# Model-Based Slippage Estimation to Enhance Planetary Rover Localization with Wheel Odometry
## 2 Methods  
### Dynamical Model and Slippage Estimation  
The motion of a wheeled vehicle on unprepared ground, such as planetary surfaces, requires an accurate modeling of the compliant wheel on compliant ground. The resistance to motion encountered by the wheels is due to the energy dissipation in the wheels and the ground. Figure 1a shows a scheme of a compliant wheel that moves on a compliant ground. According to the model introduced in [19; 20; 28], we assumed that the deformation in the tire, \(f\), occurs only if the maximum pressure exerted on the ground, \(\sigma_{z}\), exceeds the pressure, \(p_{gr}\), needed to deform the tire. As represented in Figure 1a, this value is exceeded at point A of the contact zone. We assumed that, in the first part of the contact, i.e., in the interval \((\theta_{1},\ \theta_{0}]\) reported in Figure 1a, the wheel behaves as a rigid body and the pressure is linked to the sinkage. For \(\theta\in[-\theta_{1},\ \theta_{1}]\), we assumed a constant pressure equal to \(p_{gr}\), which stands for an equivalent stiffness of flexible wheels. The normal stress, \(\sigma_{z}\), is defined as follows:  
\[\sigma_{z}=\left\{ \begin{array}{c}\left(\frac{k_{c}}{b}+k_{\phi}\right)h( \theta)^{n}\ for\ \theta\in(\theta_{1},\ \theta_{0}]\\ p_{gr}\ for\ \theta\in[-\theta_{1},\ \theta_{1}]\end{array} \right. \tag{1}\]  
where \(k_{c}\) and \(k_{\phi}\) are the cohesive and the frictional moduli that define the stiffness of the soil, \(h(\theta)\) is the sinkage, \(b\) is the width of the wheel and \(n\) is the exponent of terrain deformation that is \(\leq\)\(\frac{1}{2}\) and \(\approx\)\(1\) for cohesive and frictional soils, respectively. The wheel-soil interaction results in a vertical force, \(F_{z}\), a longitudinal force, \(F_{x}\), a lateral force, \(F_{y}\), and an aligning moment, \(M_{z}\), acting on each vehicle's wheel (Figure 1b). The vertical force \(F_{z}\) is:  
\[F_{z}=R_{w}b\int_{\theta_{1}}^{\theta_{0}}[\sigma_{z}(\theta)\cos(\theta)+ \tau_{x}(\theta)\sin(\theta)]\;d\theta+2 bx_{A}p_{gr}=\frac{mg}{n_{w}}\cos(i) \tag{2}\]  
where \(R_{w}\) is the radius of the wheel; \(\tau_{x}\) is the longitudinal shear stress and \(x_{A}\) is the x-coordinate of point A; \(m\) is the mass of the vehicle, \(n_{w}\) is the number of the wheels, \(g\) is the gravitational acceleration and \(i\) is the inclination of the plane. The first contribution to \(F_{z}\) in Equation (2) represents the vertical force due to the normal load, \(\sigma_{z}\), directed radially along arc AB, which depends on the sinkage, \(h(\theta)\), and to the vertical projection of the longitudinal shear stress \(\tau_{x}\), acting in the contact area delimited by \((\theta_{1},\;\theta_{0}]\). The second contribution to \(F_{z}\) is due to the constant pressure, \(p_{gr}\), which acts in the \(z\) direction of the wheel reference frame _[W]_ along line A'A. Solving the nonlinear Equation (2), one obtains the values of \(\theta_{0}\) and \(\theta_{1}\). The longitudinal force (\(F_{x}\)) is:  
\[F_{x}=R_{w}b\int_{\theta_{1}}^{\theta_{0}}[-\sigma_{z}(\theta)\sin(\theta)+ \tau_{x}(\theta)\cos(\theta)]d\theta+R_{w}b\int_{-\theta_{1}}^{\theta_{1}}\tau _{x}(\theta)d\theta \tag{3}\]  
where the first integral term of Equation (3) is the contribution of the radial pressure \(\sigma_{z}\) and the longitudinal shear stress \(\tau_{x}\) acting along arc AB and projected along the \(x\) direction of the wheel's reference frame _[W]_; the second integral term is due to the action of \(\tau_{x}\) along line A'A. The expression of \(\tau_{x}\) at a generic point of the contact zone is assumed to be:  
\[\tau_{x}(\theta)=sgn(\sigma)[c+\sigma_{z}\tan(\phi)]\left[1-e^{-\frac{R_{w}[c ]}{K_{x}}(\theta_{0}-\theta)}\right] \tag{4}\]  
and \(c\), \(K_{x}\), \(\phi\) are the cohesion, the modulus of the shear deformation in the \(x\) direction and the internal friction angle, respectively.