FM04.txt

                                                            Paper: ASAT-13-FM-04

13th International Conference on
AEROSPACE SCIENCES & AVIATION TECHNOLOGY,
ASAT- 13, May 26 ­ 28, 2009, E-Mail: asat@mtc.edu.eg
Military Technical College, Kobry Elkobbah, Cairo, Egypt
Tel : +(202) 24025292 ­ 24036138, Fax: +(202) 22621908

Trajectory Prediction for a Typical Fin Stabilized Artillery Rocket

                                M. Khalil*, H. Abdalla* and O. Kamal*

Abstract: This paper investigates the trajectory prediction and dispersion for unguided fin
stabilized artillery rocket in order to explain the importance of the rocket production accuracy
and the benefit of using guided rockets. The total dispersion results mainly from three effects.
The first is the dispersion due to rocket production inaccuracy, which includes propellant
mass, composition inaccuracy, rocket total mass, axial and lateral moments of inertia and
resultant center of gravity. The second dispersion during boosting phase which includes
launcher deflection, missile tip-off from the launcher, thrust and fin misalignments, and
atmospheric disturbances such as tail wind, cross wind, and gusts. While the third is the
dispersion during free-flight phase that is due to the fluctuations in wind profile.

In this study, a trajectory calculation using a 6-DOF model was developed and applied for a
typical artillery rocket, the 122 mm artillery rocket, at different mass and flight properties to
predict the trajectory parameters and dispersion.

Keywords: Dispersion Analysis, Nominal Trajectory, Trajectory Prediction, 6-DOF Model,

               122mm unguided artillery rocket.

Nomenclature   Acceleration vector acting on the body.
               Total axial force coefficient.
[aN aE aD]T    Total roll moment coefficient.
CA             Rolling moment coefficient derivative with roll rate.
Cl             Rolling moment coefficient derivative with yaw rate.
Clp            Pitching moment coefficient derivative with pitch rate.
Clr            Pitching moment derivative with rate of change of angle of attack.
Cmq            Pitching moment derivative with rate of change of angle of attack.
Cm             Normal force coefficient derivative with angle of attack
Cm             Normal force coeff. derivative with rate of change of angle of attack
CN
CN             Normal gravity on the ellipsoidal surface.
g              Body axial moment of inertia, [kg.m2].
Ix             Body mass products of inertia, [kg/m2].
Ixy, Iyz, Izx

* Egyptian Armed Forces
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                  Paper: ASAT-13-FM-04

Iy, Iz            Body transverse moment of inertia, [kg.m2].
LBE               Transformation matrix from earth frame FE to body frame FB.
M                 Mach number.
m                 Total mass of the body, [kg].
p                 Body spin rate, [rad/s].
[P Q R]T          Body angular velocity, [rad/s].
q                 Body pitch rate, [rad/s].
r                 Body yaw rate, [rad/s].
[Tx Ty Tz]T       Components of the resultant external force acting on the body in the
                  body fixed reference frame, [N].
[u v w]T          Components of the velocity vector of the body in the body fixed
                  reference frame, [m/s].
[ V N V E V D ]T  Total acceleration vector acting on the body.
                  Angle of attack.
                  Angle of sideslip.

                  Specific heat ratio ( = 1.4 for isentropic flow).
                  Roll angle (Bank angle) [deg].
                  The latitude of body C.G w.r.t. Earth fixed reference
                  frame [rad].
                  Longitude of body C.G w.r.t. earth fixed reference frame [rad].
                  Pitch angle (inclination angle), [deg].
                  Yaw angle (azimuth angle), [deg].
E                 Angular velocity of the earth [pE qE rE]T about the inertia reference
                  frame [rad/s].

1. Introduction

Simulation of the trajectory of unguided rockets is a combination between all characteristics
of the rocket (e.g. mass properties and configuration) and atmospheric conditions (predicted
values). In practice, there are some differences between the real and the nominal trajectory
due to manufacturing, measurement and atmospheric modeling errors. These differences
make the rocket deviates from its nominal trajectory and miss its target.
Therefore, studying sources of error between the real and predicted trajectories is a must, in
order to help rocket designers to optimize the manufacturing tolerances and restrictions to
avoid these sources of error, [8].

The total dispersion results mainly from three phases [1, 2, 3]:

     Dispersion due to rocket production inaccuracy, which includes:
             - The propellant mass and composition inaccuracy.
             - The rocket total mass, axial and lateral moments of inertia and resultant
                    center of gravity inaccuracies.

