Ment yk ; for i = 1 . . . Np do propagate by way 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 eight 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.five. Remark on an Existing Function As pointed out in Section 1, from a mathematical perspective, the proposed algorithm (STC-PF) is equivalent to scPF (soft-constrained Particle Filter) [35]. Similar to STC-PF, scPF is depending on the SIR particle filter; having said that, the two Tacalcitol Epigenetic Reader Domain differ within 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 really 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 is also 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 may be identified by equating the elevation model using the pseudo-measurement. Consequently, scPF could be lowered to STC-PF as long as the assumption for target motion holds.Sensors 2021, 21,9 ofFigure 3. Implementation of Elevation Model Propagation.4. Simulation 4.1. Situation and Parameter Settings To evaluate STC-PF, numerical experiments are performed 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 CC-90011 Biological Activity tracks a single target that moves along the surface at a speed of 25 m/s. (see Figure four) The simulation runs for one hundred s. Moreover, 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 and every data entry in DTED. Because it is reasonable to bound the uncertainty of DTED, sampled noise is clipped to 50 m if its absolute value exceeds 50 m.Figure 4. 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 regarding the choice of GP hyper-parameters is in the Appendix B. The simulations are performed with two settings that differ within the value of DTED . The reasonable value for DTED is three.77 m, which is inferred from [37]. On the other hand, a different setting whose DTED is 1.89 m is also used to observe the sensitivity on the important parameter.Table 1. Parameter Setting.Name DTED (m) (deg-2 ) L ( arcsec) t (s) Initial Cov. Np Q R 4.2. Baseline Approaches diagValue 3.77, 1.891 (2.78e-4)2 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 2(m/s) 0 0 0 five(m/s) 2 diag ten(m) 0.1(deg) 0.1(deg) 1e2(m2 ) I3 0 3To compare STC-PF with other filters that may incorporate nonlinear constraints, the Smoothly Constrained Kalman Filter (SCKF) is implemented as well [30]. Note that `Smoothly Constrained’ within the name of SCKF doesn’t imply soft constraint. Due to the fact SCKF can incorporate only deterministic constraints, it calls for approximations of ground-truth terrain elevation that call for h and h to become fixed to specific values. 1 method applied for the comparison will be to ignore the noise inherent in DTED and use bilinear interpolation to retrieve the terrain elevation at arbitrary p.