The Calibration of theta-phi Fiber Positioners Based on the Differential Evolution Algorithm

Robotic fiber positioner (RFP) arrays are commonly adopted in multiobject spectroscopic instruments. The positioning accuracy is a common but vital issue for RFP as inaccurate fiber placement may heavily affect the observation performance. The calibration of RFP can effectively improve the positioning accuracy. Least-square is a widely used calibration method. However, it has disadvantages, such as sensitivity to the initial values and calculation complexity. To improve the positioning accuracy and reduce the iteration moves, we propose a new calibration method based on the differential evolution algorithm and verify it by calibrating the RFP of the Large Sky Area Multi-Object Fiber Spectroscopy Telescope. We first build the kinematic models of the RFP based on the Denavit–Hartenberg matrix and geometry relationship. Then, we analyze the error components and present the proposed calibration algorithms. The experiments are done with the digital universal tool microscope 19JC and the errors are calculated using the distance between the positions of achieved and target. Results show that the proposed algorithm can achieve higher accuracy than the least-square method and the average positioning accuracy is improved by 78.94% after calibration. Combined with the “pulse reduction” strategy and close-loop compensation, after two moves, the positioners can place the fiber ends within 40 μm of the intended location. The proposed calibration method is also suitable for other similar theta-phi positioners.


