path: root/apps/examples/kalman_demo/KalmanNav.cpp

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/**
 * @file KalmanNav.cpp
 *
 * kalman filter navigation code
 */

#include <poll.h>

#include "KalmanNav.hpp"

// constants
// Titterton pg. 52
static const float omega = 7.2921150e-5f; // earth rotation rate, rad/s
static const float R0 = 6378137.0f; // earth radius, m
static const float g = 9.806f; // gravitational accel. m/s^2, XXX should be calibrated

KalmanNav::KalmanNav(SuperBlock *parent, const char *name) :
	SuperBlock(parent, name),
	// ekf matrices
	F(9, 9),
	G(9, 6),
	P(9, 9),
	P0(9, 9),
	V(6, 6),
	// attitude measurement ekf matrices
	HAtt(6, 9),
	RAtt(6, 6),
	// position measurement ekf matrices
	HPos(5, 9),
	RPos(5, 5),
	// attitude representations
	C_nb(),
	q(),
	// subscriptions
	_sensors(&getSubscriptions(), ORB_ID(sensor_combined), 5), // limit to 200 Hz
	_gps(&getSubscriptions(), ORB_ID(vehicle_gps_position), 1000), // limit to 1 Hz
	_param_update(&getSubscriptions(), ORB_ID(parameter_update), 1000), // limit to 1 Hz
	// publications
	_pos(&getPublications(), ORB_ID(vehicle_global_position)),
	_att(&getPublications(), ORB_ID(vehicle_attitude)),
	// timestamps
	_pubTimeStamp(hrt_absolute_time()),
	_fastTimeStamp(hrt_absolute_time()),
	_slowTimeStamp(hrt_absolute_time()),
	_attTimeStamp(hrt_absolute_time()),
	_outTimeStamp(hrt_absolute_time()),
	// frame count
	_navFrames(0),
	// miss counts
	_missFast(0),
	_missSlow(0),
	// accelerations
	fN(0), fE(0), fD(0),
	// state
	phi(0), theta(0), psi(0),
	vN(0), vE(0), vD(0),
	lat(0), lon(0), alt(0),
	// parameters for ground station
	_vGyro(this, "V_GYRO"),
	_vAccel(this, "V_ACCEL"),
	_rMag(this, "R_MAG"),
	_rGpsVel(this, "R_GPS_VEL"),
	_rGpsPos(this, "R_GPS_POS"),
	_rGpsAlt(this, "R_GPS_ALT"),
	_rAccel(this, "R_ACCEL"),
	_magDip(this, "MAG_DIP"),
	_magDec(this, "MAG_DEC"),
	_faultPos(this, "FAULT_POS"),
	_faultAtt(this, "FAULT_ATT"),
	_positionInitialized(false)
{
	using namespace math;

	// initial state covariance matrix
	P0 = Matrix::identity(9) * 0.01f;
	P = P0;

	// initial state
	phi = 0.0f;
	theta = 0.0f;
	psi = 0.0f;
	vN = 0.0f;
	vE = 0.0f;
	vD = 0.0f;
	lat = 0.0f;
	lon = 0.0f;
	alt = 0.0f;



	// initialize quaternions
	q = Quaternion(EulerAngles(phi, theta, psi));

	// initialize dcm
	C_nb = Dcm(q);

	// HPos is constant
	HPos(0, 3) = 1.0f;
	HPos(1, 4) = 1.0f;
	HPos(2, 6) = 1.0e7f * M_RAD_TO_DEG_F;
	HPos(3, 7) = 1.0e7f * M_RAD_TO_DEG_F;
	HPos(4, 8) = 1.0f;

	// initialize all parameters
	updateParams();
}

void KalmanNav::update()
{
	using namespace math;

	struct pollfd fds[3];
	fds[0].fd = _sensors.getHandle();
	fds[0].events = POLLIN;
	fds[1].fd = _param_update.getHandle();
	fds[1].events = POLLIN;
	fds[2].fd = _gps.getHandle();
	fds[2].events = POLLIN;

	// poll 20 milliseconds for new data
	int ret = poll(fds, 2, 20);

	// check return value
	if (ret < 0) {
		// XXX this is seriously bad - should be an emergency
		return;

	} else if (ret == 0) { // timeout
		return;
	}

	// get new timestamp
	uint64_t newTimeStamp = hrt_absolute_time();

