Unsupervised Calibration of Wheeled Mobile Platforms

Maurilio Di Cicco, Bartolomeo Della Corte, Giorgio Grisetti

Abstract— This paper describes an unsupervised approach toretrieve the kinematic parameters of a wheeled mobile robot.The robot chooses which action to take in order to minimizethe uncertainty in the parameter estimate and to fully explorethe parameter space.

Our method explores the effects of a set of elementary motionon the platform to dynamically select the best action and to stopthe process when the estimate can be no further improved.

We tested our approach both in simulation and with realrobots. Our method is reported to obtain in shorter timeparameter estimates that are statistically more accurate thanthe ones obtained by steering the robot on predefined patterns.

I. INTRODUCTION

Calibrating a robot is a tedious and time consumingmandatory procedure, whose outcome influences the overallsystem performances. A small error in calibration mightlead to severe inconsistencies in tasks that rely on sensorinformation such as localization, mapping and navigation ingeneral.

The task of calibrating a mobile robot has been addressedsince many years. In this paper we focus on the so calledkinematic calibration. In a wheeled mobile robot this consistsof estimating the odometry parameters, that are required toconvert wheel encoder ticks in a relative motion of the mobilebase on a local plane, and of estimating the position of oneor more sensors on the mobile base.

Several approaches to address this task have been proposedin the literature. All of them are regarded as passive pro-cesses, where the user moves the platform in an environment,recording the data from the encoder and from the extero-ceptive sensors (like laser scanners or cameras) mounted onthe robot. The calibration procedure consumes these dataand seeks for the parameters that better explain the sensorperceptions, given a calibration pattern.

The quality of the calibration is greatly influenced by thetrajectory taken by the robot, and for specific combination ofplatforms/sensors there are common motion patterns that arelikely to produce good results. As an instance, calibrating awheeled platform moving on a plane and equipped with alaser scanner can be done by moving the robot on a squareor along an eight-shaped path in a known room so to explorethe entire parameter space.

All authors are with Department of Computer, Control, and ManagementEngineering Antonio Ruberti of Sapienza University of Rome, Via Ariosto25, I00185 Rome, Italy.

Email: lastname@dis.uniroma1.it.This work has partly been supported by the European Commission under

FP7-600890-ROVINA.

0 50 100 150 2000.1

0.2

0.3

0.4

0.5

0.6

0.7

seconds

mete

rs

0 50 100 150 2000.38

0.39

0.4

0.41

0.42

0.43

seconds

mete

rs

Fig. 1: On the top of the image two different kind of mobilerobots are shown, a differential drive and an omnidirectionalbase. In the bottom the respective baseline estimates are shownduring the execution of different calibration strategies along withtheir uncertainties. The green dashed line represents the evolutionobserved during the unsupervised procedure. The blue dotted lineshows the evolution during the calibration following the square-path. Lastly, the magenta dash-dotted line shows the evolution of theobserved parameter following the eight-shaped path. In the bottomright image, even the ground truth value (black solid line) is shownsince the experiment was conducted in simulation.

Yet, this procedure needs to be typically carried on by anexpert. To produce sufficient data to guarantee a reasonablecalibration, the data acquisition usually lasts some minutes.

In this paper we present a fully automatic approachfor calibrating the kinematic parameters of a mobile robotequipped with a sensor suitable for motion tracking, such asa depth camera or a laser scanner. The system relies onlyon the knowledge of the kinematic model of the robot, andchooses how to move in the environment to minimize theuncertainty of the estimate and to fully explore the parameterspace.

We performed extensive tests on robots having differentkinematic models both in real world scenarios and in simu-lation. We compared our results with the ones obtained bymanually executing fixed calibration patterns. In our teststhe proposed approach provided statistically more accurateparameter estimates and required less time than the manualprocedures.

Figure 1 shows a motivating example of our procedure,used to calibrate two types of mobile base. The plots inthe second line show the evolution of the estimate and its

uncertainty as more training data are gathered.

