Guohu ZHAO, Dn WANG, Lingo LIU, Jingzhen GUO,*,Wuyi CHEN, He LI

a School of Mechanical Engineering and Automation, Beihang University, Beijing 100083, China

b School of Electric and Automatic Engineering, Changshu Institute of Technology, Changshu 215500, China

c AVIC Aviation High-Technology Co., Ltd, Nantong 226011, China

KEYWORDS Compensation effect;Kinematic calibration;Parallel mechanism;Positioning accuracy;Structure parameters

Abstract Compared with serial mechanisms, the parallel mechanism (PM) theoretically exhibited higher positioning accuracy, dynamic performance, strength-to-weight ratio, and lower manufacturing cost,but they had not been widely used in the practical application.One key issue,positioning accuracy, which directly affected their performance and was greatly influenced by the errors of kinematic structure parameters was analyzed. To effectively enhance the positioning precision of PMs, a novel modeless kinematic calibration method, namely the split calibration, was presented and its compensation effect of the positioning error was comprehensively compared with that of an integrated method on two different types of PMs. A strange phenomenon-correct and incorrect identified results were derived from two different PMs by the same integrated method,respectivelywhich had not been reported yet was discovered, and the origin of it was revealed utilizing numerical simulations. Finally, respective merits and drawbacks of these two methods obtained in this paper provided underlying insights to guide the practical application of the kinematic calibration for PMs.

1. Introduction

Majority of machines widely used at present were serial mechanisms, wherein the actuators acted in serials. To pursue higher positioning accuracy1, dynamic performance2,strength-to-weight ratio3, and lower manufacturing cost1,PM was firstly proposed by Gough4. A generalized PM could be defined as a manipulator whose mobile platform possessed n degrees of freedom (DOF) and were linked to the fixed base by several closed-loop kinematic chains1.Since the PM’s structure was proposed, PM was applied to many different fields,such as flight simulator, precise assembly, pose adjustment,novel aircraft design, multidimensional loading device, and parallel machine tool (PMT)1,5. Wu6and Jia7et al. pointed out that PM based simulator was ideal for flight training and finding the causal factors in an accident. A 6-DOF PM was proposed to be used as a flight simulator by Stewart8.Franklin institute built a helicopter and a driving simulator for the Rock Island Arsenal and Benz4.In addition,PM was an ideal choice for the assembly of large and high-precision device,such as aircrafts, rockets, and satellites9-11. Wire-driven PM were also used as the aircraft suspension system in low-speed tunnel tests12-15. An16utilized the cable-driven PM to analyze the stealth performance in the compact range. Zhu17proposed a multi-rotor aircraft based on the Stewart platform. Meanwhile, Tian et al.18desired a morphing wing with the 3-DOF PM. Lu et al.19filed a patent which was named ‘‘A parallel drive mechanism of vectorial nozzle”. Furthermore, we proposed a novel multidimensional loading device based on the Stewart platform to investigate real mechanical properties of the helicopter thrust bearing under different loading conditions20. Results of another two papers of our research group demonstrated that material usually exhibited new mechanical properties under different loading conditions21,and PM based loading device were very suitable to perform these multidimensional loading experiments22, respectively. Besides, PM were also used to manufacture PMTs. Obvious advantages of the PM indicated that it should be suitable to machine hard materials in the aerospace field with high speed23. Li et al.24established a semi-analytical stiffness model for a novel 4-DOF PMT which was applied to fabricate large components. A novel type of PMT with large tilting capacity was introduced and fabricated by our research group25,26. Tian proposed a new calibration method for the inertial parameters of a 6-DOF parallel manipulator27.

Although PM possessed so many merits, it was not widely used in the practical application.One key issue for the PM was the positioning accuracy which directly affected the performance of them1,28,29. According to the research of Merlet1,PM usually possessed much better repeatability compared with the pose accuracy. Hence, majority of scholars laid considerable emphases on the latter. References et al.1,29pointed out that the geometrical transformation errors were the main factors of decreasing the positioning precision, and these errors could be reduced via various kinematic calibrations and compensations. Another way of enhancing the positioning accuracy was directly heightening the components’ quality of manufacturing and assembly, which would be costly and time-consuming30.Therefore,many scholars dedicated to measuring and utilizing the actual values of the manufactured components and assembled situations to improve the positioning accuracy of the PM.

