Reference Frames
Navigation systems require the transformation of measured and computed quantities between various frames-of-reference. The purpose of this topic is to define the various frames and the methods for transforming the coordinates of points and representation of vectors between frames. Before proceeding to the main body of the topic, the next paragraph steps through the measurements and computations of a strapdown inertial navigation system (INS). The goals of this brief introduction are to define notation and to illustrate how the different frames-of-reference come into play in navigation applications.
Figure 2.1: High level block diagram of an inertial navigation system. For a vector, the superscript defines the frame-of-reference in which the vector is represented. For a matrix transformations between two frames-of-reference, the subscript defines the origination frame and the superscript defines the destination frame. The encircled symbol n represents a product.
A high level block diagram for a typical strapdown INS is shown in Figure 2.1. Before discussing its operation, it is necessary to briefly discuss notational conventions. As will be discussed in this topic, vectors have distinct coordinate representations in distinct reference frames. Therefore, a notation is required to record the frame of the representation. The reference frame in which a vector is represented is indicated by a superscript, for example v" is the velocity vector represented in the n or navigation frame. In the text of the topic, vectors and matrices will be represented in boldface. Rotational transformations will be represented by an R with a subscript and superscriptindicating the origin and destination frames-of-reference. For example,
represents the rotational transformation from the p or platform frame to the n or navigation frame. The various reference frames are defined in Section 2.2. Finally, certain quantities such as angular rates measure the rotational rate of one frame relative to another in addition to being represented in a specific reference frame. For example,
should be read as the angular rate of frame p with respect to frame i represented in frame p. With this notation, it is true that
Throughout the topic, all angular rates are defined to be positive in the right-handed sense. This means that if the thumb of the right hand points along the direction of the angular rate vector, then the fingers of the right hand will indicate the physical sense of the rotation. Alternatively, this can be stated that looking up along the angular rate vector, from the tail toward the head, the sense of the rotation is clockwise.
The INS in Figure 2.1 uses variables related to five different coordinate systems. The accelerometers measure
which is the platform acceleration relative to an inertial frame-of-reference, resolved in the accelerometer frame-of-reference (i.e., along the accelerometer sensitive axis). The ac-celerometer measurements are transformed into the platform frame using the (usually) constant calibration matrix
The gyros measure
which is the platform angular rate relative to the inertial frame-of-reference, as resolved in gyro frame (i.e., along the gyro sensitive axis). The gyro measurements are also transformed into the platform frame using the (usually) constant calibration matrix
The platform frame gyro measurements are processed to maintain the platform-to- navigation frame rotation matrix 
Finally, the platform-to-navigation frame rotation matrix
is used to transform the accelerometer measurements into the navigation frame where they are processed to determine navigation frame velocity and position.
This topic discusses the definition of, properties of, and transformation between frames-of-reference. Section 2.1 provides a short summary of some properties of reference frames and coordinate systems to simplify the discussion of the subsequent sections. Section 2.2 defines each frame-of-reference along with at least one definition of a suitable coordinate frame. Section 2.3 focuses on definition of and transformations between common coordinates systems for the important Earth Centered Earth Fixed frame-of-reference. Section 2.4 discusses the general approach to transforming points and vectors between rectangular coordinate systems. Section 2.5 discusses plane rotations with derivations of specific rotation matrices that are useful in navigation applications. Since navigation systems are freely moving, one navigation system may rotate relative to another. Therefore, Section 2.6 discusses the properties of time derivatives of vectors represented in and rotation matrices representing transformations between rotating reference frames. Section 2.7 discusses methods to compute the direction cosine matrix based on measurements of the relative angular rate between two reference frames.
As previously motivated, navigation systems involve variables measured in various frames-of-reference. To insure system interoperability and clear communication between engineers working with the system each frame and its properties should be clearly defined.
Unless otherwise stated, all rectangular frames-of-reference are assumed to have three axes defined to be orthonormal and right-handed. Let the unit vectors along the coordinate axes of a reference frame be I, J, and K. Orthonormality requires that
where K • I denotes the inner product between K and I. Right-handed implies that I x J = K.
Due to the gravitational effects near the surface of the Earth, it is often convenient to consider ellipsoidal coordinate systems in addition to rectangular coordinate systems. The Earth’s geoid is a hypothetical equipotential surface of the Earth’s gravitational field that coincides with the Earth’s mean sea level. By definition the gravity vector is everywhere perpendicular to the geoid. For analytic tractability the Earth’s geoid is usually approximated by a reference ellipsoid that is produced by rotating an ellipse around its semi-minor axis. Figure 2.2 depicts an ellipse that has much greater flatness than the that of the Earth, but the ellipse is useful for making a few important points. Most importantly, the normal to the ellipse, when extended towards the interior of the ellipse does not intersect the center of the Earth. See exercise 2.1. Since the gravity vector is (nominally) perpendicular to the ellipsoid, the gravity vector does not point to the center of the Earth. Also, as shown in the left portion of the figure, the latitude of a point P can be defined in two different ways. The geodetic latitude $ is the angle from the equator to the outward pointing normal vector with positive angles in the northern hemisphere. The geocentric latitude is the angle from the equator to the vector from the center of the Earth to P. Therefore, we can have geodetic or geocentric ellipsoidal coordinates. Also, we must be careful to clearly distinguish whether a local coordinate systems aligns with the geodetic or the geocentric normal.
There are also two common classes of methods for defining the origin O of a reference system on the surface of the Earth. As shown in the left portion of Figure 2.2, the origin of a reference frame can be defined as the projection of the point P onto the reference ellipsoid. In this case, the reference frame origin O moves as dictated by the horizontal portion of the motion of P. Alternatively, as shown in the right portion of the figure, the origin O may be fixed to the surface of the Earth at a point convenient for local reference, e.g., the end of a runway. In this case, the origin of the local reference frame does not move with P.
Figure 2.2: Left — Exaggerated ellipse depicting the difference between geodetic and geocentric latitude. Right — Depiction of a local frame-of-reference.