	// check updated subscriptions
	if (_param_update.updated()) updateParams();

	bool gpsUpdate = _gps.updated();
	bool sensorsUpdate = _sensors.updated();

	// get new information from subscriptions
	// this clears update flag
	updateSubscriptions();

	// abort update if no new data
	if (!(sensorsUpdate || gpsUpdate)) return;

	// if received gps for first time, reset position to gps
	if (!_positionInitialized &&
	    gpsUpdate &&
	    _gps.fix_type > 2 &&
	    _gps.counter_pos_valid > 10) {
		vN = _gps.vel_n;
		vE = _gps.vel_e;
		vD = _gps.vel_d;
		setLatDegE7(_gps.lat);
		setLonDegE7(_gps.lon);
		setAltE3(_gps.alt);
		_positionInitialized = true;
		printf("[kalman_demo] initializing EKF state with GPS\n");
		printf("phi: %8.4f, theta: %8.4f, psi: %8.4f\n",
		       double(phi), double(theta), double(psi));
		printf("vN: %8.4f, vE: %8.4f, vD: %8.4f, lat: %8.4f, lon: %8.4f, alt: %8.4f\n",
		       double(vN), double(vE), double(vD),
		       lat, lon, alt);
	}

	// fast prediciton step
	// note, using sensors timestamp so we can account
	// for packet lag
	float dtFast = (_sensors.timestamp - _fastTimeStamp) / 1.0e6f;
	_fastTimeStamp = _sensors.timestamp;

	if (dtFast < 1.0f / 50) {
		predictFast(dtFast);
		// count fast frames
		_navFrames += 1;
	} else _missFast++;

	// slow prediction step
	float dtSlow = (_sensors.timestamp - _slowTimeStamp) / 1.0e6f;

	if (dtSlow > 1.0f / 100) { // 100 Hz
		_slowTimeStamp = _sensors.timestamp;
		if (dtSlow < 1.0f / 50) predictSlow(dtSlow);
		else _missSlow ++;
	}

	// gps correction step
	if (gpsUpdate) {
		correctPos();
	}

	// attitude correction step
	if (_sensors.timestamp - _attTimeStamp > 1e6 / 1) { // 1 Hz
		_attTimeStamp = _sensors.timestamp;
		correctAtt();
	}

	// publication
	if (newTimeStamp - _pubTimeStamp > 1e6 / 50) { // 50 Hz
		_pubTimeStamp = newTimeStamp;
		updatePublications();
	}

	// output
	if (newTimeStamp - _outTimeStamp > 10e6) { // 0.1 Hz
		_outTimeStamp = newTimeStamp;
		printf("nav: %4d Hz, misses fast: %4d slow: %4d\n",
				_navFrames/10, _missFast/10, _missSlow/10);
		_navFrames = 0;
		_missFast = 0;
		_missSlow = 0;
	}
}

void KalmanNav::updatePublications()
{
	using namespace math;

	// position publication
	_pos.timestamp = _pubTimeStamp;
	_pos.time_gps_usec = _gps.timestamp;
	_pos.valid = true;
	_pos.lat = getLatDegE7();
	_pos.lon = getLonDegE7();
	_pos.alt = float(alt);
	_pos.relative_alt = float(alt); // TODO, make relative
	_pos.vx = vN;
	_pos.vy = vE;
	_pos.vz = vD;
	_pos.hdg = psi;

	// attitude publication
	_att.timestamp = _pubTimeStamp;
	_att.roll = phi;
	_att.pitch = theta;
	_att.yaw = psi;
	_att.rollspeed = _sensors.gyro_rad_s[0];
	_att.pitchspeed = _sensors.gyro_rad_s[1];
	_att.yawspeed = _sensors.gyro_rad_s[2];
	// TODO, add gyro offsets to filter
	_att.rate_offsets[0] = 0.0f;
	_att.rate_offsets[1] = 0.0f;
	_att.rate_offsets[2] = 0.0f;

	for (int i = 0; i < 3; i++) for (int j = 0; j < 3; j++)
			_att.R[i][j] = C_nb(i, j);

	for (int i = 0; i < 4; i++) _att.q[i] = q(i);

	_att.R_valid = true;
	_att.q_valid = true;
	_att.counter = _navFrames;

