## 3 Method
### Motion Planning for Jumping Maneuver  
In the original control system in Figure 4, under the joint constraint of Equation (6), we could only ensure that the knee joint torque matched the ultimate torque. To ensure that all three joint torques yield the torque speed curve profile simultaneously as much as possible, we removed the joint constraint and commanded each motor output torque to the torque speed curve. Driven by these torques, the robot will turn over. Thus, we added the ZMP constraint to help the robot to be stable in the push-off phase. If the ZMP constraint is violated, we will fine-tune these torques through optimization. If the ZMP constraint is satisfied, we will directly apply these torques to the joints. Figure 5: With the same motor and model, different reduction ratio leads to different jump height. When the same initial state \(X_{0}=(-75^{\circ},150^{\circ},\)\(-75^{\circ},\)\(0,\)\(0,\)\(0)\) is determined, the relationship between \(h\) and \(i_{r}\) can be presented. We substitute \(\dot{q}_{1t},q_{2t},q_{3t}\) into Equations (2)-(4) to calculate the peak ankle, knee and hip joint torques \(\tau_{1t},\tau_{2t},\tau_{3t}\), respectively, at time \(t\). The maximum angular accelerations of the three joints can then be obtained as follows:  
\[\ddot{q}_{t}=M(q_{t})^{-1}(\tau_{t}-V(q_{t},\dot{q}_{t})-G(q_{t})), \tag{13}\]  
where \(\tau_{t}=(\tau_{1t},\tau_{2t},\tau_{3t})\) represents the peak torques of the three joints within the limitations of the motor characteristics and \(M,C,G\) can be calculated with state \(X_{t}(q_{t},\dot{q}_{t})\). Using Equations (5) and (11), the ZMP of the robot at time \(t\) can be given as below [24]:  
\[\left( \begin{array}{c}p_{xt}\\ p_{yt}\\ p_{zt}\end{array} \right)=\left( \begin{array}{c}x_{t}-\frac{(z_{t}-p_{zt}) \dot{x}_{t}}{z_{t}+g}\\ y_{t}-\frac{(z_{t}-p_{zt})\dot{y}_{t}}{z_{t}+g}\\ 0\end{array} \right), \tag{14}\]  
where \((x_{t},y_{t},z_{t})\) is the coordinate of the COM that can be calculated with \(q_{t}\). Generally, if the joints move with maximum angular acceleration calculated by Equation (13), the \(p_{yt}\) may move out of the supporting polygon formed by the feet and the ZMP constraint will be violate. Assuming that the limitation to \(p_{yt}\) is \(p_{ylim}\in(-0.05,\ 0.15)\) (m), which is smaller than the distances from the projection of ankle joint on the ground to the heel \(d_{1}\) (0.10 (m)) and to the tiptoe \(d_{2}\) (0.20 (m)) as Figure 7 shows. At time \(t\), according to the state of the robot \(X_{t}(q_{t},\dot{q}_{t})\) and Equation (5), (11) and (14), the ZMP can be expressed as a function of \(\ddot{q}_{t}=(\ddot{q}_{1t},\ddot{q}_{2t},\ddot{q}_{3t})\). By taking the total derivative, we can get the \(\Delta p_{yt}\) as below:  
\[\Delta p_{yt}=\frac{\partial p_{yt}}{\partial\dot{q}_{1t}}\Delta\dot{q}_{1t}+ \frac{\partial p_{yt}}{\partial\dot{q}_{2t}}\Delta\dot{q}_{2t}+\frac{\partial p _{yt}}{\partial\dot{q}_{3t}}\Delta\dot{q}_{3t}=\sum_{i=1}^{3}(\frac{\partial p _{yt}}{\partial\dot{q}_{it}}\Delta\dot{q}_{it}). \tag{15}\]  
Figure 6: All three joints are equipped with the same K044100-6Y-4.98 Ams motor and the same reduction ratio of \(i_{r}=100\). (**a**) shows the torque of each joint; and (**b**) represents relationships between the motor torque and speed. If \(p_{yt}\) is located beside \(p_{ylim}\), then the \(\Delta\dot{q}_{t}\) added to \(\dot{q}_{t}\) should satisfy the following equation:  
\[-0.05-p_{yt}\leq\sum_{i=1}^{3}(\frac{\partial p_{yt}}{\partial\dot{q}_{it}} \Delta\dot{q}_{it})\leq 0.15-p_{yt}. \tag{16}\]  
The same method can also be applied to the joint torque constraint. Due to the dynamic equation and \(X_{t}(q_{t},\dot{q}_{t})\), the joint torque \(\tau_{t}\) can be expressed as a function of \(\ddot{q}_{t}=(\ddot{q}_{1t},\ddot{q}_{2t},\ddot{q}_{3t})\).