Vehicle Localization Using In-Vehicle Pitch Data and Dynamical Models

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Vehicle Localization Using In-VehiclePitch Data and Dynamical Models

Emil I. Laftchiev, Student Member, IEEE, Constantino M. Lagoa, Member, IEEE, andSean N. Brennan, Member, IEEE

Abstract—This paper describes a dynamical model-basedmethod for the localization of road vehicles using terrain datafrom the vehicle’s onboard sensors. Road data are encoded usinglinear dynamical models and then, during travel, the locationis identified through continuous comparison of a bank of linearmodels. The approach presented has several advantages overprevious methods described in the literature. First, it createscomputationally efficient linear model map representations of theroad data. Second, the use of linear models eliminates the needfor metrics during the localization process. Third, the localizationalgorithm is a computationally efficient approach that can have abounded localization distance in the absence of noise, given certainuniqueness assumptions on the data. Fourth, encoding road datausing linear models has the potential to compress the data, whileretaining the sensory information. Finally, performing only linearoperations on observed noisy data simplifies the creation of noisemitigation algorithms.

Index Terms—Dead reckoning (DR), feature extraction, timeseries, vehicle navigation.

I. INTRODUCTION

R ECENT years have seen rapid development of ad-vanced vehicle safety systems, navigation systems, and

intelligent transportation systems capable of optimizing traf-fic patterns, tracking resources, and identifying infrastructureproblems as they arise. All of these systems heavily dependon the ability to localize a vehicle and, in many cases, aresignificantly more effective with meter-scale resolution of thevehicle’s location. In addition, any safety system must be highlyreliable to prevent potential accident causing failures [1].

The Global Positioning System (GPS) has been the standardbearer for vehicle localization for some time. Current single-frequency GPS receivers can achieve a minimum error of about10 m [2], [3]. Achieving meter resolution requires a two-frequency GPS receiver, which is currently cost prohibitivefor consumers. Furthermore, even with the three independent

Manuscript received September 21, 2013; revised April 2, 2014; acceptedMay 29, 2014. This work was supported in part through instrumentation fundedby the National Science Foundation under Grant OCI0821527. Additionalsupport was provided by the National Science Foundation under Grants ECCS-1201973 and CNS1329422. The Associate Editor for this paper was A. Amditis.

E. I. Laftchiev and C. M. Lagoa are with the Department of ElectricalEngineering, The Pennsylvania State University, University Park, PA 16801USA (e-mail: [email protected]; [email protected]; [email protected]).

S. N. Brennan is with the Department of Nuclear and Mechanical Engi-neering, The Pennsylvania State University, University Park, PA 16801 USA(e-mail: [email protected]).

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TITS.2014.2330795

GPS, the U.S.’s GPS, the European Gallileo, and the RussianGLONASS, there are still geographic locations that will expe-rience insufficient coverage or multipath effects due to signalocclusions [3], [4]. These situations are sufficiently frequentlyencountered on roadways to necessitate the development ofalternate localization technologies.

To address these issues, manufacturers have focused ondeveloping systems that augment the GPS position estimate.The augmentation is typically performed using a combinationof vehicle sensors, road maps, and motion models. Regardlessof what type of augmentation is performed, each augmentationstrategy is dependent on vehicle sensors that provide informa-tion for the vehicle maps and models to estimate short-termposition changes during GPS dropouts. However, costs canincrease significantly when including additional sensors. Thisis particularly true as the field of research moves to visualodometry sensors such as cameras and LIDAR [5]–[8].

Sensors that have proven to be reliable for localizationand navigation include steering encoders, odometers, wheeltachometers, and inertial sensors. Published results have shownthat the first three are capable of accurately tracking a vehicle’slocation [9], [10]. In addition, recent work by Jo et al. [11]has demonstrated the fusion of these sensors with low-costGPS for effective localization. Inertial sensors in the form ofinertial measurement units (IMUs) have been previously usedin commercial and military-grade applications but suffer fromintegration errors that grow unbounded with respect to oper-ation time. Nonetheless, many IMUs are sufficiently accuratefor localization during brief dropouts [12].

Inertial measurements are particularly attractive when aug-menting GPS data because they provide a self-contained sourceof measurements that is not susceptible to the environment.This complements the attributes of the GPS and creates asystem more robust to environmental factors such as signalblocking and vehicle condition [1]. To address IMU outputdrift, the sensor output can be transformed into a set of featuresthat neutralize integrative error. These features are stored in areference map that is compared with new data during vehicletravel. An example of this is the work from Penn State that usesvehicle pitch to measure road grade, a disturbance measured inthe vertical IMU’s measurements. In this work, new featuresare correlated to the previously stored map [13]–[18]. Theuse of reference maps is the main limitation in using IMUsbecause it requires substantial computational power and storageto support the map-matching [19] infrastructure. Therefore,new approaches to the creation and structuring of referencemaps are necessary to improve the viability of using IMU data

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Fig. 1. Data collection process.

for localization when augmenting GPS, with map and featurerepresentation research being a key area of ongoing research.

To this end, this paper presents a dynamical model-based ap-proach to the problem of localizing a road vehicle using inertialmeasurement data. In particular, the method described hereinuses vehicle pitch data in a self-contained dead reckoning (DR)approach that does not require the addition of a GPS signal.Creating self-contained approaches is useful, even if used inconjunction with a GPS, because it creates both hardware andsoftware redundancy in the localization process and therebyincreases the reliability of the overall navigation system.

A. Problem Formulation

Previous localization work has addressed the implementationproblems by either setting the localization problem in theprobabilistic domain, using tools such as the Kalman filter,the Bayes filter, or the particle filter to obtain the locationestimates, or by evaluating map location using a comparisonmetric. In contrast, this paper and the previous work presentedby Laftchiev et al. [20] choose to frame localization as adeterministic problem by expressing the road data using linearmodels. Linear models address the problem of data compres-sion by expressing large portions of the data using a small num-ber of model coefficients. During localization, linear modelssimplify computations by eliminating the need for metrics tomatch incoming data. Instead, classically defined error boundscan be used to define data agreement. Relative to the featurematching approaches, this further reduces the complexity of thetransformations performed on the observed noise and simplifiesnoise mitigation algorithms.

Vehicle pitch, as recorded by the IMU, is the road gradefiltered by the wheelbase of the vehicle. The collection of roaddata is illustrated in Fig. 1. In tandem with this illustration,Fig. 2 shows a block diagram representation of the mappingof vehicle position into pitch. In these depictions, the variablev(t) represents the vehicle velocity, s(t) is the displacementof the vehicle, p(t) is the position of the vehicle, relative toits initial position p(to), and m(t) represents vehicle pitchthat is mapped from the vehicle’s position. Typically, duringlocalization, the variables v(t), s(t), and m(t) are available; andthe localization problem is reduced to identifying the positionp(t). The challenge of localization can be further reduced todetermining initial position constant p(to).

In this paper, the localization of the vehicle is accomplishedby a two-step process. First, a specially instrumented testvehicle collects position and pitch road data.1 These data are

1The instrumentation of the vehicle is described in detail in Section III-A.

Fig. 2. Sensor output diagram.

compressed and stored as a road map for use in another vehicle.The compression and storage of the data is performed byextracting linear models, which represent patterns from thedata that are robust to noise. When a different vehicle travelsalong the same road, the data observed by its sensors arecompared with the predetermined map. Regions of the mapare continuously eliminated until the correct vehicle locationis identified. This approach to localization is referred to as themap-matching approach.

1) Implementation Issues: Similar to other forms of sensor-based vehicle localization, localization based on vehicle pitchsuffers from several common implementation problems. First,the volume of data generated by the vehicle’s IMU over anentire roadway map is very large, requiring any localizationalgorithm to tackle the two-part task of efficiently storing theIMU data and then parsing through the data to localize thevehicle. Second, because of the large amount of data to be pro-cessed, the localization algorithm must be carefully chosen tominimize the in-vehicle computational requirements. Third, thesignal-to-noise ratio (SNR) of the IMU data is poor, requiringapproaches that are robust to noise and imposing an uncertaintyradius on the location estimates.

B. Contributions

This paper introduces linear models as a means of storingdata patterns. These linear models are structured in a tree-like fashion to simplify the online localization process. Duringlocalization, this paper proposes to eliminate the need for thecomputation of an agreement metric and instead compares eachmodel output to its modeling error bound. Agreement or dis-agreement with this bound determines the segment feasibility.Thus, the introduction to linear models is advantageous becauseit greatly reduces the complexity of both the mapping and thelocalization mechanisms while enabling the creation of simplernoise mitigation schemes.

While the development of this paper is focused on vehicle lo-calization, the applicability of the presented data modeling andsegment identification is broad. In particular, this paper con-tributes to the field of data mining, where large databases needto be efficiently represented for subsequence identification. Theidea to use linear models in database management is not new[19], [21], [22]. However, while most researchers believe thisfield to be mature, the simplicity of linear modeling lends itselfnaturally to large databases, streaming data, multidimensionaldata, and limited computational power in mobile technology.This paper contributes the following:

• Linear models of order greater than 2 that more faithfullydescribe the data. This preserves a greater portion of theinformation in the data.


LAFTCHIEV et al.: VEHICLE LOCALIZATION USING IN-VEHICLE PITCH DATA AND DYNAMICAL MODELS 3

• The elimination of a matching metric during the local-ization process, which reduces both the computationalcomplexity and the algorithm design uncertainty.

