Use of Micro Spoilers for Control of Finned Projectiles

Transcription

1 Ue of Miro Spoiler for Control of Finned Projetile John Dyke 1 and Mark Cotello Georgia Intitute of Tehnology, Atlanta, GA Daniel L. Cler 3 U.S. Army Reearh, Development and Engineering Command, Piatinny, NJ 786 Robert Caron 4, and Robert Dillon 5 U.S. Army Reearh, Development and Engineering Command Benét Laboratorie, Watervliet, NY 193 Relative to other air vehile, the phyial ontrol mehanim on a mart projetile play muh more of a entral role in the overall ytem deign. Many different mart projetile ontrol mehanim have been reated, inluding aerodynami baed mehanim uh a movable anard, propellant baed mehanim uh a quib, and inertia baed mehanim uh a internal moving mae. The work reported here onider mall miro poiler loated between rear fin to reate aerodynami fore hange to enable projetile ontrol. In partiular, boundary layer hok interation between the projetile body, fin, and miro poiler provide a multipliative effet on ontrollable fore and moment. A parametri tudy varying the miro poiler onfiguration i onduted to examine the level of ontrol authority poible for thi ontrol mehanim onept. Reult indiate that relatively mall miro poiler loated between fin generate ubtantial ontrol authority that i apable of eliminating impat error aued by muzzle jump, aerodynami unertainty, and atmopheri wind. Thee onluion are baed on CFD predition of the effet of miro poiler on air load oupled to a rigid 6-degree-of-freedom projetile trajetory imulation. Nomenlature CFD = omputational fluid dynami C NA = normal fore aerodynami oeffiient C MQ = pith damping moment oeffiient C X = zero yaw aerodynami axial fore oeffiient parallel to projetile motion C Y,C Z = aerodynami trim oeffiient perpendiular to projetile axi of ymmetry C LDD = roll moment oeffiient from fin ant C YPA = magnu fore oeffiient C LP = roll damping oeffiient D = projetile referene diameter g = aeleration due to gravity (9.81 m/ ) I = projetile inertia matrix K P = ontroller proportional gain K I = ontroller integral gain K D = ontroller derivative gain L,M,N = external moment omponent on the projetile body expreed in the projetile referene frame L SA,M SA,N SA = teady aerodynami moment omponent on the projetile body expreed in the projetile referene frame 1 Graduate Reearh Aitant, Shool of Mehanial Engineering, Member AIAA. Sikorky Aoiate Profeor, Shool of Aeropae Engineering, Aoiate Fellow AIAA. 3 Senior Engineer, Weapon Tehnology Branh, Small & Medium Cal Diviion, RDAR-WSW-F. 4 Mehanial Engineer, Eletro-Mehani & Control Branh, RDAR-WSB-DA. 5 Siene Advior, Diretor Offie, RDAR-WSB. 1

