Basically, gimbal lock is caused by the nonexistence of any map SO(n) —> T^N such as I described. In fact, such a map would have to be of degree = 1. A continuous map SO(n) —> T^k of degree 1 cannot exist unless n = k = 1. —Dan
On May 18, 2016, at 5:24 PM, Fred Lunnon <fred.lunnon@gmail.com> wrote:
Gimbal lock is another relevant issue I have never understood to my own satisfaction, although at one point I remember discussing it with a software engineer and coming to the conclusion that it constitutes an excellent reason for religiously avoiding the Euler angles he had been unsuccessfully attempting to deploy. Especially if you're designing (say) a transmission systems for a helicopter ...
But it seems this business is something different again. I followed up the Wikipedia reference to Anderson (2000) at http://www.netlib.org/lapack/lawnspdf/lawn150.pdf where it appears that in an attempt to avoid numerical overflow, software adopts a more complicated method of determining the angle which results in the square root acquiring an unpredictable sign, so that the rotation matrix computed is a discontinuous function of the input isometry.
Algorithm 4 on page 10 employs a 4-way branch to correct this in turn --- nothing is simple!
Fred Lunnon
On 5/18/16, Dan Asimov <asimov@msri.org> wrote:
I think that refers to finding a continuous map from SO(n) to the torus
T^N = (S^1)^N
(N = (n^2-n)/2) of the same dimension (where S^1 is the unit circle), such that the angles in the torus, when plugged into Givens rotations in a fixed order of indices, give the same element of SO(n) that one started with. (The problem, when this does not happen in 3D, is called gimbal lock.)
—Dan
On May 18, 2016, at 2:29 PM, Fred Lunnon <fred.lunnon@gmail.com> wrote:
The Wikipedia page paragraph "Stable calculation" discusses tinkering with the sign of the rotation angle in order to ensure "continuity". Does anyone understand what this is about?
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