On the observability of indirect filtering in vehicle tracking and localization using a fixed camera

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1 On the observability o indiret iltering in vehile traking and loalization using a ixed amera Perera L.D.L. and Elinas P. Australian Center or Field Robotis The University o Sydney Sydney, NSW, Australia {l.perera, p.elinas}@ar.usyd.edu.au Abstrat In several vehile traking and loalization appliations, the initial position o a vehile may be given by GPS measurements or other means. However, the inormation required or aurate traking ater initialization may only be available intermittently or not at all. In this paper, we demonstrate that the indiret or error orm o state variables an be used in aurate bearing only traking o a vehile when the GPS measurements o its loation are disontinued or some reason. Using Piee-wise Constant Systems Theory o Observability Analysis and the indiret orm o the state variables or a onstant veloity model we show that an objet moving in a two-dimensional environment traked by bearing only measurements using a ixed monoular amera is ully observable. We experimentally veriy the theoretial results with simulations and real data rom a ixed monoular amera traking a pedestrian onsistently when GPS measurements are disontinued. Keywords: Traking, loalization, observability. 1 Introdution We onsider the problem o loalizing and traking a moving objet by using data rom a GPS sensor and a single amera. Our motivation is rom an appliation in open pit mining where the aurate loalization o vehiles and mining personnel as they go about the pit is important or both ontrolling the proess aurately and inreasing saety. In this paper, we show that given an initial estimate o an entity s position using GPS, we an improve our estimate o its loation by integrating data rom a remote amera. We also show that when the GPS signal is lost and given that the system has been initialized, we an ontinue loalizing the entity using only a single amera until GPS traking is restored. We perorm the neessary observability analysis to show that our method is valid. In addition, we perorm an experimental evaluation using both simulated data and real data loalizing a pedestrian. 2 Problem statement and previous work It is ommon pratie in open pit mining that vehiles are loalized using GPS [9]. Aurately knowing the position and veloity o any vehile as it goes about the mine is important or supporting automation systems. Due to the geometry o an open pit mine, there are several loations where GPS ails [7] beause o an insuiient number o visible satellites, multi-path eets o loal terrain and periodi signal blokage due to oliage or plaes having restrited view o the sky. We onsider solving the problem o ontinuous objet loalization using a ombination o GPS and a monoular amera. A visual traking system an be used to detet entities in video [15]. This inormation an be used to provide ontinuous loalization in 3D. We show that when an objet s loation is initially known we an aurately estimate its position by observing it rom a stationary amera. Moreover, we show that when data rom both sensors, GPS and amera, are available, we an get a more aurate estimate o the objet s loation. Even though our motivation is based on the automation o open pit mining operations, there are several other vehile traking and loalization appliations [3] where you know the initial position o the vehile by GPS or by other means but adequate inormation ater initialization essential or aurate traking is oten hard to obtain. Bearing only traking and loalization o a missile rom a known initial position [13] and airrat traking in and around airports [4] to improve trai apaity and saety are some example appliations our work also addresses. Several sensors suh as Inertial Measurement Units (IMU) [11], laser, vision and radar are oten used to omplement or aid GPS systems in vehile traking and loalization using sensor usion algorithms. A omprehensive and detailed review o the use o suh multisensor usion systems is available in [3]. In partiular, use o omputer vision in vehile traking and loaliza-

