## 4 Results
### Minimize the Power Consumption  
In this second algorithm, we optimize the suitable GP combinations for which the sphere contained in the useful workspace is at least 0.5-S\({}_{max}\). The suitable GP combinations that fulfill this restriction are the _StudyVariables_ that will be considered for the power consumption optimization. In our case there are 325 _StudyVariables_ combinations in total. We fix the radius of the limbs and the thickness of the mobile platform. Even if these two parameters affect power consumption, in this method we only optimize the _StudyVariables_ obtained by solving the optimization of the workspace: The optimization variables in our case will continue being H, L and R. In case we wanted to optimize the thickness of the mobile platform and the radius of the legs, the procedure would be similar, but the computational cost would increase considerably. To solve the dynamics, we also \begin{table}
\begin{tabular}{l c c c} \hline \hline  & **H (m)** & **L (m)** & **R (m)** \\ \hline
1 & 0.35 & 0.25 & 0.2 \\
2 & 0.4 & 0.3 & 0.2 \\
3 & 0.45 & 0.35 & 0.2 \\
4 & 0.5 & 0.45 & 0.2 \\
5 & 0.55 & 0.5 & 0.2 \\
6 & 0.55 & 0.55 & 0.2 \\
7 & 0.6 & 0.55 & 0.2 \\
8 & 0.6 & 0.6 & 0.2 \\
9 & 0.65 & 0.6 & 0.35 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Best GP combinations for S\({}_{max}\) in maximum number of positions in the useful workspace. have to define the density of each component of the manipulator. We consider the material of the mobile platform to be aluminum and the material of the limbs to be steel. We chose the radius of the limbs to be 0.005 m and the thickness of the mobile platform to be 0.004 m.  
The motors are Maxon DC Motors, with graphite brushes and a maximum power of 150 Watt. We attach to them a Maxon Planetary GP 42 C with a reduction ratio of 15. The guides we use are IGUS toothed belt linear guides. A GP combination is suitable in the optimization of the power consumption for a given trajectory when it fulfills all the restrictions of the linear guides, motors and the gear head. According to this, in order to be a suitable GP combination for the optimization of the power consumption for a given trajectory, a GP combination has to fulfill the following conditions:  
1. The displacement of the linear guides has to be in their displacement range: \(0<\rho_{i}\) \(<\) 0.3 m.
2. The velocity of the linear guides has to be lower than their velocity limit: \(\mathrm{d}\rho_{i}<\) 5 m/s. 3. The radial load in the linear guides has to be lower than the maximum radial load: \(\mathrm{F}_{radial_{i}}<\) 300 N.
4. The axial load in the linear guides has to be lower than the maximum belt tension: \(\mathrm{F}_{axial}<\) 200 N.
5. The speed of the motors have to be lower than the maximum speed allowed: \(\mathrm{Speed}_{max}<\) 12,000 rpm. The expression that gives the speed of the motors is Equation (12): \[Speed_{i}=\frac{d\rho_{i}}{R_{gearhead}}\cdot\frac{60}{2\pi}\cdot Reduction\ Ratio\ \ \ \ \ (r.p.m)\] (12) where \(\mathrm{R}_{gearhead}\) is the radius of the gearhead and we obtain it by applying Equation (13). \[R_{gearhead}=\frac{Displacement\ Ratio}{2\pi}\ \ \ \ \ (m) \tag{13}\]
6. The power required by the motors has to be lower than the maximum power: \(\mathrm{Pow}_{i}<\) 150 W. We calculate the power that the motors consume by applying Equation (14). \[Pow_{i}=d\rho_{i}\cdot F_{axial_{i}}\ \ \ \ (W) \tag{14}\]
7. The torque supported by the motors cannot exceed the maximal possible torque: \(\mathrm{T}_{i}<\) 0.177 Nm. We obtain the torque in the motors by solving Equation (15). \[T_{i}=\frac{mean(F_{axial_{i}})\cdot R_{gearhead}}{Reduction\ Ratio}\ \ \ \ \ (N \cdot m) \tag{15}\]  
We check these criteria for each _StudyVariables_ and obtain the _suitable_ GP combinations for the power optimization process. We calculate the total power consumption during the studied trajectory for each _suitable_ GP combination. The power consumption of one motor over the trajectory is obtained by integrating the power required by the motor overtime, as shown in Equation (16).