II. RELATED WORK

Calibrating a mobile robot has been addressed since morethan two decades. In this section we provide an overviewof approaches addressing this topic and we relate them withour contribution.

Borenstein and Feng [4] developed a method namedUMBmark. It requires to drive a differential drive robotalong a square path both clockwise and counterclockwise.Following this cyclic path, the robot has to come back tothe starting position. The distance between the initial poseand the final pose measured by the odometry represents theerror. Minimizing such error leads to the calibration of thedirect kinematics of the wheeled base. This method relies ona pre-constructed trajectory and it neglets the possibility tocalibrate nothing but the odometry parameters.

Subsequently Kelly [7] proposed a generic odometry cal-ibration approach that does not require the robot to travelon a squared path. Such method practically requires to drivethe robot along any trajectories for a couple of dozens oftimes while predefined waypoints along the paths are usedas ground truth. The quality of results is highly dependanton the trajectories involved in the calibration as well as thenumber of ground truth points.

Chenavier and Crowley [6] worked on a Kalman filter tocalibrate both the odometry parameters and the camera poseon the mobile robot. A known map of fixed objects, deployedinto the environment, is exploited by the camera, which istreated as bearing sensor.

The approach of Martinelli et al. [2] simultaneusly es-timates the systematic and non systematic odometry errorof a mobile robot. By using this statistics they enhance theeffectiveness of a gaussian filter for SLAM [8].

Censi et al. [1] proposed a max likelihood approach whichdoes not rely on specific trajectories. The method providesa simultaneous calibration of both the intrinsics odometryparameters as well as the pose of a 2D laser scanner usedegomotion estimator.

Kummerle et al. [9] augment the state of a graph basedSLAM method with unknown calibration parameters. Thisapproach allows to simultaneously calibrate both odometryand position of sensors while the robot operates. It isparticularly effective in case of non-stationary situations thatmight occur during a mission (e.g. changes in the kinematicsdue do varying load distributions).

The last two approaches are among the few that addressthe issue of calibrating the odometry and the sensor positions.Yet, they operate as passive processes consuming the data asthey are gathered by the moving robot.

The path taken by the robot, however has a great influenceon the quality of the calibration. Each particular motion alongthe trajectory might depend only on a subset of parameters.

III. LEAST SQUARES CALIBRATION OF A MOBILE ROBOT

Calibrating the kinematic parameters k of a wheeledmobile platform means estimating the coefficients ko that

k b

ks

zi

oi

Fig. 2: This figure illustrates the relation between the mobile baseand the sensor reference frame. The relative pose of the sensor, withrespect to the robot frame, is ks. While the mobile base moves alongthe trajectory oi, the sensor will move along the path zi in its ownreference frame. Lastly, kb represents the robot baseline.

allow us to compute the relative motion of the base fromthe ticks measured by the encoders. If the robot is equippedwith one or more exteroceptive sensor, these parameters alsoinclude the relative position of the sensor ks w.r.t. the mobilebase, as shown in Figure 2.

Knowing the mobile base parameters ko and the ticksof the wheel encoders ui we can compute the motion ofthe mobile base during the ith interval, through a directkinematics function f(ko,ui). Let oT

i = (xoi , y

oi , θ

oi )

T bethis motion. If the robot is equipped with an exteroceptivesensor such as a laser scanner, and it operates on a plane,the sensor parameters are augmented with the sensor positionks = (xs ys θs)T . Estimating the full kinematics of the robotthus means estimating the vector kT = (koT ksT ). If thereis an algorithm that allows to estimate the ego-motion of anexteroceptive sensor uniquely from the sensor’s observations,we can use the output of this algorithm as a reference forcalibration. In particular, if the robot is equipped with alaser scanner such an estimate can be obtained by a scan-matcher. Also RGBD cameras can be used for this purpose,as their data can be fed to tracking algorithm such asKINFU[3] or NICP [11] that provides the trajectory of thesensor. In the remainder of this section we focus on laserscanners and 2D motion, however the results of this papercan be straightforwardly generalized to the 3D case. Letzi = (xi yi θi)

T be the estimated motion of a sensor duringthe ith interval.

