Locating error analysis for workpieces with general fixture layouts and parameterized tolerances

Abstract

In manufacturing engineering, locating error analysis for workpieces is a common issue in predicting the quality of machined features, computer-aided fixture design and tolerance allocation. This article presents a locating error analysis approach for workpiece with general fixture layouts and parameterized tolerances. Initially, a fixture is transformed into six locating points in the space. A linearized model is then derived to convey the relationship between the errors of locating points and the locating errors, by using the implicit differentiation and the first-order Taylor expansion. Then, the tolerances for the features of plane, cylinder and free-form surface are parameterized using 4 × 4 homogeneous transformation matrices. Finally, two formulas are derived to calculate the errors in the locating points in surface-to-surface mating and hole-to-pin mating. Two workpieces are presented to demonstrate and validate the proposed method. The relative errors between the results calculated by the proposed method and the results calculated by 3DCS are less than 3%.

Introduction

In manufacturing engineering, designers need to accomplish two basic tasks: machining surface accuracy check and locator tolerance allocation. ¹ In recent years, aided by developments in computer-aided technology, these two tasks are performed more simply and efficiently. ² With computer-aided design (CAD)/computer-aided manufacturing (CAM) platforms, information such as geometrical features and tolerances is easily gathered, allowing a three-dimensional (3D) fixture layout for a workpiece to be designed rather quickly. Despite this, designers still encounter significant; for example, knowing how to best use the 3D information efficiently to predict the quality of a workpiece, and to verify the reasonability of a tolerance assignment. To remedy this, a mathematical model related to tolerances, geometrical errors and the quality of workpieces is necessary. The interactions between workpieces and fixtures in the manufacturing processes are quite complex, and two important factors, datum errors and fixture errors, affect the quality of machined features. In this article, we primarily consider these two factors to present a method for locating error analysis of a workpiece, which must be completed before performing the two main tasks aforementioned.

Previous research on locating error analysis for workpieces has focused chiefly on fixture layout optimization. According to the configuration of the fixture, these methods can be divided into two categories: “3-2-1”-oriented and “general fixture layout”-oriented. In the “3-2-1”-oriented methods, Cai et al. ³ established a mathematical model to express the relationship between the errors of locating pin and the locating errors of workpiece. Zhang et al. ⁴ proposed an approach for setup planning with a combination of process planning and computer-aided fixture design. In this article, the workpiece locating errors and the machining errors were used as evaluation indexes for fixture design. With consideration of 3-2-1, Pin-Hole and V block, Rong et al. ⁵ proposed a locating error analysis method for three fixture layouts. In Raghu and Melkote, ⁶ Qin et al., ⁷ Vishnupriyan et al., ⁸ Zhu et al. ⁹ and Wang, ¹⁰ factors including geometric errors of workpiece and fixture are considered. In Raghu and Melkote ⁶ three major error sources, including fixture geometric errors and the elastic deformation of fixture and workpiece due to fixturing forces, were considered. Qin et al. ⁷ and Vishnupriyan et al. ⁸ presented a general analysis methodology that is able to characterize the effects of localization error sources based on the position and orientation of the workpiece. Zhu et al. ⁹ presented a prediction model of workpiece locating error caused by the setup error and geometric inaccuracy of locaters for the fixtures with one locating surface and two locating pins. Wang ¹⁰ established a mathematical model to convey the relationship between the geometric errors and locating errors. Tolerance analysis for workpiece and fixture is also considered by researchers. Asante ¹¹ presented a methodology for modeling and analyzing the combined effect of workpiece geometric errors, locator geometric error and clamping error on a machined feature based on small displacement torsors. Loose et al. ¹² integrated Geometric Dimensioning & Tolerancing (GD&T) factors into locating error analysis. Choudhuri and De Meter ¹³ analyzed the tolerances of profile and angle for the machined features with consideration of datum errors, geometric errors. More recently, Zhou et al. ¹⁴ presented a state-space model and its modeling strategies to describe the variation stack-up in multi-operational machining process. In the method, surface deviation was expressed in terms of deviation from nominal orientation, location and dimensions, and a part’s deviation was expressed in terms of the deviation of its surfaces and is stored in a state vector x(k). In “general fixture layout”-oriented methods, Carlson ¹⁵ presented a quadratic sensitivity analysis for locating errors with the consideration of locator errors and the geometric errors of workpiece. Wang ¹⁶ presented an analysis method to describe the impact of locating error sources on the geometric errors of machined features. The method has shown the importance to consider the errors in the multiple critical locating points (LPs) of fixture. Loose et al. ¹⁷ developed a linear model to describe the dimensional variation propagation of machining processes with general fixture layouts. However, in the method, the locator was treated as a point, so it could not deal with plane/plane contact or cylinder/cylinder floating contact when positioning the workpiece.

As established by previous researchers, it is clear that locating error analysis for a workpiece plays an important role in fixture layout optimization and tolerance allocation. However, the types of methods presented in much of this literature have shortcomings. First, the methods for locating error analysis are chiefly for 3-2-1 fixture layout, which limits the application of those methods. Also, the key pieces of information in the process of workpiece locating, geometric features and tolerances, are not integrated into the current popular methods. This creates difficulties in applying said methods to the tolerance allocation of workpieces and fixtures, or to integrate the methods into current CAD/CAM platforms. This article is organized as follows. In section “Linear locating error analysis model for workpiece,” a linear deviation model for workpiece locating is derived. In section “Tolerance modeling based on variational feature,” a parameterized tolerance model is presented for planar, cylindrical, free-form surface features. Two formulas are derived to calculate the errors of LPs in surface-to-surface mating and hole-to-pin mating based on the parameterized tolerance in section “Calculating the deviations of LP.” Two workpieces are illustrated to validate the proposed method in section “Numerical experiments.” Finally, some conclusions of this article are presented in section “Conclusion.”

Linear locating error analysis model for workpiece

In a typical process of workpiece–fixture assembly, a workpiece is constrained by a fixture. The constraints between the workpiece and fixture are characterized by two features, which are carriers of datum errors and fixture errors. As illustrated in Figure 1, the block is constrained by four geometrical constraints (or matings): the pair f₁ and f₄ is a plane-to-plane constraint; the pair f₁ and f₄, the pair f₂ and f₅, the pair f₂ and f₆ and the pair f₃ and f₇ are plane-to-point constraints. It is noted that f₅, f₆ and f₆ are small planes, so they are considered as points. According to the theory of virtual LPs proposed by Huang et al.,^18,19 a geometrical constraint can be transformed into several LPs according to certain principles. In this article, the following set of principles are adopted: first, three LPs are selected on the primary datum surface, which make the area as large as possible; second, two LPs are selected on the secondary datum surface, which make the distance as long as possible; third, an LP is selected in the center of the tertiary datum surface. By these standards, we can obtain a set of LPs interactively on the CAD model. As shown in Figure 1(b), f₁ is the primary datum, allowing the designers to select three LPs (i.e. LP1, LP2 and LP3) from the workpiece’s CAD model and creating a large area with the three LPs; similarly, f₂ is the secondary datum, so two LPs (i.e. LP4 and LP5) are selected from the CAD model, and creating a lengthy distance; finally, an LP (i.e. LP6) is selected on feature f₃. It is noted that the constraint is constructed by a pair of features, so when selecting LP1–LP6 on the workpiece, it is necessary to ensure corresponding points on the fixture, as with LP1′–LP6′ in Figure 1(c). In other words, the LPs are selected as a pairing: one is on the workpiece and another is on the fixture. The coordinates of the two points are identical.

