Abstract:
Disclosed is a technique for obtaining an estimate and variance of each variable based on a constraint manifold. Particles (or samples) are sampled in order to filter and fuse ambiguous data or information on at least one state variable of a system using the particles. The sampling is carried out in consideration of an influence which non-linearity of the constraint manifold of a system model, an observation model or another system model exerts on a probability distribution of the state variable. With this construction, it is possible to reduce decrease of fusion and filtering performance, decrease a Gaussian approximation error, and detect mismatched information.

Description:
BACKGROUND OF THE INVENTION 
   1. Field of the Invention 
   The present invention relates to a particle sampling method and a sensor fusion and filtering method, and more particularly, to a technique for obtaining an estimate and variance of each variable based on a constraint manifold. 
   2. Description of the Conventional Art 
   In general, sensor fusion refers to integrated processing for combining or fusing different types of sensor information into one representative information data. 
     FIG. 15  illustrates a pattern of general sensor fusion. 
   In  FIG. 15 , outputs X 1  and X 2  obtained from a first two sensors are fused into anew upper representative value X 1,2 . An output X 3  obtained from a third sensor is fused with the fused node X 1,2 , thereby generating a new representative value X 1,2,3  to be fused with an upper node. 
   As illustrated in  FIG. 1 , this sensor fusion and filtering method performs fusion and filtering  103  through two stages: prediction  101  using an equation for a system model and observation  102  using an equation for observation. Conventional filters using the sensor fusion and filtering method include a Kalman filter, an extended Kalman filter, an unscented Kalman filter, a particle filter, and so on. 
   Among these filters, the Kalman filter for basic sensor fusion and filtering is a linear filter suitable for a Gaussian error model, but it has a problem when it comes to non-linear system processing. For the purpose of non-linear system processing, an extended Kalman filter has been developed. However, the extended Kalman filter has a drawback that an error is caused by linearization, and thus its performance can vary according to non-linearity. As an alternative to the extended Kalman filter, an unscented Kalman filter, which treats non-linear and Gaussian error model systems, has been developed. The unscented Kalman filter has been actively studied up to present, but it has a problem in that its performance can be lowered due to inaccurate Gaussian approximation. 
   In order to cope with the problems of these Kalman filters, a particle filter has been developed. The particle filter has the advantage of employing a method of indicating results fused by particles having various weights, and thus it can be applied to fusion and filtering of a system having non-linear and arbitrary error modes as well as ambiguous information. However, the particle filter creates a problem in real-time processing due to failure and long processing time resulting from inaccurate prediction of particles. 
   The following references are related to sensor fusion and filtering: 
   [1] M. S. Grewal and A. P. Andrews, “Kalman Filtering,” 2nd ed. New York: Wiley, 2001. 
   [2] H. W. Sorenson, “Recursive Estimation for Nonlinear Dynamic Systems,” New York: Marcel Dekker, 1988. 
   [3] A. H. Jazwinski “Stochastic Processes and Filtering Theory,” New York: Academic, 1970. 
   [4] T. Ghirmai N. F. Bugallo, J. Miguez and P. P. Djuri, “A Novel Particle Filtering Approach to Blind Symbol Detection and Timing Estimation,” IEEE 58th Vehicular Technology Conference (VTC2003), vol. 2, pp. 1147-1151, October 2003. 
   [5] D. Crisan and A. Doucet, “A Survey of Convergence Results on Particle Filtering Methods for Practitioners,” IEEE Transactions on Signal Processing, vol. 50, pp. 736-746, March 2002. 
   [6] J. H. Kotecha and P. M. Djuric´, “Sequential Monte Carlo Sampling Detector for Rayleigh Fast-fading Channels,” in Proc. Int. Conf. Acoust., Speech, Signal Process., 2000. 
   [7] A. Doucet, S. Godsill, and C. Andrieu, “On Sequential Monte Carlo Sampling Methods for Bayesian Filtering,” Statist. Comput., vol. 10, no. 3, pp. 197-208, 2000. 
   SUMMARY OF THE INVENTION 
   The present inventors have developed an integrated sensor fusion and filtering method by finding a joint probability on a constraint manifold defined in a unified fusion and filtering space unlike, thereby simplifying a conventional sensor fusion and filtering method of performing two steps of prediction using a model equation and observation using an observation equation. Thereby, application of various constraints is possible, and the geometrically expressed constraint is directly treated, so that it can be applied to various models. In addition, it can be applied to fusion and filtering of a system having a non-linear, non-Gaussian error model, as well as ambiguous information. 
   Accordingly, the present invention has been made to solve the above-mentioned problems, and an objective of the present invention is to provide a sensor fusion and filtering method and system based on a constraint manifold, in which a technique of finding the joint probability on the constraint manifold defined in the unified fusion and filtering space is applied, thereby enabling application of various constraints, and directly treating the geometrically expressed constraint to permit application to various models. 
