# Stability Criterion for Dynamic Gaits of Quadruped Robot
## 5 Results and Discussion
### Simulation 2
The magnitude of the forces that the robot can bear differs in the positive and negative directions of \(y_{c}\) if the positions of \(ZMP_{0}\) and the next support line are fixed. For example, in the case shown in Figure 21, the impact forces in the positive and negative directions of \(y_{c}\) can both drive \(ZMP_{0}\) closer to the boundaries of the virtual-support quadrilateral. However, the force in the positive direction can reduce the distance between \(ZMP\) and the support lines, whereas the force in the negative direction increases this distance. Thus, the robot can endure a greater positive impact force along \(y_{c}\). \begin{table}
\begin{tabular}{c c c c} \hline  & **Method 1** & **Method 2** & **Method 3** \\ \hline Test 1 & 0.0225 & 0.7899 & 0.1185 \\ Test 2 & 0.0365 & 0.8181 & 0.1130 \\ Test 3 & 0.0277 & 0.8630 & 0.1402 \\ \hline \end{tabular}
\end{table} 
Table 4: Average values of stability evaluations. Figure 20: Stability measure of: (**a**) Method 1; (**b**) Method 2; and (**c**) Method 3. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Timing (s)** & **Magnitude (N)** & **Duration (s)** \\ \hline
5.1 & 100 & 0.2 \\
5.1 & \(-\)80 & 0.2 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Impact forces applied in Test 1 and Test 2. The changes of the posture angles of these two tests under impacts are shown in Figure 22.  
The robot in Test 2 tumbled to the ground after those two impact forces acting in different directions, while the robot in Test 3 could sustain the impacts without losing its stability. This means that the stability of the motion in Test 1 was better than Test 2, thus the method proposed in this paper could evaluate the stability of the quadruped robot with trotting gait more accurately than LAR. Three impact forces parallel to \(y_{c}\) were applied to the CoM of the body in the three trotting tests, as described above, to further illustrate the accuracy of the proposed stability criterion and measurement. The magnitudes, timings, and durations of these forces are shown in Table 6.  
The simulation results of these three tests after impacts are shown in Figure 23.  
Test 2 was the least able to bear the impact force in the positive direction of \(y_{c}\), since the positions of the next set of support feet differed with respect to \(ZMP_{0}\) (as can be seen simply from the positions of the next support lines with respect to CoG), tumbling after the first impact (Figure 23b). The robot \begin{table}
\begin{tabular}{c c c} \hline
**Timing (s)** & **Magnitude (N)** & **Duration (s)** \\ \hline
5 & 100 & 0.2 \\
9.2 & \(-\)82 & 0.2 \\
14.1 & 130 & 0.2 \\ \hline \end{tabular}
\end{table} 
Table 6: Impact forces applied to the body. Figure 23: Tumbling moments of the three tests: (**a**) Test 1: 9.76 s; (**b**) Test 2: 5.45 s; and (**c**) Test 3: 14.56 s.  
Figure 22: Posture angles of the two tests with impact forces: (**a**) 100 N; and (**b**) \(-\)80 N.  
walked stably in Test 1 until the second impact was applied to the body (Figure 23a), and the robot in Test 3 lost its balance after the last impact (Figure 23c). The three stability-evaluation methods were also used for the analysis of the three tests with external impacts. To illustrate this intuitively, for Method 2, the stability measurement was set to \(-0.3\) if \(ZMP_{0}\) was outside the virtual-support quadrilateral and the robot lost its balance according to Equation (40). As shown in Figure 24, the distance between \(ZMP\) and the current support line was not zero for most of the time, making it very difficult to determine when the robot would lose its balance. Using Method 2, it was hard to decide when to update the value of LAR if the robot tips over, because one of the feet would never touch the ground. From the results of Method 3, the robot was considered to lose its balance at 9.753 s (Test 1), 5.109 s (Test 2), and 14.2 s (Test 3), which was consistent with the actual motions of the robot. The simulation data from Test 3 were analyzed separately.