px4-firmware/EKF/mag_fusion.cpp

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/**
 * @file heading_fusion.cpp
 * Magnetometer fusion methods.
 *
 * @author Roman Bast <bapstroman@gmail.com>
 * @author Paul Riseborough <p_riseborough@live.com.au>
 *
 */
#include "ekf.h"
#include "mathlib.h"

void Ekf::fuseMag()
{
	// assign intermediate variables
	float q0 = _state.quat_nominal(0);
	float q1 = _state.quat_nominal(1);
	float q2 = _state.quat_nominal(2);
	float q3 = _state.quat_nominal(3);

	float magN = _state.mag_I(0);
	float magE = _state.mag_I(1);
	float magD = _state.mag_I(2);

	// XYZ Measurement uncertainty. Need to consider timing errors for fast rotations
	float R_MAG = fmaxf(_params.mag_noise, 0.0f);
	R_MAG = R_MAG * R_MAG;

	// intermediate variables from algebraic optimisation
	float SH_MAG[9];
	SH_MAG[0] = 2*magD*q3 + 2*magE*q2 + 2*magN*q1;
	SH_MAG[1] = 2*magD*q0 - 2*magE*q1 + 2*magN*q2;
	SH_MAG[2] = 2*magD*q1 + 2*magE*q0 - 2*magN*q3;
	SH_MAG[3] = sq(q3);
	SH_MAG[4] = sq(q2);
	SH_MAG[5] = sq(q1);
	SH_MAG[6] = sq(q0);
	SH_MAG[7] = 2*magN*q0;
	SH_MAG[8] = 2*magE*q3;

	// rotate magnetometer earth field state into body frame
	matrix::Dcm<float> R_to_body(_state.quat_nominal);
	R_to_body = R_to_body.transpose();

	Vector3f mag_I_rot = R_to_body * _state.mag_I;

	// compute magnetometer innovations
	_mag_innov[0] = (mag_I_rot(0) + _state.mag_B(0)) - _mag_sample_delayed.mag(0);
	_mag_innov[1] = (mag_I_rot(1) + _state.mag_B(1)) - _mag_sample_delayed.mag(1);
	_mag_innov[2] = (mag_I_rot(2) + _state.mag_B(2)) - _mag_sample_delayed.mag(2);

	// Note that although the observation jacobians and kalman gains are decalred as arrays
	// sequential fusion of the X,Y and Z components is used.
	float H_MAG[3][24] = {};
	float Kfusion[24] = {};

	// Calculate observation Jacobians and kalman gains for each magnetometer axis
	// X Axis
	H_MAG[0][0] = SH_MAG[7] + SH_MAG[8] - 2*magD*q2;
	H_MAG[0][1] = SH_MAG[0];
	H_MAG[0][2] = -SH_MAG[1];
	H_MAG[0][3] = SH_MAG[2];
	H_MAG[0][16] = SH_MAG[5] - SH_MAG[4] - SH_MAG[3] + SH_MAG[6];
	H_MAG[0][17] = 2*q0*q3 + 2*q1*q2;
	H_MAG[0][18] = 2*q1*q3 - 2*q0*q2;
	H_MAG[0][19] = 1.0f;

	// Y axis

	H_MAG[1][0] = SH_MAG[2];
	H_MAG[1][1] = SH_MAG[1];
	H_MAG[1][2] = SH_MAG[0];
	H_MAG[1][3] = 2*magD*q2 - SH_MAG[8] - SH_MAG[7];
	H_MAG[1][16] = 2*q1*q2 - 2*q0*q3;
	H_MAG[1][17] = SH_MAG[4] - SH_MAG[3] - SH_MAG[5] + SH_MAG[6];
	H_MAG[1][18] = 2*q0*q1 + 2*q2*q3;
	H_MAG[1][20] = 1.0f;

	// Z axis

	H_MAG[2][0] = SH_MAG[1];
	H_MAG[2][1] = -SH_MAG[2];
	H_MAG[2][2] = SH_MAG[7] + SH_MAG[8] - 2*magD*q2;
	H_MAG[2][3] = SH_MAG[0];
	H_MAG[2][16] = 2*q0*q2 + 2*q1*q3;
	H_MAG[2][17] = 2*q2*q3 - 2*q0*q1;
	H_MAG[2][18] = SH_MAG[3] - SH_MAG[4] - SH_MAG[5] + SH_MAG[6];
	H_MAG[2][21] = 1.0f;

	// Perform an innovation consistency check on each measurement and if one axis fails
	// do not fuse any data from the sensor because the most common errors affect multiple axes.
	bool mag_fail = false;
	for (uint8_t index = 0; index <= 2; index++) {
		_mag_test_ratio[index] = sq(_mag_innov[index]) / (sq(math::max(_params.mag_innov_gate, 1.0f)) * _mag_innov_var[index]);

		if (_mag_test_ratio[index] > 1.0f) {
			mag_fail = false;
			_innov_check_fail_status.value |= (1 << (index + 3));
		} else {
			_innov_check_fail_status.value &= !(1 << (index + 3));
		}
	}

	if (mag_fail) {
		return;
	}

	// update the states and covariance using sequential fusion of the magnetometer components
	for (uint8_t index = 0; index <= 2; index++) {
		// Calculate Kalman gains
		if (index == 0) {
			// intermediate variables
			float SK_MX[5] = {};
			// innovation variance
			_mag_innov_var[0] = (P[19][19] + R_MAG + P[1][19]*SH_MAG[0] - P[2][19]*SH_MAG[1] + P[3][19]*SH_MAG[2] - P[16][19]*(SH_MAG[3] + SH_MAG[4] - SH_MAG[5] - SH_MAG[6]) + (2*q0*q3 + 2*q1*q2)*(P[19][17] + P[1][17]*SH_MAG[0] - P[2][17]*SH_MAG[1] + P[3][17]*SH_MAG[2] - P[16][17]*(SH_MAG[3] + SH_MAG[4] - SH_MAG[5] - SH_MAG[6]) + P[17][17]*(2*q0*q3 + 2*q1*q2) - P[18][17]*(2*q0*q2 - 2*q1*q3) + P[0][17]*(SH_MAG[7] + SH_MAG[8] - 2*magD*q2)) - (2*q0*q2 - 2*q1*q3)*(P[19][18] + P[1][18]*SH_MAG[0] - P[2][18]*SH_MAG[1] + P[3][18]*SH_MAG[2] - P[16][18]*(SH_MAG[3] + SH_MAG[4] - SH_MAG[5] - SH_MAG[6]) + P[17][18]*(2*q0*q3 + 2*q1*q2) - P[18][18]*(2*q0*q2 - 2*q1*q3) + P[0][18]*(SH_MAG[7] + SH_MAG[8] - 2*magD*q2)) + (SH_MAG[7] + SH_MAG[8] - 2*magD*q2)*(P[19][0] + P[1][0]*SH_MAG[0] - P[2][0]*SH_MAG[1] + P[3][0]*SH_MAG[2] - P[16][0]*(SH_MAG[3] + SH_MAG[4] - SH_MAG[5] - SH_MAG[6]) + P[17][0]*(2*q0*q3 + 2*q1*q2) - P[18][0]*(2*q0*q2 - 2*q1*q3) + P[0][0]*(SH_MAG[7] + SH_MAG[8] - 2*magD*q2)) + P[17][19]*(2*q0*q3 + 2*q1*q2) - P[18][19]*(2*q0*q2 - 2*q1*q3) + SH_MAG[0]*(P[19][1] + P[1][1]*SH_MAG[0] - P[2][1]*SH_MAG[1] + P[3][1]*SH_MAG[2] - P[16][1]*(SH_MAG[3] + SH_MAG[4] - SH_MAG[5] - SH_MAG[6]) + P[17][1]*(2*q0*q3 + 2*q1*q2) - P[18][1]*(2*q0*q2 - 2*q1*q3) + P[0][1]*(SH_MAG[7] + SH_MAG[8] - 2*magD*q2)) - SH_MAG[1]*(P[19][2] + P[1][2]*SH_MAG[0] - P[2][2]*SH_MAG[1] + P[3][2]*SH_MAG[2] - P[16][2]*(SH_MAG[3] + SH_MAG[4] - SH_MAG[5] - SH_MAG[6]) + P[17][2]*(2*q0*q3 + 2*q1*q2) - P[18][2]*(2*q0*q2 - 2*q1*q3) + P[0][2]*(SH_MAG[7] + SH_MAG[8] - 2*magD*q2)) + SH_MAG[2]*(P[19][3] + P[1][3]*SH_MAG[0] - P[2][3]*SH_MAG[1] + P[3][3]*SH_MAG[2] - P[16][3]*(SH_MAG[3] + SH_MAG[4] - SH_MAG[5] - SH_MAG[6]) + P[17][3]*(2*q0*q3 + 2*q1*q2) - P[18][3]*(2*q0*q2 - 2*q1*q3) + P[0][3]*(SH_MAG[7] + SH_MAG[8] - 2*magD*q2)) - (SH_MAG[3] + SH_MAG[4] - SH_MAG[5] - SH_MAG[6])*(P[19][16] + P[1][16]*SH_MAG[0] - P[2][16]*SH_MAG[1] + P[3][16]*SH_MAG[2] - P[16][16]*(SH_MAG[3] + SH_MAG[4] - SH_MAG[5] - SH_MAG[6]) + P[17][16]*(2*q0*q3 + 2*q1*q2) - P[18][16]*(2*q0*q2 - 2*q1*q3) + P[0][16]*(SH_MAG[7] + SH_MAG[8] - 2*magD*q2)) + P[0][19]*(SH_MAG[7] + SH_MAG[8] - 2*magD*q2));