• An approach that can be easily extended to streaming dataand large data sets.

• An easily extendable approach for multidimensional data.These contributions are as meaningful in the data mining com-munity as they are here.

II. STATE OF RESEARCH

The goal of this paper is to introduce the approach of linear-model-based data compression and, in parallel, linear-model-based localization. The problem discussed herein is commonin the fields of data mining, pattern detection, and mobilerobotics/vehicle navigation. This section will review relevantwork in each of these fields. It is not the intent of this paper todevelop a localization algorithm that outcompetes all other pre-viously developed algorithms. Instead, the focus of this paper isto introduce the approach and to demonstrate its benefits in theprocess.

A. Sensor-Based Localization

The three sensors commonly used for DR localization arethe odometer, the wheel tachometer, and the IMU. A series ofreview papers can be found in [1], [9], and [23]. Perhaps, thesimplest method of navigation is odometry. To keep track ofodometry, manufacturers install a sensor that records a pulseeach time it passes by a specific point on the wheel. The numberof pulses is then multiplied by a scale factor that is relatedto the wheel diameter, tire pressure, temperature, and anyenvironmental factor that changes the wheel radius. Referencesciting the use of odometry and error-correcting techniques invehicle localization can be found in [9] and within the mobilerobotics community [24], [25].

A similar set of sensors is the wheel tachometers. Thesesensors are typically employed in antilock brake systems todetect the differences in wheel speeds. Using the same frame-work, the difference in wheel speeds can reveal the direction ofthe vehicle. Like the odometer, the critical piece of knowledgenecessary to make use of velocity encoders is wheel radius. In[10] and [26], velocity encoders are used in a dual GPS/DRsystem. Here, while GPS is available, the signal is used toprovide a heading and a wheel radius estimate. When GPS be-comes unavailable, wheel tachometers are calibrated to provideheading information.

Odometers and velocity encoders suffer from well-knownsources of terrain error, such as wheel slips, uneven roadsurfaces, and skidding. In addition, these same sources sufferfrom vehicle structure errors such as wheel diameter changes,wheelbase uncertainty, and low resolution of the encodingsensors. These errors have been observed and characterizedboth in the vehicle navigation community [9], [10] and themobile robotics community [24].

In contrast to odometry and tachometers, gyroscopes and ac-celerometers in IMUs measure rates of acceleration and rates ofchange in displacement of the vehicle. A typical IMU consists

of three gyroscopes and three accelerometers, each directedin one of the axes of motion. To obtain vehicle heading andvelocity, the IMU outputs are integrated in each direction. Thisintegration leads to the cumulative growth of small bias andnoise sources in the IMU as a function of the operation time.These noise sources will be further discussed in Section II-A2.However, because IMUs are self-contained and thereby notsusceptible to the vehicular and environmental noise sources,IMUs are a good complement to other vehicle sensors and GPS[14], [18]. For this reason, GPS and IMU data are frequentlyintegrated in augmented systems [1], [27], [28].

1) Map Matching: The DR localization approaches de-scribed above can be used to estimate vehicle location givena known starting point for the vehicle. However, if the locationis unknown, then the vehicle position can be estimated througheither triangulation using active beacons (the GPS is a form ofactive beacon) or map matching using a preexisting referencemap [19], [29]–[32]. A third alternative is under development,where building the reference map and localization are simulta-neously carried out [33]–[35].

The linear modeling approach to reference map creation andlocalization in this paper combines DR using inertial measure-ments and map matching. A state-of-the-art review of mapmatching is found in [30]. The authors subdivided the field ofmap matching into early techniques that take into account theroad geometry, such as arcs and lines; topological approaches,which take into account road geometry and the interconnectionbetween each geometric feature; probabilistic approaches thatassign a region of error within which a likely road segmentis found; and more recent approaches termed advanced mapmatching using Kalman filters, fuzzy logic, particle filters, etc.The references therein provide an in-depth look of the field ofmap matching.

As a general outline, the map-matching process begins byextracting or observing features from the data [36]. The choiceof features during the extraction process has a pivotal rolein the subsequent localization performance of the algorithm.Early algorithms use arcs and lines as features that modeledthe road shape [19], [37]. In advanced algorithms, the featuresare extracted through a nonlinear transform that is used toidentify the most noise-robust portions of the data. A fewexamples are local extrema, geometric features such as arcsand lines, or geometric beacons. Further examples can be foundin the literature on pattern matching and data mining [13],[38]–[41].

Then, during localization, the features are reextracted fromnew vehicle data. The location is determined by comparingthese features to the map. The most common comparisonapproach is the threshold. Here, a distance is computed from theroad data to each possible map location. An empirical thresh-old is used to determine the correct segment. More advancedmethods of matching include those presented in the roboticscommunity where the probability of the vehicle’s location andof the detected features is calculated using Kalman filteringin a formal framework methodology established in [42]–[44],or a Bayesian comparison approach such as the one usedby Levinson and Thrun [45], or a particle filtering approachin [46].



The motivation to develop advanced map-matching algo-rithms stems from the need to handle increasingly complexroad networks that easily overwhelm earlier algorithms. Un-fortunately, to the best of our knowledge, there does not exista unifying publication that elucidates the limitations of eachmap-matching approach in the vehicle navigation community.In contrast, several review papers in the data mining communityhave concluded that more work is necessary to design efficientalgorithms to handle large databases [47].

Choosing the features and the extraction process plays apivotal role in the subsequent performance of the algorithm.Previous research in our own group has focused on IMU pitchdata for localization [13]–[15], [17], [18], [48]. In particular,Dean [48] demonstrated a particle-filter-based approach usinginertial measurements. A more noise-robust approach was pre-sented in [15], where an optimal filter was derived to processthe data prior to extracting extremum features. Combiningprevious work, the work by Kadetotad et al. [17] employedboth the particle filtering and feature-based approaches to allowlocalization and tracking of a vehicle over a greater area.

2) Noise Characterization of the IMUs: Recent decreasesin the cost of microelectromechanical systems (MEMS) IMUshave contributed to a rise in the number of publications citingaugmented GPS/IMU systems [14], [18], [49], [50]. Given thisdevelopment and the potential for further research in the future,it is important to understand how sensor limitations affectthe localization approach. In particular, El-Sheimy et al. [51]characterized IMU noise in terms of Allan variance and powerspectral densities. The same authors then offered approachesto mitigating IMU noise [52]. A MEMS-specific analysis isdemonstrated by Aydemir and Saranli [53], whereas the workby Jerath and Brennan [16] evaluated the noise performance ofseveral grades of IMUs.

Extensive research has been also presented on the calibrationof IMUs [54]–[57]. An in-depth discussion of IMU noisecomponents is shown in Section IV-C. The described noise isthen used to corrupt the IMU data for localization simulations.

B. Applications of Sets of Linear Systems

In addition to applicability in vehicle localization, the refer-ence map creation algorithm can be thought of as a databaseconstruction algorithm whose goal is to rapidly and efficientlyparse data of many kinds for easy retrieval. For a single seriesof data described by a sequence of models, this reference mapis also an application of switched linear systems, which arefrequently used to describe nonlinear data. If the models arenot in a sequence, they can be thought of as a set of modelssimilar to those used in texture classification and tracking [21],[58]–[62], shape detection [63], fault detection [64], acousticfiltering [65]–[67], and fingerprint identification [68].

This section will discuss texture detection because it providesthe closest analog to identifying vehicle pitch during vehicletravel. Texture detection is subdivided into dynamic texturedetection, i.e., the ability to track an evolving texture acrossseveral frames, and single-frame detection, i.e., the ability tofind a texture in one frame [69]–[74]. Using dynamical modelsin these applications is advantageous because many textures

exhibit linear dynamical behavior and allow the use of well-developed tools from the control system literature [59].

The general approach when using linear dynamical systemsfor texture detection is to extract a set of linear models fromtraining data that represent the desired textures. Some authorseven employ switched linear systems to obtain this referenceset [73], [75]. Then, during detection, new models are extractedfrom testing data. The new models are compared with the refer-ence set of models through the use of a metric, typically in thespace of linear dynamical systems [74]. A common metric is theLp norm, but a second notable metric is the Martin metric [76].This metric measures distance in the cepstral domain, wherethe cepstrum of the models is an additive function of the polesand zeros of the system. Other techniques include robust modelvalidation techniques [59] and optimization techniques, whichdetermine model feasibility in the presence of uncertainty [71].

C. Previous Work in Data Mining

Finally, the problem of vehicle localization using IMU datais analogous to the problem of subsequence identification inthe data mining community [77]. Subsequence identificationis the identification of a newly collected subset of data froma previously stored database. Because the volume of datacollected for each database is large, typically, the data are notcompared on a one-to-one basis but instead are transformed intorepresentations to provide an easier means of comparison andstorage. Published representations include the discrete Fouriertransform coefficients [78], segment averages [79], first-orderlinear regressions [22], wavelet transformations [80], principalcomponent analysis and singular value decomposition [81], andsymbols [82]. When comparing the stored representations withthe newly acquired or user-specified query, a metric is used todetermine agreement. Some of the more popular metrics in-clude Lp norms [83], dynamic time warping [84], thresholdingqueries [85], and the longest common distance [86].

