## 3 Dynamic Balance Control Method
In (**a**), it’s a view from the positive to the negative direction of the y axis. In (**b**), it’s a view from the positive to the negative direction of the x axis. The diagonal line of void points represents the moment when the robot's diagonal legs fall from air to ground, and the diagonal line of solid points represents the moment when the robot's diagonal legs rise from support ground to air. \(O\) is the intersection of two diagonal lines, \(a\) is the intersection of the centerline and the 1-4 leg diagonal legs, \(b\) is the intersection of the centerline and the 2-3 leg diagonal, \(O_{1}\) is the center of gravity, \(l_{1}\) is the distance from the gravity center of robot to the 1-4 leg diagonal, \(l_{2}\) is the distance from the gravity center of robot to the 2-3 leg diagonal, \(d_{1}\) is the distance from the gravity center of robot to \(a\), \(d_{2}\) is the distance from the gravity center of robot to \(b\), \(\varphi\) represents the angle between two diagonals, \(\varphi_{1}\) represents the angle between the centerline and the 1-4 leg diagonal. \(\varphi_{2}\) represents the angle between the centerline and 2-3 leg diagonal. According to the movement characteristics of the quadruped robot's diagonal gait, there is no displacement in the \(y\)-axis during movement, so the movement of the robot's gravity center is always on the centerline. If the center of gravity is not on the line of the diagonal leg that is about to fall, one of the swinging legs touches the ground first, and the impact from the ground causes instability of the trunk [20]. Set the touchdown time of legs 1, 2, 3, and 4 as \(t_{1}\), \(t_{2}\), \(t_{3}\), and \(t_{4}\) respectively. When legs 1 and 4 are in a swing state, the time difference between them touching the ground is expressed as:  
\[\Delta t_{14}=t_{1}-t_{4} \tag{9}\]  
When legs 2 and 3 are in a swing state, the time difference between them touching the ground is expressed as:  
\[\Delta t_{23}=t_{2}-t_{3} \tag{10}\]  
The values of \(l_{1}\) and \(l_{2}\) are estimated according to the landing time difference. The greater \(l_{1}\) is, the greater the torque of gravity on 1-4 leg diagonal, and the greater the absolute value of the time difference \(\Delta t_{23}\) between leg 2 and 3. Similarly, the greater \(l_{2}\) is, the larger \(\Delta t_{14}\) is. \(l_{1}\) and \(l_{2}\) can be estimated as follows:  
\[ \begin{array}{l}l_{1}=k_{1}t_{23}=k_{1}(t_{2}-t_{3})\\ l_{2}=k_{2}t_{14}=k_{2}(t_{1}-t_{4})\end{array}  \tag{11}\]  
where, \(k_{1}\) and \(k_{2}\) are proportional coefficients with values greater than 0. Analyze the four conditions of the gravity center in Figure 6 respectively:  
1. The projection point of the gravity center is in front of the 2-3 leg diagonal. When the robot's gravity center is projected in front of the 2-3 leg diagonal, the distance \(d_{2}\) between the gravity center on centerline and the 2-3 leg diagonal is as follows: \[d_{2}=\frac{k_{2}(t_{2}-t_{3})}{\sin\ \varphi_{2}}\] (12)  
Figure 6: Four cases of gravity center distribution. In (**a**), the gravity center is before the 2–3 diagonal. In (**b**), the gravity center is behind the 2–3 diagonal. In (**c**), the gravity center is before the 1–4 diagonal. In (**d**), the gravity center is behind the 1–4 diagonal. Since the center of gravity is in front of the 2-3 leg diagonal, leg 2 falls first and leg 3 falls later, \(t_{2}<t_{3}\). It can be known from the above formula that \(t_{2}-t_{3}<0\), \(d_{2}<0\). (b) The projection point of the gravity center is behind the 2-3 leg diagonal. When the robot's gravity center is projected behind the 2-3 leg diagonal, the distance \(d_{2}\) between the gravity center on the centerline and the 2-3 leg diagonal can be calculated by Formula (12). Since the center of gravity is behind the 2-3 leg diagonal, leg 3 falls first and leg 2 falls later, \(t_{2}>t_{3}\). It can be known from Formula (12) that \(t_{2}-t_{3}>0\), \(d_{2}>0\).