An Augmented State Vector Approach to GPS-Based ... - IEEE Xplore
Proceedings of the 2007 IEEE/RSJ International Conference on Intelligent Robots and Systems San Diego, CA, USA, Oct 29 - Nov 2, 2007
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An Augmented State Vector Approach to GPS-Based Localization Francesco Capezio, Antonio Sgorbissa, Renato Zaccaria DIST – University of Genova, Italy Abstract - The paper focuses on the localization subsystem of ANSER, a mobile robot for autonomous surveillance in civilian airports and similar wide outdoor areas. ANSER localization subsystem is composed of a non-differential GPS unit and a laser rangefinder for landmark-based localization (inertial sensors are absent). An augmented state vector approach and an Extended Kalman filter are successfully employed to estimate the colored components in GPS noise, thus getting closer to the conditions for the EKF to be applicable. I. INTRODUCTION The work described in this paper is part of the ANSER1 project, an ongoing project for autonomous surveillance in civilian airports and similar wide outdoor areas. In the last years several autonomous surveillance systems based on a mobile platform have been presented (see for example [4][7][6]): autonomous navigation and selflocalization capabilities are fundamental prerequisites in all these systems. This is the reason why, starting from a minimal configuration in which self-localization relies on an Inertial Measurement Unit (IMU) and a Carrier Phase, Differential GPS receiver (CP-DGPS) [3][5][1][2], a very common approach is to equip the mobile platform with a large set of sensors (video cameras, Passive Infra-Red sensors, sonar, laser range finders etc.), thus consequently requiring a high computational power and complex data filtering techniques. In partial contrast with this “over equipping” philosophy, the ANSER self-localization sub-system relies only on a standard (non-differential) GPS unit, and on a laser rangefinder. Unfortunately, GPS data are known to be affected by low-frequency errors that cannot be modeled as zero mean, Additive White Gaussian Noise (AWGN), thus making simple state estimation approaches (e.g., Kalman Filter) unfeasible (Figure 1). Under these conditions, the filter is very likely to underestimate the actual positioning error, and possibly to diverge without the system being aware of that: this problem is faced in [8] by purposely increasing the estimated covariance, through fuzzy rules acting on the basis of the current innovation. This work proposes to estimate the low-frequency components of GPS noise through an augmented state vector approach, similar to [2][9]. The paper shows that, by combining laser-based localization and GPS measurement, it is possible to estimate both the robot’s position and the non1 ANSER is an acronym for Airport Night Surveillance Expert Robot, and the Latin name for “goose” (referring to the Capitoline Geese which – according to tradition - neutralized a nighttime attack by the Gauls during the siege of Rome).
1-4244-0912-8/07/$25.00 ©2007 IEEE.
AWG components of GPS noise. A particularity of our approach is that, by making assumptions on the spectral characteristics of the GPS noise, it is possible to temporarily accept the system not to be fully observable. Section II briefly discusses about the properties of GPS noise, and describes the localization techniques adopted; Section III theoretically investigates the properties of the approach, and carries out an observability analysis; Section IV presents experimental results obtained so far with a realistic simulator, and in a field set-up at the Villanova d’Albenga Airport. Conclusions follow.
Figure 1: FFT of GPS latitude data
Figure 2: The estimated GPS bias
II. GPS- AND LASER-BASED SELF-LOCALIZATION A.. GPS-Based Localization A single non-differential GPS receiver provides the mobile robot with absolute position measurements that, under the assumption of unicycle kinematics, allow to correct the x, y, and θ−components of the estimate provided by odometry (details in the following). However, it is known that GPS measurements are corrupted by different error sources, which degrade the positioning accuracy: 1) ionospheric effects, i.e., changes in the speed of the GPS signals as they
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pass through the ionosphere, as a consequence of changing atmospheric conditions; 2) inaccurate computation of the satellites position due to ephemeris errors; 3) inaccurate synchronization between the satellite and the receiver clock; 4) tropospheric effects, i.e., changes in the speed of the GPS signals depending on humidity; 5) multipath distortion, due to reflecting surfaces in the immediate surroundings of the receiver; 6) numerical errors. In literature, many experiments have been performed trying to characterize the contribution of each error source on the final positioning accuracy. In spite of some differences, it is commonly agreed that, in the 95% of cases, the error is approximately within 15 meters: the biggest contribution is given by ionospheric effects, introducing an error that varies slowly with time (but is usually less than 510 meters), and by multipath distortion, introducing an error that is small in open areas, but increases in presence of occlusions (e.g., buildings, tree canopy, etc.), especially when satellites are low on the horizon. Many techniques are available to reduce these errors: e.g., it is possible to consider two different transmission frequencies L1 and L2 at the same time to estimate and reduce the error, since atmospheric conditions affect the speed of radio waves differently based on frequency, or even applying special antennas and filtering techniques to ignore GPS signals that have been reflected or emitted by low elevation satellites. This is not the place for an exhaustive investigation. In general, the error introduced by 1) 2), 3) and 4) is referred to as common-mode error, and it is almost invariant within a range of many kilometers: whenever this is the most significant error source, exploiting Differential GPS measurements is the most powerful technique to substantially increase accuracy. The underlying idea is simple: a ground reference station - whose georeferenced position is known - receives the GPS signal from all the satellite in sight, and compare actual and expected pseudorange measurements to estimate the current common -mode error. This information is sent to the DGPS receiver whose position must be computed, therefore enabling it to significantly improve its accuracy: experiments return an error of about 20cm in the neighborhoods of the reference station, with an error growth of about 25cm per 100km from the broadcasting site (since multipath distortion depends on the specific location of the GPS receiver, obviously it cannot be reduced through DGPS measurements). Unfortunately, the DGPS signal is not worldwide available, since it depends on the presence of a reference station in the neighborhoods; whereas United States are almost completely covered, the same is not true for many areas in Europe, and the situation is even worse in less developed countries of Africa and Asia. The situation improves when considering Satellite-Based Augmentation Systems (e.g., the WAAS - Wide Area Augmentation System in USA and Canada, the EGNOS - European Geostationary Navigation Overlay Service in Europe, MFSAS – Multi Functional Satellite Augmentation System, in Japan, etc.), that share the same principle of DGPS but are based on ground stations and geostationaries satellites to broadcast the correction messages to GPS receivers.