Introduction
Massive observational surveys have gained burgeoning interest in astronomical research (Martell et al. 2016).They have considerably revolutionized our understanding of the formation and evolution of the universe (Dong et al. 2018).Their success has motivated the proposal and construction of new large-scale multiplexed spectroscopic instruments (Miszalski et al. 2006;Zhang et al. 2020aZhang et al. , 2020b)).Currently, they are primarily conducted at multiobject spectroscopy (MOS) facilities on large-aperture telescopes (Morales et al. 2012;Makarem et al. 2014).The MOS can mainly be divided into multislit and fiber-fed spectrographs.Multislit spectrographs were originally attached directly to the telescope structure, thus experiencing a varying gravity vector during observation and placing rigid constraints on the mechanical design (Wang et al. 2015;Zhang et al. 2016).Furthermore, the mask had to be prepared and exchanged before each observation.Contrarily, fiber-feed spectrographs can be remotely mounted at a stable location for the telescope by routing thousands of fibers to a spectrograph.For each pointing, the reconfiguration of the fibers at the focal plane is convenient (Jovanovic et al. 2020).Thus, the fiber-feed approach is considered more versatile (Azzaro et al. 2010).
Different approaches for fiber feeding have been proposed.The plug-plate method was adopted by the Sloan Digital Sky Survey project (SDSS; Smee et al. 2013).640 fibers were manually plugged into the predrilled holes in the prepared single-used aluminum focal plate (Gunn et al. 2006).The plugging process takes several hours (Horler et al. 2018).The pick-and-place robotic systems were first used at the Twodegree Field (2dF) facility at the prime focus of the Anglo-Australian Telescope (Lewis et al. 2002).Before observation, the robot arms are manipulated to pick and place fibers at the target position on the focal plane sequentially.The reconfiguration timescales with the number of fibers are thus inefficient for next-generation massive spectroscopy surveys (Cui et al. 2012).The most efficient devices are the robotic fiber positioner (RFP) arrays because thousands of fibers can be positioned simultaneously (Martinez-Garcia et al. 2019) at the cost of numerous actuators (Smith et al. 2004).
For the RFP introduced above, the fiber ends must be accurately positioned at the focal surface to capture the light from targets (Fahim et al. 2015).The positioning accuracy is a common problem in MOS facilities (Azzaro et al. 2010).To achieve the desired precision, one focus is on the image-guided closed-loop control (Baltay et al. 2019;Gillingham 2020).For example, DESI relies on the Fiber View Camera System (FVC) to measure the locations of the back-illuminated fiber tips and then calculates the next move of RFP.SDSS-V deploys an FVC to measure the positions of the metrology fibers illuminated by the fiber back-illumination system (Engelman et al. 2022).PFS performs iterations based on the fiber position measured using a metrology camera (Sugai et al. 2015).The above instruments recognize the fiber tips in the focal plane by setting a light source inside the spectrometer at the other end of the fiber (called the back-illuminated method).This method has advantages such as high stability and accuracy.However, it needs to switch to the observation mode after positioning, increasing the time needed for each observation.To overcome the drawbacks, front-illumination is proposed.It detects the center of the ferrule rather than directly detecting the fiber position, eliminating complicated lighting in the spectrometer and saving switching time.However, currently, the measurement accuracy with front-illumination is lower than that with back-illumination (Zhou et al. 2021b).
The other focus is on the calibration of RFP.Here, we mainly focus on the theta-phi positioners.Owing to the unavoidable tolerance of the manufacturing and assembly processes, parameters transforming between the commended angular motions of the motors and the measured fiber positions deviate from the nominal ones (details about these parameters are presented in Section 2).Inaccurate parameters impact collision avoidance, fiber assignment, and survey speed (DESI Collaboration et al. 2022).The calibration can be done with randomly distributed points or by using independent arcs of points on the theta and phi axes (Silber et al. 2023).Locations of 210 points on each arc (obtained by theta or phi motion only) of LAMOST RFP were measured and then the geometric parameters such as arm length were calculated using the leastsquare method (LSM; Liu et al. 2011).Based on a dualtelecentric measurement system and ignoring the error caused by lens distortion, researchers of LAMOST also measured the positions of 36 randomly and 76 uniformly distributed points within the patrol area and obtained six calibrated initial parameters using LSM (Zuo et al. 2020).To calibrate the RFP of DESI, the positions of 16 calibration points per arc and that of a rectilinear grid of 24 calibration points are measured and the six geometric parameters are calibrated using LSM (Silber et al. 2023).The LSM method is also adopted by SDSS-V to calibrate parameters such as arm length (Kronig et al. 2020c).However, LSM has disadvantages, such as sensitivity to the initial values and calculation complexity.Recently, based on the images obtained using the front-illumination method, the U-net architecture was adopted to detect the ceramic ferrule of LAMOST RFP by calibrating only the initial angles (Zhou et al. 2021a).However, even after calibration, several iterations were required to achieve the required accuracy.For example, for the positioners of PFS, less than 60% of the points converge within the required accuracy in the first four steps.This percentage is less than 95% in the first five steps (Sugai et al. 2015).To improve the positioning accuracy and decrease the number of iteration steps, saving valuable observation time and enhancing the survey speed, we propose a new calibration method based on the differential evolution algorithm.We take LAMOST RFP as an example.The calibration is performed annually and the accuracy required is within 40 μm.We aim to achieve the desired accuracy with less iterations.
The remainder of this study is organized as follows.In Section 2, we describe the theta-phi positioners and establish the related kinematic models based on the Denavit-Hartenberg (DH) matrices and geometry.The error models are established.In Section 3, we introduce the DE algorithm and explain how to calibrate and compensate for the parameters using the DE.We describe the experimental procedure and present the results.We summarize and conclude in Section 4.

