate transformation matrixes between the N-theodolites. For the traditional theodolite, there are a considerable number of related studies on the calibration methods of extrinsic parameters. Usually, the extrinsic parameters calibration of traditional theodolite measurement system is achieved based on high requirements for theodolite leveling and accurate mutual aiming [12]. Bundle adjustment is a representative calibration method without leveling [13], which can be achieved by measuring the known standard length in several positions [14]. In the perspective projection model, the essential matrix between the theodolites can be calculated linearly by using 8 projection points, and extrinsic parameters are obtained from the essential matrix by SVD decomposition [15]. Besides, a global calibration method proposed recently is solving position results based on the angular distance of control points and then computing orientation parameters linearly [16]. However, there is no common rotary center for the laser axes under different motion states due to the nonorthogonality of N-theodolite, which means that the laser axes would not emit from the same point as the traditional theodolite. Thus, the calibration methods for the traditional theodolite are not applicable to the bistatic N-theodolite system. At present, the unresolved problem related to the extrinsic parameters calibration limits the measurement precision of the bistatic N-theodolite system in engineering application.

To refine and improve the extrinsic parameters calibration of the bistatic N-theodolite system, a two-step calibration method is proposed in this paper. It is known that the initial values play an important role in the optimization problem, so acquisition of the initial values within reasonable range is the key step of the proposed method. Meanwhile, the reference values provided by PSD sensors [17,18] can effectively optimize the calibration results in the right direction. The main contents of this paper are as follows:

(1) The proposed calibration method is designed to acquire the initial extrinsic parameters, and then substitute them into the subsequent optimization to obtain the high-accuracy results.
(2) In the first step, the calculation of the initial extrinsic parameters can be simplified by setting the approximate emitted point during the motion of the laser axis.
(3) According to the analysis and simulation of the laser axis motion, the feasibility of setting the approximate emitted point is verified.
(4) In the second step, the unconstrained optimal objective function is constructed with PSD sensors, and the final extrinsic parameters with high accuracy are obtained.
(5) The bistatic N-theodolite system calibrated by the proposed method is applied to 3D measurement, and the practicability and accuracy is verified by the results.

2. System modeling

With the introduction of the bistatic N-theodolite system, the related coordinate measurement model and the required parameters are clarified in this section.

2.1 Kinematic model of N-theodolite

The structure of N-theodolite is shown in Fig. 1(a). It can be simplified into two rotation axes and one laser axis, which are non-orthogonal to each other. The calibration of intrinsic parameters can refer to the method of [10]. As shown in Fig. 1(b), the perpendicular points between the rotation axes are P₁ and P₂, and the unit direction vectors of the rotation axes are V₁ and V₂ respectively. As shown in Fig. 1(c), the coordinate system of N-theodolite is established based on the internal shafting parameters, the z axis is parallel to the vertical rotation axis, the x axis is coplanar with the horizontal rotation axis, and P₁ is the coordinate origin.

Fig. 1. The structure of N-theodolite. (a) N-theodolite. (b) Structural simplification. (c) Coordinate system establishment.

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The laser axis rotates around the horizontal rotation axis firstly, and the corresponding pose transformation matrix is (R₁, T₁). Then the laser axis rotates around the vertical rotation axis, the corresponding pose transformation matrix is (R₂, T₂). The initial pose of laser axis is represented by point P_Laser and unit direction vector V_Laser. The laser axis pose after rotary motion can be represented by point P_New-laser and unit direction vector V_New_-laser:

(1)$$\left[ {\begin{array}{{cc}} {{P_{\textrm{New - laser}}}}&{{{\boldsymbol V}_{{\mathbf {New - laser}}}}}\\ 1&0 \end{array}} \right] = \left[ {\begin{array}{{cc}} {{R_2}}&{{T_2}}\\ 0&1 \end{array}} \right]\left[ {\begin{array}{{cc}} {{R_1}}&{{T_1}}\\ 0&1 \end{array}} \right]\left[ {\begin{array}{{cc}} {{P_{\textrm{Laser}}}}&{{{\boldsymbol V}_{{\mathbf {Laser}}}}}\\ 1&0 \end{array}} \right]$$

2.2 Measurement model and extrinsic parameters

The coordinates of the measured points are calculated according to the extrinsic parameters and rotation angles when the N-theodolites aim at the same point together. As shown in Fig. 2, the extrinsic parameters required to be calibrated are the coordinate transformation matrix between the left and right N-theodolite coordinate systems. The measured point P_M can be calculated by Eq. (2), P_L and V_L are the pose parameters of left laser axis, P_R and V_R are the pose parameters of right laser axis, k₁ and k₂ are spatial lengths.

