Optimization Method of Assembly Tolerance Types Based on Degree of Freedom

Abstract

The automatic generation of tolerance specifications is an important aspect of achieving digital product design. An obvious feature of the current automatic generation of tolerance based on rule reasoning is that all tolerance types will be inferred for the same assembly feature. However, when labelling part tolerance information, designers need to further screen based on the geometric function of the assembly, which may result in prioritizing tolerance types that do not meet the geometric requirements of the assembly. This paper presents an assembly tolerance type optimization method based on the degree of freedom (DOF) of tolerance zone for the optimization and screening problem after reasoning all possible tolerance types. Firstly, we define the DOF of tolerance zones and their representations, while also define the control parameter degrees of freedom (CPDF) of assemblies, and analyze the CPDF of typical geometric functional tolerances of assemblies; Secondly, the Boolean operation relationship between sets is used to construct a Boolean operation preference method for the CPDF. Then, an algorithm for the optimal selection of the shape and position tolerance items of the assembly is established based on the DOFs of tolerance zone. Finally, the proposed method is verified by an engineering example, and the result shows that the method can optimize and screen the geometric tolerance types of assemblies.

1. Introduction

Product precision design is an important task in mechanical product design [1,2,3,4,5], and product precision affects the quality and performance of mechanical products [2,3,4]. Reasonable and accurate tolerance specifications for parts can help improve product quality and reduce manufacturing costs [3,4,5,6,7]. At present, with the development of digital and intelligent product design, the automatic generation of tolerance specifications has become an important research content and made an important breakthrough. The multi-color set method [8,9], hierarchical tolerance information model method [10,11,12,13,14], adjacency matrix method [15,16] and other methods [17] are combined with the computer to realize automatic reasoning of tolerance types of assembly parts. With the above method of automatic generation of tolerance types, the assembly features of the part are analyzed in order to obtain all the tolerance terms matching the assembly features. However, it generates a large number of tolerance terms, and designers need further manual screening and processing when marking part tolerance information. In the screening process, the designers mainly rely on their own experience in tolerance application to select the dimensioned tolerance type [13]. Because the designers are not thoughtful, the selected tolerance types may not meet the geometric tolerance functional requirements of mechanical products. Therefore, it is necessary to study a method that can realize screening optimization of the automatically recommended tolerance items under the circumstance of meeting the demand of the geometrical function tolerance of assemblies.

The tolerance zone is usually used to limit the allowable variation range of the geometric shape, geometric position and geometric dimension of the geometric entity [18,19]. To determine different tolerance zones, the tolerance zones can be determined and distinguished by the control parameters of the shape, size, direction, and position of the different tolerance zones [19].

The DOF can effectively represent the position and possible motion of assembly feature elements in three-dimensional space [20,21]. In an assembly, the different directions of the tolerance zones of the assembly feature elements and their positional variations are ultimately transferred to the target parts of the assembly [7,21,22,23]. Khodaygan [21] constructed a cumulative error transfer equation between contact displacement variations and assembly geometric-functional tolerances using small tolerance band displacement degrees of freedom to enable assembly tolerance analysis. In the ideal assembly connection without considering contact deformation, the position variation of the assembly connection mainly comes from the errors in the four dimensions of shape, size, direction, and position of the assembly feature surfaces. Assembly feature elements marked with different shape and position tolerances will have different tolerance zones. Different tolerance zones will constrain different variation characteristics of assembly feature surfaces, and the degree of influence on the geometric function of the assembly will be different [18,19,21]. Logically, using the DOF to describe the change direction of the tolerance zone and the change direction that affects the geometric functional error of the assembly, and studying the influence relationship between the above two types of change displacement is helpful to optimize the tolerance type of the assembly feature elements.

Therefore, by introducing the concept of DOF to define the change direction of tolerance zone, the DOFs of tolerance zones with different shapes are obtained. By analyzing the level of influence of the variation displacement of typical assembly joint feature surfaces on the geometric function of typical assemblies, the mapping relationship between the geometric function requirements of typical assemblies and the control parameter freedom degrees (CPDFs) is constructed. These relationships provide the theoretical basis for optimizing tolerance types with DOF of the tolerance zone. To summarize, based on the analysis of the geometric functional requirements of the assembly, the relationship between the CPDF that affects the geometric functional index value and the DOF of the tolerance zone of different tolerance types on the assembly feature surface is obtained. This article proposes a method for the selection and optimization of assembly tolerance types based on degrees of freedom; studies the Boolean relationship between the degree of freedom vector of the tolerance zone and the control parameter degree of freedom vector of the assembly body under different tolerance types and tolerance zones of assembly feature elements, and achieves the optimization of assembly tolerance projects.

The remainder of this paper is organized as follows: Section 2 introduces the relevant work of tolerance specification design and preferred tolerance types. Section 3 defines DOF of tolerance zones and vector representation of DOF, and introduces the meaning of CPDF of assembly and its vector representation. Section 4 introduces the process of obtaining the control parameter degrees of freedom of an assembly and summarizes the mapping relationship between typical tolerance bands and the control parameter degrees of freedom obtained. Section 5 introduces Boolean operations to construct a Boolean operation method for obtaining CPDF, and obtains an automatic tolerance item generation algorithm for optimizing the selection of assembly body shape and position tolerances based on DOF. Section 6 conducts engineering verification. Section 7 discusses the method of selecting and optimizing tolerance types based on DOF.

2. Related Work

The optimization of tolerance type is an important work after the tolerance specification is automatically generated. This section firstly introduces the current status of automatic generation of tolerance specifications; secondly, it reviews the methods of tolerance type optimization.

In terms of automatic generation of tolerance types, many researchers have conducted a lot of research and summarized it into four categories: Technology and topology related surface (TTRS) based methods [24], assembly positioning constraint-based methods [25,26,27,28], case-based learning methods [29,30,31], and rule-based reasoning methods [8,9,10,11,12,13,14,15,16,32,33,34].

