Digital Commons @ George Fox UniversityFaculty Publications - Department of Mechanicaland Civil Engineering Department of Mechanical and Civil Engineering

2008

Predictive Control of a Munition Using Low-SpeedLinear TheoryNathan SlegersGeorge Fox University, nslegers@georgefox.edu

Follow this and additional works at: http://digitalcommons.georgefox.edu/mece_fac

Part of the Applied Mathematics Commons, and the Mechanical Engineering Commons

This Article is brought to you for free and open access by the Department of Mechanical and Civil Engineering at Digital Commons @ George FoxUniversity. It has been accepted for inclusion in Faculty Publications - Department of Mechanical and Civil Engineering by an authorized administratorof Digital Commons @ George Fox University. For more information, please contact arolfe@georgefox.edu.

Recommended CitationSlegers, Nathan, "Predictive Control of a Munition Using Low-Speed Linear Theory" (2008). Faculty Publications - Department ofMechanical and Civil Engineering. Paper 24.http://digitalcommons.georgefox.edu/mece_fac/24

Engineering Notes _________ _ ENGINEERING NOTES are short manuscripts describing new developments or important results of a preliminary nature. These Notes should not exceed 2500 words (where a figure or table counts as 200 words). Following informal review by the Editors, they may be published within a few months of the date of receipt. Style requirements are the same as fo r regular contributions (see inside back cover).

Predictive Control of a Munition Using Low-Speed Linear Theory

Nathan Slegers* University of Alabama in Huntsville,

Huntsville, Alabama 35899

A,B, C, E, F , G, H,K CDD

e LL• cMM• c NN

C LP, C MQ• C NR

cmax CNA Cxa cx2 D e emax g I I xx. I yy, Izz Ix y, I yz, Ixz K

'snax... L,M,N LA,MA,NA

f:.c,Mr;,Nc LuA• MuA• NuA

m p,q,r

R $AX

s u, v, w - - -Uw , Vw, Ww

=

=

Nomenclature longitudinal and lateral equation coefficients

fin cant rolling aerodynamic coefficient roll, pitch, and yaw control aerodynamic coefficients roll, pitch, and yaw damping aerodynamic coefficients pitch and yaw control limit normal force aerodynamic coefficient zero yaw drag aerodynamic coefficient yaw squared drag aerodynamic coefficient reference diameter impact-error vector impact-error stopping criterion gravitational constant munition inertia matrix moments of inertia products of inertia predictive control gain matrix control iteration limit external moments in the nonrolling frame aerodynamic moments in the nonrolling frame control moments in the nonrolling frame unsteady aerodynamic moments in the nonrolling frame mass angular velocity components in the nonrolling frame distance from the mass center to the aerodynamic center along the station line. nondimensional arc length velocity components in the nonrolling frame wind velocity components in the nonrolling frame

*Assistant Professor, Department of Mechanical and Aerospace Engineering. Member AIAA.

v vn vz Wo, iio. Bo

x,y,z ~. :i\ z -~A> Y_~,. , Z.J. Xw,Yw, Zw Xf, Yh Z[

XT, J T ,ZT a,j3 A, CTj, WI

p CTz , Wz

rp, e,

total velocity longitudinal velocity components vertical velocity, pitch rate, and pitch in nonrolling frame at the beginning of an update interval inertial positions of the mass center external forces in the nonrolling frame aerodynamic forces in the nonrolling frame weight in the nonrolling frame predicted-impact position components target position components aerodynamic angle of attack and side slip longitudinal equation's pole, real, and imaginary components atmospheric density lateral equation's real and imaginary components Euler roll, pitch, and yaw angles

I. Introduction

E XISTING tactical unmanned aerial systems (UAS) provide improved capability in performing a diverse set of military

missions such as reconnaissance and targeting. The potential of tactical UAS is impressive, yet use on the battlefield is limited because existing UAS have no capability to engage a time-critical identified target. One solution is to add small munitions that can be released by the UAS. Adding weaponized payloads to existing UAS is limited by many practical issues. The small payload capabilities of existing systems require the munitions to be small, lightweight, and easily integrated into existing platforms. To be effectively integrated into a UAS mission, the munition must be allowed to be released from a general area. A practical system would receive only target information, initial states, and a release command. The munition would then be required to compensate for errors in release position, velocity, and orientation during its trajectory. Possible solutions include existing 40- and 60-mm munitions with modified tail sections for guidance, stability, and control. A small blast radius requires the small munitions to have guidance to reduce dispersion induced from release.

