Preliminary study on estimation of 12.7×99 mm caliber projectile motion using Extended Kalman Filter method

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Preliminary study on estimation of 12.7×99 mm caliber projectile Preliminary study on estimation of 12.7×99 mm caliber projectile motion using Extended Kalman Filter method motion using Extended Kalman Filter method Riska Apriliaaa, Erna Aprilianiaa, Hendro Nurhadibb* Riska Aprilia , Erna Apriliani , Hendro Nurhadi *

a Department Of Mathematics, Institut Teknologi Sepuluh Nopember, Surabaya, 60111, Indonesia a Department Of Mathematics, Teknologi Sepuluh Nopember, Surabaya, 60111, Indonesia Department Of Mechanical Engineering, Center of Institut Excellence for Mechatronics and Industrial Automation (PUI-PT MIA-RC ITS), Institut b Department Of Mechanical Engineering, Center ofSepuluh Excellence for Mechatronics and Industrial Automation (PUI-PT MIA-RC ITS), Institut Teknologi Nopember, Surabaya, 60111, Indonesia Teknologi Sepuluh Nopember, Surabaya, 60111, Indonesia b

Abstract Abstract A Projectile is a shot object which fired from firearms or air rifles. There are various sizes of projectile. One of them is 12.7 x A Projectile is a shot object whichspeed fired of from firearms or air rifles. There are various of projectile. of them is hit 12.7thex 99 mm caliber projectile. The higher 12.7 x 99 mm caliber projectile makes thesizes less accurate, it is One for the shot to 99 mm caliber higher speed of 12.7 x 99 reaches mm caliber projectile makes the less accurate, it is are for modelled the shot tobyhitthree the right target. It isprojectile. necessaryThe to estimate that the projectile the right target. The projectile movements right target. It is necessary to and estimate the projectile reaches target.used The isprojectile movements areFilter, modelled three degrees of freedom (3-DOF) have that a non-linear equation. So the right estimation The Extended Kalman but itbywould degrees of freedom (3-DOF) andMethod. have a non-linear equation. the estimation used is The Extended Filter, but it would be combined with Kalman Filter Measurement data is So assumed to be linier. And we have all dataKalman measurement of projectile be combined with Kalman Filterare Method. Measurement dataThe is assumed to beoflinier. we have data measurement of Filter projectile motion. The estimation results simulated with Matlab. final result theseAnd studies showall that Extended Kalman has motion. Thethan estimation are simulated with Matlab. final result of these show97.38% that Extended Filter has small error Kalmanresults Filter with error estimation of EKFThe is 92.02% smaller in thestudies x-position, smallerKalman in the z-position, small error than Kalman Filter with error of EKFinisgamma 92.02%angle smaller thecomputational x-position, 97.38% in faster the z-position, 94.99% smaller in velocity projectile, andestimation 95.45% smaller andinthe time issmaller 0.925% than KF 94.99% smaller in velocity projectile, and 95.45% smaller in gamma angle and the computational time is 0.925% faster than KF method. method. © 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, Kalman Filter, Extended Kalman Filter; Keywords: Projectile Motion, Kalman Filter, Extended Kalman Filter;

A. Introduction A. Projectile Introduction is a shot object which is fired from firearms or air rifles. It is made of metal, generally from lead. A Projectile is a shot objectbywhich is fired from firearms or airgenerated rifles. It is of metal, from context, lead. A projectile destroys the target piercing it with the kinetic energy bymade very high speed.generally In the modern projectile destroys the target by piercing it with the kinetic energy generated by very high speed. In the modern context, a projectile along with sleeves, gunpowder, reams, and primers is a part of a unit of ammunition. How it works when a projectile along with sleeves, gunpowder, reams, and primers is a part of a unit of ammunition. How it works when

* Corresponding author. * Corresponding author. E-mail address: [email protected], [email protected] 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.2018.10.515

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fired is to push a projectile with the kinetic energy produced by a propellant explosion, which is usually a gunpowder. This explosive is lit by a small detonator called a primer. There are various size of projectile used on firearms seen from the diameter of projectile. In general, the size of a projectile is called a calibre. One of them is 12.7 x 99 mm caliber projectile. 12.7 x 99 mm caliber projectile is projectile design for heavy M2 Browning machine gun, these are also used for sniper rifles and machine guns with .50 caliber projectile. In the military, the projectile used is 12.7 x 99 mm caliber. Because it can paralyze, and damage the target. 12.7 x 99 mm caliber projectile has high speed velocity. Because of the very rapid movement of projectile, we are necessary to estimate the projectile to reach the target properly. Estimation is done because a problem can sometimes be solved by using previous information or data related to the problem [1] [2] . One of the estimation algorithm is Kalman Filter. Kalman Filter is an estimation method for the linear dynamic stochastic system [3]. Kalman Filter is an optimal linear estimator while the process noise and the measurement noise can be modelled by white Gaussian noise [4]. Kalman Filter can be implemented largely in software and run on the radar’s digital processor [5]. Kalman filter also can be applied at cuff-less blood pressure on android platform [6], 3D relative position and orientation for robot control [7] and many more. The other side, there are many modifications of Kalman Filter algorithm. These modifications have been done to avoid the convergence of algorithm, to reduce the computation time, to decrease the error of estimation and other [8]. One of the modifications of Kalman Filter is the Extended Kalman Filter. Extended Kalman Filter Method is the modified estimation of Kalman Filter method that can be used to estimate the state variable of the stochastic dynamical nonlinear. Predictive target tracking is one of the processes of estimating the location of the target at a given time in the future using Extended Kalman Filter [9]. The extended Kalman Filter can be applied as a near-optimal solution to the adaptive Multiplicative Distortion (MD) and channel parameter estimation problem [10]. In another side, Extended Kalman Filter algorithm can be used for Life Prediction of L70 life for LEDs [11]. The motion of the projectile is necessary to be predicted or estimated to reach the right target. The Projectile motion can be modelled by three degrees of freedom (3-DOF) [12]. So, the dynamics model of the projectile has a non-linear form. To estimate the projectile motion which is modelled three degrees of freedom (3-DOF), we use Kalman Filter and Extended Kalman Filter method. As far as we know, Kalman Filter is used to a linear system. So, before we apply equations of projectile motion to the Kalman Filter, it is necessary to do a linearization the dynamical model of projectile motion [4]. So, in this paper, we want to apply the non-linear system to the Kalman Filter and Extended Kalman Filter to get to know how the result of estimation the 12.7 x 99 mm caliber projectile motion are. Therefore, we want to calculate the time computation and we need accurate position estimation [13] [14] using Kalman Filter that is compared with Extended Kalman Filter. This is simulated with Matlab. B. Modelling 2.1 Dynamic Model of Projectile Motion Projectile’s movements consist of six degrees of freedom (6-DOF). There are translational movement along 𝑋𝑋, 𝑌𝑌, and 𝑍𝑍 axes and rotational movement around them. There are defined by pitch (𝛾𝛾), yaw (𝜑𝜑), roll (𝜓𝜓). The projectile’s

