Patent Application: US-201414465212-A

Abstract:
an apparatus for multilateration of a plurality of p objects , each having a transmitter device , and each transmitter device transmitting a signal , the apparatus including n sensors , a processing unit , each sensor receiving the transmitted signals and determining receive times of each signal . the processing unit performing steps : a . combining receive times for signals into a chronologically ordered series of message sets , wherein each message sets comprises the receive times of one of the transmitted signals , b . collecting all valid message sets that occur within a predetermined time interval , wherein k denotes a time step , and wherein each valid message set contains receive times from at least n sensors , c . estimating a time offset vector o k comprising time offsets o k i of the receive times of the n sensors at step k and i = 1 to n by processing message sets , and d . outputting a position of the objects .

Description:
the following notation will be used throughout the explanations given hereinafter : x k j : position of object j at time step k y k i , j : measurement at sensor i , that comes from object j at time step k as illustrated in fig1 , in multilateration tracking , the positions of p objects , for example aircraft a 1 . . . a p , located at unknown positions x k j for j = 1 , . . . , p have to be calculated at different time steps k . several sensors s 1 . . . s n located at known positions s i for i = 1 , . . . , n are measuring time events . a measured event , which stems from the object j , at sensor i at time step k is defined as y k i , j . the sensors are in communication with a processing unit pu which receives information from the sensor and produces output information according to techniques set forth below . the relationship between the measured time event and the position of the objects is given by a non - linear measurement equation y k i , j = h i , j ( x k j , s i , v k i , j ), ( 6 ) where the measured quantities are corrupted with noise v k i , j . furthermore , the dynamical behaviour of the system is described by a system equation where the state variable z k comprises the position of the objects and further state variables like velocity , etc . in order to estimate the position of the objects , the bayesian framework is used . it consists of the prediction and filter step . in the prediction step , the estimated density f e ( z k ) of the previous filter step is propagated from time step k to ( k + 1 ) by means of the chapman - kolmogorov equation f p ( z k + 1 )=∫ f ( z k + 1 | z k )· f e ( z k ) d z k , ( 8 ) where f ( z k + 1 | z k ) is the transition density defined by eq . ( 7 ). in the filter step , the current measurement value ŷ k is used for updating the result of the prediction step f p ( z k ) according to bayes &# 39 ; rule f e ( z k )= c k · f ( ŷ k | z k )· f p ( z k ), ( 9 ) where c k = 1 /(∫ f ( ŷ k | z k )· f p ( z k )· d z k ) is a normalization constant and f ( ŷ k | z k ) is the likelihood defined by eq . ( 6 ). in order to derive a measurement model for the method for multilateration according to the invention , i . e ., a method for multilateration in the case of unsynchronized sensor clocks , the different influences in the measurement process are described . first , only the time of arrival is considered , then this is extended to the case that the emission time is unknown . after that , the sensor offset and the influence of the noise are described . the four different models are shown in fig2 . 1 ) time of flight : the time of flight describes the travelling time of a signal from an object j to a sensor i . the relationship between the measured time of arrival d k i , j and the unknown object position x k j is then given by the euclidean distance according to where c is the wave propagation speed and ∥•∥ 2 is the l 2 norm . 2 ) time of flight + unknown emission time : in the second scenario , the sensor cannot directly determine the travelling time due to the fact that the emission time of the object for a certain message is not known . this unknown emission time b k j leads to an unknown offset where this emission time is equal for all sensors for this certain message . in the following , the emission time is referred to as object offset . 3 ) time of flight + unknown emission time + sensor offset : the third scenario is a situation , where in addition to the above described situation of scenario 2 ), also the sensor clocks are not synchronized to each other . thus , every sensor has a sensor offset o k j and this results in this sensor offset is equal to all messages , which are measured at the same time . 4 ) time of flight + unknown emission time + sensor offset + noise : in a real scenario , disturbances occur . in order to cope with these uncertainties , a noise process v k i , j is used , which leads to y k i , j = d k i , j + b k j + o k i + v k i , j , ( 13 ) d k i , j , the spatial distance between the affected object and the affected sensor divided by the wave propagation speed , which results in the net travelling time , b k j , the offset corresponding to the unknown transmitting time in the object , o k i , the offset corresponding to the unknown receiving time — or equivalently the lack of lack of synchronization — in the sensor , according to one aspect of the invention reference is just made to the offsets in the sensors as these are the ones to be estimated . the travelling times between sensors and objects is used for calculating the object positions simultaneously to the sensor offsets . the system model describes the evolution of the state over time . in the following , the motion model for an aircraft and the system model for the sensor offset are discussed . finally , a discussion for the aircraft offset is made . 