Description and Distribution of theta-phi Positioners
A theta-phi positioner can be considered a planar Selective Compliance Assembly Robot Arm (SCARA) robot with two serial rotational motors.The fiber ends can be placed at different positions by controlling the rotation angles of the two motors, which is the nomenclature of the theta-phi (also known as the double-revolving fiber positioning unit).Here, different motors are used for theta-phi positioners.For example, the classic stepper model is adopted by the LAMOST and MOONS, whereas a brushless direct current (BLDC) motor is used by the SDSS-V and LAMOST second stage (Kronig et al. 2020c;Guo et al. 2022).The motors are controlled using an electronic board with pulse-width modulation through an open loop (LAMOST second stage; Guo et al. 2022) or a closed loop (SDSS-V with Hall sensors; Kronig et al. 2020b).One of the main challenges for the positioners is the tiny maximal allowable positioning error, which is telescopespecific and related to the fiber core diameter and size of the objects (Horler et al. 2018;Kronig et al. 2020c).Typically, the positioning requirement is proportional to the fiber core diameter (approximately 4% of the actual core diameter; Horler et al. 2018).With the tendency of the compact positioner, the tolerance of the positioning errors decreases, and the number of positioners increases.Generally, the high-precision encoders help to achieve the required precision as they can measure rotation angles as feedback.However, because of the limited available space, affordable sensors with no high accuracy exist (Kronig et al. 2020c).Instead of encoders, reduction gears are used to guarantee the positioning precision as the positioning resolution is multiplied by the reduction ratio (Horler et al. 2018;Kronig et al. 2020bKronig et al. , 2020c)).The reduction ratio is the motor input divided by the actuator output.Thus, if the motor inaccuracy is 5°and the reduction ratio is 1:500, the accuracy at the output is 0°.01.However, the accuracy is limited by the nonlinearities and backlashes in the gearboxes (Horler et al. 2018).
The theta-phi RFP arrays are adopted by numerous MOS facilities to place fiber ends at the desired locations within the required accuracy in parallel and obtain thousands of spectra simultaneously.Each positioner can only place the fiber end within its patrol area.The maximum radius of the patrol area is calculated as the sum of the alpha and beta arm lengths.The alpha and beta arms are the links from the first to the second rotation axis and the second rotation axis to the fiber end, respectively.Notably, the alpha and beta arm lengths may not be equal.For LAMOST and DESI, equal lengths were chosen for both arms, whereas unequal arms were selected for MOONS and SDSS-V.Figure 1 shows the patrol area of the positioners with equal and unequal arms.The positioners with unequal arms cannot reach their centers.To avoid blind areas in the focal plane, they are hexagonally and densely distributed such that the adjacent patrol area overlaps to a certain extent (Zhang et al. 2020).This is especially important for unequal-arm positioners because their central exclusion zone must be achieved by adjacent positioners.Overlapping the patrol area not only helps to maximize the available space for the mechanical design of the positioners (Kronig et al. 2020c) but also makes full use of the positioner arrays.Thus, a single target can be achieved by at least one positioner.The desired location can be reached by at most three positioners for equal-arm positioners, and by four for unequal-arm positioners (Figure 2).The assignment of the targets to positioners is important for improving the efficiency of surveys.However, because of the overlapping patrol areas, adjacent positioners may collide with their neighbors, thereby damaging the positioners and affecting the efficiency of the MOS.The possibility of collision is related to the geometry of the positioners and the pitch between them (Zhang et al. 2021).Currently, fiber assignment and collision avoidance are key issues in large-scale spectroscopic surveys.Compared to equal-arm positioners, unequal-arm positioners provide more flexibility for fiber assignment.However, collision avoidance is more complex.An overview of theta-phi positioners for large-scale spectroscopic surveys is presented in Table 1.

