Asymmetrical Gating with Application on Maneuvering Target Tracking

Abstract

A new asymmetrical gate with application in target tracking is proposed. Proposed gate has asymmetric shape that has large probability of target detection in the gate and has more advantages compared with elliptical gate. The gate is defined as the region in which the tracked target is expected to exist and just observation vectors in the gate are used as target detection. An analytical method to compute optimal size of gate is proposed and recursive estimation of asymmetric parameters of gate are studied. Comparison between proposed gate and conventional elliptical gate showed the efficiency of the proposed method in maneuvering target tracking applications and simulation results showed the proficiency of the proposed method.

1. Introduction

Tracking is meant to be the estimation of the true values of specifications of target motion, such as the position and velocity, based on the n-dimensional observational vector by the sensors [1–3]. One-dimensional observation vectors are obtained in a direction-finding setup observing just the azimuth angle [4]. Two-dimensional observation vectors are obtained in optical sensors observing the azimuth angle and elevation angle, or in radars observing the range and azimuth angle. Historically, PDA (Probabilistic Data Association) [5], JPDA (Joint PDA) [6], and MHT (Multiple Hypothesis Tracking) [7–9] have drawn attention as target tracking methods in an environment in which false signals from objects other than the target such as clutter exist [10, 11]. In target tracking in such an environment, just the region is considered for each target in which the target is expected to exist at the next sampling time [12]. This region is called the gate. The observation data within the gate are used for tracking. Various gate shapes are conceivable, such as a rectangle, circle, and ellipse. However, it is not known which shape of gate is optimum. If the gate is enlarged, many observation vectors from clutter or objects other than the target fall within the gate, resulting in some difficulties in target tracking. On the other hand, if the gate is made smaller, there is a danger that the observation data from the target to be tracked may not fall within the gate. It is more desirable that the position of the observation vector detected from the target be closer to the center of the gate compared with location of the false signals in the gate [13]. A method of determining the in-gate probability has been proposed in [10].

In this paper, asymmetrical gaussian distribution is introduced. Cross-surface of asymmetrical gaussian distribution is considered as asymmetrical gate. Parameters of asymmetrical Gaussian distribution should be estimated to estimate parameters of asymmetrical gate. Kalman filter as a standard method to estimate parameters of symmetrical distribution is used to estimate parameters of asymmetrical distribution. Standard deviation of asymmetrical distribution in opposite to the movement direction is estimated using estimation of acceleration in movement direction in centric coordinate system. In this paper, analytical preference of asymmetrical gate compared with the elliptical gate is studied. In this analytical study, volumes of gates are considered as a comparison criteria subjected to the assumption that in-gate probability of distribution for both asymmetric and elliptical gates are equal.

In the following of this paper, Section 2 introduces asymmetrical gaussian distribution. Section 3 is about estimation of parameters of gate. In Section 4, analytical comparison between asymmetrical gate and elliptical gate is considered. In Section 5, optimal size of gate is calculated and in Section 6, preference of this new method is shown by simulation result.

2. Asymmetric Gaussian Gate

Maneuvering target that has nonzero acceleration in direction of movement could be modeled with almost-constant-acceleration target motion model in which process and measurement noise on direction of movement have asymmetric distribution and process and measurement noises on orthogonal to the movement direction have gaussian distribution.

In this paper, it is assumed that distribution of location of maneuvering target in the next time is asymmetric gaussian in centric coordinate system. Center of centric coordinate system is in the location of target and directions of axis of it are in the movement direction and orthogonal to the movement direction.

One-dimensional asymmetric Gaussian distribution in direction of movement is illustrated as follows:𝑓𝑥_𝑡=1𝑘−1exp2𝑥_𝑡−𝑚²𝜎²_𝑥1𝑥_𝑡1>𝑚,𝑘−1exp2𝑥_𝑡−𝑚²𝜎²_𝑥2𝑥_𝑡<𝑚, (1) where 𝑘 is normalization factor and can be calculated as follow:𝑘=^𝑚_−∞−1exp2𝑥_𝑡−𝑚²𝜎²_𝑥1+_𝑚^+∞−1exp2𝑥_𝑡−𝑚²𝜎²_𝑥2=2𝜋𝜎²_𝑥12+2𝜋𝜎²_𝑥22=√𝜎2𝜋^𝑥1+𝜎_𝑥22. (2)

In (2),  𝜎_𝑥1 and  𝜎_𝑥2  are standard deviation of distribution of 𝑥_𝑡 in positive and negative directions, respectively, and 𝑚 is median of distribution of 𝑥_𝑡.