TTRS-based method [24,25]. Desrochers et al. [25] firstly defines the geometric surfaces of mechanical parts as seven basic types, and secondly proposes to represent the above seven geometric surfaces with different combinations of three minimal geometric reference elements such as point, line, and plane. These combinations are then applied, along with a given set of rules to describe the assembly relationship between two assembly parts to achieve the tolerance topology of the assembly layer. Finally, a specific tolerance design software is developed in conjunction with the modern technical standard ASMEY14.5 to achieve automatic generation of tolerance items for assembly parts. The use of this method for designing tolerance types only ensures that the shape-tolerance type corresponds to the shape characteristics of the functional surfaces of the assembly, and the directional position tolerance corresponds to the spatial position relationship between the smallest geometric elements. It does not take into account the geometric functional requirements of the assembly.

Method based on assembly positioning constraints [26,27,28,29]. Anselmetti et al. [26] proposed a positioning table to describe the assembly constraint relationships between the target part and other assembly parts, as well as to determine the positioning priority of the assembly feature surfaces of the target part. The automatic generation of tolerance specifications is achieved through CEXEL software. Armillotta [27] designed assembly constraint tables with dimensional attributes by studying the functional relationships in the product assembly process, on the basis of which interactive tolerance design presentation software is developed. Cao [28] et al. and Ma et al. [29] combined graph theory with assembly positioning constraint information to establish a part-level geometric transfer paths for tolerance items; and implement the tolerance specification design of parts step by step. However, they did not develop corresponding software for automatic tolerance generation.

Case-based learning method [30,31,32]. Cui et al. [31] normalize the part feature and annotation tolerance information from previous tolerance specification schemes to form a tolerance specification database and build a machine learning model. Qin et al. [32] use a descriptive logic language to build an ontological knowledge database, and a similarity measure is used to evaluate the similarity between the target tolerance problem and the tolerance information of the cases. On this basis, a search algorithm is built to obtain a recommended tolerance scheme. At present, however, the successful application is limited due to the lack of sufficient model data. The metric values obtained by the recommendation still require designers to make decisions based on design experience in the case of equality.

Rule-based reasoning method [8,9,10,11,12,13,14,15,16,33,34,35]. Zhang [8], Qin [9] and others introduced the theory of polychromatic sets to describe the assembly relationships, constraint types and assembly feature characteristics between parts. On this basis, a reasoning relationship matrix is established between the above research objects and tolerance types, and an automatic tolerance item generation algorithm is formed. Zhong et al. [13,33] and Qin et al. [34,35] used ontology technology to describe various tolerance terminology concepts in the field of tolerance knowledge, as well as the attribute relationships between concepts, such as assembly constraint relationships and assembly feature space constraint relationships, and use rule language to describe the mapping relationship between tolerance types and assembly features, thereby achieving automatic generation of tolerance types of assembly parts. This type of method is easy to understand, but it recommends all shape tolerances and directional position tolerances that match assembly feature features (geometric features, space constraint relationship features), resulting in a large number of tolerance types. Designers also need to further screen and optimize based on their own design experience.

When designers use the above methods to achieve the automatic generation of tolerance types, they need to make a preference for the recommended tolerance scheme at the end. Typically, the selection methods can be grouped into two main categories, one using alternative selection and the other using screening optimization.

Alternative preference is applied to the following two methods for automatic generation of tolerance types: One is the TTRS-based method [24,25], and the other is a method based on assembly positioning constraints [26,27,28,29]. They first assume that the target feature surface elements must be positionally toleranced, and then consider whether to use directional tolerance instead of positional tolerance based on the spatial relationship between the reference-based elements and the target feature elements.

Screening optimization is applied to tolerance design methods based on reasoning rules. The rule-based reasoning method has the obvious feature that all form and orientation position tolerances conforming to the assembly feature can be derived for the same assembly feature surface. This requires the designer to screen and optimize from the recommended tolerance types. At present, screening rules are generally divided into two categories: One is based on the type of tolerance domain and the visualization of tolerance symbols to optimize. For example, the cylindrical feature surface uses roundness shape tolerance instead of line profile tolerance, reflecting the visualization of tolerance symbols [31,32,33,34]. For another example, if coaxiality tolerance is selected for two assembly feature elements, parallelism tolerance is excluded to reflect the comprehensiveness of the tolerance area. The other type is optimized based on the working characteristics of the mating part. For example, the non-reference datum feature surface of the rotating motion axis, the total run-out tolerance, and the control comprehensive tolerance are generally used to replace the position tolerance in other directions [9,19].

Optimization of these tolerance types relies heavily on the design experience of the designer. In order to overcome the subjectivity and experience of designers and improve the scientific of decision making, Zhao et al. [36] proposed an optimal tolerance method based on the independent principle and entropy theory for rotating shaft parts. However, this method needs to form an independent principle according to certain rules and select a weight factor. This method is difficult to apply to the tolerance design of parts with complex structures.

In summary, the alternative optimization tolerance scheme and the screening optimization tolerance both rely heavily on designers’ experience, and in the screening process; there will be cases which the designers do not consider thoroughly, resulting in the selection of tolerance types that cannot meet the geometric tolerance functional requirements of mechanical products. Therefore, the optimization method of tolerance types based on DOF proposed in this paper not only meets the geometric tolerance requirements of assemblies, but also has good applicability and reduces the difficulty of designers’ decision-making.

3. Basic Concepts

In this section, the DOF of tolerance zones and the CPDF of the assembly are defined. Based on this, their Boolean representations are defined.

