Proportional NavigationGuidance System

Notes

Contents

Contents 1

1 Introduction 1

1.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1.1 Types of Guidance . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1.2 Guidance Laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.1.3 Literature Survey . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

2 Velocity Pursuit Method of Guidance 3

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

3 Proportional Navigation Method of Guidance 6

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

3.2 Notes on Guidance Systems[1] . . . . . . . . . . . . . . . . . . . . . . . . 8

3.2.1 Proportional Navigation Guidance . . . . . . . . . . . . . . . . . 8

3.3 Working of Proportional Navigation Guidance System[1] . . . . . . . . . 12

3.4 General Working of Guidance System[2] . . . . . . . . . . . . . . . . . . 14

4 Integrated Guidance and Control 17

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

1

4.2 Overview [3] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

4.3 Functions of Various IGC subsystems . . . . . . . . . . . . . . . . . . . . 18

4.4 Traditional Guidance and Control Architecture . . . . . . . . . . . . . . 18

References 20

2

Chapter 1

Introduction

1.1 Overview

Two important definitions are:-

(a) Guidance. Guidance can be defined as the strategy for how to steer the missile to

intercept.

(b) Control. Control can be defined as the tactics of using the missile actuators to

implement the guidance strategy.

1.1.1 Types of Guidance

Guidance can be divided into two types namely:-

(a) Target Related Guidance. In this guidance strategy the target tracking data are

provided in real time from a sensor which can be on-board the missile or off it.

(b) Non-Target Related Guidance. In this strategy the missile navigates to some

predetermined point which can be the target or the point where target related

guidance starts.

It must be noted that the performance of integrated GPS/INS navigational systems

offer precision in the order of meters. Weapons using non-target related guidance using

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the integration of sensor-to-shooter improves and near real time targeting data can be

obtained from sensors not fixed in the missile system, can hence compete with traditional

homing missiles and the distinction between the two types becomes less clear..

1.1.2 Guidance Laws

The most fundamental, and also most commonly used, guidance laws are:-

(a) Velocity Pursuit

(b) Proportional Navigation

(c) Command-to-Line-of-Sight

(d) Beam Riding

All of these guidance laws date back to the very first guided missiles developed in the

1940’s and 1950’s. The reasons that they have been so successful are mainly that they

are simple to implement and that they give robust performance.

1.1.3 Literature Survey

The presentation in this paper is mainly based on textbooks on missile guidance e.g.,

Blakelock [1], Garnell[2],and Zarchan[4].

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Chapter 2

Velocity Pursuit Method of

Guidance

2.1 Introduction

The conceptual idea behind velocity pursuit guidance is that the missile should always

head for the target’s current position. Thus this strategy requires that the missile’s

velocity is greater than the target’s, in order to result in an intercept. The required

information for velocity pursuit is limited to the bearing to the target, which can be

obtained from a simple seeker, and the direction of the missile’s velocity. Velocity

pursuit is usually implemented in laser guided bombs, where a simple seeker is mounted

on a vane, which automatically aligns with the missile’s velocity vector relative to the

wind. The mission of the guidance and control system thus becomes to steer the bomb

such that the target is centred in the seeker.

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Using a target fixed polar coordinate system, see figure 1, the equations describing

the kinematics of velocity pursuit are

dVM = dR + dVT

dR = dVM − dVT

r = −vM − (−vT cosφ)

r = vT cosφ− vM (2.1)

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and

φ =dV

r

r.φ = −vT sinφ (2.2)

Alternately,

the rate at which the distance between the target and the missile is reducing to zero

is given by the projection of instantaneous velocity vector of target on the line joining

the missile and target i.e.,vT cosφ minus the instantaneous velocity of missile. This

gives eqn.3.2. Similarly, the rate at which the angle between the line joining missile

and target and the target velocity vector reduces to zero is given by the perpendicular

distance between missile and target velocity vector given by −vT sinφ divided by the

instantaneous distance between target and missile (r). This gives eqn.3.3. Integration

gives

r = r0(1 + cosφ0)

VMVT

(sinφ0)VMVT

(sinφ)( VM

VT− 1)

(1 + cosφ)VMVT

(2.3)

where index 0 indicates the zero or initial condition. Thus it can be noted from eqn.2.3

that r becomes zero for the value of φ = 0 i.e., when the intercept is tail chase, and in

case of head-on (φ = π) the condition is unstable (1/0) condition.The velocity pursuit

guidance law results in high demanded lateral acceleration, in most cases infinite at the

final phase of the intercept. As the missile cannot perform infinite acceleration, the

result is a finite miss distance. Velocity pursuit is thus sensitive to target velocity and

also to disturbances such as wind. The velocity pursuit guidance law is not suitable for

meter precision.

