# Aerial Physical Interaction in Grabbing Conditions with Lightweight and Compliant Dual Arms
## 4 Control  
### Definition of the Control Task  
As stated in the introduction, a dual arm system allows the realization of aerial manipulation tasks in grabbing conditions, using one arm for grabbing and for estimating the position of the aerial platform relative to the grabbing point (Section 3.2), whereas the other takes care of conducting the task, for example, the installation of a sensor device [21]. Since the grabbing arm is actuated and mechanically compliant, it is possible to estimate and control the forces and torques acting over the manipulator from the deflection of the joints [17; 21]. The idea is that the arm helps the multirotor to reach the desired position when it is disturbed by an external force, exerting a pushing/pulling force in the opposite direction of the position error, as Figure 6 illustrates.  
Therefore, the grabbing arm can be used in two ways:  
* As relative position sensor, with zero torque control, so the reaction wrenches induced over the multirotor are relatively small.
* As an active impedance link, exerting a controlled force over the aerial platform to compensate external forces and guide the multirotor towards the reference position.  
In the first case, the joints of the grabbing arm implement a PI (proportional-integral) controller to maintain a zero deflection (torque) reference, acting over the servo position as follows:  
\[\theta_{j,ref}^{2}=\theta_{j}^{2}+\left(K_{p}\Big{(}\theta_{j}^{2}-q_{j}^{2} \Big{)}+K_{i}\int\Big{(}\theta_{j}^{2}-q_{j}^{2}\Big{)}dt\right) \tag{14}\]  
Here, \(\theta_{j}^{2}\) is the position of the \(j\)-th servo of the right arm, whereas the term on the right side is the incremental position correction. The proportional and integral gains, \(K_{p}\) and \(K_{i}\), can be tuned experimentally, taking into account the nominal values of the deflection (\(\approx\)5 degrees). Now, we impose that the nominal operation position of the aerial robot relative to the grabbing point is the L-shaped configuration of the arm, since this is far enough from the joint limits and the kinematic singularities (although any other could be considered):  
\[\mathbf{z}_{TCP}^{ref}=\left[\begin{array}{c}L_{2}\\ 0\\ -L_{1}\end{array}\right]=F\mathbf{K}\Big{(}q_{ref}^{2}\Big{)}=F\mathbf{K} \left(\left[\begin{array}{c}0\\ 0\\ -\pi/2\end{array}\right]\right) \tag{15}\]  
The position deviation of the aerial platform is then defined as the displacement of the TCP of the grabbing arm w.r.t. the reference position. That is,  
\[\epsilon=\left[\begin{array}{cc}\varepsilon_{X}&\varepsilon_{Y}&\varepsilon_ {Z}\end{array}\right]^{T}=F\mathbf{K}\Big{(}q_{ref}^{2}\Big{)}-F\mathbf{K} \Big{(}q^{2}\Big{)} \tag{16}\]  
Note that the maximum deviation is limited by the reach of the arm, \(\|\epsilon\|<L_{1}+L_{2}-\sqrt{L_{1}^{2}+L_{2}^{2}}\). If the influence of the grabbing arm over the attitude controller is relatively low due to the zero-torque  
Figure 6: Model considered in grabbing conditions. Nominal operation pose (left), and displacement due to external force (right). The grabbing arm compensates the disturbance exerting a reaction force.  
controller, this measurement can then be taken as input by the position controller of the multirotor platform, as represented in upper part of Figure 7, replacing the GPS (Global Positioning System) used for navigating to achieve better accuracy during the manipulation phase.