工程科学与技术   2019, Vol. 51 Issue (4): 185-191

1. 山东理工大学 农业工程与食品科学学院，山东 淄博 255049;
2. 大连理工大学 机械工程学院，辽宁 大连 116081;
3. 山东理工大学 机械工程学院，山东 淄博 255049

Parameter Optimization of Micro-gripper System for Nonlinear Contradictory Objectives
ZHANG Yanfei1, JIA Guopeng2, GONG Jinliang3, WANG Zhiwen3
1. School of Agricultural Eng. and Food Sci., Shandong Univ. of Technol., Zibo 255049,China;
2. School of Mechanical Eng., Dalian Univ. of Technol., Dalian 116081, China;
3. School of Mechanical Eng., Shandong Univ. of Technol., Zibo 255049, China
Abstract: Maximum output displacement and resolution are two contradictory performance indexes of micro positioning system. Usually, the two important indexes cannot be satisfied simultaneously by flexibility-targeting structure optimization method. A final optimal model was established considering both output displacement and resolution. And the corresponding solution was established for the special model with two non-linear contradictory objectives. Firstly, the parameters with a greater effect on the whole performance of the system were found out and the optimization variables were obtained. Constrained condition was built up according to the design requirements and relationship among these variables. Secondly, by the usage of these optimization variables, two expressions for the maximum output displacement and the resolution were built up and defined as objective function and constrained function respectively. At last, a synthetic algorithm was adopted to solve this kind of optimization model by integrating three methods, that’s the branch and bound method, interior point method and exterior point method. The optimization model was obtained by interior point method. The synthetic algorithm took any one of the possible solutions of this optimization model as the starting point. The branch and bound method were applied to narrow searching scope. Then the final optimized solution could be obtained based on the solution of exterior point method. The method was applied to a micro-griper system. Results showed that the constrained effect on the maximum output displacement was different when the resolution requirement changed within different scopes. During the process of parameter optimization for the micro-griper system, the global optimization was realized by taking into account of two contradictory objectives.
Key words: resolution    maximum output displacement    micro-gripper system    parameter optimization    optimization algorithm

1 微夹持系统结构部分正解模型

1.1 微夹持系统模型与单元划分

 图1 微夹持系统 Fig. 1 Micro-gripper system

 图2 微夹持系统结构模型 Fig. 2 Structural model of micro-gripper system

1.2 驱动力与输出点I位移关系

 $^7{{U}}_{\rm{I}}^7 = {{{C}}_{\rm{I}}}(^3{{F}}_{\rm{E}}^{\rm{w}})$ (1)

 ${{C}}_{\rm{I}}=\left({{C}}_{\rm{A}}\right)^{-1} {{C}}_{\rm{B}},$
 \begin{aligned} {{{C}}_{\rm{A}}} =\; & {{{R}}_{67}}{{V}}_{{\rm{HI}}}^6{[{{{C}}_{456}} + {{{R}}_{36}}{{P}}_{{\rm{EH}}}^3{{{C}}_{12}}{({{{R}}_{36}}{{V}}_{{\rm{EH}}}^3)^{ - 1}}]^{ - 1}} {({{{R}}_{67}}{{P}}_{{\rm{HI}}}^6)^{ - 1}} +\\ & {{{R}}_{87}}{{V}}_{{\rm{JI}}}^8{({{{C}}_{89}})^{ - 1}}{({{{R}}_{{\rm{87}}}}{{P}}_{{\rm{JI}}}^8)^{ - 1}}, \end{aligned}
 ${{C}_{\rm{B}}} =\;{{R}_{67}}{V}_{{\rm{HI}}}^6{[{{C}_{456}} + {{R}_{36}}{P}_{{\rm{EH}}}^3{{C}_{12}}{({{R}_{36}}{V}_{{\rm{EH}}}^3)^{ - 1}}]^{ - 1}} {{R}_{36}}{P}_{{\rm{EH}}}^3{{C}_{12}},$
 \begin{aligned} {{C}_{12}} =\;& \{ {V}_{{\rm{BE}}}^1{({{C}_1})^{ - 1}}{({P}_{{\rm{BE}}}^1)^{ - 1}} + \\ &{{R}_{23}}({V}_{{\rm{CE}}}^2){({{C}_2})^{ - 1}}{[{{R}_{23}}({P}_{{\rm{CE}}}^2)]^{ - 1}}{\} ^{ - 1}}, \end{aligned}
 ${{{C}}_{456}} = ({{{R}}_{46}}{{P}}_{{\rm{FH}}}^{\rm{4}}) \otimes {{{C}}_4} + ({{{R}}_{56}}{{P}}_{{\rm{GH}}}^5) \otimes {{{C}}_5} + {{{C}}_6},$
 ${{{C}}_{89}} = ({{{R}}_{98}}{{P}}_{{\rm{KJ}}}^9) \otimes {{{C}}_9} + {{{C}}_8}{\text{。}}$
1.3 驱动力与输入点M位移关系式求解

