Basic Principles of Homing Guidance - The Johns Hopkins ...

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N. F. PALUMBO, R. A. BLAUWKAMP, and J. M. LLOYD = cos –1a c a M Missileacceleration bodydefinitions: _ 1 ris a unit vector along the missile–targetLOS, _ 1 xis a unit vector along the missile centerline(missile longitudinal axis), _ a cis the guidance lawacceleration_command perpendicular to the LOS, andaM = [a Mxa Mya Mz] T is the achieved missile accelerationvector comprising components in the x/y/z axes.Two component directions serve as a basis for thedevelopment of GCP1: the direction of the LOS ( _ 1 r)and the commanded guidance direction ( _ 1 ac= _ a c/ _ a c).Assuming perfect interceptor response to commandedacceleration (i.e., no lag from commanded to achievedacceleration), the (guidance preserved) missile accelerationcan be expressed as follows:_aM = k 1_1r + k 2_1ac . (43)The design parameters, k 1and k 2, are determined subjectto the following constraints:a : 1 = aa : 1 = aM x MxMa cc.(44)Equations 44 analytically embody the following constraints:(i) the missile does not control longitudinalacceleration and (ii) the component of missile accelerationalong the guidance direction, _ 1 ac, must be thatspecified by the guidance law ( _ a c). Using Eqs. 43 and44 and given _ 1 r• _ 1 ac= 0 (i.e., the guidance commandis perpendicular to the LOS), an expression for commandedmissile acceleration that will preserve the guidancecommand is given in Eq. 45:aGLaMx – ac : 1x= e o 1 a .1 : 1r +c(45)rx1 r1 xLOSMissilecenterlinea M a c PN guidance command LOSa M Achieved missile accelerationFigure 15. GCP1. The LOS vector is shown along with the commandedacceleration from PN, which is perpendicular to the LOS.Also, the missile centerline (direction of the missile nose) andachieved missile acceleration vector are superimposed. The guidancepreservation problem is to achieve missile acceleration thathas the component along the guidance direction as specified bythe guidance law.Equation 45 is derived independently from any particularcoordinate system. It is easy to mechanize the expressionin Eq. 45 such that the computations are carried outin the missile body frame (B). We first note that missileacceleration resolved into the missile body frame cana MBBMxBMyB TMzbe expressed as = [ a a a ] and that thebody-referenced unit vector along the longitudinal axisof the missile can be written as _ 1B x= [1 0 0] T . Thequantities _ 1 r(LOS unit vector) and _ a c(guidance lawcommand) usually are defined with respect to a guidancereference _ frame. To reflect this fact, we expressthem as 1G r= [1 0 0] T _and acG= [0 aG cya G cz] T .Here, the superscript denotes the frame of referencewith which the quantities are currently expressed; inthis case, the guidance reference frame (G). Thus, wemust resolve all guidance frame quantities into themissile body frame through the coordinate transformationCB G_= c B G(i, j), i, j = 1, 2, 3; for example, 1rB=C B G[1 0 0] T ≡ [c B G(1, 1) c B G(2, 1) c B G(3, 1)] T . Note thatif we fix the guidance frame at the start of terminalhoming (t = 0), we obtain the transformation CG I=C B I(C BS t = 0). The inertial-to-body transformation, C B I,comes from the IMU, and CSBmay be computed via theseeker gimbals (see Eq. 23). Subsequently, CB G= C B I(C G I) T .If necessary, the guidance frame may be updated frominstant to instant, but this adds computational complexity.Given that CB Gis available, the following mechanizationof Eq. 45 is possible:aBGCBBaMx– ac: 1x= rBaBfrBB p +c: 1xR V RB c ( 1, 1)aa – a S W SMx cx/ f c ( 2, 1)aB pSW + ScG( 1, 1)SSc( 3, 1)W SaT X TBB G B G B cxBBG B cyBczVWW .WWX(46)In Eq. 46, [aBcxaB cya B cz] T is the guidance commandvector expressed in the missile body frame. This mechanizationwill generate guidance commands in the missilebody frame that satisfy the constraints in Eq. 44.MIDCOURSE GUIDANCESo far, this article has been chiefly concerned withthe requirements of terminal homing, wherein targetmeasurements are provided by one or more onboard terminalsensors and minimizing miss distance at interceptis the primary objective. For completeness’ sake, we nowwill briefly discuss issues and requirements associatedwith the midcourse phase of flight.During the midcourse guidance phase of a multimodemissile (see Fig. 16), target tracking is performedby an external sensor to support the engagement. 6 Externaltracking relaxes the requirements for the onboardsensor to point at the target and detect it at large ranges.38JOHNS HOPKINS APL TECHNICAL DIGEST, VOLUME 29, NUMBER 1 (2010)
