The quadcopter model here is a higher dimensional, more realistic implementation of actual flight using multi-rotorcraft than the Planar Bi-Rotor Helicopter model. Though it is still considered to be an underactuated model relative to the full $SE(3)$ state dynamics model, if the output states are restricted to be the 3D position and the yaw, then the model becomes fully actuated. The pitch and roll dynamics become passive variables implicitly determined by the trajectory taken or being followed. Many physically motivated mobile robot systems have this property, which is known as differential flatness. Control designers and roboticists have taken advantage of the differential flatness property to design nonlinear controllers for quadcopters that have better stabilization properties than their linear counterparts.

The simplest general form of the equations use the standard second order equations for the linear states $(q, \dot q)$ and the Euler equations with dissipation for the angular states $(R, \omega)$: \begin{equation} \begin{split} m \ddot q & = -D_t \dot q + R B_t \vec u - m \vec g \\ \dot R & = R \cdot \omega \\ J \dot \omega & = -D_a \omega + (J\omega) \times \omega + B_a \vec u \end{split} \end{equation} where $J$ is the intertia matrix, $D_t$ and $D_a$ are the dissipation matrices, and $B_t$ and $B_a$ are the control matrices. The control matrices take the form: \begin{equation} B_t = b_t \begin{bmatrix} 0 & 0 & 0& 0 \\ 0& 0 & 0 & 0 \\ 1 & 1 & 1 & 1 \end{bmatrix} \qquad \text{and} \qquad B_a = \text{diag}(b_x,b_y, b_z) \begin{bmatrix} 0 & 0 & 1 & -1 \\ -1 & 1 & 0 & 0 \\ -1 & -1 & 1 & 1 \end{bmatrix}. \end{equation} The constant $b_t$ is the input to thrust factor, and $b_x$, $b_y$, and $b_z$ are input to torque factors for rotation about the body $x$, $y$, and $z$ axes. Note that this configuration is based on the first and second rotor inputs aligning with the body $x$-axis, and the third and fourth aligning with the body $y$-axis, and with both sets being counter-rotating. The first and second rotate counter in the positive sense and the third and fourth rotate in the negative sense. Thus spin-ups will induce the opposing angular acceleration as per the signs in $B_a$'s third row.

If the rotor geometry is different from the one described above, then the $B_t$ and $B_a$ matrices may require adjusting. If the rotor axes are in the body $z$-axis but do not form a plus, then the $B_a$ matrix will differ. They could form a cross with all equal angles or an x shape with two different angles, or possibly even have some asymmetric form. Likewise, the rotor axes could be tilted, which would also impact the $B_t$ matrix. There is a design like that which permits forward acceleration without significant pitching, and can achieve a steady state velocity at zero pitch. Due to drag forces, most quad-copters will need to pitch some in order to move at a constant velocity.

How to implement the equations is a function of one's comfort level with them. For those familiar with Lie groups and control on Lie groups, then the best option is to keep them in geometric form by using rotation matrices or quaternions to represent the rotation $R$, and their corresponding Lie algebra representations for the angular velocities $\omega$. A common representation for those unfamiliar with, or new to rigid body mechanics, is to employ Euler angle representations for the angular states. Which to use is up to you, though the most common will be the Tait-Bryan angles, described using the notation $X_1 Y_2 Z_3$ for the first coordinate being rotation about $x$, second about $y$ and third about $z$. The order of application is from right to left since the multiplication happens from the left side. Sometimes it is written as $R_x R_y R_z$ where $R_\cdot$ is the canonical rotation about the subscripted axis. Picking the right representation choice can simplify the nonlinearity in the equations of motion. The value of doing so is such that adaptive controllers might benefit from the simpler nonlinearity by having dependence on a sub-state of the system instead of the full state, or by having easy to describe nonlinear basis elements for adaptive cancellation.

NEED TO ADD ACTIVITIES, BUT AT LEAST WE KNOW THAT STEP 1 IS THE LINEAR AT HOVER VERSION.

Linearize the equations of motion about hover, so that the linearized state and control inputs have an equiblirium at the origin with zero linearized control input. Establish performance specifications for the system and design a linear feedback controller that will stabilize the system and meet the performance specifications.

Modify by 10-20% some of the model parameters of the system and compare the outcomes under a traditional linear controller. Incorporate linear adaptive control and show the resulting outcomes. Confirm how well the adaptive system meets the performance specifications versus the static controller.

Here, you should consider two cases. One is the initial case, where tracking a particular reference signal will lead to an adaptation transient. Simulate as normal, however pick a time post-transient and grab the adaptive gains from the output signals. Prepare a second simulation that starts with these gains. The second simulation would act like a second or subsequent deployment post-adaptation. Show that the system better meets the performance objectives.

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For sure you need to be careful with these references. In addition to different reference frame conventions between Aerospace engineers and others, plus different ways of representing the $SE(3)$ Lie group state, there are actually slightly different dynamical equations for the quadcopter. The references below just give some idea for how static controllers are synthesized for these systems. The model chosen depends on what the authors wish to demonstrate or what kinds of constraints their control method has.

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