3.1. Definition and Representation of DOF of Tolerance Zones

There are 10 types of typical tolerance zone shapes [9], as shown in Figure 1. The difference tolerances have four elements: shape, size (tolerance value), orientation and position [19,26,27]. According to the model of DOF [20], it is known that in a three-dimensional coordinate system, when a feature element changes along the direction of the X, Y, and Z axes, the change in the shape, size, and position of the element is called DOF, and vice versa, is called a degree of invariance (DOI) [20]. Similarly, define the DOF or DOI of the tolerance zone in the X, Y, and Z axes directions.

The variation of geometric features of a part within the tolerance domain can be expressed as a combination of three translational and three rotational DOFs in a Cartesian coordinate system [20,36]. Then the DOF vector of the tolerance band is denoted as V, as shown in Equation (1).

$$V=\left(\begin{array}{cccccc}{T}_Z& T_Y& T_Z& R_X& R_Y& R_Z\end{array}\right)$$

(1)

where $T_X$, $T_Y$, and $T_Z$, respectively, represent the translational DOF of the X-axis, Y-axis, and Z-axis, $R_X$, $R_Y$, and $R_Z$, respectively, representing the rotational DOF around the X-axis, Y-axis, and Z-axis.

Then, Boolean representations are given, where “0” represents DOI, meaning it does not affect the design structure of the tolerance zone in that direction, and “1” represents a DOF. The tolerance zones and DOFs relationship matrix is shown in

Table 1. Tolerance Zone and DOF Relationship Matrix.

Tolerance Code	TS1	TS2	TS3	TS4	TS5	TS6	TS7	TS8
TZ form
DOI	TZ	TX, TY	TX, TY	TZ	TZ	TZ		TZ
	RX, RY, RZ	RX, RZ	RZ	RX, RY, RZ	RZ	RZ	RX, RY, RZ
DOF	TX, TY	TZ,	TZ,	TX, TY,	TX, TY,	TX, TY	TX, TY, TZ	TX, TY,
		RY	RX, RY		RX, RY,	RX, RY		RX, RY, RZ
Representation	Ti (1,1,0)	Ti (0,0,1)	Ti (0,0,1)	Ti (1,1,0)	Ti (1,1,0)	Ti (1,1,0)	Ti (1,1,1)	Ti (1,1,0)
	Ri (0,0,0)	Ri (0,1,0)	Ri (1,1,0)	Ri (0,0,0)	Ri (1,1,0)	Ri (1,1,0)	Ri (0,0,0)	Ri (1,1,1)

(The two tolerance zones for free curves and free-form surfaces are not investigated in this paper because free-form surfaces are seldom used as assembly locating connections in typical assemblies). From the definition of the DOF of the tolerance zone, it can be seen that the DOF of the tolerance zone affects the direction of changes in assembly feature surfaces, and subsequently affects the changes in target components in the assembly.

3.2. Definition and Representation of CPDF

The CPDF is defined as follows: If the geometric functional requirements of the assembly body (such as parallelism, perpendicularity, inclination, coaxiality, runout, and other directional position tolerances) are S, the measured feature element is P, and the measured datum element is F, then the measured feature element is P, the measured datum element is F, and the vector of displacement for the change in the connection of the two assembled parts of the assembly is $v$,$v_{}=\left(\begin{array}{cccccc}{T}_x& T_y& T_z& R_x& R_y& R_z\end{array}\right)$. The variable displacement elements $T_x$, $T_y$, and $T_z$ in vector $v$ represent translational displacements along the X, Y, and Z directions, respectively; $R_x$, $R_y$, and $R_z$ represent rotational variable displacements around the X-axis, Y-axis, and Z-axis, respectively. Then, the displacement vector $T_k$(k = x, y, z) or $R_k$(k = x, y, z) in the displacement vector causing the position change of the measured feature F in the assembly and the change of the geometric function demand S index value is called the CPDF, which is represented by the Boolean value “1”. On the contrary, it is called DOI, which is represented by “0”. Therefore, the vector $v_0$ containing the CPDFs and the DOIs is called the CPDF vector.

4. CPDF for Geometric Functional Tolerancing of Assemblies

In practice, the typical geometric functional requirements of an assembly are parallelism, perpendicularity, inclination, and position of the target part with respect to the measurement datum. Due to the fact that the positionality is a comprehensive tolerance [20], any displacement change in any direction of the assembly feature surface will have an impact on it [21,23], and the assembly connection positioning feature surfaces all need to be of the positional tolerance type [27,28,29]. For this reason, only typical geometric functional requirements (except for positionality)—directional positional tolerances—were selected for analysis to obtain their CPDFs. In the following, based on the definition of CPDF, face-to-face perpendicularity was selected for analysis, and the acquisition process of determining the CPDFs was introduced.

4.1. CPDF Acquisition

The geometric functional requirements of typical assemblies have four cases of directional positional tolerances for face-to-face, face-to-line, line-to-line, and line-to-opposite, as shown in

Table 2. The geometric functional requirements of typical assemblies.

Reference Elements	Measured Feature Elements
	Line							Plane
Line
Plane

(where Symbol  indicates parallelism, Symbol  indicates verticality, Symbol  indicates inclination, Symbol  indicates degree of coaxiality, Symbol  indicates total run-out, Symbol  indicates run-out, and Symbol  indicates positionality). In the following, the perpendicularity requirement of face-to-face is selected as the research object, and the acquisition process of CPDF is introduced.

Each assembly consists of a number of parts assembled and connected to each other [28] and can be reduced to two main parts: the measuring part and the measuring reference part. When analyzing the geometric function of the assembly body, which is face-to-face perpendicularity, construct a simple assembly model as shown in Figure 2. The model consists of two parts, Part₁ and Part₂. The geometric functional requirement is the perpendicularity. Using this simple assembly model with perpendicularity tolerance requirements as an example, the process of obtaining CPDF is introduced.

