[Axiom-developer] address@hidden: Denavit-Hartenberg Matrices, Solid Modeling and CAS]

[root]

Journaling some comments made at ECCAD 2008:

Re: our post-panel discussion on "killer apps"....

<http://axiom.axiom-developer.org/axiom-website/dhmatrix.spad.pdf>
This is an axiom literate program from the Axiom algebra library
that implements DH-matrices.

The basic idea is described in terms of robots. You attach a matrix to
the base of the robot (called the world frame) and then attach a
matrix at each joint that describes the kind of motion the joint can
make (such as a sliding joint which might update the length along a
single axis). This is computed by multiplying the two matrices.

Instead of using a robot you use a steel beam. Attach the identity
matrix to the fixed end of a steel beam projecting out from a wall.
Attach another matrix at the other end of the beam.  Then a twist
applied to the beam is just a rotation around the long axis which
would be shown by multiplying the two matrices.

By applying these DH matrices in sequence you can compute the final
position of any sequence of beams, for instance. Not shown in the
literate pamphlet, but described in Richard Paul's book, are things
like the Jacobian DH matrix which will allow you to model static
and dynamic loads such as gravity and wind.

Combining the DHMatrices and a CAS one could symbolically model the
wing of a aircraft, for example, and then supply numeric values for
the variables in the resulting equations looking for where the values
exceed limits like the tensile strength of beam members. Thus, you
could symbolically predict failures.

So far I have not seen anyone apply a computer algebra system
to generate models of solid geometric structures. The usual method
of analysis is to use finite element analysis, a numeric technique.

Tim