This readme has become outdated w.r.t my recent developments. The model presented here was sufficient for my ME 762 report (discussed below), but is being re-evaluated as I have some free time at the moment. The new model is capable of establishing stability via a simple PD controller, unlike that which is discussed here.
This repository serves as the testing grounds for my project in the Boston University class ME 762. I will be attempting to simulate and control a simple hummingbird robot - modelled as a 3-D point-mass with assorted dynamics.
The control paramters are a single lift force pushing through the bird's center of mass $z$-axes, and two rotations angles; $\theta$ which defines the birds displacement from the world frame $x$-axis, and $\delta$ which defines the bird's displacement from the $z$-axis.
The state space form, represented by a second-order system of five continuous parameters is shown below.
$$
\begin{aligned}
\begin{bmatrix}
x_1 \\
x_2 \\
x_3 \\
x_4 \\
x_5
\end{bmatrix} = \begin{bmatrix}
x \\
y \\
z \\
\delta \\
\theta
\end{bmatrix},
&&
\begin{bmatrix}
x_6 \\
x_7 \\
x_8 \\
x_9 \\
x_{10}
\end{bmatrix} = \begin{bmatrix}
\dot x \\
\dot y \\
\dot z \\
\dot \delta \\
\dot \theta
\end{bmatrix}
\end{aligned}
$$
Before describing the equation of motions we can characterize the inputs in terms of the lift force on the COM, and the torque about the two principle axes.
$$
u = \begin{bmatrix}
u_1 \\
u_2 \\
u_3
\end{bmatrix} = \begin{bmatrix}
F \\
\tau_z \\
\tau_{xy}
\end{bmatrix}
$$
Making the dynamics for the hummingbird...
$$
\dot x = \begin{bmatrix}
\dot x_1 \\
\dot x_2 \\
\dot x_3 \\
\dot x_4 \\
\dot x_5 \\
\dot x_6 \\
\dot x_7 \\
\dot x_8 \\
\dot x_9 \\
\dot x_{10}
\end{bmatrix} = \begin{bmatrix}
\dot x \\
\dot y \\
\dot z \\
\dot \delta \\
\dot \theta \\
\frac{1}{m} (F \cos(x_4) \cos(x_5) - w x_6) \\
\frac{1}{m} (F \cos(x_4) \sin(x_5) - w x_7) \\
\frac{1}{m} (F \sin(x_4) - w x_8 - mg) \\
\tau_z \\
\tau_{xy}
\end{bmatrix} = \begin{bmatrix}
x_6 \\
x_7 \\
x_8 \\
x_9 \\
x_{10} \\
\frac{1}{m} (u_1 \cos(x_4) \cos(x_5) - w x_6) \\
\frac{1}{m} (u_1 \cos(x_4) \sin(x_5) - w x_7) \\
\frac{1}{m} (u_1 \sin(x_4) - w x_8 - mg) \\
u_2 \\
u_3
\end{bmatrix}
$$
Where the scalar coefficient $w$ describes the air coefficient of friction.