BASIC PRINCIPLES OF HOMING GUIDANCEBoost (Inertial)Midcourse (Guided)Terminal (Guided)• Onboard inertial guidanceprocessing• Safe launch/separation• Boost to flight speed• Establish flight path• Arrive at pre-calculated pointat end of boost (EOB)EOBIn-flighttargetupdatesHandoverSeekeracquisitionPredicted interceptpoint (PIP)EndgameFuzing• Off-board target tracking• Onboard or off-board guidance processing• One or more additional booster stages• Maintain a desired course• Bring the missile “close” to the target• Trajectory shaping/energy management• Can be active endo through exo• Onboard seeker/guidance processing• Requires high degree of accuracy and fastcommand response• Can require maneuvering to maximumcapability to intercept fast-moving,evasive targets• Active endo or exo, typically not bothPre-launch• Targeting• Targeting• Engageability • Engageability determinationdetermination• Missile • Missile initializationinitializationFigure 16. Missile guidance phases. The weapon control system first decides whether the target is engageable. If so, a launch solution iscomputed and the missile is initialized, launched, and boosted to flight speed. Inertial guidance typically is used during the boost phaseof flight during which the missile is boosted to flight speed and roughly establishes a flight path to intercept the target. Midcourse guidanceis an intermediate flight phase whereby the missile receives information from an external source to accommodate guidance to thetarget. During the midcourse phase, the missile must guide to come within some reasonable proximity of the target and must providedesirable relative geometry against a target when seeker lock-on is achieved (just prior to terminal homing). The terminal phase is thelast and, generally, the most critical phase of flight. Depending on the missile capability and the mission, it can begin anywhere fromtens of seconds down to a few seconds before intercept. The purpose of the terminal phase is to remove the residual errors accumulatedduring the previous phases and, ultimately, to reduce the final distance between the interceptor and target below some specified level.However, the accuracy of sensors generally degrades withdistance. It is therefore unlikely that a standoff sensorwill have sufficient accuracy in tracking both the targetand missile to guide a missile close enough for intercept.The objectives during midcourse guidance are insteadto guide the missile to a favorable geometry with respectto the target for both acquisition by the onboard sensorand handover to terminal homing.During the midcourse guidance phase, the weaponcontrol must provide information to the missile aboutthe threat. This information may be nothing more thanestimates of the threat kinematic states or may include apredicted intercept point (PIP). Predictions of the futuretarget trajectory, whether calculated on board the missileor by the weapon control system, are based on assumptionsof what maneuvers the threat is likely to do andwhat maneuvers are possible for the threat to perform.These assumptions typically are sensitive to the type ofthreat assumed but may include booster profiles, aerodynamicmaneuverability, or drag coefficient(s). The PIPis therefore highly prone to errors. Occasional in-flighttarget updates may be needed to improve the accuracy ofthe PIP before handover to terminal homing.A form of PN might be used during the midcourseguidance phase but generally results in excessive slowdown,however, as a result of the added atmosphericdrag generated by maneuvers. Instead, it is desirable touse a guidance law that will maximize missile velocityduring the endgame such that missile maneuverabilitywill be maximized when called for during stressingendgame maneuvers. Lin 11 describes an approach thatapplies optimal control theory to derive efficient, analyticalsolutions for a guidance law that approximatelymaximizes the terminal speed while minimizing themiss distance. The guidance commands are expressed inthe form given in Eq. 47:K1 a V2 K2 sin( ) cos( ) – V2= sin( ) . (47)RRHere, V is the missile speed, R is the range to thePIP, is the angle difference between the current anddesired final velocity, and s is the heading error. Thefirst term shapes the trajectory to achieve a desiredapproach angle to the intercept, and the second termminimizes miss. The gains K 1and K 2are time-varyingJOHNS HOPKINS APL TECHNICAL DIGEST, VOLUME 29, NUMBER 1 (2010)39
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