Geometric accuracy design and error compensation of a one-translational and three-rotational parallel mechanism with articulated traveling plate

Abstract

The demands for advanced and flexible docking equipment are increasing in the fields of aerospace, shipbuilding and construction machinery. Position and orientation accuracy is one of the most important criteria, which would directly affect the docking quality. Taking a novel one-translational and three-rotational docking equipment, referred to as PaQuad parallel mechanism as example, this article proposed an accuracy improvement strategy by geometric accuracy design and error compensation. Drawing mainly on screw theory, geometric error modeling of PaQuad parallel mechanism was first carried out via four independent routes. Joint perturbations and geometric errors were included in each route error twist. Wrenches due to articulated traveling plate were applied to eliminate joint perturbations. Then, geometric accuracy design was implemented at component and substructure levels. The basic principle was to transfer geometric errors into dimensional or geometric tolerance. High-precision machining/assembling techniques were applied to satisfy the tolerance. Finally, error compensation resorting to kinematic calibration was implemented at mechanism level. It can be summarized as identification modeling, measurement planning, and parameter identification and modification. Maximum deviations of PaQuad parallel mechanism before calibration experiment were 0.01 mm, $-0.{197}^{\circ }$ , $-0.{522}^{\circ }$ , and $2.{908}^{\circ }$ . And they become 0.01 mm, $0.{027}^{\circ }$ , $0.{046}^{\circ }$ , and $0.{521}^{\circ }$ after kinematic calibration. Orientation accuracy of PaQuad parallel mechanism has improved one order of magnitude. It proves the effectiveness of accuracy improvement in terms of geometric accuracy design and error compensation.

Introduction

Advanced and flexible docking equipment is highly required for high-precision assembling of large-scale components. This increasing demand has become a primary concern in the fields of aerospace, shipbuilding, and construction machinery. ¹ Among various topology structures of docking equipment, parallel mechanisms (PMs) have been recognized as the most promising solutions due to their advantages in stiffness, dynamics, and accuracy.^2–5 Geometric accuracy is one of the fundamental and challenging problems for PMs to the application of docking equipment. Taking a one-translational and three-rotational (1T3R) PM (named “PaQuad PM”) as example, the efficient method to improve geometric accuracy was investigated in this article.

Geometric accuracy of PMs has been intensively studied in the past few decades. It can be roughly divided into two strategies, that is, geometric accuracy design ⁶ and error compensation.^7,8 Geometric accuracy design tries to obtain precise component and substructure through high-precision machining and assembling techniques. Resorting to kinematic calibration, error compensation is to identify and modify geometric errors after the PM was built. ⁹ There have been large numbers of researches focusing on kinematic calibration because of its economical and efficient features.^10–12 However, it is found that kinematic calibration cannot achieve the best compensation if component or substructure was not properly designed. Thus, improving geometric accuracy of PMs from both accuracy design and error compensation is necessary.

Prior to geometric accuracy design and error compensation, geometric error modeling is required. Up to present, three main methods have been adopted for geometric error modeling, that is, Denavit–Hartenberg (D-H) convention, product of exponential (PoE) formula, and screw theory. D-H convention and its variants are popular among serial manipulators due to its simplicity and convenience.^13–15 However, it cannot solve near-singular problem and little structural change may lead to the failure of error modeling.^16,17 On the basis of Lie group theory, PoE formula applies line geometry to describe joint axes. The smooth exponential mapping from Lie algebra se(3) to Lie group manifold SE(3) can deal with kinematic singular problems. ¹⁸ But the formulation of error modeling becomes complicated when PM was taken into account. On the contrary, joint axis can be expressed by screw theory in a clear and concise manner,^19–22 which is convenient for describing geometric error transmission. And geometric error model can be easily formulated through reciprocal product of twist and wrench.^23,24 Hence, screw theory–based approach has been widely accepted in the error modeling of PMs, for instance, 5 degree-of-freedom (DoF) machine tools.^8,25 Herein, screw theory is adopted for the geometric error modeling of PaQuad PM. However, different from single platform of common PMs, articulated traveling plate (ATP) of PaQuad PM is composed of two in-parts and one out-part. ²⁶ Additional rotation can be obtained by relative movements among in-parts. How to properly consider the effect of ATP in error modeling was one of the difficulties in this article.

Based on geometric error model, geometric errors can be restrained by geometric accuracy design at component and substructure levels. Some basic principles would be proposed to guide the machining and assembling processes in this article. With the acceptable accuracy provided by geometric accuracy design, kinematic calibration is then implemented for compensating geometric errors at mechanism level.^7–12 Kinematic calibration is the combination of theoretical analysis and experimental technique. It can be divided into three steps, identification modeling, measurement planning, and parameter identification and modification.

The main obstacle in identification modeling is ill-conditioning problem caused by approximately linear dependent identification vectors. To solve this problem, various methods have been proposed, such as, least square algorithm, ²⁷ Levenburg–Marquardt algorithm, ²⁸ genetic algorithm, ²⁹ or Kalman filtering approach. ³⁰ Although these methods are effective in dealing with ill-conditioning problem analytically, they are not efficient enough for mechanisms similar to PaQuad PM. An intuitive way is to directly solve the ill-conditioning equations. With this concept, Huang et al. ³¹ successfully found the solution for inverse identification equation of 3-PRS PM by regularization method. Herein, P, R, S denote actuated prismatic joint, revolute joint, and spherical joint. Based on Huang’s work, ridge estimation combining with L-curve selection ³² was adopted in this article for more stable and efficient identification modeling.

Measurement planning tries to achieve the best calibration results with least measuring configurations. For this purpose, external measurements by means of high-precision measuring equipment have been widely accepted: for instance, ball-bar system Hexapod ³³ and redundant 2-DoF PM, ³⁴ magnetic tooling balls for planar 5R PM, ³⁵ visual method for Delta PM ³⁶ and industrial robot, ³⁷ and laser tracker for 5-DoF machine tool. ⁹ External measurement can be divided into partial and complete measuring schemes.^38–40 Partial measuring scheme is able to derive optimized measurements by ignoring some DoFs. But additional algorithm might be required for identifying all geometric errors. And the measuring process is not general since different PMs correspond to different partial measurements. Comparing with partial measuring scheme, complete measuring scheme is easier to implement and the comprehensive measuring data simplifies identification process. It is applicable for all sorts of PMs, including PaQuad PM.

Parameter identifying and modifying is to carry out calibration experiments resorting to identification model and measurement plan. ⁴¹ By going through all measuring configurations, geometric errors are identified by identification model. Finally, they are compensated by adjusting inputs or modifying kinematic parameters in the control system. ⁴²

Overall, accuracy improvement strategy proposed in this article can be summarized as (1) geometric error modeling between all possible geometric errors and errors of PM based on screw theory, (2) geometric accuracy design at component and substructure levels, and (3) error compensation resorting to kinematic calibration at mechanism level, including identification modeling, measurement planning, and parameter identification and modification (see Figure 1).

Figure 1.

Accuracy improvement procedure of PaQuad PM.

Taking PaQuad PM as an example to demonstrate the accuracy improvement method, this article is organized as follows. Section “Geometric error modeling” formulated geometric error modeling of PaQuad PM. Some principles for geometric accuracy design were proposed in section “Geometrical accuracy design,” while kinematic calibration was implemented in section “Error compensation.” Section “Experimental verification” carried out experimental verification. And conclusions were drawn in section “Conclusion.”

