Ment yk ; for i = 1 . . . Np do propagate through the dynamic model i , i , vi , vi,k P(k , k , v,k , v ,k |xi -1 ); k k ,k kNppropagate by means of the elevation model h, h | DTED N ( ) hi = h ( i , i ) k k k ; T vi vi = h ( i , i ) vih,k k k ,k j ,k7 8 9 10^k update the weight wi wi -1 P(yk |i , i , hi ); k k k kp ^k ^ normalize wi = wi /( j=1 wk ); kNend (Optional) Resampling (e.x. multinomial resampling); end3.5. Remark on an Current Function As talked about in Section 1, from a mathematical point of view, the proposed algorithm (STC-PF) is equivalent to scPF (soft-constrained Particle Filter) [35]. Comparable to STC-PF, scPF is depending on the SIR particle filter; however, the two differ in the sense that scPF utilizes generalized likelihood. ^k w i w i – 1 P ( y k | xi ) P ( C k | xi ) (23) k k k exactly where P(Ck |xi ) is often a pseudo-measurement that represents how much the offered state xi k k satisfies the constraint. If Equation (21) is replaced byi i q(xi |x0:k-1 , y1:k ) = P(i , i , vi , vi,k |xi -1 ), k k k k ,k(24)then the weight update rule can also be changed. wi wi -1 P(yk |i , i , hi ) P(hi |i , i ) P(vi |i , i , vi , vi,k ) k k k k k k k k h,k k k ,k (25)Hence, the generalized likelihood function can be identified by equating the elevation model using the pseudo-measurement. Because of this, scPF could be decreased to STC-PF so long as the assumption for target motion holds.Sensors 2021, 21,9 ofFigure three. Implementation of Elevation Model Propagation.4. Simulation four.1. Thapsigargin web Situation and Parameter Settings To evaluate STC-PF, numerical experiments are performed together with the following situation: The radar is mounted on an aircraft that flies at a speed of 70 m/s at a height of 2500 m. The radar tracks a single target that moves along the surface at a speed of 25 m/s. (see Figure four) The simulation runs for 100 s. Moreover, to reflect the uncertainty in DTED, a noisy version of DTED is designed. Additional especially, iid zero-mean Gaussian noise with variance DTED is sampled and added for each data entry in DTED. Because it is affordable to bound the uncertainty of DTED, sampled noise is clipped to 50 m if its absolute worth exceeds 50 m.Figure four. Trajectory in WGS84 LLA (0.05 degree interval).Sensors 2021, 21,ten ofValues of parameters used in the simulation are listed in Table 1. Detailed explanation about the selection of GP hyper-parameters is within the Appendix B. The simulations are performed with two settings that differ within the worth of DTED . The affordable value for DTED is three.77 m, that is PF 05089771 Epigenetic Reader Domain inferred from [37]. Having said that, yet another setting whose DTED is 1.89 m is also applied to observe the sensitivity on the crucial parameter.Table 1. Parameter Setting.Name DTED (m) (deg-2 ) L ( arcsec) t (s) Initial Cov. Np Q R 4.two. Baseline Approaches diagValue 3.77, 1.891 (2.78e-4)two 1 (2.78e-4)13 ( 390m) 1.0 0 three ten(m2 /s2 ) I3 1e4 20(m) I3 0 three two(m/s) 0 0 0 three 0 two(m/s) 0 0 0 five(m/s) 2 diag 10(m) 0.1(deg) 0.1(deg) 1e2(m2 ) I3 0 3To examine STC-PF with other filters that can incorporate nonlinear constraints, the Smoothly Constrained Kalman Filter (SCKF) is implemented also [30]. Note that `Smoothly Constrained’ inside the name of SCKF does not imply soft constraint. Simply because SCKF can incorporate only deterministic constraints, it needs approximations of ground-truth terrain elevation that call for h and h to become fixed to distinct values. A single strategy applied for the comparison would be to ignore the noise inherent in DTED and use bilinear interpolation to retrieve the terrain elevation at arbitrary p.
