Robust Sensor Characterization via Max-Mixture Models: GPS Sensors
Ryan Morton and Edwin Olson

Abstract— Large position errors plague GNSS-based sensors
(e.g., GPS) due to poor satellite configuration and multipath
effects, resulting in frequent outliers. Due to quadratic cost
functions when optimizing SLAM via nonlinear least square
methods, a single such outlier can cause severe map distortions.
Following in the footsteps of recent improvements in the
robustness of SLAM optimization process, this work presents
a framework for improving sensor noise characterizations by
combining a machine learning approach with max-mixture
error models. By using max-mixtures, the sensor’s noise distribution can be modeled to a desired accuracy, with robustness
to outliers. We apply the framework to the task of accurately
modeling the uncertainties of consumer-grade GPS sensors.
Our method estimates the observation covariances using only
weighted feature vectors and a single max operator, learning
parameters off-line for efficient on-line calculation.

I. I NTRODUCTION
Most state-of-the-art Simultaneous Localization and Mapping (SLAM) algorithms require sensor uncertainties to be
characterized probabilistically. Such characterization is a
straightforward task for many common robot sensors, e.g.,
LIDARs. However, the noise distribution of other sensors
provide a much more challenging characterization task. The
cause of these erroneous observations stems from the inability of the sensor to observe the complete state of the
environment. For example, ground robots’ wheels lose traction resulting in a slip-or-grip problem and vertical structures
block Global Positioning System (GPS) signals, leading to
multipath effects. Generally, the sensor return is assumed to
reflect the most likely state of the quantity being measured,
e.g., latitude and longitude for a GPS sensor1 . The task of
defining the uncertainty of these individual observations is
termed sensor characterization.
The typical approach to sensor characterization uses Maximum Likelihood (ML) optimization to find the parameters
that best describe the training data. This off-line process
returns parameters that maximize the average performance
over ground-truthed training data. These parameters can then
be used on-line to calculate observation noise estimates for
use within SLAM.
A single outlier can cause a severe, even irrecoverable,
map distortion due to the quadratic cost surface used by
the nonlinear least squares optimization within many SLAM
algorithms. In this work, we take the view that outliers
arise from overly-simple and optimistic noise models; a
better noise model would assign a higher probability to
The authors are with the Computer Science and Engineering, University
of Michigan, 2260 Hayward Street, Ann Arbor, Michigan, 48109, USA.
{rmorton, ebolson}@umich.edu.
1 In this work GPS means any Global Navigation Satellite System (GNSS)

(a) SLAM

(b) GPS

Fig. 1: Robot Trajectories for 12 Exploration Robots. Colored to reflect elapsing time (red-green-blue as t increases).

an “outlier”, thus limiting its effect. Simple Gaussian error
models often suffer from this problem, since the tails drop
off exponentially fast. We use max-mixtures of Gaussians to
improve the richness of the sensor characterization models.
To highlight our framework, we show noise models for
GPS sensors, which are particularly prone to large errors;
it is common to receive observations from consumer grade
sensors that are tens to hundreds of meters from the true
location. Specifically, the contributions of this paper include:
• A machine-learning approach for sensor uncertainty
estimation.
• An extension of sensor uncertainty estimation using
max-mixtures.
• A new metric for evaluating the impact of outliers on a
SLAM system.
• Evaluation of a variety of models for consumer-grade
GPS sensor characterization.
In the next section, we give a short overview of related
advancements in SLAM, robust estimation, and GPS error
characterization. We describe the mathematical formulation
of sensor characterization in Section III and show the robustness improvements of the max-mixture models in Section IV.
Then in Section V, we describe features from GPS sensor
data and how to combine them to robustly estimate GPS
uncertainty. In Section VI, we show empirical results for the
noise models on a real-world dataset comprised of 45 hours
of GPS data from a 14-robot team.
II. BACKGROUND
Many modern approaches formulate SLAM in terms of
two components: a front-end that builds a factor graph and
a back-end that optimizes it [1]. For example, the front-end
might add an odometry constraint (edge) representing a rigidbody-transformation between two sequential robot positions

