A Trajectory Simulation Model of the Short-range

8/11/2019 A Trajectory Simulation Model of the Short-range

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A trajectory simulation model of the short-range

anti-ship missile based on considering curvature of the earth

Weimin Lv

Dep. of System Engineering of Engineering Technology

Beijing University of Aeronautics and Astronautics

Beijing, China

E-mail: [email protected]

Weimin Lv, Liang Wang, Shiwei Jiang

Dep. of Airborne Vehicle Engineering

Naval Aeronautical Engineering Institute

Yantai, China

E-mail: [email protected]

AbstractThis paper analyzes how to set up a trajectory

simulation of the short-range anti-ship missile, considering

curvature of the earth. According to the system working

mechanism of some missile and curvature of the earth, a

motion simulation model of the whole trajectory is built. A lot

of shooting simulation tests are conducted, and the results

show that various ranges have different influences on

acquisition probability, homing probability and hitting

probability of the missile. The validity of this method is

verified by the tests.

Keywords-curvature of the earth; anti-ship missile; trajectory

simulation

I. INTRODUCTION

With the development of computers, the computertrajectory simulation is used more and more widely. Thereasonableness of a trajectory simulation model directlyrelates to the reliability of simulation results. In previoustrajectory simulation tests, we usually considered thatcurvature of the earth had influences on long-range missiles.But for short-range missiles, we assumed that the earth wasflat. That was to say, we didnt take the influences ofcurvature of the earth into consideration, and was thisassumption reasonable?

The mathematical model of over-the-horizon targetindication is analyzed on the basis of the system workingmechanism of some missile. A motion simulation model ofthe whole trajectory is built, considering curvature of theearth. Then lots of simulation tests are conducted to verifythe method by using Monte Carlo Method.

II. THE NECESSITY OF CONSIDERING CURVATURE OF THE

EARTH

First, we define the short-range anti-ship missile in thispaper as follows [1]: It is the missile whose directional shaft

of autopilot gyro maintains a constant angle with thelaunching horizon rather than to track along the earth surface.Thats to say, it doesnt maintain a constant angle with thelocal horizon. Whether the directional shaft needs to trackalong the earth surface is mainly determined by the rangeand the characteristic of the rudder loop. For those anti-shipmissiles whose ranges are mostly inside 200km, thedirectional shaft usually doesnt track along the earth surface.

The ground reference frame in this paper is defined asfollow [2]: Take the position of centroid of the missile whenit is launched as the origin. Take the direction of missile

spindle as X-axis. Take the vertical direction as Y-axis. Z-axis is determined by the right-hand rule.

Regard the earth as a sphere whose radius is 6370km.Take some short-range anti-ship missile whose range is120kmfor example. If without considering curvature of theearth, what influence does the model has on simulationresults? Because of taking no account of curvature of theearth, the altitude signal of the missile that measured by the

radio altimeter is ordinate value y rather than the height hfrom the missile to the earth surface, which is shown in Fig.1.If without considering curvature of the earth, when the

missile flies 120km, the error between ordinate value of thesimulation model and the actual ordinate value is 1130.4m. Itis patently obvious that this kind of simulation model cantreflect the actual flight of the missile well and truly. Ifcurvature of the earth isnt taken into consideration, the

trajectory obliquity is and , but the actual trajectory

obliquity is

0

and180d

R

, where d is the flight

distance [3]. When the flight distance reaches 120km, and

. According to related data, the pitch angle of

the missile in level flight isn

1.08

and . So the elevation

angle of the missile tents to be bigger with the flight distancebecoming longer in actual flight. But the elevation anglesmust be the same if there are the same lifts. In order toreduce the elevation angle, the rudder angle should bedecreased in actual flight. Its obvious that this point couldnt

be reflected if the earth is assumed to be flat, and the rudderloop couldnt be simulated correctly.

