## 4 Optimization of These Two Attachment Geometries  
With fixed input energy from the hydraulic system, the higher hopping height implies a more efficient hopping process of the robot. To verify the effectiveness of the FMAG, the hopping heights of the robot body under the DAG and FMAG are optimized and compared. The objective function is formulated as  
\[H=\frac{v_{final}^{2}}{2g}+y_{0}\Big{(}\theta_{knee\_final}\Big{)} \tag{16}\]  
where \(v_{final}\) is the take-off speed of the robot body and \(\theta_{knee\_final}\) is the angle at the end of the take-off phase. The robot is set to hop from the maximum compression position, and the control valve opens from zero displacement to the maximum displacement under the maximum input voltage. Given the initial states as \([\theta_{knee}=40^{\circ},\ \dot{\theta}_{knee}=0,\ p_{1}=p_{T},p_{2}=p_{T}]\), the take-off phase can be simulated using the Runge-Kutta solver. The simulation process will be stopped when the ground reaction force decreases to zero. A commercial hydraulic cylinder is used in this paper. The supply pressure of the hydraulic system is set to 10 MPa. The control valve is a four-way proportional directional valve with 4 L/min flow rate at maximum valve opening and 1 MPa corresponding pressure drop. The parameters of the hydraulic system are estimated and listed in Table 2.  
Before the optimization, there are some geometry constraints that should be satisfied. According to the previous literature [48; 49; 50], the range of the knee joint of the kangaroo is more than 100\({}^{\circ}\). Therefore, the legged robot should have a similar motion range of the knee joint to mimic the motion of the kangaroo. To ensure the force transmission quality of the cylinder, the minimum transmission angle of the cylinder rod should be larger than 30\({}^{\circ}\). In the DAG, the equivalent lever length \(r\) should be constrained to escape the collision between the cylinder rod and the knee joint. Thus, the constraints of the DAG are given as  
\[\left\{ \begin{array}{l}\max\theta_{knee}\geq 140^{\circ}\\ r_{N}\geq r_{knee}+r_{cy2}+0.005\\ 30^{\circ}\leq\theta_{\angle KCH_{1}}\leq 150^{\circ}\end{array} \right. \tag{17}\]  
where \(r_{N}\) is the distance between the cylinder axis and the knee joint when the cylinder fully stretches, \(r_{knee}=0.005\) m and \(r_{cy2}=0.006\) m are the radius of the knee joint and cylinder rod, and 0.005 is set as the safe distance. Besides the constraints of the transmission angle and the range of the knee joint, there are more geometric constraints in the FMAG. The lengths of \(l_{AB}\), \(l_{BC}\), \(l_{CD}\), and \(l_{AD}\) should satisfy the motion of the four-bar mechanism. When the cylinder fully stretches, the distances between the knee joint and linkages \(CD\) and \(AD\) as \(h_{2N}^{\prime}\) and \(h_{3N}^{\prime}\) should be constrained to escape the collision. The distances \(h_{4}^{\prime}\) and \(h_{5}^{\prime}\) should also be considered to \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Parameters** & **Value** & **Parameters** & **Value** \\ \hline \(\beta_{\varepsilon}\) (MPa) & 700 & \(l_{cy\_init}\) (m) & 0.256 \\ \(V_{10}\) (m\({}^{3}\)) & \(4.2\times 10^{-5}\) & cylinder stroke (m) & 0.07 \\ \(V_{20}\) (m\({}^{3}\)) & \(5\times 10^{-5}\) & \(A_{1}\) (m\({}^{2}\)) & \(3.14\times 10^{-4}\) \\ \(k_{q}\)(m\({}^{3}\)/\(\left(\mathrm{s}\sqrt{\mathrm{MPa}}\right)\)) & \(6.64\times 10^{-6}\) & \(A_{2}\) (m\({}^{2}\)) & \(2.01\times 10^{-4}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Parameters of the hydraulic system. avoid the collision between joint \(D\) and the thigh and shank. Therefore, the constraints of the FMAG are given as  
\[\left\{