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Let a, b, c, d, e, and f all be nonzero integers Suppose you start with a /b and divide it by c /d, and then divide that result by e /f Write an expression for the final quotient

2 va is Gaussian white noise with PSD equal to va = 25 10 3 m2 s3 ;

When you want to add or subtract two integers, the process is straightforward Fractions are more involved You ve probably had plenty of practice adding and subtracting fractions in arithmetic courses Let s look at these problems from a point of view a little closer to algebra, using variables instead of specific numerical examples

b is the random walk process b = b (t)

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Getting a common denominator When you have two fractions that you want to add or subtract, you should be sure that neither fraction has a negative denominator If one of them does, convert it into the equivalent form that has a positive denominator Then you can modify the fractions so they have the same denominator, called the common denominator Suppose you have two fractions a /b and c /d in which a and c are integers, and b and d are positive integers You want to add them, so you write

the distribution of the initial bias is b(0) N (0, Pb0 ) The plant state vector is x = [p, v, b] The implemented navigation equations are p(t) = v (t) (t) = a(t) v (4143)

Now multiply a/b by d/d, which is equal to 1 Then multiply c/d by b/b, which is also equal to 1 (Remember, anything multiplied by 1 is equal to itself) The above expression turns into this: (a /b)(d /d ) + (c /d )(b /b) Next, multiply the products of the fractions on each side of the plus sign, getting ad /bd + cb /db Now apply the commutative law for multiplication to the numerator cb and the denominator db, morphing the above expression into ad /bd + bc /bd This produces a sum of two fractions with the common denominator bd These two fractions have the same numerical values as the original ones, but they are in higher terms

A fraction plus another fraction Once you ve found a common denominator for a sum of two fractions, adding the fractions is easy Simply add the numerators, and put them over the common denominator In the above situation, then

where a(t) = u(t) + and is the best estimate of the accelerometer b(t) b(t) bias b that is available at time t These equations are integrated forward in time starting from the expected initial conditions where we assume that p(0) N (p0 , Pp0 ) and v(0) N (v0 , Pv0 ) The navigation system state vector is x = [ , v , where the di erential equation for is = 0 because p b] b b the mean of b is zero

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ad /bd + bc /bd = (ad + bc)/bd An equation such as this, which describes a general solution to a math problem, is called a formula This particular formula is worth homogenizing into your brain! If a and c are integers, and b and d are positive integers, then a /b + c /d = (ad + bc)/(bd ) If you d rather see it in words, the process goes like this: Multiply the numerator of the first fraction by the denominator of the second Multiply the denominator of the first fraction by the numerator of the second Add these two products together Divide this sum by the product of the denominators

CHAPTER 4 STOCHASTIC PROCESSES states p = p p, v = 0 1 0 (459) in Section 461 00 00 00 25 10 3 00 00 00 00 10 10 6 0 0 1

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