# Optimization of a Truss Structure Used to Design of the Manipulator Arm from a Set of Components
## 3 Application  
The main goal of the article is to design a manipulator arm that was inspired by [12]. At present, the arm of the manipulator is designed without optimization as a tube of appropriate dimensions, but in [12], it was designed by topological optimization and created using 3D printing. When designing a manipulator, several alternative solutions often arise. A component system for the construction of manipulator arms enables an experimental comparison of the selected variants and thereby enhances the product or reduces the cost of its design. On the other hand, a predefined system of components limits the resulting topology. In the case of truss structures, the limitation is mainly determined by the length of the components. The topology of the truss structure and the basic length of the components are therefore not subject to optimization. The arm of the manipulator consists of three parts. The arm body is represented by a block between two flanges, where the first flange is fixed and the second flange is loaded by a force, as shown in Figure 3.  
These assumptions were used for the following solutions:  
* The flange 1 (100 \(\times\) 100 \(\times\) 10 mm) is fixed on eight nodes;
* The flange 2 (100 \(\times\) 100 \(\times\) 10 mm) is loaded on four nodes;
* The flanges element size is 10 \(\times\) 10 mm;
* The load \(P=1000\) N is applied to the axis of the arm, \(L_{\mathrm{F}}=100\) mm and the load is evenly distributed to the selected nodes;
* The flanges are not included in the optimization;
* Fasteners such as screws are not analyzed here and are believed to comply;
* FEM was used for analysis;  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & **Method** & \begin{tabular}{c} **Type of** \\ **Variables** \\ \end{tabular}  & \ \begin{tabular}{c} \(A^{min}\) \\ **(mm2)** \\ \end{tabular}  & \begin{tabular}{c} \(A^{max}\) \\ **(mm2)** \\ \end{tabular}  & \(N_{c}\) \\ \hline This study & GA & discrete & 1045 & 21613 & 48 \\
[26] & Adaptive & discrete & 1045 & 21613 & 42 \\  & Elitist DE & & & & \\  & Fully & & & & \\
[42] & Constrained & discrete & 1045 & 21613 & 42 \\  & Design & & & & \\
[42] 1 & Constrained & discrete & 65 & 21743 & 30 \\  & Design & & & & \\
[24] & GA, adaptive & continuous & 65 & 22575 & - \\ \hline \multirow{3}{*}{**Member no. **} & \begin{tabular}{c} **This Study** \\ **Area** \\ **(mm2)** \\ \end{tabular}  & \begin{tabular}{c} **[26]** \\ **Area** \\ **(mm2)** \\ \end{tabular}  & \begin{tabular}{c} **[42]** \\ **Area** \\ **(mm2)** \\ \end{tabular}  & \ \begin{tabular}{c} 1 \\ **Area** \\ **(mm2)** \\ \end{tabular}  & \begin{tabular}{c} **[24]** \\ **(mm2)** \\ \end{tabular}  \\ \hline
1 & 21,613 & 21,614 & 21,614 & 21,743 & 19,762 \\
2 & 1045 & 1045 & 1045 & 65 & 65 \\
3 & 14,194 & 14,775 & 17,098 & 15,278 & 15,381 \\
4 & 10,000 & 9161 & 9  
There are three optimization criteria for Equations (1) and (2):  
* The weight of the arm body (\(a_{B}\times a_{B}\times l_{B}\) = 100 \(\times\) 100 \(\times\) 300 mm) is minimized;
* The equivalent stress of the arm body must not exceed the value of the yield stress of the material;
* The displacement of the loaded flange must not exceed the maximum allowable displacement of 2.5 mm in the direction of the loading force. A linear elastic material model (Appendix A, [43]) was used for all simulations. Five materials were selected for the design of the arm. The material properties are shown in Table 3, where \(E\) is [PERSON]'s modulus, \(\mu\) is [PERSON]'s ratio, \(\sigma^{Y}\) is yield stress and \(\rho\) is density. The arm was solved three times:  
* Model 1 is a truss structure with a rod element type. It was used for an optimization;
* Model 2 is a component structure, where the components are simulated by a quadratic shell element. It represents the behavior of the real structure;
* Model 3 is a solid structure designed for additive manufacturing, which used a quadratic solid element [12]; The rest of this chapter is arranged as follows: Section 3.1 shows the truss structure that replaces the body of the arm.