Knowing the position of the sensor on the robot ks andthe motion of the base we can estimate the motion of thesensor zi from the encoder measurements ui as

zi = hi(k) (1)= oi ⊕ ks (2)= f(ko,ui)⊕ ks. (3)

Here ⊕ denotes the motion composition operator as de-scribed by Smith and Cheesman [12], and h(·) is theprediction function depending on the parameters and theencoder measurements.

The goal of our least squares calibration procedure isthus to find the parameters k∗, that minimize the differencebetween the predicted and measured displacements of thesensor:

k∗ = argmin∑i

‖hi(k)− zi‖Ωi. (4)

Here ‖v‖ denotes the Ω norm of a vector v, as

‖v‖Ω = vTΩv. (5)

The matrix Ωi denotes the information matrix of the solutionreported by the scan matcher and it can be obtained byusing the method of Censi [5]. Considering the informationmatrices it allows to adjust the contribution of the differenterror terms based on the quality of the scan matcher solution.Since our measurement zi ∈ SE2, performing the vectordifference in Eq. (4) might result in non-valid configurationsdue to angular wraparounds. This effect can be lessened bysubstituting the − with a more appropriate motion decom-position, denoted by the symbol . Thus, our function tominimize becomes:

k∗ = argmin∑i

‖hi(k) zi︸ ︷︷ ︸ei(k)

‖Ωi. (6)

Here ei(k) is the error of the ith measurement and denotesthe relative transform that would bring the estimate of thesensor motion obtained from the odometry to match themotion estimated by the scan matcher.

Using least squares the problem in Eq. (6) can be solvediteratively, assuming a reasonable estimate of the nominalparameters is available. At each iteration the algorithmrefines the current estimate k by computing a perturbation∆k that minimizes a quadratic approximation of Eq. (6) inthe form:

∆Tk H∆k + bT∆k + c. (7)

The minimum of Eq. (7) can be found by solving thefollowing linear system:

H∆k = −b. (8)

The matrix H and the vector b are obtained from Eq. (6) bylinearizing each error ei(k) around the current guess as:

ei(k + ∆k) ' ei(k)︸ ︷︷ ︸ei

+∂ei(k)

∂k︸ ︷︷ ︸Ji

∆k. (9)

Applying the definition of Ω norm in Eq. (5), substitutingEq. (9) in Eq. (6) and grouping the terms leads to thefollowing expression for H and b:

H =∑i

JTi ΩiJi (10)

b =∑i

JTi Ωiei (11)

k s

z i

k b1

kb2

o i

Fig. 3: This figure illustrates the relation between the youbot baseand the sensor reference frame. The relative pose of the sensor,with respect to the robot frame, is ks. While the mobile base movesalong the trajectory oi, the sensor will move along the path zi inits own reference frame. As expressed by the kinematics of therobot, kbb is given by the average between the two baseline, i.ekbb = (kb1+kb2)/2

At each iteration the current guess is refined by adding thecomputed increments as:

k← k + ∆k. (12)

Under Gaussian assumptions on the measurement noise,the H matrix represents the inverse covariance of the pa-rameter estimate. Thus by analyzing H at the equilibriumwe can evaluate the quality of the calibration.

In the remainder of this section we provide the detailsof the two different kinematic models we used in ourexperiments.

