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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.

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

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

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:

In (

Considering (

For

If

In Figure

Distribution of

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:

In this paper,

Probability distribution of

In (

Considering (

Probability distributions of

(a) Probability distribution of

As it can be shown from Figures

If maneuvering target has positive acceleration in direction of movement, it is better to select

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 [

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

Using Kalman filter as a standard method in parameter estimation, median and covariance matrixes could be calculated as follows:

Typical elliptical tracking gate at time

Diagonal terms of innovation covariance

Parameters of typical elliptical gate at time

In Figure

Elliptical gate in coordinate systems

Coordinate systems

In the following, relation between parameters of

After some simplification, relations between parameters of elliptical gate and elements of innovation covariance could be achieved as follows:

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

If

If

If

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

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:

In tracking with asymmetric gate, it is assumed that probability of target distribution in centric coordinate system is asymmetric gaussian as follows:

Because of the fact that probability distributions in (

To compare between asymmetric gate and elliptical gate, it is assumed that the target has positive acceleration in movement direction. Parameters of distributions (

In-gate symmetric and asymmetric probability of distribution could be calculated as follows, respectively [

Symmetric gate volume

It is assumed that, gate probabilities are equal in symmetric and asymmetric gate. Therefore, regarding (

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.

Elliptical gates in [

If

If

If

It is easy to see from (

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 [

Suppose that whole points inside of the gate are defined as follows:

In-gate probability could be calculated by:

After some simplifications, in-gate probability simplifies as follows:

From (

Volume of gate could be considered as

After simplification, volume of gate reduced to

Assuming that probability of false alarm observation at time

With the assumption that target observation probability is constant

Larger

To simulate the preferences of the new method, in first we generate samples of a track using model (

By applying time discretization to (

State transition matrix

Measured position of the target in centric coordinate system is denoted by

In Figure

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

It is easy to understand from Figure

Kalman filter is used to estimate parameters of asymmetric gate that are considered in part 3. Actual and estimated values of

Actual (blue solid) and estimated (red dash) values of

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

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

Gate volume | Rectangular gate | Circular gate | Elliptical gate | Asymmetrical gate |
---|---|---|---|---|

0.0025 | 51% | 65% | 73% | 92% |

0.005 | 68% | 78% | 84% | 96% |

0.01 | 90% | 98% | 100% | 100% |

In Table

For different volumes of gates, it could be seen from Table

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.