Ment yk ; for i = 1 . . . Np do propagate by means of the dynamic model i , i , vi , vi,k P(k , k , v,k , v ,k |xi -1 ); k k ,k kNppropagate through 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 described in Section 1, from a mathematical point of view, the proposed algorithm (STC-PF) is comparable to scPF (soft-constrained Particle Filter) [35]. Related to STC-PF, scPF is according to the SIR particle filter; having said that, 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 a pseudo-measurement that represents how much the given 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)Therefore, the generalized likelihood function is usually identified by equating the elevation model using the pseudo-measurement. Because of this, scPF might be decreased to STC-PF provided that the assumption for target motion holds.Sensors 2021, 21,9 ofFigure 3. Implementation of Elevation Model Propagation.4. Simulation four.1. Situation and Parameter Settings To evaluate STC-PF, numerical experiments are performed together with the following scenario: 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 4) The simulation runs for one hundred s. In addition, to reflect the uncertainty in DTED, a noisy version of DTED is produced. Additional especially, iid zero-mean Gaussian noise with variance DTED is sampled and added for each information 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 Moxifloxacin-d4 Description degree interval).Sensors 2021, 21,ten ofValues of parameters used in the simulation are listed in Table 1. Detailed explanation concerning the decision of GP hyper-parameters is in the Appendix B. The simulations are performed with two settings that differ in the value of DTED . The affordable worth for DTED is 3.77 m, which can be inferred from [37]. Nevertheless, another setting whose DTED is 1.89 m can also be employed to observe the sensitivity in the essential parameter.Table 1. Parameter Setting.Name DTED (m) (deg-2 ) L ( arcsec) t (s) Initial Cov. Np Q R 4.two. Baseline Procedures diagValue three.77, 1.891 (2.78e-4)two 1 (2.78e-4)13 ( 390m) 1.0 0 3 ten(m2 /s2 ) I3 1e4 20(m) I3 0 three 2(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 will incorporate nonlinear constraints, the Smoothly Constrained Kalman Filter (SCKF) is implemented too [30]. Note that `Smoothly Constrained’ within the name of SCKF will not imply soft constraint. For the reason that SCKF can incorporate only Difamilast Protocol deterministic constraints, it needs approximations of ground-truth terrain elevation that call for h and h to become fixed to precise values. One method applied for the comparison should be to ignore the noise inherent in DTED and use bilinear interpolation to retrieve the terrain elevation at arbitrary p.