Guidance techniques for munitions are well established, with details on many standard control techniques provided by Zarchan [1] . Alternative control techniques have also been investigated, such as pulse jets on a spinning projectile using proportional navigation [2] . Later, Jitpraphai and Costello [3] also investigated pulse jets for trajectory tracking of direct-fire rockets. More recently, dynamic inversion with neural networks and control Lyapunov functions have been used for guided munitions [4,5].

The work reported here develops a predictive guidance strategy to reduce dispersion of a small statically stable munition using tail surfaces for control. Predictive control uses a dynamic model to predict the future state, which is then used to determine suitable control. Model predictive control has been successfully applied over a range of systems such as guided parafoils and unmanned systems [6,7]. Model predictive control has also been successfully applied to projectiles by Burchett and Costello [8]. They investigated pulse jets located near the nose on a slow-spinning fin-stabilized rocket with pulse-jet firing logic based on a projectile linear model. Ollerenshaw and Costello [9) later applied projectile linear theory to a canard-

controlled projectile and formed an optimal predictive controlsolution. In both cases, the authors employed a set of assumptionssuch as small aerodynamic angles, large roll rate comparedwith pitchand yaw rates, and near-flat-fire conditions, resulting in a set ofanalytically solvable equations commonly referred to as projectilelinear theory [10]. Conventional projectile linear theory has beenextended to account for various specialized cases such as fluidpayloads [11], lateral impulses [12], and dual-spin projectiles [13].Recently, Hainz and Costello [14] developed a modified lineartheory that accounts for large pitch while retaining the small-aerodynamic-angle assumption of conventional linear theory.

A munition released at low speeds provides unique problems forprojectile linear theory, because pitch ranges from near zero atrelease to near vertical at impact and vertical velocities and pitch ratesmay become significant. Modified linear theory accounts for theformer, but large impact-prediction error occurs due to influence ofvertical velocities and pitch rates during prediction of yaw, pitch, andtotal velocity. This work considers a statically stable low-speedmunition with low spin rate. Modified linear theory is changed toaccount for vertical velocities, pitch rates, and control surfaces toimprove impact prediction. Furthermore, the roll rate is consideredsmall, resulting in separation of the epicyclic equations intolongitudinal and lateral equations. Low-speedmodified linear theoryis used to rapidly compute impact predictions that are fed to theguidance system. The guidance system uses tail control surfaces toreduce the predicted-impact errors. Simulations are used todemonstrate the effective dispersion reduction of the predictivecontrol strategy.

II. Dynamic Model

Amunition can bemodeled as a rigid body possessing six degreesof freedom (6DOF), including three inertial position components ofthe system mass center as well as the three Euler orientation angles,as shown in Fig. 1.Abody reference frame isfixed to the body,whereIB is aligned with the axis of revolution. A nonrolling frame with IN ,also aligned with the axis of revolution, is defined so that it is fixed tothe body but does not roll. The dynamic equations of motion arederived in the nonrolling frame and provided in Eqs. (1–4).8<

:_x_y_z

9=;�

c�c �s s�cc�s c s�s�s� 0 c�

24

358<:

~u~v~w

9=; (1)

8<:

_�_�_

9=;�

1 0 t�0 1 0

0 0 1=c�

24

358<:

~p~q~r

9=; (2)

8<:

_~u_~v_~w

9=;�

1

m

8<:

~X~Y~Z

9=;�

8<:

~r ~v� ~q ~w�t� ~r ~w� ~r ~u~q ~u�t� ~r ~v

9=; (3)

8<:

_~p_~q_~r

9=;� �I��1

8<:

~L~M~N

9=; �

0 � ~r ~q~r 0 t� ~r� ~q �t� ~r 0

24

35�I�

8<:

~p~q~r

9=;

0@

1A (4)

In Eqs. (1–4), the common notations sin��� s�, cos��� s� andtan��� t� are used. Forces appearing in the previous equationshave contributions fromweight and aerodynamic loadswithMagnusforces neglected because of the munition’s slow spin rates. Thecombined forces are provided in Eq. (5).8<

:~X~Y~Z

9=;�

8<:

~XW~YW~ZW

9=;�

8<:

~XA~YA~ZA

9=; (5)

Weight and aerodynamic forces in a nonrolling reference frame aregiven in Eqs. (6) and (7), with aerodynamic angles defined in Eqs. (8)and (9). 8<

:~XW~YW~ZW

9=;�mg

8<:�s�0

c�

9=; (6)

8<:

~XA~YA~ZA

9=;��

�

8�V2D2

8<:Cx0 � Cx2��2 � �2�

CNA�CNA�

9=; (7)

�� tan�1�� ~w � ~ww�=� ~u � ~uw�� (8)

�� tan�1�� ~v � ~vw�=� ~u � ~uw�� (9)

Moments acting on the munition have contributions formaerodynamic forces, unsteady aerodynamic moments, and controlmoments. 8<

:~L~M~N

9=;�

8<:

~LA~MA~NA

9=;�

8<:

~LUA

~MUA

~NUA

9=;�

8<:

~Lc~Mc~Nc

9=; (10)

Moments due to aerodynamic forces and unsteady aerodynamicmoments are8<

:~LA~MA~NA

9=;�

0 0 0

0 0 �R�AX0 R�AX 0

24

358<:

~XA~YA~ZA

9=; (11)

8<:

~LUA~MUA

~NUA

9=;�

�

8�V2D3

8<:CDD � ~pCLP�D=2V�

~qCMQ�D=2V�~rCNR�D=2V�

9=; (12)

Control is achieved through deflection of tail surfaces and ismodeledas the injection of control moment coefficients.8<

:~LC~MV~NC

9=;�

�

8�V2D3

8<:CLLCMMCNN

9=; (13)

III. Trajectory Solution

The previous munition dynamic model is highly nonlinear andmust be numerically integrated to estimate a trajectory. Historically,simplifications have been applied such as assuming symmetry, smallEuler pitch and yaw angles, small aerodynamic angles of attack,velocity along the axis of revolution that is equal to the total velocityV, and a roll rate that is large when compared with pitch and yawrates. These assumptions result in standard projectile linear theory[10], which yields accurate analytic solutions for flat-fire projectileFig. 1 Munition coordinate systems.

trajectories. In the case of large pitch angles, standard linear theoryresults in significant impact-prediction errors. Recently, modifiedlinear theory has been developed by relaxing the small-Euler-pitch-angle assumption [14]. The result is a reasonably accurate analyticsolution that usually requires some updates during the trajectory toaccount for pitch changes. Although modified linear theoryaddresses large pitch angles, low-speed munitions require twoadditional assumptions to be addressed. The vertical velocity andpitch rate may be significant and the roll rate is small. To developsuitable linear theory equations for a statically stable low-speedmunition, the following assumptions are made.

1) A change of variable is made from the body velocity ~u to totalvelocity V.

2) Velocity along the axis of revolution V is large compared withthe side velocity ~v, ~vw, and the roll and yaw rates ~p and ~r. Higher-order terms involving ~v, ~vw, ~p, and ~r are neglected. Nothing isassumed about the vertical velocity ~w, ~ww, and pitch rate ~q.

3) Yaw angle is small, allowing the following small-angleapproximations.

sin� � ; cos� � 1 (14)

4) Aerodynamic angles are small, allowing the following small-angle approximations.

� � ~w � ~ww�=V; � � ~v � ~vw�=V (15)

5) Munition is both geometrically and aerodynamicallysymmetric, resulting in the following simplifications.

