Multi-Source Error Coupling and Tolerance Optimization for Improving the Precision of Automated Assembly of Aircraft Components

Abstract

In automated aircraft assembly, achieving high-precision alignment is essential due to the presence of multiple coupled error sources that significantly affect final product quality. This study proposes an integrated framework to model multi-source errors via a directed coupling network and to quantify their impact using Monte Carlo simulations. To reduce the complexity of tolerance allocation, Sobol-based global sensitivity analysis is applied to identify dominant contributors to assembly deviations. The most influential parameters are retained for multi-objective optimization using the non-dominated sorting genetic algorithm II (NSGA-II). This framework enables the minimization of key assembly deviations while maintaining computational efficiency. Experimental validation on a typical helicopter ring assembly demonstrates that the proposed optimization approach increases the position pass rate from 67.4% to 100.0% and the coaxiality pass rate from 93.5% to 100.0%. The corresponding process capability indices (CPK) also improve significantly, from 0.31 to 2.19 for position and from 0.62 to 1.06 for coaxiality. These improvements not only satisfy high-precision assembly requirements but also exceed common industry benchmarks, demonstrating the method’s practical effectiveness under multi-source uncertainty.

1. Introduction

Modern military and commercial aircraft demand extremely tight assembly tolerances and efficiency. For example, automated drilling and riveting systems for aircraft panels have achieved a positioning accuracy of ±0.05 mm [1], far higher than typical manual assembly. As a result, original equipment manufacturers such as Boeing and Airbus are rapidly expanding the use of robotics and digital assembly [2]. For example, Airbus has opened a highly automated fuselage assembly line for the A320 aircraft in Hamburg [3,4], reflecting the industry’s trend of adopting robotics in high-volume aircraft production. These automation measures have brought significant benefits: (a) With increasing precision requirements in aerospace manufacturing, automated drilling and riveting cells have demonstrated positional accuracies of approximately ±0.05 mm [5] in real-world panel assembly operations, which significantly exceeds the typical ±0.2 mm [6] accuracy achievable through manual assembly or basic industrial robotic systems. Automated assembly can provide more reliable and efficient precision control [7]. (b) Automated assembly can adapt to wall panels with different curvatures through flexible tooling, such as multi-matrix vacuum suction cups [8], overcoming the limitations of traditional rigid tooling in adapting to multiple models of products [9]. (c) The application of the automatic drilling and riveting system E5000 ASAT4 has shortened the assembly cycle by 40%, while reducing the human error rate to below 0.1%.

However, despite the significant progress in precision and efficiency of automated assembly technology, the complexity [10] and multi-layer [11] nested structure of aircraft still pose great challenges to the assembly process. Compared with traditional manual assembly, although automated assembly eliminates subjective errors in manual operation [12], it introduces new error sources [13] and forms a more complex “equipment–process–environment” coupled error chain, and is sensitive to cumulative nonlinear and dynamic error chains [14]. For example, long-term use can cause robot positioning to drift due to temperature effects [15], while coupled errors such as robot positioning uncertainty and system vibration introduce non-Gaussian distributions that are difficult to predict and control [16]. Therefore, the nonlinear propagation of such errors invalidates traditional static tolerance analysis [17]. Traditional methods assume that the error sources are linear and independent, so they often fail in this dynamic environment.

At present, although digital assembly technology has made certain progress, many studies have not been able to fully solve the unique error sources and optimization problems in automated assembly. Existing research faces several gaps: (a) Most existing studies focus on the analysis of traditional manufacturing tolerances, while ignoring the error sources unique to automation, such as nonlinear friction of robot joints and electromagnetic interference in visual measurement [18]. (b) The application of traditional optimization methods to multi-objective problems is still limited. Most tolerance allocation methods still use the weighted summation method, which is difficult to find a reasonable balance between multiple conflicting objectives [19]. For example, single-objective methods achieve only 40% Pareto frontier coverage compared to over 85% with NSGA-II [20]. (c) Most existing error modeling methods are based on static assumptions. For example, the error sources in Monte Carlo simulation are assumed to be independent and static. However, in actual automated assembly, the correlation and time-varying nature of the error sources cannot be ignored. Taking the robot positioning error as an example, it exhibits obvious Markov characteristics, that is, the accumulation of errors depends not only on the current error, but also on the previous error state [21]. Studies have shown that dynamic tolerance allocation methods have been shown to improve accuracy by more than 30% and reduce prediction errors by more than 10% [17].

To address these challenges, this study proposes a new aircraft automatic assembly accuracy optimization method to solve the accuracy control challenges brought by the coupling of multi-source error chains. Different from the traditional tolerance analysis method, this method combines error modeling based on directed graphs, global sensitivity analysis based on Sobol index, and NSGA-II [22] optimization algorithm to balance the tolerances of multiple error sources while meeting different assembly accuracy requirements, thereby improving the overall performance of automatic assembly, as shown in Figure 1.

The remainder of this paper is structured as follows: Section 2 outlines the theoretical framework for automated assembly precision optimization and introduces the modeling approach for multi-source error chains. Section 3 presents the construction of a trade-off model among key assembly features using a multi-objective optimization strategy. Section 4 evaluates the effectiveness of the proposed method through experimental validation and comparison with existing approaches. Section 5 discusses the generalizability, limitations, and future prospects of the proposed method in broader manufacturing contexts. Section 6 concludes the paper by summarizing the main findings and implications.

2. Error Propagation and Coordination Modeling in Automated Assembly

In the assembly process of complex aircraft components, the accumulation and propagation of errors directly determine the geometric accuracy and functional coordination performance of the final product. Significant differences exist between traditional manual assembly and modern automated assembly in terms of error formation mechanisms and propagation paths [23], as shown in Figure 2.