4.1.1. Analyze the Geometric Function of the Assembly

According to the engineering semantics of perpendicularity error measurement [19], the measurement plane P₃ is constructed as shown in Figure 3a. The measurement plane P₃ is not only perpendicular to the geodetic datum plane P₁, but also parallel to the ideal feature plane P₂ to be measured. During actual measurement, within the measurement plane, the measuring pointer is perpendicular to the ideal feature plane P₂ being measured, contacts the actual feature plane F₁₂, and moves back and forth along the measurement plane. The maximum change in pointer reading is the verticality error. The verticality error is expressed as the variation in the actual feature element being measured relative to the measurement plane P₃.

4.1.2. Establish an Analytical Coordinate System

In order to analyze the impact of assembly connection features on verticality error, the overall coordinate system and local coordinate system are established as shown in Figure 3a. The establishment principle of the overall coordinate system is as follows: Take the geometric center of the ideal plane of the assembly connection plane F₁₁ as the origin O, define the Z-axis as the normal direction of the ideal plane of the assembly connection feature over the origin O, determine the X-axis as the normal direction of the measured ideal feature plane P₂ over the origin O, and establish the Y-axis perpendicular to the O-XZ plane over the origin O within the ideal feature plane of the assembly, so as to obtain the overall coordinate system G (O-XYZ) based on the ideal assembly connection feature plane. The principle for establishing the local coordinate system is as follows: Take the geometric center of the measured ideal feature plane P₂ as the coordinate origin w, which is convenient for subsequent analysis and processing. The directions of the m-axis, l-axis, and n-axis of the local coordinate system are, respectively, consistent with the directions of the X-axis, Y-axis, and Z-axis of the G (O-XYZ) coordinate system, so as to obtain the local coordinate system B (w-m l n) based on the measured ideal feature elements.

4.1.3. Analysis of the Impact of Contact Change Displacement on the Geometric Function

Based on the dimensions marked in Figure 2 of the assembly model, the vector of the coordinate origin of the local coordinate system B (w-mln) in the global coordinate system is expressed as $r_o$ = (b 1 k), and the position vector of the measured point P in the ideal feature plane P₂ is denoted as $r_p$ in the local coordinate system B (w-mln), as shown in Equation (2).

$$r_p=\left(\begin{array}{cccc}0& l& n& 1\end{array}\right)$$

(2)

When the actual assembly joint changes in displacement relative to the ideal assembly joint, the coordinate system G (O-XYZ) of the ideal feature plane of the assembly joint changes, and the coordinate system is transformed into G’(O-X’Y’Z’), as shown in Figure 3b, whose displacement change is described by the displacement vector V, as shown in Equation (3).

$$v_{}=\left(\begin{array}{cccccc}{T}_x& T_y& T_z& R_x& R_y& R_z\end{array}\right)=\left(\begin{array}{ccc}\begin{array}{cccc}\Delta X& \Delta Y& \Delta Z& \alpha \end{array}& \beta & \lambda \end{array}\right)$$

(3)

At the same time, the position of the measured point P also changes. In the global coordinate system G (O-XYZ), the position vector of the measured point P after change of its position is expressed as $r_G$, $r_G$ is as shown in Equation (4).

$$r_G=(\begin{array}{cccc}{X}_P& Y_P& Z_P& 1\end{array})$$

(4)

When the global coordinate changes, the conversion relation between the position vectors $r_G$ and $r_p$ is as shown in Equation (5) [3].

$$r_G=T_B{r}_p$$

(5)

In Formula (5), $T_B$ is the homogeneous transformation matrix, and the expression is shown in Formula (6) [37]:

$$T_B=\left(\begin{array}{cccc}{r}_{11}& r_{12}& r_{13}& X_0\\ r_{21}& r_{22}& r_{23}& Y_0\\ r_{31}& r_{32}& r_{33}& Z_0\\ 0& 0& 0& 1\end{array}\right)=\left(\begin{array}{c}\begin{array}{cc}\begin{array}{cc}{C}_B& \end{array}& \begin{array}{cc}& d_G\end{array}\end{array}\\ \begin{array}{cccc}0& 0& 0& 1\end{array}\end{array}\right)$$

(6)

The value of the element $r_{ij}$ in the matrix $T_B$ is related to the rotational coordinate axis; $C_B$ is the rotation matrix. When the G-coordinate system is rotated around the X-axis only and the rotational displacement is α, the expression $C_B$ is as shown in Formula (7) [38].

$$C_B=C_X=\left(\begin{array}{ccc}1& 0& 0\\ 0& \cos \alpha & \sin \alpha \\ 0& -\sin \alpha & \cos \alpha \end{array}\right)$$

(7)

The $d_G$ is the relative position vector between the local coordinate system and the global coordinate system. In this paper, $d_G$ is as shown in Equation (8).

$$d_G=(\begin{array}{ccc}{X}_0& Y_0& Z_0\end{array})=r_0$$

(8)

When there is only rotational displacement around the X-axis, Equation (9) is obtained from Equations (6) and (7).

$$T_B=T_X=\left(\begin{array}{cccc}1& 0& 0& X_0\\ 0& \cos \alpha & \sin \alpha & Y_0\\ 0& -\sin \alpha & \cos \alpha & Z_0\\ 0& 0& 0& 1\end{array}\right)$$

(9)

In the global coordinate system, the measured point P position changes $\Delta r$ is determined by Formula (10):

$$\Delta r=\left(\begin{array}{c}\Delta X_P\\ \Delta Y_P\\ Z{\Delta }_P\end{array}\right)=r_{G_{}}-(r_p+d_G),$$

(10)

The element values $\Delta X_p$, $\Delta Y_p$, and $\Delta Z_p$ in the position change vector $\Delta r$ represent the changes of the measured point P in the X-axis direction, the Y-axis direction, and the Z-axis direction, respectively. They have different effects on the functional geometric error of the assembly body. In the assembly model shown in Figure 2, only the displacement change in the X-axis direction $\Delta X_p$ affects the perpendicularity error value.