IXY � IXZ � IYZ � 0 (16)

IYY � IZZ (17)

CMQ � CNR (18)

6) Total velocity can be represented as a point mass.

_V x ���

1

8m��VD2Cx0

�Vx (19)

_V z ���

1

8m��VD2Cx0

�Vz � g (20)

V �������������������V2x � V2

z

q(21)

7) Independent variable is changed from time t to dimensionlessarc length s, measured in calibers of travel.

s� 1

D

Zt

0

V dt (22)

The common notation of a prime superscript represents aderivative with respect to dimensionless arc length s. The followingrelationship between the arc length and time derivatives exists.

0 � �D=V� _ (23)

Application of the previous seven assumptions yields the followingsimplified equations.

x0 � c�D (24)

y0 � c�D ��D

V

�~v�

�s� D

V

�~w (25)

z0 � �Ds� ��Dc�V

�~w (26)

�0 �DV

~q (27)

0 � D

Vc�~r (28)

~v 0 � ���D3

8mCNA� ~v � ~vw� �

�1� t� ~w

V

�D ~r (29)

~w 0 � ���D3

8mCNA� ~w � ~ww� �D ~q�Dg

Vc� (30)

~q 0 � ��D3R�AX

8IYYCNA� ~w � ~ww� �

��D5

16IYYCMQ ~q�

��D4V

8IYYCMM

(31)

~r0 � ���D3R�AX

8IYYCNA� ~v � ~vw� �

��D5

16IYYCMQ ~r

� ~q0t�D

V~r� ��D

4V

8IYYCNN (32)

V 0x ���

1

8m��D3Cx0

�Vx (33)

V 0z ���

1

8m��D3Cx0

�Vz �

Dg

V(34)

Approximate closed-form solutions can be found by making theadditional assumptions that aerodynamic coefficients are constantand total velocity V is slowly changing, being treated only as adynamic variable in Eqs. (21), (33), and (34). As with conventionaland modified linear theory, these assumptions largely decouple theequations into velocity, roll rate, Euler angle, and swerve equations.A difference occurs in the epicyclic equations in which they can befurther decoupled into longitudinal and lateral equations, with Eulerpitch included in the former.

A. Longitudinal Solution

Expanding the cosine of Euler pitch in Eq. (30) as a first-orderTaylor series and treating the total velocity V as constant, Eqs. (27),(30), and (31) can be written in the form

8<:

~w0

~q0

�0

9=;�

�A D BC=D E 0

0 D=V 0

24

358<:

~w~q�

9=;�

8<:

G� A ~wwKCMM � �C=D� ~ww

0

9=;

(35)

The coefficients appearing in Eq. (35) are

A� ���D3=8m�CNA (36)

B��Dgs�0=V (37)

C� ���D4R�AX=8IYY�CNA (38)

E� ���D5=16IYY�CMQ (39)

G� �Dg=V��c�0 � s�0�0� (40)

K � ���D4V=8IYY� (41)

The longitudinal eigenvalues are a simple pole � and a complex pair�1 � i�1. Laplace transformations are used to solve the coupleddifferential equations in Eq. (35), with the solution shown next.Coefficients in Eqs. (42–44) are easily found by solving algebraicequations; however, they are extensive and not included. Thelongitudinal equations for the low-speed solution are fundamentallydifferent from modified linear theory because of the inclusion of adecaying exponential and only one damped sinusoid.

~w�s� � Cw0 � Cwee��s � e�1s�Cwc cos��1s� � Cws sin��1s��(42)

~q�s� � Cq0 � Cqee��s � e�1s�Cqc cos��1s� � Cqs sin��1s��(43)

��s� � C�0 � C�ee��s � e�1s�C�c cos��1s� � C�s sin��1s�� (44)

B. Lateral Solution

Treating the total velocity V and longitudinal states as constant,Eqs. (29) and (32) can be written in the form

�~v0

~r0

�� �A �H�C=D F

� ��~v

~r

���

A ~vwKCNN � �C=D� ~vw

�(45)

F� E�D ~q0Vt�0 (46)

H �D�1�

t�0 ~w0

V

�(47)

A difference in the low-speed lateral solution and the modified lineartheory lateral solution is the inclusion of the second components ofthe coefficients in Eqs. (46) and (47). The lateral eigenvalues are acomplex pair, �2 � i�2, leading to the approximate solution inEqs. (48) and (49). Similar to the longitudinal solution, thecoefficients in Eqs. (48) and (49) are found by taking the Laplacetransform of Eq. (45) and solving the resulting algebraic equations.

