PX4-Autopilot/docs/en/contribute/notation.md

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 (planes).
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}}}	Hamiltonian attitude quaternion (see 1 below)
\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.

• 1 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}})

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.