Distribution of Miss Distance for Pursuit-Evasion Problem

Shengwen Xiang,Hongqi Fan,and Qiang Fu

Abstract—Miss distance is a critical parameter of assessing the performance for highly maneuvering targets interception(HMTI). In a realistic terminal guidance system, the control of pursuer udepends on the estimate of unknown state, thus the miss distance becomes a random variable with a prior unknown distribution.Currently,such a distribution is mainly evaluated by the method of Monte Carlo simulation. In this paper, by integrating the estimation error model of zero-effort miss distance(ZEM)obtained by our previous work,an analytic method for solving the distribution of miss distance is proposed, in which the system is presumed to use a bang-bang control strategy. By comparing with the results of Monte Carlo simulations under four different types of disturbances(maneuvers), the correctness of the proposed method is validated. Results of this paper provide a powerful tool for the design,analysis and performance evaluation of guidance system.

I.Introduction

with the increasing maneuverability of evader, the pursuit-evasion problem has drawn an intensive attention recently in the guidance community[1]–[4].For highly maneuvering target interception(HMTI),the problem of homing guidance is a typical stochastic optimal control problem with the objective to minimize the expected miss distance.Therefore,as a critical performance parameter of guidance system,miss distance is directly related to the success or failure of an interception, which is determined by the used control strategy,sensor noise,maneuver of evader,and other factors.

Under the assumption of perfect information pattern,a series of deterministic control strategiesu=u(t,z(t)) (wherez(t) denotes ZEM,zero-effort miss distance,representing the miss distance without control effort until the final time)have already been proposed.See[5]–[7]for the differential game based bang-bang strategies as well as several linear/nonlinear strategies.

However,in a realistic interception scenario, the perfect information assumption does not always hold.On the one hand, there is a random measurement error in the sensor observations,and it is very difficult for currently used sensors to measure the evader’s acceleration(especially,the lateral acceleration)directly;On the other hand,the traditional guidance laws based on optimal control theory assume the pursuer has a much better maneuverability than the evader,however the acceleration of pursuer is bounded in reality and the evader often holds a similar maneuverability to the pursuer for HMTI problems.All these facts impede significantly the implementation of theoretically robust transferring strategies.In order to use these noise-corrupted measurements,an estimator, restoring and filtering the state variables, becomes an indispensable component of the guidance system[8].Therefore,the control functionureceives,instead of the accurate value ofz(t), a random outputz?(t)=z(t)+η(t),where η(t)is the estimation error of ZEM.Thus,miss distance becomes a random variable with an unknown prior.

To evaluate the extent of performance deterioration caused by the noise-corrupted measurements and guarantee the guidance performance,it is a necessary work to investigate the distribution of miss distance,especially in the stage of system design.Currently,such a distribution is mainly evaluated by a large set of Monte Carlo simulations given the specific system dynamics,estimator/control strategy combination,disturbance(i.e.,the acceleration command of evader)and noise model.However,this posterior test is not very suitable for the stage of system design.Thus,as a part of guidance system design,an analytical method of evaluating m iss distance is dramatically demanded.

Gilzeret al.[9]–[12]conducted a series researches of the terminal state distribution with a saturated linear control strategy.But in their work the distribution of ZEM error was acquired by Monte Carlo simulations and the initial statez0was set to a known amount. Notice that the control function actually received is an estimated value of ZEM,thus analyzing the estimation error distribution of ZEM is a basic work for calculating the distribution of miss distance.Through the shaping filter,Moldavskaya and Shinar[13]modelled the acceleration commandvof evader as an output of the first order system driven by the white noise holding a same autocorrelation with the actual command.For the linear Gaussian case, they showed the error of ZEM was subject to Gaussian distribution with zero mean and its state covariance matrix was governed by the Raccati equation.For the integrated estimation and guidance(IEG)system, by introducing a separate mode decision-maker and modelling the evader’s lateral acceleration as a jump-Markov process,Xianget al.[14],[15]studied the error distribution of ZEM in the presence of mode mismatch,the obtained ZEM error followed a biased Gaussian distribution.This paper studies the distribution of miss distance by integrating the ZEM error distribution in[14],[15]into the analysis of miss distance.Four different disturbances are taken into account,and the bang-bang control strategy is used in our analysis as it can guarantee a maximum capture zone of the pursuer[16].The results of this paper provide a tool for the design and performance evaluation of guidance system.

The rest of paper is organized as follows:Section II formulates the pursuit-evasion problem.Section III devotes to the derivation of miss distance distribution.Section IV validates the above derivations by comparing with the Monte Carlo simulations where four different types of maneuvers are taken into account.Conclusions are drawn in the last section.

II.Problem Formulation

III.Probability Density Function of zn+1

V.Conclusion

Fig.3.Simulative and theoretical distribution of | zN|for Case 1.

Fig.4.Simulative and theoretical distribution of | zN|for Case 2.

Fig.5.Simulative and theoretical distribution of | zN|for Case 3.

In this paper,an analytic method for calculating the distribution of miss distance is proposed.As a forwarding work of[14],[15],this paper integrates the estimation error model of ZEM into the analysis of miss distance.In this problem, the system is subject to a bang-bang control strategy,and it is assumed that the disturbancevnis independent with the stateznand controlun,four different types of disturbances are taken into account specifically.The correctness of theoretical derivation is proved through the comparison with the results of Monte Carlo simulations under a typical interception scenario.The proposed analytic method serves as a powerful tool to analyze and evaluate the guidance performance in the stage of guidance system design.

Fig.6.Simulative and theoretical distribution of | zN|for Case 4.