﻿ 基于免疫遗传算法的冗余并联机构多目标优化
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 哈尔滨工程大学学报  2018, Vol. 39 Issue (12): 2033-2039  DOI: 10.11990/jheu.201706007 0

### 引用本文

DAI Xiaolin, WANG Peng, GONG Dawei. Multi-objective optimization of redundant parallel mechanism based on immune genetic algorithm[J]. Journal of Harbin Engineering University, 2018, 39(12), 2033-2039. DOI: 10.11990/jheu.201706007.

### 文章历史

1. 电子科技大学 机械与电气工程学院, 四川 成都 611731;
2. 电子科技大学 机器人中心, 四川 成都 611731

Multi-objective optimization of redundant parallel mechanism based on immune genetic algorithm
DAI Xiaolin 1,2, WANG Peng 1, GONG Dawei 1,2
1. School of Mechanical and Electrical Engineering, University of Electronic Science and Technology of China, Chengdu 611731, China;
2. Center for Robotics, University of Electronic Science and Technology of China, Chengdu 611731, China
Abstract: Considering the inefficiencies in the driving capability of electric parallel mechanism, a parallel manipulator with three redundant legs is proposed to balance gravity. In addition, an immune genetic algorithm is used to design the mechanism by multi-objective optimization to address the negative impact of the maximum velocity, driving force, and power in actual working conditions. The simulation result shows that compared with the typical six degree of freedom parallel mechanism, the maximum velocity, driving force, and power reduced by 13.83%, 24.50%, and 45.68%, respectively, after optimization. The redundant structure and multi-objective optimization scheme can reduce the load of electric-driven legs, and it is suitable for the test bed of heavy load simulation.
Keywords: parallel mechanism    redundant structure    power optimization    speed optimization    driving force optimization    immune genetic algorithm    multi-objective optimization

1 数学建模与动力学分析 1.1 冗余并联机构结构

1.2 坐标系的建立

 $\mathit{\boldsymbol{R}} = \left[ {\begin{array}{*{20}{c}} {{\rm{c}}{q_5} \cdot {\rm{c}}{q_6}}&{ - {\rm{c}}{q_4} \cdot {\rm{s}}{q_6} + {\rm{s}}{q_4} \cdot {\rm{s}}{q_5} \cdot {\rm{s}}{q_6}}&{{\rm{s}}{q_4} \cdot {\rm{s}}{q_6} + {\rm{c}}{q_4} \cdot {\rm{s}}{q_5} \cdot {\rm{c}}{q_6}}\\ {{\rm{c}}{q_5} \cdot {\rm{s}}{q_6}}&{{\rm{c}}{q_4} \cdot {\rm{c}}{q_6} + {\rm{s}}{q_4} \cdot {\rm{s}}{q_5} \cdot {\rm{s}}{q_6}}&{ - {\rm{s}}{q_4} \cdot {\rm{c}}{q_6} + {\rm{c}}{q_4} \cdot {\rm{s}}{q_5} \cdot {\rm{s}}{q_6}}\\ { - {\rm{s}}{q_5}}&{{\rm{s}}{q_4} \cdot {\rm{c}}{q_5}}&{{\rm{c}}{q_4} \cdot {\rm{c}}{q_5}} \end{array}} \right]$

 $\begin{array}{*{20}{c}} {\mathit{\boldsymbol{L}} = \left[ {\begin{array}{*{20}{c}} {{\mathit{\boldsymbol{L}}_1}}&{{\mathit{\boldsymbol{L}}_2}}&{{\mathit{\boldsymbol{L}}_3}}&{{\mathit{\boldsymbol{L}}_4}}&{{\mathit{\boldsymbol{L}}_5}}&{{\mathit{\boldsymbol{L}}_6}} \end{array}} \right] = }\\ {\mathit{\boldsymbol{R}} \cdot \mathit{\boldsymbol{A}} + \mathit{\boldsymbol{P}} + {\mathit{\boldsymbol{P}}_0} - \mathit{\boldsymbol{B}}} \end{array}$ (1)

