Kinematic calibration and error compensation of a hexaglide parallel manipulator

Abstract

A calibration method of a hexaglide parallel manipulator is presented to improve its accuracy. A prototype of the hexaglide parallel manipulator is first proposed and its kinematics is analyzed. Through differentiating kinematic equations, 54 geometric error parameters are generated to present the pose error of the moving platform, on which an iterative algorithm for the calibration is based. The experiment starts with the data acquisition. All of measuring poses are newly selected based on the orthogonal design, and the deviations in each pose are measured by a laser tracker. Subsequently, 54 actual geometric parameters are identified by least squares method and compensated to the nominal kinematic model, which is assessed by 25 configurations to obtain the accuracy of the calibrated hexaglide parallel manipulator. It is discovered that the pose errors of the calibrated hexaglide parallel manipulator are significantly reduced and illustrate the validity of the calibration method to improve its accuracy. Finally, we discussed the feasibility of implementing this method in high-accuracy calibration of variant-scale parallel mechanisms.

Introduction

Besides the well-known Gough–Stewart platform, the hexaglide parallel manipulator (HPM) is another popular type of 6-degree-of-freedom (DOF) parallel mechanism, which is constructed by six prismatic-universal-spherical (PUS) parallel kinematic limbs. ¹ Its six actuators are all positioned on the fixed base, which significantly relieves the inertia effect of the end-effector and improves its dynamic characteristics. Many achievements with respect to HPMs have been presented,^2–5 and their positioning accuracy is often the emphasis among them.

The ultimate accuracy of the HPMs will be inevitably affected due to the existing errors in the processing and assembling, which is also known as geometric parameter errors. ⁶ This influence can be minimized by kinematic calibration and compensation, which mainly contains four steps: system modeling, error measurement, parameter identification and error compensation.^7,8 Among them, the accuracy of error measurement will directly influence the final accuracy of a calibration. A laser tracker is the typical external measuring device used in the calibration to improve the mechanism accuracy and especially suitable for a mechanism with spatially moving freedoms due to its three-dimensional (3D) tracking capability.^9,10 Researches on calibration of parallel mechanisms using laser tracker have been widely reported,^6,10–12 but few are with respect to the HPMs.

Besides using the laser tracker, scholars focused on calibrating the HPM by other measuring devices and related papers have been published. Ryu and Cha ¹³ established a comprehensive volumetric error model related to all geometric errors and optimized the design to achieve the required accuracy of end-effector. Rauf et al. ¹⁴ designed a device to measure the pose error of the end-effector, and its translational and rotational errors were decreased to less than 2 mm and 1°, respectively. Gong et al. ¹⁵ presented an error compensation method considering the elastic deformation and the clearance between the joints. Abtahi et al.^16,17 proposed a calibration method based on the relative measurement and simplified the measuring equipment using only three gauges, but the error model only consisted of 42 parameters omitting the orientation errors of each rail, and its maximum positioning error reached to around 1 mm. Ota et al. ¹⁸ proceeded a calibration for the HPM by a double ball bar (DBB) and obtained a positioning accuracy of 0.029 mm, but using the DBB is less flexible than using laser trackers to calibrate variant-scale parallel mechanisms. Reviewing these literatures, we reveal that the calibration of the HPM by laser trackers, which improves the positioning accuracy of the HPM to less than 0.03 mm, is rarely reported.

Another issue in the measurement, besides measuring devices, is the measuring pose selection, because the accuracy of an end-effector is differently influenced by its geometric errors in various poses. ¹⁹ Many scholars dedicate to optimizing the process of selecting poses to improve the final accuracy of a calibration,^20–22 but its effect is still controversial^23,24 and optimizing the pose selection makes the calibration complex. To best represent the diverse relationship between the geometric error and the pose error of the end-effector in different configurations, actuating it to reach each pose in its 6-DOF workspace is obviously effective, but it is time-consuming and impractical. Therefore, a practical method to select poses to represent the characteristics of all poses in 6-DOF workspace is significantly required. The orthogonal design, which has been widely used in the manufacturing,^25–27 is an effective way to deal with multi-factor experimental designs. The critical benefit of orthogonal design is that we can evaluate characteristics of large samples that are generated by the method of exhaustion with far fewer experimental units. The selected sample by the orthogonal design best represents the whole samples to be evaluated. Applying the orthogonal design to select poses in calibration process is rarely reported.

In this article, a calibration method of a proposed HPM by a laser tracker was presented. A prototype of the HPM was first proposed with its architecture description and coordinate establishment. Based on its kinematic analysis, its 54-parameter error model, which represents all geometric errors, ¹⁴ was established by differential method. An iterative calibration algorithm was subsequently designed, and the measuring poses selection based on the orthogonal design was originally proposed in the calibration process. Finally, a calibration experiment for the HPM was proceeded, and a Leica laser tracker was implemented to acquire the 6-DOF pose errors in each selected configuration. According to the proposed algorithm, the geometric errors were identified by the least squares method and utilized to compensate the nominal geometric parameters. Moreover, the pose errors of moving platform before and after compensation were compared in another 25 configurations to evaluate the effectiveness of the calibration and compensation.

