Air Object Height Estimation with 2-D Radars using Fuzzy Logic

Multi-sensor tracking is a widely used technique in aerospace applications to precisely estimate the target kinematics. Particularly naval-based tracking systems utilise different types of radars (2-D, 3-D) in multi-sensor tracking scenarios for robust estimation. As the supplied information from 2-D radar contains only range and azimuth values, it is difficult to estimate the height of an air object using 2-D radar. To overcome the limitation, a geometric method is considered to combine the information obtained from two 2-D radars located as two different locations. As the solution of the geometric method depends upon certain geometric features, it is not possible to get good results with one pair of sensors. However, to obtain better results, the proposed method combines geometric as well as fuzzy logic-based validation and experimented with more than two 2-D radars. The issues related to 2-D radar tracking and the method comprising triangulation geometry and a fuzzy logic-based selection method to improve the height estimation accuracy in real time have been discussed.

As radar-target geometric values are affecting the accuracy of the solution, it is not possible to get good results always. Gai-Ming-jiu2, et al. and Hakl10, et al. provided a solution for tracking 3-D targets with a single 2-D radar, but this solution works in some specialised conditions that include: (i) air object speed is known, (ii) air object flies at a constant altitude. However, it is observed that the accuracy of the geometric solution is a function of three parameters, i.e., angle making target with the radar stand points and slant ranges from the radar stand points. In order to obtain optimal geometric solution, it is proposed that more than two 2-D radars located at different locations be deployed and a pair of radars be selected, which produces good results computed using fuzzy logic.

Consider a 3-D object (air target) located at E tracked by two 2-D radars located at A and B. For each iteration, radars at A and B (Radar 1, Radar 2) are supplying range and azimuth information of the target E (R1, Az1 and R2, Az2). Using the geometric solution described in Fig. 2, air object height is estimated. Performance of triangulation geometry is highly dependent on the accuracies of the observed measurements and geometric values.

The following diagram describes the geometry of air object tracking from two 2-D radars (Radar 1, Radar 2) located on two ships separated geographically.

By considering the radar positions in geodetic coordinate frame, located at A and B, and the obtained ranges (R1, R2) and azimuths (Az1, Az2) in each iteration, elevation angles wrt to each radar located at A and B are computed. C is a point of intersection of two azimuths (Az1, Az2) observed of A and B and ∠ACE and ∠BCE are right angles. The elevation angle obtained with the second radar, that is angle ∠CBE is converted wrt the Radar1 position. In turn, this value is combined with angle ∠CAE (obtained with Radar 1) to compute the mean elevation. The algorithm explains the process in detail for a single iteration as follows:

Step 1: Compute the distance between radar stand points A and B using the following equation.

$AB=\sqrt{{\left({x}_{1}-{x}_{2}\right)}^{2}+{\left({y}_{1}-{y}_{2}\right)}^{2}+{\left({z}_{1}-{z}_{2}\right)}^{2}}$

where A = (x1, y1, z1) and B = (x2, y2, z2).

Step 2: Compute the angles ∠BAC and ∠ABC using the observed azimuth values Az1 and Az2.

Step 3: Compute the angle ∠ACB = π– ∠ABC – ∠BAC.

Step 4: Compute the side lengths BC and AC using the following trigonometric equations and the computed values in steps as discussed.

$\begin{array}{l}BC=\left(sin\left(\angle BAC\right)*AB\right)/sin\left(\angle ACB\right)\\ AC=\left(sin\left(\angle ABC\right)*AB\right)/sin\left(\angle ACB\right)\end{array}$

Step 5: Using BC and AC, compute the target elevations El1 and El2 wrt Radar 1 and Radar 2 using the following trigonometric equations.

$\begin{array}{l}E/1={\mathrm{cos}}^{-1}\left(AC/R1\right)\\ E/2={\mathrm{cos}}^{-1}\left(BC/R2\right)\end{array}$

Step 6: Converting the two polar coordinates (range, azimuth, elevation) wrt a point of interest either (Radar 1 or Radar 2 stand point) and average the elevation value.

