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# lecture20 - Attitude parameters There are several dierent...

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Attitude parameters There are several different parameter sets to describe attitude, among them Euler angles (3–1–3, 3–2–1, etc.) direction cosines, Euler parameters/quaternions. Much as there are 6 quantities to minimally specify the orbital state of a satellite, there are 3 quantities to minimally specify the relation of one set of coordinate axes to another with common origin. This will describe the attitude of a rigid spacecraft. Rates are an additional three. L. Healy – ENAE404 – Spring 2007 – Lecture 20 (Apr. 10) 1

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Euler angle description The relative orientation of the body can be described with Euler angles. We’ve seen the “classical,” or 3–1–3 Euler angles, where ro- tation is first about the z axis, then about the x axis, then about the z axis, when we were do- ing transformations of the perifocal plane into IJK space, for conversion of orbital elements to IJK Cartesian. In fact, they are classical because they were in- vented by Euler in the early 1700s for precisely this purpose. There are other sets of angles, however, that are more suited to other pur- poses: you can use any of twelve possible sets 1–2–1, 1–2–3, 1–3–1, 1–3–2, 2–1–2, 2–1–3, 2–3–1, 2–3–2, 3–1–2, 3–1–3, 3–2–1, 3–2–3. Some people (like Curtis) reserve “Euler an- gles” for the 3–1–3 set, others use the term generically. L. Healy – ENAE404 – Spring 2007 – Lecture 20 (Apr. 10) 2
3–1–3 Euler angles Ref. Curtis Sec. 9.9 Previously, we saw the transformation from perifocal frame to inertial as a sequence of three rotations - ω , - i , - Ω about the third axis, the new first axis, then the new third axis. Generalize this to any orientation description with angles φ , θ , ψ . For attitude applications, there is another an- gle set that is more natural in some applica- tions. L. Healy – ENAE404 – Spring 2007 – Lecture 20 (Apr. 10) 3

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3–2–1 Euler, or Tait-Bryan, angles Ref.: Curtis Sec. 9.10. We can use another set of three rotations, this time about the third axis, then the new second axis, then about the new first axis. The most common application is for a plane-like craft for which the body frame would correspond to its symmetry axes as described before, and the reference frame would be RSW. We must do some reordering and sign changes; no real convention, but this form is common. You must figure out how rotations are defined when looking at something new; even within a given book, convention will change. L. Healy – ENAE404 – Spring 2007 – Lecture 20 (Apr. 10) 4
Yaw, pitch, roll Each of the angles has a name: Yaw rotation φ about ˆ k up-down axis, or steer- ing. Pitch rotation θ about the ˆ j cross-body “wing” axis. Roll rotation ψ about the ˆ i nose-tail axis. For “straight and level” flying, the most natu- ral external axis set satellite-centered RSW in which all angles are zero. The usual conven- tion is ˆ k = - ˆ R (positive to steer right) ˆ j = - ˆ W (positive if nose up) ˆ i = ˆ S (positive clockwise) Picture.

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lecture20 - Attitude parameters There are several dierent...

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