px4_autopilot/EKF/covariance.cpp

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/**
 * @file covariance.cpp
 * Contains functions for initialising, predicting and updating the state
 * covariance matrix
 *
 * @author Roman Bast <bastroman@gmail.com>
 *
 */

#include "ekf.h"

#include <ecl.h>
#include <math.h>
#include <mathlib/mathlib.h>

// Sets initial values for the covariance matrix
// Do not call before quaternion states have been initialised
void Ekf::initialiseCovariance()
{
	P.zero();

	_delta_angle_bias_var_accum.setZero();
	_delta_vel_bias_var_accum.setZero();

	const float dt = FILTER_UPDATE_PERIOD_S;

	resetQuatCov();

	// velocity
	P(4,4) = sq(fmaxf(_params.gps_vel_noise, 0.01f));
	P(5,5) = P(4,4);
	P(6,6) = sq(1.5f) * P(4,4);

	// position
	P(7,7) = sq(fmaxf(_params.gps_pos_noise, 0.01f));
	P(8,8) = P(7,7);

	if (_control_status.flags.rng_hgt) {
		P(9,9) = sq(fmaxf(_params.range_noise, 0.01f));

	} else if (_control_status.flags.gps_hgt) {
		float lower_limit = fmaxf(_params.gps_pos_noise, 0.01f);
		float upper_limit = fmaxf(_params.pos_noaid_noise, lower_limit);
		P(9,9) = sq(1.5f * math::constrain(_gps_sample_delayed.vacc, lower_limit, upper_limit));

	} else {
		P(9,9) = sq(fmaxf(_params.baro_noise, 0.01f));
	}

	// gyro bias
	P(10,10) = sq(_params.switch_on_gyro_bias * dt);
	P(11,11) = P(10,10);
	P(12,12) = P(10,10);

	// accel bias
	_prev_dvel_bias_var(0) = P(13,13) = sq(_params.switch_on_accel_bias * dt);
	_prev_dvel_bias_var(1) = P(14,14) = P(13,13);
	_prev_dvel_bias_var(2) = P(15,15) = P(13,13);

	// record IMU bias state covariance reset time - used to prevent resets being performed too often
	_last_imu_bias_cov_reset_us = _imu_sample_delayed.time_us;

	resetMagCov();

	// wind
	P(22,22) = sq(_params.initial_wind_uncertainty);
	P(23,23) = sq(_params.initial_wind_uncertainty);

}

Vector3f Ekf::getPositionVariance() const
{
	return P.slice<3,3>(7,7).diag();
}

Vector3f Ekf::getVelocityVariance() const
{
	return P.slice<3,3>(4,4).diag();
}

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

	const float dax = _imu_sample_delayed.delta_ang(0);
	const float day = _imu_sample_delayed.delta_ang(1);
	const float daz = _imu_sample_delayed.delta_ang(2);

	const float dvx = _imu_sample_delayed.delta_vel(0);
	const float dvy = _imu_sample_delayed.delta_vel(1);
	const float dvz = _imu_sample_delayed.delta_vel(2);

	const float dax_b = _state.delta_ang_bias(0);
	const float day_b = _state.delta_ang_bias(1);
	const float daz_b = _state.delta_ang_bias(2);

	const float dvx_b = _state.delta_vel_bias(0);
	const float dvy_b = _state.delta_vel_bias(1);
	const float dvz_b = _state.delta_vel_bias(2);

	// Use average update interval to reduce accumulated covariance prediction errors due to small single frame dt values
	const float dt = FILTER_UPDATE_PERIOD_S;
	const float dt_inv = 1.0f / dt;

	// convert rate of change of rate gyro bias (rad/s**2) as specified by the parameter to an expected change in delta angle (rad) since the last update
	const float d_ang_bias_sig = dt * dt * math::constrain(_params.gyro_bias_p_noise, 0.0f, 1.0f);

	// convert rate of change of accelerometer bias (m/s**3) as specified by the parameter to an expected change in delta velocity (m/s) since the last update
	const float d_vel_bias_sig = dt * dt * math::constrain(_params.accel_bias_p_noise, 0.0f, 1.0f);

	// inhibit learning of imu accel bias if the manoeuvre levels are too high to protect against the effect of sensor nonlinearities or bad accel data is detected
	const float alpha = math::constrain((dt / _params.acc_bias_learn_tc), 0.0f, 1.0f);
	const float beta = 1.0f - alpha;
	_ang_rate_magnitude_filt = fmaxf(dt_inv * _imu_sample_delayed.delta_ang.norm(), beta * _ang_rate_magnitude_filt);
	_accel_magnitude_filt = fmaxf(dt_inv * _imu_sample_delayed.delta_vel.norm(), beta * _accel_magnitude_filt);
	_accel_vec_filt = alpha * dt_inv * _imu_sample_delayed.delta_vel + beta * _accel_vec_filt;

	if (_ang_rate_magnitude_filt > _params.acc_bias_learn_gyr_lim
	    || _accel_magnitude_filt > _params.acc_bias_learn_acc_lim
	    || _bad_vert_accel_detected) {

		// store the bias state variances to be reinstated later
		if (!_accel_bias_inhibit) {
			_prev_dvel_bias_var = P.slice<3,3>(13,13).diag();
		}

		_accel_bias_inhibit = true;

	} else {
		if (_accel_bias_inhibit) {
			// reinstate the bias state variances
			P(13,13) = _prev_dvel_bias_var(0);
			P(14,14) = _prev_dvel_bias_var(1);
			P(15,15) = _prev_dvel_bias_var(2);

		} else {
			// store the bias state variances to be reinstated later
			_prev_dvel_bias_var = P.slice<3,3>(13,13).diag();

		}

		_accel_bias_inhibit = false;
	}

	// Don't continue to grow the earth field variances if they are becoming too large or we are not doing 3-axis fusion as this can make the covariance matrix badly conditioned
	float mag_I_sig;

	if (_control_status.flags.mag_3D && (P(16,16) + P(17,17) + P(18,18)) < 0.1f) {
		mag_I_sig = dt * math::constrain(_params.mage_p_noise, 0.0f, 1.0f);

	} else {
		mag_I_sig = 0.0f;
	}

	// Don't continue to grow the body field variances if they is becoming too large or we are not doing 3-axis fusion as this can make the covariance matrix badly conditioned
	float mag_B_sig;

	if (_control_status.flags.mag_3D && (P(19,19) + P(20,20) + P(21,21)) < 0.1f) {
		mag_B_sig = dt * math::constrain(_params.magb_p_noise, 0.0f, 1.0f);

	} else {
		mag_B_sig = 0.0f;
	}

	float wind_vel_sig;

	// Calculate low pass filtered height rate
	float alpha_height_rate_lpf = 0.1f * dt; // 10 seconds time constant
	_height_rate_lpf = _height_rate_lpf * (1.0f - alpha_height_rate_lpf) + _state.vel(2) * alpha_height_rate_lpf;

	// Don't continue to grow wind velocity state variances if they are becoming too large or we are not using wind velocity states as this can make the covariance matrix badly conditioned
	if (_control_status.flags.wind && (P(22,22) + P(23,23)) < sq(_params.initial_wind_uncertainty)) {
		wind_vel_sig = dt * math::constrain(_params.wind_vel_p_noise, 0.0f, 1.0f) * (1.0f + _params.wind_vel_p_noise_scaler * fabsf(_height_rate_lpf));

	} else {
		wind_vel_sig = 0.0f;
	}

	// compute noise variance for stationary processes
	matrix::Vector<float, _k_num_states> process_noise;

	// Construct the process noise variance diagonal for those states with a stationary process model
	// These are kinematic states and their error growth is controlled separately by the IMU noise variances
	process_noise.slice<10,1>(0,0) = 0.0f;

	// delta angle bias states
	process_noise.slice<3,1>(10,0) = sq(d_ang_bias_sig);
	// delta_velocity bias states
	process_noise.slice<3,1>(13,0) = sq(d_vel_bias_sig);
	// earth frame magnetic field states
	process_noise.slice<3,1>(16,0) = sq(mag_I_sig);
	// body frame magnetic field states
	process_noise.slice<3,1>(19,0) = sq(mag_B_sig);
	// wind velocity states
	process_noise.slice<2,1>(22,0) = sq(wind_vel_sig);

	// assign IMU noise variances
	// inputs to the system are 3 delta angles and 3 delta velocities
	float daxVar, dayVar, dazVar;
	float dvxVar, dvyVar, dvzVar;
	float gyro_noise = math::constrain(_params.gyro_noise, 0.0f, 1.0f);
	daxVar = dayVar = dazVar = sq(dt * gyro_noise);
	float accel_noise = math::constrain(_params.accel_noise, 0.0f, 1.0f);

	if (_bad_vert_accel_detected) {
		// Increase accelerometer process noise if bad accel data is detected. Measurement errors due to
		// vibration induced clipping commonly reach an equivalent 0.5g offset.
		accel_noise = BADACC_BIAS_PNOISE;
	}

	dvxVar = dvyVar = dvzVar = sq(dt * accel_noise);

	// Accelerometer Clipping
	// delta velocity X: increase process noise if sample contained any X axis clipping
	if (_imu_sample_delayed.delta_vel_clipping[0]) {
		dvxVar = sq(dt * BADACC_BIAS_PNOISE);
	}
	// delta velocity Y: increase process noise if sample contained any Y axis clipping
	if (_imu_sample_delayed.delta_vel_clipping[1]) {
		dvyVar = sq(dt * BADACC_BIAS_PNOISE);
	}
	// delta velocity Z: increase process noise if sample contained any Z axis clipping
	if (_imu_sample_delayed.delta_vel_clipping[2]) {
		dvzVar = sq(dt * BADACC_BIAS_PNOISE);
	}

	// predict the covariance

	// intermediate calculations
	float SF[21];
	SF[0] = dvz - dvz_b;
	SF[1] = dvy - dvy_b;
	SF[2] = dvx - dvx_b;
	SF[3] = 2*q1*SF[2] + 2*q2*SF[1] + 2*q3*SF[0];
	SF[4] = 2*q0*SF[1] - 2*q1*SF[0] + 2*q3*SF[2];
	SF[5] = 2*q0*SF[2] + 2*q2*SF[0] - 2*q3*SF[1];
	SF[6] = day/2 - day_b/2;
	SF[7] = daz/2 - daz_b/2;
	SF[8] = dax/2 - dax_b/2;
	SF[9] = dax_b/2 - dax/2;
	SF[10] = daz_b/2 - daz/2;
	SF[11] = day_b/2 - day/2;
	SF[12] = 2*q1*SF[1];
	SF[13] = 2*q0*SF[0];
	SF[14] = q1/2;
	SF[15] = q2/2;
	SF[16] = q3/2;
	SF[17] = sq(q3);
	SF[18] = sq(q2);
	SF[19] = sq(q1);
	SF[20] = sq(q0);

	float SG[8];
	SG[0] = q0/2;
	SG[1] = sq(q3);
	SG[2] = sq(q2);
	SG[3] = sq(q1);
	SG[4] = sq(q0);
	SG[5] = 2*q2*q3;
	SG[6] = 2*q1*q3;
	SG[7] = 2*q1*q2;

	float SQ[11];
	SQ[0] = dvzVar*(SG[5] - 2*q0*q1)*(SG[1] - SG[2] - SG[3] + SG[4]) - dvyVar*(SG[5] + 2*q0*q1)*(SG[1] - SG[2] + SG[3] - SG[4]) + dvxVar*(SG[6] - 2*q0*q2)*(SG[7] + 2*q0*q3);
	SQ[1] = dvzVar*(SG[6] + 2*q0*q2)*(SG[1] - SG[2] - SG[3] + SG[4]) - dvxVar*(SG[6] - 2*q0*q2)*(SG[1] + SG[2] - SG[3] - SG[4]) + dvyVar*(SG[5] + 2*q0*q1)*(SG[7] - 2*q0*q3);
	SQ[2] = dvzVar*(SG[5] - 2*q0*q1)*(SG[6] + 2*q0*q2) - dvyVar*(SG[7] - 2*q0*q3)*(SG[1] - SG[2] + SG[3] - SG[4]) - dvxVar*(SG[7] + 2*q0*q3)*(SG[1] + SG[2] - SG[3] - SG[4]);
	SQ[3] = (dayVar*q1*SG[0])/2 - (dazVar*q1*SG[0])/2 - (daxVar*q2*q3)/4;
	SQ[4] = (dazVar*q2*SG[0])/2 - (daxVar*q2*SG[0])/2 - (dayVar*q1*q3)/4;
	SQ[5] = (daxVar*q3*SG[0])/2 - (dayVar*q3*SG[0])/2 - (dazVar*q1*q2)/4;
	SQ[6] = (daxVar*q1*q2)/4 - (dazVar*q3*SG[0])/2 - (dayVar*q1*q2)/4;
	SQ[7] = (dazVar*q1*q3)/4 - (daxVar*q1*q3)/4 - (dayVar*q2*SG[0])/2;
	SQ[8] = (dayVar*q2*q3)/4 - (daxVar*q1*SG[0])/2 - (dazVar*q2*q3)/4;
	SQ[9] = sq(SG[0]);
	SQ[10] = sq(q1);

	float SPP[11] = {};
	SPP[0] = SF[12] + SF[13] - 2*q2*SF[2];
	SPP[1] = SF[17] - SF[18] - SF[19] + SF[20];
	SPP[2] = SF[17] - SF[18] + SF[19] - SF[20];
	SPP[3] = SF[17] + SF[18] - SF[19] - SF[20];
	SPP[4] = 2*q0*q2 - 2*q1*q3;
	SPP[5] = 2*q0*q1 - 2*q2*q3;
	SPP[6] = 2*q0*q3 - 2*q1*q2;
	SPP[7] = 2*q0*q1 + 2*q2*q3;
	SPP[8] = 2*q0*q3 + 2*q1*q2;
	SPP[9] = 2*q0*q2 + 2*q1*q3;
	SPP[10] = SF[16];

	// covariance update
	matrix::SquareMatrix<float, _k_num_states> nextP;