In traditional manual assembly, assembly errors are primarily composed of the following multi-source factors: inherent manufacturing errors of components, human factors, and fixture positioning deviations [24]. These errors propagate relatively along dimensional chains based on physical positioning datums, exhibiting linear superposition characteristics. The error distribution can be statistically analyzed using Monte Carlo simulations and compensated through empirical adjustments or post-assembly rework [25,26]. The error chains are localized and discrete, mainly concentrated at critical mating features such as misalignment in hole-shaft fits or local gaps caused by step differences [27].

Compared to traditional manual assembly, automated assembly demonstrates significant advantages in precision, efficiency, and repeatability but introduces new error mechanisms and coupling paths [27]. Automated assembly integrates multi-source errors—component-induced errors, automation-induced errors, and process-execution errors—into predefined process flows to form error chains. Since automated systems lack traditional rigid physical positioning references, they rely on virtual point cloud registration [28] or vision-guided [29] for positioning. However, these errors interact nonlinearly in both spatial and temporal dimensions, making it difficult to accurately assess assembly accuracy [30]. Therefore, the error coupling model must be introduced during the assembly process design phase, and improved nonlinear optimization algorithms must be developed for predictive analysis.

2.1. Multi-Source Error Classification in Automated Assembly

In automated assembly environments, part errors extend beyond traditional manufacturing tolerances to include novel deviations introduced by automated equipment and process execution. To systematically capture these multi-source errors, this study categorizes them into three classes: component-induced errors, automation-induced errors, and process-execution errors, as illustrated in Figure 3.

Critical error terms require mathematical formalization and parameterization to establish explicit model inputs for tolerance analysis and optimization prior to constructing multi-source error coupling models. This section defines mathematical representations and distribution patterns for distinct error types.

2.1.1. Component-Induced Errors

Part geometric deviations originate from the manufacturing stage, including dimensional errors and form errors caused by processes such as computer numerical control machining, stamping or additive manufacturing [31], and profiles or roughness caused by surface machining and grinding processes that do not meet design requirements [32]. These errors are usually obtained by sampling key geometric features of parts using a three-dimensional coordinate measurement system and comparing them with theoretical models to obtain deviations. Typical evaluation methods include statistical analysis of dimensional deviations, calculation of surface roughness power spectral density, and statistical regression analysis of material test samples [33].

• Manufacturing geometric deviation

Component geometric deviations represent discrepancies between manufactured features and CAD models, typically expressed as vector differences between measured points and nominal surfaces. Let $d_i^{meas}$ and $d_i^{nom}$ denote the measured and nominal coordinates of the i-th sampling point respectively. Then the manufacturing geometric deviation can be written as Equation (1).

$$\delta d_i=∥d_i^{meas}-d_i^{nom}∥$$

(1)

The collective deviation follows a normal distribution $N\left({\mu }_d,{\sigma }_d^2\right)$, where ${\mu }_d$ and ${\sigma }_d^2$ represent the mean and standard deviation of sampled deviations [34,35].

2.

Surface topography errors

Surface topography is quantified using arithmetic average roughness $Ra$, as shown in Equation (2).

$$Ra={\displaystyle \frac{1}{L}}{\int }_0^1\left|z\left(x\right)\right|dx$$

(2)

where $z\left(x\right)$ denotes profile deviation from the mean line and L the evaluation length. Statistically, $Ra$ values follow a log-normal distribution $Log-N\left({\mu }_r,{\sigma }_r^2\right)$ [36].

2.1.2. Automation-System Errors

Automation-system errors arise from inherent characteristics of robotic equipment, actuators, and measurement systems. Robotic manipulators exhibit repeatability drift influenced by joint clearances and control system precision, with displacement ranges determined through repetitive positioning experiments [37]. Vision and laser measurement systems contain both systematic offsets and stochastic noise components, quantifiable using ISO-specified uncertainty evaluation protocols for coordinate measuring systems [38].

• Repeat positioning error

The repeatability error of automated equipment is obtained by performing multiple positioning experiments on the same target. Assuming that the translation error between the end position and the reference position of the i-th experiment is $\Delta p_i$, and the rotation error is $\Delta {\theta }_i$, the repeatability error can be described by the statistical distribution of these two sets of data, usually using a normal distribution, that is, $\Delta p\sim N\left(0,{\sigma }_p^2\right)$, $\Delta \theta \sim N\left(0,{\sigma }_p^2\right)$. This mathematization provides the initial conditions for the device-level error in the coupling network.

2.

Sensor/vision measurement uncertainties

Sensor uncertainties combine systematic bias and random noise [39]. If $\tilde{y}$ is the measured value and y is the true value, then it can be written as Equation (3).

$$\tilde{y}=y+b+ϵ$$

(3)

where b is the systematic deviation and $ϵ$ is the zero-mean normal noise. Parameterizing this error allows the coupled model to distinguish between correctable errors and random perturbations and to assign tolerances accordingly in the optimization.

2.1.3. Process-Execution Errors

Distinct from manual assembly errors, automated assembly systems exhibit programmatically induced temporal and synchronization errors. Control latency manifests as statistically distributed time delays between command signals and physical responses, measurable through command–response timestamp analysis. Multi-axis and multi-robot operations demonstrate spatial–temporal misalignment during coordinated motion sequences, quantifiable through a synchronization log analysis of actuator initiation/termination timestamps. These temporal errors directly impact process sequence integrity and component fitment accuracy, necessitating compensation through time-series analysis and synchronization correction algorithms [40,41].