Kinematic Model Based on Geometry
Because the geometry of theta-phi positioners is simple, the geometry-based kinematic model is commonly used.A schematic of this process is shown in Figure 3.
From Figure 3, the forward and inverse kinematic models can be derived, and given by Equations (1) and (2) respectively.
where l is the distance between the rotation center of the positioner and fiber end (CEL 2 in Figure 3).Equations (1) and (2) are ideal without considering the errors.Practically, the lengths of the alpha and beta arms differ from the ideal values because of the tolerance during manufacturing and assembly.
When mounting a positioner onto the focal plane, the center of the positioner deviates from the desired location because of inevitable hole drilling and mounting errors.Furthermore, the orientation of the positioner differs from that of the reference frame of the focal plane.Considering these errors, Equation (1) (the forward kinematic model) can be written as q q q q j j q q q q j j = + D + D where Δx ce and Δy ce are the errors of the coordinates of the positioner center, Δl 1 and Δl 2 the deviations of the alpha and beta arm lengths, respectively, and θ off and j off the initial angles of the alpha and beta arm, respectively.Δx ce , Δy ce , Δl 1 , and Δl 2 affect the patrol area of the positioner, thereby resulting in wrong fiber assignment and possible positioner collision.For example, for equal-arm positioners, owing to Δl 1 and Δl 2 , the alpha and beta arms become unequal.Hence, the center of the positioner cannot be reached by either itself or the adjacent positioners (Figure 1).The abovementioned errors are usually calibrated parameters for the fiber positioners in the focal plane.Except for the factors mentioned above (as described in Section 2.1), the reduction ratios of the gearboxes of the alpha and beta arms may differ from the designed ratios.
Because the kinematic errors such as the length of arms account for 90% of the total errors (Jiang et al. 2020), we assume that the real reduction ratio is proportional to the designed one, which does not considerably affect the accuracy.If the ideal rotation angles of the alpha and beta arms are represented by θ r and j r , respectively, then the real rotation angles can be expressed as where θ i and j i are the ideal rotation angles of the alpha and beta arms, respectively, k rθ the coefficient of the reduction ratio of the alpha arm, and k rj the coefficient of the reduction ratio of the beta arm.Ideally, k rθ and k rj are equal to one.Because both motors of one positioner are installed parallel at a rather small distance, the movement of one motor may cause the movement of the other (inductive effects).For example, the beta arm may rotate by an angle of θ rc when the alpha arm rotates by θ r .The actual rotation angle of the beta arm can be expressed as where k c is the coefficient of the beta arm coupled to the alpha arm.The real positions of the fiber ends can be obtained by combining Equations (3)-( 5)

Kinematic Model Based on the DH Method
Theta-phi positioners are essentially SCARA-like serial manipulators.The Denavit-Hartenberg convention method (the DH method), proposed by Denavit and Hartenberg (Denavit & Hartenberg 1955), is a widely adopted modeling method for describing robot kinematics (Conte et al. 2016;Jiang et al. 2020).The geometry model can be seen as a simplification of the DH method.As the key calibration  parameters are those used to transform between the input (θ, j) and fiber position (x l2 , y l2 ), we established the kinematic model based on the DH method to see whether higher accuracy can be achieved with more parameters involved.The DH method establishes coordinate systems for joints and models the joints with parameters represented by a i , d i , α i , and θ i (Figure 4). a i is the distance from Z i−1 to Z i (the length of links), d i the distance between X i−1 and X i , α i is the rotation angle around X i−1 from Z i−1 to Z i , and θ i is the rotation angle around Z i−1 (the rotation angle of joints).The relationship between the joints can be considered the transformation of the related coordinates described by the DH parameters.The coordinate transformation of both adjacent joints (represented by T d z a x Trans 0, 0, Rot , Trans , 0, 0 Rot , .6 ) is a translation matrix, which translates the distances l, m, and n along the x-, y-, and z-axis, respectively.z Rot , i ( ) q and x Rot , i ( ) a are rotation matrices representing rotations of θ i and α i angels around the z-and x-axis, respectively.The four transformation operations in Equation (6) can be expressed using Equations (7)-(10).
From Equations ( 6)-( 10) The end position and pose of the manipulator with n degrees of freedom can be derived stepwise as where R is the 3 × 3 submatrix representing the orientation of the end position and P = [p x , p y , p z ] T the position matrix.Equation (12) can be written as The parameters in Equation (13) are ideal.Owing to manufacturing and assembly errors, these parameters deviate from the desired values.If Δa, Δd, Δα, and Δθ are the corresponding errors, Equation (13) can be rewritten as The DH parameters of the positioner are listed in Table 2. Considering the center as the origin, the coordinates of the positioners are given by Equations ( 15) and ( 16) p a a d a cos cos cos sin sin cos sin sin , 15 q q q a q a q q = + -+ Assuming that the differences between the ideal and actual parameters are small, the coordinate error (Equation ( 17) can be written in a matrix form Equation (18), with A representing the coefficient matrix and Δω the parameter error matrix.