Using the elevation value obtained from the Step 6 and existing range and azimuth values, the air object height is estimated in ENV (East, North, Vertical) frame.

In 2-D radar tracking, multiple 2-D radars track the 3-D targets from the ships and it is observed that certain target geometric values are affecting the accuracy of the solution. A fuzzy selection4 method is considered to obtain the pair of 2-D radars, which gives optimal accuracy.

Fuzzy logic is an extension of Boolean logic supports in between values between true and false. The concept of fuzzy logic was introduced by Zadeah5. The goal was to develop a model that could be close to natural language process.

3.1 Fuzzification

Fuzzification is the first step in a fuzzy system. In this process, all crisp inputs are converted into fuzzy variables. The fuzzification process partitions the universe of discourse spanned by each variable into a number of fuzzy sets, assigning a linguistic variable6. Fuzzy sets are represented by fuzzy membership functions (MF). Different types of membership functions can be used for fuzzification process.

3.1.1 Membership Function

In fuzzy systems, fuzzy sets are represented by fuzzy membership functions (MF). Triangular, trapezoidal and Gaussian, are some of the membership functions5, used as per the convenience and nature of the problem. Figure 3 describes the considered membership functions for the fuzzification process of input ‘angle’ for the present problem.

For every crisp value, membership functions represent a degree, which indicates the strength of its concern to the fuzzy set. The degree of membership is always in the range [0,1]. For example, any crisp value has a membership degree (or degree of support) 0, indicates that the value is not in the fuzzy set. If the degree is 1, it means, the value is completely within the fuzzy set. The following equations explain how to calculate a membership value (m (x)) for a crisp value x, using a triangular membership function.

$\mu \left(x\right)=\left\{\text{\hspace{0.17em}}\begin{array}{ll}\hfill 0\hfill & \text{\hspace{0.17em}}if\text{\hspace{0.17em}}\text{\hspace{0.17em}}x<{a}_{1}\hfill \\ \left(x-{a}_{1}\right)/\left({a}_{2}-{a}_{1}\right)\text{\hspace{0.17em}}\hfill & if\text{\hspace{0.17em}}\text{\hspace{0.17em}}{a}_{1}\le x\le {a}_{2}\hfill \\ \left({a}_{3}-x\right)/\left({a}_{2}-{a}_{3}\right)\hfill & if\text{\hspace{0.17em}}\text{\hspace{0.17em}}{a}_{2}\le x\le {a}_{3}\hfill \\ \hfill 0\hfill & if\text{\hspace{0.17em}}\text{\hspace{0.17em}}x>{a}_{3}\hfill \end{array}\text{\hspace{0.17em}}$

where a1, a2, and a3 are the x coordinates of the start, middle, and end points of a triangular fuzzy membership function.

3.2 Fuzzy Inference

Fuzzy inference is a process of formulating a mapping from inputs to output using fuzzy logic. In fuzzy inference, knowledge is expressed in terms of fuzzy if-then rules. The computation of fuzzy inference consists of two components: aggregation (computation of if-part) and composition (computation of then-part). The most common inference methods AND, OR, and NOT are used in majority of fuzzy logic applications.

In this case, a numerical output is required from the inference process, a defuzzification step must follow the fuzzy inference. Mamdani type fuzzy inference is considered. A weight factor is used to tune the fuzzy logic system for optimal performance, the weight factor represents the strength of the rule belongs to the interval [0,1]. It is multiplied with the aggregation result in the composition step. Table 1 some of the fuzzy rules considered for the present problem are described.

3.3 Defuzzification

Defuzzification process translates fuzzy values to crisp value. Most of the fuzzy systems required this process to visualise the result in a numerical form rather than fuzzy. Defuzzification step also requires membership functions as shown in Fig. 4. The following equation represents computation of crisp value U using proposed centre of area defuzzification method for the present problem. Defuzzification method computes a crisp value U, which is in range of [0, 100].

