卡尔曼滤波简介+ 算法实现代码

    技术2022-05-11  20

    最佳线性滤波理论起源于40年代美国科学家Wiener和前苏联科学家Kолмогоров等人的研究工作,后人统称为维纳滤波理论。从理论上说,维纳滤波的最大缺点是必须用到无限过去的数据,不适用于实时处理。为了克服这一缺点,60年代Kalman把状态空间模型引入滤波理论,并导出了一套递推估计算法,后人称之为卡尔曼滤波理论。卡尔曼滤波是以最小均方误差为估计的最佳准则,来寻求一套递推估计的算法,其基本思想是:采用信号与噪声的状态空间模型,利用前一时刻地估计值和现时刻的观测值来更新对状态变量的估计,求出现时刻的估计值。它适合于实时处理和计算机运算。

    现设线性时变系统的离散状态防城和观测方程为:

    X(k) = F(k,k-1)·X(k-1)+T(k,k-1)·U(k-1)

    Y(k) = H(k)·X(k)+N(k)

    其中

    X(k)和Y(k)分别是k时刻的状态矢量和观测矢量

    F(k,k-1)为状态转移矩阵

    U(k)为k时刻动态噪声

    T(k,k-1)为系统控制矩阵

    H(k)为k时刻观测矩阵

    N(k)为k时刻观测噪声

    则卡尔曼滤波的算法流程为:

    预估计X(k)^= F(k,k-1)·X(k-1) 计算预估计协方差矩阵C(k)^=F(k,k-1)×C(k)×F(k,k-1)'+T(k,k-1)×Q(k)×T(k,k-1)'Q(k) = U(k)×U(k)' 计算卡尔曼增益矩阵K(k) = C(k)^×H(k)'×[H(k)×C(k)^×H(k)'+R(k)]^(-1)R(k) = N(k)×N(k)' 更新估计X(k)~=X(k)^+K(k)×[Y(k)-H(k)×X(k)^] 计算更新后估计协防差矩阵C(k)~ = [I-K(k)×H(k)]×C(k)^×[I-K(k)×H(k)]'+K(k)×R(k)×K(k)' X(k+1) = X(k)~C(k+1) = C(k)~重复以上步骤

    其c语言实现代码如下:C++实现代码如下:

    ============================ kalman.h ================================ //  kalman.h: interface for the kalman class. // / / #if  !defined(AFX_KALMAN_H__ED3D740F_01D2_4616_8B74_8BF57636F2C0__INCLUDED_) #define  AFX_KALMAN_H__ED3D740F_01D2_4616_8B74_8BF57636F2C0__INCLUDED_ #if  _MSC_VER > 1000 #pragma once #endif   //  _MSC_VER > 1000 #include  < math.h > #include  " cv.h "   class  kalman  { public void  init_kalman( int  x, int  xv, int  y, int  yv); CvKalman *  cvkalman; CvMat *  state;  CvMat *  process_noise; CvMat *  measurement;  const  CvMat *  prediction; CvPoint2D32f get_predict( float  x,  float  y); kalman( int  x = 0 , int  xv = 0 , int  y = 0 , int  yv = 0 );  // virtual ~kalman(); }; #endif   //  !defined(AFX_KALMAN_H__ED3D740F_01D2_4616_8B74_8BF57636F2C0__INCLUDED_) ============================ kalman.cpp ================================ #include  " kalman.h " #include  < stdio.h > /*  tester de printer toutes les valeurs des vecteurs */ /*  tester de changer les matrices du noises  */ /*  replace state by cvkalman->state_post ???  */ CvRandState rng; const   double  T  =   0.1 ;kalman::kalman( int  x, int  xv, int  y, int  yv){         cvkalman  =  cvCreateKalman(  4 4 0  );    state  =  cvCreateMat(  4 1 , CV_32FC1 );    process_noise  =  cvCreateMat(  4 1 , CV_32FC1 );    measurement  =  cvCreateMat(  4 1 , CV_32FC1 );     int  code  =   - 1 ;         /*  create matrix data  */       const   float  A[]  =  {     1 , T,  0 0 ,    0 1 0 0 ,    0 0 1 , T,    0 0 0 1   };           const   float  H[]  =  {      1 0 0 0 ,     0 0 0 0 ,    0 0 1 0 ,    0 0 0 0   };             const   float  P[]  =  {    pow( 320 , 2 ), pow( 320 , 2 ) / T,  0 0 ,   pow( 320 , 2 ) / T, pow( 320 , 2 ) / pow(T, 2 ),  0 0 ,    0 0 , pow( 240 , 2 ), pow( 240 , 2 ) / T,    0 0 , pow( 240 , 2 ) / T, pow( 240 , 2 ) / pow(T, 2 )    };      const   float  Q[]  =  {   pow(T, 3 ) / 3 , pow(T, 2 ) / 2 0 0 ,   pow(T, 2 ) / 2 , T,  0 0 ,    0 0 , pow(T, 3 ) / 3 , pow(T, 2 ) / 2 ,    0 0 , pow(T, 2 ) / 2 , T   };         const   float  R[]  =  {    1 0 0 0 ,    0 0 0 0 ,    0 0 1 0 ,    0 0 0 0    };           cvRandInit(  & rng,  0 1 - 1 , CV_RAND_UNI );    cvZero( measurement );        cvRandSetRange(  & rng,  0 0.1 0  );    rng.disttype  =  CV_RAND_NORMAL;    cvRand(  & rng, state );    memcpy( cvkalman -> transition_matrix -> data.fl, A,  sizeof (A));    memcpy( cvkalman -> measurement_matrix -> data.fl, H,  sizeof (H));    memcpy( cvkalman -> process_noise_cov -> data.fl, Q,  sizeof (Q));    memcpy( cvkalman -> error_cov_post -> data.fl, P,  sizeof (P));    memcpy( cvkalman -> measurement_noise_cov -> data.fl, R,  sizeof (R));     // cvSetIdentity( cvkalman->process_noise_cov, cvRealScalar(1e-5) );         // cvSetIdentity( cvkalman->error_cov_post, cvRealScalar(1));  // cvSetIdentity( cvkalman->measurement_noise_cov, cvRealScalar(1e-1) );      /*  choose initial state  */     state -> data.fl[ 0 ] = x;    state -> data.fl[ 1 ] = xv;    state -> data.fl[ 2 ] = y;    state -> data.fl[ 3 ] = yv;    cvkalman -> state_post -> data.fl[ 0 ] = x;    cvkalman -> state_post -> data.fl[ 1 ] = xv;    cvkalman -> state_post -> data.fl[ 2 ] = y;    cvkalman -> state_post -> data.fl[ 3 ] = yv; cvRandSetRange(  & rng,  0 , sqrt(cvkalman -> process_noise_cov -> data.fl[ 0 ]),  0  );    cvRand(  & rng, process_noise );    }     CvPoint2D32f kalman::get_predict( float  x,  float  y){         /*  update state with current position  */     state -> data.fl[ 0 ] = x;    state -> data.fl[ 2 ] = y;         /*  predict point position  */      /*  x'k=A鈥k+B鈥k       P'k=A鈥k-1*AT + Q  */     cvRandSetRange(  & rng,  0 , sqrt(cvkalman -> measurement_noise_cov -> data.fl[ 0 ]),  0  );    cvRand(  & rng, measurement );          /*  xk=A?xk-1+B?uk+wk  */     cvMatMulAdd( cvkalman -> transition_matrix, state, process_noise, cvkalman -> state_post );         /*  zk=H?xk+vk  */     cvMatMulAdd( cvkalman -> measurement_matrix, cvkalman -> state_post, measurement, measurement );         /*  adjust Kalman filter state  */      /*  Kk=P'k鈥T鈥?H鈥'k鈥T+R)-1       xk=x'k+Kk鈥?zk-H鈥'k)       Pk=(I-Kk鈥)鈥'k  */     cvKalmanCorrect( cvkalman, measurement );     float  measured_value_x  =  measurement -> data.fl[ 0 ];     float  measured_value_y  =  measurement -> data.fl[ 2 ];      const  CvMat *  prediction  =  cvKalmanPredict( cvkalman,  0  );     float  predict_value_x  =  prediction -> data.fl[ 0 ];     float  predict_value_y  =  prediction -> data.fl[ 2 ];     return (cvPoint2D32f(predict_value_x,predict_value_y));} void  kalman::init_kalman( int  x, int  xv, int  y, int  yv){ state -> data.fl[ 0 ] = x;    state -> data.fl[ 1 ] = xv;    state -> data.fl[ 2 ] = y;    state -> data.fl[ 3 ] = yv;    cvkalman -> state_post -> data.fl[ 0 ] = x;    cvkalman -> state_post -> data.fl[ 1 ] = xv;    cvkalman -> state_post -> data.fl[ 2 ] = y;    cvkalman -> state_post -> data.fl[ 3 ] = yv;}  

