Error modeling and tolerance design of a parallel manipulator with full-circle rotation

Abstract

A method for improving the accuracy of a parallel manipulator with full-circle rotation is systematically investigated in this work via kinematic analysis, error modeling, sensitivity analysis, and tolerance allocation. First, a kinematic analysis of the mechanism is made using the space vector chain method. Using the results as a basis, an error model is formulated considering the main error sources. Position and orientation error-mapping models are established by mathematical transformation of the parallelogram structure characteristics. Second, a sensitivity analysis is performed on the geometric error sources. A global sensitivity evaluation index is proposed to evaluate the contribution of the geometric errors to the accuracy of the end-effector. The analysis results provide a theoretical basis for the allocation of tolerances to the parts of the mechanical design. Finally, based on the results of the sensitivity analysis, the design of the tolerances can be solved as a nonlinearly constrained optimization problem. A genetic algorithm is applied to carry out the allocation of the manufacturing tolerances of the parts. Accordingly, the tolerance ranges for nine kinds of geometrical error sources are obtained. The achievements made in this work can also be applied to other similar parallel mechanisms with full-circle rotation to improve error modeling and design accuracy.

Introduction

The industrialization of high-speed parallel robots can be greatly promoted by improving their accuracy. As the foundation of accuracy problems, error modeling can provide a theoretical basis for accurate design and kinematic calibration by establishing the mapping relationship between the pose errors in the end-effector and the geometric error sources of the mechanism.

In recent years, a number of error models for parallel manipulators have been established. The matrix perturbation method based on Denavit–Hartenberg (D-H) homogeneous transformation is a classical approach for error modeling.^1,2 Yao et al. ³ applied the matrix differential method to establish a static position and orientation error model and analyzed their contributions to the pose error of end-effector for a wheeled train uncoupling robot with 4 degrees of freedom (DOF). And the input motion planning method is adopted to improve the position and orientation accuracy. Wang et al. ⁴ established an error model for a redundant hybrid robot, which had the ability to account for static error sources. However, this method has the problem that the parameters become discontinuous when the adjacent axes are parallel. Thus, many modified approaches have been proposed. Judd ⁵ introduced additional parameters into the parallel axis problem and established a modified four-parameter model. However, the corresponding D-H error representation is ill-conditioned when the adjacent axes are vertical. Zhuang et al. ⁶ proposed a complete and parametrically continuous model, which is able to avoid mutation of the parameters. Besides this, they pointed out that the change in pose is very small as a result of a change in the parameters. Subsequently, the parameters are guaranteed to be complete and continuous.

Differential geometry is another powerful tool for modeling errors in parallel manipulators. Tao et al. ⁷ established a kinematic model for robot manipulators using a “product of exponentials” (POE) approach. This has the advantage of avoiding parameter discontinuity and also simplifies the calibration process somewhat. Using Lie groups and Lie algebra, Chen et al. ⁸ established an error model for a serial robot with 6 DOF according to the transformation principles of Lie algebra. They then integrated the links’ geometric errors and the joints’ offset errors together using the quotient manifold of the Lie group so that all errors could be identified simultaneously.

Recently, “screw theory” has been widely applied to error modeling. Zhao et al. ⁹ applied screw theory to a 2-DOF mechanism and simulated a 5-axis blade milling machine. They also applied their method to evaluate the physical effects of the geometric error sources. Coincidentally, Frisoli et al. ¹⁰ adopted screw theory to determine the position accuracy of parallel manipulators under the influence of joint clearances. They applied the method to two 3-DOF translational manipulators and computed the angular- and linear-positioning accuracy. Kumaraswamy et al. ¹¹ adopted screw theory to analyze the kinematic accuracy of mechanisms with varying link lengths and joint clearances and demonstrated that it can be applied conveniently for closed- or open-loop serial manipulators.

As these matrix methods often involve a large number of matrix differential operations, the vector chain method (based on first-order perturbation theory) is widely used. Briot and Bonev ¹² developed a kinematic error model for a parallel robot to analyze the maximum orientation and position output errors. Cheng et al. ¹³ established a pose error model of a symmetrical parallel robot and obtained a statistical model for the sensitivity coefficients. Although the above approaches are widely used for error models, they cannot separate compensatable and uncompensatable error sources. As the pose error of lower-mobility parallel manipulators cannot be compensated, ¹⁴ these approaches are not suitable for such manipulators. Bearing this in mind, Huang et al., ¹⁵ taking a 3-DOF parallel kinematic machine with parallelogram struts as an example, proposed a modeling method to separate the geometric error sources which affect the position and orientation errors in the end-effector. Using the homogeneous transformation matrix method and screw theory, Liu et al. ¹⁶ proposed a general and systematic error model method which allowed the error sources affecting the compensatable and uncompensatable pose accuracy of a platform to be identified in an explicit manner.

Accuracy analysis is a powerful method of mitigating uncompensatable error. For a 6-DOF parallel mechanism, Patel and Ehmann ¹⁷ proposed an error model that divides the error sources into those affecting leg lengths and those affecting joint positions. Then, the tolerance allocations of the parts were studied with the aid of a sensitivity analysis. Wang and Ehmann ¹⁸ presented first- and second-order error models for a 6-DOF Stewart platform. By means of a sensitivity analysis, the contribution of each error component to the total position and orientation error of the mechanism was thus determined. Kim and Choi ¹⁹ proposed an “error range” method, which established a relationship between the pose error of the end-effector and the pose errors of the joint. This was then used to instruct tolerance allocation. Soon et al. ²⁰ constructed an error-mapping model for a Hexapod based on the vector chain method. In this work, a sensitivity analysis was again applied to analyze the errors. Together with the aid of an experiment that measured errors, this work’s results suggest that the pose error is equal to the mean value of each error source. Chen et al. ²¹ established an error model for a novel “selective compliance assembly robot arm” parallel robot with parallelogram structures. The model was able to reflect the process of error transmission in detail and show the influence of each geometric error on the pose error of the moving platform via a sensitivity analysis. Maurine and Domber ²² proposed an error-mapping model for a mechanism with characteristic parallelogram structures and used the result of a sensitivity analysis to estimate the pose accuracy of the end-effector. Although the above approaches are widely used in accuracy analyses, they cannot filter out the effect of uncompensatable error sources on pose accuracy. For a translational parallel mechanism with parallelogram struts under constraint, Huang and colleagues^23,24 considered that the relative and coplanar errors of the parallelogram have a uncompensatable effect on the pose of the end-effector. In order to reduce these two kinds of errors, they subsequently proposed an assembly process which was effectively able to suppress the pose error of the end-effector.

The above achievements constitute important steps in improving error modeling and design accuracy of parallel mechanisms. However, they are mainly focused on planar parallel mechanisms and ordinary spatial parallel mechanisms. In addition, they do not propose a comprehensive and universally applicable method of integrating error analysis and accuracy design into lower-mobility parallel mechanisms. This article considers a novel parallel manipulator with full-circle rotation. The novel manipulator enables kinematic analysis, error modeling, sensitivity analysis, and tolerance allocation to be integrated into a comprehensive framework. Therefore, the work presented addresses the disadvantages of the methods in the literature discussed above by separating pose errors, determining the influence of each error on the pose accuracy of the end-effector, and reducing the uncompensatable error. The method provides tolerance ranges for the uncompensatable error sources which are required in the manufacturing and assembling stages with the aid of a sensitivity analysis.

The work of this article is organized as follows. Section “Kinematic analysis” describes the manipulator system and presents an inverse kinematics analysis, which provides the foundation for the rest of the work. Section “Error-mapping model and error separation” establishes an error model for the manipulator based on the vector chain method. Then, the position and orientation error sources that affect the accuracy in the end-effector are effectively separated by application of a mathematical transformation with parallelogram structure characteristics. Section “Sensitivity analysis” adopts a global sensitivity evaluation index to carry out a sensitivity analysis of all the error sources. The law governing the effect of the geometric parameter errors on the accuracy in the end-effector is proposed, which lays the foundation for the subsequent work on accuracy design. Section “Tolerance allocation” performs tolerance allocation using a genetic algorithm (GA), which provides fundamental guidelines for component manufacture and assembly. Section “Conclusion” presents the conclusions of this article.

Kinematic analysis

System description

The object considered in this study is shown in Figure 1. The parallel manipulator with full-circle rotation,^25,26 which has two translational and one rotational (2T1R) DOF, consists of a fixed frame, servo motors, three branched chains, and a moving platform. In the first and second chains, the active links are connected to their servo motors by a transmission mechanism, but in the third chain, the active link is directly connected to the servo motor. One end of each passive link is connected to its active link by a spherical joint, while the end side is connected to the moving platform. A parallelogram structure is employed in each passive link. To ensure the follow-up characteristic, the passive link of the third chain is fixed using three tie rods that are equal in length to a rod that contains the revolute joints on both sides. The third chain can realize two-dimensional translations on the $x_2{o}_2{z}_2$ plane and rotation about z₂-axis (Figure 2). All three chains contain revolute joints that rotate around the same axis, and all the ends of the passive links are connected to the moving platform. Thus, one translation and two rotations are realized and a larger workspace and broader transmission range acquired.

