## 2 Quadruped Bounding Model with a Flexible-Articulated Spine
### Hybrid Dynamics of Quadruped Bounding
#### 2.2.2 Trigger of Events and Renewal of Discrete states  
That the trigger of events alternates with the renewal of related discrete states constantly is an important characteristic of the quadruped motion. Based on Figure 2, the event functions and renewal formulas of discrete states of quadruped bounding are to be presented in this section. \(\bullet\)Trigger of the touchdown-event and renewal of related discrete states  
When the vertical distance between foot-end with ground decreases to zero monotonously, the touchdown-event is triggered. The touchdown-event functions of the anterior and posterior legs are denoted by \(f_{a}^{01}\) and \(f_{p}^{01}\), respectively. The corresponding trigger condition can be expressed as  
\[f_{i}^{01}=0,\ \dot{f}_{i}^{01}<0,\ \mathit{phase}_{i}=1,\ i\in[a,p], \tag{4}\]  
where  
\[f_{a}^{01}=(z_{s}+d\sin(\theta_{a}))-l_{0}\cos(\gamma_{a}^{td}) \tag{5}\]  
\[f_{p}^{01}=(z_{s}-d\sin(\theta_{p}))-l_{0}\cos(\gamma_{p}^{td}) \tag{6}\]  
where \(\gamma_{a}^{td}\) and \(\gamma_{p}^{td}\) are the touchdown angle of the anterior and posterior legs, respectively. \(\mathit{phase}_{a}\) and \(\mathit{phase}_{p}\) denote the current phase of the anterior and posterior legs, respectively. Figure 2: Phase transitions of every leg. The anterior and posterior legs are distinguished by subscript ‘\(a\)’, ’\(p\)’, respectively. The motion of the whole period of every leg is divided into two stages (stance stage and swing stage) or three phases (flight phase, stance phase, and climbout phase). The leg-phase (blue blocks) which can be identified by ’1’, ’2’, and ’3’ transits with the event-trigger (purple words) which can be identified by ’01’, ’02’, and ’03’. The related discrete states including the phase and the horizontal coordinate of touchdown position of leg renew at the moment the touchdown event is triggered. The renewal formulas corresponding to the anterior and posterior legs are  
\[phase_{a}\Rightarrow 2\,\ x_{a}^{td}\Rightarrow x_{s}+d\cos(\theta_{a})+l_{0} \sin(\gamma_{a}^{td}) \tag{7}\]  
\[phase_{p}\Rightarrow 2\,\ x_{p}^{td}\Rightarrow x_{s}-d\cos(\theta_{p})+l_{0} \sin(\gamma_{p}^{td}) \tag{8}\]  
respectively, where \(x_{a}^{td}\) and \(x_{p}^{td}\) denote the horizontal coordinate of touchdown position of anterior and posterior legs, respectively. The symbol '\(\Rightarrow\)' represents the transition of discrete states, and the following are all the same. * Trigger of liftoff-event and renewal of related discrete states  
When the gap between the current length with the nominal length of a leg increases to zero monotonously, the liftoff-event is triggered. The liftoff-event functions of the anterior and posterior legs are denoted by \(f_{a}^{02}\) and \(f_{p}^{02}\), respectively. The corresponding trigger condition can be expressed as  
\[f_{i}^{02}=0,\ \dot{f}_{i}^{02}>0,\ \mathit{phase}_{i}=2,\ i\in[a,p] \tag{9}\]  
where  
\[f_{a}^{02}=(z_{s}+d\sin(\theta_{a}))/\cos(\gamma_{a})-l_{0} \tag{10}\]  
\[f_{p}^{02}=(z_{s}-d\sin(\theta_{p}))/\cos(\gamma_{p})-l_{0} \tag{11}\]  
The related discrete states including the phase and the termination time of following climbout phase of leg renew at the moment the liftoff event is triggered. The renewal formulas corresponding to anterior and posterior leg are  
\[phase_{a}\Rightarrow 3\,\ \mathit{ts}_{a}\Rightarrow tr_{a}+\Delta tr \tag{12}\]  
\[f_{a}^{01}=(z_{s}+d\sin(\theta_{a}))-l_{0}\cos(\gamma_{a}^{td}) \tag{5}\]0  
respectively, where \(ts_{a}\) and \(ts_{p}\) are the termination times of climbout phase of the anterior and posterior legs, respectively. \(tr_{a}\) and \(tr_{p}\) are the termination times of stance phase of the anterior and posterior legs, respectively. \(\Delta tr\) is the duration of climbout phase of every leg. \(\bullet\) Trigger of flying-event and renewal of related discrete states  
When the gap between the current time and the termination time of climbout phase of a leg increases to zero monotonously, the flying-event is triggered.