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!
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