The most recent publication surveying the field of datamining by Ding et al. [47] has compared the published databaserepresentations and argued that using a lowest bounding dis-tance for the representations reveals the ability to accurately de-tect subsequences. The lowest bounding distance is a Euclideandistance metric that describes the distance between the twoclosest representations in a database. In the paper, Ding et al.argued that the published representations are almost identicalin their performance. The authors also demonstrated that, fora growing database, the proposed metrics above performed aswell as the Euclidean metric. Due to the relative simplicity ofthe Euclidean metric, this metric is preferred during implemen-tations that require computational efficiency.

III. ALGORITHM DESCRIPTION

A. Road Map Model Extraction

The first step in the proposed approach is to collect roadpitch data using a specially instrumented vehicle. This vehiclehas been equipped with the Honeywell HG1700 IMU, whichis mounted to the console near the vehicle’s center of gravitybetween the driver and passenger seats. The vehicle pitch



TABLE IOPTIMAL GREEDY ALGORITHM

data are estimated using both the longitudinal gyroscope andlongitudinal accelerometer readings and are internally filteredin the IMU using a factory-integrated Kalman filter that reducesthe effects of IMU noise.

The data are then modeled using a set of autoregressivemodels with an exogenous input (ARX) [87], which describesnonoverlapping segments of the data. ARX models are partic-ularly suitable for this application because the model outputis a prediction of the incoming data. In this application, theexogenous input is the error between the estimate and thecollected data; it is assumed that this error is bounded by abound ε. The general form of an ARX model is

m[d] = a1m[d− 1] + · · ·+ aNm[d−N ] + ε[d]

=Ax[d] + ε[d], |ε[d]| ≤ ε (1)

where d is the data index, m[d] represents the current sample ofdata, A = [a1, . . . , aN ] is a vector that contains the coefficientsof the linear model of order N , the vector x[d] = [m[d− 1],. . . ,m[d−N ]] contains the previous N samples of data, i.e.,the so-called regressor vector, and ε[d] is the model output errorbounded by the aforementioned bound ε.

Models are extracted using a greedy algorithm developed in[88]. This algorithm simultaneously obtains the ARX modelcoefficients and breaks the road data vector into nonoverlappingsegments. The algorithm is shown in Table I.

The algorithm begins at the (N + 1)th data point, whichis labeled d0. Starting at this initial data point, the algorithmsearches for the largest interval for which it is possible toobtain a single ARX system that satisfies the error bound forevery point.2 Once this is not possible, a transition is declared,indicating the end of the model fit and start of the next model;the corresponding data index is labeled τo. Henceforth, τo iscalled a transition point, and it is the point at which one modelsegment ends and the next begins. The model-fitted segment isremoved, and this process is repeated until the final data point(dmax) is reached. The resulting set of data segments spansthe values of consecutive transition points: (d0, τ0), (τ0, τ1),

2The search itself is performed solving a linear feasibility problem.

Fig. 3. Model fit demonstration.

etc.3 The optimality of this algorithm is described below inProposition I. This proposition was proven in [88].

Proposition I: Given a bound on the error, i.e., ε, and amodel of order N , the algorithm described in Table I breaksthe collected data set into the smallest possible number ofsegments.

The segmentation of the data set is illustrated in Fig. 3. Inthis figure, the horizontal axis represents the distance in metersfrom the initial map position. The vertical axis represents theangular pitch rate output of the IMU. In each plot, the solidline represents the query data, the solid line with circle markersdisplays the model output, and the dashed lines represent theerror bounds. In each case, the model is only valid for a portionof the plotted data. For example, the first plot shows a modelvalid from 300 to 500 m. Then, the next plot shows the modelthat belongs to the segment from 500 to 600 m. Finally, the thirdplot shows a model that is valid for the remaining distance. Notethat each model is within its error bound for the given segment.Outside of this segment, the model may or may not agree withits bound. It is the unique sequence of these models, even if theythemselves are not unique, that will be used in the localizationalgorithm development.

B. Locating a Vehicle Using the Extracted Models

Vehicle localization can be subdivided into two distinctproblems: locating the vehicle without a priori knowledge ofthe start location and tracking a vehicle given the knowledgeof its initial position. The set of ARX models extracted in theprevious section can be used to perform both functions. At first,segments are continuously eliminated until a single feasiblesegment remains. This segment represents a possible set ofvehicle locations. Then, when a transition between models isfound, the precise location of the vehicle is detected. From thispoint forward, the vehicle’s location can be tracked through val-idation of the current segment and detection of future transitionpoints. The iterative process by which this is accomplished isdescribed in Table II. In this algorithm, the variable L is setto one.

3The model coefficients that correspond to each segment are not unique sincemultiple sets of coefficients may result in a model that satisfies the error bound.



TABLE IIVEHICLE LOCALIZATION ALGORITHM FOR MODEL STRUCTURES

Similar to the greedy algorithm in Table I, the localizationprocedure in Table II begins with the collection of N + 1pitch data points. The data index is set to d = N + 1, and theacquired data are sequentially tested in each of the extractedlinear models, denoted by the index n. The output error ofthe models is compared with their error bound, i.e., εn. Moreprecisely, this comparison is shown in (2).

If, for a given set of data, the model error is smaller thanthe bound, then the current segment is labeled as a possible,or a feasible, set of vehicle positions; otherwise, the segmentis labeled as infeasible. At each time step, only segments thatwere previously labeled as feasible are tested. Thus, under someassumptions to be discussed below, the number of feasible seg-ments monotonically decreases to one after some iterations, i.e.,

|m[d]−Anx[d]| ≷ εn. (2)

Two critical assumptions must hold to facilitate the localiza-tion of the vehicle. First, the pitch map must not be periodicto within an epsilon bound equivalent to the model error. Forreal-world pitch maps that have been measured to date by theauthors, roughly 10 000 km or 6000 mi, this assumption is quitevalid. Second, the vehicle must travel a distance sufficientlygreat such that there exists a unique observed sequence of

models from among all possible sequences of models on themap. The length of necessary travel distance is dependent onthe starting location of the vehicle on the map.

As an illustrative example, consider a vehicle that beginstraveling on a path that contains two hills and, otherwise, flatroad in between and around the hills. Suppose that the hills aresimilar in their pitch profiles. Then, no single segment of themap alone is sufficiently unique to identify the location of thevehicle. For a small travel distance, the algorithm would haveat least two possible feasible paths during localization. In thisexample, travel distances smaller than the distance between thehill peaks would result in this localization ambiguity.

However, if the vehicle travel is extended to at least thedistance between peaks of the hills, then there is only oneunique profile that can be identified. The unique profile in thiscase is the flat road ahead of the vehicle followed by two hillsand then flat road again. Similar to this example, each time avehicle begins travel, there exists some distance within which aunique localization sequence of models can be found.

1) Model Transitions: In the vehicle is traveling inside ofa model segment and a data point is collected that does notagree with the model, then a transition test is performed.Transition points contain more information with regard to thevehicle location because they must satisfy the following threeinequalities:

|m[d− 1]−Anx[d− 1]| ≤ εn|m[d]−Anx[d]| >εn

|m[d]−An+1x[d]| ≤ εn+1. (3)

The first inequality shows that the vehicle was in segment nat the data point immediately preceding point d. The nextinequality shows that the vehicle is no longer in segment n,and the third inequality shows that the vehicle is now insegment n+ 1. The latter two inequalities have domains thatonly overlap at the transition point and therefore provide strongevidence about the location of the vehicle.

When segmenting the model map, the error associated withthe first model is also recorded. This provides an additionalpoint of reference when determining the validity of a transition.Thus, transition points contain information about the vehi-cle’s location that is significantly larger than ordinary points.Segmenting a map with as many transition points as possibleimproves the localization speed and increases the robustness ofthe map to noise.

C. Tracking a Vehicle Using the Extracted Models

Once the vehicle’s starting point has been identified, theset of models at the bottom of the model tree, or a switchedlinear system, can be used to track the vehicle’s progress. Thisis a critical function of the algorithm because it provides amethod of detecting changes in direction, sudden maneuvers,etc. Tracking is performed in a similar manner to localization,but only a single feasible segment is tested at a time. While thevehicle is located in a segment, its location can be verified usingthe simpler test shown in (2), and its position on the map can beupdated using odometry readings. This process continues untila data point that does not agree with the model is found.



The data point that does not agree with the current modelis assumed to be a transition point and is evaluated using (3).Additionally, the vehicle location is verified by comparing thedistance since the last transition point to the previous segmentlength. If the data satisfy both checks, then the models areiterated, and the process is repeated. Otherwise, the initiallocalization is assumed to be erroneous. In this case, thevehicle localization loop is repeated, and a new location isselected.

In general, once a correct transition point is located, onlylarge changes in the road surface or unexpected maneuverswill lead to errors in the tracking phase of the algorithm.For example, if the driver of the vehicle executes a suddenbraking or accelerating maneuver, large oscillations will beintroduced in the data. This will lead to a rapid eliminationof the feasible models. When this occurs, the tracking codewill restart all levels of the model tree that have no feasiblemodels. Practically, this means that relocalization is occurringin a neighborhood of the last known vehicle location, ignoringthe data produced during the unexpected maneuver.

IV. PRACTICAL CONSIDERATIONS

A. Advantages of Linear-Model-Based Localization

The dynamical models extracted from the vehicle pitch datarepresent a mapping of the underlying road surface. It is notpossible to claim that these models are uniquely correlated toa location on a global scale; indeed, there are many situationswhere road profiles in one location are quite similar to roadprofiles in a very different location. However, as discussed inthe previous section, given a sufficient travel distance, a uniquesequence of models can be found for localization; in otherwords, two roads may be similar to each other for a small modelsegment, but they will not be similar to each other over manysegments in a series.