Measurement of the Fiber Ends
The calibration parameters are determined by the following process: (1) select a set of nominal points, (2) calculate the pulses needed using the nominal parameters according to the kinematic models, and (3) measure the actual fiber position for each point and calculate the best-fit parameters.Note that RFPs are driven by pulses.For LAMOST, the range limitations of the central arm (360°) and eccentric arm (180°) are uniformly divided into 63,000 pulses.Precisely measuring the fiber end is one of the key issues.To ensure the accuracy of the measurements, the measuring system was based on a digital universal tool microscope 19JC (Figure 5).Its overall measurement precision is 1 μm with the instability and uncertainty of the measurement no greater than 1 and 1.5 μm, respectively.Thus, the measurement accuracy is sufficiently high to perform calibration.However, this method is not sufficiently automated, thereby resulting in the need for manual measurements, adjustment, and visual imaging.When identifying and measuring the fiber ends, we first fixed the RFP with the universal tool microscope.The focal length of 19JC was adjusted to clarify the image shown in the relevant software.After each positioning operation of RFP, the sliding rails were adjusted so that the camera could image the fiber ends.The image was transferred into a computer and the coordinate positions were identified and recorded by relevant software.

DE Algorithm
Considering the disadvantages of LSM, we treated the calibration problem as an optimization problem.DE is one of the simplest and most effective global optimization algorithms with few parameters and high identification accuracy and optimal rate (Das & Suganthan 2011;Zhang et al. 2020b).Thus, we adopted the DE algorithm to calibrate the positioner parameters.DE is a population-based stochastic optimizer.It begins with a randomly initialized population of the NP vectors.Each vector contains D components that represent a candidate solution of the optimization problem, which can be expressed as where X i,G is the ith vector at the Gth generation and x i G j , is the jth component of the vector.In this study, is the compensation value of the calibration parameters.If x j,min and x j,max represent the minimum and maximum bounds, respectively, then x i j ,1 is initialized as ´-Three mutually exclusive integers (r 1 , r 2 , r 3 ä [1, NP]) are randomly generated to select the vectors for the mutation operation based on the following equation: where M i,G is the mutant vector and F a scalar parameter whose typical value is in the range [0.4,1].Here, we adjusted F as.
where N denotes the number of iterations.After mutation, a crossover operation is performed to enhance the potential diversity of the population as follows: , are the components of the trial vectors T i,G (generated by the crossover operation), CR is the possibility of crossover, and randj is a randomly chosen index.A selection operation is performed to select the target vector or trial vector in the next generation based on their fitness as follows: where (x aj , y aj ) and (x tj , y tj ) are the coordinates of the actual fiber and target positions, respectively, and n is the number of points used for calibration.The pseudocode for the DE algorithm is presented in Table 3.

Calibration Algorithm
The main idea of the proposed calibration algorithm is to iteratively achieve the best compensation values for the kinematic parameters by minimizing the designed fitness function.As positioners are driven by pulses, a series of pulses and the corresponding actual arrival positions must be obtained.The DE algorithm generates compensation values and the nominal values are compensated.Thus, the position calculated based on the kinematic models with the calibrated parameters is constantly approximated to the actual arrival position.The detailed calibration process is as follows: Step 1: The coordinates of the center and length of the arms are calculated using the five-point algorithm (Figure 6 Step 2: 84 evenly distributed points within the patrol area are selected (Figure 6(b)), and the corresponding rotation angles are calculated using Equation (2).
Step 3: Based on the angles obtained in Step 2, the corresponding pulses are calculated and used to motivate the motor rotation.The actual positions of the positioners are measured, and the position errors of the fiber ends are obtained by calculating the distance between the nominal points and fiber positions.
Step 4: A population of randomly initialized vectors is established.Each vector comprised compensation values for the corresponding kinematic parameters.For example, if we only take the length of arms as the calibration parameters, the vector can be represented as l l ,  15) and (16) for the DH models or Equation (3) for the geometry models).Hence, the sum of errors between the real and calculated positions can be used as the fitness function (Equation ( 25)).
Step 5: Steps 3 and 4 are repeated until the stop criterion is satisfied.
The workflow of the proposed calibration algorithm is shown in Figure 7.