For the assembly body shown in Figure 2, the influence of the rotational displacement of the assembly feature surface around the X, Y, and Z coordinate axes on the perpendicularity error is analyzed using Formulas 1 and 5, respectively.

Only when the assembly feature surface F₁₁ is rotated around the X-axis, the rotation displacement is α_2-1. In the local coordinate system, the position vector of measured point P in the target feature surface F₁₂ is as follows: $r_p=\left(\begin{array}{cccc}0& l& n& 1\end{array}\right)$, $-k\le n\le k$, $d_G=r_o=\left(\begin{array}{cccc}0& b& k& 1\end{array}\right)$. Set the position vector of the measured point P after the position is changed as $r_{G,m}$. From Equations (5) and (9), $r_{G,m}$ can be obtained:

$$r_{G,m}=\left(\begin{array}{c}{X}_P\\ Y_P\\ Z_P\\ 1\end{array}\right)=\left(\begin{array}{cccc}1& 0& 0& 0\\ 0& \cos {\alpha }_{2-1}& -\sin {\alpha }_{2-1}& b\\ 0& \sin {\alpha }_{2-1}& \cos {\alpha }_{2-1}& k\\ 0& 0& 0& 1\end{array}\right)\left(\begin{array}{c}0\\ l\\ n\\ 1\end{array}\right)=\left(\begin{array}{c}0\\ l\cos {\alpha }_{2-1}-n\sin {\alpha }_{2-1}+b\\ l\sin {\alpha }_{2-1}+n\cos {\alpha }_{2-1}+k\\ 1\end{array}\right)$$

(11)

When part₁ undergoes a rotational displacement around the X-axis at the assembly feature surface F₁₁, the displacement change vector $\Delta r_m$ at point P within the ideal surface is obtained from Equations (10) and (11):

$$\Delta r_m=\left(\begin{array}{c}\Delta X_P\\ \Delta Y_P\\ Z{\Delta }_P\end{array}\right)=\left(\begin{array}{c}0\\ l(\cos {\alpha }_{2-1}-1)-n\sin {\alpha }_{2-1}\\ l\sin {\alpha }_{2-1}+n\cos {\alpha }_{2-1}-n\end{array}\right)$$

(12)

Similarly, according to Formulas (5) and (6), calculate $\Delta r_l$ when Part₁ rotates around the Y-axis and $\Delta r_n$ when Part₁ rotates around the Z-axis on the assembly feature surface F₁₁, and the results are as follows:

$$\Delta r_l=\left(\begin{array}{c}\Delta X_p\\ \Delta Y_p\\ \Delta Z_p\end{array}\right)=\left(\begin{array}{c}-n\sin {\beta }_{2-1}\\ 0\\ n\sin {\beta }_{2-1}-n\end{array}\right),\Delta r_n=\left(\begin{array}{c}\Delta X_P\\ \Delta Y_P\\ \Delta Z_P\end{array}\right)=\left(\begin{array}{c}-l\sin {\gamma }_{2-1}\\ l\cos {\gamma }_{2-1}-l\\ 0\end{array}\right)$$

It can be seen from the calculation results that in the global coordinate, when the ideal assembly connection has a rotation shift change, the influence on the measurement of the perpendicularity error is as follows: When the assembly feature surface F₁₁ rotates around the X-axis, it can be seen from the change vector $\Delta r_m$ that the displacement variation of the measured point P in the X-axis measurement direction is 0. When rotating around the Y-axis, it can be seen from the change vector $\Delta r_l$ that the displacement variation of the measured point P in the measurement direction of the X-axis is $n\sin {\beta }_{2-1}$. Based on the perpendicularity measurement method, the ideal feature plane P₂ to be measured is always parallel to the measurement plane P₃. The measurement plane P₃ rotates around the Z-axis synchronously with the ideal feature plane P₂ to be measured, and the distance between the measurement point and the placement point of the measurement instrument does not change, then the relative variation of the measurement point and the placement point of the measurement instrument is constant” 0”, so when the measurement value of the perpendicularity error rotates around the Z-axis, the X-axial variation of the measurement value under the overall coordinate is also constant at “0”.

In summary, when the measurement reference element is a plane and the measured element is also a plane, the rotational displacement that affects the perpendicularity error is only a rotational variation around the Y-axis, and the direction of this displacement variation is the CPDF of the assembly shown in Figure 2, and the displacement variations in the other five directions do not affect the assembly geometrical function, and together they form the CPDF vector $V_0$, denoted as $V_0=(0 0 0 0 1 0)$.

4.2. Summarize the CPDF for Typical Assembly Geometry Functions

In a triple datum system, the DOF of the assembly joint feature that affects the single datum orientation tolerance must affect the measurement of the positional tolerance of the dual datum and triple datum [19]. In addition, the positional tolerance can be applied to various positional tolerances under different spatial relations, and the form tolerance domain and orientation tolerance domain of the same feature being measured are within the location tolerance domain. Therefore, this paper only analyzes and summarizes the parallelism, perpendicularity, coaxiality, and run-out of a single datum. Through the analysis of the typical single datum direction position tolerance, the control parameter freedom $v_0$ of the target part under different direction position tolerance is obtained, and its mapping relationship is shown in

Table 3. The mapping relationship between DOF of measurement elements and V₀.