Using dynamical models offers several advantages. First,given a sufficiently unique map and a noise-free environment,the vehicle can be accurately localized for every experiment.While this localization may be to within a neighborhood ofthe correct position, it is not a probabilistic neighborhood, butrather a deterministic range determined by collection sensorcharacteristics. Second, in contrast to previous approaches tolocalization, subsequence identification, and pattern recogni-tion, the approach presented here does not require a metric formatching. Instead, the modeling output error is compared withthe modeling error bound, i.e., (2), which is determined for eachmodel during the map creation. This simultaneously reducesthe need for a metric function, calculating the metric function,and empirically determining a threshold for the metric output.Third, following above, the algorithm is most computationallyexpensive during the extraction of the model map. This isadvantageous because the largest computational constraints arein-vehicle, where computational power is constrained by costand energy efficiency. Finally, the linear treatment of data al-lows for the implementation of simpler noise mitigation strate-gies. This again reduces the computational power necessaryduring vehicle travel.

Fig. 4. Example model structure.

B. Computational Burden

Because computational power of in-vehicle computers limitsthe usefulness of most self-contained localization strategies,it is critical to design algorithms that are efficient in theironline execution [1]. In particular, the research community inmobile robotics has invested significant efforts in the real-timeimplementations of the simultaneous localization and mapping(SLAM) algorithms [46], [89]–[91]. Here, the computationalcost of the presented algorithm is compared with the parti-cle filtering approach in [48]. This comparison is particularlyrelevant because the particle filtering approach is similar tothe prevalent approach used to solve the SLAM problem, andsecond, the approach in [48] utilized the same database as thework presented here.

The measure used to assess computation cost in [48] is thenumber of floating point operations (FLOPs) required duringthe solution of the algorithm. The number of FLOPs requiredfor particle filter convergence is dependent on the number ofparticles used per mile. For 1000 particles/mi, the particle filterapproach required 37 009 FLOPs/mi for each iteration duringthe localization process. Given this number of operations, thein-vehicle computer was only able to scan several miles of theroad map during localization.

For the model-based localization approach in this paper, anadequate vehicle map covering several miles would requiremany thousands of models. This is because, for a map to beconsidered adequate, it must cover a region large enough tospan the uncertainty in vehicle position. The size of this mapwould result in a very large initial computational burden.

One way to reduce the initial computational burden is tocreate a tiered tree structure. At the top level of the structure,there are “coarse” models, which describe large segments ofthe map. At the bottom level of the structure, there are “fine”models, which describe small sections of the map. Fig. 4 showsan example of this type of model structure. The models aredenoted by Ak,n, where k is the level in the model structure,and n is the segment index on the kth level. Each modelstructure will have L levels and Mk models per level. At eachsuccessive model level, the ε region is contracted so that eachsegment from the previous level is both bisected and modeledwith a tighter error bound. This leads to an increase in thenumber of segments for each consecutive level.

The model structure extracted for this paper uses bisection atsuccessive levels, i.e., at each consecutive level, the segmentsfrom the previous level are bisected. Practically, bisection isadvantageous because eliminating large segments at the topof the model structure eliminates a large number of models atlower levels and reduces the computational cost. Additionally,there is little difference between finding a single transition ormultiple transition points at a time. This is because there exist



Fig. 5. Example FLOPs count.

naturally occurring transition points between different models,which result from changes in the data. These transition pointsoccur for all segmentation types and consequently result inequivalent overall performance.

The localization process discussed in Section III-B can bealso extended to the model structure. In short, the feasiblemodels on each level of the structure are sequentially tested.In this case, a model is considered feasible when both themodel and its parent have not been invalidated. For example,testing begins at the top level with a single model that describesthe map. Then, at the second level, two segments are tested.Because the first segment describes the entire map, all segmentson level two begin as feasible. Then, on the third level, only seg-ments that were described by a feasible model in the previouslevel are tested. This process is iterated to the bottom of themodel structure. At each successive level, the models becomemore precise because the error bound ε becomes tighter and,therefore, a greater number of segments can be eliminated. Thelocalization procedure is described by the algorithm in Table II.

Fig. 5 demonstrates the reduction of the computational re-quirements with respect to distance traveled. In this figure,the vertical axis represents the number of FLOPs performedper iteration, and the horizontal axis represents the vehicle’stravel distance in meters. The sample point spacing duringvehicle travel is 0.5 m, which is a decimation similar to thatin [48]. Note that, to process the whole map, the initial numberof FLOPs is 36 000, but within several meters, the majorityof the models in the structure have been eliminated, and thesteady-state number of FLOPs per iteration is around 9600 foreach incremental map query. The periodic spikes that occur insteady state are transition points that require a larger number ofoperations to verify.

The results presented in this figure are typical for this map butcan vary with respect to the starting point of the simulation. As apoint of comparison with [48], note that testing was performedon a 6-km (3.73-mi) stretch of road. At the time, 6 km wasthe largest stretch of road that was tractable to store and parsein a vehicle using the particle filtering approach. Comparingthis same map, the algorithm presented in this paper required9600 FLOPs per mile and per iteration. After convergence, thealgorithm requires 2574 FLOPs per mile and per iteration. Inthis case, we note a fourfold reduction in FLOPs. The steady-

state number of FLOPs required can be further reduced byoptimizing the code and using model similarities. Therefore,the size of the map that can be computed with this approach ismany times larger than the map that is feasible using a particlefilter. It is important to note that the approach presented hereis extensible and capable of accommodating any data set size.The main limitations are the in-vehicle computation power andstorage capacity, both of which are already within the domainof present desktop computing.

C. Measurement Noise and Pitch Profile Variance

There are two significant sources of error that affect changesin the road pitch profile that may occur between road traversals:sensor noise and changes in the road pitch profile. The latteris a significant problem because the observed pitch profileis dependent on environmental and mechanical circumstancesthat surround the vehicle. Even for the same vehicle, differentcharacteristics can alter the observed road profile. These char-acteristics include the number of passengers, the weight in thetrunk, the level of fill in the fuel tank, the state of the systemshock absorbers, the amount of wind resistance, the pressurein the tires, and even pot holes. This could be interpreted asa variance in the pitch profile of the road. However, repeatedmeasurements of actual roads show that the road grade profileis invariant or at least can be bounded by a fixed measurementerror. Because the goal of this paper is to demonstrate theapproach of compressing data using linear models and thenefficiently parsing through these data online, the pitch profileis assumed to be invariant and repeatable with every vehiclerun. The remainder of this section discusses IMU noise.

1) IMU Characterization: An IMU contains three gyro-scopes that provide an angular rate measurement for each ofthe 3-D coordinate axes and three accelerometers that outputvelocity rate information. Although the averaged rate measure-ments are accurate over time, they are integrated with respectto time to obtain orientation, position, and velocity estimates.This integration leads to the accumulation and propagation ofsmall errors.

IMU noise is characterized by the manufacturers in terms ofAllan variance of the angular rate output. The primary compo-nents of this noise are angle random walk noise and bias noise.Angle random walk and bias are added to the angular pitchrate before it is integrated to obtain the orientation, position,and velocity estimates. Thus, even small errors are indefinitelypropagated through the integration process.

There is a wide range of commercially available IMU sensorsthat each exhibit varying noise characteristics. Fig. 6 uses thecode developed in [16] to demonstrate the noise characteristicsfor a noisy IMU, ADIS16367; a mid-grade IMU, Crossbow440; and a low-noise IMU, Honeywell HG1700. The top plotshows the angle random walk component of the IMU noise, andthe bottom plot shows the bias noise component. In both plots,the horizontal axis represents vehicle travel distance in meters.

Note that the larger component of noise is the angle randomwalk noise. The angle random walk noise can be describedas white noise added to the angle rate measurements of thegyroscopes prior to integration. The standard deviation of the



Fig. 6. IMU noise by sensor price. (Top) Angle random walk noise. (Bottom)Bias noise.

TABLE IIISNR GIVEN SAMPLING FREQUENCY AND SENSOR CHOICE

white noise is specified by the manufacturer for a 1-Hz samplerate. The specification at 1 Hz means that the variance of noiseis scaled by the sampling frequency at the output of the IMU.Table III shows the maximum and average SNRs using themanufacturer specifications and the test data for this paper. SNRis calculated using (4) [92], where σm is the standard deviationof the map data, σn is the standard deviation of white noiseprovided by the IMU manufacturer, and fs is the sampling rateat the IMU output. Thus

SNR = 20 log10

(σm√fsσn

). (4)

There is an important observation that can be made fromthe preceding discussion; as the IMU sampling frequency in-creases, the average SNR monotonically decreases. Becausethe test map data have been resampled to place equal spacingbetween samples, the effect of noise on vehicle localization canbe reformulated as a question of localization resolution. Forexample, a 100-Hz sample rate corresponds to a 0.05-m mapdecimation, whereas a 1-Hz sample rate corresponds to 5 mbetween samples. According to (4), reducing the sample ratefrom 100 to 1 Hz improves the SNR by a factor of 2. Thus,the sampling frequency can be decreased to improve SNR andalgorithm performance. This can be done for low-cost sensors,provided a coarse map resolution is acceptable.

