Preliminary study on estimation of 12.7 x 99 mm caliber projectile using Unscented Kalman Filter method

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Preliminary study on estimation of 12.7 x 99 mm caliber projectile Preliminary study on estimation of 12.7 x 99 mm caliber projectile using Unscented Kalman Filter method using Unscented Kalman Filter method Luh Ayu B.Laa, Erna Apriliania* , Hendro Nurhadibb* a* Luh Ayu B.L , Erna Apriliani , Hendro Nurhadi *

a Department of Mathematics, Institut Teknologi Sepuluh Nopember, 60111, Indonesia ofCenter Mathematics, Institutfor Teknologi Sepuluh Indonesia Department of MechanicalaDepartment Engineering, of Excellence Mechatronics andNopember, Industrial60111, Automation (PUI-PT MIA-RC ITS), Institut b Department of Mechanical Engineering, Center of Excellence Mechatronics andIndonesia Industrial Automation (PUI-PT MIA-RC ITS), Institut Teknologi Sepuluhfor Nopember, 60111, Teknologi Sepuluh Nopember, 60111, Indonesia b

Abstract Abstract The projectile dynamics model is non-linear system. The specifications of projectile used are 12.7 x 99 mm caliber. In the The projectile dynamics model is non-linear system. The specifications of projectile used 99 estimation mm caliber. In thex projectile motion, to achieve the target can be estimated by using some estimation methods. In are this 12.7 study,x the of 12.7 projectile motion, to achieve the target can be estimated by using some estimation methods. In thisIn study, the estimation 12.7 x 99 mm caliber projectile motion is computed by using Unscented Kalman Filter (UKF) method. this simulation, weof estimate 99 mmofcaliber projectile motion is each computed by using Unscented Kalman results Filter (UKF) method. In this of simulation, model the system by measuring variables position. The estimation show the performance UKF andwe KFestimate in nonmodel of the system by measuring position. The estimation results show the study performance of UKF and KF in nonlinear systems is different, based oneach theirvariables respective characteristics. The final result of this shows that the error estimation linear systems different, on their respective characteristics. The-position final result studyinshows that the error estimation of UKF is 81 %issmaller in based -position , 85 % smaller in -position and , 84of%this smaller -position. UKF performance -position 85 % smaller with in -position and has -position , 84 %result. smaller in -position. UKF performance of very UKFgood is 81for % non-linear smaller in system is and, the approach KF algorithm less optimal is very good for non-linear system and the approach with KF algorithm has less optimal result. © 2018 The Authors. Published by Elsevier Ltd. © 2018 The Authors. Published by Ltd. © 2018 The Authors. by Elsevier Elsevier Ltd. This is an open accessPublished article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0/) This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0/) This is an and openpeer-review access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0/) Selection under responsibility of the INNS Conference Big Data and Deep Learning 2018. Selection and peer-review under responsibility of the INNS Conference ononBig Data and Deep Learning 2018. Selection and peer-review under responsibility of the INNS Conference on Big Data and Deep Learning 2018. Keywords: Projectile Motion, Estimation, Unscented Klaman Filter (UKF) Keywords: Projectile Motion, Estimation, Unscented Klaman Filter (UKF)

Nomenclature Nomenclature Horizontal Coordinate (m) Horizontal Coordinate Vertical Coordinate (m)(m) Vertical Coordinate (m)

State Estimation State Estimation Estimated State Covariance Estimated State Covariance

*Corresponding author. E-mail address: [email protected], [email protected] *Corresponding author. E-mail address: [email protected], [email protected] 1877-0509 © 2018 The Authors. Published by Elsevier Ltd. 1877-0509 © 2018 Thearticle Authors. Published by Elsevier Ltd. This is an open access under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0/) This is an and openpeer-review access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0/) Selection under responsibility of the INNS Conference on Big Data and Deep Learning 2018. Selection and peer-review under responsibility of the INNS Conference on Big Data and Deep Learning 2018.

