UKF-GPS-IMU-MATLAB

组合导航系统的计算程序代码 function yy=ukf_IMUgps() %function ukf_IMUgps() % UKF在IMU/GPS组合导航系统中应用 % % 以IMU中的位置、速度、姿态误差角、陀螺漂移常值为状态量； % 以GPS中的位置、速度为观测量。 % % 7,July 2008. clc % Initialise state global we RN RM g fl deta wg Tt wt d ww v u W Rbl Ta wa d=0; %验证循环次数 %地球自转角速度we(rad/s): we=7.292115e-5; g=9.81; %地球重力加速度（m/s^2） a=6.378137e+6; %地球长半轴 e2=0.0066944; %地球第一偏心率的平方 %姿态角初始值（r,p,y） zitai=(pi/180).*[0 2.0282 196.9087]; %姿态误差角 fai=(pi/180).*[1/36 1/36 5/36]; %(100'',100'',500'') r=zitai(1)+fai(1); p=zitai(2)+fai(2); y=zitai(3)+fai(3); %当地坐标系（l）相对于载体坐标系（b）的转换矩阵：Rbl(在e,n,u坐标系下） Rbl=[cos(r)*cos(y)-sin(r)*sin(y)*sin(p) -sin(y)*cos(p) cos(y)*sin(r)+sin(y)*sin(p)*cos(r) cos(r)*sin(y)+sin(r)*cos(y)*sin(p) cos(y)*cos(p) sin(y)*sin(r)-cos(y)*sin(p)*cos(r) -cos(p)*sin(r) sin(p) cos(p)*cos(r)]; %由转换矩阵Rbl计算四元数的初始值Q0=[q1,q2,q3,q4]' QQ=[0.5*sqrt(1+Rbl(1,1)+Rbl(2,2)+Rbl(3,3)) 0.25*(Rbl(3,2)-Rbl(2,3))/(0.5*sqrt(1+Rbl(1,1)+Rbl(2,2)+Rbl(3,3))) 0.25*(Rbl(1,3)-Rbl(3,1))/(0.5*sqrt(1+Rbl(1,1)+Rbl(2,2)+Rbl(3,3))) 0.25*(Rbl(2,1)-Rbl(1,2))/(0.5*sqrt(1+Rbl(1,1)+Rbl(2,2)+Rbl(3,3)))]; %IMU系统给出的初始值：速度（ve,vn,vu），位置（l,m,h）＝（经度，纬度,高度），姿态误差角fai(e,n,u) vIMU=[-21.775011 -71.631027 3.10094]; point_IMU=[0.147022986 0.81837365 3122.826]; T=1; %UKF滤波的采样周期（s） %陀螺常值漂移初始值tuo(e,n,u)(单位：。/s） tuo=[0 0 0]; %陀螺一阶相关时间Tt(s) Tt=[60 60 60]; %加速度计常值漂移初始值jiasu(e,n,u)(单位：m/s^2） jiasu=[0 0 0]; %加速度计一阶反相关时间Ta(s) Ta=[60 60 60]; %IMU系统的状态向量X x=[vIMU point_IMU fai tuo jiasu]'; %观测量噪声V的斜方差矩阵 R=[]; %GPS系统的量测值Z(vn,ve,vd,m,l,h) [z Rz]=gps_tiqushuju; %Vk的噪声序列方差为：Rk Rz=(Rz./T); %陀螺常值漂移wt(e,n,u) wt=(pi/180).*[1/(3600) 1/(3600) 1/(3600)]; % 1 (。/h) %陀螺常值漂移误差wtt(e,n,u) wtt=(pi/180).*[0.08/(3600) 0.08/(3600) 0.08/(3600)]; % 0.08 (。/h) %加速度计常值漂移wa(e,n,u) wa=[2e-6 2e-6 2e-6]; % 200μg (2e-6 m/s^2) %加速度计常值漂移误差waa(e,n,u) waa=[2e-6 2e-6 2e-6]./4; % 50μg (0.5e-6 m/s^2) %姿态误差角噪声wg wg=wt; %状态向量X的斜方差矩阵 P = eye(length(x))*eps; % note: for stability, P should never be quite zero %速度误差：（0.1m/s) 位置误差：水平（1m),高度（5m) u=[0.1 0.1 0.1 0.000474429 0.000474429 2 wg wtt waa]'; %IMU系统在载体坐标系下的比力值输出值fb fb=[]; %IMU系统中陀螺输出量 wib=[]; %为载体坐标系（b）相对于惯性坐标系（i）的角速度向量 [f w]=IMU_tiqushuju; %IMU系统输出的比力值和角速度 %%%%%%%%%通过GPS观测值计算得到的姿态角 zitaijiao_gps=xlsread('D:\myfile\UKF\kalmanfilter_MATLAB\germany_ukf\原始数据\zitiajiao-gps.xls'); %%%------------------------------------------------------------ %% 循环开始 ：for 1:n outx=[]; outp=[]; detajiao=zeros(3,1); NN =1000; for i=1:NN outx=[outx;x']; outp=[outp;diag(P)']; %大地子午圈曲率半径：RM RM=a*(1-e2)/(1-e2*(sin(x(5)))^2)^(2/3); %地球卯酉圈的曲率半径：RN RN=a/sqrt(1-e2*(sin(x(5)))^2); %当地坐标系下的比力值fl %IMU系统在载体坐标系下的比力值输出值fb转换到当地坐标系（l）下fl fb=f(i,:)'; %fl=Rbl*(fb+[x(13) x(14) x(15)]'); %初始对准，比力值加上加速度计的测量偏差 fl=Rbl*fb; %计算IMU的速度、位置输出减去GPS相应的输出值：deta(ve,vn,l,h) zd=z(i,:)'; deta=x([1 2 5 6],:)-zd([1 2 5 6],:); %观测值zz,及观测噪声R zz=z(i+1,:)'; v=Rz(i+1,:)'; %zitai_v=[0.001 0.0266 0.9664]'; %GPS观测值姿态角的误差 zitai_v=[0.00001 7.0983e-004 0.0006]'; %GPS观测值姿态角的误差 v=[v;zitai_v]; R=diag(v.