Research on Decoupling of Assembly Errors and Optimization of Positioning Accuracy for Precision Linear Platforms at Micro-scale

doi:10.3969/j.issn.1004-132X.2026.09.019

Abstract

Abstract:

To improve positioning accuracies of precision linear platforms at micro-stroke scales， based on their mechanism under different displacement scales and assembly errors， an optimization method for assembly error decoupling and positioning accuracy improvement was proposed. First， using a refined probability density function transfer operator and the many-to-many error distribution coupling mappings of “components–assembly error items-geometric accuracies”， a comprehensive prediction model for precision motion platforms was established， including positioning accuracy variation trends during assembly and assembly error transfer laws； second， with assembly sequences as variables， error measurement verifications and coupling mapping characterizations were performed by indirectly changing assembly error couplings； then， based on assembly error transfer laws， platform assembly optimizations and positioning accuracy improvements were realized.Optimized assembly results show that， per ISO standards， at a 1 mm stroke， bidirectional positioning error decreases from 20 μm to 6 μm， absolute distribution means from 3.9 μm to 0.5 μm， and standard deviations from 4.9 μm to 2.4 μm， indicating that this method is effective for improving precision linear platform positioning accuracies.

Key words: micro-distance, probability density function, assembly error, error modeling, positioning accuracy optimization

摘要：

为提高微行程尺度下精密线性平台定位精度，根据不同位移尺度定位精度机理，基于装配误差，提出一种装配误差解耦与定位精度提高优化方法。首先，采用提炼的概率密度函数传递算子，通过“组件-装配误差项-几何精度”多对多误差分布耦合映射关系，建立包括装配过程定位精度变化趋势，以及装配误差传递规律的精密运动平台综合预测模型；然后，以装配次序为变量，通过间接改变装配误差的耦合关系，进行误差测量验证和耦合映射关系表征；最后，依据误差传递规律，实现了平台装配优化和定位精度的提高。优化后的装配实验结果表明：依照ISO标准，在1 mm行程下双向定位误差由20 μm降低到6 μm，分布均值绝对值从3.9 μm降低至0.5 μm，标准差从4.9 μm降低至2.4 μm，说明该方法对精密线性平台定位精度的提高行之有效。

关键词: 微行程, 概率密度函数, 装配误差, 误差建模, 定位精度优化

CLC Number:

TH115

YAN Dingwei, TANG Hao, ZHENG Yu, LI Heting. Research on Decoupling of Assembly Errors and Optimization of Positioning Accuracy for Precision Linear Platforms at Micro-scale[J]. China Mechanical Engineering, 2026, 37(9): 2268-2277.

严定伟, 唐皓, 郑煜, 李鹤葶. 微行程尺度下精密线性平台装配误差解耦与定位精度优化研究[J]. 中国机械工程, 2026, 37(9): 2268-2277.