Satellite-based augmentation systems usually guarantee a position accuracy of approximately 7 meters 95% of the time, with a horizontal accuracy of less than 2 meters recorded in many locations. However, also in this case, the signal is available only in some areas of the world, thus not providing a general solution to the positioning problem. A possible solution is to install – if absent - an ad hoc DGPS reference station; however, this obviously has a cost, even if it is affordable in most surveillance applications. This work proposes an alternative solution especially suited for autonomous rover navigation: since one can realistically assume that a laser rangefinder is already available on the rover for different purposes (e.g., navigation & obstacle avoidance, surveillance, etc.), the proposed solution does not imply – in theory – any additional costs. In the following, it is assumed that the rover operates in a wide, open area (e.g., far from tall buildings and not under tree canopy). This partially limits the application field of the system; notice however that the conditions are met in many interesting domains, from agricultural, to humanitarian demining, to surveillance in airports, etc. As anticipated, both theoretical and experimental analyses in literature show that the common-mode error introduces into the GPS measurement a colored noise with a significant lowfrequency component. That is, when averaged over a short time period, errors in latitude and longitude have a non-zero mean value: in the following, this will be referred to as a “bias” in GPS measurements. The analysis of longitude and latitude data collected at a fixed location during 24 hours shows this effect: by considering the Fast Fourier Transform (FFT) of GPS longitude and latitude data (the latter shown in Figure 1), low-frequency components can be noticed, corresponding to slow variation of the signal in time. When dealing with colored noise, a common approach is to consider the noise signal as a component to be estimated in the state vector. An augmented state vector x is defined; x comprises both (x, y, θ) components of the robot’s position, and the (xGPS , yGPS) components of the low-frequency bias in GPS measurements. By estimating the bias in GPS measurements (Figure 2), one can expect that – at least in theory – the precision of GPS data should improve, therefore making localization more accurate. Moreover, by separating the colored components from AWG components of GPS noise, the system gets closer to the conditions for an Extended Kalman Filter to be applicable. This means that the divergence problems considered in [8] should not manifest. Unfortunately, the increased dimensions of the state vector usually requires to consider additional information sources to keep all the state components observable. In this work, a laser range-finder is proposed as the candidate sensor to provide such additional information. B. Laser Based Localization In indoor applications, it is very common to provide mobile robots with an a-priori map of the environment. If the robot is equipped with a laser rangefinder, standard techniques are available to updates the robot’s position by comparing this map with the features detected by the laser
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rangefinder. A similar system, relying on an EKF estimator to correct the robot pose, has been implemented on board the ANSER rover. In particular, [11] describes in details how segment-like features are extracted from raw laser data and compared with the a-priori model: 1) line extraction produces a set of lines { l j }; 2) the Mahalanobis distance associated to each couple of line ( l j , mi ) is computed
It includes the robot’s position and orientation with respect to a fixed frame Fw , and the two components (with respect to the same frame) of the bias in GPS measurements. After integrating the dynamic equations of the system through a standard Euler approximation with step size ∆t=0.1sec, the system can be described with the following finite difference Equations:
(where { mi } is a set of oriented segment lines that define the a-priori map); 3) for each l j , the line mi for which such distance is minimum is selected and fed to the EKF. In the outdoor scenario, lines in the a-priori map could possibly correspond to the external walls of buildings. When new measurements are available (i.e., both GPS data and the features detected by a laser rangefinders), the full state vector can be estimated through observations. Obviously, it is reasonable to assume that features for laser-based localization are not always available, since the robot mostly traverses areas where no buildings are present at all. If features for laser-based localization would be available most of the time, they would be sufficient for accurately computing the rover’s pose, and the GPS would not be required at all. With respect to other augmented-state systems dealing with colored noise components, the particularity of this work is that, even when additional measurement are not always available (and hence the augmented-state vector is not fully observable), the spectral characteristics of common-mode error allows to assume that the bias in GPS errors changes slowly. That is, when features for laser-based localization are detected, the bias is correctly estimated, and the estimate can be used to reduce commonmode error (for a given time) even in areas where the rover car rely exclusively on GPS. Unfortunately, the external walls of buildings are not the best candidate feature for laser-based localization; in fact, in the proximity of buildings, GPS measurements are worse, and the system is more likely to estimate the error due to multipath distortion and the occlusion of low elevation satellites, instead of the slowly changing common-mode error. As a result, the estimated bias could not be sufficient to reduce the error when laser measurements are no longer available. To avoid this, ad hoc landmarks for laser-based localization can be installed in chosen locations: e.g., along a pre-defined route, or even along the border of a field (as in the agricultural or in the humanitarian de-mining domain). First experiments in this sense show that if the rover is able to estimate the GPS bias every 200meters, this is sufficient to keep the average error in (x,y) below 80cm. III. THE AUGMENTED STATE APPROACH As anticipated, the approach relies on the idea of “guessing” the bias that affects GPS measurements at a given time, by including it in the state to be estimated. The resulting augmented state vector is shown in Equation 1. xT = [ x
y θ
xGPS
yGPS ]
(1)
xk = xk −1 + dsk −1 ⋅ cos θ k −1 yk = yk −1 + dsk −1 ⋅ sin θ k −1 with ds = (dr + dl ) 2 dω = (dr − dl ) D θ k = θ k −1 + dω k −1 xGPS ,k = xGPS ,k −1 yGPS ,k = yGPS , k −1
(2)
The first three equations represent the discrete approximation of the robot’s inverse kinematics, by assuming a unicycle differential drive model. As usual, dr and dl indicate the linear displacements of the right and left driving wheels, and D is the distance between them. In the last two Equations, the dynamic of the bias xGPS T = [ xGPS yGPS ] is modeled. Notice that a constant dynamic is assumed for x GPS T , since no cues are available to make more accurate hypotheses. This means that, when predicting the new state in the time-update phase of the EKF, xGPS T is left unchanged. However, the predicted value
xGPS T is updated whenever new measurement are available (i.e., in the correction phase of the EKF), thus finally producing an estimate that varies in time, and hopefully approximates the actual bias in GPS measurements. The approach seems reasonable whenever a component of the state vector changes slowly in time with respect to the other components, which is exactly the case. When considering the remaining noise that affects x, the process can be described as governed by a non-linear stochastic difference Equation in the form of
x k = f (xk −1 , u k −1 , w k −1 )
with
x ∈ ℜ5
w = N (0,W )
(3)
where w represents the process noise, and u is the driving function; i.e., the 2-dimensional vector describing the current wheels displacements dr and dl ( uT = [dr dl ] ). For what concerns errors in the process, they are currently modeled through a vector w T = [wr wl wg ] . The first two element sums up to dl and dr (e.g., when the left encoder returns dl, the actual path traveled by the left wheel is dl+wl), whereas wg represents the error made in assuming that the bias has not changed since the last iteration of the Filter. By assuming that w has a zero-mean Gaussian distribution with covariance matrix W, systematic errors in odometry due to the approximate knowledge of the robot’s geometric characteristics are not explicitly considered (in theory, geometric parameters should be included in the augmented state vector as well, as proposed in [9]). Observations are provided both by the GPS and, when available, by the laser rangefinder. The measurements provided by the GPS are a non-linear function of the state:
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zlong = CLONG ( zlat ) ⋅ ( x + xGPS ) zlat = CLAT ⋅ ( y + yGPS )
(4)
Where z=(zlong, zlat) is a 2-dimensional vector representing the longitude and latitude of the mobile robot, by supposing the X-axis of Fw lying on the parallel passing through Fw’s origin, and Fw’s Y-axis lying on the meridian. The measurement model is not linear, mainly because the relationship between measured longitude and the estimated x-coordinate varies with the latitude, as a consequence of the non planarity of the earth surface (according to CLONG (zlat)). For each line l j observed by the laser rangefinder, the line
one GPS measurement). However, this is assumed only to investigate the state’s observability; during experiments, the laser rangefinder and the GPS returns observations asynchronously, and each observation is used to update the state as soon as available. The observability analysis demonstrates that, even if each measurement is able to correct the state only partially, the state is fully observable when more measurements are considered in cascade.