Architectural description and inverse kinematics

The proposed HPM has a 6-PUS parallel mechanism, as depicted in Figure 1, which consists of a fixed base, a moving platform and six electric servo-motor-driven limbs; each limb includes a linear actuation, a universal joint, a length-fixed linkage, a uniaxial force sensor and a spherical joint. A servo motor connected with a ball screw is implemented as a prismatic joint in the linear actuation; two guide rails are fabricated to guaranty the linear motion of the slider that is fixed with the nut of the screw. The spherical joint is composed of three revolute joints whose rotation axes are non-coplanar; this structure aims to eliminate the clearance existed in socked-and-ball joints without introducing considerable friction. The fixed base has three slopes distributed symmetrically, and two linear actuations are parallelly arranged on each slope at given intervals.

Figure 1.

Prototype of the HPM.

In each limb, as shown in Figure 2, ${\mathit{C}}_i$ indicates the rotation center of the universal joint fixed on the slider of the prismatic joint, whose initial position is ${\mathit{A}}_i$ on the fixed base, and the unit vector of the prismatic joint is defined as ${\mathit{s}}_i$ ; ${\mathit{B}}_i$ is the center of spherical point fixed on the moving platform, and ${\mathit{k}}_i$ denotes the unit vector of the linkage vector $\overrightarrow{{\mathit{C}}_i{\mathit{B}}_i}$ . A fixed coordinate $O\{x,y,z\}$ is established at the center of fixed base, given the center of hexagon, O , as the origin. The x-axis coincides with the vector $\overrightarrow{{\mathit{A}}_{34}\mathit{O}}$ , where ${\mathit{A}}_{34}$ is the midpoint of ${\mathit{A}}_3{\mathit{A}}_4$ ; z-axis is normal to the fixed base surface, and y-axis can be generated by the right-hand rule. Similarly, a moving coordinate $O\prime \{x\prime ,y\prime ,z\prime \}$ is set up at the center of the moving platform; its origin, $\mathit{O}\prime$ , locates at the center point. The $x\prime$ -axis coincides with the vector $\overrightarrow{{\mathit{O}}^{\prime }{\mathit{B}}_{16}}$ , where ${\mathit{B}}_{16}$ is the midpoint of ${\mathit{B}}_1{\mathit{B}}_6$ ; $z\prime$ -axis is perpendicular to the moving platform, and $y\prime$ -axis can be defined then.

Figure 2.

Frame diagram and coordinate definition.

Let us define a configuration of the HPM as $\mathit{x}={\left[{\mathit{p}}^T,{\theta }^T\right]}^T={\left[p_x,p_y,p_z,{\theta }_x,{\theta }_y,{\theta }_z\right]}^T$ , where the vector p and $\mathit{\theta }$ , as two inputs of the inverse kinematics, represent the position and orientation of the moving coordinate with respect to the fixed coordinate, respectively. Implementing the orientation $\mathbf{\theta }$ , a rotation matrix is derived as

$$\mathit{R}=\left[\begin{array}{ccc}c{\theta }_{y}c{\theta }_z& c{\theta }_{z}s{\theta }_{y}s{\theta }_x-c{\theta }_{x}s{\theta }_z& s{\theta }_{x}s{\theta }_z+c{\theta }_{x}c{\theta }_{z}s{\theta }_y\\ c{\theta }_{y}s{\theta }_z& c{\theta }_{x}c{\theta }_z+s{\theta }_{x}s{\theta }_{y}s{\theta }_y& c{\theta }_{x}s{\theta }_{y}s{\theta }_z-c{\theta }_{z}s{\theta }_x\\ -s{\theta }_y& c{\theta }_{y}s{\theta }_x& c{\theta }_{x}c{\theta }_y\end{array}\right]$$

where s and c represent the sine and cosine functions, for example, $c{\theta }_y=\cos \left({\theta }_y\right)$ . In the ith closed-loop kinematic chain $\mathit{O}{\mathit{A}}_i{\mathit{C}}_i{\mathit{B}}_i\mathit{O}\prime$ , it is obvious that

$${\mathit{a}}_i+d_i{\mathit{s}}_i+{\mathit{L}}_i{\mathit{k}}_i=\mathit{p}+\mathit{R}{\mathit{b}\prime }_i$$

(1)

where ${\mathit{a}}_i=\overrightarrow{\mathit{O}{\mathit{A}}_i}$ , and $\mathit{b}{\prime }_i$ describes $\overrightarrow{\mathit{O}\prime {\mathit{B}}_i}$ in the moving coordinate $O\prime \{x\prime ,y\prime ,z\prime \}$ . $L_i$ denotes the length of the linkage in ith limb, and $d_i$ indicates the actuation value used in ith motor driver to actuate the moving platform to reach input configurations, cooperating with other motors. Rearranging equation (1) derives a quadratic equation with respect to $d_i$

$${\left(d_i{\mathit{s}}_i-{\mathit{h}}_i\right)}^2={\left({\mathit{L}}_i{\mathit{k}}_i\right)}^2$$

(2)

$${\mathit{h}}_i=\mathit{p}+\mathit{R}{\mathit{b}\prime }_i-{\mathit{a}}_i$$

Therefore, the actuation $d_i$ , which is the output of the inverse kinematics, can be derived from equation (2), and from the perspective of motion continuity, one solution is obtained

$$d_i={\mathit{h}}_i{\mathit{s}}_i-\sqrt{{({\mathit{h}}_i{\mathit{s}}_i)}^2-({\mathit{h}}_i^2-{\mathit{L}}_i^2)}$$

