From: dwells@nrao.edu (Don Wells)
Newsgroups: sci.math.num-analysis
Subject: Re: Freebody problem
Date: 25 Dec 1996 20:31:57 GMT
Mike McDermott <100447.764@CompuServe.COM> wrote:
>> I need to solve the equations of motion of a 3 dimensional free
>> body,
>> eg a tetrahedron, positively located by a total of six links from
>> its
>> vertices to six fixed points. Any ideas, references, etc? It has
>> been
>> applied to the platform of a flight simulator, so numerical
>> solutions must be known, but I can't find anything
>> published in that area.
This type of system is often called a Stewart(sp?) Platform. There is
an extensive literature on the subject, as you guessed, primarily in
areas of robotics and precision machine tool control. I am at home, so
I can't include sample citations in this followup.
The problem is solved by starting from the desired orientation of the
platform and solving for the actuator lengths. I.e., tilt and
translate the platform, get the new coordinates of the U-joint
connection points and compute the distances from those points to the
fixed U-joint points at the other end of the actuators; these
distances give you the lengths to command to get the desired platform
orientation. If you need to solve the problem in the other (backward)
direction, i.e. to get the platform orietation for a set of 6 lengths,
you find it by Newton-Raphson iterative numerical inversion of the
"forward" algorithm. It appears that, in many (most? all?) practical
cases, you don't need to compute the partial derivatives for each
iteration, that you can, in fact, use derivatives for the "home"
positional of the platform. I implemented such a solution for control
of an ellipsoidal mirror 8 meters in diameter which is moved by six
actuators. See:
ftp://fits.cv.nrao.edu/pub/gbt_actuators.tar.gz
--
Donald C. Wells Associate Scientist dwells@nrao.edu
http://fits.cv.nrao.edu/~dwells
National Radio Astronomy Observatory +1-804-296-0277
520 Edgemont Road, Charlottesville, Virginia 22903-2475 USA