Figure 1.

A block’s LPs: (a) a block’s locating and (b) fixture.

In this article, to derive a general model for locating error analysis, a fixture is turned into six LPs, which make up a general fixture layout. As shown in Figure 2, the black line indicates the theoretical position of the workpiece and the blue line indicates the real position of the workpiece. L₁–L₆ are the six LPs. $\mathrm{OXYZ}$ and $O\prime X\prime Y\prime Z\prime$ represent the global coordinate system and the workpiece coordinate system, respectively. ${\mathit{r}}_p$ is the position vector of $O\prime X\prime Y\prime Z\prime$ ’s original point in $\mathrm{OXYZ}$ . ${\mathit{r}}_i$ and ${{\mathit{r}}_i}^{\prime }$ are the position vectors of $\mathrm{LP}$ in $\mathrm{OXYZ}$ and $O\prime X\prime Y\prime Z\prime$ , respectively. Unless otherwise indicated, vectors with ′ indicate the vectors in $O\prime X\prime Y\prime Z\prime$ and others are the vectors in $\mathrm{OXYZ}$ . If ${\mathit{r}}_i$ exists deviations ( $\delta {\mathit{r}}_i$ ), then this will cause deviations ( $\delta \mathit{u}$ ) in the workpiece. $\delta \mathit{u}$ is defined as the workpiece locating error.

Figure 2.

General fixture layout of a 3D part.

It is assumed that $L_i$ always contacts the surface ${S_i}^{\prime }(x\prime ,y\prime ,z\prime )=0$ . Based on this assumption, the first-order Taylor expansion of ${S_i}^{\prime }(x\prime ,y\prime ,z\prime )=0$ at ${{\mathit{r}}_{\mathit{i}}}^{\prime }$ is used to present the constraint between the LP and the part

$$n{\prime }_i^T(\mathit{r}\prime -{\mathit{r}\prime }_{\mathit{i}})=0$$

(1)

In which, $n{\prime }_i={[n{\prime }_{\mathrm{ix}},n{\prime }_{\mathrm{iy}},n{\prime }_{\mathrm{iz}}]}^T$ is the unit normal vector of ${S_i}^{\prime }(x\prime ,y\prime ,z\prime )=0$ at ${\mathit{r}\prime }_{\mathit{i}}$ ; ${\mathit{r}\prime }_{\mathit{i}}={[{x_i}^{\prime },{y_i}^{\prime },{z_i}^{\prime }]}^T$ . By translating $r\prime$ into the reference coordinate system, it can be found that

$${\Phi }_i={\mathit{n}\prime }_i^T\mathit{R}{\left(\varphi \right)}^T({\mathit{r}}_i-{\mathit{r}}_p)-{\mathit{n}\prime }_i^T{\mathit{r}\prime }_i=0$$

(2)

where $\varphi ={[\begin{array}{ccc}\hfill \alpha ,\hfill & \hfill \beta ,\hfill & \hfill \gamma \hfill \end{array}]}^T$ are the Euler angles of $O\prime X\prime Y\prime Z\prime$ in $\mathrm{OXYZ}$ ; ${\mathit{r}}_p={[\begin{array}{ccc}\hfill x_p,\hfill & \hfill y_p,\hfill & \hfill z_p\hfill \end{array}]}^T$ is the position vector of the original point of $O\prime X\prime Y\prime Z\prime$ in $\mathrm{OXYZ}$ and $\mathit{R}(\varphi )$ is the rotational transformation matrix from $O\prime X\prime Y\prime Z\prime$ to $\mathrm{OXYZ}$ . As shown in equation (3), $c\theta =\cos \theta ,s\theta =\sin \theta$

$$\mathit{R}(\varphi )=\left[\begin{array}{ccc}\hfill \mathrm{c}\alpha \mathrm{c}\beta \hfill & \hfill \mathrm{c}\alpha \mathrm{s}\beta \mathrm{s}\gamma -\mathrm{s}\alpha \mathrm{c}\gamma \hfill & \hfill \mathrm{c}\alpha \mathrm{s}\beta \mathrm{c}\gamma +\mathrm{s}\alpha \mathrm{s}\gamma \hfill \\ \multicolumn{1}{c}{\mathrm{s}\alpha \mathrm{c}\beta }& \hfill \mathrm{s}\alpha \mathrm{s}\beta \mathrm{s}\gamma +\mathrm{c}\alpha \mathrm{c}\gamma \hfill & \hfill \mathrm{s}\alpha \mathrm{s}\beta \mathrm{c}\gamma -\mathrm{c}\alpha \mathrm{s}\gamma \hfill \\ \multicolumn{1}{c}{-\mathrm{s}\beta }& \hfill \mathrm{c}\beta \mathrm{s}\gamma \hfill & \hfill \mathrm{c}\beta \mathrm{c}\gamma \hfill \end{array}\right]$$

(3)

Combine all ${\Phi }_i$ ( $i=1,2,\dots ,6$ ) to form the constraint equations as follows

$$\Phi ={\left[\begin{array}{cccc}\hfill {\Phi }_1,\hfill & \hfill {\Phi }_2,\hfill & \hfill \dots ,\hfill & \hfill {\Phi }_6\hfill \end{array}\right]}^T$$

(4)

By defining $\mathit{u}={[\begin{array}{cc}\hfill {{\mathit{r}}_p}^T,\hfill & \hfill {\varphi }^T\hfill \end{array}]}^T$ and $\mathit{r}={[\begin{array}{cccc}\hfill {\mathit{r}}_1^T,\hfill & \hfill {\mathit{r}}_2^T,\hfill & \hfill \dots ,\hfill & \hfill {\mathit{r}}_6^T\hfill \end{array}]}^T$ , equation (4) can be written as

$$\Phi (\begin{array}{cc}\hfill \mathit{u},\hfill & \hfill \mathit{r}\hfill \end{array})=0$$

(5)

