Point-by-Point-Contact-Based Approach to Compute Position and Orientation between Parts Assembled by Multiple Non-Ideal Planes

Abstract

Position and orientation deviations (PODs), being affected by surface deviations, occur after parts are assembled, which directly affects the performance of mechanical products. Moreover, mechanical parts are generally assembled with multiple constraint planes, and the generated PODs are influenced by the type of positioning. Therefore, the PODs of multiple planes should be computed in the design stage according to the predicted surface deviations, to control the product performance. However, even though the POD computation of multiple planes has been researched, the effects of surface deviations and multiple types of positioning cannot be considered simultaneously. To address this problem, this study proposes a point-by-point-contact-based approach. The six-point positioning principle is employed to determine the possible number of contact points on each mating plane. The surface deviations are modeled from the perspective of manufacturing errors. Furthermore, the contact points on each mating plane are determined successively using both the strategies of progressively approaching position and of the orientation and recursion of contact points. As a result, the PODs are acquired. The feasibility and usefulness of the proposed approach are verified through a case study. Herein, effects of surface deviations and multiple types of positioning on PODs are unified as contact point variations. Consequently, this approach is expected to assist with accurately controlling the POD influence on the performance of mechanical products in the design stage.

1. Introduction

Position and orientation described by distances and angles between two assembled parts are determined after assembly. They can affect the product performance to a large extent, considering that assembly is generally the final step in manufacturing mechanical products [1,2]. Furthermore, surface deviations of the manufactured surfaces result in position and orientation deviations (PODs) between the acquired position and orientation and the ideal circumstances, directly affecting the product performance [3,4]. For instance, the machining accuracy of the machining center is affected by PODs generated after the guide rail planes of a workbench and base are assembled [5,6]. Therefore, to ensure the performance of mechanical products, PODs should be evaluated using the predicted surface deviations in the design stage.

Mechanical parts can be assembled according to a single mating plane. However, additional mating planes are generally required to constrain the degrees of freedom (DOFs) that are unconstrained by a single plane. For instance, the column and base of the machining center are assembled only through a single mating plane, whereas the workbench and the base are assembled through two planes: the bearing and guiding planes of the rail. These examples are shown in Figure 1.

The position and orientation deviations of multiple planes are caused by surface deviations and are also affected by multiple possible types of positioning. An example is illustrated in Figure 2. After two parts are assembled by two mating planes, five DOFs are constrained, and the position and the orientation corresponding to the constrained DOFs are determined. Because the non-ideal morphologies of the planes at the five contact points deviate from the ideal circumstances, PODs occur, as shown in Figure 2a,b. Because of the multiple mating planes, completely constrained positioning or incompletely constrained positioning may occur. In addition, over-constrained positioning will exist if several DOFs are constrained by more than one contact point. When incompletely constrained positioning occurs, the assembly positions corresponding to the unconstrained DOFs are uncertain. Different contact points will emerge after the uncertain positions are changed, which inherently results in different PODs, as shown in Figure 2b,c. When over-constrained positioning also exists, it is uncertain whether the over-constrained DOFs will be constrained and through which contact points. Thus, the number of contact points on each constraint plane can vary, bringing about different PODs, as shown in Figure 2b,d.

As shown, multiple mating planes are generally required when mechanical parts are assembled. Consequently, the problem of the POD computation of multiple non-ideal planes should be investigated. Furthermore, the effects of not only surface deviations but also multiple types of positioning should be considered.

1.1. Related Works

Computing the position and orientation between assembled parts is a key issue of tolerance analysis [7] and has received much research attention.

Various research of tolerance analysis has been carried out assuming that manufactured surfaces only have size and position errors, and some approaches to compute PODs were proposed. These approaches are generally based on establishing and solving models of two- or three-dimensional assembly constraints of motion and rotation [8,9,10]. For instance, two-dimensional assembly constraint models can be established according to the moving distances between discrete points or vertices if the manufactured surfaces are simplified as curves or lines deviating from the ideal circumstances [11,12]. In addition, three-dimensional constraint assembly models can be acquired by analyzing the constrained DOFs if the following two conditions are met. First, the manufactured surfaces should be simplified as the surfaces of ideal geometric features that deviate from the ideal circumstances. Second, the manufactured surfaces should be described by tolerance models of the deviation domain, Jacobian-torsor, or small displacement torsor [13,14,15]. Because the non-ideal morphologies of the manufactured surfaces are ignored by the approaches mentioned above, the acquired PODs differ from the actual circumstances even though the size and position errors are simultaneously considered, which directly affects the accuracy of tolerance analysis.

