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--- ece6554:project_planarheli [2023/03/21 10:59] – [Parameters and Limits] classes
+++ ece6554:project_planarheli [2024/08/20 21:38] (current) – external edit 127.0.0.1
@@ Line 6: / Line 6: @@
 ==== Equations of Motion ====
-Defining $q = (x, y)^T$ to be the center of mass of the ducted fan, and $\theta$ to be the orientation of the ducted fan,
+Defining $q = (x, y)^T$ to be the center of mass of the ducted fan, and $\theta$ to be the orientation of the ducted fan, the most general form of the equations is
+\begin{equation}
+\begin{split}
+  m \ddot q & = -d \dot q + R(\theta) \left[ \begin{matrix} 0 & 0 \\ 1 & 1 \end{matrix} \right] \vec f - m \vec g \\
+  J \ddot \theta & = r \left[ \begin{matrix} 1 & -1 \end{matrix} \right] \vec f
+\end{split}
+\end{equation}
+while a more specialized form is
 \begin{equation}
 \begin{split}
@@ Line 14: / Line 21: @@
 \end{equation}
 where the force vector $f$ is in the body frame of the ducted fan, generated from the two fans' thrust.  Each coordinate of $f \in \mathbb{R}^2$ can be independently controlled (but can never really go negative due to the nature of the fan blades used).
-Though not used explicitly in the equation above, let $R$ be a planar rotation matrix for which $e_2$ is the vector generated from the second column of the rotation matrix,
+Regarding the functions of $\theta$ used in the equations of motion, $R$ is a planar rotation matrix for which $e_2$ is the vector generated from the second column of the rotation matrix,
 \begin{equation}
   R(\theta) = \left[ \begin{matrix} \cos(\theta) & -\sin(\theta) \\ \sin(\theta) & \cos(\theta) \end{matrix}  \right]