1877-0509 © 2018 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0/) Selection and peer-review under responsibility of the INNS Conference on Big Data and Deep Learning 2018. 10.1016/j.procs.2018.10.518

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Velocity (m/s) Pitch Angle (rad) Mass of Projectile (kg) Lift Coefficient Drag Coefficient Surface of Projectile (m2) Atmosphere Density (kg/m3) Gravitational Acceleration (m/s2) Gaussian Noise with Covariance Q Gaussian Noise with Covariance R Function handle for Constants used to determine the distribution State Correction State Prediction Matrix Controller

185

Current Measurement Process Noise Covariance Measurement Noise Covariance Matrix Input Matrix Output Unobserved State of The System The Exogenous Input The Observed Measurement Signal Noise Measurement The Observation Noise Noise System Scaling Parameters Dimensions Matrix Noise Covariance noise process Covariance noise measurement

1. Introduction Bullets generally consist of various parts, namely projectiles, sleeves, gunpowder, and lighter. The projectile is part of a bullet that moving in the air due to the thermal expansion occurring inside the sleeve [1]. In general, projectiles have many sizes (diameter) used on firearms declared as caliber. One of the most commonly used caliber of war or armed vehicles by Indonesian National Army (TNI) is a 12.7 x 99 mm caliber projectile, where a 12.7 x 99 mm caliber projectile is a projectile designed for the M2 Browning Machine Gun rifle, also used for sniper rifles and machine guns with a .50 caliber projectile. The projectiles fired from these rifles can travel at a speed of 900 meters per second. That is, projectiles move toward the target with a distance of 2,000 meters can be taken in less than 3 second. The projectile can move in the air by giving initial velocity to its initial state. After it moves in the air the motion of the projectile is completely affected by gravity. A projectile destroys the target by piercing it with the kinetic energy generated by its very high speed. In very fast projectile movements, we require an estimation to reach the target correctly. Some research concering to projectiles and position are “Estimating 3D Positions and velocities of Projectiles from Monocular Views”, and “Estimasi Posisi Mobile Robot Menggunakan Akar Kuadrat Unscented Kalman Filter (AKUKF)” [2],[3]. In estimating the projectile motion, we can use several estimation methods. One of the estimators that can be used is an estimator according to the results of a research by Piotr Smagowski and Piotr Kaniewski entitled “Estimation of Ballistic Object Trajectory Using Non-Linear Kalman Filtering”. The model used is a nonlinear dynamic system with three degrees of freedom (3-DOF) motion in a vertical plane. This 3-DOF motion model considering the position where this model only includes translations from the normal plane to the OXZ plane, is indicated by the gamma angle. The authors estimate the Ballistic Object path Using a Non-Linear Kalman Filter, KF estimation is based on a linearized system [4],[5]. Some development of Kalman filter methods for position estimation are Ensemble Kalman Filter (EnKF) and Ensemble Kalman Filter Square Root (EnKF-SR) and Fuzzy Kalman Filter (FKF) applied to AUV [2],[6]. Kalman filter is an estimation method for the linear dynamic stochastic system and the first introduced model by Rudolph E. Kalman [7],[8],[9]. However, many problems appear not only in linear dynamic models but also appear in non-linear dynamic models. From the problems that often arise, the need to develop algorithms that can not only be implemented in linear dynamic system but can also be implemented in a non-linear dynamic system.The algorithm has been developed from Kalman Filter algorithm modification to estimate non-linear dynamic system that has been equipped with Extended Kalman Filter (EKF), Ensemble Kalman Filter (EnKF), and Unscented Kalman Filter (UKF). One the results of research by S. Konatowski and A. T. Pieniezny with the title "A comparison of estimation accuracy by the use of KF, EKF, and UKF filters", in the case of the model system used is

