Aerodynamic Parameter Identification for Symmetric
Projectiles: Comparing Gradient Based and
Evolutionary Algorithms
Bradley T. Burchett∗
Rose-Hulman Institute of Technology, Terre Haute, IN, 47803
Two methods for finding aerodynamic coefficients from free flight range data are compared—
a gradient based method and an evolutionary algorithm each using a linear theory solution.
The inputs are limited to swerve and yaw measurements such as would be available from
spark range or yaw card tests. Each method is tested with minimal inputs. Methods are
compared for computational effort and accuracy. Numerical results for two typical finned
projectiles are presented.
Nomenclature
x T , y T , zT
CM Q
CN A
CX0
CLP
CLDD
CM A
D
Ixx , Iyy
J
L, M, N
V
X, Y, Z
x, y, z
position vector components of the target expressed in the inertial reference frame
pitch rate damping moment aerodynamic coefficient
normal force aerodynamic coefficient
axial force aerodynamic coefficient
roll rate damping moment aerodynamic coefficient
fin rolling moment aerodynamic coefficient
pitch moment due to AOA aerodynamic coefficient
projectile characteristic length
roll and pitch inertia
Jacobian matrix
total external applied moment on the projectile about the mass center
expressed in the projectile reference frame
projectile mass
angular velocity vector components expressed in the fixed plane reference frame
stationline of the projectile c.g. location
stationline of the projectile c.p. location
residual vector
translation velocity components of the projectile center of mass resolved
in the fixed plane reference frame
magnitude of the mass center velocity
total external applied force on the projectile expressed in the projectile reference frame
position vector components of the projectile mass center expressed in the inertial reference frame
Greek
θαf , θαs
λf , λs
ξi , ηiT
ρ
Ψf , Ψs
ψ, θ, φ
Ωαf , Ωαs
phase angles of the fast and slow epicyclic modes
fast and slow epicyclic mode damping factors
ith epicyclic mode right and left eigenvectors
air density
fast and slow epicyclic mode damped natural frequencies
Euler yaw, pitch, and roll angles
amplitudes of the fast and slow epicyclic modes
m
p, q, r
SLcg
SLcp
T
u, v, w
∗ Associate
Professor, Department of Mechanical Engineering, [email protected], Associate Fellow, AIAA
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ϑ
vector of unknown parameters
I.
Introduction
F
ree flight ballistic range data has been used to characterize the flight of small projectiles for more than
a century. Early in the 20th century, F. W. Mann used yaw card data to discover the epicyclic yawing
and pitching motion of spinning bullets, and quantify the geometry of typical trajectories.1 By 1920, Fowler,
Gallop, Lock and Richmond characterized the motion as a superposition of two oscillatory frequencies.2
Here we seek to apply recent methods of optimization to reducing data from the century–old technique of
yaw cards. Virtual yaw card data is generated by a high–fidelity six degree–of–freedom simulation. Launch
position, azimuth and elevation are assumed to be well known. Launch velocity is assumed to be perfectly
measured by a chronometer. The virtual yaw cards record altitude, cross range, pitch, and yaw at 24 stations
evenly spaced from 200 to 4800 calibers from the muzzle. Over this limited distance, velocity decreases
slightly due to drag, however with the slight variation in mach number, the aerodynamic parameters can be
modeled as constants. The data is assumed to be collected without error such that any limitations found in
these methods are a consequence of the algorithm, and not erroneous data collection. This paper applies the
method to two dis-similar fin stabilized projectiles. Spin stabilized projectiles are to be addressed in future
efforts.
II.
Projectile Dynamic Model
The nonlinear trajectory simulation used in this study is a standard six-degree-of-freedom model typically
used in flight dynamic modeling of projectiles. A schematic of the projectile configuration is shown in Figure
1. The six degrees of freedom are the three inertial components of the position vector from an inertial frame
to the projectile mass center and the three standard Euler orientation angles. The equations of motion are
provided in Eqs. (1-4).6

