卡尔曼滤波与SINS/GNSS组合导航

用于Kalman滤波的函数有：
psinstypedef(nnm)：KF状态维数nn/量测维数m定义（一般可直接用于SINS/GNSS组合），或用户自定义字符串，在kfinit/kffk/kfhk/kfplot等函数中作为类型区分标识；
kfinit：滤波器主要参数初始化，再用kfinit0更多的参数初始化；

function kf = kfinit(ins, varargin)
% Kalman filter initializes for structure array 'kf', this precedure
% usually includs the setting of structure fields: Qt, Rk, Pxk, Hk.
%
% Prototype: kf = kfinit(ins, varargin)
% Inputs: ins - SINS structure array, if not struct then nts=ins;
%         varargin - if any other parameters
% Output: kf - Kalman filter structure array
%
% See also  kfinit0, kfsetting, kffk, kfkk, kfupdate, kffeedback, psinstypedef.% Copyright(c) 2009-2014, by Gongmin Yan, All rights reserved.
% Northwestern Polytechnical University, Xi An, P.R.China
% 09/10/2013
global glv psinsdef
[Re,deg,dph,ug,mg] = ... % just for shortsetvals(glv.Re,glv.deg,glv.dph,glv.ug,glv.mg);
o33 = zeros(3); I33 = eye(3);
kf = [];
if isstruct(ins),    nts = ins.nts;
else                 nts = ins;
end
switch(psinsdef.kfinit)case psinsdef.kfinit153,psinsdef.kffk = 15;  psinsdef.kfhk = 153;  psinsdef.kfplot = 15;[davp, imuerr, rk] = setvals(varargin);kf.Qt = diag([imuerr.web; imuerr.wdb; zeros(9,1)])^2;kf.Rk = diag(rk)^2;kf.Pxk = diag([davp; imuerr.eb; imuerr.db]*1.0)^2;kf.Hk = kfhk(0);case psinsdef.kfinit156,psinsdef.kffk = 15;  psinsdef.kfhk = 156;  psinsdef.kfplot = 15;[davp, imuerr, rk] = setvals(varargin);kf.Qt = diag([imuerr.web; imuerr.wdb; zeros(9,1)])^2;kf.Rk = diag(rk)^2;kf.Pxk = diag([davp; imuerr.eb; imuerr.db]*1.0)^2;kf.Hk = kfhk(0);case psinsdef.kfinit183,psinsdef.kffk = 18;  psinsdef.kfhk = 183;  psinsdef.kfplot = 18;[davp, imuerr, lever, r0] = setvals(varargin);kf.Qt = diag([imuerr.web; imuerr.wdb; zeros(9+3,1)])^2;kf.Rk = diag(r0)^2;kf.Pxk = diag([davp; imuerr.eb; imuerr.db; lever]*1.0)^2;kf.Hk = zeros(3,18);case psinsdef.kfinit186,psinsdef.kffk = 18;  psinsdef.kfhk = 186;  psinsdef.kfplot = 18;[davp, imuerr, lever, r0] = setvals(varargin);kf.Qt = diag([imuerr.web; imuerr.wdb; zeros(3,1); imuerr.sqg; imuerr.sqa; zeros(3,1)])^2;kf.Rk = diag(r0)^2;kf.Pxk = diag([davp; imuerr.eb; imuerr.db; lever]*1.0)^2;kf.Hk = zeros(6,18);case psinsdef.kfinit193psinsdef.kffk = 19;  psinsdef.kfhk = 193;  psinsdef.kfplot = 19;[davp, imuerr, lever, dT, r0] = setvals(varargin);kf.Qt = diag([imuerr.web; imuerr.wdb; [1/Re;1/Re;1]*glv.mpsh; ...[1;1;1]*0*glv.dphpsh; [1;1;1]*0*glv.ugpsh; [1;1;1]*0.*glv.mpsh; 0])^2;kf.Rk = diag(r0)^2;kf.Pxk = diag([davp; imuerr.eb; imuerr.db; lever; dT]*1.0)^2;kf.Hk = zeros(3,19);case psinsdef.kfinit196psinsdef.kffk = 19;  psinsdef.kfhk = 196;  psinsdef.kfplot = 19;[davp, imuerr, lever, dT, r0] = setvals(varargin);kf.Qt = diag([imuerr.web; imuerr.wdb; [1/Re;1/Re;1]*0*glv.mpsh; ...[1;1;1]*0*glv.dphpsh; [1;1;1]*0*glv.ugpsh; [1;1;1]*0*glv.mpsh; 0])^2;kf.Rk = diag(r0)^2;kf.Pxk = diag([davp; imuerr.eb; imuerr.db; lever; dT]*1.0)^2;kf.Hk = zeros(6,19);case psinsdef.kfinit331,psinsdef.kffk = 33;  psinsdef.kfhk = 331;  psinsdef.kfplot = 33;[davp, imuerr, r0] = setvals(varargin);kf.Qt = diag([imuerr.web; imuerr.wdb; zeros(9+15+3,1)])^2;kf.Rk = diag(r0)^2;kf.Pxk = diag([davp; imuerr.eb; imuerr.db; imuerr.dKga; imuerr.KA2])^2;kf.Hk = kfhk(ins);kf.xtau(1:psinsdef.kffk,1) = 0;case psinsdef.kfinit346,psinsdef.kffk = 34;  psinsdef.kfhk = 346;  psinsdef.kfplot = 34;[davp, imuerr, lever, dT, r0] = setvals(varargin);kf.Qt = diag([imuerr.web; imuerr.wdb; zeros(9+3+1+15,1)])^2;kf.Rk = diag(r0)^2;kf.Pxk = diag([davp; imuerr.eb; imuerr.db; lever; dT; imuerr.dKga])^2;kf.Hk = kfhk(ins);kf.xtau(1:psinsdef.kffk,1) = 0;case psinsdef.kfinit376,psinsdef.kffk = 37;  psinsdef.kfhk = 376;  psinsdef.kfplot = 37;[davp, imuerr, lever, dT, r0] = setvals(varargin);kf.Qt = diag([imuerr.web; imuerr.wdb; zeros(9+3+1+15+3,1)])^2;kf.Rk = diag(r0)^2;kf.Pxk = diag([davp; imuerr.eb; imuerr.db; lever; dT; imuerr.dKga; davp(4:6)]*10)^2;kf.Hk = kfhk(ins);kf.xtau(1:psinsdef.kffk,1) = 0;otherwise,kf = feval(psinsdef.typestr, psinsdef.kfinittag, [{ins},varargin]);
end
kf = kfinit0(kf, nts);

