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1 AN ABSTRACT FOR THE THESIS OF Bradley T. Burhett for the degree of Dotor of Philosophy in Mehanial Engineering presented on May Title: Robust Lateral Pul"e let Control of an Atmospheri Roket. Redated for Privay Abstrat approved: Mark F. Costello Unontrolled diret fire rokets exhibit high impat point dispersion, even at relatively short range, and as suh have been employed as area weapons on the battlefield. In order to redue the dispersion of a diret fire roket, feedbak ontrol is employed to fire short-duration solid roket pulses mounted ne~.r the nose of the projetile and ating perpendiular to the projetile axis of symmetry. The feedbak law is developed by first determining a piee wise linear model of the projetile swerving motion, subsequently using this linear model to predit the projetile impat point both with and without ontrol, and using the results to ommand pulses at appropriate times to drive the impat point loser to the speified target. Candidate optimal ontrol laws are formed using rules based on deision grids, and a global ontrol strategy searh algorithm. The global searh ontrol law proves to be prohibitively omputationally expensive for on-line implementation. The performane of the baseline ontrol law is found to be omparable ~o the rule based and global searh optimal ontrol laws. The ontrol gains of the baseline ontrol law are optimized in the presene of parametri plant unertainty using a Monte Carlo simulation. Performane of the system in the presene of parametri plant unertainty using the optimized gains is deemed omparable to performane of the baseline ontroller with no piant unertainty. The level of unertainty

2 of several plant parameters is varied in order to ompare robustness of the ontroller when optimized with unertainty viz. without unertainty.

3 Copyright by Bradley T. Burhett May 2, 2001 All Rights Reserved

4 Robust Lateral Pulse Jet Control of an Atmospheri Roket By Bradley T. Burhett A THESIS Submitted to Oregon State University In partial fulfillment of the requirements for the degree of Dotor of Philosophy Presented May 2, 2001 Commenement June 2001

5 Dotor of Philosophy thesis of Bradley T. Burhett presented on May APPROVED: Redated for Privay Major Professor, representing Mehanial Engineering Redated for Privay Head of Department of Mehanial Engineering Redated for Privay I understand that my thesis will beome part of the permanent olletion of Oregon State University libraries. My signature below authorizes release of my thesis to any reader upon request. Redated for Privay

6 ACKNOWLEDGMENTS I want to thank Dr. and Mrs. Mark Costello for their friendship, mentorship, and support beyond the all of duty during this projet. I wish to thank my family. Carol, without your enouragement, I never would have pursued a Ph.D. Your patient endurane through this proess has been inredible. Tabitha, Claudia, Lydia, Emmy, Davis and Nate, thank you for letting me pursue my goal and work harder than ever before. You make it all worthwhile. I wish to thank the Northwest Hills Baptist Tuesday morning men's devotion group, and the Gideons International, Corvallis Camp. You have been a steadfast support. Finally, I thank the God of Abraham, Isaa and Jaob, who has authored my life, and 'worked together all things for good'. Romans 8:28, Psalm 19:1. There are two books laid before us to Shldy, to prevent our falling into error. First, the volume of the Sriptures, whih reveal the will of God; then the volume of the reation, whih expresses His power. -Franis Baon To be so oupied in the investigation of the serets of nature, as never to tum the eyes to its author, is a most perverted study; and to enjoy everything in nature without aknowledging the author of the benefit, is the basest ingratitude. -John Calvin, 1554

7 11 TABLE OF CONTENTS ~ 1. INTRODUCTION LITERATURE REVIEW Spaeraft Pulse Jet ControL Optimal Control of Pulse-Width and Pulse Frequeny Modulated Systems Projetile Pulse Jet ControL Assessment of Related Work DIRECT FIRE ROCKET MATHEMATICAL MODELING Diret Fire Roket Non-Linear Dynami ModeL Diret Fire Roket Linear Dynami ModeL Comparison of Non-Linear and Linear Models PREDICTIVE FLIGHT CONTROL SYSTEM Baseline Preditive Flight Control System Non-Parametri Statistis and Kolmogorov Bands Baseline Control System Parametri Trade Studies An Analogous One-Dimensional Optimal Control Problem Rule Based Optimal Controller Limit of Ahievable Performane Choie of Controller for Robustness Testing and Optimization ROBUST PREDICTIVE PULSE JET FLIGHT CONTROL SYSTEM DESIGN CONCLUSIONS AND FUTURE WORK REFERENCES APPENDIX: FORTRAN SUBROUTINES TO IMPLEMENT THE CONTROL LAWS IN THE BOOM CODE

8 iii LIST OF FIGURES 3.1 Shemati of a Diret Fire Roket with Lateral Pulse Jets Cross Range vs. Range, (Linear Model Update Rate = 1 aliber) Altitude vs. Range, (Linear Model Update Rate = 1 aliber) Total Veloity vs. Range, (Linear Model Update Rate = 1 aliber) Body Side Veloity J Component vs. Range, (Linear Model Update Rate = 1 aliber) Body Side Veloity K Component vs. Range. (Linear Model Update Rate = 1 aliber) Body Roll Rate vs. Range, (Linear Model Update Rate = 1 aliber) Pith Rate vs. Range, (Linear Model Update Rate = 1 aliber) Yaw Rate vs. Range, (Linear Model Update Rate = 1 aliber) Pith Angle vs. Range, (Linear Model Update Rate = 1 aliber) Yaw Angle vs. Range, (Linear Model Update Rate = 1 aliber) Cross Range vs. Range, (Linear Model Update Rate = 1965 alibers) Altitude vs. Range, (Linear Model Update Rate = 1965 alibers) Total Veloity vs. Range, (Linear Model Update Rate = 1965 alibers) Body Side Veloity JComponent vs. Range, (Linear Model Update Rate = 1965 alibers) Body Side Veloity K Component vs. Range, (Linear Model Update Rate = 1965 Calibers) Body Roll Rate vs. Range, (Linear Model Update Rate = 1965 alibers) Pith Rate vs. Range, (Linear Model Update Rate = 1965 alibers) Yaw Rate vs. Range, (Linear Model Update Rate = 1965 alibers) Pith Angle vs. Range, (Linear Model Update Rate = 1965 alibers) Yaw Angle vs. Range, (Linear Model Update Rate =1965 alibers)... 47

