# Modeling of Walking-Gait Parameters and Walking Strategy for Quadruped Robots
## 3 Results and Discussion
### The Gait Planning of the Bionic-Robot Dog
#### 3.3.2 Dynamic-Gait Planning  
A gait with a duty cycle \(\beta_{i}\leq 0.5\) is called a dynamic gait. Dynamic gait is the gait taken by pedestrians when they require a high speed of movement. The planning is mainly divided into two parts. One part is to determine the coordination relationship between the limbs, that is, to determine the duty cycle and phase difference, and the other part is to plan a reasonable foot-trajectory curve. The coordination relationship between the limbs of a quadruped robot is determined by phase difference and duty cycle. When the movement speed of a biological dog exceeds 6.0 km/h, the gait adopted is the dynamic gait. Firstly, we calculated the gait duty cycle. Figure 10 shows the displacement diagram of the right-hind-limb foot end in the \(Z\)-direction at 6.0 km/h, and the diagram shows a period. The displacement change in the \(Z\)-direction reflects the state of the limb. When the displacement remains constant, it indicates that the limb is in contact with the ground and in the contact phase. When the displacement changes, it indicates that the limb is swinging in the air and in the swinging phase. By combining the time point of landing in the captured video, the duty cycle \(\beta\) at 6.0 km/h can be calculated, that is, \(\beta=\frac{0.75-0.5}{10.55-0.5}=0.455\). Similarly, the duty cycles of 6.5 km/h and 7.0 km/h are 0.417 and 0.420, respectively. When designing the gait of a bionic-robotic dog, in order to ensure the stability of the dog during high-speed motion, take the duty cycle \(\beta\) = 0.5 to ensure that both limbs are on the ground at all times. The order of swinging the limbs is to swing both the left-front limb and the right-hind limb  
Figure 10: Right-hind-limb \(Z\)-axis displacement at 6.0 km/h. simultaneously, followed by swinging both the left-hind limb and the right-front limb simultaneously. Using the left-front limb as the reference limb, the phase difference between each limb is as follows: \(\varphi_{LF}=0\), \(\varphi_{RF}=0.5\), \(\varphi_{LH}=0.5\), \(\varphi_{RH}=0\). In order to achieve bionic gait, it is necessary to first study the foot-trajectory characteristics of biological dogs during rapid gait. Figure 11 shows the movement of the right-hind-limb foot end relative to the hip during the test of the Labrador dog on a treadmill at speeds of 6.0 km/h, 6.5 km/h, and 7.0 km/h. Observations have shown that the foot trajectory of a biological dog can be described in two ways, one in the form of a compound cycloid and the other in the form of a polynomial. This article simulates the cycloid and polynomial foot-trajectory curves separately, and then derives the simulation results data from ADAMS post-processing. Because the bionic-robot dog enters the simulation state from zero when it starts moving in the simulation environment, its position undergoes a sudden change. At the end of the exercise, it suddenly enters the zero position and the position undergoes a sudden change. So, this study selected the second period in the middle to observe its motion, and it compared the two curves in terms of stability and foot-end force. Figure 12 shows the motion of the centroid of a bionic-robotic dog. Figure 13 shows a comparison of foot-end forces between the two trajectories. Figure 11: Movement of the right-hind-limb foot end relative to the hip of the biological dog during dynamic gait. Figure 12: Comparison of centroid motion between two trajectories. (**a**) Cycloid-trajectory centroid displacement; (**b**) polynomial-trajectory centroid displacement. From the perspectives of stability and foot force, both trajectory curves have advantages and disadvantages. This study ultimately chose polynomial foot trajectory. On the one hand, stability is the first factor to consider in the overall motion of the machine, and the stability of the polynomial foot trajectory is greater than that of the composite cycloid.