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The paper addresses a problem of ballistic object tracking with the use of the cinetheodolite electro-optical tracking system. Electro-optical systems are applied for acquiring the trajectory data of missiles, satellites, and rockets used for delivery of satellites to their prevised orbits. Despite the importance of such systems and their applications, in the open literature there are no publications describing tracking algorithms processing data from cinetheodolites. The paper describes a model-based algorithm of estimation of position and parameters of target motion for such a system, developed by the authors. The model of the system, nonlinear both in its description of the target dynamics and the measurement equations, is presented in detail. The proposed algorithm of estimation is also described, and chosen simulation results are included in the paper. Furthermore, a comparison of the proposed estimation algorithm with other possible, but simpler algorithms is presented.

The main motivation for the work presented in this paper was a necessity to obtain possibly accurate and continuous data on position and other parameters of motion of ballistic objects in practical tests of weaponry conducted by the Military Institute of Armament Technology (MIAT), Zielonka, Poland. Among the areas of expertise of the Institute, tests of anti-aircraft missiles are of great importance. One of the methods to assess the capabilities of the missile is the live firing with trajectory recording. It allows estimating the position and velocity of the missile and the target and eventually the mutual geometrical relationship between them, particularly the miss distance.

To estimate the position of a missile or a target, either an accurate on-board positioning system or an external tracking system is necessary. To assess the miss distance, an aerial target imitator fulfilling demanding requirements, between the others in terms of velocity, must be used. At MIAT, a special aerial target, ICP-89, based on the S-5 unguided missile, was developed. Its feasibility of deploying, high initial velocity, and market availability make it adequate for rocket missile tests. ICP-89 has proved its usefulness in numerous experiments with weapons so far.

In practice, an accurate on-board positioning system with an appropriate downlink is unlikely to be installed on a combat missile (particularly a small one) or an aerial target such as ICP-89. Therefore, to assess the performance of the missile during live tests, electro-optical tracking systems are widely used. A cinetheodolite-based electro-optical tracking system (EOTS), acquiring precise angular data to assess flight paths of disparate objects, is currently used at MIAT. It is a core system to determine the performance of the tested missiles.

In the open literature, there are no publications describing tracking algorithms for processing EOTS data; therefore, custom solutions were elaborated at MIAT. So far, straightforward geometrical computations have been used for position calculations of the missile and the aerial target. Such a method, however, has several significant drawbacks. Incomplete measurement data, occasionally occurring during some experiments, require supplementary techniques to attain only approximate and reduced-accuracy results. Moreover, the applied algorithm is not model-based and does not involve any data filtering; thus, its accuracy is not optimal. In this paper, a new algorithm of estimation of the position and parameters of ICP-89 target motion, based on the extended Kalman filter (EKF) [

The layout of the further part of this paper is as follows. The EOTS measurement system is presented in Section

The EOTS measurement system used for fire tests consists of at least two cinetheodolite stations, located at positions measured before the experiments, and providing for azimuth

EOTS cinetheodolite on the operating post.

Positions of the stations and tracked object as well as all the parameters of motion are expressed in a local horizontal frame of reference which must be defined at the beginning of the experiments. The geometrical relationships in the EOTS system with two cinetheodolite stations

Geometrical relationships in the EOTS measurement system [

The frame of reference is established as follows. Firstly, the positions of

After such a setting of the local frame of reference, the local coordinates of

The cinetheodolite stations are synchronized by a common registration-triggering signal and perform acquisition of the images of the observed object as well as the angles of its observation from separate locations. The angles provided by two stations include azimuths

A pair of measurements of azimuth

Based on the above equations, the following set of four equations with three unknowns

This is an overdetermined set of equations, which can be approximately solved for

Equation (

In practice, disturbing factors and an inaccurate setting of the reference frame may result in a nonzero value of

Estimation of the ICP-89 position can be realized with the use of various algorithms, which can be generally divided into two groups, i.e., non-model-based algorithms and model-based ones. The simplest are single-point solution algorithms, such as ordinary-least-squares (OLS) and weighted-least-squares (WLS) methods, which iteratively estimate the target position [

The primary measurements realized in the EOTS measurement system are azimuth and elevation angles, which are obtained simultaneously, with a period of about 33 msec. For every time instance, they can be grouped into the primary measurement vector

As the relationship between the original measurement vector

The above equation, apart from the geometrical relationships, includes also a vector of measurement errors

