Reference Frames | Axis Systems

Reference Frames (Axis Systems)

Introduction

Reference frames or axis systems consist of three orthogonal vectors of unit length. Some reference frames are fixed with the earth, others with the aircraft, etc., depending on whatever turns our to be most useful.

List of Reference Frames

This list of reference frames contains some of the most frequently used axis systems. More axis systems exists, and some of these reference frames come in several flavors. The principal axis system not mentioned below is, for instance, an alternate body axis system, where the axes are aligned with the principal axes of the mass distribution and not the nominally defined axes of the aircraft.

Inertial Frame (World/Earth Frame)

The inertial frame is the axis system fixed with the world (in flat earth approximation), with axes defined as follows:

  • \(x_E\)-axis: always points horizontally North.
  • \(y_E\)-axis: always points horizontally East.
  • \(z_E\)-axis: always points down, perpendicularly towards the Earth’s surface.

The above axes form a standard, orthogonal, right-handed axis system. (Right handed: take your right hand and 1) stretch out your thumb and point it North, 2) stretch out your index finger and point it East, 3) stretch out your middle finger down (perpendicularly to the plane spanned by the other two fingers).)

Defining the \(z_E\)-axis as pointing down, instead of up, is convention in aerospace engineering, to obtain a right-handed axis system (otherwise we would have obtained a left-handed axis system – try in out with your hands). Right-handed axis systems are preferred for calculations by mathematical convention.

Gravity is most naturally expressed in the inertial frame: the weight force vector always points in direction of the \(z_E\)-axis.

Body Frame

The body frame is a reference frame fixed with the aircraft, with axes defined as follows:

  • \(x_B\)-axis: always points forward towards the nose of the aircraft.
  • \(y_B\)-axis: always points to the right wing.
  • \(z_B\)-axis: always points down towards the feet of the pilot.

The way the above is actually constructed is by first defining the \(x_B\)-axis, and then demand that the \(z_B\)-axis be perpendicular to the \(x_B\)-axis and lie in the plane of symmetry of the aircraft (for left-right symmetric aircraft). The \(y_B\)-axis is then defined to be perpendicular to the other two, such that \(x_B\), \(y_B\), \(z_B\) form a right-handed coordinate system.

The \(x_Bz_B\)-plane is the plane of symmetry of the aircraft.

Note that the body frame coincides with the inertial frame (i.e. axes are the same) for an aircraft flying north in straight-and-level flight (up to a translational offset of the origin and neglecting angle of attack). But for an arbitrary aircraft attitude, the \((x_B, y_B, z_B)\) axes of the Body Frame point in different directions than those of the Inertial Frame \((x_E, y_E, z_E)\).

The Euler angles φ, θ, ψ, translate between the body frame and the inertial frame (see article “Aircraft Attitude and Euler Angles”).

Thrust force from the engines is most naturally expressed in the body frame. Assuming that the thrust line is aligned with the longitudinal axis for the aircraft, it points in \(x_B\) direction of the body frame.

Wind Frame

The wind frame is a reference frame focusing on the relative wind or velocity vector of the aircraft. It will be useful, when we discuss aerodynamic forces. Its axes are defined as follows:

  • \(x_W\)-axis: always points along the velocity vector of the aircraft (i.e. in the direction from which the relative wind is coming from, assuming the air is still (no wind blowing over earth surface).
  • \(y_W\)-axis: perpendicularly to the other two axes, such that a right-handed coordinate system is formed.
  • \(z_W\)-axis: perpendicular to the xW-axis and in the plane of symmetry of the aircraft (i.e. in \(x_Bz_B\)-plane), in such a fashion that the axis is positive if pointing towards the belly of the airplane.

The way the above is actually constructed is by first defining the \(x_W\)-axis, and then demanding that the \(z_W\)-axis be perpendicular to the \(x_W\)-axis and lie in the plane of symmetry of the aircraft (for left-right symmetric aircraft). The \(y_W\)-axis is then defined to be perpendicular to the other two, such that \(x_W\), \(y_W\), \(z_W\) for a right-handed axis system.

The Euler angles μ, γ, ξ, translate between the wind frame and the inertial frame (see article “Aircraft Attitude and Euler Angles”).

Angle of attack (AOA) α and sideslip angle (AOS) β translate between the wind frame and the body frame. They are introduced formally in the article “Aircraft Attitudes and Euler Angles”.

The resultant aerodynamic force (RAF) is most naturally expressed in the wind frame. Its components lie along the three axes of the wind frame: drag D is by definition always along the \(x_W\)-axis, but in opposite direction; lift L is by definition always along the \(z_W\)-axis in opposite direction; and sideforce Y is by definition along the \(y_W\)-axis.

Velocity Frame

The velocity frame is obtained the same way as the wind frame, except that the \(z_V\)-axis is required to lie in the same plane as the \(z_E\) axis.