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99 lines
12 KiB
Markdown
99 lines
12 KiB
Markdown
# Terminology
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The following terms, symbols, and decorators are used in text and diagrams throughout this guide.
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## Notation
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- Bold face variables indicate vectors or matrices and non-bold face variables represent scalars.
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- The default frame for each variable is the local frame: $\ell{}$.
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Right [superscripts](#superscripts) represent the coordinate frame.
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If no right superscript is present, then the default frame $\ell{}$ is assumed.
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An exception is given by Rotation Matrices, where the lower right subscripts indicates the current frame and the right superscripts the target frame.
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- Variables and subscripts can share the same letter, but they always have different meaning.
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## Acronyms
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| Acronym | Expansion |
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| ----------- | ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ |
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| AOA | Angle Of Attack. Also named _alpha_. |
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| AOS | Angle Of Sideslip. Also named _beta_. |
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| 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. |
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| FW | Fixed-wing (planes). |
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| MC | MultiCopter. |
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| MPC or MCPC | MultiCopter Position Controller. MPC is also used for Model Predictive Control. |
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| 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. |
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| PID | Controller with Proportional, Integral and Derivative actions. |
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## Symbols
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| Variable | Description |
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| --------------------------------- | --------------------------------------------------------------------------------------------------------------------------------------------------------------- |
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| $x,y,z$ | Translation along coordinate axis x,y and z respectively. |
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| $\boldsymbol{\mathrm{r}}$ | Position vector: $\boldsymbol{\mathrm{r}} = [x \quad y \quad z]^{T}$ |
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| $\boldsymbol{\mathrm{v}}$ | Velocity vector: $\boldsymbol{\mathrm{v}} = \boldsymbol{\mathrm{\dot{r}}}$ |
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| $\boldsymbol{\mathrm{a}}$ | Acceleration vector: $\boldsymbol{\mathrm{a}} = \boldsymbol{\mathrm{\dot{v}}} = \boldsymbol{\mathrm{\ddot{r}}}$ |
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| $\alpha$ | Angle of attack (AOA). |
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| $b$ | Wing span (from tip to tip). |
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| $S$ | Wing area. |
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| $AR$ | Aspect ratio: $AR = b^2/S$ |
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| $\beta$ | Angle of sideslip (AOS). |
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| $c$ | Wing chord length. |
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| $\delta$ | Aerodynamic control surface angular deflection. A positive deflection generates a negative moment. |
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| $\phi,\theta,\psi$ | Euler angles roll (=Bank), pitch and yaw (=Heading). |
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| $\Psi$ | Attitude vector: $\Psi = [\phi \quad \theta \quad \psi]^T$ |
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| $X,Y,Z$ | Forces along coordinate axis x,y and z. |
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| $\boldsymbol{\mathrm{F}}$ | Force vector: $\boldsymbol{\mathrm{F}}= [X \quad Y \quad Z]^T$ |
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| $D$ | Drag force. |
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| $C$ | Cross-wind force. |
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| $L$ | Lift force. |
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| $g$ | Gravity. |
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| $l,m,n$ | Moments around coordinate axis x,y and z. |
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| $\boldsymbol{\mathrm{M}}$ | Moment vector $\boldsymbol{\mathrm{M}} = [l \quad m \quad n]^T$ |
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| $M$ | Mach number. Can be neglected for scale aircraft. |
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| $\boldsymbol{\mathrm{q}}$ | Vector part of Quaternion. |
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| $\boldsymbol{\mathrm{\tilde{q}}}$ | Hamiltonian attitude quaternion (see `1` below) |
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| $\boldsymbol{\mathrm{R}}_\ell^b$ | Rotation matrix. Rotates a vector from frame $\ell{}$ to frame $b{}$. $\boldsymbol{\mathrm{v}}^b = \boldsymbol{\mathrm{R}}_\ell^b \boldsymbol{\mathrm{v}}^\ell$ |
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| $\Lambda$ | Leading-edge sweep angle. |
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| $\lambda$ | Taper ratio: $\lambda = c_{tip}/c_{root}$ |
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| $w$ | Wind velocity. |
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| $p,q,r$ | Angular rates around body axis x,y and z. |
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| $\boldsymbol{\omega}^b$ | Angular rate vector in body frame: $\boldsymbol{\omega}^b = [p \quad q \quad r]^T$ |
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| $\boldsymbol{\mathrm{x}}$ | General state vector. |
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- `1` 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: $\boldsymbol{\mathrm{\tilde{v}}}^\ell = \boldsymbol{\mathrm{\tilde{q}}} \, \boldsymbol{\mathrm{\tilde{v}}}^b \, \boldsymbol{\mathrm{\tilde{q}}}^*{}$ (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: $\boldsymbol{\mathrm{\tilde{v}}} = (0,\boldsymbol{\mathrm{v}})$
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### Subscripts / Indices
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| Subscripts / Indices | Description |
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| -------------------- | ---------------------------------------------------------------- |
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| $a$ | Aileron. |
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| $e$ | Elevator. |
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| $r$ | Rudder. |
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| $Aero$ | Aerodynamic. |
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| $T$ | Thrust force. |
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| $w$ | Relative airspeed. |
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| $x,y,z$ | Component of vector along coordinate axis x, y and z. |
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| $N,E,D$ | Component of vector along global north, east and down direction. |
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<a id="superscripts"></a>
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### Superscripts / Indices
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| Superscripts / Indices | Description |
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| ---------------------- | ----------------------------------------------- |
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| $\ell$ | Local-frame. Default for PX4 related variables. |
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| $b$ | Body-frame. |
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| $w$ | Wind-frame. |
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## Decorators
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| Decorator | Description |
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| ------------ | ------------------ |
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| $()^*$ | Complex conjugate. |
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| $\dot{()}$ | Time derivative. |
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| $\hat{()}$ | Estimate. |
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| $\bar{()}$ | Mean. |
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| $()^{-1}$ | Matrix inverse. |
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| $()^T$ | Matrix transpose. |
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| $\tilde{()}$ | Quaternion. |
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