• Actuator dynamic response error

End-effector dynamic errors manifest as force–displacement hysteresis during loading–unloading cycles. For applied force F and displacement $\delta$, the hysteresis loop is modeled by $\delta \left(F\right)$; its hysteresis curve can be fitted by piecewise linear or polynomial function and combined with the damping coefficient c to represent the energy dissipation [41].

2.

Control latency and synchronization offsets

Sequence control delay $\tau$ refers to the time difference between command issuance and execution response [42]; multi-axis synchronization offset $\Delta t_{ij}$ refers to the start/stop time difference of different execution units under the same instruction [43]. Both usually present log-normal distribution $Log-N\left({\mu }_{\tau },{\sigma }_{\tau }^2\right)$ and normal distribution $N\left(0,{\sigma }_{\Delta t}^2\right)$.

2.2. Multi-Source Error Coupling Modeling

Table 1. Definitions and mathematical expressions of various error sources involved in the automated assembly process.

Error Sources	Mathematical Expressions	Distribution Type
Manufacturing geometric deviation	$\delta d_i=∥d_i^{meas}-d_i^{nom}∥$	$N\left({\mu }_d,{\sigma }_d^2\right)$
Surface topography errors	$Ra={\displaystyle \frac{1}{L}}{\int }_0^1\left|z\left(x\right)\right|dx$	$Log\text{-}N\left({\mu }_r,{\sigma }_r^2\right)$
Repeat positioning error	$\Delta p,\Delta \theta$	$N\left(0,{\sigma }_p^2\right),N\left(0,{\sigma }_p^2\right)$
Sensor/vision measure uncertainties	$\overline{y}=y+b+\epsilon$	$N\left(0,{\sigma }_{\epsilon }^2\right)$
Actuator dynamic response error	$\delta \left(F\right)$	—
Control latency and synchronization offsets	$\tau ,\Delta t_{ij}$	$N\left(0,{\sigma }_{\epsilon }^2\right),N\left(0,{\sigma }_{\Delta t}^2\right)$

shows the error sources in automated assembly, which provides a clear input source for the subsequent error transmission chain model and a quantitative basis for error sensitivity analysis and weight allocation in multi-objective tolerance optimization.

In Table 1, all kinds of errors are discrete and do not have the prerequisite of linear superposition in the same coordinate system, so the traditional linear accumulation method of dimensional chain cannot be directly used. Errors such as manufacturing geometric deviation, positioning drift of automation equipment, and program control delay present different dimensions and distribution characteristics in space. If they are linearly accumulated, not only will the cross-coupling effect between errors be ignored, but it will also be difficult to reflect the actual situation of error amplification or offset. In order to accurately describe the error propagation and cumulative effects, it is necessary to first model the three types of errors separately, then build a coupling network at the key nodes of the assembly process and recursively propagate them.

2.2.1. Multi-Source Error Modeling

• Component-induced error modeling

Let the i-th component-induced errors $\Delta M_i$ be confined to the tolerance interval $\left[-T_i/2,+T_i/2\right]$ and assume a truncated normal distribution, then the geometric deviation can be expressed as Equation (4).

$$\Delta M_i\sim N_{trunc}(0,{\sigma }_{Gi}^2;-T_i/2,+T_i/2)$$

(4)

Using the Jacobian matrix $J_G$, map each $\Delta M_i$ into an assembly pose error $\Delta p_{comp}$, as shown in Equation (5).

$$\Delta p_{comp}=J_G{[\Delta M_1,\Delta M_2,\dots ,\Delta M_n]}^{\mathsf{T}}$$

(5)

2.

Automation-system error modeling

The robot repetitive positioning error and measurement error are uniformly regarded as automation factor errors. The end linear error and angular error of the industrial robot can be approximately described by truncated normal distribution and mixed distribution respectively. The end linear position error of the robot is $\Delta L={(\Delta x,\Delta y,\Delta z)}^{\mathsf{T}}$, and its distribution model is shown as Equation (6).

$$\Delta L\sim N_{trunc}(0,{\sigma }_L^2{I}_3;-L_{max},+L_{max})$$

(6)

where $I_3$ denotes the 3 × 3 identity matrix, ensuring isotropic error distribution across all three spatial dimensions.

As multiple independent factors (e.g., servo resolution, kinematic compliance, sensor quantization) contribute to the robot’s positioning uncertainty, their aggregate effect is well approximated by a normal distribution according to the central limit theorem. Moreover, empirical studies of industrial robots have observed that repeatability errors (e.g., joint position deviations) follow a Gaussian pattern [44]. Therefore, we assume a zero-mean normal distribution for the robot’s repeatability error. We truncate this distribution at $\pm L_{max}$ (the manufacturer-specified maximum positional error) to enforce the physical bound on extreme errors, ensuring no simulated error exceeds the robot’s known repeatability limits [45].

The end angle deviation $\Delta \theta ={(\Delta \alpha ,\Delta \beta ,\Delta \gamma )}^{\mathsf{T}}$ then it approximately obeys a mixture of normal and uniform distribution, as shown in Equation (7).

$$\Delta {\theta }_j\sim w_{1}N(0,{\sigma }_{\theta }^2)+w_2·U(-{\theta }_{max},+{\theta }_{max}),j=1,2,3$$

(7)

where $w_1+w_2=1$, ${\theta }_{max}$ and $L_{max}$ are obtained according to the equipment manual and experimental calibration.

