# Dynamic Modeling and Simulation of a Robotic Lander Based on Variable Radius Drums
## 2 Dynamic Modeling of the Robotic Lander
### Dynamic Model  
The dynamic model of the system can be obtained by considering only a third of the lander thanks to its axial symmetry. A bidimensional model is considered and the displacement of the lander is constrained to be only along the vertical axis. As a starting point for the definition of the dynamic model, the kinetic energy of the system is computed as follows,  
\[\mathcal{K}=\frac{1}{6}m\dot{x}^{2}+\frac{1}{2}I_{GA}\dot{\vartheta}^{2}+\frac {1}{2}m_{a}\left(\dot{x}^{2}+\dot{x}\vartheta l\cos\vartheta+\frac{l^{2}}{4} \dot{\vartheta}^{2}\right) \tag{7}\]  
where \(m\) is the mass of the lander, \(x\) is the vertical displacement of its center of mass, \(I_{GA}\) is the moment of inertia of the leg computed in its center of mass, \(m_{a}\) is the mass of the leg, and \(l\) is the leg length.  
The potential energy of the system \(\mathcal{U}\) accounts for the gravitational terms of the leg and the structure and the nonlinear response of the VRD, modeled as a nonlinear torsional spring. It results as  
\[\mathcal{U}=-m_{a}\boldsymbol{g}\cdot(\boldsymbol{G}_{A}-\boldsymbol{K})-mg \cdot(\boldsymbol{G}-\boldsymbol{K})+\frac{2}{\pi}M_{0}\left(\theta\arctan{(f \theta)}-\frac{1}{2f}\log{\left(1+(f\theta)^{2}\right)}\right)+\frac{1}{2}k \theta^{2} \tag{8}\]  
where \(\boldsymbol{g}\) is the gravitational force field. The considered model is a single-degree-of-freedom model. The generalized coordinate \(\theta\) is chosen to describe the entire mechanical system. Then, by defining the Lagrangian function as \(\mathcal{L}=\mathcal{K}-\mathcal{U}\), the Lagrangian equation can be written. This results in a second-order ordinary differential equation that represents the equation of motion of the system, as follows,  
\[\frac{\partial}{\partial t}\left(\frac{\partial\mathcal{L}}{\partial\dot{ \theta}}\right)-\frac{\partial\mathcal{L}}{\partial\theta}=\sum_{i=1}^{n}Q_{ c,i}+\sum_{j=1}^{l}Q_{n.c,j} \tag{9}\]  
where \(Q_{c,i}\) and \(Q_{n.c,j}\) are the Lagrangian components of the \(i\)-th conservative and \(j\)-th non-conservative forces, respectively, which are not directly included in the Lagrangian function. In this specific case, those forces contains the components of the contact forces that take place between the lander pad and the ground. The contact force is composed by the normal and frictional components whose modeling is explained in detail below. Note that the generalized force \(Q_{c,q}\) for a given force \(\boldsymbol{F}_{c}\), with point of application \(\boldsymbol{x}_{c}\), and a generalized coordinate \(q\) is expressed as follows,  
\[Q_{c,q}=\boldsymbol{F}_{c}\cdot\frac{\partial\boldsymbol{x}_{c}}{\partial q} \tag{10}\]  
Equation (9) is obtained using Equations (7), (8), and (10), and represents the dynamic model of the lander in the phase of impact with ground. The dynamic model of the lander for the phase between the release of the lander and the touchdown is the one for a free falling object. This model is used to compute the state variables of the lander at the end of the free falling stage, i.e., the beginning of the impact problem. These computed states becomes the boundary initial conditions for the ordinary differential equations resulting from Equation (9). The main effect of the VRD is to generate a nonlinear torque applied to the leg joint in order to keep constant the force acting on the pads of the lander. The nonlinear torque obtained from the VRD synthesis is shown in Figure 3.