Considering (2) and (1), one-dimensional asymmetric gaussian distribution of 𝑥_𝑡 can be achieved as follows:𝑓𝑥_𝑡=1√2𝜋𝜎_𝑥1+𝜎_𝑥2−1/2exp2𝑥_𝑡−𝑚²𝜎²_𝑥1𝑥_𝑡1>𝑚,√2𝜋𝜎_𝑥1+𝜎_𝑥2−1/2exp2𝑥_𝑡−𝑚²𝜎²_𝑥2𝑥_𝑡<𝑚. (3)

For 𝑥_𝑡>𝑚, 𝑓(𝑥_𝑡) is same as typical Gaussian distribution with standard deviation 𝜎_𝑥1 and for 𝑥_𝑡<𝑚, it is the same as typical Gaussian distribution with standard deviation 𝜎_𝑥2.

If 𝜎_𝑥1=𝜎_𝑥2, (3) is standard Gaussian distribution and if 𝜎_𝑥1≠𝜎_𝑥2, (3) is asymmetrical Gaussian distribution. It means that distribution is not symmetrical around median 𝑚, and dampings of distribution are different around of the median.

In Figure 1, distribution of 𝑓(𝑥_𝑡) with zero median and two different standard deviations is shown.

Figure 1 

Distribution of 𝑓(𝑥𝑡) with zero mean and two variant variances.

It is assumed that Probability Distribution Function (PDF) of target in orthogonal to the direction of movement is zero mean gaussian with standard deviation as follows:𝑓𝑦_𝑡=12𝜋𝜎_𝑦−1exp2𝑦_𝑡²𝜎²_𝑦. (4)

In this paper, 𝑧_𝑡=[𝑥_𝑡𝑦_𝑡]^𝑇 is defined in centric coordinate system, in which 𝑥_𝑡 and 𝑦_𝑡 are measured values of target in centric coordinate system.

Probability distribution of 𝑧_𝑡 in centric coordinate system is illustrated as follows:𝑓𝑥_𝑡,𝑦_𝑡=1𝑘−1exp2𝑧_𝑡−𝑚_𝑧Σ₁⁻¹𝑧_𝑡−𝑚_𝑧^𝑇𝑥_𝑡1>𝑚,𝑘−1exp2𝑧_𝑡−𝑚_𝑧Σ₂⁻¹𝑧_𝑡−𝑚_𝑧^𝑇𝑥_𝑡<𝑚, (5) where𝑚_𝑧=𝑚0,Σ₁=𝜎²_𝑥100𝜎²_𝑦,Σ₂=𝜎²_𝑥200𝜎²_𝑦. (6)

In (5), 𝑘 is normalization factor and can be calculated as𝑘=^𝑚_−∞^+∞_−∞−1exp2(𝑥−𝑚)²𝜎²_𝑥2+𝑦²𝜎²_𝑦+𝑑𝑦𝑑𝑥_{𝑚+∞+∞−∞}−1exp2(𝑥−𝑚)²𝜎²_𝑥1+𝑦²𝜎²_𝑦𝜎𝑑𝑦𝑑𝑥=2𝜋^𝑥1𝜎_𝑦2𝜎+2𝜋^𝑥2𝜎_𝑦2=2𝜋𝜎_𝑦𝜎_𝑥1+𝜎_𝑥22=𝜋Σ₁^1/2+Σ₂^1/2. (7)

Considering (5) and (7), probability distribution of 𝑓(𝑥_𝑡,𝑦_𝑡) in centric coordinate system can be achieved as follows:𝑓𝑥_𝑡,𝑦_𝑡=12𝜋𝜎_𝑦𝜎_𝑥1+𝜎_𝑥2−1/2×exp2𝑧_𝑡−𝑚_𝑧Σ₁⁻¹𝑧_𝑡−𝑚_𝑧^𝑇𝑥_𝑡1>𝑚,2𝜋𝜎_𝑦𝜎_𝑥1+𝜎_𝑥2−1/2×exp2𝑧_𝑡−𝑚_𝑧Σ₂⁻¹𝑧_𝑡−𝑚_𝑧^𝑇𝑥_𝑡<𝑚. (8)

Probability distributions of 𝑓(𝑥_𝑡,𝑦_𝑡) in (8) with surfaces of distribution are illustrated in Figure 2.