			// check for a badly conditioned covariance matrix
			if (_mag_innov_var[0] >= R_MAG) {
				// the innovation variance contribution from the state covariances is non-negative - no fault
				_fault_status.flags.bad_mag_x = false;

			} else {
				// the innovation variance contribution from the state covariances is negative which means the covariance matrix is badly conditioned
				_fault_status.flags.bad_mag_x = true;
				// we need to reinitialise the covariance matrix and abort this fusion step
				initialiseCovariance();
				return;
			}

			// Calculate X axis Kalman gains
			SK_MX[0] = 1.0f / _mag_innov_var[0];
			SK_MX[1] = SH_MAG[3] + SH_MAG[4] - SH_MAG[5] - SH_MAG[6];
			SK_MX[2] = SH_MAG[7] + SH_MAG[8] - 2*magD*q2;
			SK_MX[3] = 2*q0*q2 - 2*q1*q3;
			SK_MX[4] = 2*q0*q3 + 2*q1*q2;

			Kfusion[0] = SK_MX[0]*(P[0][19] + P[0][1]*SH_MAG[0] - P[0][2]*SH_MAG[1] + P[0][3]*SH_MAG[2] + P[0][0]*SK_MX[2] - P[0][16]*SK_MX[1] + P[0][17]*SK_MX[4] - P[0][18]*SK_MX[3]);
			Kfusion[1] = SK_MX[0]*(P[1][19] + P[1][1]*SH_MAG[0] - P[1][2]*SH_MAG[1] + P[1][3]*SH_MAG[2] + P[1][0]*SK_MX[2] - P[1][16]*SK_MX[1] + P[1][17]*SK_MX[4] - P[1][18]*SK_MX[3]);
			Kfusion[2] = SK_MX[0]*(P[2][19] + P[2][1]*SH_MAG[0] - P[2][2]*SH_MAG[1] + P[2][3]*SH_MAG[2] + P[2][0]*SK_MX[2] - P[2][16]*SK_MX[1] + P[2][17]*SK_MX[4] - P[2][18]*SK_MX[3]);
			Kfusion[3] = SK_MX[0]*(P[3][19] + P[3][1]*SH_MAG[0] - P[3][2]*SH_MAG[1] + P[3][3]*SH_MAG[2] + P[3][0]*SK_MX[2] - P[3][16]*SK_MX[1] + P[3][17]*SK_MX[4] - P[3][18]*SK_MX[3]);
			Kfusion[4] = SK_MX[0]*(P[4][19] + P[4][1]*SH_MAG[0] - P[4][2]*SH_MAG[1] + P[4][3]*SH_MAG[2] + P[4][0]*SK_MX[2] - P[4][16]*SK_MX[1] + P[4][17]*SK_MX[4] - P[4][18]*SK_MX[3]);
			Kfusion[5] = SK_MX[0]*(P[5][19] + P[5][1]*SH_MAG[0] - P[5][2]*SH_MAG[1] + P[5][3]*SH_MAG[2] + P[5][0]*SK_MX[2] - P[5][16]*SK_MX[1] + P[5][17]*SK_MX[4] - P[5][18]*SK_MX[3]);
			Kfusion[6] = SK_MX[0]*(P[6][19] + P[6][1]*SH_MAG[0] - P[6][2]*SH_MAG[1] + P[6][3]*SH_MAG[2] + P[6][0]*SK_MX[2] - P[6][16]*SK_MX[1] + P[6][17]*SK_MX[4] - P[6][18]*SK_MX[3]);
			Kfusion[7] = SK_MX[0]*(P[7][19] + P[7][1]*SH_MAG[0] - P[7][2]*SH_MAG[1] + P[7][3]*SH_MAG[2] + P[7][0]*SK_MX[2] - P[7][16]*SK_MX[1] + P[7][17]*SK_MX[4] - P[7][18]*SK_MX[3]);
			Kfusion[8] = SK_MX[0]*(P[8][19] + P[8][1]*SH_MAG[0] - P[8][2]*SH_MAG[1] + P[8][3]*SH_MAG[2] + P[8][0]*SK_MX[2] - P[8][16]*SK_MX[1] + P[8][17]*SK_MX[4] - P[8][18]*SK_MX[3]);
			Kfusion[9] = SK_MX[0]*(P[9][19] + P[9][1]*SH_MAG[0] - P[9][2]*SH_MAG[1] + P[9][3]*SH_MAG[2] + P[9][0]*SK_MX[2] - P[9][16]*SK_MX[1] + P[9][17]*SK_MX[4] - P[9][18]*SK_MX[3]);
			Kfusion[10] = SK_MX[0]*(P[10][19] + P[10][1]*SH_MAG[0] - P[10][2]*SH_MAG[1] + P[10][3]*SH_MAG[2] + P[10][0]*SK_MX[2] - P[10][16]*SK_MX[1] + P[10][17]*SK_MX[4] - P[10][18]*SK_MX[3]);
			Kfusion[11] = SK_MX[0]*(P[11][19] + P[11][1]*SH_MAG[0] - P[11][2]*SH_MAG[1] + P[11][3]*SH_MAG[2] + P[11][0]*SK_MX[2] - P[11][16]*SK_MX[1] + P[11][17]*SK_MX[4] - P[11][18]*SK_MX[3]);
			Kfusion[12] = SK_MX[0]*(P[12][19] + P[12][1]*SH_MAG[0] - P[12][2]*SH_MAG[1] + P[12][3]*SH_MAG[2] + P[12][0]*SK_MX[2] - P[12][16]*SK_MX[1] + P[12][17]*SK_MX[4] - P[12][18]*SK_MX[3]);
			Kfusion[13] = SK_MX[0]*(P[13][19] + P[13][1]*SH_MAG[0] - P[13][2]*SH_MAG[1] + P[13][3]*SH_MAG[2] + P[13][0]*SK_MX[2] - P[13][16]*SK_MX[1] + P[13][17]*SK_MX[4] - P[13][18]*SK_MX[3]);
			Kfusion[14] = SK_MX[0]*(P[14][19] + P[14][1]*SH_MAG[0] - P[14][2]*SH_MAG[1] + P[14][3]*SH_MAG[2] + P[14][0]*SK_MX[2] - P[14][16]*SK_MX[1] + P[14][17]*SK_MX[4] - P[14][18]*SK_MX[3]);
			Kfusion[15] = SK_MX[0]*(P[15][19] + P[15][1]*SH_MAG[0] - P[15][2]*SH_MAG[1] + P[15][3]*SH_MAG[2] + P[15][0]*SK_MX[2] - P[15][16]*SK_MX[1] + P[15][17]*SK_MX[4] - P[15][18]*SK_MX[3]);
			Kfusion[16] = SK_MX[0]*(P[16][19] + P[16][1]*SH_MAG[0] - P[16][2]*SH_MAG[1] + P[16][3]*SH_MAG[2] + P[16][0]*SK_MX[2] - P[16][16]*SK_MX[1] + P[16][17]*SK_MX[4] - P[16][18]*SK_MX[3]);
			Kfusion[17] = SK_MX[0]*(P[17][19] + P[17][1]*SH_MAG[0] - P[17][2]*SH_MAG[1] + P[17][3]*SH_MAG[2] + P[17][0]*SK_MX[2] - P[17][16]*SK_MX[1] + P[17][17]*SK_MX[4] - P[17][18]*SK_MX[3]);
			Kfusion[18] = SK_MX[0]*(P[18][19] + P[18][1]*SH_MAG[0] - P[18][2]*SH_MAG[1] + P[18][3]*SH_MAG[2] + P[18][0]*SK_MX[2] - P[18][16]*SK_MX[1] + P[18][17]*SK_MX[4] - P[18][18]*SK_MX[3]);
			Kfusion[19] = SK_MX[0]*(P[19][19] + P[19][1]*SH_MAG[0] - P[19][2]*SH_MAG[1] + P[19][3]*SH_MAG[2] + P[19][0]*SK_MX[2] - P[19][16]*SK_MX[1] + P[19][17]*SK_MX[4] - P[19][18]*SK_MX[3]);
			Kfusion[20] = SK_MX[0]*(P[20][19] + P[20][1]*SH_MAG[0] - P[20][2]*SH_MAG[1] + P[20][3]*SH_MAG[2] + P[20][0]*SK_MX[2] - P[20][16]*SK_MX[1] + P[20][17]*SK_MX[4] - P[20][18]*SK_MX[3]);
			Kfusion[21] = SK_MX[0]*(P[21][19] + P[21][1]*SH_MAG[0] - P[21][2]*SH_MAG[1] + P[21][3]*SH_MAG[2] + P[21][0]*SK_MX[2] - P[21][16]*SK_MX[1] + P[21][17]*SK_MX[4] - P[21][18]*SK_MX[3]);
			Kfusion[22] = SK_MX[0]*(P[22][19] + P[22][1]*SH_MAG[0] - P[22][2]*SH_MAG[1] + P[22][3]*SH_MAG[2] + P[22][0]*SK_MX[2] - P[22][16]*SK_MX[1] + P[22][17]*SK_MX[4] - P[22][18]*SK_MX[3]);
			Kfusion[23] = SK_MX[0]*(P[23][19] + P[23][1]*SH_MAG[0] - P[23][2]*SH_MAG[1] + P[23][3]*SH_MAG[2] + P[23][0]*SK_MX[2] - P[23][16]*SK_MX[1] + P[23][17]*SK_MX[4] - P[23][18]*SK_MX[3]);