^2); %%通过GPS观测值，计算得到roll=0 ，pitch=atan(ve/vn)，yaw=asin(h12/s12) ，将姿态误差角加入观测量中进行滤波计算 zz=[zz;detajiao]; %%%%GPS计算得到的姿态角 z_zitai=zitaijiao_gps(i+1,:); %%IMU系统的力学编排方程和姿态误差方程离散化之后的截断误差： ve=x(1);vn=x(2);vu=x(3);l=x(4);m=x(5);h=x(6);faie=x(7);fain=x(8);faiu=x(9);tuo1=x(10);tuo2=x(11);tuo3=x(12); fl1=fl(1);fl2=fl(2);fl3=fl(3);deta1=deta(1); deta2=deta(2); deta3=deta(3); deta4=deta(4);wg1=wg(1);wg2=wg(2);wg3=wg(3); jiasu1=x(13);jiasu2=x(14);jiasu3=x(15);Ta1=Ta(1);Ta2=Ta(2);Ta3=Ta(3);wa1=wa(1);wt2=wa(2);wt3=wa(3); Q=diag((u).^2); % predict [x,P] = predict(x, P,u, Q, T); % update [x,P] = update(x, P, zz, R); %% xx=x; x=xx(1:15,1); u=xx(16:30,1); %u(1)=u(1)+x(13); %u(2)=u(2)+x(14); %u(3)=u(3)+x(15); %u(7)=x(10); %u(8)=x(11); %u(9)=x(12); PP=P; P=PP(1:15,1:15); %利用四元数Q计算转换矩阵Rbl %首先计算在当地坐标系（l）下的角速度向量wbl %地固坐标系（e）相对于当地坐标系（l）的转换矩阵：Rel=Rle' %Rel=[-sin(x(4)) cos(x(4)) 0 % -sin(x(5))*cos(x(4)) -sin(x(5))*sin(x(4)) cos(x(5)) % cos(x(5))*cos(x(4)) cos(x(5))*sin(x(4)) sin(x(5))]; wie=[0 0 we]'; %wie 为地球自转角速度向量 %%%%%%% omiga12=[]; for k=0:1 wib=w(i+k,:)'+T.*[x(10) x(11) x(12)]'; %初始对准，角速度加上陀螺漂移 wiel=[0 we*cos(x(5)) we*sin(x(5))]'; %wiel=Rel*wie; well=[-x(2)/(RM+x(6)) x(1)/(RN+x(6)) x(1)*tan(x(5))/(RN+x(6))]'; will=wiel+well; wlb=wib-Rbl'*will; %四元数的更新 %计算反对称矩阵omiga omiga=[0 wlb(3) -wlb(2) wlb(1) -wlb(3) 0 wlb(1) wlb(2) wlb(2) -wlb(1) 0 wlb(3) wlb(1) wlb(2) wlb(3) 0]; omiga12=[omiga12,omiga]; end omiga1=omiga12(1:4,1:4); omiga2=omiga12(1:4,5:8); %采用四阶龙格－库塔法数值积分解Q(K+1) QQ=QQ+(0.5*T.*(omiga1+omiga2)+8^(-1)*T^2.*(omiga1^2+omiga1*omiga2+omiga2^2)+48^(-1)*T^3.*(omiga1^2*omiga2+omiga1*omiga2^2)+384^(-1)*T^4.*(omiga1^2*omiga2^2))*QQ; %由Q(k+1)可得Rbl(k+1) Rbl=[QQ(1)^2+QQ(2)^2-QQ(3)^2-QQ(4)^2 2*(QQ(2)*QQ(3)-QQ(1)*QQ(4)) 2*(QQ(2)*QQ(4)+QQ(1)*QQ(3)) 2*(QQ(2)*QQ(3)+QQ(1)*QQ(4)) QQ(1)^2-QQ(2)^2+QQ(3)^2-QQ(4)^2 2*(QQ(3)*QQ(4)-QQ(1)*QQ(2)) 2*(QQ(2)*QQ(4)-QQ(1)*QQ(3)) 2*(QQ(2)*QQ(1)+QQ(4)*QQ(3)) QQ(1)^2-QQ(2)^2-QQ(3)^2+QQ(4)^2]; %由Rbl(k＋1）可得姿态角，（翻滚角r，俯仰角p，航向角y）k+1 %实际导航系（p系）相对于理想导航系（n系）存在误差角fai(e,n,u),Cpn为校正矩阵 %当方位角为大失准角时 Cpn=[cos(x(9)) -sin(x(9)) x(8)*cos(x(9))+x(7)*sin(x(9)) sin(x(9)) cos(x(9)) x(8)*sin(x(9))-x(7)*cos(x(9)) -x(8) x(7) 1]; %%当三个方向的失准角为小角度时 Cpnn=[1 -x(9) x(8) x(9) 1 -x(7) -x(8) x(7) 1]; if