(a)

(b)

Figure 2 

(a) Probability distribution of 𝑓(𝑥𝑡,𝑦𝑡) for 𝑚=7, 𝜎𝑦=5, 𝜎𝑥1=10, 𝜎𝑥2=4, (b) probability distribution of 𝑓(𝑥𝑡,𝑦𝑡) for 𝑚=7, 𝜎𝑦=5, 𝜎𝑥1=4, 𝜎𝑥2=10.

As it can be shown from Figures 2(a) and 2(b), with variation of ratio of 𝜎_𝑥1 and 𝜎_𝑥2, it is easy to stretch distribution surface in movement direction or opposite of movement direction. Application of tuning of variances is that, for maneuvering targets that have positive or negative accelerations on direction of movement, it is possible to modify surface of distribution, optimally.

If maneuvering target has positive acceleration in direction of movement, it is better to select 𝜎_𝑥1>𝜎_𝑥2 to stretch gate in positive direction of movement as Figure 2(a) and if maneuvering target has been negative acceleration in direction of movement, it is better to select 𝜎_𝑥1<𝜎_𝑥2 to stretch gate in the negative direction of movement as Figure 2(b). Therefore, with appropriate choosing of 𝜎_𝑥1 and 𝜎_𝑥2, it is possible to stretch the gate in direction that probability of target detection inside of the gate increases, whereas volume of gate remains constant.

3. Gate’s Parameters Estimation

In this section, parameter estimation of target position distribution is considered. If process and measurement noise have Gaussian distribution, Kalman filter would be optimal recursive method to estimate parameters of distribution. Therefore, typical elliptical gate would be optimal tracking gate in such cases [14].

In this paper, maneuvering target with positive or negative acceleration in movement direction is considered. Kalman filter as a standard recursively estimation method is used to recursively estimate of asymmetrical gate’s parameters.

Let us assumed that parameters of asymmetrical gate on time 𝑡−1 are 𝑚_𝑡−1, 𝜎_{𝑥1,𝑡−1}, 𝜎_{𝑥2,𝑡−1}, 𝜎_{𝑦,𝑡−1}, 𝜃_𝑡−1, where 𝑚_𝑡−1 is center of gate, 𝜎_{𝑥1,𝑡−1}, 𝜎_{𝑥2,𝑡−1} shows asymmetric stretching of gate in the direction of movement, 𝜎_{𝑦,𝑡−1} is the stretching parameter of gate in orthogonal to the movement direction and 𝜃_𝑡−1 is rotation angle of centric coordinate system. The goal is to estimate parameters of gate at time 𝑡.

Using Kalman filter as a standard method in parameter estimation, median and covariance matrixes could be calculated as follows:𝑚_{𝑡∣𝑡−1}=𝐴𝑚_{𝑡−1∣𝑡−1},Σ_{𝑡∣𝑡−1}=𝐴Σ_{𝑡−1∣𝑡−1}𝐴^𝑇𝑚+𝑄,_𝑡∣𝑡=𝑚_{𝑡∣𝑡−1}+𝐾_𝑡𝑧_𝑡−𝐶𝑚_{𝑡∣𝑡−1},Σ_𝑡∣𝑡=Σ_{𝑡∣𝑡−1}−𝐾_𝑡𝐶Σ_{𝑡∣𝑡−1},𝑆_𝑡=𝐶Σ_{𝑡∣𝑡−1}𝐶^𝑇𝐾+𝑅,_𝑡=Σ_{𝑡∣𝑡−1}𝐶^𝑇𝑆_𝑡⁻¹. (9)

Typical elliptical tracking gate at time 𝑡, could be formulated using median 𝑚_𝑡∣𝑡 and innovation covariance 𝑆_𝑡 as follows:𝑧_𝑡−𝑚_𝑡∣𝑡^𝑇𝑆_𝑡⁻¹𝑧_𝑡−𝑚_𝑡∣𝑡≤𝛾, (10) where 𝑧 is the validated measurement and 𝛾 is a parameter to adjust gate size.

