In one design of inertial guidance system, three mechanical spinning gyroscopes with orthogonal input axes are employed to stabilize a platform that remains fixed in inertial space even as the craft rotates around it. Gimbals isolate the platform from the craft. Angle encoders on the gimbals provide orientation information to the navigational controller. The gyros output a signal when they sense a rotation about their input axes, and the gimbals can be torqued to null out that signal and maintain the platform stability. Nominally, three gimbals, initially along orthogonal axes, suffice to compensate for arbitrary orientation. However, if the middle gimbal rotates 90 degrees, the inner and outer gimbal axes become coincident, and control is lost. This is called gimbal lock. Systems that must navigate through gimbal lock make use of a forth gimbal to maintain control. A completely different methodology is strapdown inertial guidance. Here, the inertial platform is strapped to, and rotates with, the craft, so gimbal lock can't happen. With mechanical gyros as the rotation sensor, the gyros themselves are torqued to maintain a null output signal, and the torque signal provides the rotational information to the navigation controller. There are also optical gyros that use counter-rotating light beams (Sagnac interferometer). Strapdown inertial navigation is necessarily dependent on the use of a computer, but this allows for clever designs. One such redundant system uses six gyros and six accelerometers with axes along the face normals of a dodecahedron. The unit could function with the loss of three of either type of sensor. -- Gene From: Fred Lunnon <fred.lunnon@gmail.com> To: math-fun <math-fun@mailman.xmission.com> Sent: Wednesday, May 18, 2016 5:24 PM Subject: Re: [math-fun] Orthogonal group generators 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?