# Dynamic Modeling and Simulation of a Robotic Lander Based on Variable Radius Drums
## 3 Simulated Test Bed  
In this section, the modeling of the multi-body system (MBS) representing the lander and the setup for the numerical simulations are presented. Both the entire lander model and the soil are modeled, respectively, as a MBS and a rigid body, and the result of this modeling process can be seen in Figure 5.  
The MBS representing the lander model is constructed by connecting, with appropriate joints, each single rigid body which is functional to the correct behavior of the global system. The functional bodies, which globally compose the MBS of the lander, are the lander hexagon structure, its three legs, the three ratchet mechanisms and finally the three covers, respectively, indicated in yellow, gray, red, and green in Figures 5 and 6. Their geometrical properties are summarized in Table 2, whereas the joints between the parts are summarized in Table 3. It is important to report that the non-functional parts of the lander together with the payload are included in the model by adding their mass properties to the lander structure body. Moreover, the joints used are considered as ideal joints, i.e., without considering friction and deformations. Finally, for what concerns the soil rigid body, it is modeled as a box having dimensions \((10\times 0.05\times 1)\) m\({}^{3}\) and positioned below the lander structure center mass. Figure 5: View of the lander MBS and the soil modeled in ADAMS: (**a**) the initial condition of the simulation; (**b,c**) the lander configuration at the beginning and at the end of an impact simulation on a flat surface, respectively. Once all the bodies and their joints are defined, forces acting on the model should be added. These forces can be split up between external forces and contact forces. The external forces implemented in the simulation environment are a gravitational force field, a linear torsional spring between each ratchet and the structure, and a nonlinear torque applied between each leg and the lander structure. The linear torsional spring between each ratchet and the structure is introduced in order to simulate the elastic deflection of the ratchet from its rest position. In order to provide values that describe the torsional spring, the ratchet is considered as a cantilever, which deflects under the action of the contact force between the ratchet and the leg. Therefore, by using the elastic theory, it is possible to define an equivalent torsional stiffness coefficient \(k_{r,eq}\) for the spring, whose expression is \(k_{r,eq}=EI/l\). \(E\) is the Young elastic modulus of the PLA, \(l\) is the moment of inertia of the resistant cross-area of the ratchet, and \(l\) is the length of the cantilever. Finally, a stiffness coefficient of \(k=288\,\mathrm{N\,mm\,rad^{-1}}\) is set for the springs. The nonlinear torsional spring between each leg and the structure is introduced in order to simulate the resistant torque applied to the leg joint generated by the action of the VRD pulley, as stated in Section 2. For the implementation in ADAMS, instead of using directly Equation (1), a cubic interpolating polynomial function is used in order to approximate the response of the VRD pulley, with coefficients \(a_{0}=1147\), \(a_{1}=18.6761\), \(a_{2}=-0.2175\), and \(a_{3}=-1.0823\times 10^{-4}\). \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Bodies** & \(m\)**[kg]** & \(\mathbf{I_{xx}}\) & \(\mathbf{I_{yy}}\) & \(\mathbf{I_{zz}}\) \\ \hline Structure & 0.5 & 430.14 & 684.78 & 430.14 \\ Leg & 3.7\(\times 10^{-2}\) & 374.29 & 10.67 & 364.37 \\ Ratchet & \(5\times 10^{-4}\) & \(9.40\times 10^{-2}\) & 0.11 & 3.49 \(\times 10^{-2}\) \\ Cover & \(1\times 10^{-3}\) & 17.00 & 11.49 & 6.45 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Geometrical properties of the main functional rigid bodies composing the lander MBS model. All the inertia components are expressed in kg mm\({}^{2}\).