Diagonal terms of innovation covariance 𝑆_𝑡 are diameters of elliptical gate which are standard deviations of gaussian distribution on centric coordinate system. Off-diagonal terms of 𝑆_𝑡 represent rotation angle of centric coordinate system.

Parameters of typical elliptical gate at time 𝑡 are 𝜎_𝑥,𝑡, 𝜎_𝑦,𝑡 and 𝜃_𝑡.

In Figure 3, elliptical gate in 𝑥₁𝑜𝑥₁ is represented as 𝑥^𝑇Σ₂⁻¹𝑥=1 and in 𝑋₁𝑂𝑋₁ is represented as 𝑋^𝑇Σ₁⁻¹𝑋=1, whereΣ₂=𝜎²_𝑥,𝑡00𝜎²_𝑦,𝑡,Σ₁=𝑆_𝑡,𝑋=𝑋₁𝑋₂^𝑇,𝑥=𝑥₁𝑥₂^𝑇. (11)

Figure 3 

Elliptical gate in coordinate systems 𝑥1𝑜𝑥2 and 𝑋1𝑂𝑋2.

Coordinate systems 𝑥₁𝑜𝑥₁ is transferred to the 𝑋₁𝑂𝑋₁ using simple transfer matrix 𝑇 as follows:𝑋=𝑇𝑥where𝑇=cos𝜃−sin𝜃sin𝜃cos𝜃. (12)

In the following, relation between parameters of Σ₂ and Σ₁ are shown:𝑥^𝑇Σ₂⁻¹𝑥=𝑋^𝑇Σ₁⁻¹Σ𝑋,₂₋₁=𝑇^𝑇Σ₁⁻¹𝑇. (13)

After some simplification, relations between parameters of elliptical gate and elements of innovation covariance could be achieved as follows:𝜃_𝑡=12𝑡𝑔⁻¹2𝑆_12,𝑡𝑆_11,𝑡−𝑆_22,𝑡,𝜎²_𝑥,𝑡=−𝑆_11,𝑡cos²𝜃_𝑡−1+𝑆_22,𝑡sin²𝜃_𝑡−1,𝜎²_𝑦,𝑡=𝑆_11,𝑡sin²𝜃_𝑡−1−𝑆_12,𝑡sin²𝜃_𝑡−1+𝑆_22,𝑡cos²𝜃_𝑡−1, (14) where 𝑆_11,𝑡, 𝑆_12,𝑡, 𝑆_21,𝑡, 𝑆_21,𝑡 and 𝑆_22,𝑡 are elements of matrix 𝑆_𝑡.

In this paper, Klaman filter is used to estimate parameters of asymmetric gate. The median that is achieved from Kalman filter is set as median of asymmetric Gaussian distribution which is the center of asymmetrical gate 𝑚_𝑡. Using (14), 𝑆_11,𝑡, 𝑆_12,𝑡, 𝑆_21,𝑡, 𝑆_21,𝑡, and 𝑆_22,𝑡 are used to estimate 𝜃_𝑡, 𝜎_𝑥,𝑡, and 𝜎_𝑦,𝑡. It is assumed that 𝜎_𝑥1,𝑡=𝜎_𝑥,𝑡 as a standard deviation of asymmetrical gate on movement direction. To calculate parameter 𝜎_𝑥2,𝑡, estimated acceleration in movement direction, 𝑎_𝑥,𝑡 is used which could be estimated from constant acceleration model and Kalman filter.

If 𝑎_𝑥,𝑡=0, probability of existence of target in both side of median in movement direction are equal. Therefore, 𝜎_𝑥1,𝑡 is set equal to 𝜎_𝑥2,𝑡 and asymmetrical gate is converted to elliptical gate.

If𝑎_𝑥,𝑡>0, probability of target existence in movement direction is greater than negative side. It means that 𝜎_𝑥1,𝑡 should be greater than 𝜎_𝑥2,𝑡 which is resulted to following proposed equation:𝜎_𝑥1,𝑡𝜎_𝑥2,𝑡1=1+2𝑎_𝑥,𝑡𝑇², (15) where 𝑇 is sampling time and 𝑎_𝑥,𝑡 is estimation of acceleration in movement direction.

