# Modeling of Walking-Gait Parameters and Walking Strategy for Quadruped Robots
## 3 Results and Discussion
### The Gait Planning of the Bionic-Robot Dog
#### 3.3.3 Static-Gait Planning  
A gait with a duty cycle \(\beta_{i}>0.5\) is called a static gait. Static gait is the gait taken by a pedestrian when the speed requirement is not high. Its planning was also divided into two parts, the coordination relationship between the four limbs and the trajectory curve of the foot ends. To determine the duty cycle and phase difference during static gait, we observed the static-gait movement of the biological dog first. Figure 14 shows the gait of a biological dog at a speed of 3.5 km/h. When a biological dog moves at low speed, its four limbs are stepped out one by one in sequence, transitioning from having no less than two limbs swinging at any time to having more than two limbs landing on the ground at the same time. Figure 15 shows the displacement of the right-hind limb in the \(Z\)-direction at 4.0 km/h, and its duty cycle can be calculated based on the time point of landing in the video, that is, \(\beta=\frac{115-0.65}{1.35-0.65}=0.74\). Similarly, it can be calculated that the duty cycle at 3.5 km/h was 0.722. Figure 14: Gait of biological dogs at low speed. Figure 13: Comparison of foot-end forces between two trajectories. To ensure that three limbs land at any one time, take the duty cycle \(\beta\) = 0.75. Observing the gait of biological dogs at low speeds, the order of limb swinging used by the Labrador dog was left front-right hind-right front-left hind. Research has found that when a biological dog adopts this limb-swinging sequence, its stability is the greatest, and the gait at this time is called a wave gait. Taking into account stability and the bionics of the robotic dog, this experiment determined the order of swinging the limbs to be the waveform-gait sequence of left front-right hind-right front-left hind. When using a wave gait, the phase difference between the limbs is shown in Equation (28):  
\[ \begin{array}{l}\varphi_{2n-1}=F(m\beta)\\ \varphi_{2n-1}=F(\varphi_{2n-1}+0.5)\ m=1,2,\ldots n-1,\ \frac{3}{2n}\leq\beta<1\end{array}  \tag{28}\]  
In the formula, \(F(x)\) represents the decimal part of the real number \(x\). \(2n+1\) represents the limb on the left side of the body. Because it was a quadruped robot, it had limb 1 and limb 3 from front to back, and the corresponding right limbs were limb 2 and limb 4. Equation (29) can be obtained:  
\[ \begin{array}{l}\varphi_{1}=0\\ \varphi_{2}=0.5\\ \varphi_{3}=\beta\\ \varphi_{4}=F(\beta+0.5)\end{array}  \tag{29}\]  
Take \(\beta\) = 0.75, then \(\varphi_{1}\) = 0, \(\varphi_{2}\) = 0.5, \(\varphi_{3}\) = 0.75, and \(\varphi_{4}\) = 0.25. Follow the order of limb swings 1-4-2-3, with phases increasing by 0.25 in sequence. At this point, the gait is an equal-phase-wave gait, and when the robot walks in this gait, its energy consumption fluctuates minimally. Comparative analysis showed that the polynomial foot-trajectory curve is better than the cycloid, so the polynomial curve was proposed for the static-gait foot trajectory. In order to plan a reasonable trajectory curve for the bionic foot, this study analyzed the movement of a biological dog's body relative to the ground during movement. Following the movement of a biological dog, when designing a static landing phase, in order to ensure uniform movement of the body relative to the ground, it was necessary to ensure that the three limbs in the landing phase had the same speed relative to their respective hips. Therefore, a straight-line form was chosen for the landing phase on the \(X\)-axis. The curve equation combining straight lines and polynomials was used to describe the foot trajectory, as shown in Equation (30):  
\[ \begin{array}{l}X_{w}=\left\{\begin{array}{c}X_{sw}=at^{3}+bt^{2}+ct+d\ (0 \leq t<\frac{T}{4})\\ X_{st}=et+f\ (\frac{T}{4}\leq t\leq T)\end{array}\right.\\ Z_{w}=\left\{\begin{array}{c}Z_{sw}=At^{4}+B^{3}+CI^{2}+Dt+E\ (0\leq t<\frac{T}{4})\\ Z_{st}=-H\ (\frac{T}{4}\leq t\leq T)\end{array}\right.\end{array}