# Stability Criterion for Dynamic Gaits of Quadruped Robot
## 4 Dynamic-Stability Measurement  
The next problem to be solved is how to evaluate stability if the dynamic motion of the robot is stable. It is clear that the robot rarely tips over if the difference between ground-reaction forces at the support feet is small. The following three quantities are taken into account to quantify the stability of the robot:  
1. Distances between \(ZMP_{0}\) and the boundaries of the virtual-support quadrilateral in the virtual-support plane. These distances not only show the stability of the robot with its current support line but also consider the stability after the next support line appears. The vertical distance between \(ZMP_{0}\) and each boundary is a comprehensive representation of the distances between \(ZMP_{0}\) and the two support feet forming the boundary. From Equation (25), it can be seen that, in a certain direction, larger minimum distances between \(ZMP_{0}\) and the two support feet located at both ends result in smaller differences between the ground-reaction forces at these support feet. The instability of the robot is mainly caused by unbalanced rotation in two mutually perpendicular directions. Therefore, the distances between \(ZMP_{0}\) and the two sets of opposite boundaries (\(d_{11}\) and \(d_{12}\), and \(d_{21}\) and \(d_{22}\)) should be considered when evaluating the stability of the robot (Figure 8). The two support points forming a certain boundary of the quadrilateral do not necessarily exist at present. Hence, the distances between \(ZMP_{0}\) and boundaries should be multiplied by their corresponding coefficients \(\omega_{i}\), which are defined as follows:  
\[\omega_{i}= \begin{cases}1,&\text{\emph{two}\ support\ feet\ both\ exist}\\ 0.5,&\text{\emph{only}\ one\ support\ foot\ exists}\\ 0.2,&\text{\emph{otherwise}}\end{cases}  \tag{26}\]  
The integrated minimum distance between \(ZMP_{0}\) and the boundaries is defined as  
\[d_{edge}=\min\{\omega_{11}d_{11},\ \omega_{12}d_{12}\}+\min\{\omega_{21}d_{21}, \ \omega_{22}d_{22}\}. \tag{27}\]  
\(d_{edge}\) is expected to be as large as possible, and it is the most important quantity for evaluating the robot's stability. 2. Angle between the vector pointing from CoM to \(ZMP_{0}\) and the normal vector of the virtual-support plane. The ground-reaction force can be divided into two parts: support and friction forces. The support force is perpendicular to the support plane, and, in theory, it can be provided sufficiently to guarantee the force balance in this direction. The direction of the friction force is opposite to that of the trend of the foot's motion with respect to the support plane if the foot does not slip; the largest static-friction force that the ground can provide is proportional to the magnitude of the support force. Under a constant equivalent force \(F_{e}\), larger angles, \(\tau\), between the vector pointing from CoM to \(ZMP_{0}\) (the direction is the same as \(F_{e}\)) and the normal vector of the virtual-support plane (Figure 9)  
Figure 8: Distances between \(ZMP_{0}\) and each boundary of the virtual-support quadrilateral. result in lower support forces and greater required friction forces; thus, the feet are more likely to slip, and the stability of the robot becomes worse. \(|\sin\tau|\) is used as a key quantity to measure the robot's stability, in which \(u\) represents the vector pointing from CoM to \(ZMP_{0}\):  
\[\cos\tau=\frac{u\cdot x_{s}^{i}}{\|u\|\|x_{s}^{i}\|}; \tag{28}\]  
\[|\sin\tau|=\sqrt{1-\cos\tau^{2}}. \tag{29}\]  
For dynamic gaits, the support lines are alternately formed so that the ground-reaction force can push the robot to move forward and prevent it from falling. What really interacts with the ground are the support legs; therefore, the distance between \(ZMP_{0}\) and the support lines is also very important for the evaluation of robot stability. The larger is the distance between \(ZMP_{0}\) and a certain support line, the easier it is for the robot to tip over around this support line.