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Flight Stability of an Asymmetric Projectilewith Activating Canards
Gene Cooper and Frank Fresconi
U.S. Army Research Laboratory, Aberdeen Proving Ground, Maryland 2101
and
Mark Costello
Georgia Institute of Technology, Atlanta, Georgia 30332
DOI: 10.2514/1.A32022
Driven by the creation of new smart projectile concepts with maneuver capability, projectile configurations with
large aerodynamic asymmetries are becoming more common. Standard linear stability theory for projectiles
assumes the projectile is symmetric, both from aerodynamic and mass properties perspectives. The work reported
here extends standard projectile linear theory to account for aerodynamic asymmetries caused by actuating canards.
Differences between standard linear and extended linear theories reported here are highlighted. To validate the
theory, time simulation of the extended linear theory and a fully nonlinear trajectory simulation are made for a
representative scenario, with excellent agreement noted. The extended linearprojectile theory offers a tool to
address flight stability of projectiles with aerodynamic configuration asymmetries.
Nomenclature
Ci = projectile aerodynamic coefficientsCic = projectile aerodynamic coefficients due to a
canardD = projectile characteristic length (diameter), ftf, , = Euler roll, pitch, and yaw angles of projectileg = gravitational constant, ft s2In, Jn,Kn = unit vectors for a coordinate system designated
by nIX, IT = mass moments of inertia, slug ft2m = projectile mass, slugp, q, r = angular velocity components vector of projectile,
s1
r = atmosphere density, slug ft3S, S = complex and conjugate projectile yaw ratesSLCg = stationline center of gravity, ftSLCOP = stationline center of pressure, ftSLCOPC = stationline center of pressure of canard, ftSLMAG = stationline center of Magnus, ftu, v, w = mass center velocity components in the body
reference frame, ft s1jV A=Ij V = projectile centerofmass velocity, ft s1xc, yc = displacement of canard center of pressure relative
to mass center, ft = canard deflection angle with respect to projectile
axis
Introduction
T HERE is no doubt that there are many ways to create controllable forces and moments on a projectile to enable sufficientlylarge changes to its trajectory for flight control purposes. A traditional and powerful way to control a projectile is through aerodynamic force and moment changes. Examples of the type ofmechanism include canards [1]. Since the flight control systemusually commands an inertial position change, the control mechanism fixed on the body will oscillate at the roll rate of the projectile.Many aerodynamic control mechanisms (like canards) introduce anasymmetry into the basic configuration.
Projectile linear theory has long been an analytical workhorse inthe ballistics community. Over time, projectile linear theory has beenused for stability analysis, aerodynamic coefficient estimation usingrange data, and fast trajectory prediction. Basic projectile lineartheory has been extended by various authors to handle moresophisticated aerodynamic models [2], asymmetric configurationsincludingmass properties [3,4],fluid payloads [5,6], moving internalparts [79], dual spin projectiles [10,11], ascending and descendingflight [12], and lateral force impulses [1316]. The work reportedhere is along these lines and develops an extended linear theoryapplicable to aerodynamically asymmetric projectile configurations.Moreover, the paper focuses on projectiles with dithering canardssimilar to that shown in Fig. 1. The extended linear theory is validatedagainst nonlinear sixdegreeoffreedom (6DOF) calculations for anexample projectile.
The paper begins with a description of the basic projectilemathematical model followed by the development of the extendedlinear theory. The theory is then applied to an example configuration.
Trajectory Equations of Motion
To facilitate the mathematical description of the dynamic model,we define the vector component operator Cn and the vectorproductoperator S as a skew symmetric matrix applied to any vector A as
A InAx JnAy KnAz CnA
8>>>:Ax
Ay
Az
9>>=>>;
SA 0 Az AyAz 0 AxAy Ax 0
264
375 (1)
Presented as Paper 20107636 at the AIAA Guidance, Navigation, andControl, Atmospheric Flight Mechanics, Modeling and SimulationTechnologies, AIAA/AAS Astrodynamics Specialist , and Atmosphericand Space Environments Conferences, Toronto, 25 August 2010; received20December 2010; revision received 26 April 2011; accepted for publication19May 2011. This material is declared a work of the U.S. Government and isnot subject to copyright protection in the United States. Copies of this papermay be made for personal or internal use, on condition that the copier pay the$10.00 percopy fee to the Copyright Clearance Center, Inc., 222 RosewoodDrive, Danvers, MA 01923; include the code 00224650/12 and $10.00 incorrespondence with the CCC.
Research Physicist. Senior Member AIAA.Mechanical Engineer. Member AIAA.Sikorsky Associate Professor, School of Aerospace Engineering.
Associate Fellow AIAA.
JOURNAL OF SPACECRAFT AND ROCKETSVol. 49, No. 1, JanuaryFebruary 2012
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http://dx.doi.org/10.2514/1.A32022

These definitions provide a succinct method of writing the mathematics in this paper. Also, the standard shorthand for sine and cosineare employed: sin s and cos c.
