# Inverse Kinematics of a Serial Manipulator with a Free Joint for Aerial Manipulation
## 1 Introduction  
Aerial manipulation has emerged as a critical research domain, with applications spanning industrial maintenance, inspection, and assembly tasks in challenging environments. Examples of application of these systems include the inspection of infrastructure like wind turbines, nuclear facilities, and bridges, as well as archaeological site exploration and sensor deployment in hard-to-reach locations [1]. These tasks leverage the maneuverability of Unmanned Aerial Vehicles (UAVs) equipped with robotic manipulators, collectively referred to as aerial manipulators or Unmanned Aerial Manipulators (UAMs). UAMs combine the capabilities of UAVs, which provide mobility, with robotic arms that enable physical interaction with the environment. The most common UAV configuration is a multicopter, due to its hovering capabilities and high maneuverability inthree-dimensional space. These platforms are often coupled with serial robotic arms for enhanced workspace and dexterity, making them suitable for complex tasks such as grasping, assembly, and inspection [2]. Despite their potential, UAMs face significant challenges, particularly because the UAV base is not fixed and is subject to external disturbances, such as wind gusts. These disturbances complicate the control of the UAM and can compromise the precision of the manipulator's end-effector positioning. Additionally, the dynamic coupling effect between the manipulator and the UAV base further exacerbates these issues. The movement of the manipulator generates reaction forces and torques that destabilize the UAV, affecting its attitude and position [3]. Two main control approaches have been explored to address these challenges: coupled and decoupled control. In the coupled approach, the entire dynamic system, including the UAV base and the manipulator, is modeled and controlled as a single entity [4]. Decoupled control, on the other hand, treats the UAV and the manipulator as separate systems, relying on robust controllers or disturbance observers to compensate for interaction forces [5]. A third approach, known as the partially coupled strategy, combines elements of both: the UAV and manipulator are controlled independently, but interaction forces are estimated or measured to improve coordination. This strategy leverages additional sensing to mitigate disturbances while maintaining computational simplicity [6; 7]. Initial research on free-flying robotic systems, originally conducted for space robotics applications, has laid a robust foundation for methodologies that are highly applicable also to aerial manipulation. The use of these methods with UAMs suggests a promising pathway for addressing challenges in this field. Researchers have came up with methodologies such as the Generalized Jacobian matrix to develop inverse kinematics control schemes for floating-base manipulators [8; 9]. These strategies have proven instrumental in enabling precise positioning for floating-base systems and have been effectively adapted to manage the challenges of dynamic coupling in aerial platforms [10]. The Generalized Jacobian approach was further developed in [6], where it was extended for aerial manipulation in order to consider also gravity and UAV control forces. Another possibility that was explored to counter the dynamic coupling effect was to employ redundant arms. For example, ref. [11] presents a method for the solution of the inverse kinematics that leverages redundancy to locally minimize the torque exerted on the base by the manipulator. Closed-Loop Inverse Kinematics (CLIK) [12] algorithms incorporate a feedback loop in the solution of the inverse kinematics of manipulators, improving the trajectory tracking capabilities even in the presence of external disturbances. These algorithms were applied to aerial manipulators in [13; 14]. In [15], CLIK approach for UAMs is further developed to operate at the acceleration level, addressing real-world disturbances like wind and sensor noise.