Patent Publication Number: US-11387883-B2

Title: Estimator for determining updates to an estimated local state vector

Description:
INTRODUCTION 
     The present disclosure relates to an estimator for a communication network. More particularly, the present disclosure is directed towards an estimator that determines updates for an individual node based on local measurements and collaborative measurements between other nodes that are part of the communication network. 
     BACKGROUND 
     A collaborative position, navigation, and timing (PNT) system includes a group of users that are connected to one another by a wireless communication network. Each user may be a vehicle or an individual that includes PNT devices and sensors, where each user is referred to as a node. Each node that is part of the PNT system may be located in a different geographical region. As a result, some nodes may be situated in a location where global navigation satellite systems (GNSS) based signals are widely available, while other nodes may receive very limited or no signals at all. For example, a group of high-rise or tall buildings in a city, which are referred to as an urban canyon, tend to block GNSS signals. In another example, some nodes may be located in an area with a significant amount of radio frequency interference or jamming. 
     Some of the nodes may only be equipped with a GNSS receiver and therefore may not be able to receive signals in some situations. However, other nodes may include a combined GNSS and inertial measurement unit. Some other nodes may include celestial navigation systems or vision based navigation systems that are able to provide PNT solutions even when GNSS signals are unavailable. However, a collaborative PNT system helps alleviate some of these issues by allowing a node to leverage not only its own information, but also information that is available over the wireless network. 
     SUMMARY 
     According to several aspects, an estimator that is part of a communication network including a plurality of nodes is disclosed. A centralized portion of the estimator comprises one or more processors in wireless communication with the plurality of nodes that are part of the communication network, wherein the communication network includes one or more pairs of collaborating nodes, and a memory coupled to the one or more processors, the memory storing data into a database and program code that, when executed by the one or more processors, causes the centralized portion of the estimator to receive a respective estimated local state vector, a respective estimated local measurement, a respective local residual, and a respective local measurement from each of the plurality of nodes that are part of the communication network. The centralized portion of the estimator determines a total error covariance matrix for the communication network based on the respective estimated local state vector for each of the plurality of nodes that are part of the communication network. The centralized portion of the estimator determines a respective local update based on the total error covariance matrix and the local residual for each of the plurality of nodes. The respective local update is applied to the respective estimated local state vector for a particular node. The centralized portion of the estimator also predicts an estimated collaborative measurement based on a collaborative measurement between a pair of collaborating nodes that are part of the communication network, where a collaborative residual is associated with the estimated collaborative measurement. The centralized portion of the estimator also determines a collaborative update based on the estimated collaborative measurement and the collaborative residual, where the collaborative update is applied to the respective estimated local state vector for both nodes of the pair of collaborating nodes. 
     In another aspect, a method for updating an estimated local state vector for a plurality of nodes that are part of a communication network by a centralized portion of an estimator is disclosed. The method includes receiving, by the centralized portion of the estimator, a respective estimated local state vector, a respective estimated local measurement, a respective local residual, and a respective local measurement from each of the plurality of nodes that are part of the communication network. The method also includes determining, by the centralized portion of the estimator, a total error covariance matrix for the communication network based on the respective estimated local state vector for each of the plurality of nodes that are part of the communication network. The method also includes determining, by the centralized portion of the estimator, a respective local measurement sensitivity matrix representing an amount of change that local measurements for a particular node undergo based on a change in a respective local state vector for a particular node. The method further includes determining, by the centralized portion of the estimator, a respective local measurement variance matrix that represents an uncertainty in the local measurements for the particular node. The method also includes determining, by the centralized portion of the estimator, a respective local residual covariance matrix for each of the plurality of nodes that are part of the communication network based on the total error covariance matrix for the communication network and the respective local measurement variance matrix. The method also includes determining, by the centralized portion of the estimator, a respective local gain matrix for the particular node based on the respective local residual covariance matrix for the particular node, the total error covariance matrix for the communication network, and the respective local measurement sensitivity matrix for the particular node. The method also includes combining, by the centralized portion of the estimator, the respective local gain matrix with the local residual of the particular node to create a respective local update. The method also includes predicting, by centralized portion of the estimator, an estimated collaborative measurement based on a collaborative measurement between a pair of collaborating nodes that are part of the communication network, where a collaborative residual is associated with the estimated collaborative measurement. The method also includes determining, by the centralized portion of the estimator, a collaborative measurement sensitivity matrix representing an amount of change that the collaborative measurement undergoes based on a corresponding change in the respective local state vector for the particular node that the collaborative measurement is taken with respect to. The method also includes determining, by the centralized portion of the estimator, a compound covariance matrix that characterizes an uncertainty in the collaborative measurement when an impact of one or more states of a collaborative node of the pair of collaborating nodes are modeled as random noise. The method includes determining, by the centralized portion of the estimator, a collaborative error covariance matrix of the collaborative residual based on at least the collaborative measurement sensitivity matrix and the compound covariance matrix. The method also includes determining, by the centralized portion of the estimator, a collaborative gain matrix based on the collaborative error covariance matrix of the collaborative residual and the collaborative measurement sensitivity matrix. Finally, the method includes combining, by the centralized portion of the estimator, the collaborative gain matrix with the collaborative residual to create the collaborative update. 
     The features, functions, and advantages that have been discussed may be achieved independently in various embodiments or may be combined in other embodiments further details of which can be seen with reference to the following description and drawings. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       The drawings described herein are for illustration purposes only and are not intended to limit the scope of the present disclosure in any way. 
         FIG. 1  is a schematic diagram of the disclosed communication network having a plurality of nodes that are in wireless communication with one another, according to an exemplary embodiment; 
         FIG. 2  is a schematic diagram of the communication network according to a local with immediate neighbor (LWIN) approach for updating an estimated state vector of an individual node, according to an exemplary embodiment; 
         FIG. 3  is a schematic diagram of an individual node that updates the estimated state vector based on local measurements as well as collaborative measurements based on the LWIN approach, according to an exemplary embodiment; 
         FIG. 4  illustrates a collaborative positioning, navigation, and timing (PNT) system, according to an exemplary embodiment; 
         FIG. 5A  illustrates one example of a collaborative measurement between the individual node and a collaborative node that includes relative distance measurements combined with relative line-of-sight (LOS) measurements, according to an exemplary embodiment; 
         FIG. 5B  illustrates another example of a collaborative measurement between the individual node and the collaborative node that includes a relative LOS direction measurement, according to an exemplary embodiment; 
         FIG. 5C  illustrates yet another example of a collaborative measurement between the individual node and the collaborative node including a relative range measurement, according to an exemplary embodiment; 
         FIGS. 6A-6B  illustrate a process flow diagram illustrating a method for determining updates for the estimated state vector of the individual node based on the LWIN approach, according to an exemplary embodiment; 
         FIG. 7  is a schematic diagram of the communication network according to an integrated total network solution (ITNS) approach for updating the estimated state vector, according to an exemplary embodiment; 
         FIG. 8  is a schematic diagram of a centralized portion of an estimator shown in  FIG. 7  that determines the local updates for each node that is part of the communication network and collaborative updates, according to an exemplary embodiment; 
         FIGS. 9A-9B  illustrate a process flow diagram illustrating a method for determining updates for the estimated state vector of the individual node based on the ITNS approach, according to an exemplary embodiment; and 
         FIG. 10  is the computing system for the disclosed estimator, according to an exemplary embodiment. 
     
