基于雅可比嵌入神经网络的装配误差建模方法

doi:10.11996/JG.j.2095-302X.2026030653

摘要/Abstract

摘要：

在航空航天等领域，复杂产品的装配精度直接影响其核心性能与可靠性。传统装配误差建模方法在刻画“几何-力学”耦合非线性行为时存在明显不足，尤其在处理接触变形、动力学耦合效应等问题上难以满足高精度要求。针对这一挑战，提出一种创新的雅可比旋量嵌入神经网络方法，通过在神经网络架构中融入雅可比旋量理论的物理约束，构建“雅可比旋量损失函数”与网络预测误差的联合优化机制，实现对接触变形、动力学耦合等非线性机制的精准建模。既借助神经网络的多层非线性映射能力捕捉隐含物理关联，又利用雅可比旋量理论的运动学约束对网络训练进行正则化，有效提升了模型在小样本场景下的泛化能力与物理可解释性。以双轴转台为实验对象，选取方位角与俯仰角作为模型输入，预测其空间指向误差。结果表明，相较于无雅可比嵌入的传统神经网络，该方法在x，y，z方向的预测精度分别提升了46.51%，37.44%和50.86%；经五折交叉验证，其在均方误差、均方根误差、平均绝对误差及决定系数4项关键指标上均优于传统方法，不仅预测误差显著降低，且稳定性与泛化能力大幅提升。由物理机制约束与数据驱动拟合相融合的跨域框架，为解决高精度装配场景中的非线性传递难题提供了新途径。

关键词: 装配误差, 雅可比旋量模型, 非线性误差, 神经网络, 数据-物理融合

Abstract:

In high-precision manufacturing domains such as aerospace, the assembly accuracy of complex products fundamentally determines their core performance and operational reliability. Traditional assemblyerror modeling methods exhibit significant limitations in capturing the nonlinear behaviors associated with “geometry-mechanics” coupling—particularly in accurately describing contact deformation and dynamic coupling effects, which are critical for high-precision applications. To address these challenges, an innovative Jacobian-Torsor Embedded Neural Network (JENN) approach was proposed. Physical constraints derived from Jacobian-Torsor theory were embedded into the neural network architecture, establishing a joint optimization mechanism that simultaneously minimized the “Jacobian-Torsor loss function” and the network’s prediction error, enabling precise modeling of complex nonlinear mechanisms such as contact deformation and dynamic coupling. The proposed method leveraged the multilayer nonlinear mapping capability of neural networks to capture latent physical correlations, while using kinematic constraints from Jacobian-Torsor theory to regularize network training. As a result, the model’s generalization ability and physical interpretability were significantly enhanced, particularly in small-sample scenarios. Experimental validation on a dual-axis turntable used azimuth and elevation angles as inputs to predict spatial pointing errors, demonstrating substantial improvements: compared with conventional networks without Jacobian embedding, prediction accuracy improved by 46.51%, 37.44%, and 50.86% in the x, y, and z directions, respectively. Five-fold cross-validation further showed that the proposed method consistently outperformed the conventional approach across four key metrics (MSE, RMSE, MAE, and R2), yielding lower prediction errors as well as markedly enhanced stability and generalization. This cross-domain fusion framework, integrating physics-based mechanism constraints with data-driven fitting, provided a novel pathway for addressing nonlinear transfer challenges in high-precision assembly applications.

Key words: assembly error, Jacobian-Torsor model, nonlinear error, neural network, data-physics fusion

中图分类号:

TP391

赵罡, 曾远志, 刘亚醉, 申皓东. 基于雅可比嵌入神经网络的装配误差建模方法[J]. 图学学报, 2026, 47(3): 653-660.

ZHAO Gang, ZENG Yuanzhi, LIU Yazui, SHEN Haodong. Assembly error modeling method based on Jacobian-Torsor embedded neural network[J]. Journal of Graphics, 2026, 47(3): 653-660.

