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to rst order In vector form (using eqns (223) and (B15)), eqn (1054) is (1055) = Rn b n b ib in where b is the error in the gyro measurement of the body-frame inertialib ib relative angular rate vector By its de nition as b = b b , the ib ib

jBL = j[1/(628f L)] = j[1/(628 400 600)] = j000663 Next, calculate the susceptance of the capacitor (converting its value to microfarads) at 400 MHz, as follows: jBC = j(628f C ) = j(628 400 0000150) = j000377 Finally, add the inductive and capacitive susceptances to obtain the net susceptance: jB = jBL + jBC = j000663 + j000377 = j000286 This is a pure inductive susceptance

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ib de nition of b in eqn (1025), and the de nition of the gyro measurement in eqn (108), the gyro measurement error is modeled as b = xg g ib with xg = Fg xg + g Substituting eqn (1056) into eqn (1055) yields = Rn xg Rn g Rn n b b b in (1057) (1056)

The vector n , derived on p 52 and reprinted in eqn (1051), is the in rotation rate of the navigation frame relative to the inertial frame represented in navigation frame Eqn (1057) shows that the lack of position and velocity information a ects the AHRS accuracy by the accumulation of attitude error at rates related to and In the description that follows, n in is treated as a white noise term driving the error state model In fact, this term is not white or even stationary Improved handling of this term could lead to improved performance The accumulated error is intended to be estimated and removed by the accelerometer and magnetometer aiding sensors

Problem 16-12 What is the net susceptance of the above parallel-connected inductor and capacitor at a frequency of f = 531 MHz

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Based on the above analysis, with the error state vector de ned as x = [ , xg , xa ] , we combine eqns (109), (1011), and (1057) to obtain the dynamic model for the state vector as 0 Rn 0 b xg = 0 0 xg Fg xa xa 0 0 Fa n Rn 0 Rn 0 in b b g + 0 I 0 0 g 0 0 0 I a We combine eqns (1041) and (1043) to obtain the error state measurement models as gn m

First calculate the susceptance of the inductor at 531 MHz, as follows: jBL = j[1/(628f L)] = j[1/(628 531 600)] = j000500 Next calculate the susceptance of the capacitor (converting its value to microfarads) at 531 MHz, as follows: jBC = j(628f C ) = j(628 531 0000150) = j000500 Finally, add the inductive and capacitive susceptances to obtain the net susceptance: jB = jBL + jBC = j000500 + j000500 = j0 This means that the circuit has no susceptance at 531 MHz The situation in which there is no susceptance in an LC circuit is known as parallel resonance It is discussed in the next chapter

(1058) (1059)

The Kalman lter measurement update computations require the variance of the discrete-time aiding measurements This is straightforward for the magnetometer measurements:

Adding Admittance Vectors In real life, there is a small amount of conductance, as well as susceptance, in an ac parallel circuit containing a coil and capacitor This occurs when the capacitor lets a little bit of current leak through More often, though, it is the case because a load is connected in parallel with the coil and capacitor This load can be an antenna, the input to an amplifier circuit, a test instrument, a transducer, or some other device When the conductance in a parallel circuit containing inductance and capacitance is significant, the admittance vectors do not point straight up and down Instead, they run off toward the northeast (for the capacitive part of the circuit) and southeast (for the inductive part) This is illustrated in Fig 16-6

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