EP2374117B1

EP2374117B1 - Vehicle tracking system, vehicle infrastructure provided with vehicle tracking system and method for tracking vehicles

Info

Publication number: EP2374117B1
Application number: EP09771423.2A
Authority: EP
Inventors: Zoltan Papp; Gerardus Johannes Nicolaas Doodeman; Martin Willem Nelisse; Joris Sijs; Johannes Adrianus Cornelis Theeuwes; Bart Driessen
Original assignee: Nederlandse Organisatie voor Toegepast Natuurwetenschappelijk Onderzoek TNO
Current assignee: Nederlandse Organisatie voor Toegepast Natuurwetenschappelijk Onderzoek TNO
Priority date: 2008-12-12
Filing date: 2009-12-11
Publication date: 2017-08-30
Anticipated expiration: 2029-12-11
Also published as: EP2196972A1; EP2374117A1; WO2010068106A1

Description

Vehicle tracking system, vehicle infrastructure provided with vehicle tracking system and method for tracking vehicles.

BACKGROUND OF THE INVENTION

Field of the invention

The present invention relates to a vehicle tracking system.
The present invention relates to a vehicle infrastructure provided with a vehicle tracking system.
The present invention further relates to a method for tracking vehicles.

Related Art

In "Measuring Traffic", Statistical Science, 2007, Vol. 22, No. 4, pp. 581-597, Peter J. Bickel et al. describe a traffic performance measurement system, PeMS, that functions as a statewide repository for traffic data gathered by thousands of automatic sensors. It has integrated data collection, processing and communications infrastructure with data storage and analytical tools. This system provides for global information relating to the average traffic speed and the traffic density.
There is a need for a system that provides information relating to the state of individual traffic participants. This information can be used in advanced cruise controllers for traffic management or for access control for example.
US5801943 describes a wide area surveillance system for application to large road networks. The system employs smart sensors to identify plural individual vehicles in the network. These vehicles are tracked on an individual basis, and the system derives the behavior of the vehicle. Furthermore, the system derives traffic behavior on a local basis, across roadway links, and in sections of the network. Processing in the system is divided into multiple processing layers, with geographical separation of tasks.

SUMMARY OF THE INVENTION

The vehicle tracking system according to the present invention comprises

a plurality of sensor nodes that each provide a message indicative for an occupancy status of a detection area of a vehicle infrastructure monitored by said sensor node,
a message interpretator including a vehicle database facility with state information of vehicles present at the vehicle infrastructure, and a database updating facility for updating the database facility on the basis of messages provided by the sensor nodes, characterized in that said sensor nodes are arranged in the vehicle infrastructure at a density of at least 0.2 per square meter.

Contrary to the system known from US5801943 , in the vehicle tracking system according to the present invention vehicles can be tracked with relatively simple and cheap means. Smart sensors are not necessary. It is sufficient that the sensor nodes sense an occupancy state, i.e. whether a detection area associated with the sensor node is occupied by a vehicle or not and that they merely provide a message that indicates whether the occupancy state is changed. The relatively cheap and simple construction of the sensor nodes contributes to an economically feasible application in vehicle tracking systems for large vehicle infrastructures. Optionally, the message may additionally include the value of the occupancy state after the change was detected. The plurality of sensor nodes arranged in the vehicle infrastructure having at least the above-mentioned density provide a course image of the vehicles present at the vehicle infrastructure. However, as compared to an image provided by cameras, the image provided by the plurality of sensor nodes is always captured from the same perspective. This facilitates further processing.
In an embodiment the database update facility comprises

an association facility for associating the messages provided by the sensor nodes with the state information present in the vehicle data base facility,
a state updating facility for updating the state information on the basis of the messages associated therewith.

The association facility selects for which state information the received messages are relevant, and provides the selected messages to the state updating facility. In this way the state updating facility can operate more efficiently, than in case no selection takes place.
The present invention is in particular suitable for tracking vehicles. For tracking vehicles, suitable sensor elements to be used in the sensor nodes are for example magneto restrictive sensors. These sensors determine whether their associated detection area is occupied by detection of a perturbation of the earth magnetic field. Alternatively, magnetic loop sensors, may be used, which detect a change of inductance caused by the presence of ferromagnetic material.
Preferably each sensor node is provided with a wireless transmission facility that transmits the preprocessed data, e.g. the occupance status or an indication of a change thereof to a data to a receiver facility coupled to the message interpreter. The absence of wiring towards the message interpreter makes the installation easier and cost effective. Furthermore it is attractive if the sensor nodes provide their message at an event basis, e.g. if a perturbation of the earth magnetic field exceeds a threshold value. This reduces communication load of the message interpreter and minimizes power consumption of the sensor nodes.
In an embodiment the density of the sensor nodes are at least 0.6 per square meter. Due to the relatively high density of the sensor nodes in the traffic infrastructure in said embodiment an individual failure of a sensor or of an individual sensor does not have serious consequences for the estimation of the states of the traffic participants. Accordingly, any return transmission from the message interpreter to the sensor nodes to actively verify the occupancy state is superfluous, which is also favorable for a low power consumption of the sensor nodes.
The vehicle tracking system may in addition to the plurality of sensor nodes arranged in the traffic infrastructure comprise one or more cameras. A camera may be used for example if a perturbation of the earth magnetic field can not be measured. This is the case for example if (parts of) the infra structure comprises metal components e.g. a bridge.
It is not necessary that the detection areas of the sensor elements are complementary. The detection areas may overlap, or spaces may exist between the detection areas, but it is required that the detection area of the sensor be smaller than the vehicles to be tracked. Ideally the sensor elements are point detectors.
The sensor nodes can be either randomly distributed over the vehicle infrastructure or placed in a pattern optimized for the vehicle tracking problem in hand.
In an embodiment the vehicle tracking system comprises a plurality of system modules, each module comprising a respective subset of the plurality of sensor nodes for monitoring a respective section of the vehicle infrastructure and a respective message interpreter, the vehicle tracking system has a communication facility for enabling system modules of mutually neighboring sections to exchange state and detection information. In this way the vehicle tracking system can be easily expanded if required. A new system module need only to communicate with the system modules arranged for neighboring sections. Neighboring sections may be arranged in one dimensional scheme, e.g. in case of a narrow road. For example if a certain road is already provided with an vehicle tracking system, it is sufficient to provide for a communication facility between the system module for the last section of said vehicle tracking system and the new system module for the appended section. In case of infrastructures for unconstrained vehicle movements the new module may communicate with other modules neighboring in various directions. As the system modules merely need to exchange state information and vehicle-detection information (i.e. the unprocessed sensor signals) in a limited subarea of the respective sections, the amount of communication between the system modules is modest resulting in a scalable vehicle tracking system.
The association facility associates the messages provided by the sensor nodes or neighboring system modules with the state information present in the vehicle data base facility. In other words the association facility determines the probability that the detections are caused by a particular vehicle for which state information is present in the vehicle data base facility. If the messages cannot be associated with state information of an already identified vehicle here or in the neighboring system module, a new entry may be added to the database. Alternatively, the entry for the new vehicle may be added by a separate procedure. For example the vehicle infrastructure may have an access with a vehicle identification facility that provides for an identification of every vehicle that enters the infrastructure.
The individual sensor nodes do not need to provide other information than an occupancy status of their associated detection area. However, optionally the sensor node may associate its own signal with a color, shape, or other signature of the tracked vehicles to facilitate or obviate association by the message interpreter. An association facility for associating the detection signals obtained on asynchronous basis with state information of a particular vehicle may be based on one of the following methods.

Gating,
Nearest Neighbor (NN),
(Joint) Probabilistic Data Association ((J)DPA),
Multiple Hypothesis Tracker (MHT), and
MCMCDA.

Gating comprises forming a gate around the predicted measurement of a vehicle. The size and shape of the gate are chosen in such a way that unlikely messages are precluded to be associated with this particular vehicle-track. The method determines a statistical, quadratic distance $d_{oi}$ $_{2}$
from vehicle i. A measurement y is associated with the state-vector x^oi of vehicle i if $d_{oi}^{} \leq G,$
with G some constant threshold and $d_{oi}^{}$
equal to:

Various methods can be used for finding the Threshold G. However, this data association method is not suitable for associating event based messages. Moreover problems arise when two gates overlap.

The Nearest Neighbor method also uses a gate, but it can handle overlapping gates. The sum of all possible combinations to associate a certain measurement to a certain track is analyzed. The chosen combination associates the most measurements to a track for a minimum sum of distances.
A (Joint) Probabilistic Data Association method is described in Multitarget-Multisensor Tracking: Principles and Techniques. YBS, 1995, by Y. Bar-Shalom and R. Li, for example. The (Joint) Probabilistic Data Association methods (J)PDA is unsuitable for event-based sampling because it assumes that one target can give rise to at most one measurement and one measurement is a result of at most one vehicle. This cannot be assumed with event based sampling. An extension to the JPDA can be found in O. Songhwai, S. Sastry, and L. Schenato, "A Hierarchical Multiple-Target Tracking Algorithm for Sensor Networks," in Proc. of the 2005 IEEE International Conference on Robotics and Automation, Barcelona, Spain, 2005. Where the original JDPA assumes that a probability density function (PDF) of state-vector x_k+1|jk is a Gaussian function, in the extended JPDA the PDF can be a non-Gaussian function.
A Multiple Hypothesis Tracker (MHT) allows that the state-vector of a single vehicle can has multiple tracks. This method resembles to the Particle filter as described in B. Ristic, S. arulampalam, and N. Gordon, "Beyond the Kalman filter: Particle filter for tracking applications", 2002. Therein, each state is estimated by simulating N states with each a different probability. It is a drawback of this method that it requires a high computational power.
A further data association method is the Markov chain Monte Carlo data association (MCMCDA). All observations are used to classify and cluster them. To that end the whole set of observations is divided into a number of partitions represented by the set w. This is done n_mc times resulting in n_mc sets of w, i.e. possible partitions. The set of w with the highest probability, given the number of vehicles in the previous sample instant, is chosen and the state-vectors of the tracks a are updated according the partitioned observation. The computational time can be decreased by not using the total history of observations, but by using a moving horizon. A downside of this method is that each observation can belong to at most one vehicle and, making this method unsuitable for event-based state-estimation.
Although these data association methods can be applied to associate messages obtained by (a)synchronous sampling, they are not suitable for association messages obtained by event-based sampling. For then the main issue with present association methods is that they assume either that one target results in at most one observation or that one observation comes from at most one vehicle. According to a preferred embodiment of the present invention messages received from the sensor nodes are first associated with a particular vehicle. In other words it is determined which vehicle most probably caused the observation by the sensor resulting in the message. Subsequently the state of said vehicle is (re-)estimated using that message. The estimation is based on the latest observation and the current state of the said vehicle. The state of the vehicle that is most probable in view of all available observations is calculated.
More in particular the step of associating may comprise

initializing (S40) a vehicle index (i),
retrieving (S41) the current state known for the vehicle with that index from a vehicle database facility,
determining (S42) a probability that the vehicle with that index caused the detection reported by the message D,
incrementing (S43) the vehicle index,
determining (S44) whether the vehicle index is less than the number of vehicles,
if the outcome of the determination is positive repeating steps S41 to S43 with the incremented vehicle index, and
if the outcome of the determination is negative, determining (S45) which vehicle caused the detection reported by the message D with the highest probability.
returning (S46) the index of the vehicle identified in step S45.

If no message is received, e.g. because the transmission of the message was disturbed by external causes, or because the sensor node is defect, there can accordingly be no association. However, due to the relatively high density of sensor nodes, it is not necessary to verify whether the absence of a message is due to a failure of the message creation and transmission or due to the absence of a vehicle. I.e. the message interpreter does not need to send verification messages to the sensor nodes to verify correct operation of the transmission. Accordingly a one-way message traffic from the sensor nodes to the message interpreters is sufficient for a correct operation of the method.
The state of a vehicle can also be estimated at a point in
time later than the last message, but before a new message has arrived. In that case the error covariance matrix is bounded, as it is known that the state change of the vehicle must be within the detection boundaries of the sensor node.

BRIEF DESCRIPTION OF THE DRAWINGS

These and other aspects are described in more detail with reference to the drawing. Therein:

Figure 1 shows a first view of an embodiment of a vehicle infrastructure provided with a vehicle tracking system according to the invention,
Figure 2 shows a second view of an embodiment of a vehicle infrastructure provided with a vehicle tracking system according to the invention,
Figure 3 shows another view of an embodiment of a vehicle tracking system according to the invention,
Figure 4 schematically shows a part of the vehicle infrastructure that is provided with a plurality of sensor nodes,
Figure 5 schematically shows a signal flow in a sensor node,
Figure 6 schematically shows a possible hardware implementation of a sensor node,
Figure 7 shows a possible method carried out by a sensor node,
Figure 8 shows a signal flow in a message interpretor,
Figure 9 shows a possible hardware implementation of a message interpretor,
Figure 10 shows neighboring infrastructure regions, with specially handled subregions marked,
Figure 11 shows an overview of a method carried out by the message interpreter,
Figure 12 shows a first detail of the method of Figure 11,
Figure 13 shows a second detail of the method of Figure 11,
Figure 14 shows an example of a vehicle to be detected at a reference position and orientation and at a different position and orientation,
Figure 15 shows a definition of a set S and the equidistant sampled set A,
Figure 16 shows detection of a vehicle at multiple detection points,
Figure 17 shows a definition of the set On of possible positions oⁱ _k for a single detection point,
Figure 18 shows a definition of the set ON of possible positions oⁱ _k for multiple detection points,
Figure 19 shows a derivation of ON(q) given 2 detections and 2 different samples of q,
Figure 20 shows (Left) determination of -A, (right) the vehicle's possible position set ^On given d_n and q,
Figure 21 shows (left) the mean of all Gaussians from f(o |z₁, θ) and f(o|z₂, θ); (right) The selection of means of the Gaussians from f(o |z₁, θ) and f(o |z₂, θ), of which their mean ô_ni is close or in C_N(θ),
Figure 22 shows an association result with event-based data-association,
Figure 23 shows an association result with Nearest Neighbor data-association,
Figure 24 shows time sampling of a signal y(t),
Figure 25 shows event sampling of a signal y(t),
Figure 26 shows event sampling: Send-on-Delta,
Figure 27 shows the Gaussian function,
Figure 28 shows a top view of the Gaussian function,
Figure 29 shows an approximation of Λ _Hke (y_n ) as a sum of Gaussian functions,
Figure 30 shows position, speed and acceleration of a simulated vehicle,
Figure 31 shows a position estimation error for various methods,
Figure 32 shows a speed estimation speed for various methods,
Figure 33 shows a factor of increase in estimation error after z_ke, or y _ka .

