## 3 Omnidirectional Robotic Platform. Hardware Design
### Locomotion Calculation
#### 3.1.1 Nominal Torque and Power  
Equation (1) relates the desired linear velocity of the robot, \(V_{nom}\), with the angular velocity of the wheels in r.p.m., \(n_{nom}\). Here, a factor \(\eta\) is applied to ensure good performance even with the inevitable presence of friction.  
\[n_{nom}=\frac{60}{\pi}\cdot\frac{V_{nom}}{d\cdot\eta} \tag{1}\]  
On the other hand, (2) calculates the nominal torque, \(M_{nom}\), to move the entire robot at nominal velocity. This torque depends proportionally on the force necessary to apply, \(F_{nom}\), which is determined in (3), and inversely on the bearing factor, \(\delta\).  
\[M_{nom}=\frac{d}{2}\cdot\frac{F_{nom}}{\delta} \tag{2}\]  
The nominal force is the needed force to keep the robot at the nominal velocity. Typically, three main friction forces appear during the displacement of a vehicle: (a) rolling resistance, (b) aerodynamic resistance, and (c) slope resistance. Both aerodynamic and slope resistances may be neglected in our case. Aerodynamic resistance does not contribute because of the following reasons: (a) the air in the tunnel is static, without the presence of air currents, (b) the robot moves at low velocity, and (c) the air is clean and dry. Sloperesistance does not contribute because the tunnel is mainly flat. Thus, the rolling force is calculated in (3) through (a) the rolling resistance coefficient, or the coefficient of rolling friction (CRF), \(C_{rr}\), and (b) the normal force, \(F_{N}\), this is, the perpendicular force to the surface on which the wheel is rolling. The coefficient of rolling friction is given in (4) [18] by the sinking depth, \(z\), which depends on the elastic features of the material of the rollers, and the wheel diameter, \(d\). On the other hand, \(F_{N}\) depends on the robot weight, \(m_{R}\).  
\[F_{nom}=C_{rr}\cdot F_{N}=C_{rr}\cdot m_{R}\cdot g \tag{3}\]  
\[C_{rr}=\frac{z}{d} \tag{4}\]  
The calculated nominal torque has to be split in the number of traction wheels, \(N\), in the robot, as expressed in (5), resulting in the needed nominal torque in each motor set, \(M_{nom/m}\).  
\[M_{nom/m}=\frac{M_{nom}}{N} \tag{5}\]  
So far, the torque when the nominal velocity is reached has been calculated. However, in order to calculate the acceleration, average and maximum torques, a velocity profile has to be created, as shown in Figure 1, where the time to reach the nominal velocity, \(t_{acc}\), and the time of nominal velocity, \(t_{nom}\), are indicated. For simplicity, we have assumed constant acceleration, \(a\), and equal acceleration and deceleration time. Thus, acceleration is calculated in (6).  
\[a=\frac{V_{nom}}{t_{acc}} \tag{6}\]  
The last variable to figure out is the power that the motor set has to provide while the robot is driving at nominal velocity. Thus, in (7) we calculate the necessary nominal power of the motor set, \(P_{nom/m}\), where a confidence factor, \(\kappa\), and a power factor, \(PF\), have been set to slightly oversize the set. Anyway, the chosen devices have to provide more power than calculated during the nominal stage.  
\[P_{nom/m}=PF\cdot\frac{m_{R}\cdot V_{nom}\cdot(a+g\cdot\sin\theta)}{2\cdot \pi\cdot\kappa\cdot N} \tag{7}\]