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These error equations have the form of eqn (479) with Gaussian white noise inputs This model structure is important for the optimal state estimation methods of 5 This example has considered a hypothetical one dimensional INS with only two sensors The state of the original INS was two and three error states were appended In realistic navigation systems in a three dimensional world with many more sensed quantities, it should be clear that the dimension of the state of the error model can become quite large When designing a system that will be implemented in a real-time application, there is usually a tradeo required between reasonable cost and computation time and the desire for accurate modeling Even in the above single axis example, the dimension of the augmented error state vector x is large (ie, 5) relative to the dimension of the original state x = [p, v] of the INS (ie, 2), and several more error states could be included in the quest for modeling accuracy In realistic applications, the dimension of the augmented state vector is potentially quite large, often too large for a realtime system Therefore, the design may result in two models The most complex model that accounts for all error states considered to be signi cant is referred to as the truth model A simpli ed Design model may be constructed from the truth model by eliminating or combining certain state variables The art is to develop a design model small enough to allow its use in practical realtime implementations without paying a signi cant performance penalty in terms of state estimation accuracy The methodology for analyzing the performance tradeo s is discussed in 6

Another major rule that applies to addition involves how the addends are grouped when you have three or more numbers You can lump the addends together any way you want, and you ll always end up with the same result The simplest case of this rule, called the associative law for addition, involves sums of three integers

From eqn (465), if the mean of the state vector is known at some time k0 and E wk = 0, then the mean can be propagated forward according to E xk+1 E xk+1 = E k xk + E wk = k E xk + E wk = k E xk (497)

It works when you add Here s how a mathematician would state the associative law for three addends For any three integers a, b, and c

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Value 1,000 pF /001 F 1,500 pF /0015 F 2,000 pF /002 F 2,200 pF /0022 F 4,700 pF /0047 F 5,000 pF /005 F 5,600 pF /0056 F 6,800 pF /0068 F 01 015 02 022 033

Intuitively, this formula states that since nothing is known a priori about the speci c realization of the process noise for a given experiment, the mean of the state is propagated according to the state model: xk+1 = k xk yk = Hk xk (498)

This works whether the numbers are positive, negative, or 0 It also works if there are four or more numbers in a sum

Pk = E (xk k )(xk k ) ,

It fails when you subtract In subtraction, the grouping does matter It s easy to see why you can t apply the associative law to subtraction and get away with it Consider this:

Type Ceramic / Mylar Ceramic / Mylar Ceramic / Mylar Ceramic / Mylar Ceramic / Mylar Ceramic / Mylar Ceramic / Mylar Ceramic / Mylar Ceramic / Mylar Mylar Mylar Mylar Mylar Mylar Mylar Mylar Mylar Mylar Mylar Mylar Mylar Mylar Mylar Mylar Mylar

(3 5) 7 = 2 7 = 9 but 3 (5 7) = 3 ( 2) =3+2=5 In formal terms, we would say that for any three integers a, b, and c, it is not always true that (a b) c = a (b c) The associative law hardly ever works with subtraction

where k = E xk The state covariance at time k + 1 Pk+1 = E (xk+1 k+1 )(xk+1 k+1 )

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