TERMINAL GUIDANCE USING A DOPPLER BEAM SHARPENING RADAR Jeremy A. Hodgson *, BSc(Hons), MSc, MIEE The MathWorks Ltd, Cambridge, UK

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1 TERMINAL GUIDANCE USING A DOPPLER BEAM SHARPENING RADAR Jeremy A. Hodgso *, BSc(Hos, MSc, MIEE The MathWorks Ltd, Cambridge, UK David W. Lee, BSc(Hos,MSc QietiQ Limited, Farborough, UK Abstract This paper ivestigates the guidace issues relatig to the use of a form of Sythetic Aperture Radar (SAR to locate a static target durig the termial phase of a air-to-surface missile egagemet. Sythetic Aperture Radars geerate photographic images of the groud below through the trasmissio ad receptio of electromagetic eergy. The clarity of the images will deped o the resolutio both alog, ad perpedicular, to the lie of sight betwee the observer ad a poit o the groud. The resolutio alog the lie of sight is cotrolled through the trasmitted pulsewidth, ad the cross rage resolutio through the icremetal Doppler shift of adjacet poits o the groud. This paper focuses o the use of a particular form of SAR kow as Doppler Beam Sharpeig (DBS to geerate images that ca be used to guide a missile oto a target. For a successful egagemet the missile must fly a trajectory that satisfies a umber of requiremets. The geerated images must be of sufficietly high resolutio that the target ca be idetified; the iformatio extracted from successive images must be sufficiet for the guidace filters to accurately determie the positio of the target; ad fially, the missile must be able to hit the target at a desired speed, flightpath agle, ad with a low agle of attack to achieve a kill. I this paper a trajectory optimizatio techique is used to shape the trajectory to achieve these requiremets, ad a cadidate guidace law is demostrated where the gais are determied so that the missile follows the omial optimal trajectory, ad achieves a low miss distace at impact.. Itroductio The basis for the use of a DBS radar for geeratig images is that a lower cross rage resolutio ca be obtaied for a give atea legth by makig use of the Doppler shift i the frequecy of the retured sigal due to the relative motio betwee the target object ad the missile. The magitude of the Doppler shift that ca be resolved is a fuctio of the radar illumiatio time (dwell time. The cross rage resolutio that ca the be achieved is depedet upo both the resolvable Doppler shift, ad the offset agle betwee the velocity vector ad the lie of sight. I order to image the target the missile must therefore fly with a headig offset * Seior Egieer, Cosultig Team Leader, Guidace ad Imagig Solutios from the lie of sight, util a poit is reached where imagig will o loger be possible, ad the missile will fly towards the best estimate of the target locatio. A modified form of velocity pursuit guidace for use with a SAR sesor is outlied i Zipfel. The guidace law shapes the trajectory to follow aalytical fuctios (e.g. circular, or spiral shapes i order to maitai the agular offset, ad brig the missile alog a desired lie of approach at impact. The limitatio of this form of guidace law is that it does ot accout for costraits imposed by limited acceleratio capabilities of the missile airframe, or costraits imposed by gimbal limits restrictig the maximum look agle betwee the lie of sight ad the velocity vector. It is also uable to accout for the missile drag characteristics, which is eeded to meet requiremets o the velocity at impact. Trajectory optimizatio methods have bee employed i a umber of applicatios 3,7 to determie omial trajectories that miimize certai objectives, costrai the cotrol demads, ad impose iterior poit costraits o fuctios of the states. This paper demostrates the use of optimizatio methods to determie trajectories which aim to miimize the dwell time to achieve a desired cross rage resolutio durig the imagig period, satisfy cotrol ad look agle costraits, ad which meet the flight requiremets at impact. Geeratig images with the required resolutio so that the target object ca be idetified does ot o its ow guaratee good guidace loop performace. Measuremets of the rage to the target, ad the Doppler shift at the target locatio ca be extracted from each image. The target positio relative to the missile ca the be derived usig these measuremets, ad the velocity ad height estimates from a Iertial Navigatio System (INS. Ay errors i the INS estimates will lead to iaccurate estimates of the target locatio, hece a estimator is required to provide a filtered value of the relative locatio of the target to the missile, ad a estimate of the error i the velocity output from the INS. A compariso of the filter methods that ca be applied to this problem is give i Rollaso 8 ad Ristic 4. For this paper the expected accuracy to which the target locatio ca be estimated is quatified by determiig the Cramer-Rao Lower Boud (CRLB at the ed of the imagig period. The CRLB is determied for a o-liear least squares batch filter utilizig a stored set of measuremets take at itervals durig the egagemet. The effect o the optimal trajectories of icludig the magitude of the

