Two USC engineer's find a "simple equation" that brings Newton's laws
of motion into the real world.


Answering a challenge more than two centuries old, two USC scientists
have discovered a simple equation that expands Newton's familiar laws
of motion into the real world of complex, "constrained" mechanical
systems.

Their discovery is reported in the November issue of the Proceedings
of the Royal Society of London first series.

Firdaus E. Udwadia, professor of mechanical engineering, civil
engineering, and decision systems, who discovered the relation in
collaboration with Robert E. Kalaba, professor of biomedical
engineering, economics, and electrical engineering, notes that the
discovery offers "a simple, aesthetic and thought-provoking
description of the world at a very fundamental level. This is about
the way systems of bodies move." The pair believe the equation will
have applications in fields such as manufacturing and industrial
design, including such problems as controlling the motion of robot
arms.

Kalaba and Udwadia say their work provides a complete and definitive
solution to a problem that has been attacked by the best scientific
and mathematical minds of the last two centuries, since the
publication of the classic Mecanique Analytique (Analytical
Mechanics) in 1788, by the the great French physicist Joseph Louis
Lagrange. In that work, Lagrange offered a brilliant but
frustratingly limited solution to the difficult problem of predicting
the motion of complex systems of multiple bodies mutually
constraining each others' freedom of movement.

Lagrange's system ingeniously solved one large class of these
problems, the motion of systems with so-called "holonomic
constraints" - with complete success, by a system still used
unchanged today. But for "non-holonomic constraints," his solution
was more problematic. In his system, success in predicting these
complex motions depended on finding the value of a set of parameters
he introduced, called "Lagrange multipliers." The technique for
finding the multipliers is "problem-specific" - it must be
individually tailored to each problem - and in systems of even modest
complexity, the values of the multipliers are difficult if not
impossible to obtain.

In ensuing years, such great scientists as Jacobi, Hamilton, Volterra
and Boltzmann labored to improve the system, or find an alternative.
The most successful alternative method, and the one most used
presently, was perfected in 1902 by the Frenchman Paul Appell, but
anticipated 20 years earlier by the American J. Willard Gibbs. It
replaces Lagrange multipliers with "quasi-coordinates" but has the
same drawback. It remains problem-specific; and complicated real
world systems in such areas as robotics and motion-tracking control
of machine tools are still for all practical purposes unsolveable.



Udwadia and Kalaba took an entirely different tack. They went back to
the problem in its original form, as it was when Lagrange attacked
it. Their analysis ignores the distinction between holonomic and
non-holonomic constaints. Instead, it projects what the system would
look like at each succeeding instant of time in the future if no
constraints at all were in place and each individual part of the
system were free to take the simple path Newton's second law predicts
for it.

Udwadia and Kalaba then pictured the difference between the predicted
free motion at each instant and the actual, constrained motion at the
same time and uncovered a central principle which they state in
following way in their Proceedings paper: "...at each instant, the
force of constraint acting on the particles of a system is directly
proportional to the extent to which the accelerations corresponding
to its unconstrained motion, at that instant, do not satisfy the
constraints. The matrix of proportionality is the weighted
Moore-Penrose generalized inverse of the weighted constraint matrix."
(The Moore-Penrose generalized inverse is a mathematical operator
invented by the two scientists whose name it bears.)

Kalaba and Udwadia say that their solution represents a major advance
over existing methods in that it does not need to be custom fitted to
each problem individually. At least in principle, it would be
possible to program a computer to analyze any given mechanical system
automatically, and accurately predict its motion. In fact, one
application of the Udwadia/Kalaba relation is likely to be easier and
more accurate computer simulation of the motion of complex mechanical
systems.



It is particularly satisfying," notes Udwadia, "that despite the
highly non-linear behavior of even the simplest mechanical system,
Nature behaves at each instant in time in a remarkably simple and
linear fashion, one that can be understood with the tools of linear
algebra. I find the simplicity with which nature emerges from the
equations to be deeply satisfying."

"We feel privileged to have made this discovery," says Kalaba.

The Proceedings of the Royal Society of London is, appropriately, the
journal which published Newton's "Principia Mathematica" more than
300 years ago.



[Photo:] Firdaus Udwadia: The equation offers "a simple, aesthetic
and thought-provoking description of the world at a very fundamental
level. This is about the way systems of bodies move."