	// update publications
	SuperBlock::updatePublications();
}

void KalmanNav::predictFast(float dt)
{
	using namespace math;
	Vector3 w(_sensors.gyro_rad_s);

	// attitude
	q = q + q.derivative(w) * dt;

	// renormalize quaternion if needed
	if (fabsf(q.norm() - 1.0f) > 1e-4f) {
		q = q.unit();
	}

	// C_nb update
	C_nb = Dcm(q);

	// euler update
	EulerAngles euler(C_nb);
	phi = euler.getPhi();
	theta = euler.getTheta();
	psi = euler.getPsi();

	// specific acceleration in nav frame
	Vector3 accelB(_sensors.accelerometer_m_s2);
	Vector3 accelN = C_nb * accelB;
	fN = accelN(0);
	fE = accelN(1);
	fD = accelN(2);

	// trig
	float sinL = sinf(lat);
	float cosL = cosf(lat);
	float cosLSing = cosf(lat);

	// prevent singularity
	if (fabsf(cosLSing) < 0.01f) {
		if (cosLSing > 0) cosLSing = 0.01;
		else cosLSing = -0.01;
	}

	// position update
	// neglects angular deflections in local gravity
	// see Titerton pg. 70
	float R = R0 + float(alt);
	float LDot = vN / R;
	float lDot = vE / (cosLSing * R);
	float rotRate = 2 * omega + lDot;
	float vNDot = fN - vE * rotRate * sinL +
		      vD * LDot;
	float vDDot = fD - vE * rotRate * cosL -
		      vN * LDot + g;
	float vEDot = fE + vN * rotRate * sinL +
		      vDDot * rotRate * cosL;
	//printf("vNDot: %8.4f, vEDot: %8.4f, vDDot: %8.4f\n",
	//double(vNDot), double(vEDot), double(vDDot));
	//printf("LDot: %8.4f, lDot: %8.4f, rotRate: %8.4f\n",
	//double(LDot), double(lDot), double(rotRate));

	// rectangular integration
	//printf("dt: %8.4f\n", double(dt));
	//printf("fN: %8.4f, fE: %8.4f, fD: %8.4f\n", double(fN), double(fE), double(fD));
	//printf("vN: %8.4f, vE: %8.4f, vD: %8.4f\n", double(vN), double(vE), double(vD));
	vN += vNDot * dt;
	vE += vEDot * dt;
	vD += vDDot * dt;
	lat += double(LDot * dt);
	lon += double(lDot * dt);
	alt += double(-vD * dt);
}

void KalmanNav::predictSlow(float dt)
{
	using namespace math;

	// trig
	float sinL = sinf(lat);
	float cosL = cosf(lat);
	float cosLSq = cosL * cosL;
	float tanL = tanf(lat);

	// prepare for matrix
	float R = R0 + float(alt);
	float RSq = R * R;

	// F Matrix
	// Titterton pg. 291

	F(0, 1) = -(omega * sinL + vE * tanL / R);
	F(0, 2) = vN / R;
	F(0, 4) = 1.0f / R;
	F(0, 6) = -omega * sinL;
	F(0, 8) = -vE / RSq;

	F(1, 0) = omega * sinL + vE * tanL / R;
	F(1, 2) = omega * cosL + vE / R;
	F(1, 3) = -1.0f / R;
	F(1, 8) = vN / RSq;

	F(2, 0) = -vN / R;
	F(2, 1) = -omega * cosL - vE / R;
	F(2, 4) = -tanL / R;
	F(2, 6) = -omega * cosL - vE / (R * cosLSq);
	F(2, 8) = vE * tanL / RSq;

	F(3, 1) = -fD;
	F(3, 2) = fE;
	F(3, 3) = vD / R;
	F(3, 4) = -2 * (omega * sinL + vE * tanL / R);
	F(3, 5) = vN / R;
	F(3, 6) = -vE * (2 * omega * cosL + vE / (R * cosLSq));
	F(3, 8) = (vE * vE * tanL - vN * vD) / RSq;

	F(4, 0) = fD;
	F(4, 2) = -fN;
	F(4, 3) = 2 * omega * sinL + vE * tanL / R;
	F(4, 4) = (vN * tanL + vD) / R;
	F(4, 5) = 2 * omega * cosL + vE / R;
	F(4, 6) = 2 * omega * (vN * cosL - vD * sinL) +
		  vN * vE / (R * cosLSq);
	F(4, 8) = -vE * (vN * tanL + vD) / RSq;

