Position, Navigation, and Timing Technologies in the 21st Century. Группа авторов

conditioned on the parameter set j. As mentioned previously, these likelihood values are based on the following evaluation of a normal density function:

      (36.46)

      where z_k is the measurement realization at time k. This likelihood is equivalent to the likelihood of the residual from a Kalman filter tuned to the j‐th parameter vector, a^[j].

Practically speaking, the parameter pdf consists of the discrete (fixed) parameter set and the associated weights (likelihood) at each epoch. The parameter density update shown in Eq. 36.45 shows the evolution of each parameter weight as a function of time, which can be rewritten as

      (36.47)

Our final task is to determine the overall posterior joint pdf of the system. Substituting Eq. 36.44 into Eq. 36.35, we obtain

      (36.48)

which, when combined with knowledge of the delta function and implementing a straightforward rearrangement of terms produces the joint posterior density function

      (36.49)

      This pdf is clearly a weighted sum of Gaussian densities, each of these densities corresponding to the posterior state estimate of an individual Kalman filter, tuned to the parameter vector a^[j]. The blended posterior state estimate and covariance are given by

      (36.50)

      (36.51)

The MMAE filter can be visualized in block diagram form in Figure 36.2.

      Additional forms that are very similar conceptually to the MMAE filter are known as interactive mixture model (IMM) estimators [8] and Rao‐Blackwellized particle filters (RB‐PFs) [9, 10], to name a few.

      In the next section, we present a simple example to illustrate a potential application for Gaussian sum filters derived in this section.

      36.3.3 MMAE Example – Integer Ambiguity Resolution

      The benefits of the Gaussian sum filter can be illustrated using a simple example. Consider the following one‐dimensional navigation scenario. A radio transmitter broadcasts a ranging signal from a fixed location, x_t. A ranging receiver is mounted on a vehicle that is free to move in the x‐direction. The vehicle motion can be represented using a first‐order Gauss–Markov velocity model [2] with uncertainty σ_v and time constant τ_v. The resulting state vector is given by

      (36.52)

      where p_k and v_k are the position and velocity of the vehicle at time k. The dynamics of the vehicle are given by

      (36.53)

      where

      (36.54)

      and w_k is a zero‐mean Gaussian random vector with

      (36.55)

The ranging signal consists of both a noise‐corrupted measurement of the true range along with a measurement of the integrated carrier phase. The integrated carrier phase is a high‐precision measurement, but is corrupted by an unknown integer ambiguity. The observation model is

Figure 36.2 MMAE filter implementation. The MMAE filter constructs the state estimate by combining results from individual Kalman filters tuned to a parameter realization [7].

      (36.56)

      (36.57)

      where λ is the carrier wavelength, and N is the integer ambiguity. Both observations are corrupted by zero‐mean white Gaussian noise sequences with

      (36.58)

      (36.59)

      (36.60)

      Our goal is to use the MMAE estimator to accurately represent the (non‐Gaussian) posterior pdf, thus maintaining a consistent overall state estimate and uncertainty, while incorporating all available information.

      In this example, the integer ambiguity is the unknown parameter set, which in the previous development we designated as the vector a. We choose a range of J plausible integers based upon any a priori knowledge or even the initial range observation itself, which results in the following unknown parameter vector:

      (36.61)

      with overall joint probability density

      (36.62)

      From this point forward, the implementation proceeds as outlined in the previous section. A total of J weighted Kalman filters are constructed, each with the assumption that N^[j] is the correct integer ambiguity. The joint posterior density is given by

      (36.63)

In order to demonstrate the performance of the Gaussian sum filter, the above scenario was implemented in a simulation environment. A trajectory and measurement set is randomly generated using the parameters specified in Table 36.1.

      Note the carrier phase wavelength is 0.2 m, and the carrier phase measurement uncertainty is 0.1 cycles, which results in a measurement precision of 0.02 m, which is an improvement of 50 times over the pseudorange measurement errors.

The resulting trajectory, range observations, and phase observations are shown in Figure

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