# Hybrid Locomotion Evaluation for a Novel Amphibious Spherical Robot
## 3 The Hybrid Locomotion
### Forward Kinematic Model of ASR-III  
To describe the locomotion of the leg-wheeled robot simply, we term the four legs LF, LH, RH, and RF (the left foreleg, left hind leg, right hind leg, and right foreleg, respectively; Figure 4). The Figure 4 shows the coordinate system \(\{O_{B}\}\) of the trunk and assumes that the coordinate lies in the geometrical center of the trunk. The \(X_{B}\) axis represents the forward direction, the \(Z_{B}\) axis the vertical direction (perpendicular to the trunk's horizontal plane), and the \(Y_{B}\) axis the direction to the right of the trunk. Figure 4 shows the coordinates of the legs \(\{O_{0}^{i}\}\) where \(i\) represents a leg number (\(i=1\) for LF, \(i=2\)  
Figure 3: The circuit system of ASR-III. DC: direct current; GPS: Global Positioning System; INS: inertial navigation system. for LH, \(i=3\) for RH, and \(i=4\) for RF). The coordinates of each leg are similar; we describe only the coordinates of the left foreleg in Figure 5. Each leg has a hip joint, a knee joint, and a toe, corresponding to the coordinates {\(O_{1}\)}, {\(O_{2}\)}, and {\(O_{3}\)}, respectively. For on-land locomotion of the leg-wheeled robot, it can be known that two parameters (the stride and frequency) influence the velocity of the robot. In order to get a high velocity, the stride length and frequency need to be increased. Considering the compact structure of ASR-III, the design using servo motors is a nice choice to the mechanical leg. Every leg has two degrees of freedom with two servo motors. The bottom and center of the middle plant is a good site to install the communication module. In order to hold the communication module and lower the barycenter of the robot, the height of the LSWM was decided to 12.1 cm. As shown in Figure 2a, the distance from the upper state to the lower state was 2.5 cm. With the increasing of the under holder, the robot gets a higher barycenter, which decreases the stability of ASR-III. Therefore, to obtain a lower barycenter and make the robot stand up at the lower state of LSWM, the length of the under holder shown in Figure 2b was decided to 9.5 cm. In our robot, the longer the upper holder, the longer the stride length. However, for keeping the spherical boundary, the longest length of the upper holder was 4 cm. The robot was designed using Solidworks 2017. In addition, avoiding interference of parts in assembly, all the parameters of the leg were optimized to get a higher velocity of sliding locomotion and walking locomotion, and a lower barycenter for ASR-III. Table 1 lists all of the Denavit-Hartenberg (D-H) parameters of the left foreleg. \(\theta_{j}^{i}\) and \(d_{j}^{i}\) are the joint angle and the distance between the joint \(j-1\) and \(j\) of leg \(i\); and \(a_{j-1}^{i}\) and \(a_{j-1}^{i}\) are the torsional  
Figure 4: The coordinates of the robot and the legs (top view). Figure 5: Kinematics model of the left foreleg of ASR-III. angle and the length of the bar \(j-1\) of leg \(i\), respectively. The five principal physical parameters of the leg are shown in Table 1. The parameter \(l_{1}\), the horizontal distance from the center of the plant to the hip joint, is 70 mm. Using the D-H homogeneous transformation formula in the driven-axis context, we obtain Equation (3). \[{}^{B}\mathbf{T}_{3}^{1}=\left[ \begin{array}{cccc}s_{1}^{1}c_{1}^{1}&c_{1}^{ 1}&s_{1}^{1}s_{2}^{1}&l_{3}s_{1}^{1}c_{2}^{1}+r_{3}s_{1}^{1}s_{2}^{1}+l_{2}s_{ 1}^{1}+l_{1}\\ -c_{1}^{1}c_{2}^{1}&s_{1}^{1}&-c_{1}^{1}s_{2}^{1}&-l_{3}c_{1}^{1}c_{2}^{1}-r_{3 }c_{1}^{1}s_{2}^{1}-l_{2}c_{1}^{1}-l_{1}\\ -s_{1}^{1}&0&c_{2}^{1}&-l_{3}s_{2}^{1}+r_{3}c_{2}^{1}+r_{1}\\ 0&0&0&1\end{array} \right] \tag{3}\]  
where, \(s_{j}^{i}=\sin\theta_{j}^{i}\) and \(c_{j}^{i}=\cos\theta_{j}^{i}\). The position of the toe of the left foreleg in terms of the trunk coordinates can be obtained as follows Equation (4). \[\mathbf{p}^{1}=\left[ \begin{array}{c}p_{x}^{1}\\ p_{y}^{1}\\ p_{z}^{1}\end{array} \right]=\left[