1 ) motion models for aircraft : different types of motion models for aircraft can be applied , depending on the expected motion to be performed . in one embodiment , a standard motion model , the constant velocity model , is applied [ 7 , 8 ]. the constant velocity model is described in state space as where t is the sampling time and { dot over ( x )} k j the velocity of the aircraft j . the process noise w k x , { dot over ( x )} is assumed as gaussian distributed , with zero - mean and covariance matrix given by the covariance matrix c { dot over ( x )} depends on the continuous time system model this is a specific object motion model , a so - called “ constant velocity model ”, with generic noise parameters for the acceleration that is modelled as white noise . c x , c y , c z denote the spatial variances of the acceleration in a cartesian space . for a concrete application , these values have to be determined based on practical considerations . here reference can be made to ref [ 8 ]. 2 ) sensor offset models : due to the clock drift , the clock of each of several sensors drift apart and a sensor / clock offset o k j of sensor i occurs . the clock offset is modelled as a continuous time process where the ρ i ( t ) is modelled as a noise process with time characteristics of the drift , given by { dot over ( ρ )} i ( t )= w { dot over ( o )} ( t ), ( 19 ) ö i ( t )= w { dot over ( o )} ( t ). ( 20 ) and the process w k o ,{ dot over ( o )} is assumed as gaussian distributed , with covariance c { dot over ( o )} is the variance of the continuous time process . 3 ) aircraft offset models : in the proposed approach it is assumed that the evolution of the emission time over time cannot be described by an adequate system model . the emission times are assumed as a noise process b k j which is modelled later as a full correlated noise process in the measurement equation . in the following , the resulting measurement equation and system equation for the method for multilateration according to the invention with unsynchronised clocks are shown for a complete message set . furthermore , an inequality for one time step is given . based on eq . ( 13 ), the resulting measurement equation for p aircraft and n sensors is given by y k = h ( x k 1 , . . . , x k p )+ h b · b k + h o · o k + v k ( 24 ) h b = i p 1 n , h o = 1 p i n . ( 25 ) stands for the kronecker product . in the non - linear equation , the times of flight are listed according to h ( x k 1 , . . . , x k p )=[ d k 1 , 1 . . . d k n , 1 . . . d k i , j . . . d k n , p ] t ( 26 ) using eqs . ( 25 ) and ( 26 ), eq . ( 24 ) can also be written in the form as described before , the emission time is considered as fully correlated noise due to the fact that no system model for the emission time can be assumed . in this case , the new noise process v k f in y k = h ( x k 1 , . . . , x k p )+ h o · o k + v k f ( 28 ) the first two moments of noise process are then given by the system state consists of the positions , the velocities , the sensor offsets , and the sensor drifts , according to z k =[( x k 1 ) t ( { dot over ( x )} k 1 ) t . . . ( o k 1 ) t ( { dot over ( o )} k 1 ) t . . . ] t . ( 31 ) the system equation can be modelled as a linear system equation based on eqs . ( 14 ) and ( 21 ), given by a = blkdiag ( a x , { dot over ( x )} , . . . , a x , { dot over ( x )} , a o ,{ dot over ( o )} , . . . , a o ,{ dot over ( o )} ) ( 33 ) c w = blkdiag ( c x , { dot over ( x )} , . . . , c x , { dot over ( x )} , c o ,{ dot over ( o )} , . . . , c o ,{ dot over ( o )} ) ( 34 ) if p aircraft are emitting signals , which are received by n sensors , n · p possible measurements can be used for one time step . the measurement equation consists of n + p ·( d + 1 ) unknown variables , where d is the considered dimension , in general d = 3 . a unique solution can be given if the condition n · p ≧ n + p ·( d + 1 ) is fulfilled . for estimation , a gaussian assumed density filter [ 9 ] is used , where the involved densities are described by the first two moments . due to the linear motion model , eq . ( 8 ) can be solved with the kálmán predictor equation . however , due to the non - linear measurement equation , eq . ( 9 ) cannot be solved directly . in order to solve the filter step , it is assumed that state and measurement are jointly gaussian distributed . due to the non - linear measurement equation , a gaussian assumed density filter is used . the filter step is solved based on the first two moments and it is assumed that the measurement and state are jointly gaussian distributed . for a given measurement ŷ k , the conditional gaussian is given by where μ k p and c k p are the predicted mean and covariance . in order to calculate the estimated mean μ k e and covariance c k e , the three unknown moments , i . e ., the cross - covariance between state and measurement c k z , y , the covariance of the measurement process c k y , and the predicted measurement μ k y have to be calculated . it should be noted that for the calculation of the predicted measurement μ k y the values of the aircraft offsets are not considered . the values for the aircraft offsets are determined in a separate step , which will be explained below . in the method for multilateration according to the invention , decomposition methods as proposed in ref [ 10 ] are applied in order to reduce the mathematical effort . when using these decomposition methods in conditional linear models , it is possible to approximate the above outlined calculations , i . e ., in particular the calculation of the expressions of eqs . ( 35 ), with regard to the positions by a sample - based approach . the sample points are propagated through the measurement equation in eq . ( 28 ). these sample points and the decomposed part are then used in order to calculate the mean and the covariance of the measurement process , as well as the cross - covariance between state and measurement . until now the aircraft offset is still unknown . in order to determine the aircraft offset , the mahalanobis distance of the measurement density is used . for any time step k , the mahalanobis distance can be expressed