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  • Bold face variables indicate vectors or matrices and non-bold face variables represent scalars.
  • The default frame for each variable is the local frame: . Right superscripts represent the coordinate frame. If no right superscript is present, then the default frame 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.


AOAAngle Of Attack. Also named alpha.
AOSAngle Of Sideslip. Also named beta.
FRDCoordinate 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.
FWFixed-wing (planes).
MPC 或 MCPCMultiCopter Position Controller. MultiCopter Position Controller. MPC is also used for Model Predictive Control.
NEDCoordinate 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.
PIDController with Proportional, Integral and Derivative actions.


x,y,zTranslation along coordinate axis x,y and z respectively.
rPosition vector: r=[xyz]T
vVelocity vector: v=r˙
aAcceleration vector: a=v˙=r¨
αAngle of attack (AOA).
bWing span (from tip to tip).
SWing area.
ARAspect ratio: AR=b2/S
βAngle of sideslip (AOS).
cWing chord length.
δAerodynamic control surface angular deflection. A positive deflection generates a negative moment. A positive deflection generates a negative moment.
ϕ,θ,ψEuler angles roll (=Bank), pitch and yaw (=Heading).
ΨAttitude vector: Ψ=[ϕθψ]T
X,Y,ZForces along coordinate axis x,y and z.
FForce vector: F=[XYZ]T
DDrag force.
CCross-wind force.
LLift force.
l,m,nMoments around coordinate axis x,y and z.
MMoment vector M=[lmn]T
MMach number. Can be neglected for scale aircraft.
qVector part of Quaternion.
q~Hamiltonian attitude quaternion (see 1 below)
RbRotation matrix. Rotates a vector from frame to frame b. vb=Rbv
ΛLeading-edge sweep angle.
λAspect ratio. $$AR = b^2/S$$.
wWind velocity.
p,q,rAngular rates around body axis x,y and z.
ωbAttitude vector. $$\Psi = [\phi \quad \theta \quad \psi]^T$$.
xGeneral state vector.
  • 1 Hamiltonian attitude quaternion. Hamiltonian attitude quaternion. $$\boldsymbol{\mathrm{\tilde{q}}} = (q_0, q_1, q_2, q_3) = (q_0, \boldsymbol{\mathrm{q}})$$. To represent a vector in local frame given a vector in body frame, the following operation can be used: v~=q~v~bq~ (or q~1 instead of q~ if q~ is not unitary). v~ represents a quaternionized vector: v~=(0,v)

Subscripts / Indices

Subscripts / Indices描述
TThrust force.
wRelative airspeed.
x,y,zComponent of vector along coordinate axis x, y and z.
N,E,DComponent of vector along global north, east and down direction.

Superscripts / Indices

Superscripts / Indices描述
Local-frame. Local-frame. Default for PX4 related variables.


()Complex conjugate.
()˙Time derivative.
()1Matrix inverse.
()TMatrix transpose.