Measurement Reference Elements Vf(Tx,Ty,Tz)(Rx,Ry,Rz)		Measured Feature Elements
		Line			Plane
		Tolerance Zone	DOFs Vt (Tx,Ty,Tz,Rx,Ry,Rz)	CPDF V0 (Tx,Ty,Tz,Rx,Ry,Rz)	Tolerance Zone	DOFs Vt (Tx,Ty,Tz,Rx,Ry,Rz)	CPDF V0 (Tx,Ty,Tz,Rx,Ry,Rz)
(0,0,1)(1,1,0)			0,0,1,1,1,0	0,0,0,1,0,0		0,0,1,1,1,0	0,0,0,1,1,0
			1,0,1,1,0,1	0,0,0,1,1,0
			1,1,0,1,1,0	0,0,0,1,1,0
			0,1,0,1,0,1	0,0,0,1,0,0		0,1,0,1,0,1	0,0,0,1,0,0
			1,0,0,1,1,0	0,0,0,0,1,0		1,0,0,0,1,1	0,0,0,0,1,0
(1,1,0)(1,1,0)			1,1,0,1,1,0	0,0,0,1,1,0		0,0,1,1,1,0	0,0,0,1,1,0
						0,0,1,1,1,0	0,0,0,1,1,0
			1,1,0,1,1,0	0,0,0,1,1,0
			1,1,0,1,1,0	0,0,0,1,1,0		0,1,0,1,0,1	0,0,0,1,0,0
			0,1,0,1,0,1	0,0,0,1,0,0		1,0,0,0,1,1	0,0,0,0,1,0
			1,0,0,0,1,1	0,0,0,0,1,0
			1,0,1,1,0,1	0,0,0,1,0,0		0,0,1,1,1,0	0,0,0,1,1,0
			0,0,1,1,1,0	0,0,0,1,1,0

. Symbol  indicates parallelism, Symbol indicates verticality, Symbol  indicates inclination, Symbol  indicates degree of coaxiality, Symbol  indicates total run-out, and Symbol  indicates run-out.

5. A DOF-Based Method for Screening and Optimizing Tolerance Items

From the analysis in Section 4, it can be seen that the six variable displacements generated by the assembly joint feature surfaces of the assembly affect the geometric function index value of the assemblies. Different tolerance zones do not necessarily have exactly the same DOFs. In the case of a consistent coordinate system, the screening optimization tolerance types can be realized by comparing the relation between the DOF of the tolerance zone and the CPDF. In order to facilitate the automatic generation of assembly tolerances, it is necessary to explore the process of obtaining the CPDFs by using the DOFs of the tolerance zone of the geometric functional tolerance of assembly and the DOFs of the features of the reference datum, and to form the automatic generation algorithm of tolerance items for the screening optimization of the assembly’s shape and positional tolerances based on the DOFs.

5.1. Algorithm for Acquiring CPDF

The Boolean operation is widely used in set operations, and the DOF vector of the tolerance zone can be regarded as a form of set. Therefore, by analyzing the data in Table 3, the flow of algorithm to find the CPDF vector can be obtained, as shown in Figure 4.

• Step 1: Analyze the geometric functional requirements of the assembly body, specify the reference datum feature features and the measured feature features of the geometric functional tolerance of the assembly body, and determine the global coordinate system.
• Step 2: Determine the DOF vector $V_t$ of the tolerance zone of the measured feature element and the DOF vector $V_f$ of the reference datum feature element in the assembly.
• Step 3: Calculate the common DOF $V_c$; $V_c$ is defined as the result of the Boolean operation between the vector $V_t$ and $V_f$, and the Boolean operation formula is shown in Equation (13):

  $$V_c=V_t\cap V_f$$

  (13)

Calculation rule of Formula (8): The elements of the two vectors, if the value of the element is “1”, then the Boolean operation value will be “1”; otherwise, the value will be “0”. From the Boolean rules, it can be seen that the DOF in a certain direction in vector $V_c$ is “1”, which means that the DOF in a certain direction in the vector $V_c$ affects the geometrical functional requirements of the assembly, and needs to be taken into account when choosing the type of tolerance for the assembly feature elements.

• Step 4: Determine the CPDF vector $V_0$. When the geometric functional tolerance of the assembly is the orientation tolerance, set an auxiliary calculation vector $V_p=\left(\begin{array}{cccc}0& 0& 0& \begin{array}{ccc}1& 1& 1\end{array}\end{array}\right)$, solving for CPDF $V_0$, the expression is shown in Formula (14):

  $$V_0=V_c\cap V_p$$

  (14)

When the geometric functional requirement of the assembly is the position tolerance, the expression for the CPDF vector $V_0$ is given in Equation (15):

$$V_0=V_c$$

(15)

5.2. Process for Selecting Tolerance Types of Assembly Features of Assembly Parts

According to the process of obtaining the control parameter degrees of freedom of the assembly in Section 4, it is known that a certain tolerance type is selected for the assembly feature surface, the DOFs of the tolerance zone of the tolerance indicates that it cannot constrain the displacement changes in the corresponding direction of the assembly feature surface, and its displacement changes may affect the geometric functional requirements of the assembly, which provides a basis for using the DOFs of the tolerance zone to screen and optimize tolerance types. For this purpose, the algorithmic optimizing of tolerance types based on DOFs is designed, as shown in Figure 5. The steps are as follows:

• Step 1: Determine the tolerance zones of different tolerance types of the assembly feature features and their DOFs vector $V_i$ (subscript i is the tolerance mark serial number).
• Step 2: Calculate the Boolean operation value between the DOFs vector Vi and the CPDF vector $V_0$ which can be represented by the vector $V_i^{\prime }$, $V_i^{\prime }$ is called the comparison freedom vector. Its calculation formula is shown in Formula (16):

  $$V_i^{\prime }=V_i\cap V_o$$

  (16)
• Step 3: Select tolerance types. Judge the relationship between the comparative DOF vector $V_i^{\prime }$ and the CPDF vector $V_0$. If the CPDF vector $V_0$ is equal to the comparative DOF vector $V_i^{\prime }$, it indicates that the i-th tolerance type of an assembly feature surface of the assembly part is the preferred tolerance type.