Based on the comprehensive analysis of the PM,Majarena29reviewed the geometrical calibration method (internal or external sensors based methods,and constraint based method) to improve PM’s positioning accuracy. Chanal et al.31carried out the geometrical calibration experiment for a PMT by machining a dedicated part, but this calibration method could not be conducted for a general PM, on which machining was unable to be done. Using relative measurements of the joint vectors under different orientations of the mobile platform, Abtahi et al.32conducted the kinematic calibration experiment. Zhang et al.33established a geometric error model and analyzed the sensitivity of an overconstrained PM. Ren18proposed and performed a new calibration method based on the orientation constraint.This work was under the assumption that there was no clearance and deformation between each joint. However, for actual components, this assumption was hardly satisfied. Hence calibration results might vary toward different poses of the mobile platform, and corresponding errors might be generated due to the assumption.34Meanwhile, Wu et al.34pointed out that pose selections greatly affected the precision of the calibration effect (CE), and should be laid considerable emphases. Noted here that the phrase‘‘CE”denoted the positioning error of the PM after kinematic calibration experiments.Since the position workspace of the PM usually exhibited irregular shapes and orientations, the symmetrical measured poses used in his research might be unreasonable. Lots of researches had been carried out to optimize the selection of the poses,but,unfortunately, there had been no satisfied method yet.35Olarra36and Nategh37et al.pointed out that extreme calibration poses were more sensitive to geometrical parameters. However, interference problems might arise when the calibrated PMs were driven to the places near the boundary of the workspace,especially for those PMs whose zero switches were settled with relatively large initial errors. Noted here that installation errors of these switches could be compensated after the geometrical identification.35Additionally, to maximum consider the influence of the assembly clearance on the identified results,the selected poses used in the calibration experiments should be evenly distributed in the workspace.The orthogonal design method was adopted to select the desired calibration poses in this paper because this method could use the least possible poses of the end-effector representing characteristics of the whole configuration space.35Rauf proposed a general and adaptive measurement device, and performed the kinematic calibration experiment of a Hexaglide mechanism via this instrument.38Zhang et al.39carried out two different kinds of kinematic calibration experiments, namely the differential method and linear perturbation method, on a 2-DOF PM.

Reviewing the commonality of these literatures above, we might name them the integrated calibration (IC) method because it need not to disassemble the PM during the whole calibration process. Different from the IC method, our research group proposed a novel split calibration(SC)method to simplify the calibration algorithm, and conducted corresponding calibration experiments in this paper. Noted here that the SC and IC methods, mentioned in this paper, were two independent calibration methods. That was to say, the SC method presented in this paper was not used as the preliminary procedure of the IC method.1One main problem, which was not discovered and investigated by Guo et al.35, was the influence of the ill-conditioned matrix on the identified geometrical parameter errors.The other one that he had not considered was the impact of random (caused by the assembly clearance or the precision of the measuring devices) and nonrandom errors, occurred during the calibration experiments,on the accuracy of calibrated geometrical parameters. Huang et al.40used a modified regularization method to deal with the identification of structural errors with ill-conditioned identified matrix.However,an appropriate regularization parameter, α, might greatly affect the accuracy of identified results,and might not easy to be determined.41-43Sun et al.41introduced five different optimization methods for the kinematic parameter identification, and compared their identification accuracy by simulations. Moreover, using the regularization method they solved the ill-conditioned identification matrix problem, and obtained the calibration results. They pointed out that the hybrid genetic algorithm optimization method was more accurate and efficient.However,algorithm of these optimization methods were usually comprehensive. Therefore, we adopted the balance method, which was easier to be implemented and was ideal for severely ill-conditioned matrix, to solve the ill-conditioned problem.44

In this paper, a new modeless kinematic calibration method, namely the SC, were proposed to simplify the complex algorithm, and to enhance the stability and reliability in practical applications. Using the balance method solved the problem of the ill-conditioned calibration matrix, which made the solution of its inverse matrix much easier than other methods. Furthermore, the problem of wrong calibration results acquired by the IC method in situation of relatively large random errors was discovered in this paper,and reasons of it were investigated and validated by utilizing numerical simulations and experiments, respectively. Finally, the CE, merits, and defects of the SC and IC methods were obtained by comparing these two methods on two different types of PMs, which provided some instructions of using them to maximize the positioning accuracy of PMs in practical applications.