DETAILED DESCRIPTION OF EMBODIMENTS

In the following detailed description numerous specific details are set forth in order to provide a thorough understanding of the present invention. However, it will be understood by one skilled in the art that the present invention may be practiced without these specific details. In other instances, well known methods, procedures, and components have not been described in detail so as not to obscure aspects of the present invention.
The invention is described more fully hereinafter with reference to the accompanying drawings, in which embodiments of the invention are shown. This invention may, however, be embodied in many different forms and should not be construed as limited to the embodiments set forth herein. Rather, these embodiments are provided so that this disclosure will be thorough and complete, and will fully convey the scope of the invention to those skilled in the art. In the drawings, the size and relative sizes of layers and regions may be exaggerated for clarity.
It will be understood that when an element or layer is referred to as being "on", "connected to" or "coupled to" another element or layer, it can be directly on, connected or coupled to the other element or layer or intervening elements or layers may be present. In contrast, when an element is referred to as being "directly on," "directly connected to" or "directly coupled to" another element or layer, there are no intervening elements or layers present. Like numbers refer to like elements throughout. As used herein, the term "and/or" includes any and all combinations of one or more of the associated listed items.
It will be understood that, although the terms first, second, third etc. may be used herein to describe various elements, components, and/or sections, these elements, components, and/or sections should not be limited by these terms. These terms are only used to distinguish one element, component or section from another element, component, and/or section. Thus, a first element, component, and/or section discussed below could be termed a second element, component, and/or section without departing from the teachings of the present invention.
Unless otherwise defined, all terms (including technical and scientific terms) used herein have the same meaning as commonly understood by one of ordinary skill in the art to which this invention belongs. It will be further understood that terms, such as those defined in commonly used dictionaries, should be interpreted as having a meaning that is consistent with their meaning in the context of the relevant art and will not be interpreted in an idealized or overly formal sense unless expressly so defined herein. In addition, the materials, methods, and examples are illustrative only and not intended to be limiting.
Figure 1 and 2 show a first and a second view of an embodiment of a vehicle infrastructure 80 provided with a vehicle tracking system. The vehicle infrastructure is intended to allow stationary and/or moving vehicles 70 thereon, e.g. a road or a parking place. The vehicle infrastructure may be part of a public or private space, e.g. a recreational park.
The vehicle tracking system comprises a plurality of sensor nodes 10 that each provide a message indicative for an occupancy status of a detection area of the vehicle infrastructure monitored by said sensor node 10. As shown therein the sensor nodes are randomly distributed over the vehicle infrastructure.
The vehicle tracking system comprises a message interpretator MI, each comprising a vehicle database facility, an association facility and a state updating facility. Each message interpretator is responsible for handling messages D from a respective section 80A, 80B, 80C, 80D of the vehicle infrastructure 80.
Figure 3 is another schematic view of the vehicle tracking system. Figure 3 shows how sensor nodes 10 transmit (detection) messages to a message interpreter MI in their neighborhood. The message interpreters MI may also communicate to each other via a communication channel 60 to indicate that a vehicle crosses a boundary between their respective sections and to exchange a status of such a vehicle. As shown in Figure 3, the vehicle tracking system comprises a plurality of system modules MD1, MD2, MD3. Although three modules are shown in this example, any number of
system modules is possible, dependent on the application. For example for an isolated vehicle infra structure, e.g. an intersection of roads a single module may be applicable, while on a long road thousands of modules may be present. Each module MD 1, MD2, MD3 comprises a respective subset of the plurality of sensor nodes 10 for monitoring a respective section of the vehicle infrastructure and a respective message interpreter MI. The vehicle tracking system further has a communication facility 60 for enabling system modules MD1, MD2, MD3 of mutually neighboring sections to exchange state information. In the embodiment shown, messages from the sensor nodes are directly transmitted to a message interpretor. Alternatively, the sensor nodes may form a network that routes the messages to the message interpreters. In that case the transmitters may have a short transmission range.
Figure 4 schematically shows a part of the vehicle infrastructure that is provided with a plurality of sensor nodes j having position c_j. The sensor nodes have a detection area with radius R. A vehicle i is present at the infrastructure having a position (vⁱ _x, vⁱ _y). In this case if the vehicle substantially covers the detection area the sensor node indicates that the detection area is occupied as indicated in gray. Otherwise the sensor node indicates that the detection area is not occupied (white). In practice the fraction of the detection area that should be covered before an occupied status is detected may deviate from the above-mentioned 50% depending on the type of vehicle.
Figure 5 schematically illustrates the signal flow for the sensor node 10, having sensor element 12, a processing unit 14 (with memory), and a radio link 16.
The sensor element 12 is capable of sensing the proximity of the vehicles to be tracked. The processing unit 14 determines if a vehicle is present or absent on the basis of the signals from the sensor element 12. If an occupancy status of the detection area of the sensor changes, the processing unit 14 initiates a transmission of a message D indicating the new occupancy status. In an alternative embodiment the message may include a time stamp indicative of the time t at which the new occupancy status occurred. Alternatively the sensor nodes may transmit occupancy status information on a periodical basis for example. However, an event-based transmission enables a lower power use. The message D sent should reach at least one message interpreter MI. In a concrete implementation of the sensor node 10 for tracking vehicles on a road the sensor element 12 is a magnetoresistive component, which measures the disturbance on the earth magnetic field induced by the vehicles. Alternatively, a magnetic rod or loop antenna may be used to detect the occupancy by a vehicle.
Figure 6 shows a possible implementation of the hardware involved for the sensor node 10 of Figure 5. The sensor element 12 is coupled via an A/D converter 13 to a microcontroller 14 that has access to a memory 15, and that further controls a radio transmitter 16 coupled to an antenna 17.
Figure 7 schematically shows a method performed by a sensor node to generate a message indicative for occupancy status of a detection area of the sensor node.
Starting (Step S1: initialization) from an off-state of the sensor node, input from the A/D converter is received (Step S2). In a next step S3, offset is removed from the sensed value.
In step S4 it is determined whether the occupancy state of the detection area as reported by the last message transmitted by the sensor node was ON (vehicle was present in the detection range) or OFF (no vehicle present in the detection range. This occupancy state is internally stored in the sensor node.
In the first case, program flow continues with step S5. In the second case processing flow continues with step S9. In step S5 it is determined whether a signal value v obtained from the A/D converter, and indicative for an occupied status of the detection area is below a first predetermined value T_L. If this is not the case program flow continues with step S2. If however the value is lower than said first predetermined value then program flow continues with step S6. In step S6 it is verified whether the signal value v remains below the first predetermined value T_L for a first predetermined time period. During step S6 the retrieval of input from the A/D convertor is continued. If the signal value v returns to a value higher then said predetermined value T_L before the end of said predetermined time-period then processing flow continues with step S2. Otherwise the value for the occupancy state is internally saved as unoccupied in step S7, and a message signaling this is transmitted in step S8.
In step S9 it is determined whether the signal value v obtained from the A/D converter, and indicative for an occupied status of the detection area is above a second predetermined value T_H. The second predetermined value T_H is higher than the first predetermined value T_L. If this is not the case program flow continues with step S2. If however the value is higher than said second predetermined value T_H then program flow continues with step S10. In step S10 it is verified whether the signal value v remains above the second predetermined value T_H for a second predetermined time period, which may be equal to the first predetermined time period. During step S10 the retrieval of input from the A/D convertor is continued. If the signal value v returns to a value lower then said predetermined value T_H before the end of said predetermined time-period then processing flow continues with step S2. Otherwise the value for the occupancy state is internally saved as occupied in step S11, and a message signaling this is transmitted in step S12.
Figure 8 illustrates the signal flow in a message interpreter MI. As shown in more detail in Figure 8, a radio receiver 20 receives the binary "vehicle present" signals D (optionally with timestamp) from the sensor nodes 10 via the radio link and runs a model based state estimator algorithm to calculate the motion states of the vehicles individually (i.e. each vehicle is represented in the message interpreter). The sensor density may be chosen dependent on the required accuracy of the estimation. If a very accurate vehicle tracking is desired multiple sensors per vehicle area may be present.
The message interpreter MI has a vehicle database facility 32, 34 that comprises state information of vehicles present at the vehicle infrastructure.
The message interpreter MI further has a sensor map 45describing the spatial location of the sensor nodes 10. Alternatively, the sensor nodes may transmit their location, or their position could even be derived by a localization method for wireless sensor networks.
The message interpreter MI further has an association facility 40 for associating the messages D provided by the sensor nodes 10 with the state information present in the vehicle data base facility 32, 34. The association facility 40 may associate the messages received with state information for example with one of the methods Gating, Nearest Neighbor (NN), (Joint) Probabilistic Data Association ((J)DPA), Multiple Hypothesis Tracker (MHT) and the MCMCDA.
The message interpreter further has a state updating facility 50 for updating the state information on the basis of the messages D associated therewith by the association facility 40. Once the messages D are associated with a particular vehicle the state of that vehicle in a local vehicle data base is updated by the state updating facility 50.
The association facility 40 and the state updating facility 50 together form a database updating facility DBU.
In the embodiment shown a global map builder 65 may exchange this updated information with global map builders of neighboring message interpreters via network interface 60 (wired or wireless) and to receive close to border detections. Other uses are also possible to exchange the motion state of crossing vehicles (e.g. to calculate system level features like vehicle density and average velocity, but these are independent from the motion state estimation).
A message interpreter MI, shown in Figure 9, consists of a radio receiver 20, coupled to antenna 22, a processing unit 24 (with memory 28) and a network interface 65, as well as a real-time clock 26. In other embodiments a real-time clock may be part of the sensor node, and the sensor node may embed a time-stamp indicative for time at which an event was detected in the message. However, generally a message interpretor will have a more reliable clock, as it can be more reliable synchronized with a reference clock. The network interface 65 couples the message interpreter MI via the communication channel 60 to other message interpreters.
In the embodiment shown the microcontroller 24 of Figure 9 processes the received messages D. The memory 28 stores the local and global vehicle map and the sensor map as well as the software for carrying out the data association and state estimation tasks. In an alternative embodiment separate memories may be present for storing each of these maps and for storing the software. Likewise dedicated hardware may be present to perform one or more of these tasks.
There is no communication or any other interaction between the vehicles tracked. The result of the processing (i.e. the estimation of the motion states of all sensed vehicles) is present in the memory of the message interpreters in a distributed way.
Message interpreters may run additional (cooperative) algorithms to deduct higher level motion characteristics and/or estimate additional vehicle characteristics (e.g. geometry).
For applications in relative small area, e.g. a parking place, the vehicle tracking system may comprise only a single message interpreter MI. In that case the global map builder is superfluous, and local vehicle map is identical to the global vehicle map.
In the embodiment shown in Figure 3, each message interpreter MI for a respective module comprises hardware as described with reference to Figure 8 and 9.
Operation of the message interpreter is further illustrated with respect to Figures 10-13
Figure 10 schematically shows a part of a vehicle infrastructure having sections R_j-1, R_j, R_j+1. By way of example it is presumed that a vehicle moves in a direction indicated by arrow X from R_j-1, via R_j, to R_j+1.
Figure 11 shows an overview of a method for detecting the vehicle performed by the message interpreter for section R_j, using the messages obtained from the sensor nodes.
In step S20 the method waits for a message D from a sensor node. At the moment that a message D is received, program flow continues with step S21, where the time t associated with the message is registered. The registered time t associated with the message may be a time-stamp embedded in the message or a time read from an internal clock of the message interpreter.
In step S22, it is verified whether the detection is made by a sensor node in a location of section Rj that neighbors one of the neighboring sections R_j-1 or R_j+1. If that is the case, then in step S23 the event is communicated via the communication network interface to the message interpreter for that neighboring section. In step S24 it is determined which vehicle O in the vehicle data base facility is responsible for the detected event. An embodiment of a method used to carry out step S24 is described in more detail in Figure 12. After the responsible vehicle O is identified in Step 25, i.e. an association is made with existing vehicle state information, it is determined in Step 26 whether it is present in the section Rj. If that is the case, control flow continues with Step S27, where the state of vehicle O is estimated. Otherwise control flow returns to step S20. A procedure for estimating the state is described in more detail with reference to Figure 13. In step S28 it is determined whether the state information implies that the vehicle O has a position in a neighboring section R_j-1 or R_j+1. In that
case the updated state information is transmitted in step S29 to the message interpreter for the neighboring section and control flow returns to step S20. Otherwise the control flow returns immediately to Step S20.
A method to associate a message D at time t, with a vehicle O is now described in more detail with reference to Figure 12.
In a first step S40, a vehicle index i is initialized (e.g. i=1). In a next step S41, the current state known for the vehicle with that index i is retrieved from the vehicle database facility. In the next step S42 a probability is determined that the vehicle O caused the detection reported by the message D at time t. The vehicle index i is incremented in step S43 and if it is determined in step S44 that i is less than the number of vehicles, the steps S41 to S43 are repeated. Otherwise in step S45 it is determined which vehicle caused the detection reported by the message D at time t with the highest probability. In step S46 the index of that vehicle is returned as the result if the method.
A method to estimate (update the present estimation of) the state of a vehicle is now described in more detail with reference to Figure 13.
In step S60 the messages D₁,...,D_n associated with vehicle O are selected.
In step S61 a probability density function is constructed on the basis of the associated messages D₁,...,D_n.
In step S62 the current state S₀ and time to for vehicle O is retrieved from the vehicle database.
In step S63 it is determined whether the time for which the state S of the vehicle O has to be determined is greater than the time to associated with the current state S₀.
If that is the case, the state S (determined by the estimation method) is the state update of S0 to t, performed in step S65. If that is not the case, then the message D relates to a detection preceding the detection that resulted in the earlier estimation for state S0. In that case the state S0 is updated using the detection D by the state estimation method in step S64
In the claims the word "comprising" does not exclude other elements or steps, and the indefinite article "a" or "an" does not exclude a plurality. A single component or other unit may fulfill the functions of several items recited in the claims. The mere fact that certain measures are recited in mutually different claims does not indicate that a combination of these measures cannot be used to advantage. Any reference signs in the claims should not be construed as limiting the scope. Further, unless expressly stated to the contrary, "or" refers to an inclusive or and not to an exclusive or. For example, a condition A or B is satisfied by any one of the following: A is true (or present) and B is false (or not present), A is false (or not present) and B is true (or present), and both A and B are true (or present).
More details relevant for the present invention are described in the following

