Thoughtful posts from alf, ATC Watcher and FullWings.
I'll respond to alf and ATC Watcher separately from FullWings, since I am having trouble getting the SW to let me jump between pages to copy quotes, without losing my reply window.
Originally Posted by alf5071h
your response related to GPWS
That's right. FullWings posed the GPWS warning as a comparison. I believe the decision problem with GPWS is fairly straightforward; also, one has apparently only about 12 seconds to respond, so even if there are uncertainties, there is no time to formulate a decision: your reaction must have been formulated in advance.
I am also talking about the algorithms, assuming that the equipment functions as intended. And I realise that can be a big "if". ATC Watcher's point about maturity refers partly to this issue: can we get the kit to function as intended?
If so, let me call the kit "refinable".
Let me call a system "adequate" if, when it functions as intended, its goal is achieved.
Let me call a aid system "optimal" if pilots can adequately follow the required behavior.
Then the GPWS algorithm is refinable, adequate, but not optimal. Whereas TCAS is refinable, not adequate, and questionably optimal if at all.
Comparing with EGPWS is a different kettle of fish. For example, Capt Pit Bull's point about map-shifts show that it is not adequate (in the terminology above). I haven't worked on EGPWS long enough to have as good a grasp of the issues as I would like.
Originally Posted by alf5071h
As for the solution to the three aircraft ACAS problem, this is done reasonably successfully by combat pilots in 1 vs 2 situations.
Yes, but they are using 3-dimensional avoidance, whereas TCAS is a 1-dimensional algorithm. The ACAS problem and the TCAS problem are not identical.
Originally Posted by alf5071h
Considering three aircraft avoidance, x, y and z, then a solution could be in the form of x^4 = y^4 + z^4.
This form of equation for powers greater than 2 has been proven to have no solution (Fermat’s conjecture).
What are x, y, and z here?
Three aircraft follow trajectories in 3-d Euclidean space, so
their position at time t is given by functions
f(t), g(t) and h(t), and the values of these functions are points in 3-D space. So there are coordinate functions (in your favorite coordinate system) f1, f2, f3, g1, g2, etc. Easiest is to pick one coordinate as the direction of the TCAS movement: say the third. Then either there are some trajectories f,g,h such that, with accelerations as specified by TCAS in direction f3, g3, h3, at least two of those trajectories come sufficiently close, or for all trajectories f,g,h, none of them come sufficiently close.
The quartic equation you gave has no *integer* solutions. That was shown by Wiles and Taylor. If has lots of solutions; indeed, its solutions form nice shapes in 3-space.
Originally Posted by alf5071h
I claim my PhD !
Sorry, not quite there yet
PBL