2. Research method

Two different calibration methods,SC and IC,toward the kinematic structure parameters of the PM were analyzed. Noticed here that the essential difference of these two calibration methods was whether the PM needed to be separated into different parts during the whole kinematic calibration process.To achieve higher positioning precision of PM, comprehensive comparisons between the SC and IC methods were analyzed using two types of PMs. One PM mentioned in this paper was the Stewart-Gough PM namely 6UPS while the other one was a 6PUS which was known as the Hexaglide PM.The detailed flow chart,shown in Fig.1,illustrated the analysis method and main procedures. Comparisons of these two independent methods were supposed to be performed on the 6UPS PM,but the identified results derived from the IC experiment of the 6UPS PM was unreasonable. Therefore, numerical simulations were performed to investigate the influences of random and nonrandom errors on the IC method. Afterward, another IC experiment conducted on a 6PUS PM with high manufacturing and assembly accuracy was used to validated the simulation results.Meanwhile,to investigate the calibrated positioning accuracy of these two methods,SC experiment was also performed on the 6PUS PM.Finally,merits and defects of the SC and IC methods were acquired though all of these experiments and simulations, and then selection principles of these two calibration methods, in the practical applications,were attained and drawn.

3. Kinematic analysis

Kinematic analysis mentioned here was the basis of the SC and IC methods. Hence, corresponding contents were presented here.

3.1. Architectural description

Fig. 1 Flow chart of analysis method.

One prototype of the PM, calibrated in this paper, was composed of a fixed platform,a mobile platform,and six identical motor-driven limbs,shown in Fig.2(a).Each chain,possessing a ball-screw prismatic joint, connected the lower base with a universal joint while linked the upper platform via a spherical joint, see Fig. 2(b). The schematic diagram of the PM was described in Fig. 3. To calibrate the kinematic parameters of the PM, a fixed Cartesian frame, Op-XYZ, was established at the centroid of the six universal joints and a mobile Cartesian frame, Ob-xyz, was settled at the center of the six spherical joints (known as the tool center point). Based on the two Cartesian frames,shown in Fig.3,coordinates of the universal joints Ai(i=1, 2, ..., 6) in the fixed frame and the spherical joints Bbiin the mobile frame could be obtained and described in Eq.(1) and Eq.(2), respectively. Bbiis Biexpressed with respect to the mobile frame.

Fig. 2 Prototype of PM.

Fig. 3 Schematic diagram of 6UPS PM.

3.2. Inverse kinematics

Took any leg as the research object and made up a closed vector loop, see the bold lines in Fig. 3(a). Utilizing the vector algebra, vector Licould be obtained from Eq.(5).

3.3. Forward kinematics

To simplify the differential operation toward the trigonometric function, rotational matrix T, shown in Eq. (3), could be expressed as the unit quaternion45form, see Eq. (9).

where q0, q1, q2, and q3were the components of the unit quaternion; meanwhile, we had an equation expressed as below.

Rearranging Eq. (8) derived a function with respect to px,py, pz, q0, q1, q2, and q3, shown in Eq. (11).

4. Calibration experiments and results

To maximize the positioning accuracy of PMs, two different kinematic calibration methods, namely the SC and IC methods,were proposed and comprehensively analyzed via two different types of PM in the following sections.

4.1. SC and IC experiments for the 6UPS PM

Detailed principles, experiments and CE of the SC and IC methods toward the 6UPS PM were studied in the following sections, respectively. Noted here that SC and IC, mentioned in this paper,were two independent calibration methods.That was to say, the SC method here was not used as the preliminary procedure to the IC method1.

4.1.1. SC method and experiment

Principle of the SC method:The external measuring device,AT 901-B Leica laser tracker (Leica Geosystems, Switzerland)whose accuracy was (15+6) μm/m, was used to measure the actual poses of PMs in this paper. Figs. 4 and 5 described the overall flow chart and schematic diagram of the SC method, respectively.