Annexes:

A1: Estimation and association for multiple target tracking based on spatially, distributed detections
A2: On Event Based State Estimation

A1: Estimation and association for multiple target tracking based on spatially, distributed detections

Summary. In this paper we consider the multiple object tracking problem with event-based observations. For that we predefine a number detection points which are spatially distributed along the road. Whenever the edge of an object crosses one of the detection points, the position of that detection point together with the time of the event are received by our tracking algorithm. We assume that objects can cover multiple detection points and propose a method to estimate the object's position and orientation from these detections using the shape of the object. Beside that another method is designed which associates newly received detections with a known object. The objects are tracked with an event-based state-estimator that uses the estimated position and orientation, although its design is out of the scope of this paper. Finally our tracking algorithm is critically assessed in a simulation of a parking lot.

1 Introduction

In multiple target tracking [1-3] one aims to track all the objects/targets, which are moving in a certain area. Three basic problems arise from tracking objects. The first one is how to measure the object's position. The second one is to associate a certain measurement with its correct object and the third one is a state-estimator to keep track of all the objects. This paper considers the first 2 issues when objects are not measured but detected.
Consider a system in which objects are detected when they cross a predefined 'detection' point. These detectors are triggered by the event that the object's edge crosses its position. However, they cannot distinguish between the objects. This paper describes a method in which a new detection is associated with the object that most probable generated it. Also, a method is described which estimates the position and orientation of the object given the observations in position and time due to the detections. Other examples in which sensor-data is generated due to an event can be found in [4, 5].
This paper is organized as follows. Section 2 defines background knowledge such as the notation of (object) variables and functions that are used throughout this paper. After that the problem is formulated in section 3 together with existing methods. Section 5 describes the approach which is taken in the design. A more detailed description of the estimation and associated is presented in Section 5 and 6 respectively. Finally both methods are tested in a small application example presented in Section 6 and conclusions are drawn in section 7. But let's start with the background information.

2 Background

In order to be clear about notations and variables this section describes those that can be found throughout this paper.

2.1 Variables

R defines the set of real numbers whereas the set R₊ defines the non-negative real numbers. R _xy defines the set spanned by the vectors e_x and e_y , the point p := x·e_x + y·e_y is shortly denoted as p = (x,y) ^T. The set Z defines the integer values and Z₊ defines the set of non-negative integer numbers. The variable 0 is used either as null, the null-vector or the null-matrix. Its size will become clear from the context.
Vector x(t) ∈ R ⁿ is defined as a vector depending on time t and is sampled using some sampling method. The time t at sampling instant k ∈ Z₊ is defined as t_k ∈ R. The variables τ_k ∈ R , x_k ∈ R ⁿ and x _0:k ∈ R^n×k+1 are defined as: $τ_{k} : = t_{k} - t_{k - 1},$
$x_{k} : = x (t_{k}),$
$x_{0 : k} : = (\begin{matrix} x (t_{0}) & x (t) (_{1}) & \dots & x (t_{k}) \end{matrix}) .$
The matrix A(t ₂ -t ₁ )∈ R^a×b depends on the difference between two time instants t₂ > t ₁ and is shortly denotes as A _{t 2-t 1}

2.2 Functions

The transpose, inverse and determinant of a matrix A∈ R^n×n are denoted as A^T , A ^-1 and |A| respectively.
Let us define the probability of the random vector x∈R ⁿ as the scalar Pr(x) ∈ {0,1} and the conditional probability of x given the vector u ∈ R ^m the scalar Pr(x I u) ∈ {0,1}. The probability density function (PDF), as defined in [6] section B2, of the vector x∈R ⁿ is denoted as p(x) and the conditional PDF of x given u ∈ R ^q is denoted as p(x|u). The expectation and covariance of x are denoted as E[x] and cov(x) respectively. The conditional expectation of x given a vector u is denoted as E[x|u]. The definitions of E[x], E[x|u] and cov(x) can be found in [6] sections B4 and B7.
The Gaussian function, shortly noted as Gaussian, depending on vectors x ∈ R ⁿ and u ∈ R ⁿ and on matrix P ∈ R^n×n is defined as: $\begin{matrix} G (x, u, P) & : R^{n} \times R^{n} \times R^{n \times n} \to R, \\ = \frac{1}{{(2 π)}^{n / 2} |P|} e^{- 0.5 {(x - u)}^{T} P^{- 1} (x - u)} . \end{matrix}$
If p(x) = G(x,u,P), then by definition it holds that E[x] = u and cov(x) = P.
Assume we have the set C ⊂ R ^q and the vectors x ∈ R ^q and y ∈ R ^q . Then the function ∥ x - y ∥∈ R is defined as the distance between vectors x and y. The function |〈x-C〉 |∈ R is defined as the shortest distance from vector x to set C : $\begin{matrix} |< x - C >| & : R^{q} \to R, \\ : = \min (‖ x - c ‖), \forall c \in C . \end{matrix}$

2.3 Object variables

Assume there exist an object which is moving in the 3D world space. This object is observed with, for example, a camera or sensors in the road. Meaning that the object is projected to a 2D space, i.e. R _xy . If we assume that the shape of the projected object is constant and known, then we can draw a smallest, rectangular box around the object. For the box we define a position-vector o = (x, y) ^T ∈ R _xy , equal to the center of the box, and an orientation-vector θ ∈ R. In the case of o = 0 and θ = 0 the corners of this box, as shown in Figure 14, are defined in the set C₀ : $C_{0} : = \{c_{1}, c_{2}, c_{3}, c_{4}\}, {with c}_{i} \in R_{xy} .$
Notice that for an object having a certain o and θ the new corner-positions of the object's box are calculated with C₀. For that a rotation matrix T ∈ R ^q×q is used as defined in (7). An example of the object's box for a certain o and θ is graphically depicted in Figure 14. $T_{k} = (\begin{matrix} \cos (θ_{k}) & \sin (θ_{k}) \\ - \sin (θ_{k}) & \cos (θ_{k}) \end{matrix}) .$
Beside the positions o and θ each object also has a certain shape or geometry which covers a certain set of positions in R _xy , i.e. the grey area of Figure 14. This closed set is denoted with S ⊂ R _xy and is defined as the union of the open set of the object's body S_B ⊂ R _xy and the closed set of the object's edge S_E ⊂ R _xy , i.e. S := S_B ∪S_E. The set S is approximated by a set of sampled position-vectors A = [λ ₁,λ ₂,···,λ_K ], with λ_i ∈ R _xy . To define the vectors λ _i we equidistant sample the rectangular box defined by C₀ using a grid with a distance r. Each λ _i is a grid point within the set S as graphically depicted in Figure 15.
The aim is to estimate position, speed and rotation of the object in the case that its acceleration and rotational speed are unknown. Therefore the object's state-vector s(t) ∈ R⁵ and process-noise w(t) ∈ R² are defined as: $s (t) : = {(\begin{matrix} o (t) & θ (t) & \frac{δo (t)}{δt} \end{matrix})}^{T}, w (t) : = {(\frac{δ o^{2} (t)}{δ t} \frac{δθ (t)}{δt})}^{T} .$
Next the problem is formulated using this background knowledge.

3 Problem formulation

A total of E objects are observed within the set R _xy . The vectors oⁱ = (xⁱ, yⁱ ) ^T and θⁱ are the i^th object's position- and rotation-vector respectively. Tⁱ represents the i^th object's rotation-matrix dependent on θⁱ. The dynamical process of object i with state-vector sⁱ , process-noise wⁱ and measurement-vector mⁱ is defined with the following state-space model: $s_{k}^{i} : = A_{τ_{k}} s_{k - 1}^{i} + B_{τ_{k}} w_{k - 1}^{i},$
$m_{k}^{i} : = (\begin{matrix} o_{k}^{i} \\ θ_{k}^{i} \end{matrix}) = {Cs}_{k}^{i} + v_{k}^{i},$
with $C : = (\begin{matrix} I^{3 \times 3} & 0 \end{matrix}), p (w_{k}^{i}) : = G (w_{k}^{i}, 0, Q^{i}), p (v_{k}) : = f (v_{k}) .$
The definition of the elements of state-vector sⁱ (t), also shown in Figure 16, are: $s^{i} (t) : = {(\begin{matrix} x^{i} (t) & y^{i} (t) & θ^{i} (t) & \frac{{dx}^{i} (t)}{dt} & \frac{d y^{1} (t)}{dt} \end{matrix})}^{T} .$
The objects are observed in R _xy by a camera or a network of sensors. For that M 'detection' points are marked within R _xy and collected in the set D ⊂ R _xy . The position of a detection point is denoted as d ∈ D. The k^th detection of the system generates the observation vector $z_{k}^{i} \in \{R_{xy}, R\}$
if the edge of the i^th object covers one of the detection points d_k ∈ D at time t_k : $z_{k}^{i} : = (d_{k}, t_{k}), if {(T_{k}^{i})}^{- 1} (d_{k} - o_{k}^{i}) \in S_{E}^{i} .$
However, the system does not know which object was detected for it can be any object. As a result the system will not generate $z_{k}^{i}$
but a general observation vector z_k ∈ {R _xy , R}, which is yet to be associated with an object. Therefore, due to the k^th detection, the observation vector z_k is generated whenever one of the E object covers a detection point d_k ∈ D at time t_k : $z_{k} : = (d_{k}, t_{k}), if \{\exists i : {(T_{k}^{i})}^{- 1} (d_{k} - o_{k}^{i}) \in S_{E}^{i}\}$
From equations (9) and (10) we conclude that $z_{k}^{i}$
of (9) is the result after the received observation vector z_k (14) is associated with object i. Notice that both definitions of z_k and $z_{k}^{i}$
assume that the object's edge is detected exactly at a detection point d. In reality the detection will be affected by noise. The object therefore has some probability to be detected at a position υ ∈ R _xy which is close to d. This is modeled by defining that the object's position at the instant of the detection, i.e. υ, is a random vector with mean d and covariance ε ∈ R : $p (υ) : = G (υ, d, εI)$
Figure 16 shows an example of object i which is detected by multiple detection points. The covariance ε of each detection point is also indicated.
The sampling method of the observation vectors z _0:k is a form of event sampling [4, 5, 7]. For a new observation vector is sampled whenever an event, i.e. object detection, takes place. With these event samples all N objects are to be tracked. To accomplish that three methods are needed. The first one is the association of the new observation-vector z_k to an object i and therefore denote it with $z_{k}^{i} .$
Suppose that all associated observation-vectors $z_{n}^{i}$
are collected in the set $Z_{k}^{i} \in \{z_{0 : k}\} .$
Then the second method is to estimate $m_{k}^{i}$
from the observation-set $Z_{k}^{i} .$
This is used in the third method, which is a state-estimator.
Present association methods are: Gating and Nearest Neighbor (NN) [2], (Joint) Probabilistic Data Association ((J)DPA) [2, 8], Multiple Hypothesis Tracker (MHT) [9] and the MCMCDA [10]. Although these can be transformed for associating the event samples z _0:k, this paper will show that the estimation of $m_{k}^{i}$
results in a probability that z_k is in fact $z_{k}^{i},$
i.e. $\Pr (z_{k} = z_{k}^{i}) .$
Therefore the problem which is covered in this paper is the estimation of $m_{k}^{i}$
from the set $Z_{k}^{i},$
which also results in the probability $\Pr (z_{k} = z_{k}^{i}) .$
For that we assume that the shape of the object is known and that it is samples as shown in Section 2.3. The state-estimation is not covered in this paper, although it is used in the application example. Before going into the mathematical details of the estimation we will first describe the approach that is taken.

