A Multi-Constraint Assembly Registration Method Based on Actual Machined Surfaces

Abstract

Modern manufacturing places increasing demands on assembly accuracy, revealing the limitations of conventional tolerance-based methods and studies that oversimplify multi-surface constraints into single-surface problems. To address this challenge, it is crucial to account for geometric distribution errors on multiple surfaces and constraints from multiple mating surfaces, analyzing their coupling effects in assembly. This paper presents a model that incorporates the effects of machining-induced geometric distribution errors and the constraints arising from multiple mating surfaces. The model determines contact points between two pairs of mating surfaces and calculates the spatial pose of the assembled part to predict assembly accuracy. The model validation was conducted in two stages: initial verification of fundamental principles through a two-dimensional simulation, followed by experimental validation. The experimental study involved mating surfaces with distinct geometric distribution errors manufactured by different machine tools. Assembly tests were performed under two distinct orientations of applied external forces. Results show close agreement between predicted and measured values, with a root mean square error (RMSE) below 2%, confirming the method’s effectiveness. The proposed method offers a solution to the assembly registration problem involving coupled multi-constraints and geometric distribution errors.

1. Introduction

As the cornerstone of manufacturing, machine tools are crucial for processing various components. The assembly accuracy of a machine tool itself, as well as that of other key rigid components such as guideways and shafts, directly determines the precision of the final product. In assembly processes, components are often constrained by two or more mating surfaces simultaneously. Moreover, with ever-increasing demands for higher assembly precision, the impact of geometric distribution errors has become critical, especially for products characterized by these complex multi-surface constraints. Geometric distribution error refers to the deviation of a real surface from its ideal geometric form, where this deviation exhibits an inconsistent spatial distribution [1,2,3]. Figure 1 illustrates the spatial distribution of these form errors and their mechanism affecting assembly accuracy. As shown in Figure 1a, geometric distribution errors can induce angular deflection in an assembled component. Furthermore, the applied external force influences the final assembly pose (Figure 1b). This study focuses on stable assembly conditions where the force magnitude is sufficient to maintain contact stability. Moreover, the coupling effects arising from multiple mating constraints further complicate the prediction of assembly accuracy (Figure 1c).

This multi-constraint problem with geometric distribution errors is fundamentally a spatial surface-registration problem. Due to these errors, mating surfaces form discrete point contacts rather than ideal full-surface contact. Additionally, constraints between paired surfaces exhibit coupling effects: a change in one pair’s contact state alters the other pair’s state. External forces may further modify part positioning. In this study, we define this process as multi-constraint coupled surface registration, where parts progressively conform under geometric distribution errors and the applied forces until a stable contact-point set is established. Its primary objective is to address the spatial surface-registration problem by integrating the influence of geometric distribution errors, multi-constraint coupled effect, and applied external forces, thereby providing a basis toward predictive assembly pose determination.

The assembly of constrained structures with multiple mating surfaces has been widely studied. The foundational work for assembly sequence analysis was established by Mantripragada et al. [4]. Subsequent studies have traced assembly error sources and proposed error propagation models for multi-station assembly systems [5], while others have investigated the influence of locating features and datum sequences [6,7]. Further research has explored criteria for assembly positioning priority [8], developed clearance variation algorithms based on prioritizing assembly feature constraints [9], and analyzed the coupled effects of multiple error sources in complex assemblies [10]. However, most of the studies are tolerance-based, where tolerances merely describe the permissible variation range of geometric errors. thereby neglecting the spatial distributions of geometric distribution errors resulting from machining. Meanwhile, the true registration state between mating surfaces remains obscured. Furthermore, some computational methods, such as the deviation domain method [11], Jacobian-Torsor model [12,13], and point-based techniques [14,15], and iterative closest point method [16,17], also neglect the critical impact of process-induced geometric distribution errors and the contact states at the mating surfaces.

To address these gaps, recent efforts such as the method proposed by Shen et al. [18] have been made to incorporate assembly process parameters and to adopt distinct methods for different mating surface types. However, their method suffers from some limitations: its reliance on Gaussian-distributed random data, rather than on measured data from actual machined surfaces, and it does not determine contact points. These points are critical because geometric distribution errors on part surfaces cause the actual contact area to be much smaller than the nominal area, resulting in only limited points or localized contact. Ultimately, it is these discrete contact points (not the nominal surfaces) that determine the final assembly pose. Meanwhile, explorations into geometric distribution errors have been conducted, such as their effects on shaft-hole and cylindrical part assemblies using Newton-Raphson techniques [19], and systematic analyses of multifactorial impacts (including geometric distribution errors and waviness) on assembly precision [20,21,22]. Computer-aided methods have also been employed to demonstrate how machining variations affect product functionality [23], and deep learning algorithms have shown high accuracy in predicting coaxiality in aero-engine assembly by integrating geometric distribution error models [24]. Despite these valuable contributions, a key limitation persists: the predominant focus remains on single mating surfaces as assembly datums. Moreover, approaches such as differential surface methods [25], constrained registration techniques [26], and the quaternion-based method [27] overlook the critical influence of applied external forces. Furthermore, the method proposed by Samper et al. [28] may yield non-unique solutions.

To address these challenges, this study proposes a multi-constraint coupled registration method that is driven by actual measured data (geometric distribution errors) and accounts for critical assembly parameters (applied external forces). The proposed framework utilizes the NURBS method to characterize geometric distribution errors and integrates a hierarchical strategy to construct a multi-constraint coupled model. Subsequently, a penalty-function-based optimization is solved via a hybrid particle swarm algorithm. Finally, the method is validated on a two-dimensional (2D) model to verify its capability in determining both the contact points and the assembled pose, and is then extended to three-dimensional (3D) assemblies with experimental confirmation of its accuracy.

2. Multi-Constraint Coupled Registration Method

This study assumes all components as rigid bodies, neglecting deformation effects, due to the high stiffness of the parts in the target systems (e.g., machine tools). Under this assumption [29], this paper proposes a hierarchical Multi-Constraint Coupled model to resolve the multi-constraint coupling problem, providing contact points and assembly poses that better reflect actual assembly conditions. The following subsections present the details of this model.

