Method and system for simultaneous receiver calibration and object localisation for multilateration

1. A processing unit apparatus for use in a system for multilateration of at least one of a plurality of P objects, each of the plurality of P objects having a transmitter device, and each of the transmitter devices being configured to transmit a respective signal, each of a plurality of N sensors adapted to receive the transmitted signals from each of the objects and to measure respective receive times of each of the signals, the processing unit apparatus adapted to be operatively coupled to the plurality of sensors and to perform steps comprising:

A. combining the receive times for the signals into a chronologically ordered series of message sets, wherein each of the message sets comprises the receive times of one of the transmitted signals as determined by each sensor;

B. collecting all valid message sets that occur within a predetermined time interval ΔT k , wherein the index kε{1, . . . , N ΔT } 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 time step k and iε{1, . . . , N} by processing at least one of the message sets; and

D. outputting a position of said at least one of the plurality of objects.

2. The apparatus of claim 1 , wherein the step of estimating comprises:

a prediction step that is performed according to a Chapman-Kolmogorov equation; and

a filter step that is performed using Bayes' rule.

3. The apparatus of claim 2 , wherein an emission time vector b k is at least one of (i) modelled as a full-correlated noise process and (ii) estimated by using a distance expression depending on an expectation value of b k , wherein b k comprises emission times b k j of each of the P objects at time step k, wherein jε{1, . . . , P}.

4. The apparatus of claim 2 , wherein the filter step is performed in accordance with a measurement equation

y k = {tilde over (h)} ( x k , S , b k , o k , v k ),

that, for any time step k, describes a relation between a message set y k , and wherein {tilde over (h)} is a function depending on:

an object position vector x k at time step k, comprising positions x k j of each of the P objects at time step k, wherein jε{1, . . . , P};

a sensor position vector S comprising positions S i of the N sensors, wherein iε{1, . . . , N};

the time offset vector o k ;

a general noise vector v k comprising general noises v k i,j at time step k.

5. The apparatus of claim 4 , wherein the measurement equation is given by

y k = h ( x k 1 , . . . , x k P )+ H b · b k +H o · o k + v k

with

H b =I P 1 N , H o = 1 P I N .

6. The apparatus of claim 2 , wherein the prediction step is performed in accordance with a model for describing the time evolution of states, the time is described by a series of discrete time steps kε{1, . . . , N ΔT }, wherein for each of the time steps kε{1, . . . , N ΔT } the state is given by a vector z k comprising:

the object position vector x k ;

an object velocity vector {dot over (x)} k comprising the velocity of each of the objects;

the sensor offset vector o k ;

a sensor drift vector {dot over (o)} k comprising the drift of each of the sensor offsets;

and wherein the model comprises a recursive system equation

z k+1 = a ( z k , w k ),

wherein a is a function depending on the state at time step k and w k denotes a noise process over the time steps kε{1, . . . , N ΔT }.

7. The apparatus of claim 6 , wherein the function a is a linear, affine function, of the state z k and the noise w k .

8. The apparatus of claim 3 , wherein the noise process is modelled as a Gaussian distributed process.

9. The apparatus of claim 2 , wherein the prediction step is performed using the Kálmán predictor equations.

10. The apparatus of claim 6 , wherein:

the receive times and the states are jointly Gaussian distributed; and

an estimated density of the states is described by an expectation value and a covariance.

11. The apparatus of claim 2 , wherein a dependency for a component of a height of one of the plurality of objects is reduced by applying a rotation operator on coordinates of positions of the sensors.

12. The apparatus of claim 11 , wherein dynamics in values for the j'th objects are reduced by subtracting a minimal value of the message set from each of the components of the measurement vector y k , and by adding, to the result thereof, an expected distance of the j'th object, where jε{1, . . . , P}.

13. The apparatus of claim 1 , wherein

the N sensors are located at known positions ( S 1 , . . . S N ), respectively;

the transmitter devices in the objects are at unknown positions ( x k 1 , . . . , x k P ), respectively,

the transmitter device of each of the objects transmits signals at unknown times of transmissions;

the steps further comprising

providing a measurement equation for the P objects and the N sensors of

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that, for any time step k, describes a relation between a message set y k , and wherein y k is depends on:

c, which is the propagation speed of the transmitted signals in the considered medium;

an object position vector x k at a time step k, comprising the positions x k j of each of the P objects at time step k;

a sensor position vector S comprising the positions S i of each of the N sensors, wherein ∥·∥ is the standard Euclidian norm;

the time offset vector o k , wherein o k =(o k 1 , . . . , o k N );

a general noise vector v k comprising general noises v k i,j at time step k, wherein each of the sensors is identified by an index i and each of the objects is identified by index j and the receive times are identified by the index y k i,j ;

wherein the numbers of N sensors and P objects fulfil the following condition:

N·P≧N+P·(D+1); and, where D is the considered dimension; and

wherein the transmission time vector is designated by b k comprising the transmission times b k j of each of the P objects at time step k;

the calculated object positions are identified by x k 1 , . . . , x x P .

14. The apparatus of claim 13 , wherein transmission time vector b k is multiplied with a matrix H b and the resulting term is modelled as a fully correlated noise process by using a distance expression depending on an expectation value of b k , where

H b =I P 1 N , H o = 1 P I N .

15. A method for multilateration of at least one of a plurality of P objects, each of the plurality of P objects having a transmitter device, and each of the transmitter devices being configured to transmit a respective signal, the method using a plurality of N sensors,

each sensor adapted to receive the transmitted signals from each of the objects and to determine respective receive times of each of the signals, the method comprising:

A. combining the receive times for the signals into a chronologically ordered series of message sets, wherein each of the message sets comprises the receive times of a single transmitted signal as determined by each sensor;

B. collecting all valid message sets that occur within a predetermined time interval ΔT k , wherein the index kε{1, . . . , N ΔT } denotes a time step, and wherein a 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 time step k and iε{1, . . . , N} by processing at least one of the message sets; and

D. outputting the position of said at least one of the plurality of objects.