## 3 Method
### Selection of an Appropriate Reduction Ratio  
[PERSON] et al. designed a nonlinear gear ratio profile with the aim of performing a high jump using direct current motors [19]. However, this profile led to major difficulties in mechanical design, processing and assembly. Our purpose in this work is to determine an appropriate reduction ratio for a specific motor without consideration of the energy transfer efficiency of the gear reducer. Because of the nonlinearity of the dynamic equation, i.e., Equation (1), it is difficult to determine three appropriate reduction ratios for three motors simultaneously. With reference to the work of [PERSON] and [PERSON] [19; 21], we designed a simple jumping pattern in which the trunk remains upright and above the ankle joint during the push-off phase, as shown in Figure 2. In this simple pattern, we can then estimate that of the three joints, and the knee joint will output the maximum torque and speed. The BH6 is equipped with the same motor K044100-6Y-4.98 Ams (Parker Hannifin Corp, Clifford, OH, USA) on the ankle, knee and hip joint. In Figure 3, according to the instructions from Parker Hannifin, we showed the torque speed curve and properties of motor K044100-6Y-4.98 Ams when the voltage supplied to this motor is 120 V. The blue line is the peak output torque of the motor and the red line is the continue output torque of the motor. Because jumping is a fast and dynamic motion, the peak output torque is used in the push-off phase. By the using of the motor driver, the motor can work at an arbitrary point in the right trapezoid formed by the blue line and axis. From Figure 3, the peak torque speed curve of this motor can be modeled as a segmented function. When the motor speed \(\omega_{motor}\leq\omega_{break}(\omega_{break}=10634\) (rpm)), the peak motor torque \(\tau_{motor}\) is \(\tau_{peak}=1.17\) (Nm). When \(\omega_{motor}\in[\omega_{break},\omega_{max}]\) (\(\omega_{max}=18835\) (rpm)), \(\tau_{motor}\) decreases linearly with increasing \(\omega_{motor}\). Given these parameters, we can then obtain the relationship between \(\omega_{motor}\) and \(\tau_{motor}\), as shown below:  
Figure 2: The joint constraint for appropriate reduction ratio selection. \[\tau_{motor}= \begin{cases}\tau_{peak}&\omega_{motor}\leq\omega_{break},\\ \tau_{peak}-\frac{\tau_{peak*}\omega_{motor}}{(\omega_{max}-\omega_{break})}& \omega_{motor}\in[\omega_{break},\omega_{max}].\end{cases}  \tag{2}\]  
We used the harmonic reducer to deliver the motor torque. If the reduction ratio applied to the motor is \(i_{r}\), then the joint speed \(\omega_{joint}\) will be \(\frac{1}{i_{r}}\) of the \(\omega_{motor}\). If the transfer efficiency of reducer is assumed as 100%, the output power of motor and joint are identical, \(\omega_{joint}\tau_{joint}=\omega_{motor}\tau_{motor}\). In addition, the peak joint torque \(\tau_{joint}\) will be \(i_{r}\) times as \(\tau_{motor}\):  
\[\tau_{joint}=i_{r}\tau_{motor}\;(\text{Nm}), \tag{3}\]  
\[\omega_{joint}=\frac{\omega_{motor}}{i_{r}}(\text{rpm})=\frac{2\pi\omega_{ motor}}{60i_{r}}\;(\text{rad}/\text{s}). \tag{4}\]  
It is assumed that the model state at time \(t\) is \(X_{t}(q_{t},q_{t})\). \(q_{t}=(q_{1t},q_{2t},q_{3t})\) (rad) and \(q_{t}=(q_{1t},q_{2t},q_{3t})\) (rad/s). Parameter \(C_{t}=(x_{t},y_{t},z_{t})\) (m) represents the coordinate of the COM. The translational velocity of the COM \(\dot{C}_{t}\) can then be calculated using the COM Jacobian matrix \(J(q_{t})\):  
\[\dot{C}_{t}=J(q_{t})q_{t}. \tag{5}\]  
Using the parameters given in Table 1, where \(l_{1}=l_{2}\), and the simple jumping pattern, we can obtain the joint constraint equations as shown below:  
\[ \begin{cases}q_{2t}=&-2q_{1t}=-2q_{3t},\\ \dot{q}_{2t}=&-2\dot{q}_{1t}=-2\dot{q}_{3t},\\ \ddot{q}_{2t}=&-2\ddot{q}_{1t}=-2\ddot{q}_{3t}.\end{cases}  \tag{6}\]  
We can estimate that, of the three joints, the knee joint will output the maximum torque and speed.