by a function of the mean value μ k b of the aircraft offset b k . then , the minimum of the mahalanobis distance corresponds to the best estimation value for the aircraft offset . in order to find the best estimation value for the aircraft offset , the derivation of the mahalanobis with respect to the mean value of the aircraft offset is taken , due to the linear models in eq . ( 32 ), the kálmán predictor equations are used , which are given by c k p = a · c k e ·( a ) t + c w . ( 39 ) with reference to fig5 , the method can be sketched as follows . at step 510 , for an arbitrary time step , with the predicted density of the state described by its two first moments , i . e ., the expectation value μ p and the covariance matrix c p , the predicted measurement μ y , the covariance of the measurement process c y , and the cross - covariance between the state and the measurement c z , y are calculated . then , at step 520 , the calculated quantities are used to obtain an estimation for the aircraft offset μ b according to eq . ( 38 ). next , at step 530 , in the filter step , the estimated density of the state described by its expectation value μ e and c e is calculated according to eq . ( 35 ) using the results from the previous steps . then , at step 540 , according to the kálmán predictor equations ( 39 ), the expressions μ e and c e are used to calculate the predicted density of the state for the next time step , which is again given by the expectation value μ p and the covariance matrix c p . embodiments of the invention may also comprise a feature that allows for an improvement of the accuracy of the method for multilateration . this optional feature is related to the implementation of the method for multilateration and will be discussed in the following . the estimation is performed in a cartesian coordinate system . due to the fact that the aircraft is moving on a certain barometric height , this leads to the fact that the aircraft moves in a certain cartesian space on a plane . in order to reduce dependencies for the component in the z axis , a transformation can be used . it is assumed that the certain area lies at latitude α and longitude β , and so a rotation matrix is given by the sensor positions are rotated , in such a way , that the dependencies in the z axis are removed according to in order to reduce the dynamics in the measurement values for the j &# 39 ; th aircraft , the minimal value can be subtracted from the message set y k 1 . . . n , j =[ y k i , j . . . y k n , j ] t ( 42 ) coming from the j &# 39 ; th aircraft , and to reduce the dynamic in the aircraft offset for the j &# 39 ; th aircraft , a possible distance can be added according to y k 1 . . . n , j − min ( y k 1 . . . n , j )+ d k argmin ( y k 1 . . . n , j ), j . ( 43 ) for the proposed estimator it is useful to have a certain number of measurements over time . if the drift is slowly varying , the message sets from different aircraft can be collected over some short time horizon . according to the embodiments of the invention described above , the method for multilateration has been evaluated with simulated and real data . in the simulation , the method of the invention is compared to the case , when the sensors are accurately synchronized . in the experiment , the estimated positions are compared to the ads - b position of the aircraft . in the simulation , the method according to the invention is compared to the case when the sensor offset is perfectly known , where both algorithms make use of a state estimator ( gaussian assumed density filter - gadf ) and in the latter one the sensor offset is omitted from the state space . for the simulation , 8 sensors are placed according to ref [ 11 ] so as to measure the receiving times of messages from different aircraft . the number of aircraft was selected to 10 and 30 , respectively . the measured receiving times comprise the time of flight , an aircraft and sensor offset . furthermore , the receiving times are corrupted with gaussian noise varying from zero to 0 . 1 km . the simulation time is 360 seconds , with a sampling time of 0 . 1 seconds . furthermore , the algorithms are compared between full measurement , i . e ., all sensors received data from all aircraft at every time step , to part measurements , i . e ., only a part of the possible measurement are measured . in fig3 ( a ), the root - mean - square error ( rmse ) for all aircraft ( 30 aircraft ) are shown for the case of calibrated and uncalibrated data , as well for full and part measurements . the ground truth corresponds to when the sensor offset is perfectly known . for uncalibrated data and full measurement , the method according to the invention has a higher rmse than the ground truth , when the noise level increases . this deviation increases , when only part measurements are taken . if the number of aircraft is decreased to 10 , the accuracy for the method according to the invention compared to calibrated clocks is decreased as shown in fig3 ( b ). this can be explained by the fact that less aircraft serve for synchronization . in the real world experiment , uncalibrated data was used in order to estimate the trajectory of the aircraft . the proposed approach was compared to the ads - b data from the aircraft . the whole dataset contains a recording about 13 minutes . in the evaluation , data over a 0 . 1 second horizon are collected to a data set for every measurement step . furthermore , preferably a validation gate is used in order to cope with outliers . outliers can cause poor estimates for the localization and synchronisation procedure . the validation gate is used for checking the result of the calculation before updating the estimated state of the system . more specifically , in order to detect outliers , the predicted measurement is compared to actual measurement depending on a given uncertainty . in fig4 the results for the proposed approach and the ads - 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