## 2 Materials and Methods
### The Crawler Unit: Overview
Each module has an average weight of 1.7 Kg. kinematics. For convenience, the local frame \(\mathcal{B}\) is fixed on the central module center of mass (CoM). The Crawler Unit is modeled assuming its modules as belonging to two serial kinematics originating from frame \(\mathcal{B}\). Applying the Denavit-Hartenberg convention (D-H), the \((i-1)\)-th frame is attached to the \(i\)-th joint, with the local \(z\) axis parallel to the joint rotation axis. Finally, frames \(\mathcal{E}_{f}\) and \(\mathcal{E}_{r}\) identify the two Crawler Unit end effectors. Table 3 lists the D-H parameters used for modeling these two serial kinematics. In particular, the two transformations from frame \(\mathcal{B}\) to the frames \(0_{f}\) and \(0_{r}\) are static transformations. The central module length is represented by \(l_{c}\). The variable \(l_{l}\) corresponds to the length of the link among two consecutive pitch joints, e.g., the distance between frame \(0_{f}\) and \(1_{f}\). The variable \(l_{\rho}\) denotes the radii sum of successive pitch and yaw joints, e.g., the distance between frame \(1_{f}\) and \(2_{f}\). The parameters \(l_{f}\), \(l_{r}\) and \(l_{d}\) represent the length of the central, front, rear and docking modules, respectively. Each variable \(q_{i}\) denotes the corresponding \(i\)-th joint angle. The vector \(\mathbf{q}_{j}=\left[q_{1},q_{2},q_{3},q_{4},q_{5},q_{6},q_{7},q_{8},q_{9} \right]^{T}\) collects the robot-generalized variables. Figure 5 illustrates the kinematic chain that connects the central module to the rear one. Each joint is driven by a servomotor, which is equipped with a dual stage transmission in order to increase the output torque. Here, timing belts and pulleys were used to implement all the transmissions; in this way it was possible to design compact light-weight reducers. In particular, the transmissions of both pitch joints have a reduction ratio \(\eta_{p}=5.6\):\(1\), while the transmission of the yaw joint has a reduction ratio \(\eta_{y}=3.8\):\(1\). Referring to Figure 5, the yaw joint is completely enclosed in the module chassis. This joint assembles with the middle pitch joint with a C flange. Due to the short distance, \(l_{\rho}=0.05\) [m], among the axis of these two joints, such connection can be intended as a Cardan joint. The middle pitch  
Figure 4: Schematic representation of the Crawler Unit kinematics. The inertial frame is denoted by \(\mathcal{W}\), and the local frame \(\mathcal{B}\) is attached to the central module center of mass (CoM). The \((i-1)\)-th local frame is attached to all the \(i\)-th joint, and frames \(\mathcal{E}_{f}\) and \(\mathcal{E}_{r}\) identify the Crawler Unit extremes. The \(x\), \(y\) and \(z\) axis of each frame are denoted by the red, green and blue colors, respectively. Orange arrows represent the joint degree of freedom (DoF). \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Reference Frame** & \(\mathbf{a}_{i}\) & \(\mathbf{\alpha}_{i}\) & \(\mathbf{d}_{i}\) & \(\mathbf{\theta}_{i}\) \\ \hline \(\mathcal{B}\to 0_{f}\) & \(l_{c}/2\) & \(\pi/2\) & \(0\) & \(0\) \\ \(0_{f}\to 1_{f}\) & \(l_{l}\) & \(0\) & \(0\) & \(q_{1}\) \\ \(1_{f}\to 2_{f}\) & \(l_{\rho}\) & \(-\pi/2\) & \(0\) & \(q_{2}\) \\ \(2_{f}\to 3_{f}\) & \(l_{f}\) & \(0\) & \(0\) & \(q_{3}\) \\ \(3_{f}\to 4_{f}\) & \(l_{\rho}\) & \(\pi/2\) & \(0\) & \(q_{4}\) \\ \(4_{f}\to 5_{f}\) & \(l_{l}\) & \(0\) & \(0\) & \(q_{5}\) \\ \(5_{f}\to\mathcal{E}_{f}\) & \(l_{d}\) & \(-\pi/2\) & \(0\) & \(q_{6}\) \\ \hline \(\mathcal{B}\to 0_{r}\) & \(l_{c}/2\) & \(-\pi/2\) & \(0\) & \(\pi\) \\ \(0_{r}\to 1_{r}\) & \(l_{l}\) & \(0\) & \(0\) & \(q_{7}\) \\ \(1_{r}\to 2_{r}\) & \(l_{\rho}\) & \(\pi/2\) & \(0\) & \(q_{8}\) \\ \(2_{r}\to\mathcal{E}_{r}\) & \(l_{r}\) & \(0\) & \(0\) & \(q_{9}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: The Denavit–Hartenberg parameters of the Crawler Unit kinematics. joint is partially encapsulated into the linking element to the consecutive pitch joint.