## 2 Background Theory  
### Unmanned Aerial Manipulator Dynamics and Control Model  
In this research, we select a fixed-rotor multicopter configuration as our UAV base, with the manipulator as a serial multiple link arm mounted on the base. We model the unmanned aerial manipulator (UAM) as a dynamically coupled system, i.e., the base and manipulator are regarded as two distinct systems, but the motion of either system affects the other, in addition to the external forces and torques acting on the whole system. The UAV is modeled as a rigid body subjected to net thrust/attitude torques from the propellers and gravity. The UAV controller tries to keep altitude and inclination angle in hover conditions (i.e., maintaining elevation and minimizing attitude changes). We model the UAV and manipulator dynamics separately, taking into account the interaction forces, and employ separate control algorithms: (1) PID control to maintain the UAV in hover conditions and (2) inverse kinematic control of the manipulator to perform the desired task. The reference frames for the UAV are shown in Figure 1, where \(I\) denotes the inertial frame and \(B\) is the UAV body-fixed frame. We use the conventional simplification for the controller forces and torques for UAVs, where the body of the UAV moves vertically along \(z_{B}\) through a net thrust force and rotates about all three local body axes \(x_{B}\), \(y_{B}\), and \(z_{B}\) through attitude torques. In a real-world application, a low-level controller calculates the rotor speeds that result in these control forces and torques. The general dynamic equations relating rotor speeds and simplified thrust/attitude control inputs for a multicopter system in 3D are summarized in [12]. In our analysis, we only consider UAM translations in the 2D plane \(x_{I}z_{I}\) and rotation about the \(y_{I}\)-axis orthogonal to that plane. The control actions on the UAV base in the \(x_{I}z_{I}\) plane are simplified into a vertical thrust force \(U_{1}\) and roll moment \(U_{2}\). The mechanical design of a fixed-rotor multicopter does not permit horizontal displacement to be controlled independently, hence horizontal motion is controlled by applying a combination of thrust force and roll moment on the UAV. We model the UAV center of gravity (CoG) in the inertial frame with the Cartesian position in the 2D \(x_{I}z_{I}\) plane \([x,z]^{T}\) and the UAV rotation with the roll angle \(\phi\) about the UAV body \(y_{B}\)-axis (centered at the CoG) parallel to the inertial \(y_{I}\)-axis. The net thrust force \(U_{1}\) acts directly upwards from the UAV CoG at the roll angle \(\phi\). \(U_{2}\) is the net roll moment about the UAV CoG. With \(m_{B}\) as the mass and \(I_{B}\) as the rotational moment of inertia about the CoG, the UAV dynamic equations are the following:  
\[ \begin{array}{c}m_{B}\ddot{x}=-U_{1}\sin(\phi)\\ m_{B}\ddot{z}=-m_{B}g+U_{1}\cos(\phi)\\ I_{B}\ddot{\phi}=U_{2}\end{array}  \tag{1}\]  
The manipulator is mounted on the base with an offset vector \(r_{m}\) with respect to the UAV CoG, and the manipulator motion exerts reaction forces along the \(x_{I}\) and \(z_{I}\) axes, \(R_{x}\) and \(R_{z}\), as well as a reaction torque \(\tau_{m}\) on the interface point. The UAM model is illustrated in Figure 2. In this paper, we focus on tracking a desired end-effector trajectory while the UAV maintains hover, and on locally minimizing the manipulator reaction torques while the manipulator executes a task. Figure 1: Illustration of reference frames for the UAV system (DJI S1000) [12]. Figure 2: Coordinates and forces of the UAM. UAV forces/torques are in red and manipulator reaction forces/torques are in orange. The UAV is shown in blue and manipulator in green. Indeed, as demonstrated in [12], a torque transferred to the UAV CoG causes an undesired large horizontal displacement of the UAV, which in turn jeopardizes the end-effector precision.