9 iv LIST OF FIGURES (Continued) 3.22 Target Predition Error vs. Model Update IntervaL Potential Change in Impat Point versus Range from Target at Impulse Firing Preditive Impat Point Flight Control System Geometry of Current Predited Impat PoiPt versus Desired Impat Point Sw Versus Time Pulse Jet Firing Logi Set of Ahievable Impat Points Due to a Single Pulse Geometry of Allowable Overshoot Region Cross Range vs. Range, Typial Controlled Trajetory Altitude vs. Range - Typial Controlled Trajetory Total Veloity vs. Range, Typial Controlled Trajetory Body Side Veloity J Component vs. Range, Typial Controlled Trajetory Body Side Veloity K Component vs. Range, Typial Controlled Trajetory Body Roll Rate vs. Range, Typial Controlled Trajetory Pith Rate vs. Range, Typial Controlled Trajetory Yaw Rate vs. Range, Typial Controlled Trajetory Pith Angle vs. Range, Typial Controlled Trajetory Yaw Angle vs. Range, Typial Controlled Trajetory Roll Angle vs. Range, Typial Controlled Trajetory Evolution of Projetile Impat Point Evolution of Projetile Impat Point, with Predition of Controlled Impat Point Unontrolled Dispersion (CEP = ft)... 73

10 v LIST OF FIGURES (Continued) 4.21 Controlled Dispersion (CEP = 1.15 ft) Typial Cumulative Distribution Funtion with Kolrnogorov Bands Trade Study of Sw. (Constant Overshoot Radius) Trade Study of Controller Sampling Period Trade Study of Constant Overshoot Radius Trade Study of Sw. Allowable Overshoot Radius Equals Current Miss Distane Median Miss Distane versus Number of Pulse Jets for 1, 10, 15, 20, and 30 N-s pulses % Upper Confidene Band on Sample Median versus Number of Pulse Jets for 1, 10, 15, 20 and 30 N-s pulses Zoom plot of 95% Upper Confidene Band on Sample Median for 1, 10, 15, 20, and 30 N-s pulses Median Miss Distane versus Number of Pulse Jets Constant Energy Ring, Ring Energy = 40, 60, 80, 120 and 160 N-s % Upper Confidene Band on Sample Median versus Number of Pulse Jets for Constant Energy Ring, Ring Energy = 40, 60, 80, 120 and 160 N-s Zoom plot of 95% Upper Confidene Band on Sample Median versus Number of Pulse Jets for Constant Energy Ring, Ring Energy = 80, 120 and 160 N-s Set of Ahievable Impat Points from a Single Pulse when Using Preditive Alignment Control Sheme Set of Ahievable Impat Points from Pulse jet Array when Using Preditive Alignment Control Sheme Analogous Optimal Control Problem Deision grid for Solving the Analogous Optimal Control Problem... 92

11 VI LIST OF FIGURES (Continued) 4.37 Deision Grid Showing Optimal Paths only Unwrapped Euler Roll Angle for Typial Controlled Trajetory Distane Until Repetition of Roll Angle as A Funtion of Distane to Target Median Miss Distane versus Number of Pulse Jets, Rule Based Optimal Controller, 1, 5, 10, 15, and 20 N-s pulses % Upper Confidene Band on Sample Me.ian versus Number of Pulse Jets, Rule Based Optimal Controller, 1, 5, 10, 15 and 20 N-s pulses Zoom plot of 95% Upper Confidene Band on Sample Median versus Number of Pulse Jets, Rule Based Optimal Controller, 1,5, 10, 15 and 20 N-s pulses Number of possible ontrol strategies versus Number of Pulses Remaining for a Trajetory with 32 Pulsing opportunities Miss Distanes for Global Searh Optimal Controller, Single Pulse Magnitude = 20 N-s Approximate length of Kolmogorov 95% Confidene Band on Population Median versus Sample Size for Typial Dispersion Data Histogram of SLCOP, Sample size = 200 (Bin Width = 0.48) Example Randomized Aerodynami Coeffiient Curve Non-Robust Control System Performane Measure as a Funtion of Control Parameters Robust Control System Performane Measure as a Funtion of Control Parameters.. 114

12 vii LIST OF TABLES Table Page 4.1 Parameters for Typial Controlled Trajetory Kolmogorov Confidene Band Numbers Non-linear model properties and assoiated unertainties

13 viii LIST OF APPENDIX FIGURES A.1a Alb A1 AId A.2a A2b A.3a A3b A3 A.3d A.3e A.3f A3g A.3h A.3i A.3j FORTRAN Subroutine detontroi.f, Main SL'broutine Implementing the Robust Controller FORTRAN Subroutine detontrol.f Continued FORTRAN Subroutine detontrol.f Continued FORTRAN Subroutine detontrol.f Continued FORTRAN Subroutine pre_splash.f, Subroutine to Predit the Unontrolled Impat Point FORTRAN Subroutine pre_splash.f Continued FORTRAN Subroutine pieslin2.f, Subroutine to Compute Gliding, Unontrolled Trajetory FORTRAN Subroutine pieslin2.f Continued FORTRAN Subroutine pieslin2.f Continued FORTRAN Subroutine pieslin2.f Continued FORTRAN Subroutine pieslin2.f Continued FORTRAN Subroutine pieslin2.f Continued FORTRAN Subroutine pieslin2.f Continued FORTRAN Subroutine pieslin2.f Continued... l34 FORTRAN Subroutine pieslin2.f Continued... 13S FORTRAN Subroutine pieslin2.f Continued A.4 FORTRAN Subroutine initbod3.f, Subroutine to Initialize Projetile Mass Properties for Gliding Flight A.Sa A.Sb A.S FORTRAN Subroutine powrlin.f, Subroutine to Compute Powered Portion of Unontrolled Trajetory FORTRAN Subroutine powrlin.f Continued FORTRAN Subroutine powrlin.f Continued