   It is another objective of the present invention to provide a sensor fusion and filtering method and system based on a constraint manifold, in which system and observation models are expressed in the fusion and filtering space without linearization using the constraint manifold as they are, thereby being capable of reducing decrease of fusion and filtering performance caused by wrong linearization. 
   It is still another objective of the present invention to provide a sensor fusion and filtering method and system based on a constraint manifold, in which an error model is expressed and calculated by a sum of Gaussian thereby being capable of decreasing a Gaussian approximation error, detecting mismatched information, and being applied to the fusion and filtering problem of a system having ambiguous information. 
   In order to accomplish these objectives, according to an aspect of the present invention, there is provided a particle sampling method for sampling particles in order to filter and fuse ambiguous data or information on at least one state variable of a system using the particles, characterized in that the sampling is carried out in consideration of an influence which non-linearity of the constraint manifold of a system model, an observation model or another system model exerts on a probability distribution of the state variable. 
   Further, the particle sampling method may be characterized by the steps of: uniformly sampling the particles on an axis of a previous state variable in order to find, from the previous state variable, a probability distribution of a current state variable estimated by an equation for the system model or other system model; finding estimated weights of particles of the current state variable according to the equation of the system model or other system model and of each particle of the current state variable, from a weight allocated by the sampled particles of the previous state variable and information on a prior probability of the previous state variable; finding a prior probability distribution of a current state estimated from the estimated weight of the sampled particles of the current state variable and from the estimated weight of each particle of the current state variable; and obtaining particles of an observation variable estimated by an equation for the observation model from the current state variable in order to obtain the particles of the observation variable. 
   Furthermore, the particle sampling method may be characterized by the steps of: when the sampling performed on a manifold in a geometrical space of an arbitrary model equation at regular intervals is defined as “Uniform Sampling on Constraint Manifold,” previously performing the sampling in the geometrical space from numerous samples meeting an equation for the system model or other system model at regular intervals, and obtaining particles of a previous state variable and a current state variable; when an interval divided uniformly on an axis of the previous state variable is defined as a bucket, finding a weight of each particle of the current state variable estimated through a weight allocated by the particles of the previous state variable and prior probability information of the previous state variable and through a number of the particles existing in the bucket; finding a prior probability distribution of a current state estimated from the estimated weight of the sampled particles of the current state variable and from the estimated weight of each particle of the current state variable; and previously performing the sampling in the geometrical space from the numerous samples meeting an equation for the observation model at regular intervals, and obtaining the particles of the previous state variable and particles of an observation variable. 
   In addition, the particle sampling method may be characterized by the step of obtaining samples which meet the non-linearity of the constraint manifold of the system model, the observation model or another system model as well as the constraint manifold by a Monte Carlo method. 
   Further, the particle sampling method may be characterized in that, when the number of the samples existing in each bucket is denoted by m, the weight of the particles of the current state variable is found by: 
   Weight of particles of the current state variable=Weight of samples existing in each bucket/m. 
   According to another aspect of the present invention, there is provided a sensor fusion and filtering method for fusing and filtering ambiguous data or information on at least one state variable of a system using particles, which is characterized by the steps of: setting a data or information fusing and filtering space composed of state and observation variables; defining a constraint manifold having various constraints in the fusing and filtering space; calculating a joint probability distribution on the constraint manifold using a prior probability distribution of the variables; calculating a marginal probability distribution of the variables from the joint probability distribution; and obtaining estimate and variance of each of the variables. 

