Jtam-A4.dvi


JOURNAL OF THEORETICAL

AND APPLIED MECHANICS

54, 4, pp. 1183-1195, Warsaw 2016
DOI: 10.15632/jtam-pl.54.4.1183

COMPARISON OF EXPLICIT AND IMPLICIT FORMS OF THE MODIFIED
POINT MASS TRAJECTORY MODEL

Leszek Baranowski
Military University of Technology, Faculty of Mechatronics and Aeropace, Warszawa, Poland

e-mail: leszek.baranowski@wat.edu.pl

Błażej Gadomski, Jacek Szymonik
PIT-RADWAR S.A., Warszawa, Poland

e-mail: blazej.gadomski@pitradwar.com; jacek.szymonik@pitradwar.com

Przemysław Majewski
PIT-RADWAR S.A., Warszawa, Poland and

University of Warsaw, Chair of Mathematical Methods in Physics, Warsaw, Poland

e-mail: przemek.majewski@gmail.com

The article compares the results of trajectory computation for a 35mm projectile using
two forms (explicit and implicit) of themodified point-mass trajectorymodel. All necessary
ammunition parameters (aerodynamic coefficients, dimensions, mass etc.) and initial con-
ditions for differential equations are provided. The results of numerical integration (using
non-stiff fourth-order Runge-Kutta solver) are presented in form of projectile trajectories
projections onto vertical andhorizontal planes.Data tables comparingbothmodels in terms
of projectile position and velocity in chosen time steps are also attached.

Keywords: exterior ballistics, projectile motion, MPMTM, equations of motion, projectile
trajectory

1. Introduction

One of the requirements for algorithms implemented in Fire Control Systems (e.g. probability
calculation via initial condition error propagation, finding fire-control solution) is limited runti-
me. Some of FCS algorithms require the use of a certain model of the projectile trajectory. The
basic approach to projectile trajectory calculations is to use a Modified Point-Mass Trajectory
Model (MPMTM), also knownas the four degree-of-freedommodel or theLieskemodel (after an
American R. Lieske, who initiated its widespread usage), whose foundations were introduced by
Lieske andReiter (1966). One of theMPMTM implementations is presented in (STANAG4355,
2009) and exemplary calculations based on it were published by Baranowski (2013), McCoy
(1999), Coleman et al. (2003).
A significant weakness of this MPMTM representation is the iterative method for determi-

ning the yaw of repose, which makes this approach time consuming. In order to overcome this
limitation, a new explicit form of theMPMTMwas proposed by Baranowski et al. (2016).
The aim of this paper is to compare both implicit and explicit forms of MPMTM. The

two model variations are analyzed in terms of the obtained results and runtime. Firstly, we
characterize the differential equations describing motion of the projectile for both forms. We
also provide initial conditions for the equations of motion as well as other necessary parameters
of the mathematical model such as:

• physical parameters of the chosen 35mm ammunition;



1184 L. Baranowski et al.

• values of aerodynamic coefficients of the projectile;

• atmospheric conditions used in simulation tests.

Calculations of 35mmprojectile trajectories are conducted for standard atmospheric conditions
(STANAG 4119, 2007) considering three different scenarios, i.e.:

• no wind present along the projectile trajectory;

• homogeneous cross wind introduced along the projectile trajectory;

• homogeneous tail wind introduced along the projectile trajectory.

The runtimes of integration of the projectile path for different elevation angles for both the
explicit and implicit forms have beenmeasured and later compared. In order to enhance model
analysis, we have usedMATLAB scripts and built-in functions.