The robot’s orientation error is bounded (cannot exceed $\pm {\theta }_{max}$ by design) and yet can be unpredictably anywhere within that range. To capture this bounded yet random behavior, we model the angular error as a mixture of a Gaussian component and a uniform component. The zero-mean Gaussian part represents the typically small, frequent deviations about the commanded angle (due to controller precision and encoder noise), whereas the uniform part accounts for any larger bias or free-play within $\pm {\theta }_{max}$ (e.g., due to gear backlash or calibration drift) that could cause an arbitrary offset in orientation [46]. This mixed distribution ensures that most error instances are clustered around zero, as expected for a high-precision robot, while still allowing the errors to be distributed with moderate probability throughout the allowed range, which reflects the non-Gaussian orientation error behavior observed in practice.

The measurement error of the visual measurement system for posture can be divided into position measurement error $\Delta M_p$ and attitude measurement error $\Delta M_o$. The position error is described by uniform distribution superimposed on normal distribution, as in Equation (8).

$$\Delta M_p\sim u(-u_m,-u_m)+N(0,{\sigma }_m^2{I}_3),\Delta M_o\sim N_{trunc}(0,{\sigma }_o^2{I}_3;-{\theta }_o,{\theta }_o)$$

(8)

The chosen distributions for measurement errors reflect sensor characteristics and physical constraints. The position measurement error $\Delta M_p$ is modeled as a uniform distribution (range $\pm u_m$) superimposed on a normal distribution $N(0,{\sigma }_m^2)$. The uniform term represents a bounded systematic offset or quantization uncertainty (for instance, due to finite camera pixel resolution or an unknown calibration bias, assumed equally likely anywhere in $\pm u_m$), while the Gaussian term captures random noise in the sensor measurement around the true value [47]. Meanwhile, the orientation measurement error $\Delta M_o$ is assumed to follow a truncated normal distribution $N_{\mathrm{trunc}}(0,{\sigma }_o^2;-{\theta }_o,+{\theta }_o)$, reflecting that angular sensing errors are approximately Gaussian in distribution but cannot exceed a maximum angle ${\theta }_o$. In practice, the vision system’s field-of-view and algorithm impose hard limits on detectable orientation error; by truncating at $\pm {\theta }_o$, we account for the physical limit beyond which the sensor would not provide a valid reading [48]. These distribution models thus incorporate realistic sensor noise profiles while respecting known bounds of the measurement system.

When the robot performs closed-loop correction under vision guidance, the terminal output deviation can be approximately written as Equation (9).

$$\Delta p_{auto}=J_R\Delta \theta +K_v\left[\begin{array}{c}\Delta M_p\\ \Delta M_o\end{array}\right]$$

(9)

where $J_R$ is the robot Jacobian matrix, $\Delta \theta$ is the joint angle error vector, and $K_v$ is the visual measurement mapping coefficient matrix. The statistical distribution of $\Delta p_{auto}$ is obtained through Monte Carlo sampling, which provides a data basis for the “automation factor error input” of each node in the subsequent coupling network.

3.

Process-execution error modeling

Process-execution errors are mainly caused by the dynamic response of end-effectors such as clamps or actuators during load–unload cycles. This manifests as a force–displacement hysteresis effect. Let the applied normal force be F, and the resulting displacement be $\delta \left(F\right)$. The loading–unloading path forms a closed-loop hysteresis curve, which can be represented by a piecewise polynomial model as shown in Equation (10).

$$\delta \left(F\right)=\left\{\begin{array}{c}{a}_{1}F+b_1+c_1\dot{F},\mathrm{Loading}\mathrm{branch}\hfill \\ a_{2}F+b_2+c_2\dot{F},\mathrm{Unloading}\mathrm{branch}\hfill \end{array}\right.$$

(10)

where $\dot{F}$ is the rate of force application, and the coefficients $a_i,b_i,c_i$ capture the stiffness, preload offset, and damping-related energy loss, respectively. The system damping ratio $\zeta$ can be expressed via a viscous damping coefficient c, and the estimation method is shown in Equation (11).

$$\zeta ={\displaystyle \frac{c}{2\sqrt{km}}}$$

(11)

where k is the effective stiffness of the contact interface, and m is the equivalent mass of the actuator–part system. The net displacement deviation (Equation (12)) due to hysteresis is thus defined as the maximum deviation between loading and unloading paths at a given force level:

$$\Delta \delta \sim {\delta }_{load}\left(F\right)-{\delta }_{unload}\left(F\right)$$

(12)

This residual deviation contributes to positional assembly error in the direction of the contact normal, and is mapped into the 6-DOF assembly error space via a transformation matrix $T_p$.

$$\Delta p_{\mathrm{proc}}=T_p·\left[\begin{array}{c}\Delta s_x\\ \Delta s_y\\ \Delta s_z\\ 0\\ 0\\ 0\end{array}\right]$$

(13)

Equation (13) preserves the physically observed nonlinear, rate-dependent, and asymmetric behaviors of tool–part interaction, ensuring that the process-induced positional errors are realistically captured and accurately integrated into the downstream error coupling model.

2.2.2. Multi-Source Error Coupling Network

To capture the adjacency-aware propagation of multi-source errors, we define a coupling network model as shown in Figure 4. Each part introduces three types of errors: manufacturing ($\Delta M$), alignment ($\Delta A$), and process execution ($\Delta P$). These errors propagate through inter-part adjacency weights to produce overall assembly deviations. The model incorporates not only linear and quadratic terms but also cross-coupling interactions among distinct error sources, as shown in Equation (14). The index l in the term $k<l$ denotes the second variable in a distinct error pair, ensuring each cross-term ${ϵ}_k{ϵ}_l$ is counted once. This structure reflects the directed graph in Figure 4, where nodes represent error inputs and links represent weighted coupling relations.

The notations in the figure above are defined as follows:

• $\Delta M_i$: Component-induced errors;
• $\Delta A_i$: Automation-related pose errors (including robot and sensing);
• $\Delta P_i$: Process-execution errors from operations.