   
     BRIEF DESCRIPTION OF THE DRAWINGS 
     The above and other objects, features and advantages of the present invention will be more apparent from the following detailed description taken in conjunction with the accompanying drawings, in which: 
       FIG. 1  is a block diagram illustrating a sensor fusion and filtering process; 
       FIG. 2  is a flowchart illustrating a sensor fusion and filtering method based on a constraint manifold in accordance with the present invention; 
       FIG. 3  illustrates a concept of sensor fusion based on a linear constraint manifold in accordance with an embodiment of the present invention; 
       FIG. 4  illustrates a concept of sensor fusion based on a non-linear constraint manifold in accordance with an embodiment of the present invention; 
       FIG. 5  illustrates results of implementing uniform sampling on a geometrical manifold constituting a system model of Equation 10 of the present invention; 
       FIG. 6  illustrates a sample distribution of x n  when x n−1  of Equation 10 of the present invention is composed of Gaussian samples; 
       FIG. 7  illustrates results of implementing uniform sampling on a geometrical manifold constituting an observation model of Equation 10 of the present invention; 
       FIG. 8  is a graph illustrating average mean square errors according to embodiment 1 of the present invention and comparative example 1; 
       FIG. 9  is a graph illustrating average mean square errors according to embodiment 2 of the present invention and comparative example 2; 
       FIG. 10  is a graph illustrating average mean square errors according to embodiment 3 of the present invention and comparative example 3; 
       FIG. 11  is a graph illustrating average mean square errors according to embodiment 4 of the present invention and comparative example 4; 
       FIG. 12  is a graph illustrating average mean square errors according to embodiment 5 of the present invention and comparative example 5; 
       FIG. 13  is a graph illustrating average mean square errors according to embodiment 6 of the present invention and comparative example 6; 
       FIG. 14  is a graph illustrating average mean square errors according to embodiment 7 of the present invention and comparative example 7; and 
       FIG. 15  illustrates a pattern of general sensor fusion. 
   

   DETAILED DESCRIPTION OF PREFERRED EMBODIMENTS 
   Hereinafter, exemplary embodiments of the present invention will be described with reference to the accompanying drawings. In the following description and drawings, the same reference numerals are used to designate the same or similar components, and such components will be no longer described. 
   First, a concept of the present invention will be described. 
   As illustrated in  FIG. 2 , a sensor fusion and filtering method based on a constraint manifold in accordance with the present invention includes a step S 1  of setting a sensor fusion and filtering space composed of each variable, a step S 2  of setting the constraint manifold in the sensor fusion and filtering space, a step S 3  of finding a prior probability distribution of the variables, a step S 4  of calculating a joint probability distribution on the constraint manifold from the prior probability distribution, and a step S 5  of performing back propagation from the joint probability distribution toward each variable axis and calculating a marginal probability distribution. These steps are executed by a plurality of sensors and a processor for processing data collected from the plurality of sensors. For example, the plurality of sensors used in the present invention may be composed of, but not limited to, an ultraviolet sensor, an infrared sensor, a heat sensor, and so on. The processor for processing data collected from the plurality of sensors has a function capable of executing steps S 1  to S 5 , which is accomplished by a typical computer system. 
   In order to explain the sensor fusion and filtering approach according to the present invention, assuming that the observations be z 1 , z 2 , and z 3  respectively, that estimates or of the observations be v 1 , v 2 , and v 3  respectively, and that correlation, constraint, between the respective estimates be f(v 1 , v 2 , v 3 )=0, it can be considered that the marginal probability distribution of the variables is obtained as in Equation 1 as follows. 
   
