[ODE] Re: Iterative solution

[Aaron Dwyer]

    Concerning the 6 DOF or less 2-body substeps in
the iterative solver and whether LCP is necessary at
that point... (warning, I am not a math major and may
be using some terms wrong, hopefully I can convey my
idea).  
    Would you agree that we are working with a known
set of joint types, each with their own fixed jacobian
setup, and only coefficients in that jacobian are
varied?  If so, we are no longer working with an
arbitrary matrix, as we are with a general articulated
RB sytem.  Then for each joint type, it seems that we
could do math-on-paper to reduce each joint's small,
isolated linear system to a simple set of dependent
expressions to solve for the unknowns for each joint
type.  Only variables important to each joint's
particular jacobian will change in value, but not
placement in the equations (and therefore no complex
solving work should be necessary).  LCP or as I
mention later maybe conjugate gradient (?) or some
other linear system solver could still be used for
arbitrary constraints, but known ones should be
reduceable, if I'm not completely wrong.

    I am studying dynamics intensively (but the above
shows that I may still be thrashing in the dark
somewhat) and would like to know where to read about
iterative methods, both the general math side of
things and the application specific side.  Is this a
form of "relaxation"?  I have only encountered
relaxation in the past with Dijkstra's Shortest Path
algorithm.  Is constraint enforcing in general a form
of optimization?

    I have a good understanding of single rigid
bodies, ordinary differential equations, and diff eq
solution approximation using Runge Kutta 4 (I wish I
paid better attention to Taylor Series in Calc 3!!)
and have created my own working implementation of
these.  I am now trying to learn about constrained
systems and enforcing constraints by first starting
with a survey of the methods.  I've learned about
reduced coordinate (fewer, more tightly coupled
equations, harder to reconfigure at runtime) and
vector formulations (many loosely coupled equations,
solvable by sparse matrix methods such as conjugate
gradient, easy to reconfigure at runtime, more
numerical error prone than reduced coord). 
Generalized coordinates make sense, and I'm about
halfway through Shabana's Computational Dynamics book,
with about 70% understanding.  Concerning ODE, where
does LCP fit into all of these categories?  From what
I can tell, Linear Complimentary Problem seems to be
yet another way to solve a certain class of Ax=b
problems where A and b must be of a certain format,
and x will have some unneeded byproducts in the end,
but I haven't seen a good general explanation of it so
I'm not sure.  It seems that eventually everybody uses
Lagrange Multipliers somewhere in the constraint
linear system.  I've also wondered if using Lagrange
Multipliers implies that you're using Lagrangian
Dynamics.  

BTW, hello Nate W, I had a great discussion with you
on runryder.com about blade moments, whether dynamic
balance matters, and flap/lead/lag :)

-Aaron Dwyer


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