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Figure 63: Position covariance for the position aided INS in Example 62 a diagonal matrix of the following elements [00, 25 10 3 , 10 10 6 ], and M0 is a diagonal matrix with elements [100, 10, 001] The position covariance sequence resulting from this simulation is shown in Figure 63 and Figure 64 as the solid line 2 Uncertain Initial Conditions This simulation uses the Kalman gain vector of simulation 1, with R and Q set equal to appropriately dimensioned zero matrices The M0 matrix assumes its original value as de ned in Simulation 1 Therefore, this simulation shows the e ect of the uncertain initial conditions if both the INS and position sensors were perfect The results of the simulation are shown in Figure 64 as the dashed-dotted line 3 Position Measurement Noise This simulation uses the Kalman gain vector of simulation 1 The M0 and Q matrices were set equal to appropriately dimensioned zero matrices The R matrix assumes its original value as de ned in Simulation 1 Therefore, this simulation shows the e ect of the position measurement noise if there were no uncertainty in the initial conditions or INS sensors The results of

Question 7-2

Value 15 pF 33 pF 10 pF 15 pF 20 pF 30 pF 33 pF 47 pF 56 pF 68 pF 75 pF 82 pF 91 pF 100 pF 120 pF 130 pF 150 pF 180 pF 220 pF 330 pF 470 pF 560 pF 680 pF 750 pF 820 pF

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Figure 64: Total and component portions of the position covariance for the position aided INS in Example 62 the simulation are shown in Figure 64 as the dashed line 4 INS Sensor Error This simulation uses the Kalman gain vector of simulation 1 The M0 and R matrices were set equal to appropriately dimensioned zero matrices The Q matrix assumed its original value as de ned in Simulation 1 Therefore, this simulation shows the e ect of the INS sensor noise and bias drift if there were no uncertainty in the initial conditions or position sensor The results of the simulation are shown in Figure 64 as the dotted line The individual components of Q were not analyzed separately As in any correct application of the principle of superposition, each (error) source is accounted for exactly once and the results corresponding to each source individually sum to the total result with all sources forcing the system at the same time Figure 64 illustrates this fact Figure 64 shows that during the rst 100 seconds, the system has essentially reached steady state The error due to the initial condition M(0) is asymptotically approaching zero as expected in a stable system The main factor limiting performance during the initial 100 seconds is the additive noise on the position measurement For the time interval t (100, 200] seconds, no position measurements

Answer 7-2

An order of magnitude is a way to express how many times larger or smaller a certain quantity is, in terms of absolute value, compared to another quantity In the base-10 system: When quantities differ by one order of magnitude, then the absolute value of one quantity is 10 times larger than the absolute value of the other When quantities differ by two orders of magnitude, then the absolute value of one quantity is 102, or 100, times larger than the absolute value of the other When quantities differ by n orders of magnitude, then the absolute value of one quantity is 10n times larger than the absolute value of the other

Figure 65: Total and component portions of the position covariance for the position aided (improved) INS in Example 62

Type Ceramic Ceramic Ceramic Ceramic Ceramic Ceramic Ceramic Ceramic Ceramic Ceramic Ceramic Ceramic Ceramic Ceramic Ceramic Ceramic Ceramic Ceramic Ceramic Ceramic Ceramic Ceramic Ceramic Ceramic Ceramic

Question 7-3

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