# Hybrid Locomotion Evaluation for a Novel Amphibious Spherical Robot
## 4 The On-Land Locomotion Experiments
### Two On-Land Locomotion Experiments  
To measure sliding mode velocities, we evaluated the relationships between robot velocity, and the swing frequency and angle of the legs. For sliding locomotion and walking locomotion of ASR-III, two key parameters affecting velocity are the stride length and frequency. To evaluate performance on land, we conducted sliding and walking experiments exploring these parameters. When walking, the velocity is given by Equation (15). \[v=s_{L}\times f \tag{15}\]  
where, \(s_{L}\) is the length of one stride in the \(x\) direction, and \(f\) is the stride frequency. Using the ASR-III kinematic model, the stride in the \(x\) direction is given by Equation (16). \[s_{L}=l\times\sin\theta_{1}^{i} \tag{16}\]  
where \(l\) is the horizontal length of the leg (related to the extent of rotation of the knee joint), and can be obtained by Equation (17). \[l=l_{2}+\Delta l=l_{2}+R\sin(\theta_{2}^{i}+\alpha) \tag{17}\]  
Figure 11: Extent of rotation of the knee joint by LSWM descent distance. The blue state (\(\Delta h=11\) mm) indicates that the toe only just touches the ground, and the red state (\(\Delta h=36\) mm) indicates the vertical stance. The green arrow indicates that the knee joint rotates from the blue to the red state. Figure 12: The sliding velocities at different descent distances. where, \(R\) is the distance from the knee joint to the toe, and \(\alpha\) is the leg angle (\(\alpha=18.43^{\circ}\) in Figure 11). The stride frequency depends on the rotation rate of the servo (\(w_{s}\)); this was 0.18 s/60\({}^{\circ}\) at 7.4 V. A short time \(\Delta t\) elapses between the support and the transfer phase of the leg, maintaining stability. Therefore, the stride frequency is as Equation (18). \[f=\frac{1}{2\theta_{1}^{i}w_{s}+\Delta t} \tag{18}\]  
Finally, the velocity is given by Equation (19). \[v=\frac{[l_{2}+R\sin(\theta_{2}^{i}+\alpha)]\sin\theta_{1}^{i}}{2\theta_{1}^{ i}w_{s}+\Delta t} \tag{19}\]  
From Equation (19), as the angle \(\theta_{2}^{i}\) increases, the velocity increases when \(\theta_{2}^{i}+\alpha<90^{\circ}\). When \(\theta_{2}^{i}\) is \(37.97^{\circ}\), the leg length is maximal (up to 52.68 mm). Then, the stride length depends only on the angle \(\theta_{1}^{i}\). To maximize velocity, we sought to define the optimal angle \(\theta_{1}^{i}\) of the horizontal servo. As the rotatory range of the servo is 0-90\({}^{\circ}\), we evaluated nine angles (10, 20, 30, 40, 50, 60, 70, 80, and 90\({}^{\circ}\)). In Figure 13, the green curve represents the simulated results; the maximum velocity was about 12.0 cm/s when the rotary angle was less than 50\({}^{\circ}\). The deep-sky-blue asterisks indicate the experimental results; the red curve fits these results. The maximum simulated and experimental velocities differed, perhaps because of slipping between the toe and the ground. However, the trends of the two curves were similar. Servo wear was more severe at higher frequencies. We set the optimal angle \(\theta_{1}^{i}\) to 55\({}^{\circ}\). The other parameter affecting velocity is the stride frequency. Ideally, the higher the frequency, the greater the velocity. However, stride frequency cannot be infinite, and slipping will develop at higher frequencies. To evaluate the sliding gaits, we measured velocities when the horizontal servo (set to 55\({}^{\circ}\)) operated at various frequencies. On smooth flat terrain, we tested three gaits (i.e., a synchronous gait using two legs, a trot, and a tripod walk). Given the limited performance of the servo, the frequencies of the synchronous sliding gait with two legs (SSGTL) and the trotting walking gait (TTWG) ranged from 0-2.5 Hz, and that of the tripod walking gait (TPWG) from 0-1.7 Hz. Figure 14 shows the velocities at different frequencies on smooth flat terrain. With the SSGTL and TPWG, as the gait frequency increased, the velocity initially increased linearly at low frequencies and then declined at high frequencies. For the TTWG, at relatively low frequencies (under 0.5 Hz), the robot could not maintain balance.