4 Approach measurement estimation

In the problem formulation we stated that $Z_{k}^{i}$
is defined as the set with all observation-vectors from z _0:k that were associated with object i. We will first redefine this set before continuing with the approach for estimating $m_{k}^{i} .$
The set $Z_{k}^{i} \in [z_{0 : k}]$
is defined as the set of all observation-vectors z_n which were associated with object i , from which their detection point is still covered by the object. We will first show how this is done. At time step k we have the observation-set $Z_{k - 1}^{i}$
and the observation z_k was associated to object i, i.e. $z_{k}^{i} .$
Now if the object's edge is detected at d_k for the first time, then $z_{k}^{i}$
is added to the set $Z_{k - 1}^{i} .$
However, if the object's edge is detected at d_k for the second time, then $z_{k}^{i}$
is not added to the set $Z_{k - 1}^{i}$
and the vector $z_{n}^{i},$
for which holds that d_n = d_k. is removed from $Z_{k - 1}^{i} .$
This because in the second case, it means that object i drove off the detection point positioned at d_n = d_k. Therefore $Z_{k}^{i}$
is defined as: $Z_{k}^{i} : = {\begin{matrix} Z_{k - 1}^{i} \cap z_{k}, {if d}_{k} \neq d_{n}, \forall z_{n}^{i} \in Z_{k - 1}^{i}, \\ Z_{k - 1}^{i} \ z_{n}, if \{\exists n | d_{k} = d_{n} \land z_{n}^{i} = (d_{n}, t_{n}) \in Z_{k - 1}^{i}\} . \end{matrix}$
With this definition of $Z_{k}^{i}$
the approach for estimating $m_{k}^{i},$
i.e. $p (m_{k}^{i} | Z_{k}^{i}),$
is given. For clarity we assume that the object's shape is rectangular and that all its detection points are denoted with d_n , with n ∈ N ⊂ [0,k].

1. The first step is to position the object on each detection point d_n and mirror its set S into the set O_n, as shown in Figure 17 for a single detection. This way we transform the points that are covered by the object, into possible vectors of the object's position $o_{k}^{i} \in O_{n}$
given that it is detected at the detection point d_n.
2. The second step, graphically depicted in Figure 18, is to turn all sets O_n simultaneously around their detection point d_n. This way, each possible orientation $θ_{k}^{i}$
of the object results in a corresponding possible object's position $o_{k}^{i} .$
For $o_{k}^{i}$
must be inside all the sets O_n , ∀n ∈ N, and therefore thus inside the intersection of all sets O_n , ∀n ∈ N, which is denoted as O_N.

Therefore if we apply these two steps for a number of orientations $θ_{k}^{i},$
then at each orientation we have a set O_N which has to contain the object's position $o_{k}^{i} .$
From all these orientations we can calculate $p (m_{k}^{i} | Z_{k}^{i})$
as shown in the next section.

5 Measurement estimation

Estimation of the measurement-vector $m_{k}^{i}$
given the observation set $Z_{k}^{i}$
results in calculating $p (m_{k}^{i} | Z_{k}^{i}) .$
Because both $m_{k}^{i}$
and $Z_{k}^{i}$
always belong to the same object and at sample instant k throughout this section we will remove the sub- and superscripts i and k in the rest of this section. Therefore we have; $m_{k}^{i} \to m$
and $Z_{k}^{i} \to Z .$
The set Z consists of the observation vectors z_n, for all n ∈ N ⊂ [0,k], that were associated to the same object.
Although the measurement vector is defined as m = (o, θ) ^T, with o ∈ R _xy and θ ∈ R, the detection point at time-step n are defined as d_n ∈ R _xy . Meaning that the objects orientation is not directly. However, because every observation vector z_n ∈ Z detects the object for one and the same θ, the PDF p(m|Z) is approximated by sampling in θ, i.e.: $p (m | Z) \approx {(\sum_{l = 1}^{L} α_{i})}^{- 1} \sum_{l = 1}^{L} α_{l} p (o | Z, θ = l Δ),$
with $Δ : = \frac{2 π}{L} {and α}_{l} : = \Pr (θ = l Δ | Z) .$
The main aspect of equation (17) is to determine p(o|Z,θ). To do that we define the set O_n (θ) ∈ R _xy to be equal to all possible object positions o, given that the object is detected at position d_n ∈ z_n (∈ Z) and that the object's rotation is equal to θ. The determination of O_n (θ) ∈ R _xy is presented in the n the next section.
Therefore, if one object is detected at multiple detection points d_n, ∀n ∈ N, then the set of all possible object positions o given a certain θ equals O_n (θ) : $O_{N} (θ) : = \cap_{n \in N}^{} O_{n} (θ) .$
Equation (19) is graphically explained in Figure 19 for two different values of θ and N={1,2}.
Both p(o|Z,θ) and α_l are related to the set O_N (θ) due to the fact that it O_N (theta) defines the set of possible object positions o for a given θ. To calculate p(o|Z,θ) and α_l we define the functions f(o|Z,θ) and g(o|Z,θ): $\begin{array}{l} f (o | z_{n}, θ) : = {\begin{matrix} 0 & if o \notin O_{N} (θ), \\ 1 & if o \in O_{n} (θ), \end{matrix} \\ g (o | Z, θ) : \prod_{n \in N} f (o | z_{n}, θ) = {\begin{matrix} 0 & if o \notin O_{N} (θ), \\ 1 & if o \in O_{N} (θ), \end{matrix} \end{array}$
Therefore the PDF p(o|Z,θ) and probability α_l are: $p (o | Z, θ) : = \frac{g (o | Z, θ)}{\int_{- \infty}^{\infty} g (o | Z, θ) do}, α_{l} : = \frac{\int_{- \infty}^{\infty} g (o | Z, θ) do}{\int_{- \infty}^{\infty} g (o | z_{n}, θ) do} .$
With (21) both p(m|Z) is calculated according to (6). The rest of this section is divided into two parts. The first part derives the probability function based on a single detection, i.e. f(o|z_n,θ). While the second part derives the probability function based on a multiple detections, i.e. g(o|Z,θ).

5.1 Single event detection

In order to derive f(o|z _n,θ) we will use the set Λ, defined in 2.3, which contains the sampled positions λ_i that are covered by the object if o = 0 and θ = 0. Notice that if the object covers the origin, i.e. (x, y) ^T = 0, then the possible values of the object position o are given by the set - Λ. This is graphically depicted in Figure 20 (left). From that we can conclude that if the object covers the detection point d_n , given a certain orientation θ and rotation-matrix T, the sampled set Λ can be transformed into a sampled set of O_n , denoted with O _n : ${\hat{O}}_{n} : = [{\hat{o}}_{1}, {\hat{o}}_{2}, \dots, {\hat{o}}_{K}], with {\hat{o}}_{i} : = d - {Tλ}_{i} \in O_{n} .$
Figure 20 (right) graphically depicts the determination of Ô_n from the set Λ for a given θ and detection point d_n .
The function f(o|z_n,θ), as defined in (15), is approximated by placing a Gaussian function at each sampled position ô _i ∈ Ô_n with a certain covariance dependent on the grid-size r : $f (o | z_{n}, θ) \approx 2 π γ^{2} \sum_{i = 1}^{K} G (o, {\hat{o}}_{i}, γ I^{2 \times 2}),$
with, $γ : = {(\frac{2 Δ}{K})}^{2} (0.25 - 0.05 e^{- \frac{4 (K - 1)}{15}} - 0.08 e^{- \frac{4 (K - 1)}{180}}) .$
The approximation of (18) assumes that the object is detected exactly at d_n. In Section 4 we stated that the detection can be a bit of a detection point. The PDF that the object is detected at position υ ∈ R _xy given the detection point d_n is defined in (15). Inserting this uncertainty into (18) results in the final f (o | z_n , θ): $f (o | z_{n}, θ) \approx 2 π γ^{2} \sum_{i = 1}^{K} \int_{- \infty}^{\infty} G (υ, d, εI) G (o, υ - {Tλ}_{i}, γI) dυ .$
Equation (19) is solved with the following Proposition and the fact that G(x,a+b,C) = G(x-b,a,C) :

Proposition 1. Let there exist two Gaussian functions of the random vectors x ∈ R ⁿ and m ∈ R ^q and the matrix Γ ∈ R^q×n ; G(x,u,U) and G(m,Γx,M). Then they have the following property: $\int_{- \infty}^{\infty} G (x, u, U) G (m, Γ x, M) dx = G (Γ u, m, Γ U Γ^{T} + M) .$
Proof. The proof can be found in Section 9.

Applying Proposition 1 to (19) results in: $f (o | z_{n}, θ) \approx 2 π γ^{2} \sum_{i = 1}^{K} G (o, {\hat{o}}_{i}, R), with R : = (ε + γ) I^{2 \times 2} .$
From f(o|z _n,θ) based on a single detection, the next step to multiple detections, i.e. g(o|Z,θ), is taken.

5.2 Multiple event detections

The aim of this section is to calculate the function g(o|Z,θ) by substituting equation (27) in the definition of g(o|Z,θ) as shown in (20): $g (o | Z, θ) \approx \prod_{n \in N} 2 π γ^{2} \sum_{i = 1}^{K} G (o, {\hat{o}}_{i}^{n}, R), with {\hat{o}}_{i}^{n} : = d_{n} - {Tλ}_{i} .$
If N contains m elements, then calculating equation (22) would result in K^m products of m Gaussian functions and sum them afterwards. This would take too much processing power if m is large. That is why equation (22) is calculated differently.
Instead of using all detection points d_n we will use a subset of them. The derivation of this subset is graphically depicted in Figure 21 for N = {1,2}. For that consider the rectangular set C₀ ∈ R _xy of Section 2.3 defined by its corners [c ₁,c ₂,c ₃,c ₄]. For each detection point d_n we define the set C _n (θ) ⊂ R _xy with corner-points [c _{n 1} (θ),c _{n 2} (θ),c_n3 (θ),c _{n 4} (θ)] defined as: $c_{n_{i}} (θ) : = {Tc}_{i} + d_{n} .$
Let us define the rectangular set C _N (θ) ⊂ R _xy as the intersection of the sets C _n (θ), ∀n ∈ N, i.e.: $C_{N} (θ) : = \underset{n \in N}{\cap} C_{n} (θ) .$
Meaning that each detection point d_n defines a rectangular set denoted with C _n (θ) dependent on rotation θ. The intersection of all these rectangular sets is defined with the set C _N (θ).
In the beginning of this section we defined two different sets shown in Figure 17 and 18. The first set, O_n (θ),shown in Figure 17 defines all possible objet positions o based on a single detection at d_n . The second set, i.e. O_N (θ),shown in Figure 18, defines all possible object positions o based on all detections at d_n , ∀n ∈ N. Notice that as a result O_n (θ) ⊂ C _n (θ) and O_N (θ) ⊂ C _N (θ). Meaning that only within the set C_N (θ) all the functions f (o | z_n,θ) have an overlapping area in which they are 1. Outside C_N (θ) there is always at least one f(o|z _n, θ) which is 0 and therefore makes g(o|Z,θ) outside C _N (θ) equal to 0. Therefore g(o|Z,θ) of (22) can be approximated by taking only those Gaussians of the functions f (o | z _n ,θ) into account of which their mean, i.e. ${\hat{o}}_{i}^{n},$
is close or in the set C _N (θ). We define that close to C _N (θ) means a distance of at most γ+ε, which defined R in (27). The function g(o | Z,θ) of (22) is therefore approximated as: $g (o | Z, θ) \approx \prod_{n \in N} 2 π γ^{2} \sum_{i \in I^{n}}^{} G (o, {\hat{o}}_{i}^{n}, R),$
with $\{|^{n} : | 〈 {\hat{o}}_{i}^{n} - C_{N} (θ) 〉 | \leq (γ + ε), \forall i \in |^{j}\} .$
We can even decrease the number of Gaussians of (25) even further. This because if for a certain detection point d_n it holds C _N (θ) ⊂ O_n (θ), it means that when we remove the detection point d_n it will not affect the set C _N (θ). Therefore equation (25) is reduced to: $g (o | Z, θ) \approx \prod_{n = N} 2 π γ^{2} \sum_{i \in I^{n}}^{} G (o, {\hat{o}}_{i}^{n}, R),$
$with \{N \subset N : C_{N} (θ) \subset O_{n} (θ), \forall n \in \{N \ N\}\} .$
The calculation of (26) is done by applying the following two propositions. The first one, i.e. Proposition 2, shows how to rewrite a product of a summation of Gaussians into a summation of a product of Gaussians. The second one, i.e. Proposition 3, proofs that a product of Gaussians results in a single Gaussian.

Proposition 2. The product of α summation of Gaussians can be written into α summation of a product of Gaussian: $\prod_{j = 1}^{C} ρ_{j} \sum_{i = 1}^{C_{j}} G (x, x_{j}^{i}, R) = (\prod_{j = 1}^{C} ρ_{j} C_{j}) \sum_{m = 1}^{\prod_{j = 1}^{C} C_{j}} \prod_{j = 1}^{C} G (x, x_{j}^{f (j, m)}, R), with$
$\begin{array}{l} f (j, m) : ⌈ \frac{m - \sum_{r = 1}^{j} (f (r - 1, m) - 1) (\prod_{r}^{C + 1} ρ_{r})}{\prod_{r = j + 1}^{C + 1} ρ_{r}} ⌉, \\ f (0, m) : = 1, ρ_{j + 1} : = 1. \end{array}$
The proof is given by writing out the left hand side of (27a) and restructuring it.
Proposition 3. The product of Gaussians is again a Gaussian: $\prod_{j = 1}^{C} G (x, x_{j}, R) = βG (\frac{x, \sum_{j = 1}^{C} x_{j}}{C}, \frac{R}{C}) and β = \prod_{j = 2}^{C} G (x_{j}, \frac{\sum_{n = 1}^{j - 1} x_{n}}{j - 1}, \frac{jR}{j - 1}) .$
The proof is given in Section 10.

