## 1 Introduction
However, adapting this approach from space to aerial manipulators requires several modifications to the inverse kinematic control algorithm, as aerial manipulators do not operate in an environment free of external forces. In this research, we utilize the generalized Jacobian and extended Jacobian [18] approaches (i) to track the desired end-effector trajectory even if the UAV moves in the 3D space, and (ii) to minimize the dynamic disturbances transferred to the UAV while the manipulator completes the desired tracking/picking task, in the case of a kinematically redundant arm. The UAV is hovering during the above-mentioned maneuvers. We also consider the change in momentum resulting from external forces on the system such as gravity and UAV controller forces in the inverse kinematic control computation. This is a novel contribution required to adapt the generalized Jacobian formulation from its original assumption of conservation of momentum. The use of an extended generalized Jacobian for trajectory tracking and simultaneous reaction torque minimization in aerial manipulation is another novel contribution. The proposed algorithms are validated in simulation for two types of task: trajectory tracking and load picking. The UAV base is a multicopter, while the arm is a 2- or 3-degree-of-freedom (DOF) serial link arm. The motions of the base and manipulator are restricted to the 2D plane to simplify the problem and better compare the impact of different Jacobian formulationsin terms of end-effector position error, UAV base translations, and reaction torques. We consider three different Jacobian formulations:  
1. Generalized Jacobian for an ideal-trolley model (Section 4): the UAV base controller maintains perfect hover conditions vertically and rotationally, with only base horizontal translations permitted. A linear trajectory tracking task is completed by a 2-DOF serial link manipulator utilizing a simplified generalized Jacobian formulation. This provides an initial exploration of the problem and demonstrates some of the limitations of purely using the generalized Jacobian for inverse kinematic control without inclusion of external forces. 2. Generalized Jacobian with external forces (Section 5): the Jacobian includes the impact of all base motions, and the inverse kinematics control equation takes into account the external forces and torques such as gravity and UAV controller forces. Both 2-DOF (nonredundant) and 3-DOF (redundant) serial link manipulator cases are simulated for the trajectory tracking tasks outlined in Section 3. 3. Extended generalized Jacobian (Section 6): the generalized Jacobian is extended to include an additional constraint, which is the minimization of the reaction torques on the UAV caused by the manipulator motion for a redundant 3-DOF serial link manipulator. We validate this algorithm with two trajectory tracking tasks as outlined in Section 3. Section 2 contains the background for the proposed algorithms, including the dynamic model for planar UAV motions, a general outline of inverse kinematic algorithms for manipulator joint control, and the generalized Jacobian formulation. Section 3 describes the simulation setup, including the trajectory tasks and the kinematic and dynamic parameters used. Sections 4-6 derive the Jacobian formulations and show results for the trajectory tracking tasks. Section 7 considers the scenario where the end-effector is picking a load and compares the generalized Jacobian and extended Jacobian formulations for completing this task. Computational times of the inverse kinematics solution through the generalized and extended generalized Jacobian were measured on a machine featuring an Intel i7 eighth generation exa-core (12 thread) processor with 16 GB RAM, SSD mass storage and using Windows 11 and MATLAB version 2022a. The measured computational times are mentioned in the corresponding Simulation Results subsections.