To avoid the influence of ignoring form errors on tolerance analysis, the problems of mating between two non-ideal planes have been extensively studied. For instance, inspired by the idea that the distances between the corresponding points on two mating planes will converge and the interference will never occur if two non-ideal planes are mated [16], the approach of constrained registration (CR) has been proposed [17]. Constrained registration can determine PODs by continuously adjusting the assembly position of the mating plane until the signed projected distance, signed normal distance, or convex hull volume is minimized, and the constraint condition of a non-interference is simultaneously ensured. Consequently, CR is often employed to determine the “best-fit” assembly position between two parts [18,19]. Several approaches based on determining contact points have been proposed under the condition that the assembly of two non-ideal planes is judged according to the six-point positioning principle. For instance, Samper has proposed an approach of difference surface (DS), wherein three contact points of a mating plane are determined successively via searching for the points with limit coordinates on the DS [20]. Difference surface is well suited for computing or analyzing the position and the orientation between two parts of 3-2-1 positioning [21,22,23]. Moreover, Zhang [24] and Wang [25] have also proposed approaches that determine PODs of a plane by searching for three contact points successively. The main principle is that the position and the orientation between two assembled parts result from the continuous adjustment of the assembled part position from the ideal circumstances until the six-point positioning principle is met. The adjustment magnitude of the position and orientation is the key in determining contact points. This magnitude can be acquired by computing distances between the corresponding points on the mating planes based on the methods of projecting [26], raying [27], or gap surface [28]. Although the studies mentioned above lay a foundation for the tolerance analysis considering surface deviations, they fail to handle the assembly of the multiple mating planes that are generally present in mechanical products.

To enable the actual assembly circumstances of mechanical parts to be covered by tolerance analysis, attention has been paid to computing the PODs of multiple non-ideal surfaces. For instance, Benjamin [29] considers that the assembly of multiple mating surfaces is a process over which each mating surface mates in order iteratively, and the Euclidean distance between reference points picked on two assembled parts converges progressively. However, the mating surfaces cannot interfere with each other. Consequently, a computing framework with the convergence of the Euclidean distance between reference points as the search target has been proposed. According to displacement conditions (e.g., non-interpenetration and displacement boundary conditions) and load conditions (e.g., force balance and moment balance conditions) that should be satisfied if parts are assembled with multi-mating surfaces, Yan and Ballu proposed a linear-complementarity-condition-based computing approach [30]. Under the condition of deviations of multiple mating surfaces, both of the approaches have been verified in the tolerance analysis based on non-ideal surfaces. However, the concepts of assembly employed by the two approaches do not cover different types of positioning (e.g., incomplete positioning and over-constrained positioning). Consequently, they cannot solve the problem of POD computation considering both the effects of surface deviations and different types of positioning.

As shown, to avoid the effects on the accuracy of tolerance analysis caused by only considering the size and position errors but ignoring the form errors, the problems of POD computation considering surface deviations have gained much attention. However, two problems remain unsolved in the existing literature:

• Although much attention has been paid to the problems of the POD computation of a single mating plane, the problem of the POD computation of multiple mating planes has still been rarely researched.

• Two types of approaches have been proposed to compute the PODs of multiple surfaces, but they only take into account surface deviations and neglect multiple types of positioning, which are not included in the employed concepts of assembly.

1.2. Main Contribution and Overview

To address the problem of the POD computation of multiple non-ideal planes, this study proposes a point-by-point-contact-based approach. The six-point positioning principle is employed to determine the cases in which DOFs are constrained by the contact points on each mating plane. The surface deviations of each constraint plane are modeled from the perspective of manufacturing errors. Next, the DOFs corresponding to each mating plane are constrained successively to determine each contact point by progressively approaching position and orientation and by recursion of the contact points. As a result, the PODs corresponding to the given surface deviations are acquired.

The remainder of this paper is organized as follows. Section 2 discusses the principle of POD computation by successively determining contact points. Based on the discussion in Section 2, the process of computing the POD of multiple mating planes is introduced in Section 3. Next, this approach is verified by computing the PODs of an assembly with multiple mating planes and a sliding workbench, and the efficiency of the approach is discussed. Finally, conclusions are drawn.

2. Position and Orientation Deviation Computation Principle by Successively Determining Contact Points

2.1. Framework of the Approach

Several possible assembly states may exist after the parts are assembled by multiple mating planes because the number of contact points on each plane may be multiple. In addition, each possible assembly state corresponds to a type of POD. Thus, a problem in POD computation is to determine all possible assembly states. Moreover, surface deviations of the mating planes, which act as inputs, have direct impacts on the results of POD computation. Consequently, modeling surface deviations is also a problem. The core problem of POD computation is to successively determine each contact point by continuously adjusting position and orientation. As a result, the framework of POD computation by successively determining contact points is established, as shown in Figure 3.

Assembly state determination. The six-point positioning principle is used to determine cases in which the constrained DOFs correspond to each mating plane. Consequently, each assembly state is determined. Considering the situation in which each assembly state corresponds to a type of POD, the approach based on successively determining contact points should be separately employed to determine the PODs corresponding to each assembly state.

