## 3 Dynamic Model and Control  
### Air Dynamics Modeling and Ground Dynamics Modeling  
The position of the robot in the world coordinate system is \(p_{w}=(x,y,z)^{T}\) and the attitude angle is \(\Omega=(\varphi,\theta,\psi)^{T}\). The joint angle of the two servos is \(\eta=(\eta_{1},\eta_{2})^{T}\). The rotation matrix \(R_{b}^{w}\) between \(O_{w}\) and \(O_{b}\) is denoted as:  
\[R_{b}^{w}=\left[ \begin{array}{ccc}c\varphi c\theta&c\psi s\theta s\varphi-s \varphi c\varphi&c\psi s\theta c\varphi+s\psi s\varphi\\ s\varphi c\theta&s\psi s\theta s\varphi+c\varphi c\varphi&s\psi s\theta c \varphi-c\psi s\varphi\\ -s\theta&c\theta s\varphi&c\theta c\varphi\end{array} \right] \tag{15}\]  
where \(c\theta\) denotes \(cos(\theta)\) and \(s\theta\) denotes \(sin(\theta)\). \(c\varphi\) denotes \(cos(\varphi)\) and \(s\varphi\) denotes \(sin(\varphi)\). \(c\psi\) denotes \(cos(\psi)\) and \(s\psi\) denotes \(sin(\psi)\). \[ \begin{array}{c}\phi,\theta\in(-\pi/4,\pi/4)\\ \psi\in(-\pi,\pi]\\ \eta_{1},\eta_{2}\in[-\pi/2,\pi/2]\end{array} \]  
The dynamics model of the robot can be represented using the Euler-Lagrange equations. \[\text{M}(q)\ddot{q}+C(q,\dot{q})\dot{q}+G(q)=B\tau \tag{16}\]  
where \(q=[{p_{w}}^{T},\Omega^{T},\eta^{T}]^{T}\) is the vector containing all the generalized coordinate variables. \(M(q)\) is the inertia matrix, \(C(q,\dot{q})\) is the Coriolis matrix, \(G(q)\) is the gravity vector, \(\tau\) is the generalized moment input, and \(B\) is the input transformation matrix. The inertia matrix \(M(q)\) is shown as follows [24]:  
\[\text{M}(q)=M_{t,b}^{T}m_{b}M_{t,b}+M_{r,b}^{T}RI_{b}R^{T}M_{r,b}+\sum_{i=1}^{ 2}M_{t,i}^{T}m_{i}M_{t,i}+M_{r,i}^{T}(RR_{i})I_{i}(RR_{i})^{T}M_{r,i} \tag{17}\]  
where \(M_{t,b}=\frac{\dot{p}}{q},M_{r,b}=\frac{\omega}{q},M_{t,i}=\frac{\dot{p}}{q},M_{r,i}=\frac{\omega_{i}}{q}\) are the matrices that form a velocity relationship. \(\dot{p}\) and \(\omega\) are the translational and angular velocity of body in the inertial frame. \(\dot{p}_{i}\) and \(\omega_{i}\) are the translational and angular velocity of \(i\)th joint in the inertial frame. \(m_{b}\) and \(m_{i}\) are the mass of the body and the \(i\)th joint. \(I_{b}\) and \(I_{i}\) are the rotational inertia of the body and the \(i\)th joint. \(R=R_{b}^{w}\). \(R_{i}\) denotes the rotation matrix from the body frame to the \(i\)th joint. Figure 9: Influence of the support bar angles \(\theta_{\text{s}}\) to the support force \(N_{2}\). The Corillis matrix \(C(q,\dot{q})\) is shown as follows [24; 25]:  
\[c_{kj}=\sum_{i=1}^{8}\frac{1}{2}\Bigg{\{}\frac{\partial m_{kj}}{\partial q_{i}}+ \frac{\partial m_{ki}}{\partial q_{j}}-\frac{\partial m_{ij}}{\partial q_{k}} \Bigg{\}}\dot{q}_{i} \tag{18}\]  
where \(c_{kj}\) indicates the element of \(C(q,\dot{q})\) at the \(k\)th row and \(j\)th column. \(m_{ab}\) indicates the element of \(M(q)\) at the ath row and bth column. The gravity vector \(G(q)\) is shown as follows:  
\[\left\{ \begin{array}{l}G(q)=\frac{\partial U}{\partial q}\\ U=m_{b}ge_{g}^{T}p_{w}+\sum\limits_{i=1}^{2}m_{i}ge_{g}^{T}(p_{w}+R_{i}p_{i}^{ b})\end{array} \right. \tag{19}\]  
where \(e_{g}\) is the unit vector \(\left[ \begin{array}{cc}0&0&1\end{array} \right]^{T}\). \(p_{i}^{b}\) denotes the position of the \(i\)th joint in the body coordinate system. \(U\) is the potential energy of the robot. Although the Euler-Lagrange Equation (16) can describe the motion of the robot, the accumulated dynamic model is too complicated to design an efficient controller. Because the range of motion of the joints during adsorption is limited, the joint firstly needs to be adjusted to the preparation state and the mass of the joints is small compared to the rest of the robot. Thus, we decouple the joint and the robot body, use the motion at the end of the suction cup as a perturbation to the robot body, and improve the efficiency of the control. The rotor lift is expressed by the following equation.