movements can be simplified to three degrees of freedom (3-DOF) motion in a vertical plane [10]. There are translational motion along the 𝑋𝑋 and 𝑍𝑍 axes and rotational surrounding the normal plane to the 𝑂𝑂𝑂𝑂𝑂𝑂 plane, represented by pitch angle and it can be illustrated in fig.1. Fig. 1. Object dynamics with reduced degrees of freedom

Generally, three degrees of freedom (3-DOF) motion equation of projectile are a non-linear system. The equations consist of distance (horizontal coordinate), height (vertical coordinate), velocity, and pitch angle. The equation of



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projectile motion as follows [12]:

𝑥𝑥̌̇ = 𝑣𝑣̌ cos 𝛾𝛾̌ ℎ̌̇ = 𝑣𝑣̌ sin 𝛾𝛾̌ 1 𝑣𝑣̌̇ = − 𝐶𝐶 𝑆𝑆𝑆𝑆𝑣𝑣̌ 2 − 𝑔𝑔 sin 𝛾𝛾̌ 2𝑚𝑚 𝑑𝑑 1 𝑔𝑔 𝛾𝛾̌̇ = 𝐶𝐶𝑙𝑙 𝑆𝑆𝑆𝑆𝑣𝑣̌ 2 − cos 𝛾𝛾̌

(1) (2) (3) (4)

𝑣𝑣̌

2𝑚𝑚

The projectile motion causes aerodynamic force effect in the air. The influencing force can be seen in Figure 1. There are the drag force(𝐹𝐹𝑑𝑑 ), lift force (𝐹𝐹𝑙𝑙 ), gravitation force (𝐹𝐹𝑔𝑔 ), and the thrust force (𝐹𝐹𝑡𝑡 ). There are four variables in the dynamical projectile motion. They represent the projectile position, velocity of projectile and pitch angle. It is necessary to predict projectile motion in order to achieve the target properly. 2.2 Kalman Filter Method The Kalman Filter Methods was first introduced by R.E. Kalman in 1960 [15]. Kalman Filter is an optimal data processing algorithm as well as a linear dynamic system estimator. Kalman Filter is able to estimate the dynamic state variables of the system with two stages. They are the prediction stage and the correction stages. The prediction stages (time update) is the estimation stage of the dynamic model system, while the correction stage (measurement update) is the estimation stage of the measurement model [16]. The algorithm of Kalman Filter is written as follows : (5) System model : 𝑥𝑥𝑘𝑘+1 = 𝑨𝑨𝑥𝑥𝑘𝑘 + 𝑩𝑩𝑢𝑢𝑘𝑘 + 𝑮𝑮𝑤𝑤𝑘𝑘 Measurement model : 𝑧𝑧𝑘𝑘 = 𝑯𝑯𝑥𝑥𝑘𝑘 + 𝑣𝑣𝑘𝑘 (6) With assumption : 𝑥𝑥0 ~𝑁𝑁(𝑥𝑥̅0 , 𝑃𝑃𝑥𝑥0 ), 𝑤𝑤𝑘𝑘 ~𝑁𝑁(0, 𝑄𝑄𝑘𝑘 ), 𝑣𝑣𝑘𝑘 ~𝑁𝑁(0, 𝑅𝑅𝑘𝑘 )

Where 𝑤𝑤𝑘𝑘 is noise system and 𝑣𝑣𝑘𝑘 is noise measurement. Before we estimate using Kalman Filter, we must define Jacobi matrix to linearize the nonlinear equation [17]. It can be defined by : 𝜕𝜕𝑓𝑓1

𝐴𝐴 =

𝜕𝜕𝑓𝑓1

𝜕𝜕𝑓𝑓1

𝜕𝜕𝑓𝑓1

̌𝑘𝑘 𝜕𝜕𝑥𝑥

̌ 𝜕𝜕 ℎ 𝑘𝑘

̌𝑘𝑘 𝜕𝜕𝑣𝑣

̌𝑘𝑘 𝜕𝜕𝛾𝛾

𝜕𝜕𝑓𝑓3 ̌𝑘𝑘 𝜕𝜕𝑥𝑥

𝜕𝜕𝑓𝑓3 ̌ 𝜕𝜕 ℎ

𝜕𝜕𝑓𝑓3 ̌𝑘𝑘 𝜕𝜕𝑣𝑣

𝜕𝜕𝑓𝑓3 ̌𝑘𝑘 𝜕𝜕𝛾𝛾

𝜕𝜕𝑓𝑓2

̌𝑘𝑘 𝜕𝜕𝑥𝑥

𝜕𝜕𝑓𝑓4 [𝜕𝜕𝑥𝑥̌𝑘𝑘

𝜕𝜕𝑓𝑓2

̌ 𝜕𝜕 ℎ 𝑘𝑘 𝑘𝑘

𝜕𝜕𝑓𝑓4 ̌ 𝜕𝜕 ℎ

𝑘𝑘

𝜕𝜕𝑓𝑓2

̌𝑘𝑘 𝜕𝜕𝑣𝑣

𝜕𝜕𝑓𝑓4

̌𝑘𝑘 𝜕𝜕𝑣𝑣

𝜕𝜕𝑓𝑓2

̌𝑘𝑘 𝜕𝜕𝛾𝛾

(7)

