# The Design and Building of a Hexapod Robot with Biomimetic Legs
## 4 Locomotion  
The locomotion of this hexapod can be analyzed in terms a simplified model (Figure 7), a biped robot with absolute stability, by regarding the left and right legs of the biped version as the legs of the upper and lower deck of the hexapod, respectively. Following this, the locomotion of the robot can be characterized by the motion of its COG (Center of Gravity). The cranks of the left and right legs will keep apart by 180\({}^{\circ}\) during rotation, and Figure 8 illustrates how the motion of the COG relates to the foot trajectory where (x,y), (x, y, y), \(L_{s}\), \(H_{s}\), and \(M_{g}\) stand for the foot trajectory, the motion of the COG, the step size, the step height, and the fluctuated amplitude, respectively. The robot is initially at rest with one foot standing at point A, while another foot stands at point B. The coordinates of point A are denoted as (x, y), which is associated with the crank angle \(\theta_{s}\) \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Case** & \(L_{1}\) **(mm)** & \(L_{2}\) **(mm)** & \(L_{3}\) **(mm)** & **Max. Step Size (mm)** & **Max. Step Height (mm)** \\ \hline
1 & 45 & 160 & 210 & 223.6 & 26.2 \\
2 & 45 & 172 & 200 & 223.8 & 18.1 \\
3 & 30 & 172 & 200 & 148.8 & 15.4 \\
4 & 30 & 150 & 164 & 139.8 & 37.8 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Four sets of design data for \(L_{1}\), \(L_{2}\), and \(L_{3}\). Figure 8: (a) Foot trajectory. (b) Motion of COG. Figure 6: Actions: (a) inclination decreased, (b) step size reduced, and (c) step height increased. Figure 7: A simplified model of the robot: (a) Initial configuration and (b) motion of the center of gravity Projection (COG). while the crank angle of point B is 180\({}^{\circ}\) apart from \(\theta_{s}\), i.e., \(\theta_{s}+\pi\). Hence, point A can be obtained by determining the crank angle \(\theta_{s}\), where the height of point A is equal to the height of point B, using Equations (1)-(16). Following, the step size \(L_{s}\), step height \(H_{s}\), and fluctuated amplitude \(M_{g}\), are calculated and listed in Table 2.  
As the foot engages on point A, it pushes the robot forward, as well as upward. Hence, by mirroring the part of foot trajectory ABC, with respect to the horizontal and vertical axes, the locus of the COG is derived, yielding the following equations:  
\[x_{g}(\theta)=nL_{s}+x_{s}-x(\theta-n\pi) \tag{17}\]  
\[y_{g}(\theta)=y_{s}-y(\theta-n\pi) \tag{18}\]  
\[n=\left[\frac{\theta-\theta_{s}}{\pi}\right] \tag{19}\]  
where, the bracket \([\quad]\) yields the maximum integer, no greater than the value it contains. The locomotion of the robot, i.e., rhythmically alternating one foot with another, can be well characterized by the locus of its COG, using Equations (17)-(19). The loci of the COG pertaining to four different cases are as shown in Figures 9-12. In Figure 9, the locus manifests that the robot will wobble significantly, while moving. This adverse effect arises from the fact that the foot trajectory, generated by the TJL in Case 1, exhibits large obliqueness. In an effort to improve this situation, the inclination of the orbit was reduced, causing a side effect \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Case** & \(\theta_{s}\) (\({}^{\circ}\)) & \(L_{s}\) (mm) & \(H_{s}\) (mm) & \(M_{g}\) (mm) & \(H_{s}\)/\(M_{g}\) \\ \hline
1 & 317.4 & 131.3 & 8.0 & 18.2 & 0.44 \\
2 & 234.3 & 201.4 & 3.8 & 14.3 & 0.27 \\
3 & 241.8 & 141.8 & 4.6 & 10.8 & 0.43 \\
4 & 246.9 & 135.4 & 12.3 & 25.5 & 0.48 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Characteristics of foot trajectories related to four sets of design data. Figure 11: Locus of center of gravity for Case 3. Figure 12: Locus of center of gravity for Case 4. Figure 9: Locus of center of gravity for Case 1.  
on the actual step size, i.e., an increase by 1.5 times (Figures 9 and 10). The maximum step sizes in Cases 1 and 2 (Table 1) are almost the same and the respective orbits (Figure 6a) are similar.