A multi-objective differential evolution approach based on ε-elimination uniform-diversity for mechanism design

Abstract

In this paper, a new multi-objective uniform-diversity differential evolution (MUDE) algorithm is proposed and used for Pareto optimum design of mechanisms. The proposed algorithm uses a diversity preserving mechanism called the ε-elimination algorithm to improve the population diversity among the obtained Pareto front. The proposed algorithm is firstly tested on some constrained and unconstrained benchmarks proposed for the special session and competition on multi-objective optimizers held under IEEE CEC 2009. The inverted generational distance (IGD) measure is used to assess the performance of the algorithm. Secondly, the proposed algorithm has been used for multi-objective optimization of two different combinatorial case studies. The first case contains a two-degree of freedom leg mechanism with springs. Three conflicting objective functions that have been considered for Pareto optimization are namely, leg size, vertical actuating force, and the peak crank torque. The second case is a two-finger robot gripper mechanism with two conflicting objectives which are the difference between the maximum and minimum gripping force and the transmission ratio of actuated and experienced gripper forces. Comparisons of obtained Pareto fronts using the method of this work with those obtained in other references show significant improvements.

Appendixes

1.1 Appendix 1

1.1.1 A. robot leg constraints

$$ {g}_1(x)\equiv 2{X}_{J_1}-{s}_h\ge 0 $$

(8)

$$ {g}_2(x)\equiv \frac{Y_J\Big|{}_{\alpha =\pi }-{Y}_J{\Big|_{\alpha =}}_0}{s_h}-{H}_{C2}\ge 0 $$

(9)

$$ {g}_3(x)\equiv 2{ \sin}^{-1}\left(\frac{x_2-{x}_1}{2{x}_3}\right)-{H}_{C3}\ge 0 $$

(10)

$$ {g}_4(x)\equiv -2{ \sin}^{-1}\left(\frac{x_2+{x}_1}{2{x}_3}\right)+{H}_{C4}\ge 0 $$

(11)

$$ {g}_5(x)\equiv { \cos}^{-1}\left(\frac{2{\left({x}_4+{x}_5\right)}^2-{l^2}_{\min }}{2{\left({x}_4+{x}_5\right)}^2}\right)-{H}_{C5}\ge 0 $$

(12)

$$ {g}_6(x)\equiv -{ \cos}^{-1}\left(\frac{2{\left({x}_4+{x}_5\right)}^2-{l^2}_{\max }}{2{\left({x}_4+{x}_5\right)}^2}\right)+{H}_{C6}\ge 0 $$

(13)

$$ {g}_7(x)\equiv {\theta}_5-{H}_{FC1}\ge 0,\forall \alpha \in {R}_p $$

(14)

$$ {g}_8(x)\equiv -\frac{d^2{Y}_J}{d{\alpha}^2}\ge 0, over{J}_1{J}_2{J}_3 $$

(15)

$$ {g}_9(x)\equiv \frac{ \min \left({l}_1\right)}{l_{01}}-{H}_{SC1}\ge 0 $$

(16)

$$ {g}_{10}(x)\equiv -\frac{ \max \left({l}_1\right)}{l_{01}}+{H}_{SC2}\ge 0 $$

(17)

$$ {g}_{11}(x)\equiv \frac{ \min \left({l}_2\right)}{l_{02}}-{H}_{SC3}\ge 0 $$

(18)

$$ {g}_{12}(x)\equiv -\frac{ \max \left({l}_2\right)}{l_{02}}+{H}_{SC4}\ge 0 $$

(19)

$$ {g}_{13}(x)\equiv -{s}_2+ \min {Y}_J\ge 0 $$

(20)

In which,

$$ {l}^2={\left({X}_J-{X}_E\right)}^2+{\left({Y}_J-{Y}_E\right)}^2. $$

(21)

$$ {\theta}_5={ \cos}^{-1}\left(\frac{X_E}{U}\right)-{ \cos}^{-1}\left(\frac{U}{2{x}_4}\right), $$

(22)

$$ {U}^2={X^2}_E+{Y^2}_E+{Y^2}_H-2{Y}_E{Y}_H. $$

(23)

$$ {l^2}_1={X^2}_E+{\left({Y}_E-{s}_1\right)}^2 $$

(24)

$$ \min \left({l}_2\right)={s}_2-\left({Y}_H+\frac{s_v}{2\left(1+\frac{x_5}{x_4}\right)}\right), $$

(25)

$$ \max \left({l}_2\right)={s}_2-\left({Y}_H-\frac{s_v}{2\left(1+\frac{x_5}{x_4}\right)}\right). $$

(26)

Where l _min and l _max are the minimum and maximum value of link EJ. In addition in the above equations, it is assumed that Y_H is at its mid-range position. The constant parameters of the constraints are as follows, H_SC1 = 1, H_SC2 = 1.3, H_SC3 = 1, H_SC4 = 1.3, H_SC5 = −0.3, H_SC6 = 0.1, H_SC7 = 0.1, H_SC8 = 0.6.

1.2 Appendix 2

1.2.1 B.1. robot gripper geometric equations

$$ g=\sqrt{{\left(l-z\right)}^2+{e}^2} $$

(27)

$$ \alpha = \arccos \left(\frac{a^2+{g}^2-{b}^2}{2.a.g}\right)+\varphi $$

(28)

$$ \beta = \arccos \left(\frac{b^2+{g}^2-{a}^2}{2.b.g}\right)-\varphi $$

(29)

$$ \varphi = \arctan \left(\frac{e}{l-z}\right) $$

(30)

$$ {a}^2={b}^2+{g}^2-2bg \cos \left(\beta +\varphi \right) $$

(31)

$$ {b}^2={a}^2+{g}^2-2 ag \cos \left(\alpha -\phi \right) $$

(32)

1.2.2 B.2. robot gripper force equations

$$ R.b. \sin \left(\alpha +\beta \right)={F}_k.c $$

(33)

$$ R=\frac{P}{2. \cos \left(\alpha \right)} $$

(34)

$$ {F}_k=\frac{P.b. \sin \left(\alpha +\beta \right)}{2.c. \cos \left(\alpha \right)} $$

(35)

where P is the actuating force, R is the reaction force on link a, and F _k is the gripping force which is experienced at the gripper ends.

1.2.3 B.3. robot gripper constraints

$$ {g}_1(x)={Y}_{\min }-y\left(x,{Z}_{\max}\right)\ge 0 $$

(36)

$$ {g}_2(x)=y\left(x,{Z}_{\max}\right)\ge 0 $$

(37)

$$ {g}_3(x)=y\left(x,0\right)-{Y}_{\max}\ge 0 $$

(38)

$$ {g}_4(x)={Y}_G-y\left(x,0\right)\ge 0 $$

(39)

$$ {g}_5(x)={\left(a+b\right)}^2-{l}^2-{e}^2\ge 0 $$

(40)

$$ {g}_6(x)={\left(l-{Z}_{\max}\right)}^2+{\left(a-e\right)}^2-{b}^2\ge 0 $$

(41)

$$ {g}_7(x)=l-{Z}_{\max}\ge 0 $$

(42)

$$ {g}_8(x)= \min {F}_k\left(x,z\right)-FG\ge 0 $$

(43)

where y(x, z) = 2. [e + f + c. sin(δ + β)] is the displacement of gripper ends. Y_min and Y_max are maximum and minimum dimensions of gripping object. Y_G is the maximum range of gripper ends displacement and FG is the assumed minimal gripping force. More description of constraints could be found in (Osyczka 2002; Datta and Deb 2011).