Position, Navigation, and Timing Technologies in the 21st Century

(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.

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 vand time constant τ v. The resulting state vector is given by

(36.52)

where p kand v kare 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 kis 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 36.3.

Table 36.1 Simulation parameters

The MMAE global state estimate and density function of position after one observation ( t = 1 s) are shown in Figure 36.4. The probability density function is clearly multi‐modal, which accurately represents the range of solutions associated with the phase observation. As expected, the peaks are located at integer multiples of the carrier wavelength which corresponds to the most likely values of the unknown integer ambiguity. These peaks indirectly indicate the relative likelihood of the associated ambiguity being correct by exhibiting influence on the overall position density.

After 22 cycles, the position density shows a reduced number of peaks (see Figure 36.5). This indicates that the filter is incorporating sensor observations and the statistical dynamics model to effectively eliminate a number of potential ambiguity possibilities.

After 100 cycles ( Figure 36.6), the filter has converged to a single ambiguity.