$U=\sum _{i=1}^{n}\left({A}_{i}*{L}_{i}\right)/\sum _{i=1}^{n}{A}_{i}$

where Ai represents the area covered under each fuzzy set and Li represents mid-point of the fuzzy set on x-axis and n = 5 (number of fuzzy sets in the defuzzification process).

To obtain optimal results, it is considered that more than two 2-D radars are deployed on different ships, which are separated geographically. Each 2-D radar deployed on the ship tracks the air object, if it is within the coverage area. So that for a particular air object, tracked radar supplies range and azimuth information. For estimating air object height, there is a need to process the received information by considering the geometric solution mentioned in Section 2. During the experimental phase, it is observed that angles and the slant ranges play an important role in the height estimation process and identifying the best pair based on the angles and slant ranges is difficult part with conventional approach. It is well known that fuzzy systems perform well in nonlinear dynamic applications9, where mathematical models are complex. To get optimal accuracy, fuzzy logic based selection is considered for the optimal pair from the available pairs. For the present study, three 2-D radars are considered for tracking the target, so that three data sets, i.e., T1, R1, Az1 (time, range, azimuth) from Radar 1 T2, R2, Az2, from Radar 2 and T3, R3, Az3 from Radar 3 are obtained for the process in each process. The flow chart as shown in Fig. 5 describes the process for each iteration.

Using this method, a software has been developed using MATLAB 7.0. To test the software, aircraft trajectories were simulated by considering target dynamics. Software was tested with different data sets obtained from the simulation models and the results are shown in MATLAB graphs. To compare the performance of the proposed method, mean error (ME) was computed with the following equation.

$ME=\frac{1}{n}\sum _{i=1}^{n}|{H}_{R}-{H}_{E}|$

where HR represents real height of air object and HE is the height computed by one pair of radars or optimal pair, obtained by fuzzy selection method. Considering different dynamics of aerodynamic targets, the software has been tested and mean errors of all the experiments are given in Table 5.

In Experiment 1, a simulated maneuvering target was considered. In this simulation, the sampling time t =1(s). The initial position of the target is given by [x(0), y(0), z(0)] = [2,500 (m),14,000 (m),8,000 (m)] with an initial speed of [vx,vy,vz] = [150(ms-1), -60(ms-1), 4(ms-1)]. The target moved with a constant acceleration of [ux(0), uy(0), uz(0)] = [–0.5(ms-2), –0.5(ms-2), 0(ms-2)] until t = 100(s), Then it started to maneuver with acceleration [ux(101),uy(101),uz(101)] = [2(ms-2), 10(ms-2), –3(ms-2)]. This acceleration continued to t = 200s. Then, the target started another maneuver with acceleration [ux (201), uy (201), uz (201)] = [–10 (ms-2), –30(ms-2), 5(ms-2)]. This acceleration continued to t = 300 (s) and at t = 301 (s) target took one more maneuver with acceleration [ux(301), uy (301), uz(301)] =[–10 (ms-2), 20 (ms-2), –6 (ms-2)] and it continues up to t = 600 (s). Ships/Radars locations are represented in ENV frame. Ships/radars positions considered for the experiment are given in Table 2. Figures 6, 7 and 8 describe the estimation errors of available pairs with fuzzy selection method.

In Experiment 2, the trajectory generated for Experiment 1 was considered and the ship locations were changed as mentioned in Table 3. Figures 9, 10 and 11 describe the estimation errors of available pairs with fuzzy selection method. In Experiment 3, a simulated medium-maneuvering target, which was far from all the ships, was considered. In this simulation, the sampling time t = 1 (s). The initial position of the target is given by [x(0), y(0), z(0)] = [50,500(m), 34,000(m), 10,000(m)] with an initial speed of [vx,vy,vz] = [150 (ms-1), -60 (ms-1), 4(ms-1)]. The target moved with a constant acceleration of [ux(0),uy(0),uz(0)] = [-0.5(ms-2), -0.5(ms-2), 0(ms-2)] until t = 100(s), Then it started to maneuver with acceleration [ux(101), u (101), u (101)] = [2(ms-2), 3 (ms-2), -3(ms-2)]. This acceleration continues to t = 200s. Then, the target starts another maneuver with acceleration [ux(201), uy(201), uz(201)] = [5 (ms-2), -3 (ms-2), 5 (ms-2)]. This acceleration continues to t = 300(s) and at t = 301(s) target took one more manoeuver with acceleration [ux (301), uy (301), uz (301)] =[6 (ms-2), -5(ms-2), -5(ms-2)] and it continued up to t = 600(s). Ships/radars locations are represented in ENV frame are given in Table 4. Figures 12, 13 and 14 describe the estimation errors of available pairs with fuzzy selection method.