    
    
     DETAILED DESCRIPTION 
     The present disclosure is directed towards an estimator that determines corrections or updates that are applied to an estimated local state vector for an individual node. The updates are based on local measurements as well as collaborative measurements between the individual node and a collaborative node. The present disclosure describes two different approaches for updating the estimated local state vector for an individual node. In the first example, the individual node includes its own estimator, and the estimated local state vector is updated based on the local measurements as well as a collaborative measurement between the individual node and the collaborative node. This decentralized approach is referred to as a local with immediate neighbor (LWIN) approach. In a second approach, a portion of the local updates for each node are performed locally, while the collaborative updates for each pair of collaborating nodes are performed at a centralized portion of the estimator. This is a more centralized approach and therefore is referred to as an integrated total network solution (ITNS) approach. Both approaches improve the accuracy of the estimated local state vector for each node, since the updates are based on not only the local measurements, but also collaborative measurements between nodes. 
     The following description is merely exemplary in nature and is not intended to limit the present disclosure, application, or uses. 
     Referring to  FIG. 1 , an exemplary communication network  10  having a plurality of nodes  18  is illustrated. The nodes  18  are in wireless communication with one another by network connections  20 . Each node  18  that is part of the communication network  10  is configured to collect local measurements and collaborative measurements that are measured between two collaborating nodes  18 . Some examples of the local measurements and the collaborative measurements include, but are not limited to, time transfer and synchronization, three-dimensional images, range measurements between nodes  18 , line-of-sight (LOS) measurements between nodes  18 , angle of arrival (AOA) measurements between nodes  18 , or direction of arrival (DOA) between nodes  18 . The nodes  18  represent any device configured to estimate a local state vector such as, for example, a machine, a vehicle, or an individual such as a soldier. For example, in one embodiment, the nodes  18  may each represent a machine, where the nodes  18  are located in a manufacturing facility. As explained below and shown in  FIG. 4 , in another embodiment the nodes  18  each represent users that are part of collaborative positioning, navigation, and timing (PNT) network  26 . 
     Referring to  FIGS. 2 and 3 , in one approach an individual node  18   i  includes an estimator  40  that determines updates to an estimated local state vector. The updates are determined based on the local measurements as well as collaborative measurements that are measured between the individual node  18   i  and a collaborating node  18   j . This approach is referred to as a local with immediate neighbor (LWIN) approach because all of the calculations are performed locally at the individual node  18   i . Alternatively, in the embodiment as shown in  FIGS. 7 and 8 , the communication network  10  includes a distributed approach. Specifically, this approach includes an estimator  140  that is distributed throughout the communication network  10 . Specifically, a portion  98  of the estimator  140  is located locally at each node  18  that is part of the communication network  10  and determines some of the local updates. A centralized portion  100  of the estimator  140  determines a remaining portion of the local updates for each node  18  that is part of the communication network  10  as well collaborative updates based on the collaborative measurements between two collaborating nodes  18 . This approach is referred to as an integrated total network solution (ITNS) because some of the calculations are performed at the centralized portion  100  of the estimator  140 . 
     Turning back to  FIG. 2 , in the LWIN approach each node  18  is in wireless communication with at least one other neighboring node  18  that is part of the communication network  10 . For purposes of the present disclosure, a neighboring node  18  refers to a logical or topographical relationship between two nodes  18 . Furthermore, although  FIG. 2  illustrates node  18   i  and node  18   j , node  18   k  and node  18   n , and node  18   j  and node  18   n  in electronic communication with one another, it is to be appreciated that each node  18  may be in communication with all of the remaining nodes  18  that are part of the communication network  10 . Specifically, if there are n number of nodes that are part of the communication network  10 , then each node  18  may be in wireless communication with up to n−1 number of nodes. 
     Each node  18  includes a computing system  30 , a measurement device  32 , a transceiver  34 , and an antenna  36 , where the nodes  18  are in wireless communication with one another by their respective antennas  36 . The computing system  30  is in electronic communication with the measurement device  32 , the transceiver  34 , and the antenna  36 . The measurement device  32  represents any device or combination of devices configured to collect the local measurements and the collaborative measurements for a respective node  18 . For example, in one embodiment, if the local measurements are three-dimensional images, then the measurement device  32  includes one or more cameras. In another embodiment, the measurement device  32  is a PNT system that determines a position of the respective node  18  in a frame of reference of the Earth. It is to be appreciated that the PNT system is not limited to a specific type of system. For example, some nodes  18  that are part of the communication network  10  may include only a global navigation satellite systems (GNSS) receiver as the PNT system. However, other nodes  18  may include a GNSS receiver combined with an inertial measurement unit as the PNT system. Alternatively, in another approach, some nodes  18  may include a vision based navigation system or a celestial navigation system as the PNT system. 
       FIG. 3  is a schematic diagram of the computing system  30  of the individual node  18   i  shown in  FIG. 2  including the estimator  40 , which is configured to determine local updates  46  based on the local measurements that are collected from the measurement device  32  ( FIG. 2 ) of the individual node  18   i . The local measurements represent information that is based on only the individual node  18   i . The estimator  40  is also configured to determine collaborative updates  48  based on collaborative measurements between the individual node  18   i  and the collaborative node  18   j  (as shown in  FIG. 2 ). The collaborative measurements between the individual node  18   i  and the collaborative node  18   j  are measured with respect to the individual node  18   i.    
     The estimator  40  includes a local state propagation block  50 , a local measurement block  52 , and a local update block  54  for determining the local update  46 . The local state propagation block  50  determines a plurality of local state propagators. The plurality of local state propagators include an estimated local state vector for the individual node  18   i  and a local error covariance matrix of the estimated local state vector. The local state propagation block  50  receives as input a local state vector x i (k), a deterministic input vector u i (k), and a local measurement vector z i ( k ), where k represents a specific point in time. The local state vector x i (k), the deterministic input vector u i (k), and the local measurement vector z i (k) are based on the local measurements collected from the measurement device  32  of the individual node  18   i  ( FIG. 2 ). 
     The local state propagation block  50  includes a local state propagation block  60  and a local error covariance block  62 . The local state propagation block  60  estimates the estimated local state vector for the individual node  18   i  based on the local state vector x i (k) and the deterministic input vector u i (k) according to Equation 1, which is as follows:
 
 {circumflex over (x)}   i ( k|k− 1)= f   i ( {circumflex over (x)}   i ( k− 1| k− 1), u   i ( k− 1), k− 1))  Equation 1
 
where {circumflex over (x)} i (k|k−1) is the estimated local state vector for the individual node  18   i  up to a point in time of (k−1), and f i  represents a function that models a dynamic behavior of the communication network  10  ( FIG. 1 ) at the individual node  18   i.  
 
     The local error covariance block  62  determines the local error covariance matrix for the estimated local state vector {circumflex over (x)} i (k|k−1). The local error covariance matrix characterizes error of the estimated local state vector {circumflex over (x)} i (k|k−1). The local error covariance matrix is determined based on the local measurements and the estimated local state vector {circumflex over (x)} i (k|k−1), and is expressed in Equation 2:
 
 P   i ( k|k− 1)=Φ i ( k− 1) P   i ( k− 1| k− 1)Φ i   T ( k− 1)+ Q   i ( k )  Equation 2
 
where P i (k|k−1) is the local error covariance matrix when the time is equal to (k−1) based on measurements up to the point of (k−1), P i  (k−1|k−1) is the covariance matrix for the individual node  18   i  when the time is equal to k based on measurements up to the point of (k−1), Φ i  is a state transition matrix for the node  18   i, Φ   i   T  represents a transposed state transition matrix, and Q i (k) is a process noise covariance matrix for the node  18   i . It is to be appreciated that the state transition matrix Φ i  and the process noise covariance matrix Q i (k) are both block-diagonal matrices, which may result in reduced computation.
 
