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

BASIC PRINCIPLES OF HOMING GUIDANCElevel of (sensor) measurement uncertainties, errors inthe assumptions used to model the engagement (e.g.,unanticipated target maneuver), and errors in modelingmissile capability (e.g., deviation of actual missile speedof response to guidance commands from the designassumptions). Nevertheless, the selection of a guidancestrategy and its subsequent mechanization are crucialdesign factors that can have substantial impact onguided missile performance. Key drivers to guidance lawdesign include the type of targeting sensor to be used(passive IR, active or semi-active RF, etc.), accuracy ofthe targeting and inertial measurement unit (IMU) sensors,missile maneuverability, and, finally yet important,the types of targets to be engaged and their associatedmaneuverability levels. We will begin by developing abasic model of the engagement kinematics. This willlay the foundation upon which PN, one of the oldestand most common homing missile guidance strategies,is introduced.Engagement Kinematics: The Line-of-SightCoordinate SystemThe development presented here follows the onegiven in Ref. 1. In the sequel, the following notation isused: X = n 3 m (read n-by-m) matrix of scalar elementsx i,j, i = 1 … n, j = 1 … m; _ x = n 31 vector of scalar elementsx i , i = 1 … n; _ n 2x = / i = 1x i = Euclidean vectornorm of _ x; _ 1 x= _ x/ _ x = n 31 unit vector (e.g., _ 1 x = 1)in the direction of _ x; /t = time derivative with respectto a fixed (inertial) coordinate system; and d/dt = timederivative with respect to a rotating coordinate system.Consider the engagement geometry shown in Fig. 2,where _ r Mand _ r Tare the position vectors of the missileinterceptor and target with respect to a fixed coordinateframe of reference (represented by the triad { _ 1 x, _ 1 y, _ 1 z}).Consequently, we define the relative position vector ofthe target with respect to the missile as shown in Eq. 2:_ r =_rT – _ r M. (2)The relative position vector can be written as _ r = R _ 1 r,where R = _ r is the target–missile range, and _ 1 ris theunit vector directed along _ r (we refer to _ 1 ras the LOSunit vector). Differentiating the relative position vector,_ r = R_1r , with respect to the fixed coordinate system, weobtain the following expression for relative velocity _ v:v /r = Ro 1 R 1 . t r + t r(3)From Eq. 3, one can see that the rate of change of therelative position vector (i.e., relative velocity) comprisestwo components: (i) a change in _ r as a result of a changein length (Ṙ) and (ii) a change in direction (a rotation)as a result of the rate of change of the LOS unit vector.r1 tn =r tLOS coordinate 1 = r frame rrMissile r LOS1 = 1 r 1 n rMFixed coordinateframe1 x–1 zrT1 yTargetFigure 2. In the LOS coordinate frame, the LOS rate is perpendicularto the LOS direction and rotation of the LOS takes placeabout the – 1 .We define this change in direction by the vector _ n asgiven in Eq. 4:n / . t 1 rConsequently, a second unit vector, _ 1 n, is defined tobe in the direction of _ n as shown:1n(4)1r/t= =n.(5)1/ tnrFinally, to complete the definition of the (righthanded)LOS coordinate system, a third unit vector, _ 1 ,is defined as the cross product of the first two:_1 = _ 1 r3 _ 1 n. (6)In general, the angular velocity of the LOS coordinatesystem with respect to an inertial reference frame isgiven by o = o r1r + o n1n + o 1,where the componentsof the angular velocity are given in Eq. 7:or = o: 1ron = o: 1no= o: 1 .(7)Thus, upon reexamination of Eq. 4, we note that theLOS rate, _ n, can be expressed as shown in Eq. 8:n =d 1 1 .dt r + o # r(8)On the right-hand side of Eq. 8, the expression d _ 1 r/dtrepresents the time derivative of the LOS unit vectorwith respect to a rotating coordinate frame, and o isJOHNS HOPKINS APL TECHNICAL DIGEST, VOLUME 29, NUMBER 1 (2010)27­

