# Model-Based Slippage Estimation to Enhance Planetary Rover Localization with Wheel Odometry
## 4 Wheel Odometry Correction for Rover's Localization
### Rover's Localization  
The reference trajectory was then used to run numerical simulations for the localization system presented in this study. The sensors that are modeled in our software provide WO encoder's measurements of the wheels' angular rate and IMU measurements of the heading direction and the yaw rate [36; 37]. This dead-reckoning system is affected by non-systematic effects on demanding terrains since WO yields the travelled distance without considering the slippage between the wheels and the ground. Systematic errors due to the integration of these data in a simplified kinematic model accumulate in time leading to poor reconstructed trajectories. By applying our dynamical model presented in Section 2, we compute a slippage correction for WO to enhance the rover's localization with dead-reckoning measurements only. Figure 11: Rover’s speed vs. time. Figure 12: Steering angle of wheel 1R. Its time evolution points out the adopted steering strategies. To determine the errors induced by WO and IMU data, we implemented a dead-reckoning estimator based on a kinematic model. The noise models for WO and IMU measurements are zero mean Gaussian. WO data are perturbed with a variable standard deviation \(\sigma_{\omega}\) that increases with the wheel's angular rate. A maximum standard deviation value of 0.25 cm/s was adopted for a wheel's angular rate of 0.0779 rad/s (i.e., \(\overline{V}=\omega R_{w}=1.11\) cm/s). The IMU yaw rate measurements are affected by \(\sigma_{\psi}=0.013\) deg/s. Both instruments are assumed to collect data with a 60-s sampling rate. Errors associated with the terrain properties and the simplified motion equations lead to significant discrepancies between the simulated and reconstructed trajectory with uncorrected WO+IMU observations. The final estimated position of the rover is \(\sim\)60 m away from the target location, confirming the poor performance of this dead-reckoning system. Our localization system is based on the modeling of the dynamical equations to significantly reduce systematic effects. The wheels angular rate \(\widetilde{\omega}\) from WO and the vehicle's yaw rate \(\widetilde{\dot{\psi}}\) from IMU enable an update of the rover's state \(\widetilde{q}=\left[\widetilde{X},\,\widetilde{Y},\,\widetilde{\dot{\psi}},\, \widetilde{\dot{\psi}},\,\widetilde{u},\,\widetilde{v}\right]^{T}\) through Equations (10) and (11). The initial rover's coordinates and heading angle are assumed to be perfectly known as in the previous case for uncorrected WO+IMU. The initial components of the rover's speed are assumed to be \(\widetilde{u}_{0}=\widetilde{\omega}_{0}R_{w}\) and \(\widetilde{v}_{0}=0\). At each time step, \(\Delta t=60\) s, the measured value \(\widetilde{\omega}\) and \(\widetilde{\dot{\psi}}\) are integrated in the dynamical equations to enable a slippage correction. The estimated \(\widetilde{X},\,\widetilde{Y},\,\widetilde{\dot{\psi}},\,\widetilde{u}\) and \(\widetilde{v}\) are then propagated for 60 s until new measurements of WO and IMU are available. To account for mismodeling in the terrain properties that significantly affect the wheel-soil interaction, we introduced a 10% error in the soil parameters. We assumed to detect drift soil with cohesion, \(c=583\) Pa, friction angle, \(\phi=23.76^{\circ}\), and density, \(\rho=1035\) kg/m\({}^{3}\). A major error contribution is related to our knowledge of the slope angle over the site. We then perturbed the slope in our dynamical model by including a variable slope angle between \(0.1^{\circ}\) and \(-0.1^{\circ}\) and a white Gaussian noise with a standard deviation of \(\sigma_{i}=0.1^{\circ}\). The slippage correction method provides an accurate estimation of the rover's trajectory (Figure 13). This method allows us to compensate WO and IMU errors. Furthermore, the use of the dynamical equations reduces systematic errors leading to better estimates of the vehicle speed and yaw rate.