Rotation about primary (X) axis

X₁ = 0
Y₁ = r cos a
Z₁ = -r sin a

Rotate above system about secondary (Z) axis

X₂ = Y₁ sin b
Y₂ = Y₁ cos b
Z₂ = Z₁

Add translation along X axis

X = A(t) + X₂
Y = Y₂
Z = Z₂
Where  A = some constant

Total equation of motion for particle as function of time (t):
For the case under consideration: 

A = 1
r  = .5
b = 22.5(t) degrees
a = 2b

Thus: 

X = t + .5 cos(45t)sin(22.5t)
Y = .5 cos(45t)cos(22.5t)
Z = -.5 sin(45t)