1) Differential Drive: The odometry parameters are thefollowing

ko =(kl kr kb

)T. (13)

Here kl and kr denote respectively the coefficient betweenthe circumference of the left and right wheel and the encoderticks per revolution of the wheels’ shaft. kb is the distancebetween the wheels (baseline). ui =

(tli t

ri

)consists of the

tick increments reported by the left and the right wheels,denoted respectively as tli and tri . The direct kinematicsf(k,ui) can be approximated as:

oi = f(ko,ui) =

(krtri+kltli)20

krtri−kltli2kb

(14)

2) Omnidirectional Robot: The odometry parameters are

ko =(kfl kfr kbl kbr kbb

)T, (15)

where the first four parameters are the coefficients forrespectively front left, front right, back left and back rightwheels. kbb is a coefficient defined as the average betweenbaseline of the wheels and the distance between the front

and rear axes, as shown in Figure 3. The encoder ticks arestored in a 4 dimensional vector uT

i = (tfli tfri t

bli tbr

i )T as

oi =

−1/4 1/4 −1/4 1/41/4 1/4 −1/4 −1/4

1/4kbb 1/4kbb 1/4kbb 1/4kbb

kfltflikfrtfrikbltbl

i

kbrtbri

(16)

IV. ISSUES OF CALIBRATION USING LEAST SQUARES

Most of the calibration procedures in the literature rely onleast squares minimization. The general procedure consistsin moving the robot along a predefined trajectory, whilerecording its encoder ticks. During the acquisition, a groundtruth of the position of the mobile base or its sensors isobtained through some external observer or by algorithmsthat process only the exteroceptive sensor input.

By knowing the kinematic model is then possible toestimate the motion of the mobile base or its sensors fromthe measured encoder ticks.

The trajectory is chosen according to some common senserule in order to provide the least squares engine with suffi-cient information to produce a reasonable estimate. Thesetrajectories are chosen to be rather long to minimize the riskof having insufficient information and thus having to repeatthe task. However, the longer an experiment the higher arethe chances that something goes wrong. Additionally, in caseof different kinematic models the choice of a good trajectoryto follow is left to the user’s experience.

Often the parameters are not fully observable while fol-lowing a specific motion pattern. This leads the system toconverge at a value that, while minimizing the error metricin Eq. (6), produces invalid results. In such situations thecalibration worsens with the length of the trajectory. Thisfact is shown in Figure 4, where we show the evolution ofthe estimate of the baseline and the encoder coefficients of arobot that is constantly rotating on the spot. Note, howeverthat the ratio between baseline and ticks is approaching thenominal value.

In presence of non-linearities, the success of a leastsquares procedure greatly depends on the initial guess.However, due to non-observabilities this initial guess mightbecome inconsistent if the trajectory does not allow toexplore sufficiently the parameter space.

A solution to this problem is to continuously estimate thequality of the calibration, and choose the motion patterns thatimprove the estimate. A good calibration is characterized bybeing

• accurate: it is highly consistent with respect to thetraining data. This can be measured by a low value ofthe error function in Eq. (6).

• complete: it explores all the parameter space, withoutoverfitting the training data. This can be measured byanalyzing the ratio between the smaller and biggereigenvalue of the inverse covariance H at equilibrium.The higher this value the more “balances” is the uncer-tainty, thus there are no directions where the uncertainty

0 5 10 15 20 250.28

0.285

0.29

0.295

0.3

0.305

iterations

Ratio between encoders over baseline parameters

Even if the pure Rotation cannot guar-antee a full observability on the wholeko, the ratio between the sum of kl andkr over kb tends to nominal value.

0 5 10 15 20 250.25

0.3

0.35

0.4

0.45

iterations

me

ters

Baseline lenght estimation

Pure Rotation motion does not led to aconsistent baseline estimation withoutperforming Forward or Arc motion atbeginning.

0 5 10 15 20 250

2e-07

4e-07

6e-07

8e-07

iteration

Ratio between min and max eigenvalues

In this figure we show the effect ofloss of confidence along at least onedirection. This brings to an incrementof uncertainty and to a inaccurate pa-rameter estimantion.

Fig. 4: The figures show the effects of the pure Rotate motion onthe ko parameters.

is substantially higher than others. The eigenvectorsof small eigenvalues typically denote a non-observableparameter subspace.