	F(5, 0) = -fE;
	F(5, 1) = fN;
	F(5, 3) = -2 * vN / R;
	F(5, 4) = -2 * (omega * cosL + vE / R);
	F(5, 6) = 2 * omega * vE * sinL;
	F(5, 8) = (vN * vN + vE * vE) / RSq;

	F(6, 3) = 1 / R;
	F(6, 8) = -vN / RSq;

	F(7, 4) = 1 / (R * cosL);
	F(7, 6) = vE * tanL / (R * cosL);
	F(7, 8) = -vE / (cosL * RSq);

	F(8, 5) = -1;

	// G Matrix
	// Titterton pg. 291
	G(0, 0) = -C_nb(0, 0);
	G(0, 1) = -C_nb(0, 1);
	G(0, 2) = -C_nb(0, 2);
	G(1, 0) = -C_nb(1, 0);
	G(1, 1) = -C_nb(1, 1);
	G(1, 2) = -C_nb(1, 2);
	G(2, 0) = -C_nb(2, 0);
	G(2, 1) = -C_nb(2, 1);
	G(2, 2) = -C_nb(2, 2);

	G(3, 3) = C_nb(0, 0);
	G(3, 4) = C_nb(0, 1);
	G(3, 5) = C_nb(0, 2);
	G(4, 3) = C_nb(1, 0);
	G(4, 4) = C_nb(1, 1);
	G(4, 5) = C_nb(1, 2);
	G(5, 3) = C_nb(2, 0);
	G(5, 4) = C_nb(2, 1);
	G(5, 5) = C_nb(2, 2);

	// continuous predictioon equations
	// for discrte time EKF
	// http://en.wikipedia.org/wiki/Extended_Kalman_filter
	P = P + (F * P + P * F.transpose() + G * V * G.transpose()) * dt;
}

void KalmanNav::correctAtt()
{
	using namespace math;

	// check for valid data, return if not ready
	if (_sensors.accelerometer_counter < 10 ||
	    _sensors.gyro_counter < 10 ||
	    _sensors.magnetometer_counter < 10)
		return;

	// trig
	float cosPhi = cosf(phi);
	float cosTheta = cosf(theta);
	float cosPsi = cosf(psi);
	float sinPhi = sinf(phi);
	float sinTheta = sinf(theta);
	float sinPsi = sinf(psi);

	// mag measurement
	Vector3 zMag(_sensors.magnetometer_ga);

	// mag predicted measurement
	// choosing some typical magnetic field properties,
	//  TODO dip/dec depend on lat/ lon/ time
	float dip = _magDip.get() / M_RAD_TO_DEG_F; // dip, inclination with level
	float dec = _magDec.get() / M_RAD_TO_DEG_F; // declination, clockwise rotation from north
	float bN = cosf(dip) * cosf(dec);
	float bE = cosf(dip) * sinf(dec);
	float bD = sinf(dip);
	Vector3 bNav(bN, bE, bD);
	Vector3 zMagHat = (C_nb.transpose() * bNav).unit();

	// accel measurement
	Vector3 zAccel(_sensors.accelerometer_m_s2);
	float accelMag = zAccel.norm();
	zAccel = zAccel.unit();

	// ignore accel correction when accel mag not close to g
	Matrix RAttAdjust = RAtt;

	bool ignoreAccel = fabsf(accelMag - g) > 1.1f;

	if (ignoreAccel) {
		RAttAdjust(3, 3) = 1.0e10;
		RAttAdjust(4, 4) = 1.0e10;
		RAttAdjust(5, 5) = 1.0e10;

	} else {
		//printf("correcting attitude with accel\n");
	}

	// account for banked turn
	// this would only work for fixed wing, so try to avoid
	//Vector3 zCentrip = Vector3(0, cosf(phi), -sinf(phi))*g*tanf(phi);

	// accel predicted measurement
	Vector3 zAccelHat = (C_nb.transpose() * Vector3(0, 0, -g) /*+ zCentrip*/).unit();

	// combined measurement
	Vector zAtt(6);
	Vector zAttHat(6);

	for (int i = 0; i < 3; i++) {
		zAtt(i) = zMag(i);
		zAtt(i + 3) = zAccel(i);
		zAttHat(i) = zMagHat(i);
		zAttHat(i + 3) = zAccelHat(i);
	}