5.3. Establish Algorithm for Automatic Generation of Tolerance Items Based on DOF

As can be seen from the content of Section 2, the tolerance design method based on rule-based reasoning will obtain all the tolerance items that meet the assembly characteristics of the part, the number is large, and further screening and optimization is needed. For this reason, this paper combines the algorithmic process of using DOF to screen and optimize tolerance types with the above method to establish an automatic assembly tolerance generation algorithm based on DOF, whose algorithm is shown in Figure 6:

6. Case Study

The simplified assembly model of a mechanical product is shown in Figure 7a. The product consists of four main parts, and its geometric tolerance requirement is perpendicularity. The part P₄ is movably connected to the hydraulic cylinder drive unit, which drives the part P₄ in an up and down motion. Part P₄ moves up and down relative to part P₁. When designing the tolerance items of each part, it is necessary to select a reasonable tolerance type to meet the geometric functional requirements of the assembly. In this section, with reference to the automatic generation algorithm of tolerance items shown in Figure 7, the working part P₄ in the assembly model is selected as the research object to explain in detail the implementation steps of optimizing tolerance items by screening with DOFs of the tolerance zone, and to verify the validity of optimizing the assembly tolerance types by using the DOFs of the tolerance zone.

• Step 1: Construct a CAD three-dimensional assembly model according to the model in Figure 7a.
• Step 2: Select the global coordinate system, and select the target part and measurement reference part of the assembly. Considering the convenience of determining the degrees of freedom of the tolerance band, the principle to be followed when establishing the overall coordinate system is as follows: take the measurement reference datum of the geometric function of the assembly as the XY plane, define the X-axis, Y-axis, and the direction perpendicular to the XY plane as the Z-axis. This is shown in Figure 8a. From Figure 8a of the model, it can be seen that the geometric functional requirement of the assembly is the perpendicularity, part P₄ is the target part of the assembly, and P₁ is the measurement reference part. The measured feature element is the center axis of the cylindrical surface P₄S₂, and the reference datum element is the plane P₁S₁ of part P₁ as shown in Figure 7b.
• Step 3: According to Figure 5, the CPDF vector $V_0$ of the assembly body is determined.

First, according to the global coordinate system, determine the DOF vectors $V_t$ and $V_f$, the feature being measured is the center axis of the P₄S₂ cylindrical surface, the tolerance type is perpendicularity, and the shape of the tolerance zone is a cylinder with diameter size t. According to Table 1, the DOF of the tolerance zone vector $V_t$ = (1 1 0 1 1 0), and according to Figure 4, the feature DOF vector $V_f$ of the reference feature P₁S₁ plane can be obtained as (0 0 1 1 1 0).

Secondly, determine the common DOF vector $V_c$. From Equation (8), it can be seen that

$$V_c=V_t\cap V_f=(1 1 0 1 1 0 )\cap (0 0 1 1 1 0 )=(0 0 0 1 1 0 )$$

(17)

Finally, determine the CPDF vector $V_0$. The geometric functional tolerance of the assembly is the perpendicularity, which belongs to the direction tolerance. Therefore, by introducing Equation (12) into Equation (9) and performing a Boolean operation, we obtain $V_0$ = (0 0 0 1 1 0).

• Step 4: Select the automatic tolerance generation method based on the rule reasoning algorithm, and deduce the tolerance item $TS{S}_{i(j)\_k}$ of the assembly feature elements of each part. The recent ontology-based automatic reasoning algorithm of Qin et al. [33,34] is a representative automatic generation method of tolerance specifications. In this paper, we choose the automatic generation algorithm of tolerance items proposed in the literature by Qin [33], use Protégé software to construct the tolerance domain knowledge ontology, input the SWRL inference rules, and obtain the automatically recommend tolerance items of each part of the assembly. The assembly constraint relationship of the assembly and the assembly feature surface of each part are marked before automatic reasoning, as shown in Figure 8b. Select part P₄ as the representative to carry out tolerance project reasoning, and obtain the markable shape tolerance of the assembly feature cylindrical surfaces P₄S₁ and P₄S₂ of workpiece P₄. With the central element of feature plane P₄S₂ as the reference datum, the optional directional position tolerance of the target assembly feature plane P₄S₁ can be obtained, as shown in Table 3.
• Step 5: Determine the DOF vector $V_{4(j)\_k}$ of the tolerance zone for all recommended tolerance types of part P₄ based on the global coordinate system. Then obtain the DOF vector for all recommended tolerance types based on Table 1 (Note the order of exchanging the DOF and DOI in the DOF vector of the tolerance zone when the coordinate system in the tolerance band in the table is oriented differently from the overall coordinate system.), as shown in

  Table 4. Pre-selected form tolerances and associated degrees of freedom for mating feature features.

  Global Coordinate System	AFS 1	Serial No	Tolerance Types	TZ Code	DOF of TZ V4(j)_k	Comparison Freedom Vector V′4(j)_k
  	P4S1	1	Straightness in any direction ()	TS4	0,1,0,0,1,0	0,0,0,0,1,0
  		2	Straightness in X-Y plane ()	TS1	1,1,0,0,0,0	0,0,0,0,0,0
  		3	Roundness ()	TS3	1,1,0,1,1,0	0,0,0,1,1,0
  		4	Cylindricity ()	TS5	1,1,0,1,1,0	0,0,0,1,1,0
  		5	Coaxiality ()	TS4	0,1,0,0,1,0	0,0,0,0,1,0
  		6	Position ()	TS4	1,1,0,0,0,0	0,0,0,0,0,0
  		7	Run-out ()	TS3	1,1,0,1,1,0	0,0,0,1,1,0
  		8	Full runout ()	TS5	1,1,0,1,1,0	0,0,0,1,1,0
  	P4S2	1	Straightness in any direction ()	TS4	1,1,0,1,1,0	0,0,0,1,1,0
  		2	Straightness in X-Y plane ()	TS1	0,1,0,0,1,0	0,0,0,0,1,0
  		3	Roundness ()	TS3	1,1,0,0,0,0	0,0,0,0,0,0
  		4	Cylindricity ()	TS5	1,1,0,1,1,0	0,0,0,1,1,0

  ¹ AFS represents assembly feature surface.

