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Integrate fourth- or sixth-order point mass equations of motion in coordinated flight

**Library:**Aerospace Blockset / Equations of Motion / Point Mass

UAV Toolbox / Algorithms

The Fixed-Wing Point Mass block integrates fourth- or sixth-order point mass equations of motion in coordinated flight.

The flat Earth reference frame is considered inertial, an approximation that allows the forces due to the Earth's motion relative to the "fixed stars" to be neglected.

The block assumes that there is fully coordinated flight, that is, there is no side force (wind axes) and sideslip is always zero.

The integrated equations of motion for the point mass are:

$$\begin{array}{l}\dot{V}=(T\mathrm{cos}\alpha -D-W\mathrm{sin}{\gamma}_{ai})/m\\ {\dot{\gamma}}_{a}=((L+T\mathrm{sin}\alpha )\mathrm{cos}\mu -W\mathrm{cos}{\gamma}_{ai})/(mV)\\ {\dot{X}}_{e}={V}_{a}+{V}_{w}\end{array}$$

6th order equations:

$$\begin{array}{l}{\dot{X}}_{a}=((L+T\mathrm{sin}\alpha )\mathrm{sin}\mu )/(mV\mathrm{cos}{\gamma}_{a})\\ {{\dot{X}}_{a}|}_{East}=V\mathrm{cos}{\chi}_{a}\mathrm{cos}{\gamma}_{a}\\ {{\dot{X}}_{a}|}_{North}=V\mathrm{sin}{\chi}_{a}\mathrm{cos}{\gamma}_{a}\\ {{\dot{X}}_{a}|}_{Up}=V\mathrm{sin}{\gamma}_{a}\end{array}$$

4th order equations:

$$\begin{array}{l}{\dot{\chi}}_{a}=0\\ {{\dot{X}}_{a}|}_{East}=V\mathrm{cos}{\gamma}_{a}\\ {{\dot{X}}_{a}|}_{North}=0\\ {{\dot{X}}_{a}|}_{Up}=V\mathrm{sin}{\gamma}_{a}\end{array}$$

where:

*m*— Mass.*g*— Gravitational acceleration.*W*— Weight (*m***g*).*L*— Lift force.*D*— Drag force.*T*— Thrust force.α — Angle of attack.

μ — Angle of bank.

γ

_{ai}— Input port value for the flight path angle.*V*— Airspeed, as measured on the aircraft, with respect to the air mass. It is also the magnitude of vector V_{a}.*V*_{w}— Steady wind vector.Subscript

*a*— For the variables, denotes that they are with respect to the steadily moving air mass:*γ*_{a}— Flight path angle.*χ*_{a}— Heading angle.*X*_{a}— Position [East, North, Up].

Subscript

*e*— Flat Earth inertial frame such that so*X*_{e}is the position on the Earth after correcting*X*_{a}for the air mass movement.

Additional outputs are:

$$\begin{array}{l}G=\sqrt{({{V}_{e}|}_{Eas{t}^{2}}+{{V}_{e}|}_{Nort{h}^{2}})}\\ \gamma ={\mathrm{sin}}^{-1}\left(\frac{{{V}_{e}|}_{Up}}{\Vert \underset{\xaf}{{V}_{e}}\Vert}\right)\\ \chi ={\mathrm{tan}}^{-1}\left(\frac{{{V}_{e}|}_{North}}{{{V}_{e}|}_{East}}\right)\end{array}$$

where:

The four-quadrant inverse tangent (

`atan2`

) calculates the heading angle.The groundspeed,

*G*, is the speed over the flat Earth (a 2-D projection).

4th Order Point Mass (Longitudinal) | 4th Order Point Mass Forces (Longitudinal) | 6th Order Point Mass (Coordinated Flight) | 6th Order Point Mass Forces (Coordinated Flight) | 6DOF (Euler Angles) | 6DOF (Quaternion) | 6DOF ECEF (Quaternion) | 6DOF Wind (Wind Angles) | Custom Variable Mass 6DOF (Euler Angles) | Custom Variable Mass 6DOF (Quaternion) | Custom Variable Mass 6DOF ECEF (Quaternion) | Custom Variable Mass 6DOF Wind (Quaternion) | Custom Variable Mass 6DOF Wind (Wind Angles) | Simple Variable Mass 6DOF (Euler Angles) | Simple Variable Mass 6DOF (Quaternion) | Simple Variable Mass 6DOF ECEF (Quaternion) | Simple Variable Mass 6DOF Wind (Quaternion) | Simple Variable Mass 6DOF Wind (Wind Angles)