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

T HIS Note presents a methodology for determining circular and spherical position-error bounds for Gaussian random-error processes, such as are commonly encountered in covariance-basednavigation system performance analysis. Such analyses typically require an assessment of the achievable position accuracy in the form of a planaror spatial-error characterization. Planar position error is conveniently characterized by establishing the radius of a circle that encompasses a prescribed percentage of all possible outcomes in a horizontal (or sometime vertical) plane. When the percentage of outcomes is specified at 50%, the corresponding radial error is the familiar circular error probable (CEP). Spatial position error is conveniently characterized by establishing the radius of a sphere that encompasses a prescribed percentage of all possible 3-D positionerror outcomes. When the percentage is specified at 50%, the corresponding radial error is the spherical error probable (SEP). The problem of determining CEP and SEP is of long standing and has been the subject of numerous earlier works (see [1–8], for example). Three approaches have traditionally been taken in determining CEP and SEP. The first approach is to numerically integrate a classic probability integral, thereby providing a tabular definition for the CEP or SEP [2,5,7]. Once the results of the numerical integration are available, they may be used as the basis of a closed-form expression for CEP or SEP. This constitutes the second approach, which has traditionally been used in arriving at approximations for CEP and in which the CEP function is represented over one or more ranges by polynomials chosen to achieve a desired accuracy [8]. This is also the approach taken in the present Note. The third approach is to simplify the probability integral, thereby allowing an approximate closedform expression for the CEP or SEP [1,4,6]. The best known of the closed-form expressions for CEP and SEP is due to Grubbs [6], with further elaboration and discussion provided in [2,3,8]. This Note deals with the determination of circular and spherical error bounds in covariance-based performance analyses. In this type of analysis, the 3-D statistically distributed position errors are theoretically known via the position-error covariance matrix. It remains only to convert this information intomore useful forms, such as CEP or SEP. This Note addresses the question of how this type of conversion may be accurately and conveniently carried out. A set of polynomial functions is numerically derived that accurately characterizes spatial position-error bounds for four distinct cases in which a sphere encompasses 50, 90, 95, and 99% of all possible outcomes. Determination of the corresponding circular error bounds is treated as a special case of the more general problem of determining the spherical error bounds. The accuracies associated with the derived circular and spherical error bounds are defined, and comparisons are made with respect to existing approximations for CEP and SEP.