     
       
         
           
             
               
                 
                   
                     
                       
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   where i=1, 2 and 3, j=mod(i+1), and k=mod(i+2). 
   This process of the present invention can be compared with estimate of Bayesian concept used in general sensor fusion and filtering, as follows. 
   Estimates of v 1 , v 2 , and v 3  of the Bayesian concept can be obtained by finding a posterior probability distribution of Equation 2 below.
 
 p ( v   i|   z   1   , z   2   , z   3   , f ( v   1   , v   2   , v   3 )=0)   [Equation 2]
 
   where i=1, 2 and 3. 
   Accordingly, it can be seen in the present invention that the process of finding the marginal probability distribution of each variable v 1  is equal to the process of finding the posterior probability distribution of Equation 2 above. 
   A process of finding the marginal probability distribution of Equation 1 in the present invention is as follows. 
   First, a sensor fusion and filtering space consisting of v 1 , v 2 , and v 3  is defined as (v 1 , −v 2 , −v 3 ), and a constrict is set as f(v 1 , v 2 , v 3 )=0. 
   At this time, the observations z 1 , z 2  and z 3  and variances are given by a prior probability of each of v 1 , v 2 , and v 3  axes, and a joint probability distribution p(v 1 , −v 2 , −v 3 |z 1 , z 2 , z 3 ) is expressed in the sensor fusion and filtering space (v 1 , −v 2 , −v 3 ) by a product of each prior probability value. 
   The constraint, f(v 1 , v 2 , v 3 )=0, is given, thus being geometrically denoted in this space, and a joint probability distribution on the constraint, p(v 1 , −v 2 , −v 3 |z 1 , z 2 , z 3 , f(v 1 , v 2 , v 3 )=0), is found. 
   The estimate of each variable and an updated variance is obtained by finding the marginal probability distribution that performs back propagation on the joint probability distribution on the constraint toward each axis. Thus, the sensor fusion and filtering method may be treated equal to the process of finding the marginal probability distribution as in Equation 1. 
     FIG. 3  illustrates a concept of sensor fusion based on a linear constraint manifold in accordance with an embodiment of the present invention. The sensor fusion and filtering method according to the present invention will be described below with reference to  FIG. 3 . 
   In general when two observations z 1  and z 2  are given in the Bayesian concept, a formula for estimating a fused result, v, is given as the following Equation 3. 
   
     
       
         
           
             
               
                 
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                                         , 
                                         
                                           v 
                                           2 
                                         
                                         , 
                                       
                                     
                                   
                                   
                                     
                                       
                                         
                                           v 
                                           1 
                                         
                                         = 
                                         
                                           v 
                                           1 
                                         
                                       
                                     
                                   
                                 
                                 ) 
                               
                             
                           
                           
                             p 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               ( 
                               
                                 
                                   z 
                                   1 
                                 
                                 , 
                                 
                                   z 
                                   2 
                                 
                                 , 
                                 
                                   
                                     v 
                                     1 
                                   
                                   = 
                                   
                                     v 
                                     2 
                                   
                                 
                               
                               ) 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         
                           = 
                             
                           ⁢ 
                           
                             
                               k 
                               2 
                             
                             ⁢ 
                             
                               p 
                               ⁡ 
                               
                                 ( 
                                 
                                   
                                     
                                       
                                         
                                           z 
                                           2 
                                         
                                         ❘ 
                                       
                                     
                                   
                                   
                                     
                                       
                                         
                                           z 
                                           1 
                                         
                                         , 
                                         
                                           v 
                                           2 
                                         
                                         , 
                                         
                                           
                                             v 
                                             2 
                                           
                                           = 
                                           
                                             v 
                                             1 
                                           
                                         
                                       
                                     
                                   
                                 
                                 ) 
                               
                             
                             ⁢ 
                             p 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               ( 
                               
                                 
                                   
                                     
                                       
                                         z 
                                         1 
                                       
                                       ❘ 
                                     
                                   
                                 
                                 
                                   
                                     
                                       v 
                                       1 
                                     
                                   
                                 
                               
                               ) 
                             
                           
                         
                         ⁢ 
                         
                             
                         
                       
                     
                   
                 
                 ⁢ 
                 
                   
 
                 
                 ⁢ 
                 
                   
                     where 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       k 
                       2 
                     
