This section defines various frames-of-reference that are commonly used in navigation system applications.

## Inertial Frame

**An inertial frame is** a reference frame in which Newton’s laws of motion apply. An inertial frame is therefore not accelerating, but may be in uniform linear motion. The origin of the inertial coordinate system is arbitrary, and the coordinate axis may point in any three mutually perpendicular directions. All inertial sensors produce measurements relative to an inertial frame, resolved along the instrument sensitive axis.

**For discussion purposes it** is sometimes convenient to define an Earth centered inertial (ECI) frame which at a specified initial time has its origin coincident with the center of mass of the Earth, see Figure 2.3. At the same initial time, the inertial x and z axes point toward the vernal equinox and along the Earth spin axis, respectively. The y-axis is defined to complete the right-handed coordinate system. The axes define an orthogonal coordinate system. Note that the ECEF frame, defined in Section 2.2.2, rotates with respect to this ECI frame with angular ratetherefore, in the ECI frame the angular rate vector is

**Figure 2.3: Rotation of the ECEF frame with respect to an Earth-centered inertial frame. The vectors****define the axes of the ECI frame.**

The vectordefines the x-axis of the ECEF frame.

## Earth Centered Earth Fixed (ECEF) Frames

This frame has its origin fixed to the center of the Earth. Therefore, the axes rotate relative to the inertial frame with frequency

due to the 365.25 daily Earth rotations per year plus the one annual revolution about the sun. Relative to inertial frame, the Earth rotational rate vector expressed relative to the ECEF axes is

**The Earth’s geoid is usually** approximated as an ellipsoid of revolution about its minor axis. A consistent set of Earth shape (i.e., ellipsoid) and gravitation model parameters must be used in any given application. Therefore, the value for in eqn. (2.2) should only be considered as an approximated value. Earth shape and gravity models are discussed in Section 2.3. Due to the Earth rotation, the ECEF frame-of-reference is not an inertial reference frame. Two common coordinate systems for the ECEF frame-of-reference are discussed in Section 2.3.

## Geographic Frame

**The geographic frame is defined locally,** relative to the Earth’s geoid. The origin of the geographic frame moves with the system and is defined as the projection of the platform origin P onto the reference ellipsoid, see the left portion of Figure 2.2. The geographic z-axis points toward the interior of the ellipsoid along the ellipsoid normal. The x-axis points toward true north (i.e., along the projection of the Earth angular rate vectoronto the plane orthogonal to the z-axis). The y-axis points east to complete the orthogonal, right-handed rectangular coordinate system.

**Since the origin of the** geographic frame travels along with the vehicle, the axes of the frame rotate as the vehicle moves either north or east. The rotation rate is discussed in Example 2.6. Because the geographic frame rotates with respect to inertial space, the geographic frame is not an inertial frame.

**Two additional points** are worth specifically stating. First, true north and magnetic north usually are distinct directions. Second, as illustrated by the exaggerated ellipse in the left portion of Figure 2.2, the normal to a reference ellipsoid (approximate Earth geoid) does not pass through the center of the ellipsoid, unless the platform origin P is at the equator or along the Earth spin axis.

## Geocentric Frame

**Closely related to the geographic** frame-of-reference is the geocentric frame. The main distinction is that the geocentric z-axis points from the system location towards the Earth’s center. The x-axis points toward true north in the plane orthogonal to the z-axis. The y-axis points east to complete the orthogonal, right-handed rectangular coordinate system. Like the geographic frame, the axes of the geocentric frame rotate as the vehicle moves north or east; therefore, the geocentric frame is not an inertial frame.

## Local Geodetic or Tangent Plane

**The local geodetic frame is the north,** east, down rectangular coordinate system we often refer to in our everyday life (see Figure 2.4). It is determined by fitting a tangent plane to the geodetic reference ellipse at a point of interest. The tangent plane is attached to a fixed point on the surface of the Earth at some convenient point for local measurements. This point is the origin of the local frame. The x-axis points to true north. The z-axis points toward the interior of the Earth, perpendicular to the reference ellipsoid. The y-axis completes the right-handed coordinate system, pointing east.

**For a stationary system,** located at the origin of the tangent frame, the geographic and tangent plane frames coincide. When a system is in motion, the tangent plane origin is fixed, while the geographic frame origin is the projection of the platform origin onto the reference ellipsoid of the Earth. The tangent frame system is often used for local navigation (e.g., navigation relative to a runway).