2 CRLB as a cost to be miimized is ivestigated i this paper. Fially, a modified form of the guidace law outlied i Zipfel is developed which shapes the trajectory to follow the omial optimal trajectories. The followig Sectios iclude a descriptio of the model to represet the dyamics of the missile; a derivatio of the relatioship betwee the DBS cross rage resolutio, the dwell time, ad the missile to target geometry; a derivatio of the CRLB for the batch filter; the optimizatio method employed ad the resultig trajectories, ad fially the closed loop guidace law ad a compariso betwee the ope ad closed loop trajectories that are followed.. Missile dyamic model The equatios that represet the dyamics of a poit mass missile model, i the urolled coordiate system i Figure, are give i Equatio (.. The states x, yh,, the pitch cosist of the positio of the missile ( ad yaw ( γψ, flightpath agles, ad the specific V eergy E h+. At ay poit the velocity ( V g ca be derived directly from the algebraic relatioship with the specific eergy. The cotrol variables are the a, a applied ormal to the demaded acceleratios ( y z velocity vector, ad the thrust level ( T, which is assumed to be variable but fixed durig the fial stages of the egagemet. h V x& V cosγ cosψ y& V cosγ siψ h& Vsiγ ( az cos γ. g & γ V (. ay. g ψ& V cosγ V. ( T D E& mg I Equatio (., the missile drag (D is calculated usig Equatios (. ad (.3, where the drag coefficiet ( CD is a fuctio of the zero icidece drag ( C D0, ad the iduced drag coefficiet( k. D ρv SRef C D (. ρ g LR T0 + Lh ρ 0 T0 D D0 + L / C C kc ( z + ay (.3 mg a CL ρv SRef A estimate for the total agle of attack ( α ca be obtaied by assumig a approximately liear relatioship with the lift coefficiet ( C L i Equatio (.4. CL α (.4 CL α Characteristics for a represetative air-to-surface missile are give i Farooq 7, ad are preseted i Table. a z a y y γ ψ x Figure : Coordiate System 3. Determiig the achievable cross rage resolutio The Doppler shift from a poit o the groud ca be determied by cosiderig the differece betwee trasmitted ad received frequecies whe the rage to the object is varyig. The Doppler shift from poit A i Figure is give by Equatio (3., where - r r σ v cos ( u v ulos is the total agle betwee the velocity vector ad the lie of sight, u r v is a uit vector alog the velocity vector, ad u r LOS is a uit vector alog the lie of sight defied by the pitch ad yaw sightlie agles i Equatio (3.3. fd fretur ftrasmitted R& V cos( σ v (3. A λ λ