	// calculate variances and upper diagonal covariances for quaternion, velocity, position and gyro bias states
	nextP(0,0) = P(0,0) + P(1,0)*SF[9] + P(2,0)*SF[11] + P(3,0)*SF[10] + P(10,0)*SF[14] + P(11,0)*SF[15] + P(12,0)*SPP[10] + (daxVar*SQ[10])/4 + SF[9]*(P(0,1) + P(1,1)*SF[9] + P(2,1)*SF[11] + P(3,1)*SF[10] + P(10,1)*SF[14] + P(11,1)*SF[15] + P(12,1)*SPP[10]) + SF[11]*(P(0,2) + P(1,2)*SF[9] + P(2,2)*SF[11] + P(3,2)*SF[10] + P(10,2)*SF[14] + P(11,2)*SF[15] + P(12,2)*SPP[10]) + SF[10]*(P(0,3) + P(1,3)*SF[9] + P(2,3)*SF[11] + P(3,3)*SF[10] + P(10,3)*SF[14] + P(11,3)*SF[15] + P(12,3)*SPP[10]) + SF[14]*(P(0,10) + P(1,10)*SF[9] + P(2,10)*SF[11] + P(3,10)*SF[10] + P(10,10)*SF[14] + P(11,10)*SF[15] + P(12,10)*SPP[10]) + SF[15]*(P(0,11) + P(1,11)*SF[9] + P(2,11)*SF[11] + P(3,11)*SF[10] + P(10,11)*SF[14] + P(11,11)*SF[15] + P(12,11)*SPP[10]) + SPP[10]*(P(0,12) + P(1,12)*SF[9] + P(2,12)*SF[11] + P(3,12)*SF[10] + P(10,12)*SF[14] + P(11,12)*SF[15] + P(12,12)*SPP[10]) + (dayVar*sq(q2))/4 + (dazVar*sq(q3))/4;
	nextP(0,1) = P(0,1) + SQ[8] + P(1,1)*SF[9] + P(2,1)*SF[11] + P(3,1)*SF[10] + P(10,1)*SF[14] + P(11,1)*SF[15] + P(12,1)*SPP[10] + SF[8]*(P(0,0) + P(1,0)*SF[9] + P(2,0)*SF[11] + P(3,0)*SF[10] + P(10,0)*SF[14] + P(11,0)*SF[15] + P(12,0)*SPP[10]) + SF[7]*(P(0,2) + P(1,2)*SF[9] + P(2,2)*SF[11] + P(3,2)*SF[10] + P(10,2)*SF[14] + P(11,2)*SF[15] + P(12,2)*SPP[10]) + SF[11]*(P(0,3) + P(1,3)*SF[9] + P(2,3)*SF[11] + P(3,3)*SF[10] + P(10,3)*SF[14] + P(11,3)*SF[15] + P(12,3)*SPP[10]) - SF[15]*(P(0,12) + P(1,12)*SF[9] + P(2,12)*SF[11] + P(3,12)*SF[10] + P(10,12)*SF[14] + P(11,12)*SF[15] + P(12,12)*SPP[10]) + SPP[10]*(P(0,11) + P(1,11)*SF[9] + P(2,11)*SF[11] + P(3,11)*SF[10] + P(10,11)*SF[14] + P(11,11)*SF[15] + P(12,11)*SPP[10]) - (q0*(P(0,10) + P(1,10)*SF[9] + P(2,10)*SF[11] + P(3,10)*SF[10] + P(10,10)*SF[14] + P(11,10)*SF[15] + P(12,10)*SPP[10]))/2;
	nextP(1,1) = P(1,1) + P(0,1)*SF[8] + P(2,1)*SF[7] + P(3,1)*SF[11] - P(12,1)*SF[15] + P(11,1)*SPP[10] + daxVar*SQ[9] - (P(10,1)*q0)/2 + SF[8]*(P(1,0) + P(0,0)*SF[8] + P(2,0)*SF[7] + P(3,0)*SF[11] - P(12,0)*SF[15] + P(11,0)*SPP[10] - (P(10,0)*q0)/2) + SF[7]*(P(1,2) + P(0,2)*SF[8] + P(2,2)*SF[7] + P(3,2)*SF[11] - P(12,2)*SF[15] + P(11,2)*SPP[10] - (P(10,2)*q0)/2) + SF[11]*(P(1,3) + P(0,3)*SF[8] + P(2,3)*SF[7] + P(3,3)*SF[11] - P(12,3)*SF[15] + P(11,3)*SPP[10] - (P(10,3)*q0)/2) - SF[15]*(P(1,12) + P(0,12)*SF[8] + P(2,12)*SF[7] + P(3,12)*SF[11] - P(12,12)*SF[15] + P(11,12)*SPP[10] - (P(10,12)*q0)/2) + SPP[10]*(P(1,11) + P(0,11)*SF[8] + P(2,11)*SF[7] + P(3,11)*SF[11] - P(12,11)*SF[15] + P(11,11)*SPP[10] - (P(10,11)*q0)/2) + (dayVar*sq(q3))/4 + (dazVar*sq(q2))/4 - (q0*(P(1,10) + P(0,10)*SF[8] + P(2,10)*SF[7] + P(3,10)*SF[11] - P(12,10)*SF[15] + P(11,10)*SPP[10] - (P(10,10)*q0)/2))/2;
	nextP(0,2) = P(0,2) + SQ[7] + P(1,2)*SF[9] + P(2,2)*SF[11] + P(3,2)*SF[10] + P(10,2)*SF[14] + P(11,2)*SF[15] + P(12,2)*SPP[10] + SF[6]*(P(0,0) + P(1,0)*SF[9] + P(2,0)*SF[11] + P(3,0)*SF[10] + P(10,0)*SF[14] + P(11,0)*SF[15] + P(12,0)*SPP[10]) + SF[10]*(P(0,1) + P(1,1)*SF[9] + P(2,1)*SF[11] + P(3,1)*SF[10] + P(10,1)*SF[14] + P(11,1)*SF[15] + P(12,1)*SPP[10]) + SF[8]*(P(0,3) + P(1,3)*SF[9] + P(2,3)*SF[11] + P(3,3)*SF[10] + P(10,3)*SF[14] + P(11,3)*SF[15] + P(12,3)*SPP[10]) + SF[14]*(P(0,12) + P(1,12)*SF[9] + P(2,12)*SF[11] + P(3,12)*SF[10] + P(10,12)*SF[14] + P(11,12)*SF[15] + P(12,12)*SPP[10]) - SPP[10]*(P(0,10) + P(1,10)*SF[9] + P(2,10)*SF[11] + P(3,10)*SF[10] + P(10,10)*SF[14] + P(11,10)*SF[15] + P(12,10)*SPP[10]) - (q0*(P(0,11) + P(1,11)*SF[9] + P(2,11)*SF[11] + P(3,11)*SF[10] + P(10,11)*SF[14] + P(11,11)*SF[15] + P(12,11)*SPP[10]))/2;
	nextP(1,2) = P(1,2) + SQ[5] + P(0,2)*SF[8] + P(2,2)*SF[7] + P(3,2)*SF[11] - P(12,2)*SF[15] + P(11,2)*SPP[10] - (P(10,2)*q0)/2 + SF[6]*(P(1,0) + P(0,0)*SF[8] + P(2,0)*SF[7] + P(3,0)*SF[11] - P(12,0)*SF[15] + P(11,0)*SPP[10] - (P(10,0)*q0)/2) + SF[10]*(P(1,1) + P(0,1)*SF[8] + P(2,1)*SF[7] + P(3,1)*SF[11] - P(12,1)*SF[15] + P(11,1)*SPP[10] - (P(10,1)*q0)/2) + SF[8]*(P(1,3) + P(0,3)*SF[8] + P(2,3)*SF[7] + P(3,3)*SF[11] - P(12,3)*SF[15] + P(11,3)*SPP[10] - (P(10,3)*q0)/2) + SF[14]*(P(1,12) + P(0,12)*SF[8] + P(2,12)*SF[7] + P(3,12)*SF[11] - P(12,12)*SF[15] + P(11,12)*SPP[10] - (P(10,12)*q0)/2) - SPP[10]*(P(1,10) + P(0,10)*SF[8] + P(2,10)*SF[7] + P(3,10)*SF[11] - P(12,10)*SF[15] + P(11,10)*SPP[10] - (P(10,10)*q0)/2) - (q0*(P(1,11) + P(0,11)*SF[8] + P(2,11)*SF[7] + P(3,11)*SF[11] - P(12,11)*SF[15] + P(11,11)*SPP[10] - (P(10,11)*q0)/2))/2;
	nextP(2,2) = P(2,2) + P(0,2)*SF[6] + P(1,2)*SF[10] + P(3,2)*SF[8] + P(12,2)*SF[14] - P(10,2)*SPP[10] + dayVar*SQ[9] + (dazVar*SQ[10])/4 - (P(11,2)*q0)/2 + SF[6]*(P(2,0) + P(0,0)*SF[6] + P(1,0)*SF[10] + P(3,0)*SF[8] + P(12,0)*SF[14] - P(10,0)*SPP[10] - (P(11,0)*q0)/2) + SF[10]*(P(2,1) + P(0,1)*SF[6] + P(1,1)*SF[10] + P(3,1)*SF[8] + P(12,1)*SF[14] - P(10,1)*SPP[10] - (P(11,1)*q0)/2) + SF[8]*(P(2,3) + P(0,3)*SF[6] + P(1,3)*SF[10] + P(3,3)*SF[8] + P(12,3)*SF[14] - P(10,3)*SPP[10] - (P(11,3)*q0)/2) + SF[14]*(P(2,12) + P(0,12)*SF[6] + P(1,12)*SF[10] + P(3,12)*SF[8] + P(12,12)*SF[14] - P(10,12)*SPP[10] - (P(11,12)*q0)/2) - SPP[10]*(P(2,10) + P(0,10)*SF[6] + P(1,10)*SF[10] + P(3,10)*SF[8] + P(12,10)*SF[14] - P(10,10)*SPP[10] - (P(11,10)*q0)/2) + (daxVar*sq(q3))/4 - (q0*(P(2,11) + P(0,11)*SF[6] + P(1,11)*SF[10] + P(3,11)*SF[8] + P(12,11)*SF[14] - P(10,11)*SPP[10] - (P(11,11)*q0)/2))/2;
	nextP(0,3) = P(0,3) + SQ[6] + P(1,3)*SF[9] + P(2,3)*SF[11] + P(3,3)*SF[10] + P(10,3)*SF[14] + P(11,3)*SF[15] + P(12,3)*SPP[10] + SF[7]*(P(0,0) + P(1,0)*SF[9] + P(2,0)*SF[11] + P(3,0)*SF[10] + P(10,0)*SF[14] + P(11,0)*SF[15] + P(12,0)*SPP[10]) + SF[6]*(P(0,1) + P(1,1)*SF[9] + P(2,1)*SF[11] + P(3,1)*SF[10] + P(10,1)*SF[14] + P(11,1)*SF[15] + P(12,1)*SPP[10]) + SF[9]*(P(0,2) + P(1,2)*SF[9] + P(2,2)*SF[11] + P(3,2)*SF[10] + P(10,2)*SF[14] + P(11,2)*SF[15] + P(12,2)*SPP[10]) + SF[15]*(P(0,10) + P(1,10)*SF[9] + P(2,10)*SF[11] + P(3,10)*SF[10] + P(10,10)*SF[14] + P(11,10)*SF[15] + P(12,10)*SPP[10]) - SF[14]*(P(0,11) + P(1,11)*SF[9] + P(2,11)*SF[11] + P(3,11)*SF[10] + P(10,11)*SF[14] + P(11,11)*SF[15] + P(12,11)*SPP[10]) - (q0*(P(0,12) + P(1,12)*SF[9] + P(2,12)*SF[11] + P(3,12)*SF[10] + P(10,12)*SF[14] + P(11,12)*SF[15] + P(12,12)*SPP[10]))/2;
	nextP(1,3) = P(1,3) + SQ[4] + P(0,3)*SF[8] + P(2,3)*SF[7] + P(3,3)*SF[11] - P(12,3)*SF[15] + P(11,3)*SPP[10] - (P(10,3)*q0)/2 + SF[7]*(P(1,0) + P(0,0)*SF[8] + P(2,0)*SF[7] + P(3,0)*SF[11] - P(12,0)*SF[15] + P(11,0)*SPP[10] - (P(10,0)*q0)/2) + SF[6]*(P(1,1) + P(0,1)*SF[8] + P(2,1)*SF[7] + P(3,1)*SF[11] - P(12,1)*SF[15] + P(11,1)*SPP[10] - (P(10,1)*q0)/2) + SF[9]*(P(1,2) + P(0,2)*SF[8] + P(2,2)*SF[7] + P(3,2)*SF[11] - P(12,2)*SF[15] + P(11,2)*SPP[10] - (P(10,2)*q0)/2) + SF[15]*(P(1,10) + P(0,10)*SF[8] + P(2,10)*SF[7] + P(3,10)*SF[11] - P(12,10)*SF[15] + P(11,10)*SPP[10] - (P(10,10)*q0)/2) - SF[14]*(P(1,11) + P(0,11)*SF[8] + P(2,11)*SF[7] + P(3,11)*SF[11] - P(12,11)*SF[15] + P(11,11)*SPP[10] - (P(10,11)*q0)/2) - (q0*(P(1,12) + P(0,12)*SF[8] + P(2,12)*SF[7] + P(3,12)*SF[11] - P(12,12)*SF[15] + P(11,12)*SPP[10] - (P(10,12)*q0)/2))/2;
	nextP(2,3) = P(2,3) + SQ[3] + P(0,3)*SF[6] + P(1,3)*SF[10] + P(3,3)*SF[8] + P(12,3)*SF[14] - P(10,3)*SPP[10] - (P(11,3)*q0)/2 + SF[7]*(P(2,0) + P(0,0)*SF[6] + P(1,0)*SF[10] + P(3,0)*SF[8] + P(12,0)*SF[14] - P(10,0)*SPP[10] - (P(11,0)*q0)/2) + SF[6]*(P(2,1) + P(0,1)*SF[6] + P(1,1)*SF[10] + P(3,1)*SF[8] + P(12,1)*SF[14] - P(10,1)*SPP[10] - (P(11,1)*q0)/2) + SF[9]*(P(2,2) + P(0,2)*SF[6] + P(1,2)*SF[10] + P(3,2)*SF[8] + P(12,2)*SF[14] - P(10,2)*SPP[10] - (P(11,2)*q0)/2) + SF[15]*(P(2,10) + P(0,10)*SF[6] + P(1,10)*SF[10] + P(3,10)*SF[8] + P(12,10)*SF[14] - P(10,10)*SPP[10] - (P(11,10)*q0)/2) - SF[14]*(P(2,11) + P(0,11)*SF[6] + P(1,11)*SF[10] + P(3,11)*SF[8] + P(12,11)*SF[14] - P(10,11)*SPP[10] - (P(11,11)*q0)/2) - (q0*(P(2,12) + P(0,12)*SF[6] + P(1,12)*SF[10] + P(3,12)*SF[8] + P(12,12)*SF[14] - P(10,12)*SPP[10] - (P(11,12)*q0)/2))/2;
	nextP(3,3) = P(3,3) + P(0,3)*SF[7] + P(1,3)*SF[6] + P(2,3)*SF[9] + P(10,3)*SF[15] - P(11,3)*SF[14] + (dayVar*SQ[10])/4 + dazVar*SQ[9] - (P(12,3)*q0)/2 + SF[7]*(P(3,0) + P(0,0)*SF[7] + P(1,0)*SF[6] + P(2,0)*SF[9] + P(10,0)*SF[15] - P(11,0)*SF[14] - (P(12,0)*q0)/2) + SF[6]*(P(3,1) + P(0,1)*SF[7] + P(1,1)*SF[6] + P(2,1)*SF[9] + P(10,1)*SF[15] - P(11,1)*SF[14] - (P(12,1)*q0)/2) + SF[9]*(P(3,2) + P(0,2)*SF[7] + P(1,2)*SF[6] + P(2,2)*SF[9] + P(10,2)*SF[15] - P(11,2)*SF[14] - (P(12,2)*q0)/2) + SF[15]*(P(3,10) + P(0,10)*SF[7] + P(1,10)*SF[6] + P(2,10)*SF[9] + P(10,10)*SF[15] - P(11,10)*SF[14] - (P(12,10)*q0)/2) - SF[14]*(P(3,11) + P(0,11)*SF[7] + P(1,11)*SF[6] + P(2,11)*SF[9] + P(10,11)*SF[15] - P(11,11)*SF[14] - (P(12,11)*q0)/2) + (daxVar*sq(q2))/4 - (q0*(P(3,12) + P(0,12)*SF[7] + P(1,12)*SF[6] + P(2,12)*SF[9] + P(10,12)*SF[15] - P(11,12)*SF[14] - (P(12,12)*q0)/2))/2;
