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Unformatted text preview: 9 KINEMATICS OF MOVING FRAMES 67 9 KINEMATICS OF MOVING FRAMES An understanding of inertial guidance systems for navigation of vehicles and robots requires some background in kinematics. Central in our discussion is the use of multiple reference frames. Such frames surround us in our daily lives: • Earth latitude and longitude • Forward, backward motion relative to current position • Right, left motion • Axes printed on an inertial measurement unit • Vehicle-referenced coordinates, e.g., relative to the centroid We first describe how to transform vectors through changes in reference frame. Considering differential rotations over differential time elements gives rise to the concept of the rotation vector, which is used in deriving inertial dynamics in a moving body frame. 9.1 Rotation of Reference Frames A vector has a dual definition: It is a segment of a a line with direction, or it consists of its projection on a reference system xyz , usually orthogonal and right handed. The first form is independent of any reference system, whereas the second (in terms of its components) depends directly on the coordinate system. Here we use the second notation, i.e., x is meant as a column vector, whose components are found as projections of an (invariant) directed segment on a specific reference system. We denote through a subscript the specific reference system of a vector. Let a vector ex- pressed in the inertial (Earth) frame be denoted as x , and in a body-reference frame x b . For the moment, we assume that the origins of these frames are coincident, but that the body frame has a different angular orientation. The angular orientation has several well-known descriptions, including the Euler angles and the Euler parameters (quaternions). The former method involves successive rotations about the principal axes, and has a solid link with the intuitive notions of roll, pitch, and yaw. One of the problems with Euler angles, however, is that for certain specific values the transformation exhibits discontinuities (as will be seen below). Quaternions present a more elegant and robust method, but with more abstraction. We will develop the equations of motion here using Euler angles....
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This note was uploaded on 11/29/2011 for the course CIVIL 1.00 taught by Professor Georgekocur during the Spring '05 term at MIT.
- Spring '05