Two laser scanners raw sensory data fusion for objects tracking using Inter-Rays uncertainty and a Fixed Size assumption.

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1 th International Conference on Information Fusion Seattle, WA, USA, July 6-9, 9 Two laser scanners raw sensory data fusion for objects tracking using Inter-Rays uncertainty and a Fixed Size assumption. PawełKmiotek a,b and Yassine Ruichek a (a) Systems and Transportation Laboratory University of Technology of Belfort-Montbeliard - France pawel.kmiotek,yassine.ruichek@utbm.fr (b) Department of Computer Science AGH University of Science and Technology - Kraków, Poland Abstract This paper presents a fusion method for objects tracking using two Laser Range Finders (LRF). The tracking is based on the Extended Kalman filter. Tracked objects are represented by Oriented Bounding Box (OBB). To improve the objects state estimation, two paradigms are introduced. The first one concerns Inter-Rays (IR) uncertainty, which considers the fact that the raw data points representing the extremities of an extracted OBB do not coincide with the real objects extremities. The second paradigm, called Fixed Size assumption, assumes that objects size does not change during the tracking. This is expressed by the fact that a track representing an object change its size depending on the IR uncertainty. The fusion technique benefits of the increased perception angular resolution obtained by using two LRFs. The fusion technique takes place in the early stage of the measurement extraction from the raw data points. Experimental results are presented to demonstrate the reliability of the two-lrf based fusion method, especially for far objects. INTRODUCTION In the last decade, many research programs have been launched to study the concept of intelligent vehicles and their integration in the city of the future. In this framework, the Systems and Transportation Laboratory of the University of Technology of Belfort-Montbéliard (France) is working to develop a vehicle having the ability to navigate autonomously in various urban environments. The research developments are based on an experimental platform consisting of an electrical vehicle with an automatic control, equipped with several sensors and communication interfaces. To reach the objective, the first primary task is to develop a perception system for detecting, localising and tracking objects in this type of environments. In this paper, the emphasis is put on tracking of compact dynamic objects using laser sensory data. Representation of dynamic objects is crucial for tracking and trajectory planning. In the literature concerning tracking, points with elliptical uncertainty are used for representing objects position [][]. This representation is good ISIF 997 pawel.kmiotek@agh.edu.pl enough for obstacle detection, collision warning or driving assistance systems in well structured environments like highways [][3]. In the urban areas, there are less constraints on the objects movements. Thus, for the task of autonomous navigation in demanding urban areas, these representation methods are not sufficient. Oriented Bounding Box (OBB) [4][5][6] provides a good approximation of the size, shape and orientation angle of dynamic objects, with a good data compression ratio. In general, laser range sensors have a small angular resolution. To overcome the limitation, the authors propose to enriche the OBB model by an Inter-Rays (IR) uncertainty and a Fixed Size (FS) assumption. The IR uncertainty is developed to handle the fact that the raw data points representing the extremities of the extracted OBB do not coincide with the real object s extremities. The idea of the FS assumption is to consider that objects size does not change during the tracking. The introduction of these two paradigms allows to increase the tracking system reliability by better object s size and center position estimation. Even with the improvement thanks to the IR uncertainty and FS assumption, a system equipped with a single LRF does not perform well for far objects, because of the limited spatial resolution. One cannot obtain correct estimation of the object s size and the other information such as orientation, speed and position are very uncertain. Multi-sensor configuration is often used to increase robustness and accuracy of the environment representation [][3]. Multiple sensors can provide redundant, complementary or both kind of information. By increasing the perception resolution, a two LRF configuration (complementary aspect) allows to obtain better estimation of objects state at long ranges. The proposed tracking system is based on the Extended Kalman Filter (EKF) with Discrete White Noise Acceleration Model (DWNA) []. There are two main approaches of fusion using KF: measurements fusion and tracks fusion. In the first approach, one can extract two variants: Weighted Measurement Fusion (WMF), Merge Measurement Fusion (MMF). In the second approach, one can find: Track-To- Track Fusion, Modified Track-To-Track Fusion, Track-To-