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Chapter 3

Proportional Navigation Method of

Guidance

3.1 Introduction

The conceptual idea behind proportional navigation is that the missile should keep

a constant bearing to the target at all time.The guidance law that is used to implement

this concept is given by

γM = c.φ (3.1)

where γM is the direction of the missiles velocity vector, φ is the bearing of missile to

target, and c is a constant. Both of the angles γM and φ are measured relative to some

fixed reference.

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Figure 3.1: Proportional navigation, kinematics and definition of angles.

With angles as defined in figure 4 and using polar coordinates the following kinematic

equations in the 2D case can be obtained:-

dVM = dR + dVT

dR = dVM − dVT

r = −vMcos(φ− γ)− (−vT cosφ)

r = vT cosφ− vMcos(φ− γ) (3.2)

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φ =dV

r

r.φ = vMsin(φ− γ)− vT sinφ (3.3)

3.2 Notes on Guidance Systems[1]

The lateral autopilot in a missile receives an acceleration command as its input and the

aim of the autopilot is to track this commanded acceleration. The acceleration command

is provided by an appropriate guidance system. Two such guidance systems namely

proportional navigation and line-of-sight command guidance systems are discussed.

(a) In proportional navigation, the missile is guided either by the reflected radio fre-

quency (RF) or the radiant infrared (IR) energy from the target. In case of active

homing missiles, the illuminating radar is in the missile itself whereas in case of

semiactive homing missiles, a separate illuminating radar which is the fire control

radar of the launch aircraft in case of air-to-air missiles and a ground radar at the

launch site in case of surface-to-air missiles, is used.

(b) In case of command guidance which is generally used in case of surface-to-air

missiles, two tracking radars, one to track the target and the other to track the

missile are located at the launch site.

3.2.1 Proportional Navigation Guidance

In proportional navigation guidance method, the missile turns at a rate proportional

to the angular velocity of the line-of-sight (LOS). The LOS is defined as an imaginary

line from the missile to the target. The ratio of the missile turning rate to the angular

velocity of the LOS is called the proportional navigation constant denoted by N . The

value of N is usually greater than one and usually ranges from 2 to 6. This means that

the missile will always be turning at a rate faster than the LOS rate and thus build up

a lead angle with respect to the LOS. Thus for a constant velocity missile and target

(target not maneuvering), anuglar velocity of line of sight is zero and the lead angle

generated can put the missile on a collisionm course with the target.

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(a) If N=1, then the missile is turning at the same rate as the LOS and thus homing

on to the target.

(b) If N < 1, the missile will be turning slower than the LOS thus continuously falling

behind the target and making an intercept impossible.

The seeker, in case of proportional navigation guidance system, while tracking the target

establishes the direction of the LOS. Thus the output of the seeker is the angular velocity

of the LOS with respect to inertial space as measured by rate gyros mounted on the

seeker.

The guidance geometry for proportional navigation guidance is as shown in fig.3.2.1

where initially the turning angle of the target γT is considered to be zero while that of

the missile is given by γM . The angle of LOS is denoted by φ. The target and missile

velocities are denoted by VT and VM .

Determination of Angular Velocity

The magnitude of the angular velocity of the LOS which generates the angular veloc-

ity of the seeker ωSK is determined by the components of missile and target velocity

perpendicular to the LOS from the fig.3.2.1.

(a) The component of missile velocity perpendicular to the LOS is given by

vMsin(φ− γM)

This generates a positive LOS rotation.

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(b) The component of target velocity perpendicular to the LOS is given by

vT sin(φ− γT )

This generates a negative LOS rotation.