2 L UA,M UA,N UA = unteady aerodynami moment omponent on the projetile body expreed in the projetile referene frame L C,M C,N C = miro poiler moment omponent on the projetile body expreed in the projetile referene frame m = ma of the projetile p,q,r = omponent of the angular veloity vetor of the projetile body expreed in the projetile referene frame SoS = peed of ound for air t a = ontrol ytem maneuver ativation time u,v,w = tranlation veloity omponent of the projetile enter of ma expreed in the projetile referene frame V = veloity magnitude of projetile enter of ma X,Y,Z = total external fore omponent on the projetile body expreed in the projetile referene frame X G,Y G,Z G = gravitational fore omponent on the projetile body expreed in the projetile referene frame X A,Y A,Z A = aerodynami fore omponent on the projetile body expreed in the projetile referene frame X C,Y C,Z C = miro poiler fore omponent on the projetile body expreed in the projetile referene frame x,y,z = poition vetor omponent of the projetile body enter of ma expreed in the inertial referene frame α = total aerodynami angle of attak γ = projetile inertial trajetory error ommand angle Γ = total projetile inertial trajetory error δ = ontrol ytem roll angle ativation window ρ = denity of air ϕ,θ,ѱ = Euler roll, pith and yaw angle I. Introdution IRECT fire projetile are fired by line-of-ight aiming from ground baed platform, heliopter, and fixed Dwing airraft. A number of ondition an aue round to mi an intended target. Thee ondition inlude manufaturing inauraie of the gun tube, propellant, and projetile, along with variable atmopheri ondition, firing platform motion, and aiming error. With the advent of low ot, mall, rugged, miro-eletro-mehanial ytem, dramati redution in diperion for diret fire projetile equipped with a flight ontrol ytem i poible. The ontrol mehanim mut be apable of altering the trajetory of the projetile in uh a way that impat point error indued at launh and in flight an be orreted. At the ame time, the ontrol mehanim mut be rugged to withtand high aeleration load at launh, mall o that payload pae i not ompromied, and inexpenive for ot onideration, a explained by Roger 1. Many different ontrol mehanim are being developed with the previou requirement in mind, but that all onept eentially fall into three ategorie: aerodynami load mehanim, jet thrut mehanim, and inertial load mehanim. Example of aerodynami ontrol mehanim, inluding rotation of aerodynami lifting urfae appendage, defletion of the noe, and defletion of ram air to ide port. Example of jet thrut ontrol mehanim inlude ga jet thruter and exploive thruter. Example of inertial ontrol mehanim inlude internal tranlation of a ontrol ma and internal rotation of an unbalaned part. The ontrol mehanim invetigated in thi paper fall into the ategory of an aerodynami mehanim, where mall miro poiler are ued to indue an aerodynami fore and moment perturbation on high-peed projetile due to hok wave interation between the fin, body, and miro poiler. Maey 4-6 a well a Bell 7 onidered protuberane on fin-body interation of high-peed projetile. Both omputational and experimental analyi howed that ignifiant fore multipliation an exit by modifying the boundary layer hok interation of the projetile fin and body. The devie ued in thee previou tudie were relatively large (on the order of one third the fin height) and protruded beyond the boundary layer. Large and ontrollable aerodynami load perturbation were noted. A imilar phyial ontrol mehanim i onidered here. A number of mall miro poiler are inerted between the rear fin of the projetile. The miro poiler are ativated in a oordinated fahion to generate maximal ontrol authority. A et of parametri trade tudie are onduted on miro poiler onfiguration and it effet on ontrol authority i determined. Furthermore, ative ontrol performane of a nominal miro poiler onfiguration i preented to etablih the promie of thi projetile ontrol mehanim. Reult provided in thi paper are baed on flight dynami imulation uing a rigid body repreentation. Predition of the effet of miro poiler defletion on aerodynami load i produed uing omputational fluid dynami analyi. Detail on thee model are given below, followed by preentation of reult.

3 3 II. Projetile Dynami Model The non-linear trajetory imulation ued in thi tudy i a tandard 6-degree-of-freedom model typially ued in flight dynami modeling of projetile. Thee 6 degree of freedom inlude three inertial omponent of the poition vetor and three tandard Euler projetile orientation angle, referened to an earth-fixed inertial frame. The equation of motion are provided in Eq. (1)-(4), a derived by MCoy 8 and Murphy 9. w v u z y x (1) r q p t t / / 1 () w v u p q p r q r m Z m Y m X w v u / / / (3) r q p I p q p r q r N M L I r q p 1 (4) In Eq. (1) and (), the tandard horthand notation for trigonometri funtion i ued: α = in(α), α = o(α), and t α = tan(α). The fore appearing in Eq. (3) ontain ontribution from gravity (G), body aerodynami (A), and miro poiler ontrol (C). C C C A A A G G G Z Y X Z Y X Z Y X Z Y X (5) The dynami equation are expreed in a body fixed referene frame, thu all fore ating on the body are expreed in the projetile referene frame. The fore ating due to gravity i hown in Eq. (6). mg Z Y X G G G (6) The body aerodynami fore ating at the enter of preure of the projetile i given by Eq. (7). ( ) / / 8 / A X X A NA A NA X C C v w V Y V D C v V Z C w V (7) The applied moment about the projetile ma enter ontain ontribution from teady aerodynami (SA), unteady aerodynami (UA), and the miro poiler ontrol (C).