		} else if (index == 1) {
			// intermediate variables - note SK_MY[0] is 1/(innovation variance)
			float SK_MY[5];
			_mag_innov_var[1] = (P[20][20] + R_MAG + P[0][20]*SH_MAG[2] + P[1][20]*SH_MAG[1] + P[2][20]*SH_MAG[0] - P[17][20]*(SH_MAG[3] - SH_MAG[4] + SH_MAG[5] - SH_MAG[6]) - (2*q0*q3 - 2*q1*q2)*(P[20][16] + P[0][16]*SH_MAG[2] + P[1][16]*SH_MAG[1] + P[2][16]*SH_MAG[0] - P[17][16]*(SH_MAG[3] - SH_MAG[4] + SH_MAG[5] - SH_MAG[6]) - P[16][16]*(2*q0*q3 - 2*q1*q2) + P[18][16]*(2*q0*q1 + 2*q2*q3) - P[3][16]*(SH_MAG[7] + SH_MAG[8] - 2*magD*q2)) + (2*q0*q1 + 2*q2*q3)*(P[20][18] + P[0][18]*SH_MAG[2] + P[1][18]*SH_MAG[1] + P[2][18]*SH_MAG[0] - P[17][18]*(SH_MAG[3] - SH_MAG[4] + SH_MAG[5] - SH_MAG[6]) - P[16][18]*(2*q0*q3 - 2*q1*q2) + P[18][18]*(2*q0*q1 + 2*q2*q3) - P[3][18]*(SH_MAG[7] + SH_MAG[8] - 2*magD*q2)) - (SH_MAG[7] + SH_MAG[8] - 2*magD*q2)*(P[20][3] + P[0][3]*SH_MAG[2] + P[1][3]*SH_MAG[1] + P[2][3]*SH_MAG[0] - P[17][3]*(SH_MAG[3] - SH_MAG[4] + SH_MAG[5] - SH_MAG[6]) - P[16][3]*(2*q0*q3 - 2*q1*q2) + P[18][3]*(2*q0*q1 + 2*q2*q3) - P[3][3]*(SH_MAG[7] + SH_MAG[8] - 2*magD*q2)) - P[16][20]*(2*q0*q3 - 2*q1*q2) + P[18][20]*(2*q0*q1 + 2*q2*q3) + SH_MAG[2]*(P[20][0] + P[0][0]*SH_MAG[2] + P[1][0]*SH_MAG[1] + P[2][0]*SH_MAG[0] - P[17][0]*(SH_MAG[3] - SH_MAG[4] + SH_MAG[5] - SH_MAG[6]) - P[16][0]*(2*q0*q3 - 2*q1*q2) + P[18][0]*(2*q0*q1 + 2*q2*q3) - P[3][0]*(SH_MAG[7] + SH_MAG[8] - 2*magD*q2)) + SH_MAG[1]*(P[20][1] + P[0][1]*SH_MAG[2] + P[1][1]*SH_MAG[1] + P[2][1]*SH_MAG[0] - P[17][1]*(SH_MAG[3] - SH_MAG[4] + SH_MAG[5] - SH_MAG[6]) - P[16][1]*(2*q0*q3 - 2*q1*q2) + P[18][1]*(2*q0*q1 + 2*q2*q3) - P[3][1]*(SH_MAG[7] + SH_MAG[8] - 2*magD*q2)) + SH_MAG[0]*(P[20][2] + P[0][2]*SH_MAG[2] + P[1][2]*SH_MAG[1] + P[2][2]*SH_MAG[0] - P[17][2]*(SH_MAG[3] - SH_MAG[4] + SH_MAG[5] - SH_MAG[6]) - P[16][2]*(2*q0*q3 - 2*q1*q2) + P[18][2]*(2*q0*q1 + 2*q2*q3) - P[3][2]*(SH_MAG[7] + SH_MAG[8] - 2*magD*q2)) - (SH_MAG[3] - SH_MAG[4] + SH_MAG[5] - SH_MAG[6])*(P[20][17] + P[0][17]*SH_MAG[2] + P[1][17]*SH_MAG[1] + P[2][17]*SH_MAG[0] - P[17][17]*(SH_MAG[3] - SH_MAG[4] + SH_MAG[5] - SH_MAG[6]) - P[16][17]*(2*q0*q3 - 2*q1*q2) + P[18][17]*(2*q0*q1 + 2*q2*q3) - P[3][17]*(SH_MAG[7] + SH_MAG[8] - 2*magD*q2)) - P[3][20]*(SH_MAG[7] + SH_MAG[8] - 2*magD*q2));

			// check for a badly conditioned covariance matrix
			if (_mag_innov_var[1] >= R_MAG) {
				// the innovation variance contribution from the state covariances is non-negative - no fault
				_fault_status.flags.bad_mag_y = false;

			} else {
				// the innovation variance contribution from the state covariances is negtive which means the covariance matrix is badly conditioned
				_fault_status.flags.bad_mag_y = true;
				// we need to reinitialise the covariance matrix and abort this fusion step
				initialiseCovariance();
				return;
			}

			// Calculate Y axis Kalman gains
			SK_MY[0] = 1.0f / _mag_innov_var[1];
			SK_MY[1] = SH_MAG[3] - SH_MAG[4] + SH_MAG[5] - SH_MAG[6];
			SK_MY[2] = SH_MAG[7] + SH_MAG[8] - 2*magD*q2;
			SK_MY[3] = 2*q0*q3 - 2*q1*q2;
			SK_MY[4] = 2*q0*q1 + 2*q2*q3;