Each step of the assembly process is mapped to a network node $v_i$ in sequence. Each directed edge $e_{ij}$ represents the error transmission direction from node $v_i$ to the subsequent node $v_j$ and is assigned a weight $w_{ij}\in \left[0,1\right]$ to quantify the impact of the upstream error on the downstream step. At node $v_i$, a semi-parametric polynomial coupling function $F_{v_i}$ is defined to map the three types of input errors to output deviations, as shown in Equation (14).

$${\Delta }_{out}^{\left(i\right)}=\sum _{k=1}^3{a}_k{ϵ}_k+\sum _{k=1}^3{b}_k{ϵ}_k^2+\sum _{k<l}{c}_{kl}{ϵ}_k{ϵ}_l$$

(14)

where $ϵ\in \left[\Delta M_i,\Delta A_i,\Delta P_i\right]$; $a_i$ is the linear sensitivity coefficient, reflecting the primary contribution of each single-error source to the current node; $b_i$ is the secondary amplification coefficient, which is used to describe the error accumulation or nonlinear growth effect; and $c_i$ is the cross-coupling coefficient, which is used to characterize the interaction or amplification effect between different error sources, such as the additional deviation generated when the geometric deviation and the actuator residual displacement interact with each other. On this basis, along each assembly path $p=\left\{v_0\to v_1\to \dots \to v_n\right\}$, the error recursive transmission can be expressed by Equation (15).

$${\Delta }_p=F_{v_n}\left(F_{v_{n-1}}\left(\dots F_{v_1}\left({\Delta }_{v_0}\right)\dots \right)\right)$$

(15)

For each key matching feature, what is finally obtained is an error vector, whose components correspond to different quality indicators in turn, such as gap deviation, position deviation, etc., as shown in Equation (16).

$${\Delta }_{feature}=\left[{\Delta }_{gap},{\Delta }_{pos},\dots \right]$$

(16)

where ${\Delta }_{feature}$ is the final error vector. Each characteristic component is predicted and checked separately to ensure that each component meets the corresponding tolerance threshold $E_j^{*}$, as shown in Equation (17).

$${\Delta }_{feature}\le E_j^{*},j=1,2,\dots ,m$$

(17)

The coupled network model system integrates the statistical characteristics and nonlinear cross-effects of three types of error sources, clarifies the error transmission path and amplification mechanism along the assembly process, and provides a rigorous mathematical foundation for feature-level precision control. Subsequent multi-objective tolerance optimization can obtain the quantitative impact of each tolerance parameter on the deviation of different assembly features based on this model, so as to allocate tolerances while meeting all precision thresholds. This model also supports critical path identification, which is used to determine the error node that has the greatest impact on the final assembly accuracy, providing a basis for subsequent process improvements or equipment calibration.

3. Multi-Objective Tolerance Optimization

3.1. Uncertainty Analysis and Optimization Model Construction

The multi-objective tolerance optimization is founded on a multi-source coupling model that captures how individual part tolerances propagate to assembly errors. In this framework, each geometric quality metric is expressed as a function of the relevant tolerances. These relationships are formally stated in Equations (14)–(17), which define the objective functions and constraints in terms of the coupled variation model; the optimization problem is formulated as Equation (18).

$$\underset{\theta }{min}{\Delta }_{feature}\left(\theta \right)=[{\Delta }_{gap},{\Delta }_{position},\dots ]\mathrm{s}.\mathrm{t}.{\theta }_i^{min}\le {\theta }_i\le {\theta }_i^{max},i=1,\dots ,n$$

(18)

where $\theta ={[{\theta }_1,{\theta }_2,\dots ,{\theta }_n]}^T$ represents the vector of tolerance parameters to be optimized, and ${\Delta }_{feature}\left(\theta \right)$ are the feature deviations derived from the multi-source coupling network.

To account for the uncertainty in the automated assembly process, a Monte Carlo simulation of the assembly process is performed. Random samples are drawn from a specified distribution of part tolerances, and each sample is propagated through the coupled model to calculate the resulting feature deviations. Monte Carlo simulation is well suited for nonlinear tolerance analysis because it generates statistical samples of assembly responses by evaluating the response function for many random inputs. The Monte Carlo process is as follows:

• Sample generation: Randomly generate sets of part dimensions according to their tolerance distributions;
• Assembly evaluation: For each sample, apply the multi-source coupling model to compute the assembly feature deviations such as gap, coaxiality, etc.;
• Statistical estimation: Compute performance metrics such as the mean and standard deviation of each error from the ensemble of simulated samples.

These simulated statistics serve as the basis for the optimization. The Monte Carlo outputs populate the objective functions in Equation (18), ensuring that the optimization model accounts for the actual distribution of each error under the given tolerances.

3.2. Sensitivity-Guided Multi-Objective Optimization via NSGA-II

To efficiently manage the complexity of the tolerance optimization problem, a global sensitivity analysis based on Sobol indices is conducted to identify the most influential tolerance parameters. The Sobol index for each input quantifies the proportion of output variance attributable to that parameter alone [49], as computed by Equation (19).

$$S_{Ti}={\displaystyle \frac{Va{r}_{{\theta }_i}\left(E\left[{\Delta }_{feature}\right|{\theta }_i]\right)}{Var\left({\Delta }_{feature}\right)}}$$

(19)

where $S_{T_i}$ represents the total Sobol sensitivity index for the i-th tolerance parameter ${\theta }_i$; ${\Delta }_{feature}$ represents the vector of assembly deviation metrics such as ${\Delta }_{gap}$, ${\Delta }_{position}$, etc.; $Va{r}_{{\theta }_i}\left(E\left[{\Delta }_{feature}\right|{\theta }_i]\right)$ is the variance of the conditional expectation of feature deviations, given ${\theta }_i$; and $Var\left({\Delta }_{feature}\right)$ is the total variance of the deviations.