2 Track Fusion with fused predictions [7][]. In [8], the authors used two LRF and the WMF method was chosen. This approach, however, takes only into account the redundancy aspect of the two-lrf configuration, and, thus, does not perform well for far objects. This paper presents a two-lrf based fusion method, which takes into account the complementary aspect of the multi-sensor configuration in terms of angular resolution. In the context of KF, the method takes place in the stage of measurements extraction (raw data points fusion). The paper is organized as follows. Section presents the OBB representation for dynamic objects, with the IR uncertainty paradigm and the FS assumption. The data association technique is described in section 3. The tracking model is explained in section 4. In section 5 the fusion method is presented. Before concluding, experimental results are presented in section 6. OBJECT REPRESENTATION. OBB based model for object representation Urban environments are characterised by limited spaces available for navigation and there are little objects movement constraints. In these conditions, geometrical representation of dynamic objects is necessary. Oriented bounding box (OBB) is a way of representing objects geometry with sufficient approximation for the means of navigation. The OBB based representation is described by two vectors z () and σz (). The first one represents the OBB geometry and includes the centre coordinations cx, cy, the orientation angle θ and the size dx, dy. The second vector represents uncertainties on the components of the vector z. z = [cx, cy, θ, dx, dy] T () σ z = [σ cx, σ cy, σ θ, σ dx, σ dy] T () To construct the OBB based measurement, a specific method is used. The OBB construction method consists of the four following main steps. The first step is to find a contour of the tracked objects using a semi convex-hull technique [9]. In the second step, a method based on Rotating Calipers (RC) technique [] is used to construct an OBB, which is best aligned to the object s contour. The third step consists of the uncertainty computation. Finally, the forth step concerns the application of the IR uncertainty paradigm and the FS assumption. In this paper, we will focus on the forth step. The previous steps are described in details in [4].. Inter-Rays uncertainty An important aspect of OBB extraction is the fact that the raw data points representing the extremities of the extracted OBB do not coincide with the real object s extremities (see Figure ). In the Figure, minx, miny, maxx, maxy are respectively the minimum x coordinate, the minimum y coordinate, the maximum x coordinate and the maximum y 998 Lr+n d (r+n) IRy Pr+n (r+n) IRy maxy minx y Extracted OBB Real Object x (r) IRx r+n r+ r+n r Laser rays maxx miny Pr Lr (r) d IRx Figure : Inter-Rays uncertainty paradigm. coordinate of the extracted OBB. The line Lr (respectively Lr + n) is crossing the point maxx (respectively maxy ) and is perpendicular to the OBB side to which maxx (respectively maxy ) belongs. The Inter-Rays (IR) real object s extremities position estimation and their variances are added to the OBB s size and OBB s size uncertainty. The real object s extremities are situated between the raw data points delimiting the OBB (maxx, maxy ) and the points Pr and Pr+n. Pr (respecitvely Pr+n) is the intersection point between the ray r (respecitvely r+n) with the line Lr (respectivelylr+n). The introduction of the IR uncertainty, the measurement vector z is augmented by quantities: d IRx, µ IRx, σirx (for the OBB s local X axis), d IRy, µ IRy, σiry (for the OBB s local Y axis). Considering the OBB s local X axis, the real object s extremity position is uniformly distributed with the mean µ IRx, which is equal to the half of the IR line segment length d IRx. The IR line segment is defined by the point maxx and Pr. To fulfil Kalman Filter assumption, the distribution of the real object s extremity position is approximated by a normal distribution with the mean µ IRx, and the variance σirx dirx, which is set to ( N σ ). N σ is the number of sigmas and represents the confidence interval of the approximated distribution which is equal to the IR line segment length d IRx (see Figure ). The N σ is set to 6. Standard deviations from the mean Cumulative %.%.3% 5.9% 5% 84.% 97.7% 99.9% Figure : Normal distribution approximation. The measurement Inter-Rays values z[µ IRx ] and z[σ IRx ] are used in each iteration of the tracking algorithm to correct the size of the OBB measurement [4]. The correction equations are expressed as follows: w4 IR d IR