2. Projectile motion models

2.1. Modified point-mass trajectory model (MPMTM) – implicit form

The basic MPMTM is a conventional point-mass model but, in addition, the instantaneous
equilibrium yaw is calculated at each time step along the trajectory. It provides estimates of
yaw and drag, lift, andMagnus force effects resulting from the yaw of repose. The assumptions
made at the stage of model derivation require that the projectile is dynamically stable, only
the most essential forces andmoments are taken into account, transition processes in projectile
oscillatory motion around its center of mass are ignored due to replacement of the total angle
of attack αt with the yaw of repose αe.
Themathematical model of artillery projectile 3Dmotion, according to the implicit form of

theMPMTM, contains the following equations (Baranowski, 2013; Coleman et al., 2003):

— dynamic differential equation of motion of the projectile center of mass

mu̇=DF+LF+MF+mg (2.1)

— dynamic equation for rotation around the projectile axis of symmetry

dp

dt
=
πρd4vCspin

8Ix
p (2.2)

— equation of the yaw of repose vector

αe =−
8Ixp(v× u̇)

πρd3(CMα +CM
α
3
α2e)v

4
(2.3)

whereDF,LF,MF, g are the drag, lift, Magnus and gravity force vector, respectively

DF=−
πρid2

8
[CD0 +CD

α
2
(QDαe)

2]vv LF=
πρd2fL

8
(CLα +CL

α
3
α2e)v

2
αe

MF=−
πρd3QMpCmag−f

8
(αe×v) mg=−mg





0
1
0






(2.4)

Themeaning of letter symbols used in equations (2.1)-(2.4) is as follows: ρ–density of air,d–
caliber of the projectile,m –mass of the projectile, v – velocity of the projectile with respect to
the air: v=u−w,u – velocity of the projectilewith respect to a ground– fixed reference system,



Comparison of explicit and implicit forms... 1185

w – velocity of the wind, p – angular velocity of the spinning motion, Ix – moment of inertia
along the axis of the projectile, Cspin – spin damping coefficient, CMα – overturning moment
coefficient,CM

α
3
– cubicoverturningmoment coefficient,CD0 –drag force coefficient,CD

α
2
–yaw

drag coefficient,CLα – lift force coefficient,CL
α
3
– cubic lift force coefficient,Cmag−f –Magnus

force coefficient, i,fL,QM,QD – fitting factors for the drag, lift, Magnus force and yaw drag,
respectively, g – gravitational acceleration.
In the above equations of motion of the projectile, the Coriolis force is neglected. In our

calculations we assumed (unlike in the model described in STANAG 4355) constant gravity
force along the projectile trajectory as we considered only ground firing trajectories. It is worth
mentioning that we also assumedCM

α
3
=CL

α
3
=0 during the simulations (as a result of linear

dependency ofCL andCM on α).

2.2. Modified point-mass trajectory model – explicit form

In the form presented in the previous Section, the vector αe depends on u̇, which results
in a differential equation being defined by an implicit function. The derivation of the explicit
MPMTM was introduced by Baranowski et al. (2016). In the equations of motion of the pro-
jectile, the Coriolis force is neglected and the gravitational force is constant. The differential
equations for the final form of the explicit model are as follows:

ẋ=v+w (2.5)

where x is the three-dimensional position vector

ṗ=
ρv2

2Ix
SdCspinp̂ S =

πd2

4

v̇=−
ρv2

2m
S
[
CD0+ ĈDα2

(2mg
ρv2S

)2 Î2xp̂
2cos2γa

(1− Îxp̂2Ĉmag−f)2+(Îxp̂ĈLα)2

]
−gsinγa

(2.6)

and1

[
γ̇a

χ̇acosγa

]

=−
g

v

cosγa
(1− Îxp̂2Ĉmag−f)2+(Îxp̂ĈLα)2

[
1− Îxp̂2Ĉmag−f
Îxp̂ĈLα

]

(2.7)

Dimensionless coefficients used in equations are

Îx =
Ix

md2
p̂=
pd

v
(2.8)

The above equations describe the case in which the wind is homogeneous within the interval
of integration, i.e. ẇ = 0. Please note, that rigorously it is not equivalent to a constant wind,
merely that thewind is constant along theflight path.Nevertheless, in practice, suchphenomena
are almost impossible to occur in ballistics. The dimensionless coefficients are given as

ĈDα2 =
CDα2

(CMα)2
ĈLα =

CLα

CMα
Ĉmag−f =

Cmag−f

CMα
(2.9)

Let us recall that from (Baranowski et al., 2016) it follows that the explicit and implicit forms
of theMPMTM are equivalent.

