import sympy as sm
import sympy.physics.mechanics as me
me.init_vprinting()

theta, beta, alpha, l = me.dynamicsymbols('theta, beta, alpha, l')
A, B, C, D = sm.symbols('A, B, C, D', cls=me.ReferenceFrame)

B.orient_axis(A, theta, A.z)
C.orient_axis(B, beta, -B.x)
D.orient_axis(C, alpha, C.z)

r = l*D.x
r

r.express(C)

r.express(B)

r.express(A)

r.dot(C.x).diff(alpha)

r.dot(C.y).diff(alpha)

r_alpha_C = r.dot(C.x).diff(alpha)*C.x + r.dot(C.y).diff(alpha)*C.y
r_alpha_C

r.diff(alpha, C).express(C).simplify()

theta, beta, alpha, l

t = me.dynamicsymbols._t

theta.diff(t)

r

r.diff(t, D)

r.diff(t, A).simplify()

r.dt(A)

r.dt(D)

q = t*A.y
q

q.dt(A).dt(B)

q.dt(B).dt(A)