# Dynamic Modeling and Simulation of a Robotic Lander Based on Variable Radius Drums
## 2 Dynamic Modeling of the Robotic Lander
### Soil Characterization  
In order to completely define the dynamic model of the lander, the properties of the soil need to be analyzed. Figure 3: Nonlinear, softening, spring response obtained from VRD synthesis. In this work, we consider a single type of soil whose characteristics are determined experimentally; while mission-specific conditions of the ground can differ greatly, in this preliminary phase of experimentation we have elected to consider only one type. In particular, the normal contact force between the pads of the lander and the soil is modeled as a non-linear spring damper system as depicted in Figure 2, and is described by the following [13],  
\[ \begin{cases}F_{k}=-k_{eq}x^{\delta}-c_{eq}\dot{x},&x\leq 0\\ F_{k}=0,&x>0\end{cases}  \tag{11}\]  
where \(k_{eq}\) is the equivalent stiffness of the soil, \(c_{eq}\) is the equivalent damping factor of the soil, \(\delta\) the force exponent responsible of the nonlinear behavior of the spring, and \(x\) and \(\dot{x}\) are the depth and the speed of penetration of the pad into the soil, respectively. In contacts problems a further force that plays an important role is the frictional force acting between the pad and the soil. In this context, the Coulomb friction model is used, whose expression is reported in  
\[ \begin{cases}F_{f}=-\mu_{s}N,&\dot{y}=0\\ F_{f}=-\mu_{k}N,&\dot{y}\
eq 0\end{cases}  \tag{12}\]  
where \(\dot{y}\) is the speed acting along a plane parallel to both the surfaces in contact, \(N\) is the normal force acting between the parts in contact and having direction coincident with the normal of the previously defined plane, \(\mu_{s}\) is the static friction coefficient, and \(\mu_{k}\) is the dynamic friction coefficient. In order to conduct accurate and comparable simulations, the parameters \(k_{eq}\), \(c_{eq}\), \(\mu_{s}\), and \(\mu_{k}\) need to be estimated. For what concern the parameters that describe the nonlinear spring damper system, some preliminary tests have been performed to estimate the modulus of subgrade reaction \(k\). This can be considered directly related to the soil equivalent stiffness and is defined as follows,  
\[k=\frac{q}{\Delta} \tag{13}\]  
where \(q\) is the applied pressure and \(\Delta\) is the penetration depth. More precisely, the test is conducted as follows; a circular plate with dimensions comparable with the lander pad, is positioned on the flat sample soil surface. Subsequently, a 1 kg steel weight is positioned above the circular plate and the deflection at static conditions is measured. The measures, as the deflections are truly little and submillimeter, are taken by mean of image analysis. Comparison between the pictures taken before applying the load and the one taken after the load is applied is done, as can be seen in Figure 4b. The aforementioned test is repeated 13 times, and the measured penetration depth is found to be \(\Delta=(0.141\pm 0.049)\) mm. Therefore, choosing the penetration depth to be the mean value and by using Equation (13), it is possible to estimate the modulus of subgrade reaction \(k\) and then by geometric considerations estimate the soil equivalent stiffness \(k_{eq}\)[22]. The equivalent stiffness \(k_{eq}\) is then estimated to be \(k_{eq}\approx 70\) N mm\({}^{-1}\). Finally, the equivalent damping factor \(c_{eq}\) instead is estimated as \(0.1\div 1\%\) of the equivalent stiffness factor [22]. Thus, in this work we have assumed \(c_{eq}=0.7\) N s mm\({}^{-1}\). For what concerns the second kind of parameters which describe the friction model, in a first analysis the static friction coefficient can be estimated as \(\mu_{s}\approx\tan\phi\), where \(\phi\) is the internal friction angle of the soil. The angle has been measured experimentally by conducting 15 tests. More precisely, the test is conducted as follows; a small quantity of the sample soil is slowly poured on a flat surface, and a picture of the heap vertical profile is taken.