The SC method introduced in this paper mainly possessed two steps, shown in Fig. 4. In the first step, the PM was separated into two parts and coordinates of the spherical and universal joints, in the world frame, were measured via the laser tracker, see Fig. 5(a) and (b). Then using these obtained joint coordinates, the mobile frame, Ob-xyz, settled on the mobile platform and the fixed frame,Op-XYZ,fixed on the base could be established. In the second step, three things would be done to solve the initial length of each leg. Before and after the mobile platform was assembled onto the base through the legs,the four pivots fixed on the mobile platform would be measured to calculate the transfer matrix TT. Then, spherical joints’ coordinates, described in the fixed frame, could be obtained via TT.Finally,initial length of each leg could be calculated using the vector algebra. So far, all kinematic parameters of the 6UPS PM were acquired.

SC experiment for 6UPS PM: The PM had been separated into two parts and fixed on the floor using the hot-melt adhesive, shown in Fig. 6(a) and Fig. 6(b), before carrying out the SC experiment. Pivot 5 ～Pivot 7, settled on three different places of the base, were used to check whether the base had been moved or not during the whole calibration progress.The experiment could be failed and should be redone if any movement of the two separated parts occurred before the mobile platform was assembled onto the base.

Each universal joint rotary center coordinate in the world frame were fitted by using 50 points’ data, which were located upon a spherical surface and were obtained by the laser tracker via turning the limb, shown in Fig. 6(a). It was worth noting here that the distribution of these data should be as large and uniform as possible. Using the six rotary centers of the universal joint (A4, A5, A6), a circle center Opwas obtained by the least squares method and was set as the origin of the fixed frame,Op-XYZ,described in Fig.6(a).X axis of the fixed frame pointed from Opto the midpoint of the line segment,which was constructed by centers of universal joint 4 and 5.Z axis was perpendicular to the plane formed by three rotary centers of universal joint 4 to 6.Finally,Y axis could be determined based on the right-hand rule.Rotary center coordinates of spherical joints and the mobile frame,Ob-xyz,could be gotten by utilizing the same method as that for the universal joint and the fixed frame, see Fig. 6(b) and (c). Noticing here that the x axis of the frame Ob-xyz coincide with the line ObB45.B45was the midpoint of the segment B4B5, see Fig. 3 and Fig. 6(b).

Additionally, four non-collinear points, which were named by Pivot 1 ～Pivot 4 and fixed on the mobile platform, were used to calculate the transformation matrix(TT)of the mobile platform when it moved from the ground to the place where it assembled onto the base, depicted in Fig. 5. Coordinates of these four fixed points should also be measured before the mobile platform was assemble onto the base via six limbs.After all the work mentioned above were finished, the mobile platform was installed on the base and each limb was driven to shortest using the limit switches mounted on each limb.Then the coordinates of Pivot 1 to Pivot 4 were obtained by measuring the corresponding points. Since the spherical joints were fixed on the mobile platform, there was no displacement between them. Consequently, central coordinates of the six spherical joints,which were described in the fixed frame,could be achieved via the transformation matrix (TT). Utilizing the spatial SA software,lengths of each limb and the initial height between plane x-Ob-y and X-Op-Y were obtained. Hereto, all actual values of the kinematic parameters were acquired. Calibrated and theoretical structure parameter values of the 6UPS PM were summarized in Tables 1 and 2, respectively.

Fig. 4 Flow chart of the SC method.

Assessment of the SC method: 25 groups of typical poses were selected from the PM’s configuration space using the orthogonal experiment design, L25(5)6, to assess the CE. It was worth noting here that the distribution of these poses should be as large as possible, because the kinematic accuracy of the PM was usually worse near the boundary than those in the middle of the pose space.36,37Detailed poses, analyzed in this paper, were summarized in Table A1.

The PM was driven to the given 25 groups of poses,shown in Table A1, under the theoretical and calibrated structure parameters, respectively. Their actual poses with respect to each configuration were measured, using the Pivot 1 to 4, via the laser tracker, see Fig. 5(a) and Fig. 6(b). To illustrate the compensation effect of the calibration experiment, absolute errors (using the term ‘‘error” to represent the ‘‘a(chǎn)bsolute error”) of the position and rotation angle were plotted, based on the theoretical and compensation parameters, respectively.Fig.7 described the position errors along each axis in the fixed frame while Fig.8 depicted the rotary angle errors toward axes of the fixed frame.