  .
• Step 6: Calculate the comparison freedom vectors $V_{4(j)\_k}^{\prime }$. Use Equation (11) to perform a Boolean operation on the DOF vector $V_{4(j)\_k}$ of the tolerance zones and the CPDF vector $V_0$ for different tolerance types; the operation results are shown in Table 3.
• Step 7: Screening optimization of tolerance types. Based on the decision rule of the optimization, the comparison freedom vector $V_{4(j)\_k}^{\prime }$ of different tolerance types is compared with the CPDF vector $V_0$ =(0 0 0 1 1 0) to obtain the optimization of tolerance types. Through comparison, it can be seen that in the form tolerance class of reference datum feature plane P₄S₂, the comparison freedom vector $V_{4(j)\_k}^{\prime }$ of straightness and roundness in the X-Z plane is not equal to the CPDF vector $V_0$, straightness, or cylindricity in any direction, and the other two form tolerances meeting the geometric functional requirements of the assembly are obtained through screening optimization. The non-reference feature plane P₄S₁ is optimized to obtain three directional position tolerances, including total run-out, position, and coaxiality, and two form tolerances, including straightness and cylindricity in any direction. It can be seen from the above optimization results of tolerance types that the number of tolerance types is optimized, which not only reduces the recommended number of tolerance items, but also avoids marking unreasonable tolerance types.
• Step 8: Label the detailed tolerance information of the part. From the optimization results, it can be seen that when the detailed tolerance specifications for part P₄ are finally completed, further screening of tolerance types and adding tolerance symbols, such as material conditions and tolerance domain feature symbols [39,40], are needed. According to the fit characteristics and kinematic characteristics of workpiece P₄, the assembly feature surface P₄S₂ is the first assembly datum, and the cylindricity shape tolerance is selected [41]. The feature surface P₄S₁ is selected as cylindricity shape tolerance and coaxiality tolerance [42], and the final labeling of the tolerances of part P₄ is shown in Figure 8. Refer to the determination steps of part P₄ for detailed tolerance specifications of other assembly parts to obtain the marked tolerance as shown in Figure 9.

7. Discussion

From the content of the related works in Section 2, it is clear that the tolerance design method based on rule-based reasoning makes it easy to achieve the automatic generation of tolerance specifications. In particular, the ontology-based rule-based reasoning is more widely used in the application of the automatic generation of tolerance specifications. However, the above methods recommend multiple tolerance types for the same assembly feature surface of the assembly body. Taking the part P4 shown in Figure 8 as an example, Armillotta’s [27] method, Zhang’s [8] method, and Qin’s [33] method are selected for tolerance specification design. With different tolerance design methods, the assembly feature element P₄S₂ will receive different numbers of recommended tolerance types as shown in

Table 5. Reasoning results of different automatic tolerance generation methods on the same feature.

Methods	AFS 1
		In The Plan	In Any Direction
Armillotta [26]	P4S1 (Cylinder)		√	√	√	√	√	√	√
Zhang et al. [8]		√	√	√	√	√	√	√	√	√
Qin et al. [33]		√	√	√	√	√	√	√	√
Proposed			√		√		√	√	√

¹ AFS represents assembly feature surface.

. The initial tolerance types obtained by the above methods are 7, 9, and 8, respectively. After screening by the tolerance band freedom preference method proposed in this paper, five types of tolerance design solutions are finally obtained. The four shape-tolerance types obtained by the rule-based reasoning method are reduced to two, and the excluded tolerance types have obvious characteristics and their tolerance zones belong to two-dimensional tolerance zones. When defining the DOF of the two-dimensional tolerance bands, the variation of the third dimension is defined as a constant degree, and for this reason it is preferentially excluded from the Boolean screening due to the small number of DOFs. Of the various tolerance-band generation algorithms mentioned above, the preferred method proposed in this paper has obvious implications and increases the decision difficulty for designers.

8. Conclusions

The method of optimizing tolerance types using the DOF of the tolerance zone has a theoretical basis. This article analyzes the influence of the displacement of the assembly contact surface on the geometric tolerance of the assembly, and obtains the CPDF. It can be considered a sensitive factor in assembly tolerance analysis. In order to facilitate the optimization of tolerance types for the assembly feature surfaces of parts through the tolerance zone degrees of freedom, an algorithm flow for selecting tolerance types, the DOF of the tolerance zone vectors, and CPDF vectors was constructed using the Boolean operation relationship between vector sets.

The example verifies the feasibility of the preferred model of assembly body shape and position tolerance elements based on the DOF of the tolerance zone. The algorithmic process of selecting tolerance types using the DOF of the tolerance zone and CPDF can improve the automatic generation method of tolerance specifications based on rule-based reasoning. The new method can reduce the number of recommended tolerances and the difficulty of designers’ decision making: for example, in combination with Qin’s [33] method, the number of tolerance types is reduced from eight to five. At the same time, it can ensure that the selection of tolerance types meets the geometric functional requirements of assemblies: for example, there will not be a single circularity or straightness item to control the shape tolerance, and there will not be a combination of circularity and straightness selected to control the shape tolerance.

The method proposed in this article has practical application value in the tolerance design of mechanical products, and this example can serve as a simplified model for a simple stamping machine, providing reference for practical applications. However, in terms of the breadth of the research, there are some limitations in the application scope, as shown in the process of using Boolean operations to find CPDFs, where they are only used for the geometric function requirements of components with a single benchmark. The future research direction will explore how to use the DOF vector to derive the CPDF for assembly under multiple benchmarks, and complete the tolerance specification design of high-precision mechanical products with multiple degrees of freedom [43,44]; Additionally, future research will explore how to use ontology technology to achieve intelligent screening and improve design efficiency.