			Kfusion[0] = SK_MY[0]*(P[0][20] + P[0][0]*SH_MAG[2] + P[0][1]*SH_MAG[1] + P[0][2]*SH_MAG[0] - P[0][3]*SK_MY[2] - P[0][17]*SK_MY[1] - P[0][16]*SK_MY[3] + P[0][18]*SK_MY[4]);
			Kfusion[1] = SK_MY[0]*(P[1][20] + P[1][0]*SH_MAG[2] + P[1][1]*SH_MAG[1] + P[1][2]*SH_MAG[0] - P[1][3]*SK_MY[2] - P[1][17]*SK_MY[1] - P[1][16]*SK_MY[3] + P[1][18]*SK_MY[4]);
			Kfusion[2] = SK_MY[0]*(P[2][20] + P[2][0]*SH_MAG[2] + P[2][1]*SH_MAG[1] + P[2][2]*SH_MAG[0] - P[2][3]*SK_MY[2] - P[2][17]*SK_MY[1] - P[2][16]*SK_MY[3] + P[2][18]*SK_MY[4]);
			Kfusion[3] = SK_MY[0]*(P[3][20] + P[3][0]*SH_MAG[2] + P[3][1]*SH_MAG[1] + P[3][2]*SH_MAG[0] - P[3][3]*SK_MY[2] - P[3][17]*SK_MY[1] - P[3][16]*SK_MY[3] + P[3][18]*SK_MY[4]);
			Kfusion[4] = SK_MY[0]*(P[4][20] + P[4][0]*SH_MAG[2] + P[4][1]*SH_MAG[1] + P[4][2]*SH_MAG[0] - P[4][3]*SK_MY[2] - P[4][17]*SK_MY[1] - P[4][16]*SK_MY[3] + P[4][18]*SK_MY[4]);
			Kfusion[5] = SK_MY[0]*(P[5][20] + P[5][0]*SH_MAG[2] + P[5][1]*SH_MAG[1] + P[5][2]*SH_MAG[0] - P[5][3]*SK_MY[2] - P[5][17]*SK_MY[1] - P[5][16]*SK_MY[3] + P[5][18]*SK_MY[4]);
			Kfusion[6] = SK_MY[0]*(P[6][20] + P[6][0]*SH_MAG[2] + P[6][1]*SH_MAG[1] + P[6][2]*SH_MAG[0] - P[6][3]*SK_MY[2] - P[6][17]*SK_MY[1] - P[6][16]*SK_MY[3] + P[6][18]*SK_MY[4]);
			Kfusion[7] = SK_MY[0]*(P[7][20] + P[7][0]*SH_MAG[2] + P[7][1]*SH_MAG[1] + P[7][2]*SH_MAG[0] - P[7][3]*SK_MY[2] - P[7][17]*SK_MY[1] - P[7][16]*SK_MY[3] + P[7][18]*SK_MY[4]);
			Kfusion[8] = SK_MY[0]*(P[8][20] + P[8][0]*SH_MAG[2] + P[8][1]*SH_MAG[1] + P[8][2]*SH_MAG[0] - P[8][3]*SK_MY[2] - P[8][17]*SK_MY[1] - P[8][16]*SK_MY[3] + P[8][18]*SK_MY[4]);
			Kfusion[9] = SK_MY[0]*(P[9][20] + P[9][0]*SH_MAG[2] + P[9][1]*SH_MAG[1] + P[9][2]*SH_MAG[0] - P[9][3]*SK_MY[2] - P[9][17]*SK_MY[1] - P[9][16]*SK_MY[3] + P[9][18]*SK_MY[4]);
			Kfusion[10] = SK_MY[0]*(P[10][20] + P[10][0]*SH_MAG[2] + P[10][1]*SH_MAG[1] + P[10][2]*SH_MAG[0] - P[10][3]*SK_MY[2] - P[10][17]*SK_MY[1] - P[10][16]*SK_MY[3] + P[10][18]*SK_MY[4]);
			Kfusion[11] = SK_MY[0]*(P[11][20] + P[11][0]*SH_MAG[2] + P[11][1]*SH_MAG[1] + P[11][2]*SH_MAG[0] - P[11][3]*SK_MY[2] - P[11][17]*SK_MY[1] - P[11][16]*SK_MY[3] + P[11][18]*SK_MY[4]);
			Kfusion[12] = SK_MY[0]*(P[12][20] + P[12][0]*SH_MAG[2] + P[12][1]*SH_MAG[1] + P[12][2]*SH_MAG[0] - P[12][3]*SK_MY[2] - P[12][17]*SK_MY[1] - P[12][16]*SK_MY[3] + P[12][18]*SK_MY[4]);
			Kfusion[13] = SK_MY[0]*(P[13][20] + P[13][0]*SH_MAG[2] + P[13][1]*SH_MAG[1] + P[13][2]*SH_MAG[0] - P[13][3]*SK_MY[2] - P[13][17]*SK_MY[1] - P[13][16]*SK_MY[3] + P[13][18]*SK_MY[4]);
			Kfusion[14] = SK_MY[0]*(P[14][20] + P[14][0]*SH_MAG[2] + P[14][1]*SH_MAG[1] + P[14][2]*SH_MAG[0] - P[14][3]*SK_MY[2] - P[14][17]*SK_MY[1] - P[14][16]*SK_MY[3] + P[14][18]*SK_MY[4]);
			Kfusion[15] = SK_MY[0]*(P[15][20] + P[15][0]*SH_MAG[2] + P[15][1]*SH_MAG[1] + P[15][2]*SH_MAG[0] - P[15][3]*SK_MY[2] - P[15][17]*SK_MY[1] - P[15][16]*SK_MY[3] + P[15][18]*SK_MY[4]);
			Kfusion[16] = SK_MY[0]*(P[16][20] + P[16][0]*SH_MAG[2] + P[16][1]*SH_MAG[1] + P[16][2]*SH_MAG[0] - P[16][3]*SK_MY[2] - P[16][17]*SK_MY[1] - P[16][16]*SK_MY[3] + P[16][18]*SK_MY[4]);
			Kfusion[17] = SK_MY[0]*(P[17][20] + P[17][0]*SH_MAG[2] + P[17][1]*SH_MAG[1] + P[17][2]*SH_MAG[0] - P[17][3]*SK_MY[2] - P[17][17]*SK_MY[1] - P[17][16]*SK_MY[3] + P[17][18]*SK_MY[4]);
			Kfusion[18] = SK_MY[0]*(P[18][20] + P[18][0]*SH_MAG[2] + P[18][1]*SH_MAG[1] + P[18][2]*SH_MAG[0] - P[18][3]*SK_MY[2] - P[18][17]*SK_MY[1] - P[18][16]*SK_MY[3] + P[18][18]*SK_MY[4]);
			Kfusion[19] = SK_MY[0]*(P[19][20] + P[19][0]*SH_MAG[2] + P[19][1]*SH_MAG[1] + P[19][2]*SH_MAG[0] - P[19][3]*SK_MY[2] - P[19][17]*SK_MY[1] - P[19][16]*SK_MY[3] + P[19][18]*SK_MY[4]);
			Kfusion[20] = SK_MY[0]*(P[20][20] + P[20][0]*SH_MAG[2] + P[20][1]*SH_MAG[1] + P[20][2]*SH_MAG[0] - P[20][3]*SK_MY[2] - P[20][17]*SK_MY[1] - P[20][16]*SK_MY[3] + P[20][18]*SK_MY[4]);
			Kfusion[21] = SK_MY[0]*(P[21][20] + P[21][0]*SH_MAG[2] + P[21][1]*SH_MAG[1] + P[21][2]*SH_MAG[0] - P[21][3]*SK_MY[2] - P[21][17]*SK_MY[1] - P[21][16]*SK_MY[3] + P[21][18]*SK_MY[4]);
			Kfusion[22] = SK_MY[0]*(P[22][20] + P[22][0]*SH_MAG[2] + P[22][1]*SH_MAG[1] + P[22][2]*SH_MAG[0] - P[22][3]*SK_MY[2] - P[22][17]*SK_MY[1] - P[22][16]*SK_MY[3] + P[22][18]*SK_MY[4]);
			Kfusion[23] = SK_MY[0]*(P[23][20] + P[23][0]*SH_MAG[2] + P[23][1]*SH_MAG[1] + P[23][2]*SH_MAG[0] - P[23][3]*SK_MY[2] - P[23][17]*SK_MY[1] - P[23][16]*SK_MY[3] + P[23][18]*SK_MY[4]);