14 ix LIST OF APPENDIX FIGURES (Continued) A.5d A.5e A.5f A.5g A.5h A.5i A.5j A.5k A.6a A.6b A.6 FORTRAN Subroutine powrlin.f Continued FORTRAN Subroutine powrlin.f Continued FORTRAN Subroutine powrlin.f Continued FORTRAN Subroutine powrlin.f Continued FORTRAN Subroutine powrlin.f Continued FORTRAN Subroutine powrlin.f Continued FORTRAN Subroutine powrlin.f Continued FORTRAN Subroutine powrlin.f Continued FORTRAN Subroutine loadaero2.f, Subroutine to Load Aerodynami Coeffiients into the Linear Model FORTRAN Subroutine loadaero2.f Continued FORTRAN Subroutine loadaero2.f Continued A.7 FORTRAN Subroutine loadrap.f, Subroutine to Load Roket Motor Data into the Linear Model A.8a A.8b A.9a A.9b A.9 A.9d A.ge A.9f FORTRAN Subroutine rapal.f, Subroutine to Compute Roket Thrust Fores for Linear Model FORTRAN Subroutine rapal.f Continued FORTRAN Subroutine pulslin.f, Subroutine to Compute Controlled Gliding Trajetory FORTRAN Subroutine pulslin.f Continued FORTRAN Subroutine pulslin.f Continued FORTRAN Subroutine pulslin.f Continued FORTRAN Subroutine pulslin.f Continued FORTRAN Subroutine pulslin.f Continued

15 x LIST OF APPENDIX FIGURES (Continued) A.9g A.9h A.9i A.9j A.9k FORTRAN Subroutine pulslin.f Continued FORTRAN Subroutine pulslin.f Continued FORTRAN Subroutine pulslin.f Continued FORTRAN Subroutine pulslin.f Continued FORTRAN Subroutine pulslin.f Continued A.10a FORTRAN Subroutine powrpuls.f, Subroutine to Compute Powered, Controlled Portion of Trajetory A.10b A.10 FORTRAN Subroutine powrpuls.f Continued FORTRAN Subroutine powrpuls.f Continued A.1Od FORTRAN Subroutine powrpuls.f Continued A.1Oe A.10f A.1Og FORTRAN Subroutine powrpuls.f Continued FORTRAN Subroutine powrpuls.f Continued FORTRAN Subroutine powrpuls.f Continued A.10h FORTRAN Subroutine powrpuls.f Continued A.10i A.1Oj FORTRAN Subroutine powrpuls.f Contillued... l72 FORTRAN Subroutine powrpuls.f Continued A.1Ok FORTRAN Subroutine powrpuls.f Continued

16 ROBUST LATERAL PULSE JET CONTROL OF AN ATMOSPHERIC ROCKEI' CHAPTER 1: INTRODUCTION Unontrolled diret fire rokets exhibit high impat point dispersion, even at relatively short range, and as suh have been employed as area weapons on the battlefield. Army Field Manual [1] defines diret fire as "Gunfire delivered on a target, using the target itself as a point of aim for either the gun or the diretor. Fire inludes gun, missile, or roket fire. Fire direted at a target that is visible to the aimer or firing unit." Therefore, a diret fire roket is essentially a roket that is fired by line-of-sight aiming, and using no mid-ourse guidane. An example of a diret fire roket is the HYDRA-70. It is perhaps the most widely used heliopter weapon system in the world today. [2] Diret fire rokets are typially fired from heliopters suh as the AH-64A Apahe or OH-5SD Kiowa, fighter airraft suh as the U.S. Air Fore F-16, and vehile mounted pods suh as the M270 MLRS Self-Propelled Loader/Launher. Aording to U.S. Army Field Manual , "The 2.75 inh folding fin aerial roket and the 20-mm annon ommon to some attak heliopters are good area weapons to use against enemy fores in the open or under light over. They are usually ineffetive against a large masonry target." [3] Beause diret fire rokets exit the launher with low veloity, any aerodynami disturbanes presented to the roket near the launher reate relatively large aerodynami angles of attak, leading to inreased target dispersion. Furthermore, main roket motor thrust during the initial portion of flight tends to amplify the effet of transverse and angular veloity perturbations on initial disturbanes as the roket enters atmospheri flight. The U.S. Army is urrently developing the Advaned Preision Kill Weapon System (APKWS) to fill the gap between very high ost preision guided missiles, suh as the

17 2 HELLFIRE missile, and muh lower ost, but very poor preision diret fire rokets suh as the HYDRA 70. [4] [5] Improved target dispersion would make diret fire rokets more effetive against harder targets, and ultimately provide more hits per number of rokets fired. Positive side effets of inreased auray are redued ollateral damage as well as improved effiieny of military operations. The urrent primary andidate for APKWS is a semi-ative laser guided munition that is dependent upon an external lasing soure, and therefore suseptible to laser ountermeasures. The work reported in this thesis seeks to redue dispersion of an atmospheri diret fire roket using ative feedbak ontrol. Feedbak is provided by an onboard inertial measurement unit (IMU) whih is insensitive to enemy laser, infrared, or eletroni ountermeasures. A ring of lateral pulse jets mounted on the skin of the roket near the nose is used as the ontrol atuator. Eah individual pulse jet imparts a relatively large, short duration lateral fore on the projetile body. Due to the limited number of very short ontrol fore inputs, this flight ontrol system design problem falls out of the realm of onventional ontinuous flight ontrol. A new flight ontrol system design methodology is developed that uses an internal dynami model of projetile motion to aid pulse jet firing logi. Simulation results to establish the utility of the new flight ontrol system design methodology are generated for an exemplar diret fire roket whih is similar to the HYDRA 70 roket. Parametri trade studies are onduted that onsider the effet of the number of pulse jets, impulse magnitude, and modeling unertainty on impat point dispersion. The thesis is organized as follows. Chapter 2 provides an overview of previous published researh involving pulse jet ontrol of projetiles and spaeraft. Chapter 3 desribes, in detail, the six degree of freedom non-linear dynami model that serves as the basis for our numerial results. Also, Chapter 3 develops a losed-form solution of projetile motion whih is used in the feedbak ontroller. In Chapter 4, a pulse jet flight ontrol system is developed whih is fundamentally based on diretly improving impat point errors in flight. Resulting improvements of impat point dispersion are shown as well as several

18 3 parametri studies that provide for the refinement of ontrol system parameters. In Setion 4.4, we demonstrate that the pulse queuing sheme, and linear preditor used in the suboptimal sheme ombine to make ontrol of the projetile impat point ompletely analogous to a simple one-dimensional optimal ontrol problem. In Setion 4.5, we employ deision grids to develop a rule-based optimal ontroller, and show dispersion results. We show a novel global searh optimal ontroller with limited results in Setion 4.6. These results serve as an upper limit on ahievable performane. In Setion 4.7, we selet a andidate ontroller for final optimization design and robustness testing. We present our optimization results in Chapter 5, where repeated time simulations are used to ompute a ost funtion whih we then optimize by a global searh of the parameter spae.