2) Bias Noise: The first noise component to be addressed isbias noise. The bottom plot in Fig. 6 shows that bias is relativelyconstant in neighboring data points. Such a low-frequency or dcsignal can be rejected through the use of the signal derivative.Therefore, assuming that bias remains constant in neighboringpoints, a map of differences can be generated, where each pointis the difference between two neighboring pitch values. This isillustrated by

Δm[d] = m[d] + β[d]−m[d+ 1]− β[d+ 1] (5)

Fig. 7. (Top) Pitch values versus difference map. (Bottom) Difference map.

Fig. 8. Approximate data collection route in State College, Pennsylvania,PA, USA.

where m[d] and m[d+ 1] are adjacent pitch values with biasβ[d] and β[d+ 1], respectively. Because bias is relatively con-stant, β[d] ≈ β[d+ 1], the variable Δm[·] is insensitive tobias noise. Hence, Δm[·] can be used during the extractionof the road map models and structure and during the onlinelocalization to mitigate the effects of bias noise. The two datasets, i.e., pitch data and angular pitch rate data, are illustratedin Fig. 7. To provide context for the data, the approximateroute along which the data were collected is included in Fig. 8.This figure demonstrates that the test set of data included bothmeasurements from highways and arterial roads.

In Fig. 7, the horizontal axis represents the distance oftravel in meters. The top half of the plot shows the pitch dataprofile used in this paper. The bottom half of the plot showsthe difference map profile of the data. These plots clearlyshow that any bias integrated into the pitch signal is greatlyreduced in the difference map and is very small when comparedwith the terrain signal size. In particular, when using the biascharacteristics of the mid-grade sensor, the residual bias termswere found to be, on average, four orders of magnitude smallerthan the terrain signal size.

3) Angle Random Walk Noise: The second mitigation strat-egy addresses angle random walk noise. This noise is modeledas white noise added to the angular rate output of the IMU.



Otherwise stated, angle random walk noise adds small pertur-bations to each acquired data point. These perturbations cannotbe eliminated by subtraction such as the bias noise above. Analternative mitigation strategy is to introduce tolerance for eachdata point.

Assume that the perturbations can be bounded by some ηB .Let each data point have a perturbation |η[d]| ≤ ηB . Then, amodel is said to be feasible or agrees with the data, containedin vector x[d], if a set of constants η̄ = [η[d], . . . , η[d+N − 1]]can be found such that

|Δm[d]−A (x[d] + η̄)| ≤ ε. (6)

The choice of the bound ηB is simplified because we knowfrom the manufacturer that the added noise can be modeled aswhite Gaussian noise with a standard deviation σ specified inthe IMU datasheet at a 1-Hz sampling rate. Thus, the bound ηBis set to two standard deviations of the additive white Gaussiannoise for the particular sensor and sampling frequency.

One important observation is that, because pitch is integratedfrom angular pitch rate, the perturbations at each pitch datapoint are not independent. Testing a large horizon of data pointssimultaneously allows for the greatest ability to detect andmitigate the correlation between the points. Practically, thisscales the computations of the model validation step, and it istherefore not feasible to test more than a sequential few pointsat a time.

D. Simulation Setup

When setting up the numerical experiments, the two criticalparameters are model order N and error bound ε. Unfortu-nately, there do not exist optimal values for either parameter.Instead, these parameters can be varied with respect to oneanother to create different model structures with similar local-ization properties. For this paper, the model order was heldconstant at five, and the error bound was allowed to vary foreach model.

The initial value for the error bound is 0.2556◦ of pitch. Thisis the smallest value of ε such that, at the top of the modelstructure, a five-coefficient model can be used to describe theentire data segment. For each successive level, the error boundwas contracted by a factor of 0.85.

V. NUMERICAL RESULTS

A. Localization in the Noise-Free Case

In the ideal case, the traveling vehicle records noise-freepitch measurements from an invariant pitch profile. In this idealcase, given all available data, the algorithm presented in this pa-per will always converge to the correct location. Because of this,the presentation here is focused on illustrating the underlyingmethod and demonstrating that the algorithm is coded correctly,rather than analyzing the localization algorithm’s convergenceproperties in the presence of noise.

Fig. 9 demonstrates one approach to illustrating the algo-rithm’s convergence. This figure shows the convergence of

Fig. 9. Feasible segments versus iteration number.

100 trials using random start points in a noise-free environment.The horizontal axis in the plot shows the distance traveled bythe vehicle in meters. The vertical axis shows the number oftrials that have converged. Note here that more than 90% ofthe trials have converged by 25 m of travel distance. All trialshave converged within 60 m of travel, which corresponds tothe length of the largest segment on the lowest level of themodel structure. Thus, for the map used in this work, in a noise-free case, the maximum travel distance to convergence can bebounded by the length of this segment.

The variation in convergence distance can be also explainedusing transition points. Because these points provide a greateramount of information for convergence, the intersection ofmultiple transition points on the vehicle path leads to shorterconvergence distances. This can be observed on the plot where amajority of segments are eliminated within 25 m. The paths thatrequire longer convergence distances lie in more informationsparse regions of the map and need a longer time to acquire theinformation-rich points needed to localize the vehicle.

The figure underscores two of the fundamental advantagesof the algorithm developed in this paper. The first is that thereexists an upper bound on the convergence time of localization.The second is that the convergence distance can be predictedfrom the observation of models along the vehicle route.

The processes of segment elimination and transition pointidentification are illustrated in Figs. 10 and 11. In these figures,the horizontal axes show the distance traveled from the maporigin. The top plots show the model output plotted against thereference map data, with the error bounds ε plotted in red dottedlines. The vertical axis in these plots represents the angle rate. Inthe bottom plots, the model error, in degrees, is plotted againstthe distance traveled in meters. In this case, the vertical axisrepresents the model errors.

Suppose that the vehicle begins traveling at 30 m on themap. Then, comparing the output of model 1 in Fig. 10 withits error bound shows that the segment is a feasible set ofvehicle locations. While the vehicle is traveling, all modeloutput errors remain below the modeling bound ε. However,when the vehicle reaches the end of the segment at 110 m,the output error is significantly higher than the bound ε. Thisoutput error is compared with the known transition point error,which was recorded during model identification, for Model 1,



Fig. 10. (Top) Map data versus Model 1 output. (Bottom) Model error.

Fig. 11. (Top) Map data versus Model 2 output. (Bottom) Model error.

and the input data are also compared with Model 2. The math-ematical description of a transition point is described in (3).If all three equations are simultaneously valid and the errormatches the known transition point error, then the point at110 m is identified as a transition point, and Model 2 becomesthe feasible vehicle location model.

The comparison of errors is performed for all feasible seg-ments each time a new data point is collected. Because only fea-sible segments are evaluated and because only segments whoseparents are feasible are reactivated, the number of feasiblesegments monotonically decreases as the number of iterationsrises. Thus, the number of feasible segments in Fig. 9 convergesto one as the number of iterations increases.

B. Localization Using Noisy Data

To test the algorithm in the presence of noise, the map dataused to create the reference map are corrupted with the noisecharacteristics corresponding to the Gaussian random walkcharacteristics of the Crossbow 440 IMU. The corrupted dataare then used for the localization experiments here. The mapdata are plotted along with the road pitch profile in Fig. 7, andthe approximate real-world map location is shown in Fig. 8. Ata 1-Hz sampling rate, or a 5-m map decimation, the Crossbowsensor has noise characteristics that are an order of magnitudehigher than the collection sensor but sufficiently low to allow

Fig. 12. Distance traveled until one feasible path using Crossbow 440 IMU.

Fig. 13. Convergence distance and error versus start point.

for accurate localization. The algorithm performance is evalu-ated over a 2-km travel distance from a random start point thatis uniformly chosen.

Fig. 12 shows a histogram of the vehicle travel distancerequired for algorithm convergence in over 1100 trials witha random start point. As before, the convergence to a singlepath corresponds to the first estimate of vehicle location. Thehorizontal axis in the figure represents the vehicle travel inmeters, and the vertical axis represents the number of trials.The mean convergence time for this set of trials is 172 m.Furthermore, the maximum convergence distance is about280 m of travel.

For trials in the presence of noise, the plot of localiza-tion errors [m] versus the trial start points [m] in Fig. 13helps to illustrate the algorithm’s performance across the map.For example, note that the localization distance is different inthe beginning of the map when compared with the end of themap. This difference comes from the physical nature of theroads from which the data were collected. In this case, the roadvehicle began traveling on a secondary road and merged ontoa highway. Thus, the beginning half of the map contains datawhose variance is high, and the latter portion of the mapcontains highway data with low variance. In the presence ofhigh-variance data, models have larger modeling error boundsand are eliminated at a slower rate. This fact is illustrated in



the convergence error, which is highest in the beginning of themap. Examining the convergence errors more closely revealsthat, when errors are measured in terms of map decimations(5 m), all errors fall within a few decimations.

The accuracy achieved in these simulations is comparablewith the findings of Dean [48], who found that the limitingfactor in his particle filtering approach was the map decimation.The results also favorably compare with the accuracy foundin recent map-based localization work [31], [32], which hadlocalization accuracies of about 1 and 15 m, respectively. While[31] and [32] did not use IMU data for localization, the researchpresented in these papers is map-based localization researchusing real-world noisy sensors. These references show that thework presented here is at least as accurate as the state of theart in localization, with the additional benefits discussed in thispaper. A strength of the approach presented here is that local-ization is a function of the map decimation, as pointed out inSection IV-C1. In particular, map decimation is proportional tothe size of the observed sensor noise, and thus, choosing higherfidelity sensors will allow finer resolutions and correspondinglysmaller localization errors.