Placing two canards, diametrically positioned on a symmetricprojectile, results in additional aeroloads that must be accounted forin the equations of motion. For this study, two canards are located atprojectile roll angles 0 and ; see Fig. 1. This angularseparation > 2=3 introduces rotational asymmetry [3];therefore, it is preferable to write the dynamic equations in thebody reference frame. In fact, the analysis is more tractable if wework in the body frame instead of the wellknown nonrolling frame,particularly if the asymmetry is fluctuating. The angular velocity ofthe body frame is
! B=I pIB qJB rKB (2)
The equations ofmotion relative to the inertial and body frames are
CI _VA=I cc ssc cs csc ss cs sss cc css sc s sc cc
24
35CB _VA=I
(3)
8=>>; T1
8>>>:
IB
JB
KB
9>>=>>;
c 0 s0 1 0
s 0 c
264
3758>>>:
IB
JB
KB
9>>=>>;; 08>>>:Ic2
Jc2
Kc2
9>>=>>; T2
8>>>:
IB
JB
KB
9>>=>>;
c 0 s0 1 0
s 0 c
264
375
1 0 0
0 1 00 0 1
264
3758>>>:
IB
JB
KB
9>>=>>;; (17)
Therefore, the relative velocities of the lifting surfaces written inlocal canard coordinates have the form
CC1VC1=I T1CBVC1=I; CC2VC2=I T2CBVC2=I(18)
Construct the angle of attack for each canard j as
tan1 KC1 CC1VC1=I
k1 IC1CC1VC1=Ik
tan2 KC2 CC2VC2=I
k1 IC2CC2VC2=Ik(19)
The axial Xj and normal Nj aerodynamic forces acting on thelifting surfaces of the canards are modeled as
CCjFAj
8>>>>>>>:
_v
_w
_q
_r
9>>>>>=>>>>>;
AVD
p 0 Vp VV2A
DVV3 1 0
BVD2
M2CVD2
M3EVD
FVpDD
CVD2
BVD2
FVpDD
EVD
2666664
3777775
8>>>>>>>>>:
v
w
q
r
9>>>>>=>>>>>;
8>>>>>>>>>:
sinpt gD
cospt gD D2V2Cnac
4m
xcD2V2Cnac4IT
yc2pD2VCnac4IT
9>>>>>=>>>>>;
(22)
The full expressions for the matrix are lengthy, and they areprovided in the Appendix, which also shows the contributions due toasymmetry are (V2,V3,M2,M3). When these contributions are equalto zero, no asymmetry exists, and the system in Eq. (22) returns to thefamiliar linear theory describing symmetric projectiles written in thebody frame. The eigenvalues of the system above split into twoconjugate pairs: a fast mode and a slow mode denoted as SF and SS(SF;S F;S i _F;S), respectively. Dynamic stability of Eq. (22) isdetermined by the real parts of these eigenvalues. The linearcombination of damped and undamped sinusoid terms forming thesolution of Eq. (22) are obtained using the Laplace transformfollowed with employing the Mellin inverse formula [18]. Theparticular canard timedependent deflection angle used to generatein this report has the form
sinkt; 10 (23)
Model Validation
To establish the utility of the extended projectile linear theorydeveloped above, time domain solutions from the linear model[Eq. (16)] are compared with time domain calculations of thenonlinear model [Eqs. (47)]. The particular example given here hasmass properties m 1:2680 slug, moments of inertia IX 2:449 102 slug ft2 and IT 0:51569 slug ft2, and air density 2:38 103 slug ft3. The spin rate p 75:4 s1, and the forwardvelocity V 558:0 ft s1. This projectiles geometry characteristics areD 0:343521 ft, SLCOP 0:7284 ft, SLMAG 0:2390 ft,SLCG 1:17787 ft, xc 1:2434 ft, and yc 0:134 ft. Theprojectile aerocoefficients are CNA 5:8150, CNAC 0:4875,CYPA 28, and CMQ 151:91, and all reference areas arebased on diameter D.
Comparisons of linear theory to nonlinear 6DOF calculations,along with data from flight experiments, are shown in Figs. 36.Figure 3 compares linear theory to 6DOF canard moment ratiosMCY=MCYMAX as functions of time for 5:0 s< t < 8:0 s. The scale
Fig. 3 Comparison ofMCY=MCYMAX vs time.
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factor MCYMAX is the maximum value of MCY of the linear theorycalculations of the duration of the flight 0 s< t < 30 s, for which thecanard activation frequency is chosen as k p to effectivelymaximize projectile range (glide).
A similar chart showing the canard force ratio FCZ=FCZMAX isgiven in Fig. 4, and the scale factor FCZMAX is the maximum value ofthe linear FCZ over the 30 s flight.