19 4 CHAPTER 2: LITERATURE REVIEW A signifiant body of literature related to lateral pulse jet ontrol of smart diret fire munitions is available. The reported literature an loosely be organized into three ategories, namely, spaeraft pulse ontrol, general pulse modulation theory, and projetile pulse ontrol. The following three setions outline representative and relevant previous work in these ategories. 2.1 Spaeraft Pulse Jet Control Bennighof, Chang, and Subramaniam [6] and Bennighof, and Subramaniam [7] have demonstrated minimum time optimal ontrol shemes for flexible strutures using pulse response. Their work involves determining the optimal ontinuous ontrol input from pulse response using onvolution sums, and results in a linear system of equations whose dimension is determined by the number of time steps required for ontrol. Bennighof et al [6, 7] use the pulse response in lieu of an infinite-dimensional model of the flexible struture dynamis, thus simplifying the modeling proess, while inluding all pratial vibration modes. They onluded that the minimum time ontrol profile for a flexible struture an be omputed with a modest amount of omputation. Singhose, Bohlke, and Seering [8] have proposed fuel-effiient pulse ommand profiles for flexible spaeraft in whih the ommand profiles onsist only of positive or negative onstant-amplitude pulses. They apply their tehnique to a very simple mehanial model and obtain optimal ommand profiles for the rest-to-rest slewing of spaeraft. Their resulting ommand profiles are muh more fuel-effiient that ompeting time-optimal profiles, with an insignifiant time penalty. They also investigated the robustness to modeling errors of both the time-optimal and fuel-effiient ommands onluding that fueleffiient profiles are always as robust as the time-optimal profiles.

20 5 More reently, Phillips and Malyeva [9] developed a tehnique for pulse motor optimization using mission harts for an exoatmospheri intereptor. They were able to optimize the pulse split and interpulse delay for the pulse motor of an exoatmospheri midourse stages. They developed the relationships for the earliest and latest time-to-go to interept for burnout of the seond pulse. This work has some relevane to our urrent problem in that, their objetive was a diret hit on a target of military importane, and the optimization objetive was to determine the best firing times for a pair of propulsive pulses. 2.2 Optimal Control of Pulse-Width and Pulse Frequeny Modulated Systems The topi of optimal ontrol of pulse width and pulse frequeny modulated systems has been studied for many years due to appliations in eletronis. One partiularly good referene regarding pulse frequeny modulated systems is Pavlidis [10]. Through heuristi arguments, he derives the ontrol law for minimum time and minimum fuel problems applied to a simple linear seond-order system. The result is a swithing surfae. Onyshko and Noges [11] formulated the optimal ontrol funtion for pulse-frequeny modulated systems using a Modified Maximum Priniple. They found that the first variation of the ontrol funtion is of a speial form, namely the removal, addition, polarity reversal, or time shift of the pulses. Thus, extremizing the ost funtion with respet to the ontrol is not allowable beause the ontrol funtion is disontinuous. thus, the ost funtion must be extremized with respet to the ontrol by a hoie of: 1) inspetion, 2) an iterative proedure, or 3) trial and error. They present results for a simple first-order linear system. Vander Stoep and Alexandro [12] extended the optimal ontrol theory regarding pulse frequeny modulated systems to inlude those with ost funtions onsisting of weighted sums of quadrati terminal state error. They optimized the performane index by enumerating and omparing the 3 N possible sequenes of ontrols orresponding to positive, zero, and negative

21 6 polarity of possible pulses. We investigate the limits of performane using a similar tehnique in Setion 4.6. Onyshko and Noges [13] also investigated the appliation of dynami programming to pulse-frequeny modulated systems. They found it neessary to impose a restrition on the ontrol funtion, namely that pulses may only our at predetermined instants of time whih are separated by a time interval equal to or larger than the pulse width plus dead time. The searh proess for general systems of medium to large dimension is awkward and lengthy. If the augmented system equations are separable and linear with respet to the state variables, the searh proess is simplified beause the optimum ontrol is independent of initial values of the state variables. Niolleti and Raioni [14] proposed a omputer algorithm to optimize the ost funtions derived by Onyshko and Noges [13]. Their work redued the optimization problem to a Boolean linear programming algorithm. They onluded that even problems with ompliated inequality onstraints an be treated by a Boolean linear programming ode, the linear programming approah has a signifiantly smaller omputer storage requirement when ompared to dynami programming, and redution to the Boolean linear programming problem allows the problem to be solved with ommerially available odes. Elgazzar and Onyshko [15] demonstrated that for a ertain lass of pulse-frequeny modulated systems, omputation of the performane index may be simplified by a similarity transform to a anonial form. This simplified algorithm provides for effiient omputation of the optimal ontrol by a global searh over the set of all possible ontrols. 2.3 Projetile Pulse Jet Control Srivastava has published several works fousing on omputational fluid dynami (CFD) modeling of supersoni missiles with pulse jet ontrol. [16, 17] He points out that lateral jet ontrol has been a researh interest for many years partiularly in the field of

22 7 high-altitude, atmospheri, super-soni air-to-air missiles, however, his fous is primarily on modeling the flow fields and aero-oeffiient onsequenes from pulse jet ontrol. He found that urrent CFD methodology does provide aurate preditions of the fore and moment oeffiients at widely varying flow onditions. A more relevant ontribution was his disovery that windward oriented jets have a low amplifiation fator. Linear Projetile theory is a method of simplifiation whereby the non-linear six degree of freedom equations of motion for a symmetri projetile are linearized and partially deoupled. We reserve detailed disussion of the proedure itself to Chapter 3. As early as 1909, [18] ballistiians reognized that symmetri projetile yawing and pithing motion onsists of two damped sinusoids. Kelley and MShane first formalized notion of projetile linear theory in [19] More reently, MCoy [20] summarized the linear projetile theory for epiyli yawing and pithing motion of the projetile as well as swerve in his exhaustive book on the history and theory of exterior ballisti modeling. Guidos and Cooper [21] extended the linear theory solution to obtain a losed form solution of projetile motion subjeted to a simple in-flight lateral pulse. They demonstrated that the singular impulse or Dira delta model of impulse fore is suffiient to model the response due to finite duration pulses. Their simulation results were speifi to a non-rolling finned projetile with negligible drag and gravity. The above three referenes on linear theory all use a omplex yaw angle, and oordinate systems at variane with typial airraft dynamis oordinate systems. Burhett, Peterson, and Costello [22] extended the linear theory for a seven-degree of freedom projetile with singular impulse ontrol, inluding drag and gravity, and demonstrated good agreement with a non-linear numerial simulation that inluded finite duration pulses. The seven degree of freedom projetile has two body setions onneted by a visous bearing suh that the fore and aft setions of the projetile may have different roll rates.