(2)$$\left[ {\begin{array}{{c}} {{P_\textrm{M}} - {P_\textrm{L}}}\\ {{P_\textrm{M}} - {P_\textrm{R}}} \end{array}} \right] = \left[ {\begin{array}{{c}} {{k_1}{{\boldsymbol V}_{\mathbf L}}}\\ {{k_2}{{\boldsymbol V}_{\mathbf R}}} \end{array}} \right]$$

Fig. 2. The working mode of bistatic N-theodolite system.

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To realize the precise measurement of bistatic N-theodolite system, the extrinsic parameters of the bistatic N-theodolite system are required to be calculated accurately in this paper.

3. Two-step calibration method of extrinsic parameters

3.1 First step: Acquisition of initial parameters in local space

The planar checkerboard for visual measurement is used to obtain initial extrinsic parameters in this step. The visual calibration method is shown in Fig. 3. the camera is calibrated by Zhang's calibration method [19]. The laser axis of N-theodolite is incident on the planar checkerboard, and the image is collected by the camera. The planar checkerboard is regarded as world coordinate system (WCS), and the corresponding 2D pixel coordinates (u, v) of laser points in image plane can be expressed as

(3)$$s\left[ {\begin{array}{{c}} u\\ v\\ 1 \end{array}} \right] = A{M_c}\left[ {\begin{array}{{c}} {{X_\textrm{W}}}\\ {{Y_\textrm{W}}}\\ {{Z_\textrm{W}}}\\ 1 \end{array}} \right]$$

where A is the intrinsic parameters of camera, and M_c is the coordinate transformation matrix between the planar checkerboard and camera coordinate system.

Fig. 3. The visual calibration method.

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The 3D coordinates (X_LW, Y_LW, 0) of laser points in WCS can be calculated by Eq. (3). The relationship between the pose parameters of spatial laser axis and laser point coordinates in WCS can be expressed as

(4)$${P_l} + k{{\boldsymbol V}_l} = \left[ {\begin{array}{{c}} {{x_l}}\\ {{y_l}}\\ {{z_l}}\\ 1 \end{array}} \right] + k\left[ {\begin{array}{{c}} {{u_l}}\\ {{v_l}}\\ {{w_l}}\\ 0 \end{array}} \right] = \left[ {\begin{array}{{cccc}} {{r_1}}&{{r_2}}&{{r_3}}&t \end{array}} \right]\left[ {\begin{array}{{c}} {{X_{\textrm{LW}}}}\\ {{Y_{\textrm{LW}}}}\\ 0\\ 1 \end{array}} \right]$$

where P_l and V_l are the specific point and unit direction vector of the laser axis respectively, and k is the spatial length of laser axis.

The laser axes of different poses are considered to approximately emit from the same point, so the above P_l component can be omitted, and the approximate emitted point will be analyzed and calculated in section 4. As shown in Fig. 4, a virtual camera image plane is constructed, and above equation can be simplified and rewritten into the form as

(5)$$\left[ {\begin{array}{{@{}c@{}}} {{u_{{\mathop{\rm Im}\nolimits} g}}}\\ {{v_{{\mathop{\rm Im}\nolimits} g}}}\\ 1 \end{array}} \right] = \left[ {\begin{array}{{@{}c@{}}} {({{f / {\textrm{dx}}}} )({ - {{{w_l}} / {{v_l}}}} )- {u_0}}\\ {({{f / {\textrm{dy}}}} )({{{{u_l}} / {{v_l}}}} )- {v_0}}\\ 1 \end{array}} \right] = \left[ {\begin{array}{{@{}ccc@{}}} {{h_1}}&{{h_2}}&{{h_3}} \end{array}} \right]\left[ {\begin{array}{{@{}c@{}}} {{X_{\textrm{LW}}}}\\ {{Y_{\textrm{LW}}}}\\ 1 \end{array}} \right] = \left[ {\begin{array}{{ccc}} {{h_{11}}}&{{h_{12}}}&{{h_{13}}}\\ {{h_{21}}}&{{h_{22}}}&{{h_{23}}}\\ {{h_{31}}}&{{h_{32}}}&{{h_{33}}} \end{array}} \right]\left[ {\begin{array}{{c}} {{X_{\textrm{LW}}}}\\ {{Y_{\textrm{LW}}}}\\ 1 \end{array}} \right]$$

Fig. 4. Construction of virtual image plane.

