# Quadrupedal Robots' Gaits Identification via Contact Forces Optimization
## 4 Results  
### Optimal Gait Solution  
For certain optimization variable sets \(\textbf{\emph{p}}_{GA}\), the solver may not converge, or the solution may be unrealistic or impractical. A maximum foot stride \(s_{max}\) has been imposed to avoid solutions that required unrealistic leg dimensions. The condition was applied to the upper bound of the duty factor: \(\beta_{i_{max}}=\frac{s_{max}}{VT}\), where \(V\) is the mean target speed of the simulation:  
\[0<\beta_{i}\leq\beta_{i_{max}} \tag{32}\]  
The optimization finds an efficient gait (in the sense specified in the previous section) that moves the body at an average speed of 1.35 m/s, with a gait period \(T\) of 1 s. The mass properties of the body are selected considering the characteristic parameters of quadrupeds in nature, in particular horses (Table 1, see reference [34]). In addition, some simplification but reasonable assumptions are made, namely: \(\beta_{FR}=\beta_{FL}=\beta_{F}\), \(\beta_{HR}=\beta_{HL}=\beta_{H}\), \(F_{z_{FR}}=F_{z_{FL}}\), \(F_{z_{HR}}=F_{z_{HL}}\), \(F_{x_{FR}}=F_{x_{FL}}\), \(F_{y_{FR}}=-F_{y_{FL}}\), and \(F_{y_{HR}}=-F_{y_{HL}}\). The obtained solutions are plotted in Figure 8, where a correlation between the energy cost and contact-forces jerk is shown (Pareto frontier). The resulting gaits present a specific leg sequence classified as walk and trot gaits. In nature, walking gaits involve overlapping contact phase such that \(\beta>0.5\). In particular, in a trot gait, the diagonal forelimb and hindlimb move in phase, while in a walking gait the limbs have a lag of 0.25 between the two legs. The best solution according to the Pareto frontier is presented in Figure 9. It can be seen how the body keeps a steady longitudinal speed \(\dot{x}\), about the target speed of \(V=1.35\) m/s, maintaining a stable and restrained attitude and assuring the periodicity constraints. Figure 8: Pareto curve of optimal gaits for \(V=1.35\) m/s and \(T~{}=~{}1\) s.  
Figure 9: Trajectory and attitude of the selected optimal solution. This case is classified as a walking trot since it presents a duty cycle \(\beta_{FR}=\beta_{FL}=\beta_{HR}=\beta_{HL}=0.55\), equal for the hind and forelimbs, and \(t_{i}=\ [0.7;0.2;0.2;0.7]\) (Figure 10). The contact forces are instead shown in Figure 11, and the friction coefficients in Figure 12. It can be observed how the normal forces \(F_{z_{i}}\) have a flat double-hump time history, typical of quadrupeds' locomotion [2; 35]. All other significant parameters are shown in Table 1.  
Figure 11: Contact forces normalized over the body weight. Figure 10: Gait sequence of the optimal walking trot solution: \(FL\) (front left), \(FR\) (front right), \(HL\) (hind left), \(HR\) (hind right), CP (contact phase), and SP (swing phase). In Figure 13 we can see the trajectory performed by the foot FR during one complete cycle. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameter** & **Value** & **Unit** \\ \hline \(m\) & 520 & kg \\ \(m_{p}\) & 26 & kg \\ \(I_{x}\) & 103 & kg m\({}^{2}\) \\ \(I_{y}\) & 26.2 & kg m\({}^{2}\) \\ \(I_{z}\) & 103 & kg m\({}^{2}\) \\ \(L\) & 1.44 & m \\ \(h\) & 1 & m \\ \(T\) & 1 & s \\ \(V\) & 1.35 & m/s \\ \(\Delta x_{FR,FL_{mn}}:\Delta x_{FR,FL_{mn}}\) & 1.08; 0.27 & m \\ \(\Delta x_{HR,HL_{mn}}:\Delta x_{HR,HL_{mn}}\) & \(-0.36\); \(-1.17\) & m \\ \(s_{FR}=s_{FL}=s_{HR}=s_{HL}\) & 0.81 & m \\ \(x_{0}\) & 0.36 & m \\ \(y_{0}\) & 0.0027 & m \\ \(z_{0}\) & 1 & m \\ \(\beta_{FR}=\beta_{FL}=\beta_{HR}=\beta_{HL}\) & 0.55 & \\ \(t_{FR}=t_{HL}\) & 0.7 & s \\ \(t_{FL}=t_{HR}\) & 0.2 & s \\ \hline \end{tabular}
\end{table} 
Table 1: Simulation setting parameters and results. Figure 12: The trend of the coefficient of adhesion for each foot. To better assess the variability of the results, a statistical analysis on 10 runs of the optimization algorithm was performed.