When talking of a “balanced” covariance, the reader mightnotice that the units of measure strongly affect the shape ofa covariance. As an example, expressing the the baseline kb

in cm. would result in a standard deviation of the estimatethat is 100 times bigger than if the baseline is expressed inmeters. This phenomena, if we need to estimate one singleparameter is not relevant. However, when multiple valuesneed to be simultaneously estimated (e.g. the baseline andthe coefficients kl and kr), the effect becomes dominant: astandard deviation of one centimeter for the baseline mightbe acceptable, while for the encoder coefficients is reallybad.

As a rule of thumb, one should “prescale” the units of theparameters to a factor that reflects the desired accuracies. Asan example, if we want to estimate the odometry coefficientsat an accuracy of 1e − 5 meters and the baseline at 1e − 3meters of accuracy, we should express the respective unitsaccording to these scales. In this way, a standard deviationalong the ticks of 1e− 5 will be mapped with an eigenvalueof 1. The same happens for a standard deviation along thebaseline of 1e− 3, that results in an eigenvalue of 1 turningthe uncertainty ellipse into a circle.

having two quantities expressed one incentimeters and onein meters w

V. AUTONOMOUS CALIBRATION

In this section we sketch our method that allows a robotequipped with an exteroceptive sensor to autonomously

choose its trajectory to quickly reduce the uncertainty aboutits parameters. We remark that the approach is not specificto a differential drive robot and can be easily extendedto other classes of wheeled robots such as Hackermann-steering, synchro-drive or omnidirectional platforms.

Our procedure works as follows: The robot is providedwith a set of elementary motions motionj ∈ m . Each ofthese motions obtained by setting a set of constant velocitiesto each robot wheel, measured in encoder ticks. Typically,each motion allows to observe only a subspace of theparameter set.

When started, the system computes the effect that eachmotion has on the observability of the parameters. This isdone by executing each motion while recording the encoderticks and the estimated motion from the scan-matcher. Afterone motion motionj is executed, our procedure estimatesan optimal parameter kj set for that motion, together withits inverse covariance Hj . This is done by solving theleast squares problem in Eq. (6), by considering only themeasurements gathered during the execution of the motion.We call this stage “exploration”, as the system estimates theeffect that each particular motion has to the parameters.

After the system is done with exploration, the real trainingbegins. The idea is to initialize the calibration procedurefrom scratch, and compose a trajectory by selecting at eachtime the motion that improves the estimate of the parametersbased on an approximation of the two criteria describedabove.

The tuple 〈z1:t,Ω1:t,u1:t〉 constitutes our motion historyup to the current time t, and at the beginning is empty. Herez1:t and Ω1:t are respectively the relative motion observed bythe exteroceptive sensor and its inverse covariance, while u1:t

are the encoder differences. Let kt and Ht be the parameterestimate obtained by solving Eq. (6) with the measurementsup to time t.

Assuming the current parameter estimate kt to be reason-ably close to the optimum, and considering the cumulativenature of the H matrix highlighted by Eq. (10), than selectingthe motion motionj would result in a new Hj

t obtained as:

Hjt = Ht + Hj . (17)

This means that we can estimate the effect on our cali-bration of augmenting the current history of measurementswith the ones obtained by motion motionj . The determinantof the predicted information matrix denotes the cumulativeinformation in the estimate. The higher det(Hj

t ), the moreconvenient will be executing the motion motionj .

Thus we select the motion motionl such that

l = argmaxj

det(Ht + Hj) (18)

and we perform this motion with the robot by adding to thehistory of measurements the ones gathered during the currentexecution of the movement.

This procedure is repeated until the system has reachedconvergence and the parameter space is sufficiently explored.The first condition is checked by monitoring the evolution

Fig. 5: This image shows the environment, and its map, wherethe mobile base executed the unsupervised calibration procedure.The corridor environment has not been augmented with any kindof additional structure

of the normalized χ2 error of the estimate, measured as thevalue of Eq. (6) analyzed at the current estimate. The secondcondition is evaluated by monitoring the ratio between thesmaller and the larger eigenvalue of the current H.

