# 卡尔曼滤波简介＋ 算法实现代码（转）

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

Y(k) = H(k)·X(k)+N(k)

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

1. 预估计X(k)^= F(k,k-1)·X(k-1)
2. 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)'
3. 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)'
4. 更新估计
X(k)~=X(k)^+K(k)×[Y(k)-H(k)×X(k)^]
5. 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)'
6. X(k+1) = X(k)~
C(k+1) = C(k)~
重复以上步骤

#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);  }
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 */
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 */
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;
}

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