Figure 1.

Solid model of the parallel manipulator with full-circle rotation.

Figure 2.

Schematic diagram of the parallel manipulator.

A schematic diagram of the parallel manipulator with full-circle rotation is shown in Figure 2. We take the coaxial hinge point of active links 1 and 2 as the origin O and their coaxial rotation axis as the Z-axis. As the manipulator can realize full-circle rotation, the X- and Y-axes can be selected randomly following the right-hand rule. A fixed coordinate system $O-\mathrm{XYZ}$ can thus be established and all vectors described within the $O-\mathrm{XYZ}$ frame. Rotating by ${\theta }_0$ around the Z-axis of the fixed coordinate system in a counterclockwise direction, the moving coordinate system $o_1-x_1{y}_1{z}_1$ is established. In the axis hinge point of active link 3, we also establish the moving coordinate system $o_2-x_2{y}_2{z}_2$ with $o_2$ being the origin and z₂-axis being coincident with the Z-axis. The x₂-axis is in the direction of rotation axis of active link 3, and the y₂-axis is defined by the right-hand rule. The moving coordinate system $o_2-x_2{y}_2{z}_2$ can be obtained by rotating the frame anticlockwise about the Z-axis by ${\theta }_0+\pi /2$ .

Inverse kinematic analysis

Inverse kinematic analysis ²⁷ is considered to determine the active link position angle ${\theta }_i(i=1,2,3)$ (the range of ${\theta }_1$ and ${\theta }_3$ is $0-\pi$ , and the range of ${\theta }_2$ is $\pi -2\pi$ ) according to the position vector $\mathit{r}=(\begin{array}{ccc}x\hfill & y\hfill & z\hfill \end{array}{)}^T$ of the reference point A on the moving platform (which is $2l_0$ and $2b_0$ in the length and width directions, respectively). The offset displacement of kinematic chain 3 is h along the direction of the z-axis. The distance between the location of the center of chain 3 and the external end-surface of the moving platform is s. The projection of the position vector $\mathit{r}=(\begin{array}{ccc}x\hfill & y\hfill & z\hfill \end{array}{)}^T$ is $u_0=(\begin{array}{ccc}\cos {\theta }_0\hfill & \sin {\theta }_0\hfill & 0\hfill \end{array}{)}^T$ on the $O-\mathrm{XY}$ plane. ${\theta }_0$ is the angle between the projection and the positive direction of the x-axis in the coordinate system $O-\mathrm{XYZ}$ and may be determined from $\arctan (y/x)={\theta }_0$ .

Inverse kinematic analysis of chains 1 and 2

As chains 1 and 2 have the same structure and form of motion, we can consider them together. In the moving coordinate system $o_1-x_1{y}_1{z}_1$ , ${\theta }_1$ is defined as the angle between the active link 1 and the positive direction of the x₁-axis, ${\theta }_2$ is the angle between the active link 2 and the positive direction of the x₁-axis. Thus, ${\mathit{u}}_1^{o_1}=[\begin{array}{ccc}\cos {\theta }_1\hfill & \sin {\theta }_1\hfill & 0\hfill \end{array}{]}^T$ , ${\mathit{u}}_2^{o_1}=[\begin{array}{ccc}\cos {\theta }_2\hfill & \sin {\theta }_2\hfill & 0\hfill \end{array}{]}^T$ are measured in the moving coordinate system $o_1-x_1{y}_1{z}_1$ .

In the fixed reference coordinate system, the closed-loop vector equation of chains 1 and 2 can be written as

$$r-l_1{u}_1+l_0{u}_v=l_2{w}_i$$

(1)

where $l_1$ , $l_2$ , ${\mathit{u}}_i$ , ${\mathit{w}}_i$ represent the length and unit vector of the active and driven arms of chain $i(i=1,2)$ , respectively, and $u_1={(\cos {\theta }_0\cos {\theta }_1-\sin {\theta }_0\sin {\theta }_1\sin {\theta }_0\cos {\theta }_1+\cos {\theta }_0\sin {\theta }_{1}0)}^T,w_i={(\cos {\alpha }_i\cos {\beta }_i\cos {\gamma }_i)}^T$ .

The vector ${\mathit{u}}_v=(\begin{array}{ccc}-\sin {\theta }_0\hfill & \cos {\theta }_0\hfill & 0\hfill \end{array})$ is the normal vector of the projection r on the $O-\mathrm{XY}$ plane.

Taking the modulus of both sides of equation (1) gives

$$A_i\sin {\theta }_i+B_i\cos {\theta }_i+C_i=0$$

(2)

where

$$A_i=2l_{1}x\sin {\theta }_0-2l_{1}y\cos {\theta }_0+(-1{)}^{i}2l_0{l}_1$$

$$B_i=-2l_{1}x\cos {\theta }_0-2l_{1}y\sin {\theta }_0$$

$$C_i=x^2+y^2+z^2+l_1^2-l_2^2+l_0^2+{\left(-1\right)}^{i}2l_0\left(-x\sin {\theta }_0+y\cos {\theta }_0\right)$$

According to the assembly mode, inverse position analysis gives

$${\theta }_i=2\arctan \frac{-A_i+\sqrt{A_i^2-C_i^2+B_i^2}}{C_i-B_i}$$

(3)

Accordingly, ${\mathit{u}}_i$ can be determined. In addition, ${\mathit{w}}_i$ can be determined from

$${\mathit{w}}_i=\frac{\mathit{r}-l_1{\mathit{u}}_i+{(-1)}^{i+1}{l}_0{\mathit{u}}_v}{l_2}$$

(4)

Inverse kinematic analysis of chain 3

Here, we establish a reference coordinate system $o_2-x_2{y}_2{z}_2$ for chain 3 involving rotating around the Z-axis of the fixed coordinate system in a counterclockwise direction by ${\theta }_0+\pi /2$ . The z₂-axis coincides with the Z-axis of $O-\mathrm{XYZ}$ , the rotation axis of the active link 3 is the x₂-axis, and the y₂-axis is vertical according to the right-hand rule (Figure 2). We define ${\theta }_3$ as the angle between the active link 3 and the positive direction of the x₂-axis.

Next, we define the vector ${\mathit{u}}_3^{o_2}=(\begin{array}{ccc}0\hfill & \cos {\theta }_3\hfill & \sin {\theta }_3\hfill \end{array}{)}^T$ . Transformation of ${\mathit{u}}_3^{o_2}$ into the fixed coordinate system $O-\mathrm{XYZ}$ yields the vector ${\mathit{u}}_3=(\begin{array}{ccc}-\cos {\theta }_0\cos {\theta }_3\hfill & -\sin {\theta }_0\cos {\theta }_3\hfill & \sin {\theta }_3\hfill \end{array}{)}^T$ and the closed-loop vector equation for chain 3 can be written as

$$\mathit{r}-l_1{\mathit{u}}_3-h{\mathit{e}}_0+b_0{\mathit{e}}_0-s{\mathit{u}}_0=l_2{\mathit{w}}_3$$

(5)

where $\mathit{e}$ is the offset of the kinematic chain 3 with ${\mathit{e}}_0=h{\mathit{e}}_0$ and ${\mathit{e}}_0$ is a unit vector the along Z-axis.

The following trigonometric expression is obtained by taking the modulus of both sides of equation (5)

$$A_3\sin {\theta }_3+B_3\cos {\theta }_3+C_3=0$$

(6)

where

$$\begin{array}{l}{A}_3=2l_{1}x\sin {\theta }_0-2l_{1}y\cos {\theta }_0+(-1)i2l_0{l}_1\\ B_3=-2l_{1}x\cos {\theta }_0-2l_{1}y\sin {\theta }_0\\ C_3=x^2+y^2+z^2+l_1^2-l_2^2+l_0^2\\ +(-1)i2l_0(-x\sin {\theta }_0+y\cos {\theta }_0)\end{array}$$

Considering the assembly mode gives the solution

$${\theta }_3=2\arctan \frac{-A_3+\sqrt{A_3^2-C_3^2+B_3^2}}{C_3-B_3}$$

(7)

Thus, ${\mathit{u}}_3$ can be determined and ${\mathit{w}}_{\mathit{3}}$ can then be determined from

$${\mathit{w}}_3=\frac{\mathit{r}-l_1{\mathit{u}}_{\mathit{3}}-h{\mathit{e}}_0+b_0{\mathit{e}}_0-s{\mathit{u}}_0}{l_2}$$

(8)

Note that ${\theta }_1$ and ${\theta }_2$ values obtained should be added $({\theta }_0=(({\theta }_1\prime +{\theta }_2\prime )/2)-\pi )$ . Then, they are changed into ${\theta }_1\prime ={\theta }_1+{\theta }_0$ , ${\theta }_2\prime ={\theta }_2+{\theta }_0$ . They are the driven angle of the active arm with respect to the fixed coordinate system.