4 L L M M N N SA SA SA L M N UA UA UA L M N C C C (8) The moment omponent due to teady aerodynami fore and ontrol fore are omputed with a ro produt between the ditane vetor from the ma enter to the loation of the peifi fore and the fore itelf. The unteady body aerodynami moment provide a damping oure for projetile angular motion and i given by Eq. (9). L M N SA SA SA V 8 C 3 D LDD pdc qdc V qdc V V MQ MQ LP (9) The projetile ma, enter of gravity, and inertial propertie are all aumed to be ontant throughout the duration of the flight. The enter of preure loation and all aerodynami oeffiient depend on loal Mah number and are omputed during imulation uing linear interpolation. The dynami equation given by Eq. (1) (4) are numerially integrated forward in time uing a fourth-order, fixed tep Runge-Kutta algorithm. Cotello and Anderon 8 preent orrelation of thi dynami model againt range data for a fin tabilized projetile. The fore and moment generated by the miro poiler are deribed below. III. Miro Spoiler CFD Analyi A. CFD Model The grid utilize a tetrahedral meh in the far-field with trutured prim meh at the near-wall urfae with a tarting ell height of.3 mm. The grid onit of. million ell. The CFD olution wa run uing ANSYS Fluent on 16 proeor on an IBM Sytem x365/x355 luter. The 3-D olution utilize the denitybaed expliit teady-tate double-preiion olver. The fluid medium i modeled a ideal ga air with propertie at tandard ea-level atmopheri ondition. The k-omega SST turbulene model i ued to model near-wall and farfield turbulene. The near-wall boundary-layer meh i on the order of y-tar equal to one. The k-omega SST turbulene model utilize a blended approah that allow the near-wall effet to modeled diretly with the grid without ompromiing far-field effet. Thi i done with a blending funtion that model the near-wall diretly (or with a k-omega model if uffiient grid doe not exit to model it diretly with y-tar above 1) and then model the far-field with a k-epilon turbulene model. Thi i aomplihed through the ue of a blending funtion that gradually turn the k-omega model into a k-epilon model in the far-field. The benefit to thi type of turbulene model in high-peed flow i better predition of boundary-layer interation and eparation a i the ae with the urrent onfiguration. Seond order diretization i utilized for the flow equation, turbulent kineti energy, and peifi diipation rate. The olution wa run until olution reidual were redued three order of magnitude. A baeline validation ae i ontruted baed on the geometry develop by Maey 1. The geometry eleted mimiked the onfiguration where a pin diameter of. inhe and height of.5 inhe i loated.6 inhe from the trailing edge of a tapered fin mounted on a flat plate. The fin i expoed to Mah 1.7 flow from a round jet in the experimental etup. The experimental reult produed 6.61 lb of net ide fore on the fin and the CFD predition produed 8.39 lb of net ide fore on the fin. The experimental fore wa determined by aigning area to eah of the 8 preure tap on the pin ide of the fin and then integrating to determine the fore on that ide of the fin. In addition, fore were likely not alulated on the harp rear and front edge of the fin due to the fat no intrumentation wa loated here. How the fore wa determined for the lee-ward ide of the fin wa not doumented in Maey report a well. Beaue preure data wa not available on the lee-ward ide of the fin, ome experimental error ould have reulted. In addition, the entire fin may not have been ompletely ubmerged in the Mah 1.7 flow tream a wa the ae for the CFD model. Overall, CFD reult ompared favorably to experimental reult. 4