The variables in Figs. 3 and 4 are chosen for comparison since thenonlinear effects due to the 6DOF are most dissimilar for the canardforce andmoment. A chart comparing linear and 6DOF calculationsof the similarly scaled q=q
MAXratios is shown in Fig. 5
One way to view the angular motion of this asymmetric projectileis to consider the time dependence of , , defined as follows [3]:
8>>>:tan1
cosptvsinptw
V
tan1
cosptwsinptvV
9>>=>>; (24)
Figure 6 has a chart for flight times t 30 s comparing lineartheory and 6DOF calculations along with flighttest data of theprojectile modeled in the above calculations [19].
Evidently, linear theory capturesmost of the physics governing theangular motion of this projectile when subjected to two activatingcanards configured at 0 and . The magnitude of transitoryconing behavior for t 30 s is caused by the canard moments MCand the body moment B. Essentially, the canards induce the angle ofattack in the body. At the angle of attack, the Magnus moment of thebody contributes to the transient coning along with the canardactuation. In the plot of Fig. 6, theMagnusmoment appears as nearlycircular orbits that complete a full revolution at the projectile yaw
rate. The perturbations in the orbits of Fig. 6 are due to the fastercanard actuation frequency. The canard actuationinduced angle ofattack is near 3, as seen in Fig. 6. The experimental and lineartheorybased transient coning magnitude is around 3, and theangular motion perturbations due to canard actuation at the roll rateare evident in Fig. 6.
The steadystate coning motion corresponding to the particularsolution of Eq. (22) is attained for flight times 200 s due to the sizeof jF;Sj. This steadystate motion is an offset ellipse centered at the
Fig. 4 Comparison of FCZ=FCZMAX vs time.
Fig. 5 Comparison of q=qMAX
vs time.
Fig. 6 Comparisons of transient coning motions for time 30 s.
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geometric center of Fig. 6 with vanishing small radii (analysis notpresented here since the expressions are judged to be too long).
Stability Analysis
Numerical values of fast and slow eigenvalues SF;S are presentedas functions of projectile spin rate p, given in Figs. 7 and 8. The slowmode SS is real and positive for 10:2 s
1 p 13:3 s1, whichindicates dynamic instability; see Fig. 5 of [3]. As p increases,p 79:0 s1, causes the fast mode contribution to grow unstable.
Another way to present the numerical values of fast and sloweigenvalues SF;S is to form a root locus plot for changing projectilespin rate p, shown in Fig. 9.
These eigenvalues provide a tool for addressing dynamic stabilityof the asymmetric canard configuration modeled in this effort.
Sinusoidal driving terms representing canard actuation [Eq. (23)]cause component solutions of Eq. (23) to have terms with form
Gt _F;S k2 2F;S
Gt represents a timedependent numerator (25)
Thus, in the event jF;Sj 1, k! j _F;Sjmaypotentially generateflight instability when the canards are actuated as sinusoids. Ametricto demonstrate this potential problem is defined in Eq. (26):
Max2 2
p8 t 2 0; 30 s (26)
Figure 10 shows a plot of this behavior where the spin rate is heldconstant p 75:4 s1 for varying values of k.
Apparently, when the frequency k is in the neighborhood of the
fast mode, the frequency _F causes the strongest destabilizing effect.
Fig. 7 Fast and slow real roots FAST;SLOW as functions of spin rate p.
0
20
40
60
80
100
120
0 20 40 60 80 100 120
Spin Rate p (s1 )
Imag SF,S vs Spin Rate p
P
.
FAST
.
SLOW
. (s
1 )
Fig. 8 Fast and slow real roots FAST;SLOW as functions of spin rate p.
Fig. 9 Fast and slow roots S parameterized by projectile spin rate p.
0
20
40
60
80
100
120
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0
Deg
rees
( 2 + 2)1/2 at time t=30s, p = 75.4 s1
Slow Mode
Fast Mode
Fk
Fig. 10 Metric resonance vs k= _F.
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Conclusions
An extension of the familiar linearprojectile theory for symmetricand asymmetric projectiles has been formulated and solved. Thisextension accounts for configuration asymmetries caused by twoactuating canards placed at roll angles 0 and . Thedocumentation shows that a wide class of configuration asymmetriescan be addressed by the techniques given in this report. One of theresults obtained in this paper presents a way to analyze the dynamicstability of projectiles exhibiting asymmetries. This can have animportant impact on the design of projectiles. The importance ofcanard actuation focuses on a potential problem causing flightinstability when the actuation frequencies are too close to one of themodal yawing frequencies. A spinning projectile with two canardsactivated as sinusoids was shown to contribute to transient coningmotion when this additional moment was sufficiently large. Theclosedform solutions discussed in this report were shown tocompare well with standard 6DOF calculations. Postprocessingsolutions given here to obtain all state components of asymmetricprojectiles may prove useful in developing firmware. The asymmetric linear theory reduces to the familiar linear symmetric theorywhen the asymmetries vanish.
Appendix
A D3CNA
8 mB pD
5CYPASLMAG SLCG16ITV
C D4CNASLCOP SLCG
8ITE D
5CMQ16IT
F pDIxITV
V2 D3CNAC
4mV3
D2xcCNAC4m
M2 D4xcCNAC
4ITM3
D3xc2CNAC4IT
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