More formally, our approach stops when

1

t

t∑i=1

‖hi(kt) zi‖Ωi− ‖hi(kt−1) zi‖Ωi

< ε (19)

andλmint

λmaxt

> γ. (20)

In practice at each step our system decides which motionto execute in order to improve its knowledge based onthe past experience and on the predicted behavior of theinformation gathered during the execution of a movement. Itstops when it has sufficient knowledge about the parameterspace, both in term of completeness and consistency. In ourexperiments we set the following values for the terminationcriterion: ε = 1e− 9 and γ = 1e− 2.

VI. EXPERIMENTS

We compared our approach with manual calibration meth-ods, both in real world and simulated experiments usingtwo types of mobile bases having different kinematics. Weconducted real world experiments using the Kobuki Turtlebotdifferential drive robot in the department corridors (Figure 5),while we performed simulated experiments on V-REP [10]where we used an ActivMedia Pioneer 2 DX (differentialdrive) and a KUKA YouBot (omnidirectional). All platformswere equipped with a Hokuyo-URG scanner.

We performed two types of manual calibration for eachplatform, by executing paths that depend on the kinematicsof the robot. Subsequently, we processed the data acquired

along the path by running incrementally the least squaresestimation described in Section III. To monitor the evolutionof the estimate as more data are gathered, we computed anew solution each time a certain number of new training databecome available.

We then executed our automatic calibration for eachplatform. We selected the basic motions based on the type ofkinematics, and let the system learn the effects of each mo-tion. Subsequently, we let the system run and automaticallychoose the sequence of motions to execute, while recordingthe training data. After a new motion is executed, ourapproach provides a new estimate that can be compared withthe one achieved by manual calibration after incorporatingthe same number of samples.

In the remainder of this section we describe the manualcalibration paths taken for the different platforms and thebasic motions used to start our automatic calibration.

3) Differential Drive: The calibration paths taken by themanual procedure for the differential drive robots are:

• The standard “square” path, where the robot movesalong a square having a side of 1 meter, and at theend of each corner rotates of 90. After a full squarehas been completed, the robot continues by revertingthe path.

• The standard “8” path, where the robot moves alongan 8 shape covering an area of approximately 4 by 2meters.

The basic motions taken by the differential drive robot arethe following:

• Forward: obtained by setting the speed of right and leftwheel to the same value.

• Rotate: obtained by setting the speed of right and leftwheel to opposite values.

• Arc: obtained by setting the speed of right and left wheelto different values.

Figures 6 and 7 show the results of these experiments forthe real world experiments with the turtlebot and for thesimulated experiment with the Pioneer 2. In the simulatedexperiments we had access to the ground truth, thus weperform a direct estimation of the accuracy, whereas in thereal world experiments we could only evaluate indirectly theestimate by plotting the reconstructed motion of the sensorunder different calibration estimates, as shown in Figure 9

In all cases our approach provided more consistent esti-mates, and required a lower number of samples to produceusable values.

4) Omnidirectional Robot: The calibration paths taken bythe manual procedure for the omnidirectional platform are:

• Double “square” path, where the robot moves along asquare having a side of 1 meter. The first time the robottraverses the square, it does not rotate at the corners,while the second time it performs a 90 rotation as ifit was a differential drive, and the procedure restarts.

• Double “8” shaped path, where the robot moves alongan 8 shape covering an area of approximately 4 by 2meters. The first revolution is done without rotation,

0 50 100 150 200-0.2

-0.1

0

0.1

0.2

0.3

0.4

seconds

mete

rs

0 50 100 150 200-0.08

-0.06

-0.04

-0.02

0

0.02

0.04

seconds

mete

rs

Fig. 6: This image shows the evolution of the parameters estimatefor the real differential drive platform (Turtlebot). On the left sideof the image the evolution of xs is shown, while on the right sidethe same evolution for ys is shown. In both cases, the uncertaintiesin the parameter estimate are highlighted by coloured strips. Thegreen dashed line represents the evolution observed during theunsupervised procedure. The blue dotted line shows the evolutionduring the calibration following the square-path. Lastly, the magentadash-dotted line shows the evolution of the observed parameterfollowing the eight-shaped path.