(nodes); the constraint’s mean represents the raw encoder
observation while the edge weight is a function of the a
priori uncertainty of the encoder. The back-end periodically
optimizes the graph, producing a ML estimate for all nodes
in the graph, i.e., the best position of the nodes given all the
constraints. When errors are Gaussian this optimization can
be viewed as a least squares method with each constraint
incurring a quadratic cost.
A known problem in SLAM and other nonlinear least
squares optimization problems stems from these quadratic
costs, namely that a single outlier can have significant and
detrimental impact on the solution [2], [3], [4]. The quadratic
costs come directly from assuming constraints have Gaussian
error, which brings computational efficiency.
Improving the robustness of SLAM to these outliers is
currently an active area of research. Some approaches use
switching variables, which can ‘turn off’ constraints and,
thus, handle outliers by modifying the graph structure in
the back-end [2], [5]. Another related approach detects and
rejects outliers via loop consistency checks on small clusters
of nodes [6]. Both of these approaches show promise at
dealing with outliers in the back-end of SLAM. In this paper,
we take the view that outliers arise from modeling errors and
that richer error models can result in systems that are robust
to ouliers and do not require specialized “outlier rejection”
methods.
While mixture models have long been used to approximate complex distributions, they typically result in high
computational costs. Our system uses max-mixtures, which
allows both flexible error models and fast inference [4]. Some
sensors could be characterized with robust cost functions, [7],
[8], but these have been previously shown to be subsumed
by max-mixtures [4].
This paper focuses on models for GPS data, which can be
used both to improve the accuracy of the map and to register
the relative frame to Earth. Even with many loop closures,
SLAM maps can have significant distortion compared to
ground truth, e.g., long hallways may erroneously bend and
GPS sensors can offer the needed constraints even when
global registration is not necessarily needed [9]. Unmanned
Aerial Vehicles (UAVs) and agricultural robots, which generally have unobstructed satellite line-of-sight, can rely heavily
upon GPS sensors for positioning. In environments with
an unobstructed view of the sky, GPS sensors offer costeffective and computationally efficient position estimates
that are, when supplemented with other sensors, sufficient
for many systems [10]. Additional sensors, such as inertial
measurement units or visual odometry have been shown to
reduce errors caused by short blackouts in GPS data, i.e.,
obstruction of satellite view [11], [12], [13].
Early error models for GPS assumed a constant variance
for all GPS returns [11], [14]. Many GPS sensors provide
their own uncertainty estimates, but they can be misleading.
For example, they may report erroneous readings for a period
of time before detecting the loss of satellite locks. This
problem stems from the discrete nature of individual satellite
locks that can cause position estimate discontinuities [15], a

problem that is blamed for a crash during the DARPA Grand
Challenge [16].
Rather than accurately modeling the noise of GPS sensors, a common approach has been binary classification of
(in)valid sensor returns [13], [17], [16]. These systems stop
adding GPS data to SLAM upon detection of a GPS dropout
and/or multipath effects. Some approaches use additional
sensors to improve multipath detection via satellite lineof-sight calculations by building 3D-models of the local
structure [18], [19].
Some approaches circumvent the GPS sensor’s position
estimate and calculate positions directly from the trilateration
of pseudo-ranges to individual satellites [20]. The most
closely related work uses robust cost function and switch
variables to modify the graph, but their method also requires
a GPS sensor with pseudo-range capabilities [21]. Although
these approaches using sensors equipped with pseudo-ranges
and/or sensor utilizing the local structure have shown signs of
success, we desire a robust approach that uses observations
directly from standard consumer-grade GPS sensors.
III. S ENSOR C HARACTERIZATION
Sensor characterization consists of estimating the distribution of observations, zi , returned from the sensor around
the true location, xi . The goal of sensor characterization is a
distribution best defining the zero-mean P (zi |xi ). Designers
generally choose the form of the distribution, e.g., Gaussian,
and estimate the parameters of the model using training data,
or by hand-tuning, to maximize log-likelihood. The output
of the characterization is a covariance matrix Σi , which may
be constant or a function of the observation.
A. Simple Gaussian Models
Let ei be the observation error, i.e., ei = (zi − xi ).
Assuming a zero-mean uni-modal Gaussian distribution,
parametrized by Σi , the distribution becomes:
P (ei ) = N (0, Σi ) =