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The above equation can be converted to following form:

(6)$$\left[ {\begin{array}{{ccccccccc}} {{X_{\textrm{LW}}}}&{{Y_{\textrm{LW}}}}&1&0&0&0&{ - {u_{{\mathop{\rm Im}\nolimits} g}}{X_{\textrm{LW}}}}&{ - {u_{{\mathop{\rm Im}\nolimits} g}}{Y_{\textrm{LW}}}}&{ - {u_{{\mathop{\rm Im}\nolimits} g}}}\\ 0&0&0&{{X_{\textrm{LW}}}}&{{Y_{\textrm{LW}}}}&1&{ - {v_{{\mathop{\rm Im}\nolimits} g}}{X_{\textrm{LW}}}}&{ - {v_{{\mathop{\rm Im}\nolimits} g}}{Y_{\textrm{LW}}}}&{ - {v_{{\mathop{\rm Im}\nolimits} g}}} \end{array}} \right]\left[ {\begin{array}{{c}} {{h_{11}}}\\ {{h_{12}}}\\ {{h_{13}}}\\ {{h_{21}}}\\ {{h_{22}}}\\ {{h_{23}}}\\ {{h_{31}}}\\ {{h_{32}}}\\ {{h_{33}}} \end{array}} \right] = 0$$

The constraints involved can be expressed as

(7)$$\left\{ {\begin{array}{{ccc}} {\left[ {\begin{array}{{ccc}} {{r_1}}&{{r_2}}&t \end{array}} \right] = \lambda \left[ {\begin{array}{{ccc}} {{h_1}}&{{h_2}}&{{h_3}} \end{array}} \right]}\\ {{r_1}^T{r_2} = 0}\\ {|{{r_1}} |= |{{r_2}} |= 1} \end{array}} \right.$$

The results obtained above are substituted into Eq. (4), and the unconstrained optimal objective function can be constructed as

(8)$$F = \sum\limits_{i,j = 1}^n {(||{{P_{li}} + {k_i}{{\boldsymbol V}_{li}} - {P_{lj}} - {k_j}{{\boldsymbol V}_{lj}}} ||} - ||{{P_i} - {P_j}} ||{)^2}$$

where P_i and P_j represent the coordinates of the laser points i and j in the camera coordinate system.

Based on the results above, the coordinates of laser points on the planar checkerboard are obtained in N-theodolite coordinate system. Then the coordinate transformation matrix between WCS and N-theodolite is solved by SVD method [20]. The initial extrinsic parameters M of bistatic N-theodolite system are expressed as

(9)$$M = {M_L}M_R^{ - 1} = \left[ {\begin{array}{{cc}} {{R_L}}&{{T_L}}\\ 0&1 \end{array}} \right]{\left[ {\begin{array}{{cc}} {{R_R}}&{{T_R}}\\ 0&1 \end{array}} \right]^{ - 1}}$$

where M_L and M_R are the coordinate transformation matrixes from WCS to left and right N-theodolites.

3.2 Second step: Extrinsic parameter further optimization based on PSD

The measurement space of the bistatic N-theodolite system is not covered by the calibration space in the first step, and the accuracy of image processing is limited, so the extrinsic parameters are required to be further optimized in global space. In the second step, the position sensitive detector (PSD) is used to provide the high-accuracy reference distance.

As shown in Fig. 5, both the left and right N-theodolites are controlled to aim at same point on photosensitive surface of PSD, and the laser intersection points are the calibration points. The PSDs are pre-calibrated by high-precision measuring instrument, the related calculation method of the distances between spatial laser points can be referred to [21]. The calibration space is required to cover the measurement space of the bistatic N-theodolite system. The laser axes of left and right N-theodolite aiming at the same point satisfy the Eq. (10).

