7/30/2019 Differential Geometric Guidance Based on the Involute of the Target's Trajectory 2-D Aspects

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Proceeding of the2004American Control ConferenceBoston,Massachusetts une30-July2,2004 ThP12.3

Differential geometric guidance based on the involute of thetarget's trajectory: 2-D aspects0.Ariff, R. Zbikowski, A. Tsourdos, and B. A. WhiteDepartment of Aerospace, Power & SensorsCranfield University (RMCS Shrivenham)Swindon SN6 8LA , ENGLA ND

Absfracf-This paper presents a novel approach to missileguidance utilising the differential geometry of curves and notrelying on the line of sight (LOS) information. The target'strajectory is treated asa smooth curve of known curvature andthe new algorithm is based onthe involuteof the target's curve.The missile's trajectory uses the concept of virtual target togenerate the correct involute trace. I t is shown that the missileis either on the trace immediately or can reach it througha flying in procedure. Following the trace may require a 3-D manoeuvre, not described here, while the 2-Daspects ofthe algorithm lead to very simple formulae. The referencescenario sa planar air-to-air engagementof point masseswitha manoeuvrable target of the F-16 fighter class. Simulationsforperfect target information show intercepts both for the involutelaw andPN guidance. PN based intercepts occur quicker,butthe involute based trajectoriesaremore diff icult to evade andalways result in a side impact

I. INTRODUCTIONThis paper proposes a fresh look at the generation ofintercept trajectories for missile guidance. The referencescenario is a planar air-to-air engagement of point masseswith a manoeuvrable target of the F- 16 fighter class. Themotivation is to overcome, without the complexity of Opti-

mal Control [I], the t wo shortcomings of the proportionalnavigation (PN) algorithm [I41 family: (i) reliance on theline of sight ( LOS) for derivation of intercept geometryand kinematics, (ii) effectiveness against non-manoeuvringtargets only. The second limitation can be, to some extent,alleviated through various modifications of the original PNalgorithm, e.g. augmented PN. Still, the kinematic rules ofthe PN algorithm family aim at compensating the deviationof the missile velocity vector from the LOS. In particu-lar, the target manoeuvres manifest themselves indirectlythrough themotion of theLOS. Such motion becomes veryrapid when the missile and target are close to each other,even for benign evasive manoeuvres. This is an inescapableconsequence of using straight lines (intercept triangles)to express curved trajectories (manoeuvres). I ndeed theoriginal PN was conceived for a target moving with aconstant speed along a straight line. A generalisation toscenarios with manoeuvres leads naturally to freeing theintercept geometry of the rectilinear framework.This paper is organised as follows. This introductorysection continues by presenting a synoptic account of theessential facts from Differential Geometry in Section I -A ,followed by asummaryof the relevant prior work in SectionI-B.Thenovelty of the proposed approach iscontrastedwith

0-7803-8335-4104/$17.00 02004A A C C

that work in Section I-C. Brief information on involutes isgiven in Section I1 which leads to the heart of the paper,Section 111 where the new algorithm is derived in detail.The assumptions on the scenarios used to test the new laware collected in Section I V. The actual results follow inSection V and the paper ends with conclusions in SectionVI.A. Background

How can the curvilinear engagement geometry be refor-mulated and endowed with appropriate intercept kinemat-ics? We believe that the differential geometry of spatialcurves [9], [IO], [ 5 ] offers an attractive setting for thistask. Indeed, it has long been a tool for kinematic analysisof material points, especially in mechanism and machinetheory [4].The essence of differential geometric description ofsmooth curves inB3 s the use of Calculus to quantify howa curve deviates locally from its linear approximation. Thepoint mass velocity vectorv=v(t)over time t is expressedas a multiple U =u ( t ) of the unit tangent vector T =T(t),i.e. v(t) = v(t)T(t). At the same moment in time, thenormal vector k =k(t) sobtained by differentiatingT andis therefore' orthogonal to T. Again, k(t)=n(t)N, whereN is the unit normal vector and n is the curvature. Thecurvature measures the local deviation of the curve fromthe rectil inear progression along the tangent line. Indeed,its inverse p = l /n is the radius of best fitting circle inthe plane spanned by the tangent and normal vectors. Theright-handed local coordinate frame is completed by thebinormal vector b(t)=7(t)B,where B '2 T x N is theunit binormal vector. The torsion T measures locally howmuch the curve deviates from the plane spanned by thetangent and normal vectors. It i s convenient to replace thetime parameter t with the arc length s = so v(l)dt, for thenthe reparameterised curve has unit speed, as ds /d t =U. Theabove definitions and relationships are summarised2 by theFrenet-Serret equations:[(s) = [ n(S) &(') &] , (1)

def t

0 - 7( 5) 0 B(s)'Since T .T =1, di f ferent iat ion gives T .T =0.'Note that each of T,N. B i s a vector in W3, n general.

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where means differentiation with respect to s. Parameten-sations of (1) differ by the factor of speedU =ds / d t , as forany vector x , we obviously have d x l d t =( d x / d s ) ( d s / d t ) .It should be emphasised that differential geometry ofcurves has nothing new to say about straight lines, asK = T = 0 then. Indeed, such case can be handled byPN, and hence ow focus on manoeuvring targets.B. Pr i or work

There have not been many attempts to use the differentialgeometric formulation for missile guidance.The notable work [3], [6], [7] is a recent example. Inthose papers the 3-D kinematics of missile-target pointmasses were resolved with respect to the LOS. The mis-sile trajectory arc length s,, was.used as the commonparameter for both curves. The targedmissile speed ratiom=ut /u, =const2 was introduced

If A >2, then the LOS rate w decreases, provided T