PROPORTIONAL NAVIGATION

Introduction

1. Proportional navigation guidance is used in a majority of tactical radar, infrared (IR) and TV guided missiles. It gained popularity because of its simplicity, effectiveness and ease of implementation.

Historical Background

2. The Lark missile which was first tested in December, 1950 successfully was the first missile to use proportional navigation. However proportional navigation dates back to World War II and was apparently known to the Germans though they did not apply it practically in their missiles. Apparently proportional navigation was studied by C.Yuan and others at the RCA Laboratories during World War II sponsored by the U.S.Navy. It was implemented first by Hughes Aircraft Company in a tactical missile using a pulsed radar system. Raytheon further developed proportional navigation and implemented it in a tactical continuous wave radar homing missile.

Definition

3. The proportional navigation guidance law basically generates acceleration commands which are perpendicular to the instantaneous missile-target line-of-sight (LOS). These acceleration commands are proportional to the LOS rate and closing velocity and expressed mathematically as

_____________________________(1)

where is the acceleration command (m/sec2),

k is the effective navigation ratio (usually in the range 3 to 5),

is the missile-target closing velocity (m/sec), and

is the time derivative of the line-of-sight angle, , or simply the line-of-sight rate (rad/sec).

4. The acceleration commands thus generated are given to the missile autopilot (pitch or yaw) and by the movement of the control surfaces (or other means of control such as thrust vector control or lateral divert engines or squibs in case of exo-atmospheric strategic interceptors) the missile is made to move in the desired direction towards the target.

5. The line-of-sight rate is usually measured by the seeker. The closing velocity is measured by a Doppler radar in case of tactical radar homing missiles whereas in tactical IR missiles or TV guided missiles, the closing velocity is “guestimated”.

Two-Dimensional Engagement Geometry for Proportional Navigation

6. Consider a two-dimensional point mass missile-target engagement geometry as shown in figure below:-

The co-ordinate system used is that of inertial co-ordinates fixed to the surface of a flat-Earth model. Thus the components of acceleration and velocities along the two axes or directions can be integrated without having to worry about the additional terms due to the Coriolis effect. Axis 1 represents the down range whereas axis 2 may represent the altitude or cross-range.

7. Assumptions

(a) Both the missile and target are assumed as point masses travelling at constant velocity.

(b) The gravitational and drag effects are neglected for simplicity.

8. Derivation

(a) Consider that the missile is heading towards the target with a velocity, Vm, and lead angle, L, with respect to the line-of-sight. The lead angle is theoretically the angle at which the missile must be oriented to be on a collision triangle with the target. If the missile is on the correct lead angle, no further acceleration commands are required for the missile to hit the target.

(b) In practice, the missile is launched towards an approximate intercept point since we do not know in advance what the target will do in future. Thus the missile will not be exactly on a collision triangle initially. The initial angular deviation of the missile from the collision triangle is known as heading error (HE).

(c) The imaginary line connecting the missile and target is known as line-of-sight (LOS). The angle the LOS makes with respect to the fixed reference axis 1 is denoted as λ.

(d) The instantaneous separation in range between the missile and target is denoted as RTM.

(e) The guidance will be considered proper if and only if the range between the missile and the target at the expected time of intercept is as small as possible or zero. The point of closest approach of missile and target is known as miss distance.

(f) The closing velocity, vc, is defined as the negative rate of change of the distance from the missile to the target, i.e.,

_______________________________(2)

At the end of the engagement, i.e., when the missile and target are in closest proximity, the sign of vc will change. From calculus, a function is either minimum or maximum when its derivative is zero. Thus when RTM is minimum, closing velocity velocity will be zero.

(g) The velocity of the target is denoted as VT and the target acceleration perpendicular to the target velocity vector is denoted as nT. Thus the angular velocity of the target can be expressed as

_________________________________(3)

where β is the flight path angle of the target which can be obtained by integrating eqn.(3). Thus the target velocity components with respect to the two axes 1 and 2 is given as

_________________________(4)

__________________________(5)

Since velocity is rate of change of position, it follows that

____________________________(6)

____________________________(7)

where RT1 and RT2 are the components of target position along axis 1 and 2 respectively.

(h) The missile acceleration components with respect to the two axes are given by aM1

and aM2. The missile velocity and position components can be expressed as differential equations involving the missile acceleration components as

___________________________(8)

___________________________(9)

__________________________(10)

__________________________(11)

9. Calculation of Missile Acceleration Components

The missile acceleration components can be found from the components of relative missile-target separation as follows:-

(a) The components of relative missile-target separation along axis 1 and 2 can be defined as

___________________(12)

___________________(13)

(b) Using trigonometry, the line of sight angle in terms of the relative separation components can be found as

_____________________(14)

(c) In a similar manner, the components of relative velocity along axis 1 and 2 are defined as

___________________(15)

___________________(16)

(d) The relative separation between the target and missile RTM can be expressed in terms of its inertial components along axis 1 and 2 by application of distance formula as

______________(17)

(e) Differentiating equation (17) and adding a negative sign gives the closing velocity vc

as

__________________(18)

Since the derivative of position is velocity, eqn.(18) can be rewritten as

________________(19)

(f) Differentiating eqn.(14) gives the line of sight rate as

__________________________(20)

The missile acceleration command (nc) can be found by substituting equations (19) and (20) in equation (1). The missile acceleration components in the Earth co-ordinates are given from trigonometry as

______________________________(21)

_______________________________(22)

10. Initial Conditions

A missile employing proportional navigation is fired in a direction to lead the target and thus forms a collision triangle with the line-of-sight. The angle between the instantaneous missile path or its velocity vector and the line-of-sight is called the missile lead angle, L. The theoretical missile lead angle can be found by application of law of sines as

________________________(23)

Since there is an initial angular deviation of the missile from collision triangle given by the heading error, HE, the initial missile velocity components in terms of the theoretical lead angle, L, and actual heading error, HE, can be expressed as

_______________(24)

________________(25)

Reference

Paul Zarchan, “Tactical & Strategic Missile Guidance”, Vol 199, Progress in Astronautics & Aeronautics, A Volume in AIAA Tactical Missile Series.