## 4 The Foot Trajectory Analysis  
The SCalf team has already done some research on the gait trajectory planning [18; 19; 20], but they only considered the situations under the duty cycle as 50%. In this paper, a more general trot gait trajectory is proposed. In the trot gait, \(S\) is the step length, \(T\) is the gait cycle, \(H\) is the step height, and \(H_{0}\) is the standing height, as illustrated in Figure 6. \(\beta\) is the duty cycle of the trot gait. The velocity of the robot \(v_{r}\) can be obtained by the step length \(S\) and the gait cycle \(T\) by using Equation (38). \[v_{r}=2S/T \tag{38}\] \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Joint** & \(a_{i}\) **(mm)** & \(b_{i}\) **(mm)** & **Angles (\({}^{\circ}\))** \\ \hline Pitching hip joint & 289.26 & 56.87 & \(e_{1}=0.094\) & \(e_{2}=0.96\) \\ Pitching knee joint & 289.26 & 56.89 & \(e_{3}=0.14\) & \(e_{4}=0.041\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: The mechanical parameters used in Figure 5.  
Figure 5: The mechanical structure parameters of the leg. Figure 6: The illustration of the gait parameters. During the movements of the robot, the legs will swing periodically with respect to the trunk. In order to make the quadruped robot walk at a steady speed, this work realizes the trajectory planning of the robot by simulating the trajectory of the quadruped animals' feet. Based on the motions of the horses, a foot trajectory including the cubic spline interpolation curve and the straight line is studied. The phases of the four legs are shown in Figure 7, where the two feet on the diagonal have the same movements. Here, the trajectory of the RF leg is analyzed. During the stance phase, the trunk of the robot is moving forward. According to Equations (16), (22), and (34), in order to minimize the energy consumption, joint torques have to be as small as possible. Then, the first component of \(M\), which is \(F_{x}=Ma\), have to be small. Therefore, the trunk acceleration \(a\) is set as zero, and the motions of the stance phase become uniform motions in the \(x\) direction. The positions in the \(z\) direction are equal to \(-H_{0}\). The trajectory equation of the stance phase in the coordinate frame \(\{O_{b}\}\) can be obtained in Equation (39). \[ \begin{cases}x(t)=\dfrac{S}{2}(1-\dfrac{2t}{\beta T})\,,\quad 0<t\leq\beta T\\ z(t)=-H_{0}\end{cases}  \tag{39}\]  
In order to obtain the parameters of the foot trajectory during the swing phase, the initial and terminated conditions of the stance phase are listed in Table 3, including the positions and velocities of the foot endpoint in the \(x\) and \(z\) directions. In the swing phase, the cubic spline interpolation and the cosine curve are used in the \(x\) and \(z\) directions respectively. In the trajectory expression of the \(x\) direction, the coefficients of the cubic equation are obtained from the data in Table 3. For the cosine curve in the \(z\) direction, the foot endpoint reaches the curve's top in the middle of the swing phase. According to the above conditions, the foot trajectory can be written in Equation (40). \[ \begin{cases}x(t)=\dfrac{S}{2}(a_{3}t^{3}+a_{2}t^{2}+a_{1}t+a_{0})\\ z(t)=-H_{0}+\dfrac{H}{2}\Big{\{}1-cos\Big{[}\dfrac{2\pi t}{(1-\beta T)}-\dfrac{ 2\pi\beta}{(1-\beta)}\Big{]}\Big{\}}\,,\quad\beta T<t\leq T\end{cases}  \tag{40}\] \begin{table}
\begin{tabular}{c c c} \hline \hline
**Items (time)** & \(x\) **Direction** & \(z\) **Direction** \\ \hline position (0) & \(\dfrac{S}{2}\) & \(H_{0}\) \\ position (\(\beta T\)) & \(-\dfrac{S}{2}\) & \(H_{0}\) \\ velocity (0) & \(-\dfrac{S}{\beta T}\) & 0 \\ velocity (\(\beta T\)) & \(-\dfrac{S}{\beta T}\) & 0 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: The initial and terminated conditions of the stance phase. Figure 7: The leg phases of the trot gait. where  
\[