     The local measurement block  52  predicts an estimated local measurement {circumflex over (z)} i (k|k−1) of the individual node  18   i  and a local residual μ i . Specifically, the local measurement block  52  includes a local measurement prediction block  64  that predicts an estimated local measurement {circumflex over (z)} i (k|k−1) of the individual node  18   i  based on the estimate of the local state vector {circumflex over (x)} i (k|k−1) for the individual node  18   i , the local measurement vector z i (k), and the deterministic input vector u i (k) according to Equation 3, which is as follows:
 
 {circumflex over (z)}   i ( k|k− 1)= g   i ( {circumflex over (x)}   i ( k|k− 1), u   i ( k− 1), k )  Equation 3
 
where g i  denotes a function that represents a local measurement model at the individual node  18   i . The local residual μ i  is associated with the estimated local measurement {circumflex over (z)} i (k|k−1) of the individual node  18   i . Specifically, the local measurement block  52  includes a local residual block  66  that determines the local residual μ i  for the individual node  18   i  based on a difference between the local measurement vector z i (k) and the estimated local measurement {circumflex over (z)} i (k|k−1), and is expressed in Equation 4 as:
 
μ i =( z   i ( k )− {circumflex over (z)}   i ( k|k− 1))  Equation 4
 
     The local update block  54  includes a local measurement sensitivity block  68 , a local residual covariance matrix block  70 , and a local gain matrix block  72 . The local update block  54  determines the local update  46  that is applied to the estimated local state vector {circumflex over (x)} i (k|k−1) and the local covariance matrix P i (k|k−1) for the individual node  18   i . Specifically, the local measurement sensitivity block  68  first determines a local measurement sensitivity matrix H zi (k) as well as a local measurement variance matrix R zi (k). The local measurement sensitivity matrix H zi (k) represents an amount of change that the local measurements undergo based on a change in the local state vector x i (k). The local measurement variance matrix R zi (k) represents an uncertainty in the local measurements, where the uncertainty may also be referred to as the error. The local measurement sensitivity matrix H zi (k) is expressed in Equation 5 as: 
                       H     z   ⁢   i       ⁡     (   k   )       ⁢     =   Δ     ⁢           ∂     g   i         ∂     x   i         |     x   i       =         x   ^     i     ⁡     (     k   |     k   -   1       )                 Equation   ⁢           ⁢   5               
where
 
                   ∂     g   i         ∂     x   i         |     x   i       =         x   ^     i     ⁡     (     k   |     k   -   1       )             
represents a partial derivative of the local state vector x i (k). It is to be appreciated that both the local measurement sensitivity matrix H zi  (k) as well as the local measurement variance matrix R zi (k) are block-diagonal, which simplifies computation. The local residual covariance matrix block  70  determines a local residual covariance matrix P μi  that represents an uncertainty of the μ i  for the individual node  18   i . The residual covariance matrix P μi  of the local residual μ i  is determined based on the local error covariance matrix P i (k|k−1) and the local measurement variance matrix R zi (k), and is expressed in Equation 6 as:
 
                     P     μ   i       =         (       ∂     g   i         ∂     x   i         )     ⁢         P   i     ⁡     (       ∂     g   i         ∂     x   i         )       T       +     R     z   ⁢   i                 Equation   ⁢           ⁢   6               
where
 
             (       ∂     g   i         ∂     x   i         )         
represents a partial mapping matrix. The local gain matrix block  72  determines a local gain matrix K zi (k) based on the covariance matrix P μi , the local error covariance matrix P i (k|k−1), and the local measurement sensitivity matrix H zi (k), which is expressed in Equation 7 as:
 
 K   zi ( k )= P   i ( k|k− 1) H   zi   T ( k ) B   μi   Equation 7
 
where an inverse P μi   −1  of the covariance matrix P μi  is expressed as B μi . The local gain matrix K zi (k) is combined with the local residual μ i , to create the local update  46 . Specifically, the local update  46  is a product of local gain matrix K zi (k) and the local residual μ i . The local update  46  is applied to the local state propagation block  60  and the local residual covariance matrix block  70 . Specifically, the local state propagation block  60  adds the local update  46  to the estimated local state vector {circumflex over (x)} i (k|k−1) to determine an updated estimated local state vector {circumflex over (x)} i (k|k), which is expressed in Equation 8 as:
 
 {circumflex over (x)}   i ( k|k )= {circumflex over (x)}   i ( k|k− 1)+ K   zi μ i ( k )  Equation 8
 
The local update  46  is also applied to the local residual covariance matrix P μi . Specifically, the local residual covariance matrix block  70  adds the local update  46  to the local residual covariance matrix P μi  to determine an updated residual covariance matrix P i (k|k), which is expressed in Equation 9 as:
 
 P   i ( k|k )=( I−K   zi ( k ) H   zi ( k )) P   i ( k|k− 1)  Equation 9
 
where I represents the identity matrix.
 
     The collaborative updates  48  for the individual node  18   i  are now described. It is to be appreciated that the estimator  40  may determine the collaborative updates  48  either immediately after the local error covariance block  62  determines the local error covariance matrix P i (k|k−1) for the individual node  18   i  or, alternatively, immediately after the local gain matrix block  72  determines the local gain matrix K zi (k). However, if the estimator  40  determines the collaborative updates  48  after the local updates  46 , then the covariance matrix P μi  is first updated, and then the estimator  40  may determine the collaborative measurement. 
     The estimator  40  includes a collaborative measurement block  76  and a collaborative update block  78  for determining the collaborative update  48 . The collaborative measurement block  76  includes a collaborative measurement prediction block  80  and a collaborative residual block  82 . As explained below, the collaborative measurement prediction block  80  predicts an estimated collaborative measurement ŷ ij (k|k−1) based on the collaborative measurement between the individual node  18   i  and the collaborative node  18   j . It is to be appreciated that the collaborative measurement between the individual node  18   i  and the collaborative node  18   j  is measured with respect to the individual node  18   i . The collaborative node  18   j  may send its local measurements to the individual node  18   i  over the communication network  10  ( FIG. 1 ) or, alternatively, the measurement device  32  of the individual node  18   i  may collect local measurements of the collaborative node  18   j.    
     The collaborative measurement prediction block  80  predicts the estimated collaborative measurement ŷ ij (k|k−1) based on a collaborative measurement vector y ij (k), the local state vector x i (k) of the individual node  18   i , and a local state vector y i (k) of the collaborative node  18   j , and is expressed by Equation 10 as:
 
 ŷ   ij ( k|k− 1)= h   ij ( {circumflex over (x)}   i ( k|k− 1), {circumflex over (x)}   i ( k|k− 1), k )  Equation 10
 
where h ij  represents a function that models the collaborative measurement at the individual node  18   i  as measured by the collaborative node  18   i  through the collaboration of node  18   i  with node  18   j . The collaborative measurement vector y ij (k) represents the collaborative measurement between the individual node  18   i  and the collaborative node  18   j.  
 
     A collaborative residual υ is associated with the estimated collaborative measurement ŷ ij (k|k−1). Specifically, the collaborative residual block  82  determines the collaborative residual υ. The collaborative residual υ represents a difference between the collaborative measurement vector y ij (k) and the estimated collaborative measurement ŷ ij (k|k−1), and is determined according to Equation 11 as:
 
υ=( y   ij ( k )− ŷ   ij ( k|k− 1))  Equation 11
 
     The collaborative update block  78  includes a collaborative measurement sensitivity block  84 , a collaborative covariance matrix block  86 , and a collaborative gain matrix block  88 . The collaborative measurement sensitivity block  84  determines a collaborative measurement sensitivity matrix H yij (k) and a compound covariance matrix  R   yij  (k). The collaborative measurement sensitivity matrix H yij (k) represents an amount of change that the collaborative measurement undergoes based on a corresponding change in the local state vector x i (k) of the individual node  18   i . The compound covariance matrix  R   yij (k) characterizes an uncertainty in the collaborative measurement when an impact of one or more state of the collaborative node  18   j  are modeled as random noise. The collaborative measurement sensitivity matrix H yij (k) is determined based on the estimated collaborative measurement ŷ ij (k|k−1), which is expressed in Equation 12 as: 
                       H   yij     ⁡     (   k   )       =     [                       …       0         ⁢                     ∂     h   i         ∂     x   i             0             …         ⁢           ]             Equation   ⁢           ⁢   12               
The collaborative measurement sensitivity block  84  also determines the compound covariance matrix  R   yij (k) based on the collaborative measurement variance matrix
 
                 R   yij     ⁡     (   k   )       ,     (       ∂     h   j         ∂     x   i         )           
represents an uncertainty of the state, and
 
                 P   j     ⁡     (     k   |     k   -   1       )       ⁢       (       ∂     h   j         ∂     x   i         )     T           
represents an uncertainty of the noise, and is expressed in Equation 13 as:
 
     
       
         
           
             
               
                 
                   
