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

The global state estimate and associated standard deviation result for this simulation are shown in Figure 36.7. The shape of the uncertainty bound clearly shows the effects described above. As the likelihood of each integer ambiguity realization changes, the overall uncertainty changes and eventually collapses to the centimeter level.

Finally, the associated normalized filter weights for a subset of the integer ambiguity realizations are shown in Figure 36.8. As expected, the highly unlikely edge integers quickly collapse. The integers closer to the mean take longer to resolve. It is important to note that the resulting uncertainty is dependent on the actual measurement realization sequence received; thus, each realization would produce a different uncertainty ( Table 36.2). This is a notable difference from the standard linear Kalman filter, where the uncertainty is independent of the observed measurements. Finally, it is important to note that, in this example, the state estimate and uncertainty of the MMAE filter are truly optimal (i.e. minimum mean square error). This would not be the case if the integer ambiguity were resolved using a more traditional approach (e.g. float estimate with an ad hoc fixing stage). This is an interesting property of the Gaussian sum filter and sets the stage for us to investigate additional nonlinear estimation techniques.

Table 36.2 Summary of filter classes

Figure 36.3 Sample vehicle trajectory and observations. Note that the range observations are accurate but not precise and the phase observations are precise but not accurate. Our goal is to accurately estimate the joint pdf of this system.

Figure 36.4 MMAE initial state estimate and position density function. Note the position density function is extremely multi‐modal due to the limited information available at this point.

Figure 36.5 MMAE state estimate (after 22 observations). Range observations combined with the vehicle dynamics model are eliminating unlikely integer ambiguity values.

Figure 36.6 MMAE state estimate (after 100 observations). Note the state estimate is almost completely unimodal and has converged to the correct integer ambiguity.

As mentioned in Section 36.3, the key requirement of a nonlinear filter is the ability to accurately represent arbitrary probability density functions. Particle filters accomplish this by representing density functions by using collections of discrete, weighted state vectors instances. These state vectors and associated weights are referred to as particles.

Figure 36.7 MMAE position error and one‐sigma uncertainty. Note that the error uncertainty collapses once sufficient information is available to resolve the integer ambiguity.

The development of the theory related to a particle‐based representation of density functions begins by reviewing the essential properties of both the probability density function and the cumulative distribution function. An example cdf and pdf are shown in Figure 36.9. The cumulative distribution function is a monotonically increasing function which represents the probability of a random variable realization that is less than the operand and can be defined as the integral of the density function [11]:

(36.64)

(36.65)

Additionally, the probability of a random variable realization between a range x aand x bis expressed by

(36.66)

(36.67)

As a result, the density and cumulative distribution functions must have the following properties:

(36.68)

(36.69)

(36.70)