                   
                   = 
                   
                     
                       
                         p 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           ( 
                           
                             
                               
                                 
                                   
                                     v 
                                     1 
                                   
                                   , 
                                   
                                     v 
                                     2 
                                   
                                   , 
                                 
                               
                             
                             
                               
                                 
                                   
                                     v 
                                     1 
                                   
                                   = 
                                   
                                     v 
                                     2 
                                   
                                 
                               
                             
                           
                           ) 
                         
                       
                       
                         p 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           ( 
                           
                             
                               
                                 
                                   
                                     z 
                                     1 
                                   
                                   , 
                                   
                                     z 
                                     2 
                                   
                                   , 
                                 
                               
                             
                             
                               
                                 
                                   
                                     v 
                                     1 
                                   
                                   = 
                                   
                                     v 
                                     2 
                                   
                                 
                               
                             
                           
                           ) 
                         
                       
                     
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     the 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     normalization 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       coefficient 
                       . 
                     
                   
                 
               
             
             
               
                 [ 
                 
                   Equation 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   7 
                 
                 ] 
               
             
           
         
       
     
   
   If the observations z 1  and z 2  are independent of each other, Equation 7 is simplified as in Equation 8 below. 
   
     
       
         
           
               
           
           ⁢ 
           
             
               
                 
                   
                     
                       
                         
                           p 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             ( 
                             
                               
                                 
                                   
                                     
                                       v 
                                       1 
                                     
                                     , 
                                     
                                       
                                         v 
                                         2 
                                       
                                       ❘ 
                                     
                                   
                                 
                               
                               
                                 
                                   
                                     
                                       z 
                                       1 
                                     
                                     , 
                                     
                                       z 
                                       2 
                                     
                                     , 
                                     
                                       
                                         v 
                                         1 
                                       
                                       = 
                                       
                                         v 
                                         2 
                                       
                                     
                                   
                                 
                               
                             
                             ) 
                           
                         
                         = 
                           
                         ⁢ 
                         
                           
                             k 
                             2 
                           
                           ⁢ 
                           p 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             ( 
                             
                               
                                 
                                   
                                     
                                       z 
                                       1 
                                     
                                     , 
                                     
                                       
                                         z 
                                         2 
                                       
                                       ❘ 
                                     
                                   
                                 
                               
                               
                                 
                                   
                                     
                                       v 
                                       1 
                                     
                                     , 
                                     
                                       v 
                                       2 
                                     
                                     , 
                                     
                                       
                                         v 
                                         1 
                                       
                                       = 
                                       
                                         v 
                                         2 
                                       
                                     
                                   
                                 
                               
                             
                             ) 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                         ⁢ 
                         
                           
                             k 
                             2 
                           
                           ⁢ 
                           p 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             ( 
                             
                               
                                 
                                   z 
                                   2 
                                 
                                 ❘ 
                                 
                                   v 
                                   2 
                                 
                               
                               , 
                               
                                 
                                   v 
                                   2 
                                 
                                 = 
                                 
                                   v 
                                   1 
                                 
                               
                             
                             ) 
                           
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           p 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             ( 
                             
                               
                                 z 
                                 1 
                               
                               ❘ 
                               
                                 v 
                                 1 
                               
                             
                             ) 
                           
                         
                       
                     
                   
                 
               
               
                 
                   [ 
                   
                     Equation 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     8 
                   
                   ] 
                 
               
             
           
         
       
     
   
   In this manner, Equations 7 and 8 derived from the sensor fusion and filtering of the present invention are matched with Equations 3 and 4 derived from the general sensor fusion approach. 
   Letting the constraint manifold according to the present invention be regarded as an arbitrary non-linear function and extend through a more general sensor fusion and filtering method, a state estimate problem will be described as follows. 
   It is assumed that an actual value and estimate of the variable at times k−1 and k be (v k−1 , x k−1 ) and (v k , x k ), and that a state transition function be f(v k−1 , v k )=0. 
   Further, it is assumed that an output value of a sensor observing u k  is z k , while actual state and observation u k  and v k  meet u k =v k , and that uncertainties related to x k−1  and z k  be given as p(x k−1 |v k−1 ) and p(z k |u k ). 
   At this time, the estimate v k  according to the present invention is obtained in the following procedures. 
   The sensor fusion and filtering space is set to (v k−1 , −u k ), and the constraint manifold is f(v k−1 , u k )=0. The estimate v k  is given as p(v k |x k− , z k , f(v k−1 , u k )=0, and is calculated by the following Equation 9.
 