Error-mapping model and error separation

Description of the coordinate systems and error analysis

There are some structural similarities between the parallel manipulator with full-circle rotation and existing parallel manipulators with parallelogram strut structures. ²⁸ Therefore, the following coordinate systems for the parallel manipulator with full-circle rotation can be defined (see Figure 3):

$\{O\}$ is defined with reference to the machine frame. The origin O is the point at which the active link ideal axis crosses the xy plane which passes through the midpoint of the two joints associated with the active link and which is normal to the line passing through the centers of the two joints. O is located in the xy plane. ${\theta }_o$ is the angle between the projection of position vector $\mathit{r}$ onto the xy plane and the x-axis. The y-axis is defined by the right-hand rule. The angle of the intermediate coordinate system $\{O_g\}$ relative to $\{O\}$ is ${\theta }_o+\pi /2$ through the z-axis.

$\{B_i\}$ is defined with reference to the active links. For $i=1,2$ , $B_i$ is the point of intersection of a vertical line from $C_i$ on the active link to the actual axis $Z_{\mathrm{Bi}}$ of the relevant revolute pair. The x_Bi-axis is parallel to the xy plane and the y_Bi-axis is then defined according to the right-hand rule. When $i=3$ , $B_3$ is the intersection point from $C_3$ on the active link to the corresponding actual axis. The x_B3-axis is coincident with the axis of the revolute pair of the active link. The y_B3-axis is parallel to the xy plane. The z_B3-axis is defined according to the right-hand rule.

$\{C_i\}$ is placed on the spherical joint of the active link. When $i=1,2$ , the origin is $C_i$ . The z_Ci-axis is coincident with a line connecting the centers of the corresponding two spherical pairs. The x_Ci-axis is parallel to the xy plane. The y_Ci-axis is defined by the right-hand rule. When $i=3$ , the origin is $C_3$ . The x_C3-axis is coincident with a line connecting the centers of the corresponding two spherical pairs. The z_C3-axis is vertical to the xy plane. The y_C3-axis is defined by the right-hand rule.

$\{A_i\}$ is placed on the spherical joint of the passive link. The origin $A_i$ is the midpoint of the line connecting the moving platform with the line of centers of two spherical pairs of passive links. When $i=1,2$ , the z_Ai-axis is coincident with the line connecting the centers of two spherical pairs. The x_Ai-axis is parallel to the xy plane. The y_Ai-axis is defined by the right-hand rule. When $i=3$ , the x_Ai-axis is coincident with the line connecting the centers of two spherical pairs. The y_A3-axis is parallel to the xy plane. The $l_2$ -axis is defined by the right-hand rule.

$\{O\prime \}$ is placed on the moving platform with the origin point $O\prime$ being the point where the center line crosses the ligature of the right and left spherical joints that connect the moving platform and the z-direction vertical plane of the moving platform. The $x\prime y\prime$ plane is parallel to the moving plane of the upper and lower active links. The x′-axis is parallel to the x-axis, and the y′-axis can be defined by the right-hand rule. The intermediate coordinate system $\{O_g\prime \}$ can be defined by rotating $\{O\prime \}$ about the z′-axis by ${\theta }_o+\pi /2$ .

Figure 3.

Coordinate system of chain i (i = 1, 2, 3): (a) Coordinate system of chain 1; (b) Coordinate system of chain 2 and (c) Coordinate system of chain 3.

The errors can be described as follows. In chain i, the theoretical value of $B_i$ , which is the position vector of the point in $\{O\}$ , and its error are given by ${\mathit{b}}_i$ (the theoretical values of ${\epsilon }_r$ and ${\mathit{b}}_2$ are zero) and $\Delta {\mathit{b}}_i$ . In $\{B_i\}$ , the theoretical value of the unit vector ${\mathit{u}}_i$ from point $B_i$ to point $C_i$ and its error are given by ${\mathit{u}}_{\mathrm{io}}$ and $\Delta {\mathit{u}}_i$ , respectively. The theoretical value of the length $B_i{C}_i=l_{1i}$ and its error are given by $l_{1i}$ and $\Delta l_{1i}$ , respectively. The theoretical value of ${\mathit{w}}_{\mathrm{ij}}$ , which is the unit vector of link j in the passive link, and its error are given by ${\mathit{w}}_i$ and $\Delta {\mathit{w}}_{\mathrm{ij}}$ , respectively. The theoretical value of the length $l_{2\mathrm{ij}}$ and its error are given by $l_{2i}$ and $\Delta l_{2\mathrm{ij}}$ , respectively. In $\{O\prime \}$ , the theoretical value of the length vector of the moving platform and its error are given by ${\mathit{l}}_o$ and $\Delta {\mathit{l}}_o$ , respectively. The theoretical value of the width vector of the moving platform and its error are given by ${\mathit{b}}_o$ and $\Delta {\mathit{b}}_o$ , respectively. The theoretical value of the height vector of the moving platform and its error are given by ${\mathit{s}}_o$ and $\Delta {\mathit{s}}_o$ , respectively. The theoretical value of $c_{\mathrm{ij}}$ , which is the distance of point $C_i$ to the center $C_{\mathrm{ij}}$ of the spherical joint, and its error are given by $e/2$ and $\Delta c_i$ , respectively. The theoretical value of $a_{\mathrm{ij}}$ , which is the distance of point $A_i$ to the center $A_{\mathrm{ij}}$ of the spherical joint, and its error are given by $e/2$ and $\Delta a_i$ , respectively. The orientation error vector of $\{O\prime \}$ referenced to $\{O\}$ is given by $\theta$ .

Error-mapping model

According to the definitions of the coordinate systems and the description of the geometric error sources, in the kinematic chain $O-B_i-C_i-C_{\mathrm{ij}}-A_{\mathrm{ij}}-A_i-O\prime$ , the position vector $\mathit{r}$ (see Figure 3) of point $O\prime$ can be expressed in $\{O\}$ by^15,26

$$\begin{array}{cc}\multicolumn{1}{c}{\mathit{r}=}& {\mathit{R}}_{O{O}_g}\left({\mathit{R}}_{O{B}_1}(l_{11}{\mathit{u}}_1+{(-1)}^{j+1}{\mathit{R}}_{B_1{C}_1}{c}_{1j}{\mathit{e}}_3)\right)\hfill \\ \multicolumn{1}{c}{}& +l_{21j}{\mathit{w}}_{1j}-{\mathit{R}}_{\mathrm{OO}\prime }{\mathit{R}}_{O\prime O_{{}_g}\prime }({\mathit{l}}_0+{(-1)}^{j+1}{\mathit{R}}_{O\prime A_1}{a}_{1j}{\mathit{e}}_3)\hfill \end{array}$$

(9)

$$\begin{array}{cc}\multicolumn{1}{c}{\mathit{r}=}& {\mathit{R}}_{O{O}_g}\left({\mathit{R}}_{O{B}_2}(l_{12}{\mathit{u}}_2+{(-1)}^{j+1}{\mathit{R}}_{B_2{C}_2}{c}_{2j}{\mathit{e}}_3)\right)+l_{22j}{\mathit{w}}_{2j}\hfill \\ \multicolumn{1}{c}{}& -{\mathit{R}}_{OO\prime }{\mathit{R}}_{O\prime O_g\prime }(-{\mathit{l}}_0+{(-1)}^{j+1}{\mathit{R}}_{O\prime A_2}{a}_{2j}{\mathit{e}}_3)\hfill \end{array}$$

(10)

$$\begin{array}{cc}\multicolumn{1}{c}{\mathit{r}=}& {\mathit{R}}_{O{O}_g}({\mathit{b}}_3+{\mathit{R}}_{O_g{B}_3}(l_{13}{\mathit{u}}_3+{(-1)}^{j+1}{\mathit{R}}_{B_3{C}_3}{c}_{3j}{\mathit{u}}_v))+l_{23j}{\mathit{w}}_{3j}\hfill \\ \multicolumn{1}{c}{}& -{\mathit{R}}_{OO\prime }{\mathit{R}}_{O\prime O_g\prime }(-{\mathit{s}}_0-{\mathit{b}}_0+{(-1)}^{j+1}{\mathit{R}}_{O\prime A_3}{a}_{3j}{\mathit{u}}_v)\hfill \end{array}$$

(11)

Considering the first-order linear perturbations on both sides of the above equation, we have

$$\begin{array}{cc}\multicolumn{1}{c}{\Delta \mathit{r}=}& {\mathit{R}}_g(-\Delta {\mathit{b}}_1+{\mathbf{\theta }}_{B_1}\times (l_{11}{\mathit{u}}_1+{(-1)}^{j+1}e/2{\mathit{e}}_3))\hfill \\ \multicolumn{1}{c}{}& +{\mathit{R}}_g(\Delta l_{11}{\mathit{u}}_1+{\mathbf{\theta }}_{C_1}\times {(-1)}^{j+1}e/2{\mathit{e}}_3+{(-1)}^{j+1}\Delta c_1{\mathit{e}}_3)\hfill \\ \multicolumn{1}{c}{}& +(\Delta l_{21j}{\mathit{w}}_1+l_{21}\Delta {\mathit{w}}_{1j})-\mathbf{\theta }\times {\mathit{R}}_g({\mathit{l}}_0+{(-1)}^{j+1}e/2{\mathit{e}}_3)\hfill \\ \multicolumn{1}{c}{}& -{\mathit{R}}_g(\Delta {\mathit{l}}_0+{\mathbf{\theta }}_{A_1}\times ({(-1)}^{j+1}e/2{\mathit{e}}_3)+{(-1)}^{j+1}\Delta a_1{\mathit{e}}_3)\hfill \end{array}$$