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a non-linear dynamic model by comparing the three methods of estimation, the UKF method is able to provide the best performance [10]. The UKF method is a modified estimation of the Kalman Filter algorithm that can be used to estimate nonlinear system models by using unscented transformation [11]. In this paper we investigated the motion of 12.7 x 99 mm caliber projectiles in the target shooting mission using a non-linear 3 DOF model where we measured all state variables to determine the accuracy of the estimation and real results of each variable, the goal is to help the soldiers learn to make it easier to shoot while shooting the target. And we compare the Unscented Kalman filter methods and Kalman filter method by approaching a non-linear system by using linearization to find Jacobi matrix [12]. Therefore, based on the background described we took a title Preliminary study on estimation of 12.7 x 99 mm caliber projectile motion using Unscented Kalman Filter method. 2. Modelling 2.1. Projectile Dynamics Model A full description of Newton’s laws consists of six degrees of freedom (6-DOF) equations of motion (EOM) [5],[13]. The 6-DOF EOM describes translational movement along X, Y and Z axes, and a rotational movement around them, represented by γ, φ, and ψ angles and accompanying moments Mγ, Mφ, and Mψ [5],[14]. In this model be simplified into three degrees of freedom (3-DOF) motion in the vertical plane. Motion models only include translational movements along the X and Z axes, and rotational motion around the normal plane to the OXZ plane, represented by the gamma angle so that the dynamical model of the projectile motion is obtained as follows [5]:

Figure 1. Object dynamics with reduced degrees of freedom

(1) (2) (3) (4) As the projectile moves in the air, it causes an influence of aerodynamic forces affecting the motion of the , lift force , gravitation force , and thrust force projectile. The forces that influence it are the drag force . Thus, projectile motion needs to be in the estimation to be able to dash right on target. The model is nonlinear, so one of the estimators that can be used is a modification of Kalman filter algorithm for non-linear dynamic system. 2.2 Kalman Filter Method Kalman Filter was first introduced by R.E Kalman in 1960. Kalman Filter is a method of estimating state variables of a linear discrete-time dynamical system that minimizes the estimated covariance error. Generally, a discrete-time linear dynamical system is as follows [15],[16]: System model

:

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Measurement model Inisialization

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: :

is noise system and is noise measurement. It has a random vector drawn for Gaussian distribution Where, with mean 0 and covariance Q. In Kalman Filter, estimation is done in two stages that are by predicting state variable based on a dynamical system called prediction stage (time update) and correction stage (measurement update) to improve estimation result according to measured data. Kalman Filter algorithm is presented in Table 1. Table 1. Kalman Filter Algorithm System Model Measurements Model Assumption Initialization Prediction Stage

Estimation : Error Covariance :

Correction Stage

Kalman Gain Estimation

: :

Error Covariance:

2.3 Unscented Kalman Filter Method The Unscented Kalman Filter (UKF) method is a modified estimation method of the Kalman Filter algorithm that can be used to estimate nonlinear system models by using unscented transformation. The UKF can provide considerable improvement over the EKF, an EKF algorithm works in a way that linearizes the nonlinear system around the KF estimation, and the KF estimation is based on the linearized system [4].The general form of nonlinear dynamical systems in UKF is [17],[18]: System Model: Measurements Model: is noise system, is noise measurement, and matrix Where, algorithm presented in Table 2 [16],[12],[9].

is linear. The Unscented Kalman Filter

Table 2. Unscented Kalman Filter (UKF) Algorithm System Model Measurements Model Initialization

;

; ;

For , Calculate Sigma Points :

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Where,

Prediction Stages

Estimation

;

5

, For, :

Error Covariance : ; Correction Stages

;

2.4 Flowchart Algorithm The following is a flowchart of the algorithm of the Kalman Filter method and the algorithm of the Unscented Kalman Filter method, where there are process differences from both methods. In Kalman Filter, the estimation is done by two stages: the prediction stage (time update) and the correctiont stages (measurement update). Prediction stage is predicting state variable and accuracy level calculated using covariance error equation or norm of error covariance. In the correction stage, the state variable estimation results are corrected using the measurement model. One part of this stage is to determine the Kalman Gain matrix used to minimize error covariance. The prediction and , is an correction stage are performed recursively by minimizing the estimated error covariance: assessment of the state variable[15]. At the process prediction stage and the correction stage will continue to repeat until the -time. In the algorithm of the Unscented Kalman Filter method, there is an unscented transformation. Unscented Transformation is a method used to calculate the mean and covariance of a random variable undergoing non-linear has a random variable of a transformation. For example, a discrete opportunity density function is approached with an non-linear model with dimensions , mean and covariance . The function sigma points unscented transform. Then the mean and covariance are used to determine the spread of around . The sigma points in the form of the sigma vector , can be obtained using the following equation [12], [20]. ,

(5) ,

(6)

with, The same as the process on the Kalman filter method, at the process prediction stage and the correction stage will continue to repeat until the -time. As the process of coherence, both methods are described in Figure 2 and Figure 3.