abs(x(9))*180*60/pi < 10 %当姿态误差角（u），即方位角失准角小于10’时的情况； Rbl=Cpnn*Rbl; %小失准角 else Rbl=Cpn*Rbl; %大失准角 end zitai0=[atan(-Rbl(3,1)/Rbl(3,3)) asin(Rbl(3,2)) atan(-Rbl(1,2)/Rbl(2,2))]; if Rbl(2,2)<0 & zitai0(3)<0 zitai0(3)=zitai0(3)+pi; end if Rbl(2,2)<0 & zitai0(3)>0 zitai0(3)=zitai0(3)-pi; end % detajiao=zitai0-z_zitai'; % detajiao(1)=0; % detajiao=Rbl*detajiao; % if abs(detajiao(2)) < abs(x(8)) % x(8)=detajiao(2); % %zitai0(2)=z_zitai(2); % end %if abs(detajiao(3)) < abs(x(9)) % x(9)=detajiao(3); %zitai0(3)=z_zitai(3); %end zitai=[zitai;zitai0']; end %将协方差转换成 标准 excel标准偏差excel标准偏差函数exl标准差函数国标检验抽样标准表免费下载红头文件格式标准下载 差 outp=(outp).^(1/2); %将位置误差中的经度和纬度形式转换成当地坐标系（e,n,u)形式 for k=1:NN %大地子午圈曲率半径：RM RM=a*(1-e2)/(1-e2*(sin(outx(k,5)))^2)^(2/3); %地球卯酉圈的曲率半径：RN RN=a/sqrt(1-e2*(sin(outx(k,5)))^2); outp(k,4)=(RN+outx(k,6))*cos(outx(k,5))*outp(k,4); outp(k,5)=(RM+outx(k,6))*outp(k,5); end %绘制高度图 figure plot(outx(:,6),'r-.'); title('UKF计算的高度(germany-data)'); figure plot((180/pi)*outx(:,8),'r-'); title('UKF姿态误差角pitch（度）(germany-data)'); % 绘制图：水平运动轨迹 figure plot(outx(:,4),outx(:,5),'b-'); title('UKF Level of Movement(Germany-data)'); xlabel('Longitude（rad）'); ylabel('Latitude（rad）'); %hold on %gpsweizhi=xlsread('原始数据\sudu-weizhi-gps.xls'); %plot(gpsweizhi(:,4),gpsweizhi(:,5),'r-'); %hold off %绘制图；UKF速度误差 figure subplot(3,1,1) plot(outp(:,1),'b-'); title('UKF Velocity Error(Germany-data)'); ylabel('Ve(m/s)'); subplot(3,1,2) plot(outp(:,2),'b-'); ylabel('Vn(m/s)'); subplot(3,1,3) plot(outp(:,3),'b-'); xlabel('t(s）'); ylabel('Vu(m/s)'); %绘制图；UKF位置误差 figure subplot(3,1,1) plot(100.*outp(:,4),'b-'); title('UKF Location Error(Germany-data)'); ylabel('x(cm)'); subplot(3,1,2) plot(100.*outp(:,5),'b-'); ylabel('y(cm)'); subplot(3,1,3) plot(100.*outp(:,6),'b-'); xlabel('t(s）'); ylabel('z(cm)'); %绘制图；UKF姿态角误差 figure subplot(3,1,1) plot(3600*(180/pi)*outp(:,7),'b-'); title('UKF Attitude Error(Germany-data)'); ylabel('roll(second)'); subplot(3,1,2) plot(3600*(180/pi)*outp(:,8),'b-'); ylabel('pitch(second)'); subplot(3,1,3) plot(3600*(180/pi)*outp(:,9),'b-'); xlabel('t(s）'); ylabel('yaw(second)'); xlswrite('计算结果\output_x.xls',outx); xlswrite('计算结果\output_p.xls',outp); xlswrite('计算结果\output_zitai.xls',zitai); % % function [x,P] = predict(x, P,u, Q, T) %进行无迹变换 P = blkdiag(P,Q); x=[x;u]; % Perform unscented transform [x,P] = unscented_transform(@predict_model, [], x, P, T); % notice how the additional parameter 'T' is passed to the model %%%function [y, Y] = unscented_transform(func, dfunc, x,P,varargin):注意其中'dfunc'不知道？ P(1:15,1:15)=P(1:15,1:15)+Q; % % function [x,P] = update(x, P, z, R) [x,P] = unscented_update(@observe_model,@observe_residual, x, P, z, R); % function x = predict_model(x, T) global wt Tt Ta wa Rbl %计算预报值 % 由于UKF使用的是离散时间非线性模型， % 因此需要对IMU/GPS组合系统模型进行离散化处理； % 这里采用四阶Runge-kuta法以数值积分的形式实现。 % Y(k+1)=Y(k)+(y1+2*y2+2*y3+y4)/6 % 其中：y1=f(Y(k))*T; y2=f(Y(k)+y1/2)*T; y3=f(Y(k)+y2/2)*T; y4=f(Y(k)+y3)*T; [n,N]=size(x); y=[]; for k=1:N xx=x(:,k); %陀螺漂移常值方程 xx(10)=xx(10)*exp(-1/Tt(1))+wt(1); xx(11)=xx(11)*exp(-1/Tt(2))+wt(2); xx(12)=xx(12)*exp(-1/Tt(3))+wt(3); %加速度计漂移常值方程 xx(13)=xx(13)*exp(-1/Ta(1))+wa(1); xx(14)=xx(14)*exp(-1/Ta(2))+wa(2); xx(15)=xx(15)*exp(-1/Ta(3))+wa(3); x1=detaf(xx,T); % T为采样周期 x2=detaf(xx+x1./2,T); x3=detaf(xx+x2./2,T); x4=detaf(xx+x3,T); xx=xx+(x1+2.*x2+2.*x3+x4)./6+[xx(16:30,1);zeros(15,1)]; y=[y,xx]; %由K状态到K＋1状态的预估计 end x=y; %%% function dta=detaf(x,T) global we RN RM fl g deta wg Rbl wt Tt %IMU系统的力学编排方程 zhongli=9.7803267714*(1+0.00527094*(sin(x(5)))^2+0.0000232718*(sin(x(5)))^4)-(0.3086*10^(-5))*x(6); dve=(2*we*sin(x(5))+x(1)*tan(x(5))/(RN+x(6)))*x(2)-(2*we*cos(x(5))+x(1)/(RN+x(6)))*x(3)+fl(1)+fl(2)*x(9)-fl(3)*x(8)+[Rbl(1,1) Rbl(1,2) Rbl(1,3)]*[x(13) x(14) x(15)]'; dvn=-(2*we*sin(x(5))+x(1)*tan(x(5))/(RN+x(6)))*x(1)-x(2)*x(3)/(RM+x(6))+fl(2)+fl(1)*x(9)-fl(3)*x(7)+[Rbl(2,1) Rbl(2,2) Rbl(2,3)]*[x(13) x(14) x(15)]'; dvu=(2*we*cos(x(5))+x(1)/(RN+x(6)))*x(1)+(x(2))^2/(RM+x(6))-zhongli+fl(3)+fl(2)*x(7)-fl(1)*x(8)+[Rbl(3,1) Rbl(3,2) Rbl(3,3)]*[x(13) x(14) x(15)]'; dl=x(1)/(cos(x(5))*(RN+x(6))); dm=x(2)/(RM+x(6)); dh=x(3); %IMU系统的姿态误差方程 %dfaie=(we*sin(x(5))+x(1)*tan(x(5))/(RN+x(6)))*x(8)-(we*cos(x(5))+x(1)/(RN+x(6)))*sin(x(9))-x(2)*(1-cos(x(9)))/(RM+x(6))-deta(2)/(RM+x(6))+x(2)*deta(4)/(RM+x(6))^2+[Rbl(1,1) Rbl(1,2) Rbl(1,3)]*[x(10) x(11) x(12)]'; %dfain=-(we*sin(x(5))+x(1)*tan(x(5))/(RN+x(6)))*x(7)-x(2)*sin(x(9))/(RM+x(6))+(we*cos(x(5))+x(1)/(RN+x(6)))*(1-cos(x(9)))+deta(1)/(RN+x(6))-x(1)*deta(4)/(RN+x(6))^2-we*sin(x(5))*deta(3)+[Rbl(2,1) Rbl(2,2) Rbl(2,3)]*[x(10) x(11) x(12)]'; %dfaiu=-(we*cos(x(5))+x(1)/(RN+x(6)))*(x(8)*sin(x(9))-x(7)*cos(x(9)))+x(2)*(x(8)*cos(x(9))+x(7)*sin(x(9)))/(RM+x(6))+deta(1)*tan(x(5))/(RN+x(6))-x(1)*deta(4)*tan(x(5))/(RM+x(6))^2+(we*cos(x(5))+x(1)*(sec(x(5)))^2/(RN+x(6)))*deta(3)+[Rbl(3,1) Rbl(3,2) Rbl(3,3)]*[x(10) x(11) x(12)]'; dfaie=(we*sin(x(5))+x(1)*tan(x(5))/(RN+x(6)))*x(8)-(we*cos(x(5))+x(1)/(RN+x(6)))*x(9)+deta(2)/(RM+x(6))-x(2)*deta(4)/(RM+x(6))^2+x(10); dfain=-(we*sin(x(5))+x(1)*tan(x(5))/(RN+x(6)))*x(7)-x(2)*x(9)/(RM+x(6))-deta(1)/(RN+x(6))+x(1)*deta(4)/(RN+x(6))^2+we*sin(x(5))*deta(3)+x(11); dfaiu=(we*cos(x(5))+x(1)/(RN+x(6)))*x(7)+x(2)*x(8)/(RM+x(6))-deta(1)*tan(x(5))/(RN+x(6))+x(1)*deta(4)*tan(x(5))/(RM+x(6))^2-(we*cos(x(5))+x(1)*(sec(x(5)))^2/(RN+x(6)))*deta(3)+x(12); dta=T.