∫ f(Vk−1, Uk)=0   [p ( V   k−1   , u   k   |x   k−1   , z   k )] dv   k−1  
 
or
 
∫[ p ( v   k−1   , u   k   |x   k−   , z   k   ,f ( v   k−1   , u   k )=0 ]dv    k−1   [Equation 9]
 
   Here, a joint probability p(v k− , u k |x k− , z k ), is obtained from p(x k− , v k− ) and p(z k |u k ).which has been already known. 
   In this manner, a fusion problem of the estimate v k , which is obtained from propagation of x k− by the sensor observation value z k  and the constraint manifold f(v k− , u k )=0, can be solved using the problem of meeting the constraint, as in Equation 9 above. 
     FIG. 4  illustrates a concept of sensor fusion in non-linear constraint manifold  401  according to an embodiment of the present invention. Referring to  FIG. 4 , a joint probability distribution  404  on a non-linear constraint and marginal probability distributions  402  and  406  can be represented in a fusion and filtering space consisting of two variables using prior probability distributions  403  and  406 . 
   In this case, although the prior probability distributions  403  and  406  comply with a Gaussian distribution, the marginal probability distributions  402  and  405  obtained by back propagation from the joint probability distributions  404  on the constraint  401  has a non-Gaussian distributions. 
   Therefore, the present invention makes use of a sum of Gaussian in order to express a non-Gaussian probability model. 
   According to the present invention when mismatched information is fused, the joint probability distribution on the constraint manifold itself becomes flat, so that it is possible to detect a mismatch. Further, ambiguous information is expressed and calculated as a probability having several peaks by the sum of Gaussian, so that it can be processed without a single Gaussian approximation error. 
   Now, embodiments of the present invention will be described below. 
   In order to compare sensor fusion and filtering performance of the present invention with that of the conventional art, there is introduced an UNGM (Univariate Nonstationary Growth Model) having a system model of the following Equation 10 and an observation model of the following Equation 11. A uniform sampling is made on a constraint manifold, therby reflecting geometrical non-linearity of a constraint on the filtering. Samples corresponding to samples on the manifold in a geometrical space of the constraint exist on each axis establishing the space, and it can be seen that these samples are disposed according to the non-linearity as a geometrical characteristic. A state estimation simulation is performed, which compares a sensor fusion and filtering method of the present invention with conventional extended Kalman filter, unscented Kalman filter, and particle filter. 
   The following test method is based on the sampling method according to the present invention. 
   1. Sampling is performed on a manifold in the geometrical space of a given system model equation at regular intervals. This sampling is called “Uniform Sampling on Constraint Manifold,” which is shown in  FIG. 5 . 
   2. Samples x n-1  corresponding to the “Uniform Sampling on Constraint Manifold” are not uniformly disposed. The samples are uniformly divided on x n-1  axis at an interval wider than that of the “Uniform Sampling on Constraint Manifold,” wherein each interval is called “bucket,” The number of samples existing in each bucket is counted and denoted by m. Original weights are varied according to m, each of which is referred to as “modified weight.” The modified weight is obtained as follows.
 
Modified Weight=Original Weight/ m  
 
   3. A probability distribution of x n  corresponding to x n-1  is represented through the modified weight obtained from the above step 2. Results of the probability distribution of x n  (when n=1) are shown as in  FIG. 6 . As illustrated in  FIG. 6 , it can be seen that the probability distribution is shown by the non-linearity of the system model equation. 
   4. The probability distribution of x n  is found through the above steps 1 to 3 in the given observation model equation. 
   5. The probability distribution p(x n ) of the above step 3 is multiplied by the probability distribution p(x n ) of the above step 4, and then primary moments of each sample value and weight corresponding to each sample are obtained. Thereby, a final value is obtained. 
   