	nextP(0,4) = P(0,4) + P(1,4)*SF[9] + P(2,4)*SF[11] + P(3,4)*SF[10] + P(10,4)*SF[14] + P(11,4)*SF[15] + P(12,4)*SPP[10] + SF[5]*(P(0,0) + P(1,0)*SF[9] + P(2,0)*SF[11] + P(3,0)*SF[10] + P(10,0)*SF[14] + P(11,0)*SF[15] + P(12,0)*SPP[10]) + SF[3]*(P(0,1) + P(1,1)*SF[9] + P(2,1)*SF[11] + P(3,1)*SF[10] + P(10,1)*SF[14] + P(11,1)*SF[15] + P(12,1)*SPP[10]) - SF[4]*(P(0,3) + P(1,3)*SF[9] + P(2,3)*SF[11] + P(3,3)*SF[10] + P(10,3)*SF[14] + P(11,3)*SF[15] + P(12,3)*SPP[10]) + SPP[0]*(P(0,2) + P(1,2)*SF[9] + P(2,2)*SF[11] + P(3,2)*SF[10] + P(10,2)*SF[14] + P(11,2)*SF[15] + P(12,2)*SPP[10]) + SPP[3]*(P(0,13) + P(1,13)*SF[9] + P(2,13)*SF[11] + P(3,13)*SF[10] + P(10,13)*SF[14] + P(11,13)*SF[15] + P(12,13)*SPP[10]) + SPP[6]*(P(0,14) + P(1,14)*SF[9] + P(2,14)*SF[11] + P(3,14)*SF[10] + P(10,14)*SF[14] + P(11,14)*SF[15] + P(12,14)*SPP[10]) - SPP[9]*(P(0,15) + P(1,15)*SF[9] + P(2,15)*SF[11] + P(3,15)*SF[10] + P(10,15)*SF[14] + P(11,15)*SF[15] + P(12,15)*SPP[10]);
	nextP(1,4) = P(1,4) + P(0,4)*SF[8] + P(2,4)*SF[7] + P(3,4)*SF[11] - P(12,4)*SF[15] + P(11,4)*SPP[10] - (P(10,4)*q0)/2 + SF[5]*(P(1,0) + P(0,0)*SF[8] + P(2,0)*SF[7] + P(3,0)*SF[11] - P(12,0)*SF[15] + P(11,0)*SPP[10] - (P(10,0)*q0)/2) + SF[3]*(P(1,1) + P(0,1)*SF[8] + P(2,1)*SF[7] + P(3,1)*SF[11] - P(12,1)*SF[15] + P(11,1)*SPP[10] - (P(10,1)*q0)/2) - SF[4]*(P(1,3) + P(0,3)*SF[8] + P(2,3)*SF[7] + P(3,3)*SF[11] - P(12,3)*SF[15] + P(11,3)*SPP[10] - (P(10,3)*q0)/2) + SPP[0]*(P(1,2) + P(0,2)*SF[8] + P(2,2)*SF[7] + P(3,2)*SF[11] - P(12,2)*SF[15] + P(11,2)*SPP[10] - (P(10,2)*q0)/2) + SPP[3]*(P(1,13) + P(0,13)*SF[8] + P(2,13)*SF[7] + P(3,13)*SF[11] - P(12,13)*SF[15] + P(11,13)*SPP[10] - (P(10,13)*q0)/2) + SPP[6]*(P(1,14) + P(0,14)*SF[8] + P(2,14)*SF[7] + P(3,14)*SF[11] - P(12,14)*SF[15] + P(11,14)*SPP[10] - (P(10,14)*q0)/2) - SPP[9]*(P(1,15) + P(0,15)*SF[8] + P(2,15)*SF[7] + P(3,15)*SF[11] - P(12,15)*SF[15] + P(11,15)*SPP[10] - (P(10,15)*q0)/2);
	nextP(2,4) = P(2,4) + P(0,4)*SF[6] + P(1,4)*SF[10] + P(3,4)*SF[8] + P(12,4)*SF[14] - P(10,4)*SPP[10] - (P(11,4)*q0)/2 + SF[5]*(P(2,0) + P(0,0)*SF[6] + P(1,0)*SF[10] + P(3,0)*SF[8] + P(12,0)*SF[14] - P(10,0)*SPP[10] - (P(11,0)*q0)/2) + SF[3]*(P(2,1) + P(0,1)*SF[6] + P(1,1)*SF[10] + P(3,1)*SF[8] + P(12,1)*SF[14] - P(10,1)*SPP[10] - (P(11,1)*q0)/2) - SF[4]*(P(2,3) + P(0,3)*SF[6] + P(1,3)*SF[10] + P(3,3)*SF[8] + P(12,3)*SF[14] - P(10,3)*SPP[10] - (P(11,3)*q0)/2) + SPP[0]*(P(2,2) + P(0,2)*SF[6] + P(1,2)*SF[10] + P(3,2)*SF[8] + P(12,2)*SF[14] - P(10,2)*SPP[10] - (P(11,2)*q0)/2) + SPP[3]*(P(2,13) + P(0,13)*SF[6] + P(1,13)*SF[10] + P(3,13)*SF[8] + P(12,13)*SF[14] - P(10,13)*SPP[10] - (P(11,13)*q0)/2) + SPP[6]*(P(2,14) + P(0,14)*SF[6] + P(1,14)*SF[10] + P(3,14)*SF[8] + P(12,14)*SF[14] - P(10,14)*SPP[10] - (P(11,14)*q0)/2) - SPP[9]*(P(2,15) + P(0,15)*SF[6] + P(1,15)*SF[10] + P(3,15)*SF[8] + P(12,15)*SF[14] - P(10,15)*SPP[10] - (P(11,15)*q0)/2);
	nextP(3,4) = P(3,4) + P(0,4)*SF[7] + P(1,4)*SF[6] + P(2,4)*SF[9] + P(10,4)*SF[15] - P(11,4)*SF[14] - (P(12,4)*q0)/2 + SF[5]*(P(3,0) + P(0,0)*SF[7] + P(1,0)*SF[6] + P(2,0)*SF[9] + P(10,0)*SF[15] - P(11,0)*SF[14] - (P(12,0)*q0)/2) + SF[3]*(P(3,1) + P(0,1)*SF[7] + P(1,1)*SF[6] + P(2,1)*SF[9] + P(10,1)*SF[15] - P(11,1)*SF[14] - (P(12,1)*q0)/2) - SF[4]*(P(3,3) + P(0,3)*SF[7] + P(1,3)*SF[6] + P(2,3)*SF[9] + P(10,3)*SF[15] - P(11,3)*SF[14] - (P(12,3)*q0)/2) + SPP[0]*(P(3,2) + P(0,2)*SF[7] + P(1,2)*SF[6] + P(2,2)*SF[9] + P(10,2)*SF[15] - P(11,2)*SF[14] - (P(12,2)*q0)/2) + SPP[3]*(P(3,13) + P(0,13)*SF[7] + P(1,13)*SF[6] + P(2,13)*SF[9] + P(10,13)*SF[15] - P(11,13)*SF[14] - (P(12,13)*q0)/2) + SPP[6]*(P(3,14) + P(0,14)*SF[7] + P(1,14)*SF[6] + P(2,14)*SF[9] + P(10,14)*SF[15] - P(11,14)*SF[14] - (P(12,14)*q0)/2) - SPP[9]*(P(3,15) + P(0,15)*SF[7] + P(1,15)*SF[6] + P(2,15)*SF[9] + P(10,15)*SF[15] - P(11,15)*SF[14] - (P(12,15)*q0)/2);
	nextP(4,4) = P(4,4) + P(0,4)*SF[5] + P(1,4)*SF[3] - P(3,4)*SF[4] + P(2,4)*SPP[0] + P(13,4)*SPP[3] + P(14,4)*SPP[6] - P(15,4)*SPP[9] + dvyVar*sq(SG[7] - 2*q0*q3) + dvzVar*sq(SG[6] + 2*q0*q2) + SF[5]*(P(4,0) + P(0,0)*SF[5] + P(1,0)*SF[3] - P(3,0)*SF[4] + P(2,0)*SPP[0] + P(13,0)*SPP[3] + P(14,0)*SPP[6] - P(15,0)*SPP[9]) + SF[3]*(P(4,1) + P(0,1)*SF[5] + P(1,1)*SF[3] - P(3,1)*SF[4] + P(2,1)*SPP[0] + P(13,1)*SPP[3] + P(14,1)*SPP[6] - P(15,1)*SPP[9]) - SF[4]*(P(4,3) + P(0,3)*SF[5] + P(1,3)*SF[3] - P(3,3)*SF[4] + P(2,3)*SPP[0] + P(13,3)*SPP[3] + P(14,3)*SPP[6] - P(15,3)*SPP[9]) + SPP[0]*(P(4,2) + P(0,2)*SF[5] + P(1,2)*SF[3] - P(3,2)*SF[4] + P(2,2)*SPP[0] + P(13,2)*SPP[3] + P(14,2)*SPP[6] - P(15,2)*SPP[9]) + SPP[3]*(P(4,13) + P(0,13)*SF[5] + P(1,13)*SF[3] - P(3,13)*SF[4] + P(2,13)*SPP[0] + P(13,13)*SPP[3] + P(14,13)*SPP[6] - P(15,13)*SPP[9]) + SPP[6]*(P(4,14) + P(0,14)*SF[5] + P(1,14)*SF[3] - P(3,14)*SF[4] + P(2,14)*SPP[0] + P(13,14)*SPP[3] + P(14,14)*SPP[6] - P(15,14)*SPP[9]) - SPP[9]*(P(4,15) + P(0,15)*SF[5] + P(1,15)*SF[3] - P(3,15)*SF[4] + P(2,15)*SPP[0] + P(13,15)*SPP[3] + P(14,15)*SPP[6] - P(15,15)*SPP[9]) + dvxVar*sq(SG[1] + SG[2] - SG[3] - SG[4]);
	nextP(0,5) = P(0,5) + P(1,5)*SF[9] + P(2,5)*SF[11] + P(3,5)*SF[10] + P(10,5)*SF[14] + P(11,5)*SF[15] + P(12,5)*SPP[10] + SF[4]*(P(0,0) + P(1,0)*SF[9] + P(2,0)*SF[11] + P(3,0)*SF[10] + P(10,0)*SF[14] + P(11,0)*SF[15] + P(12,0)*SPP[10]) + SF[3]*(P(0,2) + P(1,2)*SF[9] + P(2,2)*SF[11] + P(3,2)*SF[10] + P(10,2)*SF[14] + P(11,2)*SF[15] + P(12,2)*SPP[10]) + SF[5]*(P(0,3) + P(1,3)*SF[9] + P(2,3)*SF[11] + P(3,3)*SF[10] + P(10,3)*SF[14] + P(11,3)*SF[15] + P(12,3)*SPP[10]) - SPP[0]*(P(0,1) + P(1,1)*SF[9] + P(2,1)*SF[11] + P(3,1)*SF[10] + P(10,1)*SF[14] + P(11,1)*SF[15] + P(12,1)*SPP[10]) - SPP[8]*(P(0,13) + P(1,13)*SF[9] + P(2,13)*SF[11] + P(3,13)*SF[10] + P(10,13)*SF[14] + P(11,13)*SF[15] + P(12,13)*SPP[10]) + SPP[2]*(P(0,14) + P(1,14)*SF[9] + P(2,14)*SF[11] + P(3,14)*SF[10] + P(10,14)*SF[14] + P(11,14)*SF[15] + P(12,14)*SPP[10]) + SPP[5]*(P(0,15) + P(1,15)*SF[9] + P(2,15)*SF[11] + P(3,15)*SF[10] + P(10,15)*SF[14] + P(11,15)*SF[15] + P(12,15)*SPP[10]);
	nextP(1,5) = P(1,5) + P(0,5)*SF[8] + P(2,5)*SF[7] + P(3,5)*SF[11] - P(12,5)*SF[15] + P(11,5)*SPP[10] - (P(10,5)*q0)/2 + SF[4]*(P(1,0) + P(0,0)*SF[8] + P(2,0)*SF[7] + P(3,0)*SF[11] - P(12,0)*SF[15] + P(11,0)*SPP[10] - (P(10,0)*q0)/2) + SF[3]*(P(1,2) + P(0,2)*SF[8] + P(2,2)*SF[7] + P(3,2)*SF[11] - P(12,2)*SF[15] + P(11,2)*SPP[10] - (P(10,2)*q0)/2) + SF[5]*(P(1,3) + P(0,3)*SF[8] + P(2,3)*SF[7] + P(3,3)*SF[11] - P(12,3)*SF[15] + P(11,3)*SPP[10] - (P(10,3)*q0)/2) - SPP[0]*(P(1,1) + P(0,1)*SF[8] + P(2,1)*SF[7] + P(3,1)*SF[11] - P(12,1)*SF[15] + P(11,1)*SPP[10] - (P(10,1)*q0)/2) - SPP[8]*(P(1,13) + P(0,13)*SF[8] + P(2,13)*SF[7] + P(3,13)*SF[11] - P(12,13)*SF[15] + P(11,13)*SPP[10] - (P(10,13)*q0)/2) + SPP[2]*(P(1,14) + P(0,14)*SF[8] + P(2,14)*SF[7] + P(3,14)*SF[11] - P(12,14)*SF[15] + P(11,14)*SPP[10] - (P(10,14)*q0)/2) + SPP[5]*(P(1,15) + P(0,15)*SF[8] + P(2,15)*SF[7] + P(3,15)*SF[11] - P(12,15)*SF[15] + P(11,15)*SPP[10] - (P(10,15)*q0)/2);
	nextP(2,5) = P(2,5) + P(0,5)*SF[6] + P(1,5)*SF[10] + P(3,5)*SF[8] + P(12,5)*SF[14] - P(10,5)*SPP[10] - (P(11,5)*q0)/2 + SF[4]*(P(2,0) + P(0,0)*SF[6] + P(1,0)*SF[10] + P(3,0)*SF[8] + P(12,0)*SF[14] - P(10,0)*SPP[10] - (P(11,0)*q0)/2) + SF[3]*(P(2,2) + P(0,2)*SF[6] + P(1,2)*SF[10] + P(3,2)*SF[8] + P(12,2)*SF[14] - P(10,2)*SPP[10] - (P(11,2)*q0)/2) + SF[5]*(P(2,3) + P(0,3)*SF[6] + P(1,3)*SF[10] + P(3,3)*SF[8] + P(12,3)*SF[14] - P(10,3)*SPP[10] - (P(11,3)*q0)/2) - SPP[0]*(P(2,1) + P(0,1)*SF[6] + P(1,1)*SF[10] + P(3,1)*SF[8] + P(12,1)*SF[14] - P(10,1)*SPP[10] - (P(11,1)*q0)/2) - SPP[8]*(P(2,13) + P(0,13)*SF[6] + P(1,13)*SF[10] + P(3,13)*SF[8] + P(12,13)*SF[14] - P(10,13)*SPP[10] - (P(11,13)*q0)/2) + SPP[2]*(P(2,14) + P(0,14)*SF[6] + P(1,14)*SF[10] + P(3,14)*SF[8] + P(12,14)*SF[14] - P(10,14)*SPP[10] - (P(11,14)*q0)/2) + SPP[5]*(P(2,15) + P(0,15)*SF[6] + P(1,15)*SF[10] + P(3,15)*SF[8] + P(12,15)*SF[14] - P(10,15)*SPP[10] - (P(11,15)*q0)/2);
	nextP(3,5) = P(3,5) + P(0,5)*SF[7] + P(1,5)*SF[6] + P(2,5)*SF[9] + P(10,5)*SF[15] - P(11,5)*SF[14] - (P(12,5)*q0)/2 + SF[4]*(P(3,0) + P(0,0)*SF[7] + P(1,0)*SF[6] + P(2,0)*SF[9] + P(10,0)*SF[15] - P(11,0)*SF[14] - (P(12,0)*q0)/2) + SF[3]*(P(3,2) + P(0,2)*SF[7] + P(1,2)*SF[6] + P(2,2)*SF[9] + P(10,2)*SF[15] - P(11,2)*SF[14] - (P(12,2)*q0)/2) + SF[5]*(P(3,3) + P(0,3)*SF[7] + P(1,3)*SF[6] + P(2,3)*SF[9] + P(10,3)*SF[15] - P(11,3)*SF[14] - (P(12,3)*q0)/2) - SPP[0]*(P(3,1) + P(0,1)*SF[7] + P(1,1)*SF[6] + P(2,1)*SF[9] + P(10,1)*SF[15] - P(11,1)*SF[14] - (P(12,1)*q0)/2) - SPP[8]*(P(3,13) + P(0,13)*SF[7] + P(1,13)*SF[6] + P(2,13)*SF[9] + P(10,13)*SF[15] - P(11,13)*SF[14] - (P(12,13)*q0)/2) + SPP[2]*(P(3,14) + P(0,14)*SF[7] + P(1,14)*SF[6] + P(2,14)*SF[9] + P(10,14)*SF[15] - P(11,14)*SF[14] - (P(12,14)*q0)/2) + SPP[5]*(P(3,15) + P(0,15)*SF[7] + P(1,15)*SF[6] + P(2,15)*SF[9] + P(10,15)*SF[15] - P(11,15)*SF[14] - (P(12,15)*q0)/2);