*[dve,dvn,dvu,dl,dm,dh,dfaie,dfain,dfaiu,0,0,0,0,0,0]'; % T为采样周期 dta=[dta;zeros(15,1)]; % % function z = observe_model(x) % 观测方程：z=Hx+v,其中H为单位矩阵 global v [m,M]=size(x); h=[eye(9),zeros(9,6),zeros(9,15)]; z = h*x+repvec(v,M); % function dz = observe_residual(z1, z2) % Given two observation values, compute their normalised residual. % Notice, once again, that the model is vectorised in terms of z1 and z2. dz = z1-z2; function x = repvec(x,N) x = x(:, ones(1,N)); % % UKF的程序代码 function [y, Y] = unscented_transform(func, dfunc, x, P, varargin) %function [y, Y] = unscented_transform(func, dfunc, x, P, ...) % % Algorithm implemented as described in: % A New Method for the Nonlinear Transformation of Means and Covariances in Filters and Estimators % 3, APIRL 2008 % % INPUTS: % func - function handle (@myfunc) or string ('myfunc') for non-linear transform. % dfunc - function handle (or string) for residual between two transformed values: e = mydfunc(y1, y2). % x, P - initial mean and covariance. % ... - optional arguments such that 'func' has form: y = myfunc(x, ...). % % OUTPUTS: % y, Y - transformed mean and covariance. % % NOTES: % 1. The unscented filter has two primary advantages over the EKF: (i) it produces a more accurate % estimate of the first and second moments of a random vector transformed through a non-linear % function, and (ii) it does not require analytical Jacobians. % % 2. The function 'func' is the non-linear function itself and transforms 'x' to 'y'. This % function may be passed any number of additional parameters. % eg, y = myfunc(x, p1, p2, p3); % % 3. The function 'dfunc' is required to deal with discontinuous functions. Some non-linear % functions are discontinuous, but their residuals are not equal to the discontinuity. A classic % example is a normalised polar measurement: % y1 = myfunc(x1, p1, p2, p3); % lets say y1 == pi % y2 = myfunc(x2, p1, p2, p3); % lets say y2 == -pi % dy = y1 - y2; % dy == 2*pi -- this is wrong (must be within +/- pi) % dy = mydfunc(y1, y2); % dy == 0 -- this is correct % Thus, 'mydfunc' is a function that computes the true residual of y1-y2. If the function 'myfunc' % is not discontinuous, or has a trivial residual, just pass [] to parameter 'dfunc'. % % 4. The functions 'func' and 'dfunc' must be vectorised. That is, they must be able to accept a set of % states as input and return a corresponding set of