		} else if (index == 2) {
			// intermediate variables
			float SK_MZ[5];
			_mag_innov_var[2] = (P[21][21] + R_MAG + P[0][21]*SH_MAG[1] - P[1][21]*SH_MAG[2] + P[3][21]*SH_MAG[0] + P[18][21]*(SH_MAG[3] - SH_MAG[4] - SH_MAG[5] + SH_MAG[6]) + (2*q0*q2 + 2*q1*q3)*(P[21][16] + P[0][16]*SH_MAG[1] - P[1][16]*SH_MAG[2] + P[3][16]*SH_MAG[0] + P[18][16]*(SH_MAG[3] - SH_MAG[4] - SH_MAG[5] + SH_MAG[6]) + P[16][16]*(2*q0*q2 + 2*q1*q3) - P[17][16]*(2*q0*q1 - 2*q2*q3) + P[2][16]*(SH_MAG[7] + SH_MAG[8] - 2*magD*q2)) - (2*q0*q1 - 2*q2*q3)*(P[21][17] + P[0][17]*SH_MAG[1] - P[1][17]*SH_MAG[2] + P[3][17]*SH_MAG[0] + P[18][17]*(SH_MAG[3] - SH_MAG[4] - SH_MAG[5] + SH_MAG[6]) + P[16][17]*(2*q0*q2 + 2*q1*q3) - P[17][17]*(2*q0*q1 - 2*q2*q3) + P[2][17]*(SH_MAG[7] + SH_MAG[8] - 2*magD*q2)) + (SH_MAG[7] + SH_MAG[8] - 2*magD*q2)*(P[21][2] + P[0][2]*SH_MAG[1] - P[1][2]*SH_MAG[2] + P[3][2]*SH_MAG[0] + P[18][2]*(SH_MAG[3] - SH_MAG[4] - SH_MAG[5] + SH_MAG[6]) + P[16][2]*(2*q0*q2 + 2*q1*q3) - P[17][2]*(2*q0*q1 - 2*q2*q3) + P[2][2]*(SH_MAG[7] + SH_MAG[8] - 2*magD*q2)) + P[16][21]*(2*q0*q2 + 2*q1*q3) - P[17][21]*(2*q0*q1 - 2*q2*q3) + SH_MAG[1]*(P[21][0] + P[0][0]*SH_MAG[1] - P[1][0]*SH_MAG[2] + P[3][0]*SH_MAG[0] + P[18][0]*(SH_MAG[3] - SH_MAG[4] - SH_MAG[5] + SH_MAG[6]) + P[16][0]*(2*q0*q2 + 2*q1*q3) - P[17][0]*(2*q0*q1 - 2*q2*q3) + P[2][0]*(SH_MAG[7] + SH_MAG[8] - 2*magD*q2)) - SH_MAG[2]*(P[21][1] + P[0][1]*SH_MAG[1] - P[1][1]*SH_MAG[2] + P[3][1]*SH_MAG[0] + P[18][1]*(SH_MAG[3] - SH_MAG[4] - SH_MAG[5] + SH_MAG[6]) + P[16][1]*(2*q0*q2 + 2*q1*q3) - P[17][1]*(2*q0*q1 - 2*q2*q3) + P[2][1]*(SH_MAG[7] + SH_MAG[8] - 2*magD*q2)) + SH_MAG[0]*(P[21][3] + P[0][3]*SH_MAG[1] - P[1][3]*SH_MAG[2] + P[3][3]*SH_MAG[0] + P[18][3]*(SH_MAG[3] - SH_MAG[4] - SH_MAG[5] + SH_MAG[6]) + P[16][3]*(2*q0*q2 + 2*q1*q3) - P[17][3]*(2*q0*q1 - 2*q2*q3) + P[2][3]*(SH_MAG[7] + SH_MAG[8] - 2*magD*q2)) + (SH_MAG[3] - SH_MAG[4] - SH_MAG[5] + SH_MAG[6])*(P[21][18] + P[0][18]*SH_MAG[1] - P[1][18]*SH_MAG[2] + P[3][18]*SH_MAG[0] + P[18][18]*(SH_MAG[3] - SH_MAG[4] - SH_MAG[5] + SH_MAG[6]) + P[16][18]*(2*q0*q2 + 2*q1*q3) - P[17][18]*(2*q0*q1 - 2*q2*q3) + P[2][18]*(SH_MAG[7] + SH_MAG[8] - 2*magD*q2)) + P[2][21]*(SH_MAG[7] + SH_MAG[8] - 2*magD*q2));

			// check for a badly conditioned covariance matrix
			if (_mag_innov_var[2] >= R_MAG) {
				// the innovation variance contribution from the state covariances is non-negative - no fault
				_fault_status.flags.bad_mag_z = false;

			} else {
				// the innovation variance contribution from the state covariances is negtive which means the covariance matrix is badly conditioned
				_fault_status.flags.bad_mag_z = true;
				// we need to reinitialise the covariance matrix and abort this fusion step
				initialiseCovariance();
				return;
			}

			// Calculate Z axis Kalman gains
			SK_MZ[0] = 1.0f / _mag_innov_var[2];
			SK_MZ[1] = SH_MAG[3] - SH_MAG[4] - SH_MAG[5] + SH_MAG[6];
			SK_MZ[2] = SH_MAG[7] + SH_MAG[8] - 2*magD*q2;
			SK_MZ[3] = 2*q0*q1 - 2*q2*q3;
			SK_MZ[4] = 2*q0*q2 + 2*q1*q3;