(10)$$\left[ {\begin{array}{{c}} {{P_L}}\\ 1 \end{array}} \right] + {k_L}\left[ {\begin{array}{{c}} {{{\boldsymbol V}_L}}\\ 0 \end{array}} \right] = M\left( {\left[ {\begin{array}{{c}} {{P_R}}\\ 1 \end{array}} \right] + {k_R}\left[ {\begin{array}{{c}} {{{\boldsymbol V}_R}}\\ 0 \end{array}} \right]} \right)$$

Fig. 5. Extrinsic parameter optimization based on PSD.

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Based on the initial extrinsic parameters M, a set of spatial length data (k_L, k_R) can be obtained. The number of calibration points is n. The unconstrained optimal objective function in the left N-theodolite coordinate system can be constructed as

(11)$$F = \sum\limits_{i,j = 1}^n {(||{{P_{Li}} + {k_{Li}}{{\boldsymbol V}_{Li}} - {P_{Lj}} - {k_{Lj}}{{\boldsymbol V}_{Lj}}} ||} - L_{ij}^{PSD}{)^2}$$

where $L_{ij}^{PSD}$ represents the distance between the calibration points i and j.

Similarly, the unconstrained optimal objective function in the right N-theodolite coordinate system can be similarly constructed as

(12)$$F = \sum\limits_{i,j = 1}^n {(||{{P_{Ri}} + {k_{Ri}}{{\boldsymbol V}_{Ri}} - {P_{Rj}} - {k_{Rj}}{{\boldsymbol V}_{Rj}}} ||} - L_{ij}^{PSD}{)^2}$$

Finally, the accurate extrinsic parameters M is obtained with the optimized spatial length data substituted into Eq. (10).

4. Analysis of the approximate emitted point

4.1 Approximate emitted point of laser axes

As mentioned above, the rotating laser axes of N-theodolite cannot emit from the same point. To simplify the related calculation, the approximate emitted point is assumed to exist. As shown in Fig. 6, the new coordinate system obtained by translation from the N-theodolite coordinate system with the approximate emitted point as the origin. The Eq. (4) can be rewritten as:

(13)$$\Delta {P_i} + {k_i}{{\boldsymbol V}_i} = \left[ {\begin{array}{{c}} {{x_{li}}}\\ {{y_{li}}}\\ {{z_{li}}}\\ 1 \end{array}} \right] + {k_i}\left[ {\begin{array}{{c}} {{u_{li}}}\\ {{v_{li}}}\\ {{w_{li}}}\\ 0 \end{array}} \right] = \left[ {\begin{array}{{cccc}} {{r_1}}&{{r_2}}&{{r_3}}&{t + \Delta t} \end{array}} \right]\left[ {\begin{array}{{c}} {{X_{\textrm{LW}}}}\\ {{Y_{\textrm{LW}}}}\\ 0\\ 1 \end{array}} \right]$$

where $\Delta t$ represents the translation transformation between the new coordinate system and N-theodolite coordinate system. $\Delta {P_i}$ represents the intersection point of the laser axis and the straight line perpendicular to the laser axis through the origin.

Fig. 6. The analysis of approximate emitted point.

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4.2 Simulation and verification in local space

The factors that affect the distance between the rotating laser axis and the approximate emitted point include the initial pose state of the laser axis and the rotating angles.

As shown in Fig. 7, the boundary value of angle changing is reached when the laser axis aims at the edge point of the planar checkerboard. The model of planar checkerboard used in this paper is GP290-12 × 9, whose overall dimension is 290 mm × 230 mm. Considering the working distance of the N-theodolite, the boundary value Δα is calculated to be 5 degrees. The initial pose state of the laser axis is corresponding to the angle at which the laser axis aims at the center of the planar checkerboard, and the angle range corresponding to the initial pose state of laser axis is set to (-35°, 35°).

Fig. 7. Range of the laser axis rotating angle.

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It is preliminarily assumed that the approximate emitted point is located on the laser axis of initial state. The maximum distance from the approximate emitted point to the laser axes is selected as the observation index in the simulation, and the results are shown in Fig. 8. However, there must be an ideal approximate emitted point in the space, which satisfies the following inequality:

(14)$$\begin{array}{{cc}} {|{\Delta {P_i}^{\prime}} |\le {{|{\Delta {P_i}} |}_{\max }} < 1\textrm{mm},}&{i = 1, \cdots ,n} \end{array}$$

where $|{\Delta {P_i}^{\prime}} |$ represents the distance between the ideal approximate emitted point and the laser axes, ${|{\Delta {P_i}} |_{\max }}$ represents the maximum distance between the set approximate emitted point and laser axes.

