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 linear-projectile 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 = station-line center of gravity, ftSLCOP = station-line center of pressure, ftSLCOPC = station-line center of pressure of canard, ftSLMAG = station-line center of Magnus, ftu, v, w = mass center velocity components in the body

reference frame, ft s1jV A=Ij V = projectile center-of-mass 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 control-lable forces and moments on a projectile to enable sufficientlylarge changes to its trajectory for flight control purposes. A tradi-tional and powerful way to control a projectile is through aero-dynamic force and moment changes. Examples of the type ofmechanism include canards [1]. Since the flight control systemusually commands an inertial position change, the control mecha-nism 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 six-degree-of-freedom (6-DOF) 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 vector-productoperator 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 2010-7636 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 per-copy fee to the Copyright Clearance Center, Inc., 222 RosewoodDrive, Danvers, MA 01923; include the code 0022-4650/12 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 mathe-matics 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 well-known 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 time-dependent 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 character-istics 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 6-DOF calculations,along with data from flight experiments, are shown in Figs. 36.Figure 3 compares linear theory to 6-DOF 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 6-DOF are most dissimilar for the canardforce andmoment. A chart comparing linear and 6-DOF 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 6-DOF calculations along with flight-test 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 actuation-induced angle ofattack is near 3, as seen in Fig. 6. The experimental and lineartheory-based transient coning magnitude is around 3, and theangular motion perturbations due to canard actuation at the roll rateare evident in Fig. 6.

The steady-state coning motion corresponding to the particularsolution of Eq. (22) is attained for flight times 200 s due to the sizeof jF;Sj. This steady-state 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 time-dependent 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 (s-1 )

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 s-1

Slow Mode

Fast Mode

Fk

Fig. 10 Metric resonance vs k= _F.

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Conclusions

An extension of the familiar linear-projectile 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. Theclosed-form solutions discussed in this report were shown tocompare well with standard 6-DOF calculations. Postprocessingsolutions given here to obtain all state components of asymmetricprojectiles may prove useful in developing firmware. The asym-metric 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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M. MillerAssociate Editor

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