23 8 Jitpraphai and Costello [23] demonstrated that impat point dispersion of a diret fire roket an be drastially redued using a trajetory traking feedbak ontrol system with pulse-jet atuators. They found that dispersion redution is strongly dependent upon pulse jet size and number, and the size of allowable deviation from desired trajetory. 2.4 Assessment of Related Work Linear theory has typially been used for stability analysis, and aero oeffiient estimation from range data. In this work, we require a omputationally low-ost preditor of the projetile trajetory. Thus, we apitalize on the existing linear projetile theory with some rudimentary extensions. The result is a set of losed-form expressions that an be repeatedly solved for a piee wise linear predition of the trajetory. We are able to predit the projetile impat point to an arbitrary degree of auray by inreasing model updates. The flight ontroller uses a number of model updates that ensures omputational effiieny without undue sarifie of auray. Our urrent work is related to time-optimal problems in that we seek to minimize a ost funtion (namely some average miss distane) after exerising pulse ontrol over a finite set of ontrol opportunities. Our dynami model involves a rigid body, and has finite dimension. We are investigating a speifi ontrol atuator, however whih exerts ontrol pulses of a onstant magnitude and duration. We will investigate the pulse response of our finite-dimensional system in order to determine andidate ontrol strategies. Again, although similar, the work of Singhose et al differs from our in two key areas. 1) We are dealing with atmospheri flight, the essential effet of whih is to add damping to the system, so ontrol effort is not required to 'de-spin' the system. 2) Our ontrol ations will have fixed width as well as fixed amplitude. Signifiant differenes between the work of Phillips and Malyeva and ours are that 1) they were onsidering exoatmospheri flight, so the dynami motion equations are

24 9 somewhat simplified, 2) the ontrol authority was provided by a gimbaled roket motor, thus the ross-range omponent of pulse ould be varied from zero to 470 m/s, and 3) their target was moving with signifiant veloity with respet to that of the missile--our missile travels at speeds exeeding Mah 2, and engages ground vehiles as targets, thus the target ground speed is negligible, and assumed to be zero in out urrent work. Srivastava's work, has little relevane to the urrent task of prediting and ontrolling missile trajetories that use pulse jet ontrol low in the atmosphere. Our urrent problem involves ontrolling a projetile with fixed-width, fixedamplitude pulses. Thus, we annot formulate the problem as a pulse-width modulation problem. Essentially, we are dealt a pulse-frequeny modulation problem, where we have a finite number of pulses, thus a limited frequeny, or fuel-optimal problem. Pavlidis' approah of identifying optimal swithing surfaes is ompletely relevant to our urrent problem. Our system, however is muh more ompliated in its dimensionality, and inherent non-linearity. We will attempt to determine optimal swithing surfaes through Monte Carlo simulations, and trade studies.

25 10 CHAPTER 3: DIRECT FIRE ROCKET MATHEMATICAL MODELING 3.1 Diret Fire Roket Non-Linear Dynami Model The numerial simulation used in this study models the roket as a rigid body desribed by six degrees of freedom. This type of model is typial for atmospheri flight dynamis of projetiles, rokets, and missiles. The roket state onsists of three position omponents of the roket enter of mass and three Euler orientation angles of the body. A shemati of the diret fire roket with major elements of the system highlighted is given in Figure 3.1. The equations of motion are provided in Equations [20] SrpS(}C'If - CrpS'If n[.~.s.~ + s.s~j{"} (3.01 ) Y = (}s'if srps(}s'if + rp'if Crps(}s 'If - Srp 'If v Z -s(} srp(} rp(} w m~[~ In.'. rp -srp q Srpt(} srp / (} rp / (} r (3.02) m~~m-llq -r 0 P -;l{:} (3.03) m~[irl[{~}-uq -r 0 P -;}I1r}] (3.04) where [In Ixy [I] = Ixy Iyy I" I yz J Ixz IyZ I zz (3.05)

26 11 The applied loads appearing in Equation 3.03 onsist of roket weight (w), air loads (A), main roket thrust (R), and lateral pulse jet fores (J) omponents. (3.06) The roket weight omponent is given by Equation (3.07) The aerodynami fores for a fin stabilized projetile are given by the following equations. (3.08) (3.09) (3.10) The steady body aerodynami moment is omputed as a ross produt between the distane vetor from the enter of gravity to the enter of pressure and the steady body aerodynami fore vetor above. A WLCG - WLCOp LS BLop - BLCG]{XSA} (3.11) } [ 0 MSA = WLCOp - WLCG o SL CG - SLCOp Y SA { N SA BLCG - BLCOp SL COp -SLCG o ZSA The unsteady aerodynami moment omponents are given by Equations below.

27 12 (3.12) Tr 2 3[ qd ] MUA =-pvad --C MQ (3.13) 8 2V A (3.14) The total aerodynami moment is the sum of steady and unsteady aerodynami moments. (3.15) Lateral PUI\ Jet Ring.oIll~..._. ith Pulse Jet 5 Figure Shemati of a Diret Fire Roket with Lateral Pulse Jets.

28 13 The enter of pressure loation and all aerodynami oeffiients (CNA,CXO,CX2,CLP,CLDD,CMQ) depend on loal Mah number and are omputed by linear interpolation. The roket motor inreases the veloity of the roket by providing high thrust levels during the initial portion of the trajetory. In the urrent work, we assume a single roket motor with thrust omponent passing through the roket enter of gravity, and parallel to the roket axis of symmetry. The amplitude profile is a known funtion of time. The lateral pulse jet fores are modeled in the same way as the main roket motor with two exeptions. Sine the lateral pulse jets are ative over a very short duration of time, the thrust fore is modeled as a onstant when ative. Also, sine by definition a lateral pulse jet ats in the lb and kb plane, the ib omponent of the lateral pulse jet fore is zero. Equation 3.16 provides the lateral pulse jet fore formula. (3.16) When the roket motor is ative, the mass, mass enter loation, and inertial properties of the roket are updated ontinuously as a funtion of time Diret Fire Roket Linear Dynami Model The dynami model developed above is highly non-linear and not readily solved analytially. Through the following series of valid, yet simplifying assumptions a set of analytially solvable linear differential equations emerge. A) Change of variables from fixed plane, station line veloity, u, to total veloity, V. Equations 3.17 and 3.18 relate V and u and their derivatives.