2 tion with GPS is demonstrated in [8, 1, 14]. Hsien- Chou et al. [1] desribe a GPS-based objet detetion and traking algorithm in a ubiquitous amera environment. Subong et al. [14] desribe a real-time approah to alulate the three-dimensional loation o a ixed target deteted by a gimbaled amera in a ixed-wing experimental unmanned aerial vehile (UAV) equipped with a single-antenna GPS reeiver on board. Heimes at al. [8] desribe how to ombine passive GPS and map-based route guidane with model-based mahine vision in order to automatially assess or even exeute driving maneuvers in inner-ity trai situations. Although all o the above approahes use mahine vision as a omplementary sensor system with GPS in objet traking and loalization, none o the work validates the use o omputer vision to omplement GPS using an observability analysis whih is one o the main ontributions o this paper. More speiially, we show that the indiret or error orm o state variables an be used in aurate bearing only traking o an objet ater initialization or when the GPS measurements o its loation are disontinued. Moreover, we show that the error orm o the state variables o a onstant veloity model o an objet moving in a 2D environment traked by bearing only measurements is ully observable. The rest o this paper is strutured as ollows. In Setion 3, we provide a brie overview o piee-wise systems theory essential or the observability analysis o Setion 5.3. In Setion 4, we develop the indiret or error orm o 2D vehile traking and loalization. We dive into more details on visual traking in Setion 5. Finally, in Setion 6 we perorm a omprehensive evaluation o our method using real and simulated data and onlude in Setion 7. 3 Piee-wise onstant systems theory A system is said to be observable at time t i its state vetor at time t, x(t ) an be determined rom the measurements in t, t < t whih is inite. Several linear and nonlinear tehniques are used in observability analysis o engineering systems [1]. The piee-wise onstant systems theory in partiular assumes that systems are piee-wise onstant over suiiently small time segments and uses linear systems theory in the observability analysis. Use o linear systems theory in pieewise onstant systems theory provides advantages suh as the possibility o using simple linear state spae analysis tehniques to aess all state variables and simpliied observer design [5, 6]. Here, we briely summarize the piee-wise onstant systems theory or ontinuous time systems. Let a system be deined as ollows, ẋ(t) = Fx(t) (1) z(t) = Hx(t) (2) where x(t) R n represents the state variables being modelled and z(t) R m represents the measurements rom the system, F and H are the proess and measurement model transition matries respetively. The observability matrix O j o the system in time segment j is then deined as z j (t j ) = O j x(t 1 ) (3) where z j (t j ) is the onatenated vetor onsisting o the vetor z(t j ) and its n 1 derivatives, t 1 is the initial time segment and O j = [ (H j ) T (H j F j ) T... (H j (F j ) n 1 ) T ] T (4) where F j and H j are the proess and measurement model transition matries in time segment j. The total observability matrix O(r) or r time segments is then deined as ollows. Z(r) = O(r)x(t 1 ) (5) where O 1 O 2 e F1 1 O(r) = O 3 e F2 2 e F (6) O r e Fr 1 r 1 e Fr 2 r 2... e F1 1 and Z(r) = [ ] z T 1 z T 2 z T 3... z T T r (7) Here, e Fj j terms in O(r) aount or the transition o the state variables x(t j ) to that o the initial time segment x(t 1 ). The piee-wise onstant systems theory states that i and only i the rank o O(r) at any segment r is equal to the dimension o the state vetor x(t) then the system is ully observable. This ollows rom the at that the rank o O(r) determines the existene o a unique solution or x(t 1 ) in (5) and rom the deinition o the observability. We an use the simpliied observability matrix O s (r) = [ O T 1... O T ] T r in the observability analysis [5] i Null(O j ) Null(F j ) or all 1 j r. 4 2D vehile traking and loalization in indiret orm The problem o vehile traking and loalization where the vehile motion is not represented by a vehile kinemati model an be ormulated as ollows. ẋ(t) = (x(t)) + η 1 (t) (8) z(t) = h(x(t)) + η 2 (t) (9)

3 Y global oordinate rame X vehile (x v, y v ) β α amera (x, y ) Figure 1: Camera and vehile loations in the global oordinate rame. where η 1 (t) and η 2 (t) are unorrelated zero mean proess and measurement noise terms with ovariane Q(t) and R(t) respetively, (.) is the proess model and h(.) is the measurement model. The state vetor o the vehile is x(t) = [ x v v x y v v y ] T where xv, y v, v x and v y are the x oordinate, y oordinate, veloity in x axis diretion and veloity in y axis diretion o the vehile respetively. The indiret or error orm o the vehile traking and loalization problem exluding the noise terms rom the equations or simpliity is as ollows. δẋ(t) = Fδx(t) (1) δz(t) = Hδx(t) (11) where δx(t) and δz(t) are the errors between the true value and the estimated value o the state variables and the measurements respetively. The other terms o (1) and (11) are, δx(t) = [ δx v δv x δy v δv y ] T (12) F = x (13) H = h x (14) δx(t) = x true (t) ˆx(t) (15) where x true (t) is the true value o x(t) and ˆx(t) is the predited value o x(t). The value o ˆx(t) is usually alulated rom the estimated value, odometry or initialized value where appropriate. Hene, during a ertain interval i you know the value o ˆx(t) the estimated value o x(t) is the sum o ˆx(t) and the estimated value o δx(t). In the ollowing disussion we show that δx(t) is ully observable in the bearing only vehile traking and loalization problem i we use GPS or the initial estimation and subsequent estimated value o x(t) in determining ˆx(t) + δx(t). We use a onstant veloity model to represent the vehile kinemati model. When the diret orm o the 2D vehile traking and loalization problem is onsidered, it is ully observable when the GPS measurements o the vehile loations are available. Sine, 1 (x(t)) = 1 (16) h(x(t)) = [ 1 ] 1 O = [ (H) T (HF) T (HF 2 ) T (HF 3 ) T ] T (17) (18) and O is ull rank. Consider now the indiret representation o the 2D vehile traking and loalization algorithm 1 F = 1 (19) When the GPS ix o the vehile loation is available H is given by, [ ] 1 H = (2) 1 Hene the observability matrix in the irst time segment O 1 = [ (H) T (HF) T (HF 2 ) T (HF 3 ) T ] T (21) has a rank o 4. Hene, the vehile traking and loalization system (1)-(11) o error states is observable. 5 2D vehile traking and loalization using a monoular amera Let there be a vehile moving on a 2D horizontal plane as shown in Figure 1 with x v and y v as lateral and longitudinal oordinates with respet to a global oordinate rame. The veloities o the vehile in lateral and longitudinal diretions are v x and v y. Let the amera optial entre be at a point given by lateral and longitudinal oordinates x and y and α be the angle o the amera optial axis with the lateral diretion all with respet to the seleted global oordinate rame ( Figure 1 ). It is assumed that the amera optial axis is parallel to the ground plane. 5.1 Camera alibration The pinhole model [17] o the amera with perspetive projetion is used in the amera modelling and alibration. The method desribed in [17] and [16] was utilized in the amera alibration. This approah requires a amera to observe a planar pattern shown at a ew dierent orientations. Either the amera or the ixed

4 X C Image point P [ ] T I = x y C Camera Coordinate Frame y x C I Image oordinate rame Image Plane 9 P [ X Y Z ] T Sene point 9 Z C Camera Coordinate Frame C X C Image P [ ] T I = x y β y x Sene point P [ X Y Z] T Image Plane Image oordinate rame C I 9 9 Z C Y C Y C Figure 2: Camera alibration. pattern an be moved in making observations. This method provides the alibration parameters as a losed orm solution. Let a sene point P = [ ] T X Y Z be in amera oordinate rame with the oal point C as the origin and amera optial axis as the Z axis where X, Y and Z denote the oordinates o the sene point in the amera oordinate rame. The prinipal point (image entre) C I is the origin o the image plane whih is parallel to the X, C and Y plane. Let P I = [x y] T be the image o the point P in the image plane as shown in the Figure 2 where x and y are its X and Y oordinates with reerene to the image oordinate rame. Let [u v ] T be the oordinates o the image enter or the prinipal point on the pixel rame and [u v] T be the pixel oordinates o the point P in pixel oordinate rame (with u and v orresponding to x and y oordinates in the image oordinate rame) S x and S y be the eetive sizes o a pixel in the horizontal and vertial diretions respetively. The ollowing expression an then be derived or the homogeneous (projetive) oordinates o the sene point and the orresponding pixel point, u S v x u = S y v [ X Y Z 1 ] (22) 1 1 We now denote x = S x and y = S y with both the quantities having the unit o pixels. The 3 4 matrix at the right hand side o (22) is known as the Camera Calibration Matrix. 5.2 Monoular amera measurement model Let the amera oordinate rame and the world oordinate rame be peretly aligned so that there is no rotation between the amera and the world oordinate rames. Using the amera alibration matrix (22) and Figure 3: Bearing measurement alulation. Figure 3 it ollows that, But, y = (v v)sy that, tanβ = Y C Z C = y = (v v) Sy (23) = (v v) y. Hene, it ollows tanβ = v v y (24) From Figure 1 it ollows that, α + β = tan 1 yv y x v x Hene, ( tanβ = tan(tan 1 yv y x v x ( v v = tan(tan 1 yv y y x v x (25) ) α) (26) ) α) (27) Thereore, the monoular amera measurement model is; v = y tan(tan 1 yv y α) + v (28) x v x 5.3 Observability analysis Consider now the ase where GPS measurements o the vehile loation are not available and the vehile is observed by a single monoular amera loated at oordinates (x, y ). Hene using (9) and (28), the monoular amera measurement model is; h(x(t)) = y tan(tan 1 yv y α) + v (29) x v x Hene using (11) it ollows that; where H = [ h 1 h 2 ] (3) h 1 = y (y v y )se 2 θ v /r 2 (31)

