Signal Processing - March 2017 - 65
Algorithm 1. The PF for map-aided positioning.
Given the system
x t + 1 = f (x t ) + w t
y t = h (x t ) + e t
1: Initialization: For i = 1, f, N , x 0 | - 1 ~p x (0i) (x 0) and set t = 0.
2: pF measurement update: For i = 1, f, N , evaluate the importance
(i)
(i)
weights ~u t = p (y t | x t | t , Yt - 1) , and normalize
u t / / j ~u t using map information.
~t = ~
(i)
(j)
(i)
3: Resample N particles with replacement:
(i)
(j)
(j)
Pr (x t | t = x t | t - 1) = ~ t .
4: pF time update: For i = 1, f, N predict new particles
(i)
(i)
x t + 1 | t ~p (x t + 1 | t | X t , Yt ) .
5: Increase time and repeat from step 2.
(a)
KF [36] approximates the posterior at each step with a Gaussian density. Common for these methods is that it is not trivial to
impose hard constraints from the road map. They also do not
work particularly well unless the posterior density is very
monomodal or Gaussian. For a KF-based estimation with map
information, see [37].
The road constraints imply a kind of information that
normally leads to a multimodal posterior density (the target
can be on either this road, or another road on the given map,
etc.). Hence, a Gaussian approximation of the probability density function (PdF) is not suitable. a completely different
approach to nonlinear filtering is based on approximating the
posterior p (x t | Yt) numerically. The point mass filter (PmF)
[38] represents the state space using a regular grid of size N,
where the grid points and the related weights (x (i), w t(i) are
used as a representation of the posterior. different basis functions have been suggested, the simplest one being an impulse
at each grid, when the posterior approximation can be written p (x t | Yt) . / iN= 1 w (t i) d (x t - x (t i)), where d (x) denotes the
dirac-delta function. The PF [10] is the state-of-the-art numerical solution today. It uses a stochastic grid {w (t i), x (t i)} iN= 1 that
automatically changes at each iteration.
depending on the model, it is also possible to implement
numerical efficient filters combining KF and PF. The idea is to
divide the state space into two parts. If there is a conditionally linear Gaussian substructure with this partition, the KF can be utilized for that part and the PF for the other part. This is referred to
as the RBPF or the mPF [7]-[9], [39]-[41]. The RBPF improves
the performance when a linear Gaussian substructure is present,
e.g., in various map-based positioning applications and target
tracking applications as shown in [41]. The map-aided positioning algorithm based on the PF is summarized in algorithm 1.
PF-based map-aided positioning
In this section, the map-aided positioning method is first
illustrated on experimental data. Then the crucial map-based
observation is described in detail. Finally, the algorithm per-
(b)
Figure 6. Map-aided positioning using Wss information in combination
with road map information. The small black dots are particles, the red and
blue circles represent mean estimate and ground truth (gps) position, respectively. Represented are time instances after initialization when the filter
is still in (a) a multimodal state and (b) at convergence. (a) The illustration
of the particle cloud after some iterations. The multimodal pdF representing the position (several clusters of particles). The particles clustered
but still the mean point estimate (red circle) does not correspond to the
ground truth gps (blue circle). (b) The pF has converged to a unimodal
pdF (one cluster). The mean estimate is now close to the ground truth.
formance is presented on ten experiments conducted in the
same driving scenario.
Map-aided positioning illustrations
Figure 6 demonstrates the map-aided positioning using wheel
speed information and road map information, where GPs
information is used as a ground truth reference only; for other
map-aided positioning applications, see, e.g., [42]-[48]. First,
the PF is initialized in the vicinity of the GPs position. The initial distribution is chosen uniformly on road segments in a
region around the GPs fix. Particles are allowed slightly offroad to handle off-road situations and small map errors. In Figure 6(a), the algorithm has been active for some time. as can be
IEEE SIgnal ProcESSIng MagazInE
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March 2017
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65
Table of Contents for the Digital Edition of Signal Processing - March 2017
Signal Processing - March 2017 - Cover1
Signal Processing - March 2017 - Cover2
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