(3)

Error modeling

Expressing equation (1) in terms of small perturbations, we have

$$\mathrm{\Delta }{\mathit{a}}_i+\mathrm{\Delta }{d}_i{\mathit{s}}_i+d_i\mathrm{\Delta }{\mathit{s}}_i+\mathrm{\Delta }{L}_i{\mathit{k}}_i+L_i\mathrm{\Delta }{\mathit{k}}_i=\mathrm{\Delta }\mathit{p}+\mathrm{\Delta }\mathit{R}{\mathit{b}\prime }_i+\mathit{R}\mathrm{\Delta }{\mathit{b}\prime }_i$$

(4)

where $\mathrm{\Delta }{\mathit{s}}_i$ and $\mathrm{\Delta }{\mathit{k}}_i$ represent the orientation errors of the unit vector ${\mathit{s}}_i$ and ${\mathit{k}}_i$ , respectively. The position error of the ${\mathit{A}}_i$ is expressed as $\mathrm{\Delta }{\mathit{a}}_i$ , and that of the ${\mathit{B}}_i$ with respect to the moving coordinate is denoted as $\mathrm{\Delta }\mathit{b}{\prime }_i$ . $\mathrm{\Delta }{L}_i$ and $\mathrm{\Delta }{d}_i$ indicate the length error of the linkage and positioning error of the linear actuation, respectively. $\mathrm{\Delta }\mathit{p}$ is the position error of the moving platform, and the error of the rotation matrix, $\mathrm{\Delta }\mathit{R}$ , can be expressed by the asymmetric matrix generated by the $\mathrm{\Delta }\theta$ , the orientation error of the moving platform.

Since the displacement of the linear actuation in each limb is measured by an integrated grating sensor whose accuracy is 1 µm and is closed-loop controlled by feeding back the measured positions, the repeatability of the linear actuation measured is less than 1 µm. We omit the actuation error, $\mathrm{\Delta }{d}_i$ , since it is much less than other geometric errors of the HPM. Given two angular parameters in spherical coordinate, ${\theta }_i$ and ${\phi }_i$ , as shown in Figure 2, the unit vector $s_i$ can be defined as

$${\mathit{s}}_i={\left[\begin{array}{ccc}\sin {\theta }_i\cos {\phi }_i& \sin {\theta }_i\sin {\phi }_i& \cos {\theta }_i\end{array}\right]}^T$$

(5)

Differentiating equation (5), we derive $\mathrm{\Delta }{\mathit{s}}_i$ as

$$\mathit{\Delta }{\mathit{s}}_i={\mathit{\Delta }}_{si}{\mathit{d}}_{si}=\left[\begin{array}{cc}\cos {\theta }_i\cos {\phi }_i& -\sin {\theta }_i\sin {\phi }_i\\ \cos {\theta }_i\sin {\phi }_i& \sin {\theta }_i\cos {\phi }_i\\ -\sin {\theta }_i& 0\end{array}\right]\left[\begin{array}{c}d\theta \\ d{\phi }_i\end{array}\right]$$

(6)

Define ${\mathit{b}}_i=\mathit{R}\mathit{b}{\prime }_i$ , we have

$$\mathrm{\Delta }\mathit{R}{\mathit{b}\prime }_i=\mathrm{\Delta }\theta \times \left(\mathit{R}{\mathit{b}\prime }_i\right)=\mathrm{\Delta }\mathit{\theta }\times {\mathit{b}}_i$$

(7)

Substituting equations (6) and (7) into equation (4), the error model of the HPM can be established as

$$\mathrm{\Delta }{\mathit{a}}_i+\mathrm{\Delta }{L}_i{\mathit{k}}_i+L_i\mathrm{\Delta }{\mathit{k}}_i+d_i{\mathit{\Delta }}_{si}{\mathit{d}}_{si}-\mathit{R}\mathrm{\Delta }{\mathit{b}\prime }_i=\mathrm{\Delta }\mathit{p}+\mathrm{\Delta }\mathit{\theta }\times {\mathit{b}}_i$$

(8)

Multiplying ${\mathit{k}}_i^T$ on both sides of equation (8), ${\mathit{k}}_i^T{L}_i\mathrm{\Delta }{\mathit{k}}_i=0$ and

$${\mathit{k}}_i^T\mathrm{\Delta }{\mathit{a}}_i+\mathrm{\Delta }{L}_i+d_i{\mathit{k}}_i^T{\mathit{\Delta }}_{si}{\mathit{d}}_{si}-{\mathit{k}}_i^T\mathit{R}\mathrm{\Delta }{\mathit{b}\prime }_i={\mathit{k}}_i^T\mathrm{\Delta }\mathit{p}+\left({\mathit{b}}_i\times {\mathit{k}}_i\right)\mathrm{\Delta }\mathit{\theta }$$