	nextP(4,5) = P(4,5) + SQ[2] + P(0,5)*SF[5] + P(1,5)*SF[3] - P(3,5)*SF[4] + P(2,5)*SPP[0] + P(13,5)*SPP[3] + P(14,5)*SPP[6] - P(15,5)*SPP[9] + SF[4]*(P(4,0) + P(0,0)*SF[5] + P(1,0)*SF[3] - P(3,0)*SF[4] + P(2,0)*SPP[0] + P(13,0)*SPP[3] + P(14,0)*SPP[6] - P(15,0)*SPP[9]) + SF[3]*(P(4,2) + P(0,2)*SF[5] + P(1,2)*SF[3] - P(3,2)*SF[4] + P(2,2)*SPP[0] + P(13,2)*SPP[3] + P(14,2)*SPP[6] - P(15,2)*SPP[9]) + SF[5]*(P(4,3) + P(0,3)*SF[5] + P(1,3)*SF[3] - P(3,3)*SF[4] + P(2,3)*SPP[0] + P(13,3)*SPP[3] + P(14,3)*SPP[6] - P(15,3)*SPP[9]) - SPP[0]*(P(4,1) + P(0,1)*SF[5] + P(1,1)*SF[3] - P(3,1)*SF[4] + P(2,1)*SPP[0] + P(13,1)*SPP[3] + P(14,1)*SPP[6] - P(15,1)*SPP[9]) - SPP[8]*(P(4,13) + P(0,13)*SF[5] + P(1,13)*SF[3] - P(3,13)*SF[4] + P(2,13)*SPP[0] + P(13,13)*SPP[3] + P(14,13)*SPP[6] - P(15,13)*SPP[9]) + SPP[2]*(P(4,14) + P(0,14)*SF[5] + P(1,14)*SF[3] - P(3,14)*SF[4] + P(2,14)*SPP[0] + P(13,14)*SPP[3] + P(14,14)*SPP[6] - P(15,14)*SPP[9]) + SPP[5]*(P(4,15) + P(0,15)*SF[5] + P(1,15)*SF[3] - P(3,15)*SF[4] + P(2,15)*SPP[0] + P(13,15)*SPP[3] + P(14,15)*SPP[6] - P(15,15)*SPP[9]);
	nextP(5,5) = P(5,5) + P(0,5)*SF[4] + P(2,5)*SF[3] + P(3,5)*SF[5] - P(1,5)*SPP[0] - P(13,5)*SPP[8] + P(14,5)*SPP[2] + P(15,5)*SPP[5] + dvxVar*sq(SG[7] + 2*q0*q3) + dvzVar*sq(SG[5] - 2*q0*q1) + SF[4]*(P(5,0) + P(0,0)*SF[4] + P(2,0)*SF[3] + P(3,0)*SF[5] - P(1,0)*SPP[0] - P(13,0)*SPP[8] + P(14,0)*SPP[2] + P(15,0)*SPP[5]) + SF[3]*(P(5,2) + P(0,2)*SF[4] + P(2,2)*SF[3] + P(3,2)*SF[5] - P(1,2)*SPP[0] - P(13,2)*SPP[8] + P(14,2)*SPP[2] + P(15,2)*SPP[5]) + SF[5]*(P(5,3) + P(0,3)*SF[4] + P(2,3)*SF[3] + P(3,3)*SF[5] - P(1,3)*SPP[0] - P(13,3)*SPP[8] + P(14,3)*SPP[2] + P(15,3)*SPP[5]) - SPP[0]*(P(5,1) + P(0,1)*SF[4] + P(2,1)*SF[3] + P(3,1)*SF[5] - P(1,1)*SPP[0] - P(13,1)*SPP[8] + P(14,1)*SPP[2] + P(15,1)*SPP[5]) - SPP[8]*(P(5,13) + P(0,13)*SF[4] + P(2,13)*SF[3] + P(3,13)*SF[5] - P(1,13)*SPP[0] - P(13,13)*SPP[8] + P(14,13)*SPP[2] + P(15,13)*SPP[5]) + SPP[2]*(P(5,14) + P(0,14)*SF[4] + P(2,14)*SF[3] + P(3,14)*SF[5] - P(1,14)*SPP[0] - P(13,14)*SPP[8] + P(14,14)*SPP[2] + P(15,14)*SPP[5]) + SPP[5]*(P(5,15) + P(0,15)*SF[4] + P(2,15)*SF[3] + P(3,15)*SF[5] - P(1,15)*SPP[0] - P(13,15)*SPP[8] + P(14,15)*SPP[2] + P(15,15)*SPP[5]) + dvyVar*sq(SG[1] - SG[2] + SG[3] - SG[4]);
	nextP(0,6) = P(0,6) + P(1,6)*SF[9] + P(2,6)*SF[11] + P(3,6)*SF[10] + P(10,6)*SF[14] + P(11,6)*SF[15] + P(12,6)*SPP[10] + SF[4]*(P(0,1) + P(1,1)*SF[9] + P(2,1)*SF[11] + P(3,1)*SF[10] + P(10,1)*SF[14] + P(11,1)*SF[15] + P(12,1)*SPP[10]) - SF[5]*(P(0,2) + P(1,2)*SF[9] + P(2,2)*SF[11] + P(3,2)*SF[10] + P(10,2)*SF[14] + P(11,2)*SF[15] + P(12,2)*SPP[10]) + SF[3]*(P(0,3) + P(1,3)*SF[9] + P(2,3)*SF[11] + P(3,3)*SF[10] + P(10,3)*SF[14] + P(11,3)*SF[15] + P(12,3)*SPP[10]) + SPP[0]*(P(0,0) + P(1,0)*SF[9] + P(2,0)*SF[11] + P(3,0)*SF[10] + P(10,0)*SF[14] + P(11,0)*SF[15] + P(12,0)*SPP[10]) + SPP[4]*(P(0,13) + P(1,13)*SF[9] + P(2,13)*SF[11] + P(3,13)*SF[10] + P(10,13)*SF[14] + P(11,13)*SF[15] + P(12,13)*SPP[10]) - SPP[7]*(P(0,14) + P(1,14)*SF[9] + P(2,14)*SF[11] + P(3,14)*SF[10] + P(10,14)*SF[14] + P(11,14)*SF[15] + P(12,14)*SPP[10]) - SPP[1]*(P(0,15) + P(1,15)*SF[9] + P(2,15)*SF[11] + P(3,15)*SF[10] + P(10,15)*SF[14] + P(11,15)*SF[15] + P(12,15)*SPP[10]);
	nextP(1,6) = P(1,6) + P(0,6)*SF[8] + P(2,6)*SF[7] + P(3,6)*SF[11] - P(12,6)*SF[15] + P(11,6)*SPP[10] - (P(10,6)*q0)/2 + SF[4]*(P(1,1) + P(0,1)*SF[8] + P(2,1)*SF[7] + P(3,1)*SF[11] - P(12,1)*SF[15] + P(11,1)*SPP[10] - (P(10,1)*q0)/2) - SF[5]*(P(1,2) + P(0,2)*SF[8] + P(2,2)*SF[7] + P(3,2)*SF[11] - P(12,2)*SF[15] + P(11,2)*SPP[10] - (P(10,2)*q0)/2) + SF[3]*(P(1,3) + P(0,3)*SF[8] + P(2,3)*SF[7] + P(3,3)*SF[11] - P(12,3)*SF[15] + P(11,3)*SPP[10] - (P(10,3)*q0)/2) + SPP[0]*(P(1,0) + P(0,0)*SF[8] + P(2,0)*SF[7] + P(3,0)*SF[11] - P(12,0)*SF[15] + P(11,0)*SPP[10] - (P(10,0)*q0)/2) + SPP[4]*(P(1,13) + P(0,13)*SF[8] + P(2,13)*SF[7] + P(3,13)*SF[11] - P(12,13)*SF[15] + P(11,13)*SPP[10] - (P(10,13)*q0)/2) - SPP[7]*(P(1,14) + P(0,14)*SF[8] + P(2,14)*SF[7] + P(3,14)*SF[11] - P(12,14)*SF[15] + P(11,14)*SPP[10] - (P(10,14)*q0)/2) - SPP[1]*(P(1,15) + P(0,15)*SF[8] + P(2,15)*SF[7] + P(3,15)*SF[11] - P(12,15)*SF[15] + P(11,15)*SPP[10] - (P(10,15)*q0)/2);
	nextP(2,6) = P(2,6) + P(0,6)*SF[6] + P(1,6)*SF[10] + P(3,6)*SF[8] + P(12,6)*SF[14] - P(10,6)*SPP[10] - (P(11,6)*q0)/2 + SF[4]*(P(2,1) + P(0,1)*SF[6] + P(1,1)*SF[10] + P(3,1)*SF[8] + P(12,1)*SF[14] - P(10,1)*SPP[10] - (P(11,1)*q0)/2) - SF[5]*(P(2,2) + P(0,2)*SF[6] + P(1,2)*SF[10] + P(3,2)*SF[8] + P(12,2)*SF[14] - P(10,2)*SPP[10] - (P(11,2)*q0)/2) + SF[3]*(P(2,3) + P(0,3)*SF[6] + P(1,3)*SF[10] + P(3,3)*SF[8] + P(12,3)*SF[14] - P(10,3)*SPP[10] - (P(11,3)*q0)/2) + SPP[0]*(P(2,0) + P(0,0)*SF[6] + P(1,0)*SF[10] + P(3,0)*SF[8] + P(12,0)*SF[14] - P(10,0)*SPP[10] - (P(11,0)*q0)/2) + SPP[4]*(P(2,13) + P(0,13)*SF[6] + P(1,13)*SF[10] + P(3,13)*SF[8] + P(12,13)*SF[14] - P(10,13)*SPP[10] - (P(11,13)*q0)/2) - SPP[7]*(P(2,14) + P(0,14)*SF[6] + P(1,14)*SF[10] + P(3,14)*SF[8] + P(12,14)*SF[14] - P(10,14)*SPP[10] - (P(11,14)*q0)/2) - SPP[1]*(P(2,15) + P(0,15)*SF[6] + P(1,15)*SF[10] + P(3,15)*SF[8] + P(12,15)*SF[14] - P(10,15)*SPP[10] - (P(11,15)*q0)/2);
	nextP(3,6) = P(3,6) + P(0,6)*SF[7] + P(1,6)*SF[6] + P(2,6)*SF[9] + P(10,6)*SF[15] - P(11,6)*SF[14] - (P(12,6)*q0)/2 + SF[4]*(P(3,1) + P(0,1)*SF[7] + P(1,1)*SF[6] + P(2,1)*SF[9] + P(10,1)*SF[15] - P(11,1)*SF[14] - (P(12,1)*q0)/2) - SF[5]*(P(3,2) + P(0,2)*SF[7] + P(1,2)*SF[6] + P(2,2)*SF[9] + P(10,2)*SF[15] - P(11,2)*SF[14] - (P(12,2)*q0)/2) + SF[3]*(P(3,3) + P(0,3)*SF[7] + P(1,3)*SF[6] + P(2,3)*SF[9] + P(10,3)*SF[15] - P(11,3)*SF[14] - (P(12,3)*q0)/2) + SPP[0]*(P(3,0) + P(0,0)*SF[7] + P(1,0)*SF[6] + P(2,0)*SF[9] + P(10,0)*SF[15] - P(11,0)*SF[14] - (P(12,0)*q0)/2) + SPP[4]*(P(3,13) + P(0,13)*SF[7] + P(1,13)*SF[6] + P(2,13)*SF[9] + P(10,13)*SF[15] - P(11,13)*SF[14] - (P(12,13)*q0)/2) - SPP[7]*(P(3,14) + P(0,14)*SF[7] + P(1,14)*SF[6] + P(2,14)*SF[9] + P(10,14)*SF[15] - P(11,14)*SF[14] - (P(12,14)*q0)/2) - SPP[1]*(P(3,15) + P(0,15)*SF[7] + P(1,15)*SF[6] + P(2,15)*SF[9] + P(10,15)*SF[15] - P(11,15)*SF[14] - (P(12,15)*q0)/2);
	nextP(4,6) = P(4,6) + SQ[1] + P(0,6)*SF[5] + P(1,6)*SF[3] - P(3,6)*SF[4] + P(2,6)*SPP[0] + P(13,6)*SPP[3] + P(14,6)*SPP[6] - P(15,6)*SPP[9] + SF[4]*(P(4,1) + P(0,1)*SF[5] + P(1,1)*SF[3] - P(3,1)*SF[4] + P(2,1)*SPP[0] + P(13,1)*SPP[3] + P(14,1)*SPP[6] - P(15,1)*SPP[9]) - SF[5]*(P(4,2) + P(0,2)*SF[5] + P(1,2)*SF[3] - P(3,2)*SF[4] + P(2,2)*SPP[0] + P(13,2)*SPP[3] + P(14,2)*SPP[6] - P(15,2)*SPP[9]) + SF[3]*(P(4,3) + P(0,3)*SF[5] + P(1,3)*SF[3] - P(3,3)*SF[4] + P(2,3)*SPP[0] + P(13,3)*SPP[3] + P(14,3)*SPP[6] - P(15,3)*SPP[9]) + SPP[0]*(P(4,0) + P(0,0)*SF[5] + P(1,0)*SF[3] - P(3,0)*SF[4] + P(2,0)*SPP[0] + P(13,0)*SPP[3] + P(14,0)*SPP[6] - P(15,0)*SPP[9]) + SPP[4]*(P(4,13) + P(0,13)*SF[5] + P(1,13)*SF[3] - P(3,13)*SF[4] + P(2,13)*SPP[0] + P(13,13)*SPP[3] + P(14,13)*SPP[6] - P(15,13)*SPP[9]) - SPP[7]*(P(4,14) + P(0,14)*SF[5] + P(1,14)*SF[3] - P(3,14)*SF[4] + P(2,14)*SPP[0] + P(13,14)*SPP[3] + P(14,14)*SPP[6] - P(15,14)*SPP[9]) - SPP[1]*(P(4,15) + P(0,15)*SF[5] + P(1,15)*SF[3] - P(3,15)*SF[4] + P(2,15)*SPP[0] + P(13,15)*SPP[3] + P(14,15)*SPP[6] - P(15,15)*SPP[9]);
	nextP(5,6) = P(5,6) + SQ[0] + P(0,6)*SF[4] + P(2,6)*SF[3] + P(3,6)*SF[5] - P(1,6)*SPP[0] - P(13,6)*SPP[8] + P(14,6)*SPP[2] + P(15,6)*SPP[5] + SF[4]*(P(5,1) + P(0,1)*SF[4] + P(2,1)*SF[3] + P(3,1)*SF[5] - P(1,1)*SPP[0] - P(13,1)*SPP[8] + P(14,1)*SPP[2] + P(15,1)*SPP[5]) - SF[5]*(P(5,2) + P(0,2)*SF[4] + P(2,2)*SF[3] + P(3,2)*SF[5] - P(1,2)*SPP[0] - P(13,2)*SPP[8] + P(14,2)*SPP[2] + P(15,2)*SPP[5]) + SF[3]*(P(5,3) + P(0,3)*SF[4] + P(2,3)*SF[3] + P(3,3)*SF[5] - P(1,3)*SPP[0] - P(13,3)*SPP[8] + P(14,3)*SPP[2] + P(15,3)*SPP[5]) + SPP[0]*(P(5,0) + P(0,0)*SF[4] + P(2,0)*SF[3] + P(3,0)*SF[5] - P(1,0)*SPP[0] - P(13,0)*SPP[8] + P(14,0)*SPP[2] + P(15,0)*SPP[5]) + SPP[4]*(P(5,13) + P(0,13)*SF[4] + P(2,13)*SF[3] + P(3,13)*SF[5] - P(1,13)*SPP[0] - P(13,13)*SPP[8] + P(14,13)*SPP[2] + P(15,13)*SPP[5]) - SPP[7]*(P(5,14) + P(0,14)*SF[4] + P(2,14)*SF[3] + P(3,14)*SF[5] - P(1,14)*SPP[0] - P(13,14)*SPP[8] + P(14,14)*SPP[2] + P(15,14)*SPP[5]) - SPP[1]*(P(5,15) + P(0,15)*SF[4] + P(2,15)*SF[3] + P(3,15)*SF[5] - P(1,15)*SPP[0] - P(13,15)*SPP[8] + P(14,15)*SPP[2] + P(15,15)*SPP[5]);
	nextP(6,6) = P(6,6) + P(1,6)*SF[4] - P(2,6)*SF[5] + P(3,6)*SF[3] + P(0,6)*SPP[0] + P(13,6)*SPP[4] - P(14,6)*SPP[7] - P(15,6)*SPP[1] + dvxVar*sq(SG[6] - 2*q0*q2) + dvyVar*sq(SG[5] + 2*q0*q1) + SF[4]*(P(6,1) + P(1,1)*SF[4] - P(2,1)*SF[5] + P(3,1)*SF[3] + P(0,1)*SPP[0] + P(13,1)*SPP[4] - P(14,1)*SPP[7] - P(15,1)*SPP[1]) - SF[5]*(P(6,2) + P(1,2)*SF[4] - P(2,2)*SF[5] + P(3,2)*SF[3] + P(0,2)*SPP[0] + P(13,2)*SPP[4] - P(14,2)*SPP[7] - P(15,2)*SPP[1]) + SF[3]*(P(6,3) + P(1,3)*SF[4] - P(2,3)*SF[5] + P(3,3)*SF[3] + P(0,3)*SPP[0] + P(13,3)*SPP[4] - P(14,3)*SPP[7] - P(15,3)*SPP[1]) + SPP[0]*(P(6,0) + P(1,0)*SF[4] - P(2,0)*SF[5] + P(3,0)*SF[3] + P(0,0)*SPP[0] + P(13,0)*SPP[4] - P(14,0)*SPP[7] - P(15,0)*SPP[1]) + SPP[4]*(P(6,13) + P(1,13)*SF[4] - P(2,13)*SF[5] + P(3,13)*SF[3] + P(0,13)*SPP[0] + P(13,13)*SPP[4] - P(14,13)*SPP[7] - P(15,13)*SPP[1]) - SPP[7]*(P(6,14) + P(1,14)*SF[4] - P(2,14)*SF[5] + P(3,14)*SF[3] + P(0,14)*SPP[0] + P(13,14)*SPP[4] - P(14,14)*SPP[7] - P(15,14)*SPP[1]) - SPP[1]*(P(6,15) + P(1,15)*SF[4] - P(2,15)*SF[5] + P(3,15)*SF[3] + P(0,15)*SPP[0] + P(13,15)*SPP[4] - P(14,15)*SPP[7] - P(15,15)*SPP[1]) + dvzVar*sq(SG[1] - SG[2] - SG[3] + SG[4]);