N. F. PALUMBO, R. A. BLAUWKAMP, and J. M. LLOYDthe angular velocity of the rotating frame with respectto the inertial frame. The former component is equal tozero (i.e., the LOS unit vector is a constant). Therefore,the LOS rate and corresponding unit vectors simplify tothe following expressions:n = o# 1ro # 1r(9)1n= .o# 1Thus, from Eq. 3, the relative velocity expression is givenasrv = R o 1 + R( o # 1 ) .(10)rThe typical guided missile control variable is interceptoracceleration. Thus, taking the derivative of Eq. 10,and using Eq. 9, we obtain an expression for the relativeacceleration:v= Rp 1 + Ro 1 + R( or # r)+ R( # r)+ R #t ro 1 p 1 1t `otrj= Rp 1 + 2Ro ( o # 1 ) + R( p # 1 ) +R[ o # ( o# 1 )]. ( 11)rrNext, using the definition of o in Eq. 7 and the factthat _ 1 r= [1 0 0] T , we expand the term o # 1 r / n inEq. 11, which results in the following relation:1r1n1o # 1r= det o r o n o = o 1n – o n1. (12)1 0 0Here, det represents taking the determinant of amatrix. From Eq. 4, the direction of _ n can have no componentalong _ 1 . Thus, we conclude . n= 0 and obtainthe result in Eq. 13:rr o # 1 = o 1(13)r n .In Eq. 13, o / n . Using Eq. 13, the other crossproductterms in Eq. 11 yield the following results:o # ( o # 1 ) o 1 o o1p# 1 = p1 .2r = – r + r rnr(14)Using Eqs. 13 and 14 in Eq. 11, we obtain the desiredexpression for relative acceleration:22tr = a – aTM2= ( R p – Ro ) 1r+(2Ro o + Rp ) 1n+( R o o r) 1.(15)From Eq. 15, the components of relative acceleration inthe LOS coordinate frame can be written as shown inEq. 16:2( aT– aM) : 1r= Rp– Ro( aT– aM): 1n= 2Ro o + Rp( a – a ) : 1 = Roo.TMr(16)Development of PN Guidance LawMany guided missiles employ some version of PN asthe guidance law during the terminal homing phase.Surface-to-air, air-to-air, and air-to-surface missileengagements, as well as space applications (includingrendezvous), use PN in one form or another as a guidancelaw. 1–7 A major advantage of PN, contributing toits longevity as a favored guidance scheme over the lastfive decades, is its relative simplicity of implementation.The most basic PN implementations require low levels ofinformation regarding target motion as compared withother more elaborate schemes (some are discussed inthe next article in this issue, “Modern Homing MissileGuidance Theory and Techniques”), thus simplifyingonboard sensor requirements. Moreover, it has provento be relatively reliable and robust. As also will be seenin the next article in this issue, under certain conditionsand (simplifying) assumptions about target and missilecharacteristics, the PN law is an optimal guidancestrategy in the sense of minimizing the terminal missdistance.In order to develop the PN guidance strategy, we firstlook to the components of Eq. 16 for sufficient conditionsto achieve an intercept. Looking at the first component,sufficient conditions for an intercept are (i) theLOS rate is zero ( . = 0), (ii) interceptor capability toaccelerate along the LOS is greater than or equal totarget acceleration along the LOS ( _ a _ • M1 r _ a _ • T1 r),and (iii) the initial range rate along the LOS is negative(Ṙ(0) < 0). In this case, missile-to-target range (R) willdecrease linearly (( _ a T– _ a M) _ • 1 r= 0) or quadratically(( _ a T– _ a M) _ • 1 r< 0) with respect to time and, eventually,pass through zero.From the discussion above, the interceptor must acceleratesuch as to null the LOS rate ( . ). We look to thesecond component of Eq. 16 to determine how this canbe done. We first define closing velocity as V c≡ –Ṙ. Notethat for an approaching trajectory Ṙ < 0; thus V c > 0.If, for the moment, we treat closing velocity and rangeas constant, then taking the Laplace transform of thesecond component of Eq. 16 yields the following polynomialin s:( a ( s) – a ( s)) : 1 = ( sR – 2V ) o ( s) . (17)T M ncIn Eq. 17, s represents the Laplace transform variable.Thus, if we define interceptor acceleration perpendicularto the LOS to be _ a M(s) • _ 1 n= . (s), then, from Eq. 17,we can write the transfer function from target accelera-28JOHNS HOPKINS APL TECHNICAL DIGEST, VOLUME 29, NUMBER 1 (2010)