Now applying Propositions 2 and 3 on (26) results in a solution of g(o|Z,θ) as a summation of Gaussians of the form: $g (o | Z, θ) = \sum_{i = 1}^{H} β^{i} (θ) G (o, o^{i} (θ), R^{i} (θ)),$
Equation (29) is approximated as a single Gaussian function: $g (o | Z, θ) \approx \overline{β} (θ) G (o, \overline{o} (θ), \overline{R} (θ)), with$
$\begin{array}{l} \overline{β} (θ) : = \sum_{i = 1}^{H} β^{i} (θ), \overline{o} (θ) : \sum_{i = 1}^{H} \frac{β^{i} (θ)}{\overline{β}} o^{i} (θ), \\ \overline{R} (θ) : = \sum_{i = 1}^{H} \frac{β^{i} (θ)}{\overline{β}} (R^{i} (θ) + (\overline{o} (θ) - o^{i} (θ)) {(\overline{ο} (θ) - o^{i} (θ))}^{T}) . \end{array}$
With the result of (30) we can approximate g(o|Z,θ). In order to calculate the PDF p(m|Z), equation (30) is substituted into equation (16) together with f(o|z _n,θ) of (27) to calculate p(o|Z,θ) and α_l. Substituted these results into (13) gives: $p (m | Z) = \sum_{i = 1}^{L} \frac{\overline{β} (l Δ)}{\sum_{l = 1}^{L} \overline{β} (l Δ)} G (o, \overline{o} (l Δ), \overline{R} (l Δ)) .$
As was mentioned in the problem formulation, the PDF p(m|Z) also gives us the probability that a new observation vector is generated by an certain object i. This is discussed in the next section.

6 Detection association

The total probability that a new observation vector z_k is generated by object i is equal to the total probability of the measurement-vector $m_{k}^{i}$
given the observation set $[Z_{k - 1}^{i}, z_{k}] .$
For this probability we can use $p (m_{k}^{i} | Z_{k - 1}^{i}, z_{k})$
which is equal to equation (41). The definition of a PDF is that its total probability, i.e. its integral from -∞ to ∞, is equal to 1. To make sure that $p (m_{k}^{i} | Z_{k - 1}^{i}, z_{k})$
of equation (31) has a total probability of 1, it is divided by its true probability $\Pr (m_{k}^{i} | Z_{k - 1}^{i}, z_{k}) .$
In order to be able to compare these different measurement-vector per object, we normalize each probability with the surface covered by the object. As a result, $\Pr (z_{k} = z_{k}^{i})$
is equal to: $\Pr (z_{k} = z_{k}^{i}) = \frac{1}{2 π {(γ^{i})}^{2} K^{i}} \sum_{l = 1}^{L} \overline{β} (l Δ) .$
The variables γⁱ and Kⁱ are equal to γ and K respectively, which define the approximation of the function $f (m_{k}^{i} | z_{n}^{i}, θ_{k}^{i})$
as shown in (6.1). With the probability of (3) one can design a method which associates an observation-vector due to a new detection, to its most probable object i. Although the estimation method requires a certain amount of processing power, one can reduce this by reducing the number of samples in the set A. Meaning that association and estimation can be done with different sizes of A. Moreover, if the objects have a rectangular shape, then with some tricks one can reduce the amount of processing power to a level at which both association as well as estimation can run real-time.
Now that both the measurement estimation as well as the detection association are designed, both are tested in a multiple object tracking application.

7 Application example

As an application example we take a parking lot of 50 by 50 meters with a network of wireless sensors distributed randomly along the road's surface. Each sensor can detect a crossing vehicle. A total of 2500 sensors was used resulting in a density of one sensor per square meter. The vehicles are all assumed rectangular shaped objects with a length of 5 meters and a width of 2 meters. A total of 4 vehicles manoeuvre within the parking lot and are tracked using a data-associator followed with an event state-estimator.
The simulation case is made such that it contains two interesting situation. One in which two vehicles cross each other in parallel and one where two vehicles cross perpendicular. For comparison the objects are tracked using two different association methods. The first one is a combination of Gating and detection association of 6. The second one is a combination of Gating and Nearest Neighbor.
The result of the detection associator (DA) for both crossings is shown in Figure 22 while the result of the Nearest Neighbor (NN) associator is shown in Figure 23. In both results the real object is plotted in a thick, solid line while its estimated one is plotted in a thin, solid line. The associated detections of each object are given with a symbol which is different for each object; '□' if associated with vehicle 1, '○' if associated with vehicle 2, '∇' if associated with vehicle 3 and '*' if associated with vehicle 4.
Figure 22 shows with the DA all detections were correctly associated to the one object, while If NN is used as an association method,we see that a lot of incorrect associated detections. Therefore we can concluded that using the detection association of 7 results in less estimation-error compared to NN.
A second simulation is done to compare the percentage of incorrect associated detections. Again for the both DA as well as NN only now 4 different amount of detection points were used: 3000, 2500, 2000 and 1500. This table shows that the detection association has a better performance compared to Nearest Neighbor. Table 1: Percentage of incorrect association

amount of detection points DA NN

3000 0 % 4.5 %

2500 0 % 5.6 %

2000 0 % 7.8 %

1500 0 % 2.2 %

8 Conclusions

This paper presents a method for estimating the position- and rotation-vector of objects from spatially, distributed detections of that object. Each detection is generated at the event that the edge of an object crosses a detection point. From the estimation method a detection associator is also designed. This association method calculates the probability that a new detection was generated by an object i.
An example of a parking lot shows that the detection association method has no incorrect associated detections in the case that two vehicles cross each other both in parallel as well as orthogonal. If the association method of Nearest Neighbor was used, a large amount of incorrect associated detections were noticed, resulting in a higher state-estimation error.
The data-assimilation can be further improved with two adjustments. The first one is replacing the set S with S_E only at the time-instants that the observation vector is received. The second improvement is to take the detection points that have not detected anything also in account.

References

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8. R. Karlsson and F. Gustafsson, "Monte Carlo data association for multiple target tracking," IEE International Seminar on Target Tracking: Algorithms and Applications, 2001.
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12. Kelly, Introduction to probability. Macmillan Publishing Company, 1994.
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9 Proof of Proposition 1

Proof. Defined are two Gaussian functions with the vectors x ∈ R ⁿ , u ∈ R ⁿ , m ∈ R ^q and matrices U ∈ R^n×n, M ∈ R^q× ^q, Γ ∈ R^q×n: G(x,u,U) and G(m,Γx,M). Suppose we define the following PDFs and relation of m with some c ∈ R ^q : $m = Γ x + c, with$
$p (c) : = G (m, 0, M) and p (x) : = G (x, u, U) .$
Then from probability theory [6] p(m) is equal to: $\begin{matrix} p (m) : = \int_{- \infty}^{\infty} p (m | x) p (x) dx & (34 a) \\ = \int_{- \infty}^{\infty} G (m, Γ x, M) G (x, u, U) dx . & (34 b) \end{matrix}$
Applying theorem 3.2.1 of [9] on (44) we have that p(Γx) = G(Γx,Γu,ΓUΓ ^T ). Now if we have the random vectors a ∈ R ⁿ and b ∈ R ⁿ with p(a) = G(a,u ₁,U ₁) and p(b)=G(b,u₂,U ₂) then they have the property p(a+b)=G(a+b,u ₁+u ₂,U ₁+U ₂) as proven in [8]. Applying this on (43) results in: $p (m) = G (m, Γ u, Γ U Γ^{T} + M),$
$\Rightarrow \int_{- \infty}^{\infty} G (m, Γ x, M) G (x, u, U) dx = G (m, Γ u, Γ U Γ^{T} + M) .$

10 Proof of Proposition 3

Proof. A product of Gaussians can be written as: $\begin{matrix} \prod_{j = 1}^{N} G (x, x_{j}, R) = G (x, x_{N}, R) \prod_{j = 1}^{N - 1} G (x, x_{j}, R), & (36 a) \\ = β_{N - 1} G (x, x_{N}, R) G (x, \frac{\sum_{j = 1}^{N - 1} x_{j}}{N - 1}, \frac{R}{N -}) . & (36 b) \end{matrix}$
From Proposition 1 and the Kalman filter in Information form [13], a product of 2 Gaussians equals: $G (x, u, U) G (m, x, M) = G (m, u, U + M) G (x, d, D),$
with $D^{- 1} = U^{- 1} + M^{- 1}, d = {DU}^{- 1} u + {DM}^{- 1} m .$
Applying (11) on (50), together with the fact that G(x, y, Z) = G(y, x, Z) we have: $\prod_{j = 1}^{N} G (x, x_{j}, R) = β_{N - 1} G (x_{N}, \frac{\sum_{j = 1}^{N - 1} x_{j}}{N - 1}, \frac{NR}{N - 1}) G (x, \frac{\sum_{j = 1}^{N - 1} x_{j}}{N}, \frac{R}{N}) .$
Equation (53) is equal to (37) for: $\begin{array}{l} β_{N} = β_{N - 1} G (x_{N}, \frac{\sum_{j = 1}^{N - 1} x_{j}}{N - 1}, \frac{NR}{N - 1}), & (39 a) \\ = \prod_{i = 2}^{N} G (x_{i} \frac{\sum_{j = 1}^{i - 1}}{i - 1}, \frac{iR}{i - 1}) . & (39 b) \end{array}$

A2: On Event Based State Estimation

Summary. To reduce the amount of data transfer in networked control systems and wireless sensor networks, measurements are usually taken only when an event occurs, rather that at each synchronous sampling instant. However, this complicates estimation and control problems considerably. The goal of this paper is to develop a state estimation algorithm that can successfully cope with event based measurements. Firstly, we propose a general methodology for defining event based sampling. Secondly, we develop a state estimator with a hybrid update, i.e. when an event occurs the estimated state is updated using measurements; otherwise the update is based on the knowledge that the monitored variable is within a bounded set used to define the event. A sum of Gaussians approach is employed to obtain a computationally tractable algorithm.

1. Introduction

Different methods for state estimation have been introduced during the last decades. Each method is specialized in the type of process, the type of noise or the type of system architecture. In this paper we focus on the design of a state estimator that can efficiently cope with event based sampling. By even sampling we mean that measurements are generated only when an a priori defined event occurs in the data monitored by sensors. Such an effective estimator is very much needed in both networked control systems and wireless sensor networks (WSNs) [1]. Especially in WSNs, where the limiting resource is energy, data transfer and processing power must be minimized. The existing estimators that could be used in this framework are discussed in Section 4. For related research on event based control, the interested reader is referred to the recent works [2], [3].
The contribution of this paper is twofold. Firstly, we introduce a general mathematical description of event based sampling. We assume that the estimator does not know when new measurements are available, which usually results in unbounded eigenvalues of its error-covariance matrix. To obtain an estimator with a bounded error-covariance matrix, we develop an estimation algorithm with hybrid update, which is the second main contribution. The developed event based estimator is updated both when an event occurs, with received measurements, as well as at sampling instants synchronous in time. Then the update is based on the knowledge that the monitored variable is within a bounded set used to define the event. In order to meet the requirements of a low processing power, the proposed state estimator is based on the Gaussian sum filter [4, 5], which is known for its computational tractability.

2 Background notions and notation

R defines the set of real numbers whereas the set R₊ defines the non-negative real numbers. The set Z defines the integer numbers and Z₊ defines the set of non-negative integer numbers. The notation 0 is used to denote either the null-vector or the null-matrix. Its size will become clear from the context.
A vector x(t)∈ R ⁿ is defined to depend on time t ∈ R and is sampled using some sampling method. Two different sampling methods are discussed. The first one is time sampling in which samples are generated whenever time t equals some predefined value. This is either synchronous in time or asynchronous. In the synchronous case the time between two samples is constant and defined as t_s ∈ R₊. If the time t at sampling instant k_a ∈ Z₊ is defined as t_ka , with t_{0 _a} := 0, we define: $x_{k_{a}} : = x (t_{k_{a}}) {and x}_{0_{a} : k_{a}} : = (x (t_{0_{a}}), x (t_{1_{a}}), \dots, x (t_{k_{a}})) .$
The second sampling method is event sampling, in which samples are taken when an event occurred. If t at event instant k_e ∈ Z₊ is defined as t_ke , with t _{0 _e} := 0, we define: $x_{k_{e}} : = x (t_{k_{e}}) {and x}_{0_{e} : k_{e}} : = (x (t_{0_{e}}), x (t_{1_{e}}), \dots, x (t_{k_{e}})) .$
A transition-matrix A _{t 2 -t 1} ∈ R^a×b is defined to relate the vector u(t ₁)∈ R ^b to a vector x(t ₂) ∈ R ^a as follows: x(t ₂) = A _{t 2-t 1} u(t ₁).
The transpose, inverse and determinant of a matrix A∈ R^n×n are denoted as A^T , A ^-1 and |A| respectively. The i^th and maximum eigenvalue of a square matrix A are denoted as λ_i (A) and λ_max (A) respectively. Given that A∈ R^n×n and B∈ R^n×n are positive definite, denoted with A ≻ 0 and B ≻ 0, then A ≻ B denotes A - B ≻ 0. A±0 denotes A is positive semi-definite.
The probability density function (PDF), as defined in [6] section B2, of the vector x∈ R ⁿ is denoted with p(x) and the conditional PDF of x given u ∈ R ^q is denoted as p(x|u). The expectation and covariance of x are denoted as E[x] and cov(x) respectively. The conditional expectation of x given u is denoted as E[x|u]. The definitions of E[x], E[x|u] and cov(x) can be found in [6] sections B4 and B7.
The Gaussian function (shortly noted as Gaussian) of vectors x ∈ R ⁿ and u ∈ R ⁿ and matrix P∈ R^n×n is defined as G(x, u, P) : R ⁿ ×R ⁿ ×R^n×n → R , i.e.: $G (x, u, P) = \frac{1}{\sqrt{{(2 π)}^{n} |P|}} e^{- 0.5 {(x - u)}^{T} P^{- 1} (x - u) .}$
If p(x) = G(x,u,P), then by definition it holds that E[x] = u and cov(x) = P.
The element-wise Dirac-function of vector x ∈ R ⁿ , denoted as δ(x):R ⁿ →{0,1}, satisfies: $δ (x) = {\begin{matrix} 0 & if & x ≢ \underset{̲}{0}, \\ 1 & if & x \equiv \underset{̲}{0}, \end{matrix} and \int_{- \infty}^{\infty} δ (x) ⅆ x = 1.$
For a vector x ∈ R ⁿ and a bounded Borel set [7] Y ⊂ R ⁿ , the set PDF is defined as Λ _Y (x):R ⁿ → {0, v} with v ∈ R defined as the Lebesque measure [8] of the set Y, i.e.: $Λ_{Y} (x) = {\begin{matrix} 0 & if & x \notin Y, \\ v^{- 1} & if & x \in Y . \end{matrix}$

