# Inverse Kinematics of a Serial Manipulator with a Free Joint for Aerial Manipulation
## 6 Conclusions  
In this paper, a novel approach for the solution of the inverse kinematics of a serial manipulator with a floating base and a non-actuated joint was proposed. The method is based on the differential Kinematics resolved at the acceleration level. The motion of the base is calculated and compensated for by exploiting the Generalized Jacobian. The equations of motion are used to take into account the presence of external forces acting on the system; moreover, these are used to relate the motion of the non-actuated revolute joint with that of the actuated ones. The equations were generalized to redundant serial manipulators and characterized to the case with fixed base.  
The method was tested in Matlab for a planar aerial manipulator. In order to reduce the torque disturbance on the UAV, the non-actuated joint was placed between the UAV and the robotic arm. Two tasks were considered: the tracking of a trajectory with the end-effector, and the picking of a payload from ground; both the tasks were executed with a non-redundant and with a redundant manipulator. The novel approach of this paper showed small horizontal displacement of the UAV due to the reduction in torque disturbance. In particular, the horizontal displacement of the UAV was comparable or smaller than the vertical displacement, while it can be much higher without the minimization of the disturbance (as shown in [6]). The effectiveness of the method for a manipulator attached to a fixed base through a non-actuated joint was also tested. Our method was able to follow the desired trajectory with the advantage of a reduced torque on the base.  
A possible development of this work is the inclusion of constraints (such as joint limits) in the solution of the inverse kinematics, for instance, by defining a constrained least squares problem. Moreover, environmental disturbances that cannot be modeled (like wind gusts) or modeling errors (for instance in the contact force) were not considered in this work, and CLIK algorihtms can be developed to counteract the effect of these disturbances. Finally, in real applications to aerial manipulators the method could be integrated with sensors (such as UAV's IMUs or vision systems) to estimate the state of the UAV.  
Figure 21: Load picking for the 4-DOF manipulator attached to the UAV with a damper. (**left**) UAV rotation and (**right**) translation during the task.  
**Author Contributions:** Conceptualization, S.C.; methodology, S.C. and R.Z.; software, R.Z. and A.P.; formal analysis S.C., A.P. and M.P.; investigation, S.C. and R.Z.; writing-original draft preparation, A.P. and M.P.; writing--review and editing, S.C., A.P. and M.P.; and supervision, project administration, and funding acquisition, S.C. All authors have read and agreed to the published version of the manuscript.  
**Funding:** This work was carried out in the framework of the project \"DynAeRobot--Development and validation of a new dynamically balanced aerial manipulator\" (project no. BIRD213590), funded under the BIRD 2021 program promoted by the University of Padova.  
**Institutional Review Board Statement:** Not applicable.  
**Informed Consent Statement:** Not applicable.  
**Data Availability Statement:** The original contributions presented in the study are included in the article, and further inquiries can be directed to the corresponding author.  
**Conflicts of Interest:** The authors declare no conflicts of interest.