			Kfusion[0] = SK_MZ[0]*(P[0][21] + P[0][0]*SH_MAG[1] - P[0][1]*SH_MAG[2] + P[0][3]*SH_MAG[0] + P[0][2]*SK_MZ[2] + P[0][18]*SK_MZ[1] + P[0][16]*SK_MZ[4] - P[0][17]*SK_MZ[3]);
			Kfusion[1] = SK_MZ[0]*(P[1][21] + P[1][0]*SH_MAG[1] - P[1][1]*SH_MAG[2] + P[1][3]*SH_MAG[0] + P[1][2]*SK_MZ[2] + P[1][18]*SK_MZ[1] + P[1][16]*SK_MZ[4] - P[1][17]*SK_MZ[3]);
			Kfusion[2] = SK_MZ[0]*(P[2][21] + P[2][0]*SH_MAG[1] - P[2][1]*SH_MAG[2] + P[2][3]*SH_MAG[0] + P[2][2]*SK_MZ[2] + P[2][18]*SK_MZ[1] + P[2][16]*SK_MZ[4] - P[2][17]*SK_MZ[3]);
			Kfusion[3] = SK_MZ[0]*(P[3][21] + P[3][0]*SH_MAG[1] - P[3][1]*SH_MAG[2] + P[3][3]*SH_MAG[0] + P[3][2]*SK_MZ[2] + P[3][18]*SK_MZ[1] + P[3][16]*SK_MZ[4] - P[3][17]*SK_MZ[3]);
			Kfusion[4] = SK_MZ[0]*(P[4][21] + P[4][0]*SH_MAG[1] - P[4][1]*SH_MAG[2] + P[4][3]*SH_MAG[0] + P[4][2]*SK_MZ[2] + P[4][18]*SK_MZ[1] + P[4][16]*SK_MZ[4] - P[4][17]*SK_MZ[3]);
			Kfusion[5] = SK_MZ[0]*(P[5][21] + P[5][0]*SH_MAG[1] - P[5][1]*SH_MAG[2] + P[5][3]*SH_MAG[0] + P[5][2]*SK_MZ[2] + P[5][18]*SK_MZ[1] + P[5][16]*SK_MZ[4] - P[5][17]*SK_MZ[3]);
			Kfusion[6] = SK_MZ[0]*(P[6][21] + P[6][0]*SH_MAG[1] - P[6][1]*SH_MAG[2] + P[6][3]*SH_MAG[0] + P[6][2]*SK_MZ[2] + P[6][18]*SK_MZ[1] + P[6][16]*SK_MZ[4] - P[6][17]*SK_MZ[3]);
			Kfusion[7] = SK_MZ[0]*(P[7][21] + P[7][0]*SH_MAG[1] - P[7][1]*SH_MAG[2] + P[7][3]*SH_MAG[0] + P[7][2]*SK_MZ[2] + P[7][18]*SK_MZ[1] + P[7][16]*SK_MZ[4] - P[7][17]*SK_MZ[3]);
			Kfusion[8] = SK_MZ[0]*(P[8][21] + P[8][0]*SH_MAG[1] - P[8][1]*SH_MAG[2] + P[8][3]*SH_MAG[0] + P[8][2]*SK_MZ[2] + P[8][18]*SK_MZ[1] + P[8][16]*SK_MZ[4] - P[8][17]*SK_MZ[3]);
			Kfusion[9] = SK_MZ[0]*(P[9][21] + P[9][0]*SH_MAG[1] - P[9][1]*SH_MAG[2] + P[9][3]*SH_MAG[0] + P[9][2]*SK_MZ[2] + P[9][18]*SK_MZ[1] + P[9][16]*SK_MZ[4] - P[9][17]*SK_MZ[3]);
			Kfusion[10] = SK_MZ[0]*(P[10][21] + P[10][0]*SH_MAG[1] - P[10][1]*SH_MAG[2] + P[10][3]*SH_MAG[0] + P[10][2]*SK_MZ[2] + P[10][18]*SK_MZ[1] + P[10][16]*SK_MZ[4] - P[10][17]*SK_MZ[3]);
			Kfusion[11] = SK_MZ[0]*(P[11][21] + P[11][0]*SH_MAG[1] - P[11][1]*SH_MAG[2] + P[11][3]*SH_MAG[0] + P[11][2]*SK_MZ[2] + P[11][18]*SK_MZ[1] + P[11][16]*SK_MZ[4] - P[11][17]*SK_MZ[3]);
			Kfusion[12] = SK_MZ[0]*(P[12][21] + P[12][0]*SH_MAG[1] - P[12][1]*SH_MAG[2] + P[12][3]*SH_MAG[0] + P[12][2]*SK_MZ[2] + P[12][18]*SK_MZ[1] + P[12][16]*SK_MZ[4] - P[12][17]*SK_MZ[3]);
			Kfusion[13] = SK_MZ[0]*(P[13][21] + P[13][0]*SH_MAG[1] - P[13][1]*SH_MAG[2] + P[13][3]*SH_MAG[0] + P[13][2]*SK_MZ[2] + P[13][18]*SK_MZ[1] + P[13][16]*SK_MZ[4] - P[13][17]*SK_MZ[3]);
			Kfusion[14] = SK_MZ[0]*(P[14][21] + P[14][0]*SH_MAG[1] - P[14][1]*SH_MAG[2] + P[14][3]*SH_MAG[0] + P[14][2]*SK_MZ[2] + P[14][18]*SK_MZ[1] + P[14][16]*SK_MZ[4] - P[14][17]*SK_MZ[3]);
			Kfusion[15] = SK_MZ[0]*(P[15][21] + P[15][0]*SH_MAG[1] - P[15][1]*SH_MAG[2] + P[15][3]*SH_MAG[0] + P[15][2]*SK_MZ[2] + P[15][18]*SK_MZ[1] + P[15][16]*SK_MZ[4] - P[15][17]*SK_MZ[3]);
			Kfusion[16] = SK_MZ[0]*(P[16][21] + P[16][0]*SH_MAG[1] - P[16][1]*SH_MAG[2] + P[16][3]*SH_MAG[0] + P[16][2]*SK_MZ[2] + P[16][18]*SK_MZ[1] + P[16][16]*SK_MZ[4] - P[16][17]*SK_MZ[3]);
			Kfusion[17] = SK_MZ[0]*(P[17][21] + P[17][0]*SH_MAG[1] - P[17][1]*SH_MAG[2] + P[17][3]*SH_MAG[0] + P[17][2]*SK_MZ[2] + P[17][18]*SK_MZ[1] + P[17][16]*SK_MZ[4] - P[17][17]*SK_MZ[3]);
			Kfusion[18] = SK_MZ[0]*(P[18][21] + P[18][0]*SH_MAG[1] - P[18][1]*SH_MAG[2] + P[18][3]*SH_MAG[0] + P[18][2]*SK_MZ[2] + P[18][18]*SK_MZ[1] + P[18][16]*SK_MZ[4] - P[18][17]*SK_MZ[3]);
			Kfusion[19] = SK_MZ[0]*(P[19][21] + P[19][0]*SH_MAG[1] - P[19][1]*SH_MAG[2] + P[19][3]*SH_MAG[0] + P[19][2]*SK_MZ[2] + P[19][18]*SK_MZ[1] + P[19][16]*SK_MZ[4] - P[19][17]*SK_MZ[3]);
			Kfusion[20] = SK_MZ[0]*(P[20][21] + P[20][0]*SH_MAG[1] - P[20][1]*SH_MAG[2] + P[20][3]*SH_MAG[0] + P[20][2]*SK_MZ[2] + P[20][18]*SK_MZ[1] + P[20][16]*SK_MZ[4] - P[20][17]*SK_MZ[3]);
			Kfusion[21] = SK_MZ[0]*(P[21][21] + P[21][0]*SH_MAG[1] - P[21][1]*SH_MAG[2] + P[21][3]*SH_MAG[0] + P[21][2]*SK_MZ[2] + P[21][18]*SK_MZ[1] + P[21][16]*SK_MZ[4] - P[21][17]*SK_MZ[3]);
			Kfusion[22] = SK_MZ[0]*(P[22][21] + P[22][0]*SH_MAG[1] - P[22][1]*SH_MAG[2] + P[22][3]*SH_MAG[0] + P[22][2]*SK_MZ[2] + P[22][18]*SK_MZ[1] + P[22][16]*SK_MZ[4] - P[22][17]*SK_MZ[3]);
			Kfusion[23] = SK_MZ[0]*(P[23][21] + P[23][0]*SH_MAG[1] - P[23][1]*SH_MAG[2] + P[23][3]*SH_MAG[0] + P[23][2]*SK_MZ[2] + P[23][18]*SK_MZ[1] + P[23][16]*SK_MZ[4] - P[23][17]*SK_MZ[3]);

		} else {
			return;
		}

		// apply covariance correction via P_new = (I -K*H)*P
		// first calculate expression for KHP
		// then calculate P - KHP
		for (unsigned row = 0; row < _k_num_states; row++) {
			for (unsigned column = 0; column <= 3; column++) {
				KH[row][column] = Kfusion[row] * H_MAG[index][column];
			}

			for (unsigned column = 16; column <= 21; column++) {
				KH[row][column] = Kfusion[row] * H_MAG[index][column];
			}

		}

		for (unsigned row = 0; row < _k_num_states; row++) {
			for (unsigned column = 0; column < _k_num_states; column++) {
				float tmp = KH[row][0] * P[0][column];
				tmp += KH[row][1] * P[1][column];
				tmp += KH[row][2] * P[2][column];
				tmp += KH[row][3] * P[3][column];
				tmp += KH[row][16] * P[16][column];
				tmp += KH[row][17] * P[17][column];
				tmp += KH[row][18] * P[18][column];
				tmp += KH[row][19] * P[19][column];
				tmp += KH[row][20] * P[20][column];
				tmp += KH[row][21] * P[21][column];
				KHP[row][column] = tmp;
			}
		}

		// if the covariance correction will result in a negative variance, then
		// the covariance marix is unhealthy and must be corrected
		bool healthy = true;
		_fault_status.flags.bad_mag_x = false;
		_fault_status.flags.bad_mag_y = false;
		_fault_status.flags.bad_mag_z = false;
		for (int i = 0; i < _k_num_states; i++) {
			if (P[i][i] < KHP[i][i]) {
				// zero rows and columns
				zeroRows(P,i,i);
				zeroCols(P,i,i);

				//flag as unhealthy
				healthy = false;

				// update individual measurement health status
				if (index == 0) {
					_fault_status.flags.bad_mag_x = true;
				} else if (index == 1) {
					_fault_status.flags.bad_mag_y = true;
				} else if (index == 2) {
					_fault_status.flags.bad_mag_z = true;
				}
			}
		}

		// only apply covariance and state corrrections if healthy
		if (healthy) {
			// apply the covariance corrections
			for (unsigned row = 0; row < _k_num_states; row++) {
				for (unsigned column = 0; column < _k_num_states; column++) {
					P[row][column] = P[row][column] - KHP[row][column];
				}
			}

			// correct the covariance marix for gross errors
			fixCovarianceErrors();

			// apply the state corrections
			fuse(Kfusion, _mag_innov[index]);

		}
	}
}

void Ekf::fuseHeading()
{
	// assign intermediate state variables
	float q0 = _state.quat_nominal(0);
	float q1 = _state.quat_nominal(1);
	float q2 = _state.quat_nominal(2);
	float q3 = _state.quat_nominal(3);

	float R_YAW = 1.0f;
	float predicted_hdg;
	float H_YAW[4];
	matrix::Vector3f mag_earth_pred;
	float measured_hdg;

	// determine if a 321 or 312 Euler sequence is best
	if (fabsf(_R_to_earth(2, 0)) < fabsf(_R_to_earth(2, 1))) {
		// calculate observation jacobian when we are observing the first rotation in a 321 sequence
		float t9 = q0*q3;
		float t10 = q1*q2;
		float t2 = t9+t10;
		float t3 = q0*q0;
		float t4 = q1*q1;
		float t5 = q2*q2;
		float t6 = q3*q3;
		float t7 = t3+t4-t5-t6;
		float t8 = t7*t7;
		if (t8 > 1e-6f) {
			t8 = 1.0f/t8;
		} else {
			return;
		}
		float t11 = t2*t2;
		float t12 = t8*t11*4.0f;
		float t13 = t12+1.0f;
		float t14;
		if (fabsf(t13) > 1e-6f) {
			t14 = 1.0f/t13;
		} else {
			return;
		}