3 Event sampling

Many different methods for sampling a vector y(t)∈ R ^q can be found in literature. The one mostly used is time sampling in which the $k_{a}^{th}$
sampling instant is defined at time t_ka := t _{k_a -1} + τ_{k_a -1} for some τ_{k_a -1} ∈ R₊. Recall that if y(t) is sampled at t_a it is denoted as y_ka. This method is formalized by defining the observation vector $z_{k_{a} - 1} : = {(y_{k_{a} - 1}^{T}, t_{k_{a} - 1})}^{T} \in R^{q + 1}$
at sampling instant k _a-1 . Let us define the set H_ka (z_{k_a -1}) ⊂ R containing all the values that t can take between t _{k_a -1} and t _{k_a -1}+τ _{k_a -1}, i.e.: $H_{k_{a}} (z_{k_{a} - 1}) : = \{t \in R | t_{k_{a} - 1} \leq t < t_{k_{a} - 1} + τ_{k_{a} - 1}\} .$
Then time sampling defines that the next sampling instant, i.e. k_a , takes place whenever present time t exceeds the set H_ka (z _{k_a -1}). Therefore z_ka is defined as: $z_{k_{a}} : = (y_{k_{a}}, t_{k_{a}}) if t \notin H_{k_{a}} (z_{k_{a} - 1}) .$
In the case of synchronous time sampling τ_ka = t_s , ∀ _ka ∈ Z₊, which is graphically depicted in Figure 24. Notice that with time sampling, the present time t specifies when samples of y(t) are taken, but time t itself is independent of y(t) . As a result y(t) in between the two samples can have any value within R ^q . Recently, asynchronous sampling methods have emerged, such as, for example "Send-on-Delta" [9, 10] and "Integral sampling" [11]. Opposed to time sampling, these sampling methods are not controlled by time t, but by y(t) itself.
Next, we present a general definition of event based sampling, which recovers the above mentioned asynchronous methods, for a particular choice of ingredients. Let us define the observation vector at sampling instant k_e-1 as $z_{k_{e} - 1} : = {(y_{k_{e} - 1}^{T}, t_{k_{e} - 1})}^{T} \in R^{q + 1} .$
With that we define the following bounded Borel set in time-measurement-space, i.e. H_ke (z _{k_e-1},t) ⊂ R^q+1, which depends on both z_{k_e -1} and t. In line with time sampling the next event instant, i.e. k_e , takes place whenever y(t) leaves the set H_ke (z _{k_e-1},t) as shown in Figure 25 for q = 2. Therefore z _ke is defined as: $z_{k_{e}} : = (y_{k_{e}}, t_{k_{e}}) if y (t) \notin H_{k_{e}} (z_{k_{e} - 1}, t) .$
The exact description of the set H_ke (z_{k_e -1},t) depends on the actual sampling method. As an example H_ke (z_{k_e-1},t) is derived for the method "Send-on-Delta", with y(t) ∈ R. In this case the event instant k_e occurs whenever |y(t) - y _{k_e -1}| exceeds a predefined level Δ, see Figure 26, which results in H_ke (z _{k_e-1},t)={y∈ R|-Δ< y-y _{k_e -1} <Δ}.
In event sampling, a well designed H_ke (z _{k_e-1},t) should contain the set of all possible values that y(t) can take in between the event instants k_e-1 and k_e. Meaning that if t _{k_e -1}≤t<t_ke , then y(t)∈ H_ke (z _{k_e -1},t). A sufficient condition is that y _{k_e -1}∈ H_ke (z _{k_e -1} ,t), which for "Send-on-Delta" results in y(t)∈ [y _{k_e -1}-Δ,y _e-1 +Δ] for all t _{k_e -1}≤t<t_ke.

4 Problem formulation: State estimation based on event sampling

Assume a perturbed, dynamical system with state-vector x(t) ∈ R ⁿ , process-noise w(t) ∈ R ^m , measurement-vector y(t) ∈ R ^q and measurement-noise v(t) ∈ R ^q . This process is described by a state-space model with A_τ ∈ R^n×n, B_τ ∈ R^n×m and C ∈ R^q×n. An event sampling method is used to sample y(t). The model of this process becomes: $x (t + τ) = A_{τ} x (t) + B_{τ} w (t),$
$y (t) = Cx (t) + v (t),$
$z_{k_{e}} = {(y_{k_{e}}^{T}, t_{k_{e}})}^{T} if y (t) \notin H_{k_{e}} (z_{k_{e} - 1}, t),$
with $p (w (t)) : = G (w (t), 0, Q) and p (v (t)) : = G (v (t), 0, V) .$
The state vector x(t) of this system is to be estimated from the observation vectors z_{0 _e :} _ke. Notice that the estimated states are usually required at all synchronous time samples k_a , with t_s = t _{k_a} -t _{k_a-1}, e.g., as input to a controller that runs synchronously in time. As such our goal is to construct an event-based state-estimator (EBSE) that provides an estimate of x(t) not only at the event instants t_ke but also at the sampling instants t_ka . Therefore, we define a new set of sampling instants t_n as the combination of sampling instants due to event sampling, i.e. k_e , and time sampling, i.e. k_a : $\{t_{0 : n - 1}\} : = \{t_{0_{a} : k_{a} - 1}\} \cup \{t_{0_{e} : k_{e} - 1}\} {and t}_{n} : = {\begin{matrix} t_{k_{a}} & if & t_{k_{a}} < t_{k_{e}}, \\ t_{k_{e}} & if & t_{k_{a}} \geq t_{k_{e}} . \end{matrix}$
and $t_{0} < t_{1} < \dots < t_{n}, x_{n} : = x (t_{n}), y_{n} : = y (t_{n}) .$
The estimator calculates the PDF of the state-vector x_n given all the observations until t_n . This results in a hybrid state-estimator, for at time t_n an event can either occur or not, which further implies that measurement data is received or not, respectively. In both cases the estimated state must be updated (not predicted) with all information until t_n . Therefore, depending on t_n a different PDF must be calculated, i.e.: if $t_{n} = t_{k_{a}} \Rightarrow p (x_{n} | z_{0_{e} : k_{e} - 1}) {with t}_{k_{e} - 1} < t_{k_{a}} < t_{k_{e}},$
if $t_{n} = t_{k_{e}} \Rightarrow p (x_{n} | z_{0_{e} : k_{e}}) .$
The important parameters for the performance of any state-estimator are the expectation and error-covariance matrix of its calculated PDF. Therefore, from (9) we define: $x_{n | n} : = {\begin{matrix} E ⌊ x_{n} | z_{0_{e} : k_{e} - 1} ⌋ & if & t_{n} = t_{k_{a}} \\ E [x_{n} | z_{0_{e} : k_{e}}] & if & t_{n} = t_{k_{e}} \end{matrix} {and P}_{n | n} : = cov (x_{n} - x_{n | n}) .$
The PDFs of (9) can be described as the Gaussian G(x_n, x _n | _n ,P _n|n). The square root of the eigenvalues of P_n|n, i.e. $\sqrt{λ_{i} (P_{n | n})},$
define the shape of this Gaussian function. Together with x _n|n they indicate the bound which surrounds 63% of the possible values for x_n . This is graphically depicted in Figure 27 for the 1D case and Figure 29 for a 2D case, in a top view. The smaller the eigenvalues λ_i (P_n|n) are, the smaller the estimation-error is.
As such, the problem of interest in this paper is to construct a state-estimator suitable for the general event sampling method introduced in Section 3 and which is computationally tractable. Furthermore, it is desirable to guarantee that P _n|n has bounded eigenvalues for all n.
Existing state estimators can be divided into two categories. The first one contains estimators based on time sampling: the (a)synchronous Kalman filter [12, 13] (linear process, Gaussian PDF), the Particle filter [14] and the Gaussian sum filter [4, 5] (nonlinear process, non-Gaussian PDF). These estimators cannot be directly employed in event based sampling as if no new observation vector z_ke is received, then t_n-t_ke → ∞ and λ_i (P_{n|k_e -1})→∞. The second category contains estimators based on event sampling. In fact, to the best of our knowledge, only the method proposed in [15] fits this category. However, this EBSE is only applicable in the case of "Send-on-Delta" event sampling and it requires that any PDF is approximated as a single Gaussian function. Moreover, the asymptotic property of P _n|n is not investigated in [15].
In the next section we propose a novel event-based state-estimator, suitable for any event sampling method, together with a preliminary result on asymptotic analysis.

5 An event-based state estimator

The EBSE estimates x_n given the received observation vectors until time t_n . Notice that due to the definition of event sampling we can extract information of all the measurement vectors y_0:n. For with t_i ∈ {t_0:n} and t_je ∈ {t _{0 _e :k_e}} it follows that: ${\begin{matrix} y_{i} \in H_{j_{e}} (z_{j_{e} - 1}, t_{i}) & if & t_{j_{e} - 1} \leq t_{i} < t_{j_{e}}, \\ y_{i} = y_{j_{e}} & if & t_{i} = t_{j_{e}} . \end{matrix}$
Therefore, from the observation vectors z_{0 _e :k_e} and (16) the PDFs of the hybrid state-estimation of (5), with the bounded, Borel set Y_i ⊂ R ^q , results in: $p (x_{n} | y_{0} \in Y_{0}, y_{1} \in Y_{1}, \dots, y_{n} \in Y_{n}) with$
$Y_{i} : = {\begin{matrix} H_{j_{e}} (z_{j_{e} - 1}, t_{i}) & if & t_{j_{e} - 1} < t_{i} < t_{j_{e}}, \\ \{y_{j_{e}}\} & if & t_{i} = t_{j_{e}} . \end{matrix}$
For brevity (17) is denoted as p(x_n |y _0:n ∈ Y_0:n) and with Bayes-rule [16] yields: $p (x_{n} | y_{0 : n} \in Y_{0 : n}) : = \frac{p (x_{n} | y_{0 : n - 1} \in Y_{0 : n - 1}) p (y_{n} \in Y_{n} | x_{n})}{p (y_{n} \in Y_{n} | y_{0 : n - 1} \in Y_{0 : n - 1})} .$
To have an EBSE with low processing demand, multivariate probability theory [17] is used to make (13) recursive: $p (a | b) : \int_{- \infty}^{\infty} p (a | c) p (c | b) ⅆ c \Rightarrow$
$p (x_{n} | y_{0 : = 1} \in Y_{0 : n - 1}) = \int_{- \infty}^{\infty} p (x_{n} | x_{n - 1}) p (x_{n - 1} | y_{0 : n - 1} \in Y_{0 : n - 1}) ⅆ x_{n - 1},$
$p (y_{n} \in Y_{n} | y_{0 : n - 1} \in Y_{0 : n - 1}) = \int_{- \infty}^{\infty} p (x_{n} | y_{0 : n - 1} \in Y_{0 : n - 1}) p (y_{n} \in Y_{n} | x_{n}) ⅆ x_{n} .$
The calculation of p(x_n |y _0:n ∈ Y _0:n) is done in three steps: 1. Assimilate p(y_n ∈ Y_n |x_n ) for both t_n = t_ke and t_n =t_ka. 2. Calculate p(x_n|y _0:n ∈ Y _0:n) as a summation of N Gaussians. 3. Approximate p(x_n |y_0:n ∈ Y _0:n) as a single Gaussian function. The reason for this last step is to design an algorithm in which p(x_n |y _0:n ∈ Y _0:n) is described by a finite set of Gaussians and therefore attain computational tractability. Notice that (13) gives a unified description of the hybrid state-estimator, which makes an asymptotic analysis of the EBSE possible, as it will be shown later in this section.

