Type error in SymPy when using tensor arrays

Question

I'm fairly new to SymPy, SciPy, NumPy etc., so perhaps there are easier ways to accomplish my goal, which is to define a sequence of indexed 4x4 transformation matrices for a 4-bar linkage. I don't have any pressing need for the 3^rd dimension, except to make it easy to iterate through the sequence {A₀ ... A₃} in my code.

The code below generates the following error.

TypeError: unorderable types: slice() >= int()

On the other hand, replacing A[:,:,i] with something like A0 below does generate a 4x4 symbolic array as expected, so perhaps I'm not defining the slice correctly--however, none of the other slice permutations seems to work either. (I'm also having problems with implied line continuation on the last line of code, so I put everything inline.)

I'd appreciate any suggestions on how to solve this issue.

from sympy import *
def plg_jac():
    a_3, a_4, d_2, theta_1, theta_4 = symbols('a_3, a_4, d_2, theta_1, theta_4')

#   The D-H table is defined by the following 4 vectors
    a       = Matrix([0, 0, a_3, -a_4])
    alpha   = Matrix([pi/2, pi/2, 0, 0])
    d       = Matrix([0, d_2, 0, 0])
    theta   = Matrix([theta_1, pi, 0, theta_4])

#   Initialise a mutable 4x4x4 array
    A = tensor.array.MutableDenseNDimArray(range(64), (4,4,4))
#   Now define A0 ... A3, the transformation matrices between links i & i+1
    for i in range(a.shape[0]):
        A[:,:,i] = Array([[cos(theta[i]), -sin(theta[i])*cos(alpha[i]), sin(theta[i])*sin(alpha[i]), a[i]*cos(theta[i])], [sin(theta[i]), cos(theta[i])*cos(alpha[i]), -cos(theta[i])*sin(alpha[i]), a[i]*sin(theta[i])], [0, sin(alpha[i]), cos(alpha[i]), d[i]], [0, 0, 0, 1]])

Answer 1

The assignment to slices of SymPy arrays are missing, you can go around this lack by using numpy.

Import NumPy as:

import numpy

Convert the SymPy array to a NumPy array (of type object, so as to contain SymPy expressions):

In [18]: A_numpy = numpy.array(A.tolist()).astype(object)

Run a modified version of your for loop:

In [19]: for i in range(a.shape[0]):
    ...:     A_numpy[:, :, i] = [[cos(theta[i]), -sin(theta[i])*cos(alpha[i]), s
    ...: in(theta[i])*sin(alpha[i]), a[i]*cos(theta[i])], [sin(theta[i]), cos(th
    ...: eta[i])*cos(alpha[i]), -cos(theta[i])*sin(alpha[i]), a[i]*sin(theta[i])
    ...: ], [0, sin(alpha[i]), cos(alpha[i]), d[i]], [0, 0, 0, 1]]
    ...:     

Convert the resulting NumPy array back to a SymPy array:

In [20]: A = Array(A_numpy)

Print it's contents:

In [21]: A
Out[21]: 
⎡⎡cos(θ₁)  -1  1     cos(θ₄)  ⎤  ⎡sin(θ₁)   0  0    sin(θ₄)  ⎤  ⎡0  0   0  0⎤ 
⎢⎢                            ⎥  ⎢                           ⎥  ⎢           ⎥ 
⎢⎢   0     0   0    -sin(θ₄)  ⎥  ⎢   0      0  1    cos(θ₄)  ⎥  ⎢1  1   0  0⎥ 
⎢⎢                            ⎥  ⎢                           ⎥  ⎢           ⎥ 
⎢⎢sin(θ₁)  0   0        0     ⎥  ⎢-cos(θ₁)  1  0       0     ⎥  ⎢0  0   1  1⎥ 
⎢⎢                            ⎥  ⎢                           ⎥  ⎢           ⎥ 
⎣⎣   0     0   a₃  -a₄⋅cos(θ₄)⎦  ⎣   0      0  0  -a₄⋅sin(θ₄)⎦  ⎣0  d₂  0  0⎦ 

 ⎡0  0  0  0⎤⎤
 ⎢          ⎥⎥
 ⎢0  0  0  0⎥⎥
 ⎢          ⎥⎥
 ⎢0  0  0  0⎥⎥
 ⎢          ⎥⎥
 ⎣1  1  1  1⎦⎦