1 0 A = 0 0 0
mi that best matches l j can be expressed as:
ai ⋅ xk + bi ⋅ yk + ci zρ = ai2 + bi2 z = tan −1 − ai − θ b k α i
1 CLONG 0 ai 2 ai + bi 2 H= 0 a m a 2 +b 2 m m 0
(5)
where ai, bi and ci are the parameters characterizing the implicit equation of mi ; zρ and zα are, respectively, the distance between the line and the robot, and the angle between the line and the robot’s heading. When putting together Equations 4 and 5, the measurement model results to be a non-linear function of the state: z k = h( x k , v k )
with
v = N (0, R)
(6)
Since non-AWG components of the GPS noise are estimated in the state vector, the remaining noise can be reasonably modeled with the vector v, a zero-mean AWG noise with covariance matrix R. Equation 2 can be used to compute the a-priori state estimate at time k. Next, whenever new GPS or laser rangefinder data are available, they are fused with the apriori estimate through an Extended Kalman Filter to produce a new estimate, thus reducing errors that are inherently present in odometry and providing a new estimate for the GPS bias. Obviously, to evaluate the soundness of the previous assertion, it is necessary to perform an observability analysis of the system. The Kalman theorem requires to compute the observability matrix T T T T , where A and H are the 4 T Q = H | A H | ... | (A ) H
Jacobian matrices of the partial derivatives of f and h with respect to x (Q’s full expression is not shown for sake of brevity). The analysis shows that Q has full rank (and hence the state is fully observable) only when at least two observations l j and lm are available, corresponding to nonparallel lines mi and mn , together with a single GPS measurement. To meet this requirement, each landmark for laser-based localization must be constituted of two perpendicular surfaces. Matrix A results to be as in Equation 7, whereas H results to be as in Equation 8. By inspecting matrix H, one could infer that the filter is updated only when a triplet of observations are available (i.e., two non-parallel lines and
0 −dsk −1 ⋅ sin(θ k −1 ) 0 0 1 dsk −1 ⋅ cos(θ k −1 ) 0 0 0 1 0 0 0 0 1 0 0 0 0 1 0 1
0
1
CLONG
0
0
0
0
0
−1
0
bm
0
0
−1
0
CLAT bi
ai 2 + bi 2
am 2 + bm 2 0
(7)
1 CLAT 0 0 0 0 0
(8)
Unfortunately, when the rover is moving between two subsequent landmarks, the laser cannot detect any line mapped in the a-priori map: the H matrix fed to the KF comprises only the first two rows in Equation 8. By computing again the observability matrix Q, this yields the result in Equation 9. 1 / C LONG 0 1 / C LONG 0 1 / C LONG 0 Q= 1 / C LONG 0 1 / C LONG 0
0 1 / C LAT 0 1 / C LAT 0 1 / C LAT 0 1 / C LAT 0 1 / C LAT
0 0 1 ⋅ dsk −1 ⋅ sin(θ k −1 ) C LONG 1 ⋅ dsk −1 ⋅ cos(θ k −1 ) C LAT 2 − ⋅ dsk −1 ⋅ sin(θ k −1 ) C LONG 2 ⋅ dsk −1 ⋅ cos(θ k −1 ) C LAT 3 − ⋅ dsk −1 ⋅ sin(θ k −1 ) C LONG 3 ⋅ dsk −1 ⋅ cos(θ k −1 ) C LAT 4 − ⋅ dsk −1 ⋅ sin(θ k −1 ) C LONG 4 ⋅ dsk −1 ⋅ cos(θ k −1 ) C LAT −
1 / C LONG 0 0 1 / C LAT 1 / C LONG 0 0 1 / C LAT 1 / C LONG 0 0 1 / C LAT 1 / C LONG 0 0 1 / C LAT 1 / C LONG 0 0 1 / C LAT
(9)
When dsk −1 ≠ 0 , i.e. when the translational speed of the robot is not null [10], the rank of Q is 3. The rank is not full since Q’s first column (corresponding to the x-component of the state vector x) equals the fourth column (corresponding to the xGPS-component), and Q’s second column (i.e., the ycomponent) equals the fifth column (i.e., the yGPScomponent). On the opposite, Q’s third column (corresponding to the θ-component of x) is linearly independent from the others. Q’s analysis confirms the intuition that – when no laser data are available – the
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subspace defined by x+xGPS, y+yGPS, and θ is fully observable: the robot’s orientation is still corrected by GPS data, but the position has a permanent error that depends on the current estimate of the GPS bias. The observability of orientation is rather intuitive, since – with unicycle kinematics – orientation always corresponds to the rover’s direction of motion (it becomes unobservable when the rover is still). About the other components of the state, the innovation due to GPS measurements is distributed by the Kalman gain onto x-, y-, xGPS, and yGPS components of the state according to the current value of the state covariance matrix P. Since a constant dynamic is assumed for xGPS, and yGPS, corrections are adequately distributed onto the state components only if the actual bias in GPS changes slowly, and given that a new landmark for laser-based re-localization will soon be available.
