Navigation Reference Frames

Navigation systems require well-defined reference frames. Reference frames associate the origin of a coordinate system with a well-defined location. The location may be fixed, as in the center of the earth, or it may be movable. The orientation of a reference frame must also be specified. The axes of a frame may be defined to always point to a particular location, such as the earth’s prime meridian. A frame axis may also be defined to match the physical dimensions of a vehicle, or lie along a particular axis, such as the earth’s spin axis. Navigation systems use a set of frames to measure and track the translation and orientation of frames relative to each other. Some frames also provide the basis for common navigation values, such as longitude and latitude.

An inertial reference frame is a frame that is not accelerating. The inertial frame may, however, experience uniform linear motion [17]. The reference frame origin may be placed at any location. The axes of the reference frame will follow the right-hand rule of the standard 3-axis coordinate system. The earth centered inertial (ECI) reference frame is defined with the origin at the center of the earth. The z-axis

extends along the center of rotation of the earth, the x-axis points to the vernal equinox at a specified initial time, and the y-axis completes the right-hand rule[17]. This ECI reference frame does not rotate with the earth, hence a point on the earth’s surface will exhibit a constant angular rotation and velocity relative to this frame. Figure 2.5 illustrates the ECI frame.

y z

x

Vernal Equinox

Figure 2.5: An illustration of the ECI coordinate system.

The astute reader may note that the center of the earth undergoes constant motion, but this motion is NOT linear. The earth actually rotates around the sun once every 365 1₄ days. The requirements of a particular system will determine how much motion is relevant, and how much can be ignored. A spacecraft traveling to the outer reaches of the solar system may need to take the rotation of the solar system into account. In this paper, the ECI frame can be considered as a true inertial frame. The earth centered earth fixed (ECEF) reference frame is also defined with the origin at the center of the earth. This reference frame, however, will be defined such that the z-axis extends along the rotation axis, but the x-axis crosses the prime meridian. The y-axis is again defined to complete the right-hand rule. This reference frame is illustrated in Figure 2.6. A point on the surface of the earth will not move relative to the ECEF reference frame.

y z

x

Prime Meridian

Figure 2.6: An illustration of the ECEF coordinate system.

The ECEF reference frame exhibits a constant angular rate relative to the ECI reference frame. This angular rate is defined as ~ω_iei . The subscript indicates that this is the rotation of the ECEF frame relative to the ECI frame. The superscript indicates that the value is represented in the ECI frame. The vector representation for this vector would be ~ωi

ie = [ 0 0 ωei ] T_.

We generally think of navigation in terms of north, east, south, or west. The ECEF frame is not very convenient for calculating these navigation values. A vehicle traveling east in the ECEF frame would have a constant angular velocity around the z-axis, and varying amounts of translation in the X and Y axes, depending on the longitudinal position at a given time. Navigation frames are defined to better support more typical north, east navigation parameters.

2.2.4.1 Geographic and Geocentric Frames

The geographic and geocentric frames are closely related. Both move with the navigating vehicle. The geographic frame is aligned such that the origin sits on the surface of the earth’s geoid directly below the navigating vehicle. The z-axis points down and is normal to the earth’s surface. The geocentric frame also follows the

navigating vehicle, but the z-axis points to the center of the earth. For both frames, the x-axis points toward true north, and the y-axis points east [17].

2.2.4.2 Local Navigation Frames

For local navigation, a local geodetic or tangent frame is defined. This frame differs from the geographic frame in that its origin is fixed at some point on the earth’s surface. The frame serves as a convenient reference point for the system. The exact reference point may be selected according to the application. The point may be the end of a runway, a particular city in North America, or it may be the center of the desk you are working on. Since the tangent frame does not move, vehicle motion and position may be measured relative to this fixed point. The axes of the tangent frame are generally defined to be suitable for navigation. As such, the z-axis points downward, the x-axis points to magnetic north, and the y-axis points east. This frame is often referred to as the north east down (NED) reference frame. Common navigational thinking would mean that we travel north in the positive direction, or east in the positive direction. The choice of z-axis direction is used to make the x-axis and y-axis consistent with common navigational thinking and to maintain the right-hand rule.

2.2.4.3 Instrument and Body Frames

A strapdown IMU measures acceleration and angular rate relative to the physical instrument. The axis of the measurement is the instrument axis, and is referred to as the instrument frame. The vehicle reference frame is called the body frame. Ideally, the instrument frame would be perfectly aligned with the vehicle axes, but generally some variation in the alignment is present. The offset between the instrument frame and the body frame is usually a fixed X,Y,Z coordinate offset, and a fixed set of φ,

frame relative to the vehicle’s center of mass. The vehicle’s center of mass is the origin of the body frame.