Geometric error modeling

As is shown in Figure 2, PaQuad PM ¹ is composed of fixed base, ATP, and four identical PRS limbs. The PRS limbs link to ATP and fixed base with S joint and P joint. The axis of P joint is vertical to the plane of fixed base. ATP consists of in-part 1, in-part 2, and out-part. In-part 1 and in-part 2 connect to out-part by H joint and R joint. Herein, axes of the two joints are collinear and H denotes helical joint. PaQuad PM has 4 DoF in terms of 1T3R, and its detailed DoF analysis can be seen in Song et al. ¹ and Sun et al. ²⁶

Figure 2.

Virtual prototype and schematic diagram of PaQuad PM.

Some denotations were given in the schematic diagram as shown in Figure 2. The plane of fixed base was formed by point $A_i(i=1-4)$ , which were symmetrically distributed on a circle with center point O and radius a. Points $B_i$ , $C_i$ , and $D_i(i=1-4)$ denoted the centers of P joint, R joint, and S joint. The lengths of in-part 1 and in-part 2 were both $2b$ and the initial distance between them was $e$ . The moving distance of P joint and the length of bar were represented by $q_i$ and l. A fixed frame $O-\mathrm{xyz}$ was assigned to point O, of which the x-axis was collinear with $O{B}_2$ and z-axis was vertical to fixed base. Similarly, moving reference frame $P-\mathrm{uvw}$ was attached to point P at out-part. Its w-axis was at the same direction with H joint, and u-axis was parallel to $P{D}_2$ at initial configuration. Meanwhile, instantaneous frame $P-x\prime y\prime z\prime$ was fixed to point P, and the axe were parallel to those of $O-\mathrm{xyz}$ . For corresponding kinematic analysis, refer to Song et al. ¹

For PaQuad PM, there were four independent routes to transmit geometric errors from fixed base to end reference point, ith $(i=1,2,3,4)$ PRS limb plus ATP. On the basis of screw theory, each route error twist can be formulated by the superposition of joint perturbations and geometric errors. The concerned geometric errors can be expressed by deviations between adjacent axes. For the convenience of describing geometric errors, error reference frames at each joint axis were required. Thus, error modeling of PaQuad PM can be summarized as (1) divide joints in the same route into 1-DoF joints and establish error reference frame at the center of each joint, (2) define all possible geometric errors in the error reference frames and obtain route error twist by superposition of joint perturbations and component geometric errors, (3) eliminate joint perturbations by reciprocal product of route error twist and corresponding wrenches, and (4) combine errors of the four routes in the matrix form and achieve PM errors by deleting repeated geometric errors and modifying coefficients.

Note that error twists of the four routes were similar. First PRS limb plus ATP was selected as an example to demonstrate the error modeling process. As is shown in Figure 3, error reference frame $R_{j,i}(i=1,2,3,4;j=1,2,3,4,5)$ was located at the center of each 1-DoF joint, in which z_j,i-axis was collinear with the axis of jth 1-DoF joint, and x_j,i-axis was perpendicular to both z_j,i-axis and z_j+1,i-axis. Point $P_{j,i}$ represented the intersection of x_j−i,i-axis and z_j,i-axis. Frame $R_{0,i}$ for the ith PRS limb $(i=1,2,3,4)$ was defined by rotating frame $O-\mathrm{xyz}$ about z-axis with angle $(i-2)\pi /2$ . $P_{1,i}$ was the intersection of z_1,i-axis and the plane of fixed base. For the error reference frames assigned to H joint in in-part 1 and R joint in in-part 2, z_6,13-axis and z_6,24-axis were collinear to the axis of joints. The origin of H joint $P_{6,13}$ was the intersection of z_6,13-axis and $\overline{D_1{D}_3}$ , and x_6,13-axis was parallel to $\overline{D_2{D}_4}$ . Similarly, the origin of R joint $P_{6,24}$ was the intersection of z_6,24-axis and $\overline{D_2{D}_4}$ , and x_6,24-axis was parallel to $\overline{D_1{D}_3}$ . Meanwhile, frames and $P_{7,24}$ were established with their origins $P_{7,13}$ and $P_{7,24}$ both coincide with $O\prime$ . The axes of $P_{7,13}(R_{7,13}{R}_{7,24})$ were parallel to those of frame $R_{6,13}(R_{6,24})$ .

Figure 3.

Geometric errors of first PRS limb plus ATP.

According to the defined error reference frame, transformation matrices between frames can be calculated as

$$\{\begin{array}{c}\hfill {{}^{O\prime }\mathit{T}}_O=\mathrm{Trans}(-\mathit{r})\hfill \\ \multicolumn{1}{c}{{{}^O\mathit{T}}_{0,i}=\mathrm{Rot}(z,(i-2)\pi /2)}\\ \multicolumn{1}{c}{{{}^0\mathit{T}}_{1,i}=\mathrm{Trans}(x,b)}\\ \multicolumn{1}{c}{{{}^1\mathit{T}}_{2,i}=\mathrm{Trans}(z,q_i)\mathrm{Rot}(x,\pi /2)\mathrm{Rot}(z,{\theta }_{a,2,i})}\\ \multicolumn{1}{c}{{{}^2\mathit{T}}_{3,i}=\mathrm{Rot}(x,-\pi /2)\mathrm{Rot}(z,{\theta }_{a,3,i}+\pi /2)}\\ \multicolumn{1}{c}{{{}^3\mathit{T}}_{4,i}=\mathrm{Trans}(z,l)\mathrm{Rot}(x,\pi /2)\mathrm{Rot}(z,{\theta }_{a,4,i}+\pi /2)}\\ \multicolumn{1}{c}{{{}^4\mathit{T}}_{5,i}=\mathrm{Rot}(x,\pi /2)\mathrm{Rot}(z,{\theta }_{a,5,i}+\pi /2)}\\ \multicolumn{1}{c}{{{}^5\mathit{T}}_{6,13}=\mathrm{Trans}(x,-a)\mathrm{Rot}(x,\pi /2)\mathrm{Rot}(z,{\theta }_{a,6,13}+(2-i)\pi /2)}\\ \multicolumn{1}{c}{{{}^6\mathit{T}}_{7,13}=\mathrm{Trans}(z,e_{13})}\end{array}$$

(1)