(12)

$$\begin{array}{cc}\multicolumn{1}{c}{\Delta \mathit{r}=}& {\mathit{R}}_g(\Delta {\mathit{b}}_2+{\mathbf{\theta }}_{B_2}\times (l_{12}{\mathit{u}}_2+{(-1)}^{j}e/2{\mathit{e}}_3))\hfill \\ \multicolumn{1}{c}{}& +{\mathit{R}}_g(\Delta l_{12}{\mathit{u}}_2+{\mathbf{\theta }}_{C_2}\times {(-1)}^{j}e/2{\mathit{e}}_3+{(-1)}^j\Delta c_2{\mathit{e}}_3)\hfill \\ \multicolumn{1}{c}{}& +(\Delta l_{22j}{\mathit{w}}_2+l_{22}\Delta {\mathit{w}}_{2j})-\mathbf{\theta }\times {\mathit{R}}_g(-{\mathit{l}}_0+{(-1)}^{j}e/2{\mathit{e}}_3)\hfill \\ \multicolumn{1}{c}{}& -{\mathit{R}}_g(-\Delta {\mathit{l}}_0+{\mathbf{\theta }}_{A_2}\times ({(-1)}^{j}e/2{\mathit{e}}_3)+{(-1)}^j\Delta a_2{\mathit{e}}_3)\hfill \end{array}$$

(13)

$$\begin{array}{cc}\multicolumn{1}{c}{\Delta \mathit{r}=}& {\mathit{R}}_g(\Delta {\mathit{b}}_3+{\mathbf{\theta }}_{B_3}\times (l_{13}{\mathit{u}}_3+{(-1)}^{j+1}e/2{\mathit{u}}_v))\hfill \\ \multicolumn{1}{c}{}& +{\mathit{R}}_g(\Delta l_{13}{\mathit{u}}_3+{\mathbf{\theta }}_{C_3}\times {(-1)}^{j+1}e/2{\mathit{e}}_1+{(-1)}^{j+1}\Delta c_3{\mathit{e}}_1)\hfill \\ \multicolumn{1}{c}{}& +(\Delta l_{23j}{\mathit{w}}_3+l_{23}\Delta {\mathit{w}}_{3j})+\mathbf{\theta }\times {\mathit{R}}_g({\mathit{s}}_0-{\mathit{b}}_0-{(-1)}^{j+1}e/2{\mathit{u}}_v)\hfill \\ \multicolumn{1}{c}{}& -{\mathit{R}}_g(-\Delta {\mathit{s}}_0+\Delta {\mathit{b}}_0+{\mathbf{\theta }}_{A_3}\times {(-1)}^{j+1}e/2{\mathit{u}}_v+{(-1)}^{j+1}\Delta a_3{\mathit{u}}_v)\hfill \end{array}$$

(14)

where

$$i=1,2,3,j=1,2,{\mathit{e}}_3={\left(\begin{array}{ccc}0\hfill & 0\hfill & 1\hfill \end{array}\right)}^T$$

$R_g$ is the orientation matrix of $O_g$ referenced to O ( $O_g\prime$ referenced to $O\prime$ ) and is given by

$${\mathit{R}}_g=\left[\begin{array}{ccc}\cos ({\theta }_0+\pi /2)\hfill & -\sin ({\theta }_0+\pi /2)\hfill & 0\hfill \\ \multicolumn{1}{c}{\sin ({\theta }_0+\pi /2)}& \cos ({\theta }_0+\pi /2)\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & 1\hfill \end{array}\right]$$

• ${\mathbf{\theta }}_{B_k}(k=1,2,3)$ is the orientation error vector of $B_k$ referenced to $O_g$ .

• ${\mathbf{\theta }}_{C_k}(k=1,2,3)$ is the orientation error vector of $C_k$ referenced to $B_k$ .

• ${\mathbf{\theta }}_{A_k}(k=1,2,3)$ is the orientation error vector of $A_k$ referenced to $O{\prime }_g$ .

Let $\Delta {\mathit{a}}_i=\Delta {\mathit{l}}_0(i=1,2)$ , $\Delta {\mathit{a}}_i=\Delta {\mathit{s}}_0+\Delta {\mathit{b}}_0,i=3$ be the comprehensive error vector of the origin of $A_i$ referenced to $O_g\prime$ ; ${\mathbf{\psi }}_3={\mathbf{\theta }}_{B_i}+{\mathbf{\theta }}_{C_i}-{\mathbf{\theta }}_{A_i}$ be the orientation error of $O_g\prime$ referenced to $O_g$ ; $\Delta g_i=-2(\Delta c_i-\Delta a_i)$ the relative joint distance error on both sides of the parallelogram and take the product of ${\mathit{w}}_i^T$ on both sides of equations (12) - (14). Then, we have

$$\begin{array}{cc}\multicolumn{1}{c}{{\mathit{w}}_1^T\Delta \mathit{r}=}& (-{\mathit{w}}_1^T{\mathit{R}}_g\Delta {\mathit{b}}_1+{\mathit{w}}_1^T{\mathit{R}}_g\Delta {\mathit{a}}_1+\Delta l_{11}{\mathit{w}}_1^T{\mathit{R}}_g{\mathit{u}}_1+l_{11}{({\mathit{R}}_g{\mathit{u}}_1\times {\mathit{w}}_1)}^T{\mathit{R}}_g{\mathbf{\theta }}_{B_1})\hfill \\ \multicolumn{1}{c}{}& +{(-1)}^{j+1}(e/2{({\mathit{R}}_g{\mathit{e}}_3\times {\mathit{w}}_1)}^T{\mathit{R}}_g{\mathbf{\psi }}_1+\Delta g_1/2{\mathit{w}}_1^T{\mathit{R}}_g{\mathit{e}}_3)\hfill \\ \multicolumn{1}{c}{}& +\Delta l_{21j}-{({\mathit{R}}_g({\mathit{l}}_0+{(-1)}^{j+1}e/2{\mathit{e}}_3)\times w_1)}^T\mathbf{\theta }\hfill \end{array}$$

(15)

$$\begin{array}{cc}\multicolumn{1}{c}{{\mathit{w}}_2^T\Delta \mathit{r}=}& {\mathit{w}}_2^T{\mathit{R}}_g\Delta {\mathit{b}}_2+{\mathit{w}}_2^T{\mathit{R}}_g\Delta {\mathit{l}}_0+\Delta l_{12}{\mathit{w}}_2^T{\mathit{R}}_g{\mathit{u}}_2+l_{12}{({\mathit{R}}_g{\mathit{u}}_2\times {\mathit{w}}_2)}^T{\mathit{R}}_g{\mathbf{\theta }}_{B_2}\hfill \\ \multicolumn{1}{c}{}& +{(-1)}^{j+1}(e/2{({\mathit{R}}_g{\mathit{e}}_3\times {\mathit{w}}_2)}^T{\mathit{R}}_g{\mathbf{\psi }}_2+\Delta g_2/2{\mathit{w}}_2^T{\mathit{R}}_g{\mathit{e}}_3)\hfill \\ \multicolumn{1}{c}{}& +\Delta l_{22j}-{({\mathit{R}}_g(-{\mathit{l}}_0+{(-1)}^{j+1}e/2{\mathit{e}}_3)\times w_2)}^T\mathbf{\theta }\hfill \end{array}$$

(16)

$$\begin{array}{cc}\multicolumn{1}{c}{{\mathit{w}}_3^T\Delta \mathit{r}=}& {\mathit{w}}_3^T{\mathit{R}}_g\Delta {\mathit{b}}_3+{\mathit{w}}_3^T{\mathit{R}}_g\Delta {\mathit{a}}_3+\Delta l_{13}{\mathit{w}}_3^T{\mathit{R}}_g{\mathit{u}}_3+l_{13}{({\mathit{R}}_g{\mathit{u}}_3\times {\mathit{w}}_3)}^T{\mathit{R}}_g{\mathbf{\theta }}_{B_3}\hfill \\ \multicolumn{1}{c}{}& +{(-1)}^{j+1}(e/2{({\mathit{R}}_g{\mathit{u}}_v\times {\mathit{w}}_3)}^T{\mathit{R}}_g{\mathbf{\psi }}_3+\Delta g_3/2{\mathit{w}}_3^T{\mathit{R}}_g{\mathit{u}}_v)\hfill \\ \multicolumn{1}{c}{}& +\Delta l_{23j}-{({\mathit{R}}_g(-{\mathit{s}}_0+{\mathit{b}}_0+{(-1)}^{j+1}e/2{\mathit{u}}_v)\times {\mathit{w}}_3)}^T\mathbf{\theta }\hfill \end{array}$$

(17)

It can be seen that the position and orientation errors in the parallel manipulator with full-circle rotation are coupled. Note that the orientation errors cannot be compensated for as the parallel manipulator with full-circle rotation only has three controllable DOFs (associated with one translation and two rotations of the platform). Therefore, it is necessary to separate the orientation and position error sources which affect the accuracy in the end-effector from the sources of pose error. In order to achieve the requirement of accuracy in the end-effect, an accuracy analysis and kinematic calibration should be made of the different error sources.

