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This section considers various forms of gyro error For each form of error, this section de nes a component of an augmentation state vector and a linear model relating that component to the gyro error These gyro error state equations can be augmented to the nominal error state to model the e ect of gyro error on the overall navigation solution The derivations of this section are very closely related to those of Section 1162 The equation relating the actual platform angular rate to the angular rate measured in gyro frame is p ip = Cp g g ip (11125)

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Pulse code modulation, such as is shown at Fig 25-8D, is one form of analog-to-digital (A/D) conversion A voice signal, or any continuously variable signal, can be digitized, or converted into a train of pulses whose amplitudes can achieve only certain defined levels

The platform frame to gyro frame transformation Cg is a non-orthogonal p transformation which accounts for gyro misalignment This rotation has the form Cg p g p = I + g p 0 = gqr grq gpq gqp 0 (11126) (11127)

Resolution In A/D conversion, the number of states is always a power of 2, so that it can be represented as a binary-number code Fidelity improves as the exponent increases The number of states is called the

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where each element of g is a component of a small angle To rst order, p Cp = (Cg ) 1 can be shown by direct multiplication to yield g p Cp = I g p g ip p Cp g g ip (11128)

The computed angular rate in platform coordinates is described by = (11129)

1300 900 600 400 200 129 100 050 025

The gyro frame to platform frame transformation Cp is often assumed to be g an identity, but could also have o -diagonal elements determined through a calibration process This rotation matrix has the same structure as de ned for Cp Substituting the expression for Cp into eqn (11129) yields g g ip p = p I g g ip (11130)

ip The term g represents the actual gyro measurements accounting for all measurement errors To develop a nite dimensional error model, only bias, scale factor, noise, and g-sensitive errors will be considered Let ip g = (I SFg ) g bg kg g ip (11131)

$TRIN (Daily) 129

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where SFg is a diagonal matrix representing uncompensated gyro scale factor error, bg represents uncompensated gyro bias, kg represents uncompensated gyro g-sensitivity, and g represents random measurement noise

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Combining eqns (11130-11131) and linearizing the result yields the following equations for the actual platform frame angular rate measurement and measurement error: ip p p ip = = = p p g g ( SFg ) g ( bg + kg + g ) (11132) ip ip ip p ip ip p SFg g g + ( bg + kg + g ) p ip (11133)

where g = g g accounts for error in the alignment process In the p p p following paragraphs, to simplify the notation, we use the symbol T to denote the platform angular rate error vector, Tp = p ip The next step is to de ne parameter vectors xbg , xAg , and xkg such that eqn (11133) can be rewritten as Tp = Tp Tp Tp x bg + xAg + xkg + a bg xAg xkg (11134)

1 Buy: If the Arms Index goes above 20 on the close, this could be a signal to buy Above 40 (possible panic) could be a screaming buy 2 Sell: If the Arms Index goes below 050 on the close, this could be a signal to sell Below 030 (overexuberance) could be a screaming sell 3 Chart note: The extreme reading of 1322 on June 7 was very unusual, indicating short-term panic, and suggests, according to the Arms Index, an imminent rally (which actually occurred for the next two weeks) 4 Note: These are not actionable trades, but only guidelines Always use other indicators to confirm before buying or selling

In this equation, xbg contains the states needed to model bg , xAg contains the states needed to model SFg and g , and xkg contains the states needed p to model kg Given eqn (11134), it is straightforward to de ne the factors of eqn (11105) as F g = Tp Tp Tp , , bg xAg xkg and xg = [xbg , xAg , xkg ]

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