Surface deviation modeling. The PODs are directly affected by the surface deviations. For consistency of the surface deviations and the actual morphologies of each mating plane, a modeling method from the perspective of manufacturing errors is employed [31,32,33]. The deviation functions corresponding to manufacturing errors over the process of machining each mating plane are established, and the surface deviation model of each mating plane is acquired by accumulating the deviation functions. In addition, the coefficients of these deviation functions should be determined according to the impact of each manufacturing error on the surface deviations. Consequently, the surface deviations of each mating plane can be obtained by inputting the coefficients into the surface deviation model.

Successive contact point determination. Determining contact points starts from the assembly position under the ideal circumstances. Next, three contact points on the main mating plane are determined employing the one-step-adjusting-based strategy [24]. To determine the contact points on the auxiliary mating planes, the position and orientation of the assembled part are continuously adjusted according to the adjustment values acquired by computing the distances between the corresponding points on each mating plane. In addition, the recursive process is carried out to repair the broken contact states of the previously acquired points. As a result, the adjustment quantities converge progressively. After each type of adjustment value meets the contact condition, the expected PODs are acquired.

2.2. Assembly State Determination Based on the Six-Point Positioning Principle

According to the six-point positioning principle, the number of constrained DOFs corresponding to each mating plane is different. Thus, the main mating plane and the first and second auxiliary mating planes are defined based on the number of constrained DOFs, i.e., three DOFs are constrained by the main mating plane, and two and one DOFs are separately constrained by the first and second auxiliary mating planes, respectively.

If parts are assembled by multiple mating planes, the case in which several DOFs are simultaneously constrained by different mating planes may occur. This can result in an uncertain correspondence between the main and auxiliary mating planes and the mating planes. In addition, the case in which a DOF is still unconstrained after assembly may occur because the second auxiliary mating plane can be absent. For instance, an assembly with two mating planes is shown in Figure 4. The DOF rotating around the X-axis may be constrained by the XOY or XOZ plane after assembly. Thus, the main and first auxiliary mating planes may correspond to the XOY and XOZ planes or the XOZ and XOY planes. The two cases correspond to the two types of assembly states as well as the PODs. Consequently, determining the possible states of the assembly with multiple planes is the foundation for computing the PODs by successively determining the contact points.

All or some of the six DOFs should be constrained after assembly, depending on the geometric structure of the assembly. According to the six-point positioning principle, the main and auxiliary mating planes of the assembly are determined. If the assembly is completely or incompletely positioned, only one case in which the main and auxiliary mating planes correspond to each mating plane exists. The same concerns the assembly state. However, if over-constrained positioning exists, several DOFs will be simultaneously constrained by the contact points on different mating planes. Therefore, all the possible cases of the main and auxiliary mating planes should be considered to acquire multiple assembly states.

Owing to the correspondence between the constrained DOFs and the contact points on each mating plane as well as the multiple types of positioning cases that are established according to the six-point positioning principle, all the possible assembly states can be identified. This lays a foundation for computing PODs by successively determining the contact points.

2.3. Searching for Contact Points by Progressively Approaching Position and Orientation

When determining the nth (the value of n can be 4, 5, and 6) contact point on the auxiliary mating plane, each procedure of the position and orientation adjustment will break the contact states of the previously acquired n − 1 points because of the surface deviations. Consequently, the previously acquired n − 1 contact points should be re-determined after each adjustment by the obtained adjustment value. However, not only a new position and orientation but also a new adjustment value will be acquired if the previously acquired n − 1 contact points are re-determined, which means that the n^th contact point on the auxiliary mating plane cannot be determined through one adjustment. Therefore, a strategy for progressively approaching position and orientation is proposed.

The nth contact point on the auxiliary plane is found by performing several iterations of the approaching process of the position and orientation, as shown in Figure 5. The approaching process of the position and orientation starts by computing the adjustment value $\mathrm{\Delta }{\mathrm{\Omega }}_i^n$ of the current position and orientation $\mathrm{\Delta }{\mathrm{\Omega }}_i^n$. Subsequently, the position and the orientation are adjusted by $\mathrm{\Delta }{\mathrm{\Omega }}_i^n$ to acquire the new position and orientation ${\mathrm{\Omega }}_{i+1}^n$. Because the contact states of the previously acquired points are broken, the previous n − 1 contact points corresponding to the position and orientation ${\mathrm{\Omega }}_{i+1}^n$ should be re-determined through several adjustment procedures. Subsequently, the approaching process of the position and orientation is completed, and the new position and orientation ${\mathrm{\Omega }}_{i+1}^{update\_n-1}$ are acquired. The next approaching process will be carried out on the basis of the new position and orientation ${\mathrm{\Omega }}_{i+1}^{update\_n-1}$ because each adjustment starts from the position and orientation acquired at the previous step.

Determining the adjustment value is a key problem when searching for contact points on the auxiliary mating planes. The moving adjustment magnitude for CP4 refers to the minimum distance between the corresponding points on the first auxiliary mating plane along the direction of the constrained moving DOF. The rotating adjustment magnitude for CP5 refers to the minimum angle acquired by rotating the first auxiliary mating plane in the direction of the constrained rotating DOF around the vertical axis through contact point CP4. The moving adjustment magnitude for CP6 refers to the minimum distance between the corresponding points on the second auxiliary mating plane along the direction of the constrained moving DOF.

