## 2 Materials and Methods
### Systematic Odometry Error Sources in a Three-Wheeled Omnidirectional Mobile Robot  
The exact theoretical value of the inverse of the compact kinematic matrix \(M^{-1}\) computed with the nominal values of the kinematic parameters (\(R_{a}=R_{b}=R_{c}=0.195\) m, \(r_{a}=r_{b}=r_{c}=0.148\) m and \(\delta_{a}=60^{\circ}\), \(\delta_{b}=180^{\circ}\), \(\delta_{c}=300^{\circ}\)) evaluated using a standard double precision floating-point format (IEEE 754-1985: 4-bytes data or 64-bit data, precision \(1.11\times 10^{-16}\)[25]) is:  
\[M^{-1}=\left[ \begin{array}{cc}-0.0854478398400646&0.00000000000000&0.0854478398400 0646\\ 0.04933333333333333&-0.098666666666667&0.049333333333333333\\ 0.2529914529914529&0.2529914529914529&0.2529914529914529\end{array} \right] \tag{7}\]  
The simple numerical representation of the exact values of the coefficients of the inverse of the compact kinematic matrix \(M^{-1}\) of the mobile robot APR-02 intuitively indicates that any inaccuracy in the values of the kinematic parameters may cause a large impact in the generation of systematic errors during the cumulative evaluation of the odometry (see Equation (4)). In a real mobile robot application, the effective values of the kinematic parameters \(R_{a},\ R_{b},\ R_{c},\ r_{a},\ r_{b},\ r_{c},\delta_{a},\delta_{b},\delta_{c}\) can differ from the nominal values due to imprecisions in the manufacturing process and then originate systematic errors in the cumulative computation of the odometry of a mobile robot. The following figures are proposed to graphically illustrate the uncertainties that appear in the determination of the exact or effective value of the kinematic parameters of a real mobile robot. Figure 3 shows two views of one omni-directional wheel of the motion system of the mobile robot APR-02 and a representation of its radius \(r_{a}\). Even in this case in which the wheel cover is thin and smooth, the accurate determination of the radius of the wheel is very difficult and will have some uncertainty because of the difficulty in the estimation of the point of contact with the floor [26]. Similarly, Figure 4 shows two views of the representation of the distance to the center of the mobile robot. Figure 4a shows the effect of a vertical wheel misalignment (exaggerated in this case) and Figure 4b the effect caused by the point of contact with the floor. Finally, Figure 5 shows two views of the effect of angular wheel misalignment that can be caused by the application of non-uniform pressure to the screws that hold the motor to the wheelbase through a rubber piece designed to absorb vibrations. These figures clearly reveal that manufacturing inaccuracies and assembly imprecisions will originate crossed systematic errors in the odometry of the omnidirectional mobile robot. Figure 3: General view (**a**) and detail (**b**) of the wheel radius and the point of contact of the wheel with the floor: \(r_{a}\) depicts the nominal radius (green font and lines) and \(r_{a}^{\prime}\) the real or effective radius (red font and lines). Finally, Tables 1-3 show the analytic determination of the sensitivity of the kinematic matrix \(M^{-1}\) to the kinematic parameters of the wheels, computed form Equation (6). These tables show very high sensitivities in the array values of \(M^{-1}\), a matrix that is used to cumulatively update the position and location of the mobile robot 100 times per second (See Equation (4)). In the case of the distance from the center of the robot to each wheel, the maximum individual sensitivity of \(M^{-1}\) is as high as 0.43 mm/s for each millimeter of error in the determination of the real value of \((R_{a},R_{b},R_{c})\). In the case of the radii of the wheels, the maximum individual sensitivity of \(M^{-1}\) is as high as 1.70 mm/s for each millimeter of error in the determination of the real value of \((r_{a},r_{b},r_{c})\).