0 20 40 60 80 100-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

seconds

mete

rs

0 20 40 60 80 100

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

seconds

mete

rs

Fig. 7: This image shows the evolution of the parameters estimatefor the differential drive platform simulated in V-REP (Pioneer2). On the left side of the image the evolution of xs is shown,while on the right side the same evolution for ys is shown. In bothcases, the uncertainties in the parameter estimate are highlightedby coloured strips. The black solid line represents the ground truthvalue (available in simulation) for the observed parameter. Thegreen dashed line represents the evolution observed during theunsupervised procedure. The blue dotted line shows the evolutionduring the calibration following the square-path. Lastly, the magentadash-dotted line shows the evolution of the observed parameterfollowing the eight-shaped path.

while the second revolution is done as if the platformwas a differential drive.

The basic motions taken by the differential drive robot arethe following:

• Forward: obtained by setting tfl = tbl = −tfr = −tbr

to the same value.• Sideward: obtained by setting tfl = tfr = −tbl = −tbr

to the same value.• Rotate: obtained by setting tfl = tfr = tbl = tbr to the

same value.• Arc: obtained by setting tfl = −tbr and −tfr = tbl to

different values.

Figure 8 shows the results of this simulated experiment.We remark that performing manual calibration for the youbotrequired repeated trials due to the lack of standard calibrationstrategies and the increased dimension of the parameterspace. The same did not hold for the automatic calibrationthat quickly converged towards consistent estimates.

0 50 100 150 2000.15

0.2

0.25

0.3

0.35

0.4

seconds

mete

rs

0 50 100 150 200-0.04

-0.02

0

0.02

0.04

0.06

0.08

0.1

seconds

mete

rs

Fig. 8: This image shows the evolution of the parameters estimatefor the omnidirectional platform simulated in V-REP (Kuka Youbot).On the left side of the image the evolution of xs is shown, whileon the right side the same evolution for ys is shown. In bothcases, the uncertainties in the parameter estimate are highlightedby coloured strips. The black solid line represents the ground truthvalue (available in simulation) for the observed parameter. Thegreen dashed line represents the evolution observed during theunsupervised procedure. The blue dotted line shows the evolutionduring the calibration following the square-path. Lastly, the magentadash-dotted line shows the evolution of the observed parameterfollowing the eight-shaped path.

A. Behavior of Unsupervised Calibration on a DifferentialDrive Robot

In this section we shortly report some recurrent behaviorwe observed during the calibration of the differential driverobot, to better understand the behavior of the approach.Whereas we did not hardcode any heuristic for selecting themotions, we observed that typically the system follows aspecific pattern, especially at the beginning. Usually the firstchoice is to execute the Forward motion, since the Hforward

matrix has a higher determinant compared to the others. Thisresults in quickly reducing the uncertainty of the encoderratios kl, kr and of the laser heading θs.

After a pair of iterations, the system selects the Rotatemotion, that allows to estimate the baseline and the x − ydisplacement of the laser, as highlighed by the eigenvaluedecomposition of Hrotate.

Subsequently we were unable to identify a specific patternin the selection of the motion. We conjuncture it dependshighly on the laser position. If the laser is very close to thecenter of the platform, the x − y coordinates of the laserposition are poorly observable through a pure rotation, andthe system tends to select the Arc motions. Conversely ifthe laser is located far from the center it tends to select theRotate motion.