     Dispersion during boosting phase:
             - Launcher deflection: It is the most common factor present with long
                    launchers. It results from ignition shock and sudden high thrust build-up from
                    the booster, which will generate a pitching rate on the rocket and it can be

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                                                                                 Paper: ASAT-13-FM-04

               solved by: (1) designing a stiffer launcher, (2) accounting for this motion in
               setting the launcher.
         - Rocket tip-off from the launcher: if the supporting shoes on the missile do
               not leave the supporting rail or rails simultaneously, this will tilt the rocket
               downward under the force of gravity and cause the missile to fly a new flight
               path. With simultaneous shoe release, the rocket would have a simple
               translational instead of combined translational and rotational motion
               immediately upon leaving the launcher. This translational motion is not as
               detrimental to the dispersion as the rotational motion.
         - Launcher setting: since it is physically impossible to set the launcher
               precisely at the desired angle, error in launcher angle is always present.
         - Variation in rocket motor performance: because of the tolerance in rocket-
               motor design, propellant properties, and manufacturing, the total impulse of
               the rocket motor may vary.
         - Thrust and fin misalignments: it is an important source of dispersion in case
               of unguided rockets "flying on open loop". To minimize the dispersion due to
               this misalignment a large static stability margin is required but its not desired
               if the rocket is launched in a cross wind. Another method is used widely
               which is spinning the rocket immediately off the launcher. This initial spin
               may be produced by spin motor mounted on the missile or by using helical-
               rail launchers.
         - And atmospheric disturbances such as tail wind, cross wind, and gusts.

 Dispersion during free-flight phase, which caused from change in wind profile,
    variation in atmospheric density and variation in rocket characteristics.

2. Mathematical Model

In order to predict the trajectory of an unguided artillery rocket, six degrees of freedom
6-DOF mathematical model is presented in [1,6], where the block diagram of this model is
shown in Fig. 1.

The equationsRf motions WKDWdescribe the 6-DOF model are derived according to some
assumptions:

  a) The flying body is rigid.
  b) All equations are referred to a body fixed reference frame.
  c) The aerodynamic coefficients are calculated in body fixed reference frame.
  d) The Earth model is included (ellipsoidal shape, rotation, gravity....).
  e) The atmospheric model is included where the temperature, sonic speed, and air density

      are varying with the body altitude.

The 6-DOF equations of motions are three translational degrees describe the motion of mass
(CG), also called the trajectory, as shown in equation (1),

 u         1   TTTzxyAAAnaosxrimdiaeal l         sin        p  E    p     u
           m                                  gcos.sin      q  B    q
  v                                                                         v    (1)
                                               cos .cos        E    r          
                                                               B

w                                                            rBE          w 

And three attitude degrees orient the projectile, as shown in equation (2),

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  p                                r      p.q                                                                                                                   Paper: ASAT-13-FM-04
    L                       Izx                              Iy  Iz      .q.r                                                                                                           (2)
    Ix                              Ix                           Ix

  q M                       Izx      r2   p2       Iz         Ix .r.p 
    Iy                  
                                                  Iy                  

  r N                     I  zx  p        q.r               Ix   Iy      .p.q 
    Iz                                                           Iz
                                   Iz

         INITIAL CONDITIONS                                                         Acceleration in body axes                                                          Aerodynamic &
    - Initial mass mo, mp                                                                                                                                           Propulsive Forces &
    - Launching angles o ,o , o                                      u           1     Tx  Aaxial                    sin              p  E     p        u 
    - Muzzle velocity vxo , vyo , vzo                                            m                                                       B                                 Moments
    - Muzzle position o,o                                             v                 Ty   Aside               g  cos  . sin        q                   v  
    - Mass moment of inertia Ixx, Iyy, Izz, Izx                                                                                          E     q                  Aerodynamic forces and
    - Initial rates po, qo, ro                                                                                                           B                        moments modeled in
    - Wind velocity                                                                                                                                               body axes. Propulsive
                                                                    w                 Tz  Anormal                  cos .cos             rBE    r        w         forces modeled in body
                                                                                                                                                                  axes.