	// HMag , HAtt (0-2,:)
	float tmp1 =
		cosPsi * cosTheta * bN +
		sinPsi * cosTheta * bE -
		sinTheta * bD;
	HAtt(0, 1) = -(
			     cosPsi * sinTheta * bN +
			     sinPsi * sinTheta * bE +
			     cosTheta * bD
		     );
	HAtt(0, 2) = -cosTheta * (sinPsi * bN - cosPsi * bE);
	HAtt(1, 0) =
		(cosPhi * cosPsi * sinTheta + sinPhi * sinPsi) * bN +
		(cosPhi * sinPsi * sinTheta - sinPhi * cosPsi) * bE +
		cosPhi * cosTheta * bD;
	HAtt(1, 1) = sinPhi * tmp1;
	HAtt(1, 2) = -(
			     (sinPhi * sinPsi * sinTheta + cosPhi * cosPsi) * bN -
			     (sinPhi * cosPsi * sinTheta - cosPhi * sinPsi) * bE
		     );
	HAtt(2, 0) = -(
			     (sinPhi * cosPsi * sinTheta - cosPhi * sinPsi) * bN +
			     (sinPhi * sinPsi * sinTheta + cosPhi * cosPsi) * bE +
			     (sinPhi * cosTheta) * bD
		     );
	HAtt(2, 1) = cosPhi * tmp1;
	HAtt(2, 2) = -(
			     (cosPhi * sinPsi * sinTheta - sinPhi * cosTheta) * bN -
			     (cosPhi * cosPsi * sinTheta + sinPhi * sinPsi) * bE
		     );

	// HAccel , HAtt (3-5,:)
	HAtt(3, 1) = cosTheta;
	HAtt(4, 0) = -cosPhi * cosTheta;
	HAtt(4, 1) = sinPhi * sinTheta;
	HAtt(5, 0) = sinPhi * cosTheta;
	HAtt(5, 1) = cosPhi * sinTheta;

	// compute correction
	// http://en.wikipedia.org/wiki/Extended_Kalman_filter
	Vector y = zAtt - zAttHat; // residual
	Matrix S = HAtt * P * HAtt.transpose() + RAttAdjust; // residual covariance
	Matrix K = P * HAtt.transpose() * S.inverse();
	Vector xCorrect = K * y;

	// check correciton is sane
	for (size_t i = 0; i < xCorrect.getRows(); i++) {
		float val = xCorrect(i);
		if (isnan(val) || isinf(val)) {
			// abort correction and return
			printf("[kalman_demo] numerical failure in att correction\n");
			// reset P matrix to P0
			P = P0;
			return;
		}
	}

	// correct state
	if (!ignoreAccel) {
		phi += xCorrect(PHI);
		theta += xCorrect(THETA);
	}
	psi += xCorrect(PSI);
	// attitude also affects nav velocities
	vN += xCorrect(VN);
	vE += xCorrect(VE);
	vD += xCorrect(VD);

	// update state covariance
	// http://en.wikipedia.org/wiki/Extended_Kalman_filter
	P = P - K * HAtt * P;

	// fault detection
	float beta = y.dot(S.inverse() * y);
	if (beta > _faultAtt.get()) {
		printf("fault in attitude: beta = %8.4f\n", (double)beta);
		printf("y:\n"); y.print();
	}

	// update quaternions from euler
	// angle correction
	q = Quaternion(EulerAngles(phi, theta, psi));
}

void KalmanNav::correctPos()
{
	using namespace math;

	if (!_positionInitialized) return;

	// residual
	Vector y(5);
	y(0) = _gps.vel_n - vN;
	y(1) = _gps.vel_e - vE;
	y(2) = double(_gps.lat) - lat * 1.0e7 * M_RAD_TO_DEG;
	y(3) = double(_gps.lon) - lon * 1.0e7 * M_RAD_TO_DEG;
	y(4) = double(_gps.alt) / 1.0e3 - alt;

	// compute correction
	// http://en.wikipedia.org/wiki/Extended_Kalman_filter
	Matrix S = HPos * P * HPos.transpose() + RPos; // residual covariance
	Matrix K = P * HPos.transpose() * S.inverse();
	Vector xCorrect = K * y;