For the theoretical structure parameter: all magnitudes of errors with respect to the position toward X and Z axes exceed 1 mm, which indicated the position absolute accuracy along these two axes were very bad, shown in Fig. 7. Although the Y axis position absolute accuracy kept relatively low values,locating in the range of [-0.60 mm, 0.29 mm], it fluctuated fiercely for the given configurations. However, magnitudes and fluctuations of each axis were improved obviously after calibration, especially along X and Z axes.

Different from the position errors, presented in Fig. 7, the compensation effects with respect to the angle error along each axis of the fixed frame were not very significant,see Fig.8.The magnitude of the rotary error toward Z axis declined obviously after calibration, but the performance of the fluctuation was equivalent to each other. Additionally, there was no palpable amelioration for errors of rotation angle related to the X and Y axes.

Further analysis, using the mathematical statistics method,was conducted to investigate potential effects of the calibration, summarized in Table 3 and Table 4. Table 3 manifested the average errors of position and angle along each axis based on the theoretical structure parameters. Noticed here that the mean errors of these poses, mentioned in Table 3, were calculated using the absolute value of each error. Consequently,positive and negative signs could be eliminated,which enabled the average values be reflected more realistic.Furthermore,the average performance of each error was improved obviously after calibration, except for the position error along Y axis.Standard deviations of each error reflected the fluctuations of each index and were summarized in Table 4. All standard deviations, shown in Table 4, decreased after compensation,except for the angular error around Z axis, which indicated that the consistency of kinematic accuracy was generally enhanced.

Form the comparison results, presented in Fig. 7, Fig. 8,Table 3, and Table 4, a conclusion could be drawn that both of the mean and standard deviation values of the errors decreased after the SC compensation.

4.1.2. IC method and experiment

Although the SC experiment could effectively enhance the kinematic accuracy of the PM, measuring large number of points’coordinates to fit the rotation centers of universal joints and spherical joints increased the work load and difficulties of operation.Meanwhile,the accuracy of the calibrated structure parameters, obtained via the SC method, might not be precise because the results were merely derived from one pose (each limb was driven to the shortest via the limit switches mounted on them) of the configuration space, see Fig. 4 and Fig. 5.Moreover, other errors which were caused by the change of limbs’ length, for example, the straightness of the limb movement, were unable to be considered. Therefore, IC method was proposed to overcome the defects of the SC method.Detailed mathematical principles as well as the procedures were mentioned in the following.

Fig. 5 Schematic diagram of SC method.

Error modeling of kinematic structure parameters: Error modeling, known as establishing the error Jacobian matrix,revealed the relationship between errors of the mobile platform’s pose and structure parameters.

Differentiating Eq. (7) yielded that

Fig. 6 Separate diagram of PM.

Table 1 Calibrated results of 6UPS kinematic parameters via SC method.

Table 2 Theoretical structure parameter values of 6UPS PM.

Fig. 7 Position errors before and after SC compensation.

Fig. 8 Errors of rotation angles before and after SC compensation.

Table 3 Analysis of average errors based on theoretical and calibration structure parameter values.

Table 4 Standard deviations of errors toward theoretical and calibration structure parameter values.

where ΔL=[dL01, dL02, ...,dL06]Tdenoted the deviations of each limb’s length;ΔP=(Δpx,Δpy,Δpz,Δγx,Δγy,Δγz)Twas the pose deviations of the mobile platform under certain actuations of each limb, and it was described as the third term in Eq. (32); J2=diag[(l1TT, -l1T), (l2TT, -l2T), ..., (l6TT, -l6T)];ΔS=[dBb1, dA1, dBb2, dA2, ..., dBb6, dA6]Trepresented the coordinate deviations of the universal joints and spherical joints; while J1could be described in Eq. (34).

where J=[J1-1,-J1-1J2]6×42,which was the error model used in the IC experiment and mapped the errors of the structure parameters onto the pose errors of the mobile platform.