Surface deviations have a longer period of geometric fluctuation than surface roughness, which means that oscillations of the position and orientation will never emerge during the adjustment process. In addition, each adjustment procedure continues the previous one rather than restarts from the initial position and orientation. Therefore, the adjustment value converges rapidly during the iterative process. When the adjustment value converges to a certain value, it can be judged that the DOF corresponding to this type of adjustment value is constrained, and the expected contact point is determined. Consequently, the threshold of the adjustment value can be set to judge whether the contact point is determined, and the condition of contact judgment is summarized considering that each type of adjustment value is a six-dimensional vector: If $\left|\mathrm{\Delta }{\mathrm{\Omega }}_i^n\right|\le \left|\mathrm{\Delta }{\mathrm{\Omega }}_R^n\right|$, the contact point is determined. Otherwise, the contact point remains undetermined.

2.4. Recursion of Contact Points

The previously acquired n − 1 contact points should be re-determined when searching for the nth contact point on the auxiliary plane. The problem of determining the n − 1 contact points can be decomposed into two types of subproblems: searching for the n – 1th contact point and determining the previously acquired n − 2 contact points. In a similar way, the problem of determining the n − 2 contact points can be decomposed into two types of subproblems layerwise: one is to search for a contact point and another is to determine several previously acquired contact points. Furthermore, the same search strategy is employed to determine different contact points, which means that the process of computing the n − 1 contact points hierarchically nests the processes of computing each previous contact point. Therefore, re-determining the previously acquired n − 1 contact points when searching for the nth contact point is a recursive process.

The recursive process of contact points includes both backtracking the previous contact point layerwise and returning to the position and orientation obtained by computing the previous contact point layerwise, as shown in Figure 6. When searching for the n^th contact point, the previous n − 1 contact points should be re-determined after the position and the orientation are adjusted, which means that backtracing a new n – 1th contact point to the next layer is required. However, searching for the new n – 1th contact point requires backtracing a new n – 2th contact point as well. Consequently, the previous contact point should be backtracked layerwise in a similar manner. The backtracking process ends if the three contact points on the main mating plane are determined. The acquired position and orientation Ω³ at the bottom layer of the backtracking process should be returned to the layer above such that the next approaching process of the position and orientation can be carried out. In a similar way, the position and orientation acquired in each layer are returned to the layer above to carry out the progressive-approaching-based determination of the contact point. Consequently, the position and orientation Ωⁿ⁻¹ determined after re-determining the previously acquired n−1 contact points are considered to be the result obtained by returning to the position and orientation obtained by computing the previous contact points layerwise. According to Ωⁿ⁻¹, the next approaching process of the position and orientation is performed to search for the nth contact point. Finally, all the n contact points and the position and orientation Ωⁿ will be acquired by carrying out several approaching processes (i.e., the previous n − 1 contact points should be backtracked and returned).

The previous contact points should be re-determined after each adjustment step when searching for the contact points on each auxiliary plane, which means that all the previous contact points can be backtracked according to a similar principle layerwise. Moreover, the determination of the three contact points on the main mating plane does not depend on the strategy of progressively approaching position and orientation. Thus, the backtracking process is bounded. The position and the orientation are adjusted based on accumulating the adjustment values, which means that the position and the orientation obtained by determining the previous contact points should be the initial position and orientation of each approaching process during the process of searching for a contact point. Consequently, the results of computing the previous contact point layerwise can be returned.

3. Process of Computing POD of Multi-Mating Planes

Using the principle of successively determining contact points, the computation of the PODs can be conducted according to the flow in Figure 7.

At the beginning of the flow, the view coordinate system and the part definition coordinate systems Θ^A and Θ^B of the assembled and reference parts are established, and the initial assembly position is acquired by setting Θ^A and Θ^B to coincide with each other according to the ideal circumstances. Furthermore, Θ^A should be adjusted according to the given positions corresponding to the unconstrained DOFs if the assembly is incompletely positioned. According to the principle of modeling from the perspective of manufacturing errors, the surface deviations (S_A, S_B) of the two parts are generated via the deviation function corresponding to each manufacturing error, as shown in Equations (1) and (2):

(1)$de{v}_i=f_i({\mathbf{K}}_i,q_1,q_2)$

(2)$F(\mathbf{C},q_1,q_2)={\displaystyle \sum _{i=1}^n{f}_i({\mathbf{K}}_i,q_1,q_2)}$