BASIC PRINCIPLES OF HOMING GUIDANCENoiseLOSSeeker++ + + – – Gimbal angle1/s 1/sReceiver1LOS rate pickoff m 1 s .cLOS rate estimate^1 . f s + 1Guidance PNfilterGuidance computera McCommandedaccelerationNV cG FC (s).++Gimbal servoDish rateGimbal controlRate gyro1 .+–Flight controlsystem.Body attitude ratea A s + 1v mAchievedaccelerationBody rateaerodynamicsFigure 5. Simplified planar model of a traditional LOS rate reconstruction approach that directly supplies an LOS rate measurementto the guidance computer. Note that the LOS rate pickoff is assumed to be proportional to the boresight errormeasurement. The measured LOS rate is subsequently filtered to mitigate measurement noise and then applied to the PNhoming guidance law.Moreover, the transfer function from commanded acceleration(from the guidance law) to missile body rate (ċ)is approximated by the following aerodynamic transferfunction, where t Ais the turning rate time constant andv mis missile velocity:o A s + 1= .a vcm(26)In this approach, the fact that the LOS rate is embeddedin the tracking error ( m) is exploited. As illustrated, aLOS rate estimate is derived by appropriately filteringthe receiver tracking error scaled by the seeker tracklooptime constant.Other approaches can be used to derive LOS rate forhoming guidance purposes; these are generally referredto as either LOS reconstruction or LOS rate reconstruction.We next outline three alternative techniques (twoLOS reconstructions and one LOS rate reconstruction).LOS ReconstructionAs shown in Fig. 3, LOS reconstruction works toconstruct a measured LOS, l m, in an inertial frame ofreference. The measured LOS then is filtered (via anappropriate guidance filter) to derive an estimate of LOSrate for guidance purposes. Two different LOS reconstructionapproaches are as follows: 91. Integrate the seeker gyro output and sum it with themeasured tracking error. A block diagram of thisapproach is shown in Fig. 6. Mathematically, thisapproach can be expressed as Eq. 27:l m= m+ # . dt . (27)2. Integrate the output of the missile body rate gyro,obtained from the missile IMU, and sum the integratedIMU gyro output together with the seekergimbal angle and the measured tracking error. Weillustrate this approach in the block diagram shownin Fig. 7. This approach is expressed mathematicallyas shown in Eq. 28:l m= m+ b + # ċ dt . (28)From Fig. 3, it is clear that the two concepts are algebraicallyequivalent in the absence of noise and assumingperfect instruments (no gyro biases, drift, etc.); however,in practice, this is not the case. Moreover, we note that,in general, the guidance filters for the two approachesare not necessarily the same (e.g., for the simplified guidancefilters shown in the figures, the filter time constants,represented by t f, are not necessarily the same).The fundamentals of guidance filtering will be discussedin the companion article in this issue “Guidance FilterFundamentals.” It is shown in Ref. 9 that the two LOSreconstruction approaches can yield significantly differentresults when noise and imperfect instrumentsare used. How these differences manifest themselvesdepends on the quality of the measurements and instrumentsthat are used.LOS Rate ReconstructionA guidance signal also can be generated by differentiatingthe tracking error and adding it tothe seeker rate gyro output. For practical purposes,taking the derivative of the tracking error is accom-JOHNS HOPKINS APL TECHNICAL DIGEST, VOLUME 29, NUMBER 1 (2010)31­

