# Dynamic Modeling and Simulation of a Robotic Lander Based on Variable Radius Drums
## 1 Introduction
in 2011, used an explicit nonlinear finite element method implemented in ABAQUS to study the impact of a legged lander and aluminum honeycomb shock absorber with the lunar soil [19]. Other authors in [20; 21; 22; 23; 24] studied the impact of the lander with ADAMS considering the soil either as an elastic or a rigid body. In this paper, we present the dynamic modeling and simulation of a robotic lander, based on variable radius drums (VRDs), which inherits from the concept outlined by [PERSON] in [25]. The proposed mechanism consists of a non-reversible mechanism that allows a landing object to decelerate in the phase of impact with ground. A VRD is a device characterized by the variation of the spool radius along its profile, as defined by [PERSON] and [PERSON] in [26]. In this context, it is used to ensure a constant force and a controlled deceleration of the structure during landing. VRDs were studied for many applications, as, for instance, in gravity-compensation systems [27], to guide a load along an horizontal path, or to improve locomotion of a legged robot [26]. [PERSON] et al. in 2018, studied a cable-based robotic crane based on VRDs capable of moving a load through a planar working area [28]. Moreover, [PERSON] and [PERSON] in 2018 employed two antagonistic VRDs in a cable-suspended robot to steer the end-effector along a desired pick-and-place trajectory. The same concept was also adopted for the static balancing of a pendulum [29]. In these cases, the rationale for the use of the VRD is to generate geometrical trajectories in a mechanism. Conversely, in this work we employ a synthesized VRD to generate a specific force when connected to a linear spring-loaded mechanism. Other examples of the application of VRDs, also known as non-circular pulleys, include cable actuation in soft robot joints [30], devices for upper leg rehabilitation [31], mechanisms for energy saving [32], and force regulation [33]. When landing on celestial bodies, the ground characteristics are an important influence on the mission [34], and as such many attempts are made to determine them in advance [35; 36]. Many works detail the soil characterization, in particular concerning its granulometry [37]. In order to build the simulationson a solid baseline, we used an experimental approach to the determination of the characteristics of the soil; a basaltic sand of volcanic origin with average grain size range of 3 to 5 mm which pertains to the coarse sands as described by NASA [38; 39]. The main contributions of this paper can be summarized as follows; the design and the numerical validation of a three-legged robotic lander for space applications. The landing mechanism leverages a passive mechanism for impact energy absorption, which is based on VRDs and cables to transfer the impact energy to a preloaded spring. Moreover, the presented passive mechanism is able to ensure a controlled force and acceleration of the structure, thus preventing damage to the supported payload due to high accelerations. We believe this kind of system can find application in real and concrete scenarios considering its reusability, its high payload/lander mass ratio, and its modularity. Compared to relevant literature, and in particular the work of [PERSON] [25] which implements an ideal linear spring-damper contact and ideal Coulomb friction models, this work presents several novelties: it provides an experimental determination of the characteristics of soft basaltic sand soil for implementation in a spring-damper ground contact as well as implementing a nonlinear stiffness model; furthermore, it implements experimentally defined static and dynamic friction coefficients. In addition, the numerical implementation of the theoretical model is more stable and thus avoids the implementation of numerical damping of the legs rotation. Moreover, in this work in the 3D lander three ratchets are introduced, functioning as retention mechanisms that make the system non-reversible.