5 h 2 = y (x v x )se 2 θ v /r 2 (32) r 2 = (y v y ) 2 + (x v x ) 2 (33) θ v = tan 1 yv y α (34) x v x The observability matrix o the irst time segment rom (21) using F 1 = F and using H rom (3) is: O 1 = o 1 o 2 o 1 o 2 (35) where o 1 = ( y (y v,1 y )se 2 θ v,1 ) /r 2,1 (36) o 2 = ( y (x v,1 x )se 2 θ v,1 ) /r 2,1 (37) x v,i, y v,i, x,i and y,i represent the vehile and amera x and y oordinates in the i th time segment respetively and r 2,i = (y v,i y ) 2 + (x v,i x ) 2 (38) θ v,i = tan 1 yv,i y α (39) x v,i x Thereore, the rank o O 1 is 2. Hene, the system o error states is not observable in the irst time segment. It now ollows that we an transorm (5) into the ollowing orm. T r Z(r) = T r O(r)M r M 1 r x(t 1 ) (4) where T r and M r represent the matrix transormations on O(r). We now transorm T r O(r)M r into the ollowing orm, [ ] IR P U r = T r O(r)M r = R (41) where R is the rank o the O(r) and P R is the matrix resulting rom this transormation. P R is an identity matrix o dimension R. Hene, xv,1 x 1 y v,1 y U R = 1 (42) Let y O and y U be the observable and unobservable parts o the state spae. Then, y O = [I R P R ]M 1 r x(t 1 ) (43) y U = [ I n R ]M 1 r x(t 1 ) (44) where n is the dimension o the state vetor. Hene, rom (43) and (44) it ollows that, y O = δx xv,1 x v + y v,i y δy v xv,1 x δx y + (45) y v,i y δv y y U = [ ] δyv δv y (46) It ollows rom (35) that the observability matrix O 1 o the 1 st segment is also rank deiient by 2. Let the null vetors o the observability matrix O 1 o the 1 st segment be denoted by n 1,1 and n 1,2 n 1,1 = n 1,2 = [ xv,1 x y v,1 y [ xv,1 x y v,1 y 1 ] T (47) 1] T (48) However, sine F 1 n 1,2, Null(O j ) Null(F j ), or all j suh that 1 j r. Hene, we annot use the simpliied observability matrix in the segment wise observability analysis. Consider, now the Total Observability Matrix o the segments one and two. ō 1 = yse 2 θ v,2 r 2,2 ō 2 = yse 2 θ v,2 r 2,2 ō 3 = yse 2 θ v,2 r 2,2 [ O(2) = O 1 O 2 e F1 t o 1 o 2 o 1 o 2 O(2) = ō 1 ō 2 ō 1 ō 3 ō 1 ō 1 ō 1 ō 1 ] (49) (5) ((x v,2 x ) (y v,2 y )) (51) ( (xv,2 x ) (y v,2 y )e t) (52) ( (xv,2 x )e t (y v,2 y ) ) (53) It now ollows that rank o O(2) is equal to 4. Thereore, vehile traking using a single monoular amera is observable in two time segments. 6 Experimental evaluation In order to veriy our theoretial analysis, we perormed several experiments using simulations and real data loalizing a pedestrian. We present the results in the ollowing 2 setions starting with simulation. 6.1 Simulations For the simulation, we assume a two dimensional environment o 2 3 square meters. A vehile is moving at onstant veloities o 2ms 1 and 3ms 1 in longitudinal and lateral diretions respetively subjet to small aeleration perturbations. A wide ield o view amera pointing perpendiular to the vehile path is loated at oordinates (, 3). We assume that the bearing measurements o the vehile by the amera and the GPS measurements o the vehile obtained by a GPS sensor on board the vehile are ommuniated to a entral loation or proessing. Using the inormation available at the entral loation we use onstant veloity

6 Estimation Error [m] (a) Figure 5: Examples o visual traking using a partile ilter with 1 partiles. Shown are the ilter mean and the 2σ ovariane ellipse or a Gaussian distribution it to the partile estimates. (b) Figure 4: Filtering errors in x diretion with (a) GPS only, and (b) GPS and amera with GPS outage rom 4s to 7s. (see Equation 16) to model vehile motion and measurements rom GPS and amera to loalize the vehile using indiret iltering with the EKF. Alternatively, one an use a partile ilter or traking [2]; our observability analysis does not depend on any speii Bayesian ilter implementation. For the EKF, we use zero mean Gaussian errors o Standard Deviation (STD) 3m or the GPS sensor in both longitudinal and lateral diretions and a bearing error o 1 or the amera. Part (a) o Figure 4 shows the loalization error (shown by stars) o the vehile in longitudinal diretion with 95% onidene limits o the estimated unertainty (shown by thik dashed lines) when only GPS is used. Part (b) o Figure 4 shows the loalization error (shown by stars) in the longitudinal diretion and 95% onidene limits o the estimated unertainty (shown by thik dashed lines) when both GPS and amera measurements (bearing) are used. In this senario we assume that GPS measurements o the vehile s loation are not available rom 4s to 7s. Part (b) o Figure 4 shows that when