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For uR = 0: In this case the fth column of O is zero which shows that at least the RR is not observable Over time intervals where uL and uR are time varying, the state vector may not be observable at a given time instant; nonetheless, the state vector can be observable over the time interval This is easily demonstrated via covariance analysis

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The main purpose of this section is to present an covariance analysis demonstrating that the state can be observable over an interval of time even though the observability matrix is not full rank at any time instant during that interval The covariance analysis results are presented in Figure 94 During this simulation, GPS aiding measurements are assumed to occur at a one Hertz rate with tk = kT and T = 1s The vehicle forward speed is u(t) = 1 m for all t The angular rate is periodic with a period of 10s s For t [0, 10], the angular rate is de ned as (t) = 0

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When two impedance vectors don t lie along a single line, you must use vector addition to be sure that you get the correct net impedance In Fig 16-3, the geometry of vector addition is shown Construct a parallelogram, using the two vectors Z1 = R1 + jX1 and Z2 = R2 + jX2 as two adjacent sides of the figure The diagonal of the parallelogram is the vector representing the net complex impedance (Note that in a parallelogram, pairs of opposite angles have equal measures These equalities are indicated by single and double arcs in Fig 16-3)

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The resulting trajectory is oval-shaped with 8m long straight edges in the north-south directions connected by semicircles at each end At the measurement time instants during this simulation, the rank of the observability matrix is 4; therefore, the error state is not observable at any instant of time The parameter de nitions for this covariance analysis were L = 10m, R0 = 01m, PR = (10 10 4 m)2 , and R = 1/36000sec 1 The covariance of the measurement noise in eqn (943) was R = (03)2 I meters The ini tial error state covariance matrix was P = diag([1, 1, (1)2 , 01PR , 01PR ]) The PSD matrix for the vector (t) was de ned as Q(t) = diag([(0001u(t))2 , (0001u(t))2 , (00001u(t))2 , 2PR R , 2PR R ]) For the covariance analysis, the and Qd matrices were computed (see Section 47) and the error covariance matrix P was propagated at 100Hz The 10s period of the trajectory is most evident in the plot of the standard deviation n of the estimate of n in the top plot During the rst 8s of each 10s period, the value of n decreases The standard deviation e of the estimate of e attains a steady state oscillation Note that for t [0, 8]s or t [10, 18]s, the growth of e is much more signi cant than the growth of n During these time intervals, the vehicle is traveling either north or south and the yaw error has a much larger e ect on the lateral variable e than the longitudinal variable n When the vehicle is rotating

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Formula for Complex Impedances in Series Suppose you are given two complex impedances, Z1 = R1 + jX1 and Z2 = R2 + jX2 The net impedance, Z, of these in series is their vector sum, given by the following formula:

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Z = (R1 + jX1) + (R2 + jX2) = (R1 + R2) + j(X1 + X2) Calculating a vector sum using the formula is easier than doing it geometrically with a parallelogram The arithmetic method is also more exact The resistance and reactance components add separately Just remember that if a reactance is capacitive, then it is negative imaginary in this formula

Figure 94: Covariance analysis results for GPS aided encoder-based deadreckoning The operating scenario is described in Section 962

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