Due to the cos function, even if the equation is quiet simple, i think that you can not say that the equation is linear.
For the time dependency, we can suppress it because i observe my equation for a fixed time.
So we can considerer the equation without the time dependency as you can see below :
S1_eq5_psi := -cos(gamma+psi)*cr+sin(gamma)*lr-cos(-phi+gamma)*mr-cos(gamma)*pr+z0-zp = 0
for the unknown psi, i look for solution with psi belong to the range [-Pi/2,Pi/2].
For the parameters, cr>0, lr>0, mr>0, pr>0 are constant positive and z0 and zp are reals.
I hope my solution will help to bring new ideas on this subject.
Thank you for your help