Herein, $\mathrm{Trans}(-\mathit{r})(\mathit{r}=\overline{\mathrm{OO}\prime })$ is the transformation matrix of frame $O-\mathrm{xyz}$ with respect to frame $O\prime -x\prime y\prime z\prime$ , $\mathrm{Trans}(u,\kappa )$ is the translating matrix along “u” axis with a distance $\kappa$ , and $\mathrm{Rot}(u,\eta )$ is rotational matrix about “u” axis by an angle $\eta ·{\theta }_{a,j,i}(i=1,2,3,j=1,2,3,4)$ denotes the angle about jth joint axis of ith PRS limb. ${\theta }_{a,6,13}$ is rotational angle of H joint, and $e_{13}$ is the distance between point $P_{6,13}$ and point P. All possible geometric errors are listed as

$$\{\begin{array}{c}\hfill {{}^0\mathbf{\Delta }}_{1,i}={\left(\begin{array}{cccccc}\hfill {}^0\delta x_{1,i}\hfill & \hfill {}^0\delta y_{1,i}\hfill & \hfill 0\hfill & \hfill {}^0\delta {\alpha }_{1,i}\hfill & \hfill {}^0\delta {\beta }_{1,i}\hfill & \hfill {}^0\delta {\gamma }_{1,i}\hfill \end{array}\right)}^{\mathrm{T}}\hfill \\ \multicolumn{1}{c}{{{}^1\mathbf{\Delta }}_{2,i}={\left(\begin{array}{cccccc}\hfill {}^1\delta x_{2,i}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill {}^1\delta {\alpha }_{2,i}\hfill & \hfill 0\hfill & \hfill 0\hfill \end{array}\right)}^{\mathrm{T}}}\\ \multicolumn{1}{c}{{{}^2\mathbf{\Delta }}_{3,i}={\left(\begin{array}{cccccc}\hfill {}^2\delta x_{3,i}\hfill & \hfill 0\hfill & \hfill {}^2\delta z_{3,i}\hfill & \hfill {}^2\delta {\alpha }_{3,i}\hfill & \hfill 0\hfill & \hfill 0\hfill \end{array}\right)}^{\mathrm{T}}}\\ \multicolumn{1}{c}{{{}^3\mathbf{\Delta }}_{4,i}={\left(\begin{array}{cccccc}\hfill {}^3\delta x_{4,i}\hfill & \hfill 0\hfill & \hfill {}^3\delta z_{4,i}\hfill & \hfill {}^3\delta {\alpha }_{4,i}\hfill & \hfill 0\hfill & \hfill 0\hfill \end{array}\right)}^{\mathrm{T}}}\\ \multicolumn{1}{c}{{{}^4\mathbf{\Delta }}_{5,i}={\left(\begin{array}{cccccc}\hfill {}^4\delta x_{5,i}\hfill & \hfill 0\hfill & \hfill {}^4\delta z_{5,i}\hfill & \hfill {}^4\delta {\alpha }_{5,i}\hfill & \hfill 0\hfill & \hfill 0\hfill \end{array}\right)}^{\mathrm{T}}}\\ \multicolumn{1}{c}{{{}^5\mathbf{\Delta }}_{6,13}={\left(\begin{array}{cccccc}\hfill {}^5\delta x_{6,13}\hfill & \hfill 0\hfill & \hfill {}^5\delta z_{6,13}\hfill & \hfill {}^5\delta {\alpha }_{6,13}\hfill & \hfill 0\hfill & \hfill 0\hfill \end{array}\right)}^{\mathrm{T}}}\\ \multicolumn{1}{c}{{{}^6\mathit{P}}_{7,13}{{}^6\mathbf{\Delta }}_{7,13}=-\mathbf{A}{\mathbf{d}}_{{\mathrm{g}}_{7,13}^6}{{}^7\mathit{P}}_{6,13}{{}^7\mathbf{\Delta }}_{6,13},}\\ \multicolumn{1}{c}{{{}^7\mathbf{\Delta }}_{6,13}={\left(\begin{array}{cccccc}\hfill {}^7\delta x_{6,13}\hfill & \hfill {}^7\delta y_{6,13}\hfill & \hfill 0\hfill & \hfill {}^7\delta {\alpha }_{6,13}\hfill & \hfill {}^7\delta {\beta }_{6,13}\hfill & \hfill 0\hfill \end{array}\right)}^{\mathrm{T}}}\end{array}$$

(2)

Based on screw theory, the route error twist was obtained by the superposition of all possible geometric errors and joint perturbations in frame $O\prime -x\prime y\prime z\prime$

$${\mathit{\$}}_{\mathrm{t}}=\sum _{j-1}^5\Delta {\theta }_{\mathrm{a},j,1}{\mathit{\$}}_{\mathrm{ta},j,1}+\Delta {\theta }_{\mathrm{a},6,13}{\mathit{\$}}_{\mathrm{ta},6,13}+{\mathit{\$}}_{\mathrm{G},1}$$

(3)

$${\mathit{\$}}_{\mathrm{G},1}=\sum _{j-1}^6\mathbf{A}{\mathbf{d}}_{{\mathrm{g}}_{j-1,1}^{O\prime }}{{}^{j-1}\mathbf{P}}_{j,1}{{}^{j-1}\mathbf{\Delta }}_{\mathbf{j}\mathbf{,}1}+\mathbf{A}{\mathbf{d}}_{{\mathrm{g}}_{6,13}^{O\prime }}{{}^{6}P}_{7,13}{{}^6\mathbf{\Delta }}_{7\mathbf{,}13}$$

(4)

where $\Delta {\theta }_{a,j,1}$ denotes the perturbation of ${\theta }_{a,j,1}$ , $\mathbf{A}{\mathbf{d}}_{{\mathrm{g}}_{j,1}^{j-1}}$ is the $6\times 6$ adjoint transformation matrix of frame $R_{j,1}$ with respect to frame $R_{j-1,1}$ . ${{}^{j-1}P}_{j,1}=\left[\begin{array}{cc}{E}_3& \left[{{}^{j-1}r}_{j,1}\times \right]\\ {\mathrm{0}}_{3\times 3}& E_3\end{array}\right]$ , ${{}^{j-1}r}_{j,1}$ is the vector of point $P_{j,1}$ in frame $R_{j-1,1}$ , and ${\mathit{E}}_3$ is the third-order identity matrix. ${\widehat{\$}}_{ta,j,1}$ denotes the unit twist the jth 1-DoF joint. ${\widehat{\$}}_{ta,6,13}$ is the unit twist of H joint. For their expressions, refer to Song et al. ¹

Taking ATP into account, wrenches of PaQuad PM can be divided into two types: single wrench provided by each route and combination wrench provided by combined effects of opposite routes. ¹ In order to eliminate joint perturbations, reciprocal product of error twist and wrenches were applied to equation (3). When single wrench was applied, it turned out that

$${\widehat{\mathit{\$}}}_{\mathrm{wc},13}^{\mathrm{T}}{\mathit{\$}}_{\mathrm{t}}={\mathit{E}}_{\mathrm{ce},13}{\mathbf{\epsilon }}_{\mathrm{ce},13}=t_3{\mathit{E}}_{\mathrm{ce},1}{\mathbf{\epsilon }}_{\mathrm{ce},1}-t_1{\mathit{E}}_{\mathrm{ce},3}{\mathbf{\epsilon }}_{\mathrm{ce},3}$$

(5)

where ${\mathrm{\$}}_{\mathrm{wc},13}=t_3{\widehat{\$}}_{\mathrm{wc},1,1}-t_1{\widehat{\$}}_{\mathrm{wc},1,3}$ , ${\widehat{\$}}_{\mathrm{wc},1,i}$ denotes the unit wrench of constraints in ith $(i=1,3)$ PRS limb.

When combination wrench was applied, the reciprocal product yielded to

$${\widehat{\mathit{\$}}}_{\mathrm{wa},13,1}^{\mathrm{T}}{\mathit{\$}}_{\mathrm{t}}={\mathit{E}}_{\mathrm{ae},13,1}{\mathbf{\epsilon }}_{\mathrm{ae},13,1}$$

(6)

where ${\mathit{\$}}_{\mathrm{wa},13,1}=t_1{\widehat{\mathit{\$}}}_{\mathrm{wa},1,1}^{\mathrm{T}}-m_1{\widehat{\mathit{\$}}}_{\mathrm{wc},1,1}^{\mathrm{T}}$ , ${\widehat{\mathit{\$}}}_{\mathrm{wa},1,1}$ denotes the unit wrench of actuation in the first PRS limb.