experiment is repeated twice: A-tests are performed without bias estimation, whereas B-tests correspond to localization with GPS bias estimation. Figure 3 shows an A-test (i.e., without bias estimation): the rover moves along a circular path which is about 200 meters long. The Figure shows that the estimated position significantly diverges from the actual circular trajectory: the error decreases only when a landmark is visible. Figure 4 shows a B-test (i.e., with bias estimation) performed in the same conditions; once again, the error decreases when landmarks are visible, but in this case it increases slowly, thanks to the fact that the bias changes slowly as well. In this experiment, when the distance between landmarks (measured along the path) is about 50 meters, the average positioning error in the A-test is about 1.7m, whereas in the B-test it is below 0.3m. L3
L3
L2
L4
L2
L4
L1
L1
Figure 4: B-test: estimated path with bias estimation
Figure 3: A-test: estimated path without bias estimation
IV. EXPERIMENTAL RESULTS Many tests in MatLab with real GPS data have been performed, as well as preliminary tests with the ANSER rover at the Albenga Airport. In the first set of experiments, the rover is simulated, by adding systematic and non-systematic odometric errors, as well as errors in laser measurements. However, GPS data are real; they are taken at a fixed known location for 24 hours at 1Hz (latitude data are shown in Figure 2), which allows to record measurement errors. During tests, real recorded errors are added to ideal GPS measurement, thus allowing to validate the ability of the approach to estimate the real GPS bias. During each run, the robot is requested to move either along a circular or straight path. Landmarks for selflocalization (i.e., couples of perpendicular line segments) are distributed along the path at a distance that varies from test to test: roughly 50m, 100m, 200m. The rover is able to see landmarks only when they are closer than 5m; consequently, during most of the time, the rover relies exclusively on GPS measurement, whereas landmarks are used – when available - to correct all the components of the augmented state vector. Since we are interested in the behavior of the rover when relying only on GPS, errors in position and orientation are recorded only when landmarks are not visible. Finally, each
Figure 5: B-test: actual and estimated bias
Figure 5 shows the x-component of the actual and the estimated bias (upper lines) during the test in Figure 4. It can be noticed that the two plots overlap only when the full state vector is observable (i.e., when landmarks are visible, corresponding to peaks in the lower line). However, when the rover relies only on GPS, the trend of the two upper lines is similar, thus allowing to reduce the positioning error. Table 1 summarizes A-tests when increasing the distance between landmarks, Table 2 summarizes B-tests (averaging over 50 runs, including both circular and straight paths). By defining ex , ey , eϑ as the three components of the error vector, the first column of both Tables shows the average
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positioning error ex , y = E ( ex2 + ey2 ) , the second column shows its standard deviation
σ e , the third column shows the
average orientation error eϑ , and finally the fourth column shows its standard deviation σ e . The reduction in positioning errors due to the bias estimation is evident. Distance between landmarks
ex , y
σ x, y
eϑ
σϑ
50m
1.744
1.071
0.697
0.913
100m
1.938
0.954
0.542
0.821
200m
2.024
0.940
0.563
0.931
Figure 6: Estimated position in a real scenario
Table 1: Errors statistics in A-tests (in meters) Distance between landmarks
ex , y
σ x, y
eϑ
σϑ
50m
0.303
0.296
0.031
0.317
100m
0.760
0.631
0.157
0.906
200m
1.256
0.731
0.532
1.059
Table 2:Errors statistics in B-test (in meters)
Preliminary real world experiments have been carried out as well with the ANSER robot in the Albenga Airport. During the test, the robot is manually driven at 1.0m/s along a pre-established cyclic path that is about 500 meters long. Different A- and B-tests are been performed, by memorizing the robot’s estimated position in a finite number of selected places along the path (see Figure 6, position estimates are marked with circles in A-tests and with asterisks in B-tests). A single landmark for laser-based localization is available, roughly located in x=0,y=0. Preliminary tests show that, when estimating the GPS bias, the robot exhibits a more repeatable behavior: when looping along the path, position estimates recorded during different loops have a bigger dispersion in A-tests (circles) than in Btests (asterisks). A test campaign is currently being performed to validate these results. V. CONCLUSIONS The paper describes the localization subsystem of a mobile robot that has been designed for patrols and surveillance tasks within a civilian airport. Instead of equipping the robot with a huge amount of expensive sensors (and the computing power that is adequate to deal with them), a simple approach is chosen that relies exclusively on a non-differential GPS unit and a laser rangefinder (i.e., inertial sensors are absent). Laser measurements are exploited only in some areas of the outdoor patrol path of the robot, i.e. where it is possible detect line features and match them against an a-priori model of the environment. Along the rest of the path, the robot relies on GPS-based localization. An Extended Kalman Filter algorithm is employed to estimate an augmented state vector comprising the robot position and orientation, together with the low frequency components (bias) of the GPS error.
