def linearity_index_xyz_parameterization():
"""Linearity index of XYZ Parameterization"""
d, d1, alpha, sigma_d = sympy.symbols("d,d1,alpha,sigma_d")
alpha = sympy.symbols("alpha")
u = (d * sympy.sin(alpha)) / (d1 + d * sympy.cos(alpha))
# first order derivative of u
u_p = sympy.diff(u, d)
u_p = sympy.simplify(u_p)
# second order derivative of u
u_pp = sympy.diff(u_p, d)
u_pp = sympy.simplify(u_pp)
# Linearity index
L = (u_pp * 2 * sigma_d) / (u_p)
L = sympy.simplify(L)
print()
print("u: ", u)
print("u': ", u_p)
print("u'': ", u_pp)
# print("L = ", L)
print("L = ", L.subs(d, 0))
print()
# def derive_J_jacobian():
# r_W_C = sympy.symbols("r_W_C")
#
# y =
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