1
d
2

(2π) |Σi |

1

1
2

T

e − 2 ei Σ i

−1

ei

(1)

with d equal to the degree-of-freedom (DOF) of the observation. The log-likelihood, assuming n independent observations, can be written:
L = log

n−1
∏
i=0

=−

P (ei ) =

n−1
∑

log P (ei )

i=0
n−1
∑

nd
1
log(2π) −
2
2

log(|Σi |) + eT Σ−1 ei
i
i

(2)

i=0

The last term in (2) explicitly shows the quadratic costs
associated with deviations from the mean.
The sensor characterization task is the definition of Σi ,
a function of individual observations, to maximize L. Each
such function can take many forms, but we use a parametrization where each element is a linear combination of features
of the observation. For example, assuming a 2 DOF sensor

with independent noise in each DOF, the covariance may be
defined as:
[ 2
] [ T
]
σi,x
0
(fi,x wx )2
0
Σi =
=
(3)
2
T
0
σi,y
0
(fi,y wy )2
With feature vectors fi,x and fi,y as functions of each
observation (presumably informing about the error in x and
y respectively) and global weight vectors wx and wy . In
this context, the characterization task would simply be the
definition of the weight vectors.
IV. ROBUST S ENSOR C HARACTERIZATION USING
M AX -M IXTURES OF G AUSSIANS
Gaussian mixture models offer a richer representation by
combining multiple Gaussian components, allowing modeling of arbitrarily complex distributions. Max-mixture models
define the distribution as the max of k weighted components:
P (ei ) = ηi max(α1 P1 (ei ), . . . , αk Pk (ei ))
)
(
−1
1 T
αj
k
− 2 ei Σi,j ei
= ηi max
1 e
d
j=1
(2π) 2 |Σi,j | 2

(4)

Each components, Pj , is of the form in (1). The loglikelihood of the data becomes:
∑
∏
log P (ei )
P (ei ) =
L = log
i

=

∑
i

log

(

ηi

d

(2π) 2

)i

k

(

+ max log
j=1

(

1

T

αj e− 2 ei Σi,j ei
−1

1

|Σi,j | 2

))

(5)

Note that ηi becomes an additive constant that does not
effect the minimization and does not generally need to be
computed. The sensor characterization task for max-mixtures
is to define the covariance estimates Σi,j and mixing terms
αj that maximize L.
The most common mixture models are sum-mixtures, i.e.,
∑k
P (ei ) =
j=1 αj Pj (ei ), but our choice of max-mixtures
is motivated by the use of logarithms in (5). The ability to
push the logarithm inside the max operator brings significant
computational advantages [4].
V. GPS S ENSOR C HARACTERIZATION
We consider two families of GPS uncertainty predictors
based on uni-modal Gaussians and max-mixtures of Gaussians using non-linear optimization for parameter fitting.
The primary difference between these families is that, for
each observation, uni-modal models have a single covariance
estimate, Σi , and max-mixture models have k of them; plus
k weight vectors.
For mobile ground robots, we assume radial GPS errors,
with x and y errors independent and identically distributed,
i.e., ei = ||ei || represents the total translation error. Thus,
fi = fi,x = fi,y , w = wx = wy , and each covariance
function becomes:
]
[ T 2
[ 2
]
(fi w)
0
σi 0
=
Σi =
(6)
2
0 σi
0
(fiT w)2

The global weights are learned off-line and we next describe
some possible GPS feature vectors fi for each observation
zi .
A. GPS Observation Features
An interesting property of GPS receivers is that they
produce a wealth of data that can be used to generate
features. In this section, we explore a variety of features,
beginning with trivial (but still useful) ones and moving to
more complex features.
1) Constant Noise Model: While few real-world systems
today would attempt to characterize all GPS observations as
having the same uncertainty, such a model can serve as a
baseline method for evaluation. Using our framework, we
simply let
fiT = [ 1 ]
Using this model, the magnitude of the learned weight would
represent the standard deviation for all observations. Since all
observations have equal feature vectors, fi , all observations
will have the same noise estimate fiT w.
2) Number of Satellites Noise Model: One strategy for
determining the reliability of GPS data is to observe the
number of visible satellites, nsat,i . A simple feature vector
expressing this assumption is given by:
fiT = [ nsat,i ]
We expect to learn a negative weight for this feature since
more satellites should reduce the observation error. However,
this feature cannot be used alone, because negative σs are
prohibited. Later we’ll see how these simple feature can be
combined with others.
A more effective use of the number of satellites is to
construct the feature vector fi such that the nth element
sat,i
of fi is set to 1, i.e., a “one-hot” encoding. For example,
with 12 possible simultaneous satellite observations and a
reported observation of 5 satellites, the feature vector would
be given by:
fiT = [ 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0 ]
This representation allows a different covariance model to
be fit to each number of satellites, but requires more training
data through the whole range of possible nsat,i . We expect
to learn large weights for lower indices and small weights at
the larger indices as the number of satellites should decrease
the error.
3) Dilution of Precision Noise Model: A few standard
outputs from GPS sensors are intended to represent the
positional uncertainty in terms of geometric dilution of precision. Three such values are typically reported: horizontal
dilution of precision (hdop), positional dilution of precision
(pdop), and time dilution of precision (tdop). Each represents
a multiplicative scaling of the uncertainty as a function of the
geometric configuration of satellites, relative to the sensor,
and should be informative of the true uncertainty. We can
incorporate these values into our feature vector by letting (for
example) fiT = [ hdopi ]. We expect a positive correlation
between these values and true uncertainty.