                     
                       
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     The collaborative covariance matrix block  86  determines a collaborative error covariance matrix P υij  of the collaborative residual υ. The collaborative error covariance matrix P υij  is determined based on the collaborative measurement sensitivity matrix H yij (k), the collaborative measurement variance matrix R yij (k), the compound covariance matrix  R   yij (k), a collaborative error covariance matrix P ii (k|k−1) of the individual node  18   i , and a collaborative error covariance matrix P jj (k|k−1) of the collaborative node  18   j , and is expressed in Equation 14 as
 
 P   υij =( H   yi ( k ) P   ii ( k|k− 1) H   yi   T ( k )+ H   yj ( k ) P   jj ( k|k− 1) H   yi   T ( k )+   R     yij ( k ))  Equation 14
 
     The collaborative gain matrix block  88  determines a collaborative gain matrix K yij (k) based on the covariance matrix P υij  of the collaborative residual υ, a collaborative error covariance matrix P ii (k|k−1), and the collaborative measurement sensitivity matrix H yij (k), which is expressed in Equation 15 as:
 
 K   yij ( k )= P   ii ( k|k− 1) H   yij   T ( k ) B   υij   Equation 15
 
where an inverse P υij   −1  of the collaborative covariance matrix P υij  is expressed as B υij . The collaborative gain matrix K yij (k) is combined with the collaborative residual υ to create the collaborative update  48 . Specifically, the collaborative update  48  is a product of the collaborative gain matrix K yij (k) and the collaborative residual υ. The local state propagation block  60  then applies the collaborative update  48  to the estimated local state vector {circumflex over (x)} i (k|k−1). Specifically, the local state propagation block  60  adds the collaborative update  48  to the to the estimated local state vector {circumflex over (x)} i (k|k−1) to determine the updated estimated local state vector {circumflex over (x)} i (k|k), which is expressed in Equation 16 as:
 
 {circumflex over (x)}   i ( k|k )=( I−K   yij ( k ) H   yij ( k )) P   i ( k|k− 1)  Equation 16
 
The local residual covariance matrix block  70  determines the updated residual covariance matrix P i (k|k) based on the local update  46 , which is expressed in Equation 17 as:
 
 P   i ( k|k )=( I−K   yij ( k ) H   yij ( k ) P   i ( k|k− 1)  Equation 17
 
     It is to be appreciated that the estimators for the compound covariance matrix  R   yij (k) are decoupled. In other words, each node  18   i ,  18   j  is associated with a unique state vector. It is also to be appreciated that an estimation error of the collaborative node  18   j  is modeled as noise. Specifically, the collaborative measurement vector y ij (k) is expressed in Equation 18 as:
 
 y   ij ( k )= h   ij ( x   i ( k ), x   j ( k ), k )+ s   ij ( k )  Equation 18
 
where s ij (k) represents a random measurement noise vector for the individual node  18   i  at time k when collaborating with the collaborative node  18   j . Finally, it is to be appreciated that the collaborative measurement vector y ij (k) is the only measurement that is not locally independent. In other words, all of the remaining measurements described above are local measurements that are specific to either the individual node  18   i  or the collaborative node  18   j.  
 
       FIG. 4  is an exemplary illustration of the PNT network  26 , where each node  18  represents a user. In the exemplary embodiment as shown in  FIG. 4 , the nodes  18  represent a land vehicle, helicopter, or an aircraft. However, it is to be appreciated that the nodes  18  may also represent individuals as well. For example, in one embodiment, one or more nodes  18  represent an individual such as a soldier holding a PNT system. Referring to both  FIGS. 2 and 4 , the measurement device  32  for each node  18  of the PNT network  26  is a PNT system that determines a position  90  of the respective node  18  in a frame of reference of the Earth. One example of a frame of reference of the Earth is an Earth-centered, Earth-fixed (ECEF) frame of reference. Alternatively, in another embodiment, the latitude, longitude, and altitude may be used instead. In the embodiment as shown, the dashed lines between the nodes  18  represent a wireless communication connection  92 A. The thin solid lines between the nodes  18  represent a wireless communication connection  92 B that includes time-transfer and range measurements. The thick solid lines between the nodes  18  represent a wireless communication connection  92 C that includes time-transfer, range measurements, and (LOS) line-of-sight measurements. 
       FIGS. 5A-5C  illustrates exemplary collaborative measurements based on the PNT network  26  shown in  FIG. 4 . In the embodiment as shown in  FIG. 5A , the collaborative measurement between the individual node  18   i  and the collaborative node  18   j  includes relative distance measurements between the individual node  18   i  and the collaborative node  18   j  combined with relative LOS measurements between the individual node  18   i  and the collaborative node  18   j . As seen in  FIG. 5A , the relative distance measurements are represented by a first relative distance r ij  measured between the individual node  18   i  and the collaborative node  18   j  as measured by individual node  18   i , and a second relative distance r ji  measured between the individual node  18   i  and the collaborative node  18   j  as measured by the collaborative node  18   j . The relative LOS measurements include a first relative LOS measurement   as measured by the individual node  18   i  is represented by a first unit vector pointing from the individual node  18   i  to the collaborative node  18   j , where the superscript B i  indicates a body frame of the individual node  18   i . The relative LOS measurements further include a second relative LOS measurement   as measured by the collaborative node  18   j  is represented by a second unit vector pointing from the collaborative node  18   j  to the individual node  18   i , where the superscript B j  indicates a body frame of the collaborative node  18   j.    
     Equation 19 expresses a position  E R i  of the individual node  18   i  in the ECEF frame of reference, and Equation 20 expresses a position  E R j  of the collaborative node  18   j  in the ECEF frame of reference:
 
 E   R   j − E   R   i   =C   B     i     E ( r   ij   B     i     u   ij )  Equation 19
 
 E   R   i − E   R   j   =C   B     j     E ( r   ji   B     j     u   ji )  Equation 20
 
where C Bi   E  represents a directional cosine matrix for transforming a vector from the body frame of reference of the individual node  18   i  into the ECEF frame of reference, and C B     j     E  represents a directional cosine matrix for transforming a vector from the body frame of reference of the collaborative node  18   j  into the ECEF frame of reference. It is to be appreciated that the collaborative measurements between the individual node  18   i  and the collaborative node  18   j  for the LWIN approach are node-centric. In other words, the collaborative measurement between the individual node  18   i  and the collaborative node  18   j  is measured with respect to either the individual node  18   i  or the collaborative node  18   j.  
 
     In another embodiment as shown in  FIG. 5B , the collaborative measurement between the individual node  18   i  and the collaborative node  18   j  is a relative LOS direction measurement that indicates an angular measurement as measured with respect to either the individual node  18   i  or the collaborative node  18   j . In one example, the collaborative measurement is node-centric to the individual node  18   i  and include the position  E R i  of the individual node  18   i , an attitude r Bi  of the individual node  18   i  in the body frame of reference, a first relative LOS direction measurement u ij , and a first LOS angle Ψ LOSi . It is to be appreciated that two LOS angles are involved with these measurements, however, only a single angle is shown for purposes of simplicity. The first relative LOS direction measurement u ij  is measured between the individual node  18   i  and the collaborative node  18   j  with respect to the individual node  18   i , and the first LOS angle Ψ LOSi  is measured with respect to the individual node  18   i  and the first relative LOS direction measurement u ij . Equation 21 may be used to determine the attitude r Bi  and the position  E R i  of the individual node  18   i , and is expressed as:
 
 z   iij   =r   B     i     u   ij   =C   E   B     i   ( E   R   j − E   R   i )+ v   iij   Equation 21
 
where z iij  represents a measurement for the individual node  18   i , as measured by the individual node  18   i , through a collaboration between the individual node  18   i  and the collaborative node  18   j  and v iij  represents all measurement noise. In another example, instead of node-centric measurements, the position  E R j  of the collaborative node  18   j , an attitude r Bj  of the collaborative node  18   j  in the body frame of reference, and a second relative LOS direction measurement u ji  is used instead. The second relative LOS direction measurement u ji  is measured between the individual node  18   i  and the collaborative node  18   j  with respect to the collaborative node  18   j , a second LOS angle Ψ LOSj  is measured with respect to the collaborative node  18   j  and the second relative LOS direction measurement u ji . The position  E R j  of the collaborative node  18   j , the attitude r Bj  of the collaborative node  18   j , and the second relative LOS direction measurement u ji  are communicated to the individual node  18   i  over the communication network  10  ( FIG. 1 ). Equation 22 may be used to determine the attitude r Bi  and the position  E R i  of the individual node  18   i , and is expressed as:
 
 z   iji   =r   B     j     u   ji   +C   E   B     j     E   R   j   =C   E   B     j     E   R   i   +v   iji   Equation 22
 
where z iji  represents a measurement for the individual node  18   i , as measured by the collaborative node  18   j , through a collaboration between the collaborative node  18   j  and the individual node  18   i , and v iji  represents all measurement noise.
 