(9)

where the position and orientation errors $\mathrm{\Delta }\mathit{p}$ and $\mathrm{\Delta }\mathit{\theta }$ can be expressed by the nine geometric errors in ith limb, including six errors in $\mathrm{\Delta }{\mathit{a}}_i$ and $\mathrm{\Delta }\mathit{b}{\prime }_i$ , two errors in $\mathrm{\Delta }{\mathit{s}}_i$ and one in $\mathrm{\Delta }{L}_i$ . Considering six limbs, the pose error $\mathrm{\Delta }\mathit{x}={[\mathrm{\Delta }{\mathit{p}}^T,\mathrm{\Delta }{\mathit{\theta }}^T]}^T$ can be generated by 54 parameters, which is coincident with the conclusion drawn by Rauf et al. ¹⁴ Denoting the 9-parameter geometric error vector in each limb as $\mathrm{\Delta }{\mathbf{\epsilon }}_{\mathbf{i}}$ and the 54-parameter one as $\mathrm{\Delta }\mathbf{\epsilon }$ , the error model of the HPM can be established as

$$\underset{\mathit{A}}{\underbrace{{\left[\begin{array}{cc}{\mathit{k}}_1^T& {({\mathit{b}}_1\times {\mathit{k}}_1)}^T\\ {\mathit{k}}_2^T& {({\mathit{b}}_2\times {\mathit{k}}_2)}^T\\ ⋮& ⋮\\ {\mathit{k}}_6^T& {({\mathit{b}}_6\times {\mathit{k}}_6)}^T\end{array}\right]}_{6\times 6}}}\mathrm{\Delta }\mathit{x}\underset{\mathit{B}}{\underbrace{={\left[\begin{array}{cccc}{\mathit{B}}_{m1}& & & \\ & {\mathit{B}}_{m2}& & \\ & & \ddots & \\ & & & {\mathit{B}}_{m6}\end{array}\right]}_{6\times 54}}}\underset{\mathrm{\Delta }\mathit{\epsilon }}{\underbrace{{\left[\begin{array}{c}\mathrm{\Delta }{\mathit{\epsilon }}_1\\ \mathrm{\Delta }{\mathit{\epsilon }}_2\\ ⋮\\ \mathrm{\Delta }{\mathit{\epsilon }}_6\end{array}\right]}_{54\times 1}}}$$

(10)

$$\begin{array}{l}{B}_{mi}={\left[\begin{array}{cccc}{k}_i^T& -k_i^{T}R& 1& d_i{k}_i^T{\mathrm{\Delta }}_{si}\end{array}\right]}_{1\times 9},\\ \mathrm{\Delta }{\epsilon }_i={(\begin{array}{cccc}\mathrm{\Delta }{a_i}^T& \mathrm{\Delta }{b\prime }_{i}T& \mathrm{\Delta }{L}_i& d_{si}^T\end{array})}_{9\times 1}^T\end{array}$$

Rearranging equation (9), we obtain

$$\mathrm{\Delta }\mathit{x}={\mathit{J}}_e\mathrm{\Delta }\mathit{\epsilon }$$

(11)

where ${\mathit{J}}_e={\mathit{A}}^{-1}\mathit{B}$ is called error the Jacobian matrix, which presents the mapping between the pose error of the moving platform, $\mathrm{\Delta }\mathit{x}$ , and the 54-parameter geometric error vector $\mathrm{\Delta }\mathit{\epsilon }$ .

Calibration method of the HPM

Iterative algorithm

By analyzing the difference between the computed and the actual poses of the HPM with the same actuations ${\mathit{d}}_j$ deriving from the specified posed ${\mathit{x}}_{0j}$ , the geometric parameter errors $\mathrm{\Delta }\mathit{\epsilon }$ can be identified and compensated to the kinematic model. Through the iterative procedure shown in Figure 3, the actual geometric parameters will be finally obtained. The detailed calibration procedure can be summarized as follows, including the interpretation of each variable in Figure 3.

Figure 3.

Iterative calibration procedure.

First, the selected calibration poses are specified as ${\mathit{x}}_{0j}$ , including three position and three orientation variables at the measuring configuration j, and their actuation vectors ${\mathit{d}}_j$ are calculated by the nominal inverse kinematic model, denoted as f_inverse, with the $54\times 1$ original nominal geometric parameter vector ${\mathit{u}}_0$

$${\mathit{d}}_j=f_{inverse}\left({\mathit{x}}_{0j},{\mathit{u}}_0\right),j=1,2,\mathrm{\dots },n$$

(12)

where n is the number of measuring configurations, and the ${\mathit{d}}_j$ will be computed only once.

Second, the computed and measured poses of the HPM actuated by the same ${\mathit{d}}_j$ are respectively obtained. The calculated poses in t-th iterative loop ${\mathit{x}}_{\mathrm{cjt}}$ are derived from the forward kinematic model as expressed in equation (13)

$${\mathit{x}}_{cjt}=f_{forward}\left({\mathit{d}}_j,{\widehat{\mathit{u}}}_t\right),j=1,2,\mathrm{\dots },n$$

(13)

$${\widehat{\mathit{u}}}_t={\widehat{\mathit{u}}}_{t-1}+\mathrm{\Delta }{\mathit{\epsilon }}_{t-1},t=1,2,\mathrm{\dots },T,{\widehat{\mathit{u}}}_0={\mathit{u}}_0,\mathrm{\Delta }{\mathit{\epsilon }}_0=0$$

where ${\hat{\mathit{u}}}_t$ denotes the estimated geometric parameter vector in t-th loop compensated by the error $\mathrm{\Delta }{\mathit{\epsilon }}_{t-1}$ derived in the (t−1)-th loop. T is the number of iteration. The actual poses ${\mathit{x}}_{\mathrm{mj}}$ are measured once by an external measuring device, AT 901-B Leica laser tracker, which will be introduced in the next section.