	nextP(0,7) = P(0,7) + P(1,7)*SF[9] + P(2,7)*SF[11] + P(3,7)*SF[10] + P(10,7)*SF[14] + P(11,7)*SF[15] + P(12,7)*SPP[10] + dt*(P(0,4) + P(1,4)*SF[9] + P(2,4)*SF[11] + P(3,4)*SF[10] + P(10,4)*SF[14] + P(11,4)*SF[15] + P(12,4)*SPP[10]);
	nextP(1,7) = P(1,7) + P(0,7)*SF[8] + P(2,7)*SF[7] + P(3,7)*SF[11] - P(12,7)*SF[15] + P(11,7)*SPP[10] - (P(10,7)*q0)/2 + dt*(P(1,4) + P(0,4)*SF[8] + P(2,4)*SF[7] + P(3,4)*SF[11] - P(12,4)*SF[15] + P(11,4)*SPP[10] - (P(10,4)*q0)/2);
	nextP(2,7) = P(2,7) + P(0,7)*SF[6] + P(1,7)*SF[10] + P(3,7)*SF[8] + P(12,7)*SF[14] - P(10,7)*SPP[10] - (P(11,7)*q0)/2 + dt*(P(2,4) + P(0,4)*SF[6] + P(1,4)*SF[10] + P(3,4)*SF[8] + P(12,4)*SF[14] - P(10,4)*SPP[10] - (P(11,4)*q0)/2);
	nextP(3,7) = P(3,7) + P(0,7)*SF[7] + P(1,7)*SF[6] + P(2,7)*SF[9] + P(10,7)*SF[15] - P(11,7)*SF[14] - (P(12,7)*q0)/2 + dt*(P(3,4) + P(0,4)*SF[7] + P(1,4)*SF[6] + P(2,4)*SF[9] + P(10,4)*SF[15] - P(11,4)*SF[14] - (P(12,4)*q0)/2);
	nextP(4,7) = P(4,7) + P(0,7)*SF[5] + P(1,7)*SF[3] - P(3,7)*SF[4] + P(2,7)*SPP[0] + P(13,7)*SPP[3] + P(14,7)*SPP[6] - P(15,7)*SPP[9] + dt*(P(4,4) + P(0,4)*SF[5] + P(1,4)*SF[3] - P(3,4)*SF[4] + P(2,4)*SPP[0] + P(13,4)*SPP[3] + P(14,4)*SPP[6] - P(15,4)*SPP[9]);
	nextP(5,7) = P(5,7) + P(0,7)*SF[4] + P(2,7)*SF[3] + P(3,7)*SF[5] - P(1,7)*SPP[0] - P(13,7)*SPP[8] + P(14,7)*SPP[2] + P(15,7)*SPP[5] + dt*(P(5,4) + P(0,4)*SF[4] + P(2,4)*SF[3] + P(3,4)*SF[5] - P(1,4)*SPP[0] - P(13,4)*SPP[8] + P(14,4)*SPP[2] + P(15,4)*SPP[5]);
	nextP(6,7) = P(6,7) + P(1,7)*SF[4] - P(2,7)*SF[5] + P(3,7)*SF[3] + P(0,7)*SPP[0] + P(13,7)*SPP[4] - P(14,7)*SPP[7] - P(15,7)*SPP[1] + dt*(P(6,4) + P(1,4)*SF[4] - P(2,4)*SF[5] + P(3,4)*SF[3] + P(0,4)*SPP[0] + P(13,4)*SPP[4] - P(14,4)*SPP[7] - P(15,4)*SPP[1]);
	nextP(7,7) = P(7,7) + P(4,7)*dt + dt*(P(7,4) + P(4,4)*dt);
	nextP(0,8) = P(0,8) + P(1,8)*SF[9] + P(2,8)*SF[11] + P(3,8)*SF[10] + P(10,8)*SF[14] + P(11,8)*SF[15] + P(12,8)*SPP[10] + dt*(P(0,5) + P(1,5)*SF[9] + P(2,5)*SF[11] + P(3,5)*SF[10] + P(10,5)*SF[14] + P(11,5)*SF[15] + P(12,5)*SPP[10]);
	nextP(1,8) = P(1,8) + P(0,8)*SF[8] + P(2,8)*SF[7] + P(3,8)*SF[11] - P(12,8)*SF[15] + P(11,8)*SPP[10] - (P(10,8)*q0)/2 + dt*(P(1,5) + P(0,5)*SF[8] + P(2,5)*SF[7] + P(3,5)*SF[11] - P(12,5)*SF[15] + P(11,5)*SPP[10] - (P(10,5)*q0)/2);
	nextP(2,8) = P(2,8) + P(0,8)*SF[6] + P(1,8)*SF[10] + P(3,8)*SF[8] + P(12,8)*SF[14] - P(10,8)*SPP[10] - (P(11,8)*q0)/2 + dt*(P(2,5) + P(0,5)*SF[6] + P(1,5)*SF[10] + P(3,5)*SF[8] + P(12,5)*SF[14] - P(10,5)*SPP[10] - (P(11,5)*q0)/2);
	nextP(3,8) = P(3,8) + P(0,8)*SF[7] + P(1,8)*SF[6] + P(2,8)*SF[9] + P(10,8)*SF[15] - P(11,8)*SF[14] - (P(12,8)*q0)/2 + dt*(P(3,5) + P(0,5)*SF[7] + P(1,5)*SF[6] + P(2,5)*SF[9] + P(10,5)*SF[15] - P(11,5)*SF[14] - (P(12,5)*q0)/2);
	nextP(4,8) = P(4,8) + P(0,8)*SF[5] + P(1,8)*SF[3] - P(3,8)*SF[4] + P(2,8)*SPP[0] + P(13,8)*SPP[3] + P(14,8)*SPP[6] - P(15,8)*SPP[9] + dt*(P(4,5) + P(0,5)*SF[5] + P(1,5)*SF[3] - P(3,5)*SF[4] + P(2,5)*SPP[0] + P(13,5)*SPP[3] + P(14,5)*SPP[6] - P(15,5)*SPP[9]);
	nextP(5,8) = P(5,8) + P(0,8)*SF[4] + P(2,8)*SF[3] + P(3,8)*SF[5] - P(1,8)*SPP[0] - P(13,8)*SPP[8] + P(14,8)*SPP[2] + P(15,8)*SPP[5] + dt*(P(5,5) + P(0,5)*SF[4] + P(2,5)*SF[3] + P(3,5)*SF[5] - P(1,5)*SPP[0] - P(13,5)*SPP[8] + P(14,5)*SPP[2] + P(15,5)*SPP[5]);
	nextP(6,8) = P(6,8) + P(1,8)*SF[4] - P(2,8)*SF[5] + P(3,8)*SF[3] + P(0,8)*SPP[0] + P(13,8)*SPP[4] - P(14,8)*SPP[7] - P(15,8)*SPP[1] + dt*(P(6,5) + P(1,5)*SF[4] - P(2,5)*SF[5] + P(3,5)*SF[3] + P(0,5)*SPP[0] + P(13,5)*SPP[4] - P(14,5)*SPP[7] - P(15,5)*SPP[1]);
	nextP(7,8) = P(7,8) + P(4,8)*dt + dt*(P(7,5) + P(4,5)*dt);
	nextP(8,8) = P(8,8) + P(5,8)*dt + dt*(P(8,5) + P(5,5)*dt);
	nextP(0,9) = P(0,9) + P(1,9)*SF[9] + P(2,9)*SF[11] + P(3,9)*SF[10] + P(10,9)*SF[14] + P(11,9)*SF[15] + P(12,9)*SPP[10] + dt*(P(0,6) + P(1,6)*SF[9] + P(2,6)*SF[11] + P(3,6)*SF[10] + P(10,6)*SF[14] + P(11,6)*SF[15] + P(12,6)*SPP[10]);
	nextP(1,9) = P(1,9) + P(0,9)*SF[8] + P(2,9)*SF[7] + P(3,9)*SF[11] - P(12,9)*SF[15] + P(11,9)*SPP[10] - (P(10,9)*q0)/2 + dt*(P(1,6) + P(0,6)*SF[8] + P(2,6)*SF[7] + P(3,6)*SF[11] - P(12,6)*SF[15] + P(11,6)*SPP[10] - (P(10,6)*q0)/2);
	nextP(2,9) = P(2,9) + P(0,9)*SF[6] + P(1,9)*SF[10] + P(3,9)*SF[8] + P(12,9)*SF[14] - P(10,9)*SPP[10] - (P(11,9)*q0)/2 + dt*(P(2,6) + P(0,6)*SF[6] + P(1,6)*SF[10] + P(3,6)*SF[8] + P(12,6)*SF[14] - P(10,6)*SPP[10] - (P(11,6)*q0)/2);
	nextP(3,9) = P(3,9) + P(0,9)*SF[7] + P(1,9)*SF[6] + P(2,9)*SF[9] + P(10,9)*SF[15] - P(11,9)*SF[14] - (P(12,9)*q0)/2 + dt*(P(3,6) + P(0,6)*SF[7] + P(1,6)*SF[6] + P(2,6)*SF[9] + P(10,6)*SF[15] - P(11,6)*SF[14] - (P(12,6)*q0)/2);
	nextP(4,9) = P(4,9) + P(0,9)*SF[5] + P(1,9)*SF[3] - P(3,9)*SF[4] + P(2,9)*SPP[0] + P(13,9)*SPP[3] + P(14,9)*SPP[6] - P(15,9)*SPP[9] + dt*(P(4,6) + P(0,6)*SF[5] + P(1,6)*SF[3] - P(3,6)*SF[4] + P(2,6)*SPP[0] + P(13,6)*SPP[3] + P(14,6)*SPP[6] - P(15,6)*SPP[9]);
	nextP(5,9) = P(5,9) + P(0,9)*SF[4] + P(2,9)*SF[3] + P(3,9)*SF[5] - P(1,9)*SPP[0] - P(13,9)*SPP[8] + P(14,9)*SPP[2] + P(15,9)*SPP[5] + dt*(P(5,6) + P(0,6)*SF[4] + P(2,6)*SF[3] + P(3,6)*SF[5] - P(1,6)*SPP[0] - P(13,6)*SPP[8] + P(14,6)*SPP[2] + P(15,6)*SPP[5]);
	nextP(6,9) = P(6,9) + P(1,9)*SF[4] - P(2,9)*SF[5] + P(3,9)*SF[3] + P(0,9)*SPP[0] + P(13,9)*SPP[4] - P(14,9)*SPP[7] - P(15,9)*SPP[1] + dt*(P(6,6) + P(1,6)*SF[4] - P(2,6)*SF[5] + P(3,6)*SF[3] + P(0,6)*SPP[0] + P(13,6)*SPP[4] - P(14,6)*SPP[7] - P(15,6)*SPP[1]);
	nextP(7,9) = P(7,9) + P(4,9)*dt + dt*(P(7,6) + P(4,6)*dt);
	nextP(8,9) = P(8,9) + P(5,9)*dt + dt*(P(8,6) + P(5,6)*dt);
	nextP(9,9) = P(9,9) + P(6,9)*dt + dt*(P(9,6) + P(6,6)*dt);
	nextP(0,10) = P(0,10) + P(1,10)*SF[9] + P(2,10)*SF[11] + P(3,10)*SF[10] + P(10,10)*SF[14] + P(11,10)*SF[15] + P(12,10)*SPP[10];
	nextP(1,10) = P(1,10) + P(0,10)*SF[8] + P(2,10)*SF[7] + P(3,10)*SF[11] - P(12,10)*SF[15] + P(11,10)*SPP[10] - (P(10,10)*q0)/2;
	nextP(2,10) = P(2,10) + P(0,10)*SF[6] + P(1,10)*SF[10] + P(3,10)*SF[8] + P(12,10)*SF[14] - P(10,10)*SPP[10] - (P(11,10)*q0)/2;
	nextP(3,10) = P(3,10) + P(0,10)*SF[7] + P(1,10)*SF[6] + P(2,10)*SF[9] + P(10,10)*SF[15] - P(11,10)*SF[14] - (P(12,10)*q0)/2;
	nextP(4,10) = P(4,10) + P(0,10)*SF[5] + P(1,10)*SF[3] - P(3,10)*SF[4] + P(2,10)*SPP[0] + P(13,10)*SPP[3] + P(14,10)*SPP[6] - P(15,10)*SPP[9];
	nextP(5,10) = P(5,10) + P(0,10)*SF[4] + P(2,10)*SF[3] + P(3,10)*SF[5] - P(1,10)*SPP[0] - P(13,10)*SPP[8] + P(14,10)*SPP[2] + P(15,10)*SPP[5];
	nextP(6,10) = P(6,10) + P(1,10)*SF[4] - P(2,10)*SF[5] + P(3,10)*SF[3] + P(0,10)*SPP[0] + P(13,10)*SPP[4] - P(14,10)*SPP[7] - P(15,10)*SPP[1];
	nextP(7,10) = P(7,10) + P(4,10)*dt;
	nextP(8,10) = P(8,10) + P(5,10)*dt;
	nextP(9,10) = P(9,10) + P(6,10)*dt;
	nextP(10,10) = P(10,10);
	nextP(0,11) = P(0,11) + P(1,11)*SF[9] + P(2,11)*SF[11] + P(3,11)*SF[10] + P(10,11)*SF[14] + P(11,11)*SF[15] + P(12,11)*SPP[10];
	nextP(1,11) = P(1,11) + P(0,11)*SF[8] + P(2,11)*SF[7] + P(3,11)*SF[11] - P(12,11)*SF[15] + P(11,11)*SPP[10] - (P(10,11)*q0)/2;
	nextP(2,11) = P(2,11) + P(0,11)*SF[6] + P(1,11)*SF[10] + P(3,11)*SF[8] + P(12,11)*SF[14] - P(10,11)*SPP[10] - (P(11,11)*q0)/2;
	nextP(3,11) = P(3,11) + P(0,11)*SF[7] + P(1,11)*SF[6] + P(2,11)*SF[9] + P(10,11)*SF[15] - P(11,11)*SF[14] - (P(12,11)*q0)/2;
	nextP(4,11) = P(4,11) + P(0,11)*SF[5] + P(1,11)*SF[3] - P(3,11)*SF[4] + P(2,11)*SPP[0] + P(13,11)*SPP[3] + P(14,11)*SPP[6] - P(15,11)*SPP[9];
	nextP(5,11) = P(5,11) + P(0,11)*SF[4] + P(2,11)*SF[3] + P(3,11)*SF[5] - P(1,11)*SPP[0] - P(13,11)*SPP[8] + P(14,11)*SPP[2] + P(15,11)*SPP[5];
	nextP(6,11) = P(6,11) + P(1,11)*SF[4] - P(2,11)*SF[5] + P(3,11)*SF[3] + P(0,11)*SPP[0] + P(13,11)*SPP[4] - P(14,11)*SPP[7] - P(15,11)*SPP[1];
	nextP(7,11) = P(7,11) + P(4,11)*dt;
	nextP(8,11) = P(8,11) + P(5,11)*dt;
	nextP(9,11) = P(9,11) + P(6,11)*dt;
	nextP(10,11) = P(10,11);
	nextP(11,11) = P(11,11);
	nextP(0,12) = P(0,12) + P(1,12)*SF[9] + P(2,12)*SF[11] + P(3,12)*SF[10] + P(10,12)*SF[14] + P(11,12)*SF[15] + P(12,12)*SPP[10];
	nextP(1,12) = P(1,12) + P(0,12)*SF[8] + P(2,12)*SF[7] + P(3,12)*SF[11] - P(12,12)*SF[15] + P(11,12)*SPP[10] - (P(10,12)*q0)/2;
	nextP(2,12) = P(2,12) + P(0,12)*SF[6] + P(1,12)*SF[10] + P(3,12)*SF[8] + P(12,12)*SF[14] - P(10,12)*SPP[10] - (P(11,12)*q0)/2;
	nextP(3,12) = P(3,12) + P(0,12)*SF[7] + P(1,12)*SF[6] + P(2,12)*SF[9] + P(10,12)*SF[15] - P(11,12)*SF[14] - (P(12,12)*q0)/2;
	nextP(4,12) = P(4,12) + P(0,12)*SF[5] + P(1,12)*SF[3] - P(3,12)*SF[4] + P(2,12)*SPP[0] + P(13,12)*SPP[3] + P(14,12)*SPP[6] - P(15,12)*SPP[9];
	nextP(5,12) = P(5,12) + P(0,12)*SF[4] + P(2,12)*SF[3] + P(3,12)*SF[5] - P(1,12)*SPP[0] - P(13,12)*SPP[8] + P(14,12)*SPP[2] + P(15,12)*SPP[5];
	nextP(6,12) = P(6,12) + P(1,12)*SF[4] - P(2,12)*SF[5] + P(3,12)*SF[3] + P(0,12)*SPP[0] + P(13,12)*SPP[4] - P(14,12)*SPP[7] - P(15,12)*SPP[1];
	nextP(7,12) = P(7,12) + P(4,12)*dt;
	nextP(8,12) = P(8,12) + P(5,12)*dt;
	nextP(9,12) = P(9,12) + P(6,12)*dt;
	nextP(10,12) = P(10,12);
	nextP(11,12) = P(11,12);
	nextP(12,12) = P(12,12);

