卡尔曼滤波程序

1、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K。刀Mo等或的研究工作，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

2、,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)A= F(k,k-1) - X(k-1)1.2.计算预估计协方差矩阵C(k)A=F(k,k-1) x C(k) x F(k,k-1)+T(k,k-1) x Q(k) x T(k,k-1)Q(k) = U(k) x U(k)3.计算卡尔曼增益矩阵K(k) = C(k

3、)A x H(k) x H(k) x C(k)A x H(k)+R(k)A(-1)R(k) = N(k) x N(k)4.更新估计X(k)=X(k)A+K(k) x Y(k)-H(k) x X(k)A5.计算更新后估代协防差矩阵C(k) = I-K(k) x H(k) x C(k)A x I-K(k) x H(k)+K(k) x R(k) x K(k)6.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.cint lman(n,m,k,f,q,r,h,y,x,p,g)int n,m,k;doubl

4、e f,q,r,h,y,x,p,g;LJH ( int i,j,kk,ii,l,jj,js;double *e,*a,*b;e=malloc(m*m* sizeof(double);l=m;if (ln) 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+)十二( ii=i*l+j; aii=0.0;Ifor (kk=0; kk=n-1; kk+)I aii=aii+pi*n+kk*fj*n+kk;LI)for (i=0; i=n-1; i+)

5、for (j=0; j=n-1; j+)( ii=i*n+j; pii=qii;for (kk=0; kk=n-1; kk+)pii=pii+fi*n+kk*akk*l+j;for (ii=2; ii=k; ii+)( for (i=0; i=n-1; i+)I for (j=0; j=m-1; j+)十二( jj=i*l+j; ajj=0.0;Ifor (kk=0; kk=n-1; kk+)ajj=ajj+pi*n+kk*hj*n+kk;)for (i=0; i=m-1; i+)for (j=0; j=m-1; j+)( jj=i*m+j; ejj=rjj;for (kk=0; kk=n-1

6、; kk+)ejj=ejj+hi*n+kk*akk*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; gjj=0.0;for (kk=0; kk=m-1; kk+)gjj=gjj+ai*l+kk*ej*m+kk;)for (i=0; i=n-1; i+)( jj=(ii-1)*n+i; xjj=0.0;for (j=0; j=n-1; j+)xjj=xjj+fi*n+j*x(ii-2)*n+j;)for

7、 (i=0; i=m-1; i+) jj=i*l; bjj=y(ii-1)*m+i;for (j=0; j=n-1; j+)bjj=bjj-hi*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+)xjj=xjj+gi*m+j*bj*l;)if (iik) for (i=0; i=n-1; i+)for (j=0; j=n-1; j+) jj=i*l+j; ajj=0.0;for (kk=0; kk=m-1; kk+) ajj=ajj-gi*m+kk*hkk*n+j;if (i=j) ajj=1.0+a

8、jj;)for (i=0; i=n-1; i+)for (j=0; j=n-1; j+) jj=i*l+j; bjj=0.0;for (kk=0; kk=n-1; kk+)bjj=bjj+ai*l+kk*pkk*n+j;for (i=0; i=n-1; i+)for (j=0; j=n-1; j+) jj=i*l+j; ajj=0.0;for (kk=0; kk=n-1; kk+)ajj=ajj+bi*l+kk*fj*n+kk;:项:，直- .命-T.- S -圭-一圭.邕一朝EJ.SK一朝- HEEfor (i=0; i=n-1; i+)for (j=0; j=n-1; j+) jj=i*n

9、+j; pjj=qjj;for (kk=0; kk 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* m easurem ent;const CvMat* prediction;CvPoint2D32f get_predict( float x, float y);ka

10、lman( 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 /* tester de printer toutes les valeurs des vecteurs*/* tester de changer les matrices du noises */* replace state by cv

11、kalman- state_post ? */CvRandState rng;const double T = 0.1;kalman:kalman( int x, int xv, int y, int yv)(cvkalm an = cvCreateKalman( 4, 4, 0 );state = cvCreateMat( 4, 1, CV_32FC1 );process_noise =cvCreateMat( 4, 1, CV_32FC1 );m easurement =cvCreateMat( 4, 1, CV_32FC1 );int code = -1;/* create m atri

12、x 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) );con

13、st 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( m easurement );cvRandSetRange( &rng, 0, 0.1, 0 ); rng.disttype

14、 = CV_RAND_NORMAL;cvRand( &rng, state );mem cpy( cvkalman- transition_matrix- data.fl, A,sizeof (A);mem cpy( cvkalman-measurement_matrix- data.fl, H, sizeof (H);mem cpy( cvkalman-process_noise_cov-data.fl, Q,sizeof (Q);mem cpy( cvkalman- error_cov_post-data.fl, P,sizeof (P);mem cpy( cvkalman-mea

15、surement_noise_cov-data.fl, R, sizeof (R);/cvSetIdentity( cvkalman- process_noise_cov, cvRealScalar(1e-5);/cvSetIdentity( cvkalman- error_cov_post, cvRealScalar(1);/cvSetIdentity( cvkalm an- measurement_noise_cov, cvRealScalar(1e-1);/* choose initial state */state-data.fl0=x;state-data.fl1=xv;state-

16、data.fl2= y;state-data.fl3= yv;cvkalman- state_post-data.fl0=x;cvkalman- state_post-data.fl1=xv;cvkalman- state_post-data.fl2=y;cvkalman- state_post-data.fl3=yv;cvRandSetRange( &rng, 0, sqrt(cvkalman-process_noise_cov-data.fl0), 0 );cvRand( &rng, process_noise ); CvPoint2D32f kalman:get_pred

17、ict(float x, float y)/* update state with current position */state- data.fl0=x;state- data.fl2= y;/* predict point position */* xk=A钦 钦Pk=A #-1*AT + Q */cvRandSetRange( &rng, 0, sqrt(cvkalman-measurement_noise_cov- data.fl0), 0 );cvRand( &rng, m easurement );/* xk=A?xk-1+B?uk+wk */cvMatMulAd

18、d( cvkalman- transition_matrix, state, process_noise, cvkalman- state_post );/* zk=H?xk+vk */cvMatMulAdd( cvkalman- measurement_matrix, cvkalman- urement, m easurement );/* adjust Kalman filter state */* Kk=Pk钦 钦?H钦 钦+R) -1xk=xk+Kk # ?zk-H #Pk= (I- Kk #*/cvKalmanCorrect( cvkalman, measurement );float m easured_value_x = measurem ent- data.fl0;float m easured_value_y = measurem ent- data.fl2;const CvMat* prediction = cvKalmanPredict( cvkalman, 0 );float predict_value_x =prediction- data.fl0;float predict_value_y =prediction-data.fl2;return (cvPoint2D32f(predict