When does a Trebuchet Shoot Its Projectile? Consider the following sling trebuchet:

While researching I found that what controls the release angle of the projectile is the angle between the 'finger' and the extension of the beam $r_b$, as seen below:

The way that the release pin works is when
the arm begins to swing the sling is beginning to be thrown, however once it (the end of the arm) reaches a certain point the inertia pulls the loop off the pin causing the projectile to release at that particular instance.
With different angled pins the release point will change. So one pin will make it fire more vertical than horizontal.
Assuming that the finger is parallel to the extension of the beam $r_b$ (i.e.  = 0) then how and when can the release angle of the projectile be determined ?
 A: To summarise my comments :
Detail A is the joint between the "long arm" labelled $r_b$ and the "short arm" labelled $l$.  The short arm is the sling.  There is no mechanism which releases the sling : it releases itself either when the tension in the sling becomes zero, or when the sling (ring) slides sideways off the "finger".
You would like to find the optimum position (ie values of angles $\theta$ and $\phi$) for releasing the sling, in order to achieve the maximum range of the projectile m. 
Detail A appears to come from source [1] below.  That webpage gives the conditions for achieving maximum range (as calculated in [3]), which include releasing the projectile when the long arm makes an angle of $\theta = 45$ degrees with the vertical.  (This is described as the "initial" release position - whatever that means.)
This does not seem right to me.  Source [3] states that for the "sliding sling-shot Trebuchet" described in Detail A the optimum release angle in the author's simulation was $\theta = 18$ degrees, although this might optimise KE transfer rather than range.  The optimum release condition seems to be when the sling makes an angle of $\phi=45$ degrees with the vertical.  This seems to be correct, because the launch angle of the projectile will then be 45 degrees also, the usual condition for maximum range of a projectile.
You cite the project report in source [2] which presumes (p 1 para 1, p 4 para 2) that, in order to maximise the KE of projectile m, it should be released when angle $\theta = 0$, ie when M is at its lowest point and has minimum PE and maximum KE.  This seems plausible but I would think it needs to be verified by solving the equations of motion.  And it is not the criterion which you are using (maximum range).
So perhaps the optimum launch conditions are when $\theta=0$ degrees and $\phi=45$ degrees.  However, for given values of m, M, $r_b$ and $l$, you are unlikely to obtain these two conditions simultaneously.  I think you will need to "reverse engineer" the Trebuchet by selecting a combination of m, M, $r_b$ and $l$ which achieves $\phi=45$ degrees when $\theta=0$ degrees during simulation.
Source [1] explains that the angle $\phi$ at which the projectile is released can be altered by adjusting the angle $\delta$ of the finger in Detail A.  However, it does not explain how this value of $\phi$ can be predicted.  I think adjusting angle $\delta$ is a practical adjustment which you can make when you build a real Trebuchet.  It does not need to be included in your simulation.  You can simply stop the simulation when $\phi=45$ degrees and assume that the real Trebuchet will release the sling at this point. Then you adjust angle $\delta$ until the real Trebuchet does release m at this value of $\phi$.
Conclusion: I think that you will need to solve the equations of motion (source [1] or [2]) for the Trebuchet numerically to obtain or check the optimum launch conditions for maximum range.   
[1] "Trebuchet Physics"
http://www.real-world-physics-problems.com/trebuchet-physics.html
[2] "Trebuchet: The Dynamics of a Medieval Siege Engine" http://www.uphysicsc.com/2010-GM-B-210.pdf
[3] "Trebuchet Mechanics" by Donald B Siano
http://www.aemma.org/training/trebuchet/trebmath35.pdf