6 n

Figure 1. Sketch map of altimeter signal

2010 Second International Conference on Computer Modeling and Simulation

978-0-7695-3941-6/10 $26.00 2010 IEEE

DOI 10.1109/ICCMS.2010.444

338

2010 Second International Conference on Computer Modeling and Simulation

978-0-7695-3941-6/10 $26.00 2010 IEEE

DOI 10.1109/ICCMS.2010.444

354


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III. THE ESTABLISHMENT OF TRAJECTORY SIMULATIONMODEL

Because of the directional shaft of the gyro not tracking

along the earth surface, its state is established when themissile is launched [4]. All the attitude angles of the missileare measured with the directional shaft of the gyro as a

standard, and they are defined according to the above groundreference frame. For example, the directional shaft of the

pitch gyro maintains a constant angle with the launching

horizon, so the pitch angle is always the angle between

the directional shaft and the launching horizon instead of the

angle between the directional shaft and the local horizon,

which is shown in Fig.2. We just need to assume the ground

reference frame which is described above is an inertialreference frame. Both the altitude of the missile and itschange rate which are measured by radio altimeter are

simulated through the height h from the missile to the earth

surface and its change rate h . The gravity acceleration

should be changed into g at the same time, and then the

simulation model considering curvature of the earth is built.Suppose the missile coordinates are (x,y,z), then

22 2

h= x + y+R +z -R ; (1)

22 2

xx+ y+R y+zzh=

x + y+R +z

; (2)

22 2

xi+ y+R j+zk g =- g

x + y+R +z

. (3)

IV. SIMULATION RESULTS AND CONCLUSIONS

Simulation conditions:

(1) Distance error is taken equably from -280mto 280m.

(2) Azimuth error is taken equably from to0.34

0.34

(3) The course angle of the target is taken equably from

45to 135.

(4) The valid height of the wave is taken equably from0mto 4m.

Simulation results are shown in Tab 1. (The number of

simulation times is 10000, and the flight reliability isnttaken into account.)

K1 is the times that the missiles dont acquire the target.

X

y

O

O'

Figure 2. Sketch map of pitch angle

K2 is the times that missiles drop into the sea.

K3 is the off-target times that the missiles fly above thetarget.

K4 is the off-target times that the missiles fly beside the

target.K5 is the off-target times that the missiles fly diagonally

above the target.

Results analysis: For both the short-range targets and thelong-range targets, the acquisition probabilities arecomparatively low in theory. The low acquisition probabilityfor the short-range targets results from the pitch angle is

unstable, which enlarges the difficulty of catching the targetin longitudinal plane. This is proved by the data in Tab 1. Forthe long-range targets, the acquisition probability is low

because various interference factors expand the scattered

band [5]. However, this is not reflected from the above table.Its mainly because most of the scattered points that therandom interference factors result in are still in the

acquisition scale of the radar. That also explains why thiskind of missile has a comparatively high acquisition

probability (99%). We can also find out different targetdistances dont have appreciate impacts to homingprobability.

If a simulation model is built and simulated without

considering curvature of the earth, it will result in inaccuratecalculation of the trajectory, consequently affecting thecalculation of acquisition probability, homing probability

and hitting probability. The simulation model built by usingthe above method can better describe the actual flight of themissile and the work conditions of the rudder loop, and the

simulation results are more credible.

TABLE I. THE INFLUENCES OF DIFFERENT TARGET DISTANCES ON ACQUISITION PROBABILITY,HOMING PROBABILITY AND HITTING PROBABILITY

Targetdistance(km)

K1 K2 K3 K4 K5 Acquisitionprobability

Homingprobability

Hittingprobability

12 380 0 10 0 0 62.0% 98.4% 61.0%

20 0 10 10 0 0 100.0% 98.0% 98.0%

40 0 0 0 0 0 100.0% 100.0% 100.0%

60 0 0 0 0 0 100.0% 100.0% 100.0%

80 0 0 0 0 0 100.0% 100.0% 100.0%

100 0 0 10 0 0 100.0% 99.0% 99.0%

120 0 0 20 0 0 100.0% 98.0% 98.0%

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REFERENCES

[1] Q.J. Zhou, Theory of Missile Flight, Yantai: Naval AeronauticalEngineering Institute, 1998.

[2] Ch. Xu, G.B. Song and W.S. Zhou, Analysis of Naval MissileWeapon System, Yantai: Naval Aeronautical Engineering Institute,1999.

[3] Y.L. Xiao and Ch.J. Jin, Theory of Flight within AtmosphericDisturbance, Beijing: National Defence Industry Press, 1993.

[4] J.D. Ma and Sh.L. Zhou, Theory of Missile Control System, Beijing:Aviation Industry Press, 1996.

[5] X.M. Li, J. Sun and X.F. Xie, Theory of Fire Control System, Beijing:National Defence Industry Press, 2007

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A Trajectory Simulation Model of the Short-range