VII. CONCLUSIONS

In this paper we propose an automatic calibration proce-dure to determine the kinematic parameters of a mobile robotand its sensors. Our approach dynamically explores the spaceof possible actions, to determine the effect that each of themhas on the parameter estimate. After this exploration, oursystem chooses which action to take in order to improve thecurrent estimate. To this extent it aims at observing the fullparameter space and at reducing the error of the estimate untilno further improvements are possible. Our method comparesfavourably with procedures that require the human assistance

-5 0 5 10 15

-2

0

2

4

6

meters

mete

rs

Ground Truth

Unsupervised Calibration

Supervised Squared-Path Calibration

Supervised 8-Path Calibration

Fig. 9: In this figure we show the odometry estimation of the mobilebase. The red solid path is the ground truth. The green dashedpath is the odometry computed with the parameters estimated bythe unsupervised calibration. The blue dotted path is the odometrycomputed using the parameter estimate obtained with following asquared path, while the magenta dash-dotted path is the odometrycomputed using the parameters estimated following an 8-shapedpath.

both in terms of accuracy and efficiency and dynamicallyadapts to the robot configuration.

REFERENCES

[1] G. Oriolo A. Censi, L. Marchionni. Simultaneous maximum-likelihoodcalibration of robot and sensor parameters. In Proc. of the IEEEInt. Conf. on Robotics & Automation (ICRA), 2008.

[2] A. Tapus A. Martinelli, N. Tomatis and R. Siegwart. Simultaneouslocalization and odometry calibration for mobile robot. In IntelligentRobots and Systems, 2003. (IROS 2003). Proceedings. 2003 IEEE/RSJInternational Conference on, volume 2, pages 1499–1504 vol.2, 2003.

[3] R. A. Newcombe, S. Izadi, O. Hilliges, D. Molyneaux, D. Kim, A. J.Davison, , P. Kohli, J. Shotton, S. Hodges, and A. Fitzgibbon. Kinect-Fusion: Real-time dense surface mapping and tracking. In Proc. ofthe Int. Symposium on Mixed and Augmented Reality (ISMAR), 2011.

[4] J. Borenstein, B. Everett, and L. Feng. Navigating Mobile Robots:Systems and Techniques. A. K. Peters, Ltd., Wellesley, MA, 1996.

[5] A. Censi. An accurate closed-form estimate of ICP’s covariance. InProceedings of the IEEE International Conference on Robotics andAutomation (ICRA), pages 3167–3172, Rome, Italy, April 2007.

[6] F. Chenavier and J. L. Crowley. Position estimation for a mobilerobot using vision and odometry. In Robotics and Automation, 1992.Proceedings., 1992 IEEE International Conference on, pages 2588–2593 vol.3, 1992.

[7] A. Kelly. Fast and easy systematic and stochastic odometry calibration.In Intelligent Robots and Systems, 2004. (IROS 2004). Proceedings.2004 IEEE/RSJ International Conference on, volume 4, pages 3188–3194 vol.4, 2004.

[8] A. Martinelli and R. Siegwart. Estimating the odometry error of amobile robot during navigation. In Proc. of the European Conf. onMobile Robots (ECMR), 2003.

[9] W. Burgard R. Kuemmerle, G. Grisetti. Simultaneous calibration, lo-calization, and mapping. In Proceedings of the IEEE/RSJ InternationalConference on Intelligent Robots and Systems (IROS), San Francisco,CA, USA, September 2011.

[10] E. Rohmer, S.P.N. Singh, and M. Freese. V-rep: A versatile andscalable robot simulation framework. In Intelligent Robots and Systems(IROS), 2013 IEEE/RSJ International Conference on, pages 1321–1326, Nov 2013.

[11] J. Serafin and G. Grisetti. Nicp: Dense normal based point cloudregistration. In Intelligent Robots and Systems (IROS), 2015 IEEE/RSJInternational Conference on, pages 742–749, Sept 2015.

[12] C. R. Smith and P. Cheeseman. On the representation and estimationof spatial uncertainly. Int. J. Rob. Res., 5(4):56–68, December 1986.