2.1. Modeling of Coupled Multi-Constraints

To accurately characterize geometric distribution errors on part surfaces, the surface is scanned using a coordinate measuring machine (CMM) to acquire point cloud data, which is then reconstructed using non-uniform rational B-splines (NURBS) [30]. Machining errors on part surfaces consist of systematic and random components. When the machining process is consistent, the systematic error is deterministic and exhibits repeatable patterns, whereas it is this component that dominates the overall error distribution [31]. In contrast, the random error, which lacks discernible regularity, typically accounts for only a minor proportion. Accordingly, this study focuses on extracting the systematic error as the primary contributor to the assembly error model.

$$\left\{\begin{array}{c}{S}_a(u,v)=S_d(u,v)+e_r(u,v)\\ S_d(u,v)={\displaystyle \sum _{i=0}^m{\displaystyle \sum _{j=0}^n{{\displaystyle N}}_{i,3}(u){{\displaystyle N}}_{j,3}(v){{\displaystyle P}}_{i,j}}}\end{array}\right.$$

(1)

where $S_a(u,v)$ is the actual machined surface, $u$ and $v$ are the two direction parameters of the rectangular domain; $S_d(u,v)$ is the deterministic component of the surface that can be obtained by regression, and expressed with NURBS surface; ${{\displaystyle P}}_{i,j}$ are the control points, ${{\displaystyle N}}_{i,3}$, ${{\displaystyle N}}_{j,3}$ are the basic functions of cubic B-spline; $e_r(u,v)$ are the random errors.

Next, we construct the multi-constraint coupled model (see flowchart in Figure 2) through the following steps. For clarity in methodology presentation, planar contact is employed due to its prevalence in machine tool structures.

Step 1: Data Measurement and Processing

This case involves plane-to-plane mating surfaces (Figure 3), comprising two pairs of mating surfaces. The point clouds of mating surfaces S_A1/S_A2 (horizontal) and S_B1/S_B2 (vertical) were acquired via CMM, with geometric distribution errors characterized using NURBS method.

Step 2: Determination of Surface Priority Based on Assembly Parameters

Under applied external forces, mating surface S_A1 of Part₁ constrains three degrees of freedom (DOFs) of Part₂, while vertical surfaces S_B1/S_B2 exhibit line contacts restricting two DOFs. S_A1/S_A2 are designated as primary mating surfaces due to their dominant DOF constraints, with S_B1/S_B2 as secondary surfaces.

Step 3: Coordinate System Mapping

The assembly coordinate system X₁Y₁Z₁ was established per operational requirements. Point clouds S_A1/S_B1 (Part₁) and S_A2/S_B2 (Part₂) were transformed from measurement to assembly coordinates via mapping matrices M₁/M₂, yielding corresponding point sets A1_i ($x_{1i}^{\prime }$, $y_{1i}^{\prime }$, $z_{1i}^{\prime }$), B1_j ($x_{1j}^{\prime }$, $y_{1j}^{\prime }$, $z_{1j}^{\prime }$), A2_i ($x_{2i}^{\prime }$, $y_{2i}^{\prime }$, $z_{2i}^{\prime }$), and B2_j ($x_{2j}^{\prime }$, $y_{2j}^{\prime }$, $z_{2j}^{\prime }$) (i, j∈ Z⁺), as specified in Equation (2):

A1_i = M₁·S_A1, B1_j = M₁·S_B1; A2_i = M₂·S_A2, B2_j = M₂·S_B2;

(2)

where M₁/M₂ denote transformation matrices for Part₁/Part₂, respectively.

$$M_r=\left[\begin{array}{cccc}\cos {\theta }_{yr}\cos {\theta }_{zr}& -\cos {\theta }_{yr}\sin {\theta }_{zr}& \sin {\theta }_{yr}& {{\displaystyle k}}_{xr}\\ \cos {\theta }_{yr}\sin {\theta }_{zr}+\sin {\theta }_{xr}\sin {\theta }_{yr}\cos {\theta }_{zr}& \cos {\theta }_{xr}\cos {\theta }_{zr}-\sin {\theta }_{xr}\sin {\theta }_{yr}\sin {\theta }_{zr}& -\sin {\theta }_{xr}\cos {\theta }_{yr}& {{\displaystyle k}}_{yr}\\ \sin {\theta }_{yr}\sin {\theta }_{zr}-\cos {\theta }_{xr}\sin {\theta }_{yr}\cos {\theta }_{zr}& \cos {\theta }_{xr}\cos {\theta }_{zr}+\cos {\theta }_{xr}\sin {\theta }_{yr}\sin {\theta }_{zr}& \cos {\theta }_{xr}\cos {\theta }_{yr}& {{\displaystyle k}}_{zr}\\ 0& 0& 0& 1\end{array}\right]$$

(3)

where $({\theta }_{xr},{\theta }_{yr},{\theta }_{zr})$ and $(k_{xr},k_{yr},k_{zr})$ denote the rotational and translational parameters, respectively, of the measurement frame relative to the assembly frame. r = 1, 2.

Step 4: Modeling of Coupled Multi-Constraints

This study develops the hierarchical surface registration model incorporating geometric distribution errors, consisting of three components: (i) the primary mating surface model, (ii) the secondary mating surface model, and (iii) the constraint relationship model.

• Primary Mating Surface Model

The primary surfaces (A₁ on Part₁ and A₂ on Part₂) must satisfy: (1) the minimum distance between A₁ and A₂ (Equation (4)), (2) a non-interference condition in assembly, and (3) the existence of at least three contact points under applied external forces. As illustrated in Figure 4, potential contact points $C_h^1$, $C_h^2$, and $C_h^3$, along with the force application point/centroid P_c, are projected onto the X-Y plane (denoted as $C_1^{*}$, $C_2^{*}$, $C_3^{*}$, and $P_c^{*}$). The angular relationships between these projected points must satisfy Equations (4) and (5), where r₁ represents the number of corresponding point pairs ($A{1}_i$,$A{2}_i$), n₁ is the assembly direction vector, $n_{A{2}_i}$ denotes the vertex normal vector of $A{2}_i$, and $T_r/T_t$ are the rotation/translation matrices, respectively.

$$\left\{\begin{array}{c}\min f={{\displaystyle \sum _{i=1}^N\left(\left[\left(T_r\cdot A{1}_i+T_t\right)-A{2}_i\right]\cdot n_{\mathsf{1}}\right)}}^2\\ s.t.\left[\left(T_r\cdot A{1}_i+T_t\right)-A{2}_i\right]\cdot n_{A{2}_i}\ge 0, i=1,2,\dots r_1\\ {\theta }_{12}+{\theta }_{23}+{\theta }_{13}={360}^{\circ }\end{array}\right.$$

