Patent Document ID: 7702427
Application ID: 10914783
Patent Flag: 1

Claim One:
1. A method for estimating a minimum distance of approach of two aircraft that are airborne, the method comprising: providing information on an initial location vector r0(t=t1;n), on an initial velocity vector r1(t=t1;n) and on an initial acceleration vector r2(t=t1;n) at a selected time t=t1, for each of N aircraft, numbered n=1,. .. , N (N≧2) that are airborne; approximating said location vector r(t;n) for aircrafts number n=n1 and n=n2 (n1≠n2) over a selected time interval [t1,t2] by quadratic vector functions of time, 
 r ( t;n 1;app)= r 0( n 1)+ r 1( n 1)·( t−t 1)+ r 2( n 1)·( t−t 1) 2 , 
 r ( t;n 2;app)= r 0( n 2)+ r 1( n 2)·( t−t 1)+ r 2( n 2)·( t−t 1) 2 , 
 Δ r ( t ;app)= r ( t;n 1;app)− r ( t;n 2;app)=Δ r 0+Δ r 1( t−t 1)+Δ r 2( t−t 1) 2 , respectively, where t1 is a selected time within a selected time interval [T1,T2], each of the location vectors r(t;n1;app) and r(t;n2;app) substantially describes motion on a great circle in a plane, and the vector coefficients r0;n1), r1(n1), r2(n1), r0(n2), r1(n2) and r2(n2) are chosen to optimally match the vector functions r(t;n1;app) and r(t;n2;app) in the selected time interval [T1,T2]; and estimating a minimum distance of approach d(min) for a magnitude |r(t;n1)−r(t;n2)| of a vector difference, by identifying at least one real time t(min) for which a time derivative of the quantity |r(t;n1)−r(t;n2)| 2 is zero, 
 2 Δr 0Δ r 1+{Δ r 1·Δ r 1+2 Δr 0·Δ r 2)( t−t 1)+6 Δr 1·Δ r 2( t−t 1) 2 +4 Δr 2·Δ r 2( t−t 1) 3 =0, and by interpreting the vector magnitude |r(t=t(min);n1)−r(t=t(min);n2)| as the minimum distance d(min).