Estimation of three-dimensional radar tracking using modified extended kalman filter

Kalman filter is an estimation method by combining data and mathematical models then developed be extended Kalman filter to handle nonlinear systems. Three-dimensional radar tracking is one of example of nonlinear system. In this paper developed a modification method of extended Kalman filter from the direct decline of the three-dimensional radar tracking case. The development of this filter algorithm can solve the three-dimensional radar measurements in the case proposed in this case the target measured by radar with distance r, azimuth angle θ, and the elevation angle ϕ. Artificial covariance and mean adjusted directly on the three-dimensional radar system. Simulations result show that the proposed formulation is effective in the calculation of nonlinear measurement compared with extended Kalman filter with the value error at 0.77% until 1.15%.


Introduction
Kalman filter known as linear quadratic estimation, is an algorithm that uses a series of measurements observed over time, containing statistical noise and other inaccuracies, and produces estimates of unknown variables that tend to be more accurate than those based on a single measurement alone. Extensions to the method have also been developed, the extended Kalman filter is nonlinear version of Kalman filter which used for nonlinear systems. Algorithm is almost same with Kalman filter but in the extended Kalman filter, the state transition and observation models don't need to be linear functions of the state but may instead be differentiable functions [1]. Extended Kalman filter can be used for three-dimensional radar tracking system (one of example of nonlinear systems). Radar tracking control system that has been designed according to the needs with the level of accurate measurement there will be a noise. By reducing the noise in the measurement system and needed an approach that is more accurate than ever before. The approach taken is estimated to determine the level of noise [2]. Based on the results of research conducted by Jang Song and in 2001 on "Improved Kalman Filter Design for Three-Dimensional Radar Tracking" [3], is obtained new filter to overcome the problem of radar tracking. In that paper, the level of measurement accuracy compared to the method extended Kalman filter with new methods improved of design Kalman filter, and the results of the simulation indicated that the improved of Kalman filter is a better result. Extended Kalman Filter are also used in research Nousheen in 2015 [4] that discussed abaout it can not be used on the system parameters and / or measurement that is not linear. In the topic of this research, studied the relevant journal in which the estimate is not a linear measurement of three-dimensional radar tracking system. There are many research on Kalman filter such as control position of underwater vehicle [5], also mobile robot position [6], heat conduction and its development [7], about air pollution [8], also stirred tank reactor [9], extended preferred radar tracking [10] and the last is estimation of reduced model [11].

Dynamical model
Case of three-dimensional radar tracking is sensor measures the target in three dimension space including the range (distance) r, azimuth angle (turning angle) θ, and elevation angle (elevation angle) φ (cf. Fig.1) Range r is the measured distance of target from the center of the tracker, then the azimuth angle θ shows direction of the tracker rotation motion in the direction of the x-axis and y-axis, and the elevation angle φ indicates the direction of tracker's motion elevation in the direction of x, y to z axis (the development of 2D radar) [10]. Then the equation can be described as follows: Equation (1)-(3) describe the measurement model in this research. Notice that the three equations (1)-(3) are obtained from the spherical shape. Then we transform (1)-(3) into the shape of cartesian coordinates by using the following transformation: where x, y and z is the scalar component of the cartesian coordinate measurement that is described as a position of a moving target in three-dimensional space [1]. Using the chain rule,  the derivation of (4)-(6) w.r.t. time is as follows: Equation (7)-(9) is a dynamical system that will be estimated by using extended Kalman filter and its modification, with derive equations and made a stochastic discrete model the final results are as follow: We define the following measurement model: where: • x k : target position at x-axes • y k : target position at y-axes • z k : target position at z-axes • V x k : target velocity at x-axes • V y k : target velocity at y-axes • V z k : target velocity at z-axes Equations (10)-(15) will be used to generate the real system, whereas the model used for estimation is the linearized version. The linearized model produces a system matrix A, that will be used in modified extended Kalman filter algorithm (see the next section).

Modified Extended Kalman Filter
Extended Kalman filter method is the extension of the classical Kalman Filter [12] for nonlinear systems. In reality, many models are nonlinear. Since the model is nonlinear, we use extended Kalman filter. Given a nonlinear stochastic model: where the nonlinear measurement model , x 0 ∼ N (X 0 , P x0 ), v k ∼ N (0, R k ) has a normal distribution and assumed to be white that means there is no correlation between each other and also with initial condition X 0 [13]. Before the estimation process, linearization process is carried out since the system is not linear. Linearization process is done by defining it as follows: where A and H are the Jacobi matrix derived from a differential in f and h to the direction of X. A modification of Kalman filter algorithm is called extended Kalman filter algorithm [14]. From research conducted by Song Taek Park [1] modification extended Kalman filter is insert error covariance and mean-made into the algorithm as follows: where: σ θ : parameters of angle noise θ σ φ : parameters of angle noise φ θ m k : Error covariance R p k in Equation (16) and mean-made µ p k in Equation (17)  Model system

Results
Simulation is given initial conditions and parameters (see Table 2) with measurements model r, θ,and φ. In Figure 2-7 the green color of the graph shows the real value, the estimated value of eKf is blue and red color is value estimation results meKf. In Figure 2, the results showed the difference in value between the real value and eKf estimation is 0.0094249, and the difference in value between the real value and meKf estimation is 0.0088085. For Figure 3, the results show the difference in value between the real value and eKf estimation is 0.0150247, and the difference in value between the real value and meKf estimation is 0.0073667. In Figure 4, the results show the difference in value between the real value and eKf estimation is 0.0111491, and the difference in value between the real value and meKf estimation is 0.0093434. In Figure 5, the results show the difference in value between the real value and ekf estimation is 0.01824580, and the difference in value between the real value and meKf estimation is 0.0114886. In Figure 6, the results obtained show the difference in value between the real value and eKf estimation is 0.0107461, and the difference in value between the real value and meKf estimation is 0.01008190. In Figure 7, the results show the difference in value between the real value and eKf estimation is 0.0115598, and the difference in value between the real value and meKf estimation is 0.0105467. Figure 8 shows        In Table 3 shows that the RMSE value of each variable is relatively small in the error value at intervals of 0.00736670 < RM SE < 0.0115598, or it can be said mistakes by 0.77% up to 1.15% to modification of Extended Kalman Filter and 0.00942490 < RM SE < 0.01824580, or it can be said mistakes of 0.94% to 1.82% for the extended Kalman filter. So overall it can be said that the modified extended Kalman filter method is suitable for estimating the three-dimensional radar tracking system in this case specifically for the measurement is not linear.

CONCLUSIONS
Based on the analysis and discussion presented in the previous chapter, we can conclude: 1. The dynamic model of three-dimensional radar tracking is formed as showed in chapter 2.
2. The estimation results indicate that each variables (position and velocity x, y, and z) of the tracking radar three dimensions by modification of extended Kalman filter better than methods extended Kalman filter with indicated by the error rate of only 0.77% up to 1.15% for modification of extended Kalman filter, while the error rate of extended Kalman filter is of 0.94% to 1.82%. 3. Based on the computation time show that the combined method of eKf and its modification takes 4.5770319 seconds.