(c) Thus the magnitude of the angular velocity is the difference between the com-

ponents of missile and target velocities perpendicular to the LOS divided by the

distance between the target and missile which is denoted by r.

ωLOS =vMsin(φ− γM)− vT sin(φ− γT )

r(3.4)

If the perpendicular components of missile and target are equal and unchanging,

ωLOS will be zero and there will be no rotation of the LOS. Thus the missile will

be on a collision course with the target.

(d) If the angles are assumed to be small, then eqn.3.4 can be linearised as follows:-

ωLOS =vM(φ− γM)− vT (φ− γT )

r(3.5)

Since ωLOS is the angular rate or φ, taking Laplace transform of eqn.LOSratelin

gives

sφ(s) =vM(s)(φ(s)− γM(s))− vT (s)(φ(s)− γT (s))

r(s)

(s− vM(s)

r(s))φ(s) +

vM(s)γM(s)

r(s)=

vT (s)(φ(s)− γT (s))

r(s)

φ(s) =vT (s)(φ(s)− γT (s))/r(s)

(s− vM (s)r(s)

)− vM(s)γM(s)/r(s)

(s− vM (s)r(s)

)(3.6)

(e) From eqn.3.6, it can be seen that when N < 1, the geometry of proportional

navigation introduces a pole in the right half s plane. As N is increased from zero,

this pole slowly moves to the left and crosses the imaginary axis at N = 1.

Determination of LOS rate (r)

The magnitude of the linear rate of the LOS is determined by the components of missile

and target velocity parallel to the LOS from the fig.3.2.1.

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(a) The component of missile velocity parallel to the LOS is given by

vMcos(φ− γM)

This generates a positive LOS rate.

(b) The component of target velocity parallel to the LOS is given by

vT cos(φ− γT )

This generates a negative LOS rate.

(c) Thus the linear rate of change of LOS (r) is given by

r = vMcos(φ− γM)− vT cos(φ− γT ) (3.7)

Assumptions

(a) Missile and target are point masses.

(b) Missile and target velocities are constant.

(c) Autopilot and seeker loop dynamics are fast enough to be neglected when compared

to overall guidance loop behavior.

(d) Upper bound of target accleleration exists.

Relations for Target and Missile Accelerations

Using the above assumptions, the angular rates of target and missile can be related to

their respective accelerations by the following equations:-

γT =aT

vT

(3.8)

γM =aM

vM

(3.9)

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3.3 Working of Proportional Navigation Guidance

System[1]

A simple block diagram of a proportional navigational guidance system is shown in

fig.3.3.

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θT − θR

VT /Rs−VM/R

θR|T

θM |R

+-(rad) (rad) (rad)(rad/sec) (g) (g)

θRSEEKER

ωSK GUIDANCE

COMPUTER

az(comm) az θMazaz(comm)

9.81VMs

VM/Rs−VM/R

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The working is explained below:-

(a) θR|T is the LOS direction due to the target motion. This is given as one of the

inputs to the seeker through the summer block.

(b) The output of the seeker is ωSK which is the input to the guidance computer which

generates the missile acceleration command. It is assumed that the acceleration

command is proportional to the missile velocity times the angular velocity of the

LOS.

(c) The missile acceleration in g’s times 9.81/VM is equal to the missile pitch rate.

Integration of this value gives θM .

(d) The output of the transfer function in the feedback path is the rotation of the LOS

due to missile θR|M which is now given as the other input of the summer but with

opposite (negative) sign.

3.4 General Working of Guidance System[2]

A simplified closed-loop block diagram for a vertical or horizontal plane guidance loop

without an autopilot is as shown below: -

(a) The target tracker determines the target direction θt.

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(b) Let the guidance receiver gain be k1 volts/rad (misalignment). The guidance

signals are then invariably phase advanced to ensure closed loop stability.

(c) In order to maintain constant sensitivity to missile linear displacement from the

LOS, the signals are multiplied by the measured or assumed missile range Rm

before being passed to the missile servos. This means that the effective d.c. gain

of the guidance error detector is k1 volts/m.