 图3 支链9–4的结构 Fig. 3 Structure of branched chain 9–4

 ${}^4{{U}}_{\rm{E}}^4 = {{{C}}_{9 - 4}}\left( {{}^4{{F}}_{\rm{E}}^{34}} \right)$ (2)

 \begin{aligned} {{{C}}_{9 - 4}} =\; & {{{T}}_9} \otimes {{{C}}_9} + {{{T}}_8} \otimes {{{C}}_8} +{{{T}}_6} \otimes {{{C}}_6} + \\ & {{{T}}_5} \otimes {{{C}}_5} + {{{C}}_4}, \end{aligned}
 \begin{aligned} & {{ T}_9} = {{ R}_{94}}{ P}_{{\rm{KE}}}^9,{{ T}_8} = {{ R}_{84}}{ P}_{{\rm{JE}}}^8,\\ & {{ T}_6} = {{ R}_{64}}{ P}_{{\rm{GE}}}^6,{{ T}_5} = {{ R}_{54}}{ P}_{{\rm{FE}}}^5 {\text{。}} \end{aligned}

 图4 单元1的结构 Fig. 4 Structure of element 1

 ${}^1{{U}}_{\rm{B}}^1 = {{{C}}_1}\left( {{}^1{{F}}_{\rm{B}}^{31}} \right)$ (3)

 图5 单元2的结构 Fig. 5 Structure of element 2

 ${}^2{{U}}_{\rm{C}}^2 = {{{C}}_2}\left( {{}^2{{F}}_{\rm{C}}^{32}} \right)$ (4)

 图6 单元3的结构 Fig. 6 Structure of element 3

 ${}^3{{F}}_{\rm{E}}^{13} + {}^3{{F}}_{\rm{E}}^{23} + {}^3{{F}}_{\rm{E}}^{43} + {}^3{{F}}_{\rm{E}}^{\rm{w}} = 0\text{。}$

 ${}^3{{F}}_{\rm{E}}^{13} = - {{{R}}_{13}}{{V}}_{{\rm{BE}}}^1\left( {{}^1{{F}}_{\rm B}^{31}} \right),{}^3{{F}}_{\rm{E}}^{23} = - {{{R}}_{23}}{{V}}_{{\rm{CE}}}^2\left( {{}^2{{F}}_{\rm C}^{32}} \right),$
 ${}^3{{F}}_{\rm{E}}^{43} = - {{{R}}_{43}}\left( {{}^4{{F}}_{\rm{E}}^{{\rm{34}}}} \right), {}^1{{U}}_{\rm B}^{1} = {{{R}}_{31}}{{P}}_{{\rm{MB}}}^3\left( {{}^3{{U}}_{\rm{M}}^3} \right),$
 ${}^2{{U}}_{\rm C}^{2} = {{{R}}_{32}}{{P}}_{{\rm{MC}}}^3\left( {{}^3{{U}}_{\rm{M}}^3} \right), {}^4{{U}}_{\rm E}^{4} = {{{R}}_{34}}{{P}}_{{\rm{ME}}}^3\left( {{}^3{{U}}_{\rm{M}}^3} \right)\text{。}$

 ${}^3{{F}}_{\rm{E}}^{\rm{w}} = {{{K}}_{\rm{M}}}\left( {{}^3{{U}}_{\rm{M}}^{\rm{3}}} \right)$ (5)

 \begin{aligned} {{{K}}_{\rm{M}}} =\; & {{{R}}_{13}}{{V}}_{{\rm{BE}}}^{\rm{1}}{\left( {{{{C}}_1}} \right)^{ - 1}}{{{R}}_{31}}{{P}}_{{\rm{MB}}}^3 + \\ & {{{R}}_{23}}{{V}}_{{\rm{CE}}}^2{\left( {{{{C}}_2}} \right)^{ - 1}}{{{R}}_{32}}{{P}}_{{\rm{MC}}}^3 + {{{R}}_{43}}{\left( {{{{C}}_{9 - 4}}} \right)^{ - 1}}{{{R}}_{34}}{{P}}_{{\rm{ME}}}^3 \text{。} \end{aligned}
1.4 正解关系式求解