5 Devie Angle Squared Devie Streamwie Number Devie Spanwie Number Devie Length Devie Streamwie # by Devie Spanwie # Spaing From Fin Squared Devie Length by Streamwie Number Perentage of Sum of Square B. Flat Plate Fin Interation Deign of Experiment Model In order to undertand the key element governing fore and moment generation from the miro poiler, a multivariable deigned experiment i ontruted for the miro poiler on a flat plate onfiguration. A erie of 7 different CFD run were performed with ANSYS Fluent 1. to parametrially vary five different geometri parameter to develop a quadrati repone model. All CFD run were performed at Mah =1.7 and zero deg idelip angle of the fin. The five geometri parameter inluded the paing between the group of miro poiler and the idewall of the fin, the angle of the miro flap relative to the flat plate, the length of the miro poiler, the number of row of flap in the treamwie diretion, and the number of olumn of flap in the panwie diretion. Fig. 1 how the deign of experiment reult for the top even ontributor to a quadrati fit of the total fore generated on the fin, flat plate and miro poiler. Shown are the perentage of eah model ontributor relative to the total um of quare for the quadrati model of total fore. Thi allow relative omparion of the effet over the range of the model. A an be een, the two larget effet are the devie angle quared and the number of devie in the treamwie diretion. The devie angle wa varied from 3. deg off of the flat plate faing forward, through 9. deg to 15. deg faing rearward. The peak fore wa generated at 9. deg to the flat plate urfae with a harp fall-off at 3. and 15. deg. In addition, the number of devie in the treamwie diretion and panwie diretion had a trong effet a well, with the treamwie diretion having a lightly tronger effet. Thi wa likely due to a tronger interation with the fin in the treamwie diretion a the hok truture refleted off of the fin. In addition, a the devie length inreaed, the total fore inreaed. There were alo a ouple of maller ro interation between two effet a well a the paing of the overall pattern from the fin wall quared. Information from the deigned experiment wa utilized to optimize the miro poiler layout for appliation to the Army-Navy finner onfiguration. 3% 5% % 15% 1% 5% % Model Effet Figure 1. Deign of Experiment reult howing um of quare from a quadrati fit of the total fore generated on the fin, flat plate and miro poiler. 5

6 C. Miro Spoiler Fore and Moment Fig. below how the CFD model of the Army-Navy finner projetile with a ingle quadrant miro poiler ontrol mehanim. The model wa run at zero angle of attak and zero angle of ide lip form Mah.8 to Mah 4.. A baeline onfiguration of the Army-Navy finner without miro poiler wa run at the ame ondition. The miro poiler onfiguration wa optimized to produe the maximum fore on the fin and body. With the onfiguration hown in Fig., the fore are produed by inreaing the preure on two adjaent fin and projetile body between the fin. The forward two miro poiler produe hok truture on the forward fin truture. The rear two miro poiler are loated inide the forward miro poiler o they an reeive unimpeded treamwie flow. Thee poiler produe inreae preure near the rear of the fin and at the aft end of the projetile body. In addition to the miro flap onfiguration, Fig. 3 how the preure ontour on the rear fin truture a a funtion of Mah number. A an be een, the devie are fairly ineffetive at uboni ondition, but rather effetive at uperoni ondition. Fig. 4 how the fore and moment a a funtion of Mah number where delta fore and delta moment were alulated by ubtrating the baeline onfiguration fore and moment. Above Mah 1.5, the devie beome fairly effetive. The preure at the higher Mah number inreae dramatially on the body and fin near the forward et of miro poiler. The rear et of miro poiler, though le effetive, do offer ome inreae in preure on the rear body and fin truture. From Mah 1.5 to 4., the miro poiler total fore and moment inreae almot linearly a a funtion of Mah number. The miro poiler are effetive fore multiplier by reating ignifiant fore on both the fin and body through body-hok interation. Figure. Shemati of an Army-Navy finner projetile poeing a 1-quadrant miro poiler ontrol mehanim onfiguration to be ued in dynami imulation. 6

7 (a) Mah =.8 (b) Mah =.9 () Mah = 1.1 (d) Mah = 1. (e) Mah =. (f) Mah = 4. Figure 3. Stati preure ontour (Paal) of Army-Navy finner with miro poiler deployed at (a) Mah =.8, (b) Mah =.9, () Mah = 1.1, (d) Mah = 1., (e) Mah =., and (f) Mah = 4.. 7

8 Miro Spoiler Moment (ft-lb) Miro Spoiler Fore (lb) 15 X C 1 Y C Z C Mah No. 4.5 (a) L C M C N C Mah No. (b) Figure 4. Mah number dependent miro poiler (a) fore and (b) moment perturbation ating about the projetile enter of gravity for the referene frame hown in Fig.. Thi body referene frame i ued in the trajetory imulation. 8