where dev_i refers to the deviation function corresponding to manufacturing error i; f_i refers to the form of dev_i; and ${\mathbf{K}}_i$ refers to the set of all coefficients corresponding to dev_i, ${\mathbf{K}}_i=\left\{k_{ij}\right|j=1,2,\cdots ,N_i\}$. Each coefficient is denoted as k_ij, and the number of the coefficients is N_i. q₁ and q₂ represent the coordinates of the points on the ideal plane. F refers to the total-deviation function, and n refers to the number of manufacturing errors. $\mathbf{C}$ represents the coefficient sets of F, $\mathbf{C}=\underset{i=1}{\overset{n}{\cup }}{\mathbf{K}}_i$. According to F, the coordinates of an arbitrary point P on the actual plane can be represented as $(q_1^P,q_2^P,de{v}^P)$. However, they should be transformed to the coordinates represented within the PDCS $(x^P,y^P,de{v}^P)$, wherein the geometry of a part is described. Consequently, the generated surface deviations S are point clouds, and $\mathbf{S}=\left\{P_{m,n}\right|m=1,2,\cdots ,N_m;n=1,2,\cdots ,N_n\}$, where P_m,n represents the coordinates within the PDCS of the point on the actual plane. N_m and N_n are the row and column numbers of P_m,n.

During the computation process, contact points CP1, CP2, and CP3 on the main mating plane are determined first because the position and orientation ${\mathrm{\Omega }}^3$ are the basis for searching for contact point CP4. Next, according to the principle of searching for contact points via progressively approaching position and orientation, contact point CP4 and the position and orientation Ω⁴ (i.e., Ω^update_3) are determined through several iterations in four steps. First, the minimum distance of the first auxiliary mating plane is computed. Second, it is checked whether the adjustment value meets the distance threshold. Third, the position and the orientation are adjusted. Fourth, the previously acquired three contact points are re-determined. The redetermination of the previously acquired three contact points is a recursive process of contact point determination, which means that the approach returns to the step of determining the three contact points on the main mating plane after the position and the orientation are adjusted each iteration. Similarly, contact point CP5 and the position and orientation Ω⁵ (i.e., Ω^update_4) are also determined through several iterations in four steps. First, the minimum angle of the first auxiliary assembly plane is computed. Second, it is checked whether the adjustment value meets the angle threshold. Third, the position and the orientation are rotated. Fourth, the previously acquired four contact points are re-determined. If the second auxiliary mating plane does not exist, the acquired position and orientation Ω⁵ are the PODs. Otherwise, the position and the orientation are adjusted further to determine contact point CP6 on the second auxiliary mating plane. The process consists of four steps. First, the minimum distance of the second auxiliary mating plane is computed. Second, it is checked whether the adjustment value meets the distance threshold. Third, the position and the orientation are adjusted. Fourth, the previously acquired five contact points are re-determined. The acquired position and orientation Ω⁶ (i.e., Ω^update_5) are the expected PODs.

4. Case Study and Discussion

In this section, an assembly with three mating planes is taken as the example of POD computation to verify the proposed approach. Furthermore, the PODs of a sliding workbench are computed such that both the effectiveness and usefulness of the proposed approach are further demonstrated, and the approach efficiency is also discussed.

4.1. Position and Orientation Deviation Computation of an Assembly with Three Mating Planes

The assembly with three mating planes in Figure 8 is a typical case study as presented by Benjamin [17] and Yan [30], which is composed of a cube PA and a block PB. The mating plane sizes of PA and PB are 80 × 80 (mm) and 100 × 100 (mm), respectively.

According to the six-point positioning principle, six DOFs of PA are constrained via six contact points. There are six possible cases in which the six contact points are distributed on the XOY, XOZ, and YOZ planes, which consequently results in six types of assembly states, as shown in Table 1.

Considering that the principles of determining the contact points of these assembly states are consistent, the process of the successive determination of contact points is illustrated by taking assembly state 1 as an example in this section.

The initial assembly position is set according to the ideal circumstances. Consequently, the defined coordinate systems of PA and PB are coincident, as shown in Figure 9. In addition, the definition coordinate system of PB should always coincide with the view coordinate system, such that adjusting the assembly position can be conveniently realized by adjusting PA.

To generate surface deviations of each mating plane through the modeling method from the perspective of manufacturing error, the mating plane morphologies of PA and PB are supposed to be machined through end milling and are mainly affected by the inclination error of the spindle axis, straightness error in the vertical plane of the guide, and wear error of the milling cutter. The established deviation functions corresponding to these manufacturing errors are shown in Table 2.

The surface deviations, described by point clouds, can be acquired after the coefficients of each deviation function are given. In this study, each mating plane is discretized with the size of 2 mm in the length and width directions. Thus, the points on the mating planes of PA and PB are 41 × 41 and 51 × 51, respectively.

The thresholds for judging whether the fourth and sixth contact points come into contact are given as 1 × 10⁻³ mm, and the threshold for the fifth contact points is given as 1 × 10⁻³°.

Over the process of determining contact points, the three contact points on the main mating plane are determined through the one-step-adjusting-based strategy. Next, the contact points on each auxiliary mating plane are determined through the strategies of progressively approaching the position and orientation and recursion of the contact points. The positions and the orientations approaching progressively over the whole process are shown in Table 3, and the variations of six contact points are shown in Figure 10.

Similarly, the six contact points and PODs corresponding to the other five assembly states were determined, as shown in Figure 11 and Table 4, respectively.

