Research on assembly accuracy prediction of complex products considering rough surfaces

doi:10.11996/JG.j.2095-302X.2026010162

Abstract

Abstract:

Given that the impact of rough surfaces on assembly accuracy had been insufficiently considered in the existing assembly accuracy prediction for complex products, leading to inaccurate precision prediction and limited practical assembly applicability, an assembly-accuracy prediction method considering rough surfaces was proposed. Firstly, an assembly-accuracy information model was constructed to express mating feature, geometric tolerance, and roughness information. Based on the model, an assembly-precision knowledge graph was constructed. Secondly, a geometric-tolerance representation model was established based on the Small-Displacement Torsor (SDT) theory; a simulation method for rough surfaces of plane and cylindrical parts as well as a determination method of SDT expressions were studied. Thirdly, the error-propagation path of the assembly was determined according to the assembly sequence, and a pose-relationship graph for the assembly was constructed. Then, the assembly-precision prediction was achieved using a Jacobian-torsor model. Finally, the feasibility of the method was verified using the crank-connecting-rod mechanism of a specific construction-machine model as an example. The simulation results demonstrated that the method could achieve accurate assembly-precision prediction and provided valuable guidance for practical assembly operations.

Key words: assembly accuracy prediction, rough surface, knowledge graph, small displacement torsor, Jacobian matrix

CLC Number:

TG95

WANG Gangfeng, ZHANG Huan, YANG Yingying, LIU Yitao, GUO Yanyun, YUE Ping, SUN Yanhui. Research on assembly accuracy prediction of complex products considering rough surfaces[J]. Journal of Graphics, 2026, 47(1): 162-172.