     In yet another embodiment as shown in  FIG. 5C , the collaborative measurement between the individual node  18   i  and the collaborative node  18   j  is a relative range measurement between the individual node  18   i  and the collaborative node  18   j , which is measured with respect to either the individual node  18   i  or the collaborative node  18   j . Specifically,  FIG. 5C  illustrates a first relative range measurement d ij  measured between the individual node  18   i  and the collaborative node  18   j  as measured with respect to the individual node  18   i  and a second relative range measurement d ij  measured between the individual node  18   i  and the collaborative node  18   j  as measured with respect to the collaborative node  18   j . In one example, the first relative range measurement d ij  measured between the individual node  18   i  and the collaborative node  18   j  as measured with respect to the individual node  18   i  is expressed in Equation 23 as:
 
 d   ij =∥ E   R   i − E   R   j   ∥+v   ij   Equation 23
 
When the first relative range measurement d ij  is linearized into δd ij , Equation 23 becomes Equation 24, which is:
 
                     d   ij     =         d   ij     -       d   ^     ij       =       ⁢     (       δ   ⁢     R   i       -     δ   ⁢     R   j         )       +     v   ij                 Equation   ⁢           ⁢   24               
where {circumflex over (d)} ij  represents an estimate of the first relative range measurement, ∥ − ∥ represents a vector magnitude, δR i  represents a linearized position of the individual node  18   i , and δR j  represents a linearized position of the collaborative node  18   j . If the first relative LOS measurement   is expressed in Equation 25 as:
 
                     u   ij               E     =       (       R   i               E     -     R   j               E       )              R   i               E     -     R   j               E                      Equation   ⁢           ⁢   25               
then the linearized first relative range measurement δd ij  may be expressed in Equation 26 as:
 
δ d   ij =   E   R   i −   E   R   j   +v   ij   Equation 26
 
       FIGS. 6A-6B  illustrate an exemplary process flow diagram illustrating a method  200  for updating the estimated local state vector {circumflex over (x)} i (k|k−1) for the individual node  18   i . Referring to  FIGS. 2, 3, and 6A , the method  200  begins at block  202 . In blocks  202 - 220 , the estimator  40  determines the local update  46 . Specifically, in block  202 , the local state propagation block  60  estimates, based on the local measurements, the estimated local state vector {circumflex over (x)} i (k|k−1) for the individual node  18   i . The method  200  may then proceed to block  204 . 
     In block  204 , the local error covariance block  62  determines the local error covariance matrix P i (k|k−1) for the estimated local state vector {circumflex over (x)} i (k|k−1) based on the local measurements. The local error covariance matrix P i (k|k−1) characterizes error of the estimated local state vector {circumflex over (x)} i (k|k−1). The method  200  may then proceed to block  206 . 
     In block  206 , the local measurement prediction block  64  predicts the estimated local measurement {circumflex over (z)} i (k|k−1) of the individual node  18   i  based on the estimate of the local state vector {circumflex over (x)} i (k|k−1) and the local measurements, where the local residual μ i  for the individual node  18   i  is associated with the estimated local measurement {circumflex over (z)} i (k|k−1). The method  200  may then proceed to block  208 . 
     In block  208 , the local residual block  66  determines the local residual μ i  for the individual node  18   i . The local residual μ i  for the individual node  18   i  represents the difference between the estimated local measurement {circumflex over (z)} i (k|k−1) and the local measurement vector z i (k). The method  200  may then proceed to block  210 . 
     In block  210 , the local measurement sensitivity block  68  determines the local measurement sensitivity matrix H zi (k), which represents an amount of change that the local measurements undergo based on a change in the local state vector x i (k). The method may then proceed to block  212 . 
     In block  212 , the local measurement sensitivity block  68  determines local measurement variance matrix R zi (k), which represents the uncertainty in the local measurements. The method  200  may then proceed to block  214 . 
     In block  214 , the local residual covariance matrix block  70  determines the local residual covariance matrix P μi  of the local residual μ i  based on the local error covariance matrix P i (k|k−1) and the local measurement variance matrix R zi (k). The method  200  may then proceed to block  216 . 
     In block  216 , the local gain matrix block  72  determines the local gain matrix K zi (k) based on the local residual covariance matrix P μi , the local error covariance matrix P i (k|k−1), and the local measurement sensitivity matrix H zi (k). The method  200  may then proceed to block  218 . 
     In block  218 , the local gain matrix block  72  combines the local gain matrix K zi (k) with the local residual μ i  to create the local update  46 . The method  200  may then proceed to block  220 . 
     In block  220 , the local update  46  is applied to the estimated local state vector {circumflex over (x)} i (k|k−1) for the individual node  18   i  and the local residual covariance matrix P μi . The method  200  may then proceed to block  222 . 
       FIG. 6B  illustrates blocks  222 - 238 , where the estimator  40  determines the collaborative update  48 . Specifically, in block  222 , the collaborative measurement prediction block  80  predicts the estimated collaborative measurement ŷ ij (k|k−1) based on the collaborative measurement between the individual node  18   i  and the collaborative node  18   j , where the collaborative residual υ is associated with the estimated collaborative measurement ŷ ij (k|k−1). The method  200  may then proceed to block  224 . 
     In block  224 , the collaborative residual block  82  determines the collaborative residual υ, which represents the difference between the collaborative measurement vector y ij (k) and the estimated collaborative measurement ŷ ij (k|k−1). The method  200  may then proceed to block  226 . 
     In block  226 , the collaborative measurement sensitivity block  84  determines the collaborative measurement sensitivity matrix H yij (k) representing the amount of change that the collaborative measurement undergoes based on a corresponding change in the local state vector x i (k) of the individual node  18   i . The method  200  may then proceed to block  228 . 
     In block  228 , the collaborative measurement sensitivity block  84  also determines the compound covariance matrix  R   yij  (k) that characterizes the uncertainty in the collaborative measurement when an impact of one or more states of the collaborative node  18   j  are modeled as random noise. The method  200  may then proceed to block  230 . 
     In block  230 , the collaborative covariance matrix block  86  determines the collaborative error covariance matrix P υij  of the collaborative residual υ based on at least the collaborative measurement sensitivity matrix H yij (k) and the compound covariance matrix  R   yij  (k). The method  200  may then proceed to block  232 . 
     In block  232 , the collaborative gain matrix block  88  determines the collaborative gain matrix K yij (k) based on the covariance matrix P υij  of the collaborative residual υ. The method  200  may then proceed to block  234 . 
     In block  234 , the collaborative gain matrix block  88  determines the collaborative update  48 , which is based on the estimated collaborative measurement ŷ ij (k|k−1) and the collaborative residual υ. Specifically, the collaborative gain matrix K yij (k) is combined with the collaborative residual υ to create the collaborative update the collaborative update  48 . The method  200  may then proceed to block  236 . 
     In block  236 , the collaborative update  48  is applied to the estimated local state vector {circumflex over (x)} i (k|k−1) for the individual node  18   i . The method  200  may then terminate or return to block  202 . 
       FIG. 7  is a schematic diagram of a plurality of nodes  18   i ,  18   j ,  18   k , and  18   n  that are in wireless communication with the centralized portion  100  of the estimator  140  based on the ITNS approach. In the non-limiting embodiment as shown in  FIG. 7 , the communication network  10  includes a distributed estimator  140 , where a portion  98  of the estimator  140  is located at each node  18 . Specifically, the portion  98  of the estimator  140  at each node  18  comprises of the local state propagation block  60 , the local error covariance block  62 , and the local residual block  66 . The estimator  140  also includes the centralized portion  100 , which is in wireless communication with all of the nodes  18  that are part of the communication network  10 . However, in an alternative embodiment, the communication network  10  includes a plurality of centralized portions  100  that are in wireless communication with a sub-network or portion of the total nodes  18  that are part of the communication network  10 . In the non-limiting embodiment as shown in  FIG. 7 , the centralized portion  100  of the estimator  140  is a stand-alone component. In other words, the centralized portion  100  of the estimator  140  is not part of any of the nodes  18 . However, in an alternative embodiment, the centralized portion  100  of the estimator  140  is included with one of the plurality of nodes  18  of the communication network  10 . 
     The communication network  10  includes one or more pairs of collaborating nodes  18   i ,  18   j  as well. For example, in the non-limiting embodiment as shown, node  18   i  and node  18   j  are in wireless communication with one another and a collaborative measurement exists between the pair of collaborating nodes  18   i ,  18   j . It is to be appreciated that  FIG. 7  illustrates only one pair of collaborating nodes  18   i ,  18   j  for purposes of simplicity and ease of illustration. Each node  18  that is part of the communication network  10  may collaborate with each remaining node  18  that is part of the communication network  10 . In other words, if there are n number of nodes  18  that are part of the communication network  10 , then there may be up to n*(n−1) pairs of collaborating nodes  18  that are included in the communication network  10 . 
     Continuing to refer to  FIG. 7 , in the ITNS approach the plurality of nodes  18  each determine their own respective estimated local state vectors {circumflex over (x)}(k|k−1), estimated local measurement {circumflex over (z)}(k|k−1), and local residual μ. In other words, each node  18  includes a respective local state propagation block  60 , a respective local measurement prediction block  64 , and a respective local residual block  66 . Each node  18  sends its respective estimated local state vector {circumflex over (x)}(k|k−1), respective estimated local measurement {circumflex over (z)}(k|k−1), respective local residual μ, and respective local measurements to the centralized portion  100  of the estimator  140  over the communication network  10 . The centralized portion  100  of the estimator  140  receives the respective estimated local state vector {circumflex over (x)}(k|k−1), the respective estimated local measurement {circumflex over (z)}(k|k−1), the respective local residual μ, and the local measurements from each of the plurality of nodes  18  that are part of the communication network  10 . As explained below, the centralized portion  100  of the estimator  140  determines a local update  146  for each node  18  based on the respective estimated local state vector {circumflex over (x)}(k|k−1), the respective estimated local measurement {circumflex over (z)}(k|k−1), the respective local residual μ, and the respective local measurements. The centralized portion  100  of the estimator  140  also determines a collaborative update  148  for each pair of collaborating nodes  18   i ,  18   j  that are part of the communication network  10 . 
       FIG. 8  is a block diagram of the centralized portion  100  of the estimator  140 . The centralized portion  100  of the estimator  140  includes a total error covariance block  162 , a local measurement sensitivity block  168 , a local residual covariance matrix block  170 , and a local gain matrix block  172 . The total error covariance block  162  determines a total error covariance matrix for the entire communication network  10  based on the respective estimated local state vector {circumflex over (x)}(k|k−1) for each of the plurality of nodes  18  that are part of the communication network  10  ( FIG. 7 ). The total error covariance matrix characterizes error for the entire communication network  10 . That is, the total error covariance matrix characterizes the error based on the estimated local state vector {circumflex over (x)}(k|k−1) for each node  18  that is part of the communication network  10 . The total error covariance matrix is expressed in Equation 27 as:
 
