# Model-Based Slippage Estimation to Enhance Planetary Rover Localization with Wheel Odometry
## 2 Methods
### Dynamical Model and Slippage Estimation
The longitudinal slip, \(\sigma\), is a measure of the difference between the commanded speed of the vehicle, \(\Omega R_{w}\), controlled through the wheels' angular speed, and its actual speed, \(V\):  
\[\sigma=\left\{ \begin{array}{l}\frac{\Omega R_{w}-V}{\Omega R_{w}}\;\mbox{ in {driving}}\\ \frac{\Omega R_{w}-V}{V}\;\mbox{in {bra}}\end{array} \right. \tag{5}\]  
Figure 1: (a) Contact between compliant wheel and compliant terrain and (b) interaction forces acting on the rover’s wheel. The full set of equations of motion is:  
\[ \begin{array}{l}\dot{u}=vr-g\sin(i)\sin(\uppsi)+\frac{1}{m}\sum\limits_{i=1}^{n} \sum\limits_{j=1}^{2}\left[F_{x_{ij}}\cos\bigl{(}\delta_{ij}\bigr{)}-F_{y_{ij}} \sin\bigl{(}\delta_{ij}\bigr{)}\right]\\ \dot{v}=-ur-g\sin(i)\cos(\uppsi)+\frac{1}{m}\sum\limits_{i=1}^{n}\sum\limits_{j =1}^{2}\left[F_{x_{ij}}\sin\bigl{(}\delta_{ij}\bigr{)}+F_{y_{ij}}\cos\bigl{(} \delta_{ij}\bigr{)}\right]\\ \dot{r}=\frac{1}{I_{z}}\sum\limits_{i=1}^{n}\sum\limits_{j=1}^{2}\left\{F_{x_ {ij}}\bigl{[}-y_{ij}\cos\bigl{(}\delta_{ij}\bigr{)}+x_{i}\sin\bigl{(}\delta_{ ij}\bigr{)}\bigr{]}+F_{y_{ij}}\bigl{[}y_{ij}\sin\bigl{(}\delta_{ij}\bigr{)}+x_{i} \cos\bigl{(}\delta_{ij}\bigr{)}\bigr{]}+M_{z_{ij}}\right\}\\ \dot{\psi}=r\end{array}  \tag{10}\]  
where \(\delta_{ij}\) is the steering angle of the \(\ddot{\mathrm{\eta}}\)-th wheel; \(u\) and \(v\) are the components of the rover's velocity in the body reference frame, and \(\dot{\psi}\) is the yaw rate that is angular velocity about the \(z\) axis. The first two equations in Equation (10) provide the vehicle accelerations, \(\dot{u}\) and \(\dot{v}\), in the direction of \(x\) and \(y\) body axes, respectively; the contribution of the yaw rate, of the gravitational acceleration and the action of the longitudinal and lateral forces are also included. The third equation of Equation (10) yields the rover's angular acceleration \(\dot{r}\) produced by the action of the interaction forces and the aligning moments. To compute the rover's trajectory two more equations must be introduced:  
\[\left[ \begin{array}{c}\dot{X}\\ \dot{Y}\end{array} \right]=\left[ \begin{array}{cc}\cos(\uppsi)&-\sin(\uppsi) \\ \sin(\uppsi)&\cos(\uppsi)\end{array} \right]\left[ \begin{array}{c}u\\ v\end{array} \right] \tag{11}\]  
The slippage correction to correct the WO measurements is based on the combination of the motion Equations (10) and (11) and the interaction forces Equations (2), (3), (6) and (9). Our method consists in an iterative process that starts from the acquisition of the wheel's angular rate and the vehicle yaw rate from WO and IMU, respectively. By initially assuming a zero slippage and a first guess on the rover's state \((X,\ Y,\ \psi)\), the rover's state is propagated through iterative calculations of the slippage, the interaction forces and all the derivatives values shown in Equations (10) and (11). At each step, the effect of the soil properties, and, thus, of the interaction forces, impacts on the components of the rover's linear and angular accelerations. This allows the estimation of the actual slippage with Equation (5) and improves the estimates of the rover's speed and yaw rate.