Fig. 8. Results of the simulation.

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According to the results above, the absolute value of the direction vector component in Eq. (13) is greater than $|{\Delta {P_i}} |$ generally in limited local space. Therefore, within the allowable errors, the laser axes of different poses can be considered as emitting from the same point. Besides, when the initial laser axis state is around 0°, the maximum distance is far less than 1 mm, so a more ideal approximate emitted point can be obtained with the laser axis rotating around 0°. The approximate emitted point P_O for a finite number of laser axes can be expressed as

(15)$${P_O} = \left[ {\begin{array}{{c}} {{x_O}}\\ {{y_O}}\\ {{z_O}} \end{array}} \right] = {\left( {{{\left[ {\begin{array}{{ccc}} {{v_1}}&{ - {u_1}}&0\\ 0&{{w_1}}&{ - {v_1}}\\ \vdots & \vdots & \vdots \\ {{v_i}}&{ - {u_i}}&0\\ 0&{{w_i}}&{ - {v_i}}\\ \vdots & \vdots & \vdots \\ {{v_n}}&{ - {u_n}}&0\\ 0&{{w_n}}&{ - {v_n}} \end{array}} \right]}^T}\left[ {\begin{array}{{ccc}} {{v_1}}&{ - {u_1}}&0\\ 0&{{w_1}}&{ - {v_1}}\\ \vdots & \vdots & \vdots \\ {{v_i}}&{ - {u_i}}&0\\ 0&{{w_i}}&{ - {v_i}}\\ \vdots & \vdots & \vdots \\ {{v_n}}&{ - {u_n}}&0\\ 0&{{w_n}}&{ - {v_n}} \end{array}} \right]} \right)^{ - 1}}\left[ {\begin{array}{{c}} {{x_1}{v_1} - {y_1}{u_1}}\\ {{y_1}{w_1} - {z_1}{v_1}}\\ \vdots \\ {{x_i}{v_i} - {y_i}{u_i}}\\ {{y_i}{w_i} - {z_i}{v_i}}\\ \vdots \\ {{x_n}{v_n} - {y_n}{u_n}}\\ {{y_n}{w_n} - {z_n}{v_n}} \end{array}} \right]$$

where (x_i, y_i, z_i) is the point coordinate on the laser axis, and (u_i, v_i, w_i) is the unit direction vector of the laser axis.

5. Experiment and real data

5.1 Intrinsic parameters calibration

It is necessary to calibrate the intrinsic parameters of the camera and N-theodolites involved in this paper to facilitate subsequent experiments. The N-theodolites used in this paper consist of two-dimensional turntable and laser transmitter. The model of the two-dimensional turntable is AZ-EL GIMBAL from ALIO industries, the angular resolution is less than 0.02 arcsecond and the angular repeat abilities of the two axes are both less than ±0.5 arcsecond. The divergence angle of the laser transmitter is less than 0.6 milliradian, the divergence can be ignored within a certain range. The intrinsic parameters of theodolite are calibrated by the laser tracker Lecia AT960-LR, of which the measurement accuracy is ±7.5µm + 3 µm/m. The calibration process can refer to [5,7,10].The coordinate system of N-theodolite is established, and it is required to retain more significant bits for direction vectors. The intrinsic parameters of the N-theodolites are shown in Table 1.

Table 1. The intrinsic parameters of N-theodolites

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The first step of the proposed calibration method involves visual measurement, and the related parameters of the camera require to be calibrated by Zhang's calibration method. The camera used in this paper is Daheng MER2-503-36U3M, with the resolution of 2448 × 2048. The model of the planar checkerboard used in the camera calibration process is GP290-12 × 9. The camera is controlled to take multiple images of the planar checkerboard in different positions and postures. The calibration results of related parameters of the camera are shown in Table 2.

Table 2. The intrinsic parameters of camera

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5.2 Application of the two-step calibration method

In the first step, the planar checkerboard is fixed within the field of camera view. As shown in the Fig. 9, the whole process of the first step is as follows:

(1) Calculate the coordinate transformation relationship between the planar checkerboard and camera by the image of the planar checkerboard.
(2) The laser axes of the left and right N-theodolite are incident at different positions on the planar checkerboard, and the corresponding rotation angles and images with laser spot are recorded.
(3) The coordinates of the laser incident points in the camera coordinate system are calculated based on the centroid method for image processing.
(4) The pose parameters of laser axes are calculated by the rotation angles.
(5) The coordinate transformation relationship between the planar checkerboard and the N-theodolites are calculated.
(6) The initial extrinsic parameters of the bistatic N-theodolite system are calculated.