Therefore, geometric error model of first PRS limb plus ATP can be formulated by combining equations (5) and (6). With the same manner, the error models of other three routes were obtained.

After deleting repetitive geometric errors and modifying corresponding coefficients in the four error models, error model of PaQuad PM was achieved by combining error models of the four routes into matrix form as follow

$${\mathit{\$}}_{\mathrm{t}}={\mathit{J}}_{\mathrm{x}}^{-1}{\mathit{E}}_{\mathrm{e}}{\mathbf{\epsilon }}_{\mathrm{e}}$$

(7)

$$J_x=\left[\begin{array}{c}{J}_{xa}\\ J_{xc}\end{array}\right],E_x=\left[\begin{array}{c}{E}_{ae}\\ E_{ce}\end{array}\right],J_{xa}={\left[\begin{array}{cc}{\mathrm{\$}}_{\mathrm{wa},13,1}& {\mathrm{\$}}_{\mathrm{wa},13,3}\end{array}\begin{array}{cc}{\mathrm{\$}}_{\mathrm{wa},24,2}& {\mathrm{\$}}_{\mathrm{wa},24,4}\end{array}\right]}^{\mathrm{T}}$$

$$J_{\mathrm{xc}}={\left[\begin{array}{cc}{\mathrm{\$}}_{\mathrm{wc},13}& {\mathrm{\$}}_{\mathrm{wc},24}\end{array}\right]}^{\mathrm{T}},E_{\mathrm{ae}}=\left[\begin{array}{c}\begin{array}{c}\begin{array}{c}{E^{\prime }}_{\mathrm{ae},13,1}\\ {E^{\prime }}_{\mathrm{ae},13,3}\end{array}\\ {E^{\prime }}_{\mathrm{ae},24,2}\end{array}\\ {E^{\prime }}_{\mathrm{ae},24,4}\end{array}\right],E_{\mathrm{ce}}=\left[\begin{array}{c}{E^{\prime }}_{\mathrm{ce},13}\\ {E^{\prime }}_{\mathrm{ce},24}\end{array}\right],{\epsilon }_{\mathrm{e}}={\left(\begin{array}{cc}{\epsilon }_{\mathrm{e},13}^{\mathrm{T}}& {\epsilon }_{\mathrm{e},24}^{\mathrm{T}}\end{array}\right)}^{\mathrm{T}}$$

The modified error coefficient matrix ${\mathit{E}}_{\mathrm{ae}}$ , ${\mathit{E}}_{\mathrm{ce}}$ , and the geometric error ${\mathbf{\epsilon }}_{\mathbf{e}}$ are shown in Appendix 2. Altogether, there were 64 geometric errors in PaQuad PM.

Geometrical accuracy design

As mentioned above, geometric accuracy design was to restrain geometric errors in component machining and substructure assembling process. According to the composition of PaQuad PM, the concerned components in the machining process were fixed base, sliding saddle, bar, S joint components, in-part 1, and in-part 2. The assembling techniques were mainly for P joint substructure, P joint and fixed base assembly, R joint substructure, R joint and P joint assembly, S joint substructure, and ATP substructure. Corresponding machining/assembling tolerances and the geometric errors are listed in Table 4 in Appendix 3.

Component machining

Corresponding to geometric errors, dimensional and geometric tolerances are divided in the component machining process. Dimensional tolerance describes the position accuracy of component parameters while geometric tolerance shows the orientation accuracy of component features. By setting correct tolerance, the high-precision machining process was capable of restraining geometric errors. Then, measurement of these geometric errors was carried out to provide reference for assembling. Only typical dimensional and geometric errors in the main components are shown below, and the rest of the errors follow the same tolerance setting and measuring principles.

For the fixed base, dimensional tolerance of the distance between opposite board slots was taken into account. It was to assure radius of the fixed base. Geometric tolerance about co-planarity of assembling reference planes was required for orientation accuracy of opposite PRS limbs. The dimensions were measured by coordinate measuring machine (CMM).

For the translating component, axis 1 and axis 2 were collinear with the axis of lead screw and R joint. Geometric tolerance for perpendicularity of axis 1 and axis 2 was the priority, which corresponded to orientation error ${}^1\delta {\alpha }_{2,i}(i=1,2,3,4)$ . It can be obtained by measuring the distances of axis 2 and reference plane $({\mathrm{A}}_{11},{\mathrm{A}}_{21})$ .

For the bar, dimensional tolerance relating to the distance between axis 2 and reference plane was required. Geometric tolerance about perpendicular of axis 2 and axis 3 was asked for orientation error ${}^2\delta {\alpha }_{3,i}(i=1,2,3,4)$ . They were measured by ${\mathrm{A}}_{12}$ , ${\mathrm{A}}_{22}$ . Furthermore, dimensional measurement was carried out for the reference of R joint substructure assembling. With wedge block and screw jack, ${\mathrm{A}}_{32}$ , ${\mathrm{A}}_{42}$ were obtained, then ${\mathrm{A}}_{52}$ can be calculated by dimensional chain. Next, ${\mathrm{A}}_{62}$ was measured by plug gage, finally ${\mathrm{A}}_{72}$ was achieved by ${\mathrm{A}}_{52}$ and ${\mathrm{A}}_{62}$ .

For the S joint components, geometric tolerance about intersections of rotating axis were the main concern, which required axis 3, axis 4, and axis 5 mutually intersecting. The measurement of axis 3 and axis 4 in component 1 is shown in Figure 4(a). The distance between axis 3 and reference plane $({\mathrm{A}}_{13})$ was obtained through fixture and lifting jack. By rotating the component about axis 4 and measuring ${\mathrm{A}}_{23}$ , orientation error ${}^3\delta x_{4,i}(i=1,2,3,4)$ can be calculated. With similar manner, orientation error of axis 4 and axis 5 ${}^3\delta x_{5,i}(i=1,2,3,4)$ was derived from ${\mathrm{A}}_{33}$ , ${\mathrm{A}}_{43}$ as in Figure 4(b). The orientation of axis 5 was assured by the measurement of ${\mathrm{A}}_{53}$ , ${\mathrm{A}}_{63}$ in component 3 (see Figure 4(c)).

Figure 4.

Accuracy design and measuring of the (a–c) S joint components and (d) plate I.

For the ATP, in-part 1 was taken as example to demonstrate the setting of machining tolerance and error measurement. Dimensional tolerance about the length of in-part 1 was restricted, as well as geometric tolerance for co-planarity of locating holes and H joint. Geometric error measurement was as shown in Figure 4(d). The distance between two locating holes $({\mathrm{A}}_{14})$ and distance of one locating hole to H joint $({\mathrm{A}}_{24})$ were derived by CMM measuring machine. And radius error ${}^5\delta x_{6,13,i}(i=1,3)$ can be calculated. Then, the distances between H joint axis, locating holes, and reference plane ( ${\mathrm{A}}_{34}$ , ${\mathrm{A}}_{44}$ , ${\mathrm{A}}_{54}$ ) were measured. Finally, the position error ${}^5\delta z_{6,13,i}(i=1,3)$ was achieved.