 P ( k|k− 1)=Φ( k− 1) P ( k− 1| k− 1)Φ T ( k− 1)+ Q ( k )  Equation 27
 
where P(k|k−1) is the total error covariance matrix when the time is equal to (k−1) based on measurements up to the point of (k−1), P(k−1|k−1) is the covariance matrix for the entire communication network  10  when the time is equal to k based on measurements up to the point of (k−1), Φ is a state transition matrix for each node  18  in the communication network  10 , Φ T  represents a transposed state transition matrix, and Q (k) is a process noise covariance matrix for the entire communication network  10 .
 
     The local measurement sensitivity block  168  determines a respective local measurement sensitivity matrix H z (k) as well as a respective local measurement variance matrix R z (k) for a particular node  18  that is part of the communication network  10  ( FIG. 7 ). As mentioned above, the respective local measurement sensitivity matrix H z  (k) represents an amount of change that the local measurements for a particular node  18  undergo based on a change in the respective local state vector x(k), and the respective local measurement variance matrix R z (k) represents an uncertainty in the local measurements for the particular node  18 . The local residual covariance matrix block  170  of the estimator  140  determines a respective local residual covariance matrix P μ  for the particular node  18  based on the total error covariance matrix P i (k|k−1) for the entire communication network  10  and the local measurement variance matrix R zi (k), which is expressed in Equation 6 above. 
     The local gain matrix block  172  of the centralized portion  100  of the estimator  140  then determines the respective local gain matrix K z (k) for the particular node  18  based on the respective local residual covariance matrix P μ  for the particular node  18 , the total error covariance matrix P(k|k−1) for the entire communication network  10 , and the respective local measurement sensitivity matrix H z (k) for the particular node  18 . The respective local gain matrix K z (k) is combined with the local residual μ, to create the local update  146  for the particular node  18 . Specifically, the local update  146  for a particular node  18  is a product of the respective local gain matrix K z  (k) and the respective local residual μ for each node  18  that is part of the communication network  10 . Therefore, the local update  146  is determined based on the total error covariance matrix P(k|k−1) for the entire communication network  10  and the local residual μ. 
     Referring to  FIGS. 7 and 8 , the local update  146  is applied to the estimated local state vector {circumflex over (x)}(k|k−1) of the particular node  18  that is part of the communication network  10  to determine the corresponding updated estimated local state vector {circumflex over (x)} i (k|k). The local update  146  is also applied to the covariance matrix P μ  for the particular node  18 . It is to be appreciated that the centralized portion  100  of the estimator  140  determines a unique local update  146  for each of the plurality of nodes  18  of the communication network  10 . In other words, the centralized portion  100  of the estimator  140  is configured to determine an n number of local updates  146 , where each local update  146  corresponds to a specific one of the nodes  18  that are part of the communication network  10 . 
     The collaborative updates  148  are now described. There may be any number of pairs of collaborating nodes  18   i ,  18   j  that are part of the communication network  10 . Accordingly, there may be up to n*(n−1) number of collaborative updates  148  determined by the centralized portion  100  of the estimator  140 . Furthermore, the centralized portion  100  of the estimator  140  may determine the collaborative updates  148  either immediately after the total error covariance block  162  determines the total error covariance matrix P(k|k−1) or, alternatively, immediately after the local gain matrix block  172  applies the local update  146 . However, if the centralized portion  100  of the estimator  140  determines the collaborative updates  148  after the local updates  146 , then the total error covariance matrix P(k|k−1) is updated first, and then the centralized portion  100  of the estimator  140  may determine the collaborative measurement. 
     For purposes for explanation, nodes  18   i  and  18   j  ( FIG. 7 ) represent a collaborating pair of nodes. Specifically, the node  18   i  represents the individual node that the collaborative measurement is taken with respect to, and the node  18   j  represents the collaborative node. However, it is to be appreciated that each node  18  that is part of the communication network  10  may collaborate with each remaining node  18  that is part of the communication network  10  to determine the collaborative measurement. The centralized portion  100  of the estimator  140  includes a collaborative measurement block  176  and a collaborative update block  178 . The collaborative measurement block  176  includes a collaborative measurement prediction block  180  and a collaborative residual block  182 . The collaborative measurement prediction block  180  predicts the estimated collaborative measurement ŷ ij (k|k−1) based on the collaborative measurement vector y ij (k) and the local state vectors x i (k), y i (k) of the collaborating nodes  18   i ,  18   j , and is expressed by Equation 10 above. The collaborative residual υ, which is associated with the estimated collaborative measurement ŷ ij (k|k−1), represents the difference between the collaborative measurement and the estimated collaborative measurement ŷ ij (k|k−1), and is determined based on Equation 11 above. 
     The collaborative update block  178  includes a collaborative measurement sensitivity block  184 , a collaborative covariance matrix block  186 , and a collaborative gain matrix block  188 . The collaborative measurement sensitivity block  184  determines the collaborative measurement sensitivity matrix H yij (k) and the compound covariance matrix  R   yij (k). The collaborative measurement sensitivity matrix H yij (k) represents an amount of change that the collaborative measurement undergoes based on a corresponding change in the local state vector x i (k) of the node  18   i  that the collaborative measurement is taken with respect to. The compound covariance matrix  R   yij (k) characterizes an uncertainty in the collaborative measurement when an impact of one or more states of the collaborative node  18   j  of the pair of collaborating nodes  18   i ,  18   j  are modeled as random noise. The collaborative measurement sensitivity matrix H yij (k) is expressed in Equation 12 above, and the compound covariance matrix  R   yij (k) is expressed in Equation 13 above. 
     The collaborative covariance matrix block  186  determines the collaborative error covariance matrix P υij  of the collaborative residual υ, and is determined based on Equation 14 above. The collaborative gain matrix block  188  determines the collaborative gain matrix K yij (k) based on the collaborative error covariance matrix P υij  of the collaborative residual υ. The collaborative gain matrix K yij (k) is determined based on Equation 15 above. The collaborative gain matrix K yij (k) is combined with the collaborative residual υ to create the collaborative update  148  for the pair of collaborating nodes  18   i ,  18   j . Specifically, the collaborative update  148  is a product of the collaborative gain matrix K yij (k) and the collaborative residual υ. Referring to both  FIGS. 7 and 8 , the collaborative update  48  is applied to the to the respective estimated local state vectors {circumflex over (x)} i (k|k−1), {circumflex over (x)} j (k|k−1) for both nodes  18   i ,  18   j  of the pair of collaborating nodes. The collaborative update  148  is also applied to the total error covariance matrix P(k|k−1) of the communication network  10 . 
     In one embodiment, the communication network  10  including the centralized portion  100  of the estimator  140  is part of a PNT network  26  ( FIG. 4 ). Accordingly, in the embodiment as shown in  FIG. 5A , the collaborative measurement between the individual node  18   i  and the collaborative node  18   j  includes relative distance measurements between the individual node  18   i  and the collaborative node  18   j  combined with relative LOS measurements between the individual node  18   i  and the collaborative node  18   j . As mentioned above, the relative distance measurements are represented by the first relative distance r ij  measured between the individual node  18   i  and the collaborative node  18   j  as measured by individual node  18   i , and the second relative distance r ij  measured between the individual node  18   i  and the collaborative node  18   j  as measured by the collaborative node  18   j . The relative LOS measurements include a first relative LOS measurement  B     i   u ij  as measured by the individual node  18   i  is represented by a first unit vector pointing from the individual node  18   i  to the collaborative node  18   j . The relative LOS measurements further include a second relative LOS measurement  B     j   u ji  as measured by the collaborative node  18   j  is represented by a second unit vector pointing from the collaborative node  18   j  to the individual node  18   i.    
     Referring to  FIG. 5B , in another embodiment the collaborative measurement between the individual node  18   i  and the collaborative node  18   j  is the relative LOS direction measurement that indicates an angular measurement as measured with respect to either the individual node  18   i  or the collaborative node  18   j . In one example, the collaborative measurements include the position  E R i  of the individual node  18   i , the attitude r Bi  of the individual node  18   i  in the body frame of reference, and the first relative LOS direction measurement u ij . Equation 28 may be used to determine the attitude r Bi  and the position  E R i  of the individual node  18   i , and is expressed as:
 
 z   ij   =r   B     i     u   ij   =C   E   B     i   ( E   R   j − E   R   i )+ v   ij   Equation 28
 
where z ij  represents a measurement for the individual node  18   i  through a collaboration between the individual node  18   i . In another example, the collaborative measurements include the position  E R i  of the collaborative node  18   j , the attitude r Bj  of the collaborative node  18   j  in the body frame of reference, and the second relative LOS direction measurement u ji . Equation 29 may be used to determine the attitude r Bi  and the position  E R i  of the individual node  18   i , and is expressed as:
 
 z   ji   =r   B     j     u   ji   =C   E   B     j   ( E   R   i − E   R   j )+ v   ji   Equation 29
 
where z ij  represents a measurement for the collaborative node  18   j  through a collaboration between the collaborative node  18   j  and the individual node  18   i , and v iji  represents all measurement noise.
 
