# Terminology/Notation

The following terms, symbols, and decorators are used in text and diagrams throughout this guide.

## Notation

* Bold face variables indicate vectors or matrices and non-bold face variables represent scalars.
* The default frame for each variable is the local frame: $\ell{}$. Right [superscripts](#superscripts) represent the coordinate frame. If no right superscript is present, then the default frame $\ell{}$ is assumed. An exception is given by Rotation Matrices, where the lower right subscripts indicates the current frame and the right superscripts the target frame.
* Variables and subscripts can share the same letter, but they always have different meaning.

## Acronyms

| Acronym     | Expansion                                                                                                                                                                      |
| ----------- | ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ |
| AOA         | Angle Of Attack. Also named _alpha_.                                                                                                                                           |
| AOS         | Angle Of Sideslip. Also named _beta_.                                                                                                                                          |
| FRD         | Coordinate system where the X-axis is pointing towards the Front of the vehicle, the Y-axis is pointing Right and the Z-axis is pointing Down, completing the right-hand rule. |
| FW          | Fixed-Wing.                                                                                                                                                                    |
| MC          | MultiCopter.                                                                                                                                                                   |
| MPC or MCPC | MultiCopter Position Controller. MPC is also used for Model Predictive Control.                                                                                                |
| NED         | Coordinate system where the X-axis is pointing towards the true North, the Y-axis is pointing East and the Z-axis is pointing Down, completing the right-hand rule.            |
| PID         | Controller with Proportional, Integral and Derivative actions.                                                                                                                 |

## Symbols

| Variable                            | Description                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     |
| ----------------------------------- | --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- |
| $x,y,z$                             | Translation along coordinate axis x,y and z respectively.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                       |
| $\boldsymbol{\mathrm{r\}}$          | Position vector: $\boldsymbol{\mathrm{r\}} = \[x \quad y \quad z]^{T}$                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| $$\boldsymbol{\mathrm{v}}$$         | Velocity vector: $\boldsymbol{\mathrm{v\}} = \boldsymbol{\mathrm{\dot{r\}}}$                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    |
| $$\boldsymbol{\mathrm{a}}$$         | Acceleration vector: $\boldsymbol{\mathrm{a\}} = \boldsymbol{\mathrm{\dot{v\}}} = \boldsymbol{\mathrm{\ddot{r\}}}$                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              |
| $$\alpha$$                          | Angle of attack (AOA).                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| $$b$$                               | Wing span (from tip to tip).                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    |
| $$S$$                               | Wing area.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      |
| $$AR$$                              | Aspect ratio: $AR = b^2/S$                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      |
| $$\beta$$                           | Angle of sideslip (AOS).                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        |
| $$c$$                               | Wing chord length.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              |
| $$\delta$$                          | Aerodynamic control surface angular deflection. A positive deflection generates a negative moment.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              |
| $$\phi,\theta,\psi$$                | Euler angles roll (=Bank), pitch and yaw (=Heading).                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            |
| $$\Psi$$                            | Attitude vector: $\Psi = \[\phi \quad \theta \quad \psi]^T$                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     |
| $$X,Y,Z$$                           | Forces along coordinate axis x,y and z.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         |
| $$\boldsymbol{\mathrm{F}}$$         | Force vector: $\boldsymbol{\mathrm{F\}}= \[X \quad Y \quad Z]^T$                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                |
| $$D$$                               | Drag force.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     |
| $$C$$                               | Cross-wind force.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               |
| $$L$$                               | Lift force.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     |
| $$g$$                               | Gravity.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        |
| $$l,m,n$$                           | Moments around coordinate axis x,y and z.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                       |
| $$\boldsymbol{\mathrm{M}}$$         | Moment vector $\boldsymbol{\mathrm{M\}} = \[l \quad m \quad n]^T$                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               |
| $$M$$                               | Mach number. Can be neglected for scale aircraft.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               |
| $$\boldsymbol{\mathrm{q}}$$         | Vector part of Quaternion.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      |
| $$\boldsymbol{\mathrm{\tilde{q}}}$$ | <p>Hamiltonian attitude quaternion. $\boldsymbol{\mathrm{\tilde{q}}} = (q_0, q_1, q_2, q_3) = (q_0, \boldsymbol{\mathrm{q}})$.<br>$\boldsymbol{\mathrm{\tilde{q}}}{}$ describes the attitude relative to the local frame $\ell{}$. To represent a vector in local frame given a vector in body frame, the following operation can be used: $\boldsymbol{\mathrm{\tilde{v}}}^\ell = \boldsymbol{\mathrm{\tilde{q}}} , \boldsymbol{\mathrm{\tilde{v}}}^b , \boldsymbol{\mathrm{\tilde{q}}}^<em>{}$ (or $\boldsymbol{\mathrm{\tilde{q}}}^{-1}{}$ instead of $\boldsymbol{\mathrm{\tilde{q}}}^</em>{}$ if $\boldsymbol{\mathrm{\tilde{q}}}{}$ is not unitary). $\boldsymbol{\mathrm{\tilde{v}}}{}$ represents a <em>quaternionized</em> vector: $\boldsymbol{\mathrm{\tilde{v}}} = (0,\boldsymbol{\mathrm{v}})$</p> |
| $$\boldsymbol{\mathrm{R}}_\ell^b$$  | Rotation matrix. Rotates a vector from frame $\ell{}$ to frame $b{}$. $$\boldsymbol{\mathrm{v}}^b = \boldsymbol{\mathrm{R}}_\ell^b \boldsymbol{\mathrm{v}}^\ell$$                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               |
| $$\Lambda$$                         | Leading-edge sweep angle.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                       |
| $$\lambda$$                         | Taper ratio: $\lambda = c\_{tip}/c\_{root}$                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     |
| $$w$$                               | Wind velocity.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  |
| $$p,q,r$$                           | Angular rates around body axis x,y and z.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                       |
| $$\boldsymbol{\omega}^b$$           | Angular rate vector in body frame: $\boldsymbol{\omega}^b = \[p \quad q \quad r]^T$                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                             |
| $$\boldsymbol{\mathrm{x}}$$         | General state vector.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           |

### Subscripts / Indices

| Subscripts / Indices | Description                                                      |
| -------------------- | ---------------------------------------------------------------- |
| $$a$$                | Aileron.                                                         |
| $$e$$                | Elevator.                                                        |
| $$r$$                | Rudder.                                                          |
| $$Aero$$             | Aerodynamic.                                                     |
| $$T$$                | Thrust force.                                                    |
| $$w$$                | Relative airspeed.                                               |
| $$x,y,z$$            | Component of vector along coordinate axis x, y and z.            |
| $$N,E,D$$            | Component of vector along global north, east and down direction. |

### Superscripts / Indices

| Superscripts / Indices | Description                                     |
| ---------------------- | ----------------------------------------------- |
| $$\ell$$               | Local-frame. Default for PX4 related variables. |
| $$b$$                  | Body-frame.                                     |
| $$w$$                  | Wind-frame.                                     |

## Decorators

| Decorator      | Description        |
| -------------- | ------------------ |
| $$()^*$$       | Complex conjugate. |
| $$\dot{()}$$   | Time derivative.   |
| $$\hat{()}$$   | Estimate.          |
| $$\bar{()}$$   | Mean.              |
| $$()^{-1}$$    | Matrix inverse.    |
| $$()^T$$       | Matrix transpose.  |
| $$\tilde{()}$$ | Quaternion.        |