results. So, for 'func', the state x will not be a % single column vector, but a matrix of N column vectors. Similarly, for 'dfunc', the parameters y1 and % y2 will each be matrices of N column vectors. % % EXAMPLE USE: % [y,Y] = unscented_transform(@myfunc, @mydfunc, x,P, p1, p2, p3); % [a,B] = unscented_transform('continuous_model', [], x,P); % % Tim Bailey 2003, modified 2004, 2005. % Set up some values D = length(x); % state dimension N = D*2 + 1; % number of samples scale = 3; % want scale = D+kappa == 3 kappa = scale-D; % Create samples Ps = chol(P)'.* sqrt(scale); ss = [x, repvec(x,D)+Ps, repvec(x,D)-Ps]; % Transform samples according to function 'func' if isempty(dfunc), dfunc = @default_dfunc; end ys = feval(func, ss, varargin{:}); % compute (possibly discontinuous) transform base = repvec(ys(:,1), N); % set first transformed sample as the base delta = feval(dfunc, base, ys); % compute correct residual ys = base - delta; % offset ys from base according to correct residual % Calculate predicted observation mean idx = 2:N; y = (2*kappa*ys(:,1) + sum(ys(:,idx),2)) / (2*scale); % Calculate new unscented covariance dy = ys - repvec(y,N); Y = (2*kappa*dy(:,1)*dy(:,1)' + dy(:,idx)*dy(:,idx)') / (2*scale); % Note: if x is a matrix of column vectors, then x*x' produces the sum of outer-products. % % function e = default_dfunc(y1, y2) e = y1 - y2; % % function x = repvec(x,N) x = x(:, ones(1,N)); function [x,P] = unscented_update(zfunc, dzfunc, x,P, z,R, varargin) %[x,P] = unscented_update(zfunc,dzfunc, x,P, z,R, ...) % % 3,APIRL 2008 % % INPUTS: % zfunc - function handle (@myfunc) or string ('myfunc') for observe model. % dzfunc - function handle (or string) for observation residual: v = mydfunc(z, z_predict); % x, P - predict mean and covariance % z, R - observation and covariance (observation noise is assumed additive) % ... - optional arguments such that 'zfunc' has the form: z = myfunc(x, ...) % % OUTPUTS: % x, P - updated mean and covariance % % NOTES: % 1. This function uses the unscented transform to compute a Kalman update. % % 2. The function 'zfunc' is the non-linear observation function. This function may be passed % any number of additional parameters. % eg, z = my_observe_model(x, p1, p2); % % 3. The residual function 'dzfunc' is required to deal with discontinuous functions. Some non-linear % functions are discontinuous, but their residuals are not equal to the discontinuity. A classic % example is a normalised angle measurement model: % z1 = angle_observe_model(x1, p1, p2, p3); % lets say z1 == pi % z2 = angle_observe_model(x2, p1, p2, p3); % lets say z2 == -pi % dz = z1 - z2; % dz == 2*pi -- this is wrong (must be within +/- pi) % dz = residual_model(z1, z2); % dz == 0 -- this is correct % Thus, 'residual_model' is a function that computes the true residual of z1-z2. If the function % 'zfunc' is not discontinuous, or has a trivial residual, just pass [] to parameter 'dzfunc'. % % 4. The functions 'zfunc' and 'dzfunc' must be vectorised. That is, they must be able to accept a set of % states as input and return a corresponding set of results. So, for 'zfunc', the state x will not be a % single column vector, but a matrix of N column vectors. Similarly, for 'dzfunc', the parameters z and % z_predict will be equal-sized matrices of N column vectors. % % EXAMPLE USE: % [x,P] = unscented_update(@angle_observe_model, @residual_model, x,P, z,R, a, b, c); % [x,P] = unscented_update('range_model', [], x,P, z,R); % % Tim Bailey 2003, modified 2004, 2005. global d d=d+1 % Set up some values D = length(x); % state dimension N = D*2 + 1; % number of samples scale = 3; % want scale = D+kappa == 3 kappa = scale-D; % Create samples P=diag(diag(P)); Ps =chol(P)'.* sqrt(scale); ss = [x, repvec(x,D)+Ps, repvec(x,D)-Ps]; % Transform samples according to function 'zfunc' to obtain the predicted observation samples if isempty(dzfunc), dzfunc = @default_dfunc; end zs = feval(zfunc, ss, varargin{:}); % compute (possibly discontinuous) transform zz = repvec(z,N); dz = feval(dzfunc, zz, zs); % compute correct residual zs = zz - dz; % offset zs from z according to correct residual % Calculate predicted observation mean zm = (kappa*zs(:,1) + 0.5*sum(zs(:,2:end), 2)) / scale; % Calculate observation covariance and the state-observation correlation matrix dx = ss - repvec(x,N); dz = zs - repvec(zm,N); Pxz = (2*kappa*dx(:,1)*dz(:,1)' + dx(:,2:end)*dz(:,2:end)') / (2*scale); %Pzz = dz(:,2:end)*dz(:,2:end)' / (2*scale); Pzz = (2*kappa*dz(:,1)*dz(:,1)' + dz(:,2:end)*dz(:,2:end)') / (2*scale); % Compute Kalman gain %%采用平方根滤波算法对滤波误差协方差矩阵进行Cholesky分解， %%P(k,k-1)=S(k,k-1)*S(k,k-1)',P(k)=S(k)*S(k)'，其中S(k),S(k,k-1)为下三角矩阵， %%在任意k时刻，以S(k)的递推式代替P(k)的递推式，避免直接计算P(k)从而克服计算发散。 S = Pzz+R; Sc = chol(S); % note: S = Sc*Sc' Sci = inv(Sc); % note: inv(S) = Sci*Sci' Kc = Pxz * Sci; K = Kc * Sci'; %K=Pxz * inv(S); %[u,s,v]=svd(S);kc=Pxz*inv(v');kcc=kc*inv(s);K=kcc*inv(u); % Perform update x = x + K*(z - zm); P = P - Kc*Kc'; %P = P - K*S*K'; kk=diag(P); PSD_check = chol(P); % % function x = repvec(x,N) x = x(:, ones(1,N)); % % function e = default_dfunc(y1, y2) e = y1 - y2;