		H_YAW[0] = t8*t14*(q3*t3-q3*t4+q3*t5+q3*t6+q0*q1*q2*2.0f)*-2.0f;
		H_YAW[1] = t8*t14*(-q2*t3+q2*t4+q2*t5+q2*t6+q0*q1*q3*2.0f)*-2.0f;
		H_YAW[2] = t8*t14*(q1*t3+q1*t4+q1*t5-q1*t6+q0*q2*q3*2.0f)*2.0f;
		H_YAW[3] = t8*t14*(q0*t3+q0*t4-q0*t5+q0*t6+q1*q2*q3*2.0f)*2.0f;

		// rotate the magnetometer measurement into earth frame
		matrix::Euler<float> euler321(_state.quat_nominal);
		predicted_hdg = euler321(2); // we will need the predicted heading to calculate the innovation

		// Set the yaw angle to zero and rotate the measurements into earth frame using the zero yaw angle
		euler321(2) = 0.0f;
		matrix::Dcm<float> R_to_earth(euler321);

		// calculate the observed yaw angle
		if (_control_status.flags.mag_hdg) {
			// rotate the magnetometer measurements into earth frame using a zero yaw angle
			mag_earth_pred = R_to_earth * _mag_sample_delayed.mag;
			// the angle of the projection onto the horizontal gives the yaw angle
			measured_hdg = -atan2f(mag_earth_pred(1), mag_earth_pred(0)) + _mag_declination;
		} else if (_control_status.flags.ev_yaw) {
			// convert the observed quaternion to a rotation matrix
			matrix::Dcm<float> R_to_earth_ev(_ev_sample_delayed.quat);	// transformation matrix from body to world frame
			// calculate the yaw angle for a 312 sequence
			measured_hdg = atan2f(R_to_earth_ev(1, 0) , R_to_earth_ev(0, 0));
		} else {
			// there is no yaw observation
			return;
		}

	} else {
		// calculate observaton jacobian when we are observing a rotation in a 312 sequence
		float t9 = q0*q3;
		float t10 = q1*q2;
		float t2 = t9-t10;
		float t3 = q0*q0;
		float t4 = q1*q1;
		float t5 = q2*q2;
		float t6 = q3*q3;
		float t7 = t3-t4+t5-t6;
		float t8 = t7*t7;
		if (t8 > 1e-6f) {
			t8 = 1.0f/t8;
		} else {
			return;
		}
		float t11 = t2*t2;
		float t12 = t8*t11*4.0f;
		float t13 = t12+1.0f;
		float t14;
		if (fabsf(t13) > 1e-6f) {
			t14 = 1.0f/t13;
		} else {
			return;
		}

		H_YAW[0] = t8*t14*(q3*t3+q3*t4-q3*t5+q3*t6-q0*q1*q2*2.0f)*-2.0f;
		H_YAW[1] = t8*t14*(q2*t3+q2*t4+q2*t5-q2*t6-q0*q1*q3*2.0f)*-2.0f;
		H_YAW[2] = t8*t14*(-q1*t3+q1*t4+q1*t5+q1*t6-q0*q2*q3*2.0f)*2.0f;
		H_YAW[3] = t8*t14*(q0*t3-q0*t4+q0*t5+q0*t6-q1*q2*q3*2.0f)*2.0f;

		// Calculate the 312 sequence euler angles that rotate from earth to body frame
		// See http://www.atacolorado.com/eulersequences.doc
		Vector3f euler312;
		euler312(0) = atan2f(-_R_to_earth(0, 1) , _R_to_earth(1, 1)); // first rotation (yaw)
		euler312(1) = asinf(_R_to_earth(2, 1)); // second rotation (roll)
		euler312(2) = atan2f(-_R_to_earth(2, 0) , _R_to_earth(2, 2)); // third rotation (pitch)

		predicted_hdg = euler312(0); // we will need the predicted heading to calculate the innovation

		// Set the first rotation (yaw) to zero and rotate the measurements into earth frame
		euler312(0) = 0.0f;

		// Calculate the body to earth frame rotation matrix from the euler angles using a 312 rotation sequence
		float c2 = cosf(euler312(2));
		float s2 = sinf(euler312(2));
		float s1 = sinf(euler312(1));
		float c1 = cosf(euler312(1));
		float s0 = sinf(euler312(0));
		float c0 = cosf(euler312(0));

		matrix::Dcm<float> R_to_earth;
		R_to_earth(0, 0) = c0 * c2 - s0 * s1 * s2;
		R_to_earth(1, 1) = c0 * c1;
		R_to_earth(2, 2) = c2 * c1;
		R_to_earth(0, 1) = -c1 * s0;
		R_to_earth(0, 2) = s2 * c0 + c2 * s1 * s0;
		R_to_earth(1, 0) = c2 * s0 + s2 * s1 * c0;
		R_to_earth(1, 2) = s0 * s2 - s1 * c0 * c2;
		R_to_earth(2, 0) = -s2 * c1;
		R_to_earth(2, 1) = s1;

		// calculate the observed yaw angle
		if (_control_status.flags.mag_hdg) {
			// rotate the magnetometer measurements into earth frame using a zero yaw angle
			mag_earth_pred = R_to_earth * _mag_sample_delayed.mag;
			// the angle of the projection onto the horizontal gives the yaw angle
			measured_hdg = -atan2f(mag_earth_pred(1), mag_earth_pred(0)) + _mag_declination;
		} else if (_control_status.flags.ev_yaw) {
			// convert the observed quaternion to a rotation matrix
			matrix::Dcm<float> R_to_earth_ev(_ev_sample_delayed.quat);	// transformation matrix from body to world frame
			// calculate the yaw angle for a 312 sequence
			measured_hdg = atan2f(-R_to_earth_ev(0, 1) , R_to_earth_ev(1, 1));
		} else {
			// there is no yaw observation
			return;
		}
	}

	// Calculate the observation variance
	if (_control_status.flags.mag_hdg) {
		// using magnetic heading tuning parameter
		R_YAW = sq(fmaxf(_params.mag_heading_noise, 1.0e-2f));
	} else if (_control_status.flags.ev_yaw) {
		// using error estimate from external vision data
		R_YAW = sq(fmaxf(_ev_sample_delayed.angErr, 1.0e-2f));
	} else {
		// there is no yaw observation
		return;
	}

	// Calculate innovation variance and Kalman gains, taking advantage of the fact that only the first 3 elements in H are non zero
	// calculate the innovaton variance
	float PH[4];
	_heading_innov_var = R_YAW;
	for (unsigned row = 0; row <= 3; row++) {
		PH[row] = 0.0f;

		for (uint8_t col = 0; col <= 3; col++) {
			PH[row] += P[row][col] * H_YAW[col];
		}

		_heading_innov_var += H_YAW[row] * PH[row];
	}

	float heading_innov_var_inv;

	// check if the innovation variance calculation is badly conditioned
	if (_heading_innov_var >= R_YAW) {
		// the innovation variance contribution from the state covariances is not negative, no fault
		_fault_status.flags.bad_mag_hdg = false;
		heading_innov_var_inv = 1.0f / _heading_innov_var;

	} else {
		// the innovation variance contribution from the state covariances is negative which means the covariance matrix is badly conditioned
		_fault_status.flags.bad_mag_hdg = true;

		// we reinitialise the covariance matrix and abort this fusion step
		initialiseCovariance();
		return;
	}

	// calculate the Kalman gains
	// only calculate gains for states we are using
	float Kfusion[_k_num_states] = {};

	for (uint8_t row = 0; row <= 15; row++) {
		Kfusion[row] = 0.0f;

		for (uint8_t col = 0; col <= 3; col++) {
			Kfusion[row] += P[row][col] * H_YAW[col];
		}

		Kfusion[row] *= heading_innov_var_inv;
	}

	if (_control_status.flags.wind) {
		for (uint8_t row = 22; row <= 23; row++) {
			Kfusion[row] = 0.0f;

			for (uint8_t col = 0; col <= 3; col++) {
				Kfusion[row] += P[row][col] * H_YAW[col];
			}

			Kfusion[row] *= heading_innov_var_inv;
		}
	}

	// wrap the heading to the interval between +-pi
	measured_hdg = matrix::wrap_pi(measured_hdg);

	// calculate the innovation
	_heading_innov = predicted_hdg - measured_hdg;