BASIC PRINCIPLES OF HOMING GUIDANCENoiseLOSSeeker++ + + – – Gimbal angle1/s 1/s.Gimbal servo+Dish rate+LOS rate pickoff. . m s m + D s + 1+1 s . .c+Gimbal control – .Receiver1Rate gyro1mLOS rate estimate^1 . a Mc f s + 1NV cGuidance PN CommandedfilteraccelerationGuidance computerFlight control G FC (s)systemAchievedaacceleration. A s + 1Body attitude rate v mBody rateaerodynamicsFigure 8. Simplified planar model of an alternate LOS rate reconstruction approach that can directly supply an LOS rate measurement tothe guidance computer. Note that the LOS rate reconstruction differences between this figure and Fig. 5. As before, the measured LOS rate issubsequently filtered to mitigate measurement noise and then applied to the PN homing guidance law.1. It must convey the energy with minimum loss.2. It must convey the energy with minimum distortion,particularly angular distortion because this creates aparasitic feedback loop that can have a significantnegative impact on guidance performance (discussedin more detail below).3. It must have minimum aerodynamic drag.4. It must have satisfactory physical properties, suchas sufficient strength, resistance to thermal shock(from rapid aerodynamic heating), resistance torain erosion at high speeds, and minimum waterabsorption.111/2L/D 1/2Ideal electromagnetically5L/D 5Ideal aerodynamicallyL/D 3Compromise radomeFigure 9. Three possible radome shapes are illustrated. Forminimum angular distortion, a hemispherical shape (or hyperhemisphericalshape as in a ground-based radar) would be idealelectromagnetically (upper left), but the drag penalty is excessive.From an aerodynamic perspective, the lower left radome shape ispreferable, but it tends to have significant angular distortion characteristics.The tangent-ogive shape (on the right) is a typical compromisedesign. Nevertheless, some missiles use much blunterdome designs despite the drag penalty. L/D, lift-to-drag ratio.13As an example, Fig. 9 illustrates three conceivableradome shapes. The tangent-ogive shape (on the right)is a typical compromise design.Guidance System Design ChallengesThere are a significant number of challenges todesigning guidance systems: the design must provide thedesired performance while remaining robust to a multitudeof error sources, limited control system bandwidth,and inherent system nonlinearities. Some of these challengesare summarized below.1. The root-mean-square final miss distance from alldeleterious noise sources must be minimized.2. Guidance system stability must be maintained in theparasitic feedback loop (as mentioned above, this iscaused by angular distortion of the radome/irdome).Radome angular distortion, in particular, is a keycontributor to final miss distance but is consideredseparately from other noise sources as its impact onguidance system stability is substantial.3. System nonlinearities must be avoided as muchas possible; e.g., seeker gimbal angle and gyro ratesaturations must be avoided, as should commandingmissile acceleration beyond what is physicallyrealizable by the missile.Contributors to Final Miss DistanceWith respect to guidance challenge item 1 above,there are a number of contributors to final miss distance(other than angular distortion of the radome/irdome).For example, in either an RF or IR guidance system, measurementnoise from the various instruments (onboardJOHNS HOPKINS APL TECHNICAL DIGEST, VOLUME 29, NUMBER 1 (2010)33­

BASIC PRINCIPLES OF HOMING GUIDANCERadome bse(radome error)bseApparent LOSTrue LOSSeeker centerline (Seeker gimbal angle)Missile body axis bser (slope)Using the fact that the seeker dish angle can be writtenas = c + b, we obtain the following expression for thederivative of the measured tracking error:. m= l . – (1 – r)ḃ – ċ . (33)Using Eq. 33, we can modify the block diagram shown inFig. 5 to include the dome distortion effect (via Eq. 33).The result is shown in Fig. 11. Clearly, the boresight errorcreates another signal path in the missile guidance loop.To obtain an alternate view of this parasitic loopand how it can affect guidance performance, we revisitEq. 32. Referring to Fig. 3, we note that the seekergimbal angle can be expressed as b = – c and thatthe measured LOS can be written as l m= + m. Alsonoting that it is typical for r