Calibration Experiment and Verification
Before the calibration experiment, the room temperature is adjusted to 26°using an air conditioner, while the positioner carries out the positioning for half an hour to ensure that the experiment environment is stable and the positioner preheated.The calibration experiment was executed, comprising the collection of data using a universal tool microscope and calculations using MATLAB.The experiment was divided into four stages:  (1) The parameters were obtained using the five-point method.The arm lengths and center coordinates were calculated using MATLAB based on the coordinates of the five points measured using a universal tool microscope.The obtained parameters were regarded as nominal values.
(2) Based on Equation (2) and the parameters obtained above, the corresponding motivation angles and pulses of the 84 evenly distributed points were calculated (Figure 5).The positioner was positioned, and the corresponding coordinates were measured.
(3) The best compensation values for the nominal values were calculated using the DE algorithm.
(4) Twenty randomly distributed points were generated to verify the effectiveness of the calibration algorithm.The corresponding driving angles were then calculated based on the inverse kinematics with the modified parameters.Finally, the related coordinates were measured after positioning using a universal tool microscope, and the difference between the target and real coordinates was compared.

Results and Discussions
A LAMOST positioner was used to illustrate the calibration experiments and results.Taking positioner no.19165 as an example, the coordinates of the five points are listed in Table 4 (Appendix).
The center and arm lengths were calculated based on the data in Table 4.The best fitness values for the different kinematic models after Steps 2 to 5 in Section 3 are shown in Table 5 (Appendix).From Table 5, the calibration method considering the arm lengths, initial angles, center coordinates, and reduction ratio of the gearboxes obtains the best fitness function (GCR in Table 5).Thus, the parameters that need to be compensated are the lengths of the alpha and beta arm (l 1 and l 2 ), center coordinates (x ce and y ce ), reduction ratios (k rθ and k rj ) and initial angles (θ off and j off ).Comparing the fitness of GR and GRCP in Table 5, we find that considering the coupling of the motors results in a slight increase in the error.This suggests that the movements of the motors had no effect on each other.Simulations suggest that the fitness of the proposed algorithm is better than that of LSM (Table 5).The nominal and compensated values are listed in Table 6 (Appendix).
Twenty randomly distributed points were generated to verify the effectiveness of the proposed algorithm.The errors before and after calibration are shown in Figure 8.Before calibration, the average error and related deviation were 0.6387 ± 0.1125 mm.After calibration, the error was 0.1345 ± 0.1081 mm.This shows that, after calibration, the positioning accuracy improved by approximately 78.92%.
As described in Section 1, to achieve the required accuracy, the fiber view camera system is used to measure the actual position of the optical fiber, and the result is fed back to the control system.Thus, the target point is gradually attained by the multiple positioning of the positioner.We conducted closed-loop experiments to explore the number of steps required to reach the target points with the required accuracy based on the 20 randomly distributed points.We assumed that the maximum number of steps allowed for the positioner to arrive at the target point was four.The required accuracy for the LAMOST was 40 μm.If the required accuracy is achieved, no additional steps are required to control the positioner rotation.The results are presented in Figure 9.