1γ
a
is the elevation angle of v measured from the horizontal direction, i.e. the air-path inclination

angle and χ
a
is the azimuth angle of v, i.e. the air-path azimuth angle.



1186 L. Baranowski et al.

3. Physical model

The flight simulation computer program of artillery projectiles requires determination of the so-
called physical model (Koruba et al., 2010; Kowaleczko and Żyluk, 2009; Ładyżyńska-Kozdraś,
2012) which includes the following characteristics:

• geometries, mass and inertial, aerodynamic characteristics of the projectile,

• surrounding environment (density, viscosity, temperature, pressure, velocity and wind di-
rection depending on weather, flight altitude, etc.).

35mmTP-T ammunition parameters used in the simulation tests are as follows:

• projectile massm=0.55kg,

• projectile muzzle velocity v=1180m/s,

• projectile calibre d=0.035m,

• projectile axial moment of inertia Ix =0.97 ·10−4kgm2.

In the process of trajectory generation, we have used aerodynamic coefficients obtained for
35mm TP-T ammunition from PRODAS software (Baranowski and Furmanek, 2013). The co-
efficients of aerodynamic drag, lift force and spin dumpingmoment have been interpolated using
polynomials in the following form (Pope, 1978; Shanks andWalton, 1957)

C(Ma)= (1+s)A(r)+(1−s)B(r) (3.1)

where

A(r)= a0+a1r+a2r2 B(r)= b0+ b1r+ b2r2

s=
r

√
(1−L2)r2+L2

r=
Ma2−K2

Ma2+K2
(3.2)

where C(Ma) is an aerodynamic coefficient dependent on the Mach number and a0, a1, a2, b0,
b1, b2,K,L are parameters to be identified. Figures 1a and 1bpresent aerodynamic drag and lift
force coefficients in nodes and the result of polynomial interpolation with the use of polynomial
(3.1). Gridded data piecewise cubic Hermite interpolation (MATLAB griddedInterpolant class)
has been used to represent the induced drag andMagnus force coefficients.

Fig. 1. Drag (a) and lift (b) force coefficient as a function of theMach number

In order to compare explicit and implicit forms of theMPMTM, the projectile is “shot” from
the origin at [0,0,0] along the x axis with the quadrant elevation QE equal to 710mils – the



Comparison of explicit and implicit forms... 1187

angle of the maximum range for chosen 35mm ammunition. The initial angular velocity of the
spinningmotion is calculated using

p=
2πv
27.57d

(3.3)

where thenumber27.57 reflects the length of the revolution of the rifling in caliber units.The ini-
tial time instance is obviously gauged to zero.As itwasmentioned before, the gravitational force
is constant along the projectile trajectory with the gravitational acceleration g=9.80665m/s2.
The flight of projectile is simulated in standard atmospheric conditions (ISO 2533, 1975) both
without and with homogeneous non-zero wind.

4. Simulation tests results

In this Section we present and compare the results of flight simulation tests of the 35mmTP-T
projectile using both the implicit and the explicit form of the MPMTM. The parameters of
the model and initial conditions for the differential equations of motion used are presented in
Section 3.We also compare the time needed by bothmodels to generate trajectories for different
elevation angles.

4.1. Results for standard atmospheric conditions

Figures 2a and 2b present trajectory projections on the vertical andhorizontal plane, respec-
tively. Anassumption has beenmade that there is nowindwithin the integration interval. Based
on the plotted trajectories one cannot unambiguously decide whether there are any differences
between twomodels. In order to analyze the results in amore detailedway, differential equations
of motion where have been integrated using a non-stiff fourth-order Runge-Kutta solver.

