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function [xa_apo,Pa_apo,Rot_matrix,eulerAngles,debugOutput]...
= AttitudeEKF(approx_prediction,use_inertia_matrix,zFlag,dt,z,q_rotSpeed,q_rotAcc,q_acc,q_mag,r_gyro,r_accel,r_mag,J)
%LQG Postion Estimator and Controller
% Observer:
% x[n|n] = x[n|n-1] + M(y[n] - Cx[n|n-1] - Du[n])
% x[n+1|n] = Ax[n|n] + Bu[n]
%
% $Author: Tobias Naegeli $ $Date: 2014 $ $Revision: 3 $
%
%
% Arguments:
% approx_prediction: if 1 then the exponential map is approximated with a
% first order taylor approximation. has at the moment not a big influence
% (just 1st or 2nd order approximation) we should change it to rodriquez
% approximation.
% use_inertia_matrix: set to true if you have the inertia matrix J for your
% quadrotor
% xa_apo_k: old state vectotr
% zFlag: if sensor measurement is available [gyro, acc, mag]
% dt: dt in s
% z: measurements [gyro, acc, mag]
% q_rotSpeed: process noise gyro
% q_rotAcc: process noise gyro acceleration
% q_acc: process noise acceleration
% q_mag: process noise magnetometer
% r_gyro: measurement noise gyro
% r_accel: measurement noise accel
% r_mag: measurement noise mag
% J: moment of inertia matrix
% Output:
% xa_apo: updated state vectotr
% Pa_apo: updated state covariance matrix
% Rot_matrix: rotation matrix
% eulerAngles: euler angles
% debugOutput: not used
%% model specific parameters
% compute once the inverse of the Inertia
persistent Ji;
if isempty(Ji)
Ji=single(inv(J));
end
%% init
persistent x_apo
if(isempty(x_apo))
gyro_init=single([0;0;0]);
gyro_acc_init=single([0;0;0]);
acc_init=single([0;0;-9.81]);
mag_init=single([1;0;0]);
x_apo=single([gyro_init;gyro_acc_init;acc_init;mag_init]);
end
persistent P_apo
if(isempty(P_apo))
% P_apo = single(eye(NSTATES) * 1000);
P_apo = single(200*ones(12));
end
debugOutput = single(zeros(4,1));
%% copy the states
wx= x_apo(1); % x body angular rate
wy= x_apo(2); % y body angular rate
wz= x_apo(3); % z body angular rate
wax= x_apo(4); % x body angular acceleration
way= x_apo(5); % y body angular acceleration
waz= x_apo(6); % z body angular acceleration
zex= x_apo(7); % x component gravity vector
zey= x_apo(8); % y component gravity vector
zez= x_apo(9); % z component gravity vector
mux= x_apo(10); % x component magnetic field vector
muy= x_apo(11); % y component magnetic field vector
muz= x_apo(12); % z component magnetic field vector
%% prediction section
% compute the apriori state estimate from the previous aposteriori estimate
%body angular accelerations
if (use_inertia_matrix==1)
wak =[wax;way;waz]+Ji*(-cross([wax;way;waz],J*[wax;way;waz]))*dt;
else
wak =[wax;way;waz];
end
%body angular rates
wk =[wx; wy; wz] + dt*wak;
%derivative of the prediction rotation matrix
O=[0,-wz,wy;wz,0,-wx;-wy,wx,0]';
%prediction of the earth z vector
if (approx_prediction==1)
%e^(Odt)=I+dt*O+dt^2/2!O^2
% so we do a first order approximation of the exponential map
zek =(O*dt+single(eye(3)))*[zex;zey;zez];
else
zek =(single(eye(3))+O*dt+dt^2/2*O^2)*[zex;zey;zez];
%zek =expm2(O*dt)*[zex;zey;zez]; not working because use double
%precision
end
%prediction of the magnetic vector
if (approx_prediction==1)
%e^(Odt)=I+dt*O+dt^2/2!O^2
% so we do a first order approximation of the exponential map
muk =(O*dt+single(eye(3)))*[mux;muy;muz];
else
muk =(single(eye(3))+O*dt+dt^2/2*O^2)*[mux;muy;muz];
%muk =expm2(O*dt)*[mux;muy;muz]; not working because use double
%precision
end
x_apr=[wk;wak;zek;muk];
% compute the apriori error covariance estimate from the previous
%aposteriori estimate
EZ=[0,zez,-zey;
-zez,0,zex;
zey,-zex,0]';
MA=[0,muz,-muy;
-muz,0,mux;
muy,-mux,0]';
E=single(eye(3));
Z=single(zeros(3));
A_lin=[ Z, E, Z, Z
Z, Z, Z, Z
EZ, Z, O, Z
MA, Z, Z, O];
A_lin=eye(12)+A_lin*dt;
%process covariance matrix
persistent Q
if (isempty(Q))
Q=diag([ q_rotSpeed,q_rotSpeed,q_rotSpeed,...
q_rotAcc,q_rotAcc,q_rotAcc,...
q_acc,q_acc,q_acc,...
