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Usually it is some subset of the diagonal elements of the Pk matrices that are of interest It is straightforward to tabulate the appropriate elements of Pk versus the dominant error terms (ie, M0 , Qd, R) to determine the appropriate amount that each error source must be reduced to meet a given speci cation This analysis also indicates states that are unobservable and therefore candidates for either combination or deletion Example 62 Consider the one dimensional INS system described in Example 494 on page 159 Assume that a position measurement is available at a 1 Hz rate for the rst 100 seconds of a 200-second application and that the position measurement is corrupted by white measurement noise with variance of 10 m2 A 200 second simulation of eqns (626-627) and the Kalman lter gain calculation equation results in the position error covariance illustrated in Figure 63 During the rst 100 seconds when the position aiding is available, the position covariance converges to approximately 03m2 ( p = 055m) For t [100, 200], while the aiding signal is unavailable, the position error covariance increases to approximately 3200 m2 ( p = 565m) In this simple example, we will consider three independent error sources as performance limiting factors If the present performance is not satisfactory, it is necessary to further analyze the contribution of each of the error sources to the total error covariance so that educated decisions can be made concerning which of the system components should be improved to have the greatest impact on the total system performance Figure 64 displays the results of performing separate covariance simulations for each of the following cases: 1 Total Error This simulation produces the total error depicted in Figure 63 and produces the Kalman gain sequence that will be used for the following three simulations For this simulation, R = 10m2 , Q is.

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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

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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

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

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