29 14 (3.17) v=(uu+ vv+ ww)jv (3.18) B) Change of variables from time, t, to dimensionless ar length, s. The dimensionless ar length, as defined by Murphy [24] is given in Equation [3.19] and has units of alibers of travel. 1 t s=-fvdt (3.19) Do Equations 3.20 and 3.21 relate time and ar length derivatives of a dummy variable s. Dotted terms refer to time derivatives and primed terms denote ar length derivatives. (3.20) ~ =(VI D)2(S" + S'V' jv) (3.21) C) Euler yaw and pith angles are small so that sin(e)""e (3.22) os( e) "" 1 (3.23) sin( lfi) "" lfi (3.24) os( lfi) "" 1 (3.25) D) Aerodynami angles of attak are small so that a""wjv (3.26)

30 15 f3 z vjv (3.27) E) The projetile is mass balaned, suh that the enter of gravity lies on the rotational axis of symmetry. I xy = I xz = I yz =0 (3.28) I zz = Iyy (3.29) F) The projetile is aerodynamially symmetri suh that G) A flat fire trajetory assumption is invoked suh that the fore of gravity is negleted in the total veloity equation. Gravity is inluded in the epiyli yawing and pithing equations, and the swerve equations. H) The quantities V, and l/> are large ompared to e. lji. q, r, v, and w, suh that produts of small quantities and their derivatives are negligible. The result is equations x'= D (3.30), D D Y =Vv+tp: (3.31) D. z' =-w+ (}D V (3.32) l/>' = D p (3.33) V

31 16 (3.34) (3.35) (3.36) And 3 2 I _ psd CLP + psd V C (3.37) P - 41 P 21 LDD xx xx The matrix equation for epiyli pithing and yawing is: v' v 0 W' w G g 0 q' q r' r 0 (3.38) Where -A 0 0 -D (3.39) A D 0 C 0 E -F D -C - 0 F E D And A=pSD C NA 2m (3.40) 2 C=pSD C MA 2Iyy (3.41)

32 17 3 E=pSD C MQ (3.42) 4Iyy D (Iup) (3.43) F=--- Vo Iyy G=~ (3.44) Vo (3.45) And, D is the projetile harateristi length (or diameter). The solution to these linear equations is the sum of a partiular solution due to the gravity onstant, and a homogeneous solution. The partiular solution is given by setting the derivatives equal to zero and solving the resulting algebrai equation: Vp Wp qp rp -- --I 0 G --'"... g 0 0 (3.46) Resulting in: Vp wp qp rp Gg det(3) -FC EC+AF 2 +AE2 -(AE+ C)CjD AFCjD (3.47)

33 18 Where: det(3) = A2F2 +A2E2 +2AEC+ C 2 (3.48) This partiular solution is then subtrated from the initial onditions prior to solving for the homogeneous response: X= Vo Wo qo ro Vp Wp qp rp (3.49) The homogeneous response is governed by the mode shapes: i -l -l (3.50) [ ~l ~2 ~l* ~;] = K+{Q K-{Q R+.fS R-.fS 2D 2D 2D 2D i(k +{Q) i(k -{Q) i(r+.fs) i(r-.fs) 2D 2D 2D 2D where, K = ( E - A) + 2A + if (3.51) Q = (E - A)2 + 4AE+ 4C- F2 + 2i(F(E - A) + 2(AF+ B)) (3.52) R = ( E - A) + 2A - if (3.53)

34 19 Reognizing that {K, R} and {Q, S } are omplex onjugate pairs, we an define: K+.JQ i(k+,fqr g, ={i 1 2D 2D g; ={~i 1 R+-fS 2D i(r+fs)r 2D K-.JQ i(k ~,fq)r g2 ={i 1 2D 2D R--VS i g; ={~i (R~v'S)r 1 2D 2D Also, we define W as the orresponding matrix of left eigenvetors. iv -v 1 1 (-v+,u) (-v+,u) (-v+,u) (-v+,u) 17[ -if.1 -f.1-1 -i 1 (-v+,u) (-v+,u) (-v+,u) (-v+,u) 17I T* = 2. * IV v* -1 1 '11 T* 112 (v*-,u*) (v*-,u*) (v*-,u*) (v*-,u*) -l,u. * -,u * 1-1 (v*-,u*) (v*-,u*) (v*-,u*) (v*-,u*) (3.54) Where:

35 20 V= (K+~) (3.55) 2D And: (K-~) (3.56) f.1 = 2D The homogeneous solution is found from the dyadi deomposition of the matrix exponential: (3.57) also, (3.58) where (.)*j denotes the j th row of the rank one outer produt of left and right eigenvetors. (3.59) (3.60) And the artangent is a four-quadrant artangent (3.61)

36 21 where: a=v,w,q,r (3.62) and: j = 1,2,3,4 (3.63) or, more expliitly: (3.64) (3.65)

37 22 () wf = tan 2Re(~I1Jr) -I *2 T X J ( -2 Im( ~l1ji )*2 X (3.67)

38 23 In order to update the linear model, and provide the solution after pulsing, we require losed form solutions of If' and (J. These are provided by substituting Equations 3.66 and 3.67 into Equations 3.34 and 3.35 respetively and integrating. The results are Equations 3.68 and ' = 11'0 + ~ (r ps) + ~ ( 7: [/1' ost~fs + Oif) - os Oif ]+ ~ [/1' sin( ~fs+ Oif) - sin Oif ])nif (3.68) + ~ ( -~s [iss os( l/>fs + (Jrs) - os (Jrs] + ~: [iss sin( l/>ss + 8rs )- sin 8rs ])n rs 8 = 80 + ~ (qps) (3.69) +~(7: [/1' ost ~fs+ Oqt )-osoqt]+ ~; [/1' sin ( ~fs+ Oqt )-sinoqt])nqt + D (-l/>s [/,ss os( l/>fs + 8qs )- os 8qs ]+ As [ea-ss sin( l/>ss + 8qs )- sin 8qs ]Jn qs V ~ ~ The orresponding swerve equations are

