MISSILE POSITION ESTIMATION USING UNSCENTED KALMAN FILTER

Abstract

Missiles are military rocket weapons having an automatic control system to locate its targets or adjust its direction. Indonesia itself, which is a country of archipelago, covers air area of its largest territory, followed by sea area and land area. Logically, the existence of missile defense equipment (the main weapon system) or precisely the type of long-range missile is acceptable to support the defense and security of the Republic of Indonesia, but its consequences to be operated in the territory of Indonesia itself, in case of an occufanct of an error in targeting the target, will fall on of harm to its own national territory. Therefore, trajectory estimation for guided missiles is the basic requirement for guided missiles to be aimed at the precise targets. The trajectory is used as a guide to direct that the missile reach the target by following the given path. To maintain the accuracy of the trajectory continuously, the missile trajectory estimation was made by using Unscented Kalman Filter (UKF) Algorithm. This algorithm was used to estimate nonlinear dynamic models The simulation results showed that the UKF method was effective, showing the accuracy of 97% by the UKF method

Full text

https://doi.org/10.30598/barekengvol16iss1pp207-216
March 2022 Volume 16 Issue 1 Page 207-216
P-ISSN: 1978-7227 E-ISSN: 2615-3017
BAREKENG: Jurnal Ilmu Matematika dan Terapan
207
https://ojs3.unpatti.ac.id/index.php/barekeng/ barekeng.math@yahoo.com
MISSILE POSITION ESTIMATION USING UNSCENTED KALMAN
FILTER
Teguh Herlambang 1*, Subchan2
1Information System Department, FEBTD, Universitas Nahdlatul Ulama Surabaya
Jl. Raya Jemursari 51-57, Surabaya, Indonesia
2Mathemathics Department FSAD, Institut Tekologi Sepuluh Nopember
Jl. Teknik Kimia, Sukolilo, Surabaya, Indonesia
Corresponding author e-mail: ¹* teguh@unusa.ac.id
Abstract. Missiles are military rocket weapons having an automatic control system to locate its targets or adjust its
direction. Indonesia itself, which is a country of archipelago, covers air area of its largest territory, followed by sea
area and land area. Logically, the existence of missile defense equipment (the main weapon system) or precisely the
type of long-range missile is acceptable to support the defense and security of the Republic of Indonesia, but its
consequences to be operated in the territory of Indonesia itself, in case of an occufanct of an error in targeting the
target, will fall on of harm to its own national territory. Therefore, trajectory estimation for guided missiles is the
basic requirement for guided missiles to be aimed at the precise targets. The trajectory is used as a guide to direct
that the missile reaches the target by following the given path. To maintain the accuracy of the trajectory continuously,
the missile trajectory estimation was made by using Unscented Kalman Filter (UKF) Algorithm. This algorithm was
used to estimate nonlinear dynamic models The simulation results showed that the UKF method was effective,
showing the accuracy of 97% by the UKF method
Keywords: missile, UKF, trajectory, estimation, position, Kalman Filter
Article info:
Submitted: 8th January 2022 Accepted: 27th February 2022
How to cite this article:
T. Herlambang and Subchan, “MISSILE POSITION ESTIMATION USING UNSCENTED KALMAN FILTER”, BAREKENG: J. Il. Mat.
& Ter., vol. 16, no. 1, pp. 207-216, Mar. 2022.
This work is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License.
Copyright © 2022 Teguh Herlambang, Subchan.

208 Herlambang, et. al. Missile Position Estimation Using …..…
1. INTRODUCTION
Missiles are military rocket weapons having an automatic control system to locate its targets or adjust
its direction [1]. The first type of missiles used in an operation were German guided missiles in World War
II. The most famous were the V-1 and V-2, and both used a simple autopilot system to keep the bullets flying
following a predetermined route. In 1962 Rand Paul Barand, from the RAND company, was assigned to
develop a decentralized network system capable of controlling its bombing system and missile launching
system in a nuclear war.
Lately, several developing and developed countries, especially those with very high levels of aerospace
and military technology, have developed several destructive weapons ranging from those user friendly (easy
to use), and helpful for people, up to those dangerous and deadly. Examples of user friendly weapons are the
creation of military robots functioning to defuse bombs, unmanned monitoring aircrafts, and the likes.
Whereas, those of dangerous weapons are the ones generally developed for national defense, such as guided
missiles equipped with nuclear or biological weapons.
Indonesia itself, which is a country of archipelago, covers air area of its largest territory, followed by
sea area and land area. Logically, the existence of missile defense equipment (the main weapon system) or
precisely the type of long-range missile is acceptable to support the defense and security of the Republic of
Indonesia, but its consequences to be operated in the territory of Indonesia itself, in case of an occufanct of
an error in targeting the target, will fall on of harm to its own national territory [2].
Therefore, trajectory estimation for guided missiles is the basic requirement for guided missiles to be
aimed at the precise targets. The trajectory is used as a guide to direct that the missile reach the target by
following the given trajectory. Several studies have been carried out regarding the estimation of positions
such as the one implemented to AUV using Ensemble and Extended Kalman Filter [3, 4, 5] and Square Root
Ensemble Kalman Filter (SR-EnKF) method [6] [7]and Kalman Filter with fuzzy [8], and that implemented
on temperature steam drum estimation [9]. Position estimation has also been applied to ASV using the
Extended Kalman Filter method [10] To maintain the accuracy of the trajectory continuously, in this study,
the missile trajectory estimation was made by using Unscented Kalman Filter (UKF) Algorithm. The
contribution of this paper is numeric computation of the of missile position estimation and the missile motion
influenced by target
2. RESEARCH METHODS
2.1 Mathematical Model of Missile Trajectory
Movement equation of missile are modeled as bellow [11, 12]:
 