4.2. Position and Orientation Deviation Computation of Mechanical Product

The rectangular sliding guide of a sliding workbench is a typical assembly of multi-mating planes. It is assembled by the static guide and sliding table through the support and guide planes. Five DOFs are constrained after the support and guide planes are mated, which allows the table to slide straight along the direction of the unconstrained DOF. Affected by the surface deviations, the position and orientation between the static guide and sliding table vary when the table slides within the stroke. The variation in the sliding trajectory characterizes the guidance accuracy and is the key performance indicator in the design of a sliding workbench. Therefore, the proposed approach in this study can be employed for the POD variation curves within the sliding stroke, such that the performance of the sliding workbench can be controlled according to the predicted guiding accuracy.

4.2.1. Sliding Workbench and Computation Parameters

The sliding workbench in Figure 12 is taken as an example to verify both the effectiveness and usefulness of the proposed approach.

According to the six-point positioning principle, five DOFs of PPC are constrained via five contact points. There are two possible cases in which the five contact points are distributed on the support and guide planes, which consequently results in two types of assembly states.

Assembly state A: Three of the five contact points are distributed on the support planes of PPA and PPB, and the other two are distributed on the guide plane of PPA.

Assembly state B: Three of the five contact points are distributed on the support planes of PPA and PPB, and the other two are distributed on the guide plane of PPB.

The surface deviations are supposed to be modeled according to the deviation functions in Table 2, and are required to be limited within the ranges of ±0.015 mm and ±0.03 mm, respectively. Thus, the surface deviations of each mating plane can be generated after the coefficients of each deviation function are determined according to the required variation ranges. Each mating plane is also discretized with the size of 2 mm. Thus, the points on the support and guide planes of PPC-A1/A2/B1/B2 are 22 × 41 = 902 and 7 × 41 = 287, and the points on the support and guide planes of PPA and PPB are 21 × 501 = 10521 and 6 × 501 = 3006.

The thresholds employed to judge whether the fourth and fifth contact points come into contact are given as 1 × 10⁻³ mm and 1 × 10⁻³°.

4.2.2. Position and Orientation Deviations at the Beginning of the Sliding Stroke

The initial assembly position is determined when the PDCSs of the static guide and sliding table are set to coincide with each other because the assembly position in the direction of the X-axis is required to be the sliding stroke. Next, the surface deviations of each mating plane are input, such that determining contact points can begin.

Over the process of determining contact points, the three contact points on the support planes of PPA and PPB are determined through the one-step-adjusting-based strategy. Next, if computing PODs of assembly state A, the two contact points on the guide plane of PPA are determined through the strategies of progressively approaching the position and orientation and recursion of the contact points. Otherwise, the two contact points on the guide plane of PPB should be determined.

The five contact points and PODs corresponding to assembly state A and B were determined, as shown in Figure 13 and Table 5, respectively.

4.2.3. Guiding Accuracy Computation of the Sliding Workbench

The curves of the distance deviation along the X- and Z-axes and the angle deviation around the Y-axis specifically characterize the guiding accuracy of the sliding workbench. These curves can be acquired by computing PODs at different positions within the sliding stroke.

The acquired variation curves are shown in Figure 14, and they characterize the fluctuation of the position and orientation of PPC in the horizontal and vertical planes within the sliding stroke. These results can help to control the performance of the sliding workbench in the design stage. For instance, the upper and lower limits of the guidance accuracy can be determined based on these curves.

4.3. Discussion on the Approach Efficiency

The proposed approach is a type of search algorithm, and its effectiveness is significantly affected by the computational efficiency. Thus, the approach efficiency should be discussed. According to the case study, it is found that the approach efficiency is mainly affected by three factors: the principle of searching for contact points via progressively approaching position and orientation, the principle of computing the distances between the corresponding points on the mating planes, and the size of the point clouds.

• Effects of the principle of searching for contact points via progressively approaching position and orientation

The principle stipulates that each adjustment iteration continues from the previous assembly position rather than restarts from the initial position. Furthermore, the minimum distance or angle between two mating planes is taken as the value for each adjustment iteration, which can lead to adjustment value convergence in a few iterations. For instance, the position and the orientation are only moved twice, rotated once, and moved once when searching for contact points CP4, CP5, and CP6, as shown in Section 4.1. Therefore, each contact point can be determined efficiently.

Moreover, the efficiency is directly affected by the threshold as well. The smaller the threshold is, the more approaches are required to determine a contact point, which inevitably increases the computing cost. In contrast, a lower computing cost will be expended. However, the accuracy of computation will be affected if loose thresholds are given, because the actual contact state of the points are inaccurately described. Therefore, to acquire a reasonable threshold, the requirements of accuracy and efficiency should be balanced. Stringent thresholds should be given if high accuracy is required, even though the computing cost will be sacrificed to some extent. Otherwise, the threshold can be enlarged appropriately, such that the efficiency can be improved.

• Effects of the principle of computing the distances between the corresponding points on the mating planes.