kfinit0函数

function kf = kfinit0(kf, nts)
% Always called by kfinit and initialize the remaining fields of kf.
%
% See also kfinit, kfupdate, kffeedback, psinstypedef.kf.nts = nts;[kf.m, kf.n] = size(kf.Hk);kf.I = eye(kf.n);kf.Kk = zeros(kf.n, kf.m);if ~isfield(kf, 'xk'),  kf.xk = zeros(kf.n, 1);  endif ~isfield(kf, 'Qk'),  kf.Qk = kf.Qt*kf.nts;  endif ~isfield(kf, 'Gammak'),  kf.Gammak = 1; kf.l = kf.n;  endif ~isfield(kf, 'fading'),  kf.fading = 1;  endif ~isfield(kf, 'adaptive'),  kf.adaptive = 0;  end
%     if kf.adaptive==1if ~isfield(kf, 'b'),  kf.b = 0.5;  endif ~isfield(kf, 'beta'),  kf.beta = 1;  endif ~isfield(kf, 'Rmin'),  kf.Rmin = 0.01*kf.Rk;  endif ~isfield(kf, 'Rmax'),  kf.Rmax = 100*kf.Rk;  endif ~isfield(kf, 'Qmin'),  kf.Qmin = 0.01*kf.Qk;  endif ~isfield(kf, 'Qmax'),  kf.Qmax = 100*kf.Qk;  end
%     endif ~isfield(kf, 'xtau'),  kf.xtau = ones(size(kf.xk))*eps;   endif ~isfield(kf, 'T_fb'),  kf.T_fb = 1;   endif ~isfield(kf, 'fbstr'),  kf.fbstr = 'avped';  endif ~isfield(kf, 'xconstrain'),  kf.xconstrain = 0;  endif ~isfield(kf, 'pconstrain'),  kf.pconstrain = 0;  endkf.Pmax = (diag(kf.Pxk)+1)*1.0e10;kf.Pmin = kf.Pmax*0;kf.xfb = zeros(kf.n, 1);
%     kf.coef_fb = (1.0-exp(-kf.T_fb./kf.xtau));
%     kf.coef_fb = ar1coefs(kf.T_fb, kf.xtau);xtau = kf.xtau;xtau(kf.xtau<kf.T_fb) = kf.T_fb;  kf.coef_fb = kf.T_fb./xtau;  %2015-2-22

