The Pursuit Guidance Law

Transcription

1 Chapter 7 The Pursuit Guidance Law Module 6: Lecture 15 Pure Pursuit Guidance Law Keywords. Pure pursuit; Capture Region As discussed earlier, pursuit guidance law is one of the most logical guidance laws to be used for missile guidance. The basic philosophy behind this guidance law is, if the missile continues to point towards the target then it is guaranteed that after a finite time the missile will intercept the target. Intuitively speaking this must be true if the missile has a higher speed than the target. In this chapter we will formulate the equations of motion for the pursuit guidance law and analyze those equations to obtain a few important and significant results. First we will address the pure pursuit guidance law and then subsequently the deviated pursuit guidance law. 7.1 Pure Pursuit Guidance Law The engagement equations Consider the engagement geometry given in Figure 7.1. Note that if the missile is using a perfect pursuit guidance law, then at all instants in time, M should point towards the current target position. This is shown in Figure 7.1. The target is assumed to be a 98

2 Guidance of Missiles/NPTEL/2012/D.Ghose 99 M R T T α T M Ref Figure 7.1: Engagement geometry for pure pursuit non-maneuvering one. The equations of motion are given by, R = Ṙ = T cos(α T ) M (7.1) = R = T sin(α T ) (7.2) Note that here too R and are the two components of the relative velocity between the target and the missile, along the LOS and normal to the LOS. Dividing the first equation by the second and with some manipulation we get, where Integrating this equation, we get, 1 R dr = {cot(α T ) ν csc(α T )} d (7.3) ν = M T (7.4) R = f() (7.5) But this equation does not help us much since R is expressed as a function of and not as a function of time t. Of course, as shown in Locke (1955), it is possible to manipulate this complicated equation further and obtain some interesting results. But we will not follow this route since the complexity of the resulting equations cause a loss in clarity. As we did in Chapter 4, we will study the behaviour of the relative velocity components.

3 100 Guidance of Missiles/NPTEL/2012/D.Ghose R Interception Miss-distance - M T < T M X -2 M X > T M Figure 7.2: The (, R ) trajectory Trajectory in the (, R )-space To obtain the engagement trajectory in the (, R )-space let us rewrite (7.1) and (7.2) as, R + M = T cos(α T ) = T sin(α T ) Squaring both equations and summing we obtain, ( R + M ) = 2 T (7.6) This is the equation of a circle in the (, R )-space with center at (0, M ) and radius equal to T. It shows that the (, R ) point remains on the circumference of this circle as the engagement proceeds. The circle is shown in Figure 7.2. The arrows in Figure 7.2 denote the direction in which the (, R ) point moves from different positions in the (, R )-space with respect to time. These directions are obtained as follows: Differen-

4 Guidance of Missiles/NPTEL/2012/D.Ghose 101 tiating (7.1) and (7.2), we obtain, R = T sin(α T )( ) = (7.7) = T cos(α T )( ) = ( R + M ) (7.8) Multiplying R on both sides of both the above equations, we get, R R = 2 (7.9) R = ( R + M ) (7.10) Since R>0, (7.9) implies that R > 0 always. If we analyze (7.10) we find that, > 0if{ > 0 and R < M } OR { < 0 and R > M } < 0if{ > 0 and R > M } OR { < 0 and R < M } These two conditions are sufficient to determine the direction of movement of the (, R ) point. The points where the circle cuts the R -axis are stationary points since at these points =0and so from (7.7) and (7.8) we see that =0and R =0. The points on the R axis are interesting because those on the negative R axis correspond to the collision triangle and those on the positive R axis correspond to the inverse collision triangle. However, when we talk of the collision triangle in the case of pure pursuit, the missile is required to always point towards the current position of the target, and so collision can take place either in the tail-chase mode (missile pursuing the target with both the velocity vectors aligned along the LOS) or in the head-on mode (missile and target approaching each other with both the velocity vectors aligned along the LOS). Further, in the tail-chase mode collision occurs only if M > T, whereas in the head-on mode collision is possible for all values of T and M. The collision triangle in the pure pursuit case is actually a straight line since the missile and target velocity vectors are both aligned along the LOS. Similarly, points on the positive R axis may correspond to missile and target travelling in opposite directions away from each other, or in a tail-chase mode when M T. The main idea is that a point on the R axis essentially corresponds to the situation when both the missile and target velocities are aligned with the LOS. This we can also see by using the condition that on

5 102 Guidance of Missiles/NPTEL/2012/D.Ghose the R axis we have =0, which implies that, = T sin(α T ) =0 α T = or + π With the above discussion in mind, in Figure 7.2 we have shown two circles. The smaller one corresponds to the case when T < M and the larger one corresponds to the case when T > M. The smaller circle (corresponding to T < M ) shows that the trajectory of (, R ) ends up on a point on the negative R -axis and thus leads to collision. By the very nature of pure pursuit guidance we can see that any engagement whose initial condition does not correspond to a head-on situation will ultimately end up in the tail-chase mode. And if T < M then collision is also guaranteed. Whereas, when T > M the missile will continue to close on to the target till time t miss when the miss-distance R miss occurs. Obviously, this happens at the point where the (, R ) point crosses the axis or when R =0. Afterwards, the missile and target asymptotically approach the tail-chase configuration but without collision taking place since the target is faster than the missile. A last point I would like to mention is that if the initial point is on the negative R axis then the engagement is either a head-on or a tail-chase, whereas, if T < M and the initial point is not on the negative R axis then the engagement ends in a tail-chase collision The capture region Based on the above discussion we can now identify the capture region of the pure pursuit guidance law. By definition, the capture region is such that if the initial point lies inside it then the engagement leads to a successful capture or interception. In Figure 7.3 we show the capture region for the pure pursuit law and also the capture region for the case when the missile is unguided. For the latter case the capture results were obtained in Chapter 4 earlier. However, note that the capture region is obtained here with T as the free parameter and the initial geometry restricted to the cases where M points towards the target. The figure shows that the use of pure pursuit guidance has expanded the

6 Guidance of Missiles/NPTEL/2012/D.Ghose 103 R0 R M M (a) (b) Figure 7.3: Capture region for (a) Pure pursuit (b) Unguided missile capture region considerably over the unguided case.