Overall, the algorithm performs well using the noise charac-teristics of the Crossbow 440 sensor. It is possible to address thesource of error and decrease convergence distance by adjustingthe mechanism by which the likely vehicle path is selectedamong the remaining paths. This promising area of research isleft to future work.

VI. DISCUSSION OF ALGORITHM LIMITATIONS

The preceding sections tested the terrain-based localizationalgorithm for an invariant pitch profile with additive sensornoise. As shown in Fig. 13, the localization results using thisalgorithm are good for the mid-range IMU sensor. On average,the error observed in the location estimates is on the order ofthe map decimation.

There are two principal sources of error in the localiza-tion process: survival of too many paths in the presence ofnoise and erroneous segment elimination. The first is a naturalconsequence of using terrain data and a bounded localizationdistance in the demonstration of this algorithm. Terrain dataare unique on a local scale but not on a global scale. Thus,when the localization algorithm is working with an area of themap that has low SNR and does not display strong uniquenessin features, multiple models survive. For example, localizationmight be difficult in the Great Plains of the United States,where the terrain is largely flat, resulting in fewer road features.Conversely, localization might be more rapid in the RockyMountains (United States), where the topography is quicklychanging and offering a large number of unique terrain patterns.

Here, it is important to note that reference maps are madeup of patterns that are used for localization. These includeboth the geographical effects and the road construction effects(e.g., the sequence of concrete or asphalt pours). Thus, evenin flat terrains, there may still remain a surprising number oflocalization features due to location-specific construction ofthe road.

This paper showed a localization experiment for a set roaddistance. Making the road distance finite was chosen to makethe experiments tractable. If multiple models survived, a heuris-tic based on the number of detected transitions was invoked tochoose the correct path. This is not an optimal choice sincenoise can also trigger false transitions. In order to eliminateheuristic errors, the solution is to extend the run of eachexperiment until it is localized. However, the results above showgood performance for a fixed localization distance.

Conversely, the presence of noise can also lead to the erro-neous elimination of segments. For instance, the angle randomwalk mitigation approach presented in this paper bounds thepossible noise coefficients to two times the standard deviationof the angle random walk noise. This means that, in approx-imately 5% of the cases, it is expected that the noise willexceed the limitation and the correct segment will be elimi-nated. As travel distance is indefinitely increased, these types ofeliminations are corrected by segments in higher levels of thetree with larger error tolerances. However, for a fixed distance,the end result may appear erroneous because of one of theseeliminations. In this case, the algorithm can be reinitialized andrestarted.

VII. CONCLUSION

This paper has presented two linear model-based algorithmsfor localization. The first algorithm is used to compress a 1-Dvector of data and to represent these data as a tree of modelswith increasingly tighter modeling error bounds. The secondalgorithm is a parsing algorithm that can quickly locate anincoming stream of data in the extracted model tree. Both algo-rithms were developed and demonstrated for the application ofvehicle localization. In the simulation, the algorithms had lessthan 1% localization error, provided that the underlying dataare invariant and their noise characteristics did not exceed themodeling error bound.

The demonstrated algorithms have several advantages whencompared with existing localization work. First, the utilizationof linear models allows for the representation of large sectionsof the data using a few coefficients. Second, using linear mod-els reduces the computational requirements when comparedwith previously developed localization approaches. In additionto the relative simplicity of the models, this is also due tothe elimination of the need for a metric during localization.Third, performing linear operations on incoming data reducesthe complexity of noise mitigation algorithms because theresulting noise distribution is the result of scaling and addingnoise samples. Finally, the presented algorithms are scalableand are well suited to multidimensional data and streamingdata applications that are frequently encountered in mobileapplications.

ACKNOWLEDGMENT

The authors would like to thank P. Vemulapalli and K. Jerathfor collecting the road map data and, in general, for theirsupport and advice throughout the development of this work.



REFERENCES

[1] I. Skog and P. Handel, “In-car positioning and navigation technologies—A survey,” IEEE Trans. Intell. Transp. Syst., vol. 10, no. 1, pp. 4–21,Mar. 2009.

[2] J. Farrell and M. Barth, The Global Positioning System & InertialNavigation. New York, NY, USA: McGraw-Hill, 1999.

[3] N. Drawil, H. Amar, and O. Basir, “GPS localization accuracy classifica-tion: A context-based approach,” IEEE Trans. Intell. Transp. Syst., vol. 14,no. 1, pp. 262–273, Mar. 2013.

[4] W. Y. Ochieng et al., “GPS integrity and potential impact on aviationsafety,” J. Navigat., vol. 56, no. 1, pp. 51–65, Jan. 2003.

[5] J.-E. Källhammer et al., “Fulfilling the pedestrian protection directiveusing a long-wavelength infrared camera designed to meet both per-formance and cost targets,” in Proc. SPIE, Apr. 2006, vol. 6198, no. 09,pp. 619809-1–619809-11.

[6] H. Fang, M. Yang, R. Yang, and C. Wang, “Ground-texture-based local-ization for intelligent vehicles,” IEEE Trans. Intell. Transp. Syst., vol. 10,no. 3, pp. 463–468, Sep. 2009.

[7] I. Parra Alonso et al., “Accurate global localization using visual odometryand digital maps on urban environments,” IEEE Trans. Intell. Transp.Syst., vol. 13, no. 4, pp. 1535–1545, Dec. 2012.

[8] S. Sivaraman and M. Trivedi, “Integrated lane and vehicle detection, lo-calization, tracking: A synergistic approach,” IEEE Trans. Intell. Transp.Syst., vol. 14, no. 2, pp. 906–917, Jun. 2013.

[9] H. Abbott and D. Powell, “Land-vehicle navigation using GPS,” Proc.IEEE, vol. 87, no. 1, pp. 145–162, Jan. 1999.

[10] C. Carlson, J. Gerdes, and J. Powell, “Error sources when land vehicledead reckoning with differential wheelspeeds,” J. Inst. Navigat., vol. 51,no. 1, pp. 13–28, 2004.

[11] K. Jo, K. Chu, and M. Sunwoo, “Interacting multiple model filter-basedsensor fusion of GPS with in-vehicle sensors for real-time vehicle posi-tioning,” IEEE Trans. Intell. Transp. Syst., vol. 13, no. 1, pp. 329–343,Mar. 2012.

[12] M. Grewal, L. Weill, and A. Andrews, Global Positioning Systems,Inertial Navigation, Integration. Hoboken, NJ, USA: Wiley, 2007.

[13] A. Dean, R. Martini, and S. Brennan, “Terrain-based road vehicle local-ization using particle filters,” in Proc. Amer. Control Conf., Seattle, WA,USA, Jun. 2008, pp. 236–241.

[14] R. Martini, “GPS/INS sensing coordination for vehicle state identifica-tion and road grade positioning,” M.S. thesis, Dept. Mech. Nucl. Eng.,Pennsylvania State Univ., State College, PA, USA, Aug. 2006.

[15] P. Vemulapalli, A. Dean, and S. Brennan, “Pitch-based vehicle local-ization using time series subsequence matching with multi-scale ex-trema features,” in Proc. Amer. Control Conf., San Francisco, CA, USA,Jul. 2011, pp. 2405–2410.

[16] K. Jerath and S. Brennan, “GPS-free terrain-based vehicle tracking per-formance as a function of inertial sensor characteristics,” in Proc. Dyn.Syst. Control Conf., Arlington, VA, USA, Nov. 2011, pp. 367–374.

[17] S. Kadetodat, P. Vemulapalli, S. Brennan, and C. Lagoa, “Terrain-aidedlocalization using feature-based particle filtering,” in Proc. Dyn. Syst.Control Conf., Arlington, VA, USA, Nov. 2011, pp. 725–731.

[18] V. Gupta, “Vehicle localization using low-accuracy GPS, IMU and map-aided vision,” Ph.D. dissertation, Dept. Mech. Nucl. Eng., PennsylvaniaState Univ., State College, PA, USA, May 2009.

[19] D. Bernstein and A. Kornhauser, “An Introduction to Map Matching forPersonal Navigation Assistants,” Transp. Res. Board, Washington, DC,USA, Tech. Rep., Aug. 1996.

[20] E. Laftchiev, C. Lagoa, and S. Brennan, “Terrain-based vehicle localiza-tion from real-time data using dynamical models,” in Proc. IEEE Conf.Decision Control, Maui, HI, USA, Dec. 2012, pp. 3366–3371.

[21] T. Randen and J. Husøy, “Filtering for texture classification: A com-parative study,” IEEE Trans. Pattern Anal. Mach. Intell., vol. 21, no. 4,pp. 291–310, Apr. 1999.

[22] Y. Morinaka, T. Amagasa, and S. Uemura, “The l-index: An indexingstructure for efficient subsequence matching in time sequence databases,”in Proc. Pac.-Asia Conf. Knowl. Discovery Data Mining, Hong Kong,Apr. 2001, pp. 51–60.

[23] J. Borenstein, H. R. Everett, L. Feng, and D. Wehe, “Mobile robot posi-tioning: Sensors and techniques,” J. Robot. Syst., vol. 14, no. 4, pp. 231–249, Apr. 1997.

[24] J. Borenstein and F. Liqiang, “Measurement and correction of systematicodometry errors in mobile robots,” IEEE Trans. Robot. Autom., vol. 12,no. 6, pp. 869–880, Dec. 1996.

