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Terminology
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 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 aircrafts. |
| \(\boldsymbol{\mathrm{q}}\) |
Vector part of Quaternion. |
| \(\boldsymbol{\mathrm{\tilde{q}}}\) |
Hamiltonian attitude quaternion. $\boldsymbol{\mathrm{\tilde{q}}} = (q_0, q_1, q_2, q_3) = (q_0, \boldsymbol{\mathrm{q}})$.
$\boldsymbol{\mathrm{\tilde{q}}}{}$ 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: $\boldsymbol{\mathrm{\tilde{v}}}^\ell = \boldsymbol{\mathrm{\tilde{q}}} \, \boldsymbol{\mathrm{\tilde{v}}}^b \, \boldsymbol{\mathrm{\tilde{q}}}^{}$ (or $\boldsymbol{\mathrm{\tilde{q}}}^{-1}{}$ instead of $\boldsymbol{\mathrm{\tilde{q}}}^{}$ if $\boldsymbol{\mathrm{\tilde{q}}}{}$ is not unitary). $\boldsymbol{\mathrm{\tilde{v}}}{}$ represents a quaternionized vector: $\boldsymbol{\mathrm{\tilde{v}}} = (0,\boldsymbol{\mathrm{v}})$ |
| \(\boldsymbol{\mathrm{R}}_\ell^b\) |
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. |