View Full Version : 3-D vectors and inertial navigation systems

Herman J. Woltring
01-27-1992, 06:13 AM
Dear Biomch-L readers,

In another posting today, reference was made to RoboTech@uscvm.bitnet which,
it did appear, is not a list on robotics. Another query, onto the NA-net
list on Numerical Analysis last week was more succesful. This morning, I
received the latest issue with the response quoted below. Again, it appears
that similar things are (re)discovered in rather different fields, and that,
instead of reinventing the wheel and the gunpowder, it makes sense to look
around in other but potentially related disciplines ...

I have looked up the Bortz and Miller papers which provide some interesting
and complementary ideas. In particular, their interest is to assess attitude
angles from a measured rotation velocity vector; in addition to a concise
angular representation from a given attitude matrix, my interest is just the
opposite: to assess the rotation velocity and acceleration vectors from the
attitude matrix and/or angles.

Herman J. Woltring, Eindhoven/NL

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From: Daniel Johnson
Date: Mon, 20 Jan 1992 08:52:15 -0600
Subject: Re: 3-D Attitude "Vectors"
Sender: NA-net Numerical Analysis Digest, Vol. 92, Nr. 4

Herman J. Woltring asks about references to representing rotations as a
three-dimensional vector where the direction is the axis of rotation, and
the length is the amount of rotation. As a navigation house, we tend to
collect different ways of parametrizing rotations. We refer to his sug-
gestion as the "Bortz Rotation Vector", based on a paper by John Bortz in

A New Mathematical Formulation for Strapdown Inertial Navigation
John E. Bortz
IEEE Transactions on Aerospace and Electronic Systems
January 1971, Vol. AES-7, No. 1, pp 61-66

He refers to a report by J. Laning in 1949 which I have not looked at:

The vector analysis of finite rotations and angles
J. H. Laning, Jr.
MIT/IL Special Rept. 6398-S-3, 1949
Mass. Inst. of Tech., Cambridge

Some more recent references are

A New Strapdown Attitude Algorithm
Robin B. Miller
J. Guidance, Vol. 6, No. 4, July-Aug 1983, pp 287-291

An Accurate Strapdown Direction Cosine Algorithm
J. W. Jordan
NASA TN D-5384, Sept, 1969

The Bortz formulation is nice in that it has no singularities, has exactly
three free parameters, and has a closed form differential equation for the
rotation angle as a function of angular velocity. By taking lower order
approximations of that equation, it is possible to generate simple
algorithms of acceptable error for updating attitude given the angular
velocity. Typically it is used only for deriving the algorithms, however.
The actual attitude representations used internally in navigation systems
are still either the direction cosine matrix or quaternions.

Dr. Daniel P. Johnson Honeywell Systems and Research Center
e-mail: drdan@src.honeywell.com phone: 612-782-7427
US mail: MN65-2500, 3660 Technology Drive, Minneapolis, MN 55418