(4)

$${\theta }_{12}=\arccos {\displaystyle \frac{\overrightarrow{P_c^{*}{C}_1^{*}\cdot }\overrightarrow{P_c^{*}{C}_2^{*}}}{\left|\overrightarrow{P_c^{*}{C}_1^{*}}\right|\cdot \overrightarrow{|P_c^{*}{C}_2^{*}}|}}\hspace{1em}\hspace{1em}{\theta }_{23}=\arccos {\displaystyle \frac{\overrightarrow{P_c^{*}{C}_2^{*}\cdot }\overrightarrow{P_c^{*}{C}_3^{*}}}{\left|\overrightarrow{P_c^{*}{C}_2^{*}}\right|\cdot \overrightarrow{|P_c^{*}{C}_3^{*}}|}}\hspace{1em}\hspace{1em}{\theta }_{13}=\arccos {\displaystyle \frac{\overrightarrow{P_c^{*}{C}_1^{*}\cdot }\overrightarrow{P_c^{*}{C}_3^{*}}}{\left|\overrightarrow{P_c^{*}{C}_1^{*}}\right|\cdot \overrightarrow{|P_c^{*}{C}_3^{*}}|}}$$

(5)

2.

Secondary Mating Surface Model

Each secondary surface pair (B1_j on Part₁ and B2_j on Part₂) must satisfy two conditions: (1) the existence of at least two distinct contact points (e.g., $C_v^1$ and $C_v^2$) and (2) a non-interference condition in assembly, as defined by Equation (6). In this formulation, r₂ represents the number of corresponding point pairs, n₂ is the assembly direction vector, and $n_{B{2}_j}$ denotes the vertex normal vector of $B{2}_j$.

$$\left\{\begin{array}{c}\min f={{\displaystyle \sum _{i=1}^N\left(\left[\left(T_r\cdot B{1}_j+T_t\right)-B{2}_j\right]\cdot n_2\right)}}^2\\ s.t.\left[\left(T_r\cdot B{1}_j+T_t\right)-B{2}_j\right]\cdot n_{B{2}_j}\ge 0, j=1,2,\dots r_2\end{array}\right.$$

(6)

3.

The Constraint Relationship Model

The connection parameter $T_{\mathrm{contact}}=T_r\cdot T_t=\left[\begin{array}{cccc}1& -\delta z& \delta y& dx\\ \delta z& 1& -\delta x& dy\\ -\delta y& \delta x& 1& d\mathrm{z}\\ 0& 0& 0& 1\end{array}\right]$ between surface pairs is jointly constrained by the primary (Feature Vector Set 1: T₁) and secondary (Feature Vector Set 2: T₂) surfaces, following Equation (7) [32].

$$T_{\mathrm{contact}}=T_1{{\displaystyle \prod T}}_2=\left\{\begin{array}{c}\delta x=\delta x_1(\cap or\cup )\delta x_2\\ \delta y=\delta y_1(\cap or\cup )\delta y_2\\ \delta z=\delta z_1(\cap or\cup )\delta z_2\\ dx=d{x}_1(\cap or\cup )d{x}_2\\ dy=d{y}_1(\cap or\cup )d{y}_2\\ d\mathrm{z}=d{z}_1(\cap or\cup )d{z}_2\end{array}\right\}$$

(7)

The parameters {$\delta x$, $\delta y$, $\delta z$} and {$dx$, $dy$, $\mathrm{dz}$} represent rotational and translational vectors along the x-, y-, z- axes, while {$\delta x_1$, $\delta y_1$, $\delta z_1$}/{$d{x}_1$, $d{y}_1$, ${\mathrm{dz}}_1$} and {$\delta x_2$, $\delta y_2$, $\delta z_2$}/{$d{x}_2$, $d{y}_2$, ${\mathrm{dz}}_2$} denote vectors for Features 1 and 2, respectively. The operator $\prod$ indicates ${\displaystyle \cap }$ or ${\displaystyle \cup }$ based on Boolean operations between overlapping vectors.

Step 5: Model Solution

The iterative optimization began with initialization parameters (iteration count k = 0, initial error transformation matrices $T_{r0}=0$ and $T_{t0}=I$, where I denotes the identity matrix). To address the multi-constraint assembly problem, a penalty function was formulated to convert the constrained optimization (Step 4) into an unconstrained form (Equation (8)), thereby enhancing computational stability and convergence. The surface registration model with coupled constraints was subsequently optimized using a hybrid particle swarm algorithm until the optimal solution $T_{\mathrm{contact}}$ was achieved. The workflow is illustrated in Figure 2.

$$\left\{\begin{array}{l}f(T_r,T_t)=f(dx,dy,dz,\delta x,\delta y,\delta z)={\displaystyle \sum _{v=1}^q\left[{\left(W_F\cdot d_v{(u)}^{non\mathrm{int}er}\right)}^2+{\left(W_{\mathrm{int}}\cdot d_v{(u)}^{\mathrm{int}er}\right)}^2\right]}\\ W_F=\left\{\begin{array}{ll}1& d_v\ge 0\\ 0& d_v<0\end{array}\right.\hspace{1em}{W}_{\mathrm{int}}=\left\{\begin{array}{ll}0& d_v\ge 0\\ 50& d_v<0\end{array}\right.\end{array}\right.$$

(8)

where $d_v(u)$ represents the Euclidean distance between corresponding point pairs during iteration. $d_v{(u)}^{noninter}$ indicates the non-interference distance (when $d_v\ge 0$, surfaces A₁/B₁ are interference-free with weight factor $W_F=1$; $d_v<0$ indicates interference with $W_F=0$). $d_v{(u)}^{inter}$ measures interference penetration depth ($d_v\ge 0$: no interference, $W_{int}=0$; $d_v<0$: interference occurs, $W_{int}=50$).