Fig. 9. Acquisition of the initial extrinsic parameters.

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The initial extrinsic parameters are shown in Table 3, meanwhile the coordinates of the laser incident points in N-theodolite coordinate systems can be obtained. The laser incident points in camera coordinate system are used as the reference values, and the distances between the laser incident points are compared. As shown in Fig. 10, although the accuracy of the calibration results in the first step is limited, the acquired initial extrinsic parameters can be considered reliable to a certain extent, which is helpful for subsequent optimization in correct direction.

Fig. 10. Comparison of the distances between laser incident points.

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Table 3. The results of the first step

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The second step involves the PSD sensor, and the Hamamatsu’s S1880 pincushion type 2-D PSD is used as the core sensor for receiving laser, whose position resolution is 1.5 µm and detection error is less than 40 µm. The whole process of the second step is as follows:

(1) The laser axis of the left N-theodolite is incident on the photosensitive surface of the PSD.
(2) Control the laser axis of the right N-theodolite to aim at the same position on the PSD according to the output indication of PSD.
(3) The above steps are repeated in different calibration positions to cover the measurement space.
(4) The spatial distances between laser points are calculated according to the pre-calibrated PSDs.
(5) The spatial pose parameters of laser axes are calculated by the rotation angles.
(6) The above data are substituted into the unconstrained objective function to calculate the final extrinsic parameters.

The final extrinsic parameters are shown in Table 4.

Table 4. The results of the second step

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5.3 3D Measurement of the calibrated bistatic N-theodolite system

To verify the accuracy of the calibration results, the bistatic N-theodolite system calibrated by the proposed method is applied for 3D measurement. As shown in Fig. 11, the bistatic N-theodolite system is used to measure the spatial points distributed within 5 m. The PSD is still used to receive laser and judge the intersection of the laser axes, and the reference values of measured points are provided by the laser tracker. The 1.5 inch spherically mounted retroreflector (SMR reflector) is used to cooperate the laser tracker to measurement.

Fig. 11. Measurement experiment of the calibrated bistatic N-theodolite system.

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For better comparison, the initial extrinsic parameters obtained in the first step and the final extrinsic parameters are respectively substituted into the calculation. The relative pose relationship between the N-theodolites calibrated directly by the laser tracker can be regarded as the reference values of the extrinsic parameters. Both the 3D point coordinate error and the distance error are compared, and the results are shown in Fig. 12 and Table 5.

Fig. 12. Measurement results. (a) Point coordinate measurement. (b) Distance measurement.

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Table 5. Results of point coordinate measurement (mm)

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Table 6. Results of distance measurement (mm)

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According to the above results (Table 6), the measurement error of bistatic N-theodolite system with the initial extrinsic parameters is obvious. The main limiting factors for the calibration performance include the imaging quality of the laser spot and extraction algorithm. However, the initial extrinsic parameters can be used as initial values for subsequent optimization in the second step. Compared with the initial extrinsic parameters, the measurement accuracy of the bistatic N-theodolite system using the final extrinsic parameters has been significantly improved. Meanwhile, comparing with the reference extrinsic parameters, the measurement accuracy of the bistatic N-theodolite system is equivalent basically. The comparison has proven that the extrinsic parameter is calibrated by the proposed method with high accuracy, and the main factors causing the measurement error are other factors except the extrinsic parameters, such as the intrinsic parameter error or the angle error. Based on the above results, it is proved that the high-accuracy solution can be obtained by the proposed method.

6. Conclusion

To achieve the extrinsic parameters calibration of the bistatic N-theodolite system with high accuracy, a two-step calibration method is proposed in this paper. The study focuses on the simple acquisition of the initial extrinsic parameters before further optimization. Firstly, the initial extrinsic parameters acquisition scheme is designed, and with analyzing and setting the approximate emitted point in local space, the calculation process can be simplified. Then the precision distances between the spatial laser points in global space are provided by the PSD sensors to further optimizes the extrinsic parameters. The bistatic N-theodolite system calibrated by the proposed method is applied to 3D measurement, and the 3D point coordinate measurement average error is less than 0.4 mm, and the distance measurement average error is less than 0.3 mm within 5 m. Compared with the reference extrinsic parameters calibrated directly by the laser tracker, it is proved that the high-accuracy extrinsic parameters can be obtained by the proposed calibration method, which provides the practical significance for the engineering application of N-theodolite.

