# Modeling of Walking-Gait Parameters and Walking Strategy for Quadruped Robots
## 3 Results and Discussion  
### Kinematic Analysis of the Bionic-Robot Dog  
The bionic-robot dog was mainly composed of the back and four limbs. The four limbs had the same structure. The kinematic analysis of the bionic-robot dog was divided into the kinematic analysis of the single limb and the kinematic analysis of the whole robot mechanism. The thigh, calf, and foot end of one limb were in the same plane. Figure 5a shows the structural diagram of one limb. Translate the lower limb and foot end, and keep the foot-end position unchanged, to obtain a structural equivalent diagram as shown in Figure 5b. Consistent with the previous text, \(\varphi_{1}\) shows the angle of rotation of motor 1 driving the hip joint. \(\varphi_{2}\) is the angle of rotation of motor 2 that drives the knee joint. \(\theta_{1}\) is the rotation angle of the hip joint. \(\theta_{2}\) and \(\theta_{3}\) are the rotational angles of the knee and ankle joints, respectively. In the structural design, because the thigh and the foot of the biological dog have similar motion laws, the designed knee joint and ankle joint have the same motion angle, that is, \(\theta_{2}=\theta_{3}\). The hip-joint motion angle and the drive angle of motor 1 are mutually complementary, that is, \(\theta_{1}+\varphi_{1}=\pi/2\). In addition, in the parallelogram structure composed of the thigh, \(l_{1}\), \(l_{2}\), and calf, the motor drive angles \(\varphi_{1}\) and \(\varphi_{2}\) and the knee-joint rotation angle \(\theta_{2}\) are adjacent angles of the parallelogram and have a complementary relationship, that is, \(\theta_{2}=\pi-(\varphi_{1}+\varphi_{2})\). According to the geometric relationship of the single-limb \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline
**Parameter** & \(M\) **(kg)** & \(I_{0}\) **(mm)** & \(b_{0}\) **(mm)** & \(I_{1}\) **(mm)** & \(I_{2}\) **(mm)** & \(I_{3}\) **(mm)** & \(\theta_{1}\) **(deg)** & \(\theta_{2}\) **(deg)** \\ \hline Biological dog & 25 & 560 & 210 & 130 & 220 & 120 & [60; 90] & [90; 130] \\ \hline Bionic-robot dog & 20 & 550 & 238 & 135 & 180 & 160 & [40; 110] & [80; 150] \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Parameters of the biological dog and the bionic-robot dog. Figure 5: Structural diagram of one mechanical limb. (**a**) Structural diagram of one limb; (**b**) equivalent diagram of one limb’s mechanical structure. structure, the position of the foot end is uniquely determined by the driving angles of the hip motor and knee motor. Establish a coordinate system (\(X\), \(Z\)), which is fixed to the hip, and set the foot-end coordinates to (\(x\), \(z\)). The driving space composed of (\(x\), \(z\)) and \(\varphi_{1}\) and \(\varphi_{2}\) has a certain mathematical relationship, which can be expressed as Equation (1):  
\[ \begin{cases}x=X(\varphi_{1},\varphi_{2})\\ z=Z(\varphi_{1},\varphi_{2})\end{cases}  \tag{1}\]  
In the equivalent diagram of the single-limb mechanical structure, \(\angle BCO=\varphi_{1}\), \(\angle CDA=\varphi_{2}\), then the forward kinematic relationship can be expressed as Equation (2):  
\[ \begin{cases}x=O_{1}B-AB=O_{1}B-CE=(l_{1}+l_{3})\sin\varphi_{1}-l_{2}\sin \varphi_{2}\\ z=-DE-AE=-DE-BC=-(l_{1}+l_{3})\cos\varphi_{1}-l_{2}\cos\varphi_{2}\end{cases}  \tag{2}\]  
It can be seen from Table 1 that the range of motion of the hip joint was [40\({}^{\circ}\), 110\({}^{\circ}\)], and the range of motion of the ankle joint was [80\({}^{\circ}\), 150\({}^{\circ}\)]. The relationship between each joint and the motor drive angle is as in Equation (3):  
\[ \begin{cases}\theta_{2}=\theta_{3}\\ \theta_{1}+\varphi_{1}=\frac{\pi}{2}\\ \theta_{2}=\pi-(\varphi_{1}+\varphi_{2})\end{cases}  \tag{3}\]  
Based on the range of motion of each joint and the relationship between each joint and the motor drive angle, it can be concluded that the value ranges of \(\varphi_{1}\) and \(\varphi_{2}\) were [\(-20^{\circ}\), 50\({}^{\circ}\)] and [\(-20^{\circ}\), 120\({}^{\circ}\)], respectively.