2.2. Determination of Primary and Secondary Contact Points

Within the assembly coordinate system, the optimal solution ${T^{-1}}_{\mathrm{contact}}$ is used to calculate the initial contact point positions on the part to be assembled (Part₂) prior to registration, based on its post-registration contact point locations. Under stable assembly conditions, the proposed model provides the closest approximation to the physical solution.

$$A_2^m={T^{-1}}_{\mathrm{contact}}\cdot C_h$$

(9)

$$B_2^{\mathrm{m}}={T^{-1}}_{\mathrm{contact}}\cdot C_v$$

(10)

2.3. Assembly Pose Calculation Method with Multi-Constraint Coupled Registration

2.3.1. Two-Dimensional Assembly Case

As detailed in Section 2.2, under vertical applied external forces, the horizontal surfaces act as the primary mating surfaces in this assembly case, while the vertical surfaces serve as the secondary ones. As shown in Figure 5, the assembly coordinate system (ACS) for the 2D mating surfaces is defined with its horizontal axis x_a aligned with a pair of nominal horizontal contact lines. The vertical axis y_a originates from the midpoint of the horizontal mating surface. Within the ACS, the contact points are designated as: $C_h^1(x_1,y_1)$ and $C_h^2(x_2,y_2)$ for the primary mating surfaces’ contact, and $C_v(x_3,y_3)$ for secondary mating surfaces’ contact.

• Constraint Analysis of Primary Mating Surfaces

The equation of the realized contact line is given by:

$$y={\displaystyle \frac{y_1-y_2}{x_1-x_2}}x+[y_2-{\displaystyle \frac{x_2(y_1-y_2)}{x_1-x_2}}]$$

(11)

The angular deviation (γ_h) between this contact line and the nominal horizontal contact line (x_a-axis) is calculated as:

$${\gamma }_h=\arctan ({\displaystyle \frac{y_1-y_2}{x_1-x_2}})$$

(12)

with a minor translational vector along the y_a-axis expressed as:

$$d{y}^{2D}=y_2-{\displaystyle \frac{x_2(y_1-y_2)}{x_1-x_2}}$$

(13)

2.

Constraint Analysis of Secondary Mating Surfaces

The contact point on the vertical mating surface undergoes a slight translation along the xₐ-axis. In Figure 5, the nominal contact point is defined at coordinates $(a^{2D},b^{2D})$. However, its realized coordinates, denoted as (V_x, V_y), deviate from this nominal position due to constraints imposed on the horizontal surfaces. Here, the x-coordinate $a^{2D}$ = −25, indicating that the point lies on the nominal vertical contact line x = −25, while the y-coordinate b^2D is an unknown variable. Solving Equation (14) yields the result given in Equation (15).

$$V={(V_x,V_y,1)}^T={(V_x,y_3,1)}^T=\left[\begin{array}{c}\begin{array}{cc}\cos ({\gamma }_h)& \begin{array}{cc}-\sin ({\gamma }_h)& 0\end{array}\end{array}\\ \begin{array}{cc}\sin ({\gamma }_h)\hfill & \begin{array}{cc}\cos ({\gamma }_h)& d{y}^{2D}\end{array}\end{array}\\ \begin{array}{ccc}0& 0& 1\end{array}\end{array}\right]\times {(a^{2D},b^{2D},1)}^T$$

(14)

$$d{x}^{2D}=V_x-x_3$$

(15)

3

Relative Pose Representation

Consequently, the relative pose variation of the mating surfaces A2 and B2 on the part₂, relative to the nominal assembly, is described using a Small Displacement Torsor (SDT):

$$T^{2D}=\left[\begin{array}{c}\begin{array}{cc}d{x}^{2D}& 0\end{array}\\ \begin{array}{cc}d{y}^{2D}& 0\end{array}\\ \begin{array}{cc}0\hfill & {\gamma }_h\end{array}\end{array}\right]$$

(16)

2.3.2. Three-Dimensional Assembly Case

As detailed in Section 2.2, under vertical applied external forces, the horizontal surfaces act as the primary mating surfaces, while the vertical surfaces serve as the secondary ones. The assembly coordinate system (ACS) for the 3D mating surfaces is illustrated in Figure 6. The plane x_ao_ay_a passes through the highest point of Mating Surface 1, defining the nominal mating plane. The origin o_a is located at the geometric center of this nominal plane. The nominal position of the vertical mating surface is at x = −a^3D (where a^3D > 0), parallel to the y-z plane.

The contact points are defined as $C_h^1(x_1,y_1,z_1)$, $C_h^2(x_2,y_2,z_2)$, and $C_h^3(x_3,y_3,z_3)$ on the primary mating surfaces, and $C_v^1(x_4,y_4,z_4)$, $C_v^2(x_5,y_5,z_5)$ on the secondary mating surfaces.

1.

Constraint Analysis of Primary Mating Surfaces

The equation of the realized horizontal mating surface is given by:

$$z=Ax+By+C$$

(17)

$$(x_i,y_i,1)\left(\begin{array}{c}A\\ B\\ C\end{array}\right)=z_i$$

(18)

The surface normal vector is expressed as:

$${\overrightarrow{n}}_h=[-A,-B,1]$$

(19)

The angular deviation vectors of the realized mating surface relative to ACS (about x/y axes) are:

$${\alpha }^{3D}=\arccos {\displaystyle \frac{{\overrightarrow{n}}_h\cdot {\overrightarrow{e}}_x}{‖{\overrightarrow{n}}_h‖‖{\overrightarrow{e}}_x‖}}$$

(20)

$${\beta }^{3D}=\arccos {\displaystyle \frac{{\overrightarrow{n}}_h\cdot {\overrightarrow{e}}_y}{‖{\overrightarrow{n}}_h‖‖{\overrightarrow{e}}_y‖}}$$

(21)

The minor translational displacement along the z_a-axis is:

$$d{z}^{3D}=C$$

(22)

2

Constraint Analysis of Secondary Mating Surfaces

The spatial contact line equation for the Vertical Mating Surface is:

$${\displaystyle \frac{x-x_4}{x_5-x_4}}={\displaystyle \frac{y-y_4}{y_5-y_4}}={\displaystyle \frac{z-z_4}{z_5-z_4}}$$

(23)

The rotation angle about the z-axis (${\mathsf{\gamma }}^{3D}$) and translational displacement in the x-direction (dx^3D) are as follows:

The contact point on the vertical mating surface undergoes a slight translation along the xₐ-axis. Consistent with the principle established in the prior 2D assembly case analysis, the nominal contact point on this surface is defined by coordinates $(a^{3D},b^{3D},c^{3D})$. Here, a^3D is a known quantity along the x-axis, while b^3D and c^3D are unknown. The realized coordinates of this point deviate from their nominal values due to the constraints applied on the horizontal surfaces. Solving Equation (25) yields the solution presented in Equation (26).

