Principles of Modern Radar - Volume 2 1891121537

15.1 Review Of Tracking Concepts 673that are not updated with this particular measurement set. An extra row (on the bottom)is similarly included for measurements that fail to assign to any track. Thus, G is the costof coasting a track (also referred to as the guard value), while c I is the cost of initiating anew track.15.1.2.1 GatingBefore populating the (1,1) through (M,N) entries of the measurement-to-track costmatrix (from Table 15-1), it is prudent to identify infeasible associations. Their costs canimmediately be set to an arbitrarily large number, alleviating the computational burdens associatedwith computing their exact costs and considering them in assignment hypotheses.Given the large number of measurements and tracks that are typical in real-world applications,this step is usually necessary to achieve real-time tracking.Infeasible assignments are typically found with a series of tests called gates; as aresult, this process is known as gating. Coarse gates are applied first, to eliminate potentialmeasurement-track pairs that are grossly mis-matched. Coarse gates can take differentforms, with spherical and rectangular gates being common, numerically-efficient choices.Spherical gates keep only those measurement-track pairs whose states are within someEuclidean distance, d sphere , of each other, as described by√(px t − px m ) 2 + (py t − py m ) 2 + (pz t − pz m ) 2 < d sphere (15.2)where the track position is (px t , py t , pz t ), and the measurement position is (px m , py m , pz m ).Measurement-track pairs outside this radius are considered infeasible and are ruled out insubsequent parts of the measurement-to-track association process. Rectangular gates applyabsolute value tests to each element of the state, thus creating a rectangular acceptanceregion, rather than spherical one. An example of a rectangular gate for position terms isgiven by|px t − px m | < d px|py t − py m | < d py|pz t − pz m | < d pz(15.3)where d px , d py , and d pz are the allowable thresholds for the differences in x, y, and zposition, respectively.Having ruled out grossly infeasible measurement-track pairs via coarse, numericallyefficientgates, the next step is to apply fine gates. These are typically elliptical in nature, asthey incorporate the measurement and track covariances and assume underlying Gaussianstatistics. For this step, it is common practice to compute the full log-likelihood, ij , thatmeasurement j and track i originate from the same object, given by ij =− 1 2 ln (∣∣ 2π S ij∣ ∣) −12 ˜zT ij S−1 ij ˜z ij (15.4)where ˜z ij is the innovations and S ij is its covariance. The innovations is the differencebetween the measurement state, z i , and the predicted track state, x i , which is converted tothe measurement coordinate space through some (typically nonlinear) coordinate transformation,h( . ). Hence, it is given by˜z ij = z j − h (x i ) (15.5)