	// wrap the innovation to the interval between +-pi
	_heading_innov = matrix::wrap_pi(_heading_innov);

	// innovation test ratio
	_yaw_test_ratio = sq(_heading_innov) / (sq(math::max(_params.heading_innov_gate, 1.0f)) * _heading_innov_var);

	// set the magnetometer unhealthy if the test fails
	if (_yaw_test_ratio > 1.0f) {
		_innov_check_fail_status.flags.reject_yaw = true;

		// if we are in air we don't want to fuse the measurement
		// we allow to use it when on the ground because the large innovation could be caused
		// by interference or a large initial gyro bias
		if (_control_status.flags.in_air) {
			return;

		} else {
			// constrain the innovation to the maximum set by the gate
			float gate_limit = sqrtf((sq(math::max(_params.heading_innov_gate, 1.0f)) * _heading_innov_var));
			_heading_innov = math::constrain(_heading_innov, -gate_limit, gate_limit);
		}

	} else {
		_innov_check_fail_status.flags.reject_yaw = false;
	}

	// apply covariance correction via P_new = (I -K*H)*P
	// first calculate expression for KHP
	// then calculate P - KHP
	for (unsigned row = 0; row < _k_num_states; row++) {
		for (unsigned column = 0; column <= 3; column++) {
			KH[row][column] = Kfusion[row] * H_YAW[column];
		}
	}

	for (unsigned row = 0; row < _k_num_states; row++) {
		for (unsigned column = 0; column < _k_num_states; column++) {
			float tmp = KH[row][0] * P[0][column];
			tmp += KH[row][1] * P[1][column];
			tmp += KH[row][2] * P[2][column];
			tmp += KH[row][3] * P[3][column];
			KHP[row][column] = tmp;
		}
	}

	// if the covariance correction will result in a negative variance, then
	// the covariance marix is unhealthy and must be corrected
	bool healthy = true;
	_fault_status.flags.bad_mag_hdg = false;
	for (int i = 0; i < _k_num_states; i++) {
		if (P[i][i] < KHP[i][i]) {
			// zero rows and columns
			zeroRows(P,i,i);
			zeroCols(P,i,i);

			//flag as unhealthy
			healthy = false;

			// update individual measurement health status
			_fault_status.flags.bad_mag_hdg = true;

		}
	}

	// only apply covariance and state corrrections if healthy
	if (healthy) {
		// apply the covariance corrections
		for (unsigned row = 0; row < _k_num_states; row++) {
			for (unsigned column = 0; column < _k_num_states; column++) {
				P[row][column] = P[row][column] - KHP[row][column];
			}
		}

		// correct the covariance marix for gross errors
		fixCovarianceErrors();

		// apply the state corrections
		fuse(Kfusion, _heading_innov);

	}
}

void Ekf::fuseDeclination()
{
	// assign intermediate state variables
	float magN = _state.mag_I(0);
	float magE = _state.mag_I(1);

	float R_DECL = sq(0.5f);

	// Calculate intermediate variables
	float t2 = magE*magE;
	float t3 = magN*magN;
	float t4 = t2+t3;
	// if the horizontal magnetic field is too small, this calculation will be badly conditioned
	if (t4 < 1e-4f) {
		return;
	}
	float t5 = P[16][16]*t2;
	float t6 = P[17][17]*t3;
	float t7 = t2*t2;
	float t8 = R_DECL*t7;
	float t9 = t3*t3;
	float t10 = R_DECL*t9;
	float t11 = R_DECL*t2*t3*2.0f;
	float t14 = P[16][17]*magE*magN;
	float t15 = P[17][16]*magE*magN;
	float t12 = t5+t6+t8+t10+t11-t14-t15;
	float t13;
	if (fabsf(t12) > 1e-6f) {
		t13 = 1.0f / t12;
	} else {
		return;
	}
	float t18 = magE*magE;
	float t19 = magN*magN;
	float t20 = t18+t19;
	float t21;
	if (fabsf(t20) > 1e-6f) {
		t21 = 1.0f/t20;
	} else {
		return;
	}

	// Calculate the observation Jacobian
	// Note only 2 terms are non-zero which can be used in matrix operations for calculation of Kalman gains and covariance update to significantly reduce cost
	float H_DECL[24] = {};
	H_DECL[16] = -magE*t21;
	H_DECL[17] = magN*t21;

	// Calculate the Kalman gains
	float Kfusion[_k_num_states] = {};
	Kfusion[0] = -t4*t13*(P[0][16]*magE-P[0][17]*magN);
	Kfusion[1] = -t4*t13*(P[1][16]*magE-P[1][17]*magN);
	Kfusion[2] = -t4*t13*(P[2][16]*magE-P[2][17]*magN);
	Kfusion[3] = -t4*t13*(P[3][16]*magE-P[3][17]*magN);
	Kfusion[4] = -t4*t13*(P[4][16]*magE-P[4][17]*magN);
	Kfusion[5] = -t4*t13*(P[5][16]*magE-P[5][17]*magN);
	Kfusion[6] = -t4*t13*(P[6][16]*magE-P[6][17]*magN);
	Kfusion[7] = -t4*t13*(P[7][16]*magE-P[7][17]*magN);
	Kfusion[8] = -t4*t13*(P[8][16]*magE-P[8][17]*magN);
	Kfusion[9] = -t4*t13*(P[9][16]*magE-P[9][17]*magN);
	Kfusion[10] = -t4*t13*(P[10][16]*magE-P[10][17]*magN);
	Kfusion[11] = -t4*t13*(P[11][16]*magE-P[11][17]*magN);
	Kfusion[12] = -t4*t13*(P[12][16]*magE-P[12][17]*magN);
	Kfusion[13] = -t4*t13*(P[13][16]*magE-P[13][17]*magN);
	Kfusion[14] = -t4*t13*(P[14][16]*magE-P[14][17]*magN);
	Kfusion[15] = -t4*t13*(P[15][16]*magE-P[15][17]*magN);
	Kfusion[16] = -t4*t13*(P[16][16]*magE-P[16][17]*magN);
	Kfusion[17] = -t4*t13*(P[17][16]*magE-P[17][17]*magN);
	Kfusion[18] = -t4*t13*(P[18][16]*magE-P[18][17]*magN);
	Kfusion[19] = -t4*t13*(P[19][16]*magE-P[19][17]*magN);
	Kfusion[20] = -t4*t13*(P[20][16]*magE-P[20][17]*magN);
	Kfusion[21] = -t4*t13*(P[21][16]*magE-P[21][17]*magN);
	Kfusion[22] = -t4*t13*(P[22][16]*magE-P[22][17]*magN);
	Kfusion[23] = -t4*t13*(P[23][16]*magE-P[23][17]*magN);

	// calculate innovation and constrain
	float innovation = atan2f(magE , magN) - _mag_declination;
	innovation = math::constrain(innovation, -0.5f, 0.5f);

	// apply covariance correction via P_new = (I -K*H)*P
	// first calculate expression for KHP
	// then calculate P - KHP
	// take advantage of the empty columns in KH to reduce the number of operations
	for (unsigned row = 0; row < _k_num_states; row++) {
		for (unsigned column = 16; column <= 17; column++) {
			KH[row][column] = Kfusion[row] * H_DECL[column];
		}
	}

	for (unsigned row = 0; row < _k_num_states; row++) {
		for (unsigned column = 0; column < _k_num_states; column++) {
			float tmp = KH[row][16] * P[16][column];
			tmp += KH[row][17] * P[17][column];
			KHP[row][column] = tmp;
		}
	}

	// if the covariance correction will result in a negative variance, then
	// the covariance marix is unhealthy and must be corrected
	bool healthy = true;
	_fault_status.flags.bad_mag_decl = false;
	for (int i = 0; i < _k_num_states; i++) {
		if (P[i][i] < KHP[i][i]) {
			// zero rows and columns
			zeroRows(P,i,i);
			zeroCols(P,i,i);

			//flag as unhealthy
			healthy = false;

			// update individual measurement health status
			_fault_status.flags.bad_mag_decl = true;

		}
	}

	// only apply covariance and state corrrections if healthy
	if (healthy) {
		// apply the covariance corrections
		for (unsigned row = 0; row < _k_num_states; row++) {
			for (unsigned column = 0; column < _k_num_states; column++) {
				P[row][column] = P[row][column] - KHP[row][column];
			}
		}

		// correct the covariance marix for gross errors
		fixCovarianceErrors();

		// apply the state corrections
		fuse(Kfusion, innovation);

	}
}