**4.5.2 Optimization based on GDI**

 In this section we optimize the average of conditioning index in the workspace. Therefore, the problem is stated as:

*GDI*=max(*average* (1/( )**J** ))

*Subject to* 

$$\begin{aligned} 1. \quad 1/\kappa(\text{f}) &> 0.01\\ 2. \quad 0.01 &< \frac{d}{\Delta q} < 0.15 \quad , \quad 0.01 < \frac{h}{\Delta q} < 0.15 \quad , \quad 0.35 < \frac{Q\_z}{\Delta q} < 0.55 \end{aligned}$$

As it is expected, the volume of the workspace is very small and only includes a small neighbourhood of a point of the best conditioning index. This optimal design is given in Table 2.


Table 2. Optimization result for H4

**4.5.1 Optimization based on maximum workspace volume** 

*V\*=max(V) Subject to* 

> 1. 1*/*

workspace.

Volume of

workspace *(rad)(cm3)*

Table 1. Optimization result for H4

**4.5.2 Optimization based on GDI** 

( )**J** > 0.01

( )**J** ))

the problem is stated as: *GDI*=max(*average* (1/

Volume of workspace *(rad)(cm3)*

Table 2. Optimization result for H4

*Subject to* 

Table 2.

1. 1*/* 

( )**J** *>*0.01

conditioning index for 0

First we consider the volume of workspace as the objective function; namely,

*<sup>z</sup> dhQ 0.01 0.15 , 0.01 0.15 , 0.35 0.55 qq q* 

conditioning index

2. *<sup>z</sup> dhQ 0.01 0.15 , 0.01 0.15 , 0.35 0.55 qq q* 

conditioning index

The solution of this problem is given in Table 1. Moreover, the workspace for 0

workspace size is acceptable, the manipulator suffers from a poor dexterity throughout its

Average of parameters

12582.0 0.042 13.52484 13.88781 35.18892

As it is observed the GA reaches a convergence after 30 iterations as depicted in Fig. 7.

In this section we optimize the average of conditioning index in the workspace. Therefore,

As it is expected, the volume of the workspace is very small and only includes a small neighbourhood of a point of the best conditioning index. This optimal design is given in

Average of parameters

4.1887 0.0705 1.00008 1.09692 35.48742

Robot's

*Qz d(cm) h(cm) (cm)*

and *z*=0 are shown in Fig.5 and Fig.6, respectively. While the

Robot's

*Qz d(cm) h(cm) (cm)*

and the

Fig. 5. Workspace of H4 for 0 

Fig. 6. Conditioning index for z=0 and 0 

Optimization of H4 Parallel Manipulator Using Genetic Algorithm 415

Volume of workspace *(rad)(cm3)* 11983 11304 11008

First, the forward and inverse kinematics of H4 parallel manipulator has been studied here, in which the former problem has leaded to a univariate polynomial of degree eight. Then, the optimal design of the manipulator has been addressed. Using genetic algorithm the manipulator has been optimized based on a mixed performance index that is a weighted sum of global conditioning index and its workspace volume. It has been shown that by introducing this measure, the parallel manipulator has been improved at minor cost of its workspace volume.

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*Intelligent Robots and Systems*, Las Vegas, USA, October, 2003, pp. 3312-3317. Company, O.; Marquet, F. & Pierrot, F. (2003). A new high-speed 4DoF parallel robot

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Dai, J.S.; Jones, J.R. (2007). A Linear Algebraic Procedure in Obtaining Reciprocal Screw

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Hosseini, M.A; Daniali, H.M. & Taghirad, H.D. (2011). "Dexterous Workspace Optimization of a Tricept Parallel Manipulator", *Advanced Robotics*, Vol. 25, 2011, pp. 1697-1712. Lara-Molina, F.A.; Rosário, J.M. & Dumur, D. (2010). Multi-Objective Design of Parallel

Huxley equation by He's homotopy perturbation method, *Applied Mathematics and* 

Manipulator Using Global Indices, *The Open Mechanical Engineering Journal*, 2010, 4,

of a 4-DOFs parallel robot H4, *Proceedings of IEEE/RSJ International Conference on* 

synthesis and modeling issues. *IEEE, Transactions on Robotics and Automation*,

Table 3. Optimization results for H4

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pp. 37-47.

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**5. Conclusions** 

**6. References** 

*d(cm)* 12.14243 12.60003 10.36257 *h(cm)* 10.38564 9.06147 5.56941 *Qz (cm)* 37.25841 39.61105 40.69626 *GDI* 0.0521 0.0547 0.059

*w 0.25 <sup>d</sup> w 0.5 <sup>d</sup> w 0.75 <sup>d</sup>*

Fig. 7. Convergence of GA

#### **4.5.3 Optimization based on the mixed performance index**

For the problem at hand one needs a good fitness function, because the optimization based on maximal workspace volume, decreases the performance indices and the optimization based on GDI decreases the workspace volume. Therefore, one can optimize the manipulator based on performance index that described in subsection 5.3. The optimization results for *wd*=0.25, 0.5, 0.75 are given in Table 3.

Moreover, for different values of *wd*, the problem is solved and the GDI, SURI and the mixed performance index are calculated and plotted in Fig. 8. As the result, any value of *wd* greater than 0.74 leads to a limited workspace and for any values smaller than that has no substantial effects on GDI and the workspace. Therefore, it clearly shows that by introducing this measure, the performance of the manipulator can be improved at a minor cost its workspace volume.

Fig. 8. GDI and SURI and Performance Index versus weight parameter


Table 3. Optimization results for H4
