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ESKF.m
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function [Q2,Bg2,delta_X2,Pk2]=ESKF(Q1,Bg1,delta_X1,ImuData,t,Vm,Pk1)
% ESKF( Error State Kalman Filter)
% derivation <Quaternion Kinematics for the Error-state Kalman Filter>
% author Zhang xin
if isempty(Pk1)
Para_delta_theta0=0.00001;
Para_delta_Bg0=0.00001;
Pk1=diag([[1,1,1]*Para_delta_theta0,[1,1,1]*Para_delta_Bg0]);
end
norm_a=norm(ImuData(1,2:4));
%norm_g=norm(ImuData(1,5:7)-Bg1);
wx=(ImuData(1,5)-Bg1(1))*t;
wy=(ImuData(1,6)-Bg1(2))*t;
wz=(ImuData(1,7)-Bg1(3))*t;
Qp=[ 1 , -wx/2 , -wy/2 , -wz/2 ;...
wx/2 , 1 , wz/2 , -wy/2 ;...
wy/2 , -wz/2 , 1 , wx/2 ;...
wz/2 , wy/2 , -wx/2 , 1 ]*Q1';
Qp=Qp/norm(Qp);
delta_theta=norm([wx,wy,wz]);
if delta_theta==0
R_u_delta_theta=eye(3);
else
u= [wx,wy,wz]'/delta_theta;
R_u_delta_theta=eye(3)-Skew_symmetric(u)*sin(delta_theta)+Skew_symmetric(u)*Skew_symmetric(u)*(1-cos(delta_theta));
end
F_delta_X=[R_u_delta_theta, -eye(3)*t;...
zeros(3,3), eye(3) ];
Para_delta_theta=0.000001;
Para_delta_Bg=0.000001;
Q_delta_X=diag([[1,1,1]*Para_delta_theta,[1,1,1]*Para_delta_Bg]);
delta_Xp = F_delta_X*delta_X1';
P_k=F_delta_X*Pk1*F_delta_X'+Q_delta_X;
if abs(norm_a-9.8)<2 %&& norm_g< 2
% R1=[2*Qp(1)^2+2*Qp(2)^2-1;...
% 2*Qp(2)*Qp(3)-2*Qp(1)*Qp(4);...
% 2*Qp(1)*Qp(3)+2*Qp(2)*Qp(4)];
R2=[2*Qp(2)*Qp(3)+2*Qp(1)*Qp(4);...
2*Qp(1)^2+2*Qp(3)^2-1;...
2*Qp(3)*Qp(4)-2*Qp(1)*Qp(2)];
R3=[2*Qp(2)*Qp(4)-2*Qp(1)*Qp(3);...
2*Qp(3)*Qp(4)+2*Qp(1)*Qp(2);...
2*Qp(1)^2+2*Qp(4)^2-1 ];
% J1=2*[ Qp(1), Qp(2),-Qp(3),-Qp(4);...
% -Qp(4), Qp(3), Qp(2),-Qp(1);...
% Qp(3), Qp(4), Qp(1), Qp(2)];
J2=2*[ Qp(4), Qp(3), Qp(2) , Qp(1);...
Qp(1),-Qp(2), Qp(3) ,-Qp(4);...
-Qp(2),-Qp(1), Qp(4) , Qp(3)] ;
J3=2*[-Qp(3), Qp(4),-Qp(1) , Qp(2);...
Qp(2), Qp(1), Qp(4) , Qp(3);...
Qp(1),-Qp(2),-Qp(3) , Qp(4)] ;
h_acc= R3;
h_mag = Vm(2)*R2+Vm(3)*R3;
H_acc = [J3,zeros(3,3)];
H_mag = [Vm(2)*J2+Vm(3)*J3,zeros(3,3)];
Q_delta_theta= 1/2*[ -Qp(2), -Qp(3), -Qp(4) ;...
Qp(1), -Qp(4), Qp(3) ;...
Qp(4), Qp(1), -Qp(2) ;...
-Qp(3), Qp(2), Qp(1) ];
X_delta_x=[Q_delta_theta, zeros(4,3);...
zeros(3,3), eye(3)];
Hk= [H_acc;H_mag]* X_delta_x;
para_Rk_acc=0.1;
para_Rk_mag=0.2;
Rk=diag([[1,1,1]*para_Rk_acc,[1,1,1]*para_Rk_mag]);
Kk=P_k*Hk'*inv(Hk*P_k*Hk'+Rk);
delta_X_hat=Kk*([ImuData(1,2:4)'/norm_a;ImuData(1,8:10)'/norm(ImuData(1,8:10))]-[h_acc;h_mag]);
Q2=quaternProd(Qp, [1;delta_X_hat(1:3)/2])';
Q2=Q2/norm(Q2);
Bg2=Bg1+delta_X_hat(4:6)';
Pk2_=(eye(6)-Kk*Hk)*P_k;
delta_X2=delta_Xp'+delta_X_hat';
G=[eye(3)-Skew_symmetric(-delta_X2(1:3)/2),zeros(3,3);...
zeros(3,3) , eye(3)];
Pk2=G*Pk2_*G';
else
Q2=Qp';
Bg2=Bg1;
delta_X2=delta_Xp';
Pk2=P_k;
end
delta_X2(1:3)=[0,0,0];
if Q2(1)<0
Q2=-Q2;
end
end
function S=Skew_symmetric(u)
%Skew_Operator
S=[ 0 , -u(3) , u(2) ;...
u(3), 0 , -u(1) ;...
-u(2), u(1), 0 ];
end
function ab = quaternProd(a, b)
ab(1,1) = a(1)*b(1)-a(2)*b(2)-a(3)*b(3)-a(4)*b(4);
ab(2,1) = a(1)*b(2)+a(2)*b(1)+a(3)*b(4)-a(4)*b(3);
ab(3,1) = a(1)*b(3)-a(2)*b(4)+a(3)*b(1)+a(4)*b(2);
ab(4,1) = a(1)*b(4)+a(2)*b(3)-a(3)*b(2)+a(4)*b(1);
if ab(1)<0
ab=-ab;
end
end