[25] N. Doh, H. Choset, and W. Chung, “Relative localization using pathodometry information,” Autonom. Robots, vol. 21, no. 2, pp. 143–154,Sep. 2006.

[26] C. Carlson, J. Gerdes, and J. Powell, “Practical position and yaw rateestimation with GPS and differential wheelspeeds,” in Proc. Int. Symp.AVEC, Hiroshima, Japan, Sep. 2002, pp. 1–8.

[27] J. Borenstein and L. Feng, “Gyrodometry: A new method for combin-ing data from gyros and odometry in mobile robots,” in Proc. IEEEInt. Conf. Robot. Autom., Minneapolis, MN, USA, Apr. 1996, vol. 1,pp. 423–428.

[28] Z. Wu, M. Yao, H. Ma, and W. Jia, “Improving accuracy of the vehicleattitude estimation for low-cost INS/GPS integration aided by the GPS-measured course angle,” IEEE Trans. Intell. Transp. Syst., vol. 14, no. 2,pp. 553–564, Jun. 2013.

[29] M. A. Quddus, W. Y. Ochieng, and R. B. Noland, “Integrity of map-matching algorithms,” Transp. Res. C, Emerg. Technol., vol. 14, no. 4,pp. 283–302, Aug. 2006.

[30] M. A. Quddus, W. Y. Ochieng, and R. B. Noland, “Current map-matchingalgorithms for transport applications: State-of-the art and future researchdirections,” Transp. Res. C, Emerg. Technol., vol. 15, no. 5, pp. 312–328,Oct. 2007.

[31] A. Schindler, “Vehicle self-localization with high-precision digital maps,”in Proc. IEEE Intell. Veh. Symp., Gold Coast, QLD, Australia, Jun. 2013,pp. 141–146.

[32] J. Roth, T. Schaich, and G. Trommer, “Cooperative GNSS-based methodfor vehicle positioning,” Gyroscopy Navigat., vol. 3, no. 4, pp. 245–254,Oct. 2012.

[33] M. Dissanayke et al., “A solution to the simultaneous localization and mapbuilding (SLAM) problem,” IEEE Trans. Robot. Control, vol. 17, no. 3,pp. 229–241, Jun. 2001.

[34] D. Schleicher, L. Bergasa, M. Ocana, R. Barea, and M. Lopez,“Real-time hierarchical outdoor SLAM based on stereovision and GPSfusion,” IEEE Trans. Intell. Transp. Syst., vol. 10, no. 3, pp. 440–452,Sep. 2009.

[35] C. Hasberg, S. Hensel, and C. Stiller, “Simultaneous localization andmapping for path-constrained motion,” IEEE Trans. Intell. Transp. Syst.,vol. 13, no. 2, pp. 541–552, Jun. 2012.

[36] R. Smith, M. Self, and P. Cheeseman, “A stochastic map for uncer-tain spatial relationships,” in Proc. 4th Int. Symp. Robot. Res., 1988,pp. 467–474.

[37] M. Quddus, W. Ochieng, L. Zhao, and R. Noland, “A general map match-ing algorithm for transport telematics applications,” GPS Solutions, vol. 7,no. 3, pp. 157–167, Dec. 2003.

[38] J. Leonard and H. Durrant-Whyte, “Mobile robot localization by trackinggeometric beacons,” IEEE Trans. Robot. Autom., vol. 7, no. 3, pp. 376–382, Jun. 1991.

[39] M. Bosse, D. Koller, Z. Ghahramani, H. Durrant-Whyte, and A. Ng,“Keypoint design and evaluation for place recognition in 2D lidarmaps,” J. Robot. Autonom. Syst., vol. 57, no. 12, pp. 1211–1224,Dec. 2009.

[40] K. Mikolajczyk and H. Uemura, “Action recognition with motion-appearance vocabulary forest,” in Proc. IEEE Conf. Comput. Vis. PatternRecog., Anchorage, AK, USA, Jun. 2008, pp. 1–8.

[41] S. Thrun and R. Zlot, “Reduced SIFT features for image retrievaland indoor localization,” in Proc. Aust. Conf. Robot. Autom., Canberra,Australia, Dec. 2004, pp. 1–8.

[42] R. Smith and P. Cheeseman, “On the representation and estimation ofspatial uncertainty,” Int. J. Robot. Res., vol. 5, no. 4, pp. 56–68, Dec. 1987.

[43] R. Smith, M. Self, and P. Cheeseman, Estimating Uncertain Spatial Re-lationships in Robotics. New York, NY, USA: Springer-Verlag, 1990,ch. Autonomous Robot Vehicles.

[44] P. Moutarlier and R. Chatila, “An experimental system for incrementalenvironment modeling by an autonomous mobile robot,” in Proc. 1st Int.Symp. Exp. Robot., London, U.K., Jun. 1990, pp. 327–346.

[45] J. Levinson and S. Thrun, “Robust vehicle localization in urban environ-ments using probabilistic maps,” in Proc. IEEE Int. Conf. Robot. Autom.,Anchorage, AK, USA, May 2010, pp. 4372–4378.

[46] D. Törnqvist, T. Schön, R. Karlsson, and F. Gustafsson, “Particle filterSLAM with high dimensional vehicle model,” J. Intell. Robot. Syst.,vol. 55, no. 4/5, pp. 249–266, Aug. 2009.

[47] H. Ding, G. Trajcevski, P. Scheuermann, X. Wang, and E. Keogh, “Query-ing and mining of time series data: Experimental comparison of represen-tations and distance measures,” in Proc. VLDB Endowment, Aug. 2008,vol. 1, no. 2, pp. 1542–1552.

[48] A. Dean, “Terrain-based road vehicle localization using attitude measure-ments,” Ph.D. dissertation, Dept. Mech. Nucl. Eng., Pennsylvania StateUniv., State College, PA, USA, Oct., 2008.

[49] A. Brown and Y. Lu, “Performance test results of an integratedGPS/MEMS inertial navigation package,” in Proc. Int. Tech. Meet. Satell.Division, Long Beach, CA, USA, Sep. 2004, vol. 17, pp. 825–832.



[50] S. Godha and M. Cannon, “GPS/MEMS INS integrated system for nav-igation in urban areas,” GPS Solutions, vol. 11, no. 3, pp. 193–203,Jul. 2007.

[51] N. El-Sheimy, H. Hou, and X. Niu, “Analysis and modeling of inertialsensors using Allan variance,” IEEE Trans. Instrum. Meas., vol. 57, no. 1,pp. 140–149, Jan. 2008.

[52] N. El-Sheimy, S. Nassar, and A. Noureldin, “Wavelet de-noising for IMUalignment,” IEEE Aerosp. Electron. Syst. Mag., vol. 19, no. 10, pp. 32–39,Oct. 2004.

[53] G. Aydemir and A. Saranli, “Characterization and calibration of MEMSinertial sensors for state and parameter estimation applications,” Measure-ment, vol. 45, no. 5, pp. 1210–1225, Jun. 2012.

[54] P. Aggarwal, Z. Syed, X. Niu, and N. El-Sheimy, “Thermal calibration oflow cost MEMS sensors for integrated positioning navigation systems,”in Proc. Nat. Tech. Meet. Inst. Navigat., San Diego, CA, USA, Jan. 2007,vol. 1, pp. 343–349.

[55] I. Skog and P. Händel, “Calibration of a MEMS inertial measurementunit,” in Proc. IMEKO World Congr., Rio de Janeiro, Brazil, Sep. 2006,vol. 17, pp. 1–6.

[56] L. Wang, Y. Hao, and F. Wang, “Calibration of low cost MEMS inertialmeasurement unit for an FPGA-based navigation system,” in Proc. IEEEInt. Conf. Inf. Autom., Shenzhen, China, Jun. 2011, pp. 181–186.

[57] A. Ramanandan, A. Chen, and J. Farrell, “Inertial navigation aiding bystationary updates,” IEEE Trans. Intell. Transp. Syst., vol. 13, no. 1,pp. 235–248, Mar. 2012.

[58] K. Zimmermann, J. Matas, and T. Svoboda, “Tracking by an optimalsequence of linear predictors,” IEEE Trans. Pattern Anal. Mach. Intell.,vol. 31, no. 4, pp. 677–692, Apr. 2009.

[59] S. Al-Takrouri and A. V. Savkin, “A model validation approach to texturerecognition and inpainting,” Pattern Recognit., vol. 43, no. 6, pp. 2054–2067, Jun. 2010.

[60] A. B. Ashraf, S. Lucey, and T. Chen, “Reinterpreting the application ofGabor filters as a manipulation of the margin in linear support vector ma-chines,” IEEE Trans. Pattern Anal. Mach. Intell., vol. 32, no. 7, pp. 1335–1341, Jul. 2010.

[61] P. Turaga, R. Chellappa, and A. Veeraraghavan, “Advances in video-based human activity analysis: Challenges and approaches,” Adv.Comput., ser. Advances in Computers, vol. 80, pp. 237–290, 2010.

[62] U. Kandaswamy, S. Schuckers, and D. Adjeroh, “Comparison of tex-ture analysis schemes under nonideal conditions,” IEEE Trans. SignalProcess., vol. 20, no. 8, pp. 2260–2275, Aug. 2011.

[63] I. Sekita, T. Kurita, and N. Otsu, “Complex autoregressive model forshape recognition,” IEEE Trans. Pattern Anal. Mach. Intell., vol. 14, no. 4,pp. 489–496, Apr. 1992.

