Table 36.1 Simulation parameters
| Parameter | Value | Units |
|---|---|---|
| σ ρ | 0.5 | m |
| σ ϕ | 0.1 | cycles |
| λ | 0.2 | m |
| σ v | 0.2 | m/s |
| τ v | 500 | s |
| x t | 0 | m |
| Δt | 1.0 | s |
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.
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
| Linear and extended Kalman filter | ||
| Strengths | Weaknesses | Use case |
| Optimal for linear Gaussian systemsComputationally simple | Suboptimal approximation for nonlinear systems, can be prone to divergence | Linear, or close‐to‐linear, Gaussian problems |
| Gaussian sum filter | ||
| Strengths | Weaknesses | Use case |
| Optimal for linear Gaussian systems with discrete parameter vector | If parameter vector is not discrete, the differences must be observableConservative tuning can mask difference between models and reduce performanceIncreased computation requirements over simple Kalman filter | Linear, or close‐to‐linear, Gaussian problems with discrete parameters |
| Grid particle filter | ||
| Strengths | Weaknesses | Use case |
| Optimal solution when state space consists of discrete elementsSuitable for wide range of nonlinear conditions | Computational requirements can be excessiveProcessing requirements scale geometrically with the number of dimensionsDiscretizing continuous state space results in suboptimal performance | Nonlinear problems with lower dimensionality |
| Sampling particle filter | ||
| Strengths | Weaknesses | Use case |
| Can produce nearly optimal solution for nonlinear problemsComputational requirements can be reduced over a grid particle filter via importance sampling strategies | Maintaining good particle distribution can be difficultLack of repeatability from run to runComputational requirements can still be large | Nonlinear problems with higher dimensionality |
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