## 3 Selecting Optimal Measurement Poses by a Genetic Algorithm  
Assuming \(m\) (the measurement pose) was used to calibrate the robotic manipulator, the dimension of the corresponding identification Jacobian matrix \(J\) was \(3m\times n\)[29] (where only the position was considered) and \(n\) was the number of the identified parameters. Using the singular value decomposition method, the identification Jacobian matrix became:  
\[J=U\Sigma V \tag{3}\]  
where \(U\) and \(V\) are orthonormal matrices. We assumed that \(3m>n\), thus producing:  
\[\Sigma=\left( \begin{array}{ccccc}\sigma_{1}&0&\cdots&\cdots\\ \vdots&\vdots&\vdots&\vdots\\ 0&0&\cdots&\sigma_{n}\\ 0&0&\cdots&0\\ \vdots&\vdots&\vdots&\vdots\\ 0&0&\cdots&0\end{array} \right) \tag{4}\]  
where \(\Sigma\) is a \(3m\times n\) matrix and \(\sigma_{i}\) are singular values where \(1\leq i\leq n\). Figure 1: Sketch of a serial robot (YS100). Numerous observability indices were created using these singular values to quantify the quality of the pose selection. [PERSON] et al. [30] suggested an observability index based on the sum of all singularities. The inverse condition number was used by [PERSON] and [PERSON] [31]. Two observability indexes were suggested by [PERSON] et al. [32]. In this work, an observability index based on the product of singular values was used [33]. It is known as the best observability index for minimizing the variance of parameters [34]. The observability index corresponded with the determinant of \(J^{T}J\) and \(\det(J^{T}J)=(\sigma_{1}\cdots\sigma_{n})^{2}\). When the observability index was maximized, the errors of the parameters were best noticed. \[O=\frac{(\sigma_{1}\ldots\sigma_{n})^{\frac{1}{n}}}{\sqrt{m}} \tag{5}\]  
In [35], the observability (O) was approved to linearly correspond with the determinant of \(J^{T}J\), invariant to scaling. This was the best observability index to minimize the variance of the parameters. A GA is a well-known evolution method that is stimulated by the rule of natural selection. GAs mimic the selection process of nature. In the natural environment, individuals with the best fitness have a better chance of surviving. The genes of these individuals have a higher possibility of transferring to their successor. GAs include five phases:  
* Initial population;
* Fitness function;
* Selection;
* Crossover;
* Mutation. In this study, a GA was applied to find the best robot configuration set for the calibration process based on the observability index. The process was as follows:  
1. The pool of the configuration measurement, \(\Omega\), was initialized. 2. The population was generated, including an \(n\) individual. Each individual had an \(m\) configuration of measurements. 3. The fitness value of each individual based on the observability index, \(O\), (Equation (5)) was then calculated. 4. Every individual was evaluated and ranked based on their fitness values. 5. When the terminating condition was reached, the process was stopped. The outputs were the set of the \(m\) configuration of measurements of the individual with the highest observability index, \(O\). 6. The GA was used to produce the new population based on the observability index, \(O\). 7. Steps 3-6 were repeated until the stopping criterion was satisfied. In the steps of initializing and generating a new population by applying the genetic algorithm, each individual set had to be checked for duplicate configurations. If any robot configuration was replicated, its clone was eliminated and another configuration from the pool of configuration measurements (\(\Omega\)) was added. The result of this step was that the robot configuration set had a maximum observability index, \(\gamma\).