Figures/Tables 19

References 18

[1]	余海东, 高畅, 赵勇, 等. 机械产品装配偏差分析方法研究进展与展望[J]. 机械工程学报, 2023, 59(9): 212-229. DOI
	YU H D, GAO C, ZHAO Y, et al. Progress and prospect on assembly deviation propagation of mechanical products[J]. Journal of Mechanical Engineering, 2023, 59(9): 212-229 (in Chinese). DOI
[2]	刘检华, 夏焕雄, 巩浩, 等. 精密装配的内涵､技术体系和发展趋势[J]. 机械工程学报, 2023, 59(20): 436-450. DOI
	LIU J H, XIA H X, GONG H, et al. Connotation, technical system and development trend of precision assembly[J]. Journal of Mechanical Engineering, 2023, 59(20): 436-450 (in Chinese). DOI
[3]	王平, 刘象, 张义, 等. 基于联接面误差传递机理的高精装备装配精度预测与公差优化研究[J]. 西安理工大学学报, 2025, 41(2): 257-265.
	WANG P, LIU X, ZHANG Y, et al. Research on assembly accuracy prediction and tolerance optimization of high precision equipment based on error transfer mechanism for joint surface[J]. Journal of Xi’an University of Technology, 2025, 41(2): 257-265 (in Chinese).
[4]	ZOU Z H, MORSE E P. A gap-based approach to capture fitting conditions for mechanical assembly[J]. Computer-Aided Design, 2004, 36(8): 691-700. DOI URL
[5]	唐健钧, 田锡天, 耿俊浩, 等. 基于多维矢量环的装配偏差源敏感度分析[J]. 机械工程学报, 2015, 51(17): 156-161. DOI
	TANG J J, TIAN X T, GENG J H, et al. Sensitivity analysis of assembly deviation source based on multidimensional vector loop[J]. Journal of Mechanical Engineering, 2015, 51(17): 156-161 (in Chinese). DOI
[6]	陈华, 林观生, 唐涛, 等. 一种新的矩阵公差模型约束条件建立和优化方法[J]. 机械工程学报, 2016, 52(1): 123-129. DOI
	CHEN H, LIN G S, TANG T, et al. A new approach of constraints establishment and optimization for matrix tolerance model[J]. Journal of Mechanical Engineering, 2016, 52(1): 123-129 (in Chinese). DOI
[7]	DESROCHERS A, CLÉMENT A. A dimensioning and tolerancing assistance model for CAD/CAM systems[J]. The International Journal of Advanced Manufacturing Technology, 1994, 9(6): 352-361. DOI URL
[8]	肖欢, 朱永国, 刘春锋, 等. 基于T-Map的飞机部件交点轴线公差转化方法[J]. 中国机械工程, 2019, 30(13): 1558-1567.
	XIAO H, ZHU Y G, LIU C F, et al. Tolerance conversion of aircraft component intersection axes based on T-Map[J]. China Mechanical Engineering, 2019, 30(13): 1558-1567 (in Chinese).
[9]	位跃东, 方峻. 基于SDT模型和蒙特卡洛法的枪械抛壳机构公差分析[J]. 兵器装备工程学报, 2023, 44(4): 146-154.
	WEI Y D, FANG J. Tolerance analysis of the gun shell-throwing mechanism based on SDT model and Monte Carlo method[J]. Journal of Ordnance Equipment Engineering, 2023, 44(4): 146-154 (in Chinese).
[10]	张喜庆, 张爱梅. 加工误差和变形作用下的误差传递模型研究[J]. 现代制造工程, 2021(12): 74-78.
	ZHANG X Q, ZHANG A M. Research on error transmission model of machining and deformation error[J]. Modern Manufacturing Engineering, 2021(12): 74-78 (in Chinese).
[11]	陈振宇, 徐武彬, 李冰, 等. 滑动轴承转子系统中制造误差建模方法的研究[J]. 机械设计与制造, 2024(3): 271-276.
	CHEN Z Y, XU W B, LI B, et al. Modeling method of manufacturing error in sliding bearing rotor system[J]. Machinery Design & Manufacture, 2024(3): 271-276 (in Chinese).
[12]	王仲奇, 杨盼, 陈世洁, 等. 飞机舱门数字孪生模型构建及偏差传递分析研究[J]. 航空制造技术, 2022, 65(12): 36-47.
	WANG Z Q, YANG P, CHEN S J, et al. Research on construction of digital twin model and deviation transfer analysis of cabin door of airplane[J]. Aeronautical Manufacturing Technology, 2022, 65(12): 36-47 (in Chinese).
[13]	杨灵锋, 吴玉光. 几何要素误差传递关系图的建立方法[J]. 图学学报, 2020, 41(2): 295-303. DOI
	YANG L F, WU Y G. Construction method of error transfer graph of geometric elements[J]. Journal of Graphics, 2020, 41(2): 295-303 (in Chinese).
[14]	彭和平, 周志鹏. 多工序制造过程的几何偏差表示与变动传递建模[J]. 机械设计与制造, 2024(10): 83-88.
	PENG H P, ZHOU Z P. Geometric deviation representation and variation propagation modeling in multistage machining processes[J]. Machinery Design & Manufacture, 2024(10): 83-88 (in Chinese).
[15]	易扬, 冯锦丹, 刘金山, 等. 复杂产品数字孪生装配模型表达与精度预测[J]. 计算机集成制造系统, 2021, 27(2): 617-630. DOI
	YI Y, FENG J D, LIU J S, et al. Model expression and accuracy prediction method of digital twin-based assembly for complex products[J]. Computer Integrated Manufacturing Systems, 2021, 27(2): 617-630 (in Chinese).
[16]	DESROCHERS A, GHIE W, LAPERRIE`RE L. Application of a unified Jacobian—torsor model for tolerance analysis[J]. Journal of Computing and Information Science in Engineering, 2003, 3(1): 2-14. DOI URL
[17]	石嵩, 刘检华, 巩浩, 等. 考虑粗糙表面接触配合的航空发动机多级转子装配误差传递建模[J]. 机械工程学报, 2023, 59(17): 208-219. DOI
	SHI S, LIU J H, GONG H, et al. Modeling of error transfer in multistage rotors assembly of aero engine considering rough surface contact[J]. Journal of Mechanical Engineering, 2023, 59(17): 208-219 (in Chinese). DOI
[18]	朱林波, 梁洪瑀, 张瀚文, 等. 考虑宏微观几何形貌的螺栓连接结合面接触性能模型[J]. 西安交通大学学报, 2024, 58(7): 148-159.
	ZHU L B, LIANG H Y, ZHANG H W, et al. Contact performance model for bolted joint interfaces considering macro-micro morphology[J]. Journal of Xi’an Jiaotong University, 2024, 58(7): 148-159 (in Chinese).