In addition to the total number of iterations of adjusting position and orientation, the efficiency of each adjustment iteration also plays a vital role in approach efficiency. The distance computation between the corresponding points significantly affects the efficiency of each adjustment step.

To compute the distances between the corresponding points, the corresponding relations between the discrete points on the mating planes should be determined first because they are uncertain during the process of progressively approaching position and orientation. Thus, the method of projecting [26] is employed in this study, and its principle is displayed in Figure 15. The target patch of the current point should be determined. Next, a new point should be inserted on the target patch, which means that the point corresponding to the current point is determined.

The main factor affecting the efficiency of computing the distances between the corresponding points is the strategy used to determine the target patch of the current point. The simplest strategy is the traversal strategy, in which all patches on the target plane should be compared with the current point such that the target patch can be determined. The traversal strategy is feasible, but has low efficiency because the patches far from the current point are also traversed, even though they cannot be the target patch. To avoid unnecessary traversal computations, the adjacent-area strategy is employed in this study, and its principle is displayed in Figure 16. The mating area should be determined first such that the discrete points that cannot be in contact will never be traversed. Next, two steps are performed twice. Namely, the mating area is divided into four sub-areas, and the sub-area that corresponds to the current point is determined. As a result, the sub-area with only 1/16 of the mating area is determined and is the adjacent area of the current point. Finally, the target patch can be acquired by performing horizontal and vertical bisection iterations of the adjacent area. The adjacent-area strategy is more complex than the traversal strategy, which questions its efficiency if the density of the point clouds is not high. However, the efficiency advantage of the adjacent-area strategy will prevail with the increasing density of the point clouds.

• Effect of the mesh size of point clouds.

Point clouds are employed to describe the mating planes. Thus, the surface deviations of each mating plane are collections of discrete points. The mesh size refers to the distance between two discrete points in the length or width direction of a mating plane, and is a key parameter of point clouds. If small mesh size is given, the morphologies of the mating plane will be fully described, but the time required to compute the distances between the corresponding points will increase. As a result, the efficiency of determining contact points is low. On the contrary, if mesh size increases, some geometric fluctuations of the mating plane will be ignored. However, the efficiency of determining contact points will increase because the time required to compute the distances between the corresponding points is directly reduced.

To verify the impacts of the factors mentioned above on the approach efficiency, the PODs were computed under different sizes of point clouds. The computation time was recorded in the case of assembly state 1 (Section 4.1) as an example. The software environment of these simulations was MATLAB^® R2015b, and the hardware environment was an ordinary desktop computer (Windows 7 OS, 3.50 GHz Intel Core i5-4690 CPU, and 16 GB RAM). The results are listed in Table 6.

The computation time is only 14.1s when the size of the point clouds is 2.5 mm. Thus, the efficiency of the proposed approach is verified. Moreover, the computation time increases with decreasing point cloud size; for example, it increases by about 40 times when the point cloud size is reduced to 0.5 mm. Consequently, the appropriate size of the point clouds should be determined by weighing the accuracy and the efficiency when the proposed approach is employed. The size of the point clouds should be as large as possible to achieve the highest efficiency when the accuracy is acceptable. Otherwise, a small size of the point clouds should be given if the required accuracy is high, even though the computing cost are inevitably affected.

5. Conclusions

Aiming at the problem of computing position and orientation between two parts assembled by multiple planes in the design stage considering not only surface deviations but also multiple types of positioning, an approach based on successively determining contact points is proposed in this paper.

Benefitting from the six-point positioning principle as the theoretical basis for the concept of assembly with multi-mating planes, both the effects of surface deviations and multiple types of positioning can be considered in the proposed approach by unifying them as contact point variations. Moreover, the strategy of progressively approaching position and orientation as well as the strategy of the recursion of the contact points are proposed such that the position and orientation after multiple planes mated can be acquired by determining the contact points one by one.

This study proposes an effective approach for accurately investigating the influence of PODs on the performance of mechanical products at the design stage because both the effects of surface deviations and multiple types of positioning are considered. The limitation of the proposed approach is that the influence of the assembly force is not considered. However, PODs are directly affected by the local deformation of the mating surface caused by the assembly force. In the ongoing study, the proposed approach is extended to account for this factor.

Author Contributions

J.Z., L.Q., Z.H. and N.A. wrote the paper and supervised the work; L.Q., Z.H. and N.A. gave some suggestions on review and editing. All authors have read and agreed to the published version of the manuscript.

Institutional Review Board Statement

Not applicable.

Informed Consent Statement

Not applicable.

Data Availability Statement

Not applicable.

Conflicts of Interest

The authors declare no conflict of interest.

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Figures and Tables

Figure 1. Assembly cases of single and multiple planes.

Figure 2. Position and orientation deviations generated after multiple planes of two parts are assembled. DOF, degree of freedom. (a) Ideal circumstances, (b) Actual circumstances, (c) Positions of unconstrained DOFs under varying circumstances, (d) Number of the contact points on each mating plane under varying circumstances.