$${\gamma }^{3D}=\arctan {\displaystyle \frac{x_5-x_4}{y_5-y_4}}$$

(24)

$$\begin{array}{l}{V}_{3D}={(V_x^{3D},V_y^{3D},V_z^{3D},1)}^T={(V_x^{3D},y_4,z_4,1)}^T\\ =\left[\begin{array}{cccc}\cos {\beta }^{3D}\cos {\gamma }^{3D}& -\cos {\beta }^{3D}\sin {\gamma }^{3D}& \sin {\beta }^{3D}& 0\\ \cos {\alpha }^{3D}\sin {\gamma }^{3D}+\sin {\alpha }^{3D}\sin {\beta }^{3D}\cos {\gamma }^{3D}& \cos {\alpha }^{3D}\cos {\gamma }^{3D}-\sin {\alpha }^{3D}\sin {\beta }^{3D}\sin {\gamma }^{3D}& -\sin {\alpha }^{3D}\cos {\beta }^{3D}& 0\\ \sin {\alpha }^{3D}\sin {\gamma }^{3D}-\cos {\alpha }^{3D}\sin {\beta }^{3D}\cos {\gamma }^{3D}& \sin {\alpha }^{3D}\cos {\gamma }^{3D}+\cos {\alpha }^{3D}\sin {\beta }^{3D}\sin {\gamma }^{3D}& \cos {\alpha }^{3D}\cos {\beta }^{3D}& d{z}^{3D}\\ 0& 0& 0& 1\end{array}\right]\times {(a^{3D},{\mathrm{b}}^{3D},c^{3D},1)}^T\end{array}$$

(25)

$$d{x}^{3D}=V_x^{3D}-x_4$$

(26)

3.

Relative Pose Representation

The resulting SDT matrix is:

$$T^{3D}=\left[\begin{array}{c}\begin{array}{cc}d{x}^{3D}& {\alpha }^{3D}\end{array}\\ \begin{array}{cc}0& {\beta }^{3D}\end{array}\\ \begin{array}{cc}d{z}^{3D}& {\mathsf{\gamma }}^{3D}\end{array}\end{array}\right]$$

(27)

3. Analysis of Registration States in 2D Assembly

The proposed multi-constraint coupled registration method is demonstrated through specific case studies. This section details the registration states and results from 2D assemblies under two distinct loading conditions: a vertical applied external force and a horizontal applied external force.

3.1. Assembly Under a Vertical Applied External Force

Figure 7 displays three groups of 2D mating examples for the 2D-Case 1-3 under a vertical applied external force, demonstrating different geometric distribution errors of the part mating surfaces. Each part has two mating surfaces: one horizontal (e.g., 2D-Case 1(a) in Figure 7) and one vertical (e.g., 2D-Case 1(b) in Figure 7).

The corresponding registration results for the 2D-Case 1-3, computed using the proposed method, are depicted in Figure 8. A key observation is that under the vertical applied external force, the horizontal mating surfaces act as the primary mating surfaces, while the vertical ones are secondary. Moreover, two pairs of contact points are determined on the horizontal surfaces, whereas only one pair exists on the vertical surfaces. Based on the multi-constraint coupled model, the parts achieved a stable equilibrium. Quantitative data for this configuration are summarized in

Table 1. Registration state analysis of the 2D assembly.

Case (2D-Case)	Mapping Matrix M	Contact Points of the Assembly Coordinate System				SDT
		Horizontal Mating Surfaces A1 and A2		Vertical Mating Surfaces B1 and B2
1	${\displaystyle M}=\left[\begin{array}{cccc}1& 0& 0& 2.0010\\ 0& 1& 0& 0.0008\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]$	A1	(−0.1052, −0.0017) (25.8950, −0.00058)	B1	(−33.0016, 13.9946)	$T^{2D}=\left(\begin{array}{c}\begin{array}{cc}-0.0008& 0\end{array}\\ \begin{array}{cc}-0.0017& 0\end{array}\\ \begin{array}{cc}0& 0.00004\end{array}\end{array}\right)$
		A2	(−0.1052, 0.00004) (25.8950, 0.0012)	B2	(−33.0002, 13.9955)
2	${\displaystyle M}=\left[\begin{array}{cccc}1& 0& 0& 4.0025\\ 0& 1& 0& 0.0034\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]$	A1	(3.8969, −0.0019) (−10.1033, −0.0043)	B1	(−39.1795, 26.0174)	$T^{2D}=\left(\begin{array}{c}\begin{array}{cc}0.0032& 0\end{array}\\ \begin{array}{cc}-0.0025& 0\end{array}\\ \begin{array}{cc}0& 0.00017\end{array}\end{array}\right)$
		A2	(3.8969, 0.0016) (−10.1033, 0)	B2	(−39.1776, 26.0210)
3	${\displaystyle M}=\left[\begin{array}{cccc}1& 0& 0& 2.0019\\ 0& 1& 0& 0.0082\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]$	A1	(19.8806, −0.0038) (−10.1204, −0.0030)	B1	(−39.1889, 8.0650)	$T^{2D}=\left(\begin{array}{c}\begin{array}{cc}-0.0002& 0\end{array}\\ \begin{array}{cc}-0.0033& 0\end{array}\\ \begin{array}{cc}0& -0.00002\end{array}\end{array}\right)$
		A2	(19.8806, 0.0031) (−10.1204, 0.00070)	B2	(−39.1885, 8.0727)
4	${\displaystyle M}=\left[\begin{array}{cccc}1& 0& 0& 0.0034\\ 0& 1& 0& 0.0052\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]$	A1	(43.0756, −14.9524)	B1	(0, 8.9904) (−0.00184, −7.0136)	$T^{2D}=\left(\begin{array}{c}\begin{array}{cc}-0.0010& 0\end{array}\\ \begin{array}{cc}0.0039& 0\end{array}\\ \begin{array}{cc}0& -0.00011\end{array}\end{array}\right)$
		A2	(43.0756, −14.9496)	B2	(0.00064, 9.0662) (0.00002, −6.9341)
5	${\displaystyle M}=\left[\begin{array}{cccc}1& 0& 0& 0.0023\\ 0& 1& 0& -0.0467\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]$	A1	(15.0889, −15.0755)	B1	(0.0006, 0.9407); (0.00019, −7.0593);	$T^{2D}=\left(\begin{array}{c}\begin{array}{cc}-0.0007& 0\end{array}\\ \begin{array}{cc}-0.0004& 0\end{array}\\ \begin{array}{cc}0& 0.000056\end{array}\end{array}\right)$
		A2	(15.0889, −15.0770)	B2	(−0.00078, 0.9873); (−0.00033, −7.0127);
6	${\displaystyle M}=\left[\begin{array}{cccc}1& 0& 0& 0.0048\\ 0& 1& 0& 0.0100\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]$	A1	(57.0745, −14.0774)	B1	(−0.0019, 9.9882) (−0.00024, −2.0125)	$T^{2D}=\left(\begin{array}{c}\begin{array}{cc}-0.00052& 0\end{array}\\ \begin{array}{cc}-0.0080& 0\end{array}\\ \begin{array}{cc}0& 0.00014\end{array}\end{array}\right)$
		A2	(57.0745, −14.0717)	B2	(0, 9.9919) (0.0030, −2.0075)

(see 2D-Case 1–3).