[64] P. Potocnik, E. Govekar, and I. Grabec, “Application of psychoacousticfiltering for machine fault detection,” Int. J. Mater. Product Technol.,vol. 27, no. 3/4, pp. 229–237, 2006.

[65] M. Graciarena and H. Franco, “Unsupervised noise model estimationfor model-based robust speech recognition,” in Proc. IEEE WorkshopAutom. Speech Recog. Understanding, St. Thomas, VI, USA, Dec. 2003,pp. 351–356.

[66] A. Shenoy, Y. Wu, and Y. Wang, “Singing voice detection for karaokeapplication,” in Proc. Vis. Commun. Image Process., Beijing, China,Jul. 2005, vol. 5960, no. 1–4, pp. 752–762.

[67] M. Johnsen and A. Canterla, “Joint feature and model training for mini-mum detection errors applied to speech subword detection,” in Proc. IEEEInt. Workshop Mach. Learn. Signal Process., Santander, Spain, Sep. 2012,pp. 1–6.

[68] A. Jain, S. Prabhakar, L. Hong, and S. Pankanti, “Filterbank-based finger-print matching,” IEEE Trans. Image Process., vol. 9, no. 5, pp. 846–859,May 2000.

[69] A. Bissacco, A. Chiuso, Y. Ma, and S. Soatto, “Recognition of humangaits,” in Proc. IEEE Comput. Soc. Conf. Comput. Vis. Pattern Recog.,Kauai, HI, USA, Dec. 2001, vol. 2, pp. 52–57.

[70] A. Veeraraghavan, A. Roy-Chowdhury, and R. Chellappa, “Matching shapesequences in video with applications in human movement analysis,” IEEETrans. Pattern Anal. Mach. Intell., vol. 27, no. 12, pp. 1896–1909, Dec. 2005.

[71] C. Mazzaro, M. Sznaier, O. Camps, S. Soatto, and A. Bissacco, “A model(in)validation approach to gait recognition,” in Proc. Int. Symp. 3D DataProcess. Vis. Transmiss., Padua, Italy, Jun. 2002, pp. 700–703.

[72] P. Saisan, G. Doretto, Y. Wu, and S. Soatto, “Dynamic texture recogni-tion,” in Proc. IEEE Comput. Soc. Conf. Comput. Vis. Pattern Recog.,Kauai, HI, USA, Dec. 2001, pp. 58–63.

[73] A. Bissacco, A. Chiuso, and S. Soatto, “Classification and recognition ofdynamical models: The role of phase, independent components, kernelsand optimal transport,” IEEE Trans. Pattern Anal. Mach. Intell., vol. 29,no. 11, pp. 1958–1972, Nov. 2007.

[74] A. Ravichandran, R. Chaudhry, and R. Vidal, “Categorizing dynamictextures using a bag of dynamical systems,” IEEE Trans. Pattern Anal.Mach. Intell., vol. 35, no. 2, pp. 342–353, Feb. 2013.

[75] R. Li, T.-P. Tian, and S. Sclaroff, “Divide, conquer and coordinate: Glob-ally coordinated switching linear dynamical system,” IEEE Trans. PatternAnal. Mach. Intell., vol. 34, no. 4, pp. 654–669, Apr. 2012.

[76] R. Martin, “A metric for ARMA processes,” IEEE Trans. Signal Process.,vol. 48, no. 4, pp. 1164–1170, Apr. 2000.

[77] R. Agrawal, C. Faloutsos, and A. Swami, “Efficient similarity search insequence databases,” in Proc. Int. Conf. Found. Data Org. Algorithms,Chicago, IL, USA, Oct. 1993, pp. 69–84.

[78] C. Faloutsos, M. Ranganathan, and Y. Manolopoulos, “Fast SubsequenceMatching in Time-Series Databases,” Univ. Maryland at College Park,College Park, MD, USA, Tech. Rep., 1993.

[79] E. Keogh, K. Chakrabarti, M. Pazzani, and S. Mehrotra, “Dimensional-ity reduction for fast similarity search in large time series databases,”J. Knowl. Inf. Syst., vol. 3, no. 3, pp. 263–286, Aug. 2001.

[80] H. Zhang, T. Ho, and M. Lin, “A non-parametric wavelet feature extrac-tor for time series classification,” in Advances in Knowledge Discoveryand Data Mining, vol. 3056. Berlin, Germany: Springer-Verlag, 2004,ser. Lecture Notes in Computer Science, pp. 595–603.

[81] F. Korn, H. V. Jagadish, and C. Faloutsos, “Efficiently supporting ad hocqueries in large datasets of time sequences,” in Proc. ACM SIGMOD Int.Conf. Manage. Data, Jun. 1997, pp. 289–300.

[82] E. Keogh, J. Lin, and A. Fu, “HOT SAX: Efficiently finding the mostunusual time series subsequence,” in Proc. IEEE Int. Conf. Data Mining,Houston, TX, USA, Nov. 2005, pp. 226–233.

[83] B. Yi and C. Faloutsos, “Fast time sequence indexing for arbitraryLp norms,” in Proc. Int. Conf. Very Large Data Bases, Cairo, Egypt,Sep. 2000, pp. 385–394.

[84] D. Gusfield, Algorithms on Strings, Trees and Sequences: Computer Sci-ence and Computational Biology. Cambridge, U.K.: Cambridge Univ.Press, 1997.

[85] J. Aßfalg et al., “Similarity search on time series based on thresh-old queries,” in Advances in Database Technology, vol. 3896. Berlin,Germany: Springer-Verlag, 2006, ser. Lecture Notes in ComputerScience, pp. 276–294.

[86] H. André-Jönsson and D. Z. Badal, “Using signature files for queryingtime-series data,” in Proc. Eur. Symp. Principles Data Mining Knowl.Discov., Apr. 1997, pp. 211–220.

[87] P. Bosch and A. Van der Klauw, Modeling: Identification and Simulationof Dynamical Systems. Boca Raton, FL, USA: CRC Press, 1994.

[88] N. Ozay, M. Sznaier, C. Lagoa, and O. Camps, “A sparsification approachto set membership identification of switched affine systems,” IEEE Trans.Autom. Control, vol. 57, no. 3, pp. 634–648, Mar. 2012.

[89] M. Dissanayke, S. Williams, H. Durrant-Whyte, and T. Bailey, “Map man-agement for efficient simultaneous localization and mapping (SLAM),”Autonom. Robots, vol. 12, no. 3, pp. 267–286, May 2002.

[90] S. Thrun et al., “Simultaneous mapping and localization with sparseextended information filters,” Int. J. Robot. Res., vol. 23, no. 7/8, pp. 693–716, Jul. 2004.

[91] J. Guivant and E. Nebot, “Optimization of the simultaneous localizationand map-building algorithm for real-time implementation,” IEEE Trans.Robot. Autom., vol. 17, no. 3, pp. 242–257, Jun. 2001.

[92] IEEE Standard Specification Format Guide and Test Procedure for Single-Axis Interferometric Fiber Optic Gyros, IEEE Std. 952-1997, Aug. 1998,pp. 1–83.

Emil I. Laftchiev (S’11) received the B.S. degreein electrical engineering from The PennsylvaniaState University, University Park, PA, USA, in 2007,where he is currently working toward the Ph.D.degree in electrical engineering.

During his tenure in the Ph.D. program, heworked both in vehicle navigation and spectroscopy-based explosives detection. His prior research expe-rience includes spectroscopy experimentation, signalprocessing, and instrumentation design, as well asinstrumentation design for in situ combustion mon-

itoring. His research interests include control and signal processing, includ-ing system identification, dynamical systems, instrumentation design, andadvanced noise mitigation approaches in signal processing.



Constantino M. Lagoa (M’98) received the B.S.and M.S. degrees from Technical University ofLisbon, Lisbon, Portugal, in 1991 and 1994, re-spectively, and the Ph.D. degree from University ofWisconsin-Madison, Madison, WI, USA, in 1998.

In August 1998 he joined the Department ofElectrical Engineering, The Pennsylvania State Uni-versity, University Park, PA, USA, where he is cur-rently a Professor. He has authored or coauthoredover 20 journal papers and book chapters and over40 conference publications. He has a wide range

of research interests, including robust control, controller design under riskspecifications, system identification, control of computer networks, and discreteevent dynamical systems.

Dr. Lagoa is a member of the IFAC Technical Committee on Robust Controland has been a member of the international program committee of severalconferences. He is currently an Associate Editor of IEEE TRANSACTIONS

ON AUTOMATIC CONTROL, IEEE TRANSACTIONS ON CONTROL SYSTEMS

TECHNOLOGY, and Automatica.

Sean N. Brennan (M’13) received the B.S. degreein mechanical engineering and physics from NewMexico State University, Las Cruces, NM, USA, in1997 and the M.S. and Ph.D. degrees in mechanicalengineering from University of Illinois at Urbana-Champaign, Urbana, IL, USA, in 1999 and 2002,respectively.

He is an Associate Professor of mechanical en-gineering with The Pennsylvania State University,University Park, PA, USA. His research interestsinclude vehicle dynamics, control, mechatronics, di-

mensional analysis, and scaling theory.Dr. Brennan is an Associate Editor of ASME Journal of Dynamic Systems,

Measurement, and Control.

Documents

Vehicle Localization Using In-Vehicle Pitch Data and Dynamical Models