4) Vendor-Provided Noise Model: Many GPS units provide an uncertainty estimate computed by the sensor. To
evaluate the quality of this estimate, we set our feature vector
to contain just this estimate:
fiT = [ σvendor,i ]
If the vendor supplied estimates are correct, we would expect
to learn a weight of 1 for this feature.
5) Combination Noise Model: The uncertainty estimate σi
is a linear combination of features and, thus, the aforementioned features can be combined, with the individual feature
and weight components retaining their original meaning.
For example, combining the constant model, the simplified
number of satellites model, and the vendor-provided estimate
would produce the feature vector:
fiT = [ 1, nsat,i , σvendor,i ]
We expect this feature to outperform the individual features,
which are subsumed by the combination model, so long as
over-fitting is avoided.
B. Additional Consideration for Max-Mixture Models
The aforementioned models can be used directly as unimodal models of GPS uncertainty. However, simple Gaussian
error models for GPS tend to perform poorly, as both earlier
work and our experiments show. Yet, a k = 2 max mixture
model provides significant improvements. With k = 2,
characterization of GPS sensors using max-mixtures requires
maximizing L over a single α value and two w vectors.
Let wj be the dataset-wide weight vector associated with
the j th feature. For observation i, let fi,j be feature j and
T
standard deviation now given by σi,j = fi,j wj . For example,
the features of a 2 component mixture of the constant model
and the combination model (from previous section) are given
by:
[ T] [
]
fi,1
[1]
fi = T =
(7)
[ 1, nsat,i , σvendor,i ]
fi,2
With w now defining the k dataset-wide weight vectors wj :
]
[ T] [
[ w1,0 ]
w1
(8)
w=
T =
[ w2,0 , w2,1 , w2,2 ]
w2
The feature vectors and weights combine, as expected to
produce k standard deviations:
] [ T
[
]
f w
σi,1
= i,1 1
(9)
T
σi,2
fi,2 w2
Combining with (5), this leads to a value for σi defined as:
)
(
e2
i
(10)
σi = arg max log(αj ) − 2 log(σi,j ) − 2
2σi,j
σi,j
To illustrate the max-mixture approach, suppose that some
GPS measurements are nominal (with errors of a few meters),
while other measurements are “outliers” (with errors of tens
of meters). With a uni-modal approach using f = [ 1 ], we
might learn w = 9. However, by setting f = [ [ 1 ], [ 1 ] ]
(a mixture of two constant variances) we might expect to