3.2. Assembly with a Horizontal Applied External Force

To further validate the repeatability of the proposed method, its performance was evaluated under a rightward horizontal applied external force, a scenario distinct from the vertical loading case detailed in Section 3.1. The testing involved components with differing geometric distribution errors. Their initial poses and geometric distribution errors are provided in Figure 9.

The registration results for the three groups of parts under the horizontal applied external force are shown in Figure 10. A distinct trend is observed across all groups: the vertical mating surfaces now bear more contact pairs, with two pairs determined on them, while the horizontal surfaces exhibit one pair. By incorporating the applied force into the model, the proposed method captures this resulting shift in contact patterns and the consequent change in the part’s final pose, as evidenced by the reversal when compared to the vertical applied external force cases (Figure 8). This demonstrates its capability to predict the contact points and stable poses under varying load conditions. This outcome is quantitatively confirmed by the data in Table 1 (see 2D-Case 4–6).

4. Three-Dimensional Surface Registration Experiment

To validate the effectiveness of the proposed method based on geometric distribution errors, we designed experiments using plane–plane mating structures with machine tool tailstock mock-ups.

4.1. Experimental Setup

The apparatus primarily consists of a base and an L-shaped bracket (Figure 11). The experimental components were constructed from Grade 45 steel, a common material in mechanical structures. This material selection supports the validation of the model’s applicability to components made of other similarly rigid materials. The mating surfaces (Base: side D1 and horizontal plane D2; Bracket: side S1 and horizontal plane S2) were then milled according to the specifications given in

Table 2. Part status.

No.	Part	Machining Method	Material
1	Base 1	Milling machine 1	grade 45 steel
2	Base 2	Milling machine 2
3	Bracket 1	Milling machine 1
4	Bracket 2	Milling machine 2

.

4.2. Experimental Scheme

The experimental measurement process is illustrated in Figure 12a–c. First, prior to assembly, a coordinate measuring machine (CMM; model: Global 5.7.5, manufacturer: Hexagon Manufacturing Intelligence, Qingdao, China) with a measurement uncertainty of (1.7 + 3L/1000) μm was used to acquire surface point cloud data for the bracket’s side (S1), horizontal (S2), and top (S3) planes. A total of 12 rows with 25 points each were collected on the side surface of the bracket; 24 rows comprising 643 points were obtained on the horizontal surface; and 30 points were captured on the top surface. The number of points was determined by the sampling interval, which was set between 2 mm and 4 mm in this study. Measurements were performed using PC-DMIS 2018 software.

Subsequently, the side (D1) and horizontal (D2) planes of the base were measured. The measurement strategy for the base surfaces was consistent with that used for the bracket, with 300 points collected on the side surface and 643 points on the horizontal surface. All measurements were conducted following CMM calibration, with errors primarily dependent on the equipment’s inherent accuracy. The orientation of the measurement coordinates is shown in Figure 12d. For the base, the coordinate system origin was located at the geometric center of the horizontal surface, with the positive x-axis directed to the right and the positive Z-axis oriented vertically upward.

Following the part measurement phase, the components were assembled according to the procedure described later in this section. To determine the spatial pose of the assembled bracket, a global assembly coordinate system was established coincident with the base coordinate system. The pose of the bracket’s top surface (S3) was then measured within this assembly frame, consisting of 6 rows with 5 points each (30 points total). To ensure experimental repeatability, components from Table 2 were assembled in different combinations under two loading conditions: vertical (Figure 12d) and horizontal (Figure 12e). Three experimental trials were conducted for each loading condition using different component sets, resulting in a total of six experimental tests, as summarized in

Table 3. Assembly plan.

Case	Contact Surfaces	Pair of Assembled Samples		Force
1	D1–S1; D2–S2	Base 1	Bracket 1	vertical applied force
2	D1–S1; D2–S2	Base 1	Bracket 2
3	D1–S1; D2–S2	Base 2	Bracket 2
4	D1–S1; D2–S2	Base 1	Bracket 1	horizontal applied force
5	D1–S1; D2–S2	Base 1	Bracket 2
6	D1–S1; D2–S2	Base 2	Bracket 2

.

The assembly procedure involved: (1) The bracket was positioned on the base. Relying on gravity and rough alignment, the horizontal mating surfaces (D2–S2) were brought into initial contact, while the side surfaces (D1–S1) were approximately aligned and contact. (2) A bolt was inserted and hand-tightened until slight resistance was felt. Subsequently, a small initial torque was applied using a torque wrench, which was just sufficient to maintain part contact while still permitting fine adjustments. (3) Fine-tuning was performed by gently tapping the sides and ends of the bracket with a plastic hammer until the operator tactually and visually confirmed that both mating surface pairs were in full contact. (4) Finally, the bolt was tightened to the specified target torque using the torque wrench, locking the assembly in its final position. For the vertical loading condition, the final torque was 150 N·cm, while for the horizontal loading condition, it was 12 N·cm (within a range of 11.6–13 N·cm).

Subsequently, the proposed method is employed to predict the point cloud P′(x_s′, y_s′, z_s′) representing the pose of the top surface S3. Finally, the measured values P^a(x_s, y_s, z_s) of S3 are compared with the predicted values P′(x_s′, y_s′, z_s′).

4.3. Analysis of 3D Surface Registration States

The 3D registration results under different loading conditions are presented in Section 4.3.1 and Section 4.3.2. Detailed quantitative data supporting these observations are provided in Appendix A (

Table A1. Experimental Results and Analysis of the 3D Assembly.