If𝑎_𝑥,𝑡<0, probability of existence of target in negative side of movement direction is greater than positive sight. So 𝜎_𝑥2,𝑡 should be greater than 𝜎_𝑥1,𝑡 which is resulted to following proposed equation:𝜎_𝑥2,𝑡𝜎_𝑥1,𝑡1=1−2𝑎_𝑥,𝑡𝑇². (16)

In proposed estimation method, the ratio of standard deviations in direction of movement is proportional to sampling time and estimated acceleration in movement direction.

4. Analytical Comparison between Asymmetrical and Elliptical Gate

In this part, analytical comparison between asymmetrical gate and typical elliptical gate in tracking of targets with nonzero acceleration are studied. Volumes of gates are considered as a proper criterion in comparison. To compare volumes of gates, it is assumed that in-gate probabilities of distributions are equal for both gates.

In tracking with elliptical gate, probability of distribution in centric coordinate system is Gaussian as following:𝑓₁𝑧_𝑡=12𝜋𝜎_𝑥0𝜎_𝑦0−1×exp2𝑧_𝑡−𝑚₀^𝑇𝜎_𝑥000𝜎_𝑦0⁻¹𝑧_𝑡−𝑚₀, (17) where 𝑚₀ is median, 𝜎_𝑥0 is standard deviation of distribution on movement direction, and 𝜎_𝑦0 is standard deviation of distribution on orthogonal to movement direction.

In tracking with asymmetric gate, it is assumed that probability of target distribution in centric coordinate system is asymmetric gaussian as follows:𝑓₂𝑧_𝑡=12𝜋𝜎_𝑦1𝜎_𝑥1+𝜎_𝑥2−1/2×exp2𝑧_𝑡−𝑚₁^𝑇𝜎_𝑥𝑖00𝜎_𝑦1⁻¹𝑧_𝑡−𝑚₁𝑖=1,2, (18) where 𝑚₁ is median, 𝜎_𝑥1 and 𝜎_𝑥2 are standard deviations of distribution on movement direction, and 𝜎_𝑦1 is standard deviation of distribution on orthogonal to movement direction.

Because of the fact that probability distributions in (17) and (18) are represented in centric coordinate system, off-diagonal elements of covariance matrixes are zero.

To compare between asymmetric gate and elliptical gate, it is assumed that the target has positive acceleration in movement direction. Parameters of distributions (17) and (18) could be estimated using Kalman filter and the proposed method in (15) and (16). Therefore, the relation between parameters of asymmetric gate and elliptical gate are achieved as follows:𝜎_𝑥≜𝜎_𝑥0=𝜎_𝑥1,𝜎_𝑦≜𝜎_𝑦0=𝜎_𝑦1,𝑚≜𝑚₀=𝑚₁,𝜎_𝑥2<𝜎_𝑥1. (19)

In-gate symmetric and asymmetric probability of distribution could be calculated as follows, respectively [14],𝑃_1𝐺𝑡=_𝐺𝑡𝑓₁𝑧_𝑡=𝑑𝑥𝑑𝑦_𝑑10/2()−𝑑exp−𝑢𝑑𝑢=1−exp¹2,𝑃_2𝐺𝑡=_𝐺𝑡𝑓₂𝑧_𝑡=𝑑𝑥𝑑𝑦_𝑑20/2()−𝑑exp−𝑢𝑑𝑢=1−exp²2. (20)

Symmetric gate volume 𝑉_1𝐺𝑡 and asymmetric gate volume 𝑉_2𝐺𝑡 are considered as follows:𝑉_1𝐺𝑡=𝑑₁𝜋𝜎_𝑦0𝜎_𝑥0,𝑉_2𝐺𝑡=𝑑₂𝜋𝜎_𝑦1𝜎_𝑥1+𝜎_𝑥22. (21)

It is assumed that, gate probabilities are equal in symmetric and asymmetric gate. Therefore, regarding (19), the following relation could be achieved:𝑃_1𝐺𝑡=𝑃_2𝐺𝑡⟹𝑑₁=𝑑₂𝜎_𝑦0=𝜎_𝑦1𝜎_𝑥1+𝜎_𝑥22<𝜎_𝑥0⟹𝑉_2𝐺𝑡<𝑉_1𝐺𝑡. (22)

Using the above equation, we get that with equal in-gate probability for symmetric and asymmetric gates, volume of asymmetric gate would be smaller than volume of symmetric gate. It means that with equal probability for asymmetric and elliptical gate, computational complexity to find target inside asymmetrical gate would be less than computational complexity inside of elliptical gate. It is because of the fact that volume of asymmetrical gate is smaller than volume of elliptical gate with equal in-gate probability of distribution.