	// add process noise that is not from the IMU
	for (unsigned i = 0; i <= 9; i++) {
		nextP(i,i) += process_noise(i);
	}

	// process noise contribution for delta angle states can be very small compared to
	// the variances, therefore use algorithm to minimise numerical error
	for (unsigned i = 10; i <=12; i++) {
		const int index = i-10;
		nextP(i,i) = kahanSummation(nextP(i,i), process_noise(i), _delta_angle_bias_var_accum(index));
	}

	// Don't calculate these covariance terms if IMU delta velocity bias estimation is inhibited
	if (!(_params.fusion_mode & MASK_INHIBIT_ACC_BIAS) && !_accel_bias_inhibit) {

		// calculate variances and upper diagonal covariances for IMU delta velocity bias states
		nextP(0,13) = P(0,13) + P(1,13)*SF[9] + P(2,13)*SF[11] + P(3,13)*SF[10] + P(10,13)*SF[14] + P(11,13)*SF[15] + P(12,13)*SPP[10];
		nextP(1,13) = P(1,13) + P(0,13)*SF[8] + P(2,13)*SF[7] + P(3,13)*SF[11] - P(12,13)*SF[15] + P(11,13)*SPP[10] - (P(10,13)*q0)/2;
		nextP(2,13) = P(2,13) + P(0,13)*SF[6] + P(1,13)*SF[10] + P(3,13)*SF[8] + P(12,13)*SF[14] - P(10,13)*SPP[10] - (P(11,13)*q0)/2;
		nextP(3,13) = P(3,13) + P(0,13)*SF[7] + P(1,13)*SF[6] + P(2,13)*SF[9] + P(10,13)*SF[15] - P(11,13)*SF[14] - (P(12,13)*q0)/2;
		nextP(4,13) = P(4,13) + P(0,13)*SF[5] + P(1,13)*SF[3] - P(3,13)*SF[4] + P(2,13)*SPP[0] + P(13,13)*SPP[3] + P(14,13)*SPP[6] - P(15,13)*SPP[9];
		nextP(5,13) = P(5,13) + P(0,13)*SF[4] + P(2,13)*SF[3] + P(3,13)*SF[5] - P(1,13)*SPP[0] - P(13,13)*SPP[8] + P(14,13)*SPP[2] + P(15,13)*SPP[5];
		nextP(6,13) = P(6,13) + P(1,13)*SF[4] - P(2,13)*SF[5] + P(3,13)*SF[3] + P(0,13)*SPP[0] + P(13,13)*SPP[4] - P(14,13)*SPP[7] - P(15,13)*SPP[1];
		nextP(7,13) = P(7,13) + P(4,13)*dt;
		nextP(8,13) = P(8,13) + P(5,13)*dt;
		nextP(9,13) = P(9,13) + P(6,13)*dt;
		nextP(10,13) = P(10,13);
		nextP(11,13) = P(11,13);
		nextP(12,13) = P(12,13);
		nextP(13,13) = P(13,13);
		nextP(0,14) = P(0,14) + P(1,14)*SF[9] + P(2,14)*SF[11] + P(3,14)*SF[10] + P(10,14)*SF[14] + P(11,14)*SF[15] + P(12,14)*SPP[10];
		nextP(1,14) = P(1,14) + P(0,14)*SF[8] + P(2,14)*SF[7] + P(3,14)*SF[11] - P(12,14)*SF[15] + P(11,14)*SPP[10] - (P(10,14)*q0)/2;
		nextP(2,14) = P(2,14) + P(0,14)*SF[6] + P(1,14)*SF[10] + P(3,14)*SF[8] + P(12,14)*SF[14] - P(10,14)*SPP[10] - (P(11,14)*q0)/2;
		nextP(3,14) = P(3,14) + P(0,14)*SF[7] + P(1,14)*SF[6] + P(2,14)*SF[9] + P(10,14)*SF[15] - P(11,14)*SF[14] - (P(12,14)*q0)/2;
		nextP(4,14) = P(4,14) + P(0,14)*SF[5] + P(1,14)*SF[3] - P(3,14)*SF[4] + P(2,14)*SPP[0] + P(13,14)*SPP[3] + P(14,14)*SPP[6] - P(15,14)*SPP[9];
		nextP(5,14) = P(5,14) + P(0,14)*SF[4] + P(2,14)*SF[3] + P(3,14)*SF[5] - P(1,14)*SPP[0] - P(13,14)*SPP[8] + P(14,14)*SPP[2] + P(15,14)*SPP[5];
		nextP(6,14) = P(6,14) + P(1,14)*SF[4] - P(2,14)*SF[5] + P(3,14)*SF[3] + P(0,14)*SPP[0] + P(13,14)*SPP[4] - P(14,14)*SPP[7] - P(15,14)*SPP[1];
		nextP(7,14) = P(7,14) + P(4,14)*dt;
		nextP(8,14) = P(8,14) + P(5,14)*dt;
		nextP(9,14) = P(9,14) + P(6,14)*dt;
		nextP(10,14) = P(10,14);
		nextP(11,14) = P(11,14);
		nextP(12,14) = P(12,14);
		nextP(13,14) = P(13,14);
		nextP(14,14) = P(14,14);
		nextP(0,15) = P(0,15) + P(1,15)*SF[9] + P(2,15)*SF[11] + P(3,15)*SF[10] + P(10,15)*SF[14] + P(11,15)*SF[15] + P(12,15)*SPP[10];
		nextP(1,15) = P(1,15) + P(0,15)*SF[8] + P(2,15)*SF[7] + P(3,15)*SF[11] - P(12,15)*SF[15] + P(11,15)*SPP[10] - (P(10,15)*q0)/2;
		nextP(2,15) = P(2,15) + P(0,15)*SF[6] + P(1,15)*SF[10] + P(3,15)*SF[8] + P(12,15)*SF[14] - P(10,15)*SPP[10] - (P(11,15)*q0)/2;
		nextP(3,15) = P(3,15) + P(0,15)*SF[7] + P(1,15)*SF[6] + P(2,15)*SF[9] + P(10,15)*SF[15] - P(11,15)*SF[14] - (P(12,15)*q0)/2;
		nextP(4,15) = P(4,15) + P(0,15)*SF[5] + P(1,15)*SF[3] - P(3,15)*SF[4] + P(2,15)*SPP[0] + P(13,15)*SPP[3] + P(14,15)*SPP[6] - P(15,15)*SPP[9];
		nextP(5,15) = P(5,15) + P(0,15)*SF[4] + P(2,15)*SF[3] + P(3,15)*SF[5] - P(1,15)*SPP[0] - P(13,15)*SPP[8] + P(14,15)*SPP[2] + P(15,15)*SPP[5];
		nextP(6,15) = P(6,15) + P(1,15)*SF[4] - P(2,15)*SF[5] + P(3,15)*SF[3] + P(0,15)*SPP[0] + P(13,15)*SPP[4] - P(14,15)*SPP[7] - P(15,15)*SPP[1];
		nextP(7,15) = P(7,15) + P(4,15)*dt;
		nextP(8,15) = P(8,15) + P(5,15)*dt;
		nextP(9,15) = P(9,15) + P(6,15)*dt;
		nextP(10,15) = P(10,15);
		nextP(11,15) = P(11,15);
		nextP(12,15) = P(12,15);
		nextP(13,15) = P(13,15);
		nextP(14,15) = P(14,15);
		nextP(15,15) = P(15,15);

		// add process noise that is not from the IMU
		// process noise contributiton for delta velocity states can be very small compared to
		// the variances, therefore use algorithm to minimise numerical error
		for (unsigned i = 13; i <= 15; i++) {
			const int index = i-13;
			nextP(i,i) = kahanSummation(nextP(i,i), process_noise(i), _delta_vel_bias_var_accum(index));
		}

	} else {
		// Inhibit delta velocity bias learning by zeroing the covariance terms
		nextP.uncorrelateCovarianceSetVariance<3>(13, 0.0f);
		_delta_vel_bias_var_accum.setZero();
	}