1sin cos
g
LT
mV V
  
  
(1)
 
1cos sin V T D g
m

  
(2)
cos xV


(3)
sin hV


(4)
Where variables state such as flight path angle

, speed
V
, horizontal position
x
, and altitude
h
from
missile. A push force
T
and angle of attack

are two controlled variables (see Figure 1). The aerodynamics
force
D
and
L
are functions the altitude
h
, velocity
V
, and angle of attack

. With L is the lift
aerodynamic force and D is the drag lift aerodynamic force with respect to a body-axis frame.

BAREKENG: J. Il. Mat. & Ter., vol. 16(1), pp. 207-216, Mar. 2022. 209
Figure 1 Axes and Angle Missile Model [11, 13]
Aerodynamics Force [12]
󰇛α󰇜
ρ (5)
αα (6)
󰇛α󰇜
ρ (7)
α (8)
with

is air density given by
ρ (9)
and
ref
S
is reference area of the missile , m is mass and g is gravitational constant. The value of
1
A
,
2
A
,
3
A
,
1
B
,
2
B
,
1
C
,
2
C
and
3
C
are constant given in Table 1.
Table 1. Physical Modelling Parameter [13]
Quantity
Value
unit
m
1005
kg
g
9.81
2
/ms
ref
S
0.3376
2
m
1
A
-1.9431
2
A
-0.1499
3
A
0.2359
1
B
21.9
2
B
0
T
6000
1
C
9
3,312.10
2
kg
m
2
C
4
1,142.10
2
kg
m
3
C
1.224
2
kg
m
Because the system requires discretization, so the missile model in equation (1) – (4) must be discrete using
the finite difference method.
Equation (1) – (4) , If
k

flight path angle
kt
and identically for speed, horizontal position and
altitude h are

210 Herlambang, et. al. Missile Position Estimation Using …..…
k


;
k
VV
;
k
xx
;
k
hh
(10)
The change of state variables respect to the time are approximated by forward scheme of finite difference.
Thus we will get
1kk
d
dt t






(11)
1kk
VV
dV
Vdt t



(12)
1kk
xx
dx
xdt t


(13)
1kk
hh
dh
hdt t



(14)
from Equation (11) – (14) will be gotten the modified missile model in (15) below
 
 
 
 
1
1
1
1
1sin cos
1cos sin
cos
sin
k k k k k
kkk
k
k k k k k
k
kk k k
k k k
g
L T t
mV V
VT D g t V
xm
hV t x
V t h
  










   









   

 









(15)
2.2 Unscented Kalman Filter Algorithm
Algorithm of Unscented Kalman Filter is written as follows [14, 15, 16]:
 Initiation at k = 0:
󰇟󰇠
󰇟󰇛󰇜󰇛󰇜󰇠

󰇟󰇠

󰇟󰇛
󰇜󰇛
󰇜󰇠 
 
   (16)
For k = 1,2,3,...,:
1) Count sigma point



 

where:

󰇛󰇜 (17)
2) Time-update (prediction stage)

󰇛

󰇜

󰇛󰇜

 


󰇛󰇜



 



BAREKENG: J. Il. Mat. & Ter., vol. 16(1), pp. 207-216, Mar. 2022. 211
 󰇛

󰇜

󰇛󰇜

  (18)
3) Measurement update (correction stage):
󰇛󰇜 


  

󰇛󰇜



  




󰇛
󰇜


 (19)
3. RESULTS AND DISCUSSION
Numerical computation in this paper makes the missile model a platform in missile trajectory
estimation because the missile model is a nonlinear model. So, the use of UKF is one way to obtain a high
accuracy.
In this paper, two simulations are compared, covering a simulation on the missile trajectory for going
upwards and then plunging downwards, while the second simulation is a missile shooting a target at an
altitude of about 1000 meters. The first simulation is represented in Figure 2 – Figure 5, and the second
simulation is represented by Figure 6 – Figure 9 of error value for two simulations is in Table 2.
Figure 2. Speed Estimation of Missile in the first trajectory using UKF
Figure 2 indicates that the speed slightly increased from 250 m/s to 254 m/s in the 10th iteration and then
decreased to 240 m/s in the 40th iteration, then increased again to 270 m/s in the 80th iteration, and increased
rapidly to 295 m/s in the 100th iteration. In this case the missile speed decreases when it goes up, but when
it plunges down, its speed increases. Velocity is closely related to horizontal position (x) and altitude (h),
increasing and decreasing speed is affected by horizontal position (x) and altitude (h).
020 40 60 80 100 120
230
240
250
260
270
280
290
300 Speed Estimation
iteration
nilai kecepatan (V)
Real
UKF

