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 […]

# GPS with High Rate Sensors

## Reference Frame Definitions (GPS)

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 […]

## ECEF Coordinate Systems (GPS)

Two different coordinate systems are common for describing the location of a point in the ECEF frame. These rectangular and geodetic coordinate systems are defined in the following two subsections. Subsection 2.3.3 discusses the transformation between the two types of coordinates. Figure 2.6: ECEF rectangular coordinate system. ECEF Rectangular Coordinates The usual rectangular coordinate system […]

## Reference Frame Transformations (GPS)

This section presents methods for transforming points and vectors between rectangular coordinate systems. The axes of each coordinate system are assumed to be right-handed and orthogonal. Three dimensions are used throughout the discussion; however, the discussion is equally valid for The Direction Cosine Matrix Letrepresent a right-handed orthogonal coordinate system. Letbe a vector from the […]

## Specific Vector Transformations (GPS) Part 1

In Section 2.4, the transformation of vectors from frame a to frame b is shown to involve an orthonormal matrix denoted byThe elements of this matrix, called the direction cosine matrix, are the cosines of the angles between the coordinate axes of the two frames-of-reference. Although this appears to allow nine independent variables to defineorthonormality […]

## Specific Vector Transformations (GPS) Part 2

Transformation: Vehicle to Navigation Frame Consider the situation shown in Figure 2.13, which depicts two coordinate systems. The first coordinate system, denoted by (n, e, d), is the geographic frame. The second coordinate system, denoted by (u,v,w) is the vehicle body frame which is at an arbitrary orientation relative to the geographic frame2. The relationship […]

## Rotating Reference Frames (GPS)

As discussed relative to Figure 2.1, reference frames may be free to rotate arbitrarily with respect to one another. Consider for example the body frame moving with respect to the ECEF frame. The following subsections are concerned with frames-of-reference rotating with respect to one another. Direction Cosine Kinematics Section 2.5 showed that the transformation of […]

## Calculation of the Direction Cosine (GPS)

The previous sections have motivated the necessity of maintaining accurate direction cosine matrices. The following two subsections will consider two methods for maintaining the direction cosine matrix as the two reference frames experience arbitrary relative angular motion. Each technique relies on measuring the relative angular rate and integrating it (via different methods). Initial conditions for […]

## Continuous-Time Systems Models (GPS)

Deterministic Systems The quantitative analysis of navigation systems will require analytic system models. Models can take a variety of forms. For finite dimensional linear systems with zero initial conditions that evolve in continuous-time, for example, the ordinary differential equation, transfer function, and state space models are equivalent. The dynamics of the physical systems of interest […]

## State Augmentation (GPS)

In navigation applications, it will often be the case that we have multiple interconnected systems, each of which is modeled by a set of state space equations. For design and analysis, we want to develop a single state space model for the overall interconnected system. This is achieved by the process of state augmentation. This […]