Funding

National Natural Science Foundation of China (62071325); Open Foundation of Beijing Laboratory of Optical Fiber Sensing and System (GXKF2022001); Sichuan Province Science and Technology Support Program (2021YFSY0024).

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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	Point coordinate(mm)	Direction vector
Center horizontal rotation axis	(0, 0, 0)	(0.999999, 0, -0.001163)
Center vertical rotation axis	(0, 0.0694, 0)	(0, 0, 1)
Center laser axis	(-0.2803, 0, -18.7956)	(0.005028, 0.999855, 0.016285)
Right horizontal rotation axis	(0, 0, 0)	(0.999999, 0, -0.000372)
Right vertical rotation axis	(0, 0.0576, 0)	(0, 0, 1)
Right laser axis	(-0.3328, 0, -19.3210)	(0.005401, 0.999823, 0.018047)

	Transformation matrix
WCS to center N-theodolite	$[\begin{array}{cccc} 0.98896 & 0.00516 & - 0.14809 & 1746.86632 \\ 0.14811 & - 0.00320 & 0.98896 & 2144.70619 \\ 0.00463 & - 0.99998 & - 0.00393 & 227.37455 \\ 0 & 0 & 0 & 1 \end{array}]$
WCS to right N-theodolite	$[\begin{array}{cccc} 0.75381 & - 0.00006 & 0.65709 & - 494.60592 \\ - 0.65708 & - 0.00596 & 0.75380 & 2885.07031 \\ 0.00387 & - 0.99998 & - 0.00453 & 202.50536 \\ 0 & 0 & 0 & 1 \end{array}]$
Initial extrinsic parameters	$[\begin{array}{cccc} 0.64818 & - 0.76149 & - 0.00066 & 4264 .53221 \\ 0.76148 & 0.64818 & - 0.00071 & 651 .43277 \\ 0.00097 & - 0.00004 & 0.99999 & 25 .46596 \\ 0 & 0 & 0 & 1 \end{array}]$

	The initial extrinsic parameters	The final extrinsic parameters	The reference extrinsic parameters
Average error	1.836	0.307	0.378
Maximum error	2.608	0.338	0.455
Minimum error	1.564	0.254	0.267
RMSE	0.503	0.083	0.102

	The initial extrinsic parameters	The final extrinsic parameters	The reference extrinsic parameters
Average error	0.608	0.044	0.064
Maximum error	1.726	0.235	0.223
Minimum error	0.006	0.001	0.001
RMSE	0.078	0.007	0.009

	Point coordinate(mm)	Direction vector
Center horizontal rotation axis	(0, 0, 0)	(0.999999, 0, -0.001163)
Center vertical rotation axis	(0, 0.0694, 0)	(0, 0, 1)
Center laser axis	(-0.2803, 0, -18.7956)	(0.005028, 0.999855, 0.016285)
Right horizontal rotation axis	(0, 0, 0)	(0.999999, 0, -0.000372)
Right vertical rotation axis	(0, 0.0576, 0)	(0, 0, 1)
Right laser axis	(-0.3328, 0, -19.3210)	(0.005401, 0.999823, 0.018047)

Two-step calibration method of the extrinsic parameters with high accuracy for a bistatic non-orthogonal shafting laser theodolite system

Abstract

1. Introduction

2. System modeling

2.1 Kinematic model of N-theodolite

2.2 Measurement model and extrinsic parameters

3. Two-step calibration method of extrinsic parameters

3.1 First step: Acquisition of initial parameters in local space

3.2 Second step: Extrinsic parameter further optimization based on PSD

4. Analysis of the approximate emitted point

4.1 Approximate emitted point of laser axes

4.2 Simulation and verification in local space

5. Experiment and real data

5.1 Intrinsic parameters calibration

5.2 Application of the two-step calibration method

5.3 3D Measurement of the calibrated bistatic N-theodolite system

6. Conclusion

Funding

Disclosures

Data availability

References

Data availability

Cited By

Figures (12)

Tables (6)

Equations (15)

Optics Express