3 z[dx] = z perc [dx] + z[µ IRx ] (3) z[σ dx] = z perc [σ dx] + z[σ IRx] (4) where z perc is the perceived measurement, z is the corrected measurement used for tracking. The same process is applied for the OBB s local Y axis. It may happen that the IR line segment length d IRx is large. It may cause great overestimation and tracking instability. To avoid this situation, the IR line segment length d IRx is limited to a certain value..3 Fixed Size assumption The idea of the fixed size (FS) assumption is based on the fact that, in general cases, objects size does not change during the tracking. However, due to the LRF s limited resolution and change of the relative distance and orientation of the observed object, measurements of the object s size vary in. The principle of the FS assumption is that the size of the track representing the tracked object can change depending on the IR uncertainty. The FS algorithm takes place in each iteration of the tracking after the track prediction and measurements extraction. For the following algorithm description, we consider the local OBB s X axis. The same process is applied to the local OBB s Y axis. Having the perceived OBB measurement with the IR line segment length z perc [d IRx ], we obtain the corrected IR line segment length z[d IRx ] associated with the OBB measurement: measurement s sides relatively to perceived object s sides. The center translation is needed to adjust the measurement OBB sides to perceived object s sides. The updating of the center position is achieved as follows. Firstly, a visibility factor V F x is computed for the OBB s local X axis. The visibility factor permits to compute the center translation coefficient, which is proportional to the difference between sides normals angles (β minx and β maxx ). V F x = max(βf minx, βf maxx ) β f minx + (9) βf maxx where β minx and β maxx correspond respectively to the angles between OBB s sides minxside and maxxside normals and their radius vectors (see Figure 3). f is a smoothing parameter, which is set experimentally to 4. The visibility factor becomes less sensible to the angle difference as the smoothing parameter value increase. N minx Measurement OBB maxy side minx side y minx maxx side x N maxx miny side maxx z[d IRx ] = min(z perc [d IRx ], ˆx k [d IRx ]) (5) where ˆx k [d IRx ] is the IR line segment length associated with the track at k-. The quantity z[d IRx ] is then memorised in the track ˆx k : ˆx k [d IRx ] = z[d IRx ] (6) After using the equation (3) and (4), the next step consists of the measurement s size correction by using the following equations: If z perc [d IRx ] ˆx k [d IRx ] and z[dx] < ˆx k [dx] than z[dx] = ˆx k [dx] (7) If z perc [d IRx ] < ˆx k [d IRx ] and z[dx] < ˆx k [dx] than z[dx] = ˆx k [dx] z perc [µ IRx ] ˆx k [µ IRx ] (8) where ˆx k [dx] is the track predicted size at the k, and ˆx k [µ IRx ] is equal to ˆx k [d IRx]. The measurement s size correction allows to obtain the best object size perceived up to the k in terms of IR uncertainty. After correcting the perceived measurement s size, the measurement s center must be appropriately translated. The measurement s size correction introduces translation of 999 Laser rays Figure 3: Visibility factor associated to the OBB s local X axis. In the second step, the direction factor DF x associated to the OBB s local X axis is computed. Providing the direction of center translation, the direction factor is expressed as follows: {, if βmaxx > β minx (a) DF x =, if β maxx < β minx (b) In the last step, a difference between the perceived size z perc [dx] and the corrected size z[dx] is calculated: dx = z[dx] z perc [dx] () Finally, the measurement s center translation is expressed as follows: z[cx] = z[cx] + V F x DF x dx z[σ cx] = z[σ dx ] ()