Error model validation via simulated calibration:To validate the correctness of the error model established in this paper,the simulated calibration was conducted. Detailed procedure was described in Fig. 9. The given poses of the PM, denoted by Q1in Fig. 9, were selected via the orthogonal experiment design. Each pose of the configuration possessed six parameters, described as pxi, pyi, pzi, φ1i, φ2i, and φ3i. Six identical limbs contained 42 structure parameters.Consequently,to calibrate all the structure parameters, at least 7 different poses were required.To eliminate the unreasonable effects of certain poses, 25 sets of poses were selected via the orthogonal experiment design method. Factors and levels of the orthogonal experiment design L25(5)6were summarized in Table A1.Detailed illustrations with respect to Fig. 9 were described as below.

Firstly, 25 groups of theoretical poses, shown in Table A1 and denoted by Q1, whose dimension was 25×6, were given via the orthogonal experiment design method. Meanwhile, 25 sets of actuations (Λ1) for each limb and the error Jacobian matrix (Je1=[J1, J2, ..., J25]150×42) could be calculated using Eq.(8)and Eq.(35),respectively,based on the theoretical structure parameters E and Q1. Noted here that E=[L01, L02, ...,

L06, Bb1x, Bb1y, Bb1z, A1x, A1y, A1z, Bb2x, Bb2y, Bb2z, A2x, A2y,A2z, ..., Bb6x, Bb6y, Bb6z, A6x, A6y, A6z]T42×1.

Secondly, a sets of errors with respect to the theoretical structure parameters E were given and denoted by ΔE. Then the actual structure parameters (E+ΔE) could be used to simulate the measured poses obtained via the laser tracker in the real calibration experiment.Therefore,25 groups of actual poses, denoted by Q2, could be derived via the forward kinematics based on Λ1and E+ΔE. Meanwhile, pose errors between Q2and Q1could be calculated and expressed as ΔP1, whose dimension was 150×1.

Using the least squares identification, the identified errors of the structure parameters, ΔE′, whose dimension was 42×1,could be obtained based on Je1and ΔP1.Detailed formula was expressed as below

Fig. 9 Schematic diagram of IC simulated calibration.

Finally,identified structure parameter values were achieved and described as E+ΔE+ΔE′1. Then 25 sets of poses, Q3,could be obtained via the forward kinematics,and the norm of pose Q3and pose Q2were calculated. If ||Q2-Q3||∞＜σ, we obtained the final identified results of the structure parameters E+ΔE+ΔE′1. Otherwise, did the same calculations to achieve ΔE′1based on the pose Q3and compensated ΔE′1onto the structure parameters.Noticed here that the initial value of ΔE′1, shown in Fig. 9, was a zero vector.

According to the schematic diagram, described in Fig. 9,calculation programs was conducted and the final results which obtained by the numerical simulation were obtained after 14 iterations. Simulated calibration results, summarized in Table 5, indicated that the established error model mentioned above were correct,and its accuracy and efficiency were both high enough.

IC experiment based on the laser tracker: Procedures were basically the same as that in the simulated calibration except that the 25 sets of actual poses, denoted by Q2in Fig. 9, were directly measured via the laser tracker by actuating the mobile platform to the desired poses summarized in Table A2. The overall calibration setup system of the IC method was shown in Fig. 10(a), while Fig. 10(b) demonstrated the enlarged view of the PM.

A new mobile frame Ob1-x1y1z1was established by using the coordinates of Pivots 1, 3, and 4 which were assembled on the mobile platform via three reamed holes. Noticed that the three holes were located in a circle. Coordinates of the three pivots were measured when each limb of the PM were shortest.Then centers of the upper surface of each hole,represented by C1,C3,and C4,could be derived via the laser tracker.Meanwhile,a circle as well as its center,C,were established via C1, C3, and C4. Finally, the right hand frame Ob1-x1y1z1was built with its origin and x1axis coinciding with the circle center C and CC4,respectively.z1axis was perpendicular to the plane formed by C1, C3, and C4and pointed vertical upward. The mobile frame, named Op1-X1Y1Z1, could be established by using the same method as that of the fixed frame. According to the frames mentioned here, nominal structure parameters were summarized in Table 6.25 sets of actual poses,measuredvia the laser tracker, with respect to the target poses which were described in Table A1, were summarized in Table A2.Noticed here that the accuracy of the laser tracker,mentioned in this paper, was 15+6 μm/m and the values summarized in Table 7 were derived from the SA software using the least squares identification algorithm. Thus these values were not the absolute actual values.