                                                                                    Acceleration in geodetic frame

                                                                                              sin  sin  cos                cos  sin      cos        
                                                                      cos  cos                 cos  sin                     sin  sin     

                                                             a N                              sin  sin  sin                cos  sin      sin              a  x    Mach number
                                                             a E      cos                      cos  cos                     sin  cos                      a  y
                                                                               sin                                                                                                V
                                                                                                                                                                                    a
                                                                                                                                                                        M  
                                                             a D                                                                                          az  B
                                                                                                                                                      

                                                                            sin                   sin  cos                      cos  cos              
                                                                                                                                                      
                                                                                                                                                                      Sonic speed
                                            Euler's Equation                                                                                                           a  .R.T

                      1          .Iz.L  Izx .N  Iz.Izx 1        Ix  Iy   .p.q                                   
                                                                  Iz                                         .q.r
      p                                                                                        2      I  2
                         I 2zx                                                      IyIz    I  z         zx              Acceleration in geodetic frame taking in account
                                                                                                                                             Earth's rotation
          I  x  I  z
                                                                                                                           V N  (  2.).sin().VE   .VD  a N
      q   M                 Izx  r2   p2   Iz     Ix .r.p                                                                  V E  (  2.).sin().VN  (  2.).cos().VD  aE
          Iy                                                                                                               V D  (  2.).cos().VE  .VN  a D
                                          Iy               

                                . Izx .L  Ix .N 
          r
                1           2                      Iy   Ix    Iz  .Izx .r.q   I  2   IxIy      I  2   .q.p
             xIz            zx                                                   x                zx
          I              I

                                                                                                                           Navigation Equation                       Velocity in
                                                                                                                                                                  geodetic frame
                   Effect of Earth's Rotation
                                                                                                                                         VN                       VN 

                P p                          (E   ) cos                                                                           R meridian  H G                 VE  
                                                                                                                                                                      
                Q               q                                                                                                              VE
                                       LBE                                                                                        R normal      HG        .cos    VD 
                                                                                                Stop
                R r                          (E   ) sin                                     Simulation                                                                  

                        Kinematics Equation                                                                                                    Yes                if altitude  0
                                                                                                                                                Altitude
        1 sin  tan  cos  tan   P                                         

                      0          cos            sin          Q                                                                                                          No
                                                       
        0 sin  sec  cos  sec  R 

       Output Parameters                                                          ATMOSPHERIC MODEL

    - Summit point altitude and time                                          The air temperature and air
    - Impact point parameters                                                 density as functions of altitude
    - Total time of flight
    - Axial and Normal accelerations
    - etc...

                                                       Fig. 1 Block diagram of 6-DOF model

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The Earth rotation is taken into consideration as,

 P p                  (E   ) cos  
                                    
Q       q     L  BE                                                                        (3)

R r                   (E   ) sin 

where the transformation matrix from body axes to earth axes is shown in equation (4),

         cos cos                    cos sin                sin  
                                                          

            sin  sin cos        sin  sin sin              
             cos  sin 
L BE                                cos    cos        sin  cos                             (4)
                                                                  
                              

                                                          

            cos  sin   cos        cos sin sin                     
             sin  sin              sin  cos           cos  cos 

                                                                  

3. Case Study

In this study, a trajectory calculation using a 6-DOF model was developed and applied for
122 mm unguided artillery rocket. All aerodynamic forces and moments coefficients of the
given shell are calculated using Missile Datcom. The mass properties of the given shell is
calculated using Inventor considering the change of the rocket mass during propellant burning
till the propellant burn-out (active part), then the rocket will fly the rest of its trajectory as a
projectile of fixed mass (passive part). The 6-DOF model assumed the rocket is ideal, where
the axis of symmetry of the exterior surface coincides with the longitudinal principal axis of
inertia, and the two lateral principal moments of inertia are identical.

    3.1. Main data
The main data and firing conditions of the studied artillery rocket are summarized below:

Caliber.                                              D              = 122 mm
Overall Length.                                       Lt
Total Mass.                                           Mt             = 2870 mm
Propellant Mass.                                      mp
Propellant Burning Time.                              tk             = 66 Kg
Mean Thrust.                                          Tx
Initial Center of Gravity from the Nose Tip.          C.G_xi             20.5 Kg
Final Center of Gravity from the Nose Tip.            C.G_xf
Initial Axial Moment of Inertia.                      I_xxi              1.67 s
Final Axial Moment of Inertia.                        I_xxf
Initial Lateral Moment of Inertia.                    I_yyi = I_zzi      23600 N
Final Lateral Moment of Inertia.                      I_yyf = I_zzf
Shell muzzle velocity                                 Vo             = 1.374 m
Shell muzzle spin rate                                po
Firing Elevation Angle                                               = 1.264 m
                                                      o              = 0.1499 Kg.m2
                                                                     = 0.1238 Kg.m2
                                                                     = 41.58 Kg.m2
                                                                     = 33.83 Kg.m2

                                                                     = 26.7 m/s

                                                                     = 5.8 rps

                                                                     = 50

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3.2. Aerodynamic coefficients and derivatives
Knowing the aforementioned configuration definitions, the aerodynamic coefficients and
derivatives for 122mm unguided rocket are computed using the analytical capability of the
Missile Datcom code. The results of these calculations are shown in Table 1.