Assume an initial estimated target position vector

Calculate the expected measurement vector

Calculate the Jacobian matrix

Solve Equation (

Improve the previous estimate of

Repeat steps 2 to 5 until

The above OLS algorithm is run for every newly acquired measurement vector

Finally, the Jacobian matrix

The WLS algorithm is another single-point solution method [

As the measurement vector

Therefore, the covariance matrix

The Jacobian matrix

All the other partial derivatives in the Jacobian matrix

Assume an initial estimated target position vector

Calculate the expected measurement vector

Calculate the Jacobian matrix

Calculate the Jacobian matrix

Calculate the covariance matrix

Calculate the estimation errors vector

Improve the previous estimate of

Repeat steps 2 to 7 until

The OLS and WLS algorithms are simple, but they do not use information from the previous steps of data processing nor do they comprise any filtering. In contrast, the EKF presented in this section realizes model-based data filtering [

The use of EKF requires formulation of the dynamics model of the ICP-89 motion in the following form [

The elaboration of the dynamics model for the aerial target imitator ICP-89 requires consideration of its physical properties and its way of operation as well as properties of the atmosphere. The photography of the imitator at the initial phase of flight during firing tests is shown in Figure

ICP-89 aerial target imitator approximately 20 msec after being launched.

During tests, the imitator is launched from the firing stand due to the thrust force generated by its solid propellant. The propellant burns for approximately 0.7 seconds, and for the rest of the flight, the target moves without propulsion or control and follows a free-flight motion pattern. Therefore, the dynamics model of the considered system should be based on equations describing motion of a ballistic object (rocket) [

ICP-89 imitator outline in 2D space with essential variables describing its movement.

The 3-DOF EOM are derived from Newton’s second law of dynamics and assume that the motion of the target of a mass

The drag and lift forces depend on the squared target velocity with respect to the air

If we consider that, the first valid measurements of azimuth and elevation angles from the EOTS system become available when the target already moves without propulsion, i.e.,

Equations (

If we augment the undisturbed model of the target motion, taking into account the influence of wind, the changes of the target position are given as follows [

The changeable character of wind influences not only kinematic equations of position (Equations (

There exist various sophisticated stochastic models of wind, which generally assume existence of strong spatial and temporal correlation of wind velocity [

Instead, we propose using a simple stochastic model of wind, which takes into account the random temporal and spatial variability of wind velocity. Our model, however, is only an approximation as it does not consider a correlation which normally exists between the horizontal and vertical velocity of wind fluctuations. The proposed model assumes that the horizontal and vertical components of wind are two independent Wiener processes of unknown initial values, modelled as follows [

Based on the above assumptions and considerations, we finally obtain the following continuous dynamic model of the target motion, which has the standard form required by the theory and suitable for immediate use in the EKF algorithm:

It is worth noticing that in contrast to the OLS and WLS, the variables in the state vector

The above continuous model given by Equation (

The linearization is realized through calculation of the Jacobian matrix

The model sampling consists in calculation of two matrices: the state transition matrix

The covariance matrix of discrete process disturbances

The measurement model used in the EKF is given by Equation (

Finally, it is clear from Equation (

The covariance matrix of measurement errors

Estimation of the state vector

Assume an initial estimated state vector

Assume an initial value of the covariance matrix of filtration errors

Calculate the predicted state vector

Calculate the Jacobian matrix

Calculate the state transition matrix

Calculate the covariance matrix of discrete process disturbances

Calculate the covariance matrix of prediction errors:

Calculate the Jacobian matrix

Calculate the Jacobian matrix

Calculate the covariance matrix

Calculate the Kalman gain matrix

Calculate the corrected state vector

Calculate the covariance matrix of filtration errors:

Recursively repeat steps 3rd to 13th

The above set of steps realized during filtration can be interpreted as the initialization (steps 1st and 2nd), the time update or prediction (steps 3rd to 7th), and the measurement update or filtration (steps 8th to 13th). The initialization is executed only once at the beginning of the filter operation, whereas the steps of prediction and filtration are executed recursively for the whole required period of the EKF operation.

After formulating the dynamics and the observation model of the EOTS tracking system, it is possible to run the simulation and analyze the operation of designed estimation algorithms. For the purpose of comparison, two of the presented algorithms, i.e., OLS and EKF, were implemented and tested in the Matlab® scientific environment. The WLS algorithm was not implemented, as the standard deviations of all angle measurements realized in the EOTS are the same; thus, the accuracy offered by WLS should be the same as for OLS.

The appropriate choice of simulation parameters is crucial if we want to obtain valuable and meaningful results. These parameters include the dynamics and the observation model coefficients, as well as initialization parameters for the OLS and EKF.