Figure 7: The robot ANSER at Albenga Airport
REFERENCES [1] Dissanayake, G., Sukkarieh, S., Nebot, E., Durrant-Whyte, H., “The aiding of a low-cost strapdown inertial measurement unit using vehicle model constraints for land vehicle applications”, IEEE Trans. on Robotics and Automation, vol. 17, Issue 5, pp. 731 – 747, Oct. 2001. [2] Farrell, J. A., Givargis, T., Barth, M., “Real-time differential carrier phase GPS-aided INS”, IEEE Trans. Contr. Syst. Technol., vol. 8, pp. 709721, July 2000. [3] Schönberg, T., Ojala, M., Suomela, J., Torpo, A., and Halme, A, “Positioning an autonomous off-road vehicle by using fused DGPS and inertial navigation”, 2nd IFAC Conference on Intelligent Autonomous Vehicles, Espoo, 1995 Helsinki, Finland, 1995, pp. 226-231 [4] Heath-Pastore, T., H.R. Everett, and K. Bonner, “Mobile Robots for Outdoor Security Applications”, 8th International Topical Meeting on Robotics and Remote Systems (ANS'99), Pittsburgh, PA, 1999. [5] Panzieri, S., Pascucci, F., and Ulivi, G. “An Outdoor Navigation System Using GPS and Inertial Platform”, IEEE Trans. on mechatronics, vol. 7, no 2, june 2002. [6] Vidal, R.,, Shakernia, O., Kim, H. j., Shim, D. H., Sastry, S., “Probabilistic Pursuit–Evasion Games: Theory, Implementation, and Experimental Evaluation”, IEEE Trans. on Robotics and Automation, vol. 18, no. 5, October 2002 [7] Saptharishi M., Oliver, C. S., Diehl, C. P., Bhat, K. S., Dolan, J. M., Trebi-Ollennu, A., Khosla, P. K., “Distributed Surveillance and Reconnaissance Using Multiple Autonomous ATVs: CyberScout”, IEEE Trans. on Robotics and Automation, vol. 18, no.5, October 2002 [8] Sasiadek, J. Z., Wang, Q., “Low cost automation using INS/GPS data fusion for accurate positioning”, Robotica (2003) volume 21, pp. 255-260. 2003 Cambridge University Press [9] Martinelli, A. (2002) The Odometry Error of a Mobile Robot with a Synchronous Drive System. IEEE Trans. on Robotics and Automation, Vol 18, NO. 3 June 2002, pp 399-405. [10] Capezio, F., Sgorbissa, A., and Zaccaria, R. “GPS-based Localizzation for UGV Performing Surveillance Patrols in Wide Outdoor Areas”, Int. Conf. on Field and Service Robotics, Australia 2005. [11] Capezio, F., Mastrogiovanni, F., Sgorbissa, A., and Zaccaria, R., “Fast Position Tracking of an Autonomous Vehicle in Cluttered and Dynamic Indoor Environments”, SYROCO 2006, Bologna (Italy) September 6 - 8, 2006
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An Augmented State Vector Approach to GPS-Based Localization Francesco Capezio, Antonio Sgorbissa, Renato Zaccaria DIST – University of Genova, Italy Abstract - The paper focuses on the localization subsystem of ANSER, a mobile robot for autonomous surveillance in civilian airports and similar wide outdoor areas. ANSER localization subsystem is composed of a non-differential GPS unit and a laser rangefinder for landmark-based localization (inertial sensors are absent). An augmented state vector approach and an Extended Kalman filter are successfully employed to estimate the colored components in GPS noise, thus getting closer to the conditions for the EKF to be applicable. I. INTRODUCTION The work described in this paper is part of the ANSER1 project, an ongoing project for autonomous surveillance in civilian airports and similar wide outdoor areas. In the last years several autonomous surveillance systems based on a mobile platform have been presented (see for example [4][7][6]): autonomous navigation and selflocalization capabilities are fundamental prerequisites in all these systems. This is the reason why, starting from a minimal configuration in which self-localization relies on an Inertial Measurement Unit (IMU) and a Carrier Phase, Differential GPS receiver (CP-DGPS) [3][5][1][2], a very common approach is to equip the mobile platform with a large set of sensors (video cameras, Passive Infra-Red sensors, sonar, laser range finders etc.), thus consequently requiring a high computational power and complex data filtering techniques. In partial contrast with this “over equipping” philosophy, the ANSER self-localization sub-system relies only on a standard (non-differential) GPS unit, and on a laser rangefinder. Unfortunately, GPS data are known to be affected by low-frequency errors that cannot be modeled as zero mean, Additive White Gaussian Noise (AWGN), thus making simple state estimation approaches (e.g., Kalman Filter) unfeasible (Figure 1). Under these conditions, the filter is very likely to underestimate the actual positioning error, and possibly to diverge without the system being aware of that: this problem is faced in [8] by purposely increasing the estimated covariance, through fuzzy rules acting on the basis of the current innovation. This work proposes to estimate the low-frequency components of GPS noise through an augmented state vector approach, similar to [2][9]. The paper shows that, by combining laser-based localization and GPS measurement, it is possible to estimate both the robot’s position and the non1 ANSER is an acronym for Airport Night Surveillance Expert Robot, and the Latin name for “goose” (referring to the Capitoline Geese which – according to tradition - neutralized a nighttime attack by the Gauls during the siege of Rome).
1-4244-0912-8/07/$25.00 ©2007 IEEE.
AWG components of GPS noise. A particularity of our approach is that, by making assumptions on the spectral characteristics of the GPS noise, it is possible to temporarily accept the system not to be fully observable. Section II briefly discusses about the properties of GPS noise, and describes the localization techniques adopted; Section III theoretically investigates the properties of the approach, and carries out an observability analysis; Section IV presents experimental results obtained so far with a realistic simulator, and in a field set-up at the Villanova d’Albenga Airport. Conclusions follow.
Figure 1: FFT of GPS latitude data
Figure 2: The estimated GPS bias
II. GPS- AND LASER-BASED SELF-LOCALIZATION A.. GPS-Based Localization A single non-differential GPS receiver provides the mobile robot with absolute position measurements that, under the assumption of unicycle kinematics, allow to correct the x, y, and θ−components of the estimate provided by odometry (details in the following). However, it is known that GPS measurements are corrupted by different error sources, which degrade the positioning accuracy: 1) ionospheric effects, i.e., changes in the speed of the GPS signals as they
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pass through the ionosphere, as a consequence of changing atmospheric conditions; 2) inaccurate computation of the satellites position due to ephemeris errors; 3) inaccurate synchronization between the satellite and the receiver clock; 4) tropospheric effects, i.e., changes in the speed of the GPS signals depending on humidity; 5) multipath distortion, due to reflecting surfaces in the immediate surroundings of the receiver; 6) numerical errors. In literature, many experiments have been performed trying to characterize the contribution of each error source on the final positioning accuracy. In spite of some differences, it is commonly agreed that, in the 95% of cases, the error is approximately within 15 meters: the biggest contribution is given by ionospheric effects, introducing an error that varies slowly with time (but is usually less than 510 meters), and by multipath distortion, introducing an error that is small in open areas, but increases in presence of occlusions (e.g., buildings, tree canopy, etc.), especially when satellites are low on the horizon. Many techniques are available to reduce these errors: e.g., it is possible to consider two different transmission frequencies L1 and L2 at the same time to estimate and reduce the error, since atmospheric conditions affect the speed of radio waves differently based on frequency, or even applying special antennas and filtering techniques to ignore GPS signals that have been reflected or emitted by low elevation satellites. This is not the place for an exhaustive investigation. In general, the error introduced by 1) 2), 3) and 4) is referred to as common-mode error, and it is almost invariant within a range of many kilometers: whenever this is the most significant error source, exploiting Differential GPS measurements is the most powerful technique to substantially increase accuracy. The underlying idea is simple: a ground reference station - whose georeferenced position is known - receives the GPS signal from all the satellite in sight, and compare actual and expected pseudorange measurements to estimate the current common -mode error. This information is sent to the DGPS receiver whose position must be computed, therefore enabling it to significantly improve its accuracy: experiments return an error of about 20cm in the neighborhoods of the reference station, with an error growth of about 25cm per 100km from the broadcasting site (since multipath distortion depends on the specific location of the GPS receiver, obviously it cannot be reduced through DGPS measurements). Unfortunately, the DGPS signal is not worldwide available, since it depends on the presence of a reference station in the neighborhoods; whereas United States are almost completely covered, the same is not true for many areas in Europe, and the situation is even worse in less developed countries of Africa and Asia. The situation improves when considering Satellite-Based Augmentation Systems (e.g., the WAAS - Wide Area Augmentation System in USA and Canada, the EGNOS - European Geostationary Navigation Overlay Service in Europe, MFSAS – Multi Functional Satellite Augmentation System, in Japan, etc.), that share the same principle of DGPS but are based on ground stations and geostationaries satellites to broadcast the correction messages to GPS receivers.