Third, due to the geometric parameter error resulting from the inevitable manufacturing and assembling error, the deviation between the computed and the actual poses exists and can be generated as

$$\mathrm{\Delta }{\mathit{x}}_{jt}={\mathit{x}}_{mj}-{\mathit{x}}_{cjt}={\mathit{J}}_{ejt}\left(\mathit{u}\right)\mathrm{\Delta }{\mathit{\epsilon }}_t$$

(14)

where $\mathrm{\Delta }{\mathit{x}}_{\mathrm{jt}}$ , ${\mathit{J}}_{\mathrm{ejt}}\left(\mathit{u}\right)$ and $\mathrm{\Delta }{\mathit{\epsilon }}_t$ are, respectively, the pose error vector, error Jacobian matrix and geometric error in configuration j and loop t.

Finally, the geometric parameter error $\mathrm{\Delta }\mathit{\epsilon }$ can be identified by nonlinear least squares estimation

$$\min L={\left(\mathrm{\Delta }{\mathit{X}}_p-\mathit{H}\mathrm{\Delta }\epsilon \right)}^T\left(\mathrm{\Delta }{\mathit{X}}_p-\mathit{H}\mathrm{\Delta }\epsilon \right)$$

(15)

$$\mathrm{\Delta }{\mathit{X}}_{pt}={\left[\begin{array}{cccc}\mathrm{\Delta }{\mathit{x}}_{1t}^T& \mathrm{\Delta }{\mathit{x}}_{2t}^T& \cdots & \mathrm{\Delta }{\mathit{x}}_{nt}^T\end{array}\right]}_{6n\times 1}^T,{\mathit{H}}_t={\left[\begin{array}{cccc}{\mathit{J}}_{e1t}^T& {\mathit{J}}_{e2t}^T& \cdots & {\mathit{J}}_{ent}^T\end{array}\right]}_{6n\times 54}^T$$

Then, if the ${\mathit{H}}_t^T{\mathit{H}}_t$ is nonsingular, that is, $\mathrm{rank}\left({\mathit{H}}_t^T{\mathit{H}}_t\right)=54$ , the geometric parameter error vector in t-th loop $\mathrm{\Delta }{\mathit{\epsilon }}_t$ can be derived as

$$\mathrm{\Delta }{\mathit{\epsilon }}_t={\left({\mathit{H}}_t^T{\mathit{H}}_t\right)}^{-1}{{\mathit{H}}_t}^T\mathrm{\Delta }{\mathit{X}}_{\mathrm{pt}}$$

(16)

Meanwhile, the norm of pose error vector $\mathrm{\Delta }{\mathit{X}}_{\mathrm{pt}}$ is generated as the index to judge the termination of the iteration, which will be finished once that norm is less than the specified accuracy $\mathit{\xi }$ , the actual geometric parameters can be achieved as $\mathit{u}={\hat{\mathit{u}}}_t+\mathrm{\Delta }{\mathbf{\epsilon }}_{\mathbf{t}}$ , otherwise, it will return to the second step.

Data acquisition

Orthogonal design for selecting measuring poses

The geometric errors do not vary with the configurations of the HPM, but the latter ones result in their different influence to the position and orientation accuracy. ¹⁹ Therefore, to comprehensively discover the pose errors due to the geometric parameters, it is reasonable to specify the calibration poses covering the whole 6-DOF workspace, which will be extremely time-consuming. But the orthogonal design, a mathematic methodology to arrange multi-level and multi-factor experiments, ²⁵ provides an efficient method to select the calibration poses. For example, we divided each DOF into eight levels, as shown in Table 1, so the amount of possible poses derived by the method of exhaustion in the 6-DOF workspace is $8^6=262,144$ . But if we use the orthogonal design to represent those poses, only 64 poses are selected based on the orthogonal array for ${\mathrm{L}}_{64}{\left(8\right)}^6$ design. These 64 poses typically represent all possible poses in the workspace to evaluate the characteristics of pose errors in this 6-DOF workspace.

Table 1.

Factors and levels of orthogonal design $L_{64}{\left(8\right)}^6$ .

Factors	Level 1	Level 2	Level 3	Level 4	Level 5	Level 6	Level 7	Level 8
x (mm)	50	40	20	10	0	−20	−40	−50
y (mm)	50	40	20	10	0	−20	−40	−50
z (mm)	495	490	485	480	465	460	450	440
${\theta }_x$ (rad)	0.15	0.1	0.08	0.05	0	−0.08	−0.1	−0.15
${\theta }_y$ (rad)	0.2	0.15	0.1	0.05	0	−0.1	−0.15	−0.3
${\theta }_z$ (rad)	0.2	0.15	0.1	0.05	0	−0.1	−0.15	−0.3

The orthogonal design table $L_{64}{\left(8\right)}^6$ is established using a software named Orthogonal Designing Assistant. L represents the orthogonal table and its subscript denotes the number of experiments; the number 8 in bracket denotes the number of levels of each factor and its superscript expresses the number of factors. This experiment consists of six factors, including three translations along the Cartesian coordinate $O\{x,y,z\}$ and three rotations about these axes. There are eight levels specified for each factor to cover its feasible value space in the workspace, as shown in Table 1. It is not even distributed because the pose errors near the edges of workspace can better present the geometric errors.