 ${}^7{{U}}_{\rm{I}}^{\rm{7}} = {{{C}}_{\rm{I}}}{{{K}}_{\rm{M}}}({}^3{{U}}_{\rm{M}}^3)$ (6)

2 建立系统优化模型

2.1 优化变量的选择

 图7 结构优化参数 Fig. 7 Parameters of structural optimization

 ${k_{\rm{A}}} = 1\;000E{w^2}/(2n){\text{。}}$

 $\Delta {L_{{\rm{N}}\max }} = 3.3 \times {10^{ - 6}}n\;{\rm{ m}}{\text{。}}$

 $\Delta {L_{{\rm{N}}\min }} = 3.3 \times {10^{ - 6}}n/{2^{12}}\;{\rm{ m}}{\text{。}}$
2.2 优化变量的约束条件

NAC系列压电陶瓷截面边长 $w$ 和堆叠个数 $n$ 的取值范围为：

 \left\{ {\begin{aligned} & {0.002 \le w \le 0.025,{\rm{ }}1\;{\rm{ }}000w \in {Z;}}\\ & {1 \le n \le 1\;{\rm{ }}000w,n \in {Z}}{\text{。}} \end{aligned}} \right.

 \left\{ {\begin{aligned} & {0 \le l \le 0.2 (l \in{\mathbb R}),}\\ & {0 \le \alpha \le \frac{{\text{π}}}{2}(\alpha \in {\mathbb R})}{\text{。}} \end{aligned}} \right.

 \left\{ {\begin{aligned} & {0.00{\rm{ }}5\;5 \le l\sin\; \alpha \le 0.2,}\\ & {0.00{\rm{ }}5\;5 \le l\cos\; \alpha \le 0.2}{\text{。}} \end{aligned}} \right.

 $\left| {\Delta {U_{\min }}} \right| \leq \left| {\Delta {U_{{\rm{R}}\min }}} \right|{\text{。}}$

2.3 建立系统性能评价模型

 ${F_{\rm{m}}} = {k_{\rm{A}}}\Delta {L_{\rm{N}}}$ (7)

 图8 微动夹持系统单元划分 Fig. 8 Element partition of micro-gripper system

 ${}^2F_{\rm{M}}^{12} + {}^2F_{\rm{M}}^{32} = 0$ (8)

 ${}^2F_{\rm{M}}^{12} = {F_{\rm{m}}} - {k_{\rm{A}}}\Delta {L_{\rm{M}}}$ (9)

 ${}^2F_{\rm{M}}^{{\rm{23}}} = {k_{{\rm{M}}22}}\Delta {L_{\rm{M}}}$ (10)

 $\Delta {L_{\rm{M}}} = \frac{{{k_{\rm{A}}}}}{{{k_{\rm{A}}} + {k_{{\rm{M22}}}}}}\Delta {L_{\rm{N}}}$ (11)

 ${}^7{{U}}_{\rm{I}}^{\rm{7}} = {{{C}}_{\rm{I}}}{{{K}}_{\rm{M}}}({}^3{{U}}_{\rm{M}}^3){\text{。}}$

$\Delta L_{\rm{N}}$ 取压电陶瓷在最大驱动电压下的空载位移 $\Delta L_{\rm{Nmax}}$ 时，向量 ${}^7{{U}}_{\rm{I}}^{\rm{7}}$ 中的第2个元素为系统最大输出位移 $\Delta U_{\rm{max}}$ ，将 $\Delta U_{\rm{max}}$ 作为优化目标函数；当 $\Delta L_{\rm{N}}$ 取压电陶瓷在最小驱动电压下的空载位移 $\Delta L_{\rm{Nmin}}$ 时，向量 ${}^7{{U}}_{\rm{I}}^{\rm{7}}$ 中的第2个元素为系统最小输出位移 $\Delta U_{\rm{min}}$ ，即系统分辨率，将 $\Delta U_{\rm{min}}$ 作为约束条件函数。