kfsetting：直接设置几种特定类型IMU的滤波器参数；
kffk/kfhk：计算状态转移矩阵/量测矩阵，etm惯导误差传播函数；

function [Fk, Ft] = kffk(ins, varargin)
% Create Kalman filter system transition matrix.
%
% Prototype: [Fk, Ft] = kffk(ins, varargin)
% Inputs: ins - SINS structure array, if not struct then nts=ins;
%         varargin - if any other parameters
% Outputs: Fk - discrete-time transition matrix, = Phikk_1
%          Ft - continuous-time transition matirx
%
% See also  kfhk, kfinit, kfupdate, kfc2d, insupdate, etm, psinstypedef, ekffk.% Copyright(c) 2009-2014, by Gongmin Yan, All rights reserved.
% Northwestern Polytechnical University, Xi An, P.R.China
% 06/08/2012, 01/02/2014, 02/08/2016
global psinsdef%% get Ftif isstruct(ins),    nts = ins.nts;else                 nts = ins;endswitch(psinsdef.kffk)case 15,Ft = etm(ins);case {18,19} % psinsdef.kffkxx, xx=18,19Ft = etm(ins);Ft(psinsdef.kffk, psinsdef.kffk) = 0;case {33}Ft = etm(ins);Ft(psinsdef.kffk,psinsdef.kffk) = 0;% 15+dKg(9)+dKa(6)+KA2(3)Ft(1:3,16:24) = [-ins.wib(1)*ins.Cnb, -ins.wib(2)*ins.Cnb, -ins.wib(3)*ins.Cnb];Ft(4:6,25:33) = [ins.fb(1)*ins.Cnb, ins.fb(2)*ins.Cnb(:,2:3), ins.fb(3)*ins.Cnb(:,3), ins.Cnb*diag(ins.fb.^2)];case {34, 37}Ft = etm(ins);Ft(psinsdef.kffk,psinsdef.kffk) = 0;Ft(1:3,20:28) = [-ins.wib(1)*ins.Cnb, -ins.wib(2)*ins.Cnb, -ins.wib(3)*ins.Cnb];Ft(4:6,29:34) = [ins.fb(1)*ins.Cnb, ins.fb(2)*ins.Cnb(:,2:3), ins.fb(3)*ins.Cnb(:,3)];otherwise,
%             Ft = feval(psinsdef.typestr, psinsdef.kffktag, {ins, varargin});Ft = feval(psinsdef.typestr, psinsdef.kffktag, [{ins},varargin]);end%% discretizationFk = Ft*nts;if nts>0.1  % for large time interval, this may be more accurate.Fk = expm(Fk);else   % Fk = I + Ft*nts + 1/2*(Ft*nts)^2  , 2nd order expensionFk = eye(size(Ft)) + Fk;% + Fk*Fk*0.5; end

kfhk函数

function Hk = kfhk(ins, varargin)
% Create Kalman filter measurement matrix.
%
% Prototype: Hk = kfhk(ins, varargin)
% Inputs: ins - SINS structure array from function 'insinit'
%         varargin - if any other parameters
% Output: Hk - measurement matrix
%
% See also  kffk, kfinit, kfupdate, kfc2d, insupdate.% Copyright(c) 2009-2014, by Gongmin Yan, All rights reserved.
% Northwestern Polytechnical University, Xi An, P.R.China
% 27/03/2014, 02/02/2015
global psinsdefswitch(psinsdef.kfhk)case 153,Hk = [zeros(3,6), eye(3), zeros(3,6)];case 156Hk = [zeros(6,3), eye(6), zeros(6,6)];case 183 % glv.psinsdef.kfhkxx3Hk = [zeros(3,6), eye(3), zeros(3,6), -ins.MpvCnb];case 186Hk = [zeros(6,3), eye(6), zeros(6,6), [-ins.CW;-ins.MpvCnb]];case 193 % glv.psinsdef.kfhkxx3Hk = [zeros(3,6), eye(3), zeros(3,6), -ins.MpvCnb, -ins.Mpvvn];case 196Hk = [zeros(6,3), eye(6), zeros(6,6), [-ins.CW,-ins.an;-ins.MpvCnb,-ins.Mpvvn]];case 331Hk = zeros(1,33);case 343 % glv.psinsdef.kfhkxx3Hk = [zeros(3,6), eye(3), zeros(3,6), -ins.MpvCnb, -ins.Mpvvn, zeros(3,15)];case 373Hk = [zeros(3,6), eye(3), zeros(3,6), -ins.MpvCnb, -ins.Mpvvn, zeros(3,18)];case 346Hk = [zeros(6,3), eye(6), zeros(6,6), [-ins.CW,-ins.an;-ins.MpvCnb,-ins.Mpvvn], zeros(6,15)];case 376Hk = [zeros(6,3), eye(6), zeros(6,6), [-ins.CW,-ins.an;-ins.MpvCnb,-ins.Mpvvn], zeros(6,15), [-eye(3);zeros(3)]];
%         case {186, 196, 346} % glv.psinsdef.kfhkxx6
%             Hk = [zeros(6,3), eye(6), zeros(6,fix(psinsdef.kfhk/10)-9)];
%             Hk(1:3,16:19) = [-ins.CW, -ins.an];
%             Hk(4:6,16:19) = [-ins.MpvCnb, -ins.Mpvvn];
%         case 376
%             Hk = [zeros(6,3), eye(6), zeros(6,fix(psinsdef.kfhk/10)-9)];
%             Hk(1:3,[16:19,35:37]) = [-ins.CW, -ins.an, -eye(3)];
%             Hk(4:6,16:19) = [-ins.MpvCnb, -ins.Mpvvn];otherwise,Hk = feval(psinsdef.typestr, psinsdef.kfhktag, [{ins},varargin]);end