4 3 DATA ASSOCIATION One of the most important tasks of autonomous navigation in urban areas is tracking of dynamic objects. Data association, which is closely related to the objects representation and sensory data, is a crucial part of the tracking process. In this section, a data association methodology suitable for the OBB representation and laser scanner data is proposed. The emphasis is put on association of raw data points of coalescing objects. Since geometrical features are taken into account, only objects being previously recognised as separated ones can be tracked correctly. one, Mahalanobis distance based gating is used to associate raw data points with a track. Not associated points are processed to create new tracks. In the last case, a method based on the Nearest-Neighbour principle is used. 4 TRACKING The object s state estimation is done by the means of Extended Kalman Filter (EKF). All values of the track s state vector are expressed in the local ego-vehicle coordinate system. Tracks are represented by the augmented OBB state vector x k : tracks New track Points clustering Track to cluster correlation How many tracks correlated with a cluster track Single track association or more tracks Multiple tracks association Figure 4: Data association schema. The data association algorithm is composed with the following stages (see Figure 4): preliminary association (raw data points clustering), tracks to clusters correlation and raw data points to track association. The preliminary association is based on distance thresholding. Points belong to the same cluster if the Euclidean distance between them is below a certain threshold. Each cluster is represented by an Axis Aligned Bounding Box (AABB). The raw data points clustering uses two thresholds: a general threshold and a neighbouring points one. Applied to the points produced by neighbouring rays, the neighbouring threshold is greater than the general one, which is used for all non neighbour points. Neighbouring points threshold is used for points which are produced by neighbouring rays and is greater then general one which is used for all non neighbour points. After raw data points are clustered into an AABB, track to cluster correlation takes place. The track is correlated with the cluster if the track s OBB intersects with the cluster s AABB. If the track do not intersect any cluster, the track is correlated with the closest cluster. There are three possible outputs of track to cluster correlation. A cluster can be correlated with zero, one, two or more tracks. These cases represent respectively the following situations: appearance of a new object, tracking of separated object, and multi-object tracking (see Figure 4). Basing on the results of the previous step, raw data points to track association follows. In this stage, raw data points, positively associated with a track, create a measurement. Each measurement is in the OBB format (see () and ()). In the first situation (appearance of a new object), all the points are used to create the measurement. In the second x k = [cx, cx, cy, cy, θ, θ, dx, dy] T (3) Since tracking is done from dynamic platform, odometry information is used to increase the tracking accuracy. State change of the ego-vehicle is represented as differences of position x, y and angle γ between consecutive instants. Thus, the input to the state transition equation is defined as: u k = [ x, y, γ] (4) The Discrete White Noise Acceleration Model (DWNA) [] is used to describe objects kinematics and process noise. Thus, taking into account the odometry information, the track state transition is modelled as follows [4]: ˆx k k = A( x, y, γ)f ˆx k + Bu k + Gv k (5) where F is is the standard DWNA transition matrix, B is the odometry-input model, G represents the noise gain matrix, v k is the process noise, defined with the Gaussian distribution: where v k = [ cx, cy, θ, ˆσ dx, ˆσ dy ], v k N(, Qk) (6) Q k = Gv k G T (7) with ˆσ dx and ˆσ dy are the process errors for OBB sizes dx and dy respectively. The prediction covariance matrix is: P k k = A x (ˆx k )FP k A T x (ˆx k )F T + Q k (8) where P k is the estimation covariance matrix. The observation equation can be written as follows: z k = Hˆx k k + w k (9) where H is the observation model and w k, which represents the measurement noise, is defined with a Gaussian distribution: where I 5,5 is the identity matrix. w k N(, R)R = σ z I 5,5 ()