Coordinate system assignment

There are two calibration coordinates, including a moving fame $\mathit{O}{\prime }_c\{{\mathit{x}\prime }_c,{\mathit{y}\prime }_c,{\mathit{z}\prime }_c\}$ fixed on the moving platform and a fixed frame ${\mathit{O}}_c\{{\mathit{x}}_c,{\mathit{y}}_c,{\mathit{z}}_c\}$ attached with the fixed base. The poses of $\mathit{O}{\prime }_c\{{\mathit{x}\prime }_c,{\mathit{y}\prime }_c,{\mathit{z}\prime }_c\}$ with respect to ${\mathit{O}}_c\{{\mathit{x}}_c,{\mathit{y}}_c,{\mathit{z}}_c\}$ are determined as the actual configurations of the moving platform. The nominal fixed frame $\mathit{O}\{\mathit{x},\mathit{y},\mathit{z}\}$ in section “Architectural description and inverse kinematics” is first considered to be coincident with the calibration fixed frame ${\mathit{O}}_c\{{\mathit{x}}_c,{\mathit{y}}_c,{\mathit{z}}_c\}$ . But due to the existing geometric errors, the two moving frames $\mathit{O}\prime \{\mathit{x}\prime ,\mathit{y}\prime ,\mathit{z}\prime \}$ and $\mathit{O}{\prime }_c\{{\mathit{x}\prime }_c,{\mathit{y}\prime }_c,{\mathit{z}\prime }_c\}$ cannot be identical in each configuration. Therefore, the ultimate purpose of this work is compensating the geometric errors through the method described in section “Iterative algorithm” to make $\mathit{O}\prime \{\mathit{x}\prime ,\mathit{y}\prime ,\mathit{z}\prime \}$ be coincident with the $\mathit{O}{\prime }_c\{{\mathit{x}\prime }_c,{\mathit{y}\prime }_c,{\mathit{z}\prime }_c\}$ in every pose.

The moving calibration coordinate was established as expressed in Figure 4. There are three circularly distributed holes $R_1$ , $R_2$ and $R_3$ on the moving platform symmetric about z′-axis. Thus, the triangle $\mathrm{\Delta }{R}_1{R}_2{R}_3$ is equilateral and the its center is considered as the origin $\mathit{O}{\prime }_c$ . The x-axis is coincident with the vector $\overrightarrow{{O\prime }_c{R}_1}$ , and the z-axis is the normal to the plane $\mathrm{\Delta }{R}_1{R}_2{R}_3$ ; y-axis can be generated based on the right-hand rule. Meanwhile, four non-coplanar reference points are given as ${\mathit{R}}_{\mathrm{fi}}$ ( $i=1,2,\dots ,4$ ) to represent the configuration of moving platform in the coordinate of the laser tracker, because their relative positions to $\mathit{O}{\prime }_c\{{\mathit{x}\prime }_c,{\mathit{y}\prime }_c,{\mathit{z}\prime }_c\}$ are constant, considering the moving platform as a rigid body. Thus, implementing the measured data of reference points, the configuration of the moving platform can be derived by spatial fittings. Four magnetic pivots are placed on each reference point respectively to carry the reflector of the laser tracker.

Figure 4.

Establishment of moving calibration coordinate.

As shown in Figure 5(a), the fixed calibration coordinate was set up. Similarly, there are four prepared holes on the fixed base and symmetrically about the base center. ${\mathit{R}}_{b1}$ is the center, and ${\mathit{R}}_{b2}$ is designed on its x-axis in nominal coordinate $\mathit{O}\{\mathit{x},\mathit{y},\mathit{z}\}$ ; ${\mathit{R}}_{b3}$ and ${\mathit{R}}_{b4}$ are randomly placed. ${\mathit{R}}_{b1}$ is considered as the origin ${\mathit{O}}_c$ . The x-axis is parallel to vector $\overrightarrow{{\mathit{O}}_c{\mathit{R}}_{b2}}$ , and the z-axis is the normal to the plane $\mathrm{\Delta }{\mathit{R}}_{b1}{\mathit{R}}_{b3}{\mathit{R}}_{b4}$ ; the y-axis can be also derived. As depicted in Figure 5(b), three reference points with fixed magnetic pivots are also provided as ${\mathit{R}}_{\mathrm{fi}}$ ( $i=5,6,7$ ), which represent the configuration of the fixed base in the coordinate of the laser tracker.

Figure 5.

Fixed calibration coordinate.

Through the two coordinates, the relative pose of the moving calibration frame with respect to the fixed one can be accordingly generated in each calibration pose, which is considered as the actual pose of moving platform in the algorithm. Thus, in each calibration pose, the seven reference points, ${\mathit{R}}_{\mathrm{fi}}$ ( $i=1,2,\dots ,7$ ), are measured once in their static conditions. We established an experiment to test the reliability of this one-time measurement. One reference point was selected to be measured by the laser tracker 5 times, and the result shows that the repeatability of this experiment is less than 10 µm, which is not notable by the laser tracker since its measuring resolution is 10 µm. Therefore, measuring the seven reference points once in each pose is sufficient to calculate the actual configuration of the moving platform.