Substructure assembling

Based on the measured component errors, assembling technique is to further eliminate or minimize geometric errors. In general, assembling process can be summarized as (1) regard machining reference plane as assembling reference plane, (2) select several measuring points on the same plane or axis and then measure the distances between measuring points and reference plane, and (3) obtain required tolerance by adjusting components. The main substructures, P joint substructure, P joint and R joint assembly, S joint substructure, and ATP are chosen as example to demonstrate substructure assembling.

For the P joint substructure, translating accuracy was the primary consideration. Therefore, group assembling technique was adopted for the two sets of guide-slider. Installation of lead screw required the same height for bearing pedestals and connecting holes in sliding saddle. Three locations were selected to measure the distances between connecting hole and reference plane $({\mathrm{A}}_{15})$ . The location of lead screw was determined by mean value of ${\mathrm{A}}_{15}$ . Then, dimensions of bearing pedestals and reference plane $({\mathrm{A}}_{25},{\mathrm{A}}_{35})$ were measured. Finally, the height of lead screw was settled by adding adjusting gaskets. In this way, orientation error ${}^0\delta {\beta }_{1,i}(i=1,2,3,4)$ can be greatly reduced.

For the P joint and R joint assembly, co-planarity between axis 3 and lead screw axis was the most important, which corresponded to orientation error ${}^0\delta y_{3,i}(i=1,2,3,4)$ . First obtained the mean value of distance between reference hole and reference plane $({\mathrm{A}}_{16})$ by multiple measurements. Then, measured ${\mathrm{A}}_{26}$ and calculated ${\mathrm{A}}_{36}$ by dimensional chain. The thickness of adjusting gasket 1 was determined by ${\mathrm{A}}_{52}$ and ${\mathrm{A}}_{36}$ . Through dimensional chain, thickness of adjusting gasket 2 was derived.

For S joint substructure, dimensional tolerance of component 1 was required for determining the length of R joint substructure, which corresponded to position error $\Delta l_i(i=1,2,3,4)$ . As was shown in Figure 5(a), ${\mathrm{A}}_{17}$ , ${\mathrm{A}}_{27}$ were measured. And the length of R joint was adjusted by gasket between component 1 and bar. Geometric tolerance about intersections of three rotating axes was another concern. Regarding component 1 as the fixed reference, gasket was adopted to assure axis 3 in components 1 and 2 collinear. Assembling of component 3 relied on positions of axis 5 and location hole. ${\mathrm{A}}_{18}$ , ${\mathrm{A}}_{28}$ were measured from machining reference planes to location holes (see Figure 5(b)), which can be applied to calculate ${}^5\delta z_{6,13,i}(i=1,3)$ and ${}^5\delta z_{6,24,i}(i=2,4)$ . What’s more, the position of axis 5 was adjusted by ${\mathrm{A}}_{28}$ , ${\mathrm{A}}_{38}$ . They corresponded to the position error ${}^5\delta x_{6,13,i}(i=1,3)$ and ${}^5\delta x_{6,24,i}(i=2,4)$ .

Figure 5.

Assembling of (a and b) S joint substructure and (c) ATP.

For the ATP, position tolerance about the lengths of in-part 1 and in-part 2 is the most important. Geometric tolerance relating to concentricity of parts was demanded. In-part 2 connected to out-part through bearing (R joint). In-part 1 was fastened to out-part through screw/nut (H joint). The plate length and concentricity was guaranteed by adjusting R joint and H joint as was shown in Figure 5(c).

Error compensation

Identification modeling

Note that substantial geometric errors would lower the efficiency of kinematic calibration. Sensitivity analysis was carried out to exclude geometric errors that had little effect on the accuracy of PaQuad PM. The range of actual geometric errors was first obtained by error measurement. Considering the uncertainty of measurement, interval number was introduced to describe geometric errors as shown in Table 1. For the convenience of geometric error analysis, the upper and lower limits of geometric errors were shown by their absolute value.

Table 1.

Interval numbers of geometric errors (unit for position errors is mm and orientation errors is °).

Error no.	Geometric error	Interval number	Error no.	Geometric error	Interval number	Error no.	Geometric error	Interval number
1	${}^0\delta x_{2,i}$	[0.55, 059]	21	${}^0\delta {\beta }_{1,3}$	[0, 0.002]	41	${}^4\delta z_{5,2}$	[0.01, 0.02]
2	${}^0\delta x_{3,i}$	[0.03, 0.07]	22	${}^0\delta {\gamma }_{1,3}$	[0, 0.002]	42	${}^5\delta x_{6,24,2}$	[0.05, 0.21]
3	${}^0\delta {\alpha }_{1,1}$	[0, 0.002]	23	${}^1\delta {\alpha }_{2,3}$	[0.01, 0.05]	43	${}^5\delta z_{6,24,2}$	[0.08, 0.12]
4	${}^0\delta {\beta }_{1,1}$	[0, 0.002]	24	${}^2\delta {\alpha }_{3,3}$	[0.02,0.05]	44	${}^{7,24}\delta x_{6,24}$	[0, 0.01]
5	${}^0\delta {\gamma }_{1,1}$	[0, 0.002]	25	${}^3\delta x_{4,3}$	[0.00, 0.02]	45	${}^{7,24}\delta y_{6,24}$	[0, 0.01]
6	${}^1\delta {\alpha }_{2,1}$	[0.01, 0.05]	26	$\mathrm{\Delta }{l}_3$	[0.02, 0.06]	46	${}^{7,24}\delta {\alpha }_{6,24}$	[0, 0.02]
7	${}^2\delta {\alpha }_{3,1}$	[0.02, 0.05]	27	${}^4\delta x_{5,3}$	[0.01, 0.02]	47	${}^{7,24}\delta {\beta }_{6,24}$	[0, 0.02]
8	${}^3\delta {\alpha }_{4,1}$	[0.01,0.03]	28	${}^4\delta z_{5,3}$	[0.01, 0.02]	48	${}^0\delta x_{2,4}$	[0.63, 0.67]
9	$\mathrm{\Delta }{l}_1$	[0.03, 0.07]	29	${}^5\delta x_{6,13,3}$	[0.11, 0.17]	49	${}^0\delta y_{3,4}$	[0.11, 0.15]
10	${}^4\delta x_{5,1}$	[0.01,0.02]	30	${}^5\delta z_{6,13,3}$	[0.05, 0.09]	50	${}^0\delta {\alpha }_{1,4}$	[0, 0.002]
11	${}^4\delta z_{5,1}$	[0.01, 0.02]	31	${}^0\delta x_{2,2}$	[0.20, 0.24]	51	${}^0\delta {\beta }_{1,4}$	[0, 0.002]
12	${}^5\delta x_{6,13,1}$	[0.09, 0.15]	32	${}^0\delta y_{3,2}$	[0.07, 0.11]	52	${}^0\delta {\gamma }_{1,4}$	[0, 0.002]
13	${}^5\delta z_{6,13,1}$	[0.05, 0.09]	33	${}^0\delta {\alpha }_{1,2}$	[0, 0.002]	53	${}^1\delta {\alpha }_{2,4}$	[0.01, 0.05]
14	${}^{7,13}\delta x_{6,13}$	[0, 0.01]	34	${}^0\delta {\beta }_{1,2}$	[0, 0.002]	54	${}^2\delta {\alpha }_{3,4}$	[0.02, 0.05]
15	${}^{7,13}\delta y_{6,13}$	[0, 0.01]	35	${}^0\delta {\gamma }_{1,2}$	[0, 0.002]	55	${}^3\delta {\alpha }_{4,4}$	[0.01, 0.03]
16	${}^{7,13}\delta {\alpha }_{6,13}$	[0, 0.02]	36	${}^1\delta {\alpha }_{2,2}$	[0.01, 0.05]	56	$\mathrm{\Delta }{l}_4$	[0.05,0.09]
17	${}^{7,13}\delta {\beta }_{6,13}$	[0, 0.02]	37	${}^2\delta {\alpha }_{3,2}$	[0.02, 0.05]	57	${}^4\delta x_{5,4}$	[0.01, 0.02]
18	${}^0\delta x_{2,3}$	[0.21, 0.25]	38	${}^3\delta x_{4,2}$	[0.01, 0.03]	58	${}^4\delta z_{5,4}$	[0.01, 0.02]
19	${}^0\delta y_{3,3}$	[0.01, 0.05]	39	$\mathrm{\Delta }{l}_3$	[0.06, 0.10]	59	${}^5\delta x_{6,24,4}$	[0.09, 0.25]
20	${}^0\delta {\alpha }_{1,3}$	[0, 0.002]	40	${}^4\delta x_{5,2}$	[0.01, 0.02]	60	${}^5\delta z_{6,24,4}$	[0.08, 0.12]