5. Optimal Size of Gate

Elliptical gates in [14] are represented as follows:𝑧_𝑡−𝑚_𝑧^𝑇𝑆_𝑡⁻¹𝑧_𝑡−𝑚_𝑧≤𝑑, (23) where 𝑧_𝑡=[𝑥_𝑡𝑦_𝑡], 𝑚_𝑧 is center of gate, and 𝑆_𝑡 is innovation covariance matrix that has information about rotation direction of gate and ratio of diameters of gate. Asymmetrical gate that is proposed in this paper could be represented similar to (23) as𝑧_𝑡−𝑚_𝑧^𝑇𝑆_𝑡⁻¹𝑧_𝑡−𝑚_𝑧≤𝑑, (24) where 𝑧_𝑡=[𝑥_𝑡𝑦_𝑡], 𝑚_𝑧=[𝑚_𝑡0], and𝑆_𝑡=𝜎²_{𝑥𝑖,𝑡}00𝜎²_{𝑦,𝑡𝑖∈{1,2}}. (25)

If 𝑥_𝑡>𝑚_𝑡, we choose 𝑖=1 and if 𝑥_𝑡<𝑚_𝑡, we choose 𝑖=2.

𝑧_𝑡 would be inside the gate if two following constraints are satisfied.(a)If 𝑥_𝑡>𝑚_𝑡, then 𝑧_𝑡−𝑚_𝑧^𝑇𝜎²_𝑥1,𝑡00𝜎²_𝑦,𝑡⁻¹𝑧_𝑡−𝑚_𝑧<𝑑. (26)(b)If 𝑥_𝑡<𝑚_𝑡, then𝑧_𝑡−𝑚_𝑧^𝑇𝜎²_𝑥2,𝑡00𝜎²_𝑦,𝑡⁻¹𝑧_𝑡−𝑚_𝑧<𝑑. (27) Otherwise, 𝑧_𝑡 is out of the gate.

It is easy to see from (24) that shape of the gate could be determined by 𝑚_𝑧, 𝜎²_𝑥2,𝑡, 𝜎²_𝑥1,𝑡, and 𝜎²_𝑦,𝑡, whereas the size of the gate could be smaller or larger using just 𝑑.

As the gate is made larger, more clutter or observation vectors other than the tracked target are included within it, so that tracking becomes more difficult. On the other hand, if the gate is made smaller, there is an increasing danger that the observation vector from the tracked target may no longer fall within the gate.

So, in this paper, we define a cost function to find optimum size of gate using the cost function for elliptical gates that is proposed in [14]. In this paper, using the method proposed in [14], optimum size of asymmetrical proposed gate is calculated.

Suppose that whole points inside of the gate are defined as follows:𝐺_𝑡=𝑧_𝑡∣𝑧_𝑡−𝑚_𝑧^𝑇𝑆_𝑡⁻¹𝑧_𝑡−𝑚_𝑧≤𝑑. (28)

In-gate probability could be calculated by: 𝑃_𝐺𝑡=_𝐺𝑡𝑓𝑥_𝑡,𝑦_𝑡𝑑𝑥𝑑𝑦, (29) where 𝑓(𝑥_𝑡,𝑦_𝑡) is asymmetrical Gaussian distribution.

After some simplifications, in-gate probability simplifies as follows:𝑃_𝐺𝑡=₀^𝑑/2𝑒^−𝑢𝑑𝑢. (30)

From (30), it is visible that in-gate probability is independent from the shape of the gate which is defined using parameters 𝑚_𝑧, 𝜎²_𝑥2,𝑡, 𝜎²_𝑥1,𝑡, and 𝜎²_𝑦,𝑡, and it is just function of the gate size 𝑑.