Case (3D-Case)	Mapping Matrix M	Contact Points of the Assembly Coordinate System D1 and D2		SDT
		Horizontal Mating Surfaces A1 and A2; Side Surfaces B1 and B2
1	$M=\left[\begin{array}{cccc}1& 0& 0& 0.0066\\ 0& 1& 0& 0\\ 0& 0& 1& -0.0844\\ 0& 0& 0& 1\end{array}\right]$	A1	(−3.9990, 15.9993, −0.0276)	$T^{3D}=\left[\begin{array}{cc}-0.1788& 0.0002\\ 0& 0.0024\\ -0.0477& 0.0077\end{array}\right]$
			(−5.9988, −22.0010, −0.0394) (4.0006, 15.9989, −0.0528)
		B1	(−56.4066, 9.9989, 4.0006) (−56.2184, −14.0006, 4.0012)
		A2	(−4.1066, 16.0010, 0.0030) (−6.1066, −21.9992, 0.0028) (3.8933, 16.0009, 0.0066)
		B2	(−56.0979, 9.9989, 4.0007) (−56.0931, −14.0007, 4.0013)
2	$M=\left[\begin{array}{cccc}1& 0& 0& -0.0298\\ 0& 1& 0& 0\\ 0& 0& 1& -0.0750\\ 0& 0& 0& 1\end{array}\right]$	A1	(−3.9990, 15.9993, −0.0276) (0.0009, −22.0015, −0.0535) (12.0016, −14.0010, −0.0917)	$T^{3D}=\left[\begin{array}{cc}-0.1820& 0.0003\\ 0& 0.0030\\ -0.0215& 0.0077\end{array}\right]$
		B1	(−56.1899, −18.0005, 4.0102) (−56.5116, 23.9998, 4.0099)
		A2	(−4.1233, 15.9963, 0.0174) (−0.1218, −22.0024, 0.0117) (11.8787, −14.0026, 0.0048)
		B2	(−56.1308, −18.0005, 4.0102) (−56.1484, 23.9998, 4.0099)
3	$M=\left[\begin{array}{cccc}1& 0& 0& 0.0031\\ 0& 1& 0& 0\\ 0& 0& 1& -0.0829\\ 0& 0& 0& 1\end{array}\right]$	A1	(−7.9985, 16.0005, −0.0145) (0.0016, −16.0001, −0.0262) (−15.9982, −5.9998, −0.0167)	$T^{3D}=\left[\begin{array}{cc}-0.1202& 0.00005\\ 0& 0.0007\\ -0.0117& -0.0073\end{array}\right]$
		B1	(−56.1897, 2.0002, 16.0107) (−56.2479, −6.0002, 16.0103)
		A2	(−2.0607, 17.9965, 0.0141) (−0.0599, −16.0028, 0.0125) (−16.0603, −6.0033, 0.002)
		B2	(−56.0777, 2.0002, 16.0107) (−56.0749, −6.0002, 16.0103)
4	$M=\left[\begin{array}{cccc}1& 0& 0& 0.0366\\ 0& 1& 0& 0\\ 0& 0& 1& 0.0344\\ 0& 0& 0& 1\end{array}\right]$	A1	(−0.0839, −10.0002, 1.0230) (−0.3464, 23.9989, 1.0236) (−0.1286, −6.0009, 11.0485)	$T^{3D}=\left[\begin{array}{cc}-0.1591& 0.0001\\ 0& 0.0014\\ 0.0373& 0.0077\end{array}\right]$
		B1	(40.1562, −24.0009, −15.0786) (40.1563, 19.9992, −15.0731)
		A2	(0.0314, −10.0002, 1.023) (0.0206, 23.9989, 1.0236) (0.0061, −6.0009, 11.0485)
		B2	(40.0493, −23.9815, −14.987) (40.0488, 20.0175, −14.9663)
5	$M=\left[\begin{array}{cccc}1& 0& 0& 0.0202\\ 0& 1& 0& 0\\ 0& 0& 1& -0.0280\\ 0& 0& 0& 1\end{array}\right]$	A1	(−0.0711, −11.9999, 0.9542) (−0.2694, 14.0002, 0.9539) (−0.0348, −18.0005, −11.0463)	$T^{3D}=\left[\begin{array}{cc}-0.1618& 0.0003\\ 0& 0.0009\\ 0.0131& 0.0077\end{array}\right]$
		B1	(34.1562, 13.9988, −15.0622) (34.1561, −14.0012, −15.0731)
		A2	(0.0152, −11.9999, 0.9542) (0.0051, 14.0002, 0.9539) (0.0243, −18.0005, −11.0463)
		B2	(34.0328, 13.9977, −15.0402) (34.0331, −14.003, −15.0446)
6	$M=\left[\begin{array}{cccc}1& 0& 0& 0.0831\\ 0& 1& 0& 0\\ 0& 0& 1& 0.0233\\ 0& 0& 0& 1\end{array}\right]$	A1	(−0.286, −24.0002, −10.7048) (−0.0978, 2.0002, 1.2955) (−0.2324, −15.9998, 3.2949)	$T^{3D}=\left[\begin{array}{c}\begin{array}{cc}-0.1111& -0.00004\end{array}\\ \begin{array}{cc}0& 0.0004\end{array}\\ \begin{array}{cc}-0.0009& -0.0075\end{array}\end{array}\right]$
		B1	(40.0939, −5.9998, −14.7811) (48.0936, 16.0005, −14.7789)
		A2	(0.0237, −24.0002, −10.7048) (0.0145, 2.0002, 1.2955) (0.0128, −15.9998, 3.2949)
		B2	(40.0318, −5.9876, −14.7164) (48.0325, 16.0123, −14.6902)

).

The predicted assembled pose of the target surface was calculated according to the following procedure: First, the spatial pose transformation matrix M_P was constructed from the SDT parameters (determined in Section 2.3) using Equation (28). Subsequently, the point cloud P(x, y, z) was transformed into the predicted assembled point cloud P’ by applying the matrix operation defined in Equation (29), thereby obtaining the predicted pose.

$$\begin{array}{ll}{M}_P& =M_{translation}\cdot M_z\cdot M_y\cdot M_x\\ & =\left[\begin{array}{cccc}1& 0& 0& d{x}^{3D}\\ 0& 1& 0& d{y}^{3D}\\ 0& 0& 1& d{z}^{3D}\\ 0& 0& 0& 1\end{array}\right]\cdot \left[\begin{array}{cccc}\cos {\gamma }^{3D}& -\sin {\gamma }^{3D}& 0& 0\\ \sin {\gamma }^{3D}& \cos {\gamma }^{3D}& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]\cdot \left[\begin{array}{cccc}\cos {\beta }^{3D}& 0& \sin {\beta }^{3D}& 0\\ 0& 1& 0& 0\\ -\sin {\beta }^{3D}& 0& \cos {\beta }^{3D}& 0\\ 0& 0& 0& 1\end{array}\right]\cdot \left[\begin{array}{cccc}1& 0& 0& 0\\ 0& \cos {\alpha }^{3D}& -\sin {\alpha }^{3D}& 0\\ 0& \sin {\alpha }^{3D}& \cos {\alpha }^{3D}& 0\\ 0& 0& 0& 1\end{array}\right]\end{array}$$