Table 2.

Comparison of given orientation, orientation before calibration, and orientation after calibration.

No.	Given orientation			Orientation before calibration			Orientation after calibration
	$\alpha$	$\beta$	$\gamma$	$\alpha$	$\beta$	$\gamma$	$\alpha$	$\beta$	$\gamma$
1	−15	−15	135	−15.178	−15.493	135.519	−14.991	−14.983	134.712
2	−10	−10	165	−10.176	−10.410	167.099	−9.992	−9.997	164.697
3	5	−5	195	4.813	−5.473	195.058	5.008	−5.033	194.788
4	−5	−5	195	−5.186	−5.450	195.079	−4.9858	−5.0143	194.775
5	−5	5	195	−5.191	4.529	195.728	−5.001	4.974	195.281
6	5	5	195	4.802	4.511	195.548	4.994	4.953	195.222
7	0	−15	0	−0.140	−15.434	0.434	0.018	−14.989	−0.316
8	0	−10	0	−0.142	−10.438	0.035	0.011	−9.995	−0.312
9	0	−5	0	−0.145	−5.447	0.117	0.008	−5.004	−0.244
10	0	5	0	−0.153	4.538	0.598	−0.001	4.985	0.182
11	0	10	0	−0.163	9.520	1.287	−0.005	9.979	0.4982
12	0	15	0	−0.163	14.498	1.723	0.027	14.972	0.521
13	−15	0	0	−15.150	−0.408	1.421	−14.984	0.013	0.106
14	−10	0	0	−10.144	−0.427	0.658	−9.991	0.009	0.106
15	−5	0	0	−5.146	−0.438	0.142	−4.996	0.004	0.099
16	5	0	0	4.854	−0.460	0.162	5.000	−0.017	0.086
17	10	0	0	9.850	−0.464	0.543	10.003	−0.023	0.074
18	15	0	0	14.843	−0.470	1.175	15.002	−0.033	0.071
19	0	0	15	−0.157	−0.458	14.978	0.001	−0.009	14.962
20	0	0	75	−0.182	−0.489	75.038	−0.011	−0.013	75.041
21	0	0	135	−0.158	−0.522	135.035	−0.008	−0.023	135.027
22	0	0	195	−0.120	−0.515	194.995	0.003	−0.033	194.929
23	0	0	255	−0.107	−0.477	255.064	0.010	−0.025	255.113
24	0	0	315	−0.133	−0.450	315.043	0.017	−0.017	315.077

Referring to geometric error model in equation (7), sensitivity coefficients of geometric errors with respect to errors of PaQuad PM can be expressed as

$${\kappa }_{rm}=\sqrt{{\displaystyle \sum }_{i=1}^3{J}_{\mathrm{c}i,m}^2},{\kappa }_{\alpha m}=\sqrt{{\displaystyle \sum }_{i=4}^6{J}_{\mathrm{c}i,m}^2},m=1,2,3,\dots ,60$$

(8)

where ${\mathit{J}}_c={\mathit{J}}_x^{-1}{\mathit{E}}_e$ is the error mapping matrix. ${\kappa }_{\mathrm{rm}}$ , ${\kappa }_{\alpha m}$ denoted the effects of mth geometric error ${\epsilon }_{\mathrm{e}m}$ to position error ${\delta }_r$ and orientation error ${\delta }_{\alpha }$ . And

$${\delta }_r={\kappa }_{\mathrm{rm}}{\epsilon }_{\mathrm{e}m},{\delta }_{\alpha }={\kappa }_{\alpha m}{\epsilon }_{\mathrm{e}m}$$

(9)

By replacing ${\epsilon }_{\mathrm{e}m}$ with interval numbers, error mapping in equation (9) can be expressed as

$$[\Vert {\delta }_m\Vert ]={\kappa }_m\left[{\epsilon }_{\mathrm{e}m}\right],m=1,2,3,\dots ,60$$

(10)

where $[\mathrm{•}]$ denotes interval range and $\Vert {\delta }_m\Vert$ is the volumetric errors of PaQuad PM.

Sensitivity of geometric errors can be obtained by substituting the upper limit into equation (17). The results were shown in Figure 6. It is summarized that No. 5, 35 errors $({}^0\delta {\gamma }_{1,i})$ and No. 11, 41 errors $({}^4\delta z_{5,i})$ had little effects to errors of PaQuad PM, which can be ignored in the kinematic calibration. No. 1, 9, 12, 13, 18, 26, 29–31, 39, 42, 43, 48, 56, 59, 60 errors ( ${}^0\delta x_{2,i}$ , $\Delta l_i$ , ${}^5\delta x_{6,13,i}$ , ${}^5\delta z_{6,13,i}$ ) had greater influences on errors of PaQuad PM, and they were regarded as primary concerns for error compensation.

Figure 6.

Sensitivity analysis of the geometric errors.

After geometric error sensitivity analysis, identification modeling between geometric errors and measuring data can be implemented. Laser Tracker was adopted as the measuring device, whose measuring reference frame is assigned automatically. Closed-loop equation is formulated as

$${\mathit{r}}_{\mathrm{c}}={\mathit{r}}_{O\prime }+\mathit{R}{\mathit{l}}_{\mathrm{c}}={\mathit{R}}_{\mathrm{t}}{\mathit{r}}_{\mathrm{m}}+{\mathit{r}}_{\mathrm{M}}$$

(11)

where ${\mathit{r}}_{O\prime }$ , ${\mathit{r}}_{\mathrm{c}}$ , ${\mathit{r}}_{\mathrm{M}}$ denoted the position vectors of point $O\prime$ , P, $O_m$ in frame $O-\mathrm{xyz}$ . ${\mathit{l}}_{\mathrm{c}}$ , ${\mathit{r}}_{\mathrm{m}}$ were position vectors of point P in frame $O\prime -\mathrm{uvw}$ and frame $O_{\mathrm{m}}-x_{\mathrm{m}}{y}_{\mathrm{m}}{z}_{\mathrm{m}}$ . ${\mathit{R}}_{\mathrm{t}}$ represented the transformation matrix of frame $O_{\mathrm{m}}-x_{\mathrm{m}}{y}_{\mathrm{m}}{z}_{\mathrm{m}}$ with respect to frame $O-\mathrm{xyz}$ .

