The two previous examples enabled us to realize that the concept of control of A over B depends on the speed that A will adopt and the probability for B to find a thermal along the way.
We assumed that our pilots adopt a rational behavior. However, rationality is often a personal matter, especially as we practice a sport showing little rationality in absolute uncertainty environment. We will generalize the construction of loose control giving freedom to each pilot to adopt the speed he wants.
To point the finger to no particular person, let’s take a couple of top pilots to embody the A and B players: Hameau tries to fly away low while the Pinard monitors him from high behind. Hameau and Pinard do not care about theories on optimal speed. They find them interesting for free flight but without much interest in the context of competition. So they fly full bar and weld the pulleys in transitions. Or not. It is their analysis that guides their individual flight plan.
We keep the assumption that next thermal has the good taste to climb at the expected McCready Vz.
Our equations on time are the following:
t = D / Va
t = ( D – dD ) / Vb + dH’ / Vzmc
(1) dD = dH’ * Vb / Vzmc + D * ( 1 – Vb / Va )
Our equations on glide ratios are the following:
Fa = D / ( H – dH’ )
Fb = ( D – dD ) / ( H – dH )
By eliminating H:
(2) dH’ = dH + ( D – dD ) / Fb – D / Fa
I could not manage to simplify or reduce this somewhat complicated equation. We cannot directly determine the dD / dH ratio without additional hypothesis on D. The control range D is not a consequence but an adjustment variable. Depending on the range over which the pilot A wishes to exercise control over the pilot B, he will adapt his control glide ratio and transition speed. Come to think of if for two seconds, it seems quite logical.
Numerical simulation comes in handy for putting values on words. The following table provides control glide ratio for:
dH = 100 m
D = 2000 m
Vzmc = 1,5 m/s
We want to keep dH’ >= 0 and this means that A should not fly too fast in order not to destroy his advantage. Some boxes of the above table should be avoided accordingly.
I do not know about you but I’m not going to use either this formula or the tables in flight. If you find a trick to simplify the formula establishing the glide ratio of generalized control, I will be happy to read you. I also encourage you to check my equations, I have been misled when working on them sufficiently not to be 100% sure that no errors remain.
Tomorrow I will give you my personal conclusion on this topic of control.