## 3 Proprioceptive Sensing
### Statistical Feature and Validity Indices  
For the \(i\)-th feature, the main statistical moments are estimated: mean \(E_{i}\), variance \({\sigma_{i}}^{2}\), skewness \(Sk_{i}\) and kurtosis \(Ku_{i}\) of the feature signal for a single terrain patch. These statistics are defined according to Equations (11)-(14):  
\[E_{i}=\frac{1}{N}\sum_{n=1}^{N}x_{n} \tag{11}\]  
\[{\sigma_{i}}^{2}=\frac{1}{N}\sum_{n=1}^{N}(x_{n}-E_{i})^{2} \tag{12}\]  
\[Sk_{i}=\frac{1}{N}\frac{\sum_{n=1}^{N}(x_{n}-E_{i})^{3}}{\left(\sqrt{{\sigma_{ i}}^{2}}\right)^{3}} \tag{13}\]  
\[Ku_{i}=\frac{1}{N}\frac{\sum_{n=1}^{N}(x_{n}-E_{i})^{4}}{\left(\sqrt{{\sigma_{ i}}^{2}}\right)^{4}} \tag{14}\]  
where \(x_{n}\) is the value assumed by the feature at the time-step \(n\) and \(N\) is the total number of time-steps for the considered terrain patch.  
In addition, each terrain patch is labelled for terrain type, sand for Morocco data or rock for Utah data, and slippage level; high slip is for absolute value of slip greater than 0.5 and all other conditions are low slip.  
The statistical features are divided into clusters according to patch labels and a validity index value is associated with each statistical feature. Two validity indices are proposed to obtain a quantitative measure of \"feature goodness\" for terrain patch classification: the \(WB\) index [33] and a Pearson Coefficient based index (\(PC\)) [34]. These indices are employed in the feature selection method described in the following section. The \(WB\) index is computed as:  
\[WB=m\frac{SSW}{SSB}, \tag{15}\]  
The sum of square within cluster \(SSW\) and between clusters \(SSB\) can be computed as follows:  
\[SSW=\sum\
olimits_{k=1}^{m}{\sum\
olimits_{i=1}^{n_{k}}(x_{i}-\mu_{k})^{2}}, \tag{16}\]  
\[SSB=\sum\
olimits_{k=1}^{m}{n_{k}\left(\mu_{k}-\mu\right)^{2}}, \tag{17}\]  
where \(x_{i}\) is the generic statistical feature value for the patch \(i\), \(\mu_{k}\) is the cluster \(k\) centroid value, \(\mu\) is the overall dataset centroid value, \(n_{k}\) is the number of patches in the cluster \(k\) and \(m\) is the number of clusters. In this study \(m\) is 2 since we have two classes for the terrain classifier. \(WB\) index assumes lower values for higher distance between different cluster centroids and for lower variance within clusters. A good classifier feature allows us to obtain distant and compact clusters, and thus generally corresponds to a low value of the \(WB\) index. A \(PC\) index could be computed through the linear regression of a feature against the m classes, giving a progressive numerical value to each class. A higher value of \(PC\) potentially corresponds to a better classifier feature.  
Figures 3 and 4 show, respectively, the \(WB^{-1}\) and \(PC\) indices values for all the statistical features computed both on terrain and slippage labelled datasets. Some features are missing because the correspondent sensor suite is not available in the classification dataset (please refer to Table 1).  
Figure 4: _Cont._  
Figure 3: \(WB^{-1}\) index values for each feature (F1 to F15), for each statistic (S1 to S4, respectively, mean, variance, skewness and kurtosis) and for terrain (**a**) and slippage (**b**) classifier.