卡尔曼滤波简介与C——C++算法实现代码

卡尔曼滤波简介与算法实现代码

2007-01-13

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：

#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

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

{ 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__INCLUDE D_

#if _MSC_VER > 1000

#pragma once

#endif // _MSC_VER > 1000

#include

#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__I NCLUDED_)

============================kalman.cpp========================= =======

#include "kalman.h"

#include

/* 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;

}