Deviated Velocity Pursuit

Tal Shima∗

Technion - Israel Institute of Technology, Haifa 32000, Israel

A guidance law that enables imposing a predetermined interception angle relative to thetarget’s flight direction is developed using the sliding mode approach. The guidance lawcan be applied in head-on, tail-chase, and a novel head-pursuit engagement. Simulationresults confirm its viability in several representative engagements against maneuveringtargets. It is shown that the head-on scenario places the stringiest requirements, and thusin some cases tail-chase or head-pursuit engagements should be considered instead. Thechoice between the two is dependent on the speed ratio of the adversaries as tail-chaserequires a speed advantage of the interceptor, while in head-pursuit the target must havethis advantage.

I. Introduction

In future interception scenarios it may be necessary to intercept targets with a pre-determined angle relativeto their flight path. This requirement may be due to special kill mechanisms or tracking requirements.

The scenarios of interest are the classical head-on (HO) and tail-chase (TC) ones, but also the novel head-pursuit (HP) engagement presented in Refs. 1, 2. In HO the adversaries fly towards each other; in TC theinterceptor chases the target; while in HP an interceptor with a speed disadvantage is positioned ahead ofthe target such that both fly in the same direction with the target closing in on the interceptor.

Guidance laws are usually derived assuming small deviations from a collision course allowing lineariza-tion.3 Representative such guidance laws are proportional navigation (PN), augmented PN, and optimalguidance law. Using these guidance laws the interception angle relative to the target flight path is specifiedbased on the initial collision triangle. In contrast, some classical guidance laws, such as pure pursuit (PP)and its derivative deviated PP (DPP),4 have not been derived under this assumption. Such guidance lawscan impose a specific final interception geometry. In PP the interceptor is aimed at the target and it isintercepted from its rear.5 The DPP guidance law is an extension of PP in that the missile is aimed at aconstant lead angle to the target and it is intercepted from a constant angle, dependent on the lead angle andspeed ratio.4 A drawback of using these guidance laws is that they impose severe maneuver requirementson the interceptor. Moreover, they require a speed advantage from the interceptor and thus can not beimplemented for example in the novel HP engagement.

Previous works on intercept angle control mainly include optimal guidance laws and modified PN typelaws.6–9 Ref. 6 proposes an optimal control law for impact angle error and miss distance minimization, in ascenario where a reentry vehicle pursues a fixed or slowly moving ground target. Refs. 7 and 8 derive optimalcontrol laws for a missile with arbitrary order dynamics trying to hit a stationary target. Ref. 9 uses thesame formulation as Refs. 7 and 8, but extends the previous works by adopting a time-to-go weighted energycost function to shape the missile’s trajectory. One of the underlying assumptions in all of these works is astationary or slowly moving target.

If flight along a collision triangle can not be assumed then linearization around this trajectory mightlead to unsatisfactory performance. Such an interception problem can be analyzed using nonlinear controltools. The sliding mode control (SMC) methodology is such a robust control tool enabling dealing withhighly nonlinear systems with large modelling errors and uncertainties.10 The SMC methodology has beensuccessfully used in various guidance applications. A missile guidance law in the class of PN, derived usingthe SMC approach, was proposed in Ref. 11. The sliding surface was selected to be proportional to the lineof sight (LOS) rate and the target maneuvers were considered as bounded uncertainties. Using numerical

∗Senior Lecturer, Department of Aerospace Engineering; AIAA Senior Member; tal.shima@technion.ac.il

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simulations, the superiority of the proposed guidance law over the conventional PN was advocated. InRef. 12 an adaptive sliding mode guidance law was derived. Using analysis and simulations, robustness todisturbances and parameter perturbations was shown. In Ref. 13,14 SMC was used to design an integratedautopilot-guidance loop for missiles with single and dual control surfaces, respectively. In a recent paper2

SMC was applied to tailor a guidance law for the unique HP interception geometry. The obtained guidancelaw is not adequate for HO and TC.

In this paper we propose a SMC guidance law that enables imposing a predetermined interception anglein HO, TC, and HP engagements. In the next section the interception engagement is outlined. A slidingmode controller is then developed and its performance is examined through simulation. Concluding remarksare offered in the last section.