q_mag,q_mag,q_mag]);
end
P_apr=A_lin*P_apo*A_lin'+Q;
%% update
if zFlag(1)==1&&zFlag(2)==1&&zFlag(3)==1
% R=[r_gyro,0,0,0,0,0,0,0,0;
% 0,r_gyro,0,0,0,0,0,0,0;
% 0,0,r_gyro,0,0,0,0,0,0;
% 0,0,0,r_accel,0,0,0,0,0;
% 0,0,0,0,r_accel,0,0,0,0;
% 0,0,0,0,0,r_accel,0,0,0;
% 0,0,0,0,0,0,r_mag,0,0;
% 0,0,0,0,0,0,0,r_mag,0;
% 0,0,0,0,0,0,0,0,r_mag];
R_v=[r_gyro,r_gyro,r_gyro,r_accel,r_accel,r_accel,r_mag,r_mag,r_mag];
%observation matrix
%[zw;ze;zmk];
H_k=[ E, Z, Z, Z;
Z, Z, E, Z;
Z, Z, Z, E];
y_k=z(1:9)-H_k*x_apr;
%S_k=H_k*P_apr*H_k'+R;
S_k=H_k*P_apr*H_k';
S_k(1:9+1:end) = S_k(1:9+1:end) + R_v;
K_k=(P_apr*H_k'/(S_k));
x_apo=x_apr+K_k*y_k;
P_apo=(eye(12)-K_k*H_k)*P_apr;
else
if zFlag(1)==1&&zFlag(2)==0&&zFlag(3)==0
R=[r_gyro,0,0;
0,r_gyro,0;
0,0,r_gyro];
R_v=[r_gyro,r_gyro,r_gyro];
%observation matrix
H_k=[ E, Z, Z, Z];
y_k=z(1:3)-H_k(1:3,1:12)*x_apr;
% S_k=H_k(1:3,1:12)*P_apr*H_k(1:3,1:12)'+R(1:3,1:3);
S_k=H_k(1:3,1:12)*P_apr*H_k(1:3,1:12)';
S_k(1:3+1:end) = S_k(1:3+1:end) + R_v;
K_k=(P_apr*H_k(1:3,1:12)'/(S_k));
x_apo=x_apr+K_k*y_k;
P_apo=(eye(12)-K_k*H_k(1:3,1:12))*P_apr;
else
if zFlag(1)==1&&zFlag(2)==1&&zFlag(3)==0
% R=[r_gyro,0,0,0,0,0;
% 0,r_gyro,0,0,0,0;
% 0,0,r_gyro,0,0,0;
% 0,0,0,r_accel,0,0;
% 0,0,0,0,r_accel,0;
% 0,0,0,0,0,r_accel];
R_v=[r_gyro,r_gyro,r_gyro,r_accel,r_accel,r_accel];
%observation matrix
H_k=[ E, Z, Z, Z;
Z, Z, E, Z];
y_k=z(1:6)-H_k(1:6,1:12)*x_apr;
% S_k=H_k(1:6,1:12)*P_apr*H_k(1:6,1:12)'+R(1:6,1:6);
S_k=H_k(1:6,1:12)*P_apr*H_k(1:6,1:12)';
S_k(1:6+1:end) = S_k(1:6+1:end) + R_v;
K_k=(P_apr*H_k(1:6,1:12)'/(S_k));
x_apo=x_apr+K_k*y_k;
P_apo=(eye(12)-K_k*H_k(1:6,1:12))*P_apr;
else
if zFlag(1)==1&&zFlag(2)==0&&zFlag(3)==1
% R=[r_gyro,0,0,0,0,0;
% 0,r_gyro,0,0,0,0;
% 0,0,r_gyro,0,0,0;
% 0,0,0,r_mag,0,0;
% 0,0,0,0,r_mag,0;
% 0,0,0,0,0,r_mag];
R_v=[r_gyro,r_gyro,r_gyro,r_mag,r_mag,r_mag];
%observation matrix
H_k=[ E, Z, Z, Z;
Z, Z, Z, E];
y_k=[z(1:3);z(7:9)]-H_k(1:6,1:12)*x_apr;
%S_k=H_k(1:6,1:12)*P_apr*H_k(1:6,1:12)'+R(1:6,1:6);
S_k=H_k(1:6,1:12)*P_apr*H_k(1:6,1:12)';
S_k(1:6+1:end) = S_k(1:6+1:end) + R_v;
K_k=(P_apr*H_k(1:6,1:12)'/(S_k));
x_apo=x_apr+K_k*y_k;
P_apo=(eye(12)-K_k*H_k(1:6,1:12))*P_apr;
else
x_apo=x_apr;
P_apo=P_apr;
end
end
end
end
%% euler anglels extraction
z_n_b = -x_apo(7:9)./norm(x_apo(7:9));
m_n_b = x_apo(10:12)./norm(x_apo(10:12));
y_n_b=cross(z_n_b,m_n_b);
y_n_b=y_n_b./norm(y_n_b);
x_n_b=(cross(y_n_b,z_n_b));
x_n_b=x_n_b./norm(x_n_b);
xa_apo=x_apo;
Pa_apo=P_apo;
% rotation matrix from earth to body system
Rot_matrix=[x_n_b,y_n_b,z_n_b];
phi=atan2(Rot_matrix(2,3),Rot_matrix(3,3));
theta=-asin(Rot_matrix(1,3));
psi=atan2(Rot_matrix(1,2),Rot_matrix(1,1));
eulerAngles=[phi;theta;psi];
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