Table 5 IC simulated calibration effect under the given errors of theoretical structure parameters.

Fig.10 IC calibration setup of 6UPS PM.

Using nominal structure parameters, and 25 sets of target and measured poses,the calibration algorithm was programed according to the schematic diagram mentioned in Fig.9.After 17 iterations, ‖Q2-Q3‖1converged to 2.64318, which indicated that Q3could not approach Q2close enough. However,the result demonstrated that the calibration structure parameters could be uniquely obtained. Meanwhile, results of the identified structure parameter,approaching a certain set of values, were summarized in Table 7.

Compared with the nominal and SC experiment values of the structure parameters, it was easy to observe that results derived by the IC experiment presented relatively large deviations with respect to the former values. The maximum structure parameter errors of the PM achieved 23.9248 mm, see coordinate A2zin Table 6 and 7. Meanwhile, results of the SC experiment indicated that it was impossible for the IC method to possess such a large error for the PM. Therefore,structure parameter values of the IC experiment were wrong.The unacceptable calibration results might be caused by the large assembly clearance errors, which could be observed by the large absolute pose error fluctuations in Figs. 7 and 8.

Influence of the random errors on the identified results: The effects of the clearance and the measurement errors on the identified results were exactly the same. Therefore,unreasonable calibration results, mentioned in the section of‘‘IC experiment based on the laser tracker”, could be investigated via the IC simulations with different assumed measurement errors. These simulated results of the IC method were summarized in Table 8.

Table 8 demonstrated that maximum errors of the IC results were affected greatly by the measurement/assembly errors. The maximum errors of the identified results dramatically increased with the rising of the measurement errors,especially when they were more than 0.50 mm.Meanwhile,Table 8 indicated that the IC method might not be suitable for PMs with relatively large clearance or using the measuring equipment whose accuracy was not high enough unless appropriate measures be taken. However, this was out of the scope of this paper. The accuracy of the laser tracker, used in this paper,was 15+6 μm/m, which indicated that the precision of the measurement was high enough. Consequently, unreasonable calibration results mentioned above were caused by the relatively large assembly clearance (random errors) of the 6UPS PM.

4.2. SC and IC experiments for the 6PUS PM

To investigate the calibration accuracy of the SC and IC methods,the SC and IC experiments were conducted with respected to a new 6PUS PM which possessed high quality of manufacturing and assembly, in the following sections.

4.2.1. SC experiment for the 6PUS PM

6PUS PM and its SC experiment setup system were described in Fig. 11. Procedures of the SC experiment of the 6PUS PMwere almost the same as those in the SC experiment of the 6UPS PM,except for calibrating the prismatic vectors of each limb. Therefore, only the prismatic vector calibrations were introduced here.One pivot was settled on each slider of the linear stage,see Fig.11(c).Each slider was driven to six different positions along the guideway, and six point coordinates could be obtained by using the laser tracker. Thus prismatic vectors of each guideway were fitted via the least square method. 25 sets of target poses were selected based on the orthogonal experiment design, depicted in Table A3.

Table 6 Nominal structure parameter values of PM used in IC experiment.

Table 7 Calibrated structure parameter values of PM based on IC experiment.

Table 8 Influence of the measurement errors on the identified structure results via IC simulations.

Fig. 11 6PUS PM and its SC experiment setup system.

Fig. 12 25 groups of absolute position errors using theoretical and SC structure parameters.

Fig. 13 25 groups of absolute angle errors using theoretical and SC structure parameters.

Table 9 Compensation effects of the SC experiment.

Table 10 Positioning errors after the IC compensation.

Using the nominal and SC structure parameters, absolute position and angle errors with respect to the theoretical target poses were evaluated and plotted in Fig.12 and Fig.13,respectively. Noticed that deviations displayed in Figs. 12 and 13 were derived by their absolute values.

