# Dynamic Modeling and Simulation of a Robotic Lander Based on Variable Radius Drums
## 5 Discussion
### Comparison with the Theoretical Model  
Comparisons of the ADAMS simulations with the theoretical model are conducted for two different cases regarding the analytical-based simulation. For the first case, a length of the lander leg equal to \(l_{c}\) is chosen, which corresponds to the conservative value chosen, i.e., the distance from the lander joint axis from the pad closest contact point. For the second case, a length equal to \(l_{m}\) is chosen, which corresponds to the mean distance of the contact point from the joint axis. This is because despite the analytical model, in which the pad is considered a point, the pad in the ADAMS model is a rigid body having its own semicircular geometry. Therefore, while in the analytical model the contact point is always at a fixed distance from the lander joint axis, the geometry of the pad leads to the fact that the contact point keep varying all along the pad profile during the whole contact phase, thus changing the arm lever. The comparisons are conducted in the scope of the Case (1). Moreover, in order to catch the influence of the ratchets in the global dynamics of the lander, an additional simulation in ADAMS without ratchets is conducted and compared with the others. Figure 14 shows the comparison between the ADAMS simulations and the MATLAB-based analytical simulation for the two leg length value chosen: in the top row the acceleration of the lander is compared, in the middle row the contact force acting between the soil and the lander pad is compared, and in the bottom row the leg-joint rotation is compared. The black curves represent the ADAMS simulations with the ratchet (ADAMS-R), the blue curves represent the ADAMS simulations without ratchets (ADAMS-NR), and finally the red curves represent the simulations based on the analytical model described in Section 2. It is important to remark that the figures refer only to the first bounce of the lander. This is because the lander behavior is different between the simulations after the first bounce. In fact, the analytical model does not implement the ratchet. In ADAMS, on the other hand, this mechanism is present and it affects the behavior of the system after the deceleration phase. By comparing the curves, similarities in the trends and with the values assumed can be found. More precisely, with the theoretical model the deceleration of the lander during impact assumes values \(a\in[3.1,4.5]g\) for a leg length equal to \(l_{c}\) and values \(a\in[2.7,4.0]g\) for a leg length equal to \(l_{m}\), whereas with the simulation in ADAMS an acceleration \(a\in[3,3.5]g\) is observed. The same can be observed with the contact force and the leg joint rotation; in particular a contact force \(F_{c}\in[5.4,10]\)N has been observed using the theoretical model and the same is observed in the in ADAMS simulation. Finally, for what concerns the leg joints rotation, it can be seen that in both the theoretical model simulation as well as in the ADAMS simulations, the maximum rotation of the joints reaches a value slightly below \(30^{\circ}\). The discontinuities visible in the ADAMS simulation black curves visible from the acceleration and force graphs are due to the action of the ratchets which limits the rotation of the lander legs; in fact, in the curves representing the analytical model (which does not implement the action of the ratchet), these same discontinuities are not present. By comparing the MATLAB-based simulations of the left column with the right column of Figure 14, it can be noticed that the leg length parameter used for the theoretical model, i.e., the distance of the contact point of the pad from the joint axis, has a great influence on the curve's trend of the analytical model. In fact, by using as leg length \(l_{m}\), the analytical curves better fits the ADAMS ones, especially for the contact force.