Our calibration method improves not only the positioning precision but also the steps for the closed-loop control (Figure 9).For example, before calibration, no points are reached in the first step.However, after calibration in the first step, 15% of the points arrived within the accuracy.However, 10% of the points did not meet the required accuracy after the first four steps (Figure 9(b)).After analyzing the pulses driving the positioner, these points (the calculated pulses in the second step) were negative, that is, the motors needed to be rotated backward.The pulses in the third step are almost identical to those in the previous step.We believe that this phenomenon was caused by reverse errors owing to the backlash of the gearboxes.When the motor of the alpha or beta arm (or both) must be reversed, the actual rotation angle is less than the angle corresponding to the input pulses.11 of 20 randomly selected points need to reverse the central arm.For example, for one point, the pulses of the eccentric needed for the second, third, and fourth moves are −140, −137, and −14, respectively (− represents motor in reverse motion) while that for the eccentric arm are 1770, −117, and −135.For the central arm, the pulses of the second and third moves are almost the same.For the eccentric arm, the third and fourth moves are almost the same.At times, the fiber end seemed to not move at all.Consequently, additional steps were required to attain the desired accuracy.
To address this problem, we consider two methods.The first involved measuring the reverse errors.We can compensate for the input pulses to reduce the reverse error.However, we found that it was difficult to describe the reverse error as it is related to the input reverse pulses and reverse location.As an example, consider the reverse of the beta arm.We first made the alpha arm stand still, and the beta arm rotated at an angle of 15,000 pulses as the start position.The positions of the fiber ends were then measured and recorded.Additionally, the motor rotated forward and backward by approximately 5000 pulses each.The position was measured, and the difference in the input pulses of the beta arm between both positions was calculated.For the 10 experiments, the reverse error was approximately 122 pulses on average.However, when the motor rotated forward and backward by approximately 15,000 pulses each, the average reverse error was approximately 112 pulses.When the start position was changed to a position with 25,000 pulses of the beta arm as the start position, the reverse error was approximately 350 pulses on average.
The second method involves reducing the number of input pulses to avoid reverse motion.Based on the analysis of the experimental results, if we take the difference between the calculated pulses of the alpha arm and 300 pulses, and that between the calculated pulses of the beta arm and 100 pulses as the input pulses, then reverse motion can be avoided.This is known as the proposed "pulse-reduction strategy."For example, if the number of calculated pulses for the alpha arm is 1000, then the number of input pulses for the first step is 700.Only the first step requires the execution of the pulse-reduction strategy.The second step directly considers the calculated pulses as the input pulses.If the calibration method and pulsereduction strategy are adopted, all the points are within the desired accuracy in two steps or less (Figure 9).The number of steps and accuracy were improved by using the pulse-reduction strategy (Figure 9).
We studied the effects of the pulse-reduction strategy through simulations.The pulse reduction is equal to the reduction in the rotation angle because they can be linearly transformed into each other.For the alpha arm, the angle corresponding to 300 pulses is 0.0299 rad (1°.71), whereas for the beta arm, the angle corresponding to 100 pulses is 0.0051 rad (0°.29).The simulation results are presented in Figure 10.The largest error caused by the pulse reduction was 0.4845 mm, and the mean error was 0.3456 mm, which may cause collisions between the adjacent positioners.This is the main drawback of the proposed pulse-reduction strategy.One possible solution is to expand the arm width of the positioners when planning the trajectory.