Noted here that results before compensation referred to the left axis while those after compensation were plotted referring to the right axis,in Fig.12 and Fig.13.From these two figures we could see that both the position and angel errors declined dramatically after the SC compensation. Detailed comparison results were calculated and summarized in Table 9, where BC meant before compensation and AC denoted after compensation. The maximum position and angle errors in three axes were 19.16 mm and 2.72° before calibration while they decreased to 0.06 mm and 0.04° after compensation, respectively. Meanwhile, mean position and angle errors about each axis after calibration only accounted for 0.19%, 1.04%,0.07%,0.14%,4.81%,and 0.55%of the values before calibration, respectively. Noted here that large errors in Figs. 12 and 13 before calibration were mainly caused by the limit switches whose installation locations were not strictly controlled.Although their location errors could achieve several millimeters, their precise positions could be identified and compensated after kinematic calibration experiments.

4.2.2. CE comparison between the SC and IC experiments for 6PUS PM

Results of the IC experiments acquired from our previous paper35were summarized in Table 10.

Fig.14 Positioning mean errors of end-effector after SC and IC compensations.

To analyze the CE of the SC and IC experiments more intuitively, mean errors of the position and orientation after SC and IC experiments were plotted in Fig. 14(a) and (b), respectively. The position and angle errors of the end-effector after SC and IC compensation maintained at a very low levels.Both of the mean position and angle values toward these two calibration method were no more than 0.021 mm and 0.021°,which were almost equal to the accuracy of the laser tracker.Consequently, the accuracy of these two calibration methods were high enough, and existed at the same level.

5. Comprehensive comparisons of the SC and IC methods

To more clearly demonstrated characteristics of the SC and IC method for PMs, detailed comparisons (including the calibration accuracy,mathematical requirement,stability and reliability, and limitations) were obtained from the conducted works above, and were summarized here.

(1) Calibration accuracy. Both of the SC and IC methods could achieved the same calibration accuracy level for high manufacturing and assembly accuracy PMs with relatively large mobile space.

(2) Mathematical requirement.The SC method did not need to establish error models and to deal with the illconditioned matrix problem. Therefore, compared with the IC method, mathematical requirement of the SC method is extremely low.

Table 11 Comprehensive comparisons of the SC and IC methods.

(3) Stability and reliability.Besides the algorithms and computational accuracy, the IC method was more sensitive to the random errors that caused by the measurement accuracy or assembly clearance. However, the SC method was hardly influenced by the algorithms, computational accuracy, and the random errors. Therefore,the SC method possessed higher stability and reliability than that of the IC method.

(4) Limitations.Since the PM has to be disassembled in the SC method,it might damage the accuracy of the components. What’s more, operational security and inconvenient implementation issues might limit its applications with respect to large or very large PMs. It seemed to be not feasible to apply the IC method to calibrated PMs with relatively large random errors.Characteristics of these two methods mentioned above could be more succinctly listed in Table 11.

6. Conclusion

Two different kinematic structure parameter calibration methods,named the SC and IC,were introduced and systematically compared via two kinds of PMs in this paper.Both SC and IC experiments were effective to obviously enhance the positioning accuracy of the PMs. However, they all possessed their own merits and drawbacks. Main conclusions were obtained and drawn as below:

1) Mean positioning accuracy of PMs could be effectively enhanced as high as 0.021 mm and 0.021°, which were approximately the accuracy of the laser tracker, after the SC and IC compensations.

2) Simulated results of the IC method proved that random errors, which were cause by the clearance or measurement, greatly affected the calibrated result errors while nonrandom errors had no effect on the identified parameter values.

3) The SC method was a modeless method, with lower mathematical requirement and high stability and reliability, was more suitable for lower mobility PMs (compared with the IC method for lower mobility PMs).The only limitation of this method was that it could not be conducted on the PMs which were prohibited from disassembly.

4) The most prominent advantage of the IC experiment was unnecessary to disassembly the PMs. However, its drawbacks seemed to outweigh its merits.

Reviewing the merits and drawbacks of these two method,the SC method might be preferred unless the PMs already existed and not allowed to be disassembled. However, the IC method provided the possibility for the kinematic calibration of PMs under working loads,and it would be one of the focus in our future research.

Acknowledgement

This study was supported by the National Natural Science Foundation of China (No. 51905021).

Appendix A

Table A1 25 groups of typical configurations using the orthogonal experiment design L25(5) [6].

Table A2 25 groups of actual poses measured via the laser tracker corresponding to the target poses in Table A1.

Table A3 25 sets of target poses of the orthogonal experiment design L25(5)6.

Table A3 (continued)