learn w = [ [ 2 ], [ 30 ] ] and a value of α in relation to the
frequency of those outliers. While simplistic we will show
that this method works well.
VI. E VALUATION
In this section we evaluate our proposed GPS noise
models on a 14-robot dataset collected within a 220 x 160 m
indoor/outdoor region of the Adelaide Showgrounds in South
Australia using Garmin GPS18x-5Hz sensors during the
MAGIC competition [22].
A. Performance Metrics
To analyze the GPS sensor characterization we analyze
three primary metrics for each model: 1) likelihood of
observed data, 2) robustness to high-error observations, and,
more generally, 3) the distribution of error relative to modelpredicted error.
We use the normalized log-likelihood of the data, given
the model and parameters learned off-line, to measure the
overall fit of the model. This metric represents the expected
log-likelihood of each observation and we desire models that
maximize this metric. Note that the normalization constant
η is not computed, since it does not affect the maximization
of this metric.
In the SLAM context, or any similar non-linear optimization problem, a single erroneous measurement can wreak
havoc if not properly modeled. For uni-modal models, outliers have low probability and min log P (ei ) represents the
worst-case likelihood error.
Since SLAM computes an ML solution via non-linear
optimization and a 1st order Taylor expansion, the gradient magnitude, ||∇i ||, is another outlier metric. During
the optimization process ||∇i || represents the “pull” of the
constraint, relative to other constraints’ gradients. For a given
uni-modal Gaussian, the least likely measurement also has
the maximum gradient, i.e., arg mini Li = arg maxj ||∇j ||.
However, with max-mixtures an observation may be expected
with near zero probability, but have virtually no gradient and
thus no ‘pull’ within the optimization process. We desire
models that minimize the “pull” associated with outlier measurements, i.e., we desire small max ||∇i || = max ||Σi ei ||.
For each observation, ei /σi represents the number of
standard deviations predicted by the model. If the noise
was truly distributed according to a uni-modal Gaussian
distribution, the ei /σi errors would be distributed as a χ
distribution. Although this metric does not exactly relate
to max-mixtures, we still desire ei /σi distributions that are
quantitatively similar to a χ distribution. For example, we
hope to find models where all observations fall within the
statistically significant region, max(ei /σi ) < 6.
B. Analysis
Because our evaluation datasets were dominated by inliers,
the normalized log likelihood of all of our models, including
both simple Gaussian models and max-mixture models, falls
within a relatively small range (from -7.3 to -6.3). These

TABLE I: Training and Testing Error.
Model

Uni-Modal
1
n

constant-hdop
constant-hdop-nsat-vendor
constant-hdop-vendor
constant-nsat-vendor
constant-vendor
constant
vendor

∑

Train
Li

-6.855
-6.299
-6.312
-6.302
-6.312
-7.210
-6.329

max

ei
σi

14.503
6.866
6.342
6.785
6.331
21.106
6.177

max ||∇i ||
2.920
2.260
1.957
2.208
1.948
3.263
2.514

1
n

∑

Max-Mixture
Test
Li

-6.852
-6.318
-6.325
-6.319
-6.326
-7.321
-6.336

max

ei
σi

17.895
11.451
11.548
11.595
11.516
21.618
9.643

max ||∇i ||
2.750
2.127
1.890
2.096
1.878
3.251
2.331

1
n

∑

Train
Li

-6.587
-6.289
-6.303
-6.271
-6.305
-6.733
-6.320

max

ei
σi

8.810
5.142
4.318
6.736
4.286
12.353
4.204

max ||∇i ||
1.363
1.840
1.631
2.431
1.737
1.118
2.048

1
n

∑

Test
Li

-6.623
-6.306
-6.325
-6.286
-6.321
-6.820
-6.330

max

ei
σi

max ||∇i ||

10.946
7.261
7.932
10.770
7.292
12.653
6.563

1.300
1.836
1.664
2.275
1.791
1.114
2.024

small differences arise from dramatic differences in the log
likelihoods of the relatively-infrequent outlier measurements.
As seen in Table I, for any given feature vector, the
performance of a mixture of two components invariably outperformed the best single component (simple Gaussian) error
model, according to both training and test error. As measured
by log likelihood, the magnitude of these differences is small,
but again, this is due to the fact that the vast majority of the
measurements were inliers that every model handled well.
As expected, we find that test error is generally somewhat
higher than training error, but the magnitude of the increase
is similar between both the simple Gaussian model and the
more complex max-mixture models.
Unlike the normalized log likelihood, in which the performance of the models on outliers is masked by the large
number of inliers, the worst case standard deviation metric
clearly shows the advantages of the mixture models. The
worst-case standard deviation, max ei /σi , dropped for every
model, often dramatically, e.g., from 21.618 to 12.653 in
the constant-covariance model case. These improvements
highlight a significant improvement in modeling the sensor’s
noise. To be clear, measurements with 12 standard deviations
of error may still have too much influence during optimization, but the constant model is a very naive baseline and
ignores any uncertainty indications/features the sensor.
Histograms of the empirical χ error compared to the
model’s predicted density are also revealing (see Fig. 2). In
the case of simple Gaussian models (left column), we see
that the model is consistently conservative with respect to
the inlier data (on the left side of the plots). The outliers
have caused an increase in the covariance estimate, with the
consequence that inliers are given too little weight. Despite
the inflated covariance estimate, outliers still have a very
high gradient (see Table I) due to their distance from the
Gaussian distribution’s mean and will strongly influence the
optimization.
Conversely, the right column of Fig. 2, which plots histograms of the max-mixture models, shows a model error that
more tightly tracks the ideal distribution. Simultaneously,
outliers are shifted closer to the left, indicating that higher
probabilities have been predicted for them. These ouliers still
influence the graph, but have correspondingly smaller “pull”,
which would make a SLAM system more resilient to them.