Error separation

Taking the difference between the two closed-loop equations for chains 1, 2, and 3 yields

$$e{\left(R_g{e}_3\times w_1\right)}^T\theta =\Delta l_{d1}+\Delta g_1{w}_1^T{R}_g{e}_3+e{\left(e_3\times \left(R_g^T{w}_1\right)\right)}^T{\psi }_1$$

(18)

$$e{\left(R_g{e}_3\times w_2\right)}^T\theta =\Delta l_{d2}+\Delta g_2{w}_2^T{R}_g{e}_3+e{\left(e_3\times \left(R_g^T{w}_2\right)\right)}^T{\psi }_2$$

(19)

$$e{\left(R_g{u}_v\times w_3\right)}^T\theta =\Delta l_{d3}+\Delta g_3{w}_3^T{R}_g{u}_v+e{\left(u_v\times \left(R_{\mathrm{g}}^T{w}_3\right)\right)}^T{\psi }_3$$

(20)

where $\Delta l_{\mathrm{di}}=\Delta l_{2i1}-\Delta l_{2i2}$ are the relative length errors of the passive link in the ith parallelogram chain structure.

Rewriting equations (18)–(20) in matrix form leads to the orientation error-mapping function

$$\mathbf{\theta }={\mathit{J}}_{\theta }{\mathbf{\epsilon }}_{\theta }$$

(21)

$${\mathit{J}}_{\theta }={\mathit{A}}^{-1}\mathit{B}$$

where

$$\mathit{A}=e{\left[\begin{array}{ccc}{\mathit{R}}_g{\mathit{e}}_3\times {\mathit{w}}_1\hfill & {\mathit{R}}_g{\mathit{e}}_3\times {\mathit{w}}_2\hfill & {\mathit{R}}_g{\mathit{u}}_v\times {\mathit{w}}_3\hfill \end{array}\right]}^T$$

$$\mathit{B}=\mathrm{diag}\left[{\mathit{B}}_i\right]$$

$${\mathit{B}}_1=\left[\begin{array}{ccc}1\hfill & {\mathit{w}}_1^T{\mathit{R}}_g{\mathit{e}}_3\hfill & e{({\mathit{e}}_3\times \left({\mathit{R}}_g^T{\mathit{w}}_1\right))}^T\hfill \end{array}\right]$$

$${\mathit{B}}_2=\left[\begin{array}{ccc}1\hfill & {\mathit{w}}_2^T{\mathit{R}}_g{\mathit{e}}_3\hfill & e{({\mathit{e}}_3\times \left({\mathit{R}}_g^T{\mathit{w}}_2\right))}^T\hfill \end{array}\right]$$

$${\mathit{B}}_3=\left[\begin{array}{ccc}1\hfill & {\mathit{w}}_3^T{\mathit{R}}_g{\mathit{u}}_v\hfill & e{({\mathit{u}}_v\times \left({\mathit{R}}_g^T{\mathit{w}}_3\right))}^T\hfill \end{array}\right]$$

$${\epsilon }_{\theta }={\left(\begin{array}{ccc}{\epsilon }_{\theta 1}^T\hfill & {\epsilon }_{\theta 2}^T\hfill & {\epsilon }_{\theta 3}^T\hfill \end{array}\right)}^T$$

$${\epsilon }_{\theta i}={\left(\begin{array}{ccc}\Delta l_{\mathrm{di}}\hfill & \Delta g_i\hfill & {\mathbf{\psi }}_i^T\hfill \end{array}\right)}^T$$

It is clear that there are three kinds of geometrical error sources affecting the pose accuracy of the end-effector. These include the relative length errors $\Delta l_{\mathrm{di}}$ of the parallelogram, the relative errors $\Delta g_i$ of the distances between the centers of two spherical pairs, and the orientation errors ${\mathbf{\psi }}_i$ of $O_g\prime$ referenced to $O_g$ . The errors ${\mathbf{\psi }}_i$ are caused by errors made during assembly. Also, as $({\mathit{e}}_3\times ({\mathit{R}}_g^T{\mathit{w}}_1))\perp {\mathit{e}}_3,({\mathit{e}}_3\times ({\mathit{R}}_g^T{\mathit{w}}_2))\perp {\mathit{e}}_3$ , the projection of ${\mathbf{\psi }}_{1z}$ onto $({\mathit{e}}_3\times ({\mathit{R}}_g^T{\mathit{w}}_1))$ and ${\mathbf{\psi }}_{2z}$ onto $({\mathit{R}}_g{\mathit{e}}_3\times {\mathit{w}}_2)$ are both zero. This means that ${\mathbf{\psi }}_{1z}$ and ${\mathbf{\psi }}_{2z}$ have no effect on the orientation accuracy of the end-effector. Therefore, it is reasonable to remove the columns of ${\mathit{J}}_{\theta }$ associated with ${\mathbf{\psi }}_{1z}$ and ${\mathbf{\psi }}_{2z}$ . As a result, it can be concluded that 13 geometric errors are responsible for the orientation accuracy of the manipulator, and the most important ones are $\Delta l_{\mathrm{di}}$ , $\Delta g_i$ , ${\mathbf{\psi }}_{1x}$ , ${\mathbf{\psi }}_{1y}$ , ${\mathbf{\psi }}_{2x}$ , ${\mathbf{\psi }}_{2y}$ , ${\mathbf{\psi }}_{3x}$ , ${\mathbf{\psi }}_{3y}$ , and ${\mathbf{\psi }}_{3z}$ .

Additionally, the two closed-loop equations belonging to each chain give

$$\begin{array}{c}\multicolumn{1}{c}{{\mathit{w}}_1^T\Delta \mathit{r}=\Delta l_{a1}+{\mathit{w}}_1^T{\mathit{R}}_g\Delta {\mathit{b}}_1+{\mathit{w}}_1^T{\mathit{R}}_g\Delta {\mathit{a}}_1+\Delta l_{11}{\mathit{w}}_1^T{\mathit{R}}_g{\mathit{u}}_1}\\ \multicolumn{1}{c}{+l_{11}{({\mathit{u}}_1\times \left({\mathit{R}}_{\mathrm{g}}^T{\mathit{w}}_1\right))}^T{\mathbf{\theta }}_{B_1}-{({\mathit{R}}_g{\mathit{l}}_0\times w_1)}^T\mathbf{\theta }}\end{array}$$

(22)

$$\begin{array}{c}\multicolumn{1}{c}{{\mathit{w}}_2^T\Delta \mathit{r}=\Delta l_{a2}+{\mathit{w}}_2^T{\mathit{R}}_g\Delta {\mathit{b}}_2+{\mathit{w}}_2^T{\mathit{R}}_g\Delta {\mathit{a}}_2+\Delta l_{12}{\mathit{w}}_2^T{\mathit{R}}_g{\mathit{u}}_2}\\ \multicolumn{1}{c}{+l_{12}{({\mathit{u}}_2\times \left({\mathit{R}}_{\mathrm{g}}^T{\mathit{w}}_2\right))}^T{\mathbf{\theta }}_{B_2}-{({\mathit{R}}_g(-{\mathit{l}}_0)\times w_2)}^T\mathbf{\theta }}\end{array}$$

(23)

$$\begin{array}{c}\multicolumn{1}{c}{{\mathit{w}}_3^T\Delta \mathit{r}=\Delta l_{a3}+{\mathit{w}}_3^T{\mathit{R}}_g\Delta {\mathit{b}}_3+{\mathit{w}}_3^T{\mathit{R}}_g\Delta {\mathit{a}}_3+\Delta l_{13}{\mathit{w}}_3^T{\mathit{R}}_g{\mathit{u}}_3}\\ \multicolumn{1}{c}{+l_{13}{({\mathit{u}}_3\times \left({\mathit{R}}_{\mathrm{g}}^T{\mathit{w}}_3\right))}^T{\mathbf{\theta }}_{B_3}-{({\mathit{R}}_g(-{\mathit{s}}_0-{\mathit{b}}_0)\times w_3)}^T\mathbf{\theta }}\end{array}$$

(24)

where $\Delta l_{\mathrm{ai}}=(\Delta l_{2i1}+\Delta l_{2i2})/2$ is the mean of the length errors of the passive link in the ith parallelogram chain structure.