5 5 FUSION METHODOLOGY The proposed representation model performs well except for objects poorly represented by raw data points. This situation occurs for far objects. Indeed, the number of laser rays colliding with objects is inversely proportional to the distance and proportional to the LRF angular resolution. Since the increase of LRF angular resolution is limited, the number of laser rays colliding with objects decreases with the distance. Hence, at a certain range, the object state estimation becomes very uncertain or even impossible to obtain. To overcome this limitation more LRF sensors can be used. A multiple LRF configuration provides higher perception angular resolution, and, thus, better object state estimation can be achieved. Furthermore, interlacing rays allows additional size estimation refinement by utilizing Inter-Rays uncertainty. KF based fusion methods can be divided into two groups: measurements fusion and tracks fusion. In the case of far objects, none of general approaches fits. In [8], two LRF were fused by using Weighted Measurement Fusion (WMF) method [7]. In this method, OBB measurements are extracted from raw data points for each sensor. The OBB measurements coming from the two sensors are then fused. However, this method takes into account only the redundancy aspect of the two-lrf configuration, and does not benefit from the increased perception angular resolution. Thus, it does not perform well for far objects. A method taking into account the complementary aspect of the multisensor configuration must operate on raw sensory data. The proposed approach takes advantage of the following aspects of increased perception angular resolution: more raw data points per object and lower distance between laser rays. To benefit from the first aspect, the raw data points coming from different sensors must be merged to extract an OBB measurement. The first step of the whole tracking system consists of data association. Raw data points association is performed for each sensor separately, and raw data points are regrouped in clusters. The number of clusters correlated with a track is equal to the number of sensors. During the points clustering, the online semi convexhull construction takes place (see [9]). The points constructing each semi convex-hull are sorted according to their angular coordinate. To construct the fused semi convex-hull from the semi convex-hulls correlated with a track, the following algorithm is performed. It starts by inserting the two points with the smallest angular coordinates into a new semi convex-hull to be constructed. To choose a point with the smallest angular coordinate, we consider only the first points of all the semi convex-hulls, since points of each semi convex-hull are sorted. The point being inserted is deleted from the original semi convex-hull. In each iteration, a new point with the minimum angular coordinate is inserted into the semi convex-hull being constructed. For each point insertion, the convexity condition is checked and if violated the existing semi convex-hull recalculation occurs(see [9]). The constructed semi convex-hull serves then as an input for the Calipers based OBB extraction method. After the OBB extraction Inter-Ray (IR) based size refinement starts. In the case of a single LRF, the distance between rays increases with the distance from the sensor. In a mulit-sensor case, the inter-rays distance varies between and d LRF, where d LRF a inter-rays distance for each LRF. This IR distance variation allows to refine the size of perceived objects, where the refinement level corresponds to the relative position between objects and sensors. The IR uncertainty computation for multiple sensors is similar to the single sensor case (see.). The only difference between the two configurations is that in the multiple sensor case, the d IR values are computed for each LRF and than the smallest is chosen. To correctly choose rays r and r + n of each sensors, the coordinates of extreme point (e.g. maxy ) must be expressed in the local sensor coordination system. 