## 6 Extended Generalized Jacobian for a 3-DOF Redundant Manipulator with Full Base Movement
### Simulation Results for Trajectory Tracking  
We evaluated the reduction of torque disturbance for the linear and circular trajectories, we focused on the horizontal translation of the UAV base and on the ability to track the desired trajectory. We solved the inverse kinematics for the linear and circular trajectories in MATLAB by the procedure shown in Section 6.1 with a timestep of \(\Delta t=1\times 10^{-3}\) s. As in previous cases, we imposed the resulting joint velocities in the ADAMS model of the UAV to perform the dynamic simulation of the task operations.  
Initially, we considered the linear trajectory: the position error is below \(8\times 10^{-5}\) m and the UAV rotation is smaller than \(5\times 10^{-6}\) rad. Compared to the generalized Jacobian with external forces (Section 5.2), the torque disturbance is decreased by four orders of magnitude. The manipulator moves while minimizing the horizontal distance between the UAV and manipulator centers of gravity. This results in a drawback of the method since the workspace is reduced. The vertical UAV translation is not visibly affected by the introduction of the extended Jacobian. Figures 17 and 18 show the stroboscopic trajectories, joint angles, UAV translations, UAV rotation, and reaction torque.  
For the circular trajectory, the end-effector position error is again smaller than \(12\times 10^{-5}\) m, the base translation is mainly vertical since it reaches 9.5 mm in the \(y_{I}\) direction and it is smaller than 1 mm in the \(x_{I}\) direction. The torque disturbance transmitted from the manipulator to the UAV and UAV rotation have the same order of magnitude as for the linear trajectory. The stroboscopic trajectories, joint angles, UAV translations and rotation, and torque disturbance are shown in Figures 19 and 20.  
Figure 17: Stroboscopic trajectories for the first half (**left**) and second half (**right**) of the linear trajectory task for the 3-DOF redundant UAM using the extended Jacobian. The UAV base is the large green marker, joints are the black circles, links are the blue lines, and end-effector positions tracked are the red crosses.  
Figure 18: Manipulator (**top**) and UAV motions and torque disturbance (**middle** and **bottom**) for the linear trajectory task with the 3-DOF UAM using the extended Jacobian.  
Figure 19: Stroboscopic trajectories for the first half (**left**) and second half (**right**) of the circular trajectory task for the 3-DOF redundant UAM using the extended generalized Jacobian. The UAV base is the large green marker, joints are the black circles, links are the blue lines, and end-effector positions tracked are the red crosses.  
Figure 20: Manipulator (**top**) and UAV motions and torque disturbance (**middle** and **bottom**) for the circular trajectory task with the 3-DOF UAM using the extended generalized Jacobian.  
By using the extended generalized Jacobian, the computational time for the tracking of the linear and circular trajectories (with time duration of 5 s) by the 3-DOF manipulator is 3.89 s and 3.88 s, respectively. This result is comparable to the generalized Jacobian method computational speed and faster than the trajectory duration of 5 s.