5.1 Step1: measurement assimilation

This section gives a unified formula of the PDF p(y_n ∈ Y_n |x_n ) valid for both t_n =t_ke and t_n =t_ka. From multivariate probability theory [17] and (7b) we have: $p (y_{n} \in Y_{n} | x_{n}) : = \int_{- \infty}^{\infty} p (y_{n} | x_{n}) p (y_{n} \in Y_{n}) ⅆ y_{n} and p (y_{n} | x_{n}) = G (y_{n}, {Cx}_{n}, V) .$
The PDF p(y_n ∈ Y_n ) is modeled as a uniform distribution for all y_n ∈ Y_n . Therefore, depending on the type of instant, i.e. event or not, we have: $p (y_{n} \in Y_{n}) : = {\begin{matrix} Λ_{H_{k_{e}}} & if & t_{k_{e} - 1} < t_{n} < t_{k_{e}}, \\ δ (y_{n} - y_{k_{e}}) & if & t_{n} = t_{k_{e}} . \end{matrix}$
Substitution of (16) into (15) gives that p(y_n ∈ Y_n |x_n ) =G(y_ke ,Cx_n ,V) if t_n =t_ke . However, if t_n =t_ka then p(y_n ∈ Y_n |x_n ) equals $Λ_{H_{k_{e}}} (y_{n}),$
which is not necessarily Gaussian. Moreover, it depends on the set H_ke and therefore on the actual event sampling method that is employed. In order to have a unified expression of p(y_n ∈ Y_n |x_n ) for both types of t_n , independent of the event sampling method, $Λ_{H_{k_{e}}} (y_{n})$
can be approximated as a summation of N Gaussians, i.e. $Λ_{H_{k_{e}}} (y_{n}) \approx \sum_{i = 1}^{N} α_{n}^{i} G (y_{n}, y_{n}^{i}, V_{n}^{i}) and \sum_{i = 1}^{N} α_{n}^{i} : = 1.$
This is graphically depicted in Figure 29 for y_n ∈ R². The interested reader is referred to [4 for more details.
Substituting (17) into (16) yields the following p(y_n ∈ Y_n |x_n ) if t_n =t_ka : $p (y_{n} \in Y_{n} | x_{n}) \approx \sum_{i = 1}^{N} α_{n}^{i} \int_{- \infty}^{\infty} G (y_{n}, {Cx}_{n}, V) G (y_{n}, y_{n}^{i}, V_{n}^{i}) ⅆ y_{n} .$

Proposition 1. [12, 14] Let there exist two Gaussians of random vectors x ∈ R ⁿ and m ∈ R ^q , with Γ ∈ R^q×n: G(m,Γx,M) and G(x,u,U). Then they satisfy: $\int_{- \infty}^{\infty} G (x, u, U) G (m, Γ x, M) ⅆ x = G (Γ u, m, Γ U Γ^{T} + M),$
$\begin{array}{l} G (x, u, U) G (m, Γ x, M) = G (x, d, D) G (m, Γ u, Γ U Γ^{T} + M), \\ with D : = {(U^{- 1} + Γ^{T} M^{- 1} Γ)}^{- 1} and d : = {DU}^{- 1} u + D Γ^{T} M^{- 1} m . \end{array}$
Applying Proposition 1 ((19) to be precise) and G(x,y, Z) = G(y,x,Z) on (18) yields: $p (y_{n} \in Y_{n} | x_{n}) \approx \sum_{i = 1}^{N} α_{n}^{i} G (y_{n}^{i}, {Cx}_{n}, V + V_{n}^{i}), {if t}_{n} = t_{k_{a}} .$
In conclusion we can state that the unified expression of the PDF p(y_n ∈ Y|x_n ), at both t_n =t_ke and t_n =t_ka , for any event sampling method results in: $p (y_{n} \in Y_{n} | x_{n}) \approx \sum_{i = 1}^{N} α_{n}^{i} G (y_{n}^{i}, {Cx}_{n}, R_{n}^{i}) {with R}_{n}^{i} : = V + V_{n}^{i} .$
If t_n =t_ke the variables of (22) are: N=1, $α_{n}^{} = 1,$
$y_{n}^{} = y_{k_{e}}$
and $V_{n}^{} = \underset{̲}{0.}$
If t_n =t_ka the variables depend on $Λ_{H_{k_{e}}} (y_{n})$
and its approximation. As an example these variables are calculated for the method "Send-on-Delta" with y ∈ R.
In "Send-on-Delta"', for certain N, the approximation of $Λ_{H_{k_{e}}} (y_{n}),$
as presented in (17), is obtained with i ∈ {1,2,...,N} and: $\begin{array}{l} y_{n}^{i} & = y_{k_{e} - 1} - (\frac{N - 2 (i - 1) - 1}{2 N}) 2 Δ and \\ α_{n}^{i} & = 1 / N, V_{n}^{i} = {(\frac{2 Δ}{N})}^{2} (0.25 - 0.05 e^{- \frac{4 (N - 1)}{15}} - 0.08 e^{- \frac{4 (N - 1)}{180}}), \forall i . \end{array}$

With the result of (22), p(x_n |y _0:n ∈ Y _0:n) can also be expressed as a sum of N Gaussians.

5.2 Step2: state estimation

First the PDF p(x_n |y _0:n ∈ Y _0:n-1) of (14b) is calculated. From the EBSE we have p(x _n-1|y _0:n-1 ∈ Y _0:n-1):=G(x _n-1,x _n-1|n-1,P _n-1,n-1) and from (7) with τ_n :=t_n-t _n-1 we have $p (x_{n} | x_{n - 1}) : = G (x_{n}, A_{τ_{n}} x_{n - 1}, B_{τ_{n}} Q B_{τ_{n}}^{T}) .$
Therefore using (19) in (14b) yields: $\begin{array}{l} p (x_{n} | y_{0 : n - 1} \in Y_{0 : n - 1}) = G (x_{n}, x_{n | n - 1}, P_{n, n - 1}) with \\ x_{n | n - 1} : = A_{τ_{n}} x_{n - 1 | n - 1} {and P}_{n | n - 1} : = A_{τ_{n}} P_{n - 1 | n - 1} A_{τ_{n}}^{T} + B_{τ_{n}} {QB}_{τ_{n}}^{T} . \end{array}$
Next p(x _n|y_0:n ∈ Y _0:n), defined in (13), is calculated after multiplying (22) and (24): $p (x_{n} | y_{n - 1} \in Y_{0 : n - 1}) p (y_{n} \in Y_{n} | x_{n}) \approx \sum_{i = 1}^{N} α_{n}^{i} G (x_{n}, x_{n | n - 1}, P_{n | n - 1}) G (y_{n}^{i}, {Cx}_{n}, R_{n}^{i}) .$
Equation (25) is explicitly solved by applying Proposition 1: $p (x_{n} | y_{0 : n - 1} \in Y_{0 :}) p (y_{n} \in Y_{n} | x_{n}) \approx \sum_{i = 1}^{N} α_{n}^{i} β_{n}^{i} G (x_{n}, x_{n}^{i}, P_{n}^{i}) with$
$\begin{array}{l} x_{n}^{i} : = P_{n}^{i} (P_{n | n - 1}^{- 1} x_{n | n - 1} + C^{T} {(R_{n}^{i})}^{- 1} y_{n}^{i}), P_{n}^{i} : = {(P_{n | n - 1}^{- 1} + C^{T} {(R_{n}^{i})}^{- 1} C)}^{- 1} \\ {and β}_{n}^{i} : = G (y_{n}^{i}, {Cx}_{n | n - 1}, {CP}_{n | n - 1} C^{T} + R_{n}^{i}) . \end{array}$
The expression of p(x_n |y _0:n ∈ Y _0:n) as a sum of N Gaussians is the result of the following substitutions: (26) into (13), (26) into (14c) to obtain p(y_n ∈ Y_n |y _0:n-1 ∈ Y _0:n-1) and the latter into (13) again. This yields $p (x_{n} | y_{0 : n} \in Y_{0 : n}) \approx \sum_{i = 1}^{N} \frac{α_{n}^{i} β_{n}^{i}}{\sum_{i = 1}^{N} α_{n}^{i} β_{n}^{i}} G (x_{n}, x_{n}^{i}, P_{n}^{i}) .$
The third step is to approximate (27) as a single Gaussian to retrieve a computationally tractable algorithm. For if both p(x _n-1|y _0:n-1 ∈ Y_0:n-1) and p(y_n ∈ Y_n |x_n ) are approximated using N Gaussians, the estimate of x_n in (27) is described with M_n Gaussians. The value of M_n equals M _n-1 N, meaning that M_n increases after each sample instant and with it also the processing demand of the EBSE increases.

5.3 Step3: state approximation

p(x_n |y _0: _n ∈ Y _0:n) of (27) is approximated as a single Gaussian with an equal expectation and covariance matrix, i.e.: $p (x_{n} | y_{0 : n} \in Y_{0 : n}) \approx G (x_{n}, x_{n | n}, P_{n | n}) with$
$x_{n | n} : = \sum_{i = 1}^{N} \frac{α_{n}^{i} β_{n}^{i} x_{n}^{i}}{\sum_{i = 1}^{N} α_{n}^{i} β_{n}^{i}}, P_{n | n} : = \sum_{i = 1}^{N} \frac{α_{n}^{i} β_{n}^{i}}{\sum_{i = 1}^{N} α_{n}^{i} β_{n}^{i}} (P_{n}^{i} + (x_{n | n} - x_{n}^{i}) {(x_{n | n} - x_{n}^{i})}^{T}) .$
The expectation an covariance of (27), equal to x _n|n and P _n|n of (28), can be derived from the corresponding definitions. Notice that because the designed EBSE is based on the equations of the Kalman filter, the condition of computational tractability is met.

5.4 Asymptotic analysis of the error-covariance matrix

In this section we investigate the asymptotic analysis of the error-covariance matrix of the developed EBSE. By this we mean that we analyze lim_n→∞P_n|n, which for convenience is denoted as P _∞. Note that for the classical Kalman filter (KF) [12] such an analysis is already available. However, for any other type of estimator asymptotic analysis remains a very challenging problem, which is why in most cases it is not even considered.
Let us first recall the result on the asymptotic analysis of the Kalman filter. If x(t) of (7) is estimated, directly from y(t), with the KF at synchronous sampling times t_n :=n·t_s , then P _n|n is updated as follows: $P_{n | n} = {({(A_{t_{s}} P_{n - 1 | n - 1} A_{t_{s}}^{T} + B_{t_{s}} {QB}_{t_{s}}^{T})}^{- 1} + C^{T} V^{- 1} C)}^{- 1} .$
In [18, 19] it is proven that if the eigenvalues of A_ts are within the unit circle and (A_ts, C) is observable, then P _∞ = P_K. The matrix P_K equals the solution of: $P_{K} = {({(A_{t_{s}} P_{K} A_{t_{s}}^{T} + B_{t_{s}} {QB}_{t_{s}}^{T})}^{- 1} + V^{- 1})}^{- 1} .$
For the EBSE however, we cannot prove that P _∞ equals a constant matrix. Instead we will prove that all the eigenvalues of P _∞ are bounded, i.e. that λ_max (P _∞) <∞. As described in Section 4 this is a valid indication of an estimator's performance.
The main result of this section is obtained under the standing assumption that $Λ_{H_{k_{e}}}$
is approximated using a single Gaussian. Note that the result then also applies to the estimator presented in [15], as a particular case. We assume that the eigenvalues of the A_τn -matrix are within the unit-circle and (A_τn ,C) is an observable pair. The following technical Lemmas will be of use.

Lemma 1. Given the process model (7) and covariance matrices P ≻ 0 and Q ≻ 0, then for any 0 < τ ₁ ≤ τ ₂ we have that $A_{τ_{1}} P A_{τ_{1}}^{T} {A_{τ}} {}^{\circ}{_{2}} P A_{τ_{2}}^{T}$
and $B_{τ_{1}} Q B_{τ_{1}}^{T} {B_{τ}} {}^{\circ}{_{2}} Q B_{τ_{2}}^{T} .$
See the Appendix for the proof.
Lemma 2. Let any square matrices V ₁°V ₂ and W₁°W ₂ with V ₁±0 and W₁±0 be given. Suppose that the matrices U ₁ and U ₂ are defined as $U_{1} : = {(V_{1}^{- 1} + C^{T} W_{1}^{- 1} C)}^{- 1}$
and $U_{2} : = {(V_{2}^{- 1} + C^{T} W_{2}^{- 1} C)}^{1},$
for any C of suitable size. Then it holds that U ₁ °U ₂. Proof. From [20] we have that $V_{1}^{- 1} \pm V_{2}^{- 1}$
and $C^{T} W_{1}^{- 1} C \pm C^{T} W_{2}^{- 1} C .$
Hence, it follows that $V_{1}^{- 1} + C^{T} W_{1}^{- 1} C \pm V_{2}^{- 1} + C^{T} W_{2}^{- 1} C,$
which yields $U_{1}^{- 1} \pm U_{2}^{- 1} .$
Thus, U ₁ °U ₂, which concludes the proof.

Next, recall that H _{k_e} (y_n ) is assumed to be a bounded set. Therefore, it is reasonable to further assume that $Λ_{H_{k_{e}}}$
can be approximated using the formula (25), for N =1, and that there exists a constant matrix V such that $V_{n}^{} ° \overline{V}$
for all n.

Theorem 1. Suppose that the EBSE, as presented in Section 6, approximates $Λ_{H_{k_{e}}}$
according to (17) with N = 1 and the above assumptions hold. Then λ_max (P_∞ )<λ_max (P̃_k ), where P̃_K is equal to the solution of ${\tilde{P}}_{K} = {({(A_{t_{s}} {\tilde{P}}_{K} A_{t_{s}}^{T} + B_{t_{s}} {QB}_{t_{s}}^{T})}^{- 1} + {(V + \overline{V})}^{- 1})}^{- 1} .$
See the Appendix for the proof.