3 h Missile (x,y,h u r v σ v V u r LOS y R R ψ DBS ψ s Poit B x cr Poit A (x T,y T,h T Figure : Missile to Target Geometry cγ cψ cγs cψs r u v cγ s ψ r, u LOS cγs sψ s sγ sγ s ( cγ cos γ & sγ siγ γ T s ta h h ( x x + ( y y T T ( y y T ( x x (3. (3.3 ψ s ta T The first step to derive a expressio for the achievable cross rage resolutio is to determie the differece i the Doppler frequecy shift betwee poit A, ad a poit B that is at the same rage (R, ad a small horizotal rotatio( from poit A. ψ s The differece i the Doppler shift ( f from the poits is give by Equatio (3.4. cγ cψ cγs cψs cγ sψ cγ ssψ s V sγ sγ s f (3.4 λ cγ cψ cγ s c( ψs + ψs cγ sψ cγ s s( ψs ψs + sγ sγ s Expadig out Equatio (3.4, ad usig small agle approximatios for ψ s gives the expressio for f i Equatio (3.5. V f cos γ cos γ s si ( ψ ψs ψs (3.5 λ 443 ψ DBS If the small horizotal agular differece betwee the sightlie to poit A ad poit B is give by (3.6, ad the magitude of the miimum resolvable Doppler frequecy shift is give by (3.7, the the relatioship 3 betwee the dwell time, ad the achievable cross rage resolutio is give by Equatio (3.8. cr ψ s (3.6 ( R cosγ s f (3.7 Dwell Time cr Dwell Time crdt λ R (3.8 ( V.cos γ.siψ DBS For the purposes of this paper it is assumed that 3m is the recommeded resolutio cell size to idetify vehicles or buildigs from a image. It is also assumed that a object ca oly be see if the rage is less tha the acquisitio rage RAcquire defied by the power of the radar. 4. Determiig the ucertaity i the estimate of the target locatio The aim of this Sectio is to derive a measure of the accuracy to which the locatio of the target ca be determied from a discrete umber ( _ image of images geerated at itervals durig a egagemet whe usig a o-liear least squares batch filter. A compariso of the performace of the batch filter compared to other estimatio methods is give i Rollaso 8. It is assumed i the filter that the true relative positios ( xt xm, ad velocities ( vt vm, are related to the INS estimates ( xins, vins by Equatio (4.. I (4., the error at a time t from whe observatios commece is a fuctio of the positio errors ( δ x f at the ed of the period of observatio t t, ad costat velocity errors ( δ v. ( f_image

4 ( xt xm xins + ( δxf δv t ( True vt vm v True INS + δ v Where t tf_image t (4. δx δvx δx f δy, δv δv y δh δv h δ x f The costat parameters x are to be estimated δ v from a stored set of measuremets extracted from the images. The measuremets are assumed to be the rage to the target, ad the rage rate derived from the Doppler frequecy shift. The measuremets are related to the INS estimates, ad the ukow parameters, through Equatio (4.. z h x, v, x, t ( ( x + δx δv t R R INS INS INS ( yins δ y δvy t + + ( hins δh δvh t (,,, + + zr& hr& xins vins x t xins + δxf δv t. vins + δv z ( ( R x (4. Each measuremet is assumed to be idepedet, ad has zero mea Gaussia ucertaity with a stadard deviatio give by Equatio (4.3. σ R RRes λ cr (4.3 σ R& crdt The CRLB provides a lower boud o the achievable variace i the estimatio of a parameter. For a ubiased estimator it is give by the iverse of the Fisher Iformatio Matrix (J i Equatio (4.4. T E[( x ˆ( ˆ True x xtrue x ] J (4.4 The Fisher Iformatio Matrix for a o-liear least squares batch filter with zero mea Gaussia measuremet ucertaity is give by the iverse of the covariace matrix calculated with true values for the ucertai parameters 6. I this case that implies that the errors are all zero, ad that the INS estimates are the true values. The covariace matrix is defied i Equatio (4.5, where H is the stacked matrix of partial derivatives of each measuremet equatio with respect to each ukow parameter, ad is give i Equatio (4.6. R is the stacked ucertaity values for each measuremet, ad is give i Equatio (4.7. T P J H R H (4.5 ( 4 ( (, (,, f_image hr xins vins x t x h ( (, (,, R& xins vins x tf_image x (4.6 H : hr( xins( _ image, vins( _ image, x,0 x h ( xins ( _ image, vins ( _ image, x,0 R& x x xtrue R ( ( (.. ( _ ( _ diag σ R σr& σr image σ image R& (4.7 To quatify the magitude of the ucertaity regio from the covariace matrix we ca cosider the probability that the true values lie withi the hyperellipsoid give by Equatio (4.8. T ( ˆ ( ˆ x x P x x l (4.8 The semi-axis legths of this hyperellipsoid are defied by the square roots of the eigevalues of the covariace matrix scaled by l. For this paper, oly the 3x3 subset of the covariace matrix that relates to the positio estimates is cosidered. From Bryso 5, for a 3x3 covariace matrix the probability regio for a give ellipsoid is give i Equatio (4.9. For a value of l this relates to a probability regio of 0%, ad a value of l relates to a probability regio of 74%. l ( / l Pr erf le (4.9 π The cost o the accuracy of the positio estimates is proposed to be the sum of the squares of the ellipsoid semi-axes ( xr, yr, zr defied by the eigevalues of the covariace matrix. 5. Trajectory optimizatio A direct optimizatio approach was take which is based o a fiite-dimesioal discretizatio of the origial cotrol problem. This step reduces the problem to that of a fiite-dimesioal costraied optimizatio problem that ca be solved usig the Sequetial Quadratic Programmig (SQP solvers available i the MathWorks Optimizatio Toolbox 9. The beefit of usig a direct optimizatio approach, over the variatioal approach as described i Bryso 5, is the ease with which the problem ca be posed, ad the relaxed requiremets o the iitial guess for the solutio. A fixed flight time was chose, ad the parameters to be determied are the sequece of piecewise costat acceleratio demads over equidistat time itervals (Figure 3. I additio to determiig the time varyig cotrols, the thrust level, ad the iitial cross rage offset are set as parameters to be determied. For a give sequece of demads the dyamic states are calculated by umerically itegratig the state equatios. The scalar cost fuctio, ad the vector iequality costrait fuctios, are the evaluated ad