Figure 3. Framework of computing position and orientation deviations by successively determining contact points. DOF, degree of freedom.

Figure 4. Possible cases of the main and auxiliary mating planes of the assembly with two mating planes. DOF, degree of freedom. (a) Case 1 of the main and auxiliary ones among all the mating planes, (b) Case 2 of the main and auxiliary ones among all the mating planes.

Figure 5. Principle of searching for a contact point via progressively approaching position and orientation.

Figure 6. Principle of the recursion of contact points.

Figure 7. Flowchart of the approach based on successively determining contact points. VCS, view coordinate system. PDCS, part definition coordinate system.

Figure 8. Assembly with three mating planes.

Figure 9. Established part definition coordinate systems (PDCSs) and view coordinate system.

Figure 10. Variations of six contact points. The three contact points on the main mating plane are red, the two contacts point on the first auxiliary mating plane are green, and the contact point on the second auxiliary mating plane is yellow.

Figure 11. Determined contact points of the six assembly states. The three contact points on the main mating plane are red, the two contacts point on the first auxiliary mating plane are green, and the contact point on the second auxiliary mating plane is yellow.

Figure 12. Sliding workbench. (a) A real sliding workbench, (b) The sliding workbench model of this case study.

Figure 13. Determined contact points of assembly state A and B.

Figure 14. Three types of variation curves. (a) Variation curves of the distance deviation along the X-axis, (b) Variation curves of the distance deviation along the Z-axis, (c) Variation curves of the angle deviation around the Y-axis.

Figure 15. Principle of computing the distances between the corresponding points by projecting [26].

Figure 16. Principle of computing the distances between the corresponding points by determining the adjacent area.

Table 1

Assembly states of the assembly with three mating planes.

Assembly State	XOY Plane	XOZ Plane	YOZ Plane
1	3 contact points	2 contact points	1 contact point
2	3 contact points	1 contact point	2 contact points
3	2 contact points	1 contact point	3 contact points
4	2 contact points	3 contact points	1 contact point
5	1 contact point	2 contact points	3 contact points
6	1 contact point	3 contact points	2 contact points

Table 2

Deviation functions corresponding to the three main manufacturing errors of end milling.

Manufacturing Errors	Deviation Functions
Inclination error of the spindle axis	$de{v}_1(q_1,q_2)=R\times (1-cos(q_2))\times tan(k_{11})$
Straightness error in the vertical plane of the guide	$de{v}_2(q_1,q_2)=k_{21}\times sin(k_{22}\times q_1)+k_{23}$
Wear error of the milling cutter	$de{v}_3(q_1,q_2)=k_{31}\times q_1+k_{32}$

Table 3

Positions and orientations approaching progressively over the whole process.

Different Positions and Orientations	Position and Orientation Values	Number of approaches
Initial assembly position	${\mathrm{\Omega }}^0=(0,0,0,0,0,0)$	0
Position and orientation after three contact points are determined	${\mathrm{\Omega }}^3=(0,0,0.013,0.007,165.964,162.935)$	Once, adjustment corresponding to the Z-, X-, and Y- axes, respectively
Position and orientation after four contact points are determined	${\mathrm{\Omega }}^4=(0,0.011,0.014,0.614,90.572,87.271)$	Twice, approach of moving
Position and orientation after five contact points are determined	${\mathrm{\Omega }}^5=(0,0.008,0.014,0.481,90.737,88.741)$	Once, approach of rotating
Position and orientation after six contact points are determined	${\mathrm{\Omega }}^5=(5.37\times {10}^{-7},0.009,0.011,0.521,90.456,90.458)$	Once, approach of moving

Table 4

Acquired position and orientation deviations (PODs) of the assembly with three mating planes.

Assembly States	PODs
	Deviaitons along X-Axis(mm)	Deviaitons along Y-Axis(mm)	Deviaitons along Z-Axis(mm)	NutationAngle(°)	PrecessionAngle(°)	SpinAngle(°)
1	$5.39\times {10}^{-7}$	0.009	0.011	0.521	90.456	90.458
2	0.011	$-1.98\times {10}^{-7}$	0.014	0.621	169.859	179.864
3	0.007	$5.28\times {10}^{-7}$	0.012	0.383	178.996	179.994
4	0.009	0.012	0.008	0.865	131.475	130.474
5	0.018	0.008	0.012	1.112	136.098	156.098
6	0.013	0.011	0.008	0.964	119.716	139.716

Table 5

Acquired position and orientation deviations (PODs) of the rectangular sliding guide.

Assembly States	PODs
	Deviaitons along X-Axis(mm)	Deviaitons along Z-Axis(mm)	NutationAngle(°)	PrecessionAngle(°)	SpinAngle(°)
A	0.014	0.015	0.811	0.015	1.987
B	0.009	0.014	0.525	0.791	1.631

Table 6

Computation time of the PODs under different sizes of point clouds.

Size ofPoint Clouds (mm)	2.5	2	1	0.5
Time (s)	14.1	22.5	117.7	553.8

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