Optimization of Static Equilibrium Leveling Algorithm for Optoelectronic Pods Based on Trust Domain and Augmented Lagrangian Method

doi:10.3969/j.issn.1004-132X.2026.03.025

Abstract

Abstract:

To address the imbalance torque caused by the center-of-mass offset in electro-optical pods， a balancing optimization method was developed. This method integrated an eccentricity parameter estimation approach based on the trust region framework with an augmented Lagrangian optimization algorithm. A mathematical model of the mass offset was first established， and high-precision eccentricity parameters were estimated using trust-region-based method. The balancing problem was then formulated as a multi-objective optimization task aimed at minimizing both the imbalance torque and the total counterweight mass. The ε-constraint method was employed to convert the multi-objective problem into a single-objective one. Following each ε updated， the augmented Lagrangian method was applied to obtain a Pareto solution sets， from which the final solution was determined using a weighted-sum strategy. Experimental validation was conducted on two electro-optical pods weighing 4 kg and 10.5 kg， respectively， with 50 trials each. The estimation errors of eccentricity distance and angle reach the order of 10^-6 and 0.1°， respectively. Post-balancing， the qualified rates of residual imbalance torque were 96% and 100%. These results confirm that the proposed method may efficiently and accurately determine optimal balancing configurations， offering a reliable theoretical and experimental foundation for the static balancing of electro-optical pods.

Key words: electro-optical pods, static equilibrium leveling, unbalanced moment, trust domain, augmented Lagrangian method

摘要：

为解决因光电吊舱的质心偏移导致的不平衡力矩问题，提出了一种融合信赖域框架的偏心参数估计方法和增广拉格朗日法的配平优化算法。通过构建质心偏移的数学模型，利用信赖域偏心参数估计方法实现高精度偏心参数估计。将配平优化问题建模为最小化不平衡力矩和配重质量的多目标优化任务，利用ε约束将多目标问题转化为单目标优化问题；在每一次更新ε值后，利用增广拉格朗日法，得到Pareto解集；使用加权和的方法确定最终解。经4 kg和10.5 kg光电吊舱各50组的实验，偏心距和偏心角误差分别达10^-6量级和0.1°量级；配平后不平衡力矩合格率分别为96%和100%。结果表明，所提算法能够快速准确地找到最优配平方案，为光电吊舱静平衡配平提供了可靠的理论方法与实验依据。

关键词: 光电吊舱, 静平衡配平, 不平衡力矩, 信赖域, 增广拉格朗日法

CLC Number:

YANG Guang, WANG Yidong. Optimization of Static Equilibrium Leveling Algorithm for Optoelectronic Pods Based on Trust Domain and Augmented Lagrangian Method[J]. China Mechanical Engineering, 2026, 37(3): 752-761.

杨光, 王艺东. 信赖域与增广拉格朗日算法融合的光电吊舱静平衡配平优化[J]. 中国机械工程, 2026, 37(3): 752-761.

Figures/Tables 19

Fig.1 Simplified diagram of theoretical model

Fig.2 Flow chart of eccentricity parameter estimation algorithm

Tab.1 Centroid quadrant determination table

实测值与 $m g$ 的关系	质心象限
$p m e a, 1 < m g, p m e a, 2 < m g, p m e a, 3 > m g, p m e a, 4 > m g$	第一象限
$p m e a, 1 < m g, p m e a, 2 > m g, p m e a, 3 > m g, p m e a, 4 < m g$	第二象限
$p m e a, 1 > m g, p m e a, 2 > m g, p m e a, 3 < m g, p m e a, 4 < m g$	第三象限
$p m e a, 1 > m g, p m e a, 2 < m g, p m e a, 3 < m g, p m e a, 4 > m g$	第四象限

Tab.1 Centroid quadrant determination table

实测值与 $m g$ 的关系	质心象限
$p m e a, 1 < m g, p m e a, 2 < m g, p m e a, 3 > m g, p m e a, 4 > m g$	第一象限
$p m e a, 1 < m g, p m e a, 2 > m g, p m e a, 3 > m g, p m e a, 4 < m g$	第二象限
$p m e a, 1 > m g, p m e a, 2 > m g, p m e a, 3 < m g, p m e a, 4 < m g$	第三象限
$p m e a, 1 > m g, p m e a, 2 < m g, p m e a, 3 < m g, p m e a, 4 > m g$	第四象限

Fig.3 Flow chart of balance algorithm

Fig.4 Convergence curve of eccentricity objective function

Fig.5 Eccentricity parameter changing curves

Fig.6 Gradient variation diagram

Fig.7 Eccentricity distribution scatter plot

Fig.8 Distribution diagram of eccentricity error

Fig.9 Eccentric angle distribution scatter plot

Fig.10 Distribution diagram of eccentricity angle error

Tab.2 Model accuracy evaluation

预测参数	RMSE	MSE	MAE	R²
偏心距	1.5×10^-6cm	2.23×10^-12cm²	1.22×10^-6cm	0.9856
偏心角	0.6779°	0.4596（°）²	0.4638°	0.9966

Fig.11 Balancing objective function convergence curve

Fig.12 Gradient variation heatmap

Fig.13 Optimization diagram of counterweight quality

Fig.14 Pareto front solution set

Fig.15 Probability density plot of unbalanced torque after balancing

Fig.16 Box diagram of unbalanced torque after balancing

Fig.17 Cumulative distribution function diagram of actual unbalanced torque

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