                   Table 1 The aerodynamic coefficients and derivatives
                                     for 122mm unguided rocket

M  CA          Cl  C lp    Cl r     C m                        C mq                          C m
   A P CN C N            AP       AP                         AP                           AP

0.2 .340 .477 8.57 -52.129 .075 -8.198 0.161 -0.43 -53.8386 -61.6158 -2542.92 -2862.61 -96.593 -121.264

0.4 .305 .442 9.10 -52.139 .076 -8.312 0.219 -0.415 -54.3829 -62.2050 -2546.69 -2860.55 -99.443 -126.039

0.6 .291 .427 9.48 -52.148 .077 -8.449 0.225 -0.457 -55.1018 -62.9840 -2539.2 -2843.26 -104.587 -134.617

0.8 .290 .424 9.82 -52.157 .076 -8.448 0.437 -0.123 -54.3483 -62.1768 -2473.68 -2777.00 -112.628 -147.939

1.0 .391 .574 9.99 -52.63 .077 -8.339 0.615 0.294 -55.0595 -63.0754 -2423.07 -2735.95 -123.998 -166.608

1.1 .445 .648 10.16 -52.994 .094 -8.674 0.67 0.171 -62.072 -70.4885 -2675.52 -3000.02 -131.225 -178.402

1.2 .444 .637 10.00 -53.057 .102 -8.870 0.79 0.356 -65.6489 -74.3189 -2780.73 -3117.29 -140.089 -192.844

1.3 .349 .537 10.08 -63.529 .110 -9.304 0.871 0.551 -66.055 -75.5352 -2877.06 -3237.14 -144.48 -201.76

1.4 .345 .529 10.34 -65.882 .123 -10.17 0.974 0.793 -73.4021 -83.6276 -3068.43 -3463.90 -147.441 -206.842

1.5 .333 .512 10.44 -66.966 .128 -9.947 -0.807 -0.903 -70.0033 -80.0419 -2280.63 -2535.50 -148.807 -209.186

1.6 .322 .495 10.59 -67.717 .126 -9.836 -0.565 -1.079 -68.1121 -78.0571 -2282.44 -2535.58 -149.751 -210.807

1.8 .304 .464 10.67 -68.599 .120 -9.339 -0.688 -1.022 -63.0248 -72.6351 -2175.18 -2410.93 -150.862 -212.714

2.0 .287 .434 10.13 -68.875 .113 -8.762 -0.374 -0.536 -57.607 -66.8122 -2188.04 -2431.46 -151.209 -213.309

2.2 .271 .406 9.05 -68.672 .105 -8.15 0.064 -0.076 -37.6782 -45.2677 -1840.82 -2046.55 -150.953 -212.870

3.3. Trajectory Analysis
In order to investigate the trajectory parameters which are calculated using presented 6-DOF
model of the given unguided rocket, a single case will be chosen corresponds to firing angle
o = 50.

Figure 2 shows the trajectory of the rocket at o = 50. The rocket has very small drift in
case of no side wind due to its high stability (fin stabilized). Figure 3 shows the rocket
altitude during flight time, where the total flight time is 79 sec but the summit time is nearly
36 sec. Due to the high velocity of the rocket at the burn out point of the trajectory (at the
end of burning), the summit time is less than half of the total flight time.

                                                8000

                                                7000

                                                6000

                                  Altitude [m]  5000

                                                4000

                                                3000

                                                2000

                                                1000

                                                0

                                                      0  10  20  30  40               50  60  70             80

                                                                     Flight time [s]

   Fig. 2 The 3D trajectory path                Fig. 3 The rocket altitude during
                                                               flight time

Figure 4 shows the velocity of the rocket from firing to impact points, where the velocity at
the firing point is 26.7 m/s and it will be increased due to the thrust force acting in flight
direction, where the velocity at the burn-out point (end of burning) is 705 m/s. Then the

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velocity will be decreased as well as the rocket goes up (altitude increased) until the summit
point, then the rocket will go down to increase the rocket velocity due to a gravitational
acceleration component in the direction of the velocity vector which will accelerate it.