	// check correction is sane
	for (size_t i = 0; i < xCorrect.getRows(); i++) {
		float val = xCorrect(i);
		if (isnan(val) || isinf(val)) {
			// abort correction and return
			printf("[kalman_demo] numerical failure in gps correction\n");
			// fallback to GPS
			vN = _gps.vel_n;
			vE = _gps.vel_e;
			vD = _gps.vel_d;
			setLatDegE7(_gps.lat);
			setLonDegE7(_gps.lon);
			setAltE3(_gps.alt);
			// reset P matrix to P0
			P = P0;
			return;
		}
	}

	// correct state
	vN += xCorrect(VN);
	vE += xCorrect(VE);
	vD += xCorrect(VD);
	lat += double(xCorrect(LAT));
	lon += double(xCorrect(LON));
	alt += double(xCorrect(ALT));

	// update state covariance
	// http://en.wikipedia.org/wiki/Extended_Kalman_filter
	P = P - K * HPos * P;

	// fault detetcion
	float beta = y.dot(S.inverse() * y);
	if (beta > _faultPos.get()) {
		printf("fault in gps: beta = %8.4f\n", (double)beta);
		printf("Y/N: vN: %8.4f, vE: %8.4f, lat: %8.4f, lon: %8.4f, alt: %8.4f\n",
		       double(y(0) / sqrtf(RPos(0,0))),
		       double(y(1) / sqrtf(RPos(1,1))),
		       double(y(2) / sqrtf(RPos(2,2))),
		       double(y(3) / sqrtf(RPos(3,3))),
		       double(y(4) / sqrtf(RPos(3,3))));
	}
}

void KalmanNav::updateParams()
{
	using namespace math;
	using namespace control;
	SuperBlock::updateParams();

	// gyro noise
	V(0, 0) = _vGyro.get();   // gyro x, rad/s
	V(1, 1) = _vGyro.get();   // gyro y
	V(2, 2) = _vGyro.get();   // gyro z

	// accel noise
	V(3, 3) = _vAccel.get();   // accel x, m/s^2
	V(4, 4) = _vAccel.get();   // accel y
	V(5, 5) = _vAccel.get();   // accel z

	// magnetometer noise
	float noiseMin = 1e-6f;
	float noiseMagSq = _rMag.get() * _rMag.get();
	if (noiseMagSq < noiseMin) noiseMagSq = noiseMin;
	RAtt(0, 0) = noiseMagSq; // normalized direction
	RAtt(1, 1) = noiseMagSq;
	RAtt(2, 2) = noiseMagSq;

	// accelerometer noise
	float noiseAccelSq = _rAccel.get() * _rAccel.get();
	// bound noise to prevent singularities
	if (noiseAccelSq < noiseMin) noiseAccelSq = noiseMin;
	RAtt(3, 3) = noiseAccelSq; // normalized direction
	RAtt(4, 4) = noiseAccelSq;
	RAtt(5, 5) = noiseAccelSq;

	// gps noise
	float cosLSing = cosf(lat);
	float R = R0 + float(alt);

	// prevent singularity
	if (fabsf(cosLSing) < 0.01f) {
		if (cosLSing > 0) cosLSing = 0.01;
		else cosLSing = -0.01;
	}

	float noiseVel = _rGpsVel.get();
	float noiseLatDegE7 = 1.0e7f * M_RAD_TO_DEG_F * _rGpsPos.get() / R;
	float noiseLonDegE7 = noiseLatDegE7 / cosLSing;
	float noiseAlt = _rGpsAlt.get();
	// bound noise to prevent singularities
	if (noiseVel < noiseMin) noiseVel = noiseMin;
	if (noiseLatDegE7 < noiseMin) noiseLatDegE7 = noiseMin;
	if (noiseLonDegE7 < noiseMin) noiseLonDegE7 = noiseMin;
	if (noiseAlt < noiseMin) noiseAlt = noiseMin;
	RPos(0, 0) = noiseVel * noiseVel; // vn
	RPos(1, 1) = noiseVel * noiseVel; // ve
	RPos(2, 2) = noiseLatDegE7 * noiseLatDegE7; // lat
	RPos(3, 3) = noiseLonDegE7 * noiseLonDegE7; // lon
	RPos(4, 4) = noiseAlt * noiseAlt; // h
	// XXX, note that RPos depends on lat, so updateParams should
	// be called if lat changes significantly
}