     In yet another embodiment as shown in  FIG. 5C , the collaborative measurement is the relative range measurement between the individual node  18   i  and the collaborative node  18   j , which is measured with respect to either the individual node  18   i  or the collaborative node  18   j . Specifically,  FIG. 5C  illustrates the first relative range measurement d ij  measured between the individual node  18   i  and the collaborative node  18   j  as measured with respect to the individual node  18   i  and the second relative range measurement d ij  measured between the individual node  18   i  and the collaborative node  18   j  as measured with respect to the collaborative node  18   j.    
       FIGS. 9A-9B  illustrate an exemplary process flow diagram illustrating a method  300  for updating the estimated local state vector for the plurality of nodes  18  that are part of the communication network  10 . Referring to  FIGS. 7, 8, and 9A , the method  300  begins at block  302 . In blocks  302 - 316 , the centralized portion  100  of the estimator  140  determines the local update  146 . Specifically, in block  202 , the centralized portion  100  of the estimator  140  receives the respective estimated local state vector {circumflex over (x)}(k|k−1), the respective estimated local measurement {circumflex over (z)}(k|k−1), and the respective local residual μ from each of the plurality of nodes  18  that are part of the communication network. The method  300  may then proceed to block  304 . 
     In block  304 , the total error covariance block  162  of the centralized portion  100  of the estimator  140  determines the total error covariance matrix P(k|k−1) for the communication network  10  based on the respective estimated local state vector {circumflex over (x)}(k|k−1) for each of the plurality of nodes  18  that are part of the communication network  10 . The method  300  may then proceed to block  306 . 
     In block  306 , the local measurement sensitivity block  168  determines the respective local measurement sensitivity matrix H z (k), which represents an amount of change that the local measurements for the particular node  18  undergo based on a change in the respective local state vector x(k). The method  300  may then proceed to block  308 . 
     In block  308 , the local measurement sensitivity block  168  determines the respective local measurement variance matrix R z (k), which represents the uncertainty in the local measurements for the particular node  18 . The method  300  may then proceed to block  310 . 
     In block  310 , the local residual covariance matrix block  170  determines the respective local residual covariance matrix P μ  for each of the plurality of nodes  18  that are part of the communication network  10  based on the total error covariance matrix P(k|k−1) for the entire communication network  10  and the respective local measurement variance matrix R z (k). The method  300  may then proceed to block  312 . 
     In block  312 , the local gain matrix block  172  determines the respective local gain matrix K z (k) for the particular node  18  based on the respective local residual covariance matrix P μ  for the particular node  18 , the total error covariance matrix P(k|k−1) for the entire communication network  10 , and the respective local measurement sensitivity matrix H z (k) for the particular node  18 . The method  300  may then proceed to block  314 . 
     In block  314 , the collaborative update block  178  combines the local gain matrix K z (k) with the respective local residual μ of the particular node  18  to create the respective local update  146  for the particular node  18 . Specifically, the local update  146  for a particular node  18  is the product of the respective local gain matrix K z (k) and the respective local residual p for each node  18  that is part of the communication network  10 . The method  300  may then proceed to block  316 . 
     In block  316 , the local update  146  is applied to the estimated local state vector {circumflex over (x)}(k|k−1) of the particular node  18  that is part of the communication network  10  to determine the corresponding updated estimated local state vector {circumflex over (x)} i (k|k). The local update  146  is also applied to the collaborative error covariance matrix P μ  for the particular node  18 . The method  300  may then proceed to block  318 . 
       FIG. 9B  illustrates blocks  318 - 332 , where the collaborative update  148  is determined. Specifically, in block  318 , the collaborative measurement prediction block  180  predicts the estimated collaborative measurement ŷ ij (k|k−1) based on the collaborative measurement between the pair of collaborating nodes  18   i ,  18   j  that are part of the communication network  10 , where the collaborative residual is associated with the estimated collaborative measurement. Specifically, as mentioned above, the estimated collaborative measurement ŷ ij (k|k−1) based on the collaborative measurement vector y ij (k) and the local state vectors x i (k), y i (k) of the collaborating nodes  18   i ,  18   j . The method  300  may then proceed to block  320 . 
     In block  320 , the collaborative residual block  182  determines the collaborative residual υ, which represents the difference between the collaborative measurement vector y ij (k) and the estimated collaborative measurement ŷ ij (k|k−1). The method  300  may then proceed to block  322 . 
     In block  322 , the collaborative measurement sensitivity block  184  determine the collaborative measurement sensitivity matrix H yij (k), which represents the amount of change that the collaborative measurement undergoes based on a corresponding change in the local state vector x i (k) of the node  18   i  that the collaborative measurement is taken with respect to. The method  300  may then proceed to block  324 . 
     In block  324 , the collaborative measurement sensitivity block  184  determine the compound covariance matrix  R   yij (k) characterizes an uncertainty in the collaborative measurement when an impact of one or more states of the collaborative node  18   j  of the pair of collaborating nodes  18   i ,  18   j  are modeled as random noise. The method  300  may then proceed to block  326 . 
     In block  326 , the collaborative covariance matrix block  186  determines the collaborative error covariance matrix P υij  of the collaborative residual υ based on at least the collaborative measurement sensitivity matrix H yij (k) and the compound covariance matrix  R   yij (k). The method  300  may then proceed to block  328 . 
     In block  328 , the collaborative gain matrix block  188  determines the collaborative gain matrix K yij (k) based on the collaborative error covariance matrix P υij  of the collaborative residual υ and the collaborative measurement sensitivity matrix H yij (k). The method  300  may then proceed to block  330 . 
     In block  330 , the collaborative gain matrix block  188  combines the collaborative gain matrix K yij (k) with the collaborative residual υ to create the collaborative update  148 . Thus, it is to be appreciated that the collaborative update  148  is determined based on the estimated collaborative measurement ŷ ij (k|k−1) and the collaborative residual estimated collaborative measurement ŷ ij (k|k−1). The method  300  may then proceed to block  332 . 