Rewriting equations (22)–(24) in matrix form gives the position error-mapping function

$$\Delta \mathit{r}={\mathit{J}}_r{\mathbf{\epsilon }}_r=\left[\begin{array}{cc}{J}_{\mathrm{rr}}\hfill & J_{r\theta }\hfill \end{array}\right]{\left[\begin{array}{cc}{\epsilon }_{\mathrm{rr}}\hfill & {\epsilon }_{\theta }\hfill \end{array}\right]}^T$$

(25)

where

$${\mathit{J}}_{\mathrm{rr}}={\mathit{C}}^{-1}\mathit{D}$$

$${\mathit{J}}_{r\theta }={\mathit{C}}^{-1}\mathit{E}$$

$$\mathit{C}={\left[\begin{array}{ccc}{\mathit{w}}_1\hfill & {\mathit{w}}_2\hfill & {\mathit{w}}_3\hfill \end{array}\right]}^T$$

$$\mathit{D}=\mathrm{diag}\left[{\mathit{D}}_i\right]$$

$${\mathit{D}}_1=\left[\begin{array}{cccc}1\hfill & {\mathit{w}}_1^T{\mathit{R}}_g\hfill & \begin{array}{cc}{\mathit{w}}_1^T{\mathit{R}}_g\hfill & {\mathit{w}}_1^T{\mathit{R}}_g{\mathit{u}}_1\hfill \end{array}\hfill & l_{11}{({\mathit{u}}_1\times \left({\mathit{R}}_g^T{\mathit{w}}_1\right))}^T\hfill \end{array}\right]$$

$${\mathit{D}}_2=\left[\begin{array}{cccc}1\hfill & {\mathit{w}}_2^T{\mathit{R}}_g\hfill & \begin{array}{cc}{\mathit{w}}_2^T{\mathit{R}}_g\hfill & {\mathit{w}}_2^T{\mathit{R}}_g{\mathit{u}}_2\hfill \end{array}\hfill & l_{12}{({\mathit{u}}_2\times \left({\mathit{R}}_g^T{\mathit{w}}_2\right))}^T\hfill \end{array}\right]$$

$${\mathit{D}}_3=\left[\begin{array}{cccc}1\hfill & {\mathit{w}}_3^T{\mathit{R}}_g\hfill & \begin{array}{cc}{\mathit{w}}_3^T{\mathit{R}}_g\hfill & {\mathit{w}}_3^T{\mathit{R}}_g{\mathit{u}}_3\hfill \end{array}\hfill & l_{13}{({\mathit{u}}_3\times \left({\mathit{R}}_g^T{\mathit{w}}_3\right))}^T\hfill \end{array}\right]$$

$$\mathit{E}=-{\left[\begin{array}{ccc}{\mathit{R}}_g{\mathit{l}}_0\times {\mathit{w}}_1\hfill & {\mathit{R}}_g(-{\mathit{l}}_0)\times {\mathit{w}}_2\hfill & {\mathit{R}}_g(-{\mathit{s}}_0-{\mathit{b}}_0)\times {\mathit{w}}_3\hfill \end{array}\right]}^T$$

$${\epsilon }_{\mathrm{rr}}={\left(\begin{array}{ccc}{\epsilon }_{r1}^T\hfill & {\epsilon }_{r2}^T\hfill & {\epsilon }_{r3}^T\hfill \end{array}\right)}^T$$

$${\epsilon }_{\mathrm{ri}}={\left(\begin{array}{cccc}\Delta l_{\mathrm{ai}}\hfill & \Delta {\mathit{b}}_i^T\hfill & \begin{array}{cc}\Delta {\mathit{a}}_i^T\hfill & \Delta l_{1i}\hfill \end{array}\hfill & {\mathbf{\theta }}_{\mathrm{Bi}}^T\hfill \end{array}\right)}^T$$

Therefore, the geometrical error parameters affecting the position accuracy of the end-effector can be described as: the mean value of the length errors of two passive links of the parallelogram $(\Delta l_{\mathrm{di}})$ , the synthetic error vectors of the origin $B_i$ relative to the coordinate system $O_g\prime (\Delta {\mathit{b}}_i)$ , the length errors of the active links $(\Delta l_{1i})$ , and the orientation errors of $B_i$ relative to the coordinate system $O_g({\mathbf{\theta }}_{\mathrm{Bi}},i=1,2,3)$ . These include the error sources affecting pose accuracy. When the corresponding coefficients of two kinds of error sources $\Delta {\mathit{b}}_i$ and $\Delta {\mathit{a}}_i$ are proportional, the corresponding $D_i$ columns are linearly related. Hence, these two kinds of error sources can be attributed to a class. At the same time, because the error source $\Delta {\mathit{b}}_{3x}$ on chain 3 has no effect on the accuracy of the end-effector, ${\mathit{J}}_{\mathrm{rr}}$ needs to be modified to eliminate columns associated with the error source. Therefore, there are 36 kinds of geometrical error sources affecting the accuracy of the end-effector, including 13 kinds of orientation errors that affect the orientation accuracy of the end-effector.

Sensitivity analysis

Sensitivity analyses can guide tolerance design and assembly of every part by evaluating the degree of influence of the geometrical errors on the orientation accuracy of every part of the structure. Using a sensitivity analysis, the laws governing the effects of geometric parameter errors on the orientation error of this manipulator (with full-circle rotation) can be proposed.

Consider the position error-mapping model $\Delta \mathit{r}={\mathit{J}}_r{\mathbf{\epsilon }}_r$ as an example. The probability model for the sensitivity analysis is first constructed. Arranging ${\mathit{J}}_r$ and ${\mathbf{\epsilon }}_r$ in the order of the chains, the above equation can be rewritten as

$$\Delta \mathit{r}={\mathit{J}}_r\prime {\mathbf{\epsilon }}_r\prime$$

(26)

Taking the norm of both sides of equation (26) yields

$$\Delta r_r^2=\sum _{i=1}^3\sum _{j=1}^3\sum _{k=1}^n\sum _{l=1}^n(\sum _{m=1}^3{\mathit{J}}_{\mathrm{rimk}}\prime {\mathit{J}}_{\mathrm{rjml}}\prime ){\epsilon }_{\mathrm{rik}}\prime {\epsilon }_{\mathrm{rjl}}\prime$$

(27)

where ${\epsilon }_{\mathrm{rik}}\prime$ is the kth element in ${\epsilon }_{\mathrm{ri}}\prime$ , ${\mathit{J}}_{\mathrm{rimk}}\prime$ is the element in row m and column k of ${\mathit{J}}_{\mathrm{ri}}\prime$ , and n is the number of geometric error parameters in all chains.

We assume that all elements in ${\mathbf{\epsilon }}_r\prime$ are independent random variables and have normal distributions with zero mean. As a result, the mean value of $\Delta r_r$ is zero and

$$\begin{array}{cc}\multicolumn{1}{c}{D\left(\Delta r_r\right)}& =E\left(\Delta r_r^2\right)-{\left[E\left(\Delta r_r\right)\right]}^2=E\left(\Delta r_r^2\right)\hfill \\ \multicolumn{1}{c}{}& =\sum _{k=1}^n\sum _{i=1}^3\left(\sum _{m=1}^3{\mathit{J}}_{\mathrm{rimk}}^{\prime 2}\right)E\left({\epsilon }_{{}_{\mathrm{rik}}}^{\prime 2}\right),n=12\hfill \end{array}$$

(28)

For the new manipulator with full-circle rotation, the geometrical errors of the same type in each chain share the same variance, that is

$$E\left({\epsilon }_{\mathrm{rk}}^{\prime 2}\right)=E\left({\epsilon }_{r1k}^{\prime 2}\right)=E\left({\epsilon }_{r2k}^{\prime 2}\right)=E\left({\epsilon }_{r3k}^{\prime 2}\right)$$

(29)

Thus, the standard deviation of $\Delta r_r$ can be expressed as

$$\sigma \left(\Delta r_r\right)=\sqrt{\sum _{k=1}^n\sum _{i=1}^3\sum _{m=1}^3{\mathit{J}}_{\mathrm{rmik}}^{\prime 2}{\sigma }^2({\epsilon }_{\mathrm{rk}}\prime )}$$

(30)

For the orientation error $\theta =(\begin{array}{ccc}{\theta }_x\hfill & {\theta }_y\hfill & {\theta }_z\hfill \end{array}{)}^T$ , the standard deviation of ${\delta }_{\alpha }$ and ${\delta }_{\varphi }$ can be expressed as

$$\sigma \left({\delta }_{\alpha }\right)=\sqrt{\sum _{k=1}^n\sum _{i=1}^3\sum _{m=1}^2{\mathit{J}}_{\theta \mathrm{mik}}^2{\sigma }^2\left({\epsilon }_{\theta k}\right)}$$

(31)

$$\sigma \left({\delta }_{\varphi }\right)=\sqrt{\sum _{k=1}^n\sum _{i=1}^3{\mathit{J}}_{\theta 3\mathrm{ik}}^2{\sigma }^2\left({\epsilon }_{\theta k}\right)}$$

(32)

Similarly, the sensitivity coefficients of the orientation error with respect to geometric error ${\epsilon }_{\theta k}$ can be expressed as

$${\mu }_{\mathrm{kr}}\prime =\sqrt{\sum _{i=1}^3\sum _{m=1}^3{J}_{\mathrm{rimk}}^{\prime 2}},k=1,2,\dots ,12$$

(33)

$${\mu }_{\alpha k}=\sqrt{\sum _{i=1}^3\sum _{m=1}^2{\mathit{J}}_{\theta \mathrm{imk}}^2},k=1,2,\dots ,13$$

(34)

$${\mu }_{\varphi k}=\sqrt{\sum _{i=1}^3{\mathit{J}}_{\theta i3k}^2},k=1,2,\dots ,13$$

(35)

These sensitivity coefficients represent the standard deviations of the errors in the end-effector caused by a unit standard deviation of geometrical error. As sensitivity coefficients vary with the configuration of the mechanism, their mean values can be employed as a sensitivity evaluation index in the overall prescribed workspace. The index can be defined as

$${\overline{\mu }}_{\mathrm{kr}}\prime =(\underset{V}{\int }{\mu }_{\mathrm{kr}}\prime \mathrm{dV})/V,k=1,2,\dots ,12$$

(36)

$${\overline{\mu }}_{\alpha k}\prime =\left(\underset{V}{\int }{\mu }_{\alpha k}\mathrm{dV}\right)/V,k=1,2,\dots ,13$$

(37)

$${\overline{\mu }}_{\varphi k}\prime =\left(\underset{V}{\int }{\mu }_{\varphi k}\mathrm{dV}\right)/V,k=1,2,\dots ,13$$

(38)

where V is the volume of the prescribed workspace.