Taking the first-order perturbation on both sides of equation (11) yielded

$${\mathit{r}}_{\mathrm{c}}={\mathit{r}}_{O\prime }+\mathit{R}{\mathit{l}}_{\mathrm{c}}={\mathit{R}}_{\mathrm{t}}{\mathit{r}}_{\mathrm{m}}+{\mathit{r}}_{\mathrm{M}}$$

(12)

where $\mathrm{\delta }{\mathit{r}}_{O\prime }$ , $\mathrm{\delta }\mathit{\alpha }$ denoted the position and orientation errors, respectively. $\Delta {\mathit{r}}_{\mathrm{m}}={\hat{\mathit{r}}}_{\mathrm{m}}-{\overline{\mathit{r}}}_{\mathrm{m}}$ . ${\hat{\mathit{r}}}_{\mathrm{m}}$ , ${\overline{\mathit{r}}}_{\mathrm{m}}$ were the measured and ideal position vectors of point P in frame $O_{\mathrm{m}}-x_{\mathrm{m}}{y}_{\mathrm{m}}{z}_{\mathrm{m}}$ . $\Delta {\mathit{r}}_{\mathrm{M}}$ represented position error vector of point $O_{\mathrm{m}}$ in frame $O-\mathrm{xyz}$ .

Substituting geometric model in equation (7) into equation (12) yielded

$${\mathit{J}}_{\mathrm{N}}\Delta {\mathit{p}}_{\mathrm{N}}={\mathit{R}}_{\mathrm{t}}\Delta {\mathit{r}}_{\mathrm{m}}+\Delta {\mathit{r}}_{\mathrm{M}}$$

(13)

where

$$J_{\mathrm{N}}=\left[\begin{array}{cc}{J}_r-\left[\left(R{l}_c\right)\times \right]J_{\alpha }& R\end{array}\right],\Delta {\mathit{p}}_N={\left[\begin{array}{cc}{\epsilon }^{\mathrm{T}}& \Delta {\mathit{l}}_c^{\mathrm{T}}\end{array}\right]}^{\mathrm{T}},\mathit{\epsilon }={\left(\begin{array}{cccc}{\mathit{\epsilon }}_1& {\mathit{\epsilon }}_2& {\mathit{\epsilon }}_3& {\mathit{\epsilon }}_4\end{array}\right)}^{\mathrm{T}},i=1,2,3,4{\epsilon }_i={\left(\begin{array}{ccccc}\Delta {\mathit{q}}_i& {}^0\delta x_{2,i}& \Delta l_i& {}^5\delta x_{6,13,i}& {}^5\delta z_{6,13,i}\end{array}\right)}^{\mathrm{T}}$$

herein, ${\mathit{J}}_r$ , ${\mathit{J}}_{\alpha }$ were the first three rows and last three rows of submatrix ${\mathit{J}}_c$ .

Let ${\mathit{R}}_{\mathrm{m}0}$ , ${\mathit{R}}_{\mathrm{m}}^{\prime }$ denote the ideal and actual measuring orientation matrix. Following equation can be formulated

$$\Delta {\mathit{r}}_{\mathrm{N}}={\mathit{H}}_{\mathrm{N}}\Delta {\mathit{p}}_{\mathrm{N}}$$

(14)

where

$$\Delta r_{\mathrm{N}}=\left[\begin{array}{c}{\mathit{R}}_{\mathrm{t}}\mathbf{\Delta }{\mathit{r}}_{\mathrm{m}}^{\left(1\right)}\\ {\mathit{R}}_{\mathrm{t}}\left({\mathit{R}\prime }_{\mathrm{m}}^{\left(1\right)}{\mathit{e}}_3-{\mathit{R}}_{\mathrm{m}0}^{\left(1\right)}{\mathit{e}}_3\right)\\ {\mathit{R}}_{\mathrm{t}}\mathbf{\Delta }{\mathit{r}}_{\mathrm{m}}^{\left(2\right)}\\ {\mathit{R}}_{\mathrm{t}}\left({\mathit{R}\prime }_{\mathrm{m}}^{\left(2\right)}{e}_3-{\mathit{R}}_{\mathrm{m}0}^{\left(2\right)}{\mathit{e}}_3\right)\\ ⋮\\ R_{\mathrm{t}}\Delta {\mathit{r}}_{\mathrm{m}}^{\left(L\right)}\\ {\mathit{R}}_{\mathrm{t}}\left({\mathit{R}\prime }_{\mathrm{m}}^{\left(L\right)}{\mathit{e}}_3-{\mathit{R}}_{\mathrm{m}0}^{\left(L\right)}{\mathit{e}}_3\right)\end{array}\right],{\mathit{H}}_{\mathrm{N}}=\left[\begin{array}{c}{\mathit{J}}_{\mathrm{N}}^{\left(1\right)}\\ \begin{array}{c}{\mathit{M}}_{\mathrm{N}}^{\left(1\right)}\\ {\mathit{J}}_{\mathrm{N}}^{\left(2\right)}\\ {\mathit{M}}_{\mathrm{N}}^{\left(2\right)}\end{array}\\ ⋮\\ {\mathit{J}}_{\mathrm{N}}^{\left(L\right)}\\ {\mathit{M}}_{\mathrm{N}}^{\left(L\right)}\end{array}\right],{\mathit{M}}_{\mathrm{N}}^{\left(l\right)}=\left[\begin{array}{cc}-\left[{{\mathit{w}}^{″}}_l\times \right]\mathit{B}{\mathit{J}}_{\epsilon }^{\left(l\right)}& {\mathbf{0}}_{3\times 3}\end{array}\right]$$

herein, $\Delta {\mathit{r}}_{\mathrm{m}}^{(l)}$ is position error vector of point $O\prime$ in frame $O_{\mathrm{m}}-x_{\mathrm{m}}{y}_{\mathrm{m}}{z}_{\mathrm{m}}$ when the point $O\prime$ was coincident with point $P(P=1,2,\dots ,L)$ .

In order to obtain inverse solution for equation (14), Tikhonov regularization ²⁴ was applied to solve the ill-conditioning problem. Then, identification model can be expressed as

$$\Delta {\mathit{p}}_{\mathrm{N}}={({\mathit{H}}_{\mathrm{N}}^{\mathrm{T}}{\mathit{H}}_{\mathrm{N}}+\xi \mathit{I})}^{-1}{\mathit{H}}_{\mathrm{N}}^{\mathrm{T}}\Delta {\mathit{r}}_{\mathrm{N}}$$

(15)

herein, $\xi$ is coefficient determined by L-curve selection.

From equation (14), the relation between ${\mathit{H}}_N\mathrm{\Delta }{\mathit{p}}_N-\mathrm{\Delta }{\mathit{r}}_N$ and $\Delta {\mathit{p}}_N$ was studied numerically. A set of points can be obtained through changing their values. After fitting these points by a curve in the shape of “L” letter, $\xi$ was determined through the largest curvature.