(a) Constant

(b) MM-Constant

(c) Vendor

(d) MM-Vendor

(e) Const-HDOP-NSAT-Vendor

(f) MM-Const-HDOP-NSAT-Vendor

Fig. 2: Histogram of Empirical vs. Ideal χ-error (for select
models). Ideally, if the underlying Gaussian assumptions
hold, the normalized histogram of ei /σi would fit a 2 DOF χ
distribution (with horizontal units representing standard deviations, σi ). The relative movement of the worst-case arrows
to lower standard deviations, uni-modal models (left) versus
max-mixture models (right), highlights the improvements in
modeling capabilities, specifically robustness to outliers.

C. Learned Weights
We next present a few learned parameter settings for
discussion purposes correspond to the respective models
shown in Table I.
The constant models learned weights of [9.29] for unimodal and [[5.2], [16.2]] with α = [0.98, 0.02] for the
max-mixture model. This reflects the fact that the sensor
performs well most of the time, but a uni-modal model
must compensate for the high-error observations with an
overestimate of σ to fit the Gaussian assumption.
We were pleasantly surprised by the quality of the vendor’s
uncertainties both in terms of likelihood and in terms of
max ei /σi . The vendor model learned weights of [0.8] and
[[0.73], [1.16]] with α = [0.97, 0.03], reflecting only modest
adjustment of their estimates. However, we were still able to
improve upon them with our method.
VII. C ONCLUSION
We have described a general approach for computing sensor uncertainty estimates using a machine learning approach.
Feature vectors are constructed from observations and a

weight vector is learned from a ground-truthed data set, via
maximizing the log-likelihood of the training data set.
We showed how this approach can be extended to more
expressive error models using max-mixtures. We take the
view in this paper that “outliers” arise from mismatches
between the empirical performance of a system and its error
model: better models assign higher probabilities to outliers
and thus mitigate their impact. Versus an explicit outlier
rejection phase, the max-mixture approach both provides an
integrated Bayesian mechanism for robust estimation and
removes the need for a near-perfect outlier detector.
In evaluating the performance of these models we use
standard log-likelihood metrics and introduce a metric that
reflects the impact of an outlier on a SLAM system. Our
work was evaluated on a large multi-robot dataset and
we demonstrated significant performance improvements using our methods on the task of characterizing error-prone
customer-grade GPS sensors. Related methods attempting to
add robustness to the back-end of SLAM would also benefit
from improved robustness on the front-end provided by the
approach presented here.
ACKNOWLEDGEMENTS
This work was supported by U.S. DoD Grant FA2386-111-4024.
R EFERENCES
[1] E. Olson, “Robust and efficient robotic mapping,” Ph.D. dissertation,
Massachusetts Institute of Technology, Cambridge, MA, USA, June
2008.
[2] N. Sunderhauf and P. Protzel, “Switchable constraints for robust
pose graph SLAM,” in Intelligent Robots and Systems (IROS), 2012
IEEE/RSJ International Conference on. IEEE, 2012, pp. 1879–1884.
[3] Y. Latif, C. Cadena, and J. Neira, “Realizing, reversing, recovering: Incremental robust loop closing over time using the iRRR algorithm,” in
Intelligent Robots and Systems (IROS), 2012 IEEE/RSJ International
Conference on. IEEE, 2012, pp. 4211–4217.
[4] E. Olson and P. Agarwal, “Inference on networks of mixtures for
robust robot mapping,” in Proceedings of Robotics: Science and
Systems (RSS), July 2012.
[5] P. Agarwal, G. D. Tipaldi, L. Spinello, C. Stachniss, and W. Burgard,
“Robust map optimization using dynamic covariance scaling,” in
Proceedings of the IEEE International Conference on Robotics and
Automation (ICRA), 2013.
[6] Y. Latif, C. Cadena, and J. Neira, “Robust loop closing over time,” in
Proceedings of Robotics: Science and Systems (RSS), 2012.

