# Slip Estimation Model for Planetary Rover Using Gaussian Process Regression
## 5 Ground Tests Results and Discussion  
To ensure better stability and reliability of the model, we repeated ten iterations of hold-out validation on the models of each type. A total of 40 slip models were acquired based on the data collected on two kinds of simulated soil and two training methods. We used mean absolute error (MAE) and root mean square error (RMSE) as evaluation indices in this paper. \[\text{MAE}\_x=\frac{1}{n}\sum_{i=1}^{n}\abs{s_{xi}-s_{xi}} \tag{20}\]  
\[\text{MAE}\_y=\frac{1}{n}\sum_{i=1}^{n}\abs{\hat{s}_{yi}-s_{yi}} \tag{21}\]  
\[\text{RMSE}\_x=\sqrt{\sum_{i=1}^{n}\frac{\left(s_{xi}-s_{xi}\right)^{2}}{n}} \tag{22}\]  
\[\text{RMSE}\_y=\sqrt{\sum_{i=1}^{n}\frac{\left(s_{yi}-s_{yi}\right)^{2}}{n}} \tag{23}\]  
where MAE_x and RMSE_x are evaluation indices for slip_x, MAE_y and RMSE_y are evaluation indices for slip_y; \(\hat{s}_{xi}\) and \(\hat{s}_{yi}\) are predicted values for slip_x and slip_y, respectively; \(s_{xi}\) and \(s_{yi}\) are real observations for slip_x and slip_y, respectively; \(n\) is the amount of testing sets. Tables 3 and 4 detail the models' performance results for slip_x and slip_y, respectively. Figures 8 and 9 present the results visually using line charts and box plots. For longitudinal slip, Figure 8a,c shows that the estimation results of GPR models fluctuated greatly, and the box plots displayed in Figure 8b,d also show that GA-BP estimation was more concentrated. These all demonstrated that the estimation performance of the GPR models was unstable compared to the GA-BP models. Take MAE_x results on TYII-2 dataset as an example: the MAE_x values of the GA-BP models were between 2.57-2.70%, with an average of  
Figure 7: Flowchart of the model establishment process using GPR. 2.65%, and a maximum difference of 0.13%. The MAE_x of the GPR models were between 0.29-4.35%, with an average of 1.94% and a maximum difference of 4.06%. Although GPR models are unstable, the optimal performance of GPR models is beyond the reach of GA-BP models. MAE_x = 0.29% signified that the estimated slip_x values closely approximated the true values, which demonstrated the estimation potential of the GPR models. For lateral slip, the performance results of slip_y were roughly similar to those of slip_x. In conclusion, both GPR models and GA-BP models could complete the estimation with good accuracy. The estimation performance of the GPR models was unstable compared to the GA-BP models, but GPR models also demonstrated estimation potential, which GA-BP models cannot reach. \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline  & \multicolumn{3}{c}{**TYII-2**} & \multicolumn{6}{c}{**JLU Mars-2**} \\ \cline{2-10}  & \multicolumn{3}{c}{**GA-BP**} & \multicolumn{3}{c}{**GPR**} & \multicolumn{3}{c}{**GA-BP**} & \multicolumn{3}{c}{**GPR**} \\ \cline{2-10}
**Group** & \multicolumn{3}{c}{**MAE\_y**} & \multicolumn{3}{c}{**RMSE\_y**} & \multicolumn{3}{c}{**MAE\_y**} & \multicolumn{3}{c}{**RMSE\_y**} & \multicolumn{3}{c}{**No**} & \multicolumn{3}{c}{**MAE\_y**} & \multicolumn{3}{c}{**MAE\_y**} \\
**Number** & **(\%)** & **(\%)** & **(\%)** & **(\%)** & **(\%)** & **(\%)** & **(\%)** & **(\%)** & **(\%)** \\ \hline (1) & 1.15 & 1.5  
To further explore the estimation effects of GPR models, Figure 10 displays part of the GPR estimation results in the TYII-2 group (1) and JLU Mars-2 group (1). Since the test set data were randomly selected, there was no continuous trend between adjacent training data. In order to exhibit the results more clearly, Figure 10 only shows the estimation of 100 data points to avoid excessively dense data points. The gray band area in the figure represents the 95% confidence interval corresponding to the estimation result. As shown, no matter what kind of soil dataset utilized, and whether it was slip_x or slip_y, the estimation results were consistent with the trends of actual values, and almost all real points were located within the 95% confidence interval, which means GPR models can provide a reasonable estimation of slip value uncertainty.