II. Kinematics

A roll controlled interceptor is considered in this paper. For the relatively short time interval of theendgame (with small changes in the flight direction) the motion of such an interceptor can be separated intotwo perpendicular channels and the guidance problem can be treated as planar in each of these channels.The planar endgame geometry is shown in Fig. 1. The target is denoted as T while the interceptor as I. Thespeed, maneuvering acceleration, and flight path angle are denoted by V , a, and γ, respectively; the rangebetween the target and interceptor is r, and λ is the LOS angle relative to a fixed reference. The angles θand δ are the target’s and interceptor’s direction of flight relative to the LOS, respectively.

aTI

λ

θ

γI = λ + δ

γT = λ + θT

aI

r

λ

VI

δ

VT

Figure 1. Planar engagement geometry.

The engagement kinematics is expressed in a polar coordinate system (r, λ) attached to the target

r = Vr (1)

λ = Vλ/r (2)

where the speed Vr isVr = VI cos δ − VT cos θ (3)

and the speed perpendicular to the LOS is

Vλ = VI sin δ − VT sin θ (4)

We assume that the interceptor and target speeds, VI and VT , are constant and define the non-dimensionalparameter K as the speed ratio

K , VI/VT (5)

We assume ideal dynamics for the missile and target accelerations, aI and aT respectively; and that thetarget’s acceleration command is bounded

|aT | ≤ amaxT (6)

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These accelerations determine the interceptor and target trajectories

γI = aI/VI (7)

γT = aT /VT (8)

where the flight path angles γI and γT satisfy

γI = λ + δ (9)

γT = λ + θ (10)

From Eqs. (7)-(10) we obtainδ = aI/VI − Vλ/r (11)

θ = aT /VT − Vλ/r (12)

III. Interception Engagements

In this section several different interception engagements are discussed. They are distinguished by theirunique geometries. We first define these engagements and then provide conditions for their existence.

A. Definitions

We denote the running time as t. The engagement initiates at t = 0 with r(t = 0) < 0 and terminates att = tf where

tf = argt{r(t)r(t) = 0} (13)

The miss distance is r(tf ).

Definition 1. Perfect intercept An interception engagement with r(tf ) = 0.

Next we define the three possible interception engagements:

Definition 2. Head-on An interception engagement where ∃v > 0 : −π/2 ≤ θ(t) ≤ π/2, and π/2 ≤ δ(t) ≤3π/2 for t ∈ [v, tf ).

Definition 3. Tail-chase An interception engagement where ∃v > 0 : π/2 < θ(t) < 3π/2, and π/2 <δ(t−f ) < 3π/2 for t ∈ [v, tf ).

Definition 4. Head-pursuit An interception engagement where ∃v > 0 : −π/2 < θ(t) < π/2, and −π/2 <δ(t) < π/2 for t ∈ [v, tf ).

In Figs. 2-3 example geometries associated with the three different engagements are plotted. Thesegeometries were obtained for constant speeds and non maneuvering adversaries. Fig. 2 presents the HO andHP engagements possible when VI < VT . The interceptor can select between the two by choosing its flightdirection relative to the LOS (angle δ). In Fig. 3 the HO and TC engagements are drawn for the case whereVI > VT . Here the target’s flight direction θ defines a unique engagement. Note that if VI = VT then onlyHO is possible. Conditions for the existence of the different engagement geometries are given next.

B. Conditions

Conditions for the existence of the different engagement geometries are given next. It is assumed here thatthe vehicles follow straight lines (don’t maneuver) and that their speed is constant.

Lemma 1. A necessary and sufficient condition for perfect intercept is that ∃v > 0 : Vλ(t) = 0 for t ∈ [v, tf )

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IT

TVIVθ HPδ

HP

interception

pointHO

interception

point

HOδ

Figure 2. Head-on and head-pursuit engagements; VI < VT

IT

TV IV

HOθδ

TC

interception

point

HO

interception

point

TCθ

Figure 3. Head-on and tail-chase engagements; VI > VT

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Theorem 1. A necessary condition for perfect intercept is{|θ| ≤ sin−1 K K ≤ 1θ is arbitrary K > 1

(14)

Proof: Given K ≤ 1, if θ(0) > sin−1 K then Vλ > 0 ∀ t ∈ [0, tf ). Consequently, using Lemma 1, r(tf ) > 0.For θ(0) < − sin−1 K the condition is proved by symmetry. ¤