𝜕𝜕𝑓𝑓4

̌𝑘𝑘 ] 𝜕𝜕𝛾𝛾

𝑨𝑨 is matrix input to Kalman Filter method. In Kalman filter, the estimation is done by two stages: the prediction stage (time update) and the Correction stage (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 result is corrected using the measurement model. One part of this stage is to determine the Kalman gain matrix used to minimize the error covariance [18]. The prediction stage and the correction stage will be repeated continuously until the specified time 𝑘𝑘. The Kalman Filter algorithm is given in Table 1.

Table 1. : Algorithm of Kalman Filter Method Information

Algorithm

System Model

𝑥𝑥𝑘𝑘+1 = 𝑨𝑨𝑥𝑥𝑘𝑘 + 𝑩𝑩𝑢𝑢𝑘𝑘 + 𝑮𝑮𝑤𝑤𝑘𝑘

Measurement Model Assumption Initialization Prediction Stage

𝑧𝑧𝑘𝑘 = 𝑯𝑯𝑥𝑥𝑘𝑘 + 𝑣𝑣𝑘𝑘

𝑥𝑥0 ~𝑁𝑁(𝑥𝑥 ̅̅̅, 0 𝑷𝑷𝑥𝑥0 ), 𝑤𝑤𝑘𝑘 ~𝑁𝑁(0, 𝑸𝑸𝑘𝑘 ) , 𝑣𝑣𝑘𝑘 ~𝑁𝑁(0, 𝑹𝑹𝑘𝑘 )

𝑥𝑥̂0 = ̅̅̅ 𝑥𝑥0 𝑃𝑃0 = 𝑃𝑃𝑥𝑥 0

− Estimation : 𝑥𝑥̂𝑘𝑘+1 = 𝑨𝑨𝒌𝒌 𝑥𝑥̂𝑘𝑘 + 𝑩𝑩𝑢𝑢𝑘𝑘 Error Covariance : − 𝑷𝑷𝑘𝑘+1 = 𝑨𝑨𝑃𝑃𝑘𝑘 𝑨𝑨𝑇𝑇 + 𝑮𝑮𝑄𝑄𝑮𝑮𝑇𝑇

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Gain Kalman : − 𝑇𝑇 𝑇𝑇 −1 𝑲𝑲𝑘𝑘+1 = 𝑷𝑷− 𝑘𝑘+1 𝑯𝑯 (𝑯𝑯𝑷𝑷𝑘𝑘+1 𝑯𝑯 + 𝑹𝑹) Estimation : − − + 𝑲𝑲𝑘𝑘+1 (𝑧𝑧𝑘𝑘+1 − 𝑯𝑯𝑥𝑥̂𝑘𝑘+1 ) 𝑥𝑥̂𝑘𝑘+1 = 𝑥𝑥̂𝑘𝑘+1 Error Covariance : 𝑷𝑷𝑘𝑘+1 = [𝑰𝑰 − 𝑲𝑲𝑘𝑘+1 𝑯𝑯𝑘𝑘+1 ]𝑷𝑷− 𝑘𝑘+1 [𝑰𝑰 − 𝑲𝑲𝑘𝑘+1 𝑯𝑯𝑘𝑘+1 ]𝑇𝑇 + 𝑲𝑲𝑘𝑘+1 𝑹𝑹𝑘𝑘+1 𝑲𝑲𝑇𝑇𝑘𝑘+1

Correction Stage

2.3 Extended Kalman Filter Method In Kalman filter, the model used is linear, but in reality, many models are not linear. Therefore, a developed method of Extended Kalman Filter is used to solve the non-linear model. Suppose a non-linear stochastic model is given: (8) 𝑋𝑋𝑘𝑘+1 = 𝒇𝒇(𝑋𝑋𝑘𝑘 , 𝑢𝑢𝑘𝑘 ) + 𝑤𝑤𝑘𝑘 Where 𝑤𝑤𝑘𝑘 is a noise system. The synthetic model is used to update the parameters of the Gaussian functions [19]. It is a random vector drawn from the Gaussian distribution with mean=0 and covariance 𝑅𝑅 [20] [21]. With the non-linear measurement model 𝑧𝑧𝑘𝑘 ∈ 𝑅𝑅𝑛𝑛 that satisfie: 𝑧𝑧𝑘𝑘 = ℎ𝑘𝑘 (𝑋𝑋𝑘𝑘 ) + 𝑣𝑣𝑘𝑘 (9) ̂0 , 𝑃𝑃𝑥𝑥0 ), 𝑤𝑤𝑘𝑘 ~ 𝑁𝑁(0, 𝑸𝑸 ), and 𝑣𝑣𝑘𝑘 ~𝑁𝑁(0, 𝑹𝑹𝑘𝑘 ) have normal distribution and have no It is assumed that 𝑋𝑋0 ~𝑁𝑁(𝑋𝑋 𝑘𝑘

correlation with each other nor with first value 𝑋𝑋̂0 . Prior to the estimation process, linearization process is done first in non- linear. Linearization process is defined by: ∗ ̂𝑘𝑘 , 𝑢𝑢𝑘𝑘 ) 𝑋𝑋𝑘𝑘+1 = 𝑓𝑓(𝑋𝑋 𝑧𝑧𝑘𝑘 = ℎ𝑘𝑘 (𝑋𝑋𝑘𝑘 ) + 𝑣𝑣𝑘𝑘

