Proportional Navigation Guidance System

Proportional Navigation Guidance 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 2.1

3

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3 Proportional Navigation Method of Guidance

3 6

3.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6

3.2

Notes on Guidance Systems[?] . . . . . . . . . . . . . . . . . . . . . . . .

8

3.2.1

. . . . . . . . . . . . . . . . .

8

Working of Proportional Navigation Guidance System . . . . . . . . . . .

12

3.3

Proportional Navigation Guidance

References

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1

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 1

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 [?], Garnell[?], Lee, and Zarchan[?].

2

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.

3

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) 4

and dV φ˙ = 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 VM

r = r0

(V (1 + cosφ0 ) VT (sinφ) VMT − 1) VM

(sinφ0 ) VT

(2.3)

VM

(1 + cosφ) VT

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.

5

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 γ˙ M = 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.

6

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˙ = −vM cos(φ − γ) − (−vT cosφ) r˙ = vT cosφ − vM cos(φ − γ)

(3.2)

7

dV φ˙ = r ˙ r.φ = vM sin(φ − γ) − vT sinφ

3.2

(3.3)

Notes on Guidance Systems[?]

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 frequency (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. 8

(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 velocity 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 vM sin(φ − γM ) This generates a positive LOS rotation. 9

(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 components of missile and target velocities perpendicular to the LOS divided by the distance between the target and missile which is denoted by r. ωLOS =

vM sin(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)

˙ taking Laplace transform of eqn.LOSratelin Since ωLOS is the angular rate or φ, gives vM (s)(φ(s) − γM (s)) − vT (s)(φ(s) − γT (s)) r(s) vM (s)γM (s) vT (s)(φ(s) − γT (s)) vM (s) (s − )φ(s) + = r(s) r(s) r(s) vT (s)(φ(s) − γT (s))/r(s) vM (s)γM (s)/r(s) φ(s) = − (3.6) M (s) M (s) (s − vr(s) ) (s − vr(s) ) sφ(s) =

(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. 10

(a) The component of missile velocity parallel to the LOS is given by vM cos(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˙ = vM cos(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:γ˙ T =

aT vT

(3.8)

γ˙ M =

aM vM

(3.9)

11

3.3

Working of Proportional Navigation Guidance System

A simple block diagram of a proportional navigational guidance system is shown in fig.3.3.

12

θT − θR (rad)

VT /R s−VM /R

θ R |T +

-

θR SEEKER (rad)

ωSK

(rad/sec)

GUIDANCE COMPUTER

az(comm) az az(comm) (g)

θM |R VM /R s−VM /R

13

az (g)

9.81 VM s

θM

(rad)

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.

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References

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