Theorem 2. A necessary condition for perfect intercept in HP is K < 1Proof: Based on Lemma 1 and definition 4 perfect HP interception requires that ∃v > 0 : Vλ = 0,−π/2 < θ(t) < π/2, and −π/2 < δ(t) < π/2 for t ∈ [v, tf ). Given 0 ≤ θ(t) < π/2, if K ≥ 1 then, in order toobtain Vλ = 0, 0 ≤ δ(t) ≤ θ(t) must be chosen. Consequently Vr(t) ≥ 0 terminating the engagement withr(tf ) > 0. For −π/2 < θ(t) ≤ 0 the condition is proved by symmetry. ¤

Theorem 3. A necessary condition for perfect intercept in TC is K > 1Proof: Based on Lemma 1 and definition 3 perfect TC interception requires that ∃v > 0 : Vλ = 0,π/2 < θ(t) < 3π/2 and π/2 < δ(t) < 3π/2 for t ∈ [v, tf ). Given π/2 < θ(t) ≤ π, if K ≤ 1 then, in order toobtain Vλ = 0, π/2 < δ(t) ≤ θ(t) must be chosen. Consequently Vr(t) ≥ 0 terminating the engagement withr(tf ) > 0. For π ≥ θ(t) < 3π/2 the condition is proved by symmetry. ¤

Theorem 4. Given K > 1, then perfect intercept of a non maneuvering target can be performed by a uniquechoice of δ. Using this choice, if π/2 < θ(0) < 3π/2 then TC is obtained; else, the engagement is HO.Proof: Given K > 1 then based on Theorem 2 perfect HP intercept can not be obtained. Based on Lemma1 perfect intercept requires that ∃v > 0 : Vλ = 0 for t ∈ [v, tf ). If 0 ≤ θ(0) ≤ π/2 then in order to obtainVλ = 0 for t ∈ [v, tf ), where 0 ≤ v < tf , then π − θ < δ < π must be chosen and the target is intercepted inHO. If π/2 < θ(0) ≤ π then in order to obtain Vλ = 0 for t ∈ [v, tf ), where 0 ≤ v < tf , then θ < δ < π mustbe chosen and the target is intercepted in TC. For π < θ(t) < 2π the condition is proved by symmetry. ¤

Theorem 5. Given K < 1 and |θ| ≤ sin−1 K, then perfect intercept of a non maneuvering target is possiblewith two choices of δ. For π/2 ≤ δ ≤ 3π/2 then HO is obtained, else −π/2 ≤ δ ≤ π/2 and HP is obtained.Proof: Given K < 1 then based on Theorem 3 perfect TC intercept can not be obtained. Based on Lemma 1perfect intercept requires that ∃v > 0 : Vλ = 0 for t ∈ [v, tf ). Given K < 1 and |θ| ≤ sin−1 K then in order toobtain Vλ = 0 for t ∈ [v, tf ), where 0 ≤ v < tf , δ = δHP = sin−1[sin(θ)/K] or δ = δHO = π−sin−1[sin(θ)/K]can be chosen. Based on definition 4 for δ = δHP the engagement is HP; while, based on definition 2, forδ = δHO the engagement is HO. ¤

IV. SMC Guidance law

In SMC the controller is obtained by converting an n-th order tracking problem to a first order stabi-lization problem. The design is performed around a sliding surface commonly denoted by σ = 0, wherethe sliding variable σ is a function of the system tracking error and possibly its derivatives. The problemis then to drive the scalar quantity defining the sliding surface to zero, and maintain it there, ultimatelyachieving exact tracking. The sliding variable is defined next, followed by the derivation of the equivalentcontroller, responsible to maintain the system on the sliding surface, i.e. in sliding mode. This is followedby the derivation of the uncertainty controller responsible to bringing the system to the sliding surface.

A. Sliding Variable

The objective of the guidance law is to impose

θ(t−f ) = θr (15)

where θr is selected by the designer based on the interceptor’s speed ability and the required interceptiongeometry (HO, TC, or HP).

We denote the guidance law that can impose Eq. (15) as deviated velocity pursuit as, similar to DPP, weaim the interceptor with some angle δ relative to the LOS. This angle is dependent on the velocity vector ofthe target.