Vc /v M0.350.300.250.200.150.100.0500Boundary = 0.170.01Navigation ratio: N = 3Effective navigation ratio: Ñ (r) = N/(1 + rNV c /v M )Unstable region: Ñ (r ) N/(1 + rNV c /v M ) 20.020.03V c /v M = 3V c /v M = 2V c /v M = 1Stable region: Ñ (r ) N/(1 + rNV c /v M ) > 20.04 0.05 0.06Radome slope r (rads)Figure 13. For a navigation ratio of 3, the effect of positive radome boresight error slope (r)for increasing ratios of closing velocity over missile velocity is shown. To maintain guidancestability, the effective navigation ratio must remain greater than 2.Closest point of approach (ft)109876543210–0.10acceleration limit is more a result of maximum angle-ofattacklimitations. In either case, the missile guidanceand control system design must take the accelerationlimits into account. The most significant implicationis that the commanded acceleration from the guidancesystem must be hard-limited so as not to exceedthe acceleration limits of the missile, which also implies0.07 = 0.5 s –1 = 0.3 s –1 = 0.1 s –1 = 0.4 s –1 = 0.2 s –1–0.05 0 0.05Radome error slope (%)0.100.15Figure 14. This plot illustrates the miss performance of PNversus radome boresight error slope. The results were derived byusing a planar Simulink model assuming no error sources asidefrom radome error. Each curve represents the results using anautopilot with specified time constant t. A larger time constantindicates a larger lag in the missile response to accelerationcommands.0.080.090.10BASIC PRINCIPLES OF HOMING GUIDANCEthat the flight control system(autopilot) must be designed toproduce minimum overshoot toan acceleration step command.If/when the guidance commandsaturates, the missile guidanceloop is essentially opened. If guidancecommand saturation persistslong enough, and if this persistentsaturation occurs near intercept, asignificant final miss distance canresult.Missile seeker systems also areprone to nonlinear saturation,which can have significant deleteriouseffects on overall guidanceperformance. For example,on gimbaled platforms, the seekergimbal angle can saturate undercertain stressing conditions (e.g.,while pursuing a highly maneuveringtarget). If this saturationoccurs, the guidance loop haseffectively become open and, if this occurs near intercept,significant final miss distance can occur.Guidance Command PreservationThree-dimensional guidance laws usually generatea guidance command vector without consideration tohow the missile interceptor will effectuate the specifiedmaneuver. For example, it is typical for a guided missileto (i) have no direct control over its longitudinal acceleration(e.g., axial thrust is not typically throttleable)and (ii) maneuver in the guidance direction (specifiedby the guidance law) by producing acceleration normalto the missile body. Therefore, the question naturallyarises as to how a three-dimensional guidance command,specified by the guidance law without considerationof the aforementioned constraints, can beachieved by maneuvering the missile perpendicular tothe missile centerline. We shall refer to this process asguidance command preservation.In Ref. 1, two guidance command preservationtechniques are developed. The first approach, referredto here as GCP1, is further elaborated in Ref. 12. Thesecond guidance command preservation technique(GCP2) is a minimum norm solution. As such, GCP2could (theoretically) provide savings in commandedacceleration (or fuel use in exoatmospheric applications)as compared with the first method. The GCP1technique is discussed here.GCP1 TechniqueTo describe this guidance command preservationtechnique, we refer to Fig. 15 and make the followingJOHNS HOPKINS APL TECHNICAL DIGEST, VOLUME 29, NUMBER 1 (2010)37­