Satellite-based augmentation systems usually guarantee a position accuracy of approximately 7 meters 95% of the time, with a horizontal accuracy of less than 2 meters recorded in many locations. However, also in this case, the signal is available only in some areas of the world, thus not providing a general solution to the positioning problem. A possible solution is to install – if absent - an ad hoc DGPS reference station; however, this obviously has a cost, even if it is affordable in most surveillance applications. This work proposes an alternative solution especially suited for autonomous rover navigation: since one can realistically assume that a laser rangefinder is already available on the rover for different purposes (e.g., navigation & obstacle avoidance, surveillance, etc.), the proposed solution does not imply – in theory – any additional costs. In the following, it is assumed that the rover operates in a wide, open area (e.g., far from tall buildings and not under tree canopy). This partially limits the application field of the system; notice however that the conditions are met in many interesting domains, from agricultural, to humanitarian demining, to surveillance in airports, etc. As anticipated, both theoretical and experimental analyses in literature show that the common-mode error introduces into the GPS measurement a colored noise with a significant lowfrequency component. That is, when averaged over a short time period, errors in latitude and longitude have a non-zero mean value: in the following, this will be referred to as a “bias” in GPS measurements. The analysis of longitude and latitude data collected at a fixed location during 24 hours shows this effect: by considering the Fast Fourier Transform (FFT) of GPS longitude and latitude data (the latter shown in Figure 1), low-frequency components can be noticed, corresponding to slow variation of the signal in time. When dealing with colored noise, a common approach is to consider the noise signal as a component to be estimated in the state vector. An augmented state vector x is defined; x comprises both (x, y, θ) components of the robot’s position, and the (xGPS , yGPS) components of the low-frequency bias in GPS measurements. By estimating the bias in GPS measurements (Figure 2), one can expect that – at least in theory – the precision of GPS data should improve, therefore making localization more accurate. Moreover, by separating the colored components from AWG components of GPS noise, the system gets closer to the conditions for an Extended Kalman Filter to be applicable. This means that the divergence problems considered in [8] should not manifest. Unfortunately, the increased dimensions of the state vector usually requires to consider additional information sources to keep all the state components observable. In this work, a laser range-finder is proposed as the candidate sensor to provide such additional information. B. Laser Based Localization In indoor applications, it is very common to provide mobile robots with an a-priori map of the environment. If the robot is equipped with a laser rangefinder, standard techniques are available to updates the robot’s position by comparing this map with the features detected by the laser
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rangefinder. A similar system, relying on an EKF estimator to correct the robot pose, has been implemented on board the ANSER rover. In particular, [11] describes in details how segment-like features are extracted from raw laser data and compared with the a-priori model: 1) line extraction produces a set of lines { l j }; 2) the Mahalanobis distance associated to each couple of line ( l j , mi ) is computed
It includes the robot’s position and orientation with respect to a fixed frame Fw , and the two components (with respect to the same frame) of the bias in GPS measurements. After integrating the dynamic equations of the system through a standard Euler approximation with step size ∆t=0.1sec, the system can be described with the following finite difference Equations:
(where { mi } is a set of oriented segment lines that define the a-priori map); 3) for each l j , the line mi for which such distance is minimum is selected and fed to the EKF. In the outdoor scenario, lines in the a-priori map could possibly correspond to the external walls of buildings. When new measurements are available (i.e., both GPS data and the features detected by a laser rangefinders), the full state vector can be estimated through observations. Obviously, it is reasonable to assume that features for laser-based localization are not always available, since the robot mostly traverses areas where no buildings are present at all. If features for laser-based localization would be available most of the time, they would be sufficient for accurately computing the rover’s pose, and the GPS would not be required at all. With respect to other augmented-state systems dealing with colored noise components, the particularity of this work is that, even when additional measurement are not always available (and hence the augmented-state vector is not fully observable), the spectral characteristics of common-mode error allows to assume that the bias in GPS errors changes slowly. That is, when features for laser-based localization are detected, the bias is correctly estimated, and the estimate can be used to reduce commonmode error (for a given time) even in areas where the rover car rely exclusively on GPS. Unfortunately, the external walls of buildings are not the best candidate feature for laser-based localization; in fact, in the proximity of buildings, GPS measurements are worse, and the system is more likely to estimate the error due to multipath distortion and the occlusion of low elevation satellites, instead of the slowly changing common-mode error. As a result, the estimated bias could not be sufficient to reduce the error when laser measurements are no longer available. To avoid this, ad hoc landmarks for laser-based localization can be installed in chosen locations: e.g., along a pre-defined route, or even along the border of a field (as in the agricultural or in the humanitarian de-mining domain). First experiments in this sense show that if the rover is able to estimate the GPS bias every 200meters, this is sufficient to keep the average error in (x,y) below 80cm. III. THE AUGMENTED STATE APPROACH As anticipated, the approach relies on the idea of “guessing” the bias that affects GPS measurements at a given time, by including it in the state to be estimated. The resulting augmented state vector is shown in Equation 1. xT = [ x
y θ
xGPS
yGPS ]
(1)
xk = xk −1 + dsk −1 ⋅ cos θ k −1 yk = yk −1 + dsk −1 ⋅ sin θ k −1 with ds = (dr + dl ) 2 dω = (dr − dl ) D θ k = θ k −1 + dω k −1 xGPS ,k = xGPS ,k −1 yGPS ,k = yGPS , k −1
(2)
The first three equations represent the discrete approximation of the robot’s inverse kinematics, by assuming a unicycle differential drive model. As usual, dr and dl indicate the linear displacements of the right and left driving wheels, and D is the distance between them. In the last two Equations, the dynamic of the bias xGPS T = [ xGPS yGPS ] is modeled. Notice that a constant dynamic is assumed for x GPS T , since no cues are available to make more accurate hypotheses. This means that, when predicting the new state in the time-update phase of the EKF, xGPS T is left unchanged. However, the predicted value