Fig. 2. Projection of the trajectory of 35mmTP-T projectile on the vertical plane (a) and
horizontal plane (b); standard atmosphere without wind

Tables 1-4 contain information on projectile position and velocity in several chosen mo-
ments of time calculated for the implicit and explicit form of the MPMTM. The differen-
ces observed in these tables are the result of the precision2 used during simulations (1e-12).

2Tolerance set for the integrationmethod is ameasure of the error relative to the size of each solution
component. Roughly, it controls the number of correct digits in all solution componentsMATLAB 2014b
Documentation



1188 L. Baranowski et al.

While comparing the models presented in Section 2, it is important to remember (when
the wind velocity is non-zero) that they return projectile velocities in different reference
frames:

• explicit model returns velocity in the air-flow3 reference system,

• implicit model returns velocity in the the anti-aircraft gun reference system.

Figure 3 visualizes absolute errors of the projectile position and velocity for two forms of the
MPMTM.

Table 1.Comparison of the projectile horizontal position (x-coordinate) in chosen moments of
time; standard atmosphere without wind

Time of x-explicit x-implicit ∆x
flight [s] [m] [m] [m]

5 2.879740543380384e+03 2.879740543380467e+03 −8.276401786133647e-11
10 4.283281165179276e+03 4.283281165179690e+03 −4.138200893066824e-10
15 5.352708930364734e+03 5.352708930365055e+03 −3.210516297258437e-10
20 6.274635184339581e+03 6.274635184339474e+03 1.073203748092055e-10
30 7.827993890946567e+03 7.827993890945661e+03 9.067662176676095e-10
40 9.071956738403202e+03 9.071956738401780e+03 1.422449713572860e-09
50 1.001379407012815e+04 1.001379407012648e+04 1.669832272455096e-09

Table 2.Comparison of the projectile horizontal position (y-coordinate) in chosen moments of
time; standard atmosphere without wind

Time of y-explicit y-implicit ∆y
flight [s] [m] [m] [m]

5 −2.615179706617644 −2.615179706628132 1.048805486902893e-11
10 −10.437812368269643 −10.437812368304497 3.485389754587231e-11
15 −27.482113983932305 −27.482113983970038 3.773337198254012e-11
20 −57.329128551977114 −57.329128551940180 −3.693401140481001e-11
30 −1.609880120207460e+02 −1.609880120205749e+02 −1.710418473521713e-10
40 −3.035886033713625e+02 −3.035886033711299e+02 −2.325464265595656e-10
50 −4.443484747573776e+02 −4.443484747571376e+02 −2.399360710114706e-10

Table 3.Comparisonof theprojectile verticalh-coordinate in chosenmoments of time; standard
atmosphere without wind

Time of h-explicit h-implicit ∆h
flight [s] [m] [m] [m]

5 2.318321798332399e+03 2.318321798332298e+03 1.009539118967950e-10
10 3.260802764803827e+03 3.260802764803513e+03 3.137756721116602e-10
15 3.756859479068482e+03 3.756859479067962e+03 5.197762220632285e-10
20 3.959477637219192e+03 3.959477637218932e+03 2.596607373561710e-10
30 3.644341649231496e+03 3.644341649231876e+03 −3.801687853410840e-10
40 2.533030738575992e+03 2.533030738576817e+03 −8.244569471571594e-10
50 8.539543121131945e+02 8.539543121141693e+02 −9.747509466251358e-10

3Air-frame reference system has been chosen as the most convenient for calculations



Comparison of explicit and implicit forms... 1189

Table 4.Comparison of the projectile velocity in chosenmoments of time; standard atmosphere
without wind

Time of v-explicit v-implicit ∆v
flight [s] [m/s] [m/s] [m/s]