39 24 y(s) = Yo +(11'0 +~)S D D Vo and +[PSD] (C XO -C NA ) [ny (exp{ajs)sin( js+ OVj - n) -sin(ovj - n) - j os(ovj - n)s)j 2m Vo j +[PSD] (C xo -C NA 2m Vo s ) (n~s (exp{a:s)sin( ss + 0vs - n) - sin(ovs - n) - s os(0vs - n)s)) (3.70) z(s) = Zo +(-00+ WO)S+gD[ m ]2(exp([PSD]xos)_[PSD]CXOS-l) D D Vo psdvocxo m m +[PSD] (Cxo -C NA ) [ny (exp{ajs)sin( js+ 0wf - n) -sin(owj - n) - j os(0wf - n)s)j 2m Vo j xo -C NA +[PSD] (C 2m Vo s The roll rate solution is ) (n~s (exp{a:s)sin( ss + Ows - n) - sin(ows - n) - s os(0ws - n)s)) (3.71) (3.72) where The forward veloity solution is SD3C ) ' =exp P LP S (3.73) ( 4Ixx V(s) = voexp(_psdc xo s)+ 2X R (l_exp(_ps~cxo s)) (3.74) 2m pscxovo m for powered flight and

40 25 (3.75) for gliding flight. The linear preditor is used in the ontrol system as follows. The projetile urrent state is used to initialize the preditor. The remaining downrange distane is then broken into several segments of equal arlength distane. The predited impat point is solved by repeatedly omputing the system state using Equations 3.64, 3.65, 3.66, 3.67, 3.68, 3.69, 3.70, 3.71, 3.72 and 3.74 or 3.75 for eah segment of the flight. At the beginning of eah segment, the aerodynami oeffiients are updated based upon the predited mah number at the beginning of that segment. The segments are smaller for the powered portion of flight as the roket aelerates rapidly, and the roket mass properties are hanging. The linear model uses a linear table look-up to determine roket mass properties for eah segment. The losed-form predition of trajetory with a lateral pulse is a rather simple extension to the linear theory presented above. The impulsive fore may be expressed as a vetor in the no roll frame as follows YJnr} =[Cl/J Sl/J { ZJnr The moment due to this impulse is then given as a ross produt of the position vetor from projetile enter of mass to point of impulse appliation. { ~:nr}=[~ -~Ixl{~~r} NJnr 0 o ZJnr

41 26 Thus, the epiyli pithing and yawing equation may be rewritten to inlude the pulse as a funtion of downrange distane. Y/nr -A 0 0 -D m v' v Z/nr w' 0 -A D 0 w m q' C + -R1xZ/nr 0 E -F q r' D r Iyy C 0 F E R1xY/nr D Iyy o(s) (3.76) Where A, C, D, E, and F are as stated above. We take the LaPlae transform of Equation 3.76 to obtain Equation s+a 0 0 D V(s) W(s) 0 s+a -D 0 Q(s) C 0 s-e F R(s) D C 0 -F s-e D -1 Y/nr --+vo (3.77) m Z/nr --+WO m -R1xZ/nr +qo Iyy R1xY/nr +ro Iyy Equation 3.77 illustrates that the net effet of pulsing is to add a omponent to the system initial onditions. Thus, our trajetory solution after pulsing will be found simply by amending initial onditions in the epiyli yawing and pithing equation. Note that in aordane with engineering intuition regarding impulse-momentum problems that the system veloities will hange immediately with impulse. The projetile position represented by the Euler angles, downrange distane, and swerve variables remained unhanged during the pulse. Also, the total veloity will be unhanged sine we assume the pulses to have no body ib axis omponent.

42 27 The seond-order nature of our disrete dynami model demands that we ompute the entire projetile state using equations 3.64, 3.6S, 3.66, 3.67, 3.68, 3.69, 3.70, 3.71, 3.72, and 3.74 or 3.7S at t+ to aurately aount for the jump onditions in the epiyli solution formulae. That is, the impulse method of modeling the ontrol pulses leads to an immediate disontinuity in the veloity states, but not the position states. The entire state must then be alulated at a point losely following, in time or downrange travel, the point of impulse appliation in order to insure aurate integration of the disontinuity into the position states. Note that the auray of impat point predition will be dependent on the rate at whih aerodynami oeffiients are updated, and all model oeffiients dependent upon aerodynami oeffiients. 3.3 Comparison of Non-Linear and Linear Models Figures ompare the results generated using the non-linear and linear models with a full non-linear simulation for a non-thrusting flight trajetory. The numerial solution for the non-linear simulation was generated using a fixed step 4th order Runge Kutta algorithm with a time step of se. The linear model dynami trajetory is generated using the losed-form solution developed in the previous setion. The projetile onsidered for this omparison is a 4.S ft long fin stabilized projetile with a total weight of 23.0 lbf and a enter of gravity loation of 2.5 ft from the base. The projetile has 3 fins. The roll and pith inertia of the body is O.OOS slug-ft 2 and 1.4 slug_ft2. A lateral pulse jet loated a distane of ft from the projetile base and on the skin of the projetile along the kb axis is fired at s=439 ft. The total impulse of the jet is 4.5 lbf-se. The projetile is launhed with the following initial onditions: Xo=O ft, Yo = ft, Zo = ft, Jo = deg, 8 0 = 1.38 deg, lfio = deg, Vo = 2218 ft!se, Vo = 0.16 ft/se, Wo =-1.37ft/se, Po =-S1.6rad/se, qo = rad/se, and ro =-0.OOO9rad/se. Just prior to the pulse jet firing, the state of the projetile is: X- = 439 ft, y- = ft, Z- = -48.0