xGPS T is updated whenever new measurement are available (i.e., in the correction phase of the EKF), thus finally producing an estimate that varies in time, and hopefully approximates the actual bias in GPS measurements. The approach seems reasonable whenever a component of the state vector changes slowly in time with respect to the other components, which is exactly the case. When considering the remaining noise that affects x, the process can be described as governed by a non-linear stochastic difference Equation in the form of
x k = f (xk −1 , u k −1 , w k −1 )
with
x ∈ ℜ5
w = N (0,W )
(3)
where w represents the process noise, and u is the driving function; i.e., the 2-dimensional vector describing the current wheels displacements dr and dl ( uT = [dr dl ] ). For what concerns errors in the process, they are currently modeled through a vector w T = [wr wl wg ] . The first two element sums up to dl and dr (e.g., when the left encoder returns dl, the actual path traveled by the left wheel is dl+wl), whereas wg represents the error made in assuming that the bias has not changed since the last iteration of the Filter. By assuming that w has a zero-mean Gaussian distribution with covariance matrix W, systematic errors in odometry due to the approximate knowledge of the robot’s geometric characteristics are not explicitly considered (in theory, geometric parameters should be included in the augmented state vector as well, as proposed in [9]). Observations are provided both by the GPS and, when available, by the laser rangefinder. The measurements provided by the GPS are a non-linear function of the state:
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zlong = CLONG ( zlat ) ⋅ ( x + xGPS ) zlat = CLAT ⋅ ( y + yGPS )
(4)
Where z=(zlong, zlat) is a 2-dimensional vector representing the longitude and latitude of the mobile robot, by supposing the X-axis of Fw lying on the parallel passing through Fw’s origin, and Fw’s Y-axis lying on the meridian. The measurement model is not linear, mainly because the relationship between measured longitude and the estimated x-coordinate varies with the latitude, as a consequence of the non planarity of the earth surface (according to CLONG (zlat)). For each line l j observed by the laser rangefinder, the line
one GPS measurement). However, this is assumed only to investigate the state’s observability; during experiments, the laser rangefinder and the GPS returns observations asynchronously, and each observation is used to update the state as soon as available. The observability analysis demonstrates that, even if each measurement is able to correct the state only partially, the state is fully observable when more measurements are considered in cascade.
1 0 A = 0 0 0
mi that best matches l j can be expressed as:
ai ⋅ xk + bi ⋅ yk + ci zρ = ai2 + bi2 z = tan −1 − ai − θ b k α i
1 CLONG 0 ai 2 ai + bi 2 H= 0 a m a 2 +b 2 m m 0
(5)
where ai, bi and ci are the parameters characterizing the implicit equation of mi ; zρ and zα are, respectively, the distance between the line and the robot, and the angle between the line and the robot’s heading. When putting together Equations 4 and 5, the measurement model results to be a non-linear function of the state: z k = h( x k , v k )
with
v = N (0, R)
(6)
Since non-AWG components of the GPS noise are estimated in the state vector, the remaining noise can be reasonably modeled with the vector v, a zero-mean AWG noise with covariance matrix R. Equation 2 can be used to compute the a-priori state estimate at time k. Next, whenever new GPS or laser rangefinder data are available, they are fused with the apriori estimate through an Extended Kalman Filter to produce a new estimate, thus reducing errors that are inherently present in odometry and providing a new estimate for the GPS bias. Obviously, to evaluate the soundness of the previous assertion, it is necessary to perform an observability analysis of the system. The Kalman theorem requires to compute the observability matrix T T T T , where A and H are the 4 T Q = H | A H | ... | (A ) H
Jacobian matrices of the partial derivatives of f and h with respect to x (Q’s full expression is not shown for sake of brevity). The analysis shows that Q has full rank (and hence the state is fully observable) only when at least two observations l j and lm are available, corresponding to nonparallel lines mi and mn , together with a single GPS measurement. To meet this requirement, each landmark for laser-based localization must be constituted of two perpendicular surfaces. Matrix A results to be as in Equation 7, whereas H results to be as in Equation 8. By inspecting matrix H, one could infer that the filter is updated only when a triplet of observations are available (i.e., two non-parallel lines and
0 −dsk −1 ⋅ sin(θ k −1 ) 0 0 1 dsk −1 ⋅ cos(θ k −1 ) 0 0 0 1 0 0 0 0 1 0 0 0 0 1 0 1
0
1
CLONG
0
0
0
0
0
−1
0
bm
0
0
−1
0
CLAT bi
ai 2 + bi 2
am 2 + bm 2 0
(7)
1 CLAT 0 0 0 0 0
(8)
Unfortunately, when the rover is moving between two subsequent landmarks, the laser cannot detect any line mapped in the a-priori map: the H matrix fed to the KF comprises only the first two rows in Equation 8. By computing again the observability matrix Q, this yields the result in Equation 9. 1 / C LONG 0 1 / C LONG 0 1 / C LONG 0 Q= 1 / C LONG 0 1 / C LONG 0
0 1 / C LAT 0 1 / C LAT 0 1 / C LAT 0 1 / C LAT 0 1 / C LAT
0 0 1 ⋅ dsk −1 ⋅ sin(θ k −1 ) C LONG 1 ⋅ dsk −1 ⋅ cos(θ k −1 ) C LAT 2 − ⋅ dsk −1 ⋅ sin(θ k −1 ) C LONG 2 ⋅ dsk −1 ⋅ cos(θ k −1 ) C LAT 3 − ⋅ dsk −1 ⋅ sin(θ k −1 ) C LONG 3 ⋅ dsk −1 ⋅ cos(θ k −1 ) C LAT 4 − ⋅ dsk −1 ⋅ sin(θ k −1 ) C LONG 4 ⋅ dsk −1 ⋅ cos(θ k −1 ) C LAT −
1 / C LONG 0 0 1 / C LAT 1 / C LONG 0 0 1 / C LAT 1 / C LONG 0 0 1 / C LAT 1 / C LONG 0 0 1 / C LAT 1 / C LONG 0 0 1 / C LAT
(9)
When dsk −1 ≠ 0 , i.e. when the translational speed of the robot is not null [10], the rank of Q is 3. The rank is not full since Q’s first column (corresponding to the x-component of the state vector x) equals the fourth column (corresponding to the xGPS-component), and Q’s second column (i.e., the ycomponent) equals the fifth column (i.e., the yGPScomponent). On the opposite, Q’s third column (corresponding to the θ-component of x) is linearly independent from the others. Q’s analysis confirms the intuition that – when no laser data are available – the
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subspace defined by x+xGPS, y+yGPS, and θ is fully observable: the robot’s orientation is still corrected by GPS data, but the position has a permanent error that depends on the current estimate of the GPS bias. The observability of orientation is rather intuitive, since – with unicycle kinematics – orientation always corresponds to the rover’s direction of motion (it becomes unobservable when the rover is still). About the other components of the state, the innovation due to GPS measurements is distributed by the Kalman gain onto x-, y-, xGPS, and yGPS components of the state according to the current value of the state covariance matrix P. Since a constant dynamic is assumed for xGPS, and yGPS, corrections are adequately distributed onto the state components only if the actual bias in GPS changes slowly, and given that a new landmark for laser-based re-localization will soon be available.