 



cθ cψ sφ sθ cψ − cφ sψ cφ sθ cψ + sφ sψ 


 u 
 ẋ 


(1)
=  cθ sψ sφ sθ sψ + cφ cψ cφ sθ sψ − sφ cψ 
v
ẏ








w
ż
−sθ
sφ cθ
cφ cθ
(a) Position Coordinates
(b) Attitude Coordinates
Figure 1. Schematic of a Fin Stabilized Projectile
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






 



1 s φ tθ
c φ tθ

 φ̇ 

 p 


= 0
q
θ̇
cφ
−sφ 








r
ψ̇
0 sφ /cθ cφ /cθ

 
 




0 −r q

 u 

 X/m 
 
 u̇ 


− r
=
v
0 −p 
Y /m
v̇









 

w
−q p
0
Z/m
ẇ

 

 



0 −r q
ṗ 

 p 

 L 





− r
= [I]−1  M
q 
0 −p  [I]
q̇










r
−q p
0
N
ṙ
(2)
(3)
(4)
In Eqs. (1) and (2), the standard shorthand notation for trigonometric functions is used: sin(α) ≡
sα , cos(α) ≡ cα , and tan(α) ≡ tα . The force appearing in Eq. (3) contain contributions from weight
W and body aerodynamics A. Gliding flight is assumed.
 
 



 XW 
 XA 
 
 

 X 
=
+
(5)
YW
YA
Y






 
 


ZW
ZA
Z
The dynamic equations are expressed in a body-fixed reference frame, thus, all forces acting on the body are
expressed in the rocket reference frame. The rocket weight force is shown in Eq. (6):






 XW 
 −sθ 


= mg
(6)
YW
sφ cθ








ZW
cφ cθ
whereas the aerodynamic force acting at the center of pressure of the rocket is




 C + CX2 (v 2 + w2 )/V 2
 XA 

π 2 2  X0
= − ρV D
YA
CN A v/V



8



ZA
CN A w/V
given by Eq. (7):



(7)


The applied moments about the rocket mass center contain contributions from steady aerodynamics (SA)
and unsteady aerodynamics (UA).

 
 


 LU A 
 LSA 

 
 
 L 
(8)
+
=
MU A
MSA
M







 
 

NU A
NSA
N
The moment components due to steady aerodynamic forces are computed with a cross product between the
distance vector from the mass center of the rocket and the location of the specific force and the force itself.
The unsteady body aerodynamic moment provides a damping source for projectile angular motion and is
given by Eq. (9):




pDCLP 


 LU A 
 π

 CDD + 2V
qDCmq
= ρV 2 D3
(9)
MU A
2V




8




rDCmq
NU A
2V
In the six–degree–of–freedom simulation, the center of pressure location and all aerodynamic coefficients
(CX0 , CX2 , CN A , CDD , CLP , and Cmq ) depend on local Mach number and are computed during simulation
using linear interpolation.
The dynamic equations given in Eqs. (1-4) are numerically integrated forward in time using a fourthorder, fixed-step Runge-Kutta algorithm.
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III.
Projectile Linear Theory Trajectory Solution
The six degree of freedom projectile model just shown consists of 12 highly nonlinear differential equations
that are not directly amenable to a closed-form solution. Projectile linear theory is used to rapidly compute
projectile trajectories, to reduce aerodynamic range data, and to establish stability criteria for both fin-and
spin-stabilized projectiles. It is generally accepted as an accurate dynamic model for a wide class of fin- and
spin-stabilized projectiles. The following simplifying assumptions lead to the linear theory solution:
1. Change of variables from fixed plane, station line velocity, u, to total velocity, V .
2. Change of independent variables from time, t, to dimensionless arc length, s.
3. Euler pitch and yaw angles are small.
4. Aerodynamic angles of attack are small.
5. The projectile is mass balanced such that the center of gravity lies on the rotational axis of symmetry.
6. The projectile is aerodynamically symmetric.
7. A flat fire trajectory assumption is invoked such that the force of gravity is neglected in the total
velocity equation. Gravity is included in the epicyclic yawing and pitching equations, and the swerve
equations.
8. The quantities V , and φ are large compared to θ, ψ, q, r, v, and w, such that products of small
quantities and their derivatives are negligible.
The resulting equations are:
x′
=
y′
=
z′
=
φ′
=
θ′
=
ψ′
=
V′
=
p′
=
The matrix equation for epicyclic pitching


v′

 ′

w

q′


 ′
r
Where
D
D
v + ψD
V
D
w + θD
V
D
p
V
D
q
V
D
r
V
ρSD
−
CX0 V
2m
ρSD2 V
ρSD3 CLP
p+
CLDD
4Ixx
2Ixx
and yawing is:


 



v 









 

w
=Ξ
+



q 









 

r



Ξ=

−A
0
0
−C
D
0
0
−A D
C
E
D
0
F
0
G
0
0
−D
0
−F
E





g
(10)
(11)
(12)
(13)
(14)
(15)
(16)
(17)
(18)









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(19)
And
A
=
C
=
E
=
F
=
G
=
CM A
=
ρSD
CN A
2m
2
ρSD
CM A
2Iyy
(20)
(21)
ρSD3
CM Q
4Iyy
D Ixx p
V0 Iyy
D
V0
(SLcp − SLcg )C N A
(22)
(23)
(24)
(25)
And D is the projectile characteristic length (or diameter). The solution to these linear equations is the sum
of a particular solution due to the gravity constant, and a homogeneous solution.7
The particular solution is given by setting the derivatives equal to zero and solving the resulting algebraic
equation:




 vp 

 0 











wp
G
−1
= −Ξ
g
(26)


qp 
0 












rp
0
Resulting in:


vp



wp
 qp



rp
Where:










−F C
Gg
EC + AF 2 + AE 2
=
 −(AE + C)C/D

det(Ξ) 





AF C/D
det(Ξ) = A2 F 2 + A2 E 2 + 2ADC + C 2





(27)




(28)
This particular solution is then subtracted from the initial conditions prior to solving for the homogeneous
response:

 



vp 
v0 









 

wp
w0
(29)
−
χ=

 



 qp 
 
 q0 




rp
r0
The homogeneous response is governed by the mode shapes:
h
ξ1
ξ2
ξ1†
ξ2†
i
where



=


i
1√
−
i
1√
K+ Q
2D√ i K+ Q
−
2D
K− Q
2D√ i K− Q
2D
−i
1
√
R+ S
2D
√
i(R+ S )
2D
−i
1
√
R− S
2D
√
i(R− S )
2D






(30)
K = (E − A) + 2A + iF
(31)
Q = (E − A) + 4AE + 4C − F + 2i (F (E − A) + 2(AF + B))
(32)
2
2
R = K †,
S = Q†
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Also, we define W as the corresponding matrix of left eigenvectors.



iν
−ν
1
(−ν+µ)
(−ν+µ)
(−ν+µ)
η1T

−µ
−1
 T  1  −iµ
 η2 
(−ν+µ)
(−ν+µ)
(−ν+µ)
 T†  = 
†
†
ν
−1
iν
 η1  2 
 (ν † −µ† ) (ν † −µ† ) (ν † −µ† )
η2T †
−iµ†
(ν † −µ† )
−µ†
(ν † −µ† )
1
(ν † −µ† )
i
(−ν+µ)
−i
(−ν+µ)
i
(ν † −µ† )
−i
(ν † −µ† )
Where:






√ K+ Q
ν=
2D
And:
√ K− Q
µ=
2D
The homogeneous solution is found from the dyadic decomposition of the matrix exponential:
α = Ωαf eλf s sin (Φf s + θαf ) + Ωαs eλs s sin (Φs s + θαs )
(33)
(34)
(35)
(36)
also,
Ωαf =
r
2Re ξ1 η1T
χ
∗j
2
2
+ 2Im ξ1 η1T ∗j χ
where (•)∗j denotes the jth row of the rank one outer product of left and right eigenvectors.
r
2 2
2Re ξ2 η2T ∗j χ + 2Im ξ2 η2T ∗j χ
Ωαs =
θαf = tan
−1
2Re ξ1 η1T
−2Im ξ1 η1T
And the arctangent is a four-quadrant arctangent
θαs = tan
−1
2Re ξ2 η2T
where:
−2Im ξ2 η2T
∗j
∗j
∗j
χ
χ
χ
∗j
χ
(37)
(38)
!
(39)
!
(40)
α = v, w, q, r
(41)
j = 1, 2, 3, 4
(42)
and:
In order to predict the position state at downrange intervals, we require closed form solutions of ψ and
θ. these are provided by substituting Eq. 36 into Eqs. 15 and 14 respectively and integrating. The results
are Eqs. 43 and 44.
ψ
=
+
+
θ
=
+
+
D
ψ0 + (rp s)
V
λf λ s
D −φf λf s
f
e
cos (φf s + θrf ) − cos θrf +
e
sin (φf s + θrf ) − sin θrf Ωrf
V
βf
βf
λ s λs s
D −φs λs s
e cos (φs s + θrs ) − cos θrs +
e sin (φs s + θrs ) − sin θrs Ωrs
V
βs
βs
D
θ0 + (qp s)
V
λf λ s
D −φf λf s
e
cos (φf s + θqf ) − cos θqf +
e f sin (φf s + θqf ) − sin θqf Ωqf
V
βf
βf
λ s λs s
D −φs λs s
Ωqs
e cos (φs s + θqs ) − cos θqs +
e sin (φs s + θqs ) − sin θqs
V
βs
βs
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(43)
(44)
The corresponding swerve equations are
y(s)
y0
v0
=
+ ψ0 +
s
D
D
V0
)
(
ρSD (CX0 − CN A ) Ωvf ∗
exp(λf s) sin(φf s + θvf − π) − sin(θvf − π) − φf cos(θvf − π)s
+
2m
V0
φ2f
ρSD (CX0 − CN A ) Ωvs
∗
+
[exp(λs s) sin(φs s + θvs − π) − sin(θvs − π) − φs cos(θvs − π)s] (45)
2m
V0
φ2s
2 ρSD
z0
w0
m
ρsD
z(s)
CX0 s −
CX0 s − 1
=
+ −θ0 +
s + gD
exp
D
D
V0
ρSDV0 CX0
m
m
)
(
ρSD (CX0 − CN A ) Ωwf +
exp(λ∗f s) sin(φf s + θwf − π) − sin(θwf − π) − φf cos(θwf − π)s
2m
V0
φ2f
ρSD (CX0 − CN A ) Ωws
∗
[exp(λ
s)
sin(φ
s
+
θ
−
π)
−
sin(θ
−
π)
−
φ
cos(θ
−
π)s]
(46)
+
s
ws
ws
s
ws
s
2m
V0
φ2s
The roll rate solution is
p(s) = p0 Ψ +
2V0 CLDD
ρSDCX0
exp −
s (Ψ − 1)
DCLP
2m
where
Ψ = exp
The forward velocity solution is
ρSD3 CLP
s
4Ixx
ρSDCX0
s
V (s) = V0 exp −
2m
(47)
(48)
(49)
The linear predictor is used to generate estimates of the projectile state at each yaw card station. Since
certain parameters of the closed–form linear prediction are dependent upon p and V , the user may choose to
update p and V as often as every caliber of downrange travel. In other words, in the limit all 24 sets of data
could be estimated without repeated iteration of equations 36, 43, 44, 45, 46, 47, and 49, however accuracy
of the linear prediction can be enhanced by iterating the equations, updating all parameters dependent
on p and V at each iteration. Such improvement in accuracy comes at a cost of increased computational
effort directly proportional to the number of iterations. Aerodynamic parameters are held constant during
a trajectory prediction for this application.
IV.
Gradient Based Parameter Identification
The gradient based method assumes that yaw card data is available at 24 stations over a short downrange
interval. We seek to identify the three initial body angular rates (p0 , q0 and r0 ) and six major aerodynamic
parameters (including SLcp ) for a brief interval of flight where mach number varies only slightly. The initial
and final total velocities are assumed to be known from chronometer measurements. Launch position is
defined to be the origin, and launch Euler angles are assumed to be zero. A maximum of four measurements
are made from each yaw card—altitude, cross–range, yaw, and pitch—for a total of 96 measurements.
Thus, we may apply a Newton type search where the normal equation is over–determined (nine unknowns
and 96 measurements). The ‘extra’ measurements should insure that the Newton step is well conditioned
for each iteration. The Jacobian matrix takes the form:

 ∂y
∂y1
1
1
· · · ∂p
· · · ∂y
∂SLcp
∂r0
0

..
..
.. 