Let $\delta \mathit{u}={[\begin{array}{cccccc}\hfill \delta x_p,\hfill & \hfill \delta y_p,\hfill & \hfill \delta z_p,\hfill & \hfill \delta \alpha ,\hfill & \hfill \delta \beta ,\hfill & \hfill \delta \gamma \hfill \end{array}]}^T$ and $\delta \mathit{r}={[\begin{array}{cccc}\hfill \delta {\mathit{r}}_1^T,\hfill & \hfill \delta {\mathit{r}}_2^T,\hfill & \hfill \dots ,\hfill & \hfill \delta {\mathit{r}}_6^T\hfill \end{array}]}^T$ be the workpiece locating errors and the errors of the six LPs, respectively. The former three parameters in $\delta \mathit{u}$ represent the errors in position and the later three parameters in $\delta \mathit{u}$ represent the errors in orientation. If the errors are infinitesimal, the relationship between $\delta \mathit{u}$ and $\delta \mathit{r}$ can be obtained through an implicit differentiation operation in equation (5) as

$$\frac{\partial \Phi }{\partial \mathit{u}}\delta \mathit{u}+\frac{\partial \Phi }{\partial \mathit{r}}\delta \mathit{r}=0$$

(6)

If the workpiece is in a deterministic position, then $\partial \Phi /\partial \mathit{u}$ is inhomogeneous. Thus, formula (6) can be written as

$$\delta \mathit{u}=-{\left[\frac{\partial \Phi }{\partial \mathit{u}}\right]}^{-1}\frac{\partial \Phi }{\partial \mathit{r}}\delta \mathit{r}=-{\mathit{J}}^{-1}{\Phi }_r(\delta \mathit{r})$$

(7)

In which $\mathit{J}={[\partial {\Phi }_1/\partial \mathit{u},\partial {\Phi }_2/\partial \mathit{u},\dots ,\partial {\Phi }_6/\partial \mathit{u}]}^T$ is the Jacobian matrix, and ${\mathit{F}}_r$ is a matrix with a size of $6\times 18$ . Formula (7) is the locating error analysis model for a workpiece, in which the relationship between $\delta \mathit{u}$ and $\delta {\mathit{r}}_i$ is linearized. Using formula (7), we can obtain the errors of the workpiece at any point, as presented in formula (8)

$$\delta {\mathit{\Omega }}_{\mathit{i}}=\left[\begin{array}{c}\hfill \delta x_{{\Omega }_i}\hfill \\ \multicolumn{1}{c}{\delta y_{{\Omega }_i}}\\ \multicolumn{1}{c}{\delta z_{{\Omega }_i}}\\ \multicolumn{1}{c}{1}\end{array}\right]=\left[\begin{array}{cccc}\hfill 0\hfill & \hfill -\delta \gamma \hfill & \hfill \delta \beta \hfill & \hfill \delta x_p\hfill \\ \multicolumn{1}{c}{\delta \gamma }& \hfill 0\hfill & \hfill -\delta {\alpha }_i\hfill & \hfill \delta y_p\hfill \\ \multicolumn{1}{c}{-\delta {\beta }_i}& \hfill \delta {\alpha }_i\hfill & \hfill 0\hfill & \hfill \delta z_p\hfill \\ \multicolumn{1}{c}{0}& \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \end{array}\right]\left[\begin{array}{c}\hfill x_{{\Omega }_i}\hfill \\ \multicolumn{1}{c}{y_{{\Omega }_i}}\\ \multicolumn{1}{c}{z_{{\Omega }_i}}\\ \multicolumn{1}{c}{1}\end{array}\right]$$

(8)

The expressions of $\mathit{J}$ and ${\Phi }_r$ in the formula are derived as follows:

• Derivation of the Jacobian ( $\mathit{J}$ )

It can be known from formulas (4) and (7) that $\mathit{J}$ is a matrix with a size of $6\times 6$ . We define $\mathit{J}={[\begin{array}{cccccc}\hfill J_1,\hfill & \hfill J_2,\hfill & \hfill J_3,\hfill & \hfill J_4,\hfill & \hfill J_5,\hfill & \hfill J_6\hfill \end{array}]}^T$ , in which ${\mathit{J}}_i=\partial {\Phi }_i/\partial \mathit{u}$ ( $i=1,2,\dots ,6$ ). The expression of ${\mathit{J}}_i$ is derived as follows

$$\begin{array}{cc}\multicolumn{1}{c}{}& {\mathit{J}}_i=\hfill \\ \multicolumn{1}{c}{}& [-{{\mathit{n}\prime }_i}^T\mathit{R}{\left(\varphi \right)}^T\frac{\partial {\mathit{r}}_p}{\partial x_p},-{{\mathit{n}\prime }_i}^T\mathit{R}{\left(\varphi \right)}^T\frac{\partial {\mathit{r}}_p}{\partial y_p},-{{\mathit{n}\prime }_i}^T\mathit{R}{\left(\varphi \right)}^T\frac{\partial {\mathit{r}}_p}{\partial z_p},{{\mathit{n}\prime }_i}^T\frac{\partial \mathit{R}{\left(\varphi \right)}^T}{\partial \alpha }({\mathit{r}}_i-{\mathit{r}}_p),{{\mathit{n}\prime }_i}^T\frac{\partial \mathit{R}{\left(\varphi \right)}^T}{\partial \beta }({\mathit{r}}_i-{\mathit{r}}_p),{{\mathit{n}\prime }_i}^T\frac{\partial \mathit{R}{\left(\varphi \right)}^T}{\partial \gamma }({\mathit{r}}_i-{\mathit{r}}_p)]\hfill \end{array}$$

(9)

Without loss of generality, assume $O\prime X\prime Y\prime Z\prime$ coincides with $\mathrm{OXYZ}$ . Under this assumption, $\mathit{R}(\varphi )$ is a unit matrix and ${\mathit{r}}_p$ is a zero vector. Therefore, according to formulas (3) and (9), the ${\mathit{J}}_i$ can be derived

$$\begin{array}{cc}\multicolumn{1}{c}{}& {\mathit{J}}_i=\hfill \\ \multicolumn{1}{c}{}& [-n_{\mathrm{ix}},-n_{\mathrm{iy}},-n_{\mathrm{iz}},n_{\mathrm{iy}}{r}_{\mathrm{iz}}-n_{\mathrm{iz}}{r}_{\mathrm{iy}},n_{\mathrm{iz}}{r}_{\mathrm{ix}}-n_{\mathrm{ix}}{r}_{\mathrm{iz}},n_{\mathrm{ix}}{r}_{\mathrm{iy}}-n_{\mathrm{iy}}{r}_{\mathrm{ix}}]\hfill \end{array}$$

(10)

• Derivation of ${\Phi }_r$

According to formulas (2) and (4), ${\Phi }_r$ can be directly derived

$${\Phi }_r=\left[\begin{array}{ccc}\hfill {\mathit{n}\prime }_1^T\hfill & \hfill \cdots \hfill & \hfill 0\hfill \\ \multicolumn{1}{c}{⋮}& \hfill \ddots \hfill & \hfill ⋮\hfill \\ \multicolumn{1}{c}{0}& \hfill \cdots \hfill & \hfill {\mathit{n}\prime }_6^T\hfill \end{array}\right]\left[\begin{array}{ccc}\hfill \mathit{R}{\left(\varphi \right)}^T\hfill & \hfill \cdots \hfill & \hfill 0\hfill \\ \multicolumn{1}{c}{⋮}& \hfill \ddots \hfill & \hfill ⋮\hfill \\ \multicolumn{1}{c}{0}& \hfill \cdots \hfill & \hfill \mathit{R}{\left(\varphi \right)}^T\hfill \end{array}\right]$$

(11)

When $O\prime X\prime Y\prime Z\prime$ coincides with $\mathrm{OXYZ}$ , $\mathit{R}(\varphi )$ is a unit matrix and ${\mathit{n}\prime }_1^T={\mathit{n}}_1^T$ .