5 4.582123481454996e+02 4.582123481455023e+02 −2.671640686457977e-12
10 2.694330549712324e+02 2.694330549712641e+02 −3.177547114319168e-11
15 2.084609346355481e+02 2.084609346355003e+02 4.780531526193954e-11
20 1.738563395829701e+02 1.738563395828820e+02 8.813572094368283e-11
30 1.584999648980999e+02 1.584999648980213e+02 7.861444828449748e-11
40 1.814453824749712e+02 1.814453824749252e+02 4.595790414896328e-11
50 2.034057467881448e+02 2.034057467881248e+02 1.995204002014361e-11

Fig. 3. Absolute differences of projectile position and velocity as a function of time for the explicit and
implicit form of theMPMTM; standard atmosphere, no wind

Fig. 4. Projection of the trajectory of 35mmTP-T projectile on the vertical plane (a) and horizontal
plane (b); standard atmosphere, cross wind−10m/s



1190 L. Baranowski et al.

4.2. Results for standard atmospheric conditions with cross wind

Figures 4a and 4b present trajectory projections in the case of the presence of homogeneous
cross wind blowing with the velocity −10m/s (other conditions remain the same as in the
previous case). Aside from figures, we also publish tables with the results of integration of
the differential equations for both forms of the MPMTM (Tables 5 to 8). Absolute differences
between respective values are shown in the tables as well as in Fig. 5.

Table 5.Comparison of the projectile horizontal position (x-coordinate) in chosen moments of
time; standard atmosphere, homogeneous cross wind (−10m/s)

Time of x-explicit x-implicit ∆x
flight [s] [m] [m] [m]

5 2.879759177098334e+03 2.879759177098053e+03 −2.810338628478348e-10
10 4.283361832919431e+03 4.283361832918131e+03 −1.300577423535287e-09
15 5.352944802665610e+03 5.352944802661354e+03 −4.256435204297304e-09
20 6.275169112336884e+03 6.275169112329893e+03 −6.990376277826726e-09
30 7.829612391519409e+03 7.829612391508746e+03 −1.066291588358581e-08
40 9.075093437693744e+03 9.075093437681275e+03 −1.246917236130685e-08
50 1.001843598545247e+04 1.001843598543946e+04 −1.301486918237060e-08

Table 6.Comparison of the projectile horizontal position (y-coordinate) in chosen moments of
time; standard atmosphere, homogeneous cross wind (−10m/s)

Time of y-explicit y-implicit ∆y
flight [s] [m] [m] [m]

5 −20.786332880295095 −20.786332880260023 3.507238943711855e-11
10 −63.095834383862190 −63.095834383657966 2.042241931121680e-10
15 −1.183192056414523e+02 −1.183192056408177e+02 6.345430847431999e-10
20 −1.879744883329794e+02 −1.879744883319200e+02 1.059447640727740e-09
30 −3.744560742377584e+02 −3.744560742362193e+02 1.539149252494099e-09
40 −6.032941561003266e+02 −6.032941560987131e+02 1.613557287782896e-09
50 −8.336302688008207e+02 −8.336302687993701e+02 1.450530362490099e-09

Table 7.Comparisonof theprojectile verticalh-coordinate in chosenmoments of time; standard
atmosphere, homogeneous cross wind (−10m/s)

Time of h-explicit ] h-implicit ∆h
flight [s] [m] [m] [m]

5 2.318313372013739e+03 2.318313372014125e+03 3.865352482534945e-10
10 3.260776306927196e+03 3.260776306929160e+03 1.963144313776866e-09
15 3.756809955845482e+03 3.756809955850505e+03 5.022684490540996e-09
20 3.959406864678785e+03 3.959406864686925e+03 8.139977580867708e-09
30 3.644234110531493e+03 3.644234110544769e+03 1.327543941442855e-08
40 2.532897496773589e+03 2.532897496790511e+03 1.692160367383622e-08
50 8.538100257788867e+02 8.538100257975424e+02 1.865566900960403e-08



Comparison of explicit and implicit forms... 1191

Table 8.Comparisonof theprojectile velocity in chosenmoments of time; standardatmosphere,
homogeneous cross wind (−10m/s)

