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This is my first message in this site. I'm a mechanical engineer with, amongst others, an interest in inertial navigation. I'm currently reading the book "Principles of GNSS, Inertial and Multisensor Integrated Navigation Systems" from Paul D. Groves. On the development of one equation (eq. 5.26 in the second edition of the book) I came across this step:

$$C_b^e (t) \exp⁡[(Ω_{ib}^b-Ω_{ie}^b ) τ_i ]=C_b^e (t) \exp⁡(Ω_{ib}^b τ_i )-C_b^e (t)[\exp⁡(Ω_{ie}^b τ_i )-I_3 ]$$

The $C$s are rotation matrices. The $\Omega$s are the skew-symmetric matrices associated to rotation velocities between two reference systems. The sub- and superindexes $i$, $e$ and $b$ refer to $inertial$, $earth$ and $body$ reference systems. $t$ is time, $\tau_i$ is the time lapse between two sample/integration steps.

What I can't understand is how the matrix exponential of the difference of two matrices can convert in the difference of the matrix exponentials of those two matrices (plus the identity matrix). I've seen this operation in several places on the book, so I doubt it's an erratum. I guess one might prove it by means of the series expansion of the matrix exponential function and some property of skew-symmetric matrices, but I didn't get very far this way.

Can somebody please cast some light on this step? Thanks in advance!

Greetings

Guillermo

Rodrigo de Azevedo
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Guillermo Benito
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  • Unless there's a typo in the indices, three different $\Omega$ matrices are involved. If so, there's no chance of checking this identity without knowing what they are. – joriki Jul 09 '15 at 08:32
  • Thanks for your comment, joriki. I typed the equation as it appears in the book. But you're right, there must be a typo in one of the superscripts. I even wrote yesterday an email to the author pointing at that - just completely forget about it as I published the question here today, sorry. All $\Omega$'s should be resolved on the same reference system, in this case on the b (body) - that's what the superscripts stand for. One of them appears in the book (and in my first edition of my question) resolved on the earth, $\Omega_{ie}^e$, which should be $\Omega_{ie}^b$ instead.I've just edited it. – Guillermo Benito Jul 09 '15 at 08:44

1 Answers1

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If we proceed by orders of $\tau_i$, the zeroth and first orders come out right, and the second order yields

$${\Omega^b_{ib}}^2-\Omega^b_{ib}\Omega^b_{ie}-\Omega^b_{ie}\Omega^b_{ib}+{\Omega^b_{ie}}^2={\Omega^b_{ib}}^2-{\Omega^b_{ie}}^2\;,$$

or

$$\Omega^b_{ib}\Omega^b_{ie}+\Omega^b_{ie}\Omega^b_{ib}=2{\Omega^b_{ie}}^2\;.$$

This does not hold for skew-symmetric matrices in general (e.g. it doesn't hold for the generators of rotations about the $x$ and $y$ axes), so the equation is either wrong or depends on specific properties of the matrices.

joriki
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