GPS is used or initialization the ilter estimate o the vehile s loation is onsistent even when the GPS is disontinued or 3s. The at that vehile loation estimation is onsistent even when the GPS outages our veriies the ob- servability theory. However, it is important to note that or onsistent loalization, the vehile should ollow the onstant veloity model losely when the GPS outages our. I the vehile deviates signiiantly rom the used motion model it is neessary to obtain observations rom a seond amera; the observability analysis o suh a system is let or uture work. 6.2 Real Data The observability theory is urther veriied using several pedestrian traking experiments. Figure 5 shows examples o visually traking a pedestrian rom a ixed monoular amera. We use a Grass Hopper Gras- 2S4C-C amera manuatured by Point Grey Researh. The amera is loated at ( 8, ) pointing perpendiular to the person s path. The pedestrian is also arrying a onsumer-grade GPS sensor. The amera and the GPS measurements are timestamped and available at a entral loation or proessing. The GPS sensor has longitudinal and lateral errors o approximately 3m. For visual traking we used a Partile Filter [2, 12] with 1 partiles; partile weights are a untion o edge density within a ixed area retangle. We it a Gaussian distribution to the partile estimates. We show traking examples in Figure 5 with the mean and 2σ ovariane ellipse o the ilter estimate. We olleted 2 data sets. For the irst one, the person walked on an approximately straight line. For the seond, the person walked twie in a irle o 15-meter radius. Figure 5 shows images rom the seond data set. Figure 6 shows the estimated path and the 95% onidene limits o traking the pedestrian moving along the straight line path. It is assumed that as the pedes-

7 Y [m] pedestrian starts here Estimated Path 95% Conidene Limit X [m] Figure 6: Traking results o pedestrian loalization (in an approximately straight line path). trian moves hal o his trajetory a GPS outage ours until he stops at the end o his path. Diminishing area o unertainty ellipses representing the 95% onidene bounds o the loation estimates along the person s estimated trajetory in Figure 6 learly show that even when the GPS is not available, loalization is onsistent thus veriying the observability analysis given in Setion 5.3. Figure 7 shows the same experimental setup with the ixed amera but the pedestrian moving along the approximately irular path in anti-lokwise diretion. In this experiment, The amera is pointing towards the entre o the pedestrian s path. Part (a) o Figure 7 shows the person s estimated path (thik line) and unertainty ellipses (dashed lines) when only GPS is used or loalization. Part (b) o Figure 7 shows the same experiment when both the GPS and the amera measurements are used. Figure 8 ompares the estimated unertainties o the two ases, i.e., GPS data only and GPS plus amera data used. Figure 8 learly shows that the usion o monoular amera inormation and GPS improves the unertainty bound o the pedestrian s loation estimate. It is also important to note that the estimated path shown in part (b) o Figure 7 more aurately relets the true path taken by the pedestrian. 7 Conlusions and uture work In this paper, we onsidered the problem o objet loalization and traking using data rom a GPS sensor and a amera. Using the pieewise onstant systems theory we have shown that when the GPS signal is disontinued ater some time but the objet ontinues to be observed by a ixed monoular amera whose loation is known with a ertain auray, the error states or two dimensional point target traking is ully observable i the objet is observed or two onseutive time segments. We veriied our theoretial analysis us- Y [m] Y [m] Estimated Path 95% Conidene Limit pedestrian starts here X [m] Estimated Path 95% Conidene Limit (a) pedestrian starts here X [m] (b) Figure 7: Traking results using GPS and amera data when the pedestrian is walking in an approximately irular path. ing simulations and real data traking and loalizing a pedestrian. In uture work we intend to extend the observability analysis or senarios using multiple ameras and where GPS initialization o vehile loations is not possible. Using more than one amera, we also intend to make the vehile loalization more robust to hanging vehile dynamis and maneuvers removing the onstant veloity model assumption in the urrent ormulation. 8 Aknowledgements This work is supported by the Rio Tinto Centre or Mine Automation and the ARC Centre o Exellene programme unded by the Australian Researh Counil (ARC) and the New South Wales State Government.

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