## 3 Proprioceptive Sensing
### Proprioceptive Features  
A large amount of sensory data can be gathered from Sherpa and potentially used for classification purposes. However, only a few signals bring significant information related to terrain type. Here, only the most relevant features that we found are discussed. From the FTSs, three forces and three moments are available for each wheel. Among these, we retain the longitudinal force \(F_{X}\) and the wheel drive torque \(T_{D}\). The wheel drive torque can be also estimated indirectly from the electrical current, \(C_{D}\), whereas the wheel angular velocity \(\omega\) can be obtained from wheel encoders. The three body accelerations \(a_{x}\), \(a_{y}\) and \(a_{z}\) are estimated from the IMU. The mechanical power \(P_{M}\) and the electrical power \(P_{E}\) are computed, respectively, in Equations (2) and (3):  
\[P_{M}=T_{D}\omega, \tag{2}\]  
\[P_{E}=VId_{PWM}, \tag{3}\]  
where \(V\), \(I\) and \(d_{PWM}\) are, respectively, the motor voltage, current and PWM duty cycle of the wheel drive. Apart from this set of \"primary\" quantities, a second set of \"derivate\" features has been found \begin{table}
\begin{tabular}{c c} \hline \hline
**Sensor Modality** & **Sample Rate [Hz]** \\ \hline FTS & 100 \\ IMU & 100 \\ DGPS & 10–20 \\ Encoders & 100 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Sensor sample rates. to be effective for terrain characterization. Derivate features can be obtained by combining primary features according to well-known physics-based relationships. As an example, the friction coefficients \(\mu\) can be estimated using three different sensor data according to Equations (4)-(6):  
\[\mu_{1}=F_{X}/F_{Z}, \tag{4}\]  
\[\mu_{2}=T_{D}/RF_{Z}, \tag{5}\]  
\[\mu_{3}=C_{D}/RF_{Z}, \tag{6}\]  
where \(F_{Z}\) is the wheel vertical force and \(R\) is the wheel unloaded radius. Another key feature is the so-called wheel speed deviation, \(SD\), that is the absolute value of the difference between the angular speed of each wheel \(\omega\) and the average angular speed of the four wheels \(\overline{\omega}\) as defined by Equation (7):  
\[SD=|\omega-\overline{\omega}|, \tag{7}\]  
Note that the \(SD\) can be extended as well to turning motion, as explained in [31]. Wheel longitudinal stiffness \(LS\) is computed as the ratio between the friction coefficient and the wheel slip \(s\). In the simplified case of a linear relation between friction coefficient and wheel slip, this physical quantity should represent the slope of friction--slippage plot. Three different \(LS_{i}\) values are computed in Equation (8) using the three friction coefficients defined before:  
\[LS_{i}=\frac{\mu_{i}}{s},\quad i=1,\;2,\;3, \tag{8}\]  
The slippage \(s\) is univocally estimated for both accelerating and braking and for each wheel based on Equation (9):  
\[s=d\left(1-\left(\frac{V_{X}}{\omega R}\right)^{d}\right),\qquad d=\left\{ \begin{array}{ll}+1&\omega R-V_{X}\geq 0\\ -1&\omega R-V_{X}<0\end{array} ,\right. \tag{9}\]  
where \(V_{X}\) is estimated with the DGPS and \(d\) is positive for accelerating and negative for braking. An approximate estimate of wheel sinkage \(z\) can be obtained by Equation (10) [32]:  
\[z=R\left(1-\cos\left(2\cdot\frac{\frac{T_{D}}{R}-F_{X}}{F_{Z}}\right)\right), \tag{10}\]  
Table 3 sums up the proprioceptive feature space spanned by the above considerations.