Time of v-explicit v-implicit ∆v
flight [s] [m/s] [m/s] [m/s]

5 4.582266925378976e+02 4.582266925378934e+02 −4.206412995699793e-12
10 2.694372225018485e+02 2.694372225017880e+02 −6.048139766789973e-11
15 2.084647377666492e+02 2.084647377662564e+02 −3.927880243281834e-10
20 1.738602908045667e+02 1.738602908041290e+02 −4.376374818093609e-10
30 1.585036690668833e+02 1.585036690664340e+02 −4.492619609663962e-10
40 1.814475179841104e+02 1.814475179838305e+02 −2.798401510517579e-10
50 2.034062035223828e+02 2.034062035223219e+02 −6.087930159992538e-11

Fig. 5. Absolute differences of projectile position and velocity as a function of time for the explicit and
implicit form of theMPMTM; standard atmosphere, homogeneous crosswind (−10m/s)

4.3. Results for standard atmospheric conditions with tail wind

Figures 6a and 6bpresent trajectories projections in the case of the presence of homogeneous
tailwindblowingwith thevelocity 10m/s (other conditions remain the sameasdescribed in4.1).

Fig. 6. Projection of the trajectory on the vertical plane (a) and horizontal plane (b); standard
atmosphere, tail wind 10m/s



1192 L. Baranowski et al.

Table 9.Comparison of the projectile horizontal position (x-coordinate) in chosen moments of
time; standard atmosphere, homogeneous tail wind (10m/s)

Time of x-explicit x-implicit ∆x
flight [s] [m] [m] [m]

5 2.899745751481406e+03 2.899745751481390e+03 1.637090463191271e-11
10 4.342170882757351e+03 4.342170882757300e+03 5.093170329928398e-11
15 5.455738001516948e+03 5.455738001517138e+03 −1.891748979687691e-10
20 6.423191936662845e+03 6.423191936663049e+03 −2.037268131971359e-10
30 8.070359880147351e+03 8.070359880147523e+03 −1.718944986350834e-10
40 9.410976469705320e+03 9.410976469705542e+03 −2.219167072325945e-10
50 1.045149163528715e+04 1.045149163528757e+04 −4.165485734120011e-10

Table 10. Comparison of the projectile horizontal position (y-coordinate) in chosen moments
of time; standard atmosphere, homogeneous tail wind (10m/s)

Time of y-explicit y-implicit ∆y
flight [s] [m] [m] [m]

5 −2.614508424916781 −2.614508424914710 −2.071232074740692e-12
10 −10.447910913790327 −10.447910913786751 −3.575806317712704e-12
15 −27.533351120232908 −27.533351120240553 7.645439836778678e-12
20 −57.473911630279034 −57.473911630223405 −5.562839078265824e-11
30 −1.616578103374976e+02 −1.616578103373247e+02 −1.728892584651476e-10
40 −3.052317376261908e+02 −3.052317376259557e+02 −2.351043804083020e-10
50 −4.470705154062585e+02 −4.470705154060170e+02 −2.415276867395733e-10

Table 11.Comparison of the projectile vertical h-coordinate in chosen moments of time; stan-
dard atmosphere, homogeneous tail wind (10m/s)

Time of h-explicit h-implicit ∆h
flight [s] [m] [m] [m]

5 2.319843392772728e+03 2.319843392772747e+03 −1.955413608811796e-11
10 3.265607481624752e+03 3.265607481624821e+03 −6.821210263296962e-11
15 3.765857597754436e+03 3.765857597754508e+03 −7.230482879094780e-11
20 3.972340990237719e+03 3.972340990238237e+03 −5.184119800105691e-10
30 3.663900131717957e+03 3.663900131719271e+03 −1.313765096710995e-09
40 2.557278531390514e+03 2.557278531392392e+03 −1.878106559161097e-09
50 8.802228281796221e+02 8.802228281817677e+02 −2.145611688320059e-09