Tolerance modeling based on variational feature

Representation for feature and its deviations

In order to self-location, the workpiece establishes several matings with the fixture, such as surface-to-surface mating, hole-to-pin mating and others. These matings are composed of features, and deviations in these features are what cause locating errors. Deviations in the features used to generate matings are determined by tolerances. In this article, it is assumed that information, such as features (points, planes and surfaces) and tolerances, are captured by the CAD system. As shown in Figure 3, the cylinder with dashed line represents a nominal feature, and the cylinder with solid line represents a feature with deviations. $O\prime X\prime Y\prime Z\prime$ is the workpiece’s coordinate system. $o_i{x}_i{y}_i{z}_i$ is a feature coordinate system (FCS) attached to the feature, and $o{\prime }_{i}x{\prime }_{i}y{\prime }_{i}z{\prime }_i$ represents the coordinate system changing from $o_i{x}_i{y}_i{z}_i$ when the feature deviates from its nominal position. The nominal feature can be represented as follows

$$\mathit{f}={\left[\begin{array}{ccc}\hfill {\mathit{t}}^T,\hfill & \hfill {\mathbf{\theta }}^T,\hfill & \hfill {\mathit{w}}^T\hfill \end{array}\right]}^T$$

(12)

where $\mathit{t}={[\begin{array}{ccc}\hfill t_x,\hfill & \hfill t_y,\hfill & \hfill t_z\hfill \end{array}]}^T$ is the position vector of the origin of $o_i{x}_i{y}_i{z}_i$ in $O\prime X\prime Y\prime Z\prime$ , representing the position of a feature; $\mathit{\theta }={[\begin{array}{ccc}\hfill {\theta }_x,\hfill & \hfill {\theta }_y,\hfill & \hfill {\theta }_z\hfill \end{array}]}^T$ are the Euler angles of $o_i{x}_i{y}_i{z}_i$ in $O\prime X\prime Y\prime Z\prime$ , representing the orientation of a feature; $\mathit{w}$ represents the dimension of a feature, which can be empty, one dimension or a set of dimensions. The parameters in formula (12) are defined as freedom degrees of features.

Figure 3.

Feature and its deviations.

According to formula (12), the deviations of a feature can be defined as follows

$$\Delta \mathit{f}={\left[\begin{array}{ccc}\hfill \Delta {\mathit{t}}^T,\hfill & \hfill \Delta {\mathit{\theta }}^T,\hfill & \hfill \Delta {\mathit{w}}^T\hfill \end{array}\right]}^T$$

(13)

where $\Delta \mathit{t}={[\begin{array}{ccc}\hfill \Delta t_x,\hfill & \hfill \Delta t_y,\hfill & \hfill \Delta t_z\hfill \end{array}]}^T$ is the position vector of the origin of $O\prime X\prime Y\prime Z\prime$ in $o_i{x}_i{y}_i{z}_i$ ; $\Delta \mathit{\theta }={[\begin{array}{ccc}\hfill \Delta {\theta }_x,\hfill & \hfill \Delta {\theta }_y,\hfill & \hfill \Delta {\theta }_z\hfill \end{array}]}^T$ are the Euler angles of $o{\prime }_{i}x{\prime }_{i}y{\prime }_{i}z{\prime }_i$ in $o_i{x}_i{y}_i{z}_i$ and $\Delta \mathit{w}$ is the dimensional deviations.

Both the freedom degrees of a feature and their deviations can be expressed by a 4 × 4 homogeneous transformation matrix (HTM). As shown in Figure 3, ${}^0\mathit{H}$ represents the HTM from $O\prime X\prime Y\prime Z\prime$ to $o_i{x}_i{y}_i{z}_i$ and $\mathit{H}$ represents the HTM from $o_i{x}_i{y}_i{z}_i$ to $o{\prime }_{i}x{\prime }_{i}y{\prime }_{i}z{\prime }_i$ .

• ${}^0\mathit{H}$ represents the nominal information of the feature, which can be directly obtained from CAD systems. It is conveyed as formula (14), in which $\mathrm{c}\theta =\cos \theta$ , $\mathrm{s}\theta =\sin \theta$ , $t_x$ , $t_y$ , $t_z$ , ${\theta }_x$ , ${\theta }_y$ and ${\theta }_z$ are the parameters from formula (12)

$${}^0\mathit{H}=\left[\begin{array}{cccc}\hfill \mathrm{c}{\theta }_x\mathrm{c}{\theta }_y\hfill & \hfill \mathrm{c}{\theta }_x\mathrm{s}{\theta }_y\mathrm{s}{\theta }_z-\mathrm{s}{\theta }_x\mathrm{c}{\theta }_z\hfill & \hfill \mathrm{c}{\theta }_x\mathrm{s}{\theta }_y\mathrm{c}{\theta }_z+\mathrm{s}{\theta }_x\mathrm{s}{\theta }_z\hfill & \hfill t_x\hfill \\ \multicolumn{1}{c}{\mathrm{s}{\theta }_x\mathrm{c}{\theta }_y}& \hfill \mathrm{s}{\theta }_x\mathrm{s}{\theta }_y\mathrm{s}{\theta }_z+\mathrm{c}{\theta }_x\mathrm{c}{\theta }_z\hfill & \hfill \mathrm{s}{\theta }_x\mathrm{s}{\theta }_y\mathrm{c}{\theta }_z-\mathrm{c}{\theta }_x\mathrm{s}{\theta }_z\hfill & \hfill t_y\hfill \\ \multicolumn{1}{c}{-\mathrm{s}{\theta }_y}& \hfill \mathrm{c}{\theta }_y\mathrm{s}{\theta }_z\hfill & \hfill \mathrm{c}{\theta }_y\mathrm{c}{\theta }_z\hfill & \hfill t_z\hfill \\ \multicolumn{1}{c}{0}& \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \end{array}\right]$$

(14)