212 Herlambang, et. al. Missile Position Estimation Using …..…
Figure 3. Horizontal Position Estimation of Missile in the first trajectory using UKF
Figure 3 shows that the horizontal position of the missile advances to a distance of 1600 meters and undergoes
a slight bend during the 50th and 70th iterations and returns running straight up to the 100th iteration with a
distance of 1600 meters. In this case the horizontal position of the missile is always forward and will not
return (turn back). The horizontal position is closely related to the height at which when the horizontal
position is advancing (not turning) them the height will follow the horizontal position to get the maximum
height and plunge down.
Figure 4. Height Estimation of Missile in the first trajectory using UKF
Figure 4 shows that the missile always rise up to the turning point of 1000 meters in the 62nd iteration and
finally plunge down. In this case Figure 4 and Figure 3 are closely related because when the missile goes up
the speed is adjusted by slightly increasing and decreasing the speed, but when it is in the 62nd iteration
which is the turning point, the speed increases rapidly when it plunges down. In this case the height of the
missile represents an increase when flying, and it represents a decrease when it plunges down to the ground.
020 40 60 80 100 120
-200
0
200
400
600
800
1000
1200
1400
1600 Horizontal Position Estimation
iterasi
position)
Real
UKF
020 40 60 80 100 120
0
200
400
600
800
1000
1200 Height Estimation
iteration
Height
Real
UKF

BAREKENG: J. Il. Mat. & Ter., vol. 16(1), pp. 207-216, Mar. 2022. 213
Figure 5. Trajectory Estimation of Missile in the first trajectory using UKF
In Figure 5 it can be seen that the guided missile always rises to a height of 1000 meters with a distance of
1000 meters from the start of firing, and finally plunges down from an altitude of 900 meters to the ground
with a distance of 1600 meters from the start of firing. In this case, the relationship between the horizontal
position and the height of the missile is about how far the missile travels at what height will it strike and when
it hits the ground. Next, it is the simulation with the second trajectory shown in Figure 6 – figure 9.
Figure 6. Speed Estimation of Missile in the Second trajectory using UKF
-200 0 200 400 600 800 1000 1200 1400 1600
0
200
400
600
800
1000
1200 Estimation of Missile Position
Horizontal Position
Heigh
real
UKF
020 40 60 80 100 120
240
245
250
255
260
265 Speed Estimation
iteration
nilai kecepatan (V)
Real
UKF

214 Herlambang, et. al. Missile Position Estimation Using …..…
Figure 7. Horizontal Position Estimation of Missile in the Second trajectory using UKF
Similar to Figure 2 and 3, Figure 6 and 7, they show that the UKF method has a near perfect accuracy of
about 97%. Here can it be observed that between the actual and estimated results of trajectories coincide with
each other.
Figure 8. Height Estimation of Missile in the Second trajectory using
Figures 8 and 9 show that the red line is the real horizontal position, and the height that is determined to shoot
a target flying at a certain height and distance so that the missile hits the target aimed at, in which for instance
the target is at an altitude of 1100 meters and a distance of 1600 meters.
020 40 60 80 100 120
0
200
400
600
800
1000
1200
1400
1600
1800 Horizontal Position Estimation
iterasi
position)
Real
UKF
020 40 60 80 100 120
0
200
400
600
800
1000
1200
1400 Height Estimation
iteration
Height
Real
UKF

BAREKENG: J. Il. Mat. & Ter., vol. 16(1), pp. 207-216, Mar. 2022. 215
Figure 9. Trajectory Estimation of Missile in the Second trajectory using UKF
In Table 2 it can be seen that both the first and second simulations have almost the same accuracy when using
the UKF method, but the first simulation, in which the missile is fired up and down, has a smaller error than
the second simulation, which is a simulation of striking a target from a certain distance and altitude.
Table 2. RMSE value from Result Computational Simulation
UKF
RMSE First
Trajectory
RMSE Second
Trajectory
Angle Position
0.26501
0.39078
Speed
0.12113
0.2774
Horizontal Position
0.86408
0.9278
Height
0.91935
0.9649
4. CONCLUSION
Based on the simulation analysis, some conclusions are present: Unscented Kalman Filter (UKF)
method can be applied to estimate the missile trajectory and that both the first and second simulations have
almost the same accuracy when using the UKF method, but the first simulation, in which the missile is fired
up and down, has a smaller error than the second simulation, which is a simulation of striking a target from
a certain distance and altitude.
AKNOWLEDGEMENT
This research was supported by LPPM – Nahdlatul Ulama Surabaya of University (UNUSA)
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