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

N. F. PALUMBO, R. A. BLAUWKAMP, and J. M. LLOYDand depend on flight conditions, booster assumptions,and other factors. A critical component of implementingthis law is computing an accurate estimate of time-to-gowhen the missile is still thrusting, which depends on thecomponent of missile acceleration along the LOS. Theguidance calculations can be made on board the missileor off board, with acceleration commands passed fromthe weapon control system to the missile where they areconverted to body coordinates.For exoatmospheric flight, drag is no longer an issue,but long flight times will result in curved trajectories asa result of gravity. From the current position, the Lambertsolution (see Ref. 7 for a simple synopsis) definesthe necessary current velocity to reach a terminalintercept point at a given time. A guidance law can be“wrapped around” the Lambert solution to progressivelysteer the missile velocity vector to align with the Lambertvelocity. Then either thrust termination or a slowdownmaneuver can be used to match the magnitude ofthe velocity of the Lambert solution when the missilebooster burns out.CLOSING REMARKSThe key objective of this article was to provide arelatively broad conceptual foundation with respectto homing guidance but also of sufficient depth toadequately support the articles that follow. First, wediscussed handover analysis and emphasized that thedisplacement error between predicted and true targetposition, _ e, can be decomposed into two components:one along ( _ e ) and one perpendicular to ( _ e ⊥ ) the predictedLOS. Thus, because the relative velocity isalong the LOS to the predicted target location, the erroralong this direction alters the time of intercept but doesnot contribute to the final miss distance. It is the errorperpendicular to the LOS that must be removed bythe interceptor after transition to terminal homing toeffect an intercept.Next, we developed a classical form of PN and notedthat a primary advantage of PN, contributing to its longevityas a favored guidance scheme over the last fivedecades, is its relative simplicity of implementation. Infact, the most basic PN implementations require lowlevels of information regarding target motion as comparedwith other, more elaborate schemes, thus simplifyingonboard sensor requirements. Moreover, it hasproven to be relatively reliable and robust. This particular(and somewhat unique) treatment of PN was takenfrom a 1980 APL memorandum written by Alan J. Pue. 1We also discussed how PN can be mechanized for guidedmissile applications, with a focus on LOS reconstructionand guidance command preservation. With respect tohoming guidance, we itemized the primary contributorsto guidance performance degradation that can ultimatelylead to unacceptable miss distance.The onboard missile seeker has a limited effectiverange beyond which target tracking is not possible. Tosupport engagements that initially are beyond such arange, midcourse guidance is used to bring the missilewithin the effective range of the seeker. Thus, in contrastto terminal homing, during the midcourse guidancephase of flight, the target is tracked by an externalsensor and information is uplinked to the missile. Thekey objectives during midcourse guidance are to guidethe missile to a favorable geometry with respect to thetarget for both acquisition by the onboard missile targetingsensor and to provide acceptable handover to terminalhoming. Many of the terminal homing conceptsdiscussed here and in the subsequent articles on modernguidance and guidance filtering in this issue also areapplicable to developing midcourse guidance policies.Thus, mainly for completeness, we briefly introduced theproblem of midcourse guidance.REFERENCES1 Pue, A. J., Proportional Navigation and an Optimal-Aim Guidance