Volume of gate could be considered as𝑉_𝐺𝑡=_𝐺𝑡𝑑𝑥𝑑𝑦. (31)

After simplification, volume of gate reduced to𝑉_𝐺𝑡=𝑑𝜋𝜎_{𝑦,𝑡−1}𝜎_{𝑥1,𝑡−1}+𝜎_{𝑥2,𝑡−1}2. (32)

Assuming that probability of false alarm observation at time 𝑡 is a constant value 𝐵^𝑡_𝑓, mean number of false observed signals inside of the gate are 𝐵^𝑡_𝑓𝑉_𝐺𝑡:𝐵^𝑡_𝑓𝑉_𝐺𝑡=𝐵^𝑡_𝑓𝑑𝜋𝜎_{𝑦,𝑡−1}𝜎_{𝑥1,𝑡−1}+𝜎_{𝑥2,𝑡−1}2. (33)

With the assumption that target observation probability is constant 𝑃_𝐷, the probability that target is observed inside of the gate is 𝑃_𝐷𝑃_𝐺𝑡. In other side, mean number of false signals inside of the gate is 𝐵^𝑡_𝑓𝑉_𝐺𝑡. The goal is to find optimum size of the gate so that 𝑃_𝐷𝑃_𝐺𝑡 is maximum and 𝐵^𝑡_𝑓𝑉_𝐺𝑡 is minimum. With respect to the fact that 𝑃_𝐷𝑃_𝐺𝑡 and 𝐵^𝑡_𝑓𝑉_𝐺𝑡 are function of 𝑑, so it is possible to define ℎ(𝑑) in such a way that, with maximizing it, optimum value of 𝑑 is calculated:ℎ(𝑑)=𝑃_𝐷𝑃_𝐺𝑡−𝑐𝐵^𝑡_𝑓𝑉_𝐺𝑡,ℎ(𝑑)=𝑃_𝐷0^𝑑/2𝑒^−𝑢𝑑𝑢−𝑐𝐵^𝑡_𝑓𝑑𝜋𝜎_{𝑦,𝑡−1}𝜎_{𝑥1,𝑡−1}+𝜎_{𝑥2,𝑡−1}2, (34) where 𝑐 is a constant value: 𝜕ℎ(𝑑)𝜕𝑑=0⟹𝑑_opt=−2ln𝑐−2ln2𝜋−2ln𝐵^𝑡_𝑓−2ln𝜎_{𝑦,𝑡−1}𝜎_{𝑥1,𝑡−1}+𝜎_{𝑥2,𝑡−1}2+2ln𝑃_𝐷. (35)

Larger 𝑐 increase emphasis on 𝐵^𝑡_𝑓𝑉_𝐺𝑡 and number of false observed signals inside of the gate decreases.

6. Simulation Result

To simulate the preferences of the new method, in first we generate samples of a track using model (36). In this model, let 𝑟_𝑥(𝑡) be position, let 𝑠_𝑥(𝑡) be velocity, let 𝑎_𝑥(𝑡) be acceleration of the target in movement direction, let 𝑟_𝑦(𝑡) be position, let 𝑠_𝑦(𝑡) be velocity, and let 𝑎_𝑦(𝑡) be acceleration of the target in orthogonal to movement direction at time 𝑡 in centric coordinate system, where 𝑡 stands for continues time index. Dynamics of the target is described by system model that could be written in stochastic differential equation:𝑋̇(𝑡)=𝐴𝑋(𝑡)+𝐵𝑛(𝑡), (36) where 𝑋(𝑡) is state vector𝑋(𝑡)=𝑟_𝑥(𝑡)𝑠_𝑥(𝑡)𝑎_𝑥(𝑡)𝑟_𝑦(𝑡)𝑠_𝑦(𝑡)𝑎_𝑦(𝑡)^𝑇. (37)𝐴 is state transition matrix:𝐴=010000001000000000000010000001000000, (38) vector 𝐵 is defined as𝐵=001000000001^𝑇, (39) vector 𝑛(𝑡) is defined as𝑛(𝑡)=𝑛₁(𝑡)𝑛₂(𝑡)^𝑇, (40) where 𝑛₁(𝑡) is random noise with uniform distribution between zero and one and 𝑛₂(𝑡) is zero mean white gaussian noise.