Figures/Tables 26

Fig.1 Multi-stage system based on error flow theory

Fig.2 Assembly error coupling theory

Fig.3 Flowchart for assembly error decoupling based on error flow theory

Fig.4 Precision linear slide table structure diagram

Fig.5 Precision linear stage assembly topology diagram

Fig.6 Basic assembly form diagram

Fig.7 The error modeling mechanism

Tab.1 Summary of the matrix representation of basic assembly forms

装配类型

齐次坐标矩阵

面-面装配

E = 1 - ε z 0 δ x ε z 10 δ y 00100001

孔-轴装配

E = 1 - ε z ε y δ x ε z 1 - ε x δ y - ε y - ε x 1 δ z 0001

Tab.1 Summary of the matrix representation of basic assembly forms

装配类型

齐次坐标矩阵

面-面装配

E = 1 - ε z 0 δ x ε z 10 δ y 00100001

孔-轴装配

E = 1 - ε z ε y δ x ε z 1 - ε x δ y - ε y - ε x 1 δ z 0001

Fig.8 Analysis diagram of rail parallelism error

Fig.9 Analysis of assembly error in the position of the screw

Fig.10 Analysis diagram of coaxiality error motion

Tab.2 Summary of parameter expressions

几何误差	参数表达式
平行度	$ε z g = [(ε z g 1 + ε z t) + (ε z g 2 + ε z t)] 2$
丝杠位姿（载物台-丝杠-轴承座）	$ε s z = - ε z b - ε z s - ε z t$
丝杠位姿（轴承座-丝杠-载物台）	$ε s z = ε z t + ε z s + ε z b + ε x s' ε y s$
同轴度（丝杠-轴承座-电机轴）	$δ x = δ x m - δ y m (ε z b + ε z s) + δ z m ε y z + δ x b - ε z s δ y b + δ x s δ y = δ x m (ε z s + ε z b) + δ y m - ε x b δ z m + δ x b + ε z s δ x b + δ y s$
同轴度（电机轴-轴承座-丝杠）	$δ x = δ x s - δ y s (ε z b + ε z m) + δ z s (ε y m' + ε y m) + ε y m δ z m' + δ x b - ε z m δ y b + δ x m δ y = δ x s (ε z m + ε z b) + δ y s - δ z s (ε x m' + ε x m) - ε x m δ z m' + ε z m δ x b + δ y b + δ y m$

Tab.2 Summary of parameter expressions

几何误差	参数表达式
平行度	$ε z g = [(ε z g 1 + ε z t) + (ε z g 2 + ε z t)] 2$
丝杠位姿（载物台-丝杠-轴承座）	$ε s z = - ε z b - ε z s - ε z t$
丝杠位姿（轴承座-丝杠-载物台）	$ε s z = ε z t + ε z s + ε z b + ε x s' ε y s$
同轴度（丝杠-轴承座-电机轴）	$δ x = δ x m - δ y m (ε z b + ε z s) + δ z m ε y z + δ x b - ε z s δ y b + δ x s δ y = δ x m (ε z s + ε z b) + δ y m - ε x b δ z m + δ x b + ε z s δ x b + δ y s$
同轴度（电机轴-轴承座-丝杠）	$δ x = δ x s - δ y s (ε z b + ε z m) + δ z s (ε y m' + ε y m) + ε y m δ z m' + δ x b - ε z m δ y b + δ x m δ y = δ x s (ε z m + ε z b) + δ y s - δ z s (ε x m' + ε x m) - ε x m δ z m' + ε z m δ x b + δ y b + δ y m$

Fig.11 “Component-assembly error term - geometric precision” coupling relationship diagram

Fig.12 The processes of establishing the probability density function for the overall assembly errors

Fig.13 Measurement of positioning error

Fig.14 Measurement of component pose

Fig.15 Component pose parameter distribution diagram

Tab.3 The distribution characteristics and intervals of each

位姿参数	分布区间	分布特点
ε_z_g，ε_z_s	（ $-$ 3，3）mrad	正态分布
ε_z_t	（ $-$ 5，5）mrad	正态分布
δ_x_m，δ_y_m，δ_z_m	（ $-$ 20，20）μm	正态分布
ε_y_b，ε_z_b	（ $-$ 7，7）mrad	正态分布
δ_x_s，δ_y_s	（ $-$ 20，20）μm	正态分布

Tab.3 The distribution characteristics and intervals of each

位姿参数	分布区间	分布特点
ε_z_g，ε_z_s	（ $-$ 3，3）mrad	正态分布
ε_z_t	（ $-$ 5，5）mrad	正态分布
δ_x_m，δ_y_m，δ_z_m	（ $-$ 20，20）μm	正态分布
ε_y_b，ε_z_b	（ $-$ 7，7）mrad	正态分布
δ_x_s，δ_y_s	（ $-$ 20，20）μm	正态分布

Fig.16 Guide rail parallelism-positioningerror diagram

Tab.4 Summary of results

装配次序

定位误差均值

μ/μm

导轨参数

ε_z_g/（°）

Fig.17 Screw axis position error-coaxiality verification experimental device diagram

Fig.18 Assembly parameters for bearing seat-positioning error diagram

Fig.19 Assembly sequence optimization diagram

Fig.20 ISO standard positioning error curve diagram

Fig.21 Diagram of positioning error distribution for multiple assembly sequences

Tab.5 Summary of assembly sequence results

装配次序

误差均值

μ/μm

误差标准差

σ/μm

拟合R²

⑦-⑤-④-⑥-③-②

-

Tab.5 Summary of assembly sequence results

装配次序

误差均值

μ/μm

误差标准差

σ/μm

拟合R²

⑦-⑤-④-⑥-③-②

-

0.5±0.1

2.4±0.3

0.91

⑦-②-③-⑥-④-⑤

0.4±0.3

4.5±1.1

0.97

⑦-②-③-④-⑥-⑤

3.9±0.6

4.9±1.3

0.84

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