# Controllable Height Hopping of a Parallel Legged Robot
## 3 Dynamic Model Analysis
### Integrated Dynamic Modeling  
In integrated dynamic model, we limited the planar robot movement in one- dimensional constraint. Based on the model shown in Figure 5a, we introduced the actuator model into the robot structure. In stance phase, the foot mass point is fixed on the floor, so the GRF can act on the CoM (center of mass) directly. Additionally, the horizontal ground force \(f_{x}\) is ignored in the following analysis (i.e., set \(f_{x}=0\)). Therefore, the force-torque equation can be rewritten as:  
\[\left[ \begin{array}{c}0\\ f_{z}\end{array} \right]=\mathbf{J}^{-T}\left[ \begin{array}{c}\tau_{a1}\\ \tau_{a2}\end{array} \right] \tag{2}\]  
where \(f_{z}\) is vertical ground force, \(\mathbf{J}\) is Jacobian matrix, and \(\tau_{ai}\) is \(T_{a}\) in (1), which can be represented in vector form as:  
\[\mathbf{\tau}_{a}=\mathbf{C}\mathbf{i}_{m}+\mathbf{I}\mathbf{\theta}+\mathbf{B}\mathbf{\theta}+\mathbf{f}_{S} \tag{3}\]  
where \(\mathbf{C}_{l}\), \(\mathbf{I}\), and \(\mathbf{B}\) are the corresponding coefficients in (1) and \(\mathbf{f}_{S}\) is the static frictional term relative to the force condition. Therefore, we set up a new variable, to represent the sum of two terms in (3):  
\[\mathbf{\tau}_{S}=\mathbf{C}\mathbf{i}_{m}+\mathbf{f}_{S} \tag{4}\]  
where the variable \(\mathbf{\tau}_{S}\) describes the actuator's output torque in a static situation. We consider this linear variable to motor current as an integral part in the following analysis, and, as shown in Section 5, it can be calibrated experimentally. Next, the robot movement is analyzed based on the global coordinates \((x,z)\) in Figure 5a. To describe the integrated model, we set a generalized coordinates of robot as:\(\mathbf{q}=[\theta_{1},\theta_{2}]^{T}\). In the stance phase, the entire robot is assumed to be fixed on the floor. We then use a dynamic equation with the following general expression:  
\[\frac{d}{dt}\bigg{(}\frac{\partial L}{\partial\dot{\mathbf{q}}}\bigg{)}-\frac{ \partial L}{\partial\dot{\mathbf{q}}}+\frac{\partial\Delta}{\partial\dot{\mathbf{q}}}= \frac{\partial\prod}{\partial\dot{\mathbf{q}}} \tag{5}\]  
f where \(L\) is the Lagrange function, \(\Delta\) is [PERSON]'s dissipation function, and \(\Pi\) is the virtual power of the system. In detail, the Lagrange function is expressed as:  
\[L=T-U=\frac{1}{2}M\dot{z}^{2}+\frac{1}{2}I_{1}\dot{\theta}_{1}^{2}+\frac{1}{2} I_{2}\dot{\theta}_{2}^{2}-Mgz \tag{6}\]  
where \(M\) is the total mass of the robot, \(I_{i}\) is the actuation module inertia (including equivalent link inertia), and \(z\) is the vertical position of the CoM. [PERSON]'s dissipation function is \(\Delta=\frac{1}{2}B_{1}\dot{\theta}_{1}^{2}+\frac{1}{2}B_{2}\dot{\theta}_{2}^ {2}\) and the virtual power of the system is \(\Pi=\dot{\theta}_{1}\tau_{S1}+\dot{\theta}_{2}\tau_{S2}\). Because in our experiment the robot only moves in the vertical direction, several constraints are given as:  
\[\mathbf{P}(\mathbf{q})+\left[ \begin{array}{c}0\\ z\end{array} \right]=0,\;\mathbf{I}\dot{\mathbf{q}}+\left[ \begin{array}{c}0\\ z\end{array} \right]=0 \tag{7}\]  
where \(P\) is the position of robot foot calculated by forward kinematics. Hence, the generalized coordinate \(\mathbf{q}\) can be simplified as a unidimensional coordinate. Specifically, in our robot model, while assuming that \(L_{1}=L_{2}\), we can transform the constraint equations as follows: \(\theta_{1}+\theta_{2}=\pi\) and \(\dot{\theta}_{1}=-\dot{\theta}_{2}\), with force constraints \(\tau_{S1}=-\tau_{S2}\). Therefore, the relationship between the Cartesian coordinate \(z\) and joint angle \(\theta_{1}\) can be deduced. Then, Equations (6) and (7) are substituted into (5), and we can ignore the joint inertia \(I_{i}\) because \(I_{i}\ll M\) numerically.