5 x ( 0 t f x k + x k + f ( x k, u ( k dx f ( x, u dt u ( 0 x ( u ( x ( x ( 3 u ( ( x ( ( ( Parameters u 0, u,..., u ( u ( x ( u ( x t f t f t f t 0 f t f 3 ( Time passed to the optimizatio routies. The cost fuctio for this paper is made up of the compoets give i Equatio (5.. Jcr J P Jcotrol T JTot GTermGTerm αcr αp αcotrol Jcr Cost o the dwell time J P Cost o the CRLB (5. Jcotrol Cost o the acceleratio demads GTerm Termial costraits αcr, αp, αcotrol Weightig values. The cost o the dwell time to achieve a desired cross rage resolutio is give i Equatio (5., where t f is the time of flight, ad is the umber of discrete itervals that the flight is divided ito. Wcr ( R is a fuctio which weights the cost durig the period where imagig ca take place (Figure 4. The last three odes are ot icluded i the cost fuctio to represet the trasitio from imagig, to achievig the desired agle of attack at impact. t 3 f crdt ( k Jcr. Wcr ( R( k (5. k cr The cost o the ucertaity associated with the estimates of the target locatio is give i Equatio (5.3, where xr, yr, zr are the eigevalues of the subset of the covariace matrix i (4.6 associated with the positio error estimates. It is assumed that measuremets are take at each ode i the trajectory whe the coditios are satisfactory, ad that the dwell time is chose to obtai a cross rage resolutio of cr. JP ( xr + yr + zr (5.3 The cost o the acceleratio demads is give i Equatio (5.4, where the aim is to smooth the demads over the time of flight. Figure 3: Discretizatio of Cotrol Problem 5 (( z( z( ( y( y( t f Jcotrol a k a k + a k a k k 0 a (, a ( 0. (5.4 z y The costraits at the fial flight time are give i Equatio (5.5, ad are icorporated by augmetig the cost fuctio i (5.. There are six costraits, icludig the cross rage distace, height, speed, agles of approach, ad total agle of attack. There is o costrait o the fial dowrage locatio sice the optimizatio rus with a fixed flight time. For the purposes of calculatig the dwell time ad the CRLB it is always assumed that the target is located where the trajectory termiates. y( G 00 h( 00 VTerm V( 0 (5.5 Term γ Term γ ( ( 5 π /80 ψterm ψ ( ( 5 π /80 α ( ( 5 π /80 I additio to miimizig the cost fuctio, at each ode the resultig trajectory has to satisfy a umber of iterior poit costraits o the states. The three iequality costraits are: - i α αmax ii σ v σ v_max ( R < RAcquire 0 iii ψ 5 ( R < R & k < - DBS Acquire 6. Example optimal trajectory results Two resultig optimal trajectories are show i Figures 5 ad 6, where the requiremets are specified i Table 3. The first trajectory has o cost o the expected