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We define the deviation from the required hit angle as

e = θ − θr (16)

with its time derivatives beinge = aT /VT − Vλ/r (17)

The relative degree of e with respect to the control command is two as aI appears only in e. The slidingvariable σ is thus defined as

σ = τ e + e (18)

where τ is the sliding surface time constant, chosen by the designer.Substituting Eqs. (16) and (17) in Eq. (18) we can write the sliding variable as

σ = (aT /VT − Vλ/r)τ + θ − θr (19)

B. Equivalent Controller

The sliding mode controller aI consists of an equivalent part denoted aIeq and an uncertainly part denotedaIuc, where

aI = aIeq + aIuc (20)

The equivalent controller is designed to maintain the system on the sliding surface, by imposing σ = 0 inthe absence of modeling errors and target maneuver commands

aIeq = argaI

{σ0 = 0} (21)

where σ0 denotes the time derivative of the sliding variable with zero uncertainty. Differentiating Eq. (19)and nulling the uncertainty part, related to the target’s maneuver, we obtain

σ0 = τ(−aI cos δ/r + 2VλVr/r2)− Vλ/r (22)

Consequently, the obtained equivalent controller is

aIeq = Vλ(2τVr − r)/(rτ cos δ) (23)

C. Uncertainty Controller

The system may not initially be in sliding mode or modeling errors and target maneuvers can cause thesystem to depart from the sliding surface. The uncertainty controller is designed to drive the system, infinite time, to the sliding surface in face of these uncertainties. The design of the uncertainty controller isbased on the model of the interceptor and target dynamics and the known bounds on the uncertainties.

We select the following form of an uncertainty controller

aIuc = Rsign(σ) (24)

and analyze the convergence to the sliding surface using the following candidate Lyapunov function

L = σ2/2 (25)

By using the time derivative of this candidate Lyapunov function, the well-known reaching condition isobtained

L = σσ < 0 (26)

and R is chosen such thatL = σσ(aIuc, aIeq) < 0 (27)

is maintained in the face of the bounded modelling errors and target maneuver.Substituting Eq. (23) and (24) in Eq. (27) we obtain

L = −στ(R cos δ sgnσ − aT cos θ)/r (28)

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Using Eq. (6) this derivative is bounded by

L ≤ −|σ|τ(R cos δ − amaxT )/r (29)

ChoosingR cos δ > amax

T (30)

ensures negative definiteness of the Lyapunov function, such that finite time convergence to the slidingsurface is guaranteed.

V. Performance Analysis

In this section perfect information HO, TC, and HP engagements are studied using simulations. Thesimulation parameters are summarized in Table 1. To avoid excessive chattering the guidance law wasimplemented with a boundary layer around the sliding surface σ with a width of 2[deg]. The time constantchosen for the sliding surface was 0.1sec. The target performed a constant 5g maneuver while for theinterceptor no acceleration saturation was imposed.

Missile Target Kinematics

VI ∈ {1600, 2000}m/s VT ∈ {1600, 2000}m/s r(0) = 15km

δ(0) ∈ {45, 135}deg θ(0) ∈ {45, 135}deg λ(0) = 0 degaT = 5g

Table 1. Simulation parameters

Figs. 4-6 present the interceptor and target trajectories in an inertial coordinate system for a samplescenario with a target maneuver of 5g. The different figures are for TC, HO, and HP engagements. Therequirement in all these runs was for θr = 0. In all runs the LOS continuously rotates, due to the needto reach the required interception geometry in the face of the target maneuver. In the case of TC and HP(Figs. 4 and 6, respectively) it is evident that after an initial transient the interceptor maintains the sametrajectory as that of the target. In TC the interceptor follows the target while in HP it leads it. Althoughin all cases the target maneuvered, the missile was able to impose the required interception angle, with avery small miss distance that is less than 0.5m.

Fig. 7 presents the convergence to the sliding surface in the above three scenarios. Due to the initialheading error the initial value of σ is non-zero and the surface is reached only later in the engagement. Inthe case of HO it is reached only very close to the termination point, while for the two other scenarios it isreached much earlier (compared to the duration of the engagement). The presence of the boundary layer of2 deg is clear from the plots for TC and HP. Note the difference in the scenario duration, due to the differentinterception geometries (tf (HO) = 4.3s, tf (HP ) = 32.4s, and tf (TC) = 33.8s).

