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where Z3 is a three dimensional4 zero vector, and 0 g 0 0 0 G= g 0 0 0 From the system description and measurement matrix, it is clear that d is completely unobservable In the case where the system is nearly level, the w-axis accelerometer bias is completely observable The accuracy with which the north and east alignment errors can be estimated will depend on the accuracy of the east and north accelerometers and amount of platform motion due to disturbances In theory, portions of the disturbance motion could be modeled through state augmentation techniques In practice, this is di cult Therefore, vibration and disturbances will limit the achievable accuracy In an alternative approach, the rst two components of f could 1 be passed through a simple low pass lter with gain g The outputs are interpreted as E and N , respectively In either case, the estimates are used to adjust Rn (see eqn (1067)) until f is zero The accuracy of this b approach will also be limited by instrumentation imperfections, vibration, and disturbances At the conclusion of the process, the analytic version of the navigation frame is nominally in alignment with the actual navigation frame (ie, leveling has been achieved) Again, as with the mechanized approach, leveling accuracy is directly proportional to the uncompensated accelerometer bias (1 milli-radian per 1 g bias) Similarly, the gyro outputs can be processed to achieve azimuth alignment Assuming that the vehicle is stationary relative to the Earth, we have that ib ib = = ( in + nb ) = ( ie + d ) in ie b = b

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Complex Impedances When you want to determine the relationship among current, voltage, and resistance in an ac circuit that contains resistance and reactance, things get interesting Recall the formula for the square of the absolute-value impedance in a series RLC circuit:

where the stationary assumption implies that in = ie , in = ie , and bn b = 0 The symbol d is used to represent nb which is the disturbance angular rates of the vehicle that are in violation of the stationarity assumption The actual and computed geographic frame rotation rates are n in in n

These expressions can each be expanded as follows: n in = (I + [ ]) Rn b b b ib d n ib = ib + [ ] n n d

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Z 2 = R2 + X 2 This means that Z is equal to the square root of the quantity R 2 + X 2, as follows: Z = (R 2 + X 2 )1/2 This is the length of the vector R + jX in the complex impedance plane You learned this in Chap 15 This formula applies only for series RLC circuits The square of the absolute-value impedance for a parallel RLC circuit, in which the resistance is R and the reactance is X, is defined this way: Z 2 = R 2X 2/(R 2 + X 2) This means that the absolute-value impedance, Z, must be calculated using the rather arcane formula: Z = [R 2X 2/(R 2 + X 2)]1/2 The 1 2 power of a quantity represents the positive square root of that quantity

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Therefore, the residual gyro measurement in geographic frame is in ie b ib = n n = [ ] n Rn b n d in N 0 D E 0 N 0 Rn b d = D b ib E 0 D N 0 D 0 N 0 N E Rn b d (11154) = D b ib 0 N 0 D

sent in a digital mode; this offers another advantage over conventional FSTV Digital signals propagate better, are easier to deal with when they are weak, and can be processed in ways that analog signals cannot Some HDTV systems use interlacing in which two rasters are meshed together This effectively doubles the image resolution without doubling the cost of the hardware But it can cause annoying jitter in fast-moving or fast-changing images

Based on eqn (11154), two approaches are again possible The rst approach is Kalman lter based, with state error dynamics as previously de ned and the measurement matrix H= where Z3 Z3 W D 0 N Rn b Z3 (11155)

0 N 0

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