Notation

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. |