Table 12. Comparison of the projectile velocity in chosen moments of time; standard atmo-
sphere, homogeneous tail wind (10m/s)

Time of v-explicit v-implicit ∆v
flight [s] [m/s] [m/s] [m/s]

5 4.556196037618452e+02 4.556196037618386e+02 6.593836587853730e-12
10 2.686771069413229e+02 2.686771069413665e+02 −4.354205884737894e-11
15 2.077692047266052e+02 2.077692047266444e+02 −3.922195901395753e-11
20 1.731362186963469e+02 1.731362186963497e+02 −2.728484105318785e-12
30 1.578228372701631e+02 1.578228372701328e+02 3.029754225281067e-11
40 1.810541039539512e+02 1.810541039539235e+02 2.768274498521350e-11
50 2.033211141640507e+02 2.033211141640362e+02 1.458033693779726e-11



Comparison of explicit and implicit forms... 1193

Tables 9-12 present exemplary results of the projectile trajectory calculations using both forms
of the MPMTM. Absolute differences between the respective values are shown in the tables as
well as in Fig. 7.

Fig. 7. Absolute differences of projectile position and velocity as a function of time for the explicit and
implicit form of theMPMmodel; standard atmosphere, homogeneous tail wind (10m/s)

4.4. Calculation time

Another aspect that we have taken under consideration is the runtime needed for the ne-
cessary computations. Figure 8b shows the time needed for trajectory generation for different
elevation angles: 10 to 710mil with the interval of 20mils. As mentioned before, the differential
equations have been integrated using the non-stiff fourth-order Runge-Kutta solver ode45. The
stopping condition for numerical integration was the moment when the projectile reached the
point of fall4.

Fig. 8. (a) Number of steps for the implicit and explicit forms of theMPMTMusing the ode45 solver;
(b) comparison of the trajectory calculation time for both forms of theMPMTM

As it can be seen, the explicit model needs much less time (up to 4 times less) to calculate
the trajectory. Figure 8a shows statistics drawn from the ode45 solver – the aggregate number
of steps (both successful and failed). One can easily notice that the solver needsmuch less time

4The point of intersection between the trajectory and the weapon level surface STANAG 4119, 2007



1194 L. Baranowski et al.

for the explicit form of MPMTM despite a similar number of steps and function evaluations
(Fig. 8a).

5. Conclusions

The aim of this paper is to compare an implicit form of the MPMTM with the explicit one,
which was derived by Baranowski et al. (2016). As it is shown in Section 4, simulations of the
35mm TP-T projectile trajectory gives the same results for both model forms in the following
cases:
• standard atmosphere without the presence of wind within the whole integration interval;

• standard atmosphere with homogeneous side wind within the whole integration interval;

• standard atmosphere with homogeneous tail wind within the whole integration interval.

Furthermore, it is shownthat the explicitMPMTMcan integrate differential equations ofmotion
up to 4 times faster than the implicit form,which is solely attributed to the fact that the explicit
analytic formula for theyawof reposehasbeenderived.This freedus fromthecost ofunnecessary
FLOPSneededtoapproximateαe.Thegreat reductionof runtime in theexplicitmodel is of great
importance and can significantly enhance the process of aerodynamic coefficients identification.
This process, based on ammo data and firing tables (ground and anti-aircraft), needs time
consuming calculations, i.e. finding local minima of complicated functions (given by ODEs) in
multi-dimensional parameter spaces, with the impossible aim of finding the global minimum –
the ideal fit.Moreover, an explicit formula does not have convergence issueswhichmight appear
in the implicit method. It seems justified to apply the explicit form of the MPMTM to the
identification process of aerodynamic coefficients on which we will focus in our future work.

Acknowledgement

This paper was created as a part of the development project No. OROB004603001 financed by
Polish National Centre for Research andDevelopment in the period between 2012 and 2015.

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Manuscript received July 22, 2015; accepted for print February 18, 2016