## 3 Structural Optimization and Actuation Design
### Actuation Design  
#### 3.2.1 Theoretical Analysis  
The actuator of the hopping robot has the following requirements.  
1. In the process of energy storage, the actuator needs to provide the crank-gear-bar great driving torque to realize the relative movement between point \(P_{9}\) and \(P_{10}\), thereby realizing the elongation of the tension spring.
2. When the energy is released, the angular velocity of the crank will change dramatically within a very short period. Therefore, it is necessary to reduce the resistance from the motor.
3. When the hopping robot lands, it is necessary to absorb the landing shock to reduce the damage to the motor.  
Therefore, a clockwork spring that is capable of storing and releasing angular energy is introduced between the output shaft of the motor and the crank-gear-bar as an auxiliary energy storage element, as shown in Figure 9. There are three advantages of the proposed special actuator.  
1. The input torque of the drive motor is amplified by the clockwork spring, thereby reducing the requirement for the motor output power. At the same time, the great torque provided by the actuator can be released in a very short time.
2. The great torque releasing speed mentioned in (1) can drive the crank-gear-bar to cross the limit position of the hopping mechanism quickly, thereby realizing the explosive hop of the robot.
3. The clockwork spring also absorbs the impact of hopping on the motor.  
The crank-gear-bar \(P_{1}P_{3}\) and \(P_{2}P_{4}\) are an empty set in the shaft, the gear 1 and the gear 2 are fixed at the other end of the two shafts respectively, and mesh with each other. Gear 1 is driven by the motor, and the inner ring of the clockwork spring 1 and 2 are fixed on gear 1 and 2 respectively, which means \(\omega_{\text{in}}=\omega_{1}\). The outer rings are fixed on crank-gear-bar \(P_{1}P_{3}\) and \(P_{2}P_{4}\) respectively, which means \(\omega_{\text{out}}=\omega_{2}\). When the motor continuously rotates in the clockwise direction, the angular velocity of the motor is \(\omega\). The relationship between the actuator and the active component is shown in Figure 9.  
1. During the period \(t_{1}\)-\(t_{2}\), \(\omega_{1}=\omega\) and \(\omega_{2}=0\). At this phase, the inner and outer ring of the clockwork spring generates a difference in speed, and the clockwork spring begins to store energy  
Figure 9: Energy-stored type actuator. The clockwork spring is introduced between the motor output shaft and the crank-gear-bar as an auxiliary energy storage element.  
At the moment \(t_{2}\), the angular velocity of the crank-gear-bar changes, i.e., \(\omega_{2}\
eq 0\). During the period \(t_{2}-t_{3}\), \(\omega_{2}\
eq 0\) but \(\omega_{2}<\omega_{1}\). There is still a difference in the speed between the inner and outer rings of the clockwork spring, which means that the spring can still store energy, but the energy storage speed slows down.
3. At the moment \(t_{3}\), the clockwork spring rotates to its limit position, the angular velocity is \(\omega_{2}\) =\(\omega_{1}\). The inner and outer rings of the clockwork spring rotate at the same speed, which means that the process of energy storage of the clockwork spring is over.
4. In a very short time from \(t_{3}-t_{4}\), the energy stored in the clockwork spring is released, and the angular velocity is \(\omega_{2}>>\omega_{1}\).
5. During the period \(t_{4}-t_{5}\), the angular velocity is \(\omega_{2}<\omega_{1}\). When the time reaches \(\mathfrak{t}_{4}\), the next hopping cycle begins.