图/表 8

图1 基于雅可比嵌入神经网络的装配误差预测方法

Fig. 1 Assembly error prediction method using Jacobian-embedded neural network

图2 双轴转台模型((a) 双轴转台及坐标系的建立；(b) 双轴转台尺寸链等效模型)

Fig. 2 Dual-axis turntable model ((a) Establishment of dual-axis turntable and coordinate systems; (b) Functionally equivalent dimensional chain model of dual-axis turntable)

图3 双轴转台指向误差测量实验

Fig. 3 Pointing error measurement experiment for the dual-axis turntable

表1 不同装配误差源所对应的雅可比矩阵

Table 1 Jacobian matrices for different assembly error sources

装配误差	雅可比矩阵
${J}_{1}$	$\left[\begin{array}{cccccc}1& 0& 0& 0& 225& 0\\ 0& 1& 0& -225& 0& 0\\ 0& 0& 1& 0& 0& 0\\ 0& 0& 0& 1& 0& 0\\ 0& 0& 0& 0& 1& 0\\ 0& 0& 0& 0& 0& 1\end{array}\right]$
${J}_{2}$	$\left[\begin{array}{cccccc}\mathrm{cos}\alpha & -\mathrm{sin}\alpha & 0& 225\mathrm{sin}\alpha & 225\mathrm{cos}\alpha & -126\mathrm{sin}\alpha \\ \mathrm{sin}\alpha & \mathrm{cos}\alpha & 0& -225\mathrm{cos}\alpha & 225\mathrm{sin}\alpha & -126\mathrm{cos}\alpha \\ 0& 0& 1& 126m& -126n& 0\\ 0& 0& 0& \mathrm{cos}\alpha & -\mathrm{sin}\alpha & 0\\ 0& 0& 0& \mathrm{sin}\alpha & \mathrm{cos}\alpha & 0\\ 0& 0& 0& 0& 0& 1\end{array}\right]$
${J}_{3}$	$\left[\begin{array}{cccccc}1& 0& 0& 0& 0& 0\\ 0& \mathrm{cos}\beta & -\mathrm{sin}\beta & 0& 126\mathrm{sin}\beta & 126\mathrm{cos}\alpha \\ 0& \mathrm{sin}\beta & \mathrm{cos}\beta & 0& -126\mathrm{cos}\alpha & 126\mathrm{sin}\beta \\ 0& 0& 0& 1& 0& 0\\ 0& 0& 0& 0& \mathrm{cos}\beta & -\mathrm{sin}\beta \\ 0& 0& 0& 0& \mathrm{sin}\beta & \mathrm{cos}\beta \end{array}\right]$
${J}_{4}$	$\left[\begin{array}{cccccc}1& 0& 0& 0& 5& 0\\ 0& 1& 0& -5& 0& 0\\ 0& 0& 1& 0& 0& 0\\ 0& 0& 0& 1& 0& 0\\ 0& 0& 0& 0& 1& 0\\ 0& 0& 0& 0& 0& 1\end{array}\right]$

图4 雅可比嵌入神经网络和传统神经网络的损失函数

Fig. 4 Loss functions of the Jacobian-embedded neural network and the conventional neural network

图5 各模型预测值与真实值对比图((a) X向预测对比图；(b) Y向预测对比图；(c) Z向预测对比图)

Fig. 5 Comparison of predicted values from different models with ground truth ((a) X-direction prediction comparison; (b) Y-direction prediction comparison; (c) Z-direction prediction comparison)

图6 传统神经网络与不同权重下的雅可比嵌入网络的预测误差对比((a) X向预测误差对比图；(b) Y向预测误差对比图；(c) Z向预测误差对比图；(d) 总预测误差曲面对比图)

Fig. 6 Comparison of prediction errors between the baseline neural network and the Jacobian-embedded neural network under different loss weights ((a) X-direction prediction error; (b) Y-direction prediction error; (c) Z-direction prediction error; (d) Comparative surface plot of total prediction error dimensional chain model of dual-axis turntable)

表2 基于五折交叉验证的模型性能对比分析

Table 2 Performance comparison based on 5-fold cross-validation

方法	MSE	RMSE	MAE	${R}^{2}$
雅可比嵌入神经网络	$0.000\text{ }124\pm 0.000\text{ }026$	$0.010\text{ }665\pm 0.000\text{ }983$	$0.008\text{ }071\pm 0.000\text{ }672$	$0.869\text{ }270\pm 0.026\text{ }523$
传统神经网络	$0.000\text{ }185\pm 0.000\text{ }037$	$0.013\text{ }576\pm 0.001\text{ }264$	$0.009\text{ }429\pm 0.001\text{ }038$	$0.825\text{ }965\pm 0.037\text{ }604$

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