[7] P. J. Huber, “Robust estimation of a location parameter,” The Annals
of Mathematical Statistics, vol. 35, no. 1, pp. 73–101, 1964.
[8] R. Hartley and A. Zisserman, Multiple View Geometry in Computer
Vision, 2nd ed. Cambridge University Press, 2004.
[9] S. Thrun and M. Montemerlo, “The GraphSLAM algorithm with
applications to large-scale mapping of urban structures,” International
Journal of Robotics Research, vol. 25, no. 5-6, pp. 403–430, May-June
2006.
[10] A. Stentz, C. Dima, C. Wellington, H. Herman, and D. Stager, “A
system for semi-autonomous tractor operations,” Autonomous Robots,
vol. 13, no. 1, pp. 87–104, 2002.
[11] J. Kim and S. Sukkarieh, “SLAM aided GPS/INS navigation in
GPS denied and unknown environments,” in Proceedings of the 2004
International Symposium on GNSS/GPS, 2004, pp. 1–5.
[12] M. Agrawal and K. Konolige, “Real-time localization in outdoor environments using stereo vision and inexpensive GPS,” in Proceedings
of the International Conference on Pattern Recognition. IEEE, 2006.
[13] W. Ding, J. Wang, S. Han, A. Almagbile, M. Garratt, A. Lambert,
and J. Wang, “Adding optical flow into the GPS/INS integration for
uav navigation,” in Proc. of International Global Navigation Satellite
Systems Society Symposium. Citeseer, 2009, pp. 1–13.
[14] K. Ohno, T. Tsubouchi, B. Shigematsu, and S. Yuta, “Differential GPS
and odometry-based outdoor navigation of a mobile robot,” Advanced
Robotics, vol. 18, no. 6, pp. 611–635, 2004.
[15] D. C. Moore, A. S. Huang, M. Walter, E. Olson, L. Fletcher,
J. Leonard, and S. Teller, “Simultaneous local and global state estimation for robotic navigation,” in Robotics and Automation, 2009.
ICRA’09. IEEE International Conference on. IEEE, 2009, pp. 3794–
3799.
[16] L. Cremean, T. Foote, J. Gillula, G. Hines, D. Kogan, K. Kriechbaum,
J. Lamb, J. Leibs, L. Lindzey, C. Rasmussen, et al., “Alice: An
information-rich autonomous vehicle for high-speed desert navigation,” Journal of Field Robotics, vol. 23, no. 9, pp. 777–810, 2006.
[17] J. Wang, M. Garratt, A. Lambert, J. Wang, S. Han, and D. Sinclair,
“Integration of GPS/INS/vision sensors to navigate unmanned aerial
vehicles,” in XXI Congress of the Int. Society of Photogrammetry &
Remote Sensing, Beijing, PR China, 2008, pp. 963–970.
[18] B. Ben-Moshe, E. Elkin, H. Levi, and A. Weissman, “Improving
accuracy of GNSS devices in urban canyons.” in CCCG, 2011.
[19] D. Maier and A. Kleiner, “Improved GPS sensor model for mobile
robots in urban terrain,” Proceedings of the IEEE International Conference on Robotics and Automation (ICRA), pp. 4385–4390, 2010.
[20] E. Takeuchi, M. Yamazaki, K. Ohno, and S. Tadokoro, “GPS measurement model with satellite visibility using 3d map for particle
filter,” in Robotics and Biomimetics (ROBIO), 2011 IEEE International
Conference on. IEEE, 2011, pp. 590–595.
[21] N. Sunderhauf, M. Obst, G. Wanielik, and P. Protzel, “Multipath
mitigation in GNSS-based localization using robust optimization,” in
Intelligent Vehicles Symposium (IV), 2012 IEEE. IEEE, 2012, pp.
784–789.
[22] E. Olson, J. Strom, R. Morton, A. Richardson, P. Ranganathan,
R. Goeddel, and M. Bulic, “Progress towards multi-robot reconnaissance and the MAGIC 2010 Competition,” Journal of Field Robotics,
vol. 29, September 2012.