Main Content

Coordinate Systems for Modeling

Modeling aircraft and spacecraft are simplest if you use a coordinate system fixed in the body itself. In the case of aircraft, the forward direction is modified by the presence of wind, and the craft's motion through the air is not the same as its motion relative to the ground.

Body Coordinates

The noninertial body coordinate system is fixed in both origin and orientation to the moving craft. The craft is assumed to be rigid.

The orientation of the body coordinate axes is fixed in the shape of body.

  • The x-axis points through the nose of the craft.

  • The y-axis points to the right of the x-axis (facing in the pilot's direction of view), perpendicular to the x-axis.

  • The z-axis points down through the bottom of the craft, perpendicular to the x-y plane and satisfying the RH rule.

Translational Degrees of Freedom

Translations are defined by moving along these axes by distances x, y, and z from the origin.

Rotational Degrees of Freedom

Rotations are defined by the Euler angles P, Q, R or Φ, Θ, Ψ. They are

  • P or Φ: Roll about the x-axis

  • Q or Θ: Pitch about the y-axis

  • R or Ψ: Yaw about the z-axis

P, Q, R rotations defined by Euler angles in body coordinates, illustrated on an aircraft

Unless otherwise specified, by default the software uses ZYX rotation order for Euler angles.

Wind Coordinates

The noninertial wind coordinate system has its origin fixed in the rigid aircraft. The coordinate system orientation is defined relative to the craft's velocity V.

The orientation of the wind coordinate axes is fixed by the velocity V.

  • The x-axis points in the direction of V.

  • The y-axis points to the right of the x-axis (facing in the direction of V), perpendicular to the x-axis.

  • The z-axis points perpendicular to the x-y plane in whatever way needed to satisfy the RH rule with respect to the x- and y-axes.

Translational Degrees of Freedom

Translations are defined by moving along these axes by distances x, y, and z from the origin.

Rotational Degrees of Freedom

Rotations are defined by the Euler angles Φ, γ, χ. They are

  • Φ: Bank angle about the x-axis

  • γ: Flight path about the y-axis

  • χ: Heading angle about the z-axis

Rotational angles defined by Euler angles in wind coordinates, illustrated on an aircraft

Unless otherwise specified, by default the software uses ZYX rotation order for Euler angles.

Related Topics