Abstract
Six parameters (degrees of freedom) are required and sufficient to completely describe the movement of an object in space: three describe the 3-D position of the object, and three the 3-D orientation, often referred to as “attitude” in aeronautics. When describing movements that are less than a few kilometers, we often use space-fixed, Cartesian coordinate systems. In these systems, the orientation of each axis is the same for each point in space, and for all time.
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Notes
- 1.
An inertial frame is a frame of reference in which a body remains at rest or moves with a constant linear velocity unless acted upon by forces. An inertial reference frame does not have a single, universal coordinate system attached to it: positional values in an inertial frame can be expressed in any convenient coordinate system. In other words, an inertial frame is a frame of reference where the laws of inertia apply—there is no requirement for specific coordinates.
- 2.
Note for Matlab users: here and in the following, the dash in \(\mathbf {p}'\) does NOT mean the vector \(\mathbf {p}\) transposed, but rather the vector \(\mathbf {p}\) rotated!
- 3.
Appendix A.3.3 contains the proof that the body-fixed representation of rotations uses the inverse (i.e., the transpose) rotation matrix compared to the space-fixed representation.
- 4.
The required work to find those matrices is much reduced using the symbolic computation packages offered by many scripting languages. For Python, the implementation is shown in Appendix C.4.2.
- 5.
The expression Euler angles should be used very carefully: sometimes, these angles represent the Euler sequence, but often that expression is also applied when the nautical sequence is actually used!
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Haslwanter, T. (2018). Rotation Matrices. In: 3D Kinematics. Springer, Cham. https://doi.org/10.1007/978-3-319-75277-8_3
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DOI: https://doi.org/10.1007/978-3-319-75277-8_3
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