𝑨𝑨 = [𝐴𝐴𝑖𝑖,𝑗𝑗 ] = [

𝜕𝜕𝑓𝑓𝑖𝑖 ̂ , 𝑢𝑢 )] (𝑋𝑋 𝜕𝜕𝑋𝑋𝑗𝑗 𝑘𝑘 𝑘𝑘

𝜕𝜕𝑓𝑓𝑖𝑖 ∗ 𝑯𝑯 = [𝐻𝐻𝑖𝑖,𝑗𝑗 ] = [ (𝑋𝑋 )] 𝜕𝜕𝑋𝑋𝑗𝑗 𝑘𝑘+1

𝑨𝑨 and 𝑯𝑯 are Jacobi matrix derived from the decrease of 𝑓𝑓 and ℎ on the 𝑋𝑋 direction. The modification of the Kalman Filter algorithm is called Extended Kalman Filter [18]. The Extended Kalman Filter Algorithm is given in Table 2. Table 2. : Algorithm of Extended Kalman Filter Method Information Algorithm System Model 𝑥𝑥𝑘𝑘+1 = 𝑓𝑓(𝑋𝑋𝑘𝑘 , 𝑢𝑢𝑘𝑘 ) + 𝑤𝑤𝑘𝑘 Measurement Model 𝑧𝑧𝑘𝑘+1 = 𝒉𝒉(𝑋𝑋𝑘𝑘+1 ) + 𝑣𝑣𝑘𝑘 Assumption 𝑥𝑥0 ~𝑁𝑁(𝑥𝑥 ̅̅̅, 0 𝑷𝑷𝑥𝑥0 ), 𝑤𝑤𝑘𝑘 ~𝑁𝑁(0, 𝑸𝑸𝑘𝑘 ), 𝑣𝑣𝑘𝑘 ~𝑁𝑁(0, 𝑹𝑹𝑘𝑘 ) ̂ ̅̅̅ Initialization 𝑋𝑋0 = 𝑋𝑋0 , 𝑷𝑷0 = 𝑷𝑷𝑋𝑋 Prediction Stage

𝜕𝜕𝑓𝑓𝑖𝑖 𝑨𝑨 = [ (𝑋𝑋̂ , 𝑢𝑢 )] 𝜕𝜕𝑋𝑋𝑗𝑗 𝑘𝑘 𝑘𝑘

0

Estimation : 𝑥𝑥̂𝑘𝑘− = 𝑓𝑓(𝑋𝑋𝑘𝑘 , 𝑢𝑢𝑘𝑘 ) Error Covariance : Correction Stage

𝑇𝑇 𝑇𝑇 𝑷𝑷− 𝑘𝑘+1 = 𝑨𝑨𝑃𝑃𝑘𝑘 +𝑷𝑷𝑷𝑷 + 𝑮𝑮𝑘𝑘 𝑸𝑸𝑘𝑘 𝑮𝑮𝑘𝑘

Gain Kalman :

− 𝑇𝑇 𝑇𝑇 −1 𝑲𝑲𝑘𝑘+1 = 𝑷𝑷− 𝑘𝑘+1 𝑯𝑯 (𝑯𝑯𝑷𝑷𝑘𝑘+1 𝑯𝑯 + 𝑹𝑹𝑘𝑘+1 )

Estimation :

− − 𝑋𝑋̂𝑘𝑘+1 = 𝑋𝑋̂𝑘𝑘+1 + 𝑲𝑲𝑘𝑘+1 (𝒛𝒛𝑘𝑘+1 − ℎ(𝑋𝑋̂𝑘𝑘+1 )

Error Covariance :

𝑷𝑷𝑘𝑘+1 = [𝑰𝑰 − 𝑲𝑲𝑘𝑘 𝑯𝑯 ]𝑷𝑷− 𝑘𝑘+1



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2.4 Flowchart of Algorithm Start

Start

System model : 𝑥𝑥𝑘𝑘+1 = 𝑨𝑨𝑥𝑥𝑘𝑘 + 𝑩𝑩𝑢𝑢𝑘𝑘 + 𝑮𝑮𝑤𝑤𝑘𝑘 Linierization : 𝐴𝐴 = [𝐴𝐴𝑖𝑖,𝑗𝑗 ] = ൤

𝜕𝜕𝑓𝑓𝑖𝑖

𝜕𝜕𝑋𝑋𝑗𝑗

System model : 𝑥𝑥𝑘𝑘+1 = 𝒇𝒇(𝑋𝑋𝑘𝑘 , 𝑢𝑢𝑘𝑘 ) + 𝑮𝑮𝑤𝑤𝑘𝑘 Measurement model : 𝑧𝑧𝑘𝑘 = 𝒉𝒉(𝑋𝑋𝑘𝑘+1 ) + 𝑣𝑣𝑘𝑘