    #include "stdlib.h"   #include  "rinv.c"    int  lman(n,m,k,f,q,r,h,y,x,p,g)  int  n,m,k;  double  f[],q[],r[],h[],y[],x[],p[],g[];   int i,j,kk,ii,l,jj,js;    double *e,*a,*b;    e=malloc(m*m*sizeof(double));    l=m;    if (l<n) l=n;    a=malloc(l*l*sizeof(double));    b=malloc(l*l*sizeof(double));    for (i=0; i<=n-1; i++)      for (j=0; j<=n-1; j++)        { ii=i*l+j; a[ii]=0.0;          for (kk=0; kk<=n-1; kk++)            a[ii]=a[ii]+p[i*n+kk]*f[j*n+kk];        }    for (i=0; i<=n-1; i++)      for (j=0; j<=n-1; j++)        { ii=i*n+j; p[ii]=q[ii];          for (kk=0; kk<=n-1; kk++)            p[ii]=p[ii]+f[i*n+kk]*a[kk*l+j];        }    for (ii=2; ii<=k; ii++)      for (i=0; i<=n-1; i++)        for (j=0; j<=m-1; j++)          { jj=i*l+j; a[jj]=0.0;            for (kk=0; kk<=n-1; kk++)              a[jj]=a[jj]+p[i*n+kk]*h[j*n+kk];          }        for (i=0; i<=m-1; i++)        for (j=0; j<=m-1; j++)          { jj=i*m+j; e[jj]=r[jj];            for (kk=0; kk<=n-1; kk++)              e[jj]=e[jj]+h[i*n+kk]*a[kk*l+j];          }        js=rinv(e,m);        if (js==0)           { free(e); free(a); free(b); return(js);}        for (i=0; i<=n-1; i++)        for (j=0; j<=m-1; j++)          { jj=i*m+j; g[jj]=0.0;            for (kk=0; kk<=m-1; kk++)              g[jj]=g[jj]+a[i*l+kk]*e[j*m+kk];          }        for (i=0; i<=n-1; i++)          { jj=(ii-1)*n+i; x[jj]=0.0;            for (j=0; j<=n-1; j++)              x[jj]=x[jj]+f[i*n+j]*x[(ii-2)*n+j];          }        for (i=0; i<=m-1; i++)          { jj=i*l; b[jj]=y[(ii-1)*m+i];            for (j=0; j<=n-1; j++)              b[jj]=b[jj]-h[i*n+j]*x[(ii-1)*n+j];          }        for (i=0; i<=n-1; i++)          { jj=(ii-1)*n+i;            for (j=0; j<=m-1; j++)              x[jj]=x[jj]+g[i*m+j]*b[j*l];          }        if (ii<k)          for (i=0; i<=n-1; i++)            for (j=0; j<=n-1; j++)              { jj=i*l+j; a[jj]=0.0;                for (kk=0; kk<=m-1; kk++)                  a[jj]=a[jj]-g[i*m+kk]*h[kk*n+j];                if (i==j) a[jj]=1.0+a[jj];              }            for (i=0; i<=n-1; i++)            for (j=0; j<=n-1; j++)              { jj=i*l+j; b[jj]=0.0;                for (kk=0; kk<=n-1; kk++)                  b[jj]=b[jj]+a[i*l+kk]*p[kk*n+j];              }            for (i=0; i<=n-1; i++)            for (j=0; j<=n-1; j++)              { jj=i*l+j; a[jj]=0.0;                for (kk=0; kk<=n-1; kk++)                  a[jj]=a[jj]+b[i*l+kk]*f[j*n+kk];              }            for (i=0; i<=n-1; i++)            for (j=0; j<=n-1; j++)              { jj=i*n+j; p[jj]=q[jj];                for (kk=0; kk<=n-1; kk++)                  p[jj]=p[jj]+f[i*n+kk]*a[j*l+kk];              }          }      }    free(e); free(a); free(b);    return(js);  }

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