In this paper, a new algorithm has been presented that combines a geometric solution of 2-D radars with fuzzy logic-based selection technique for improving the accuracy of air target height estimation. Based on the proposed algorithm, software was designed and developed to estimate the 3-D target height using 2-D radar data. Software was tested with different sets of aircraft trajectories. In this experimental study, Mamdani type fuzzy logic technique was utilised to select a pair of sensors to get optimal values, among all available sensor pairs. It was observed that the proposed algorithm has considerable benefits over normal 2-D tracking methods. As all the fuzzy rules were generated and tuned manually, software may not provide optimal pair in case the solutions are relatively close. To enhance the accuracy, it is required to use genetic, neural techniques for optimal inference rules by considering various target geometries. All the fuzzy inference rules generated manually and using trial and error method, fuzzy rules are modified by tuning the weight values. Random errors caused by sensors are not considered. It is observed that height estimation error is increasing, when the distance between ships/radars is more (due to earth curvature). To control random disturbances in measurements, statistical estimation techniques3 (e.g., Kalman filter) are utilised frequently, which require target dynamic models and sensor uncertainties. For future enhancement, wavelets-based denoising techniques, either in combination of statistical approaches or alone are considered, as multi-resolution techniques do not require any pre-assumptions.

The authors express their deep sense of gratitude to University Grants commission for sanctioning DRS-III/SAP-I to the department of mathematics, Osmania University, Hyderabad whose infrastructural facilities were used during the course of our research. They are thankful to Shri K Srinivas, Director DCCT for giving necessary support.

1. Farid, Amoozegar, et al. Survey of fuzzy logic and neural network technology for multi-target tracking, SPIE, 3371, April 1998.

2. Gai-Ming-jiu, et al. An approach to tracking a 3-D target with 2-D Radar. Research Institute of Applied Mathematics. Naval Aeronautical Engineering Institute, Yantai, P.R. China, 2005.

3. Yaakov Bar-Shalom et al. Estimation with applications to tracking and navigation, John Wiley & Sons Inc. 2001.

4. Ym, Chen. Information fusion in data association applications. Applied Soft Computing, 2006, 6, 394-405, www.sciencedirect.com.

5. Zadeh, L.A. Fuzzy sets and systems. North-Holland, Amsterdam, 1978.

6. Reznik, Leonid. Fuzzy controllers, Newnes, Oxford 0x2 8DP, 1997, pp. 44-50.

7. Hall, David L. Handbook of multi-sensor data fusion. CRC Press, London, 2001, pp. 1-18.

8. Blackman, Samuels. Design and analysis of modern tracking Systems. Artech House, Boston, 1999.

9. Hazeem, N., et al. Multi-scale fuzzy Kalman filtering. Engineering applications of aritificial intelligence, 2006, 19, 439-50.

10. Hakl, H., et al. Aircraft height estimation using 2-D radar. Def. Sci. J., 2010, 60, pp. 100-105

 Mr S.G.K. Murthy obtained MSc (Mathematics) from Osmania University in 1990 and currently pursuing his PhD from Osmania University. He is presently working as Scientist E at Defence Research and Development Laboratory (DRDL), Hyderabad. His research areas are multi-sensor data fusion, soft computing, information security and software theorem proving techniques. Dr M.V.R. Murthy is currently Professor in Faculty of Mathematics and Computer Science at the University College of Science, Osmania University. He has received his PhD in Computational Fluid Mechanics in 1986 from Osmania University. His research areas include artificial neural networks, network securities, digital image processing.