Technique,Technical Memorandum F1C(2)80-U-024, JHU/APL, Laurel,MD (7 May 1980).2 Ben-Asher, J. Z., and Yaesh, I., Advances in Missile Guidance Theory,American Institute of Aeronautics and Astronautics, Reston, VA(1998).3 Locke, A. S., Principles of Guided Missile Design, D. Van NostrandCompany, Princeton, NJ (1955).4 Shneydor, N. A., Missile Guidance and Pursuit: Kinematics, Dynamicsand Control, Horwood Publishing, Chichester, England (1998).5 Shukala, U. S., and Mahapatra, P. R., “The Proportional NavigationDilemma—Pure or True?” IEEE Trans. Aerosp. Electron. Syst. 26(2),382–392 (Mar 1990).6 Witte, R. W., and McDonald, R. L., “Standard Missile: GuidanceSystem Development,” Johns Hopkins APL Tech. Dig. 2(4), 289–298(1981).7 Zarchan, P., Tactical and Strategic Missile Guidance, 4th Ed., AmericanInstitute of Aeronautics and Astronautics, Reston, VA (1997).8 Stallard, D. V., Classical and Modern Guidance of Homing InterceptorMissiles, Raytheon Report D985005, presented at an MIT Dept. ofAeronautics and Astronautics seminar (Apr 1968).9 Miller, P. W., Analysis of Line-of-Sight Reconstruction Approaches forSM-2 Block IVA RRFD, Technical Memorandum F1E(94)U-2-415,JHU/APL, Laurel, MD (22 Dec 1994).10 Nesline, F. W., and Zarchan, P., “Line-of-Sight Reconstruction forFaster Homing Guidance,” AIAA J. Guid. Control Dyn. 8(1), 3–8(Jan–Feb 1985).11 Lin, C. F., Modern Navigation, Guidance, and Control Processing, PrenticeHall, Englewood Cliffs, NJ (1991).12 Abedor, J. L., Short Range Anti-Air Warfare Engagement Simulation,Technical Memorandum F1E(86)U-3-031, JHU/APL, Laurel, MD(10 Nov 1986).40JOHNS HOPKINS APL TECHNICAL DIGEST, VOLUME 29, NUMBER 1 (2010)

BASIC PRINCIPLES OF HOMING GUIDANCEThe AuthorsNeil F. Palumbo is a member of APL’s Principal Professional Staff and is the Group Supervisor of the Guidance,Navigation, and Control Group within the Air and Missile Defense Department (AMDD). He joined APL in 1993after having received a Ph.D. in electrical engineering from Temple University that same year. His interests includecontrol and estimation theory, fault-tolerant restructurable control systems, and neuro-fuzzy inference systems.Dr. Palumbo also is a lecturer for the JHU Whiting School’s Engineering for Professionals program. He is a memberof the Institute of Electrical and Electronics Engineers and the American Institute of Aeronautics and Astronautics.Ross A. Blauwkamp received a B.S.E. degree from Calvin College in 1991 and an M.S.E. degree from the Universityof Illinois in 1996; both degrees are in electrical engineering. He is pursuing a Ph.D. from the University of Illinois.Mr. Blauwkamp joined APL in May 2000 and currently is the supervisor of the Advanced Concepts and SimulationTechniques Section in the Guidance, Navigation, and Control Group of AMDD. His interests include dynamic games,nonlinear control, and numerical methods for control. He is a member of the Institute of Electrical and ElectronicsEngineers and the American Institute of Aeronautics and Astronautics. Justin M. Lloyd is a member of the APL SeniorProfessional Staff in the Guidance, Navigation, and Control Group of AMDD. He holds a B.S. in mechanical engineeringfrom North Carolina State University and an M.S. in mechanical engineering from Virginia Polytechnic Instituteand State University. Currently, Mr. Lloydis pursuing his Ph.D. in electrical engineeringat The Johns Hopkins University. He joinedAPL in 2004 and conducts work in optimization;advanced missile guidance, navigation, and control;and integrated controller design. For furtherinformation on the work reported here, contactNeil Palumbo. His email address is neil.palumbo@Neil F. Palumbo Ross A. Blauwkamp Justin M. Lloyd jhuapl.edu.The Johns Hopkins APL Technical Digest can be accessed electronically at www.jhuapl.edu/techdigest.JOHNS HOPKINS APL TECHNICAL DIGEST, VOLUME 29, NUMBER 1 (2010)41­