9 IV. Dynami Flight Model Analyi A. Deription of Projetile In order to examine the effetivene of the miro poiler ontrol mehanim, trajetory reult were generated for an example projetile. The Army-Navy finner hown above i ued in thi tudy a an example diret fire fintabilized projetile. From Dupui 11, the projetile ma, diameter, ma enter meaured along the tation line, roll inertia, and pith inertia are lug,.9845 ft,.449 ft,.14 lug-ft, and.78 lug-ft, repetively. A fin ant angle of.5 deg wa hoen to indue a moderate amount of roll rate. In all the following ae, the projetile i traveling through a tandard atmophere without atmopheri wind. A hemati of the bai projetile i hown below in Fig. 5. B. Single Trajetory Reult Uing the flight dynami model previouly deribed, a et of typial trajetory reult are imulated. The projetile initial ondition are x =. ft, y =. ft, z = -1. ft, ϕ =. deg, θ = deg, ѱ =. deg, u = 3357 ft/, v =. ft/, w =. ft/, q =. rad/, and r =. rad/. A 4-quadrant miro poiler onfiguration i ued where a ingle miro poiler i loated between eah fin. Thi i different than the CFD model where only one quadrant wa defleted. However, it i likely only one quadrant will deflet at a time with the dynami flight model, hene the tati CFD model i a good approximately of dynami flight model. Eah fin on the projetile i 3. mm by 3. mm. Eah of the four miro poiler in a given quadrant i 3. mm wide and approximately 4.5 mm tall. The defletion angle between the miro poiler and the projetile body i 9. deg. The two forward miro poiler are. deg roll angle off of the nearet fin and the two rear miro poiler are 4. deg roll off of the near fin. The forward miro poiler are 5. mm from the rear of the projetile and the rear miro poiler are 13. mm from the rear of the projetile. Shown in Fig. 6, a imple flight ontrol ytem i ued to ompute proper miro poiler ativation to exeute a left turn. Control authority of the ytem i ignifiantly affeted by ativation time t a, ommand turn angle γ, and roll angle ativation window δ. For thi example trajetory, the maneuver ativation time i et to 1. e while the roll angle ativation window i et to 6. deg. Fig ompare ingle trajetory reult for unontrolled and full left ontrolled projetile. Fig. 7 how that for a range of 3. km the ontrolled projetile wa oberved to deflet m. Fig. 8 how that minor differene in altitude between ontrolled and unontrolled projetile are experiened. Fig. 9 and 1 how that ativating the ontrol mehanim doe introdue oillatory motion in Euler pith and yaw angle but dampen out oniderably throughout the remainder of flight. Note that while the pith angle doe follow the unontrolled pith angle, the yaw angle diverge from the unontrolled ae from nearly. deg to deg. Fig. 11 how that the roll rate i alo redued by approximately 1. rad/ after ativation. Fig. 1 and 13 how that maximal pith and yaw rate never exeed 4. rad/. In Fig. 14 the projetile total aerodynami angle of attak experiene oillation immediately after ontrol ativation to a maximum angle of deg but quikly dampen down to below. deg in jut over 1. e of ativation but not by a ignifiant amount. Fig. 15 how that the total veloity for the ontrolled projetile i redued from the unontrolled ae. Fig. 16 how that miro poiler total fore and moment perturbation in a body-fixed frame are ueively roll ativated with dereaing magnitude from lb and 1.15 ft-lb a the projetile deelerate. Figure 5. Army-Navy finner Geometry (all dimenion in aliber, 1 aliber = 3 mm). 9

10 Defletion (m) Figure 6. Blok diagram modeling of miro poiler fore and moment Controlled Trajetory Unontrolled Trajetory Range (m) Figure 7. Defletion v Range. 1

11 Euler Pith Angle (deg) Altitude (m) Controlled Trajetory Unontrolled Trajetory Range (m) Figure 8. Altitude v Range Controlled Trajetory Unontrolled Trajetory Time (e) Figure 9. Pith Angle v Time. 11

12 Roll Rate (rad/e) Euler Yaw Angle (deg) Controlled Trajetory Unontrolled Trajetory Time (e) Figure 1. Yaw Angle v Time Controlled Trajetory Unontrolled Trajetory Time (e) Figure 11. Roll Rate v Time. 1