6 ucertaity o the positio error estimates. The aim is therefore purely to achieve high-resolutio images with miimal dwell time durig the egagemet. The trajectory is see to iitially pull away from the target i order to achieve the desired agular offset betwee the lie of sight ad the velocity vector. The extet of the maeuver is oly limited by the look agle costrait. As the rage to go reduces, the agular offset is maitaied by icreasig the acceleratio demads, util they are forced to reduce to satisfy the agle of attack costrait at impact. Sice the thrust level is the maximum value, the acceleratio demads have to be lower tha their maximum values due to iduced drag causig the velocity to drop below the costrait set at impact. For this trajectory the ellipsoid derived from the covariace matrix is show i Figure 7. The shape of the ucertaity regio shows that from the sequece of geerated images the estimate of the dowrage target locatio is very poor, ad would likely result i large miss distaces beig achieved. The secod trajectory show i Figure 6 ow itroduces the cost o the positio estimate ucertaity, ad the ucertaity ellipsoid is show i Figure 8. The shape of the trajectory clearly provides the batch filter with more iformatio from which the target locatio ca be estimated. This reduced ucertaity regio is achieved by allowig the dwell time to icrease, ad effectively sweepig the target i yaw to achieve better dowrage positio estimates. It is difficult to ifer from these results a geeral strategy for a guidace law usig a DBS seeker, sice these results are for a specific airframe, ad for a sigle set of weights o the cost fuctio. They do though demostrate the high degree of couplig betwee the capabilities of the airframe, ad the expected accuracy to which the seeker ad the guidace filters ca estimate the target locatio. This is due to the icreasig acceleratio demads as the target is approached. Not oly does the airframe require the capability to achieve these high demads, the thrust capabilities have to be sufficiet to maitai the velocity if the warhead is to be effective at impact. 7. A trajectory followig guidace law A proposed guidace law desiged to achieve small miss distaces, ad which shapes the trajectory to follow the omial optimal trajectories is give i Equatio (7.. The form of the guidace law is take from Zipfel, ad is a modified form of velocity pursuit with the output a acceleratio demad vector i earth axes. The first cross product term geerates acceleratio demads that drive the velocity vector ( u r v to lie alog the lie of sight ( u r LOS, hece assurig small miss distaces. The secod cross product term beds the trajectory toward a desired Lie Of Approach (LOA. The bias gai ( G defies the shape of the trajectory followed. e r r r r r r a KV ( u u u G( R ( u u ( R u (7. ( ( zd v LOS v v LOA v To match the acceleratio demads from the ope loop trajectories, both the lie of approach, ad the bias gai ca be made fuctios of the rage to go. Sice the ope loop trajectories are costraied to a yaw flightpath agle of 90 degrees at impact, the lie of approach ca be defied by a sigle pitch attitude agle θ LOA i Equatio (7.. 0 r uloa cos( θ LOA ( R (7. si ( θ LOA ( R The bias gai, ad the lie of approach attitude, ca be derived from the ope loop trajectories usig both Equatio (7., ad the trasformatio of the acceleratio demads i velocity axes to earth axes i Equatio (7.3. I Equatio (7.3 Tv eis the trasformatio matrix give i Equatio ( e azd T v e a y (7.3 a z Optimal cosγ cosψ siψ siγ cosψ T v e cosγ siψ cosψ si γ siψ (7.4 siγ 0 cosγ The resultig aalytical expressios for GRad ( θ LOA ( R are ot trivial, ad are ot icluded i this paper. A example of the resultig gai values derived from the optimal trajectory foud i the previous sectio is show i Figure 9. The closed loop performace of the guidace law is demostrated usig the Simulik 0 model i Figure 0. The model implemets the poit mass dyamics i Equatio (., ad assumes a simple first order respose for the autopilot dyamics. The resultig closed loop trajectory is show alog with the omial ope loop values i Figure. The match is clearly very close, ad the guidace law has successfully bee demostrated i more detailed 6DoF models, ad i the presece of ucertai target locatios ad INS errors. 8. Coclusios This paper has demostrated that optimal trajectories ca successfully be geerated to achieve both the image quality required to idetify a target durig the termial stages of a egagemet, ad also the requiremets for the guidace filters to estimate the target locatio withi acceptable bouds to achieve low miss distaces. 9. Ackowledgemets This work has bee fuded by UK MOD. 6

