## 3 Dynamic Balance Control Method
(c) The projection point of the gravity center is in front of the 1-4 leg diagonal When the robot's gravity center is projected in front of the 1-4 leg diagonal, the distance \(d_{1}\) between the gravity center on the centerline and the 1-4 leg diagonal is as follows:  
\[d_{1}=\frac{k_{1}(t_{1}-t_{4})}{\sin\ \varphi_{1}} \tag{13}\]  
Since the center of gravity is in front of the 1-4 leg diagonal, leg 1 falls first and leg 4 falls later, \(t_{1}<t_{4}\). It can be known from Formula (13) that \(t_{1}-t_{4}<0\), \(d_{1}<0\). (d) The projection point of the gravity center is behind the 1-4 leg diagonal. When the robot's gravity center is projected behind the 1-4 leg diagonal, the distance \(d_{1}\) between the gravity center on the centerline and the 1-4 leg diagonal can be calculated by Formula (13). Since the center of gravity is behind the 1-4 leg diagonal, leg 4 falls first and leg 1 falls later, \(t_{1}>t_{4}\). It can be known from Formula (13) that \(t_{1}-t_{4}<0\), \(d_{1}>0\). We analyze the different projection positions of the gravity center and estimate the distance between the projection point of the gravity center and the diagonal of the landing leg on the centerline by foot landing time difference. The gravity center position is estimated according to the landing time difference of diagonal legs during the quadrupped robot movement. After obtaining the estimated position of the gravity center position, the foot position of the quadrupped robot is adjusted during the movement to make the landing difference gradually decrease to be stable. The position adjustment of the robot's gravity center is realized by the movement of the supporting phase. After one cycle of movement, the robot estimates the gravity center position according to the time difference of falling foot and adjusts the foot position. In the next cycle, the center of gravity is re-estimated and adjusted, so that the center of gravity gradually moves to the line of the landing diagonal leg, as shown in Figure 7:  
Due to the uncertain position of the gravity center in the early movement stage, one of the robot legs lands first, which leads to a reaction force from the ground. To reduce the reaction force on the leg, the attitude feedback strategy mentioned above is used to make the gravity center adjust smoothly. Gait adjustment strategy flowchart is designed as shown in Figure 8:  
Figure 7: Robot gravity center adjustment. 1. In a cycle, the time difference \(\Delta t_{14}\) can be obtained when legs 1 and 4 are in the swinging phase, and the time difference \(\Delta t_{23}\) can be obtained when legs 2 and 3 are in the swinging phase. 2. According to the landing time difference, the distance \(d_{1}\) or \(d_{2}\) between the projection point of the gravity center and the diagonal of the landing leg on the centerline is solved. Assuming that the initial position before adjustment is \(P_{i}=(x_{i},y_{0})\), the desired foot position is: \[ \begin{array}{c}x_{ie}=\left\{\begin{array}{cc}x_{0}+d_{1}&i=2.3\\ x_{0}+d_{2}&i=1.4\\ z_{ie}=z_{0}+z_{0i}^{\prime}\end{array}\right.\end{array} \] (14) where, \(P_{i}^{\prime}=\left(x_{i}^{\prime},y_{0}\right)\) is the desired foot position. 3. After the expected foot position of the quadruped robot is calculated, the landing attitude of the swinging leg is adjusted according to the data feedback of the attitude to reduce the force on the leg. Set the expected coordinate value \(Q_{i}^{\prime}=(x_{ie},y_{0},z_{ie})\), and the expected value of each coordinate axis is: \[ \begin{array}{c}x_{ie}=\left\{\begin{array}{cc}x_{0}+d_{1}+x_{0i}^{ \prime}&i=2,3\\ x_{0}+d_{2}+x_{0i}^{\prime}&i=1,4\\ z_{ie}=z_{0}+z_{0i}^{\prime}\end{array}\right.\end{array} \] (15) where, \(x_{0i}^{\prime}\) and \(z_{0i}^{\prime}\) represent the changes in x- and \(z\)-axis adjustment, respectively. 4. Repeat steps (1) to (3) until the flight rate \(\lambda\) is greater than \(\varepsilon\). \(\varepsilon\) is an adjustable threshold.