Patent Application: US-201615258736-A

Abstract:
a system and method for data tracking , processing , and analysis of multidimensional space trajectories of moving objects . the method is particularly applicable for moving objects having closely spaced final targets , such as aircraft landing at airport runways . the method may be used for air space sectorization . the method is provided for determining the number of asymptotically converged beams of the trajectories in 3d - space . points of the trajectory sample are scattered into a set of independent points of the trajectories . two - dimensional orthogonal projection of the set of points is considered and the most likelihood orthogonal linear regression of the points is defined . such linear regression represents an asymptote tangential to a beam . certain beam of trajectories is separated in reverse transition into original data space .

Description:
trajectory beam n k , k = 1 , k 0 ( k 0 is empirical parameter ) is considered as asymptotically converged with a threshold parameter ε , if for all vectors { x [ i ]∈ r 3 × l , i ∈ n k } in the beam n k , k = 1 , k 0 , a condition of asymptotic convergence of the beam is fulfilled ∀( i , j )∈ n k , ∥ x [ l i ; i ]∥ x [ l j ; j ]∥ 2 & lt ; ε ( 1 ) where (∀ i ∈ n k , x [ l i ; i ] are coordinates of final trajectory points on a runway . parameters l i , i ∈ n k are subject to determination , ∥. . . ∥ 2 is euclidean distance metric in three - dimensional coordinate space r 3 , ε is a cutoff parameter with value of no more than runway width . in considering the claimed approach to determination of the number of asymptotically converged beams , it should be taken into account that the trajectories in the beams have some typical form ( profile ) and specific geometric asymptote in the region of convergence ( 1 ). geometric asymptote in converged beam of multidimensional aircraft intent trajectories is a line in r 3 that meets the requirement ( 1 ). trajectories in asymptotically converged beam have tangent line in area around the final points ∀ i ∈ n k , x [ l i ; i ]. so , asymptotically converged trajectory beams can be identified by determination of their tangential geometric asymptotes in the points of their focuses . because discrete points of beam trajectories are tightly located around its &# 39 ; asymptote , to determine the number of asymptotically converged beams a sample of trajectory vectors { x [ i ]∈ r 3 × l , i = 1 , n } is scattered into set of the trajectories &# 39 ; points { x [ i ]∈ r 3 × l , i = 1 , n } {( x [ j , i ], y [ j , i ], z [ j , i ])∈ r 3 , j = 1 , l , i = 1 , n }. ( 2 ) set of the points ( 2 ) has to be sorted according to the values of one of the coordinates ( in ascending or descending order ). at that , other coordinates of points , representing converged trajectory beam of certain profile , are also ordered . then , for the scattered three - dimensional data { z i =( x i , y i , z i ), i = 1 , l · n } ( 2 ); orthogonal linear regression models m ( θ )={( x , y , z )∈ r 3 |( a 1 x + b 1 y + c 1 z = d 1 ) ( a 2 x + b 2 y + c 2 z = d 2 )} ( 3 ) are analyzed by ransac ( random sample and consensus ) algorithm here symbol is conjunction , θ ={ a 1 , b 1 , c 1 , d 1 , a 2 , b 2 , c 2 , d 2 } is vector of the parameters of the models ( 3 ), that is determined under given cutoff of euclidian distance ρ ⊥ ( z , m ( θ )) calculated by orthogonal projection of point z =( x , y , z ) from the set ( 2 ) onto a line m ( θ ). by such a way the model ( 3 ) is symmetrical relatively to the coordinates x , y , z . any pair of the points from ( 2 ) is sufficiently to put forward a hypothesis about the model of orthogonal linear regression ( 3 ). final model ( 3 ) is proved by the greatest relative quantity ( percent ) of scattered data { z i =( x i , y i , z i ), i = 1 , l · n } ( 2 ). algorithm mlesac ( maximum likelihood estimation sample consensus ) [ 12 ], that is probabilistic modification of ransac [ 11 ] algorithm , may be used for these purposes . the algorithm estimates likelihood of the model ( 3 ), in representing distance distribution of scattered data { z i =( x i , y i , z i ), i = 1 , l · n } from the model m ( θ ) ( 3 ) as a mixture of data distributions some of which support the model ( 3 ) ( inliers ), while the rest ones reject it ( outliers ). considering the scattered data z ( 2 ) as independent , we obtain a relation for logarithm of likelihood as following where γ is mixing parameter . distribution of the distances to the data , supporting the model ( 3 ), is represented by gaussian distribution where σ is standard deviation . distribution of distances to the data , rejecting the model ( 3 ), is described by uniform distribution where ρ max is maximal distance to data ( it is defined by the context ). likelihood logarithm ( 4 ) minimization allows to evaluate vector of the parameters θ and mixing parameter γ . estimation of the parameters is traditionally done using em - algorithm [ 13 ]. it is obvious for the specialist , that apart from the algorithms considering at the present work in determining of the geometric asymptote m ( θ )[ k ], k = 1 , k ( 3 ) of one of the beams under condition ( 1 ), other methods can be used . the most likely linear regression of the scattered data of trajectory sample defines geometric asymptote m ( θ )[ k ], k = 1 , k ( 3 ) of one of samples &# 39 ; beams under ( 1 ). the geometric asymptote received in such a manner meets the requirement ∀ k = 1 , k , ∀ i ∈ n k , ∥ m ( θ )[ l k , k ]− x [ l i ; i ]∥ 2 & lt ; ε . beams of the trajectories tangential to the geometric asymptotes are defined as the result of minimization of a cost function where { r [ i ; k ]∈{ 0 , 1 }, k = 1 , k }, i = 1 , n is set of binary indicator variables ( e . g . if vector x [ i ] was attributed to the beam k , then r [ i ; k ]= 1 and r [ i ; k ]= 0 , otherwise ). distance between each geometric asymptote and the sample &# 39 ; s trajectories is calculated according to cosine measure ρ cosin ( x , m ( θ )[ k ] )=( x · m ( θ )[ k ])/(√{ square root over (( x · x ))}√{ square root over (( m ( θ )[ k ]· m ( θ )[ k ] ))}). after elimination of the points , representing the trajectories of separated beam , from the scattered data ( 2 ), the procedure of geometrical asymptote detection is repeated and the next trajectory beam is separated . in this case , remained scattered data ( 2 ) are sorted with respect to another space coordinate ( different from the previous one ), as the model determination should be symmetric relatively to the coordinates x , y , z . possible dependence of result ( 3 ) from the coordinate directions is obviated by changing direction of data ordering in ( 2 ) from ascending to descending order or vice versa . analysis of trajectory sample is completed , when all beams in the sample are separated ( see fig4 h ). in general , the approach described in the claimed invention consists of two stages . sufficient reduction of data dimensions at the first stage simplifies the revealing of data specific features . in considering 2d - projection of scattered points of 3d trajectories , the most likelihood orthogonal linear regression of the scattered data which corresponds to geometrical asymptote of one of the beams in analyzed sample of the trajectories is defined . at the second stage of the approach , after reverse transferring into the original dimensional space , a certain beam is separated from the analyzed trajectory sample in accordance with proximity ( by cosine measure ) to defined asymptote . thus , due to such approach no information about original data is lost . detailed scheme of the claimed invention is shown in fig5 . 1 . erzberger h ., davis t . j ., green s . design of center - tracon automation system // in agard , machine intelligence in air traffic management — 1993 . 2 . williams d . h ., green s . m . flight evaluation of center - tracon automation system trajectory prediction process .— national aeronautics and space administration , langley research center , 1998 . 3 . mcfadyen a ., o &# 39 ; 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