(𝑥𝑥̂𝑘𝑘 , 𝑢𝑢𝑘𝑘 )൨

Measurement model : 𝑧𝑧𝑘𝑘 = 𝑯𝑯𝑥𝑥𝑘𝑘 + 𝑣𝑣𝑘𝑘

Input initial estimation: 𝑋𝑋̂0 = ̅̅̅ 𝑋𝑋0 , 𝑃𝑃0 = 𝑃𝑃𝑥𝑥 0

Input initial estimation: 𝑥𝑥̂0 = ̅̅̅ 𝑥𝑥0 , 𝑃𝑃0 = 𝑃𝑃

Prediction Stage :  − 𝑥𝑥̂𝑘𝑘+1

𝑇𝑇

= 𝑨𝑨𝑃𝑃𝑘𝑘 𝑨𝑨 + 𝑮𝑮𝑄𝑄𝑮𝑮

− 𝑷𝑷𝑘𝑘+1 = 𝑨𝑨𝒌𝒌 𝑃𝑃𝑘𝑘 +𝑷𝑷𝑷𝑷𝑇𝑇 + 𝑮𝑮𝑘𝑘 𝑸𝑸𝑘𝑘 𝑮𝑮𝑇𝑇𝑘𝑘

𝑇𝑇

Gain Kalman : − 𝑇𝑇 −1 𝑲𝑲𝑘𝑘+1 = 𝑷𝑷𝑘𝑘+1 𝑯𝑯𝑇𝑇 (𝑯𝑯𝑷𝑷− 𝑘𝑘+1 𝑯𝑯 + 𝑹𝑹) Estimation : − − + 𝑲𝑲𝑘𝑘+1 (𝑧𝑧𝑘𝑘+1 − 𝑯𝑯𝑥𝑥̂𝑘𝑘+1 ) 𝑥𝑥̂𝑘𝑘+1 = 𝑥𝑥̂𝑘𝑘+1 Error Covariance : 𝑷𝑷𝑘𝑘+1 = [𝑰𝑰 − 𝑲𝑲𝑘𝑘+1 𝑯𝑯𝑘𝑘+1 ]𝑷𝑷− 𝑘𝑘+1 [𝑰𝑰 − 𝑲𝑲𝑘𝑘+1 𝑯𝑯𝑘𝑘+1 ]𝑇𝑇 + 𝑲𝑲𝑘𝑘+1 𝑹𝑹𝑘𝑘+1 𝑲𝑲𝑇𝑇𝑘𝑘+1

𝑗𝑗

Estimation : = 𝑓𝑓(𝑋𝑋𝑘𝑘 , 𝑢𝑢𝑘𝑘 ) Error Covariance :

Prediction Stage : − = 𝑨𝑨𝒌𝒌 𝑥𝑥̂𝑘𝑘 + 𝑩𝑩𝑢𝑢𝑘𝑘 Estimation : 𝑥𝑥̂𝑘𝑘+1 Error Covariance : 𝑷𝑷− 𝑘𝑘+1

𝜕𝜕𝑓𝑓

Linierization : 𝑨𝑨 = ൤ 𝑖𝑖 (𝑋𝑋̂𝑘𝑘 , 𝑢𝑢𝑘𝑘 )൨ 𝜕𝜕𝑋𝑋

k-iterations

Final Result k=N iterations

Gain Kalman : − − 𝑯𝑯𝑇𝑇 (𝑯𝑯𝑷𝑷𝑘𝑘+1 𝑯𝑯𝑇𝑇 𝑲𝑲𝑘𝑘+1 = 𝑷𝑷𝑘𝑘+1 + 𝑹𝑹𝑘𝑘+1 )−1 Estimation : − − 𝑋𝑋̂𝑘𝑘+1 = 𝑋𝑋̂𝑘𝑘+1 + 𝑲𝑲𝑘𝑘+1 (𝒛𝒛𝑘𝑘+1 − ℎ(𝑋𝑋̂𝑘𝑘+1 ) Error Covariance : 𝑷𝑷𝑘𝑘+1 = [𝑰𝑰 − 𝑲𝑲𝑘𝑘 𝑯𝑯 ]𝑷𝑷− 𝑘𝑘+1

k-iterations

Final Result k=N iterations

Finish Finish (a)

Fig. 2.(a) Flowchart algorithm of KF, (b) Flowchart algorithm of EKF

(b)

This is the flowchart of Kalman Filter Algorithm and Extended Kalman Filter. There are several different stages. In Flowchart of Kalman Filter algorithm,the model system is linear. The model of projectile motion is nonlinear. So, we must linearize the dynamic model. Linearization is defined by Jacobi Matrix. While in Flowchart of Extended Kalman Filter algorithm, the model system is non-linear. The linearization is at the prediction stage. Besides that, the other differences between Kalman Filter and Extended Kalman Filter are the algorithm on error covariance at prediction stage and correction stage. In Fig. 2, we can see the difference of stages of each method. C. Result and Analysis As a study case, we applied the Kalman Filter and Extended Kalman Filter to predict the projectile motion so that the projectile reaches the right target. We assumed the target is silent or immobile and ignore the ambient conditions (ideal conditions). Here we generate the “real system” based on equation (8) and the measurement data based on

Riska Aprilia et al. / Procedia Computer Science 144 (2018) 153–162 Author name / Procedia Computer Science 00 (2018) 000–000

158 6

equation (9). The computational time of estimation is comparing between Kalman Filter and Extended Kalman Filter. The equation of projectile motion is the dynamical system that will be estimated using Kalman Filter (KF) and Extended Kalman Filter (EKF) So, we must discretize equation (1)-(4) respect to time, t, by using Forward Finite Difference Method. The final result is as follows : 𝑥𝑥̌𝑘𝑘+1 = 𝑣𝑣̌𝑘𝑘 𝐶𝐶𝐶𝐶𝐶𝐶𝛾𝛾̌𝑘𝑘 ∆𝑡𝑡 + 𝑥𝑥̌𝑘𝑘 + 𝑤𝑤1𝑘𝑘

(10)

1

(12)

ℎ̌𝑘𝑘+1 = 𝑣𝑣̌𝑘𝑘 𝑆𝑆𝑆𝑆𝑆𝑆 𝛾𝛾̌𝑘𝑘 ∆𝑡𝑡 + ℎ̌𝑘𝑘 + 𝑤𝑤2𝑘𝑘

(11)

𝛾𝛾̌𝑘𝑘+1 =

∆𝑡𝑡 (− 2𝑚𝑚 𝐶𝐶𝑑𝑑 𝑆𝑆𝑆𝑆𝑣𝑣̌𝑘𝑘2 − 𝑔𝑔 sin 𝛾𝛾̌𝑘𝑘 ) + 𝑣𝑣̌𝑘𝑘 + 𝑤𝑤2𝑘𝑘 1 𝑔𝑔 ∆𝑡𝑡 (2𝑚𝑚 𝐶𝐶𝑙𝑙 𝑆𝑆𝑆𝑆𝑣𝑣̌𝑘𝑘2 − 𝑣𝑣̌ cos 𝛾𝛾̌𝑘𝑘 ) + 𝛾𝛾̌𝑘𝑘 + 𝑤𝑤2𝑘𝑘 𝑘𝑘

