Abstract
The purpose of this chapter is to present a collection of vector and quaternion identities that are useful for control and estimation computations. Many of them used throughout this text. Several appear in Chap. 2 but are repeated here for convenience.
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Notes
- 1.
Some authors define the attitude matrix without the factor of ∥q∥−2 even if the quaternion is not normalized, with the result that the attitude matrix is not guaranteed to be orthogonal.
- 2.
These equations are true even though \(A(\mathbf {q})\ne A(\bar {\mathbf {q}})\) because A(δq)δq1:3 = δq1:3.
- 3.
An orthogonal transformation would preserve the norm, giving ∥δωI∥ = ∥δω∥, and it is not difficult to show that \(\| \boldsymbol {\delta }\boldsymbol {\omega }_{{I}}\|{ }^2=\| \boldsymbol {\delta }\boldsymbol {\omega }\|{ }^2+2{\boldsymbol {\omega }} ^T[I_3 - A(\boldsymbol {\delta }\boldsymbol {\omega })]{\bar {\boldsymbol {\omega }}}\).
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Markley, F.L., Crassidis, J.L. (2014). Quaternion Identities. In: Fundamentals of Spacecraft Attitude Determination and Control. Space Technology Library, vol 33. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-0802-8_8
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