Tolerance parameters with negligible Sobol indices are fixed at their nominal values to reduce the dimensionality of the optimization problem. With this reduced set of decision variables, the NSGA-II [22] is used for multi-objective tolerance optimization [50]. NSGA-II is a fast and elitist multi-objective genetic algorithm known for its good balance of convergence speed and diversity maintenance. It has been widely validated on engineering problems similar to ours. In particular, NSGA-II’s non-dominated sorting and crowding-distance mechanisms ensure a well-spread Pareto front, which is crucial for exploring trade-offs between positional accuracy and coaxiality in our case.

The NSGA-II algorithm operates as follows:

• Initialization: An initial population is generated by assigning random values to each retained tolerance parameter within their allowable ranges;
• Evaluation: For each candidate solution, the corresponding geometric deviations are computed using the Monte Carlo-based coupling model;
• Non-dominated sorting: All solutions are ranked into Pareto fronts based on dominance relations;
• Selection and genetic operations: Tournament selection, crossover, and mutation operators are applied to create offspring;
• Iteration: The evaluation and sorting process is repeated over multiple generations until convergence is achieved.

This approach yields a Pareto-optimal set of tolerance configurations, each representing a unique trade-off among geometric deviation objectives [51]. For example, one configuration might minimize one deviation at the expense of another, while another configuration achieves balanced control across all metrics. These trade-offs provide a basis for selecting tolerance schemes that best align with design priorities.

3.3. Optimization Procedure Flowchart

Figure 5 illustrates the complete workflow of the proposed tolerance optimization methodology, integrating multi-source uncertainty simulation, sensitivity analysis, and evolutionary optimization. The process begins with the input of initial tolerance ranges, which are modeled using the previously defined multi-source coupling network to characterize how geometric and process errors propagate through the assembly system.

The steps are detailed below:

• In the first step, a Monte Carlo simulation is conducted. This stage repeatedly samples tolerance values based on the defined uncertainty ranges and propagates them through the coupling model. By computing the resulting deviation distributions (e.g., for position and coaxiality), this step provides a statistical foundation for evaluating how tolerances impact assembly quality.
• In the second step, a Sobol sensitivity analysis is applied to the simulation results. This variance-based method quantifies the contribution of each tolerance variable to the variation in key performance indicators. By identifying the most influential tolerances, this step effectively reduces the dimensionality of the optimization problem, focusing efforts on parameters that matter most.
• In the third step, the NSGA-II multi-objective evolutionary algorithm is employed to optimize the influential tolerance parameters. NSGA-II searches for a Pareto front of solutions that balance competing objectives (e.g., minimizing positional and coaxiality deviations) while ensuring feasibility and process robustness. The algorithm iteratively evolves candidate solutions by simulating selection, crossover, and mutation across generations, guided by dominance ranking and crowding distance.

Finally, the workflow yields a set of Pareto-optimal tolerance configurations. These configurations offer flexible trade-offs that can be selected based on specific manufacturing priorities, such as cost, precision, or yield. This integrated workflow ensures that the optimized tolerances are both practically viable and statistically robust, enabling improved assembly quality under realistic manufacturing conditions.

3.4. Optimization Scenarios

To evaluate the contribution of different sources of error to the final assembly quality, three distinct optimization scenarios are defined:

• Component-only optimization: Only the tolerances related to the manufactured parts ($\Delta M$) are included as decision variables in the optimization. All process and automation-related parameters ($\Delta A$, $\Delta P$) are kept constant at their nominal values.
• Process-only optimization: Only the process-related factors, such as robotic positioning accuracy and process-execution errors ($\Delta A$ and $\Delta P$), are optimized. Component tolerances remain fixed at their initial values.
• Full optimization: All tolerance and error-related parameters ($\Delta M$, $\Delta A$, and $\Delta P$) are optimized simultaneously to achieve the best overall assembly quality.

These three optimization modes allow for a comparative analysis to isolate the impact of each error source and assess the effectiveness of targeted improvement strategies.

4. Case Study: Automatic Assembly of Helicopter Rings

4.1. Case Overview and Setup

The automatic assembly of the helicopter rotor ring (tilt mechanism) is analyzed, and its tolerance scheme is optimized. The assembly consists of a moving ring and a fixed ring, and its key quality indicators are the position alignment of the ring center and the coaxiality of its axis. Figure 6 shows the assembly setup and assembly features.

Three types of error sources are defined on the mating surface to simulate assembly uncertainty:

• $\Delta M$ for the geometric error caused by the two rings and the end fixture, initially defined as ±0.2 mm;
• $\Delta A$ for the positioning error of the industrial robot, defined as ±0.12 mm, ±0.01°, which can be found in Appendix A;
• $\Delta P$ for the jitter error caused by the gripper during the pick-and-place process, initially defined as ±0.3 mm.

Each error is applied to the corresponding mating surface in order to analyze its impact on the assembly alignment. And two key quality metrics are monitored:

• ${\Delta }_{position}$ for the positional deviation between the centers of the rings;
• ${\Delta }_{coaxiality}$ for the angular deviation between the center axes of the rings.

The realization of the fit constraints between the rings and the definition of the above errors and measurements of key quality metrics are achieved with the help of the tolerance analysis software 3DCS Variation Analyst for SOLIDWORKS (x64) Version 7.7.0.1. [52], as shown in Figure 7.