~v�s� � Cv0 � e�1s�Cvc cos��2s� � Cvs sin��2s�� (48)

~r�s� � Cr0 � e�2s�Crc cos��2s� � Crs sin��2s�� (49)

C. Euler Yaw and Swerve Solution

Following the same assumptions as modified linear theory, theremaining yaw and position states are simplified by treating yaw andpitch as constant when they appear with other independent variables.When Euler pitch is the only variable, its integral is approximated bythe trapezoid method. Integration of Eqs. (24–26) and (28) thenyields

�s� � C 0 � C 1s� e�2s�C c cos��2s� � C s sin��2s�� (50)

x�s� � x0 � 12Ds�cos���s�� � cos��0�� (51)

y�s� � Cy0 � Cy1s� Cy2s2 � Cyee��s

� e�1s�Cyc1 cos��1s� � Cys1 sin��1s��� e�2s�Cyc2 cos��2s� � Cys2 sin��2s�� (52)

z�s� � Cz0 � Cz1s� Czee��s � 12Ds�sin���s�� � sin��0��

� e�1s�Czc cos��1s� � Czs sin��1s�� (53)

The inclusion of pitch in the longitudinal equations alters the cross-range and altitude equations, when compared with modified andstandard linear theory, by the addition of a pure exponential decayingterm. Improvements in the lateral solution also directly enter thecross-range equation, as seen in Eq. (25). Equations (42–44) and(48–53) could be used directly for trajectory generation or impactprediction. However, because of the variation of coefficients inEqs. (35) and (45) over the trajectory, errors can become large if theyare treated as constant over an entire solution. Coefficients changeduring the trajectory primarily for two reasons: the total velocityinfluencing coefficientsB,G,F, andH and the changing pitch angleinfluencing coefficientsG,F, andH. In a manner similar to standardand modified linear theory, errors can be minimized by updating thecoefficients periodically over the entire trajectory.

IV. Trajectory Results

The improvements in impact prediction of low-speed linear theoryovermodified linear theory are highlighted using a low-speed releaseat 20 m=s, representing a typical loiter speed of awide range of smallUAS. Trajectories are computed using a 6DOF simulationrepresenting the true trajectory, modified linear theory, and low-speed linear theory. The munition used in all simulations is arepresentative 40-mm-diam grenade with an additional theoreticaltail section for static stability. Aerodynamic coefficients and physicalparameters are listed in Table 1. The tail section includes four controlfins that can be mixed together to produce roll, pitch, and yawmoments during guided trajectories. Both modified and low-speedlinear theory use an update interval of 250 calibers, resulting in a 10-m update interval for the example munition. The 6DOF simulation isintegrated using fourth-order Runge–Kutta with a time step of0.0001 s.

Figures 2–5 show a comparison of a 6DOF simulation, modifiedlinear theory, and the low-speed corrected modified linear theory,with no wind and the following initial conditions: x� 0:0 m;y� 0:0 m; z� 500 m; �, �, and � 0:0 deg; ~u� 50:0 m=s;~v� 0:0 m=s; ~w� 0:0 m=s; ~p� 0:0 rad=s; ~q��0:15 rad=s; and~r��0:15 rad=s. Figure 2 shows the angle of attack being largewith~w: as large as 15% of the total velocity. In this case, the modifiedlinear theory assumption of small ~w is unacceptable and significanterrors can be seen, whereas low-speed linear theory accuratelypredicts the angle of attack. Similar results are seen in the pitch andyaw angles shown in Fig. 3.Modified linear theory errors in ~w lead tolarge errors in predicted pitch and yaw angles.