• $\mathit{H}$ represents the deviations of the feature, obtained from tolerances. It is expressed as formula (15), in which $\Delta t_x$ , $\Delta t_y$ , $\Delta t_z$ , $\Delta {\theta }_x$ , $\Delta {\theta }_y$ and $\Delta {\theta }_z$ are the parameters from formula (13)

$$\mathit{H}=\left[\begin{array}{cccc}\hfill 1\hfill & \hfill -\Delta {\theta }_z\hfill & \hfill \Delta {\theta }_y\hfill & \hfill \Delta t_x\hfill \\ \multicolumn{1}{c}{\Delta {\theta }_z}& \hfill 1\hfill & \hfill -\Delta {\theta }_x\hfill & \hfill \Delta t_y\hfill \\ \multicolumn{1}{c}{-\Delta {\theta }_y}& \hfill \Delta {\theta }_x\hfill & \hfill 1\hfill & \hfill \Delta t_z\hfill \\ \multicolumn{1}{c}{0}& \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \end{array}\right]$$

(15)

Planar feature

Figure 4 is a planar tolerance zone and $o_i{x}_i{y}_i{z}_i$ is an FCS. The z_i-axis coincides with the normal vector of the plane. $T$ is the tolerance of the plane. The steps to calculate the deviations of the planar feature are as follows:

• Step 1. When the planar feature translates along the x_i-axis and y_i-axis, or rotates along the z_i-axis, no deviations will be produced. The corresponding freedom degrees can be eliminated. Therefore, the deviations of the plane feature can be represented as follows

  $$\Delta {\mathit{f}}_{\mathrm{Planar}}={\left[\begin{array}{cccccc}\hfill 0,\hfill & \hfill 0,\hfill & \hfill \Delta t_z,\hfill & \hfill \Delta {\theta }_x,\hfill & \hfill \Delta {\theta }_y,\hfill & \hfill 0\hfill \end{array}\right]}^T$$

  (16)

• Step 2. Choose four boundary points of the planar feature as control points, as the ${\mathbf{\tau }}_{\mathit{i}}$ ( $i=1,2,3,4$ ) shown in Figure 4. For a non-rectangular plane, control points maybe added along the boundary of the plane.

• Step 3. Calculate the deviations of the planar feature according to the control points.

Figure 4.

Planar tolerance zone.

According to formulas (15) and (16), the normal deviation of ${\tau }_{\mathit{i}}$ can be derived using formula (17), in which ${\tau }_{\mathit{i}}={[\begin{array}{cccc}\hfill {\tau }_{\mathrm{ix}},\hfill & \hfill {\tau }_{\mathrm{iy}},\hfill & \hfill {\tau }_{\mathrm{iz}},\hfill & \hfill 1\hfill \end{array}]}^T$ and $\mathit{n}={[\begin{array}{cccc}\hfill 0,\hfill & \hfill 0,\hfill & \hfill 1,\hfill & \hfill 1\hfill \end{array}]}^T$

$${\delta }_i={\mathit{n}}^T\left[\begin{array}{cccc}\hfill 0\hfill & \hfill 0\hfill & \hfill \Delta {\theta }_y\hfill & \hfill 0\hfill \\ \multicolumn{1}{c}{0}& \hfill 0\hfill & \hfill -\Delta {\theta }_x\hfill & \hfill 0\hfill \\ \multicolumn{1}{c}{-\Delta {\theta }_y}& \hfill \Delta {\theta }_x\hfill & \hfill 0\hfill & \hfill \Delta t_z\hfill \\ \multicolumn{1}{c}{0}& \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \end{array}\right]{\mathit{\tau }}_{\mathit{i}}-1$$

(17)

Assuming ${\delta }_i$ adheres to normal distribution, ${\delta }_i~N(\begin{array}{cc}\hfill 0,\hfill & \hfill T/6\hfill \end{array})$ can be calculated according to the principle of 6-sigma, where “0” and “ $T/6$ ” represent the mean and the variance, respectively. Three random numbers ( ${\Delta }_1$ , ${\Delta }_2$ and ${\Delta }_3$ ) need to be generated according to the normal distribution, $N(\begin{array}{cc}\hfill 0,\hfill & \hfill T/6\hfill \end{array})$ . After that, the equations to calculate the deviations of the planar feature are presented as follows

$$\{\begin{array}{c}\hfill {\delta }_i={\Delta }_i\hfill \\ \multicolumn{1}{c}{-T/2\le {\delta }_4\le T/2}\\ \multicolumn{1}{c}{i=1,2,3}\end{array}$$

(18)

If there is no solution to formula (18), then the three random numbers must continue to be generated until a solution becomes possible.

Cylindrical feature

As shown in Figure 5, an FCS, $o_i{x}_i{y}_i{z}_i$ , has the local z_i-axis along with the axis of the cylinder. $T$ is the tolerance zone of the cylindrical feature. The steps to calculate the deviations of the cylindrical feature are as follows:

• Step 1. When the cylindrical feature translates along the z_i-axis, or rotates around the z_i-axis, no deviations will be produced. The corresponding freedom degrees can be eliminated. Therefore, the deviations of the cylindrical feature can be represented as follows

  $$\Delta {\mathit{f}}_{\mathrm{cylinder}}={\left[\begin{array}{cccccc}\hfill \Delta t_x,\hfill & \hfill \Delta t_y,\hfill & \hfill 0,\hfill & \hfill \Delta {\theta }_x,\hfill & \hfill \Delta {\theta }_y,\hfill & \hfill 0\hfill \end{array}\right]}^T$$

  (19)

• Step 2. Choose two boundary points ( ${\mathbf{\tau }}_1$ and ${\mathbf{\tau }}_2$ ) on the axis as control points.

• Step 3. Calculate the deviations of the cylindrical feature according to the control points.

Figure 5.

Cylindrical tolerance zone.

As shown in Figure 6, ${\mathbf{\tau }}_i$ ( $i=1,2$ ) is restricted to the circular area with a radius of $T/2$ . The deviations of ${\mathbf{\tau }}_i$ , represented by $r$ , are a compound of deviations in the x-axis and y-axis. Assuming the deviations in the x-axis and y-axis are mutually independent, and that they adhere to normal distribution, $N(\begin{array}{cc}\hfill 0,\hfill & \hfill \sigma \hfill \end{array})$ , where “0” is the mean and “ $\sigma$ ” is the variance. According to Camelio et al. ²⁰ the $r$ adheres to Raleigh distribution. The mean ( ${\mu }_r$ ) and the variance ( ${\sigma }_r$ ) are represented as follows

$${\mu }_r={\left(\frac{\pi }{2}\right)}^{1/2}\sigma ,{\sigma }_r={\left[\frac{4-\pi }{2}\right]}^{1/2}\sigma$$

(20)

Figure 6.

Circular tolerance zone of ${\mathbf{\tau }}_i$ .

