I hope this question is appropriate here!
I and a friend at work are trying to understand Sebastian Madgwick's paper, "An efficient orientation for inertial and inertial/magnetic sensor arrays" (paper linked here.) Our goal as I understand it is to implement the algorithm for a hardware bracelet in order to tell its orientation in space with a clear reference to earth i.e. canonical x, y, z axes. (As far as I know -- I'm just trying to help out with the math :)
The article uses quaternions as a way to model rotations on R^3 as well as to embed vectors in R^3. Sedgwick gives an analysis whereby he appears to use gradient dissent to minimize error. between I beleve the computed and actual position of the sensors, thus yielding a high accuracy result. What has confused us is how he arrives at equation (39). His gradient dissent algorithm seems to no longer need to account for the prior step in its algorithm. It seems like he is making an argument that the RHS of (39) is so large that we no longer need to account for the prior step in the algorithm, but intuitively this seems strange (couldn't you overshoot the minimum you are looking for by not accounting for the last value you were at? Especially when the increment is considered to be very large?)
I'd also find any references that make thinking about this sort of thing -- quaternions as rotations, gradient dissent and Jacobians / differentiation in this context -- easier. I woke up this morning reminded of Schey's work, "div, grad, curl and all that", which I used when teaching multivariable calc once, but I think it doesn't quite get into this detail? That was 10 years ago so I may just misremember :).
Any ideas would be welcome, either about the paper specifically or just references to better understand this domain. Also if this question is too applied for MathOverflow, please let me know and I will happily move it. (Also any suggested tags would be appreciated.)
Thank you!!! -Michael
Maybe this book will be helpful: Visualizing Quaternions, by Andrew J. Hanson