[ODE] ODE's Jacobian matrices

[Henri Hakl]

I'm not entirely on the ball about this but:

The orientation matrix does contain 9 variables, but they are already
constraint regarding themselves - in other words they only possess 3
dimensions of freedom.

Keep in mind - the orientation matrix must obey the following:

- components of each row(column?) must be of unit length
- columns(rows?) are orthogonal to each other

These equate to 6 constraints, thus the 9 variables actually only
describe 3 (9-6) dimensions of freedom.

Thus everything works out as it should...

  Henri


-----Original Message-----
From: ode-bounces at q12.org [mailto:ode-bounces at q12.org] On Behalf Of
Aaron Dwyer
Sent: Sunday, September 28, 2003 9:51 AM
To: ode at q12.org
Subject: [ODE] ODE's Jacobian matrices

> "first create a position constraint equation C(q) 
> = 0, then take the time derivative of that" - 
> isn't the "time derivative" spoken of actually
> the partial derivatives wrt time of C(q)?

    I see what you mean, Gary.  That's where my head
starts hurting :)  Rigid bodies have 6 degrees of
freedom.  However, q is usually a position vector plus
an orientation quaternion (7 variables) or, as in the
case of the ball joint example in the joint creation
doc, a position vector plus an orientation matrix (12
variables).  Orientation expressed as quaternions and
matrices are examples of "redundant coordinates",
correct?

    Now, if I took the partial derivative of C wrt
these 12 variables (3 position vector and 9
orientation matrix), I'd end up with a much wider
matrix than I should have...but we need 6 lagrange
multipliers per body, not 12.  The position constraint
function C is a function of 12 variables, but dC/dt is
a function of 6.  That is where my confusion is
rooted, I think.

    Last, when you say "partial derivatives wrt time",
why would you say partial derivative when time is only
a single variable?  Or is it because time doesn't
appear explicitly in the constraint equation, but
instead, several variables which vary with time
appear?

-Aaron
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