1 0 𝑧𝑧𝑘𝑘 = [ 0 0

̌𝑘𝑘 𝑣𝑣1𝑘𝑘 0 0 0 𝑥𝑥 ̌ 𝑣𝑣2𝑘𝑘 1 0 0 ℎ ] 𝑘𝑘 + [𝑣𝑣 ] ̌𝑘𝑘 0 1 0 𝑣𝑣 3𝑘𝑘 𝑣𝑣4𝑘𝑘 0 0 1 [𝛾𝛾 ̌𝑘𝑘 ]

𝑣𝑣̌𝑘𝑘+1 =

(13)

Equation (10)-(13) is a discrete stochastic equation which has a noise system 𝑤𝑤𝑘𝑘 . In this research, we assume that measurement data is linear. It can be written as follows : (14)

Where 𝑣𝑣𝑘𝑘 is noise measurement with mean zeroes and covariance 𝑸𝑸, 𝑹𝑹 respectively

3.1 Implementation of Kalman Filter Equation (10)-(13) are nonlinear. We use Kalman Filter for estimation. System model at Kalman Filter is : 𝑥𝑥𝑘𝑘+1 = 𝑨𝑨𝑥𝑥𝑘𝑘 + 𝑩𝑩𝑢𝑢𝑘𝑘 + 𝑮𝑮𝑤𝑤𝑘𝑘

So we must linearization by defined a Jacoby Matrix which is described at equation (7). Then, we get : ̌𝑘𝑘 ̌𝑘𝑘 sin 𝛾𝛾 ̌𝑘𝑘 1 0 ∆𝑡𝑡 cos 𝛾𝛾 −∆𝑡𝑡 𝑣𝑣 ̌𝑘𝑘 ̌𝑘𝑘 cos 𝛾𝛾 ̌𝑘𝑘 0 1 ∆𝑡𝑡 sin 𝛾𝛾 ∆𝑡𝑡 𝑣𝑣 ∆𝑡𝑡 ̌𝑘𝑘 + 1 𝐶𝐶 𝑆𝑆𝑆𝑆𝑣𝑣 𝑚𝑚 𝑑𝑑 ∆𝑡𝑡 𝑔𝑔 ̌𝑘𝑘 𝐶𝐶𝑙𝑙 + ∆𝑡𝑡 2 cos 𝛾𝛾𝑘𝑘 0 0 𝑆𝑆𝑆𝑆𝑣𝑣 𝑚𝑚 ̌𝑘𝑘 [ 𝑣𝑣

𝐴𝐴 = 0 0

−

̌𝑘𝑘 − ∆𝑡𝑡 𝑔𝑔 cos 𝛾𝛾 𝑔𝑔

̌𝑘𝑘 + 1 sin 𝛾𝛾 ̌𝑘𝑘 𝑣𝑣 ] System model of Kalman Filter can be formed by state space as follows : 𝑥𝑥𝑘𝑘+1 = 𝑨𝑨𝑥𝑥𝑘𝑘 + 𝑩𝑩𝑢𝑢𝑘𝑘 + 𝑮𝑮𝑤𝑤𝑘𝑘 𝑥𝑥𝑘𝑘+1

1 0 0 1

∆𝑡𝑡

∆𝑡𝑡 cos 𝛾𝛾̌𝑘𝑘 ∆𝑡𝑡 sin 𝛾𝛾̌𝑘𝑘 ∆𝑡𝑡 − 𝐶𝐶 𝑆𝑆𝑆𝑆𝑣𝑣̌𝑘𝑘 + 1 = 0 0 𝑚𝑚 𝑑𝑑 ∆𝑡𝑡 𝑔𝑔 0 0 𝑆𝑆𝑆𝑆𝑣𝑣̌𝑘𝑘 𝐶𝐶𝑙𝑙 + ∆𝑡𝑡 2 cos 𝛾𝛾𝑘𝑘 𝑚𝑚 𝑣𝑣̌𝑘𝑘 [

−∆𝑡𝑡 𝑣𝑣̌𝑘𝑘 sin 𝛾𝛾̌𝑘𝑘 ∆𝑡𝑡 𝑣𝑣̌𝑘𝑘 cos 𝛾𝛾̌𝑘𝑘

𝑥𝑥̌𝑘𝑘 1 ℎ̌𝑘𝑘 0 − ∆𝑡𝑡 𝑔𝑔 cos 𝛾𝛾̌𝑘𝑘 +[ 0 𝑣𝑣̌𝑘𝑘 𝑔𝑔 0 [ 𝛾𝛾 ̌ ] 𝑘𝑘 ∆𝑡𝑡 sin 𝛾𝛾̌𝑘𝑘 + 1 𝑣𝑣̌𝑘𝑘 ]

0 1 0 0

0 0 1 0

0 𝑤𝑤1𝑘𝑘 0 𝑤𝑤2𝑘𝑘 ][ ] 0 𝑤𝑤3𝑘𝑘 𝑤𝑤 1 4𝑘𝑘

3.2 Implementation of Extended Kalman Filter The Extended Kalman Filter is an estimation method used for non-linear system. So the system model used as the initial input is : 𝑋𝑋𝑘𝑘+1 = 𝒇𝒇(𝑋𝑋𝑘𝑘 , 𝑢𝑢𝑘𝑘 ) + 𝑤𝑤𝑘𝑘 With 𝑤𝑤𝑘𝑘 is a noise system. 𝑥𝑥̌𝑘𝑘+1 = 𝑣𝑣̌𝑘𝑘 𝐶𝐶𝐶𝐶𝐶𝐶𝛾𝛾̌𝑘𝑘 ∆𝑡𝑡 + 𝑥𝑥̌𝑘𝑘 + 𝑤𝑤1𝑘𝑘