13 Yaw Rate (rad/e) Pith Rate (rad/e) 4 3 Controlled Trajetory Unontrolled Trajetory Time (e) Figure 1. Pith Rate v Time. 4 3 Controlled Trajetory Unontrolled Trajetory Time (e) Figure 13. Yaw Rate v Time. 13

14 Total Veloity (ft/) Total Aerodynami Angle of Attak (deg) 6 5 Controlled Trajetory Unontrolled Trajetory Time (e) Figure 14. Angle of Attak v Time Controlled Trajetory Unontrolled Trajetory Time (e) Figure 15. Total Veloity v Time. 14

15 Total Miro Spoiler Moment (ft-lb) Total Miro Spoiler Fore (lb) Time (e) (a) Time (e) Figure 16. (a) Total Miro Spoiler Fore Perturbation v Time, and (b) Total Miro Spoiler Moment Perturbation v Time after ontroller ativation. (b) 15

16 Altitude (m) C. Control Authority Parametri Trade Study Several parametri trade tudie were omputed to examine the ontrol authority of the miro poiler mehanim on the bai finner projetile, diret-fired to a range of 3. km. In all of the following trade tudie the nominal projetile initial ondition are x =. ft, y =. ft, z = -1. ft, ϕ =. deg, θ = 1.3 deg, ѱ =. deg, u = 3357 ft/, v =. ft/, w =. ft/, q =. rad/, and r =. rad/. Alo, nominal ondition for the ontrol ytem parameter are et to t a = 1. e and δ = 6. deg, where γ i panned from. to 36. deg in inrement of 5. deg to produe a omplete ontrol authority footprint, a hown in Fig. 17. Trade tudie inlude variation of ontrol mehanim onfiguration, ativation time, launh veloity, and roll angle ativation window and are ummarized in Fig Fig. 18 how that ontrol authority linearly inreae with variation of mehanim. CS1, CS, CS3, and CS4 repreent the number of miro poiler ontrol mehanim implemented between the projetile fin (from 1 up to 4 quadrant). Fig. 19 how that inreaing the roll angle ativation window ha a poitive effet on ontrol authority but reahe a point of diminihing return beyond 11. deg. In Fig., inreaing muzzle launh veloity i oberved to have a poitive effet on ontrol authority; however, onideration mut be given to the minimum value of thi parameter to enure that the projetile i expoed to uperoni ondition for the entirety of flight. Otherwie, the miro poiler mehanim will not properly funtion. Fig. 1 how that delaying ontroller ativation time ha a negative effet on ontrol authority, a expeted. 15 CS1 CS CS3 CS CroRange (m) Figure 17. Full ontrol authority diperion map for variation in ontrol ytem (CS) quadrant onfiguration at an impat range of 3. km with nominal ondition: t a = 1. e, δ = 6, and Ma i =

17 Control Authority Radiu (m) Control Authority Radiu (m) CS Configuration (number of flap) Figure 18. Trade tudy: effet of varied ontrol ytem (CS) quadrant onfiguration on full ontrol authority with nominal ondition: t a = 1. e, δ = 6, and Ma i = CS1 CS CS3 CS Ativation Angle (deg) Figure 19. Trade tudy: effet of varied ativation angle (δ) on full ontrol authority with nominal ondition: t a = 1. e and Ma i =

18 Control Authority Radiu (m) Control Authority Radiu (m) CS1 CS CS3 CS Mah Number Figure. Trade tudy: effet of varied launh Mah number (Ma i ) on full ontrol authority with nominal ondition: t a = 1. e and δ = CS1 CS CS3 CS Ativation Time (e) Figure 1. Trade tudy: effet of varied ontrol ytem ativation time (t a ) on full ontrol authority under nominal ondition: δ = 6 and Ma i =