Note that this parallel manipulator can realize full-circle rotation and has the same performance in all directions. As shown in Figure 4, the reachable workspace of reference point on moving platform is the intersection of three subspaces associated with three kinematic chains. The subspace of each chain is the region encircled by three surfaces whose circle center is $A_i$ and radius is $l_2$ , when active links reach their minimum limits $L_{min}$ and maximum limits $L_{max}$ , respectively. According to the actual engineering requirements, a torus of semidiameter 350 and height 200 is selected as task workspace which is located in reachable workspace. In addition, as the robot body in the lower part and avoiding interference, the task workspace is in the upper part of the reachable workspace as far as possible. The inner boundary of the task workspace is tangent to the inner boundary which is constraint by the angle $({\beta }_3=40\circ )$ of active and passive links 3. The outer boundary of the task workspace is tangent to the outer boundary which is constraint by the angles $({\beta }_1={\beta }_2=140\circ )$ of active and passive links 1 and 2.

Figure 4.

Cross-section of the workspace of the parallel manipulator with full-circle rotation (through the yz plane): (a) Geometric parameters in task workspace and (b) Sectional view of reachable and task workspace.

Thus, the three global sensitivity evaluation indices above can be measured in the prescribed working area in the yz plane, that is

$${\overline{\mu }}_{\mathrm{kr}}=(\underset{S}{\int }{\mu }_{\mathrm{kr}}\prime \mathrm{dV})/S,k=1,2,\dots ,12$$

(39)

$${\overline{\mu }}_{\alpha k}=\left(\underset{S}{\int }{\mu }_{\alpha k}\mathrm{dV}\right)/S,k=1,2,\dots ,13$$

(40)

$${\overline{\mu }}_{\varphi k}=\left(\underset{S}{\int }{\mu }_{\varphi k}\mathrm{dV}\right)/S,k=1,2,\dots ,13$$

(41)

A sensitivity analysis can be carried out on the parallel manipulator with full-circle rotation by utilizing the indices given in equations (39) - (41). The nominal dimensions used in the analysis are shown in Table 1. According to the working and force-transfer characteristics of the mechanism, the prescribed workspace is a cylindrical ring with an inner radius of 300 mm, outer radius 650 mm, and height 200 mm. In the yz plane, it is a rectangle, 300 mm away from the central axis, with a length of 350 mm and width of 200 mm (as shown in Figure 4).

Table 1.

Nominal dimensions of the parallel manipulator with full-circle rotation (the linear unit is mm).

$l_1$	$l_2$	$b_0$	$s_0$	$l_0$	$b_3$
250	500	27.8	250	73	150.5

Figures 5–8 show the results of global sensitivity analyses of the pose and volumetric errors ${\delta }_{\alpha }$ , ${\delta }_{\varphi }$ , and ${\delta }_r$ with respect to the geometrical error parameters. As shown in Figure 5, in chains 1 and 2, the orientation geometric errors $\Delta l_{d1}$ and $\Delta l_{d2}$ have a marked effect on the verticality error ${\delta }_{\alpha }$ (the spindle axis with respect to the xy plane). However, in chain 3, the rotational axis of the active link 3 is vertical to the spindle of the mechanism. So, it is on the non-sensitivity direction of error for ${\delta }_{\alpha }$ . Therefore, the sensitivity coefficients of the geometric errors in chain 3 with respect to error in the verticality of the spindle axis about the xy plane are zero, such as $\Delta l_{d3}$ , $\Delta g_3$ , ${\mathbf{\psi }}_{3x}$ , ${\mathbf{\psi }}_{3y}$ , and ${\mathbf{\psi }}_{3z}$ . It illustrates that some uncompensatable errors will not affect the corresponding orientation errors of the end-effector. As shown in Figure 6, the geometric parameter error $\Delta l_{\mathrm{di}}$ has a large effect on the angular error ${\delta }_{\varphi }$ (about the z-axis), and the geometric error $\Delta g_3$ also has an obvious effect on the angular error ${\delta }_{\varphi }$ (about the z-axis). If error ${\delta }_{\varphi }$ due to $\Delta l_{\mathrm{di}}$ is restricted to within $20\mu \mathrm{m}$ , in accordance with the $3\sigma$ principles, the tolerance of $l_{\mathrm{di}}$ must be held to $T(\Delta l_{d1})=T(\Delta l_{d2})=\pm (3\times 20/12.52)=\pm 4.79\mu \mathrm{m}$ and $T(\Delta l_{d3})=\pm (3\times 20/28.37)=\pm 2.11\mu \mathrm{m}$ . The position geometrical errors are sensitive to geometric parameter errors $\Delta l_{\mathrm{di}}$ , $\Delta l_{\mathrm{ai}}$ , $\Delta b_y$ , and $\Delta l_{1i}$ , and the angular geometrical errors are not as sensitive as the linear geometric errors are to position geometrical errors. Similarly, the tolerance of $\Delta l_{\mathrm{ai}}$ must be $T(\Delta l_{a1})=T(\Delta l_{a2})=\pm (3\times 20/1.953)=\pm 30.72\mu \mathrm{m}/\mathrm{m}$ and $T(\Delta l_{a3})=\pm (3\times 20/1.482)=\pm 40.48\mu \mathrm{m}/\mathrm{m}$ .

Figure 5.

Sensitivity analysis of the geometric errors with respect to error in the verticality of the spindle axis about the xy plane $(i=1,2)$ .

Figure 6.

Sensitivity analysis of the geometric errors with respect to the angular error in the z-axis $(i=1,2)$ .

Figure 7.

Sensitivity analysis of the geometric error with respect to position error in chain $i(i=1,2)$ .

Figure 8.

Sensitivity analysis of the geometric error with respect to position error in chain 3.

Tolerance allocation

For the parallel mechanism with less freedom, the uncompensatable error sources cannot be compensated for by kinematic calibration. In order to guarantee the accuracy in the position and pose of the mechanism, the manufacturing tolerances of the parts need to be suitably allocated via tolerance optimization. There are nine kinds of geometric error sources for the parallel manipulator with full-circle rotation. In order to reduce the manufacturing cost, the tolerance of the geometric parameters should be relaxed as far as possible subject to the premise that the tolerances of the orientation errors ${\delta }_{\alpha }$ and ${\delta }_{\varphi }$ are less than the allowable values in the global workspace. Thus, we assume that the nine kinds of uncompensatable error sources are design variables and that the tolerance manufacturing cost is the objective function. Then, giving the required tolerance range of the two errors ${\delta }_{\alpha }$ , ${\delta }_{\varphi }$ and taking each error source as the constraint factors, the tolerance design of each component is converted into a nonlinearly constrained optimization problem, as follows

$$minf=\sum _{k=1}^9\frac{s_k}{{\sigma }^2(\Delta {\epsilon }_{\mathrm{ck}}\prime )}$$

(42)

where $s_k$ are the process parameters which, for length and angle parameters are, respectively, 1 and 1.5

$$\{\begin{array}{c}\max \left(\sigma \left({\delta }_{\alpha }\right)\right)\le \left[\sigma \left({\delta }_{\alpha }\right)\right]\hfill \\ \multicolumn{1}{c}{\max \left(\sigma \left({\delta }_{\varphi }\right)\right)\le \left[\sigma \left({\delta }_{\varphi }\right)\right]}\\ \multicolumn{1}{c}{\inf \left(\sigma \left(\Delta \epsilon {\prime }_{\mathrm{ck}}\right)\right)\le \sigma \left(\Delta \epsilon {\prime }_{\mathrm{ck}}\right)\le \sup \left(\sigma \left(\Delta \epsilon {\prime }_{\mathrm{ck}}\right)\right)}\end{array}$$

(43)

where $\max (\sigma ({\delta }_{\alpha }))$ is the maximum of the standard deviation of the squareness error in the end-effector; $\max (\sigma ({\delta }_{\varphi }))$ is the maximum of the standard deviation of the spindle angular error; $[\sigma ({\delta }_{\alpha })]$ is the maximum allowable value of the standard deviation of the squareness error in the end-effector; $[\sigma ({\delta }_{\varphi })]$ is the maximum allowable value of the standard deviation of the spindle angular error; $\sup (\sigma (\Delta \epsilon {\prime }_{\mathrm{ck}}))$ is the upper limit of the standard deviation of the uncompensatable error sources; and $\inf (\sigma (\Delta \epsilon {\prime }_{\mathrm{ck}}))$ is the lower limit of the standard deviation of the uncompensatable error sources.