43 28 ft, l/>- =54.8 deg, (F =1.26 deg, 1jI- = deg, V- =2160 ft/se, V- =0.126 ft/se, W- = ft/se, p- =-60.6 rad/se, q- = rad/se, and r- = rad/se. The lateral pulse jet indues a jump suh that the resulting translational and rotational veloity omponents are: V+ =4.42 ft/se, W+ =-16.3 ft/se, f =-3.46 rad/se, and r+ = rad/se. Figures show the states of a linear model with parameters updated every ar length of travel. Figure 3.2 shows the ross range versus range. The agreement is exellent. 5r r ~ o ~Non-lihear.-.;-. Unear '-'"' (l) C> C ~ -15 en en e :...: ~--~----~----~--~----~----~----L---~----~ o Range (ft) Figure Cross Range vs. Range, (Linear Model Update Rate = 1 aliber) Figure 3.3 shows the altitude versus range. The agreement is slightly worse than that of ross range. Note that the y-axis saling is muh larger whih de-emphasizes the differene

44 29 between linear and non-linear values. When viewed from the rear, the projetile drifts slightly to the left prior to the pulse jet firing. After the pulse jet is fired, the projetile turns and drifts to the left. By the end of the simulation, the lateral pulse jet has hanged the ross range impat point by approximately 34 ft. The pulse jet effet on altitude response is not obvious. For this ase, trajetory bending due to gravity dominates the altitude time response. 80r ,, ~ 60 -;-Non-linear.-.:-. Linear ',' g o <ll -g -20 '" -'.p <i: , ,... '...'...' ~---L----~----~--~-----L----~----L---~----~ Range (ft) Figure Altitude vs. Range, (Linear Model Update Rate = 1 aliber)

45 30 As shown in Figure 3.4, the veloity of the projetile is redued signifiantly over the trajetory r----~--_,----_r----._--_.----~----._--_ :Non-linear.-.:-. Linear: _2000 ~ ~ u ~ 1800 > (U (5 ~ Range (ft) Figure Total Veloity vs. Range, (Linear Model Update Rate = 1 aliber)

46 31 As shown in Figures 3.5 and 3.6, prior to the pulse jet firing the rossing veloity omponents remain small and after firing fairly large exursions are indued. w -: (1) ~----~----~----~ ~----~----~!!! --Non-linearl 1 ---:' Unear: I ;S 100 : o a. E o 50 o ~.(3 0 r-- o Q) > > -g -50 CO --_...., , ~--~----~--~-----L----~--~----~----~--~ Range (ft) Figure Body Side Veloity J Component vs. Range, (Linear Model Update Rate = 1 aliber)

47 ,, ,----,,-----, l -:Non-linearl --.:-- Linear: J a o. E o 'ḡ > Q) > >- -50 '0 o CD ~ r- -~ ~~~----~--~~--~----~--~----~----~--~ Range (ft) Figure Body Side Veloity K Component vs. Range, (Linear Model Update Rate = 1 aliber)

48 33 The fins along with the inherent roll damping ombine to reate the body roll rate trae shown in Figure r ,, ~ '- - --' '- -30 en =0-35 g 2-40 <tl a:: 0-45 a:: -50,, ---_... -._---_ _ _ :Non-linear ~-.:~-. UriE!ar: _65~--~-----L----~--~-----L----~----L---~----~ o Range (ft) Figure Body Roll Rate vs. Range, (Linear Model Update Rate = 1 aliber)

49 34 Similar to the rossing veloity response, Figures 3.8 and 3.9 show small pith and yaw rate before the impulse and larger angular rate vibration after the applied impulse ~----~----~----~----~----~----~----~----~ 2 --'-Non-Iihear._.:-. Linear: 1 -~ "0 ~ (J) CiiO II:.. B 0:: ~----~----~----~----~----~----~----~-----L----~ o Range (ft) Figure Pith Rate vs. Range, (Linear Model Update Rate = 1 aliber)

50 35 2 -" Non-linearl.-.:... Linear: J 3~ r ~ l ttl a: ~ > -1 " : A : : \JWv~~~ \I: : : Range (ft) Figure Yaw Rate vs. Range, (Linear Model Update Rate = 1 aliber)

51 36 Figures 3.10 and 3.11 show that the ation of the pulse jet indues Euler angle hanges exeeding 5 degrees. 0.08r ,, ,,---~ 'Non-linear.-.:-. LInear: 0.04 ~ tu.!:: ~ C) :: «..t: o.b ::: ~ ::J UJ L '-----' '---L--.l l.----L---..l l Range (ft) Figure Pith Angle vs. Range, (Linear Model Update Rate = 1 aliber)

52 ~----~-----r----~----~ ~----~----~ ~ Non-linearl I.-.:... Linear: I ~ ~ Q) Cl «~ 0.02 ~ ~ ::::l w O~ '" -'---'---,"..'1....: '" -_... _ , _, Range (ft) Figure Yaw Angle vs. Range, (Linear Model Update Rate =1 aliber)

53 38 Figures 3.12 through 3.21 demonstrate that we an aurately ompute the state of the projetile at a disrete set of points that are widely separated in time. This is espeially evident in Figures 3.15 and 3.16 where we plot the veloities only where the linear model is updated. In this ase, the model updates are separated by 1965 alibers of down range travel ~----~----~----~----~----~ ~ o ~Non-linear.-.:-. Linear: -5 g-10,. Q)... ~ Ol.' -15 '-.., () e "'... :... (J), (J), _35~--~-----L----~--~~---L----~----L- ~~ ~ o Range (ft) Figure Cross Range vs. Range, (Linear Model Update Rate = 1965 alibers)

54 :Non-linear....--:::-::.l,.i.t:lear: _. g Q) "0 0 :J- -20.;:: <i _._-_...'... _ _... _._'_.. _--_.. ' _... "'"....., , ~--~----~----~--~~--~----~----~--~~--~ Range (ft) Figure Altitude vs. Range, (Linear Model Update Rate = 1965 alibers)

55 ,, :Non-linear._.:-. Linear: _2000 ~ ~ u ~ 1800,. (ij,. (5.... '. '". I :~ Range (ft) Figure Total Veloity vs. Range, (Linear Model Update Rate = 1965 alibers)

56 41 Figures 3.15 and 3.16 show good agreement between linear and non-linear models. 200r ,, ~ 150 en ~ 100 C a.> :: ", :'. o I :'. a. 50 E o () ~.(3 a o Q5 > > -g -50 III..., -100.!..'..'. I I I \ -:-Non-Iinear 0: LInear -1500~--~----~----~--~~--~----~----~--~~--~ Range (ft) Figure Body Side Veloity JComponent vs. Range, (Linear Model Update Rate = 1965 alibers)