# Modeling of Walking-Gait Parameters and Walking Strategy for Quadruped Robots
## 3 Results and Discussion
### The Gait Planning of the Bionic-Robot Dog
#### 3.3.4 Gait-Transition Planning  
In the exercise experiments on the biological dog, as the speed of the treadmill increased, the gait of the biological dog quickly changed from adapting to a low-speed static gait to adapting to a high-speed dynamic gait. For bionic-robot dogs, to achieve the transition from static to dynamic gait, the first consideration is overall coordination. To achieve the rapid-transition effect of a biological dog, it is necessary to shorten the time required for gait switching. Based on the above considerations, the bionic-robot dog adopted a method of simultaneously switching the phase of its four limbs and the trajectory of its feet, completing gait switching within half a period. Figure 17 shows the phase-change diagram of the planned transition process. The wireframe represents the state of each limb during gait transition. To the left of the wireframe is a period of static gait. On the right is the gait of two periods. Using the left-front limb as the reference limb, the static gait \([0,T/4]\) was called the static-gait swing phase; \([T/4,2T/4]\) was the first landing stage; \([2T/4,3T/4]\) was the second landing stage; and \([3T/4,T]\) was the third landing stage.  
Figure 16: Foot trajectory and velocity variation in static gait.  
Figure 17: Phase diagram of gait-transition process.  
Due to the difference in foot-trajectory curves between static and dynamic gait, the phase-transition process is accompanied by changes in foot-trajectory curves. Because each limb is in a different phase, it is necessary to plan the foot-trajectory transition curves for each of the four limbs separately. Firstly, this study planned the trajectory of the foot end of the right-hind limb. In the hip coordinate system \(\left\{O_{1}\right\}\), planning is on the \(X\) and \(Z\) axes, respectively, using a cubic polynomial curve on the \(X\)-axis and a cycloid on the \(Z\)-axis. Assuming that the transition curve equations for the \(X\) and \(Z\) axes are \(F(t)\) and \(G(t)\), respectively, and the start time of the static gait is \(t=0\), in order to ensure that there is no displacement mutation and no velocity mutation during the gait switching process, \(F(t)\) and \(G(t)\) must meet the following conditions:  
\[\begin{cases}F(0)=X_{\text{up}}(\frac{3T}{T})\\ F(\frac{T}{2})=X_{\text{rot}}(\frac{T}{2})\\ F(0)=X_{\text{up}}(\frac{3T}{T})\\ F(\frac{T}{2})=X_{\text{rot}}(\frac{T}{2})\\ \end{cases}\quad\begin{cases}G(0)=Z_{\text{up}}(\frac{3T}{T})\\ G(\frac{T}{2})=Z_{\text{rot}}(\frac{T}{2})\\ G(0)=Z_{\text{rot}}(\frac{3T}{T})\\ G(\frac{T}{2})=Z_{\text{rot}}(\frac{T}{2})\\ \end{cases}\]  
The right-hind limb enters the transition gait from the third stage of the static landing phase, with the \(X\)-axis entering the dynamic gait through a polynomial curve, and the \(Z\)-axis entering the dynamic gait through a cycloid. To achieve rapid transition, the four limbs are transited simultaneously, and the changes in the trajectory of each limb's foot end are shown in Figure 18.  
0-0.5 s: \(LF\) and \(RH\) enter the swing phase from the third- and second-landing stages of static gait, respectively, while \(RF\) and \(LH\) enter the transition phase from the first- and swing-landing stages of static gait, respectively.  
0.5-1.0 s: Each of the four limbs moves along the transition curve, with \(LF\) maintaining its highest point in the \(Z\)-axis, \(RF\) and \(LH\) having no movement in the \(Z\)-axis, and RH rising along the curve in the \(Z\)-axis.  
1.0-1.5 s: Limbs continue to move along their respective transition curves.  
1.5-2.0 s: At the end of 2 s, \(LF\) and RH simultaneously enter the landing phase of the dynamic gait, and the corresponding \(RF\) and \(LH\) become the swinging phase of the dynamic gait.  
After a 2 s transition process, the bionic robot achieves the transition from static gait to dynamic gait.