# PART 2: Reference Frames and Coordinate Transformations

For our purposes, a frame of reference is a Cartesian coordinate system defined by its orientation in space. It includes a set of three coordinate axes labeled, . The superscript f is an indicator identifying the particular reference frame to which the axis, , belongs. Unit vectors, ,are associated with each and point in the axis positive direction. Points in space, P, are specified either as the coordinate triplet or, equivalently, as the position vector, . In the latter case, are interpreted as components of the vector P along each of the axes (i.e., ). Vector quantities, , are expressed in this report using one of several equivalent representations. For example,

 (2-1)
and
 (2-2)
all represent the same vector.

Five distinct frames of reference and a geodetic coordinate system are defined and used in this report. The reference systems, upon which the mathematical model depends, include the

• Inertial Frame;
• Earth-Centered, Earth-Fixed Frame;
• North-Oriented, Local-Level Frame;
• Velocity-Oriented Frame; and
• Body-Fixed Frame; and
• Geodetic Coordinate System.

Reference frame definitions comprise an important element of the mathematical model. Nearly every quantity of interest (e.g., position, velocity, angular rates) must be expressed relative to a particular frame of reference. Precise definitions are crucial to a proper understanding of the model, and a correct application of model equations to the implementation of flight programs.

## Inertial and Earth-Centered, Earth-Fixed Frames of Reference

The inertial frame is fixed with its origin at the center of the earth. Its Cartesian axes, , remain fixed relative to the stars, and provide a reference frame for which the equations of motion are most simply expressed. The positive axis passes through the earths geographic North Pole. The and axes lie in the equatorial plane.

The axes of the Earth-Centered, Earth-Fixed (ECEF) frame of reference, labeled , are depicted in Figure 2.1. The coordinate axes of the ECEF frame remain fixed with respect to the earth. The origin of this Cartesian system, like the Inertial frame, is located at the mass center of the earth. The axis lies along the earths spin axis with its positive end passing through the earths north geographic pole. The and axes both lie in the equatorial plane, with the positive axis passing through the prime meridian. The ECEF frame rotates counter-clockwise about the Inertial frames axis with angular velocity,

 (2-3)
The earth rate, , is approximately equal to radians/hour.

## North-Oriented, Local-Level Frame of Reference

The equations of motion for the aircraft, in their final form, are expressed in the North-Oriented, Local-Level (NOLL) frame of reference. The origin of this system is the aircrafts center of mass. The NOLL frame is obtained through successive rotations of the ECEF coordinate axes. The sequence of operations begins by rotating the axes counterclockwise through the angle, (i.e., longitude of aircrafts position), about the axis. The resultant axes are labeled . The intermediate axes are then rotated counterclockwise through the angle about the axis, forming a new coordinate set . The angle is the latitude of the aircrafts position. In the final step, are rotated counter-clockwise through the angle into the desired NOLL axes, . The orientation of the NOLL frame of reference is illustrated in Figure 2.2. The and axes form a plane which is locally level. The positive axis is tangent to a line of longitude, and is directed toward the earths geographic North Pole. The positive axis is directed eastward along a line of latitude. The positive axis is directed upward.

The components of a vector in the ECEF system, , are related to their components in the NOLL system, , through the rotational operator . Symbolically,

 (2-4)
where is the 3x3 matrix,
 (2-5)

The angular velocity, , of the ECEF frame relative to the Inertial frame expressed in terms of NOLL frame coordinates is found by applying the transformation to the vector, , of Equation 1. That is,

 (2-6)
or more compactly as,
 (2-7)

Rotation of the NOLL frame relative to the ECEF frame is a consequence of aircraft motion. The angular velocity of the NOLL frame relative to the ECEF frame is a function of the angular rates and . Expressed in NOLL frame coordinates, this angular velocity is,

 (2-8)
or,
 (2-9)
Equations 2-7 and 2-9 will be needed in subsequent steps (Add link to Equations of Motion) to write the equations of motion for the aircraft expressed in the NOLL frame.

## Velocity-Oriented Frame of Reference

Resolution of the aircrafts aerodynamic and propulsive forces is most easily accomplished in a frame in which one of the coordinate axes is aligned with the aircrafts velocity vector. The Velocity-Oriented (VO) frame is created to provide just such a reference system. The VO frame is obtained from the NOLL frame in two successive rotations as illustrated in Figure 2.3. First, the axes are rotated clockwise about through aircraft heading, , obtaining the intermediate coordinate axes, . The axes are then rotated counter-clockwise about through the flight path angle, , to obtain the Velocity-Oriented axes, . Components of vectors expressed relative to the VO frame are related to components relative to the NOLL frame through the transformation matrix, in the equation,

 (2-10)
where,
 (2-11)

## Body-Fixed Frame of Reference

The axes of the Body-Fixed (BODY) frame of reference is fixed in the aircraft. Its origin is coincident with the origins of both the NOLL and the VO frames, the aircrafts center of mass. The orientation of its coordinate axes, , is depicted in Figure 2.4. The plane is the aircrafts plane of symmetry, with the positive axis directed toward the aircrafts nose, and the positive axis directed downward. The positive axis is directed outward, toward the aircrafts right wing. The transformation of the NOLL frame into the BODY frame is effected by rotating the NOLL axes successively through aircraft roll, pitch, and heading, (i.e., ) whose overall effect is encapsulated in the transformation equation,

 (2-12)
where,
 (2-13)

## Geodetic Coordinate System

In addition to the five frames already described, an ellipsoidal model of the earth is used to relate geodetic latitude, , and longitude, , to ECEF position coordinates. The ellipsoidal model of the earth assumes the earth to be an oblate spheroid with equatorial radius, a, and polar radius, b. The relationship between geodetic coordinates and coordinates of the ECEF frame of reference is graphically depicted in Figure 2.5.

The earth model ellipsoid is the surface formed by rotating an ellipse about its semi-minor axis. The semi-minor axis of the rotated ellipse is chosen to lie along the earths polar axis, . The intersection of the earth model ellipsoid with a plane perpendicular to the polar axis is a circle. The radius of a circular intersection is referred to as the normal radius of curvature. The normal radius of curvature is a function of latitude. It is a quantity of interest because it is needed in the conversion of coordinates between the rectilinear ECEF and the curvilinear geodetic coordinate systems.

The coordinates of a point with latitude,, longitude, , and altitude, h (in feet), relative to the ECEF frame are given by Equations 2-14 through 2-16. The conversion factor, , is used to convert nautical miles to feet (Add link to constants index).

 (2-14) (2-15) (2-16)

The normal radius of curvature, , is given by Equation 2-17,

 (2-17)
where a is the length of the earth model ellipsoids semi-major axis, and e is its eccentricity.

The ECEF coordinates of a point, , are converted to curvilinear geodetic coordinates using Equations 2-18 and 2-19,

 (2-18) (2-19)
and where,
 (2-20) (2-21) (2-22) (2-23)
These equations assume that the length of the ellipsoids semi-major and semi-minor axes are expressed in units of nautical miles. The constant, , converts lengths expressed in feet to nautical miles.