## 5 Safe and Accurate Aerial Manipulation
### Control Methods for Aerial Manipulation
#### 5.2.1 Dynamics and Actuation Properties of Multirotor UAVs  
Control strategies for aerial manipulators strictly depend on their controllable space. In this work, we mainly focus on rotary-wings UAVs. The generalized dynamics of these platforms can be expressed as follows: [38]:  
\[\begin{bmatrix}m\mathbf{I}^{3\times 3}&\mathbf{0}^{3\times 3}\\ \mathbf{0}^{3\times 3}&\mathbf{J}\end{bmatrix}\begin{bmatrix}\tilde{\mathbf{p}} \\ \dot{\omega}\end{bmatrix}=-\begin{bmatrix}mg\mathbf{e}_{3}\\ \omega\times\mathbf{J}\omega\end{bmatrix}+\begin{bmatrix}W\mathbf{R}_{ \mathbf{B}}&\mathbf{0}^{3\times 3}\\ \mathbf{0}^{3\times 3}&\mathbf{I}^{3\times 3}\end{bmatrix}u \tag{4}\]  
Figure 8: Cartesian-space trajectory of the planar aerial manipulator to the target force application point.  
Figure 7: Error signals for the controlled end effector of the planar aerial manipulator. The dashed vertical line points out the time when the position controller is switched to force controller.  
where \(m\) is the mass of the UAV, \(\mathbf{J}\in\mathbb{R}^{3\times 3}\) is the inertia tensor of the airframe, \(\mathbf{p}\in\mathbb{R}^{3}\) is the position of the UAV with respect to the world frame, \(\omega\in\mathbb{R}^{3}\) is the angular velocity of the UAV, \({}^{W}\mathbf{R_{B}}\in\mathcal{SO}(3)\) is the orientation of the airframe with respect to the world frame, \(\mathbf{0}^{3\times 3}\) is the zeros matrix of the corresponding dimensions, \(\mathbf{I}^{3\times 3}\) is the identity matrix of the corresponding dimensions, \(g\) is the gravity acceleration, \(\mathbf{e}_{i}\) is the canonical basis vector of \(i=1,2,3\) and \(u\in\mathbb{R}^{6}\) is the control vector in the generalized coordinates. For UAVs with passively or actively tilting rotors, the control vector is statically mapped from the rotors thrust, using the following:  
\[u=\mathbf{F}(\alpha,\varsigma,L,\sigma)f_{r} \tag{5}\]  
where \(f_{r}\in\mathbb{R}^{2j}\) is the vector containing the vectorized thrusts supplied by the \(j\) rotors [39], and \(\mathbf{F}(\alpha,\varsigma,L,\sigma)\in\mathbb{R}^{6\times 2j}\) is the allocation matrix that depends on the following:  
* The vector containing the tilting angles of the \(j\) rotors about their respective \(x\)-axis \(\alpha=\left[\alpha_{1},\alpha_{2},...,\alpha_{j}\right]\).
* The offset angle depending on the \(x_{B}\) axis of the UAV (for example \"X\" or \"+\" quadrotor configurations) \(\varsigma\).
* The length of the arms of the UAV \(L\).
* The relation between the rotor torque constant \(k_{\tau}\) and the rotor force constant \(k_{f}\)\(\sigma=k_{\tau}/k_{f}\).  
For fully actuated UAVs, i.e., those that are capable of moving in 6D with its number of actuators, there exists the allocation pseudoinverse \(\mathbf{F}(\alpha,\varsigma,L,\sigma)^{\dagger}\) such that \(f_{r}=\mathbf{F}(\alpha,\varsigma,L,\sigma)^{\dagger}u:\mathbb{R}^{6}\to \mathbb{R}^{2j}\). Therefore, the velocity of each rotor can be computed as follows [40]:  
\[\omega_{r_{i}}=\sqrt{\frac{\sqrt{f_{r_{i}}+f_{r_{j+i}}}}{k_{f}}},\;i=1,2,...,j \tag{6}\]  
The tilting angles to exert the desired thrust are given by the following [40]:  
\[\alpha_{i}=atan2(f_{j+i},|f_{i}|),\;i=1,2,...,j \tag{7}\]  
For the case when the rotors of the UAV are not able to tilt, i.e., \(\alpha=0\forall t\), the UAV becomes underactuated since the thrust of the rotors can be only exerted, in a first stage, to control the orientation and the \(z\) axis of the airframe; hence, a cascade control is required to lead the UAV to a desired Cartesian position.