The dynamics model presented in this paper contains several parameters characteristic for the tracked object and the environment. The geometrical dimensions of the object were chosen to be identical with the real ICP-89 projectile; thus, the calculated surface of the object

The elements of the covariance matrix of continuous process disturbances

The primary measurement error covariance matrix ^{-6} rad). The diagonal elements of

The methodology of simulations included several steps. Firstly, a reference flight trajectory was generated, using a numerical 4th order Runge-Kutta solution to the dynamic equations of motion (Equations (

The following figures represent a selection of the obtained results. In Figure

Simulated and estimated trajectories of ICP-89 aerial target imitator.

Figures

Simulated and estimated horizontal position of ICP-89.

Simulated and estimated vertical position of ICP-89.

The estimation of velocity, pitch angle, and wind components is not possible in the OLS, but only in the EKF; therefore, the estimation results for those quantities, shown in Figures

Simulated and estimated velocity of ICP-89.

Simulated and estimated pitch angle of ICP-89.

Simulated and estimated horizontal component of wind.

Simulated and estimated vertical component of wind.

To facilitate a comparison of accuracy of the OLS and EKF algorithms, errors of horizontal and vertical coordinates of the ICP-89 target imitator are presented in Figures

Error of estimation of horizontal position of ICP-89.

Error of estimation of vertical position of ICP-89.

This effect is due to changeable conditions of observation, i.e., changing geometrical dilution of the precision factor, which depends on the angle of intersection of the lines between the EOTS stations and the target. For the assumed simulation scenario, this angle decreases in the terminal phase of flight, which increases the geometrical dilution of precision and the target localization errors.

It is not possible to directly compare the two implemented algorithms with respect to the accuracy of estimation of the other state variables (velocity, pitch angle, and wind velocity components), because the OLS does not provide their estimates. Therefore, in the following Figures

Error of estimation of ICP-89 velocity.

Error of estimation of ICP-89 pitch angle.

Error of estimation of horizontal component of wind.

Error of estimation of vertical component of wind.

The quantitative comparison between the OLS and EKF can be based on the RMS estimation errors of both algorithms. The results of such a comparison for the estimates of target position coordinates, velocity, and pitch angle are gathered in Table

Comparison of OLS and EKF target localization errors.

Error | OLS | EKF | Improvement ratio |
---|---|---|---|

RMS |
21.2·10^{-3} |
7.4·10^{-3} |
2.87 |

RMS |
8.6·10^{-3} |
4.4·10^{-3} |
1.95 |

RMS |
5.67 | 1.03 | 5.50 |

RMS |
33.5·10^{-3} |
6.8·10^{-3} |
4.95 |

The paper presented a problem of estimation of a ballistic object trajectory and its solutions based on the OLS and EKF algorithms. Both algorithms designed, implemented, and tested by the authors of the paper proved to be useful tools for estimation of an aerial target position and its parameters of motion. The usefulness of the presented approach was demonstrated for a practical problem of estimation of a trajectory and parameters of motion of the aerial target imitator ICP-89, based on the measurements from the electro-optical tracking system with two cinetheodolite stations.

The simulations of such a system have shown that the target trajectory can be estimated with high accuracy. As expected, tracking errors depend on the geometry of the EOTS system and increase when the ICP-89 is far from the cinetheodolite stations. This conclusion should be considered when the shooting tests are planned and used for optimal location of the observation stations in the firing range.

The presented comparisons of the OLS and EKF accuracy show that the EKF can estimate the target position with 2-3 times better accuracy than the OLS. Moreover, it can directly estimate the target velocity, pitch angle, and components of wind, which is not possible in the OLS. Therefore, the EKF algorithm should be a preferred solution in realization of a practical tracking system.

The authors are currently continuing their works on the presented system aimed at implementation of the presented algorithms, especially the EKF, for processing real measurements from the EOTS system, recorded during live firing tests. The works are aimed also at improving the assumed ballistic model of ICP-89 as well as the wind model. It should be emphasized that the results obtained so far are very promising and it is expected that the EKF algorithm, elaborated by the authors, can be implemented in the software used by MIAT and replace algorithm currently used for fire tests.

The data used to support the findings of this study are available from the corresponding author upon request.

The authors declare no conflict of interest.

P. Kaniewski conceived the idea. P. Smagowski and S. Konatowski designed the experiments; P. Smagowski performed the experiments; P. Kaniewski and S. Konatowski analyzed the data; P. Kaniewski and P. Smagowski wrote the paper; and P. Kaniewski and S. Konatowski approved the paper.