公差带	SDT表达式	变量变动范围
圆度	$D={[\begin{array}{cccccc}{d}_{x}& {d}_{y}& 0& 0& 0& 0\end{array}]}^{\text{T}}$	$\begin{array}{l}-\frac{{T}_{c}}{2}\le {d}_{x}\le \frac{{T}_{c}}{2}\\ -\frac{{T}_{c}}{2}\le {d}_{y}\le \frac{{T}_{c}}{2}\end{array}$
平面度	$D={[\begin{array}{cccccc}0& 0& {d}_{z}& {\theta }_{x}& {\theta }_{y}& 0\end{array}]}^{\text{T}}$	$\begin{array}{cc}-\frac{{T}_{f}}{2}\le {d}_{z}\le \frac{{T}_{f}}{2}& \begin{array}{c}-\frac{{T}_{f}}{b}\le {\theta }_{x}\le \frac{{T}_{f}}{b}\\ -\frac{{T}_{f}}{a}\le {\theta }_{y}\le \frac{{T}_{f}}{a}\end{array}\end{array}$
圆柱度	$D={[\begin{array}{cccccc}0& {d}_{y}& {d}_{z}& 0& {\theta }_{y}& \theta \end{array}]}^{\text{T}}$	$\begin{array}{cc}-\frac{{T}_{c}}{2}\le {d}_{y}\le \frac{{T}_{c}}{2}& -\frac{{T}_{c}}{l}\le {\theta }_{y}\le \frac{{T}_{c}}{l}\\ -\frac{{T}_{c}}{2}\le {d}_{z}\le \frac{{T}_{c}}{2}& -\frac{{T}_{c}}{l}\le {\theta }_{z}\le \frac{{T}_{c}}{l}\end{array}$
直线度	$D={[\begin{array}{cccccc}0& {d}_{y}& {d}_{z}& 0& {\theta }_{y}& \theta \end{array}]}^{\text{T}}$	$\begin{array}{cc}-\frac{{T}_{c}}{2}\le {d}_{x}\le \frac{{T}_{c}}{2}& -\frac{{T}_{c}}{l}\le {\theta }_{x}\le \frac{{T}_{c}}{l}\\ -\frac{{T}_{c}}{2}\le {d}_{y}\le \frac{{T}_{c}}{2}& -\frac{{T}_{c}}{l}\le {\theta }_{y}\le \frac{{T}_{c}}{l}\end{array}$
平行度	$D={[\begin{array}{cccccc}0& 0& {d}_{z}& {\theta }_{x}& {\theta }_{y}& 0\end{array}]}^{\text{T}}$	$\begin{array}{cc}-\frac{{T}_{p}}{2}\le {d}_{z}\le \frac{{T}_{p}}{2}& \begin{array}{c}-\frac{{T}_{p}}{b}\le {\theta }_{x}\le \frac{{T}_{p}}{b}\\ -\frac{{T}_{p}}{a}\le {\theta }_{y}\le \frac{{T}_{p}}{a}\end{array}\end{array}$

公差带	SDT表达式	变量变动范围
圆度	$D={[\begin{array}{cccccc}{d}_{x}& {d}_{y}& 0& 0& 0& 0\end{array}]}^{\text{T}}$	$\begin{array}{l}-\frac{{T}_{c}}{2}\le {d}_{x}\le \frac{{T}_{c}}{2}\\ -\frac{{T}_{c}}{2}\le {d}_{y}\le \frac{{T}_{c}}{2}\end{array}$
平面度	$D={[\begin{array}{cccccc}0& 0& {d}_{z}& {\theta }_{x}& {\theta }_{y}& 0\end{array}]}^{\text{T}}$	$\begin{array}{cc}-\frac{{T}_{f}}{2}\le {d}_{z}\le \frac{{T}_{f}}{2}& \begin{array}{c}-\frac{{T}_{f}}{b}\le {\theta }_{x}\le \frac{{T}_{f}}{b}\\ -\frac{{T}_{f}}{a}\le {\theta }_{y}\le \frac{{T}_{f}}{a}\end{array}\end{array}$
圆柱度	$D={[\begin{array}{cccccc}0& {d}_{y}& {d}_{z}& 0& {\theta }_{y}& \theta \end{array}]}^{\text{T}}$	$\begin{array}{cc}-\frac{{T}_{c}}{2}\le {d}_{y}\le \frac{{T}_{c}}{2}& -\frac{{T}_{c}}{l}\le {\theta }_{y}\le \frac{{T}_{c}}{l}\\ -\frac{{T}_{c}}{2}\le {d}_{z}\le \frac{{T}_{c}}{2}& -\frac{{T}_{c}}{l}\le {\theta }_{z}\le \frac{{T}_{c}}{l}\end{array}$
直线度	$D={[\begin{array}{cccccc}0& {d}_{y}& {d}_{z}& 0& {\theta }_{y}& \theta \end{array}]}^{\text{T}}$	$\begin{array}{cc}-\frac{{T}_{c}}{2}\le {d}_{x}\le \frac{{T}_{c}}{2}& -\frac{{T}_{c}}{l}\le {\theta }_{x}\le \frac{{T}_{c}}{l}\\ -\frac{{T}_{c}}{2}\le {d}_{y}\le \frac{{T}_{c}}{2}& -\frac{{T}_{c}}{l}\le {\theta }_{y}\le \frac{{T}_{c}}{l}\end{array}$
平行度	$D={[\begin{array}{cccccc}0& 0& {d}_{z}& {\theta }_{x}& {\theta }_{y}& 0\end{array}]}^{\text{T}}$	$\begin{array}{cc}-\frac{{T}_{p}}{2}\le {d}_{z}\le \frac{{T}_{p}}{2}& \begin{array}{c}-\frac{{T}_{p}}{b}\le {\theta }_{x}\le \frac{{T}_{p}}{b}\\ -\frac{{T}_{p}}{a}\le {\theta }_{y}\le \frac{{T}_{p}}{a}\end{array}\end{array}$