Measurement planning

Measurement planning was to select measuring numbers and configurations. For 1T3R PaQuad PM, translation along z-axis did not affect the identification of geometric errors since it was linear dependent to P joints (see experimental verification in Appendix 2). To identify all geometric errors, PaQuad PM should go through three rotational DoFs at the measuring configurations. Therefore, measuring configurations were determined on a plane with fixed z value. And they are evenly distributed in the orientation workspace $(\alpha \in [-{20}^{\circ },{20}^{\circ }],\beta \in [-{20}^{\circ },{20}^{\circ }],\gamma \in [0^{\circ },{360}^{\circ }])$ .

Since singularity of identification matrix ${\mathit{H}}_N^T{\mathit{H}}_N$ would result in the failure of identification, the numbers of measuring points have to make ${\mathit{H}}_N^T{\mathit{H}}_N$ robust. Considering both efficiency and accuracy of measuring scheme, as least twice as many as geometric errors was required. It was known from the sensitivity analysis that there are 16 main geometric errors ( ${}^0\delta x_{2,i}$ , $\mathrm{\Delta }{l}_i$ , ${}^5\delta x_{6,13,i}$ , ${}^5\delta z_{6,13,i}$ , $i=1,2,3,4$ ), 4 actuation errors $(\mathrm{\Delta }{d}_i,i=1,2,3,4)$ , and the 3 measuring errors ( $\mathrm{\Delta }{l}_{\mathrm{cx}}$ , $\mathrm{\Delta }{l}_{\mathrm{cy}}$ , $\mathrm{\Delta }{l}_{\mathrm{cz}}$ ), altogether 23 parameters were to be identified. Hence, 48 measuring points were determined.

Parameter identification and error compensation

Geometric error identification and compensation was implemented by calibration experiment. This procedure can be summarized as (1) drive PaQuad PM going through all measuring configurations, (2) obtain measuring data through Laser Tracker, (3) identify geometric errors by identification model, and (4) modify geometric errors in the control system. In this process, establishment of fixed frame and selection of reference points were the most important preparations.

The establishment of fixed frame $O-\mathrm{xyz}$ is as follows: (1) choose 12 linear independent points on horizontal machining reference plane in fixed base and create plane AAA by Laser Tracker, (2) measure the coordinates of four assembling reference holes in frame $O_{\mathrm{m}}-x_{\mathrm{m}}{y}_{\mathrm{m}}{z}_{\mathrm{m}}$ , (3) obtain the projection of measured holes G₁, G₂, G₃, G₄ in plane AAA, and (4) define line G₂G₄ as x-axis, line G₁G₃ as y-axis, the intersection of G₁G₃ and G₂G₄ as origin, and normal direction of the plane AAA as z-axis.

Selection of reference points should be capable of calculating rotating angles of PaQuad PM. For this purpose, points $P_1$ , $P_2$ , and $P_3$ were chosen from the out-part. When PaQuad PM was at initial configuration, point $P_2$ was on the plane $y=0$ whereas points $P_1$ and $P_3$ located at the plane $x=0$ .

The orientation of PaQuad PM can be calculated by coordinates of points $P_1$ , $P_2$ , and $P_3$ . Line $P_1{P}_3$ was just the v-axis, and the normal direction of the plane spanned by line $P_1{P}_3$ and $P_1{P}_2$ was w-axis. u-axis was obtained by right-hand rule. Hence, the orientation matrix can be achieved, leading to the determination of rotating angles $\alpha$ , $\beta$ , and $\gamma$ .

The calibration experiment was set up as shown in Figure 7, in which Laser Tracker with type number Leica-AT901-LR was adopted. With the measurements of points $P_1$ , $P_2$ , and $P_3$ at each measuring configuration $(z=845\mathrm{mm})$ , geometric errors were obtained from identification model. Then, error compensation was achieved by adjusting corresponding kinematic parameters in the control system.

Figure 7.

Calibration experiment of PaQuad PM.

Experimental verification

In order to verify accuracy improvement in terms of geometric accuracy design and error compensation, 24 testing points were chosen. These testing points were different from measuring points in kinematic calibration.

Comparison of orientations before and after kinematic calibration was shown in Figure 8. It can be summarized that (1) $\gamma$ had the worst orientation accuracy before calibration, whose biggest deviation reached $2{.098}^{\circ }$ . Deviations of $\alpha$ , $\beta$ were near $-0.{18}^{\circ }$ and $-0.5^{\circ }$ . (2) Orientation accuracy about $\alpha$ , $\beta$ , $\gamma$ had improved one order of magnitude. Deviations of $\alpha$ , $\beta$ after calibration were within $\pm 0{.03}^{\circ }$ , and deviation of $\gamma$ reduced to $\pm 0{.5}^{\circ }$ .

Figure 8.

Orientation accuracy of PaQuad PM before and after kinematic calibration (a) deviation of α (b) deviation of β (c) deviation of γ

In Zhong et al., ⁴³ the assembly accuracy range of aircraft wing skin is (−0.273, +0.350) mm based on contour board. And it becomes (−0.36, +0.25) mm based on coordinate holes. Comparatively, position accuracy of PaQuad PM along z-axis is $\pm 0.01\mathrm{mm}$ . In Renders et al., ⁴⁴ position and orientation accuracy of fuselage assembly are $\pm 0.5\mathrm{mm}$ and $\pm 0{.5}^{\circ }$ . The orientation accuracy of PaQuad PM about $\alpha$ , $\beta$ , $\gamma$ is within this range. It demonstrates that the accuracy of PaQuad PM is highly sufficient to assembly requirement of air component assembling.

Experimental verification and comparison with other docking equipment indicates that error compensation by means of kinematic calibration was effective. Furthermore, it confirms the efficiency and correctness of accuracy improvement strategy proposed in this article.

Conclusion

Focusing on the high-precision assembling of large-scale components in aerospace, shipbuilding, and construction machinery, 1T3R PaQuad PM was taken as an example to investigate accuracy improvement of docking PMs. It was carried out by geometric error modeling, geometric error design, and error compensation. Corresponding conclusions are drawn as follows:

1. Based on screw theory, geometric error modeling of PaQuad PM was obtained from four independent routes. Each route error twist was formulated by superposition of geometric errors and joint perturbations. Single wrench within each route and combination wrench provided by opposite routes were applied to eliminate joint perturbation.

2. Geometric accuracy design restrained geometric errors at component and substructure levels. Geometric errors were first transferred into dimensional or geometric tolerance. High-precision machining and assembling techniques were then adopted to satisfy the tolerances. Error measurement was finally implemented to provide reference for kinematic calibration.

3. Error compensation was conducted by kinematic calibration at mechanism level. Sensitivity analysis by means of interval theory was applied to exclude unimportant geometric errors. L-curve selection was employed to solve ill-conditioning problem in the identification modeling. In all, 48 evenly distributed measuring configurations were determined by taking both efficiency and accuracy into account.

4. Maximum deviations of PaQuad PM before calibration experiment was 0.01 mm, $-0{.197}^{\circ }$ , $-0{.522}^{\circ }$ , and $2{.908}^{\circ }$ . And they become 0.01 mm, $0{.027}^{\circ }$ , $0{.046}^{\circ }$ , and $0{.521}^{\circ }$ after kinematic calibration. Deviations of $\alpha$ , $\beta$ after calibration are within $\pm 0.{03}^{\circ }$ , and deviation of $\gamma$ reduces to $\pm 0.5^{\circ }$ . It indicated the effectiveness of the proposed accuracy improvement strategy.