This readme has become outdated w.r.t my recent developments.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.

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Hummingbird Model

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.