.
.
. 

J =  ∂z1

∂z1
1 
 ∂SL
· · · ∂p
· · · ∂z
∂r0 
cp
0

..
..
..
.
.
.
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−3
2
x 10
1
0.8
LM prediction
−2
Altitude (ft)
Crossrange (ft)
0
GA prediction
Range Data
−4
0.6
0.4
−6
0.2
−8
−10
0
100
200
300
Downrange (ft)
400
0
0
500
100
(a) Crossrange
400
500
400
500
(b) Altitude
−3
1
200
300
Downrange (ft)
−3
x 10
2
x 10
1
0
Pitch (rad)
Yaw (rad)
0.5
0
−1
−2
−3
−0.5
−4
−1
0
100
200
300
Downrange (ft)
400
−5
0
500
100
(c) Yaw
200
300
Downrange (ft)
(d) Pitch
Figure 2. Army–Navy Finner Trajectory Matching, no Cant, initial condition 1
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Since we have a closed-form, albeit complicated solution in hand for the yaw and swerve predictions, the
96×9 Jacobian matrix may be formed one column at a time by finite differencing the 96 measurements with
respect to one unknown at a time. That is, ten predictions of the state history will be required to fill the
Jacobian matrix—one nominal and nine perturbed.
The difference between actual and predicted state history for the nominal case is stored as the current
iteration residual vector T .
0
1
LM prediction
GA prediction
Range Data
0.8
Altitude (ft)
Crossrange (ft)
−0.02
−0.04
−0.06
−0.08
−0.1
0
0.6
0.4
0.2
100
200
300
Downrange (ft)
400
0
0
500
100
(a) Crossrange
200
300
Downrange (ft)
400
500
400
500
(b) Altitude
−3
0.01
0
−1
Pitch (rad)
Yaw (rad)
0.005
0
−0.005
−0.01
0
x 10
−2
−3
−4
100
200
300
Downrange (ft)
400
−5
0
500
100
(c) Yaw
200
300
Downrange (ft)
(d) Pitch
Figure 3. Army–Navy Finner Trajectory Matching, no Cant, initial condition 2
Given the rectangular J matrix, and residual vector T with equal row dimension, a second order Newton
method is approximated by the Levenberg Marquart algorithm. The algorithm proceeds as follows.
1. Set Marquart parameters µ = 0.01 and β = 10
2. Form the Jacobian matrix and residual vector from data and predictions
3. Make a trial correction
ϑi+1 = ϑi − J T J + µI
4. Find the new T vector based on ϑi+1
−1
JT T
5. if the error (kT k2 ) is reduced, set µ = µ/β and repeat steps 2 and following
if not, set µ = µβ and repeat steps 3 and following.
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When µ is small, the algorithm approximates a second–order Newton step. When µ gets significantly
large, the iteration uses a small steepest–descent step. The author suggests that for this application µ is
limited such that when µ ≥ 10, the new parameters are adopted whether or not the error is reduced.
In addition to handling many measurements with a small number of adjustable parameters, the Levenberg
Marquardt algorithm can handle multiple data sets that have a few common parameters. This way multiple
trajectories for a common projectile can be used such that aerodynamic coefficients are shared (assuming
equal launch velocity) and initial angular rates are varied between the trajectories. In this case, a composite
Jacobian matrix can be formed from independently determined Jacobian matrices J 1 and J 2 . The combined
Jacobian matrix takes the form
#
"
1
1
0
J:,7:9
J:,1:6
˜
(50)
J=
2
2
J:,1:6
0
J:,7:9
Where the subscript (:, a : b) indicates all rows, columns a through b. The combined residual is merely a
iT
h
. For the Army–Navy finner, two
vertical concatenation of the two residual vectors ⇒ T̃ = T1T T2T
0
1
LM prediction
GA prediction
Range Data
0.8
Altitude (ft)
Crossrange (ft)
−0.02
−0.04
−0.06
−0.08
−0.1
0
0.6
0.4
0.2
100
200
300
Downrange (ft)
400
0
0
500
100
(a) Crossrange
200
300
Downrange (ft)
400
500
400
500
(b) Altitude
−3
0.01
0
x 10
−1
0.005
Pitch (rad)
Yaw (rad)
−2
0
−3
−4
−0.005
−5
−0.01
0
100
200
300
Downrange (ft)
400
−6
0
500
100
(c) Yaw
200
300
Downrange (ft)
(d) Pitch
Figure 4. Army–Navy Finner Trajectory Matching, with Cant, initial condition 1
configurations are tested—one with fin cant and one without. This is assumed to vary only CLDD while all
other aero coefficients are held constant. If two initial conditions are tested, then we have a total of four test
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American Institute of Aeronautics and Astronautics
trajectories. A combined Jacobian matrix can
 1
J:,1:4
 2
 J
J˜ =  :,1:4
3
 J:,1:4
4
J:,1:4
be formed to handle all four data sets simultaneously

