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Figure 16-3 Semiconductor pin-outs circuit board Solder them in place and then trim the excess leads if necessary Let s take a well deserved break and when we return we will inspect the circuit board for possible short circuits and cold solder joints Pick up the Lightning Monitor circuit board with the foil side of the board facing upwards towards you First we will inspect the PC board for possible cold solder joints Take a look at the solder joints, they should all look clean, shiny and bright If you see any solder joints that appear dull, dark or blobby, then you should remove the solder from the joint and re-solder the joint all over again until it looks good Once you are confident that all looks well, we can move on to inspecting the PC board for possible short circuits There are two main causes for short circuits, small round solder blobs laying across a PC trace can cause a circuit to short-out The second cause of short circuits is a stray component lead Often when cutting excess component leads, they will stick to the circuit board from rosin core solder residue Make sure that there are no stray leads bridging between the circuit board traces Once you are confident that there are no shorts, we can move on to finishing the project Now, let s move on to mounting the larger components and hardware on the chassis box The chassis mounted

(f1-1)

166 Review Questions and Answers Question 8-10

Figure 71: Timeline illustration of the relationship between the fast timeindex j used for state integration and the slow time-index k used for Kalman lter position aiding for the case that f2 = 1Hz

Suppose we have a nonzero number x, an integer p, and a nonzero integer q We want to take the q th root of the quantity x p How can we express this as a single power of x

Table 16-3

Answer 8-10

In the general mathematical formulation given above, the mechanization equations may not look quite as familiar as they do when written out in the component-wise discrete-time equivalent form pj+1 vj+1 u b

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Let s remember that the q th root of any quantity is the same as the (1/q)th power This lets us apply the rule from Answer 8-9, like this: (x p)1/q = x p(1/q) = x p /q We divide p by q and then raise x to that power It s important that q never be 0! If it is, we ll get an exponent of p /0, and the entire expression will become meaningless

= = = = =

9

1 2 3 4 8 7 6 5

(713)

Question 9-1

No matter how close together two rational numbers happen to be, we can always find another rational number between them How

where = exp( y dt) and dt = f11 At least the rst two equations should look very familiar from elementary physics These equations are iterated at the f1 rate The (fast) time index j = [0, f1 1] is used instead of the (slow) time index k = 0, 1, 2, to make clear the distinction The relation between the fast and slow time indices is illustrated in Figure 71 At the time of each position measurement update: prior to the update j should be (f1 1), after the measurement update j is reset to zero, and the result of the measurement update serves as the initial condition (ie, at j = 0) for the next interval of integration

1 2 3 4 8 7 6 5

Answer 9-1

For this sample problem, all the sensor models are stated in the problem statement Given the estimated state x, the aiding sensor output is predicted at discrete-time instants as yk = pk + y (k) b (714)

We can take the average of the original two numbers If those two numbers are p and q, then the rational number ( p + q)/2 is always greater than p but less than q

De ning the error state vector as x = x x, by linearizing the di erence between eqns (710) and (712), the dynamic equation for the state error vector is 0 0 0 0 1 0 0 0 0 0 1 0 0 1 0 u 0 0 x + 1 0 0 (715) x = 0 0 0 0 1 0 0 0 0 y 0 0 0 0 0 0 0 0 0 In the derivation of this model, in the standard form of eqn (457), the elements F23 , F25 , and G23 warrant additional discussion These terms are a linearization of u + x3 a 1 x5 evaluated for x near x Using the fact that for | x| the linearization proceeds as follows a u + x3 1 x5 = a (u + x5 u + x3 + x3 x5 ) a (a x5 a x3 1 + x5 u + x3 + x3 x5 ) (716) 1,

Question 9-2

.

Imagine that we take a continuous, infinitely long geometric line and make it into a number line Every rational number will then correspond to a unique point on this line Does that mean every point on the line will represent a rational number

1 1 x

Answer 9-2

(1 + x)

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