By applying time discretization to (36) with sampling time 𝑇, we have a discrete time system model:𝑋_𝑘=𝐴_𝑑𝑋_𝑘−1+𝐵_𝑑𝑛_𝑘. (41)

State transition matrix 𝐴_𝑑 is𝐴_𝑑=𝑇1𝑇²2𝑇00001𝑇0000010000001𝑇²200001𝑇000001, (42) and vector 𝐵_𝑑 is𝐵_𝑑=𝑇³3𝑇²2𝑇𝑇000000³3𝑇²2𝑇^𝑇. (43)

Measured position of the target in centric coordinate system is denoted by 𝑧_𝑘 and is assumed to be obtained by observation model:𝑧_𝑘=𝐶𝑋_𝑘+𝑤_𝑘, (44) where 𝐶 is𝐶=100000000100, (45) and 𝑤_𝑘 is defined as𝑤_𝑘=𝑤_1𝑘𝑤_2𝑘^𝑇, (46) where 𝑤_1𝑘 is random noise with uniform distribution between zero and one and 𝑤_2𝑘 is zero mean white gaussian noise.

In Figure 4, target positions are shown that is generated using the above model. Monte Carlo simulation is used to approximate probability distribution of target position in next time step. Approximated probability of distribution is shown in Figure 4.

Figure 4 

Target track and estimation of probability of distribution in next time step using Monte Carlo simulation.

It is easy to understand from Figure 4 that probability distribution in next time could be illustrated using asymmetric distribution that formulated previously.

Kalman filter is used to estimate parameters of asymmetric gate that are considered in part 3. Actual and estimated values of 𝑟_𝑥(𝑡), 𝑠_𝑥(𝑡), and 𝑎_𝑥(𝑡) in centric coordinate system are shown in Figure 5.

Figure 5 

Actual (blue solid) and estimated (red dash) values of 𝑟𝑥(𝑡), 𝑠𝑥(𝑡), and 𝑎𝑥(𝑡) in centric coordinate system.

To show proficiency of the proposed asymmetrical gating method in comparison with previous gating methods such as circular gating, elliptical gating, and rectangular gating, similar simulation scenarios are studied for circular, elliptical, and rectangular gates. The volumes of gates in each simulation are equal and the location of the target in the next time step is important in the comparison between different gates. The number of times that the target falls in a gate in the next time step is a comparison index between gates. In Table 1, comparison results between circular gate, elliptical gate, rectangular gate, and proposed asymmetrical gate are shown for different gate volumes.

Table 1 

Comparison results between circular gate, elliptical gate, rectangular gate and asymmetrical gate for different gate volumes.

In Table 1, percent falls of the target inside each gate are shown for different volumes of gate. For example, for volume of gate 0.0025, percent fall of the target inside rectangular gate is 51%, inside circular gate is 65%, inside elliptical gate is 73% and inside asymmetrical gate is 92%.

For different volumes of gates, it could be seen from Table 1 that proposed asymmetrical gate has more in-gate probability than other gating methods and the number of target falls inside of the asymmetrical gate in the next step is more than others. It is because of the asymmetrical gate ability to shape itself to have more percent falls of the target inside the gate. Although parameters of other gates can be updated in each step to have more percent falls inside the gates, the ability of shape changing in other gates is less than asymmetrical gate specially in maneuvering target tracking applications. Not that the percent falls of the target inside of the gates increases by increasing gate volume that could be seen easily from Table 1.

7. Conclusion and Future Remarks

In this paper, we introduced asymmetric gating technique for tracking maneuvering targets that have nonzero acceleration in movement direction. Asymmetric gate is defined as a cross-surface of asymmetric Gaussian distribution. Standard Kalman filter is used to estimate some parameters of asymmetrical gate. Standard deviation of asymmetrical distribution in opposite to the movement direction is estimated using a proposed method in which sampling time and estimated acceleration are used. Preference of asymmetrical gate to the elliptical gate is proved analytically subject to constant in-gate probability of distribution for both gates, in which volumes of gates are used as a comparison criterion. We derived an optimization method to finding optimum size of asymmetrical gate. Optimum size of asymmetrical gate is obtained as a function of some parameters such as probability of target observation and probability of false alarms that should be tune manually in each special applications.

As a future remarks, it could be assumed that probability of distribution in orthogonal to the movement direction in centric coordinate system is considered asymmetric. Also, estimation of gate parameters could be studied in typical coordinate system, after enough research, and proves on generalization of Kalman filter for linear systems with asymmetric gaussian probability of distributions for both process and measurement noise.