Particle ensemble sizing is generally viewed as a trade off between using enough particles
for an accurate representation of the distribution and computation time. Many times the
particle ensemble is sized without an accompanying analysis of this design choice [24], [32]. This section describes an analysis aimed at ensuring the set of discrete particles accurately reflects the assumed distribution when all state vector elements are coupled.
This problem can be explored by setting the desired covariance tolerance to a constant
value and varying the state dimension. Alternatively ensemble sizing can be explored by
varying the desired covariance tolerance and setting the state vector dimension to a constant
value. Both are important because they give the user insight into the computational cost of
the number of particles used to initialize the particle filter as the number of states increase
and the desired tolerance decreases. This technique assumes that all particles can be drawn
from the distribution at once using a command such as “mvnrnd” in MATLAB® to quickly
1. Set Np= 1
2. Draw Np particles from a given multivariate Gaussian distribution: xsample ∼
N (µtrue,Ptrue)
3. Calculate the particle ensemble covariance matrixP
sample
4. Determine if each entry inP
sampleis within the desired tolerance bounds
(a) If all entries inP
sample are within desired tolerance bounds → Go to step 5
(b) else Np= Np+ 1 → Go to step 2
5. Save Npas number of particles necessary to represent N (µtrue,Ptrue) within desired
covariance tolerance bounds
Figure 4.12: Particle Ensemble Sizing Algorithm.
The algorithm to determine the required particle ensemble size is specified in Figure
4.12. The only input is desired tolerance and the only output is the number of particles Np
necessary to ensure each entry in the n × n covariance matrix meets the desired tolerance
constraint.
This algorithm must be executed a large number of times in a Monte Carlo simulation
format to find an accurate average particle ensemble size, Np, since this value will change
each time the algorithm is executed as particles are sampled from different regions of the
distribution.
Two different tests were run to show how results varied based on the dimension n of the
state vector and also to investigate how results varied based on the covariance tolerance.
The true distribution was N (0n×1, ATA) where A was an n × n matrix drawn using the “rand”
command in MATLAB®. Note that any n × 1 vector could be used for the mean vector and
any positive-semidefinite symmetric matrix could be used as the covariance matrix.
The first test varied the state dimension n from 1 to 12 while setting the covariance tol-
10, 000 Monte Carlo trials with the average number of required particles for each dimen-
sion shown in Figure 4.13. The number of particles increased from approximately 14 to 648 as the dimension of the vector increased for each set of trials. As expected, the trajec-
tory became more smooth for 1, 000 Monte Carlo trials, and 10, 000 trial runs revealing a
linear relationship between the number of required particles and the dimension of the state
of the Gaussian random variable for a desired initial covariance tolerance.
2
4
6
8
10
12
0
200
400
600
State Dimension
Number of Particles
100 Monte Carlo Trials
1,000 Monte Carlo Trials
10,000 Monte Carlo Trials
Figure 4.13: Minimum particle ensemble size for 5% covariance tolerance of a Gaussian random variable with increasing dimension.
The second test held the state dimension n constant at 12 while varying the covariance
tolerance from 1% down to 10% to give particle filter users a good estimate of the number
of particles necessary to initialize the filter. It was run for 100, 1, 000, and 10, 000 Monte
Carlo trials with the average values shown below in Figure 4.14. The exponentially de- creasing trajectory shows that to meet 10% tolerance requires roughly 215 particles, but
this number increases 40 to 50 fold when 1% tolerance is desired. With such a large in-
crease in required number of particles for smaller than 5% tolerance, users must make a
tational resources available and as well as the application.
0.02 0.04 0.06 0.08
0.1
0
2000
4000
6000
8000
10000
Tolerance
Number of Particles
100 Monte Carlo Trials
1,000 Monte Carlo Trials
10,000 Monte Carlo Trials
Figure 4.14: Minimum particle ensemble size for 12-state Gaussian random variable as a function of covariance tolerance.
4.6
Chapter Summary
This chapter first described a strategy for relative location categorization by street level
(S L) and altitude (ALT ) features that facilitate sensor availability and covariance specifica-
tion. It then described the method used in this dissertation to generate sensor measurements
and sensor measurement error covariances for filtered sensor state information fromIMU,
ADS, GPS, LTE, and VISION-OF. It focused primarily on the GPS, LTE, and VISION sensors as each of these have dynamic environment-based measurement error covariance
matrices. There was also a discussion of sensor self-accuracy determination versus using
the estimated position in the environment to determine noise covariance values to send to
the filter. Additionally, a delayed-state filter implementation was described for GPS and
imum number of particles required to ensures the ensemble covariance reflects the true
CHAPTER 5
Accuracy of Navigation in a Homogeneous
Urban Environment
This chapter describes UAS urban environment navigation results using different sensor sets at different altitudes in a simulated homogeneous environment. These sensor sets were
selected to demonstrate the varying levels of navigation accuracy available using currently
available algorithms and hardware based on literature and manufacturer specifications. The
homogeneous environment provides a baseline where sensors are either available through-
out the duration of the simulation or unavailable throughout the duration of the simulation.
In this chapter theUASplant dynamics propagation model and state estimation filter prop- agation model are assumed to be identical (or matched) to objectively evaluate test the
sensor sets as a function of the environment.