6 RESULTS The evaluation is based on a software platform, developed to simulate the sensors and the multiple objects tracking process. The simulator permits flexible changing of all sensors parameters and mounting position. This allows to test developed algorithms with different sensor configurations. In the simulator, laser range finder (LRF), LIDAR, stereovision and odometry sensors are implemented. To test the proposed approach, a single LRF and a two- LRF configurations are evaluated and compared. In the first configuration, a Laser Range Finder (LRF) is mounted in front of the instrumented vehicle. In the second case, two LRFs are parallely mounted in front of the vehicle with horizontal interspace of m. The step angle for the LRFs is set to with an angle range of 8 (similarly to the real sensor parameters). The sensor range is set to m and the range uncertainty σ ρ is set to.5m. To show the interest of using IR uncertainty and FS assumption, a scenario where a tracking vehicle runs in circles in front of the instrumented vehicle. Figures 5 and 6 show the trajectory of the tracked vehicle without and with IR uncertainty and FS assumption, respectively. One can see that the second approach outperforms the first one in terms of object s center position estimation. The better center position estimation can be obtained thanks to the more reliable object s size estimation. This is due to the proportional relation between the size and the center of the OBB. The places when the measurements present great deviation from the real object center trajectory, correspond to the situations when only one side of the object can be perceived by the sensor. Thus, only one object s dimension information is available. One can see that the usage of the FS assumption, which stores the object size, allows to obtain reliable estimation, even in the cases when only one object s side is seen. To evaluate and compare the one LRF based tracking with the two LRFs based one, a second scenario is used. It corresponds to a vehicle which is travelling towards the instru-

6 8 Real center trajectory Measured center trajectory Estimated center trajectory Real center trajectory Measured center trajectory Estimated center trajectory Y Y X Figure 5: Tracked vehicle trajectory without IR uncertainty and FS assumption (in the instrumented vehicle s local coordinate system) X Figure 7: Vehicle trajectory (in the instrumented vehicle s local coordinate system). 8 Real center trajectory Measured center trajectory Estimated center trajectory Y X Figure 6: Tracked vehicle trajectory with IR uncertainty and FS assumption (in the instrumented vehicle s local coordinate system). mented one, according to the trajectory illustrated in Figure 7. One can see in Figures 8, 9,, 4, 5, 8 and 9 that the single LRF based tracking provides bad state estimation, when the vehicle is far. However, the performance of this method increases with the decrease of the distance between the sensor and the tracked vehicle. The two LRFs based method behaves similarly, but with better vehicle s state estimation. The fusion of the two LRFs allows to obtain the same performance as for the single LRF but for greater distances (see Figures,, 3, 6, 7, and ). One can see in Figures 4 and 5 that the IR uncertainty µ IR stays constant at the beginning of tracking (when the vehicle is far). This is due to the IR line segment length d IRx limitation, as mentioned in section.. In our test, the limit is set to meters. In Figures 8 and 9, showing the center position errors for the one LRF, one can see great oscillations. This effect is a result of the sensor s low resolution at far distances. To explain the nature of the problem, let us use the example shown in Figure. In the example, the real object moves to the right what can be seen as the change of the position in the different instants. The measurement, however, stays at the same place due to the low laser rays resolution. If the object continues it s movement, it will be detected by a new raw data points configuration and, hence, the measurement will change it s position. This effect takes place all the during the tracking of the object. It s intensity is proportional to the laser rays resolution and the velocity of the object. Lower the resolution and the velocity are more prominent the effect becomes, since the period, when the measurement is static, increases. Thus, in the beginning of the scenario, when the tracked object is far and it s speed is low, the object s movement is perceived as a jerking one. The use of KF smooths the estimated velocity. However at low speed, when the position of the measurement stays unchanged for a long, the estimated velocity resents great oscillations. The introduction of the second LRF allows to increase the laser rays resolution and thus the oscillation effect is importantly reduced (see Figures 3,4,,) Measured center position error (X coordinate) Estimated center position error (X coordinate) Figure 8: One LRF - object s center position error (X coordinate).