According to the principle of 6-sigma, the expansion of $\sigma$ can be derived as follows

$$\sigma =\frac{T}{6{\left[\frac{4-\pi }{2}\right]}^{1/2}}$$

(21)

Thus, it is known that ${\mathbf{\tau }}_i$ adheres to the normal distribution, $N(\begin{array}{cc}\hfill 0,\hfill & \hfill T/6{[(4-\pi )/2]}^{1/2}\hfill \end{array})$ , in the x-axis direction and y-axis direction. Four random numbers ( ${\Delta }_{11}$ , ${\Delta }_{12}$ , ${\Delta }_{21}$ and ${\Delta }_{22}$ ) need to be generated according to the normal distribution, $N(\begin{array}{cc}\hfill 0,\hfill & \hfill T/6{[(4-\pi )/2]}^{1/2}\hfill \end{array})$ . After that, the equations to calculate the deviations of the cylindrical feature are presented as follows

$$\{\begin{array}{cc}\multicolumn{1}{c}{}& \left(\Delta {\theta }_y\right){\tau }_{\mathrm{iz}}+\Delta t_x={\Delta }_{i1}\hfill \\ \multicolumn{1}{c}{}& -\left(\Delta {\theta }_x\right){\tau }_{\mathrm{iz}}+\Delta t_y={\Delta }_{i2}\hfill \\ \multicolumn{1}{c}{}& i=1,2\hfill \end{array}$$

(22)

If there is no solution to formula (22), then the four random numbers need to be generated again until a solution is found.

Free-form surface feature

Figure 7 is the tolerance zone of a free-form surface. $o_i{x}_i{y}_i{z}_i$ has the local z_i-axis along with the normal vector of one point on the free-form surface. $T$ is the tolerance zone of the free-form surface. The steps to calculate the deviations of the free-form surface are as follows:

• Step 1. The free-form surface can be represented by three translational and three rotational degrees of freedom. Therefore, the deviations of the free-form surface can be represented as

  $$\Delta {\mathit{f}}_{\mathrm{surface}}={\left[\begin{array}{cccccc}\hfill \Delta t_x,\hfill & \hfill \Delta t_y,\hfill & \hfill \Delta t_z,\hfill & \hfill \Delta {\theta }_x,\hfill & \hfill \Delta {\theta }_y,\hfill & \hfill \Delta {\theta }_z\hfill \end{array}\right]}^T$$

  (23)

• Step 2. Transform the free-form surface into a series of control points, illustrated as ${\mathbf{\tau }}_i$ ( $i=1,2,\dots ,n$ ) in Figure 7.

• Step 3. Calculate the deviations of the free-form feature according to the control points.

Figure 7.

Tolerance zone of free-form surface.

Let ${\mathit{n}}_i={[\begin{array}{cccc}\hfill n_{\mathrm{ix}},\hfill & \hfill n_{\mathrm{iy}},\hfill & \hfill n_{\mathrm{iz}},\hfill & \hfill 1\hfill \end{array}]}^T$ be the unit normal vector of ${\mathbf{\tau }}_i$ , then the normal deviations of ${\mathbf{\tau }}_i$ can be represented as follows

$${\delta }_i={\mathit{n}}_i^T\left[\begin{array}{cccc}\hfill 0\hfill & \hfill -\Delta {\theta }_z\hfill & \hfill \Delta {\theta }_y\hfill & \hfill \Delta t_x\hfill \\ \multicolumn{1}{c}{\Delta {\theta }_z}& \hfill 0\hfill & \hfill -\Delta {\theta }_x\hfill & \hfill \Delta t_y\hfill \\ \multicolumn{1}{c}{-\Delta {\theta }_y}& \hfill \Delta {\theta }_x\hfill & \hfill 0\hfill & \hfill \Delta t_z\hfill \\ \multicolumn{1}{c}{0}& \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \end{array}\right]{\mathbf{\tau }}_i-1$$

(24)

Suppose the normal deviation ( ${\delta }_i$ ) on the point ${\mathbf{\tau }}_i$ adheres to the normal distribution, $N(\begin{array}{cc}\hfill 0,\hfill & \hfill T/6\hfill \end{array})$ . Six random numbers need to be generated according to the distribution. After that, the equations to calculate the deviations of the free-form surface are presented as follows

$$\{\begin{array}{c}\hfill {\delta }_k={\Delta }_k\hfill \\ \multicolumn{1}{c}{-T/2\le {\delta }_i\le T/2}\\ \multicolumn{1}{c}{k=1,2,\dots ,6}\\ \multicolumn{1}{c}{i=7,8,\dots ,n}\end{array}$$

(25)

If there is no solution to formula (25), then the six random numbers need to be re-generated until a solution is possible.

Calculating the deviations of LP

According to the principle of virtual LP,^18,19 the features to generate mating can be simplified as several LPs. The deviations of these LPs will be derived based on the deviations of the features. In this article, features on workpiece and features on fixture are defined as object features (OFs) and target features (TFs), respectively. Workpiece locating is the process which establishes matings between OFs and TFs.

Surface-to-surface mating

A surface-to-surface mating is illustrated as Figure 8. In Figure 8, $\mathrm{OXYZ}$ is the global coordinate system. $f_1$ and $f{\prime }_1$ represent the nominal OF and the OF with deviations, respectively. $f_2$ and $f{\prime }_2$ represent the nominal TF and the TF with deviations, respectively. $\mathit{L}$ and $L\prime$ represent the nominal LP and the LP with deviations, respectively. $\mathit{r}$ and $\mathit{r}\prime$ are the position vectors of $\mathit{L}$ and $L\prime$ , respectively.

• Theoretical situation. There are no deviations between the OF and TF in this situation, and $\mathit{L}$ coincides with $L\prime$ .

• Real situation. There exist deviations between the OF and TF, and these two features contact at the point ${\mathit{L}}_a$ . In this situation, $\mathit{L}$ moves to $L\prime$ . Let $\delta {\mathit{r}}_{\mathrm{face}\_\mathrm{face}}$ be the deviations of $\mathit{L}$ , then the deviations of the LP in surface-to-surface mating can be written as follows

$$\delta {\mathit{r}}_{\mathit{face}\_\mathit{face}}=\mathit{r}\prime -\mathit{r}={\xi }_1{\mathit{n}}_1-{\xi }_2{\mathit{n}}_2$$

(26)

where ${\xi }_1$ and ${\xi }_2$ are the normal deviations of the OF and TF at $\mathit{L}$ , and ${\mathit{n}}_1$ and ${\mathit{n}}_2$ are unit normal vectors. ${\xi }_1$ and ${\xi }_2$ can be obtained by calculating the deviations of the OF and TF according to section “Tolerance modeling based on variational feature.”

Figure 8.

Deviation of LP in surface-to-surface mating.

Hole-to-pin mating

Hole-to-pin mating is a widely applied process in workpiece locating. The deviations of an LP in hole-to-pin mating concern the dimensional deviations and positional deviations of the hole and pin. The pin and hole are considered the OF and TF, respectively. To simplify our analysis process, we assume the axes of the hole and the pin are parallel to the z-axis. The procedure to calculate the deviations of an LP in hole-to-pin mating is as follows:

• Calculate the deviations of the LP caused by the positional deviations of the hole and the pin.