1
1
0
0
J:,7:9
J:,5:6

2
2
J:,5:6
0
J:,7:9
0 

3
3 
0
J:,6:9
0
J:,5
0
4
J:,6
4
J:,7:9
(51)
4
J:,5
The corresponding residual is a vertical concatenation of all four corresponding residual vectors.
V.
V.A.
Results
Hydra Identification
The Hydra–70 is a fin stabilized projectile approximately five feet long and 2.75 inches in diameter. The
gradient based algorithm was applied to two trajectories where initial angular rates were varied while launch
velocity, position and aerodynamic constants were held constant. The results are shown in Table 1.
Table 1. Hydra Aero Identification Results
Parameter
SLcp
CN A
CM Q
CX0
CLDD
CLP
p0 (1)
q0 (1)
r0 (1)
p0 (2)
q0 (2)
r0 (2)
V.B.
Actual
1.42
7.85
-1935.0
0.885
-0.0475
-15.0
-50.89
-0.020
0.012
-55.0
-0.0286
-0.000784
Range Estimate Value
1.61
4.69
-1838.7
0.877
-0.0273
-5.73
-38.96
-0.021
0.011
-50.9
-0.0285
-0.000772
Percent Difference
13.6
40.3
4.98
0.929
42.5
61.8
23.4
4.50
11.5
7.45
0.300
1.57
Army–Navy Finner
The Army–Navy Finner is a 1.18 inch diameter fin–stabilized projectile. Two configurations were tested
over two sets of initial angular rates for a total of four data sets. In order to test the robustness of the
algorithm, the aero coefficient initial guesses were set to typical parameters of the Hydra 70, except SLcp
which was reduced greatly to insure the initial guess represented a configuration with weather–vane stability.
After eleven Marquardt iterations, the 2–norm of the length 384 residual vector is 0.0057. The quality of
trajectory matching is depicted in Figures 2 through 5 where the yaw card data is depicted as solid lines
and the prediction based on identified parameters is shown as circles. The identification results are shown
in Table 2.
An evolutionary algorithm with a population of 40 individuals using 8 bit encoding was iterated for 50
generations using kT̃ k2 as the cost function. The best trajectory prediction from this GA optimization is
shown as ‘x’s in Figures 2 through 5. Note that the gradient based algorithm requires only 440 function
evaluations for its eleven iterations while the GA based algorithm uses 2000 to reduce kT̃ k2 to 0.0422. The
gradient based result matches the yaw card data much better, however the GA method gives a more accurate
coefficient estimate for all but three parameters. Apparently the gradient based method is getting caught in
a local minima. Both methods appear to suffer from a poorly designed cost function. For the gradient based
algorithm, convergence does not guarantee good aero parameter estimates. The evolutionary algorithm is
halted with a best residual larger than that of the gradient based algorithm yet with better parameters
estimates. The cost function design can be improved by penalizing errors further downrange more severely,
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American Institute of Aeronautics and Astronautics
and penalizing altitude errors less severely than cross–range, yaw, and pitch errors. These improvements are
under investigation as of this writing.
Table 2. Army-Navy Aero Identification Results
Parameter
SLcp
CN A
CM Q
CX0
CLDD1
CLP
p0 (1)
q0 (1)
r0 (1)
p0 (2)
q0 (2)
r0 (2)
CLDD2
Actual
0.295
14
-590
0.65
0.0
-8.75
-9.9804
-0.0286
-0.784
-4.9804
-0.286
-0.0784
-0.2
LM Estimate
0.257
10.99
-861.2
0.6625
-0.0028
-6.72
0.6069