 $\begin{array}{*{20}{c}} {{\mathit{\boldsymbol{E}}_i} = \left[ {\begin{array}{*{20}{c}} {{\mathit{\boldsymbol{L}}_{e1}}}&{{\mathit{\boldsymbol{L}}_{e2}}}&{{\mathit{\boldsymbol{L}}_{e3}}} \end{array}} \right] = }\\ {\mathit{\boldsymbol{R}} \cdot {\mathit{\boldsymbol{A}}_{ei}} + \left[ {\begin{array}{*{20}{c}} {\mathit{\boldsymbol{t}} + {\mathit{\boldsymbol{t}}_0}}&{\mathit{\boldsymbol{t}} + {\mathit{\boldsymbol{t}}_0}}&{\mathit{\boldsymbol{t}} + {\mathit{\boldsymbol{t}}_0}} \end{array}} \right] - {\mathit{\boldsymbol{B}}_{ei}},i = 1,2,3} \end{array}$ (2)

1.3 运动学、动力学分析

 $\mathit{\boldsymbol{\dot L}} = \left[ {\begin{array}{*{20}{c}} {{{\mathit{\boldsymbol{\dot L}}}_1}}&{{\mathit{\boldsymbol{L}}_2} \cdots {{\mathit{\boldsymbol{\dot L}}}_6}} \end{array}} \right] = \mathit{\boldsymbol{\dot P}} + \mathit{\boldsymbol{\hat \omega }} \times \mathit{\boldsymbol{RA}}$ (3)

$\mathit{\boldsymbol{\dot q = }}{\left[ {{\mathit{\boldsymbol{t}}^{\rm{T}}}{\mathit{\boldsymbol{\theta }}^{\rm{T}}}} \right]^T}$表示上平台的速度，整理得

 $\mathit{\boldsymbol{\dot L}} = \left[ {\begin{array}{*{20}{c}} \mathit{\boldsymbol{E}}&{ - \mathit{\boldsymbol{R}}{{\mathit{\boldsymbol{\tilde a}}}_1}{\mathit{\boldsymbol{R}}^{\rm{T}}}}\\ \vdots&\vdots \\ \mathit{\boldsymbol{E}}&{ - \mathit{\boldsymbol{R}}{{\mathit{\boldsymbol{\tilde a}}}_1}{\mathit{\boldsymbol{R}}^{\rm{T}}}} \end{array}} \right]\mathit{\boldsymbol{\dot q}} = \mathit{\boldsymbol{J\dot q}}$ (4)

 ${{\mathit{\boldsymbol{\dot D}}}_i} = \frac{1}{{\left| {{{\mathit{\boldsymbol{\dot L}}}_i}} \right|}}\mathit{\boldsymbol{\dot L}}_i^{\rm{T}} \cdot {{\mathit{\boldsymbol{\dot L}}}_i}$ (5)

 $\sum\limits_{i = 1}^6 {{\mathit{\boldsymbol{e}}_i} \cdot {f_i}} + \sum\limits_{i = 1}^3 {{\mathit{\boldsymbol{e}}_{ei}} \cdot {f_e}} - \mathit{\boldsymbol{mg}} = \mathit{\boldsymbol{m\ddot p}}$ (6)

 $\begin{array}{*{20}{c}} {\sum\limits_{i = 1}^6 {\left( {\mathit{\boldsymbol{R}} \cdot {\mathit{\boldsymbol{a}}_i}} \right) \times \left( {{\mathit{\boldsymbol{e}}_i} \cdot {f_i}} \right)} + \sum\limits_{i = 1}^3 {\left( {\mathit{\boldsymbol{R}} \cdot {\mathit{\boldsymbol{a}}_{ei}}} \right) \times \left( {{\mathit{\boldsymbol{e}}_{ei}} \cdot {f_e}} \right)} = }\\ {\mathit{\boldsymbol{I\dot \omega }} + \mathit{\boldsymbol{\omega }} \times \mathit{\boldsymbol{I\omega }}} \end{array}$ (7)

 $\begin{array}{*{20}{c}} {{\mathit{\boldsymbol{e}}_i} = \frac{1}{{\left| {{L_i}} \right|}}{L_i}}&{i = 1,2, \cdots ,6} \end{array}$

eei为第i条冗余支腿方向的单位向量：

 $\begin{array}{*{20}{c}} {{\mathit{\boldsymbol{e}}_{ei}} = \frac{1}{{\left| {{\mathit{\boldsymbol{L}}_{ei}}} \right|}}{\mathit{\boldsymbol{L}}_{ei}}}&{i = 1,2,3} \end{array}$