Calibration experiment

To verify the proposed method, calibration experiments were performed on the HPM (see Figure 6). The measuring distance of the Leica laser tracker was set at 1.5 m to ensure its stability and measuring accuracy. Meanwhile, three reference points ${\mathit{R}}_{\mathrm{ci}}$ were given with magnetic pivots fixed on the experimental platform to eliminate the accuracy degeneration due to the subtle change in the relative position between the HPM and the laser tracker, because the relative position between the ${\mathit{R}}_{\mathrm{ci}}$ and HPM is constant and the relative position between the laser tracker and HPM can be rebuilt by measuring ${\mathit{R}}_{\mathrm{ci}}$ when the subtle change happens. In total, 64 poses were selected according to the orthogonal design $L_{64}{\left(8\right)}^6$ , and the actual geometric parameters of the HPM were identified by the method in section “Iterative algorithm,” as summarized in Table 2 comparing with their nominal values.

Figure 6.

Experimental setup.

Table 2.

Actual geometric parameters.

		$\mathit{b}{\prime }_{\mathrm{ix}}$ (mm)	$\mathit{b}{\prime }_{\mathrm{iy}}$ (mm)	$\mathit{b}{\prime }_{\mathrm{iz}}$ (mm)	${\mathit{a}}_{\mathrm{ix}}$ (mm)	${\mathit{a}}_{\mathrm{iy}}$ (mm)	${\mathit{a}}_{\mathrm{iz}}$ (mm)	${\theta }_i$ (rad)	${\phi }_i$ (rad)	$L_i$ (mm)
1	N	95.101	34.200	−53.431	243.448	275.664	104.298	π/3	−2π/3	340
	A	94.796	34.153	−53.367	239.240	269.024	108.955	1.0414	−2.0964	339.816
2	N	−17.932	99.460	−53.431	117.008	348.664	104.298	π/3	−2π/3	340
	A	−18.039	99.319	−53.400	112.966	−341.515	108.212	1.0405	−2.0971	340.544
3	N	−77.168	65.260	−53.431	−360.456	73.000	104.298	π/3	0	340
	A	−77.044	65.261	−53.435	−345.858	−73.462	113.912	1.0484	0.0041	340.167
4	N	−77.168	−65.260	−53.431	−360.456	−73.000	104.298	π/3	0	340
	A	−77.079	−65.226	−53.307	−341.796	−72.226	116.785	1.0484	0.00037	340.785
5	N	−17.932	−99.460	−53.431	117.008	−348.664	104.298	π/3	2π/3	340
	A	−18.007	−99.142	−53.263	112.938	−340.372	112.367	1.0505	2.0930	340.047
6	N	95.101	−34.200	−53.431	243.448	−275.664	104.298	π/3	2π/3	340
	A	95.021	−34.090	−53.392	239.279	−268.456	111.576	1.0511	2.0947	340.337

N: nominal value; A: actual value.

It is revealed that all the geometric parameters are identified. Among them, the actual values of $\mathit{b}{\prime }_i$ , ${\theta }_i$ , ${\phi }_i$ and $L_i$ are rather close to nominal values, and the maximum deviation is less than 0.5% of its nominal value. But the deviation of ${\mathit{a}}_i$ is more considerable than others, and the maximum difference is nearly 4% of its nominal value, because the actual point ${\mathit{A}}_i$ is established by manually fixing the magnetic origins of the grating sensor in each limb, which is the most essential geometric error source of the HPM.

These 54 identified geometric parameters were then utilized to accordingly substitute the nominal parameters in the inverse kinematic model to compensate it, which will decrease the pose errors resulting from the deviation between the nominal and actual kinematics. To evaluate the accuracy of the HPM, it will be actually actuated by the compensated kinematic model and its pose errors will be measured and compared with those actuated by nominal kinematic model in the next section.

Calibration evaluation

To evaluate the calibration and analyze the accuracy of the HPM, as presented in Figure 7, the moving platform of the HPM will be actuated to 25 specified configurations containing three translational and three rotational components ( ${\mathit{x}}_{\mathrm{ek}}=[x_{\mathrm{ek}},y_{\mathrm{ek}},z_{\mathrm{ek}},{\theta }_{\mathrm{xek}},{\theta }_{\mathrm{yek}},{\theta }_{\mathrm{zek}}]$ , $k=1,2,\dots ,25$ ) twice, which are selected according to the orthogonal design $L_{25}{\left(5\right)}^6$ (see Appendix Table 4). All these 25 configurations are used to evaluate the positioning accuracy of the HPM. In the kth evaluation configuration, two actuations ${\mathit{d}}_{\mathrm{nk}}$ and ${\mathit{d}}_{\mathrm{ck}}$ will be derived from the nominal and compensated inverse kinematic models, respectively. Then, these two actuations will be input into the control system to practically drive the moving platform to the corresponding poses ( ${\mathit{x}}_{\mathrm{nk}}$ and ${\mathit{x}}_{\mathrm{ck}}$ ) and those will be measured by the laser tracker. Finally, these two poses will be compared with the kth specified pose to accordingly generate the pose errors before and after compensation ( $\mathrm{\Delta }{\mathit{x}}_{\mathrm{nk}}$ and $\mathrm{\Delta }{\mathit{x}}_{\mathrm{ck}}$ ); moreover, the accuracy of the moving platform before and after compensation can be analyzed after processing all 25 evaluation poses. Those pose errors are presented in Figure 8, and the logarithm of their magnitude in kth evaluation configuration, presented in equation (17), is utilized to present the pose errors with different orders of magnitude in the same chart with the same scale to facilitate comparing the results directly and clearly

$$\mathrm{\Delta }{\mathit{x}}_{nk}^l=\lg \left(\left|\mathrm{\Delta }{\mathit{x}}_{nk}\right|\right)={\left(\lg \left|x_{nk}\right|,\lg \left|y_{nk}\right|,\lg \left|z_{nk}\right|,\lg \left|{\theta }_{xnk}\right|,\lg \left|{\theta }_{ynk}\right|,\lg \left|{\theta }_{znk}\right|\right)}^T$$