-0.0399
-1.022
-17.67
-0.375
-0.1035
-0.141
Percent Difference
12.9
21.5
46.0
1.92
N/A
23.1
93.9
39.4
30.2
255
30.9
31.95
29.7
GA Estimate
0.3037
15.113
-695.3
0.623
-0.117
-9.718
-9.510
-0.0318
-0.750
-9.784
-0.300
0.0094
-0.0897
Percent Difference
2.94
8.0
17.9
4.2
N/A
11.1
4.72
11.1
4.36
96.5
4.90
112
55.2
For this projectile, many aerodynamic parameter predictions have a large error. Some of these errors can
be explained by coupling between parameters. For instance CM Q , SLcp and CN A are strongly coupled. Also,
the initial roll rate has little influence over the trajectory prediction since fin cant plays a much stronger
factor in roll rate further downrange. Although the percent error is high, it is clear that the algorithm
distinguishes well between the fin cant and no cant configurations.
VI.
Conclusions
We have demonstrated the use of a modern gradient–based optimization method to extract aerodynamic
coefficients from free flight yaw card data. The method ensures accurate matching between experimental and
theoretical trajectories, however estimates of the aerodynamic parameters are inaccurate due to an inability
to de–couple strongly linked parameters. The gradient based method finds parameters that match the yaw
card data while expending minimal computational effort, however the evolutionary algorithm provides more
accurate parameter estimates in exchange of greater computational effort.
Both methods could be improved by re–designing the cost function. Since the linear theory provides
a closed-form solution, future efforts may investigate the use of analytic derivatives in the gradient based
search. Eventually the method should also be tested on smaller spin–stabilized projectiles as well.
References
1 Mann,
F. W., The Bullet’s Flight from Powder to Target, Munn & Company, New York, 1909.
R. L., Modern Exterior Ballistics, Schiffer Military History, Atglen, PA, 1999.
3 Murphy, C. H., “Data Reduction for the Free Flight Spark Ranges,” Ballistid Research Laboratories Report No. 900,
1954.
4 Chapman, G. T., and Kirk, D. B., “A Method for Extracting Aerodynamic Coefficients from Free–Flight Data,” AIAA
Journal, Vol. 8, No. 4, pp. 753-758, April 1970.
5 Burchett, B. T., Robust Lateral Pulse Jet Control of an Atmospheric Rocket, Ph.D. Thesis, Oregon State University,
2001.
6 Burchett, B. T., and Costello, M., “Model Predictive Lateral Pulse Jet Control of an Atmospheric Rocket,” Journal of
Guidance, Control, and Dynamics, Vol. 25, No. 5, pp. 860-867, September-October 2002.
7 Burchett, B., Peterson, A., and Costello, M., “Prediction of Swerving Motion of a Dual-Spin Projectile with Lateral Pulse
Jets in Atmospheric Flight,” Mathematical and Computer Modelling, Vol. 35, pp. 821-834, 2002.
2 McCoy,
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−3
2
x 10
1
0
0.8
−2
GA prediction
Altitude (ft)
Crossrange (ft)
LM prediction
Range Data
−4
0.6
0.4
−6
0.2
−8
−10
0
100
200
300
Downrange (ft)
400
0
0
500
100
(a) Crossrange
400
500
400
500
(b) Altitude
−3
1
200
300
Downrange (ft)
−3
x 10
2
x 10
1
0
Pitch (rad)
Yaw (rad)
0.5
0
−1
−2
−3
−0.5
−4
−1
0
100
200
300
Downrange (ft)
400
−5
0
500
100
(c) Yaw
200
300
Downrange (ft)
(d) Pitch
Figure 5. Army–Navy Finner Trajectory Matching, with Cant, initial condition 2
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