Newbie question

[Southern.Cross]

Hello,

I'm new to sympy so please excuse me if this question is obvious to
the experienced.

In multibody kinematics there is a classic relationship between a
rotating body's angular velocity and the rotation matrices
representing the configuration at a given point in time. Basically, if
I am using a roll/pitch/yaw representation of the rotation then I have
three rotation matrices that comprise the full instantaneous rotation,

R = rotx(gamma) * roty(beta) * rotz(alpha)

Meaning, if I rotate the body by 'alpha' about a body fixed 'Z' axis,
followed by a rotation 'beta' about the once rotated body fixed 'Y'
axis, followed by a rotation 'gamma' about the twice rotated body
fixed 'X' axis... I now have a complete instantaneous rotation matrix
R.

The relationship of the angular velocity of the body is,

skew(w) = Rdot * R.transpose

so, a new skew symmetric matrix is created from the matrix product
between the time derivative of R and the transpose of R. (Since alpha,
beta, and gamma are all functions of time then R is a function of
time.)

Completing the above described equations in Mathematica does in fact
give me a skew-symmetric matrix. However, my first attempt in SYMPY
did not produce a skew-symmetric matrix. So, I'm including the code
below and asking the group if there is a simple newbie user error I
may have performed causing the problem. Or, is this simply a
limitation of SYMPY? By the way, TRIGSIMP() did not help...

<snip>
from sympy import *

t = Symbol('t')
a = Symbol('a')
b = Symbol('b')
g = Symbol('g')

rotX = Matrix([ [1, 0, 0],
[0, cos(g(t)), sin(g(t))],
[0, -sin(g(t)), cos(g(t))] ])
rotY = Matrix([ [cos(b(t)), 0, -sin(b(t))],
[0, 1, 0],
[sin(b(t)), 0, cos(b(t))] ])
rotZ = Matrix([ [cos(a(t)), sin(a(t)), 0],
[-sin(a(t)), cos(a(t)), 0],
[0, 0, 1] ])

R = rotX * rotY * rotZ
Rdot = R.diff(t)
skew = Rdot * R.T

print ''
print 'skew = '
print skew

# 'skew' should result in a skew-symmetic matrix.
# or, of the form,
# [[ 0, -w_z, w_y ],
# [ w_z, 0, -w_x],
# [-w_y, w_x, 0] ]
#
# where w = [w_x, w_y, w_z]^T
#
# looks like SYMPY fails to produce this results, yet, this same code
in
# Mathematica works as expected...
#
# is this a result of a "newbie" mistake? Or, a limitation of sympy?

print ''
print 'trigsimp(skew[0,0]) = '
print trigsimp(skew[0,0])
</snip>

Thanks in advance!

[Aaron Meurer]

Unfortunately, this is a limitation in SymPy right now, which is that
our trigonometric simplification is not very good.

For now, you can use a work-around suggested in another thread:

def mytrigsimp(expr):
expr = expr.rewrite(exp)
expr = expr.expand()
expr = expr.rewrite(cos)
expr = expr.expand()
return expr

and do skew.applyfunc(mytrigsimp). This works by rewriting the trig
functions in terms of complex exponentials, expanding, rewriting them
back as trig functions, and expanding again. In your case, it
simplifies nicely:

In [32]: def mytrigsimp(expr):
....: expr = expr.rewrite(exp)
....: expr = expr.expand()
....: expr = expr.rewrite(cos)
....: expr = expr.expand()
....: return expr
....:

In [33]: print skew.applyfunc(mytrigsimp)
[
0, -sin(g(t))*Derivative(b(t), t) +
cos(b(t))*cos(g(t))*Derivative(a(t), t),
-sin(g(t))*cos(b(t))*Derivative(a(t), t) - cos(g(t))*Derivative(b(t),
t)]
[sin(g(t))*Derivative(b(t), t) - cos(b(t))*cos(g(t))*Derivative(a(t),
t),
0, -sin(b(t))*Derivative(a(t), t) +
Derivative(g(t), t)]
[sin(g(t))*cos(b(t))*Derivative(a(t), t) + cos(g(t))*Derivative(b(t),
t), sin(b(t))*Derivative(a(t), t) -
Derivative(g(t), t),
0]

Aaron Meurer

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[Southern.Cross]

Great! I'll try this out. Thanks for the work around.

Joe