Conclusions
Astronomical and cosmological research typically rely on large-scale observational surveys.Most of these are provided by RFP to improve survey efficiency.Theta-phi is one of the most used positioners.The function of the positioner is to place the fiber end precisely at the target point.The required accuracy for fiber positioning is generally approximately 4% of the diameter of the fiber core, which is relatively small.Misplacement of the fibers not only leads to a loss of light but also negatively impacts collision avoidance.The calibration of the positioner is necessary to improve its accuracy.First, we established the kinematic models of the theta-phi positioners based on the DH method and geometry of the positioners.
Because the geometry of the positioner is simple, a model based on it is widely adopted.To the best of our knowledge, this is the first study to derive a model based on the DH method.The error models were established based on the analysis of the error sources such as the arm length and reduction ratio of the gearboxes.As the commonly used LSMs have disadvantages such as the sensitivity of the initial values and complexity of calculation, we proposed a calibration method based on the DE algorithm.DE has advantages such as fewer parameters and less sensitivity to the initial values.From the calibration experiments based on the positioners of the LAMOST and universal tool microscope, the geometry model considering the reduction ratios of the alpha and beta arms performed the best.Simulations suggest that the proposed algorithm performs better than LSM.The calibration method was experimentally verified using 20 randomly distributed points within the patrol area.The results showed that the positioning accuracy improved by 78.94% after calibration.The calibration process is not once for all but annually for theta-phi positioners.
Because it is difficult for the positioners to position the fiber ends with the required accuracy in the first step, it is necessary to execute closed-loop positioning based on a  vision measurement system.Before calibration, 75% of the points could be reached by the positioner within the required accuracy in at most four steps.After calibration, this percentage increased to 90%.With the proposed pulsereduction strategy, all the points attained the desired accuracy in one or two steps.Before calibration, 5% of the points reached their target points within 10 μm, and 10% after calibration.With the pulse-reduction strategy, it further improved to 40%.The results showed that our calibration method using pulse reduction improved accuracy and reduced the number of steps required to reach the target point within the desired accuracy.Although calibration experiments were performed on the positioners for LAMOST, our calibration method and pulse-reduction strategy can also be used for the theta-phi positioners of other surveys because they all are SCARA-like planar positioners.

Figure 1 .
Figure 1.Patrol area of positioners.Red represents the patrol area of the positioners with (a) equal and (b) unequal arms.

Figure 2 .
Figure 2. Number of positioners that can reach a certain location.Points in the yellow, green, red, and blue areas can be reached by one, two, three, and four positioners, respectively.Red dots represent the center of the positioners.(a) Coverage of equal-arm positioners; (b) coverage of unequal-arm positioners.

Figure 3 .
Figure 3. Geometry of the theta-phi positioner.CE (x ce , y ce ), L 1 (x l1 , y l1 ), and L 2 (x l2 , y l2 ) are the center of the positioner, rotation center of the eccentric arm, and fiber end, respectively.

Figure 4 .
Figure 4. Schematic of the DH parameters.

Figure 5 .
Figure 5. Measurement system based on the universal tool microscope 19JC.
(a)).Point 1 is the home position of the positioner.Points 2 and 3 are obtained by only rotating the beta arm by 2 p and π, respectively.Points 4 and 5 are obtained by rotating the alpha arm by 2 p and π.Based on the geometry of the five points, the nominal center coordinates and arm lengths of the positioner can be calculated.
The vectors of the next generation are obtained by executing the mutation, crossover, and selection operations of the DE algorithm.During the selection process, the compensation values are added to the corresponding nominal values.The compensated parameters are used to calculate the positions of the fiber ends (Equations ( Figure 6.(a) Schematic of the five-point algorithm; (b) 84 evenly distributed points used for the calibration.

Figure 7 .
Figure 7. Workflow of the proposed calibration algorithm.

Figure 8 .
Figure 8.Comparison of the positioning precision before and after calibration.

Figure 9 .
Figure 9. Steps and accuracy of the close-loop positioning.Left: Steps needed to reach the points within the required accuracy before, after, and after calibration using the pulse-reduction strategy.Right: Corresponding accuracy of positioning after at most four steps.

Figure 10 .
Figure 10.Simulation results of errors caused by pulse reduction.Left and middle: errors caused by the pulse reduction of alpha and beta arms, respectively.Right: errors caused by pulse reduction.

Table 1
Overview of theta-phi Positioners

Table 3
Pseudocode of the DE Algorithm