Monte Carlo simulations were performed to propagate uncertainty. For each output metric, 20,000 trials were conducted; ${\Delta }_M$, ${\Delta }_A$, and ${\Delta }_P$ were randomly drawn within their range; and the resulting deviations were calculated. Figure 8 summarizes the results. To visualize the statistical distribution of assembly deviations obtained through Monte Carlo simulations, the frequency histograms in Figure 8 and Figure 9 were constructed using the following binning method, as shown in Equation (20).

$$f_i={\displaystyle \frac{n_i}{n·w}},i=1,2,\dots ,k$$

(20)

where $f_i$ denotes the frequency (density) value of the i-th bin, $n_i$ is the number of deviation samples falling within the i-th bin interval, n is the total number of samples generated from the Monte Carlo simulation, and w is the bin width. The density curve overlaid in the plots was estimated using Gaussian kernel density estimation (KDE) for better visualization of distribution shapes.

CPK is a widely used statistical metric in manufacturing that is employed here to evaluate the assembly quality of positional and coaxiality deviations. CPK is calculated as Equation (21):

$$CPK=min\left({\displaystyle \frac{USL-\mu }{3\sigma }},{\displaystyle \frac{\mu -LSL}{3\sigma }}\right)$$

(21)

where $\mu$ is the process mean, $\sigma$ is the standard deviation obtained from Monte Carlo simulation, and $USL$ and $LSL$ represent the upper and lower specification limits defined by assembly tolerances.

The $USL$ and $LSL$ for each assembly quality feature were determined according to relevant aerospace industry standards and customer requirements for tolerance control [53]. Where customer specifications were available, these values were adopted directly. For features without explicit customer-provided limits, industry standards for precision aerospace assembly were applied to define conservative yet practical bounds. This ensures that the capability assessment is directly aligned with real-world production constraints.

CPK follows widely accepted benchmarks in manufacturing quality control, and a higher CPK indicates a more capable process [54,55]. The differences are detailed below:

• CPK $\ge 1.33$: The process capability is considered good, meaning it can stably and consistently produce products within specifications.
• $1.00\le$ CPK $<1.33$: The process capability is moderate, indicating that while products generally meet specifications, continuous monitoring and improvement are required to maintain quality.
• CPK $<1.00$: The process capability is insufficient, suggesting that the process is unable to reliably produce within specifications and corrective actions are necessary.

In our analysis, these benchmarks were used to contextualize the optimization results. For example, a CPK value of 1.11 indicates a capable process that meets specifications but leaves less margin than the “excellent” threshold, guiding further process refinement.

As shown in Figure 8, these results highlight that geometric variation ($\Delta M$) and gripper jitter ($\Delta P$) can cause significant misalignment, especially for the tight positional tolerance. Thus, a three-stage framework (Figure 5) is used for tolerance optimization:

Step 1:

Monte Carlo Uncertainty Propagation: The initial simulation quantified the combined effect of all error sources on the assembly deviations, as shown in Figure 8.

Step 2:

Global Sensitivity Analysis: The total Sobol index of each error source is calculated to distribute the output variance. The results show that $\Delta M$ contributes ~86.4% of the variance, $\Delta P$ ~12.8%, and $\Delta A$ only ~0.8%. This confirms that geometric variation is the dominant source, gripper jitter is secondary, and robot error is negligible. Consequently, $\Delta A$ is fixed at its nominal tolerance in the optimization to reduce complexity.

Step 3:

Multi-Objective Tolerance Optimization: The remaining tolerances $\Delta M$ and $\Delta P$ are optimized using NSGA-II. The goal was to minimize the combined standard deviation of ${\Delta }_{position}$ and ${\Delta }_{coaxiality}$ while improving CPK. Each candidate tolerance set was evaluated through 20,000 Monte Carlo runs to assess its performance. The algorithm identified the Pareto frontier of solutions that traded off between tight tolerances and achievable variability.

From the Pareto optimal frontier, a representative “inflection point” solution is selected to best balance assembly accuracy and manufacturing feasibility. This solution yields the following updated tolerances:

• $\Delta M$: Tightened from the initial ±0.20 mm to ±0.15 mm;
• $\Delta A$: Retained at ±0.12 mm and ±0.01°, due to its negligible influence (Sobol contribution < 1%);
• $\Delta P$: Tightened from ±0.30 mm to ±0.12 mm.

These adjustments conform to the Sobol ordering, focusing tolerance adjustments on the most influential error sources. The optimization scheme is then validated through subsequent Monte Carlo simulations.

4.2. Comparison of Optimization Strategies

To understand the effectiveness of different optimization approaches, we compare the results of component-only, process-only, and full optimization strategies. Each strategy uses the same multi-objective optimization framework, but with different sets of active decision variables:

• In the component-only strategy, only component-related tolerances ($\Delta M$) are adjusted, such as machining or manufacturing precision of parts.
• The process-only strategy focuses on reducing automation and execution errors ($\Delta A$, $\Delta P$), such as robotic positioning deviations and gripper jitter, while holding part tolerances constant.
• The full optimization strategy jointly considers all types of errors for a comprehensive improvement.

This classification allows for a fair evaluation of how much improvement can be achieved by targeting specific error sources. The resulting deviations and process capability indices (CPK) are shown in Figure 9, where each bar or curve corresponds to one of these four strategies.

In addition to the NSGA-II optimization method described in this article, we also performed comparative optimization using the Strength Pareto Evolutionary Algorithm 2 (SPEA2). SPEA2 shares the multi-objective evolutionary framework [56]. While SPEA2 employs a different fitness assignment and archive truncation strategy than NSGA-II [57], it is also widely used for multi-objective optimization. In our case, it serves as a comparative benchmark to assess the robustness and balance of NSGA-II’s solutions.

