## 3 UAV Kinodynamic Models
### Kinodynamic Model 2  
The rotational rates of the UAV around its rigid body and its time constant on horizontal movement in 2D space can be ignored regarding the sampling time of the trajectory design [50; 53]. Therefore, with the notation of (1), \(\mathcal{DM}\) in (5) can be written in the following forms in the state space domain [36]:  
\[\mathcal{K}\mathcal{M}2_{1}:=\left\{\begin{array}{c}\underbrace{\begin{bmatrix} \tilde{\xi}\\ \tilde{v}\\ \tilde{u}\end{bmatrix}}_{\tilde{X}_{2}}=\underbrace{\begin{bmatrix}0_{3}&I_{3} \\ 0_{3}&0_{3}\end{bmatrix}}_{A_{2}}\begin{bmatrix}\tilde{\xi}\\ \tilde{v}\\ \end{bmatrix}}_{X_{2}}+\underbrace{\begin{bmatrix}0_{3}\\ H_{1}\end{bmatrix}}_{B_{2}}\left(\underbrace{\begin{bmatrix}u_{3}\\ u_{2}\\ u_{1}\end{bmatrix}-\begin{bmatrix}0\\ 0\\ m\mathbf{g}\end{bmatrix}}_{\text{\tiny{$\mathfrak{A}$}}}\right)\end{array} \right. \tag{12}\]  
\[\mathcal{K}\mathcal{M}2_{2}:=\left\{\begin{array}{c}\underbrace{\begin{bmatrix} \tilde{\xi}\\ \tilde{v}\\ \tilde{u}\\ \tilde{u}\end{bmatrix}}_{\tilde{X}_{3}}=\underbrace{\begin{bmatrix}0_{3}&I_{3} &0_{3}\\ 0_{3}&0_{3}\end{bmatrix}}_{A_{3}}\underbrace{\begin{bmatrix}\tilde{\xi}\\ 0_{3}\\ a\end{bmatrix}}_{X_{3}}+\underbrace{\begin{bmatrix}0_{3}\\ 0_{3}\\ H_{1}\end{bmatrix}}_{B_{3}}\left(\underbrace{\begin{bmatrix}u_{3}\\ u_{2}\\ u_{1}\end{bmatrix}-\begin{bmatrix}0\\ 0\\ m\mathbf{g}\end{bmatrix}}_{\text{\tiny{$\mathfrak{A}$}}}\right)\end{array} \right. \tag{13}\]  
where \(0_{3}\in\mathbb{R}^{3\times 3}\) and \(I_{3}\in\mathbb{R}^{3\times 3}\) are 3-by-3 zero and identity matrices, respectively; \(X_{2}\in\mathbb{R}^{6}\) and \(X_{3}\in\mathbb{R}^{9}\) denote state vectors; \(A_{2}\in\mathbb{R}^{6\times 6}\) (\(A_{3}\in\mathbb{R}^{9\times 9}\)) and \(B_{2}\in\mathbb{R}^{6\times 3}\) (\(B_{3}\in\mathbb{R}^{9\times 3}\)) denote state and input matrices, respectively; and \(\mathfrak{A}\in\mathbb{R}^{3}\) is the control input that determines motion primitives (jerks) to the UAV navigation system. Compared with \(\mathcal{KM}2_{1}\), \(\mathcal{KM}2_{2}\)is better suited for denser, obstructed environments with obstacles (e.g., cluttered indoor environments) where more agility, i.e., more frequent changes in the velocity, is required, and thus acceleration is crucial and should be bounded. The solution to the state space model \(\mathcal{KM2}\) including first-order differential equations that specify states and thus system dynamics is obtained as:  
\[X(t)=X(0)e^{At}+\int_{0}^{t}e^{A(t-\tau)}B\mathbb{A}(\tau)d\tau \tag{14}\]  
where \(X(0)\) denotes the initial state. Since state matrix A is singular, the discrete form of the kinodynamic model can be obtained utilizing the Tayor series expansion, with sampling time \(h\) as follows:  
\[\mathcal{DKM2}_{1}:=\left\{\begin{array}{c}\underbrace{\begin{bmatrix}\xi[k+ 1]\\ v[k+1]\end{bmatrix}}_{X_{2}[k+1]}=\underbrace{\begin{bmatrix}I_{3}&I_{3}h\\ 0_{3}&I_{3}\end{bmatrix}\begin{bmatrix}\xi[k]\\ v[k]\end{bmatrix}}_{G_{2}(h)}+\underbrace{\begin{bmatrix}I_{3}h^{2}/2\\ hH_{1}\end{bmatrix}}_{H_{2}(h)}\mathbb{A}[k]\end{array}\right. \tag{15}\]  
\[\mathcal{DKM2}_{2}:=\left\{\begin{array}{c}\underbrace{\begin{bmatrix}\xi[k+ 1]\\ v[k+1]\\ a[k+1]\end{bmatrix}}_{X_{3}[k+1]}=\underbrace{\begin{bmatrix}I_{3}&I_{3}h&0.5I_{3} h^{2}\\ 0&I_{3}&I_{3}h\\ 0&0&I_{3}\end{bmatrix}\begin{bmatrix}\xi[k]\\ u[k]\end{bmatrix}}_{G_{3}(h)}+\underbrace{\begin{bmatrix}I_{3}h^{3}/6\\ I_{3}h^{2}/2\\ hH_{1}\end{bmatrix}}_{H_{3}(h)}\mathbb{A}[k]\end{array}\right. \tag{16}\]  
where \(G_{2}(h)\in\mathbb{R}^{6\times 6}\) (\(G_{3}(h)\in\mathbb{R}^{9\times 9}\)) and \(H_{2}(h)\in\mathbb{R}^{6\times 3}\) (\(H_{3}(h)\in\mathbb{R}^{9\times 3}\)) denote the discretized state and input matrices, respectively. Similar to \(\mathcal{DKM}\), the UAV heading is obtained on the basis of linear horizontal speed along the \(x\)-axis and \(y\)-axis as \(\psi=\tan^{-1}\bigl{(}v_{y}/v_{x}\bigr{)}\).