**Figure 2.4: Local geodetic or tangent plane reference coordinate system in relation to the ECEF frame.**

## Body Frame

**In navigation applications,** the objective is to determine the position and velocity of a vehicle based on measurements from various sensors attached to a platform on the vehicle. This motivates the definition of vehicle and instrument frames-of-reference and their associated coordinate systems.

**The body frame is rigidly attached** to the vehicle of interest, usually at a fixed point such as the center of gravity. Picking the center of gravity as the location of the body frame origin simplifies the derivation of the kinematic equations [90] and is usually convenient for control system design. The u-axis is defined in the forward direction of the vehicle. The w-axis is defined pointing to the bottom of the vehicle and the w-axis completes the right-handed orthogonal coordinate system. The axes directions so defined (see Figure 2.5) are not unique, but are typical in aircraft and underwater vehicle applications. In this text, the above definitions will be used. In addition, the notation [u, w,w] for the vehicle axes unit vectors has been used instead of [x, y, z], as the former is more standard.

**Figure 2.5: Top view of vehicle (body) coordinate system.**

**As indicated in Figure 2.5,** the rotation rate vector of the body frame relative to inertial space, resolved along the body axis is denoted by where p is the angular rate about the u-axis (i.e., roll rate), q is the angular rate about the w-axis (i.e., pitch rate), and r is the angular rate about the w-axis (i.e., yaw rate). Each angular rate is positive in the right-hand sense. The body frame is not an inertial frame-of-reference.

## Platform Frame

**This topic will only discuss** applications where the sensors are rigidly attached to the vehicle. For inertial navigation, such systems are referred to as strap-down systems. Although the sensor platform is rigidly attached to the vehicle, for various reasons the origin of the platform frame may be offset or rotated with respect to the origin of the body frame. The origin of the platform coordinate frame is at an arbitrary point on the platform. The platform frame axes are defined to be mutually orthogonal and right-handed, but their specific directions are application dependent. Often the rotation matrixis constant and determined at the design stage.

## Instrument Frames

**Typically an instrument frame-of-reference** is defined by markings on the case of the instrument. Sensors within the instrument resolve their measurements along the sensitive axes of the instruments. Ideally, the instrument sensitive axes align with the instrument frame-of-reference; however, perfect alignment will not be possible under realistic conditions. In addition, the sensitive axes may not be orthogonal. Instrument manufacturers may put considerable effort into in-factory calibration and orthogonalization routines in addition to temperature, linearity, and scale factor compensation. Such manufacturer defined compensation algorithms will be factory preset and programmed into the instrument. Depending on the desired level of performance, the navigation system designed may have to consider additional instrument calibration either at system initialization or during operation.

Even though the transformation of sensor data from the instrument frame to the platform frame is not necessarily a rotation, for consistency, we will use a similar notation. For example, the transformation of ac-celerometer data to platform frame will be denoted by

## Summary

**A natural question at the** beginning of this section is why are the various coordinate systems required? The answer is that each sensor provides measurements with respect to a given reference frame. The fact that the sensor-relevant coordinate frames are typically distinct from the navigation frame can now be made more concrete.

**The GPS system provides** estimates of an antenna position in the ECEF coordinate system, vision and radar provide distance measures in a local instrument-relative coordinate system, accelerometers and gyros provide inertial measurements expressed relative to their instrument axes. Given that different sensors provide measurements relative to different frames, the measurements in different frames are only comparable if there are convenient means to transfer the measurements between the coordinate systems. For example, a strap-down GPS aided INS system performing navigation relative to a fixed tangent plane frame-of-reference will typically:

**1.** transform acceleration and angular rate measurements to platform coordinates;

**2.** compensate the platform angular rate measurement for navigation frame rotation;

**3**. integrate the compensated platform frame angular rates to maintain an accurate vector transformation from platform to navigation coordinates;

**4.** transform platform frame accelerations to tangent plane using the transformation from step 3;

**5.** integrate the (compensated) tangent plane accelerations to calculate tangent plane velocity and position;

**6.** use the position estimate to predict the GPS observables;

**7.** make GPS measurements, compute the residual error between the predicted and measured GPS observables, and use these measurement residuals to estimate and correct errors in the sensed and calculated INS quantities;

**8.** transform the vehicle inertial measurements and state variables that are estimated above to frames-of-reference (e.g., body) that might be desired by other vehicle systems (e.g., control or mission planning).

**A similar procedure** is followed for other aided navigation systems. The necessary coordinate system transformations and the algorithms for their on-line calculation are derived in the following sections.