ℎ̌𝑘𝑘+1 = 𝑣𝑣̌𝑘𝑘 𝑆𝑆𝑆𝑆𝑆𝑆 𝛾𝛾̌𝑘𝑘 ∆𝑡𝑡 + ℎ̌𝑘𝑘 + 𝑤𝑤2𝑘𝑘 1

𝑣𝑣̌𝑘𝑘+1 = ∆𝑡𝑡 (− 2𝑚𝑚 𝐶𝐶𝑑𝑑 𝑆𝑆𝑆𝑆𝑣𝑣̌𝑘𝑘2 − 𝑔𝑔 sin 𝛾𝛾̌𝑘𝑘 ) + 𝑣𝑣̌𝑘𝑘 + 𝑤𝑤2𝑘𝑘 1

𝑔𝑔

𝛾𝛾̌𝑘𝑘+1 = ∆𝑡𝑡 ( 𝐶𝐶𝑙𝑙 𝑆𝑆𝑆𝑆𝑣𝑣̌𝑘𝑘2 − ̌ cos 𝛾𝛾̌𝑘𝑘 ) + 𝛾𝛾̌𝑘𝑘 + 𝑤𝑤2𝑘𝑘 2𝑚𝑚 𝑣𝑣 𝑘𝑘

In prediction stage carried out linearization. The linearization by defining the Jacobi Matrix and can be written as follows:



Riska Aprilia et al. / Procedia Computer Science 144 (2018) 153–162 Author name / Procedia Computer Science 00 (2018) 000–000 1 0

0 1

0

0

𝐴𝐴 = 0 [

0

̌𝑘𝑘 ∆𝑡𝑡 cos 𝛾𝛾 ̌𝑘𝑘 ∆𝑡𝑡 sin 𝛾𝛾 ∆𝑡𝑡 ̌𝑘𝑘 + 1 − 𝐶𝐶 𝑆𝑆𝑆𝑆𝑣𝑣 𝑚𝑚 𝑑𝑑 ∆𝑡𝑡 𝑔𝑔 ̌𝑘𝑘 𝐶𝐶𝑙𝑙 + ∆𝑡𝑡 2 cos 𝛾𝛾𝑘𝑘 𝑆𝑆𝑆𝑆𝑣𝑣 𝑚𝑚 ̌𝑘𝑘 𝑣𝑣

159 7

̌𝑘𝑘 sin 𝛾𝛾 ̌𝑘𝑘 −∆𝑡𝑡 𝑣𝑣 ̌𝑘𝑘 cos 𝛾𝛾 ̌𝑘𝑘 ∆𝑡𝑡 𝑣𝑣 ̌𝑘𝑘 − ∆𝑡𝑡 𝑔𝑔 cos 𝛾𝛾

∆𝑡𝑡

𝑔𝑔

̌𝑘𝑘 𝑣𝑣

̌𝑘𝑘 + 1 sin 𝛾𝛾

]

So, the apparent difference of the Kalman Filter and the Extended Kalman Filter method is initial input for Kalman Filter is system already in linearization. While Extended Kalman Filter, initial input is nonlinear system and linearization is done at prediction stage. 3 Simulation In this paper, we have parameters for the dynamic model of 12.7 x 99 mm caliber projectile as follows : Table 3. : Parameters of Projectile caliber 12.7 x 99 mm

Information

Parameters

Projectile Mass (mass)

0.125 kg 0.96

Drag Coefficient 𝐶𝐶𝑑𝑑

9.8065 m/s2

Lift Coefficient 𝐶𝐶𝑙𝑙

0.000127 m2

Gravitational Force (g)

0.89

Surface of projectile Air Density

1 kg / m3

.

In the simulation using the Extended Kalman Filter Method and Kalman Filter as a comparison, first initialization is 𝑃𝑃0 given on the motion equation of 12.7 x 99 mm caliber projectile. To obtain the covariance error in the prediction stage and the covariance error in the correction stage is given the initial condition 𝑄𝑄0 and 𝑅𝑅0 is as follows: 10−2 𝑃𝑃0 = ( 0 0 0

0 10−2 0 0

0 0 10−2 0

0 0 ) 0 10−3

10−3 , 𝑄𝑄0 = ( 0 0 0

0 10−3 0 0

0 0 10−3 0

0 0 ) , 𝑅𝑅 = 10−4 0 0 −4 10

The simulation result of Kalman Filter and Extended Kalman Filter Method on the equations of projectile motion is computed for 𝑁𝑁 = 100 iterations. This simulation compares the real value, estimated value using Kalman Filter method, and estimated value using Extended Kalman Filter method in each variable that can be seen in Figure 3-6

160 8

Riska Aprilia et al. / Procedia Computer Science 144 (2018) 153–162 Author name / Procedia Computer Science 00 (2018) 000–000

̌ Fig 3: Position Variable 𝒙𝒙

̌ Fig 5: d Position Variable 𝒗𝒗

̌ Fig 4: Position Variable 𝒉𝒉

Fig 6: Position Variable 𝒈𝒈𝒈𝒈𝒈𝒈𝒈𝒈𝒈𝒈

In this study, ten simulations were performed. Based on simulation result, In figure 3-6 show the graph comparison between the real value, the estimated value using Kalman Filter and the estimated value using Extended Kalman Filter. Figure 3 is the plot graph about distance or x-position of the projectile. From that graph, the resulting average error value between the real value with the estimated value using KF, and between the real value with the estimated value using EKF respectively are 0.2482 and 0.0262. Figure 4 is the plot graph about the height or z-position. From that graph, the resulting average error value between the real value with the estimated KF and between the real value with the estimated value using EKF respectively are 0.4892 and 0.0268. Figure 5 is the plot graph of the velocity of the projectile. From the graph, the resulting average error between the real value with the estimated value using KF and between the real value with the estimated value using EKF respectively are 0.6956 and 0.0308. Figure 6 is the plot graph about gamma angle or pitch angle of projectile. From that graph, the resulting average error value between the real value with the estimated value using KF and between the real value with the estimated value using EKF respectively 0.5948 and 0.0244. More detail, it can be seen in Table 4 ,Table 5, and Table 6 ,also device specification for simulation at Table 7.



