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10 In this equation, the reference system state vector has dimensions the reference system state transition matrix has dimensions and is the th element of an uncorrelated random sequence with covariance matrix The quantity generally is referred to as process noise. The variable is a discrete-time index representing time where is the discretization period of the system. The subscript distinguishes these quantities as being associated with the reference system (as opposed to the Kalman filter). The initial condition for (1) is denoted and is a random variable whose distribution is denoted by where is the expectation of The measurement vector has dimensions and is assumed to be a linear combination of the system states where is the reference system observation matrix and is the th element of an uncorrelated random sequence with covariance The output of the system is defined by the column matrix which is a linear combination of the system states where is a matrix of dimension The Kalman filter state vector which is an column matrix, evolves in time according to the equation where has dimensions is process noise (an uncorrelated sequence of covariance has dimensions and is a deterministic input to the filter of dimension Note that the subscript in this (and subsequent) equations denotes a quantity that is associated with the filter (as opposed to the reference system). The actual measurements read from the sensors and used by the filter are a linear combination of the reference states, the model for which is given in (3). The filter s model for the measurement vector, however, may be different from the actual measurement vector. Accordingly, the filter believes that the measurement taken is actually modeled by the equation where is the th element of a uncorrelated sequence with covariance matrix At each timestep, a measurement is taken and the Kalman filter generates an estimate of the state vector. The state estimate is recursively computed using the well-known Kalman filter algorithm [39]. The output of the Kalman filter is defined by the column matrix which is a linear combination of the estimate of the state (2) (3) (4) (5) (6) (7) where is a matrix of dimension and is the state estimate after the th measurement is processed. (The estimate of the state before the th measurement is processed is denoted.) Finally, we define the error between the output of the reference system and the output of the Kalman filter as and define the covariance of The goal of this derivation is to arrive at a recursive expression for the covariance matrix Before proceeding, however, an important constraint on the exogenous input to the Kalman filter must be established. Namely, it is required that be the sum of a linear combination of the reference states and an additive noise term that is assumed to be normally distributed and uncorrelated with any states or noise sources in the system. Mathematically, can be expressed as as (8) (9) (10) This assumption is not too restrictive but may require to be augmented with states that define the time history of Note that has dimensions and has dimensions With this restriction on in hand, we may proceed with the derivation. We begin by constructing a single combined state vector by stacking and so that and (11) (12) (13) Utilizing (1), (10), and (12), one can arrive at an expression that describes the propagation of in time from timestep to timestep (14) It is convenient to define and so that (14) can be rewritten as follows: where and (15) (16) (17) 154 PROCEEDINGS OF THE IEEE, VOL. 87, NO. 1, JANUARY 1999

15 Fig. 8. RMS cross-track position error using a 10 /h gyro. Fig. 10. off). RMS error in along-track position after a GPS loss (SA Fig. 9. off). RMS error in cross-track position after a GPS loss (SA Fig. 11. (DGPS). RMS error in cross-track position after a GPS loss track position error is not reduced significantly. Fig. 8 shows the total rms cross-track position error versus time for a system in which the rate gyro (with a bias drift of approximately 30 /h) was replaced with a better one (having a bias drift of approximately 10 /h). The uppermost curve in the figure is the total RMS cross-track position error, and the remaining curves represent the contributions that individual sensor errors make to the total cross-track position error. As the data show, the position error induced by SA is by far the largest contributor. When compared with Fig. 6, one can see that the contribution of the 10 /h rate gyro is significantly less than that of the 30 /h gyro. However, the contribution of SA remains nearly unchanged, and, ultimately, using the better gyro is of little consequence. For a system utilizing GPS position fixes with SA off, the relative contributions of the various sensor errors are similar to those when SA is on. Figs. 9 and 10 show data similar to Figs. 6 and 7. As the data show, the growth rate of the error in the cross-track direction is dominated by the rate gyro s bias drift and is higher than that for the along-track direction. For a system utilizing DGPS position fixes, results are roughly similar the rate gyro s bias and noise dominate the cross-track position error growth rate, and the odometer s bias drift dominates the along-track position error growth rate. Figs. 11 and 12 show data similar to Figs. 6 and 7. As the data show, the total contribution of the gyro bias drift to the cross-track position error is substantially less than when stand-alone GPS position fixes (with SA on) are utilized. This occurs because the rate gyro s bias is calibrated more accurately when DGPS is utilized. Also, the relative contribution of the noise in the gyro s output is larger in this case than when SA is on. Interestingly, the contribution of the gyro bias drift and the noise in the gyro output are commensurate. Nevertheless, the contribution of the drift eventually exceeds that of the noise, and the growth ABBOTT AND POWELL: LAND-VEHICLE NAVIGATION USING GPS 159

18 [40] R. B. Asher and R. M. Reeves, Performance evaluation of suboptimal filters, IEEE Trans. Aerospace Electron. Syst., vol. AES-11, pp , Mar [41] R. R. Clark and M. A. Needler, Performance and adjoint sensitivity analysis of the NAVSTAR global positioning system, NAECON Rec., pp , [42] J. A. DAppolito et al., Error analysis of hybrid aircraft inertial navigation systems, in Proc. AIAA Guidance and Control Conf., Aug. 1972, AIAA paper no , pp [43] R. E. Griffin and A. P. Sage, Large and small scale sensitivity analysis of optimum estimation algorithms, IEEE Trans. Automat. Contr., vol. AC-13, pp , Aug David Powell received the B.S. degree in mechanical engineering from Massachusetts Institute of Technology (MIT), Cambridge, in 1960 and the Ph.D. degree in aeronautical and astronautical engineering from Stanford University, Stanford, CA, in He is now a Member of the Faculty in the Department of Aeronautical and Astronautical Engineering at Stanford University. He has authored two books on modern control. Eric Abbott received the B.S. degree in mechanical engineering from University of Illinois in 1990 and the M.S. degree from Stanford University, Stanford, CA, in He received the Ph.D. degree from Stanford University in the field of land-vehicle navigation. He is currently employed at Allied Signal, Redmond, WA. 162 PROCEEDINGS OF THE IEEE, VOL. 87, NO. 1, JANUARY 1999

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