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# lec1 - 1 1 1.1 KINEMATICS OF MOVING FRAMES Rotation of...

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1 1 KINEMATICS OF MOVING FRAMES 1.1 Rotation of Reference Frames We denote through a subscript the specific reference system of a vector. Let a vector ex- pressed in the inertial frame be denoted as γx , and in a body-reference frame γx 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 principle axes, and has a solid link with the intuitive notions of roll, pitch, and yaw. One of the problems with Euler angles is that for certain specific values the transformation exhibits discontinuities. Quaternions present a more elegant and robust method, but with more abstraction. We will develop the equations of motion using Euler angles. Tape three pencils together to form a right-handed three-dimensional coordinate system. Successively rotating the system about three of its own principal axes, it is easy to see that any possible orientation can be achieved. For example, consider the sequence of [yaw, pitch, roll]: starting from an orientation identical to some inertial frame, rotate the movable system about its yaw axis, then about the new pitch axis, then about the newer still roll axis. Needless to say, there are many valid Euler angle rotation sets possible to reach a given orientation; some of them might use the same axis twice. x x’ y’=y’’ y x’’=x’’’ z’’ z’’’ y’’’ z=z’ Figure 1: Successive application of three Euler angles transforms the original coordinate frame into an arbitrary orientation. A first question is: what is the coordinate of a point fixed in inertial space, referenced to a rotated body frame? The transformation takes the form of a 3 × 3 matrix, which we now derive through successive rotations of the three Euler angles. Before the first rotation, the 0 b = γx . Now rotate the body-referenced coordinate matches that of the inertial frame: γx movable frame yaw axis ( z ) through an angle δ . We have cos δ sin

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lec1 - 1 1 1.1 KINEMATICS OF MOVING FRAMES Rotation of...

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