4.3. Optimization Results and Discussion

Figure 9 demonstrates the effectiveness of the NSGA-II-based multi-objective tolerance optimization in reducing positional and coaxiality deviations in the ring assembly process. As shown in Figure 9a, for positional deviation, the original tolerance scheme results in a broad spread and significant skewness, yielding a pass rate of only 67.4% and a CPK of 0.31. After applying the NSGA-II optimization, the distribution becomes sharply concentrated around the target, resulting in a pass rate of 100.0% and a CPK of 2.19, indicating excellent process capability. The SPEA2-optimized result achieves the same pass rate of 100.0%, with a slightly lower CPK of 1.41, suggesting slightly reduced robustness compared to NSGA-II. Component-only and process-only optimization schemes achieve pass rates of 99.2% and 82.5%, with CPK of 0.74 and 0.25, respectively. In Figure 9b, coaxiality deviation also shows clear improvements. The original process yields a pass rate of 93.5% and a CPK of 0.62. NSGA-II optimization raises the pass rate to 100.0% and improves the CPK to 1.06. SPEA2 optimization achieves a comparable pass rate of 99.8% and a CPK of 0.96. Component-only and process-only optimizations both yield pass rates of 99.8% and 99.2%, but with CPK of 0.86 and 0.79, respectively. These results confirm that NSGA-II provides the most balanced improvement across both quality indicators.

In our setting, the post-screening decision space is low-dimensional, and the feasible trade-off between position and coaxiality is narrow with a distinct knee. NSGA-II’s elitist non-dominated sorting on the combined parent–offspring population, followed by crowding-distance selection, preserves boundary and knee solutions and maintains a uniform spread along the front. Under Monte Carlo objective evaluations, the rank-then-crowding tournament used by NSGA-II is also less affected by small fitness fluctuations than the strength-based fitness and k-nearest-neighbor density estimator used in SPEA2, which can bias selection toward the center of the front. Consequently, NSGA-II converged more reliably to well-distributed boundary solutions, yielding slightly tighter deviation distributions and higher CPK values than SPEA2 in our experiments.

5. Discussion: Generalizability and Limitations

The proposed framework, which integrates multi-source error modeling, uncertainty propagation, and multi-objective tolerance optimization, is designed with adaptability in mind. While our case study focused on the assembly of coaxial helicopter rings, the underlying methodology is applicable to a broad range of complex assembly tasks across various industries. Our approach can be extended to assemblies involving different geometries and functional constraints, such as aircraft skin-to-frame panel assembly, automotive body-in-white processes, or spacecraft module alignment. The error propagation network (Section 2.2) can be adapted to model different contact sequences and component interactions. For example, flat panel assemblies may emphasize parallelism and flushness, while shaft-bearing fits may focus more on concentricity and clearance. Moreover, the use of a directed graph-based coupling model allows the inclusion of hierarchical or dependent assembly steps, enabling scalable modeling even in large and interdependent assemblies. As long as the error sources—such as manufacturing variability, fixture misalignment, or robot positioning errors—can be measured or estimated, the methodology can be recalibrated for a new context.

Despite its generality, several practical limitations remain: (a) The accuracy of the model relies on the representativeness of error distribution assumptions. Different industrial contexts may involve non-Gaussian errors, dynamic disturbances, or material deformations not captured by our current formulation. (b) The computational load of Monte Carlo simulations increases with assembly size and complexity; efficient sampling strategies or surrogate models may be needed for real-time applications. (c) The model currently assumes a static assembly sequence; extensions to adaptive or feedback-based strategies could further enhance robustness.

Future work could explore the integration of real-time data from digital twins or in-line sensors to adaptively update the error models. Additionally, testing alternative evolutionary optimization strategies or machine learning-assisted decision models may improve performance in highly nonlinear or multi-objective trade-off spaces. Our framework lays the foundation for these extensions and can serve as a blueprint for intelligent tolerance control in smart manufacturing systems.

6. Conclusions

This paper presents an integrated methodology for multi-objective tolerance optimization under multi-source uncertainty in automated robotic assembly. Addressing the limitations of conventional tolerance analysis in nonlinear, data-driven manufacturing environments, the proposed approach introduces three key innovations:

• A three-level uncertainty classification model that systematically captures geometric manufacturing deviations, robotic system positioning errors, and execution-induced process fluctuations;
• A directed graph-based semi-parametric coupling network that models the dynamic propagation of deviations across sequential assembly operations, enabling accurate quality estimation via Monte Carlo simulation;
• A tolerance optimization strategy that integrates Sobol-based global sensitivity analysis to identify dominant parameters with NSGA-II to explore Pareto-optimal tolerance configurations.

To verify the effectiveness of the proposed strategy, a representative coaxial ring assembly scenario was adopted. The results demonstrate significant quality improvement in both position and coaxiality metrics after optimization. For position deviation, the pass rate increased from 67.4% to 100.0%, and the CPK improved markedly from 0.31 to 2.19. For coaxiality deviation, the pass rate increased from 93.5% to 100.0%, and the CPK rose from 0.62 to 1.06. These results are detailed in Figure 8.

Comparative analysis with alternative optimization strategies (Figure 9) further underscores the robustness of our method: the NSGA-II-based scheme outperformed SPEA2, component-only, and process-only optimization strategies in both pass rate and CPK for all metrics. This highlights the importance of coordinated tolerance allocation and joint modeling under multi-source uncertainty.

Overall, this work demonstrates that the integration of sensitivity-guided optimization and dynamic deviation modeling provides a rigorous and effective pathway for tolerance synthesis in complex robotic assemblies. The proposed framework also provides a foundation for future integration into digital twin-based quality assurance systems and broader applications in intelligent manufacturing scenarios.