I为上平台在静坐标系下的惯性矩；

 $\begin{array}{*{20}{c}} {{W_{\max }} = \max \left( {{f_i} \cdot {{\dot D}_i}} \right)}&{i = 1,2, \cdots ,6} \end{array}$ (8)

2 结构优化分析 2.1 设计变量

 $\mathit{\boldsymbol{X}} = \left[ {\begin{array}{*{20}{c}} {{r_a}}&{{r_b}}&{{l_a}}&{{l_b}}&{{r_{ae}}}&{{r_{be}}}&{{f_e}}&h \end{array}} \right]$
2.2 目标函数

 ${V_{\max }} = \frac{{\max \left( {{{\dot D}_i}} \right) - {v_{\min }}}}{{{v_{\max }} - {v_{\min }}}}$ (9)
 ${F_{\max }} = \frac{{\max \left( {{f_i}} \right) - {f_{\min }}}}{{{f_{\max }} - {f_{\min }}}}$ (10)
 ${P_{\max }} = \frac{{\max \left( {{P_i}} \right) - {P_{\min }}}}{{{P_{\max }} - {P_{\min }}}}$ (11)

 $\min f\left( X \right) = {a_1}\left| {{V_{\mathit{max}}}} \right| + {a_2}\left| {{F_{\max }}} \right| + {a_3}\left| {{P_{\max }}} \right|$

2.3 约束条件

 $\begin{array}{*{20}{c}} {\mathit{\boldsymbol{X}} = }\\ {\left[ {\begin{array}{*{20}{c}} {{r_a}/m}&{{r_b}/m}&{{l_a}/m}&{{l_b}/m}&{{r_{ae}}/m}&{{r_{be}}/m}&{{f_e}/kN}&{h/m} \end{array}} \right] = }\\ {\left[ {\begin{array}{*{20}{c}} {\left[ {1.5,2.5} \right]}&{\left[ {2.0,3.0} \right]}&{\left[ {0.2,0.6} \right]}&{\left[ {0.2,0.6} \right]} \end{array}} \right.}\\ {\left. {\begin{array}{*{20}{c}} {\left[ {1.5,2.5} \right]}&{\left[ {2.0,3.0} \right]}&{\left[ {0.4,1.5} \right]}&{\left[ {1.3,3.0} \right]} \end{array}} \right]} \end{array}$

 $\left\{ {\begin{array}{*{20}{c}} {{\rm{Find}}\;\mathit{\boldsymbol{X}} = \left[ {\begin{array}{*{20}{c}} {{r_a}}&{{r_b}}&{{l_a}}&{{l_b}}&{{r_{ae}}}&{{r_{be}}}&{{f_e}}&h \end{array}} \right]}\\ {\min f\left( \mathit{\boldsymbol{X}} \right)}\\ {{\rm{s}}.\;{\rm{t}}.\;\;\;\;\;\;\mathit{\boldsymbol{X}} \in {\mathit{\boldsymbol{L}}_X}} \end{array}} \right.$ (12)
3 多目标优化设计 3.1 算法设计

 Download: 图 2 每代目标函数的最小值和平均值的变化规律 Fig. 2 Variation of max and average value of objective function in each Generation
3.2 优化结果

 Download: 图 3 各项特征运动速度峰值对比 Fig. 3 Peak velocity curve of each movement mode
 Download: 图 4 各项特征运动电动支腿驱动力峰值对比 Fig. 4 Driving force curve of each movement mode

4 结论

1) 冗余支腿能够很好的配合电动支腿的运动，以恒力平衡掉载荷的部分重力，对比工业中常用的典型六自由度并联机构，优化后摇摆台工作时电动支腿的速度、作用力和系统的功耗峰值分别降低了13.83%、24.50%和45.68%。

2) 使用免疫遗传优化算法，保证了种群的个体多样性。在确保优化效果的前提下，能够避免过早地收敛于局部最优解。

3) 采用综合权衡速度、驱动力和功率的多目标优化策略，使其更接近工作环境的实际需求。

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