7 0. Refereces [] S.A.Hovaessia, Itroductio to Sythetic Array ad Imagig Radars, Artech House, 980. [] Peter H. Zipfel, Squit Agle Guidace for Missiles with SAR Sesors, AIAA Missile Scieces Coferece, 3-5 December 996. [3] Rejith R. Kumar, Has Seywald, Three- Dimesioal Air-to-Air Missile Trajectory Shapig, Joural of Guidace, Cotrol, ad Dyamics, Vol. 8, No. 3, May-Jue 995. [4] Brako Ristic, Sajeev Arulampalam, James McCarthy, Target Motio Aalysis Usig Rage-oly Measuremets: Algorithms, Performace ad Applicatio to ISAR Data, Sigal Processig. [5] Arthur E. Bryso, Yu-Chi Ho, Applied Optimal Cotrol, Taylor ad Fracis. [6] Yaakov Bar-Shalom, X. Rog Li, Thiagaligam Kirubaraja, Estimatio with Applicatios to Trackig ad Navigatio: Theory Algorithms ad Software, Joh Wiley &Sos, 00. [7] Asif Farooq, David Limbeer, Trajectory Optimizatio for Air-to-Surface Missiles with Imagig Radars,Joural of Guidace, Cotrol, ad Dyamics, Vol. 5, No.5, September-October 00. [8] M. Rollaso, D. Salmod ad M. Evas, Parameter Estimatio for Termial Guidace Usig a Doppler Beam Sharpeig Radar, AIAA Guidace, Navigatio, ad Cotrol Coferece, August 003, AIAA [9] The MathWorks Ic, Optimizatio Toolbox User s Guide Versio., July 00. [0] The MathWorks Ic, Simulik Referece Versio 5, July 00. Parameter Descriptio Value Uits m Mass 500 kg S Referece area Ref π m T Thrust level 000 N to 500 CD0 Drag coefficiet k Iduced drag coefficiet ρ Air desity at sea.5 kg/m 3 0 level T Temperature at 88.6 K 0 sea level L Lapse rate - K/m R Gas costat 87.6 J/kg/K CL α Lift coefficiet 50 - derivative w.r.t. α. Table : Missile Dyamic Model Cofiguratio Parameter Descriptio Value Uits λ Radar 8 3x0 m wavelegth (35GHz c r Desired cross rage resolutio R Rage Re s resolutio R Acquisitio Acquire rage σ Maximum v_ Max look agle 3 m m 5000 m 40 deg Table : DBS Characteristics Parameter Descriptio Value Uits V ( 0 Iitial velocity 70 m/s h ( 0 Iitial height 000 m γ ( 0, ψ ( 0 Iitial 0 deg flightpath γ Term agles V Velocity at 70 m/s Term impact ψ Pitch ad yaw [-60,-90] deg Term flightpath agles at impact α Maximum 0 deg max agle of attack t Time of flight 3 sec f Number of odes 55 - Table 3: Optimizatio Criteria Figure 4: Weightig Fuctio 7

8 Figure 5: Optimal trajectory with the cost oly o the dwell time 8

9 Figure 6: Optimal trajectory with cost o the CRLB 9

10 Figure 7: Positio estimate ucertaity for trajectory with the cost oly o the dwell time Figure 8: Positio estimate ucertaity for trajectory with the cost o the CRLB 0

11 Figure 9: Gais for trajectory shapig guidace law Figure 0: Closed loop guidace model Figure : Compariso betwee ope ad closed loop trajectories