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START

189

START

System Model: 

System Model : 

Measurement Model:

Measurement Model :

Initialization:

Initialization: ,

,

Calculate Sigma Points :

Prediction Stage. Estimation : 

Error Covariance : Prediction Stages : Estimation : Error Covariance :

k-time

,

Calculate Kalman Gain : 

Correction Stages : ,

Final Result

FINISH

Figure 2. Flowchart Algorithm of Unscented Kalman Filter

Correction Stage KalmanGain: k-time

Estimation Error Covariance:

Final Result

FINISH

Figure 3. Flowchart Algorithm of Kalman Filter.

3. Result and Analysis The model is continuous-time deterministic dynamic system. To apply the Kalman Filter dan Unscented Kalman Filter (UKF), the model is transformed into a dynamic discrete-time system model using a finite difference method. The result is as follows: (7) (8)

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(9)

(10) is a discrete stochastic model which has a noise system In the equation drawn for Gaussian distribution with mean 0 and covariance Q.

. It has a random vector

3.1 Implementation of Unscented Kalman Filter At the time of the implementation of the Unscented Kalman Filter method, discretized non-linear systems can be directly applied to the UKF algorithm. With the system model:

With meassurement data is : After that the process progresses to the initialization stage, the prediction stage, and the correction stage. unsceneted However, when in the prediction stage, the sigma point is calculated based on the transformation. Having obtained the mean and covariance of the prediction stage, then go into the correction stage. At the correction stage, the estimated value of the covariance will be found. And the process will return back to the prediction stage for the desired - time. 3.2 Implementation of Kalman Filter In the implementation stage of the Kalman Filter method, the Kalman Filter method can only be applied to linear systems only. Therefore we need linearization on a non-linear system. This stage of linearization is done to find a Jacobi matrix, where the system model becomes:  with the defined matrix

is:

For the system in this study matrix A is described below:

With meassurement data is :

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After that the process can continue to the initialization, the prediction stage and the correction stage. In the prediction stage and the correction stage, the processes will continue to repeat until the -time. 3.3 Simulation In the simulation process, the simulation result of Kalman Filter and Unscented Kalman Filter on 12.7 x 99 mm caliber projectile motion is computed for = 20 iterations. Simulation is performed using the parameters in Table 3. Table 3. Parameters

No. 1. 2. 3. 4. 5. 6.

Parameters Mass of Projectile ( ) Air Density ( ) Drag Coefficient ( Lift Coefficient ( Surface of Projectile (S) Gravitational (g)

0.125 kg 1 kg/m3 0.96 0.89 0.000127 9.8065 m/s2

Value

In the simulation using Kalman Filter and Unscented Kalman Filter, the initial condition is on 12.7 x 99 mm caliber projectile motion to obtain the covariance error in the prediction stage and In order to obtain the covariance and as follows: error in the correction stage, we define initial conditions ;

, And

This simulation compares the real value with estimated value using Kalman Filter, and estimated value using Unscented Kalman Filter in each variable that can be seen in Figure 4 – 7.