kfupdate：KF时间/量测更新，采用序贯量测自适应/方差限制方法；

function kf = kfupdate(kf, yk, TimeMeasBoth)
% Discrete-time Kalman filter.
%
% Prototype: kf = kfupdate(kf, yk, TimeMeasBoth)
% Inputs: kf - Kalman filter structure array
%         yk - measurement vector
%         TimeMeasBoth - described as follows,
%            TimeMeasBoth='T' (or nargin==1) for time updating only,
%            TimeMeasBoth='M' for measurement updating only,
%            TimeMeasBoth='B' (or nargin==2) for both time and
%                             measurement updating.
% Output: kf - Kalman filter structure array after time/meas updating
% Notes: (1) the Kalman filter stochastic models is
%      xk = Phikk_1*xk_1 + wk_1
%      yk = Hk*xk + vk
%    where E[wk]=0, E[vk]=0, E[wk*wk']=Qk, E[vk*vk']=Rk, E[wk*vk']=0
%    (2) If kf.adaptive=1, then use Sage-Husa adaptive method (but only for
%    measurement noise 'Rk'). The 'Rk' adaptive formula is:
%      Rk = b*Rk_1 + (1-b)*(rk*rk'-Hk*Pxkk_1*Hk')
%    where minimum constrain 'Rmin' and maximum constrain 'Rmax' are
%    considered to avoid divergence.
%    (3) If kf.fading>1, then use fading memory filtering method.
%    (4) Using Pmax&Pmin to constrain Pxk, such that Pmin<=diag(Pxk)<=Pmax.
%
% See also  kfinit, kfupdatesq, kffk, kfhk, kfc2d, kffeedback, kfplot, RLS, ekf, ukf.% Copyright(c) 2009-2015, by Gongmin Yan, All rights reserved.
% Northwestern Polytechnical University, Xi An, P.R.China
% 08/12/2012, 29/08/2013, 16/04/2015, 01/06/2017, 11/03/2018if nargin==1;TimeMeasBoth = 'T';elseif nargin==2TimeMeasBoth = 'B';endif TimeMeasBoth=='T'            % Time Updatingkf.xk = kf.Phikk_1*kf.xk;    kf.Pxk = kf.Phikk_1*kf.Pxk*kf.Phikk_1' + kf.Gammak*kf.Qk*kf.Gammak';elseif TimeMeasBoth=='M'        % Meas Updatingkf.xkk_1 = kf.xk;    kf.Pxkk_1 = kf.Pxk; elseif TimeMeasBoth=='B'    % Time & Meas Updatingkf.xkk_1 = kf.Phikk_1*kf.xk;    kf.Pxkk_1 = kf.Phikk_1*kf.Pxk*kf.Phikk_1' + kf.Gammak*kf.Qk*kf.Gammak';elseerror('TimeMeasBoth input error!');end

其中参数含义为：
kf - Kalman filter structure array，表示卡尔曼滤波的结构数组
yk - measurement vector，量测向量
TimeMeasBoth -T表示只进行时间更新、M表示只进行量测更新，B表示两者同时进行
kffeedback：滤波状态反馈，可用部分反馈方法；

function [kf, ins, xfb] = kffeedback(kf, ins, T_fb, fbstr)
% Kalman filter state estimation feedback to SINS.
%
% Prototype: [kf, ins] = kffeedback(kf, ins, T_fb)
% Inputs: kf - Kalman filter structure array
%         ins - SINS structure array
%         T_fb - feedback time interval
%         fbstr - feedback string
% Outputs: kf, ins - Kalman filter & SINS structure array after feedback
%          xfb - feedback state value
%
% See also  kfinit, kffk, kfhk, kfplot, psinstypedef.% Copyright(c) 2009-2014, by Gongmin Yan, All rights reserved.
% Northwestern Polytechnical University, Xi An, P.R.China
% 05/10/2013, 06/02/2021

