Determination of Circular and Spherical Position-Error Bounds in System Performance Analysis

摘要

THIS Note presents a methodology for determining circular andnspherical position-error bounds for Gaussian random-errornprocesses, such as are commonly encountered in covariance-based-nnavigation system performance analysis. Such analyses typicallynrequire an assessment of the achievable position accuracy in the formnof a planar- or spatial-error characterization. Planar position error isnconveniently characterized by establishing the radius of a circle thatnencompasses a prescribed percentage of all possible outcomes in anhorizontal (or sometime vertical) plane. When the percentage ofnoutcomes is specified at 50%, the corresponding radial error is thenfamiliar circular error probable (CEP). Spatial position error isnconveniently characterized by establishing the radius of a sphere thatnencompasses a prescribed percentage of all possible 3-D position-nerror outcomes. When the percentage is specified at 50%, thencorresponding radial error is the spherical error probable (SEP). Thenproblem of determining CEP and SEP is of long standing and hasnbeen the subject of numerous earlier works (see [1–8], for example).nThree approaches have traditionally been taken in determiningnCEP and SEP. The first approach is to numerically integrate a classicnprobability integral, thereby providing a tabular definition for thenCEP or SEP [2,5,7]. Once the results of the numerical integration arenavailable, they may be used as the basis of a closed-form expressionnfor CEP or SEP. This constitutes the second approach, which hasntraditionally been used in arriving at approximations for CEP and innwhich the CEP function is represented over one or more ranges bynpolynomials chosen to achieve a desired accuracy [8]. This is also thenapproach taken in the present Note. The third approach is to simplifynthe probability integral, thereby allowing an approximate closed-nform expression for the CEP or SEP [1,4,6]. The best known of thenclosed-form expressions for CEP and SEP is due to Grubbs [6], withnfurther elaboration and discussion provided in [2,3,8].