Converted State Equation Kalman Filter for Three-dimensional Target Tracking

doi:10.12382/bgxb.2024.0182

Abstract

Abstract:

The three-dimensional spherical coordinate measurements collected by the radar and sonar system are nonlinear with the Cartesian coordinate state of the moving target, which limits the tracking accuracy, and. it is more difficult to use the Doppler measurement with strong nonlinearityefficiently. Aiming at the above problems, a state vector composed of distance, pitch angle, azimuth angle and their derivatives is constructed to linearize the measurement equation,and the ordinary differential dynamics equation is discretized in a two-dimensional time-varying polar coordinate system composed of distance and pitch angle.Then the azimuth angle is introduced based on the projection relationship, and a typical three-dimensional constant velocity and constant acceleration motion model in the spherical coordinate system are established. Combined with the standard Kalman filter, the tracking is realized to avoid the nonlinear errorsunder the linear Gaussian framework. The effectiveness and performance advantagesof the proposed method in several three-dimensional tracking scenarios are verified through simulation.

Key words: Doppler radar, three-dimensional target tracking, nonlinear filtering, converted state equation Kalman filter

CLC Number:

TN953

LIU Zengli, ZHANG Wen, CAO Qihong, ZHAO Xuanzhi, LIU Kang, ZENG Sai. Converted State Equation Kalman Filter for Three-dimensional Target Tracking[J]. Acta Armamentarii, 2024, 45(11): 3998-4010.

Figures/Tables 22

Fig.1 Distribution of measurements in the observation coordinate system

Fig.2 Distribution of measurements in Cartesian coordinate system

Fig.3 True distribution and fitted Gaussian distribution of measurements in Cartesian coordinate system

Fig.4 Decomposition diagram of target motion in three-dimensional measurement space

Fig.5 Decomposition diagram of time-varying two-dimensional polar coordinate motion

Table 1 Parameters of radar measurements in the simulation diagram

雷达参数	三坐标雷达	多普勒雷达
σ_r/m	50	50
σ_ϕ(°)	0.4	0.4
σ_θ(°)	0.5	0.5
$σ r ·$ /(m·s^-1)		0.5
相关系数ρ		0.6

Table 1 Parameters of radar measurements in the simulation diagram

雷达参数	三坐标雷达	多普勒雷达
σ_r/m	50	50
σ_ϕ(°)	0.4	0.4
σ_θ(°)	0.5	0.5
$σ r ·$ /(m·s^-1)		0.5
相关系数ρ		0.6

Fig.6 Position RMSE under the CV model

Fig.7 Velocity RMSE under the CV model

Fig.8 Position RMSE under the CA model

Fig.9 Velocity RMSE under the CA model

Fig.10 ANEES under the CV model

Fig.11 ANEES under the CA model

Fig.12 Position RMSE in the CV model of Doppler radar

Fig.13 Velocity RMSE under the CV model of Doppler radar

Fig.14 Position RMSE under the CA model of Doppler radar

Fig.15 Velocity RMSE under the CA model of Doppler radar

Fig.16 ANEES under the CV model of Doppler radar

Fig.17 ANEES under the CA model

Table 2 Performance comparison of different algorithms under different radar measurement parameters

雷达参数	方法	CV位置 RMSEs	CV速度 RMSEs	CA位置 RMSEs	CA速度 RMSEs
σ_r=20m	EKF	49.5510	1.5707	50.2996	1.5158
σ_θ=0.25/(°)	UKF	26.2767	1.4255	26.8038	1.2938
σ_ϕ=0.2/(°)	DUCMKF	25.9613	1.4255	26.7194	1.3371
	3DCSKF	22.2401	1.2559	20.9610	1.0916
σ_r=80m	EKF	67.5251	2.0311	68.4591	1.8785
σ_θ=0.8/(°)	UKF	55.2284	1.9414	53.0092	1.6435
σ_ϕ=0.65/(°)	DUCMKF	55.7659	1.9151	53.2348	1.7252
	3DCSKF	43.0970	1.6944	39.3281	1.5980

Table 3 Performance comparison of different algorithms with different Doppler radar measurement parameters

雷达参数	方法	CV位置 RMSEs	CV速度 RMSEs	CA位置 RMSEs	CA速度 RMSEs
σ_r=25m	EKF	25.4976	1.1758	25.8386	1.1334
σ_θ=0.25/(°)	UKF	24.7819	1.1366	25.7960	1.1424
σ_ϕ=0.2/(°)	DUCMKF	25.2530	1.1267	25.4917	1.1272
$σ r ·$ =0.2m/s	3DCSKF	19.1871	0.7401	20.5462	0.7712
ρ=0.2
σ_r=80m	EKF	55.6626	1.4727	54.5091	1.4931
σ_θ=0.8/(°)	UKF	52.8558	1.3891	52.4822	1.4327
σ_ϕ=0.65/(°)	DUCMKF	52.3959	1.3463	53.1714	1.3913
$σ r ·$ =1m/s	3DCSKF	36.4882	0.9053	37.7613	0.9677
ρ=0.9

Table 3 Performance comparison of different algorithms with different Doppler radar measurement parameters

雷达参数	方法	CV位置 RMSEs	CV速度 RMSEs	CA位置 RMSEs	CA速度 RMSEs
σ_r=25m	EKF	25.4976	1.1758	25.8386	1.1334
σ_θ=0.25/(°)	UKF	24.7819	1.1366	25.7960	1.1424
σ_ϕ=0.2/(°)	DUCMKF	25.2530	1.1267	25.4917	1.1272
$σ r ·$ =0.2m/s	3DCSKF	19.1871	0.7401	20.5462	0.7712
ρ=0.2
σ_r=80m	EKF	55.6626	1.4727	54.5091	1.4931
σ_θ=0.8/(°)	UKF	52.8558	1.3891	52.4822	1.4327
σ_ϕ=0.65/(°)	DUCMKF	52.3959	1.3463	53.1714	1.3913
$σ r ·$ =1m/s	3DCSKF	36.4882	0.9053	37.7613	0.9677
ρ=0.9

Fig.18 Computational loads for 3D radar tracking scenarios

Fig.19 Computational load for Doppler radar tracking scenarios

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