Let $\delta {\mathit{r}}_1^{\mathrm{pos}}={[\begin{array}{cc}\hfill \delta r_{1x}^{\mathrm{pos}},\hfill & \hfill \delta r_{1y}^{\mathrm{pos}}\hfill \end{array}]}^T$ and $\delta {\mathit{r}}_2^{\mathrm{pos}}={[\begin{array}{cc}\hfill \delta r_{2x}^{\mathrm{pos}},\hfill & \hfill \delta r_{2y}^{\mathrm{pos}}\hfill \end{array}]}^T$ be the positional deviations of the hole and the pin, respectively. The formula to calculate the deviations of the LP in surface-to-surface mating can be derived as shown in section “Surface-to-surface mating”

$$\delta {\mathit{r}}^{\mathrm{pos}}=\delta {\mathit{r}}_1^{\mathrm{pos}}-\delta {\mathit{r}}_2^{\mathrm{pos}}$$

(27)

where $\delta {\mathit{r}}^{\mathrm{pos}}$ represents the deviations of the LP caused by the positional deviations of the hole and pin.

• Calculate the deviations of the LP caused by the dimensional deviations of the hole and pin.

As illustrated in Figure 9, it is assumed that the hole and the pin are always in contact during workpiece locating, so the deviations of the LP caused by dimensional deviations of the hole and pin can be represented as follows

$$\delta {\mathit{r}}^{\mathrm{size}}={\left[\begin{array}{cc}\hfill {\Delta }^{\mathrm{size}}\cos \phi ,\hfill & \hfill {\Delta }^{\mathrm{size}}\sin \phi \hfill \end{array}\right]}^T$$

(28)

where ${\Delta }^{\mathrm{size}}=(w_2+\delta w_2)-(w_1+\delta w_1)$ ; $w_1$ and $\delta w_1$ represent the diameter of the pin and its deviations, respectively; $w_2$ and $\delta w_2$ represent the diameter of the hole and its deviations, respectively; and $0\le \phi \le 2\pi$ . The $\phi$ is considered as uniform distribution during actual calculation.

Figure 9.

Gap of hole-to-pin mating.

• Calculate the integrated deviations of the LP in hole-to-pin mating.

The integrated deviations of the LP in hole-to-pin mating are influenced by positional and dimensional deviations. By combining formulas (27) and (28), the integrated deviations of the LP in hole-to-pin mating can be derived

$$\delta {\mathit{r}}_{\mathrm{hole}-\mathrm{pin}}=\delta {\mathit{r}}^{\mathrm{pos}}+\delta {\mathit{r}}^{\mathrm{size}}$$

(29)

Numerical experiments

Case 1: a block

As shown in Figure 10(a), the 200 mm × 200 mm × 100 mm block is located by a fixture. There are six features used to construct matings here. First, $f_1$ and $f_2$ form a surface–surface mating. Second, $f_3$ and $f_4$ form a hole-to-pin mating. Third, $f_5$ and $f_6$ form a surface-to-surface mating. Here, the tolerances of the planar features are 0.2 mm, the dimensional tolerances of the hole and the pin are 0.1 mm and the positional tolerances of the hole and the pin are 0.15 mm.

Figure 10.

A block’s locating: (a) a block and its fixture and (b) general fixture layout of the block.

Parameterized tolerances

In this case, we chose the feature f₁ as an example of parameterized tolerance. As illustrated in Figure 11, there are four control points on f₁ presented as ${\mathbf{\tau }}_i$ ( $i=1,2,3,4$ ). The four control points are selected using the CAD model of the block. As seen in section “Planar feature,” a coordination system ( $\mathrm{oxyz}$ ) is established on f₁. The coordinates of ${\mathbf{\tau }}_i$ are presented in Figure 11.

Figure 11.

Feature f₁ and its control points.

According to formula (16), the deviations of f₁ are presented by three parameters ( $\Delta t_z$ , $\Delta {\theta }_x$ and $\Delta {\theta }_y$ ). To calculate these three parameters, three random numbers ( ${\Delta }_1$ , ${\Delta }_2$ and ${\Delta }_3$ ) need to be generated according to the normal distribution, $N(\begin{array}{cc}\hfill 0,\hfill & \hfill T/6\hfill \end{array})$ . $T$ is the tolerance of $f_1$ , which is 0.2 mm. The three random numbers are ${\Delta }_1=-0.0052$ , ${\Delta }_2=0.0011$ and ${\Delta }_3=0.0330$ . The equations to calculate the deviations of $f_1$ are derived according to formula (18) found in section “Planar feature”

$$\{\begin{array}{c}\hfill 100\left(\Delta {\theta }_x\right)-100\left(\Delta {\theta }_y\right)+\Delta t_z={\Delta }_1\hfill \\ \multicolumn{1}{c}{-100\left(\Delta {\theta }_x\right)-100\left(\Delta {\theta }_y\right)+\Delta t_z={\Delta }_2}\\ \multicolumn{1}{c}{-100\left(\Delta {\theta }_x\right)+100\left(\Delta {\theta }_y\right)+\Delta t_z={\Delta }_3}\\ \multicolumn{1}{c}{-0.1\le -100\left(\Delta {\theta }_x\right)-100\left(\Delta {\theta }_y\right)+\Delta t_z\le 0.1}\end{array}$$

(30)

According to formula (30), $\Delta {\theta }_x=-0.0000315$ , $\Delta {\theta }_y=0.0000095$ and $\Delta t_z=-0.0011$ . Only one set for the solution is found for the tolerance of $f_1$ . In the locating error analysis, it is necessary to find a series of solutions. To do this, we need to generate more random numbers and solve formula (30) repeatedly.

Locating error analysis

In Figure 10(b), the mating formed by $f_1$ and $f_2$ is translated into three LPs, while the mating formed by $f_3$ and $f_4$ is translated into two LPs. The mating formed by $f_5$ and $f_6$ is translated into one LP. These six LPs form the general fixture layout for the locating of the block. The positional vector and normal vector of the LPs are presented in Table 1.

Table 1.

LPs of the block.

LPs	Positional vector	Normal vector
L 1	(−98.414, 99.024, 0)	(0, 0, −1)
L 2	(−97.403, −97.570, 0)	(0, 0, −1)
L 3	(99.039, −99.101, 0)	(0, 0, −1)
L 4	(−50.000, −50.000, 0)	(−1, 0, 0)
L 5	(−50.000, −50.000, 0)	(0, −1, 0)
L 6	(92.501, 100.000, 8.484)	(0, −1, 0)

LPs: locating points.

As illustrated in Figure 10(b), the four top points, MP₁–MP₄, are selected as measuring points (MPs). Similar to section “Parameterized tolerances” and to the linearized locating error analysis model in section “Linear locating error analysis model for workpiece,” the normal deviations of MP₁–MP₄ are analyzed. The positional vector and normal vector of the MPs are presented in Table 2.

Table 2.

Measuring points of the block.

MPs	Positional vector	Normal vector
MP 1	(−97.001, −98.422, 100)	(0, 0, 1)
MP 2	(97.783, −97.783, 100)	(0, 0, 1)
MP 3	(97.343, 98.452, 100)	(0, 0, 1)
MP 4	(−99.662, 99.662, 100)	(0, 0, 1)

MPs: measuring points.

