# Kinematic Tripod (K3P): A New Kinematic Algorithm for Gait Pattern Generation
## 5 Test Results
\begin{table}
\begin{tabular}{c c c c} \hline \hline  & Hexapod & & \\ \hline \(\theta_{l}\) & \([-2\pi/\psi,2\pi/\psi]\) & \(\theta_{k}\) & \([-\pi/\psi_{4},\pi/4]\) \\ \(l_{max}\) & 0.165 m & \(\varphi_{L}\) & 15\({}^{\circ}\) \\ \hline \(l_{c}\) & 0.06 m & \(l_{f}\) & 0.16 m \\ \(l_{t}\) & 0.16 m & \(v_{B}\) & 0.15 m/s \\ \hline \multicolumn{4}{c}{\({}^{B}\mathbf{X}_{W,0}=[0,0,0,0.2,0,0,0,0,0,0,0,0]^{T}\)} \\ \hline \multicolumn{4}{c}{Controller gains} \\ \hline \multicolumn{4}{c}{\(\mathbf{K}_{p}=diag(2,2,2.5,0,0,0.9,0,0,0,0,0)\)} \\ \multicolumn{4}{c}{\(\mathbf{K}_{d}=diag(0.05,0.05,0.1,0,0,0.05,0,0,0,0,0,0)\)} \\ \hline \multicolumn{4}{c}{Lemniscate parameters} \\ \hline \(a\) & 1.75 & \(b\) & 1.15 \\ \(c\) & 0.0 & \(\epsilon\) & 30 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Simulation parameters. Figure 8a represents the first 40 s of simulation when the hexapod traverses the leminiscate trajectory; the shaded areas show when the even tripod is in the swing phase, while the white areas show where the odd tripod is performing the swing phase of the gait cycle. At every change in shading, the graph shows the criterion triggering the phase shift at the specific moment in time it occurred: number 1 for K3P\({}_{1}\), number 2 for K3P\({}_{2}\), and 3 for K3P\({}_{3}\). At the beginning of the simulation, when the hexapod robot aligns with the leminiscate, K3P\({}_{2}\) triggers the phase shift because the legs were getting too close to each other during the turning maneuver; this corresponds to the transient response in the \(\psi\) angle as displayed in Figure 6c. Then, K3P\({}_{1}\) triggers the phase shift because the traveled distance of the swinging tripod is longer than \(l_{max}\). The dotted lines correspond to the thresholds \(\frac{1}{2}l_{max}\) and \(\varphi_{m}\), for the maximum gait distance and minimum angle between two  
Figure 6: K3P driving the hexapod robot over uneven terrain, describing the leminiscate trajectory. (**a**) Walk gait around the leminiscate over uneven terrain (lighter colors indicate higher elevations of the terrain). (**b**) Desired (\(x_{d}(t)\) and \(y_{d}(t)\)) and actual (\(x(t)\) and \(y(t)\)) trajectories. (**c**) The transient \(\psi\) response. (**d**) Walking height \(z(t)\) vs. terrain elevation \(z_{d}(t)\). Figure 7: The stability of the walking gait. Consecutive orange dots over the trajectory represent the location of \(B\) where \(\rho<0.8\) m.  
consecutive legs, respectively. Moreover, to display that K3P is capable of commanding the robot to move at an arbitrary velocity through an arbitrary trajectory, the graph in Figure 8a displays a change in velocity, commanded right after the fourth phase shift at \(t\approx 22\) s. The velocity is doubled from \(v=0.02\) m/s to \(v=0.04\) m/s; the change in velocity can be observed from the duration of the swing phase, where they become narrower after \(t\approx 22\) s because it takes less time for the robot to cover the maximum traveled distance \(l_{max}\). Note, however, that the change in speed can be commanded at any given time, changing immediately the duty factor \(\beta\). After running the simulation, we estimated the torques exerted on every joint of the robot. We modeled the robot as a set of discrete masses, listed in Table 5, whereby the overall mass of the robot is approximately 1.6 kg. Figure 8b shows the resulting torques on the knee and lift joints for the odd subset of limbs during the same period of time and phase shifts as previously discussed for Figure 8a. Torques for the even subset of limbs are in the same order of magnitude, since the robot is symmetrical. Because the axis of rotation of the knee and lift joints are coaxial with the two axes \([K]_{z}\) and \([L]_{z}\), in Figure 8b, we only show \([\mathbf{\tau}_{L,i}]_{z}\) and \([\mathbf{\tau}_{K,i}]_{z}\). We omitted the torque exerted on the swing joint, because \([\mathbf{\tau}_{S,i}]_{z}\approx 0\).