## 8 Conclusions  
In this research, we proposed two control methods based on the generalized Jacobian to solve the inverse kinematics of an aerial manipulation and to minimize the UAV base reaction torque and, therefore, its horizontal motion. The main novelty of this work is related to the application of differential kinematics methods to a UAM through the inclusion of external forces and torques acting on the system in the inverse kinematics solution, and to the minimization of the torque disturbance transferred from the manipulator to the UAV. In our case, external forces and torques include weight, UAV control inputs, and forces exerted from the load to be picked to the end-effector. The method can also be extended to other cases where the forces can be modeled, for example to the case of interaction forces with the environment. The effectiveness of both methods was confirmed by the capability of tracking a linear and a circular trajectory, and in a load picking test case. In the extended generalized Jacobian case, the task was carried out with zero reaction torque transferred from the manipulator to the UAV. This is particularly important since the horizontal translation of the UAV is almost zero and, therefore, the aerial manipulator can safely operate in the proximity of walls. For the generalized Jacobian with external forces method, the base has noticeable rotations and large horizontal translations during task execution due to the UAV dynamics and the UAV controls used. By including an additional task (null manipulator reaction torque), UAV translations are minimized in the case of the extended generalized Jacobian. The simulated cases showed that using the extended generalized Jacobian method significantly reduces the horizontal translation of the base, however, this is achieved by reducing the available workspace. This is because the main contribution to the torque disturbance on the UAV is the weight of the manipulator, so it is constrained to move reducing the horizontal translation of its center of gravity. Our MATLAB implementation runs faster than real time, which indicates that the algorithm can be implemented in a real-world system with minimal optimization modifications required. The computational times could be significantly reduced by implementing the algorithm in a compiled programming language (such as C++), which would also enable inverse kinematic control with more complex models of the UAV and manipulator. Future extensions to this research will be to conduct real-world experiments on a UAM in a planar control scenario. To implement this algorithm on a physical aerial manipulator, we require precise estimation/measure of the UAV pose, end-effector position, joint positions, and contact forces. This will require integration of sensors such as IMUs, force-sensors, and vision systems such as Vicon. Although the algorithms outlined in this research are validated in 2D, this approach can easily be generalized to 3D systems and different types of manipulators. To address the problem of reduced workspace while minimizing the disturbance torques in the extended generalized Jacobian, we can also investigate the feasibility of adding a controlled counterweight for torque balancing. This would require further modifications to the dynamic modeling of the UAM as well as the inverse kinematic control algorithm. Another topic of research that will be part of future work is the investigation of different solution methods, e.g. using constrained least squares, which would allow for constraints on joint positions and velocities and faster computational times. Conceptualization, S.C.; methodology, S.C. and A.P. ; software, A.P. ; investigation, S.C., A.P. and Y.V. ; writing--original draft preparation, A.P. and Y.V. ; writing--review and editing, S.C., A.P. and Y.V. ; supervision, project administration, funding acquisition, S.C. All authors have read and agreed to the published version of the manuscript.