Table 1 Munition properties

Parameter Value

m, kg 0.278IXX , kg �m2 3:861 10�5

IYY , IZZ, kg �m2 4:685 10�4

D, m 0.04R�AX , m �0:0153Cx0 0.309Cx2 3.96CNA 4.96CDD 0.00CLP �0:081

CMQ, CNR �8:96

Altitude and cross range are shown in Figs. 4 and 5, in which it isclear that modified linear theory fails to predict the impact locationaccurately. Modified linear theory results in impact-prediction errorsof 40 and 1.9 m in range and cross range, compared with 3.4 and0.4 m for low-speed linear theory. Errors in range and cross rangefrom modified linear theory result mainly from large errors in Eulerangles, which are significantly less for low-speed linear theory.Errors in angle predictions are smaller in low-speed linear theory dueto the point-mass-velocity assumption and the addition ofconsidering ~w and ~q as having significant contributions. Low-speedlinear theory also reduces impact-prediction error by including ~wdirectly into the cross-range equation, as seen in Eq. (25).

V. Predictive Control System

The predictive control system uses low-speed modified lineartheory to predict the impact location based on the current munitionstates, constant pitch and yaw moment commands, and targetaltitude. The predicted impact location defined by xI and yI iscompared with the target locations xT and yT to generate errors in therange and cross range at impact. The control system seeks tominimize the penalty function in Eq. (54) using the two controlparameters in Eq. (56), subject to the constraint that the magnitude ofeach element of the control vector u is bounded by Cmax.

J� eTe (54)

e � f �xI � xT� �yI � yT� gT (55)

u � fCMM CNN gT (56)

The penalty function in Eq. (54) is a nonlinear function of all themunition states and the two control elements with no closed-formsolution available. An approximate numerical solution can be foundby using the rapid impact-prediction capabilities of low-speed lineartheory and a gradient descent method [15]. Figure 6 shows a diagramof the complete predictive control system. First, the approximateoptimal control vector u� is initialized to zero. The current munitionstate vector ismeasured and low-speed linear theory is used to predictthe impact error e. A perturbation to the current control is found bymultiplying a 2 2 matrix K by the current impact error. To satisfythe constraint, the perturbed control vector is then saturated withrespect to Cmax. The control vector continues to be updated untileither kek � emax or the maximum iterations kmax occur. Theapproximate optimal control is applied and the process is repeated atthe nextmeasurement update, startingwith the previous approximate

Fig. 2 Angle of attack vs range for low release velocity.

Fig. 3 Munition pitch and yaw angles for low release velocity.

Fig. 4 Altitude vs range for low release velocity.

Fig. 5 Cross range vs range for low release velocity.

optimal solution. For a fin-stabilized munition with zero roll, theimpact range and cross range will increase for positive controlcoefficients CMM and CNN , requiring K to be positive definite. Asimple proportional–derivative roll controller is used to regulate theroll angle after release and is provided in Eq. (57).

CLL � K��� Kp ~p (57)

Typical guided trajectories of the example 40-mm grenade withtail section are shown in Figs. 7–10 for kmax of 1 and 10.When kmax is1, the numerical solution for u� is stopped after just one applicationof the low-speed linear theory predictor, resulting in a rapid solutionat the expense of accuracy. The grenade is released with thefollowing initial conditions: x��180 m; y� 10:0 m; z� 500 m;�, �, , ~v, ~w, and ~p� 0:0 deg; ~u� 20:0; ~q��0:15 rad=s; and~r��0:15 rad=s. The possible singularity when � is ��=2 isavoided by stepping to either side by 0.01 rad if it occurs. The desiredtarget is xT , yT , and zT � 0 m. The control system is updated every0.5 s with the low-speed linear theory impact predictor using anupdate interval of 250 calibers. The perturbation matrix K is chosenas diagonal with 0.00005 elements, the saturation limit Cmax is0.0015, emax is 1.0 m, and the roll gains K� and Kp are �0:125 and�0:01. Figure 7 shows the altitude vs range and Fig. 8 shows thecross range. In both Figs. 7 and 8, the trajectories for kmax of 1 and 10are nearly indiscernible. Figure 9 shows the control history for bothguided trajectories and the number of iterations used in the numericalsolution. Initially, when kmax is large, multiple iterations are used.However, after 1.5 s, control trajectories for kmax of both 1 and 10 arenearly identical. Figure 10 shows a history of the low-speed lineartheory impact prediction, with the square and circle representing theimpact of the unguided and guided grenade, respectively. At release,