雅可比矩阵	矩阵值	雅可比矩阵	矩阵值
${J}_{s1P1}$	$\left[\begin{array}{cccccc}1& 0& 0& 0& -151& 0\\ 0& 1& 0& 0& 0& 0\\ 0& 0& 1& 151& 0& 0\\ 0& 0& 0& 1& 0& 0\\ 0& 0& 0& 0& 1& 0\\ 0& 0& 0& 0& 0& 1\end{array}\right]$	${J}_{s2P3}$	$\left[\begin{array}{cccccc}1& 0& 0& 0& -151& 0\\ 0& 1& 0& 0& 0& 0\\ 0& 0& 1& 151& 0& 0\\ 0& 0& 0& 1& 0& 0\\ 0& 0& 0& 0& 1& 0\\ 0& 0& 0& 0& 0& 1\end{array}\right]$
${J}_{s2P1}$	$\left[\begin{array}{cccccc}1& 0& 0& 0& -151& 0\\ 0& 1& 0& 0& 0& 0\\ 0& 0& 1& 151& 0& 0\\ 0& 0& 0& 1& 0& 0\\ 0& 0& 0& 0& 1& 0\\ 0& 0& 0& 0& 0& 1\end{array}\right]$	${J}_{s3P3}$	$\left[\begin{array}{cccccc}1& 0& 0& 0& -151& 0\\ 0& 1& 0& 0& 0& 0\\ 0& 0& 1& 151& 0& 0\\ 0& 0& 0& 1& 0& 0\\ 0& 0& 0& 0& 1& 0\\ 0& 0& 0& 0& 0& 1\end{array}\right]$
${J}_{s1P2}$	$\left[\begin{array}{cccccc}1& 0& 0& 0& -151& 0\\ 0& 1& 0& 0& 0& 0\\ 0& 0& 1& 151& 0& 0\\ 0& 0& 0& 1& 0& 0\\ 0& 0& 0& 0& 1& 0\\ 0& 0& 0& 0& 0& 1\end{array}\right]$	${J}_{s1P4}$	$\left[\begin{array}{cccccc}1& 0& 0& 0& 15& 0\\ 0& 1& 0& -15& 0& 0\\ 0& 0& 1& 0& 0& 0\\ 0& 0& 0& 1& 0& 0\\ 0& 0& 0& 0& 1& 0\\ 0& 0& 0& 0& 0& 1\end{array}\right]$
${J}_{s1P3}$	$\left[\begin{array}{cccccc}1& 0& 0& 0& -151& 0\\ 0& 1& 0& 0& 0& 0\\ 0& 0& 1& 151& 0& 0\\ 0& 0& 0& 1& 0& 0\\ 0& 0& 0& 0& 1& 0\\ 0& 0& 0& 0& 0& 1\end{array}\right]$	${J}_{s1P5}$	$\left[\begin{array}{cccccc}1& 0& 0& 0& 0& 0\\ 0& 1& 0& 0& 0& 0\\ 0& 0& 1& 0& 0& 0\\ 0& 0& 0& 1& 0& 0\\ 0& 0& 0& 0& 1& 0\\ 0& 0& 0& 0& 0& 1\end{array}\right]$