Riska Aprilia et al. / Procedia Computer Science 144 (2018) 153–162 Author name / Procedia Computer Science 00 (2018) 000–000

Table 4. : The error estimation and computational time using KF Error of Kalman Filter 𝑥𝑥̌(m) ℎ̌(m) 𝑣𝑣̌(m/s2) 𝛾𝛾̌(rad)

No.

161 9

Table 5. : The error estimation and computational time using EKF No.

Time

Error of Extended Kalman Filter ℎ̌(m) 𝑣𝑣̌(m/s2) 𝛾𝛾̌(rad)

Time

𝑥𝑥̌(𝑚𝑚)

0.0254

0.0243

0.0337

0.0196

1.572143

0.0279

0.0253

0.0292

0.0246

1.557751

1

0.2876

0.5076

0.7732

0.6145

1.630954

1

2

0.2233

0.6010

0.9862

0.6398

1.557771

2

3

0.2145

0.3216

0.6454

0.5947

1.664895

3

0.0279

0.0267

0.0316

0.0241

1.553126

4

0.4563

0.8530

0.5931

0.6118

1.543039

4

0.0263

0.0268

0.0285

0.0281

1.551524

5

0.2078

0.3450

0.3746

0.5792

1.552176

5

0.0274

0.0270

0.0315

0.0256

1.644349

6

0.2098

0.5315

1.3591

0.5872

1.531673

6

0.0253

0.0276

0.0325

0.0242

1.551209

7

0.1942

0.2999

0.5480

0.5829

1.540935

7

0.0234

0.0236

0.0316

0.0255

1.620959

8

0.2654

0.4117

0.5654

0.5697

1.530787

8

0.0251

0.0271

0.0271

0.0243

1.544015

9

0.2429

0.4389

0.5765

0.5931

1.548301

9

0.0268

0.0282

0.0314

0.0228

1.532276

10

0.1798

0.5817

0.5336

0.5746

1.558580

10

0.0264

0.0313

0.0307

0.0255

1.522510

Average

0.2482

0.4892

0.6956

0.5948

1.565911

Average

0.0262

0.0268

0.0308

0.0244

1.564986

Table 6 : The error percentage of estimation Method KF EKF

𝑥𝑥̌ (%)

0.2858 0.0228

Table 7 : Device Specification Device Name

ℎ̌ (%)

0.9971 0.0261

𝑣𝑣̌ (%)

0.4951 0.0248

𝛾𝛾̌ (%)

0.4853 0.0221

DEKSTOP-GAR23NK

Processor

Intel® Core™i3 CPU M 370 @ 2.40Ghz

Installed RAM

2.00 GB

Device ID

535EDC07-BF53-40C6-8A15-BBC9AC04A2EC

Product ID

0039-00000-00003-AA020

System Type

32-bit operating system, x64-based processor

Pen and Touch

No open or touch input is available for this display

Matlab

R2013a

From the Table 6, it is showed that the estimation by Extended Kalman Filter is more accurate than Kalman Filter with error estimation of EKF is 92.02% smaller in 𝑥𝑥̌ (x-position), 97.38% in ℎ̌ (z-position), 94.99% in 𝑣𝑣̌(velocity of projectile), and 95.45 % in 𝛾𝛾̌(gamma angle). It can be seen that estimation using Extended Kalman Filter has smaller error than Kalman Filter. Based on the computational time shows that Kalman Filter needs average on 1.565911 s and Extended Kalman Filter need average on 1.564986 s . So, The computational time of Extended Kalman Filter is 0.925 % faster than Kalman Filter, although the difference computational time between the both of them is small. Conclusion In this paper, dynamic model of 12.7 x 99 mm caliber projectile motion is non-linear with three degrees of freedom (3-DOF) then it is estimated with Extended Kalman Filter Method and compared by Kalman Filter method. The form of the equation of projectile motion is continuous, so discretization is necessary to convert a continuous equation into a stochastic form. In this paper, the motions of the projectile are estimated. We can apply the Kalman Filter and Extended Kalman Filter to estimate the motion of 12.7 x 99 mm caliber projectile. The performance of each method is compared based on the accuracy of estimation. The equations of projectile motion was performed 10 times simulation. From the simulation result, the error estimation of EKF is 92.02% smaller in 𝑥𝑥̌ (x-position), 97.38% smaller in ℎ̌ (z-position), 94.99% % smaller in 𝑣𝑣̌ (velocity of the projectile), and 95.45 % smaller in 𝛾𝛾̌(gamma angle). RMSE of EKF is smaller than KF. The computational time of Extended Kalman Filter is 0.925% faster than Kalman Filter. So, we can conclude that Kalman Filter is becoming non-optimal filter in the sense of minimizing the mean square error. Hence, the Extended Kalman Filter is better than Kalman Filter to estimate the motion of the projectile in this case.

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This research uses two estimation methods. That is Kalman Filter and Extended Kalman Filter. But the KF estimation result is not as optimal as EKF. Recommendation for next research is to apply two methods which can be used for the non-linear system. Such as the Extended Kalman Filter (EKF) and Modification Extended Kalman Filter (MEKF) to get the optimal result from both. Acknowledgements The authors would like to thank many people who has involved in this paper especially to Center of excellence for Mechatronics and Industrial Automation (PUI-PT MIA-RC ITS) and Ministry of Research Technology and Higher Education of The Republic of Indonesia (Ristekdikti) for support, also to Mrs. Siti Mushonnifah, S.Si, M.Si for her helpful discussion around the topic of the paper an Dr. Dieky Adzkiya, S.Si, M.Si for comments and suggestions in writing paper. References Herlambang, T., R.R. Amri., H.Sri, R. Dinita. (2017).“Estimasi posisi Moblie Robot Menggunakan Akar Kuadrat Unscented [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21]

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