## 3 Method
### Motion Planning for Jumping Maneuver
By taking the total derivative, we can get the \(\Delta\tau_{it}(i=1,2,3)\) as below:  
\[\Delta\tau_{it}=\frac{\partial\tau_{it}}{\partial\ddot{q}_{1t}}\Delta\dot{q}_ {1t}+\frac{\partial\tau_{it}}{\partial\ddot{q}_{2t}}\Delta\ddot{q}_{2t}+\frac {\partial\tau_{it}}{\partial\ddot{q}_{3t}}\Delta\ddot{q}_{3t}=\sum_{j=1}^{3}( \frac{\partial\tau_{it}}{\partial\ddot{q}_{jt}}\Delta\dot{q}_{jt});(i=1,2,3). \tag{17}\]  
To maintain the torque constraint, the \(\Delta\dot{q}_{t}\) added to \(\dot{q}_{t}\) must satisfy the following equation:  
\[|\sum_{j=1}^{3}(\frac{\partial\tau_{it}}{\partial\ddot{q}_{jt}}\Delta\ddot{q}_ {jt})+\tau_{it}|\leq|\tau_{it}|\quad(i=1,2,3), \tag{18}\]  
where \(\tau_{it}(i=1,2,3)\) is the maximal torque allowed by torque speed curve at speed \(\dot{q}_{it}(i=1,2,3)\). To get the optimal solution for the set of inequalities composed by Equations (16) and (18), we built a cost function  
\[Cost(\Delta\ddot{q}_{t})=\sum_{i=1}^{3}\sum_{j=1}^{3}|\frac{\partial\tau_{it}} {\partial\dot{q}_{jt}}\Delta\dot{q}_{jt}\dot{q}_{it}|. \tag{19}\]  
Since \(|\Delta\tau_{it}|\) is the decrease of each joint torque, then \(|\Delta\tau_{it}\dot{q}_{it}|\) will be the decrease of each joint power. The sum decreases of the three joint output power can be expressed as the Equation (19). A smaller value of Equation (19) leads to a more output energy that affects the jump height directly. Using the linear programming, we obtain the optimum \(\Delta\ddot{q}_{t}\) required to minimize the cost function of Equation (19) while simultaneously satisfying the the set of inequalities composed by Equation (16) and Equation (18). Figure 8 shows the control system. In this program process, there is no joint constraint any more. The torque constraint is used to ensure that all three joint torques yield the torque speed curve profile. In addition, the ZMP constraint is used to keep the dynamic balance throughout the entire pass-off phase. From the control system, we can see that the initial state \(X_{0}\) plays an important role in the jump height. We used the traversal method to obtain the jump height for different \(X_{0}(q_{10}\in[-65^{\circ},\ -30^{\circ}],\)  
Figure 7: (a) the ZMP (zero moment point) limitation in the \(y\)-direction \(p_{ylim}\in(-0.05,\ 0.15)\) (m); (b) the \(\Delta p_{yt}\) added to \(p_{yt}\) to maintain the ZMP constraint. \(q_{20}\in[-180^{\circ},~{}180^{\circ}]\), \(q_{30}\in[-180^{\circ},~{}180^{\circ}]\), \(\dot{q}_{10}=0,\dot{q}_{20}=0,~{}\dot{q}_{30}=0\)). For various initial values of ankle angle \(q_{10}\in[-65^{\circ},~{}-30^{\circ}]\), we got the optimal \(q_{20}\), \(q_{30}\) and the jump height as shown in Figure 9a-c. With the increasing \(q_{10}\), the jump height \(h\) increases while \(q_{10}<-57^{\circ}\) and subsequently decreases. When compared with the original control method, the height \(h\) obtained using the proposed method is obviously greater. In particular, at \(X_{0}=(-57^{\circ},~{}158^{\circ},~{}-126^{\circ},~{}0,~{}0,~{}0)\), we obtained a maximum jump height of 0.61 (m). The jumping height is increased by nearly 70%. Additionally, we calculated the jumping direction \(\theta\) with  
\[\theta=atan2(\hat{z}_{take\,off},\dot{y}_{takeoff}). \tag{20}\]  
Figure 9d shows that the jumping direction \(\theta\) and the initial ankle angle \(q_{10}\) have an almost linear relationship. Therefore, we can almost control the jumping direction \(\theta\) via the initial ankle angle \(q_{10}\). Assuming that \(X_{0}=(-57^{\circ},~{}158^{\circ},~{}-126^{\circ},~{}0,~{}0,~{}0)\), Figures 10 and 11 show the torques of the three joints \(\tau_{i}\) and the ZMP \(p_{y}\), respectively, during the entire push-off phase. For most of the motion time, the ankle, knee, and hip joint torques are all equal to the maximal torques allowed by motors, which greatly increases the kinetic energy. At the beginning of the jump motion, we can see the effectiveness of the control system, which reduces the ankle joint torque slightly to satisfy the ZMP constraint.