19 D. Ative Control Analyi An ative ontrol ytem i next developed to demontrate the apabilitie of the miro poiler ontrol mehanim to overome typial launh and in-flight error. Thi ontroller wa deigned to trak a uer peified ommanded balliti trajetory through ue of GPS and other onboard enor to provide feedbak of x, y, z, and ϕ for the entire flight. The algorithm illutrated in Fig. ue the ommand trajetory to develop an inertial-fixed projetile error vetor, whih i redued by a proportional-integral-derivative (PID) ontroller through variation of the roll angle ativation window. The total projetile inertial trajetory error Γ and the error ommand angle γ are defined in Eqn. (1) and (11). y e z e (1) 1 ez tan (11) ey The error ommand angle and roll angle ativation window are then input into the model predition algorithm in Fig. 3, where γ provide an inertial diretion for roll angle ativation and δ peifie a neighborhood about γ for ativation. The hoen ommand trajetory onited of 3 direte point to be linearly interpolated by the ontroller during flight and wa generated by the nominal initial ondition previouly ued. Next, the following perturbed initial ondition were ued for ontrolled and unontrolled trajetorie: x =. ft, y =. ft, z = -1. ft, ϕ =. rad, θ =.18 rad, ѱ =.1 rad, u = 336 ft/, v = ft/, w =. ft/, p =.17 rad/, q = -.79 rad/, and r =.73 rad/. For the ontrolled ae, the roll angle ativation window wa limited to a maximum value of δ max = 9. deg. Alo, t a = 1. e, K P =.61, K I =.154, and K D =.675. Fig. 4-9 how the performane of the ontrolled projetile. Fig. 4 and 5 demontrate the ontrolled projetile ability to quikly onverge to the ommand trajetory oon after ativation. Note that the unontrolled ae mie the target (onidered to be impat point of the ommanded trajetory) by m in altitude and m in defletion, while the ontrolled ae only mie the target by approximately.617 m in altitude and.373 m in defletion. Figure. Control Sytem Logi Figure 3. Miro Spoiler Logi 19

20 Defletion (m) Fig. 6 how that the total projetile angle of attak reahe a maximum value of 7.77 deg upon ontrol maneuver ativation but then quikly drop down below 3. deg within 1. e. Fig. 7 how variation in the error ommand angle after ontrol maneuver ativation. At firt thi angle i tay level at approximately 18. deg and then teadily deend. Thi implie that, one the ontroller i ativated, the projetile turn left to approah the ommand trajetory and then tart to piral about it in a lokwie manner, a it eek the ommand trajetory. Fig. 8 how that the total inertial error i rapidly redued from 1.8 m to approximately.56 m after ontrol maneuver ativation and then level off. Thi ugget that one the projetile ha napped to the ommand trajetory and ha begun to piral about it, it doe o at a nearly ontant radiu of.56 m. Fig. 9 how the variation of the roll angle ativation window after ontrol maneuver ativation, whih i initially ontant, ine maximum ontrol authority i needed, and then redue down to around 3. deg. 3 5 Command Trajetory Unontrolled Trajetory Controlled Trajetory Range (m) Figure 4. Defletion v Range.

21 Total Aerodynami Angle of Attak (deg) Altitude (m) Command Trajetory Unontrolled Trajetory Controlled Trajetory Range (m) Figure 5. Altitude v Range. 8 7 Unontrolled Trajetory Controlled Trajetory Time (e) Figure 6. Angle of Attak v Time. 1

22 Total Error (m) Error Command Angle (deg) Time (e) Figure 7. Error Command Angle v Time after ontroller ativation Time (e) Figure 8. Total Error v Time after ontroller ativation.

23 Ativation Window (deg) Time (e) Figure 9. Roll Angle Ativation Window v Time after ontroller ativation.. E. Diperion Analyi Monte Carlo diperion ae were omputed to demontrate the performane of the ontrol ytem with perturbed initial ondition (tip-off effet) and projetile ma/inertial property unertainty. To imulate tip-off effet, initial tate were perturbed from the nominal ondition whih inlude initial pith angle, initial yaw angle, initial muzzle veloity, initial roing veloitie in the y and z diretion, initial roll rate, initial pith rate, and initial yaw rate. The tandard deviation for thee value are.1 rad,.1 rad, 3. ft/, 1. ft/, 1. ft/,.15 rad/,.7 rad/, and.7 rad/ repetively. To imulate variability in projetile weight, diameter, tationline enter of gravity, moment of inertia about the x, y, and z body axe. Standard deviation for thee parameter are.837 lbf,.5 ft,.53 ft,.68 lug-ft,.189 lug-ft, and.189 lug-ft repetively. Fig. 3 and 31 how the impat point reult in a vertial plane at a range of 3. km for the unontrolled and ontrolled diperion ae. A oberved, implementation of the miro poiler ontrol ytem aue a 97.86% redution in the Cirular Error Probable (CEP) - from 37.8 m (unontrolled) down to.81 m (ontrolled). Note that the CEP wa alulated about the mean impat value for eah ae. Thee reult indiate that the miro poiler mart projetile ontrol mehanim i apable of reduing trajetory traking error down to enor auray. 3

