# Dynamic Modeling and Simulation of a Robotic Lander Based on Variable Radius Drums
## 2 Dynamic Modeling of the Robotic Lander
### Variable Radius Drum Mechanism  
A schematic view of the lander-leg mechanism at the beginning of the impact is illustrated in Figure 2. The main parts composing each lander-leg mechanism are a leg, a fixed radius pulley with radius \(r_{1}\), attached to the leg, a VRD, another fixed radius pulley with radius \(r_{2}\), attached to the VRD, and a preloaded spring. Moreover, two cables are used in the mechanism. The extremities of the first cable are attached to the VRD and to the fixed radius pulley (red line in Figure 2). The extremities of the other cable are attached to the preloaded spring, which is connected to the lander chassis, and to the fixed radius pulley connected to the VRD (green line in Figure 2). This specific layout allows the VRD to shape the elastic response of the preloaded spring. More precisely, the preloaded spring generates a reactive torque \(M\) acting on the fixed-radius pulley attached to the leg. As the response of a preloaded spring is discontinuous, the reactive torque \(M\) can be approximated by the following continuous and differentiable function [25; 40; 41],  
\[M=\frac{2}{\pi}M_{0}\arctan\left(f\vartheta\right)+k\vartheta \tag{1}\]  
where \(M_{0}\) is the preloaded torque, \(f\) an approximation factor, \(k\) the spring stiffness, and \(\vartheta\) the relative rotation of the leg from its rest position \(\vartheta_{r}\).  
Figure 1: The computer-aided design (CAD) model of the proposed three-legged lander, along with labels for its main parts and main dimensions: (**a**) lateral view and (**b**) top view.  
Figure 2: Schematic view of the lander-leg mechanism at the beginning of the impact with soil with detailed view of the model of the normal contact force as a spring–damper system.  
In the following, the profile of the VRD is synthesized as explained in [26; 28] in order to guarantee a constant force acting on the pads of the lander during the deceleration phase, that is, \(F_{\mathbf{k}}(\vartheta)=\text{cost}\). A generic point of the VRD profile \(\mathbf{P_{m}}\), expressed in its local coordinates, is defined by  
\[\mathbf{P_{m}}=\mathbf{T}(\alpha)\left\{\begin{matrix}c_{d}\\ 0\end{matrix}\right\}+\mathbf{T}(\alpha)\mathbf{T}(-\gamma)\mathbf{T}\left(-\frac{\pi}{2 }\right)\left\{\begin{matrix}l_{t}\\ 0\end{matrix}\right\} \tag{2}\]  
where \(\alpha\), with reference to Figure 2, is the relative rotation of the VRD; \(l_{t}\) is the distance between the point \(\mathbf{P_{m}}\) and the idle pulley center \(O\), expressed by  
\[l_{t}=\frac{c_{d}\sin\gamma}{1+\frac{\frac{d^{2}\pi}{d\alpha}}{\sqrt{c_{d}^{2 }-\left(\frac{d\pi}{d\alpha}\right)^{2}}}} \tag{3}\]  
where \(\mathbf{T}\in\mathbb{R}^{2\times 2}\) is a generic rotation matrix, and \(\gamma\) is defined as  
\[\gamma=\arccos\left(\frac{1}{c_{d}}\frac{dg}{d\alpha}\right) \tag{4}\]  
With reference to Figure 2, \(c_{d}\) is defined as the distance between \(O\) and \(O_{1}\), whereas \(g(\alpha)\) is the cable wound function. In order to synthesize the VRD for this application, the expression of the cable wound function \(g(\alpha)\) and its first- and second-order derivatives with respect to \(\alpha\) need to be derived. From Figure 2, the wound cable function and its derivatives assume the following expressions.  
\[g(\alpha)=r_{1}\vartheta,\qquad\frac{dg(\alpha)}{d\alpha}=r_{1}\frac{d \vartheta}{d\alpha},\qquad\frac{d^{2}g(\alpha)}{d\alpha^{2}}=r_{1}\frac{d^{2 }\vartheta}{d\alpha^{2}} \tag{5}\]  
By looking at Figure 2, and by applying the principle of virtual works, it is easy to demonstrate that the derivative of \(\vartheta\) with respect to \(\alpha\) is  
\[\frac{d\vartheta}{d\alpha}=\frac{||\mathbf{F_{s}}||r_{2}}{||(\mathbf{K}-\mathbf{O})\times \mathbf{F_{k}}||} \tag{6}\]  
where \(\mathbf{F_{s}}\) is the force exerted by the preloaded spring. Starting from Equation (6) and its second-order derivative, it is possible to synthesize the VRD, given the pad-soil contact force \(\mathbf{F_{k}}\).