This is a complicated optimization problem, one which we tackle using a GA. GA is an evolutionary algorithm. It is based on imitating the mechanism of genetic selection that occurs in nature to find the optimal solution.

GA includes three basic operators: selection, crossover, and mutation. First, the algorithm searches for a set of approximate optimal values among all the possible values, which then form the genetic population. Then, it screens for satisfactory individuals among the genetic group to form a new population. In the screening process, new individuals are continuously produced by crossover and mutation. Finally, a set of satisfactory optimal values are obtained. Iteration is the main numerical process used to solve this nonlinear optimization problem. However, iterative methods in general can easily fall into local minima which may trap them, and as a result, the phenomenon referred to as “death cycle” is observed. GA is capable of overcoming this shortcoming. As a consequence, a GA is better placed to find the global minimum error. Moreover, compared with other optimization methods, GA that has small population sizes and high mutation rates can rapidly find a good solution.

In this work, the solution process is directly performed using the GA toolbox built into the MATLAB software application. The minimum value of the objective function is the fitness function, which is determined by equation (39). The constraint conditions are determined using equation (40). Table 2 shows the allowable parameter values wherein the standard deviations are determined as per JB/T 10792.1-2007. ²⁹

Table 2.

Allowable parameter values.

$[\sigma ({\delta }_{\alpha })](\mathrm{\mu }\mathrm{m}/\mathrm{m})$	$[\sigma ({\delta }_{\varphi })](\mathrm{\mu }\mathrm{m}/\mathrm{m})$	$\sup (\sigma (\Delta {\epsilon }_{\mathrm{ck}}\prime ))(\mathrm{\mu }\mathrm{m})$	$\inf (\sigma (\Delta {\epsilon }_{\mathrm{ck}}\prime ))(\mathrm{\mu }\mathrm{m})$
40/3	30/3	6	0

Parameters needed to implement the GA include variable dimensions, number of generations, population size, mutation probability, and crossover probability. In this case, the variables are nine kinds of uncompensatable error sources and so the variable dimension is 9. The number of generations is taken to be 400, which can produce good calculated values and is relatively fast. Figure 9 shows the optimization process when the population size is set to 20 and 80, both of which have bad convergences. Using a large population size has the advantage of improving the search quality of the GA and prevents convergence before maturation. However, it also increases the number of calculations involved and reduces the rate of convergence.

Figure 9.

Optimization of the fitness function with different population sizes: (a) population size is 20 and (b) population size is 80.

Figure 10 shows the optimization processes when the crossover probability is set to 0.05 and 0.99. When the crossover probability is large, the introduction of a new structure is more likely. However, it also results in randomization. A low crossover probability may make the genetic search stagnate.

Figure 10.

Optimization of the fitness function with different crossover probabilities: (a) crossover probability is 0.05 and (b) crossover probability is 0.99.

Figure 11 shows the optimization processes when the mutation probability is set to 0.001 and 0.08. Mutation probability is an important factor that keeps the population diverse. A low mutation probability may easily generate a local extreme value, but too a high mutation probability can make the genetic search too random. Mutation probability affects the ability of the algorithm to search locally, whereas the crossover probability affects the ability of the algorithm to search globally. So, they need to be coordinated with each other in an appropriate manner.

Figure 11.

Optimization of the fitness function with different mutation probabilities: (a) mutation probability is 0.001 and (b) mutation probability is 0.08.

The selection of the parameters discussed above is dependent on the type of problem involved. Therefore, in any actual problem, one needs to change these parameters continuously in order to obtain the best results according to certain requirements. The above selection process is appropriate in special situations. Actually, in the current situation, many simulations have been done that suggest the best values to use. The final parameter values assumed are shown in Table 3. The values of the fitness function subsequently obtained using the GA are shown in Figure 12.

Table 3.

Parameters used in the genetic algorithm.

Population size	Variable dimension	Crossover probability	Mutation probability	Generations
50	9	0.4	0.04	400

Figure 12.

Optimization of the fitness function.

Based on the $3\sigma$ criterion, the standard deviation of each geometric error source is transformed into a tolerance $T=\pm 3\sigma$ . The results are shown in Table 4. It is essential to control and allocate the manufacturing tolerances according to these principles in the manufacturing and assembly processes.

Table 4.

Optimization results for the tolerances of the sources of uncompensatable error.

$T(\Delta l_{\mathrm{di}})/(\mathrm{\mu }\mathrm{m})(i=1,2)$	$T(\Delta g_i)/(\mathrm{\mu }\mathrm{m})(i=1,2)$	$T(\Delta {\mathbf{\psi }}_{\mathrm{ix}})/(\mathrm{\mu }\mathrm{m}{\mathrm{m}}^{-1})(i=1,2)$	$T(\Delta {\mathbf{\psi }}_{\mathrm{iy}})/(\mathrm{\mu }\mathrm{m}{\mathrm{m}}^{-1})(i=1,2)$	$T(\Delta l_{d3})/(\mathrm{\mu }\mathrm{m})$
±4	±9	±12	±18	±4
$T(\Delta g_3)/(\mathrm{\mu }\mathrm{m})$	$T(\Delta {\mathbf{\psi }}_{3x})/(\mathrm{\mu }\mathrm{m}{\mathrm{m}}^{-1})$	$T(\Delta {\mathbf{\psi }}_{3y})/(\mathrm{\mu }\mathrm{m}{\mathrm{m}}^{-1})$	$T(\Delta {\mathbf{\psi }}_{3z})/(\mathrm{\mu }\mathrm{m}{\mathrm{m}}^{-1})$
±4	±12	±15	±13

A photograph of a prototype parallel manipulator with full-circle rotation is shown in Figure 13. According to our measurements, it can achieve $30\circ$ elevation and $45\circ$ depression at the far end. At the close end, it can achieve $45\circ$ elevation and $20\circ$ depression (this is so it can avoid interference and collision). These guarantee that the manipulator can work in the range of $\varphi 360\times 300$ . At the same time, when the motor reaches its rated speed of 3000 r/s, the maximum speed of the end of the moving platform can reach 4.5 m/s. According to our simulation results, the absolute maximum values of the relevant errors (position error in the X, Y, and Z directions and the volume error) are 33.2, 33.5, 152.2, and 155.7 µm, and the absolute mean values are 32.7, 32.9, 150.6, and 154 µm, respectively. ²⁶

Figure 13.

Prototype of a parallel manipulator with full-circle rotation.

Conclusion

A unified error model for a novel 3-DOF parallel manipulator with full-circle rotation has been formulated. The following conclusions can be drawn:

1. The error model can be formulated using the method of vector chains. Position and orientation error-mapping models can be established via mathematical transformation of the parallelogram structure characteristics. The orientation and position error sources can then be separated effectively. There are 18 orientation and 36 position error sources. There are some common error sources within the two kinds of error sources, namely, $\Delta l_{\mathrm{di}}$ , $\Delta g_i$ , ${\mathbf{\psi }}_{1x}$ , ${\mathbf{\psi }}_{1y}$ , ${\mathbf{\psi }}_{2x}$ , ${\mathbf{\psi }}_{2y}$ , ${\mathbf{\psi }}_{3x}$ , ${\mathbf{\psi }}_{3y}$ , and ${\mathbf{\psi }}_{3z}$ .

2. A global sensitivity evaluation index can be formulated to evaluate all the error sources, and the laws governing the effects of the geometric errors on accuracy in the end-effector can thus be derived. The result indicates that the relative link length errors $\Delta l_{\mathrm{di}}$ and the center distance errors $\Delta g_i$ of spherical joints are main geometric errors. So, they should be controlled strictly in the tolerance allocation.

3. Based on the analysis results, the problem of the design accuracy of the mechanism can be converted into a nonlinearly constrained optimization problem. The tolerance allocation of each link can then be optimized using a GA. This provides the information necessary for the manufacture and assembly of each component.

4. The uncompensatable errors are separated effectively and controlled in the process of manufacturing and assembly. The compensatable errors are compensated by kinematic calibration when the parts achieve the certain accuracy. The method presented in this article is facilitated to the kinematic calibration of the mechanism in the future.

5. The method of design can be readily extended to other similar low-mobility parallel structures. As low-mobility parallel structures include uncompensatable error sources, the end-effector pose error cannot be totally compensated for using software. The model presented shows how kinematic analysis, error modeling, sensitivity analysis, and tolerance allocation can be successfully integrated into a comprehensive framework to improve the accuracy of low-mobility parallel structures.