Root-mean-square miss distance of proportional navigation missile against sinusoidal target

Abstract

The performance of a proportional navigation homing missile against a target performing a sinusoidal weave maneuver is evaluated. The missile's effectiveness is measured in terms of the root-mean-square miss distance over a set of engagements in which the initial phase of the target weave is uniformly distributed. Closed form solutions for the root-mean-square miss are derived for the case where the missile guidance system is modeled by a first-order lag and the lateral acceleration is unlimited. The analysis is then extended to include the effects of acceleration saturation and higher order missile dynamics. Comparisons are made between a first-order and a fifth-order guidance system, and the root-mean-square miss is determined numerically as a function of the interceptor's effective navigation gain, time constant and acceleration limit, and the target's weave amplitude and frequency. UTURE homing interceptor missiles will face new and unique challenges as the sophistication of the threat spectrum in- creases. Engagements against air targets can occur at both very low and very high altitudes, with the threats accidentally or intentionally performing weaving or spiraling maneuvers during their midcourse and terminal phases.14 The lateral displacement, acceleration ca- pability, and weave frequency of the target maneuver can greatly enhance the threat's ability to survive a counterattack. To counter this, the defensive missile must have sufficient lateral acceleration, guidance system time constant, and terminal homing time to achieve a high probability of intercept. Whereas the per- formance of the interceptor will be influenced by many scenario- dependent factors,58 a major consideration will be the fundamental response of the proportional navigation (PN) guidance system to the postulated target weave motion.9'10 In the general case, the target dynamics may involve arbitrary periodic motion in three dimensions. A useful starting point for analysis, however, is the response of the PN homing system to a single plane sinusoidal maneuver of constant amplitude and fre- quency. The phase angle of target weave, which is associated with initial conditions at the start of the missile's terminal guidance, can be treated as a random variable, uniformly distributed between 0 and 2n over a set of engagements. The missile's dynamics are approxi- mated by a simple first-order transfer function, and unlimited lateral acceleration capability is assumed. The miss distance can then be parameterized in terms of the effective PN navigation gain N, the missile time constant T, and the amplitude AT and frequency co of the target weave. This paper focuses on root-mean-square (rms) miss distance as a recommended measure of effectiveness in analyzing missile perfor- mance against weaving targets. This measure allows uncertainties in target phase characteristics to be accounted for in the terminal per- formance results. The weaving target problem was first addressed by Chadwick, 9 who determined analytical expressions for the rms miss distance of the single lag PN missile for values of N =2 and 3. Zarchan 10 employed adjoint theory and transfer function tech- niques to determine formulas for the peak miss distance against a weaving target for values of N between 3 and 6. The present paper derives general closed-form expressions for the rms miss distance against a sinusoidal target. New results are obtained for arbitrary