alt="Graph depicts MMAE state estimate. Range observations combined with the vehicle dynamics model are eliminating unlikely integer ambiguity values."/>
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.
36.3.4 Particle Filters
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 xa and xb is 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)
Figure 36.8 MMAE integer ambiguity particle weights (subset). The correct ambiguity particle (N=7) likelihood increases over time while the outliers are determined to be less likely.
Figure 36.9 Probability density function (PDF) and cumulative density function example (CDF).
(36.71)
The particle filter uses a collection of weighted delta functions to represent the pdf:
where w[j] is a scalar weighting value for the j‐th particle with location x[j]. As mentioned previously, the sum of weights must be unity:
(36.73)
An example pdf represented by a collection of weighted particles is shown in Figure 36.10. This importance sampling strategy allows us to represent any pdf with a desired level of fidelity, given enough particles. Additional details regarding importance sampling are provided in Section 36.3.7.
In addition to representing arbitrary pdfs of random vectors, successful nonlinear estimation requires the ability to determine the resulting pdfs after applying nonlinear transformations to random vectors. In general, this can be intractable; however, representing the pdf using the collection of weighted particles makes the transformation relatively straightforward. An example of the effect of some sample nonlinear transformations is shown in Figure 36.11.
One of the most common functions necessary for filtering applications is calculation using the expectation operator. The expectation operator is defined as
(36.74)
where E[·] is the expectation operator, g(x) is an arbitrary function of the random vector x, and p(x) is the pdf of the random vector.
Based on this definition, we can easily calculate some common expectations of the weighted particle pdf. The first is the mean, which is defined as E[x]:
(36.75)
Substituting the pdf of x from Eq. 36.72, rearranging the summation and integral, and then applying the sifting property:
(36.76)
