## 4 Mechanism Performance Indicators
### Other Performance Indicators  
As previously mentioned, performance indicators must allow solutions to be compared and, for that reason, must be based on normalized quantities. From many performance indicators that can be considered while calculating fitness functions, dexterity and the maximum feasible Euler angles have been selected as the most valuable for flight simulation. In this case, it is necessary to normalize the Jacobian matrix because the elements have different units of measurement. Certain elements are divided by the platform's radius or by the length of the lower lever to perform normalization.  
\[\mathbf{J}_{\mathbf{x}\mathbf{n}}=\left[\mathbf{i}_{\mathbf{u}\mathbf{l}}\quad\mathbf{O}_{\mathbf{p}}\mathbf{P}_{ \mathbf{i}}\times\mathbf{i}_{\mathbf{u}\mathbf{l}}/R_{\mathbf{p}}\right],\ \mathbf{J}_{\mathbf{x}\mathbf{n}}=\mathbf{J}_{\mathbf{x}}/a \tag{37}\]  
Figure 11: (**a**) Top and (**b**) bottom view of the workspace for zero orientation.  
Based on the previous two equations, the normalized Jacobian matrix is:  
\[\mathbf{J_{n}}=\mathbf{J_{xn}^{-1}}\mathbf{J_{xn}} \tag{38}\]  
In order to calculate the Local Dexterity Index (LDI) \(\eta\) as the reciprocal value of the Jacobian matrix condition number [16], a homogeneous Jacobian matrix \(\mathbf{J_{h}}\) should first be obtained. As shown in [20], the Jacobian matrix of external coordinates \(\mathbf{J_{x}}\) can be separated into two matrices that have different physical dimensions. Then some sort of normalization can be applied to obtain a homogeneous matrix. The first matrix, obtained as the first three columns, is related to forces, while the last three columns are related to torques.  
Paper [46] defines robot manipulability as the absolute value of the determinant of the Jacobian matrix, which in this case was previously normalized.  
\[w=|\text{det}(\mathbf{J_{n}})| \tag{39}\]  
Paper [34] defines robot dexterity as the reciprocal value of the condition number of the Jacobian matrix, which in this case was previously normalized. The condition number of the matrix is the product of the Euclidean norm of the matrix and the Euclidean norm of the inverse matrix.  
\[\eta=\frac{1}{k}=\frac{1}{\text{cond}(\mathbf{J_{n}})}=\frac{1}{\|\mathbf{J_{n}}\|_{2 }\cdot\left\|\mathbf{I_{n}}^{-1}\right\|_{2}} \tag{40}\]  
Manipulability and dexterity of this type of mechanism are also analyzed in papers [47; 48; 49]. For some applications, performance indicators can also be partial or full isotropy of the mechanism described in [50].  
The third and fourth performance indicators are based on the maximum absolute value of the feasible independent Euler pitch and roll angles (\(\theta_{j,max}\) and \(\varphi_{j,max}\)) about global y- and x-axes for the given position. In relation to the method with the incremental change of angle with constant step, until the maximum value (with a certain tolerance) is determined, a significantly more efficient method is implemented. This new method is based on the variable step and determining the node in which the indicator of feasibility change value, certain tolerance is met.  
An additional indicator, for high-precision robots, is the elasticity of their structure. The mechanism can be designed to withstand all loads within some allowed limits of displacement, or this can be computed or even measured and then compensated with a control algorithm. In both cases, in the design process, displacement of the structure under load has to be calculated. The interesting part of this analysis is the numerical modeling of joints. The finite element method can be applied to obtain displacements of this mechanism, while joints can be implemented using the Lagrangian-multiplier method.