# Model-Based Slippage Estimation to Enhance Planetary Rover Localization with Wheel Odometry
## 3 Traversability Analyses
### Traversability over Steep Terrains  
The contribution of the terrain properties on the dynamical forces depends on the average slope of the surface. Steeper is the terrain larger are the path discrepancies for different soil types. By assuming constant terrain slopes and vehicle's steering angles, we investigate the evolution of both uphill and downhill paths with different terrains. A proportional derivative (PD) control scheme was implemented to control the wheels' angular and linear velocity preventing from locking condition (\(\overline{V}\approx v_{min}\)) and exceeding the maximum speed (\(\overline{V}\approx v_{max}\)). If we assumed a constant commanded speed along the entire trajectory, the constraint on the maximum speed in the descending path would be violated, and the rover's motion would result in a slow uphill traverse. Therefore, at each integration step the angular acceleration of the wheels, \(\hat{\Omega}\), and the value of the commanded velocity, \(\overline{V}\), are given by:  
\[\hat{\Omega}=-K_{p}(V-V_{d})-K_{d1}u-K_{d2}\dot{v} \tag{12}\]  
\[\overline{V}=\Omega\mathrm{R_{w}} \tag{13}\]  
which enable the fulfilment of the previous motion constraints by assuming \(\hat{\Omega}=0\) if \(\overline{V}\geq v_{max}\) and \(\hat{\Omega}\geq 0\) or \(\overline{V}\leq v_{min}\) and \(\hat{\Omega}\leq 0\). In our numerical simulations, the PD control gains \(K_{p}\), \(K_{d1}\) and \(K_{d2}\) are 200, 50 and 50, respectively. A first test case is based on a 10\({}^{\circ}\)-slope and a constant steering angle \(\delta=4^{\circ}\) for the front and rear wheels. A stable rover speed \(V_{d}=40\) m/h is obtained through extra drive torque when driving uphill. This improves the action of the tractive forces and increases the actual speed of the rover, almost maintaining the desired value. On the contrary, braking is applied during rover's downhill path. In this case the commanded speed is reduced to obtain a negative value of the slip that induces a negative traction coefficient and a stronger braking action of the longitudinal forces. Figures 4-6 show the results of the trajectory integrated for 3 h by using an initial heading angle \(\psi_{0}=\pi/2\) and speed \(u_{0}=v_{c}\). At \(t=0\) the commanded speed is set equal to the desired speed (\(\overline{V}=V_{d}=40\) m/h). The rover's speed is controlled within the boundary constraints for the three terrain types. The commanded speed approaches the value \(v_{max}\) in  
Figure 3: Integration errors in position (\(\Delta x,\ \Delta y\)) and orientation (\(\Delta\psi\)) for the kinematic model. the uphill phase, and tends to lower values when the rover goes downhill. The actual speed of the rover follows the desired value \(V_{d}\). As expected, in the uphill the longitudinal slip is positive, while it tends to negative values in the downhill phases. The longitudinal slip reaches the maximum value for drift soil, \(\sigma=0.153\), indicating high-slippage conditions. Figure 4: Rover’s trajectories in the 3D space on a \(10^{\circ}\) slope with constant steering angle \(\delta=4^{\circ}\) and adaptive speed \(\
abla\). Figure 5: Rover’s speed on (**a**) cloddy soil, (**b**) mixed drift-cloddy soil and (**c**) drift soil on a \(10^{\circ}\) slope with constant steering angle \(\delta=4^{\circ}\). A second case is based on the trajectory integration with the same vehicle's steering angle and initial state conditions on a steeper surface with a 20\({}^{\circ}\) slope. This terrain inclination is an upper limit constraint on ExoMars traverse. Its onboard path planning standard will avoid slopes \(\geq\) 20\({}^{\circ}\)[25]. The rover's paths with a steeper terrain are dramatically different compared to the previous scenario (Figure 7). The rover completes one and a half loop on cloddy soil but its trajectory is along a straight line for mixed drift cloddy and drift soils. A steering angle of 4\({}^{\circ}\), therefore, is not sufficient to follow a circular path because of different terrain parameters that affect the wheel-soil interaction.