# Optimization of a Truss Structure Used to Design of the Manipulator Arm from a Set of Components
## 2 Method
### Application to a Truss Structure  
The behavior of a truss structure is given by its member-rods. The behavior of a rod is determined by its dimensions and material. The rod dimensions are represented by a cross-sectional area and a length. The material of the rod is represented by the corresponding material parameters, which are derived from the linear elastic isotropic material model. Let the rods be made only of a predetermined set of materials and cross-section areas. Then every component \(C_{j}\) is defined as:  
\[C_{j}=[A_{j},\rho_{j},E_{j},\sigma_{j}^{Y}], \tag{7}\]where \(A_{j}\), \(\rho_{j}\), \(E_{j}\), \(\sigma_{j}^{Y}\) are component parameters. The parameter \(A_{j}\) is a cross-section area and can be attached to different shapes of profile; the others are the parameters of the component material. They are determined by the equations for a rod:  
\[\sigma=E\epsilon,\sigma=\frac{P}{A},\mathsf{abs}(\sigma)=\sigma^{EQV},m=\rho I A, \tag{8}\]  
where \(\sigma\) is a normal stress, \(\epsilon\) is an elastic strain and \(P\) is a force, all in the rod axis. \(E\) is the Young's modulus, \(\rho\) is the material density and \(l\) is the length, all for the rod.  
The components in \(\mathbf{C}\) are sorted by the conditions:  
\[\mathbf{C}=[C_{j},j=1...,N_{c}], \tag{9}\] \[m_{j}>m_{j+1},m_{j}=\rho_{j}l^{s}A_{j},\] \[\Delta l_{j}<\Delta l_{j+1},\Delta l_{j}=\frac{P^{s}l^{s}}{E_{j} A_{j}},\] \[P_{j}^{Y}>P_{j+1}^{Y},P_{j}^{Y}=\sigma_{j}^{Y}A_{j}.\]  
The resulting values \(m_{j}\), \(\Delta l_{j}\) and \(P_{j}^{Y}\) describe the behavior for a rod, which is defined by a sorting length \(l^{s}\) and loaded by a sorting force \(P^{s}\). \(\Delta l_{j}\) is an extension of the rod, \(P_{j}^{Y}\) is the maximum force allowed in the rod and \(m_{j}\) is the weight of the rod. The resulting values \(m_{j}\), \(\Delta l_{j}\) and \(P_{j}^{Y}\) always contain two component parameters, see Equation (7). The value \(l^{s}\) is not the actual length of the j-th component but the sorting length, which is the same for all components. Similarly, the value of \(P^{s}\) is not a force acting on the j-th component, but a value common to all components used to classify them. The values \(l^{s}=100\) mm and \(P^{s}=1000\) N were chosen with respect to Section 3 to show where the values of \(m_{j}\) and \(\Delta l_{j}\) are placed.  
Meeting the condition of Equation (9) for \(\mathbf{C}\) significantly simplifies the optimization process. Methods for optimizing truss structures can be found in the literature, see Section 1, where the cross-sectional area can take on discrete values, and one material is considered. Then components (for example, for \(N_{c}=5\)) can be defined by Equation (7) as \(\mathbf{C}=[C_{1},C_{2},C_{3},C_{4},C_{5}]\) and is fulfilled for one material:  
\[\rho=\rho_{1}=\rho_{2}=...,E=E_{1}=E_{2}=...,\sigma^{Y}=\sigma_{1}^{Y}=\sigma_ {2}^{Y}=..., \tag{10}\]  
\(\mathbf{C}\) can be modified to \(\mathbf{C^{*}}=[A_{1},A_{2},A_{3},A_{4},A_{5}]\), and for Equation (9), the following also applies:  
\[\mathbf{C^{*}}=[A_{j},j=1...,N_{c}],A_{j}>A_{j+1}, \tag{11}\] \[m_{j}>m_{j+1},m_{j}=\rho^{s}A_{j},\] \[\Delta l_{j}<\Delta l_{j+1},\Delta l_{j}=\frac{P^{s}l^{s}}{EA_{j}},\] \[P_{j}^{Y}>P_{j+1}^{Y},P_{j}^{Y}=\sigma^{Y}A_{j},\]  
Based on this agreement, an example often used in the literature was selected for verification, which only deals with the cross-sectional areas \(\mathbf{C^{*}}\). It is also very easy to swap \(\mathbf{C^{*}}\) and \(\mathbf{C}\) and use the algorithm designed for \(\mathbf{C^{*}}\) to solve a problem with \(\mathbf{C}\) components. In this case, it would be necessary to analyze the behavior of the selected algorithm in much more detail and compare several methods. This was not the aim of this article, so a custom modification of the GA was proposed, and other methods were used only for verification.