## 3 Systematic Odometry Error Evaluation and Correction
### Odometry Calibration Strategies and Results  
The odometry calibration strategies tested in this paper are based on analysis of the calibration trajectories proposed in Figure 9. The iterative calibration procedure used has been previously described in Section 2.6. In summary, this iterative calibration gets the registered calibration trajectories and iteratively adjusts the values of the effective kinematic parameters to globally match the odometry and ground truth trajectories. Table 4 presents the results obtained: _Trajectories_ depicts which trajectories have been used in the iterative search; _Strategy_ shows the acronym of the calibration strategy applied; _Method_ describes the iterative function used (GA or fmin) and the number of trajectory repetitions used in the iterative search: (1) one or (5) five repetitions; \(M^{-1}\) shows the value of the kinematic matrix obtained as a result of the iterative search; \(CF_{CALIBRATION}\) is the value of the average cost function obtained during the iterative search or training; \(CF_{VALIDATION}\) is the average value of the cost function obtained with five additional calibration trajectories (the complete flower-shape); _Improvement_ depicts the relative improvement of \(CF_{VALIDATION}\) relative to the uncalibrated case (_None_ strategy). Figure 9: Representation of the ideal flower-shaped trajectories proposed to calibrate the odometry of the omnidirectional mobile robot APR-02. The calibration strategies shown in Table 4 have been generally labeled as _STX-GAZ_ and _STX-FminZ_, where \(X\) describes the group of trajectories considered (from \(A\) to \(E\)); \(GA\) refers to the use of the _ga.m_ iterative function and _Fmin_ to the _fmincon.m_ iterative function; and \(Z\) is the number of repetitions of each calibration trajectory considered, a value that  
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Trajectories** & **Strategy** & **Method** & \(\mathbf{M}^{-1}\) & \(\mathbf{CF_{CLLIBRATION}}\) & \(\mathbf{CF_{VALID}}\) & **Improvement** \\ \hline \hline \multirow{4}{*}{**All**} & \multirow{4}{*}{**None**} & \multirow{4}{*}{-} & \(\left[ \begin{array}{ccc}-0.0854&-0.0000&0.0854\\ 0.0493&-0.0987&0.0493\\ 0.2530&0.2530&0.2530\\ \end{array} \right]\) & \multirow{4}{*}{-}  
can be 1 or 5. The evaluation of each calibration strategy with one or five repetitions was proposed to compare the achievements obtained relative to the additional effort required to obtain 5 repetitions of each calibration trajectory. The calibration strategies evaluated in Table 4 are:  
**None**. This strategy provides a reference evaluation result of the cost function \(CF_{VALIDATION}\) using the uncalibrated kinematic parameters to compute the odometry of the 5 repetitions of all calibration trajectories registered to validate the results. **STA**. This strategy uses only the straight calibration trajectories corresponding to a forward motion and two additional motions at \(\pm\)120\({}^{\circ}\) (Figure 9, trajectories: R1, R5 and R9). For example, STA-GA1 uses one repetition of the straight calibration trajectories and genetic algorithms, whereas STA-Fmin5 uses five repetitions of the straight calibration trajectories evaluated with the nonlinear multivariable function. **STB**. This strategy uses only the curved clockwise and counterclockwise trajectories corresponding to a forward motion and two additional motions at \(\pm\)120\({}^{\circ}\) (Figure 9, trajectories: B2, B6, B10, G12, G4 and G8). **STC**. This strategy uses only the straight trajectories corresponding to a forward motion and eleven additional motions at \(\pm\)30\({}^{\circ}\) (Figure 9, trajectories from R1 to R12). **STD**. This strategy uses only the curved clockwise and counterclockwise trajectories corresponding to a forward motion and eleven additional motions at \(\pm\)30\({}^{\circ}\) (Figure 9, trajectories from B1 to B12 and from G1 to G12). **STE**.