其中参数含义为：
kf - Kalman filter structure array，卡尔曼滤波结构数组
ins - SINS structure array，SINS结构数组
T_fb - feedback time interval，反馈时间间隔
fbstr - feedback string，反馈字符串，若要返馈位置，则输入p，返馈姿态则输入a,速度则是v，具体可参见源码

kfstat：滤波误差分配与可观测度分析（协方差分析）；

滤波误差分配与可观测度分析：

做双位置对准的时候，选择了三个失准角，三个速度误差，三个陀螺漂移，三个加表零偏，对其进行初始对准；最后可以得到状态的可观测都度

kfplot/xpplot/rvpplot：滤波结果绘图；
sinsgps：典型的193或196维SINS/GNSS松组合函数；
POSProcessing：343或346维SINS/GNSS双向滤波松组合函数，POSFusion对双向滤波结果做加权平均，posplot作图

典型的SINS/GNSS组合滤波

该功能可由sinsgps函数实现，源码中给出了使用该函数的示例:

19-state SINS/GNSS integrated navigation Kalman filter.
% The 19-state includes:
%       [phi(3); dvn(3); dpos(3); eb(3); db(3); lever(3); dT(1)]
% The 3- or 6- measurements are:
%       [dvn(3)] or [dvn(3); dpos(3)]
%
% Prototype: [avp, xkpk, zkrk, sk, ins, kf] = sinsgps(imu, gps, ins, davp, imuerr, lever, dT, rk, Pmin, Rmin, fbstr, isfig)
% Inputs: imu - IMU array [wm, vm, t]
%         gps - GNSS array [vn, pos, t] or [pos, t];
%         ins - ins array, set by function 'insinit'
%         davp - AVP array for P0 setting
%         imuerr - set by function 'imuerrset', for P0 and Qk setting
%         lever - lever arm from IMU to GNSS, if lever(4)=0 then Pk(lever)=0 for no lever estimation
%         dT - time delay from IMU to GNSS, if dT(2)=0 then Pk(dT)=0 for no time delay estimation
%         rk - measurement noise std(dpos) or std([dvn;dpos])
%         Pmin - Pmin setting, Pmin<=0 for no Pmin constrain
%         Rmin - Rmin setting, Rmin<=0 for no adaptive KF, Rmin=0~1 scale for adaptive KF and Rmin = Rk*Rmin
%         fbstr - KF feedback string from 'avpedLT'
%         isfig - figure flag
%
% Example 1:
%   [avp1, xkpk, zkrk, sk, ins1, kf1] = sinsgps(imu, gps, 300);
%
% Example 2:
% ins = insinit([yaw;pos], ts);
% avperr = avperrset([60;300], 1, 100);
% imuerr = imuerrset(0.03, 100, 0.001, 1);
% Pmin = [avperrset([0.1,1],0.001,0.01); gabias(0.1, [10,30]); [0.01;0.01;0.01]; 0.0001].^2;
% Rmin = vperrset(0.001, 0.01).^2;
% [avp1, xkpk, zkrk, sk, ins1, kf1] = sinsgps(imu, gps, ins, avperr, imuerr, rep3(1), 0.01, vperrset(0.1,10), Pmin, Rmin, 'avp');
%
% Example 3:
% t0 = 1;  t1 = 916;
% avp0 = getat(avp,t0);
% ins = insinit(avp0, ts);
% avperr = avperrset([60;300], 1, 10);
% imuerr = imuerrset(0.5, 1000, 0.1, 25);
% Pmin = [avperrset([0.2,1.0],0.01,0.2); gabias(0.01, [10,10]); [0.01;0.01;0.01]; 0.001].^2;
% Rmin = vperrset(0.1, 0.3).^2;
% [avp1, xkpk, zkrk, sk, ins1, kf] = sinsgps(imu(t0/ts:t1/ts,:), gps, ins, avperr, imuerr, rep3(1), 0.1, vperrset(0.1,10), Pmin, Rmin, 'avped');
%
% See also  kfinit, kfupdate, imugpssyn, igsplot, insupdate, posprocessing.

该函数实现的是19维状态，3维量测的组合导航

POS处理

包括正反向滤波，以及利用POSFusion的正反向融合
可参见仿真例子test_POS_fusion.m

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基于PSINS工具箱的卡尔曼滤波与SINS/GNSS组合导航

文章目录

卡尔曼滤波与SINS/GNSS组合导航

典型的SINS/GNSS组合滤波

POS处理

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