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In the above analysis t represents true time, which for the purposes of this chapter is de ned to be the GPS system time as maintained by the United States Naval Observatory (USNO) Eqn (841) shows that the clock time can be modeled as true time plus a time varying error term that is determined by the initialization error (to ) plus a time-varying (clock drift) error which is the integral of the frequency error f (q) Obviously, oscillators for which the frequency error is small and stable result in better clocks A pendulum clock nicely exempli es the principle described above The pendulum swings at a resonant frequency A set of gears count the swings of the pendulum and display the cumulative phase of the oscillator using hands on the face of the clock The user may be able to tune the resonant frequency by adjusting the length of the pendulum to decrease f Modern, high accuracy clocks such as those used in GPS satellites and receivers use the same principle, but use very stable crystal oscillators with electronic counters to accumulate the oscillator phase Eqn (841) applies to both the satellite and receiver clocks: s (t) = r (t) = t + s (t) t + r (t) (842) (843)

Reciprocal of reciprocal For all nonzero real numbers, x, the reciprocal of the reciprocal is equal to the original number:

104 k

Here are some more sophisticated rules for expression morphing As with the facts in the previous section, you don t have to try to memorize these It can help if you work out a few examples using numbers in place of the variables Some of the restrictions here, in which variables are not allowed to equal 0, are a little stronger than necessary to keep things straightforward and to be sure we stay safe!

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where s (t) is the error in the clock of satellite s and r (t) is the error in the clock of receiver r Substituting eqn (842 843) into eqn (837) yields s r (t) r = = c t + r (t) t ts + s (t ts ) r r c ts + r (t) s (t ts ) r r (844)

Product of sums For all real numbers w, x, y, and z,

which shows that the measured pseudorange s r (t) is proportional to r the time of propagation ts plus the receiver clock error r (t) minus the r satellite clock error s (t ts ) at the time of transmission The time of r propagation will be further analyzed in subsequent sections

Cross multiplication For all real numbers w, x, y, and z where neither x nor z is equal to 0,

Multiplier Tolerance 01 F 10%

8 Given that the state is controllable from the driving noise and is observable from the measurements, the Kalman lter estimation error dynamics are asymptotically stable Therefore, for an observable system, the e ects of the initial conditions (x and P ) decay away and 0 0 do not a ect the solution as k Since the above properties rely on three assumptions, it is natural to consider the reasonableness of these assumptions Although most physical systems are in fact non-linear, the linearity assumption can be locally applied when the system non-linearities are linearizable and the distance from the linearizing trajectory is small These conditions are usually valid in navigation problems, especially when aiding information such as GPS is available The white noise assumption is also valid, since colored driving noise can be modeled by augmenting a linear system with white driving noise to the system model The Gaussian assumption is valid for most driving noise sources as expected based on the Central Limit Theorem [107] Even when an application involves non-Gaussian noise, it is typical to proceed as if the noise source were Gaussian with appropriately de ned rst and second moments In such cases, although a better nonlinear estimator may exist, the Kalman lter will provide the minimum variance, linear, unbiased state estimate

Reciprocal of product For all nonzero real numbers x and y,

Product of quotients For all real numbers w, x, y, and z where x 0 and z 0,

The Kalman lter algorithm as presented in Table 55 is formulated to process a vector of m simultaneous measurements The portion of the measurement update that requires the most computing operations (ie, FLOP s) is the covariance update and gain vector calculation For example, the standard algorithm K x+ P+ = P H R + HP H = x + K( H ) y x = (I KH) P

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(w /x)(y /z) = (wy)/(xz)

(572) (573) (574)

Reciprocal of quotient For all nonzero real numbers x and y,

can be programmed to require 3 n2 m + 3 nm2 + nm + m3 + 1 m2 + 1 m 2 2 2 2 FLOP s Alternatively, when R is a diagonal matrix, the measurements can be equivalently treated as m sequential measurements with a zero-width time-interval between measurements, which results in signi cant computational savings

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