when kmaxis 10, multiple iterations of the low-speed impact predictorare use to estimate u� � � 0:0011 �0:00075 �T , which is near thefinal solution at impact. The approximate u� calculated at releaseresults in a predicted impact close to the target. During the next eightmeasurement updates, controls are slightly modified, reaching anear-constant solutionwithin 5 s. At all times, the predicted impact is

Fig. 6 Predictive control guidance strategy.

Fig. 7 Altitude vs range of a typical guided trajectory.

Fig. 8 Cross range vs range of a typical guided trajectory.

Fig. 9 Control coefficient history of a typical guided trajectory.

near the desired impact. For kmax equal to 1, the impact prediction atrelease is near the unguided impact. During the next four controlupdates, the predicted impact continually moves closer to the desiredimpact location. With kmax equal to 1, the predictive control stillsuccessfully placed the grenade on a correct trajectorywithin 5 s. Thepredictive controller reduces impact error from 11.9 m for theunguided grenade to 0.6 and 0.5 m for the guided grenade, with kmax

being 1 and 10, respectively.Monte Carlo simulations were completed for the unguided and

guided example grenade, with kmax being 1 and 10. Dispersionresults are shown in Fig. 11 for 50 trajectories. The release position isnominally x��170 m, y� 0:0 m, and z� 500 m, with eachposition independently varied, having a Gaussian distribution withstandard deviation of 5.0 m. Release speeds ~u and ~v areindependently varied, having means of 20 m=s and 0 m=s andGaussian distributions with standard deviations of 1:0 m=s. Initialangular rates ~q and ~r are also independently varied, having means of0 rad=s and standard deviations of 0:1 rad=s. All other initialconditions are zero.A constant atmosphericwind is included for each

release; however, the control algorithm has no knowledge of thewind and assumes they are zero during impact prediction. The windmagnitude varies with a standard deviation of 3:0 m=s and thedirection is uniformly distributed. Gaussian noise is added to thecontrol system sensors, with a standard deviation of 1.0 m on x, y,and z; a standard deviation of 0.018 rad on �, �, and ; a standarddeviation of 0:15 m=s on ~u, ~v, and ~w; and a standard deviation of0:018 rad=s on ~p, ~q, and ~r. The circular error probable (CEP) isshown in Fig. 11 and is defined as the minimum radius circlecontaining 50% of the hits. The predictive control algorithm shows adrastic improvement in dispersion, reducing the CEP from 14.1 to2.7 and 2.2 m for kmax equal to 1 and 10. It is worth noting that astandard high-explosive 40-mm grenade such as the M460 has aburst radius that causes casualties within 5 m. For the unguidedgrenade, only 10% of impacts are within 5 m, whereas for both kmax

equal to 1 and 10, the predictive guidance increases impacts within5 m to 78%.

VI. Conclusions

Modified linear theory provides reasonable impact predictions athigh speeds. However, for typical small UAS mission speeds, lessthan 20-m/s impact errors were substantial due to large angles ofattack and pitch rates. Low-speed linear theory was developed byincluding higher-order terms involving ~w and ~q that modified lineartheory neglects. As a result, the angle of attack, pitch, and yawpredictions are significantly improved, leading to accurate impactpredictions even at very low speeds.