     In block  332 , the collaborative update  148  is applied to the respective the estimated local state vectors {circumflex over (x)} i (k|k−1), {circumflex over (x)} j (k|k−1) for both nodes  18   i ,  18   j  of the pair of collaborating nodes. The collaborative update  148  is also applied to the total error covariance matrix P(k|k−1) of the communication network  10 . The method  300  may then terminate or return to block  302  (shown in  FIG. 9A ). 
     Referring generally to the figures, the present disclosure provides various technical effects and benefits. Specifically, the present disclosure describes an estimator that determines local updates as well as collaborative updates that are applied to the estimated local state vector for an individual node. The local updates are determined based on local measurements, while the collaborative updates are based on collaborative measurements between the individual node and a collaborative node. In one decentralized approach, the estimator may be included as part of each node that is part of the communication network. This approach may require less computational power when compared to the centralized approach. Alternatively, in another approach, the estimation is done at a centralized location, which may result in improved accuracy. However, this approach may require additional computational power when compared to the decentralized approach. Both the centralized and the decentralized approaches may improve the accuracy of the estimated local state vector for each node, since the updates are based on not only the local measurements, but also collaborative measurements between nodes. 
     Referring to  FIG. 10 , the computing system  30  ( FIG. 2 ) and the centralized portion  100  of the estimator  140  ( FIG. 7 ) are implemented on one or more computer devices or systems, such as exemplary computer system  1030 . The computer system  1030  includes a processor  1032 , a memory  1034 , a mass storage memory device  1036 , an input/output (I/O) interface  1038 , and a Human Machine Interface (HMI)  1040 . The computer system  1030  is operatively coupled to one or more external resources  1042  via the network  1026  or I/O interface  1038 . External resources may include, but are not limited to, servers, databases, mass storage devices, peripheral devices, cloud-based network services, or any other suitable computer resource that may be used by the computer system  1030 . 
     The processor  1032  includes one or more devices selected from microprocessors, micro-controllers, digital signal processors, microcomputers, central processing units, field programmable gate arrays, programmable logic devices, state machines, logic circuits, analog circuits, digital circuits, or any other devices that manipulate signals (analog or digital) based on operational instructions that are stored in the memory  1034 . Memory  1034  includes a single memory device or a plurality of memory devices including, but not limited to, read-only memory (ROM), random access memory (RAM), volatile memory, non-volatile memory, static random-access memory (SRAM), dynamic random-access memory (DRAM), flash memory, cache memory, or any other device capable of storing information. The mass storage memory device  1036  includes data storage devices such as a hard drive, optical drive, tape drive, volatile or non-volatile solid-state device, or any other device capable of storing information. 
     The processor  1032  operates under the control of an operating system  1046  that resides in memory  1034 . The operating system  1046  manages computer resources so that computer program code embodied as one or more computer software applications, such as an application  1048  residing in memory  1034 , may have instructions executed by the processor  1032 . In an alternative example, the processor  1032  may execute the application  1048  directly, in which case the operating system  1046  may be omitted. One or more data structures  1049  also reside in memory  1034 , and may be used by the processor  1032 , operating system  1046 , or application  1048  to store or manipulate data. 
     The I/O interface  1038  provides a machine interface that operatively couples the processor  1032  to other devices and systems, such as the network  1026  or external resource  1042 . The application  1048  thereby works cooperatively with the network  1026  or external resource  1042  by communicating via the I/O interface  1038  to provide the various features, functions, applications, processes, or modules comprising examples of the disclosure. The application  1048  also includes program code that is executed by one or more external resources  1042 , or otherwise rely on functions or signals provided by other system or network components external to the computer system  1030 . Indeed, given the nearly endless hardware and software configurations possible, persons having ordinary skill in the art will understand that examples of the disclosure may include applications that are located externally to the computer system  1030 , distributed among multiple computers or other external resources  1042 , or provided by computing resources (hardware and software) that are provided as a service over the network  1026 , such as a cloud computing service. 
     The HMI  1040  is operatively coupled to the processor  1032  of computer system  1030  in a known manner to allow a user to interact directly with the computer system  1030 . The HMI  1040  may include video or alphanumeric displays, a touch screen, a speaker, and any other suitable audio and visual indicators capable of providing data to the user. The HMI  1040  also includes input devices and controls such as an alphanumeric keyboard, a pointing device, keypads, pushbuttons, control knobs, microphones, etc., capable of accepting commands or input from the user and transmitting the entered input to the processor  1032 . 
     A database  1044  may reside on the mass storage memory device  1036  and may be used to collect and organize data used by the various systems and modules described herein. The database  1044  may include data and supporting data structures that store and organize the data. In particular, the database  1044  may be arranged with any database organization or structure including, but not limited to, a relational database, a hierarchical database, a network database, or combinations thereof. A database management system in the form of a computer software application executing as instructions on the processor  1032  may be used to access the information or data stored in records of the database  1044  in response to a query, where a query may be dynamically determined and executed by the operating system  1046 , other applications  1048 , or one or more modules. 
     The description of the present disclosure is merely exemplary in nature and variations that do not depart from the gist of the present disclosure are intended to be within the scope of the present disclosure. Such variations are not to be regarded as a departure from the spirit and scope of the present disclosure.