## 2 Materials and Methods
### Odometry Trajectory  
Odometry is a simple and practical method that provides a periodic real-time estimation of the relative displacement of a mobile robot based on the measurement of the angular rotational speed of its wheels. Odometry estimation is valid in the case of non-slippage wheel conditions in which wheel revolutions can be translated into linear displacement relative to the floor. The advantage of an omnidirectional mobile robot using three optimal omnidirectional wheels is that the wheels do not have motion constrains and do not require slippage in order to implement any motion trajectory [20]. The kinematics and the odometry of the mobile robot APR-02 are described in [20]. Figure 2 presents the parametric definition of the omnidirectional motion system based on three omnidirectional wheels (Figure 2a) and a detail of the parameters of one wheel (Figure 2b). The main parameters are: the position of the mobile robot \((x,y,\theta)\) referred to the fixed world frame \((X_{W},Y_{W})\), the motion command of the robot \((v,\alpha,\omega)\), the angular velocities of the wheels \((\omega_{a},\omega_{b},\omega_{c})\), the linear velocities of the wheels \((V_{a},V_{b},V_{c})\), the radii of the wheels \((r_{a},r_{b},r_{c})\), the distance between the center of the robot and each wheel \((R_{a},R_{b},R_{c})\) and the angular orientation of each wheel \((\delta_{a},\delta_{b},\delta_{c})\) referred to the mobile robot frame \((X_{R},Y_{R})\). The odometry procedure uses the instantaneous estimate of the current angular velocities of the three wheels, \(a,\ b,\ c\), available as a vector sequence \((\omega_{a}(k),\omega_{b}(k),\omega_{c}(k))\) in order to estimate the instantaneous position of the mobile robot \((x(k),y(k),\theta(k))\) in the world frame \((X_{W},Y_{W})\). The relation between the instantaneous estimation of the angular velocities of the wheels \((\omega_{a}(k),\omega_{b}(k),\omega_{c}(k))\) and the instantaneous robot velocity \(\big{(}v_{x}(k),v_{y}(k),\omega(k)\big{)}\) in the world frame \((X_{W},Y_{W})\) can be summarized as [20]:  
\[\left[ \begin{array}{c}\omega_{a}(k)\\ \omega_{b}(k)\\ \omega_{c}(k)\end{array} \right]=\left[ \begin{array}{ccc}1/r_{a}&0&0\\ 0&1/r_{b}&0\\ 0&0&1/r_{c}\end{array} \right]\cdot\left[ \begin{array}{ccc}-sin(\delta_{a})& cos(\delta_{a})&R_{a}\\ -sin(\delta_{b})&cos(\delta_{b})&R_{b}\\ -sin(\delta_{c})&cos(\delta_{c})&R_{c}\end{array} \right]\cdot\left[ \begin{array} []{ccc}cos(\theta(k-1))&sin(\theta(k-1))&0\\ -sin(\theta(k-1))&cos(\theta(k-1))&0\\ 0&0&1\end{array} \right]\cdot\left[ \begin{array}{c}v_{X}(k)\\ v_{Y}(k)\\ \omega(k)\end{array} \right]_{World} \tag{1}\]  
Which includes a rotation matrix \(R(\theta)\) and a compact kinematic matrix \(M\) that defines the overall kinematics of the mobile robot:  
\[R(\theta)=\left[ \begin{array}{ccc}\cos(\theta)&\sin(\theta)&0\\ -\sin(\theta)&\cos(\theta)&0\\ 0&0&1\end{array} \right] \tag{2}\]  
\[M=\left[ \begin{array}{ccc}-sin(\delta_{a})/r_{a}&cos(\delta_{a})/r_{a}&R_{a }/r_{a}\\ -sin(\delta_{b})/r_{b}&cos(\delta_{b})/r_{b}&R_{b}/r_{b}\\ -sin(\delta_{c})/r_{c}&cos(\delta_{c})/r_{c}&R_{c}/r_{c}\end{array} \right] \tag{3}\]  
Equation (1) can be arranged to update the current position of the mobile robot \((x(k),y(k),\theta(k))\) in the world frame \((X_{W},Y_{W})\) based on the new estimate of the current angular velocities of the three wheels of the mobile robot \((\omega_{a}(k),\omega_{b}(k),\omega_{c}(k))\) and the time lapse \(\Delta t\) between the samples \(k-1\) and \(k\), which in the mobile robot APR-02 coincides  
Figure 2: Parametric definition of the omnidirectional motion system of the mobile robot APR-02: **(a)** general representation of the three omnidirectional wheels and **(b)** detail of the parameters of one omnidirectional wheel. \((X_{R},Y_{R})\) represents the mobile robot frame in which \(X_{R}\) is the front of the mobile robot. with the sampling time \(T\) of its proportional-integral-derivative (PID) motor controllers, \(\Delta t=T=10\) ms:  
\[\left[