L

obtained by substituting Eq. (11-29) into Eq. (11-19), with ax replaced by a, and ay replaced by o2. That is.

The physical interpretation of the error ellipse in Fig. 11-9 is different in several respects from that of the quantized error parallelogram of Fig. 11-7. As shown in Eq. (11-18) and Fig. 11-10, the probability density is no longer uniform, but is a maximum at the center and falls off continuously away from the center. The boundaries of the error ellipses are lines of constant probability density. The no uncertainty along any arbitrary axis is given by the perpendicular projection of the no error ellipse onto that axis. Thus, the lo uncertainty along the/' axis is the distance from the origin to the point A in Fig. 11-10; that is, the probability that the/' component of the attitude lies between A and A' is 0.68.

Although the probability of any one component being within the lo uncertainty boundary is 0.68, the probability of both attitude components in any orthogonal coordinate system being within the lo error ellipse is less than 0.68.* Specifically, the probability of the attitude lying somewhere inside the lo error ellipse is 0.39. Table 11-1 gives the probability for the attitude to lie within various error ellipses and for any one component to lie within the boundary of the error ellipse.

'This is easily visualized by considering a two-component error rectangle. If the two components, x and y, have upper limits at the boundary of the rectangle of Bx and By, then four possibilities exist: x < Bx. and y ^Bj, x>Bx and y > By, x< Bx and y > By, and x>B„ and y < By. However, only the first combination results in the point defined by (x,y) being inside the box. The probability of this occurring is clearly less than either the probability of x< Bx or the probability of y < By.

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