24 Altitude (m) Altitude (m) 15 Impat Point CEP CroRange (m) Figure 3. Monte Carlo diperion pread and CEP for an unontrolled projetile with variation in tip-off and projetile propertie at an impat range of 3. km. 4 3 Impat Point CEP CroRange (m) Figure 3. Monte Carlo diperion pread and CEP for a ontrolled projetile with variation in tip-off and projetile propertie at an impat range of 3. km. 4

25 V. Conluion Thi reearh indiate that miro poiler implementation for ative ontrol of finned projetile in uperoni flight i a viable phyial ontrol mehanim that provide high maneuverability to the round. Control authority i linearly inreaed with the number of miro poiler. Inreaing the roll angle ativation window alo inreae ontrol authority but a point of diminihing return i reahed beyond an angle of 11. deg. Inreaing projetile launh muzzle veloity and dereaing ontrol ytem ativation time are alo oberved to have poitive effet on ontrol authority. The PID ontrolled diperion reult how that the miro poiler mehanim i apable of reduing error down to enor error, a nearly a 98% redution in mean impat CEP i oberved. The onluion of thi tudy are reahed uing 6-degree-of-freedom flight dynami imulation to generate fly out, with miro poiler aerodynami load provided by omputational fluid dynami modeling. VI. Referene 1 Roger, J., Cotello, M., Control Authority of a Projetile Equipped with a Controllable Internal Tranlating Ma, AIAA Paper 8-649, AIAA Atmopheri Flight Mehani Conferene, Augut 7, Hilton Head, South Carolina. Rihardon, D., Summoning the Fire, Armada International, 7(1), p. 1, 3. 3 Gander, T. J., Artillery On Target Trajetory Corretable Munition, Armada International, 7(), p. 7, 3. 4 Maey, K. C., MMihael, J., Warnok, T., Hay, F., Mehanial Atuator for Guidane of a Superoni Projetile, AIAA Paper 5-497, AIAA Applied Aerodynami Conferene, June 5, Toronto, Ontario Canada. 5 Maey, K. C., Guthrie, K. B., Optimized Guidane of a Superoni Projetile Uing Pin Baed Atuator, AIAA Paper , AIAA Applied Aerodynami Conferene, June 5, Toronto, Ontario Canada. 6 Maey, K. C., Silton, Sidra I., Teting the Maneuvering Performane of a Mah 4 Projetile, AIAA Paper , AIAA Applied Aerodynami Conferene, June 6, San Franio, California. 7 Bell, M., Watteron, J. K., Lik, D., A Numerial Study into a Loal Protuberane Interation with a Fin on a Superoni Projetile, AIAA Paper 9-19, 47 th AIAA Aeropae Siene Meeting Inluding The New Horizon Forum and Aeropae Expoition, 5-8 January 9, Orlando, Florida. 8 MCoy, R. L., Modern Exterior Balliti, Shiffer Publihing Ltd., Atglen, PA, Murphy, C. H., 1963, Free Flight Motion of Symmetri Miile, U.S. Army Balliti Reearh Laboratorie, BRL Report No Cotello M., Anderon D., Effet of Internal Ma Unbalane on the Terminal Auray and Stability of a Projetile, AIAA Paper 1996, AIAA Flight Mehani Conferene, San Diego, California, Dupui, A. and Hathaway, W., Aeroballiti Range and Wind Tunnel Tet of the Bai finner Referene Projetile from Suboni to High Superoni Veloitie, DREV-TM-136, Otober. 1 Carlui, D., Jaobon, S., Balliti: Theory and Deign of Gun and Ammunition, CRC Pre, Boa Raton, Florida, 8. 5