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F_update.m
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function [F, G] = F_update(upd, DCMbn, imu)
% F_update: updates Kalman filter transition matrix F.
%
% Copyright (C) 2014, Rodrigo Gonzalez, all rights reserved.
%
% This file is part of NaveGo, an open-source MATLAB toolbox for
% simulation of integrated navigation systems.
%
% NaveGo is free software: you can redistribute it and/or modify
% it under the terms of the GNU Lesser General Public License (LGPL)
% version 3 as published by the Free Software Foundation.
%
% This program is distributed in the hope that it will be useful,
% but WITHOUT ANY WARRANTY; without even the implied warranty of
% MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
% GNU Lesser General Public License for more details.
%
% You should have received a copy of the GNU Lesser General Public
% License along with this program. If not, see
% <http://www.gnu.org/licenses/>.
%
% References:
% Titterton, D.H. and Weston, J.L. (2004). Strapdown
% Inertial Navigation Technology (2nd Ed.). Institution
% of Engineering and Technology, USA. Eq 12.18, p. 345.
%
% Farrell, J. (2008). Aided Navigation: GPS With High Rate
% Sensors. McGraw-Hill Professional, USA. Eq. 11.108, p. 407.
%
% R. Gonzalez, J. Giribet, and H. Patiño. NaveGo: a
% simulation framework for low-cost integrated navigation systems,
% Journal of Control Engineering and Applied Informatics, vol. 17,
% issue 2, pp. 110-120, 2015. Eq. 26.
%
% R. Gonzalez, J. Giribet, and H. Patiño. An approach to
% benchmarking of loosely coupled low-cost navigation systems,
% Mathematical and Computer Modelling of Dynamical Systems, vol. 21,
% issue 3, pp. 272-287, 2015. Eq. 22.
%
% Version: 001
% Date: 2014/09/11
% Author: Rodrigo Gonzalez <rodralez@frm.utn.edu.ar>
% URL: https://github.com/rodralez/navego
Vn = upd(1);
Ve = upd(2);
Vd = upd(3);
lat = upd(4);
% h = upd(5);
fn = upd(6);
fe = upd(7);
fd = upd(8);
if (isa(Vn,'single'))
[RM,RN] = radius(lat, 'single');
Om = single(7.292115e-5);
I = single(eye(3));
Z = single(zeros(3));
else
[RM,RN] = radius(lat, 'double');
Om = 7.292115e-5;
I = eye(3);
Z = zeros(3);
end
RO = sqrt(RN*RM);
a11 = 0;
a12 = -((Om * sin(lat)) + (Ve/(RO) * tan(lat)));
a13 = Vn/(RO);
a21 = (Om * sin(lat)) + (Ve/(RO) * tan(lat) );
a22 = 0 ;
a23 = (Om * cos(lat)) + (Ve/(RO)) ;
a31 = -Vn/(RO);
a32 = -(Om * cos(lat)) - (Ve/(RO));
a33 = 0 ;
Fee = [a11 a12 a13; a21 a22 a23; a31 a32 a33;];
a11 = 0;
a12 = 1/(RO);
a13 = 0;
a21 = -1/(RO);
a22 = 0 ;
a23 = 0 ;
a31 = 0 ;
a32 = -tan(lat)/RO ;
a33 = 0 ;
Fev = [a11 a12 a13; a21 a22 a23; a31 a32 a33;];
a11 = -Om * sin(lat);
a12 = 0;
a13 = -Ve/((RO)^2);
a21 = 0 ;
a22 = 0 ;
a23 = Vn/((RO)^2);
a31 = -(Om*cos(lat)) - (Ve/((RO)*(cos(lat))^2));
a32 = 0 ;
a33 = (Ve * tan(lat)) / (RO^2) ;
Fep = [a11 a12 a13; a21 a22 a23; a31 a32 a33;];
a11 = 0 ;
a12 = -fd ;
a13 = fe ;
a21 = fd ;
a22 = 0 ;
a23 = -fn ;
a31 = -fe ;
a32 = fn ;
a33 = 0 ;
Fve = [a11 a12 a13; a21 a22 a23; a31 a32 a33;];
a11 = Vd/(RO);
a12 = -2*((Om * sin(lat)) + ((Ve/(RO)) * tan (lat))) ;
a13 = Vn/RO ;
a21 = (2*Om * sin(lat)) + ( (Ve/(RO)) * tan(lat) );
a22 = (1/(RO)) * ((Vn * tan(lat)) + Vd) ;
a23 = 2*Om * cos(lat) + (Ve/(RO));
a31 = (-2*Vn)/(RO);
a32 = -2*(Om * cos(lat) + (Ve/(RO))) ;
a33 = 0;
Fvv = [a11 a12 a13; a21 a22 a23; a31 a32 a33;];
a11 = -Ve*( (2*Om * cos(lat)) + (Ve/((RO)*(cos(lat))^2)));
a12 = 0 ;
a13 = 1/((RO)^2)*(((Ve^2)*tan(lat)) - (Vn*Vd) );
a21 = 2*Om *((Vn*cos(lat)) - (Vd*sin(lat))) + ((Vn*Ve) / ((RO)*(cos(lat))^2)) ;
a22 = 0 ;
a23 = -(Ve/(RO^2)) * (Vn*tan(lat)+Vd);
a31 = 2 * Om * Ve * sin(lat);
a32 = 0;
a33 = 1/((RO)^2) * ((Vn^2) + (Ve^2));
Fvp = [a11 a12 a13; a21 a22 a23; a31 a32 a33;];
Fpe = zeros(3);
a11 = 1/(RO);
a12 = 0;
a13 = 0;
a21 = 0;
a22 = 1/((RO)*cos(lat));
a23 = 0;
a31 = 0;
a32 = 0;
a33 = -1;
Fpv = [a11 a12 a13; a21 a22 a23; a31 a32 a33;];
a11 = 0;
a12 = 0;
a13 = -Vn/((RO)^2);
a21 = (Ve*tan(lat))/((RO)*cos(lat));
a22 = 0;
a23 = -Ve/(((RO)^2)*cos(lat));
a31 = 0;
a32 = 0;
a33 = 0;
Fpp = [a11 a12 a13; a21 a22 a23; a31 a32 a33;];
Fva = I;
Frog = I;
if (isinf(imu.acorr))
Faa = Z;
else
Faa = diag(-1./imu.acorr);
end
if (isinf(imu.gcorr))
Fgg = Z;
else
Fgg = diag(-1./imu.gcorr);
end
F= [ Fee Fev Fep (-DCMbn*Frog) Z (-DCMbn*Frog) Z;
Fve Fvv Fvp Z (DCMbn*Fva) Z (DCMbn*Fva);
Fpe Fpv Fpp Z Z Z Z;
Z Z Z Z Z Z Z;
Z Z Z Z Z Z Z;
Z Z Z Z Z Fgg Z;
Z Z Z Z Z Z Faa;
];
% Grover 12.58
% Hasnur 6.15
% F= [ Fee Fev Fep (-DCMbn*Frog) Z Z Z;
% Fve Fvv Fvp Z (DCMbn*Fva) Z Z;
% Fpe Fpv Fpp Z Z Z Z;
% Z Z Z Z Z Z Z;
% Z Z Z Z Z Z Z;
% Z Z Z Z Z Fgg Z;
% Z Z Z Z Z Z Faa;
% ];
G = [ -DCMbn Z Z Z;
Z DCMbn Z Z;
Z Z Z Z;
Z Z Z Z;
Z Z Z Z;
Z Z I Z;
Z Z Z I;];
end