VTOL system

Consider the VTOL example. The following shortcuts ${\rm c}(\cdot)=\cos(\cdot),\quad {\rm s}(\cdot)=\sin(\cdot)$ are used. The $\Delta_0$ is given as

$$\begin{align*} \omega_{0}^{1} &: {\mathrm d}x-v_{x}{\mathrm d}t \\ \omega_{0}^{2} &: {\mathrm d}v_{x}-(u^{2}\epsilon {\mathrm c}(\theta)-u^{1}{\mathrm s}(\theta)){\mathrm d}t \\ \omega_{0}^{3} &: {\mathrm d}z-v_{z}{\mathrm d}t \\ \omega_{0}^{4} &: {\mathrm d}v_{z}-(u^{1}{\mathrm c}(\theta)+u^{2}\epsilon {\mathrm s}(\theta)-1){\mathrm d}t \\ \omega_{0}^{5} &: {\mathrm d}\theta-\omega {\mathrm d}t \\ \omega_{0}^{6} &: {\mathrm d}\omega-u^{2}{\mathrm d}t, \end{align*}$$

where $\epsilon$ is a constant parameter. It can be shown that $\Delta_{0,\mathrm{d}t}^{\perp} = \mathrm{span}\{\delta_{u^1},\delta_{u^2}\}$. Construct $\Delta_1 \in \Delta_0$ such that $v_0(\Delta_1)\in \Delta_1$ holds with $v_0=\Delta_{0,{\mathrm d}t}^{\perp}$, i.e. $\Delta_1$ is given as:

$\begin{align*} \omega_{1}^{1} &: {\mathrm d}x-v_{x}{\mathrm d}t \\ \omega_{1}^{2} &: {\mathrm c}(\theta){\mathrm d}v_{x}+{\mathrm s}(\theta){\mathrm d}v_{z}-\epsilon {\mathrm d}\omega+{\mathrm s}(\theta){\mathrm d}t \\ \omega_{1}^{3} &: {\mathrm d}z-v_{z}{\mathrm d}t \\ \omega_{1}^{4} &: {\mathrm d}\theta-\omega {\mathrm d}t \end{align*}$