(Q2)Derive the transfer function Gθ(s) using the force, F(s), as an input variable and the angle, X3(s), as the output variable. Derive the transfer function Gx(s) using the force, F(s), as an input variable and the horizontal position, X1(s).

PendulumQuestion2.py

import sympy as sym
m, ell, x1, x2, x3, x4, M, g, F, m = sym.symbols('m, ell, x1, x2, x3, x4, M, g, F, m')
# φ(F, x3, x4)
phi = 4*m*ell*x4**2*sym.sin(x3) + 4*F - 3*m*g*sym.sin(x3)*sym.cos(x3)
phi /= 4*(M+m) - 3*m*sym.cos(x3)**2
dphi_x3 = phi.diff(x3)
dphi_x4 = phi.diff(x4)
dphi_F = phi.diff(F)

psi = -3*(m*ell*x4**2*sym.sin(x3)*sym.cos(x3) + F*sym.cos(x3) - (M+m)*g*sym.sin(x3))
psi /= (4*(M+m) - 3*m*sym.cos(x3)**2)*ell
dpsi_x3 = psi.diff(x3)
dpsi_x4 = psi.diff(x4)
dpsi_F = psi.diff(F)
# Equilibrium point
Feq = 0
x3eq = 0
x4eq = 0
#------------------------------
#  adding equilibrium to x1 and x2
#------------------------------
x1eq = 0
x2eq = 0

dphi_F_eq = dphi_F.subs([(F, Feq), (x3, x3eq), (x4, x4eq)])
dphi_x3_eq = dphi_x3.subs([(F, Feq), (x3, x3eq), (x4, x4eq)])
dphi_x4_eq = dphi_x4.subs([(F, Feq), (x3, x3eq), (x4, x4eq)])

dpsi_F_eq = dpsi_F.subs([(F, Feq), (x3, x3eq), (x4, x4eq)])
dpsi_x3_eq = dpsi_x3.subs([(F, Feq), (x3, x3eq), (x4, x4eq)])
dpsi_x4_eq = dpsi_x4.subs([(F, Feq), (x3, x3eq), (x4, x4eq)])
#Definitions of positive real constants
a = dphi_F_eq
b = -dphi_x3_eq
c = 3/(ell*(4*M + m))
d = 3*(M+m)*g/(ell*(4*M + m))
# ---------------------------------------------------------------------
a, b, c, d = sym.symbols('a:d')
s, t = sym.symbols('s, t')
#Derived Transfer fucntions
transfer_function_F_to_x3 = -c/(s**2 - d)
transfer_function_F_to_x1 = (a*(s**2 - d) + b*c)/(s**2)*(s**2-d)