800Velocity magnitude [m/s]                                                        800
700                                                     Velocity magnitude [m/s]   700
600                                                                                600
500      2  4    6           8 10                                                  500      20  40   60      80
400                                                                                400
300                                                                                300
200                                                                                200
100                                                                                100

   0                                                                                  0
      0                                                                                  0

            Flight time [s]                                                                 Flight time [s]

            (a)                                                                                 (b)

         Fig. 4 The velocity magnitude of the rocket vs. flight time

Figure 5-a shows the axial acceleration of the rocket at the beginning (firing point) where it
is equal 35.4g in flight direction, due to the thrust force act in the axial direction of the
rocket. The axial acceleration is increasing till the burn out point (end of burning) where the
thrust equal zero then the axial acceleration is now 4g in direction opposite to its flight
direction, due to the aerodynamic axial force acting on the rocket and the gravitational
acceleration component in the opposite direction of its flight direction. Figure 5-b shows
that the axial acceleration will be decreased due to the decreasing of its aerodynamic axial
force (due to decreasing of free stream velocity) and decreasing of gravitational acceleration
component in the opposite direction of flight direction (elevation angle is decreasing). The
axial acceleration will be in the direction of flight again after the summit point, where the
component of gravitational force is greater than the aerodynamic axial force and then it will
be decreased again due to increasing of its aerodynamic axial force (increasing of free
stream velocity).

Axial acceleration [gees]50                                                         0.5
                                                        Axial acceleration [gees]40   0
 30
 20      2  4    6           8 10                                                  -0.5     20  40   60      80
 10                                                                                  -1

  0                                                                                -1.5
-10                                                                                  -2

     0                                                                             -2.5
                                                                                     -3

                                                                                   -3.5
                                                                                     -4
                                                                                         0

            Flight time [s]                                                                 Flight time [s]

            (a)                                                                                 (b)

         Fig. 5 The axial acceleration of the rocket vs. flight time

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Figure 6-a shows the normal acceleration from the point of firing till 10 s flight time, where
the magnitude at the firing point is 0.64g. The normal acceleration is oscillating highly
during the first 4 seconds, which may be attributed to oscillations of the aerodynamic rates.

Figure 6-b shows that the normal acceleration is increasing due to the decrease in the
elevation angle of the rocket (increasing the gravitational acceleration component
normal to rocket body) till the summit point of the rocket where the elevation angle
is equal to zero to equal 1g and then the normal acceleration will decrease due to the
increase of the elevating angle (decreasing the gravitational acceleration component
normal to rocket body).

 0.8Normal acceleration [gees]                                                      1.2
0.75                                                    Normal acceleration [gees]    1

 0.7     2  4    6           8 10                                                   0.8     20  40   60      80
0.65                                                                                0.6
                                                                                    0.4
 0.6                                                                                0.2
0.55
                                                                                      0
 0.5                                                                                     0
      0

            Flight time [s]                                                                 Flight time [s]

            (a)                                                                                 (b)

         Fig. 6 The normal acceleration of the rocket vs. flight time

Figure 7 shows the spin rate of the given rocket as function of flight time, where the spin
rate at the firing point is 36 rad/s [5-6 rps] and then it will be decreased slightly after leaving
launcher due to friction acting on the rocket body and the low velocity of the rocket. As the
velocity of the rocket increased during burning of propellant, the spin rate will be increased
due to the inclination of the rocket fins which will rotate the rocket in a positive spin
direction as it shown in Fig. 7-a. After the burn-out point the rocket velocity will be
decreased which will decrease the spin rate as shown in Fig. 7-b. Finally, the velocity will
increase again to increase the spin rate too.

Figure 8 shows the variation of the elevation angle of the rocket during flight time, where
the summit point is occurred nearly 36s. During the beginning of rocket flight there are
fluctuations in the pitch angle as shown in Fig. 8-a due to the high pitch rate as shown in
Fig. 9.