(17)

$$\mathrm{\Delta }{\mathit{x}}_{\mathrm{ck}}^l=\lg \left(\left|\mathrm{\Delta }{\mathit{x}}_{\mathrm{ck}}\right|\right)={(\lg \left|x_{\mathrm{ck}}\right|,\lg \left|y_{\mathrm{ck}}\right|,\lg \left|z_{\mathrm{ck}}\right|,\lg \left|{\theta }_{\mathrm{xck}}\right|,\lg \left|{\theta }_{\mathrm{yck}}\right|,\lg \left|{\theta }_{\mathrm{zck}}\right|)}^T$$

where the superscript l represents the logarithm. The detailed values of the pose errors are summarized in Appendix Table 5, meanwhile those with maximum magnitude in each component of evaluation configurations are shown in Table 3.

Figure 7.

Flowchart of evaluating calibration.

Figure 8.

Pose errors compared before and after compensation: (a) logarithm of position error along x-axis, (b) logarithm of position error along y-axis, (c) logarithm of position error along z-axis, (d) logarithm of orientation error about x-axis, (e) logarithm of orientation error about y-axis and (f) logarithm of orientation error about z-axis.

Table 3.

Maximum pose errors before and after compensation.

	Before compensation (BC)	After compensation (AC)	AC/BC (%)
$\mathrm{\Delta }x$ (mm)	15.792	−0.027	0.171
$\mathrm{\Delta }y$ (mm)	2.646	0.030	1.134
$\mathrm{\Delta }z$ (mm)	18.267	−0.022	0.120
$\mathrm{\Delta }{\theta }_x$ (rad)	−4.395×10−2	3.853×10−4	0.877
$\mathrm{\Delta }{\theta }_y$ (rad)	−1.555×10−2	−6.484×10−4	4.170
$\mathrm{\Delta }{\theta }_z$ (rad)	3.940×10−2	6.519×10−4	1.655

From Figure 8 and Table 3, it is shown that all the six components of poses errors in 25 evaluation configurations are significantly decreased by the calibration and compensation, and these errors before compensation are mostly 10 times as large as those after compensation due to the geometric errors. After the compensation, the maximum translational and rotational errors of the moving platform are 0.03 mm along y-axis and $6.519\times {10}^{-4}$ rad (0.037°) about z-axis, respectively, and especially the pose errors $\mathrm{\Delta }x$ and $\mathrm{\Delta }z$ after compensation have dramatically fallen to 0.17% and 0.12% of those before compensation due to the considerable geometric errors resulting from manually fixing point ${\mathit{A}}_i$ . The experimental results suggest that the orthogonal design is an efficient method to select the calibration poses and demonstrate that the calibration and error compensation are valid for improving the accuracy of the HPM.

Discussions

As the typical external measuring device, the laser tracker is often utilized to detect the mechanism accuracy due to its large measuring scale and flexible application. Although the measuring error of the Leica laser tracker is 0.015 mm, which is more than 10 times as large as that of the DBB (0.001 mm), it can still reduce the error of the HPM to 0.03 mm, approaching the results via DBB (0.029 mm), reported by of HexaM. ¹⁸ Therefore, the laser tracker has the capability to be implemented in the high-accuracy calibration for variant-scale parallel mechanisms.

Conclusion

A calibration method of the HPM, using the laser tracker, is presented to effectively improve its accuracy. At first, a prototype of the HPM is proposed and its kinematics is analyzed. Its error model is then derived by differentiating the kinematics, and 54 geometric error parameters are generated to present the pose error of its moving platform. The calibration experiment is subsequently established after designing an iterative algorithm.

Calibration poses are selected based on the $L_{64}{\left(8\right)}^6$ orthogonal design to represent the possible configurations in its workspace. The moving platform of the HPM is actuated to these poses by the actuations derived from the nominal inverse kinematic model. Measuring the selected poses, 54 geometric errors are identified by the iterative algorithm. Experimental results suggest that the orthogonal design presents an effective method to select calibration poses.

To evaluate the accuracy of the HPM, 25 evaluation configurations are specified based on orthogonal design. The maximum translational and orientation errors of the moving platform are 0.03 mm and $6.519\times {10}^{-4}$ rad ( $0.{037}^{\circ }$ ), respectively. Compared with pose errors before and after the compensation, we reveal that the pose errors of the HPM after compensation are significantly reduced to less than 5% of those before compensation. Generally, the pose errors $\mathrm{\Delta }x$ and $\mathrm{\Delta }z$ have dramatically fallen to 0.17% and 0.12% of their initial errors by the compensation. The experimental results demonstrate that the proposed calibration and error compensation are capable to improve the accuracy of the HPM.

Finally, the accuracy of the HPM is compared with a reported same-type machine tool calibrated by the DBB, and the similar accuracy after calibration approves that implementing laser trackers in high-accuracy calibrations is feasible and effective.