	// Don't do covariance prediction on magnetic field states unless we are using 3-axis fusion
	if (_control_status.flags.mag_3D) {
		// calculate variances and upper diagonal covariances for earth and body magnetic field states
		nextP(0,16) = P(0,16) + P(1,16)*SF[9] + P(2,16)*SF[11] + P(3,16)*SF[10] + P(10,16)*SF[14] + P(11,16)*SF[15] + P(12,16)*SPP[10];
		nextP(1,16) = P(1,16) + P(0,16)*SF[8] + P(2,16)*SF[7] + P(3,16)*SF[11] - P(12,16)*SF[15] + P(11,16)*SPP[10] - (P(10,16)*q0)/2;
		nextP(2,16) = P(2,16) + P(0,16)*SF[6] + P(1,16)*SF[10] + P(3,16)*SF[8] + P(12,16)*SF[14] - P(10,16)*SPP[10] - (P(11,16)*q0)/2;
		nextP(3,16) = P(3,16) + P(0,16)*SF[7] + P(1,16)*SF[6] + P(2,16)*SF[9] + P(10,16)*SF[15] - P(11,16)*SF[14] - (P(12,16)*q0)/2;
		nextP(4,16) = P(4,16) + P(0,16)*SF[5] + P(1,16)*SF[3] - P(3,16)*SF[4] + P(2,16)*SPP[0] + P(13,16)*SPP[3] + P(14,16)*SPP[6] - P(15,16)*SPP[9];
		nextP(5,16) = P(5,16) + P(0,16)*SF[4] + P(2,16)*SF[3] + P(3,16)*SF[5] - P(1,16)*SPP[0] - P(13,16)*SPP[8] + P(14,16)*SPP[2] + P(15,16)*SPP[5];
		nextP(6,16) = P(6,16) + P(1,16)*SF[4] - P(2,16)*SF[5] + P(3,16)*SF[3] + P(0,16)*SPP[0] + P(13,16)*SPP[4] - P(14,16)*SPP[7] - P(15,16)*SPP[1];
		nextP(7,16) = P(7,16) + P(4,16)*dt;
		nextP(8,16) = P(8,16) + P(5,16)*dt;
		nextP(9,16) = P(9,16) + P(6,16)*dt;
		nextP(10,16) = P(10,16);
		nextP(11,16) = P(11,16);
		nextP(12,16) = P(12,16);
		nextP(13,16) = P(13,16);
		nextP(14,16) = P(14,16);
		nextP(15,16) = P(15,16);
		nextP(16,16) = P(16,16);
		nextP(0,17) = P(0,17) + P(1,17)*SF[9] + P(2,17)*SF[11] + P(3,17)*SF[10] + P(10,17)*SF[14] + P(11,17)*SF[15] + P(12,17)*SPP[10];
		nextP(1,17) = P(1,17) + P(0,17)*SF[8] + P(2,17)*SF[7] + P(3,17)*SF[11] - P(12,17)*SF[15] + P(11,17)*SPP[10] - (P(10,17)*q0)/2;
		nextP(2,17) = P(2,17) + P(0,17)*SF[6] + P(1,17)*SF[10] + P(3,17)*SF[8] + P(12,17)*SF[14] - P(10,17)*SPP[10] - (P(11,17)*q0)/2;
		nextP(3,17) = P(3,17) + P(0,17)*SF[7] + P(1,17)*SF[6] + P(2,17)*SF[9] + P(10,17)*SF[15] - P(11,17)*SF[14] - (P(12,17)*q0)/2;
		nextP(4,17) = P(4,17) + P(0,17)*SF[5] + P(1,17)*SF[3] - P(3,17)*SF[4] + P(2,17)*SPP[0] + P(13,17)*SPP[3] + P(14,17)*SPP[6] - P(15,17)*SPP[9];
		nextP(5,17) = P(5,17) + P(0,17)*SF[4] + P(2,17)*SF[3] + P(3,17)*SF[5] - P(1,17)*SPP[0] - P(13,17)*SPP[8] + P(14,17)*SPP[2] + P(15,17)*SPP[5];
		nextP(6,17) = P(6,17) + P(1,17)*SF[4] - P(2,17)*SF[5] + P(3,17)*SF[3] + P(0,17)*SPP[0] + P(13,17)*SPP[4] - P(14,17)*SPP[7] - P(15,17)*SPP[1];
		nextP(7,17) = P(7,17) + P(4,17)*dt;
		nextP(8,17) = P(8,17) + P(5,17)*dt;
		nextP(9,17) = P(9,17) + P(6,17)*dt;
		nextP(10,17) = P(10,17);
		nextP(11,17) = P(11,17);
		nextP(12,17) = P(12,17);
		nextP(13,17) = P(13,17);
		nextP(14,17) = P(14,17);
		nextP(15,17) = P(15,17);
		nextP(16,17) = P(16,17);
		nextP(17,17) = P(17,17);
		nextP(0,18) = P(0,18) + P(1,18)*SF[9] + P(2,18)*SF[11] + P(3,18)*SF[10] + P(10,18)*SF[14] + P(11,18)*SF[15] + P(12,18)*SPP[10];
		nextP(1,18) = P(1,18) + P(0,18)*SF[8] + P(2,18)*SF[7] + P(3,18)*SF[11] - P(12,18)*SF[15] + P(11,18)*SPP[10] - (P(10,18)*q0)/2;
		nextP(2,18) = P(2,18) + P(0,18)*SF[6] + P(1,18)*SF[10] + P(3,18)*SF[8] + P(12,18)*SF[14] - P(10,18)*SPP[10] - (P(11,18)*q0)/2;
		nextP(3,18) = P(3,18) + P(0,18)*SF[7] + P(1,18)*SF[6] + P(2,18)*SF[9] + P(10,18)*SF[15] - P(11,18)*SF[14] - (P(12,18)*q0)/2;
		nextP(4,18) = P(4,18) + P(0,18)*SF[5] + P(1,18)*SF[3] - P(3,18)*SF[4] + P(2,18)*SPP[0] + P(13,18)*SPP[3] + P(14,18)*SPP[6] - P(15,18)*SPP[9];
		nextP(5,18) = P(5,18) + P(0,18)*SF[4] + P(2,18)*SF[3] + P(3,18)*SF[5] - P(1,18)*SPP[0] - P(13,18)*SPP[8] + P(14,18)*SPP[2] + P(15,18)*SPP[5];
		nextP(6,18) = P(6,18) + P(1,18)*SF[4] - P(2,18)*SF[5] + P(3,18)*SF[3] + P(0,18)*SPP[0] + P(13,18)*SPP[4] - P(14,18)*SPP[7] - P(15,18)*SPP[1];
		nextP(7,18) = P(7,18) + P(4,18)*dt;
		nextP(8,18) = P(8,18) + P(5,18)*dt;
		nextP(9,18) = P(9,18) + P(6,18)*dt;
		nextP(10,18) = P(10,18);
		nextP(11,18) = P(11,18);
		nextP(12,18) = P(12,18);
		nextP(13,18) = P(13,18);
		nextP(14,18) = P(14,18);
		nextP(15,18) = P(15,18);
		nextP(16,18) = P(16,18);
		nextP(17,18) = P(17,18);
		nextP(18,18) = P(18,18);
		nextP(0,19) = P(0,19) + P(1,19)*SF[9] + P(2,19)*SF[11] + P(3,19)*SF[10] + P(10,19)*SF[14] + P(11,19)*SF[15] + P(12,19)*SPP[10];
		nextP(1,19) = P(1,19) + P(0,19)*SF[8] + P(2,19)*SF[7] + P(3,19)*SF[11] - P(12,19)*SF[15] + P(11,19)*SPP[10] - (P(10,19)*q0)/2;
		nextP(2,19) = P(2,19) + P(0,19)*SF[6] + P(1,19)*SF[10] + P(3,19)*SF[8] + P(12,19)*SF[14] - P(10,19)*SPP[10] - (P(11,19)*q0)/2;
		nextP(3,19) = P(3,19) + P(0,19)*SF[7] + P(1,19)*SF[6] + P(2,19)*SF[9] + P(10,19)*SF[15] - P(11,19)*SF[14] - (P(12,19)*q0)/2;
		nextP(4,19) = P(4,19) + P(0,19)*SF[5] + P(1,19)*SF[3] - P(3,19)*SF[4] + P(2,19)*SPP[0] + P(13,19)*SPP[3] + P(14,19)*SPP[6] - P(15,19)*SPP[9];
		nextP(5,19) = P(5,19) + P(0,19)*SF[4] + P(2,19)*SF[3] + P(3,19)*SF[5] - P(1,19)*SPP[0] - P(13,19)*SPP[8] + P(14,19)*SPP[2] + P(15,19)*SPP[5];
		nextP(6,19) = P(6,19) + P(1,19)*SF[4] - P(2,19)*SF[5] + P(3,19)*SF[3] + P(0,19)*SPP[0] + P(13,19)*SPP[4] - P(14,19)*SPP[7] - P(15,19)*SPP[1];
		nextP(7,19) = P(7,19) + P(4,19)*dt;
		nextP(8,19) = P(8,19) + P(5,19)*dt;
		nextP(9,19) = P(9,19) + P(6,19)*dt;
		nextP(10,19) = P(10,19);
		nextP(11,19) = P(11,19);
		nextP(12,19) = P(12,19);
		nextP(13,19) = P(13,19);
		nextP(14,19) = P(14,19);
		nextP(15,19) = P(15,19);
		nextP(16,19) = P(16,19);
		nextP(17,19) = P(17,19);
		nextP(18,19) = P(18,19);
		nextP(19,19) = P(19,19);
		nextP(0,20) = P(0,20) + P(1,20)*SF[9] + P(2,20)*SF[11] + P(3,20)*SF[10] + P(10,20)*SF[14] + P(11,20)*SF[15] + P(12,20)*SPP[10];
		nextP(1,20) = P(1,20) + P(0,20)*SF[8] + P(2,20)*SF[7] + P(3,20)*SF[11] - P(12,20)*SF[15] + P(11,20)*SPP[10] - (P(10,20)*q0)/2;
		nextP(2,20) = P(2,20) + P(0,20)*SF[6] + P(1,20)*SF[10] + P(3,20)*SF[8] + P(12,20)*SF[14] - P(10,20)*SPP[10] - (P(11,20)*q0)/2;
		nextP(3,20) = P(3,20) + P(0,20)*SF[7] + P(1,20)*SF[6] + P(2,20)*SF[9] + P(10,20)*SF[15] - P(11,20)*SF[14] - (P(12,20)*q0)/2;
		nextP(4,20) = P(4,20) + P(0,20)*SF[5] + P(1,20)*SF[3] - P(3,20)*SF[4] + P(2,20)*SPP[0] + P(13,20)*SPP[3] + P(14,20)*SPP[6] - P(15,20)*SPP[9];
		nextP(5,20) = P(5,20) + P(0,20)*SF[4] + P(2,20)*SF[3] + P(3,20)*SF[5] - P(1,20)*SPP[0] - P(13,20)*SPP[8] + P(14,20)*SPP[2] + P(15,20)*SPP[5];
		nextP(6,20) = P(6,20) + P(1,20)*SF[4] - P(2,20)*SF[5] + P(3,20)*SF[3] + P(0,20)*SPP[0] + P(13,20)*SPP[4] - P(14,20)*SPP[7] - P(15,20)*SPP[1];
		nextP(7,20) = P(7,20) + P(4,20)*dt;
		nextP(8,20) = P(8,20) + P(5,20)*dt;
		nextP(9,20) = P(9,20) + P(6,20)*dt;
		nextP(10,20) = P(10,20);
		nextP(11,20) = P(11,20);
		nextP(12,20) = P(12,20);
		nextP(13,20) = P(13,20);
		nextP(14,20) = P(14,20);
		nextP(15,20) = P(15,20);
		nextP(16,20) = P(16,20);
		nextP(17,20) = P(17,20);
		nextP(18,20) = P(18,20);
		nextP(19,20) = P(19,20);
		nextP(20,20) = P(20,20);
		nextP(0,21) = P(0,21) + P(1,21)*SF[9] + P(2,21)*SF[11] + P(3,21)*SF[10] + P(10,21)*SF[14] + P(11,21)*SF[15] + P(12,21)*SPP[10];
		nextP(1,21) = P(1,21) + P(0,21)*SF[8] + P(2,21)*SF[7] + P(3,21)*SF[11] - P(12,21)*SF[15] + P(11,21)*SPP[10] - (P(10,21)*q0)/2;
		nextP(2,21) = P(2,21) + P(0,21)*SF[6] + P(1,21)*SF[10] + P(3,21)*SF[8] + P(12,21)*SF[14] - P(10,21)*SPP[10] - (P(11,21)*q0)/2;
		nextP(3,21) = P(3,21) + P(0,21)*SF[7] + P(1,21)*SF[6] + P(2,21)*SF[9] + P(10,21)*SF[15] - P(11,21)*SF[14] - (P(12,21)*q0)/2;
		nextP(4,21) = P(4,21) + P(0,21)*SF[5] + P(1,21)*SF[3] - P(3,21)*SF[4] + P(2,21)*SPP[0] + P(13,21)*SPP[3] + P(14,21)*SPP[6] - P(15,21)*SPP[9];
		nextP(5,21) = P(5,21) + P(0,21)*SF[4] + P(2,21)*SF[3] + P(3,21)*SF[5] - P(1,21)*SPP[0] - P(13,21)*SPP[8] + P(14,21)*SPP[2] + P(15,21)*SPP[5];
		nextP(6,21) = P(6,21) + P(1,21)*SF[4] - P(2,21)*SF[5] + P(3,21)*SF[3] + P(0,21)*SPP[0] + P(13,21)*SPP[4] - P(14,21)*SPP[7] - P(15,21)*SPP[1];
		nextP(7,21) = P(7,21) + P(4,21)*dt;
		nextP(8,21) = P(8,21) + P(5,21)*dt;
		nextP(9,21) = P(9,21) + P(6,21)*dt;
		nextP(10,21) = P(10,21);
		nextP(11,21) = P(11,21);
		nextP(12,21) = P(12,21);
		nextP(13,21) = P(13,21);
		nextP(14,21) = P(14,21);
		nextP(15,21) = P(15,21);
		nextP(16,21) = P(16,21);
		nextP(17,21) = P(17,21);
		nextP(18,21) = P(18,21);
		nextP(19,21) = P(19,21);
		nextP(20,21) = P(20,21);
		nextP(21,21) = P(21,21);

		// add process noise that is not from the IMU
		for (unsigned i = 16; i <= 21; i++) {
			nextP(i,i) += process_noise(i);
		}

	}

	// Don't do covariance prediction on wind states unless we are using them
	if (_control_status.flags.wind) {