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Spin rate [rad/s]  160                                  Spin rate [rad/s]  160
                   140                                                     140
                   120     2  4    6           8 10                        120                        20      40           60      80
                   100                                                     100
                                                                            80
                    80                                                      60
                    60                                                      40
                    40                                                      20
                    20
                      0                                                      0
                                                                                 0
                        0

                              Flight time [s]                                                             Flight time [s]

                              (a)                                                                             (b)

                              Fig. 7 The spin rate of the rocket vs. flight time

                   51

Pitch angle [deg]  50                                                      Pitch angle [deg]   50
                                                                                               35
                   49                                                                          20

                   48                                                                           5
                                                                                              -10
                   47                                                                         -25
                                                                                              -40
                   46                                                                         -55
                                                                                              -70
                   45
                                                                                                   0
                       0   1  2    3           4  5                                                       20       40          60      80

                              Flight time [s]                                                             Flight time [s]

                              (a)                                                                             (b)

                              Fig. 8. The pitch angle of the rocket vs. flight time

Figure 9 shows the aerodynamic angles which are angle of attack , side-slip angle , and
total angle of attack acting on the rocket during its flight time. Figure 9-a shows that the
total angle of attack is zero at the launch point (no perturbation) and then increased and
decreased in a cyclic motion with a high total angle of attack due to the rocket low velocity.
As the velocity of the rocket increased to its maximum value at the burn out point at
(t = 1.67 s), the total angle of attack fluctuation will be terminated. The total angle of attack
will be increased as shown in Fig. 9-c till the summit point of the rocket, then it will be
decreased till the end of flight.

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0.8                            Beta

0.6                            Alpha                                                                        0.03

Aerodynamic angles [deg]0.4    Alpha_bar                                                                    0.025
                                                                               Total angle of attack [deg]
0.2                                                                                                         0.02

0                                                                                                           0.015

-0.2
                                                                                                   0.01

-0.4

                                                                                                  0.005
-0.6

-0.8                                                                                                        0

      0  0.5       1           1.5        2                                                                        0  20       40          60       80

              Flight time [s]                                                                                             Flight time [s]

              (a)                                                                                                         (b)

         Fig. 9. The aerodynamic angles of the rocket vs. flight time

3.4. Dispersion Analysis
The dispersions investigated in this paper have been applied to the rocket dynamics. The
models modified in the rocket simulation to include dispersion capabilities which are firing
conditions, and rocket mass properties. Table 2., shows a list of eight uncertainty parameters
that have been used in this work. It is tried to consider all the important parameters.

The limits of uncertainties presented in table 2 are improved by individual error analysis,
where a range of values in the defined limits in table 2, was given to each parameter and
simulation was run several times. Using simulation results, it was possible to plot the impact
point distance error vs. different parameters variation. This investigation is done in order to
find a good estimation for each individual parameter uncertainty.

              Table 2 Uncertainty parameters and ranges

                      Parameter Definition                                                                            Uncertainty Range Unit
1 Launching pitch angle
2 Rocket total mass                                                                                                   [-0.4 0.4]               deg
3 Propellant mass
4 Propellant burning time                                                                                             [-1.0 1.0]               %
5 Thrust mean value
6 Air density                                                                                                         [-1.0 1.0]               %
7 Axial moment of inertia
8 Lateral moment of inertia                                                                                           [-0.1 0.1]               s
9 Rocket launch velocity
10 Rocket launch spin rate                                                                                            [-1.0 1.0]               %
11 Wind speed at zero altitude
12 Wind direction at zero altitude                                                                                    [-4.0 4.0]               %

                                                                                                                      [-2.0 2.0]               %

                                                                                                                      [-2.0 2.0]               %

                                                                                                                      [-2.0 2.0]               %

                                                                                                                      [-2.0 2.0]               %

                                                                                                                      [-2.0 2.0]               m/s

                                                                                                                      [-2.0 2.0]               deg

The rocket dispersion is obtained from calculating range and drift differences from the
projectile nominal point of impact. The effects of rocket launch velocity error on dispersion
as shown in Fig. 10, is caused by the change of gases pressure applied on the base of the
projectile due to the propellant ignition. The rocket range is inverse proportional to
atmospheric density where the drag force is increased with the increasing of atmospheric
density.

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600                  Rang e error      Drift error     Radial error                               Range error    Drift error        Radial error

                                                                                       1500

400                                                                                    1000

200                                                                                    500
Errors [m]
                                                                   Errors [m]0    0

               1. 5                                                                              3
                                                                                                 2
-200 1
                                                                              -500 1
               0.5
                                                                                                 0
-400          0                                                                                 -1
         - 0.5                                                                                 -2

             -1                                                              -1000 - 3
         - 1. 5
                                                                                                    -2 -1 0 1 2
                     -6 -4 -2 0 2 4 6

-600                                                                         -1500
       -5 -4 -3 -2 -1 0 1 2 3 4 5
                                                                                       -1.5 -1 -0.5 0                      0.5      1        1.5
                   Air density error[%]
                                                                                                  Propellant mass [%]