Figure 4. Position variable

Figure 6. Position variable

Figure 5. Position variable

Figure 7. Position variable

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In Figure , shows the combination of the real value, the estimated value of Kalman Filter (KF) and the estimated value of Unscented Kalman Filter (UKF). In this paper, the estimation results of each variable are shown in Table 4. Table 4. RMSE value UKF and KF

Experiment 1 2 3 4 5 6 7 8 9 10 Average

Variable

Variable

0.0227 0.0169 0.0168 0.0240 0.0308 0.0218 0.0362 0.0312 0.0262 0.0314 0.0258

0.0274 0.0257 0.0224 0.0360 0.0296 0.0238 0.0204 0.0281 0.0245 0.0251 0.0263

UKF Variable

Variable

0.0232 0.0205 0.0241 0.0272 0.0142 0.0237 0.0167 0.0224 0.0253 0.0244 0.0222

0.0239 0.0303 0.0383 0.0337 0.0253 0.0208 0.0179 0.0197 0.0267 0.0244 0.02613

Computational Time 1.845972 1.745576 1.754926 1.705297 1.721772 1.777591 1.700797 1.750528 1.724762 1.729981 1.748720

Variable

Variable

KF Variable

Variable

0.1406 0.0884 0.1698 0.1327 0.1842 0.1331 0.1184 0.1193 0.1108 0.1367 0.1334

0.1748 0.1586 0.2108 0.1939 0.2060 0.1883 0.1780 0.1896 0.1844 0.1695 0.1854

0.1516 0.0944 0.1733 0.1400 0.1585 0.1338 0.1398 0.1464 0.1110 0.1490 0.1398

0.1600 0.1403 0.2024 0.1728 0.1741 0.1728 0.1645 0.1664 0.1435 0.1656 0.1662

Computational Time 1.911080 1.732274 1.758592 1.737266 1.752048 1.802116 1.726105 1.761856 1.731910 1.770905 1.768415

The final result of these study shows that UKF’s performance is seen to provide better accuracy than KF performance. In figure 4 the resulting average error value between the real value with estimated value using KF is 0.1334 and average error value between real value with estimated value using UKF is 0.0258. Figure 5 the resulting average error value between the real value with estimated value using KF is 0.1854 and average error value between real value with estimated value using UKF is 0.0263. Figure 6 the resulting average error value between the real value with estimated value using KF is 0.1398 and average error value between real value with estimated value using UKF is 0.0222. and in Figure 7, the resulting average error value between the real value with estimated value using KF is 0.1662 and average error value between real value with estimated value using UKF is 0.02613. Here is the percentage of RSME of each variable listed in the table 5. Table 5. Pecentage RSME of each variable. Method UKF KF

Percentage of Variable 0.00312 0.01638

Percentage of Variable 0.00321 0.02265

Percentage of Variable 0.00268 0.01776

Percentage of Variable 0.00318 0.02033

The following is a device of specifications that we use to simulate the program shows in Table 6. Table 6. Device of Specifications Device Name Processor Installed RAM Device ID Product ID System type Pen and touch MATLAB

: DESKTOP-9BKC4FR : Intel(R) Pentium(R) CPU B950 @2.10Ghz : 2,00 GB : D8816302-2A71-4CDB-A2C6-2919B953BF91 : 00329-00000-00003-AA492 : 32-bit operating system, x64-based processor : No pen or touch input is available for this display : R2013a

4. Conclusions The performance of each method is compared based on the accuracy and computational time of estimation. The error estimation of UKF is 81 % smaller in -position , 85 % smaller in -position and -position , 84 % smaller in -position. Unscented Kalman Filter’s performance on this non-linear system is very good with Unscented Kalman Filter (UKF) computing time is 0.011 % faster than Kalman Filter (KF) method. Based on the analysis and discussion presented in the previous chapter we can conclude that dynamic model of 12.7 x 99 mm calibre projectile

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motion is non-linear.The model can be applied with Unscented Kalman Filter and can be done linear approximation by using Kalman filter method, but approach with KF algorithm has less optimal result. Acknowledgements This research was supported by Center of Excellence for Mechatronics and Industrial Automation (PUI-PT MIARC ITS) and Ministry of Riset Technology and Higher Education of Republik Indonesia (RISTEK-DIKTI). We thank our helpful colleagues. We thank Mister Dr. Dieky Adzkiya, S.Si, M.Si for comments that greatly improved the manuscript, And we also thank Miss Siti Mushonnifah, S.Si, M.Si for all the help and insight given. References [1]

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