(d) If the missile servo gain is k2 rad/volt and the control surfaces and airframe produce

a steady state lateral acceleration of k3m/s2/rad then the guidance loop has a

steady state open loop gain of k1k2k3m/s2/m or k1k2k3s

−2.

(e) The loop is closed by two inherent integrations from lateral acceleration to lateral

position. Since the error angle is always very small, one can say that the change in

angle is this lateral displacement divided by the instantaneous missile range Rm.

(f) The guidance loop has a gain which is normally kept constant and consists of the

product of the error detector gain, the servo gain and the aerodynamic gain.

Consider now the possible variation in the value of aerodynamic gain k3 due to change in

static margin. The c.g. can change due to propellant consumption and manufacturing

tolerances while changes in c.p. can be due to changes in incidence, missile speed and

manufacturing tolerances. The value of k3 can change by a factor of 5 to 1 for changes

in static margin (say 2 cm to 10 cm in a 2 m long missile). If, in addition, there can be

large variations in the dynamic pressure 12ρu2 due to changes in height and speed, then

the overall variation in aerodynamic gain could easily exceed 100 to 1.

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Chapter 4

Integrated Guidance and Control

4.1 Introduction

Traditional guidance and control design involves the use of a decoupled architecture

wherein the airframe, sensor and propulsion systems and the guidance and control sub-

systems are treated separately thus requiring three discrete algorithms. The separated

design and as a result the weak functional interaction amon the algorithms will mean a

difficult performance optimisation problem to be solved.

In the integrated guidance and control design, the three functions are designed to be

performed by a single component. This paradigm is expected to improve the synergy

between flight control and guidance functions as opposed to the decoupled architecture.

4.2 Overview [3]

Missile interceptors of the modern era are expected to engage different types of mis-

siles whose trajectories are varied. They may be required to engage tactical ballisitc

missiles (TBMs) or high performance cruise missiles. They may require a high degree of

precision and thus highest kill probability considering the payloads these missiles may

carry. The design of the missile must take care of future threats also. Hence next gen-

eration interceptors need to fly faster and longer with a high degree of efficiency, better

resolution, increased maneuverability and required lethality.

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4.3 Functions of Various IGC subsystems

The central role of guidance and control algorithm is to functionally integrate the

various sub-systems.

(a) IGC must ensure that the missile maintains stable flight and converges on the

enemy missile/target with minimum miss distance.

(b) IGC must ensure that the angle at which the missile converges is such that the

onboard seeker is able to acquire the target and track it which is another critical

function.

(c) IGC must manage the propulsion and airframe subsystems in such a way as to

maximise the range or time of flight.

(d) In case of proximity fuse based warheads, the direction of approach needs to be

controlled by IGC in such a way as to maximise the kill probability.

4.4 Traditional Guidance and Control Architecture

The traditional decoupled guidance and control architecture consists of the guidance

filter, the guidance law and autopilot.

(a) The guidance filter takes noisy target measurement data from which, for example,

target position, velocity and acceleration with respect to Cartesian reference frame

can be estimated. In general, a Lalman filter or an extended Kalman filter is used.

(b) The guidance law takes the above inputs from the filter and determines the di-

rection of travel of interceptor for a positive interception with the target/threat.

Proportional Navigation guidance is one of the oldest guidance laws.

(c) The autopilot ensures stable flight and also ensures the acceleration commands

from the guidance computer are tracked with required accuracy through proper

commands to the airframe (e.g. fins) or propulsion (e.g., thrust vector) sub-

systems.

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(d) The inertial navigation system is used for determining the instantaneous missile

position in range as well as heading (co-ordinates).

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References

[1] J. H. Blakelock, Automatic Control of Aircraft and Missiles,Second Edition. New

York: John Wiley and Sons,Inc, 1990.

[2] P. Garnell, Guided Weapon Control Systems. London: Brassey’s Defence Publishers,

1980.

[3] Palumbo, “Integrated guidance and control,” John Hopkins, vol. 79, no. 12, pp.

1736–1768, 1991.

[4] P. Zarchan, Tactical Missile Guidance. New York: John Wiley and Sons,Inc, 1990.

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