雅可比矩阵	矩阵值	雅可比矩阵	矩阵值
${J}_{s1P1}$	$\left[\begin{array}{cccccc}1& 0& 0& 0& -151& 0\\ 0& 1& 0& 0& 0& 0\\ 0& 0& 1& 151& 0& 0\\ 0& 0& 0& 1& 0& 0\\ 0& 0& 0& 0& 1& 0\\ 0& 0& 0& 0& 0& 1\end{array}\right]$	${J}_{s2P3}$	$\left[\begin{array}{cccccc}1& 0& 0& 0& -151& 0\\ 0& 1& 0& 0& 0& 0\\ 0& 0& 1& 151& 0& 0\\ 0& 0& 0& 1& 0& 0\\ 0& 0& 0& 0& 1& 0\\ 0& 0& 0& 0& 0& 1\end{array}\right]$
${J}_{s2P1}$	$\left[\begin{array}{cccccc}1& 0& 0& 0& -151& 0\\ 0& 1& 0& 0& 0& 0\\ 0& 0& 1& 151& 0& 0\\ 0& 0& 0& 1& 0& 0\\ 0& 0& 0& 0& 1& 0\\ 0& 0& 0& 0& 0& 1\end{array}\right]$	${J}_{s3P3}$	$\left[\begin{array}{cccccc}1& 0& 0& 0& -151& 0\\ 0& 1& 0& 0& 0& 0\\ 0& 0& 1& 151& 0& 0\\ 0& 0& 0& 1& 0& 0\\ 0& 0& 0& 0& 1& 0\\ 0& 0& 0& 0& 0& 1\end{array}\right]$
${J}_{s1P2}$	$\left[\begin{array}{cccccc}1& 0& 0& 0& -151& 0\\ 0& 1& 0& 0& 0& 0\\ 0& 0& 1& 151& 0& 0\\ 0& 0& 0& 1& 0& 0\\ 0& 0& 0& 0& 1& 0\\ 0& 0& 0& 0& 0& 1\end{array}\right]$	${J}_{s1P4}$	$\left[\begin{array}{cccccc}1& 0& 0& 0& 15& 0\\ 0& 1& 0& -15& 0& 0\\ 0& 0& 1& 0& 0& 0\\ 0& 0& 0& 1& 0& 0\\ 0& 0& 0& 0& 1& 0\\ 0& 0& 0& 0& 0& 1\end{array}\right]$
${J}_{s1P3}$	$\left[\begin{array}{cccccc}1& 0& 0& 0& -151& 0\\ 0& 1& 0& 0& 0& 0\\ 0& 0& 1& 151& 0& 0\\ 0& 0& 0& 1& 0& 0\\ 0& 0& 0& 0& 1& 0\\ 0& 0& 0& 0& 0& 1\end{array}\right]$	${J}_{s1P5}$	$\left[\begin{array}{cccccc}1& 0& 0& 0& 0& 0\\ 0& 1& 0& 0& 0& 0\\ 0& 0& 1& 0& 0& 0\\ 0& 0& 0& 1& 0& 0\\ 0& 0& 0& 0& 1& 0\\ 0& 0& 0& 0& 0& 1\end{array}\right]$

接触面配合	配合类型	SDT表达式
s1P1-s1P2	柱面配合	${[\begin{array}{cccccc}-0.1157& -0.2483& 0& -0.0461& -0.0232& 0\end{array}]}^{\text{T}}\times {10}^{-2}$
s2P1-s1P3	平面配合	${[\begin{array}{cccccc}0& 0& 0.1302& 0.0316& -0.1455& 0\end{array}]}^{\text{T}}\times {10}^{-2}$
s2P3-s1P2	柱面配合	${[\begin{array}{cccccc}-0.1206& -0.2357& 0& -0.0391& -0.0209& 0\end{array}]}^{\text{T}}\times {10}^{-2}$
s3P3-s1P4	柱面配合	${[\begin{array}{cccccc}-0.156& -0.2195& 0& -0.0396& -0.0301& 0\end{array}]}^{\text{T}}\times {10}^{-2}$
s1P4-s1P5	柱面配合	${[\begin{array}{cccccc}-1.498& -0.653& 0& -0.0511& -0.0326& 0\end{array}]}^{\text{T}}\times {10}^{-3}$