A predictive control scheme was developed to reduce dispersionusing control surfaces near the tail. The predictive controller useslow-speed linear theory to rapidly predict the impact error using thecurrent state and control. Based on the estimated impact error, thecontrol is iteratively found to minimize the predicted-impact error.For an example munition, it was shown that the maximum number ofiterations during the control solution only impacted the initial controlestimates. Limiting the guidance algorithm to a single iteration hadlittle impact on the final accuracy and permitted a rapid solution. Itwas shown for the example munition that the predictive guidancesignificantly reduced the CEP from 14.1 to 2.7 and 2.2 m when themaximum iterations were 1 and 10. Furthermore, for a typical high-explosive 40-mm grenade, the percentage of impacts within a lethalradius was increased from 10 to 78% when the maximum iterationswere both 1 and 10. In practical applications, errors in the targetlocationmust be includedwhen considering the probability of impactwithin a lethal range of a target.

References

[1] Zarchan, P., Tactical and Strategic Missile Guidance, AIAA, Reston,VA, 1997, pp. 11–28, 95–118, 185–205.

[2] Calise, A., “An Analysis of Aerodynamic Control for Direct FireSpinning Projectiles,” AIAA Paper 2001-4217, Aug. 2001.

[3] Jitpraphai, T., and Costello, M., “Dispersion Reduction of a Direct FireRocket Using Lateral Pulse Jets,” Journal of Spacecraft and Rockets,Vol. 38, No. 6, 2001, pp. 929–936.

[4] Calise, A., Sharma,M., and Corban, J., “Adaptive Autopilot Design forGuided Munitions,” Journal of Guidance, Control, and Dynamics,Vol. 23, No. 5, 2000, pp. 837–843.

[5] Curtis, J., “An Input-to-State Stabilizing Control Lyapunov Functionfor Autonomous Guidance and Control,” AIAA Paper 2005-6282,Aug. 2005.

[6] Slegers, N., and Costello, M., “Model Predictive Control of a Parafoiland Payload System,” Journal of Guidance, Control, and Dynamics,Vol. 28, No. 4, 2005, pp. 816–821.

[7] Slegers, N., Kyle, J., and Costello, M., “A Nonlinear Model PredictiveControl Technique for Unmanned Air Vehicles,” Journal of Guidance,Control, and Dynamics, Vol. 29, No. 5, 2006, pp. 1179–1188.doi:10.2514/1.21531

[8] Burchett, B., and Costello, M., “Model Predictive Lateral Pulse JetControl of anAtmospheric Rocket,” Journal of Guidance, Control, andDynamics, Vol. 25, No. 5, 2002, pp. 860–867.

[9] Ollerenshaw, D., and Costello, M., “Model Predictive Control of aDirect Fire Projectile Equipped with Canards,” AIAA Paper 2005-5818, Aug. 2005.

Fig. 10 Predicted impact history of a typical guided trajectory.

Fig. 11 Dispersion comparison of a guided and unguided munition.

[10] McCoy, R., Modern Exterior Ballistics, Schiffer, Attlen, PA, 1999,pp. 221–244.

[11] Murphy, C., “Angular Motion of a Spinning Projectile with a ViscousFluid Payload,” Journal of Guidance, Control, and Dynamics, Vol. 6,No. 4, 1983, pp. 280–286.

[12] Cooper, G. R., and Costello, M., “Flight Dynamic Response ofSpinning Projectiles to Lateral Impulsive Loads,” Journal of DynamicSystems, Measurement, and Control, Vol. 126, Sept. 2004,pp. 605–613.

doi:10.1115/1.1789976[13] Costello,M., and Peterson,A., “Linear Theory of aDual-Spin Projectile

in Atmospheric Flight,” Journal of Guidance, Control, and Dynamics,Vol. 23, No. 5, 2000, pp. 789–797.

[14] Hainz, L. C., III, and Costello, M., “Modified Projectile Linear Theoryfor Rapid Trajectory Prediction,” Journal of Guidance, Control, and

Dynamics, Vol. 28, No. 5, 2005, pp. 1006–1014.[15] Stengel, R. F., Optimal Control and Estimation, Dover, New York,

1994, pp. 254–270.