		// calculate variances and upper diagonal covariances for wind states
		nextP(0,22) = P(0,22) + P(1,22)*SF[9] + P(2,22)*SF[11] + P(3,22)*SF[10] + P(10,22)*SF[14] + P(11,22)*SF[15] + P(12,22)*SPP[10];
		nextP(1,22) = P(1,22) + P(0,22)*SF[8] + P(2,22)*SF[7] + P(3,22)*SF[11] - P(12,22)*SF[15] + P(11,22)*SPP[10] - (P(10,22)*q0)/2;
		nextP(2,22) = P(2,22) + P(0,22)*SF[6] + P(1,22)*SF[10] + P(3,22)*SF[8] + P(12,22)*SF[14] - P(10,22)*SPP[10] - (P(11,22)*q0)/2;
		nextP(3,22) = P(3,22) + P(0,22)*SF[7] + P(1,22)*SF[6] + P(2,22)*SF[9] + P(10,22)*SF[15] - P(11,22)*SF[14] - (P(12,22)*q0)/2;
		nextP(4,22) = P(4,22) + P(0,22)*SF[5] + P(1,22)*SF[3] - P(3,22)*SF[4] + P(2,22)*SPP[0] + P(13,22)*SPP[3] + P(14,22)*SPP[6] - P(15,22)*SPP[9];
		nextP(5,22) = P(5,22) + P(0,22)*SF[4] + P(2,22)*SF[3] + P(3,22)*SF[5] - P(1,22)*SPP[0] - P(13,22)*SPP[8] + P(14,22)*SPP[2] + P(15,22)*SPP[5];
		nextP(6,22) = P(6,22) + P(1,22)*SF[4] - P(2,22)*SF[5] + P(3,22)*SF[3] + P(0,22)*SPP[0] + P(13,22)*SPP[4] - P(14,22)*SPP[7] - P(15,22)*SPP[1];
		nextP(7,22) = P(7,22) + P(4,22)*dt;
		nextP(8,22) = P(8,22) + P(5,22)*dt;
		nextP(9,22) = P(9,22) + P(6,22)*dt;
		nextP(10,22) = P(10,22);
		nextP(11,22) = P(11,22);
		nextP(12,22) = P(12,22);
		nextP(13,22) = P(13,22);
		nextP(14,22) = P(14,22);
		nextP(15,22) = P(15,22);
		nextP(16,22) = P(16,22);
		nextP(17,22) = P(17,22);
		nextP(18,22) = P(18,22);
		nextP(19,22) = P(19,22);
		nextP(20,22) = P(20,22);
		nextP(21,22) = P(21,22);
		nextP(22,22) = P(22,22);
		nextP(0,23) = P(0,23) + P(1,23)*SF[9] + P(2,23)*SF[11] + P(3,23)*SF[10] + P(10,23)*SF[14] + P(11,23)*SF[15] + P(12,23)*SPP[10];
		nextP(1,23) = P(1,23) + P(0,23)*SF[8] + P(2,23)*SF[7] + P(3,23)*SF[11] - P(12,23)*SF[15] + P(11,23)*SPP[10] - (P(10,23)*q0)/2;
		nextP(2,23) = P(2,23) + P(0,23)*SF[6] + P(1,23)*SF[10] + P(3,23)*SF[8] + P(12,23)*SF[14] - P(10,23)*SPP[10] - (P(11,23)*q0)/2;
		nextP(3,23) = P(3,23) + P(0,23)*SF[7] + P(1,23)*SF[6] + P(2,23)*SF[9] + P(10,23)*SF[15] - P(11,23)*SF[14] - (P(12,23)*q0)/2;
		nextP(4,23) = P(4,23) + P(0,23)*SF[5] + P(1,23)*SF[3] - P(3,23)*SF[4] + P(2,23)*SPP[0] + P(13,23)*SPP[3] + P(14,23)*SPP[6] - P(15,23)*SPP[9];
		nextP(5,23) = P(5,23) + P(0,23)*SF[4] + P(2,23)*SF[3] + P(3,23)*SF[5] - P(1,23)*SPP[0] - P(13,23)*SPP[8] + P(14,23)*SPP[2] + P(15,23)*SPP[5];
		nextP(6,23) = P(6,23) + P(1,23)*SF[4] - P(2,23)*SF[5] + P(3,23)*SF[3] + P(0,23)*SPP[0] + P(13,23)*SPP[4] - P(14,23)*SPP[7] - P(15,23)*SPP[1];
		nextP(7,23) = P(7,23) + P(4,23)*dt;
		nextP(8,23) = P(8,23) + P(5,23)*dt;
		nextP(9,23) = P(9,23) + P(6,23)*dt;
		nextP(10,23) = P(10,23);
		nextP(11,23) = P(11,23);
		nextP(12,23) = P(12,23);
		nextP(13,23) = P(13,23);
		nextP(14,23) = P(14,23);
		nextP(15,23) = P(15,23);
		nextP(16,23) = P(16,23);
		nextP(17,23) = P(17,23);
		nextP(18,23) = P(18,23);
		nextP(19,23) = P(19,23);
		nextP(20,23) = P(20,23);
		nextP(21,23) = P(21,23);
		nextP(22,23) = P(22,23);
		nextP(23,23) = P(23,23);

		// add process noise that is not from the IMU
		for (unsigned i = 22; i <= 23; i++) {
			nextP(i,i) += process_noise(i);
		}

	}

	// stop position covariance growth if our total position variance reaches 100m
	// this can happen if we lose gps for some time
	if ((P(7,7) + P(8,8)) > 1e4f) {
		for (uint8_t i = 7; i <= 8; i++) {
			for (uint8_t j = 0; j < _k_num_states; j++) {
				nextP(i,j) = P(i,j);
				nextP(j,i) = P(j,i);
			}
		}
	}

	// covariance matrix is symmetrical, so copy upper half to lower half
	for (unsigned row = 1; row < _k_num_states; row++) {
		for (unsigned column = 0 ; column < row; column++) {
			P(row,column) = P(column,row) = nextP(column,row);
		}
	}

	// copy variances (diagonals)
	for (unsigned i = 0; i < _k_num_states; i++) {
		P(i,i) = nextP(i,i);
	}

	// fix gross errors in the covariance matrix and ensure rows and
	// columns for un-used states are zero
	fixCovarianceErrors(false);

}

void Ekf::fixCovarianceErrors(bool force_symmetry)
{
	// NOTE: This limiting is a last resort and should not be relied on
	// TODO: Split covariance prediction into separate F*P*transpose(F) and Q contributions
	// and set corresponding entries in Q to zero when states exceed 50% of the limit
	// Covariance diagonal limits. Use same values for states which
	// belong to the same group (e.g. vel_x, vel_y, vel_z)
	float P_lim[8] = {};
	P_lim[0] = 1.0f;		// quaternion max var
	P_lim[1] = 1e6f;		// velocity max var
	P_lim[2] = 1e6f;		// positiion max var
	P_lim[3] = 1.0f;		// gyro bias max var
	P_lim[4] = 1.0f;		// delta velocity z bias max var
	P_lim[5] = 1.0f;		// earth mag field max var
	P_lim[6] = 1.0f;		// body mag field max var
	P_lim[7] = 1e6f;		// wind max var

	for (int i = 0; i <= 3; i++) {
		// quaternion states
		P(i,i) = math::constrain(P(i,i), 0.0f, P_lim[0]);
	}

	for (int i = 4; i <= 6; i++) {
		// NED velocity states
		P(i,i) = math::constrain(P(i,i), 1E-6f, P_lim[1]);
	}

	for (int i = 7; i <= 9; i++) {
		// NED position states
		P(i,i) = math::constrain(P(i,i), 1E-6f, P_lim[2]);
	}

	for (int i = 10; i <= 12; i++) {
		// gyro bias states
		P(i,i) = math::constrain(P(i,i), 0.0f, P_lim[3]);
	}

	// force symmetry on the quaternion, velocity and position state covariances
	if (force_symmetry) {
		P.makeRowColSymmetric<13>(0);
	}

	// the following states are optional and are deactivated when not required
	// by ensuring the corresponding covariance matrix values are kept at zero

	// accelerometer bias states
	if ((_params.fusion_mode & MASK_INHIBIT_ACC_BIAS) || _accel_bias_inhibit) {
		P.uncorrelateCovarianceSetVariance<3>(13, 0.0f);

	} else {
		// Find the maximum delta velocity bias state variance and request a covariance reset if any variance is below the safe minimum
		const float minSafeStateVar = 1e-9f;
		float maxStateVar = minSafeStateVar;
		bool resetRequired = false;

		for (uint8_t stateIndex = 13; stateIndex <= 15; stateIndex++) {
			if (P(stateIndex,stateIndex) > maxStateVar) {
				maxStateVar = P(stateIndex,stateIndex);

			} else if (P(stateIndex,stateIndex) < minSafeStateVar) {
				resetRequired = true;
			}
		}

		// To ensure stability of the covariance matrix operations, the ratio of a max and min variance must
		// not exceed 100 and the minimum variance must not fall below the target minimum
		// Also limit variance to a maximum equivalent to a 0.1g uncertainty
		const float minStateVarTarget = 5E-8f;
		float minAllowedStateVar = fmaxf(0.01f * maxStateVar, minStateVarTarget);

		for (uint8_t stateIndex = 13; stateIndex <= 15; stateIndex++) {
			P(stateIndex,stateIndex) = math::constrain(P(stateIndex,stateIndex), minAllowedStateVar, sq(0.1f * CONSTANTS_ONE_G * _dt_ekf_avg));
		}

		// If any one axis has fallen below the safe minimum, all delta velocity covariance terms must be reset to zero
		if (resetRequired) {
			P.uncorrelateCovariance<3>(13);
		}

		// Run additional checks to see if the delta velocity bias has hit limits in a direction that is clearly wrong
		// calculate accel bias term aligned with the gravity vector
		float dVel_bias_lim = 0.9f * _params.acc_bias_lim * _dt_ekf_avg;
		float down_dvel_bias = 0.0f;

		for (uint8_t axis_index = 0; axis_index < 3; axis_index++) {
			down_dvel_bias += _state.delta_vel_bias(axis_index) * _R_to_earth(2, axis_index);
		}

		// check that the vertical component of accel bias is consistent with both the vertical position and velocity innovation
		bool bad_acc_bias = (fabsf(down_dvel_bias) > dVel_bias_lim
				     && ( (down_dvel_bias * _gps_vel_innov(2) < 0.0f && _control_status.flags.gps)
				     ||   (down_dvel_bias * _ev_vel_innov(2) < 0.0f && _control_status.flags.ev_vel) )
				     && down_dvel_bias * _gps_pos_innov(2) < 0.0f);

		// record the pass/fail
		if (!bad_acc_bias) {
			_fault_status.flags.bad_acc_bias = false;
			_time_acc_bias_check = _time_last_imu;

		} else {
			_fault_status.flags.bad_acc_bias = true;
		}

		// if we have failed for 7 seconds continuously, reset the accel bias covariances to fix bad conditioning of
		// the covariance matrix but preserve the variances (diagonals) to allow bias learning to continue
		if (isTimedOut(_time_acc_bias_check, (uint64_t)7e6)) {

			P.uncorrelateCovariance<3>(13);

			_time_acc_bias_check = _time_last_imu;
			_fault_status.flags.bad_acc_bias = false;
			ECL_WARN_TIMESTAMPED("invalid accel bias - resetting covariance");

		} else if (force_symmetry) {
			// ensure the covariance values are symmetrical
			P.makeRowColSymmetric<3>(13);
		}

	}

	// magnetic field states
	if (!_control_status.flags.mag_3D) {
		zeroMagCov();

	} else {
		// constrain variances
		for (int i = 16; i <= 18; i++) {
			P(i,i) = math::constrain(P(i,i), 0.0f, P_lim[5]);
		}

		for (int i = 19; i <= 21; i++) {
			P(i,i) = math::constrain(P(i,i), 0.0f, P_lim[6]);
		}

		// force symmetry
		if (force_symmetry) {
			P.makeRowColSymmetric<3>(16);
			P.makeRowColSymmetric<3>(19);
		}

	}

	// wind velocity states
	if (!_control_status.flags.wind) {
		P.uncorrelateCovarianceSetVariance<2>(22, 0.0f);

	} else {
		// constrain variances
		for (int i = 22; i <= 23; i++) {
			P(i,i) = math::constrain(P(i,i), 0.0f, P_lim[7]);
		}

		// force symmetry
		if (force_symmetry) {
			P.makeRowColSymmetric<2>(22);
		}
	}
}

void Ekf::resetMagRelatedCovariances()
{
	resetQuatCov();
	resetMagCov();
}

void Ekf::resetQuatCov(){
	zeroQuatCov();

	// define the initial angle uncertainty as variances for a rotation vector
	Vector3f rot_vec_var;
	rot_vec_var.setAll(sq(_params.initial_tilt_err));

	initialiseQuatCovariances(rot_vec_var);
}

void Ekf::zeroQuatCov()
{
	P.uncorrelateCovarianceSetVariance<2>(0, 0.0f);
	P.uncorrelateCovarianceSetVariance<2>(2, 0.0f);
}

void Ekf::resetMagCov()
{
	// reset the corresponding rows and columns in the covariance matrix and
	// set the variances on the magnetic field states to the measurement variance
	clearMagCov();

	P.uncorrelateCovarianceSetVariance<3>(16, sq(_params.mag_noise));
	P.uncorrelateCovarianceSetVariance<3>(19, sq(_params.mag_noise));

	if (!_control_status.flags.mag_3D) {
		// save covariance data for re-use when auto-switching between heading and 3-axis fusion
		// if already in 3-axis fusion mode, the covariances are automatically saved when switching out
		// of this mode
		saveMagCovData();
	}
}

void Ekf::clearMagCov()
{
	zeroMagCov();
	_mag_decl_cov_reset = false;
}

void Ekf::zeroMagCov()
{
	P.uncorrelateCovarianceSetVariance<3>(16, 0.0f);
	P.uncorrelateCovarianceSetVariance<3>(19, 0.0f);
}

void Ekf::resetWindCovariance()
{
	if (_tas_data_ready && (_imu_sample_delayed.time_us - _airspeed_sample_delayed.time_us < (uint64_t)5e5)) {
		// Derived using EKF/matlab/scripts/Inertial Nav EKF/wind_cov.py
		// TODO: explicitly include the sideslip angle in the derivation
		Eulerf euler321(_state.quat_nominal);
		const float euler_yaw = euler321(2);
		const float R_TAS = sq(math::constrain(_params.eas_noise, 0.5f, 5.0f) * math::constrain(_airspeed_sample_delayed.eas2tas, 0.9f, 10.0f));
		const float initial_sideslip_uncertainty = math::radians(15.0f);
		const float initial_wind_var_body_y = sq(_airspeed_sample_delayed.true_airspeed * sinf(initial_sideslip_uncertainty));
		const float R_yaw = sq(math::radians(10.0f));

		// rotate wind velocity into earth frame aligned with vehicle yaw
		const float Wx = _state.wind_vel(0) * cosf(euler_yaw) + _state.wind_vel(1) * sinf(euler_yaw);
		const float Wy = -_state.wind_vel(0) * sinf(euler_yaw) + _state.wind_vel(1) * cosf(euler_yaw);

		// it is safer to remove all existing correlations to other states at this time
		P.uncorrelateCovarianceSetVariance<2>(22, 0.0f);

		P(22,22) = R_TAS*sq(cosf(euler_yaw)) + R_yaw*sq(-Wx*sinf(euler_yaw) - Wy*cosf(euler_yaw)) + initial_wind_var_body_y*sq(sinf(euler_yaw));
		P(22,23) = R_TAS*sinf(euler_yaw)*cosf(euler_yaw) + R_yaw*(-Wx*sinf(euler_yaw) - Wy*cosf(euler_yaw))*(Wx*cosf(euler_yaw) - Wy*sinf(euler_yaw)) - initial_wind_var_body_y*sinf(euler_yaw)*cosf(euler_yaw);
		P(23,22) = P(22,23);
		P(23,23) = R_TAS*sq(sinf(euler_yaw)) + R_yaw*sq(Wx*cosf(euler_yaw) - Wy*sinf(euler_yaw)) + initial_wind_var_body_y*sq(cosf(euler_yaw));

		// Now add the variance due to uncertainty in vehicle velocity that was used to calculate the initial wind speed
		P(22,22) += P(4,4);
		P(23,23) += P(5,5);

	} else {
		// without airspeed, start with a small initial uncertainty to improve the initial estimate
		P.uncorrelateCovarianceSetVariance<2>(22, 1.0f);

	}
}