     
       
         
           
             
               
                 
                   x 
                   n 
                 
                 = 
                 
                   
                     α 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       x 
                       
                         n 
                         - 
                         1 
                       
                     
                   
                   + 
                   
                     β 
                     ⁢ 
                     
                       
                         x 
                         
                           n 
                           - 
                           1 
                         
                       
                       
                         1 
                         + 
                         
                           x 
                           
                             n 
                             - 
                             1 
                           
                           2 
                         
                       
                     
                   
                   + 
                   
                     γcos 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       ( 
                       
                         1.2 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           ( 
                           
                             n 
                             - 
                             1 
                           
                           ) 
                         
                       
                       ) 
                     
                   
                   + 
                   
                     u 
                     n 
                   
                 
               
             
             
               
                 [ 
                 
                   Equation 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   10 
                 
                 ] 
               
             
           
         
       
     
   
   where, u n =N(0, σ u   2 ), the system noise, and x 0 =0.1, α=0.5, β=25, γ=8, and N=100 are set. 
   
     
       
         
           
             
               
                 
                   y 
                   n 
                 
                 = 
                 
                   
                     
                       x 
                       n 
                       2 
                     
                     20 
                   
                   + 
                   
                     
                       v 
                       n 
                     
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       ( 
                       
                         
                           n 
                           = 
                           1 
                         
                         , 
                         ⋯ 
                         ⁢ 
                         
                             
                         
                         , 
                         N 
                       
                       ) 
                     
                   
                 
               
             
             
               
                 [ 
                 
                   Equation 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   11 
                 
                 ] 
               
             
           
         
       
     
   