The results are calculated by the proposed method and 3DCS software, a standard industrial software used for tolerance analysis, as shown in Table 3. It can be seen that relative errors between the results are small, in which the differences (relative errors) between the proposed method and 3DCS are <1%.

Table 3.

The results calculated by the proposed method and 3DCS software.

	MP 1	MP 2	MP 3	MP 4
The proposed method	0.2828	0.2797	0.4899	0.2831
3DCS	0.2850	0.2804	0.4934	0.2852
Relative errors	0.76%	0.26%	0.70%	0.74%

MP: measuring point.

Case 2: a rib

The location of a rib during machining can be simplified, as shown in Figure 10. L₁–L₆ are the LPs: $L_1$ , $L_2$ and $L_3$ are LPs in the primary datum; $L_4$ and $L_5$ are LPs in secondary datum and $L_6$ is LP in tertiary datum. The positional vector and normal vector of these LPs are presented in Table 4. The tolerances of the related locating datum in the rib are 0.2 mm and the tolerances of the fixture are 0.1 mm.

Table 4.

LPs of the rib.

LPs	Positional vector	Normal vector
L 1	(342.448, −56.249, 0)	(0, 0, 1)
L 2	(27.527, 87.418, 0)	(0, 0, 1)
L 3	(−318.234, −51.266, 0)	(0, 0, 1)
L 4	(−274.102, 92.053, −12.487)	(−0.054, 0.999, 0)
L 5	(334.233, 91.030, −15.060)	(0.058, 0.998, 0)
L 6	(365.940, 10.748, −15.765)	(1, 0, 0)

LPs: locating points.

Six MPs are selected for the rib, represented as MP₁–MP₆ in Figure 12. The positional vector and normal vector of these MPs are shown in Table 5.

Figure 12.

General fixture layout of a rib.

Table 5.

Measuring points of the rib.

MPs	Positional vector	Normal vector
MP 1	(332.535, −64.183, −16.422)	(0.117, −0.993, 0)
MP 2	(213.421, −75.251, −14.966)	(0.067, −0.998, 0)
MP 3	(97.037, −80.245, −12.221)	(0.018, −1, 0)
MP 4	(−10.626, −79.729, −15.710)	(−0.028, −1.000, 0)
MP 5	(−130.081, −73.363, −12.192)	(−0.079, −0.997, 0)
MP 6	(−294.343, −54.627, −13.463)	(−0.148, −0.989, 0)

MPs: measuring points.

Parameterized tolerances

For this demonstration, we chose the secondary datum as the example of parameterized tolerance, which is a free-form surface feature. It is assumed that the workpiece is designed by CATIA, which is a widely used CAD software in the industry currently. We use the tool “CAA for CATIA” to create a program when capturing control points on a surface. As illustrated in Figure 13, 100 points are retrieved for the secondary datum. The information for the LPs is presented in Table 6.

Figure 13.

The secondary datum and its control points.

Table 6.

Control points of the secondary datum.

Control points	Positional vector	Normal vector
${\mathbf{\tau }}_1$	(−324.060, 89.105, −30.000)	(−0.063, 0.998, 0)
${\mathbf{\tau }}_2$	(−324.060, 89.105, −26.667)	(−0.063, 0.998, 0)
${\mathbf{\tau }}_3$	(−324.060, 89.105, −23.333)	(−0.063, 0.998, 0)
${\mathbf{\tau }}_4$	(−324.060, 89.105, −20.000)	(−0.063, 0.998, 0)
${\mathbf{\tau }}_5$	(−324.060, 89.105, −13.333)	(−0.063, 0.998, 0)
	…	…
${\mathbf{\tau }}_{30}$	(−170.707, 96.691, 0)	(−0.035, 0.999, 0)
${\mathbf{\tau }}_{31}$	(−94.034, 98.859, −30.000)	(−0.021, 1, 0)
	…	…
${\mathbf{\tau }}_{100}$	(365.940, 89.105, 0)	(0.063, 0.998, 0.000)

As explained in section “Free-form surface feature,” to calculate the six parameters ( $\Delta t_x$ , $\Delta t_y$ , $\Delta t_z$ , $\Delta {\theta }_x$ , $\Delta {\theta }_y$ and $\Delta {\theta }_z$ ) for the secondary datum, three random numbers ( ${\Delta }_i$ , $i=1,2,\dots ,6$ ) need to be generated according to the normal distribution, $N(\begin{array}{cc}\hfill 0,\hfill & \hfill T/6\hfill \end{array})$ . $T$ is the tolerance of the secondary datum, which is 0.2 mm here. Then, according to formula (25), we can find the six parameters, similar to section “Parameterized tolerances.”

For free-form features, it may be very time-consuming to acquire control points by processing such complicated surfaces. Here, a number of experiments, capturing control points for the secondary datum, are performed and analyzed. As shown in Figure 14, as the number of control points increases, so does the computing time. When the number of control points is $2.5\times {10}^5$ , for example, the computing time reaches 0.25 s, which is still acceptable. A possible cause of this is that control points only need to be retrieved for a parameterized surface, and they do not need to be created or displayed in a CAD platform. Therefore, it is more efficient to capture control points from a surface in a CAD platform, which can meet the requirements of tolerance modeling.

Figure 14.

Computing time of capturing control points in the secondary datum.

Locating error analysis

The normal deviations of MP₁–MP₆ are calculated. The results are compared to the results calculated by 3DCS software, as shown in Table 7. The maximum relative error between the results is 2.4%, confirming the efficacy of the proposed method.

Table 7.

The results calculated by the proposed method and 3DCS software.

	MP 1	MP 2	MP 3	MP 4	MP 5	MP 6
The proposed method	0.2191	0.1830	0.1618	0.1594	0.1770	0.2262
3DCS	0.2243	0.1859	0.163	0.1602	0.1788	0.2309
Relative errors	2.37%	1.57%	0.75%	0.49%	1.01%	2.08%

MP: measuring point.

Conclusion

This article presents a locating error analysis method for workpieces based on parameterized tolerance. In this method, the fixture is turned into six LPs. A linearized model is then derived to express the relationship between the errors of the LPs and the workpiece locating errors. Then, the tolerances for planar, cylindrical and free-form surface features are parameterized using 4 × 4 homogeneous transformation matrices. Finally, two formulas are derived to calculate errors of the LPs in surface-to-surface mating and hole-to-pin mating. The proposed method provides highly accurate calculated results and shows potential to provide a basis for fixture layout optimization and tolerance allocation. However, there still exist some deficiencies in this article. For one thing, only tolerance modeling for planar features, cylindrical features and free-form surface features are considered. Additionally, it is assumed that the LPs contact with the workpiece on a rigid plane. Typical properties of contact surfaces, such as curvature and deformation, are neglected. The method in this article will be improved, as such, during the follow-up research.