## 4 Dynamics and Control of Planar One-Legged Hopping Robots
### Motion Planning for the One-Legged Hopping Robots in Steady Locomotion  
By introducing springs into the hopping robots, as shown in [29], it is possible for the robots to realize steady hop motions without using large control inputs, while the steady motions are primarily governed by the nature dynamics of the dynamical systems. As seen in the last subsection, the hopping robots are hybrid dynamical systems and the variant constraints of a locomotion system generally result in large state jumps that can easily destroy the stability of the controlled system. In order to realize energy efficient locomotion so that the control inputs can be reduced to a reasonable level and the stability of motions could be improved effectively, it is necessary to carefully plan the target locomotion trajectory of the hopping robot. A steady continuous motion process of the hopping robots is commonly a periodic motion, which is roughly composed of two harmonic oscillators with different nature frequency, and every motion cycle is composed of four phases, flight phase, touchdown, stance phase, and lift off. During flight and stance phases, the motions of the hopping systems can be controlled, while, in touchdown and lift off phases, the hopping systems can not be controlled due to the short intervals of constraint changes. During stance phases, the nature frequency of the spring leg is  
\[\omega_{l}=\sqrt{k_{l}/m}, \tag{62}\]so the resonance time period of the linear spring leg is  
\[T_{s}=2\pi/\omega_{l}=2\pi\sqrt{m/k_{l}}. \tag{63}\]  
As illustrated in Figure 2, the thick line segment represents the time-amplitude curve of the linear spring leg in stance phase of a motion cycle. Suppose the length of the leg at the instants of touchdown and lift off is the nature length \(l_{0}\), the duration of stance phase is \(T_{v}\), the initial phase-angle of the harmonic vibration of the leg is \(\phi_{l}\), and the vibration amplitude of the spring leg is \(l_{m}\), then the oscillations of the linear leg in stance phases can be approximately expressed by  
\[l(t)=\big{(}l_{0}-mg/k_{l}\big{)}+l_{m}\sin(\omega_{l}t+\phi_{l}),l(0)=l_{0}, \tag{64}\]  
and the speed of the linear leg is approximately given by  
\[\dot{l}(t)=l_{m}\omega_{l}\cos(\omega_{l}t+\phi_{l}),\ \dot{l}(0)=\dot{l}(t_{vc}^{+}), \tag{65}\]  
where \(\dot{l}(t_{vc}^{+})\) is the compress speed of the linear leg at just after the touchdown. By using the two initial conditions of Equations (64) and (65), the two undetermined parameters \(l_{m}\) and \(\phi_{l}\) can be resolved as  
\[\phi_{l}=\pi-\text{atan}\left(\left|\frac{mg\omega_{l}}{k_{l}\dot{l}(t_{vc}^{+ })}\right|\right),\ l_{m}=\frac{mg}{k_{l}\sin(\phi_{l})}. \tag{66}\]  
From Figure 2, the duration of a stance phase can be calculated as  
\[T_{v}=\frac{1}{2}T_{s}+2t_{\phi_{l}}, \tag{67}\]  
where \(t_{\phi_{l}}=\frac{1}{\omega_{l}}\text{asin}\left(\frac{mg}{k_{l}\dot{l}_{m}}\right)\). Both in stance phase and flight phase, the vibrations of the hip joint should follow the same harmonic oscillation, but with different initial phase angles. By the dynamics of the hopping robot in flight phases Equation (47), the nature frequency of the hip joint can be calculated as  
\[\omega_{h}=\sqrt{k_{h}\left(1/l_{b}+1/l_{l}\right)}. \tag{68}\]  
However, the duration of a flight phase is determined by the lift off speed of the robot, and is given as  
\[T_{f}=2\dot{z}_{lo}/g. \tag{69}\]  
Then, the time period of the hopping motions is  
\[T=T_{v}+T_{f}=\frac{2\pi}{\omega_{h}}. \tag{70}\]  
Figure 2: Diagram of harmonic vibration of the linear spring leg. Similar to Figure 2, the angular motions of the body and the leg can be analyzed in Figure 3, where the thick line segment denotes the flight phase in a hop cycle.