## 2 Motion Model Selection
### Empirical Model  
Other than the exact model, a six-degree-of-freedom motion model was proposed for governing quadrotor in a distributed setup [59]. Later, it was reduced to four-degree-of-freedom motion model [60]. Furthermore, in [61], a 4-degree of freedom (DoF) motion was used for controlling several quadrotors in a distributed setup in which nonlinear model predictive control (NMPC) and model horizon estimation (MHE) are incorporated for relative tracking, where the relative motion model was defined as:  
\[\dot{\mathbf{x}}=\mathbf{f}_{c}(\mathbf{x},\mathbf{u},\psi_{z})= \begin{bmatrix} \dot{\mathbf{p}}_{x}\\ \dot{\mathbf{p}}_{y}\\ \dot{\mathbf{p}}_{z}\\ \dot{\psi}_{z}\end{bmatrix} = \begin{bmatrix}\mathbf{v}_{x}cos(\psi_{z})- \mathbf{v}_{y}sin(\psi_{z})-\bar{\mathbf{v}}_{x}+\mathbf{p}_{y}\bar{\psi}_{z} \\ \mathbf{v}_{x}sin(\psi_{z})+\mathbf{v}_{y}cos(\psi_{z})-\bar{\mathbf{v}}_{y}- \mathbf{p}_{x}\bar{\psi}_{z}\\ \mathbf{v}_{z}-\bar{\psi}_{z}\\ \dot{\psi}_{z}-\bar{\psi}_{z}\end{bmatrix} , \tag{1}\]  
where the function \(\mathbf{f}_{c}(\cdot):\mathbb{R}^{n_{x}}\times\mathbb{R}^{n_{x}}\times\mathbb{ R}^{n_{m}}\rightarrow\mathbb{R}^{n_{x}}\) and \(n_{x}=n_{u}=n_{ru}=4\). The current control input is given by \(\mathbf{u}=[\mathbf{v}_{x},\mathbf{v}_{y},\mathbf{v}_{z},\dot{\psi}_{z}]\), whereas relative control input \(\mathbf{u}_{ru}\) is denoted by \([\bar{\mathbf{v}}_{x},\bar{\mathbf{v}}_{y},\bar{\mathbf{v}}_{z},\bar{\psi}_{z}]\). \(\mathbf{x}=[\mathbf{p}_{x},\mathbf{p}_{y},\mathbf{p}_{z},\psi_{z}]\) is the state of the motion model, where \(\mathbf{p}_{i},i\in\{x,y,z\}\) is the position of the MAV in the world frame. \(\psi_{z}\) and \(\bar{\psi}_{z}\) denote the yaw angle or heading angle around the z axis and relative yaw angle, respectively. Derivative of \(\psi_{z}\) and \(\bar{\psi}_{z}\) are denoted by \(\dot{\psi}_{z}\) and \(\bar{\psi}_{z}\), respectively. \(\mathbf{v}_{i},i\in\{x,y,z\}\) denote the velocities on each direction, whereas \(\mathbf{p}_{i},i\in\{x,y,z\}\) gives the derivatives of \(\mathbf{p}_{i}\). Since discrete space was chosen for controlling the system, [PERSON] discrete model (1) was formulated as follows:  
\[\mathbf{x}^{+}=\mathbf{f}_{d}(\mathbf{x},\mathbf{u},\psi_{z})= \begin{bmatrix} p_{x}\\ p_{y}\\ p_{z}\\ \psi_{z}\end{bmatrix} +\delta \begin{bmatrix}v_{x}cos(\psi_{z})-v_{y}sin(\psi_{z})- \bar{v}_{x}+y\bar{\psi}_{z}\\ v_{x}sin(\psi_{z})+v_{y}cos(\psi_{z})-\bar{v}_{y}-x\bar{\psi}_{z}\\ v_{z}-\bar{\psi}_{z}\\ \dot{\psi}_{z}-\bar{\psi}_{z}\end{bmatrix} , \tag{2}\]  
where \(\delta\) is the sampling period and \(\mathbf{f}_{d}(\cdot):\mathbb{R}^{n_{x}}\times\mathbb{R}^{n_{u}}\times\mathbb{ R}^{n_{m}}\rightarrow\mathbb{R}^{n_{x}}\). \(\mathbf{f}_{c}\) and \(\mathbf{f}_{d}\) denote continuous and discrete dynamics, respectively. \(\mathbf{x}^{+}\) depicts the next state given the current state \(\mathbf{x}\). Subsequently, the motion model was simplified to 4-DoF for trajectory tracking for a quadrotor [62] (Equation (1)). In this trajectory-tracking approach, planning followed by the high-level controlling paradigm was applied. Such an approach was introduced because a simplified motion model is a reasonable choice for achieving real-time performance. Quadrotor state was defined as \(\mathbf{x}=[\mathbf{p}_{x},\mathbf{p}_{y},\mathbf{p}_{z},\dot{\psi}_{z}]^{T} \in\mathbb{R}^{n_{x}}\), whereas input to the system was given by \(\mathbf{u}=[\mathbf{v}_{x},\mathbf{v}_{y},\mathbf{v}_{z},\dot{\psi}_{z}]^{T} \in\mathbb{R}^{n_{u}}\). The simplified motion model was given by:  
\[\dot{\mathbf{x}}=\mathbf{f}_{c}(\mathbf{x},\mathbf{u})= \begin{bmatrix}\dot{ \mathbf{p}}_{x}\\ \dot{\mathbf{p}}_{y}\\ \dot{\mathbf{p}}_{z}\\ \dot{\psi}_{z}\end{bmatrix} = \begin{bmatrix}\mathbf{v}_{x}cos(\psi_{z})- \mathbf{v}_{y}sin(\psi_{z})\\ \mathbf{v}_{x}sin(\psi_{z})+\mathbf{v}_{y}cos(\psi_{z})\\ \mathbf{v}_{z}\\ \dot{\psi}_{z}\end{bmatrix} , \tag{3}\]  
where \(\mathbf{f}_{c}(\cdot):\mathbb{R}^{n_{x}}\times\mathbb{R}^{n_{u}}\rightarrow \mathbb{R}^{n_{x}}\) and \(n_{x}=n_{u}=4\).