experiment is repeated twice: A-tests are performed without bias estimation, whereas B-tests correspond to localization with GPS bias estimation. Figure 3 shows an A-test (i.e., without bias estimation): the rover moves along a circular path which is about 200 meters long. The Figure shows that the estimated position significantly diverges from the actual circular trajectory: the error decreases only when a landmark is visible. Figure 4 shows a B-test (i.e., with bias estimation) performed in the same conditions; once again, the error decreases when landmarks are visible, but in this case it increases slowly, thanks to the fact that the bias changes slowly as well. In this experiment, when the distance between landmarks (measured along the path) is about 50 meters, the average positioning error in the A-test is about 1.7m, whereas in the B-test it is below 0.3m. L3
L3
L2
L4
L2
L4
L1
L1
Figure 4: B-test: estimated path with bias estimation
Figure 3: A-test: estimated path without bias estimation
IV. EXPERIMENTAL RESULTS Many tests in MatLab with real GPS data have been performed, as well as preliminary tests with the ANSER rover at the Albenga Airport. In the first set of experiments, the rover is simulated, by adding systematic and non-systematic odometric errors, as well as errors in laser measurements. However, GPS data are real; they are taken at a fixed known location for 24 hours at 1Hz (latitude data are shown in Figure 2), which allows to record measurement errors. During tests, real recorded errors are added to ideal GPS measurement, thus allowing to validate the ability of the approach to estimate the real GPS bias. During each run, the robot is requested to move either along a circular or straight path. Landmarks for selflocalization (i.e., couples of perpendicular line segments) are distributed along the path at a distance that varies from test to test: roughly 50m, 100m, 200m. The rover is able to see landmarks only when they are closer than 5m; consequently, during most of the time, the rover relies exclusively on GPS measurement, whereas landmarks are used – when available - to correct all the components of the augmented state vector. Since we are interested in the behavior of the rover when relying only on GPS, errors in position and orientation are recorded only when landmarks are not visible. Finally, each
Figure 5: B-test: actual and estimated bias
Figure 5 shows the x-component of the actual and the estimated bias (upper lines) during the test in Figure 4. It can be noticed that the two plots overlap only when the full state vector is observable (i.e., when landmarks are visible, corresponding to peaks in the lower line). However, when the rover relies only on GPS, the trend of the two upper lines is similar, thus allowing to reduce the positioning error. Table 1 summarizes A-tests when increasing the distance between landmarks, Table 2 summarizes B-tests (averaging over 50 runs, including both circular and straight paths). By defining ex , ey , eϑ as the three components of the error vector, the first column of both Tables shows the average
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positioning error ex , y = E ( ex2 + ey2 ) , the second column shows its standard deviation
σ e , the third column shows the
average orientation error eϑ , and finally the fourth column shows its standard deviation σ e . The reduction in positioning errors due to the bias estimation is evident. Distance between landmarks
ex , y
σ x, y
eϑ
σϑ
50m
1.744
1.071
0.697
0.913
100m
1.938
0.954
0.542
0.821
200m
2.024
0.940
0.563
0.931
Figure 6: Estimated position in a real scenario
Table 1: Errors statistics in A-tests (in meters) Distance between landmarks
ex , y
σ x, y
eϑ
σϑ
50m
0.303
0.296
0.031
0.317
100m
0.760
0.631
0.157
0.906
200m
1.256
0.731
0.532
1.059
Table 2:Errors statistics in B-test (in meters)
Preliminary real world experiments have been carried out as well with the ANSER robot in the Albenga Airport. During the test, the robot is manually driven at 1.0m/s along a pre-established cyclic path that is about 500 meters long. Different A- and B-tests are been performed, by memorizing the robot’s estimated position in a finite number of selected places along the path (see Figure 6, position estimates are marked with circles in A-tests and with asterisks in B-tests). A single landmark for laser-based localization is available, roughly located in x=0,y=0. Preliminary tests show that, when estimating the GPS bias, the robot exhibits a more repeatable behavior: when looping along the path, position estimates recorded during different loops have a bigger dispersion in A-tests (circles) than in Btests (asterisks). A test campaign is currently being performed to validate these results. V. CONCLUSIONS The paper describes the localization subsystem of a mobile robot that has been designed for patrols and surveillance tasks within a civilian airport. Instead of equipping the robot with a huge amount of expensive sensors (and the computing power that is adequate to deal with them), a simple approach is chosen that relies exclusively on a non-differential GPS unit and a laser rangefinder (i.e., inertial sensors are absent). Laser measurements are exploited only in some areas of the outdoor patrol path of the robot, i.e. where it is possible detect line features and match them against an a-priori model of the environment. Along the rest of the path, the robot relies on GPS-based localization. An Extended Kalman Filter algorithm is employed to estimate an augmented state vector comprising the robot position and orientation, together with the low frequency components (bias) of the GPS error.
Figure 7: The robot ANSER at Albenga Airport
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