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and (t) = 0 (cos( s t) 1) For the rst several minutes, the position g error is accurately modeled as 1 Re 0 t2 radians or g 0 t2 meters, with 2 2 m velocity error g 0 t s In steady-state, each milli-radian of tilt error results in approximately 79 m of (peak) velocity error s The discussion related to Figure 114 relates to this initial condition for the full INS The position and attitude errors of this section correspond to the rst and fourth subplots The results of the analysis for the simpli ed model is again accurate for approximately the rst hour Constant Accelerometer Error For g (t) = 0, the initial condition [v(0), (0)] = [0, 0], and constant a , the solution to eqn (1182) is v(t) (t) =

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sin s t (1 cos( s t))

Converting Complex Admittance to Complex Impedance The GB plane is, as you have seen, similar in appearance to the RX plane, although mathematically they are different Once you ve found a complex admittance for a parallel RLC circuit, you will usually want to transform this back to a complex impedance The transformation from a complex admittance G + jB to a complex impedance R + jX can be carried out using the following two formulas, one for R and the other for X:

1 s

R = G/(G 2 + B 2) X = B/(G 2 + B 2 ) If you know the complex admittance, first find the resistance and reactance components individually using the preceding formulas Then assemble the two components into the complex impedance, R + jX

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a and (t) = g 1 cos s t This shows that accelerometer bias will lead to a bias in the attitude and position errors The initial error growth for a the position and attitude errors is 2Re t2 If the position error is expressed a in meters (x = Re ), then the initial position error grows as 2 t2 which is the error growth expected for double integration of an acceleration bias; however, due to the interaction between the velocity and tilt errors, after transients decay away, each error state is oscillatory and bounded

Constant Gyro Error For a (t) = 0, the initial condition [v(0), (0)] = [0, 0], and constant g the solution to eqn (1182) is v(t) (t) = Re cos( s t) 1 1 s sin( s t)

Problem 16-17 Suppose the complex admittance of a certain parallel circuit is 0010 j00050 What is the complex impedance of this same circuit, assuming the frequency does not change In this case, G = 0010 S and B = 00050 S First find G 2 + B 2, as follows:

(1187)

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1 and (t) = g s sin( s t) t In this case, the gyro error leads to an oscillatory attitude error, but the velocity error is a biased oscillation Due to the velocity bias, the position error grows without bound The magnitude of the velocity error bias, Re g , is useful for estimating the average rate of position error accumulation due to the gyro bias The discussion related to Figure 116 relates to this initial condition for the full INS The position, north velocity, and attitude errors of this section correspond to the rst, second, and fourth subplots The results of the analysis for the simpli ed model is also accurate for approximately the rst hour

G 2 + B 2 = 0010 2 + ( 00050)2 = 0000100 + 0000025 = 0000125 Now it is easy to calculate R and X, like this: R = G/0000125 = 0010/0000125 = 80

x(t) = F(t) x(t) (1188)

When you re confronted with a parallel circuit containing resistance, inductance, and capacitance, and you want to determine the complex impedance of the combination, do these things: 1 Find the conductance G = 1/R for the resistor (It will be positive or zero) 2 Find the susceptance BL of the inductor using the appropriate formula (It will be negative or zero) 3 Find the susceptance BC of the capacitor using the appropriate formula (It will be positive or zero) 4 Find the net susceptance B = BL + BC (It might be positive, negative, or zero) 5 Compute R and X in terms of G and B using the appropriate formulas 6 Assemble the complex impedance R + jX

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