Fig. 10. The effect of air density error                              Fig. 11. The effect of propellant mass error
               on dispersion                                                             on dispersion

The range is decreased with the increasing of the propellant mass as shown in
Fig. 11 where the inertial force is less than in case of nominal trajectory during free flight
regime, but the drift is slightly increased due to the increase of the gyroscopic moment. As
the burning time of the rocket motor increased the range error increased due to the increase
of the burn-out velocity as shown in Fig. 12, but the drift error is slightly decreased.
Increasing the mean thrust value acting on the rocket will increase the rocket range error
which also increases the burn-out velocity as shown in Fig. 13.

2000                 Range error       Drift error     Rad ial error                   300        Range error    Drift error           Radial error
1500
                                                                                       200

Errors [m]1000                                                                         100
  500
                                                                           Errors [m]0 0
 -500
                                        6                                              -100                             1
-1000                                   3                                                                           0 .5
-1500                                   0                                              -200
-2000                                  -3                                                                           0
                                       -6                                              -300
      -0.15                                                                                 -1.5                    -0 .5
                                          -0.2 -0.1 0
                                                       0 .1 0 .2                                                    -1

                                                                                                                           -2   -1  0     1       2

                     -0.1 -0.05 0 0.05 0.1 0.15                                                   -1 -0.5        0             0.5     1          1.5

                      Propellant burning time [s]                                                 Mean thrust value error [%]

Fig. 12. The effect of the propellant burning                         Fig. 13. The effect of the mean thrust error on
            time error on dispersion                                                         dispersion

Since it is physically impossible to set the launcher precisely at the desired angle, error in
launcher angle is always present where the range error will be decreased or increased
depending on the nominal launch angle. In our case study the range decreased and the drift
error will be increased as shown in Fig. 14. Increasing the rocket gross mass will decrease the
burn-out velocity to decrease the rocket range as shown in Fig. 15.

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          Range error             Drift error     Radial error                        Ran ge error             Drift error        Rad ial error

150                                                                        200

100                                                                        150

Errors [m]                                                                 100
                                                                 Errors [m]
50

                                                                             50

0

 -50       2                                                                   0       0.6
-100        1                                                               -50        0.4
           0                                                               -100        0.2
          -1
          -2              123                                                              0
                                                                                      - 0.2
              -3 -2 -1 0                                                              - 0.4
                                                                                      - 0.6

                                                                                             -3 -2 -1 0 1 2 3

-150                                                                       -150

      -3  -2              -1   0     1            2              3                -3  -2  -1                   0            1     2     3

              Launch angle error[deg]                                                 Rocket gross mass error [%]

Fig. 14. The effect of rocket launch angle                                 Fig. 15. The effect of the rocket gross mass
             error on dispersion                                                         error on dispersion

Increasing the launch velocity of the rocket will increase the burn-out velocity which will
increase the range as shown in Fig. 16. as the launch spinning rate of the rocket is increased
the drift will be decreased from its nominal value due to the increasing of rocket stability as
shown in Fig. 17. The rocket range increased when a tail wind presented (the rocket Mach no
will be decreased) on the rocket body which will decrease the drag force acting on the rocket
as shown in Fig. 18. As shown in Fig. 19, if a cross wind is presented the rocket drifts to right
if the wind came from right and vise verse due to the presence of tail surfaces behind the
rocket center of gravity to make the rocket fly opposite to wind direction.
There is no effect of the axial and lateral mass moments of inertia on the rocket range. But in
case of drift, the drift will be decreased with increasing Ixx and increased with increasing of Iyy
(Izz) as shown in Fig.s 20, 21.

          Range error             Drift error        Radial error                     Range error                 Drift error        Radial error

25                                                                         0.08
20
15                                                                         0.06
10
                                                                           0.04
  5
  0Errors [m]                                                              0.02
 -5                                                            Errors [m]
-10                                                                          0
-15
-20                                   0 .06                                -0.02
-25                                   0 .04
                                      0 .02                                -0.04
     -3
                                            0                              -0.06
                                     -0 .02
                                     -0 .04
                                     -0 .06

                                               -3 -2 -1 0 1 2 3

                                                                           -0.08

          -2              -1      0            1     2              3             -3  -2  -1                   0               1     2           3

              Launch velocity error [%]                                                   Launch spin rate error [%]

Fig. 16. The effect of launch velocity error                               Fig. 17. The effect of launch spin rate error
                  on dispersion                                                               on dispersion