(28)

$${\mathrm{P}}^{\prime }=M_P\cdot \mathrm{P}$$

(29)

4.3.1. Registration Under Vertical Applied External Force

Figure 13 illustrates the initial state of three groups of 3D mating surfaces under a vertical applied external force (3D-Case 1-3), as well as the variations in geometric distribution errors on different part surfaces. In this scenario, the proposed method determined the horizontal surface to be the primary constraint, with the side surface as the secondary constraint. The red dots in Figure 13, such as 3D-Case 1(a) and 3D-Case 1(b), indicate the predicted contact points prior to registration, which are calculated from Equations (9) and (10).

Corresponding to the initial states in Figure 13, Figure 14 illustrates the resultant pose of the mating surfaces and the distribution of contact points predicted by the proposed method. It is observed that the component settles into a stable pose where the horizontal surfaces, as expected, provide the primary constraint. Meanwhile, the contact points on the side surfaces adjust to compensate for geometric distribution errors, fulfilling a secondary constraining role. This result confirms that the proposed method can determine the contact points and predict the assembly state under vertical loading.

4.3.2. Registration Under Horizontal Applied External Force

The initial states for the horizontal applied external force (3D-Case 4-6) are shown in Figure 15. This figure shares the same geometric distribution errors as Figure 13 (see Table 3), but with a differing load condition. In this scenario, the proposed method re-prioritizes the constraints, designating the side surfaces as primary and the horizontal surfaces as secondary.

As shown in Figure 16, the final assembly state differs from that under the vertically applied external force. This validates that forces acting on surfaces with geometric distribution errors directly influence part contact state. In this case, the force also alters the constraint priority of the mating surface, wherein the side surfaces act as the primary constraint. The capability to predict stable assembly states, which accounts for both geometric distribution errors and applied external forces, provides a solution for assessing assembly poses.

4.4. Experimental Results

The relative errors between predicted and experimentally measured values of S3 were calculated using Equation (30), with results presented in

Table 4. Relative errors between the predicted and measured values.

Case (3D-Case)	1 (%)	2 (%)	3 (%)	4 (%)	5 (%)	6 (%)
RMSE	1.6	1.4	1.5	1.2	1.3	1

The table demonstrates good agreement between predicted and measured values, with all relative errors remaining below 2%.

.

$$\mathrm{RMSE}=\sqrt{{\displaystyle \sum _{s=1}^k\frac{{({\mathrm{P}}^{\mathrm{a}}-{\mathrm{P}}^{\prime })}^2}{{\mathrm{P}}^{\mathrm{a}}}}}=\sqrt{{\displaystyle \sum _{s=1}^k\frac{{(x_s-x_s^{\prime })}^2+{(y_s-y_s^{\prime })}^2+{(z_s-z_s^{\prime })}^2}{{(x_s)}^2+{(y_s)}^2+{(z_s)}^2}}}$$

(30)

In the equation, k represents the number of measurement points.

4.5. Results and Discussion

Figure 13 reveals the geometrical distribution errors, characterized by spatial undulations, present on the mating surfaces under specific machining conditions. As these error surfaces are reconstructed from measurement data, their morphology closely aligns with that of the actual workpiece [30], thereby establishing a foundation for the model. Following the stable assembly of the two pairs of mating surfaces, Figure 14 demonstrates that the spatial position of the contact points influences the final pose of the assembly. This stands in stark contrast to conventional tolerance-based methods, which can only predict a range of variations but fail to ascertain the actual contact point locations or the resulting registered state. This fundamental discrepancy arises because traditional methods operate on idealized tolerance zones, which lack the topographical detail to resolve specific contact behavior. In contrast, our model, by leveraging the actual geometrical distribution, bridges this gap by simulating the physical contact interactions that ultimately define the assembly pose.

Furthermore, achieving accurate predictions for the assembly of complex, multi-constrained components necessitates considering not only these geometrical distribution errors but also assembly process parameters, particularly applied external forces. Two distinct predictions for the final part pose and contact points are obtained from a comparison of Figure 14a and Figure 16a, which use the same experimental setup, under different loading conditions. A model that neglects the influence of applied external forces would be incapable of distinguishing between these states and would produce a single prediction. The divergence in predicted poses demonstrates that the applied force alters the sequence and conditions of contact establishment, thereby guiding the assembly into different stable configurations. Consequently, capturing this force-dependent behavior is essential for predictive accuracy.

In conclusion, the proposed multi-constraint coupling model, which integrates geometrical distribution errors and critical assembly parameters, transitions assembly prediction from a purely geometric domain into a physically grounded simulation. As validated by the experimental data in Table 4, this method provides a quantitative solution for predicting the state of complex assemblies. The method could be extended to complex non-planar assemblies. Such an extension would not change the model but would depend on the data acquisition, data pre-processing, and dimensionality reduction of data from curved surfaces.

5. Conclusions

In product assembly, geometric distribution errors from machined surfaces and the coupling of multiple mating surface constraints are critical factors affecting precision. To address this challenge, this paper proposes a method for predicting key assembly outcomes, including the final pose and contact points.

Based on measured data of mating surfaces, the method utilizes the NURBS method to characterize the geometric distribution errors of parts. Integrated with a priority-based theory for multi-surface assembly, a hierarchical surface registration model was developed. This model comprehensively accounts for the effects of geometric distribution errors, multi-constraint coupling, and applied external forces, thereby enabling it to predict the spatial poses of assembled components and provide a theoretical foundation for predicting product assembly accuracy.

Model validation was conducted through progressive case studies, from 2D to 3D. The 2D cases provided an intuitive analysis of the contact state between two mating surface pairs, determining contact point locations. Subsequently, six sets of assembly experiments were conducted. These experiments determined the contact point positions and the final poses of the mating surfaces for each assembly under two distinct loading conditions. The results demonstrated that the root mean square error (RMSE) between the predicted and measured values for the critical surface parameters was less than 2%.