In Fig. 8 the interceptor’s acceleration profile is plotted. In all three cases the missile issues large initialcommands in order to null the initial heading error, evident in the previous figure. In HO the interceptor’sacceleration diverges, reaching the unrealistic value of 300g. This can be expected due to the geometryof the scenario imposing a very short engagement. In contrast, the acceleration requirements in TC andHP are much smaller. Due to the boundary layer implementation chattering is avoided. The peaks in theaccelerations occur when the sliding surface is reached. Choosing a larger boundary layer can eliminate thisphenomenon, but may produce unsatisfactory guidance performance. In HP the steady state accelerationrequirement is the smallest due to the smaller curvature of the associated target’s trajectory in this scenario,as well as the lower speed of the interceptor.

In Figs. 9-11 the interceptor relative trajectories in the target coordinate system are shown, for differentvalues of θr, in the three different engagements. Five different trajectories are plotted in each figure cor-responding to different requirements on θr. It can be seen that in the TC and HP engagements, even inthe presence of target maneuvers and initial heading errors, interception is achieved with the required θr.However, in the HO engagement θr = −40,−20, 40 deg can not be achieved, due to the geometry with a verylarge closing speed.

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−50 −40 −30 −20 −10 0 10

−15

−10

−5

0

5

10

15

20

25

30

X [km]

Y [k

m]

InterceptorTarget

t = tf

t = 0

Figure 4. Inertial trajectories in a tail-chase sample scenario; VI = 2000m/s, VT = 1600m/s

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2 4 6 8 10 12 14

−4

−2

0

2

4

6

X [km]

Y [k

m]

InterceptorTarget

t = 0t = 0

t = tf

Figure 5. Inertial trajectories in a head-on sample scenario; VI = 1600m/s, VT = 2000m/s

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0 10 20 30 40 50

0

5

10

15

20

25

30

35

X [km]

Y [

km]

InterceptorTarget

t = 0

t = tf

Figure 6. Inertial trajectories in a head-pursuit sample scenario; VI = 1600m/s, VT = 2000m/s

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0 5 10 15 20 25 30 35−50

−40

−30

−20

−10

0

10

20

t [sec]

σ [d

eg]

Head OnTail ChaseHead Pursuit

Figure 7. Convergence to the sliding surface in the different engagements; θr = 0[deg].

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0 5 10 15 20 25 30 35−50

0

50

t [sec]

a I [g]

Head OnTail ChaseHead Pursuit

Figure 8. Acceleration profiles; θr = 0[deg].

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0 5 10 15120

140

160

180

200

220

240

r [km]

θ [d

eg]

Tail Chase

θr = 220

θr = 200

θr = 180

θr = 160

θr = 140

Figure 9. Trajectories in target coordinate system; tail chase scenario.

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0 5 10 15−10

0

10

20

30

40

r [km]

θ [d

eg]

Head On

θr = 20

θr = 40

θr = −20, −40

θr = 0

Figure 10. Trajectories in target coordinate system; head on scenario.

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0 5 10 15−40

−20

0

20

40

60

r [km]

θ [d

eg]

Head Pursuit

θr = 40

θr = −20

θr = 20

θr = 0

θr = −40

Figure 11. Trajectories in target coordinate system; head pursuit scenario.

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VI. Conclusions

A guidance law that enables imposing a predetermined interception angle relative to the target’s flightdirection was presented. The guidance law can be applied in head-on, tail-chase, and a novel head-pursuitengagement. It does not require a speed advantage from the interceptor.

The new proposed guidance law was implemented using the sliding mode approach. The controllerallows bringing the system to a sliding surface that imposes the required geometric rule. Simulation resultsconfirmed that this surface can be reached and maintained in face of target maneuvers and large initialheading errors.

It was shown that head-on is the hardest scenario to obtain a required interception geometry. This isdue to the large closing speed. Thus, in cases where it is of essence to impose a predetermined interceptionangle the tail-chase or novel head-pursuit engagements should be considered. The choice between the twois dependent on the speed ratio of the adversaries, as head-pursuit can be maintained when the interceptorhas a speed disadvantage while tail-chase requires a speed advantage by the interceptor.

Acknowledgment

This work was supported in part by a Horev Fellowship through the Taub Foundation.

References

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