# Stability Criterion for Dynamic Gaits of Quadruped Robot
## 3 Dynamic-Stability Criterion  
The first problem to be solved when considering the stability of the robot is determining whether the robot can maintain balance at the current state. The stability of a robot with a dynamic gait is closely related to two factors: (1) the ground-reaction forces that act on the current support feet; and (2) the ground-reaction forces that will act on the next support feet. The robot is unstable if neither of these forces can prevent it from tumbling. In addition, the elimination of unexpected speed is also taken into account when evaluating the robot's stability by adding the velocity term into the calculation of _ZMP_. There are always two legs moving in the same phase for most dynamic gaits; hence, these two legs can be equivalent to one virtual leg from the midpoint of the support line to the robot's CoM. The stability criterion is considered in the current virtual-support plane. \(A_{L}{{}^{\prime}}\) and \(B_{L}{{}^{\prime}}\) are the projection points on the support plane of the next set of support feet, \(A_{L}\) and \(B_{L}\), respectively, as shown in Figure 4.  
It should be noted that the direction of the support force, \(F_{n}\), is perpendicular to the support plane and points outward, and its main role is to support the robot standing and to maintain the stability of the robot's rotation around the \(y_{s}\) and \(z_{s}\) axes. The frictional force \(F_{f}\) is located on the support plane, which is opposite to the direction of the feet's motion trend with respect to this plane; its function is to push the robot forward and to prevent the robot from rotating around the \(x_{s}\) axis. The contact forces act on the two support feet, \(A\) and \(B\). Hence, the current support line cannot provide the torque around the \(z_{s}\) axis; its main role is to prevent the robot from tumbling around the \(y_{s}\) axis. The torque that prevents the robot from tipping around the \(z_{s}\) axis can only be provided by the next set of support feet. The contact forces at \(A_{L}{{}^{\prime}}\) and \(B_{L}{{}^{\prime}}\) also play a supplementary role in maintaining rotational stability around \(y_{s}\). The rotation of the robot around \(x_{s}\) is stopped by the friction at the current support and next support lines, but this is not the main reason the robot loses stability. _ZMP\({}_{0}\)_ can be considered to be on the extension line of the equivalent force, \(F_{e}\), acting at the CoM of the body (Figure 5). Figure 4: Projection of the next set of support feet and contact force at the current support feet. Figure 5: Equivalent force and _ZMP\({}_{0}\)_. In the \(x_{s}Sz_{s}\) plane, \(D_{1}\) represents the point with the largest \(z_{s}\)-coordinate among the four support points (\(A\), \(B\), \(A_{L}\)', and \(B_{L}\)'), whereas the \(z_{s}\)-coordinate of \(D_{4}\) is the smallest one, as shown in Figure 6.  
For any moment,  
\[\sum_{i=1}^{4}F_{nD_{i}}=-F_{exz}\cos\tau_{xz}; \tag{23}\]  
\[M_{y_{s}}=\sum_{i=1}^{4}F_{nD_{i}}z_{D_{i}}^{s}+F_{exz}z_{ZMP_{0}}^{s}\cos\tau_ {xz}. \tag{24}\]  
\(\tau_{xz}\) is the angle between \(F_{exz}\) and the support plane; hence,  
\[M_{y_{s}}=\sum_{i=1}^{4}F_{nD_{i}}\Big{(}z_{D_{i}}^{s}-z_{ZMP_{0}}^{s}\Big{)}. \tag{25}\]  
The ground-reaction force at the current or the next set of support feet is unable to maintain the rotational balance of the robot around the \(y_{s}\) axis if \(z_{ZMP_{0}}^{s}\) is less than the minimum \(z_{s}\)-coordinate of the four support points, or larger than the maximum value, because the direction of the support forces can only be perpendicular to the support plane and point outward. Similarly, the ground-reaction force at the next set of support feet cannot prevent the robot from tumbling around the \(z_{s}\) axis if \(y_{ZMP_{0}}^{s}\) is less than the minimum or larger than the maximum \(y_{s}\)-coordinate of the four support points. However, it is insufficient to consider only the rotation around these two directions for stability analysis.