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                                                                                                                      Paper: ASAT-13-FM-04

                       Range error     Drift error        Radial error                                  Rang e error                Drift error        Radial error

            100                                                                                 80
              80
              60                                                                                60
              40
              20                                                                                40
                0
Errors [m]   -20                                                                    Errors [m]  20
             -40
             -60                                                                                0
             -80
                                        4                                                     -20           0
            -100                        2                                                     -40       -0 .1
                   -3                                                                         -60       -0 .2
                                        0                                                               -0 .3
                                       -2                                                               -0 .4
                                       -4
                                                                                                               -3 -2 -1 0 1 2 3
                                           -3 -2 -1 0 1 2 3

                       -2  -1       0               1     2          3              -80                 -2  -1                   0               1     2     3
                                                                                         -3

                           Wind velocity [m/s]                                                              Wind direction [deg]

            Fig. 18. The effect of wind velocity on                                             Fig. 19. The effect of wind direction
                             dispersion                                                                 (10 m/s) on dispersion

                       Range error  Drift error        Radial error                                     Range error                 Drift error           Radial error

            0.4                                                                     0.06

            0.3                                                                     0.04

            0.2

Errors [m]  0.1                                                         Errors [m]  0.02

            0                                                                                   0

            -0.1                                                                    -0.02

            -0.2

            -0.3                                                                    -0.04

            -0.4                                                                    -0.06

                  -3   -2  -1       0            1     2     3                                      -3  -2  -1                      0               1     2          3

                  Axial mass moment of inertia error [%]                                                Lateral mass moment of inertia error [%]

Fig. 20. The effect of axial mass moment of                             Fig. 21. The effect of lateral mass moment of
          inertia error on dispersion                                              inertia error on dispersion

4. Conclusion:

Trajectory analysis of an unguided 122mm artillery rocket using simulation software was
undertaken to show the importance of this type of analysis in order to know all parameters
acting on the rocket during its flight which may be useful to avoid flight mistakes. The
dispersion analysis is used to show the importance of identifying the design weaknesses in
margins of specific parameters. Also, it is used to find out the optimum values of the rocket
motor parameters for lowest impact point error, and the probability of flight-to-target success.
Results were presented for the selected conditions in the form of dispersion. This analysis
showed that the rocket motor parameters (burning time, propellant mass and mean thrust
value) have a great effect on the rocket range and its impact point error.

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                                                                                    Paper: ASAT-13-FM-04

5. References

 [1] Bernard Etkin, "Dynamics of Atmospheric Flight", United States of America, John
          Wiley & Sons, 1972.

 [2] Douglas O., and Mark C., "Model Predictive Control of a Direct Fire Projectile
          Equipped With Canards ", Journal of Dynamic Systems, Measurement, and Control,
          Vol. 130, NOVEMBER 2008.

 [3] Ezeddine Salem Ab. Ali, "Parametric Study Of Missile Trajectory", M.Sc. thesis,
          Military Technical College, Cairo, Egypt, March 2009.

 [4] Gagnon E., and Lauzon M., "Course Correction Fuze Concept Analysis for In
          Service 155 mm Spin-Stabilized Gunnery Projectiles", AIAA Guidance, Navigation
          and Control Conference and Exhibit, Honolulu, Hawaii, 18 - 21 August 2008.

 [5] Jankovic, S., Gallant, J., and Celens, E., "Dispersion of an Artillery Projectile due to
          Unbalance", 18th International Symposium on Ballistics, San Antonio, pp. 128-141,
          15-19 November, 1999.

 [6] Khalil M. S., "Trajectory Predection of Flying Vehicle", M.Sc. thesis, Military
          Technical College, Cairo, Egypt, 2008.

 [7] Lua, K. B., Lim, T. T., and Luo, S. C., "Helical-Groove and Circular-Trip Effects on
          Side Force", Journal of Aircraft, Vol.37, No.5, pp. 906-915, 2000.

 [8] SAGHAFI F., and KHALILIDELSHAD M., "A Monte-Carlo Dispersion Analysis
          of a Rocket Flight Simulation Software", 17th European Simulation Multi-
          Conference ESM2003, England, 9-11 June, 2003.

 [9] Sailaranta, T., Siltavuori, A., Laine, S., and Fagerstrom, B., "On Projectile Stability
          and Firing Accuracy", 20th International Symposium on Ballistics, Orlando, pp. 195-
          202, 23-27 September, 2002.

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