3 优化模型求解

Step 1　以变量 $w$ $n$ $l$ $\alpha$ 为约束条件，以 $f$ =| $\Delta U_{\rm{min}}|-$ | $\Delta U_{\rm{Rmin}}$ |为目标函数，即NP0。任取一可行解为初始点，应用内点法求得最小值点 ${{x}}_0$ ，对应目标值为 $f$ $({{x}}_0)$ ,若 $f$ $({{x}}_0)\leq 0$ ，转到Step 2；否则为无解，求解结束。

Step 2　选取合适的 $\xi$ 值，以变量 $w$ $n$ $l$ $\alpha$ $\Delta U_{\rm{min}}$ 为约束条件，以最大输出位移 $\Delta U_{\rm{max}}$ 为优化目标，即NP1。取 ${{x}}_0$ 为初始点，用内点法求得最小值 ${{x}}_1$ ，若所有 $x_{{\text{z}}i}$ 满足整数条件， ${{x}}_1$ 即为所求解，求解过程结束；否则，任取一不满足条件的 $x_{{\text{z}}i}$ ，按条件 $x_{{\text{z}}i}\geq [x_{{\text{z}}i}]+1$ $x_{{\text{z}}i}\leq [x_{{\text{z}}i}]$ ，将 $NP_1$ 分成两个子问题 $S\!NP_2$ $S\!NP_3$ ，即 $P=$ { $S\!NP_2$ , $S\!NP_3$ }，将 ${{x}}_1$ 作为子问题求解的初始点，即 $S=$ { ${{x}}_1$ , ${{x}}_1$ }。用枚举法寻找 ${{x}}_1$ 周围满足整数条件的最小可行解，若存在，即 ${{z}}_1$ ，则使 $MNP_{\rm po}={{z}}_1$ $MNP_{\rm{up}}=\;$ $\Delta U_{\rm{max}}$ $({{z}}_1)$ ；否则使 $MNP_{\rm po}=\varnothing$ $MNP_{\rm{up}}=$ ∞，转到Step 3。

Step 3　若 $P$ 为空集， $MNP_{\rm po}$ 即为所求解，求解结束；否则假设 $P$ 中第1个元素为 $S\!NP_i$ ，最后一个元素为 $S\!NP_k$ ，取 $S\!NP_i$ ，并使 $P$ $P$ –{ $S\!NP_i$ }，以 $S$ 中的对应元素 $S_i$ 作为初始点，并且 $S$ $S$ –{ $S_i$ }，应用外点法进行求解，所得结果为 ${{x}}_i$

Step 3.1　若 $\Delta U_{\rm{max}}$ $({{x}}_i)\geq$ $\;MNP_{\rm{up}}$ ，转到Step 3。

Step 3.2　若 ${{x}}_i$ 中所有 $x_{{\text{z}}i}$ 均满足整数条件，则使 $MNP_{\rm po}=\;$ $\;{{x}}_i$ $MNP_{\rm{up}}=\;$ $\;\Delta U_{\rm{max}}$ $({{x}}_i)$ ，转到Step 3。

Step 3.3　用枚举法寻找 ${{x}}_i$ 周围满足整数条件的最小可行解，若存在，即 ${{z}}_i$ ，且 $\Delta U_{\rm{max}}({{z}}_i)\leq MNP_{\rm{up}}$ ，则使 $MNP_{\rm{po}}={{z}}_i$ $MNP_{\rm{up}}=\Delta U_{\rm{max}}$ $({{z}}_i)$ 。任取一不满足条件的 $x_{{\text{z}}j}$ ，按条件 $x_{{\text{z}}j}\geq [x_{{\text{z}}j}]+1$ $x_{{\text{z}}j}\leq[x_{{\text{z}}j}]$ ，将 $S\!NP_i$ 分成两个子问题 $S\!NP_{k+1}$ $S\!NP_{k+2}$ ，并使 $P=$ $\;P$ $\{S\!NP_{k+1},S\!NP_{k+2}\}$ $S=S$ $\{{{x}}_i,{{x}}_i\}$ ，转到Step 3。

4 解析式参数分析

 图9 各变量优化结果与分辨率指标的关系 Fig. 9 Relationships between the optimization results of each variable and the resolution requirement

 图10 最大输出位移优化结果与分辨率指标关系 Fig. 10 Relationship between optimization results of maximum output displacement and resolution requirements

5 结　论

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