7 .6 Measured center position error (Y coordinate) Estimated center position error (Y coordinate). Measured center position error (Y coordinate) Estimated center position error (Y coordinate) Figure 9: One LRF - object s center position error (Y coordinate)..7.6 Measured center position error (X coordinate) Estimated center position error (X coordinate) Figure : Two LRFs - object s center position error (Y coordinate). 5 Measured orientation angle error Estimated orientation angle error Figure : Two LRFs - object s center position error (X coordinate). 7 CONCLUSIONS A two-lrf based fusion method for objects tracking is presented. An Oriented Bounding Box model is used to represent the tracked objects. Enriched by the Inter-Ray uncertainty and Fixed Size assumption paradigms, the OBB model performs well with a single LRF, except for far objects, because of the limited angular resolution of the sensor. To overcome this limitation, the authors have proposed to use two LRF in order to increase the perception angular resolution. The raw data fusion method leads to better object state estimation. Furthermore, interlacing rays allows additional size estimation refinement using the IR uncertainty. The experimental results have shown the reliability of the two-lrf based fusion system, especially for far objects, when compared with the usage of a single LRF. References [] Y. Bar-Shalom and T. E. Fortman, Tracking and data association. Academic Press Professional, Inc., 988. [] C. Blanc, L. Trassoudaine, Y. Guilloux, and R.Moreira, Track to track fusion method applied to road obstacle detection, in Proceedings of the Sev- 3 Figure : One LRF - object s orientation angle error. enth International Conference on Information Fusion, vol., pp , 4. [3] U. Hofmann, A. Rieder, and D. Dickmanns, Radar and vision data fusion for hybrid adaptive cruise control on highways, Machine Vision and Applications, vol. 4, pp. 4 49, 3. [4] P. Kmiotek and Y. Ruichek, Representing and tracking of dynamics objects using oriented bounding box and extended kalman filter., in Proceedings, th International IEEE Conference on Intelligent Transportation Systems, pp. 3 38, 8. [5] A. Petrovskaya and S. Thrun, Model based vehicle tracking for autonomous driving in urban environments, in Proceedings of Robotics: Science and Systems Conference 8, 8. [6] D. Streller, K. Frstenberg, and K. Dietmayer, Vehicle and object models for robust tracking in traffic scenes using laser range images, in Intelligent Transportation Systems,., ITSC IEEE Conference on Intelligent Transportation Systems,. [7] J. Gao and C. Harris, Some remarks on kalman filters for the multisensor fusion, Information Fusion, vol. 3, pp. 9, September.

8 5 Measured orientation angle error Estimated orientation angle error z perc [µ IRy ] x k [µ IRy ] Figure 3: Two LRFs - object s orientation angle error. z perc [µ IRx ] x k [µ IRx ]. Figure 5: One LRF - Inter-Rays uncertainty µ IR (Y coordinate) z perc [µ IRx ] x k [µ IRx ] Figure 4: One LRF - Inter-Rays uncertainty µ IR (X coordinate) [8] Y. Kmiotek, P. Ruichek, Multisensor fusion based tracking of coalescing objects in urban environment for an autonomous vehicle navigation, in Proceedings of IEEE International Conference on Multisensor Fusion and Integration for Intelligent Systems, (Seoul), pp. 5 57, 8. Figure 6: Two LRFs - Inter-Rays uncertainty µ IR (X coordinate). [9] P. Kmiotek and Y. Ruichek, Objects oriented bounding box based representation using laser range finder ensory data, in Proceedings of IEEE International Conference on Vehicular Electronics and Safety, pp. 8 84, 8. [] G. Toussaint, Solving geometric problems with the rotating calipers, in Proc. MELECON, Athens, Greece, 983. [] Y. Bar-Shalom, X. Li, and T. Kirubarajan, Estimation with applications to tracking and navigation. Wiley New York, z perc [µ IRy ] x k [µ IRy ] Figure 7: Two LRFs - Inter-Rays uncertainty µ IR (Y coordinate). 4

9 .5 Measured Y side lenght error Estimated Y side length error.5 Measured X side length error Estimated X side length error Figure 8: One LRF - object s X side size error. Figure : Two LRFs - object s Y side size error Measured Y side length error Estimated Y side length error Time: t t = t + t t 3 = t + t Laser rays, Measurement OBB, Real Object Figure : The example of the measurement OBB extraction for different object positions for greater distances (small LRF resolutions). 4 LRF - Estimated center X velocity LRF fusion - Estimated center X velocity Figure 9: One LRF - object s Y side size error Measured X side length error Estimated X side length error Figure 3: Comparison of the velocity estimation between one LRF and two LRF fusion (coordinate X) LRF - Estimated center Y velocity LRF fusion - Estimated center Y velocity Figure : Two LRFs - object s X side size error Figure 4: Comparison of the velocity estimation between one LRF and two LRF fusion (coordinate Y). 5