6 Illustrative example

In this section we illustrate the effectiveness of the developed EBSE in terms of state-estimation error, sampling efficiency and computational tractability. The case study is a 1D object-tracking system. The states x(t) of the object are position and speed while the measurement vector y(t) is position. The process-noise w(t) represents the object's acceleration. Then given a maximum acceleration of 0.5[m/s ²] its corresponding Q, according to [21], equals 0.02. Therefore the model as presented in (7) yields $A = (\begin{matrix} 1 & τ \\ 0 & 1 \end{matrix}), B = (\begin{matrix} \frac{τ^{2}}{2} \\ τ \end{matrix}),$
C = (1 0) and D = 0, which is in fact a discrete-time double integrator. The acceleration in time is shown in Figure 30 together with the object's position and speed. The sampling time is t_s = 0.1 and the measurement-noise covariance is V = 0.1.10^-3.
Three different estimators are tested. The first two estimators are the EBSE and the asynchronous Kalman filter (AKF) of [13]. For simplicity, in both estimators we used the "Send-on-Delta" method with Δ=0.1[m]. For the EBSE we approximated $Λ_{H_{k_{e}}} (y_{n})$
using (23) with N = 5. The AKF estimates the states only at the event instants t_ke . The states at t_ka are calculated by applying the prediction-step of (14b). The third estimator is based on the quantized Kalman filter (QKF) introduced in [21] that uses synchronous time sampling of y_ka. The QKF can deal with quantized data, which also results in less data transfer, and therefore can be considered as an alternative to EBSE. In the QKF y _ka is the quantized version of y_ka with quantization level 0.1, which corresponds to the "Send-on-Delta" method. Hence, a comparison can be made.
In Figure 31 and Figure 32 the state estimation-error of the three estimators is plotted. They show that the QKF estimates the position of the object with the least error. However, its error in speed is worse compared to the EBSE. Further, the plot of the AKF clearly shows that prediction of the state-estimates gives a significant growth in estimation-error when the time between the event sampling-instants increases (t > 4).
Beside estimation error, sampling efficiency η is also important due to the increased interest in WSNs. For these systems communication is expensive and one aims to have the least data transfer. We define η ∈ R₊ as $η : = \frac{{(x_{i} - x_{i | i})}^{T} (x_{i} - x_{i | i})}{{(x_{i} - x_{i | i - 1})}^{T} (x_{i} - x_{i | i - 1})},$
which is a measure of the change in the estimation-error after the measurement update with either z_ke or y _ka was done. Notice that if η < 1 the estimation error decreased after an update, if η > 1 the error increased and if η = 1 the error remained the same. For the EBSE i=k_e with i-1 equal to k_e -1 or k_a -1. For the AKF i=k_e with i-1=k_e -1. For the QKF i=k_a and i-1=k_a -1. Figure 33 shows that for the EBSE η < 1 at all instants n . The AKF has one instant, t=3.4, at which η >1. In case of the QKF the error sometimes decreases but it can also increase considerably after an update. Also notice that η of the QKF converges to 1. Meaning that for t > 5.5 the estimation error does not change after an update and new samples are mostly used to bound λ_i (P _{k_a |ka}). The EBSE has the same property, although for this method the last sample was received at t=4.9.
The last aspect on which the three estimators are compared is the total amount of processing time which was needed to estimate all state-vectors. For the EBSE, both x_ke and x_ka were estimated and it took 0.094 seconds. The AKF estimated x_ke and predicted x_ka in a total time of 0.016 seconds and the QKF estimated x_ka and its total processing time equaled 0.022 seconds. This means that although the EBSE results in the most processing time, it is computationally comparable to the AKF and QKF, while it provides an estimation-error similar to the QKF, but with significantly less data transmission. As such, it is most suited for usage in networks in general and WSNs in particular.

7 Conclusions

In this paper a general event-based state-estimator was presented. The distinguishing feature of the proposed EBSE is that estimation of the states is performed at two different type of time instants, i.e. at event instants t_ke , when measurement data is used for update, and at synchronous time sampling t_ka , when no measurement is received, but an update is performed based on the knowledge that the monitored variable lies within a set used to define the event. As a result, it could be proven that, under certain assumptions, for the error-covariance matrix of the EBSE it holds that λ_max (P _∞) <∞, even in the situation when no new observation z_ke is received anymore. Its effectiveness for usage in WSNs has been demonstrated on an application example.

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A Proof of Lemma 1

Suppose that A∈ R^n×n and B ∈ R^n×m are defined as the state-space matrices for the time-continuous counterpart of (7). Then it is known [22] that for any sampling period τ > 0, A_τ and B_τ of (7) are obtained from their corresponding continuous-time matrices A and B as follows: $A_{τ} : = e^{Aτ} : = \sum_{i = 0}^{\infty} A^{i} \frac{τ^{i}}{i!} {and B}_{τ} : = \int_{0}^{τ} e^{Aη} ⅆ η B : = \sum_{i = 0}^{\infty} A^{i} B \frac{τ^{i + 1}}{(i + 1)!} .$
Using (41) one obtains: $A_{τ_{2}} {PA}_{τ_{2}}^{T} - A_{τ_{1}} {PA}_{τ_{1}}^{T} = (\sum_{i = 0}^{\infty} A^{i} \frac{τ_{2}^{i}}{i!}) P (\sum_{i = 0}^{\infty} {(A^{T})}^{j} \frac{τ_{2}^{j}}{j!}) - (\sum_{i = 0}^{\infty} A^{i} \frac{τ_{1}^{i}}{i!}) P (\sum_{i = 0}^{\infty} {(A^{T})}^{j} \frac{τ_{1}^{j}}{j!}) = \sum_{i = 0}^{\infty} \sum_{j = 0}^{\infty} A^{i} P {(A^{T})}^{j} (\frac{τ_{2}^{i}}{i!} \frac{τ_{2}^{j}}{j!}) - \sum_{i = 0}^{\infty} \sum_{j = 0}^{\infty} A^{i} P {(A^{T})}^{j} (\frac{τ_{1}^{i}}{i!} \frac{τ_{1}^{j}}{j!}) = \sum_{i = 0}^{\infty} \sum_{j = 0}^{\infty} A^{i} P {(A^{T})}^{j} (\frac{τ_{2}^{i + j} - τ_{1}^{ι + ξ}}{i! j!})$
As for any τ > 0 the series e^Aτ converges [22], then $A_{τ_{2}} {PA}_{τ_{2}}^{T} - A_{τ_{1}} {PA}_{τ_{1}}^{T}$
also converges. Then, since 0<τ ₁≤τ ₂ and P≻0, for any fixed i, j , we have $A^{i} P {(A^{T})}^{j} (\frac{τ_{2}^{i + j} - τ_{1}^{i + j}}{i! j!}) \pm 0$
for any matrix A and thus, it follows that $A_{τ_{2}} {PA}_{τ_{2}}^{T} \pm A_{τ_{1}} {PA}_{τ_{1}}^{T} .$
The same reasoning can be used to prove that $B_{τ_{1}} {QB}_{τ_{1}}^{T} {°B}_{τ_{2}} {QB}_{τ_{2}}^{T} .$
W

1. Proof of Theorem 1

Under the hypothesis, for the proposed EBSE, P_n|n of (6.3), with τ_n :=t_n-t _n-1 and $R_{n} : = V + V_{n}^{},$
becomes: $P_{n | n} = {({(A_{τ_{n}} P_{n - 1 | n - 1} A_{τ_{n}}^{T} + B_{τ_{n}} {QB}_{τ_{n}}^{T})}^{- 1} + C^{T} R_{n}^{- 1} C)}^{- 1} .$
The upper bound on λ_max (P _∞) is proven by induction, considering the asymptotic behavior of a KF that runs in parallel with the EBSE, as follows. The EBSE calculates $P_{n | n}^{(1)}$
as (42) and the KF calculates $P_{n | n}^{(2)}$
as (39) in which V is replaced with R := V + V . Notice that for these estimators we have that τ _{n ≤} t_s and R_n °R, for all n. Let the EBSE and the KF start with the same initial covariance matrix P ₀. The first step of induction is to prove that $P_{1 | 1}^{(1)} {°P}_{1 | 1}^{(2)} .$
From the definition of $P_{1 | 1}^{(1)}$
in (42) and $P_{1 | 1}^{(2)}$
in (39) we have that $P_{1 | 1}^{(1)} = {({(A_{τ_{1}} P_{0} A_{τ_{1}}^{T} + B_{τ_{1}} {QB}_{τ_{1}}^{T})}^{- 1} + C^{T} R_{1}^{- 1} C)}^{- 1}$
and $P_{1 | 1}^{(2)} = {({(A_{t_{s}} P_{0} A_{t_{s}}^{T} + B_{t_{s}} {QB}_{t_{s}}^{T})}^{- 1} + C^{T} R^{- 1} C)}^{- 1} .$
Suppose we define $V_{1} : = A_{τ_{1}} P_{0} A_{τ_{1}}^{T} + B_{τ_{1}} {QB}_{τ_{1}}^{T},$
$V 2 : = A_{t_{s}} P_{0} A_{t_{s}}^{T} + B_{t_{s}} {QB}_{t_{s}}^{T},$
W ₁:=R ₁ and W₂:=R. Then W ₁ °W ₂ and from Lemma 6.4 it follows that V ₁°V ₂. Therefore applying Lemma 6.4, with $U_{1} : = P_{1 | 1}^{(1)}$
and $U_{2} : = P_{1 | 1}^{(2)},$
yields $P_{1 | 1}^{(2)} ° P_{1 | 1}^{(2)} .$
The second and last step of induction is to show that if $P_{n - 1 | n - 1}^{(1)} {°P}_{n - 1 | n - 1}^{(2)},$
then $P_{n | n}^{(1)} {°P}_{n | n}^{(2)} .$
Let $V_{1} : = A_{τ_{n}} P_{n - 1 | n - 1}^{(1)} A_{τ_{n}}^{T} + B_{τ_{n}} {QB}_{τ_{n}}^{T},$
$V_{2} : = A_{t_{s}} P_{n - 1 | n - 1}^{(2)} A_{t_{s}}^{T} + B_{t_{s}} {QB}_{t_{s}}^{T},$
and let W ₁ := R_n , W₂ := R . Notice that this yields W ₁°W ₂ . The second condition of Lemma 6.4, i.e. V ₁°V ₂ also holds by applying Lemma 6.4, i.e. $A_{τ_{n}} P_{n - 1 | n - 1}^{(1)} A_{τ_{n}}^{T} + B_{τ_{n}} {QB}_{τ_{n}}^{T} {}^{\circ}{A_{t_{s}}} P_{n - 1 | n - 1}^{(1)} A_{t_{s}}^{T} + B_{t_{s}} {QB}_{t_{s}}^{T} {}^{\circ}{A_{t_{s}}} P_{n - 1 | n - 1}^{(2)} A_{t_{s}}^{T} + B_{t_{s}} {QB}_{t_{s}}^{T} .$
Hence, applying Lemma 6.4, with $U_{1} : = P_{n | n}^{(1)}$
and $U_{2} : = P_{n | n}^{(2)}$
yields $P_{n | n}^{(1)} {°P}_{n | n}^{(2)} .$
This proves that $P_{\infty}^{(1)} {°P}_{\infty}^{(2)},$
which yields (see e.g., [20]) $λ_{\max} (P_{\infty}^{(1)}) {°λ}_{\max} (P_{\infty}^{(2)}) .$
As $P_{n | n}^{(2)}$
was calculated with the KF it follows from (40) that $P_{\infty}^{(2)} = {\tilde{P}}_{K},$
with P̃_K as the solution of ${\tilde{P}}_{K} = {({(A_{t_{s}} {\tilde{P}}_{K} A_{t_{s}}^{T} + B_{t_{s}} {QB}_{t_{s}}^{T})}^{- 1} + R^{- 1})}^{- 1},$
which completes the proof. □

Claims

Vehicle tracking system comprising
- a plurality of sensor nodes (10) that each provide a message (D) indicative for an occupancy status of a detection area of a vehicle infrastructure (80) monitored by said sensor node,

- a message interpreter (MI) including a vehicle database facility (32, 34) with state information of vehicles present at the vehicle infrastructure, and a database updating facility (DBU) for updating the vehicle database facility (32, 34) on the basis of messages (D) provided by the sensor nodes,
characterized in that multiple sensor nodes (10) are arranged in the vehicle infrastructure (80) at a density of at least 0.2 per square meter.
The vehicle tracking system according to claim 1, wherein said sensor nodes (10) are arranged in the vehicle infrastructure (80) at a density of at least 0.6 per square meter.
The vehicle tracking system according to claim 1 or 2, comprising a plurality of system modules (MD1, MD2, MD3), each module comprising a respective subset of the plurality of sensor nodes (10) for monitoring a respective section (80A, 80B, 80C, 80D) of the vehicle infrastructure and a respective message interpreter (MI), the vehicle tracking system further having a communication facility (60) for enabling system modules of mutually neighboring sections to exchange state information.
Vehicle tracking system according to claim 1 or 2, wherein the database updating facility (DBU) comprises
- an association facility (40) for associating the messages (D) provided by the sensor nodes with the state information present in the vehicle data base facility (32, 34),

- a state updating facility (50) for updating the state information on the basis of the messages (D) associated therewith.
Vehicle tracking system according to claim 1 or 2, wherein the sensor nodes (10) provide the messages (D) at an event basis.
Vehicle tracking system according to claim 1 or 2, wherein the sensor nodes (10) are provided with a wireless transmission facility (16) for wirelessly transmitting the message (D), and wherein the message interpreter (40) comprises a wireless reception facility (20) for receiving the message (D).
The vehicle tracking system according to claim 1 or 2, wherein the sensor nodes (10) are randomly distributed over the vehicle infrastructure (80).
The vehicle tracking system according to claim 1 or 2, wherein the state updating facility (50) is arranged for updating the state information on the basis of event-based messages (D) and on the basis of messages sampled synchronous in time.
Vehicle infrastructure (80) provided with a vehicle tracking system according to one of the previous claims.
Method for tracking vehicles at an infrastructure using the vehicle tracking system according to one of claims 1 to 8, the method comprising
a) providing a plurality of sensor nodes in the vehicle infrastructure at a density of at least 0.2 per square meter, the sensor nodes each monitoring a detection area of the vehicle infrastructure,

b) providing a message indicative for an occupancy status of their detection area,

c) storing state information of vehicles present at the vehicle infrastructure,

d) receiving the message and updating said stored information on the basis of the message.
Method according to claim 10, wherein the density of the sensor nodes is at least 0.6 per square meter.
Method according to claim 10 or 11, wherein the step of updating said stored information comprises,
d) associating the message with state information present in the vehicle data base facility,

e) updating the state information associated in step d) on the basis of said message.
Method according to claim 10 or 11, wherein the steps a) to c) are independently performed for mutually non coinciding sections of the vehicle infrastructure, the method further comprising the step of exchanging state information.
Method according to claim 12 wherein the step of associating comprises
- initializing (S40) a vehicle index (i),

- retrieving (S41) the current state known for the vehicle with that index from a vehicle database facility,

- determining (S42) a probability that the vehicle with that index caused the detection reported by the message D,

- incrementing (S43) the vehicle index,

- determining (S44) whether the vehicle index is less than the number of vehicles,

- if the outcome of the determination is positive repeating steps S41 to S43 with the incremented vehicle index, and

- if the outcome of the determination is negative, determining (S45) which vehicle caused the detection reported by the message D with the highest probability.

- returning (S46) the index of that vehicle identified in step S45.