   where v n =N(0, σ v   2 ), the observation noise. 
   The UNGM is a dynamic state space model having highly non-linear and bimodal characteristics, and shows poor estimation performance when there is inaccurate state estimation using the conventional extended Kalman filter or unscented Kalman filter, or when there is an excessive observation or non-Gaussian error using the particle filter complying with general SIS (Sequential Importance Sampling). 
   Further, the present invention obtains samples, which meet the non-linearity of the constraint manifold as well as the constraint manifold by a Monte Carlo method. 
   Embodiment 1 and Comparative Example 1  
     FIG. 8  illustrates average mean square error comparison of constraint manifold-based sensor fusion and filtering mode (embodiment 1), extended Kalman filtering mode (comparative example 1), unscented Kalman filtering mode (comparative example 1a), particle filtering mode (comparative example 1b), when simulation is performed 50 times under conditions of u n =N(0, σ u   2 =1), v n =N(0, σ v   2 =1), and N=100. It can be seen from the results of  FIG. 8  that the sensor fusion and filtering mode according to the present invention exhibited robustness to the bimodal system. 
   Embodiment 2 and Comparative Example 2  
     FIG. 9  illustrates average mean square error comparison of constraint manifold-based sensor fusion and filtering mode (embodiment 2), extended Kalman filtering mode (comparative example 2), unscented Kalman filtering mode (comparative example 2a), particle filtering mode (comparative example 2b), when simulation is performed 50 times under conditions of u n =0.5*N(0, σ u   2 =0.5)+0.5*N(0, σ u   2 =1.5), v n =N(0, σ v   2 =1), and N=100. It can be seen from the results of  FIG. 9  that the sensor fusion and filtering mode according to the present invention exhibited robustness to the non-Gaussian noise. 
   Embodiment 3 and Comparative Example 3  
     FIG. 10  illustrates average mean square error comparison of constraint manifold-based sensor fusion and filtering mode (embodiment 3) when simulation is performed 50 times under conditions of a wrong error model of u n =N(0, σ u   2 =10), v n =N(0, σ v   2 =1), and N=100, and extended Kalman filtering mode (comparative example 3), unscented Kalman filtering mode (comparative example 3a), particle filtering mode (comparative example 3b), when simulation is performed 50 times under conditions of u n =N(0, σ u   2 =1), v n =N(0, σ v   2 =1), and N=100. It can be seen from the results of  FIG. 10  that the sensor fusion and filtering mode according to the present invention exhibited robustness to the wrong error model. 
   Embodiment 4 and Comparative Example 4 
     FIG. 11  illustrates average mean square error comparison of constraint manifold-based sensor fusion and filtering mode (embodiment 4), extended Kalman filtering mode (comparative example 4), unscented Kalman filtering mode (comparative example 4a), particle filtering mode (comparative example 4b), when simulation is performed 50 times under conditions of u n =N(0, σ u   2 =100), v n =N(0, σ v   2 =1), and N=100. It can be seen from the results of  FIG. 11  that the sensor fusion and filtering mode according to the present invention exhibited excellent robustness to system noise. 
   Embodiment 5 and Comparative Example 5  
     FIG. 12  illustrates average mean square error comparison of constraint manifold-based sensor fusion and filtering mode (embodiment 5), extended Kalman filtering mode (comparative example 5), unscented Kalman filtering mode (comparative example 5a), particle filtering mode (comparative example 5b), when simulation is performed 50 times under conditions of u n =N(0, σ u   2 =100), v n =N(0, σ v   2 =1), and N=100, but with ambiguous information of p(y n )=0.5N(m y , v n )+0.5N(m y +ε, v n ), ε=0˜30 every ten times. It can be seen from the results of  FIG. 12  that the sensor fusion and filtering mode according to the present invention exhibited robustness to the ambiguous information. 
   Embodiment 6 and Comparative Example 6 
     FIG. 13  illustrates average mean square error comparison of constraint manifold-based sensor fusion and filtering mode (embodiment 6), extended Kalman filtering mode (comparative example 6), unscented Kalman filtering mode (comparative example 6a), particle filtering mode (comparative example 6b), when simulation is performed 50 times under conditions of u n =N(0, σ u   2 =100), v n =N(0, σ v   2 =1), and N=100, but with mismatched information of p(y n )=N(m y +ε, v n ), ε=30 every ten times. It can be seen from the results of  FIG. 13  that the sensor fusion and filtering mode according to the present invention exhibited robustness to the mismatched information. 
   Embodiment 7 and Comparative Example 7  
     FIG. 14  illustrates average mean square error comparison of constraint manifold-based sensor fusion and filtering modes (a) and (b) (embodiments 7 and 7a), and particle filtering modes (c) and (d) using 63 and 200 samples (comparative examples 7 and 7a), when simulation is performed 50 times on condition that an error model assumed for the filtering is u n =N(0, σ u   2 =10), u n =N(0, σ u   2 =1), v n =N(0, σ v   2 =100), and N=100, but that an actual system error model is u n =N(0, σ u   2 =,β), and N=100. It can be seen from the results of  FIG. 14  that the sensor fusion and filtering modes according to the present invention exhibited robustness to the wrong error model compared to the particle filtering modes. 
   It can be found from the above results that the sensor fusion and filtering method according to the present invention has the least average mean square error with respect to each evaluation condition, so that it is possible to realize the robust sensor fusion and filtering mode with respect to each evaluation condition. 
   As set forth above, according to the constraint manifold-based sensor fusion and filtering method and system of the present invention, a technique of finding the joint probability on the constraint manifold defined in the unified fusion and filtering space is applied. Thereby, application of various constraints is possible, and the geometrically expressed constraint is directly treated, so that it can be applied to various models. The system model and observation model are expressed in the fusion and filtering space without linearization using the constraint manifold as they are. Thereby, it is possible to reduce decrease of the fusion and filtering performance caused by wrong linearization, and decrease a Gaussian approximation error. Further, it is possible to detect mismatched information and to be applied to a fusion and filtering problem of the system having ambiguous information. 
   The present invention have been described in detail with reference to, but not limited to, exemplary embodiments thereof Those skilled in the art will appreciate that a variety of modifications can be made to the described embodiments without departing from the spirit and scope of the present invention as defined by the appended claims and their equivalents.