/*
* Copyright (C) 2011 The Android Open Source Project
*
* Licensed under the Apache License, Version 2.0 (the "License");
* you may not use this file except in compliance with the License.
* You may obtain a copy of the License at
*
* http://www.apache.org/licenses/LICENSE-2.0
*
* Unless required by applicable law or agreed to in writing, software
* distributed under the License is distributed on an "AS IS" BASIS,
* WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
* See the License for the specific language governing permissions and
* limitations under the License.
*/
#include <stdio.h>
#include <utils/Log.h>
#include "Fusion.h"
namespace android {
// -----------------------------------------------------------------------
/*
* gyroVAR gives the measured variance of the gyro's output per
* Hz (or variance at 1 Hz). This is an "intrinsic" parameter of the gyro,
* which is independent of the sampling frequency.
*
* The variance of gyro's output at a given sampling period can be
* calculated as:
* variance(T) = gyroVAR / T
*
* The variance of the INTEGRATED OUTPUT at a given sampling period can be
* calculated as:
* variance_integrate_output(T) = gyroVAR * T
*
*/
static const float gyroVAR = 1e-7; // (rad/s)^2 / Hz
static const float biasVAR = 1e-8; // (rad/s)^2 / s (guessed)
/*
* Standard deviations of accelerometer and magnetometer
*/
static const float accSTDEV = 0.05f; // m/s^2 (measured 0.08 / CDD 0.05)
static const float magSTDEV = 0.5f; // uT (measured 0.7 / CDD 0.5)
static const float SYMMETRY_TOLERANCE = 1e-10f;
/*
* Accelerometer updates will not be performed near free fall to avoid
* ill-conditioning and div by zeros.
* Threshhold: 10% of g, in m/s^2
*/
static const float FREE_FALL_THRESHOLD = 0.981f;
static const float FREE_FALL_THRESHOLD_SQ =
FREE_FALL_THRESHOLD*FREE_FALL_THRESHOLD;
/*
* The geomagnetic-field should be between 30uT and 60uT.
* Fields strengths greater than this likely indicate a local magnetic
* disturbance which we do not want to update into the fused frame.
*/
static const float MAX_VALID_MAGNETIC_FIELD = 100; // uT
static const float MAX_VALID_MAGNETIC_FIELD_SQ =
MAX_VALID_MAGNETIC_FIELD*MAX_VALID_MAGNETIC_FIELD;
/*
* Values of the field smaller than this should be ignored in fusion to avoid
* ill-conditioning. This state can happen with anomalous local magnetic
* disturbances canceling the Earth field.
*/
static const float MIN_VALID_MAGNETIC_FIELD = 10; // uT
static const float MIN_VALID_MAGNETIC_FIELD_SQ =
MIN_VALID_MAGNETIC_FIELD*MIN_VALID_MAGNETIC_FIELD;
/*
* If the cross product of two vectors has magnitude squared less than this,
* we reject it as invalid due to alignment of the vectors.
* This threshold is used to check for the case where the magnetic field sample
* is parallel to the gravity field, which can happen in certain places due
* to magnetic field disturbances.
*/
static const float MIN_VALID_CROSS_PRODUCT_MAG = 1.0e-3;
static const float MIN_VALID_CROSS_PRODUCT_MAG_SQ =
MIN_VALID_CROSS_PRODUCT_MAG*MIN_VALID_CROSS_PRODUCT_MAG;
// -----------------------------------------------------------------------
template <typename TYPE, size_t C, size_t R>
static mat<TYPE, R, R> scaleCovariance(
const mat<TYPE, C, R>& A,
const mat<TYPE, C, C>& P) {
// A*P*transpose(A);
mat<TYPE, R, R> APAt;
for (size_t r=0 ; r<R ; r++) {
for (size_t j=r ; j<R ; j++) {
double apat(0);
for (size_t c=0 ; c<C ; c++) {
double v(A[c][r]*P[c][c]*0.5);
for (size_t k=c+1 ; k<C ; k++)
v += A[k][r] * P[c][k];
apat += 2 * v * A[c][j];
}
APAt[j][r] = apat;
APAt[r][j] = apat;
}
}
return APAt;
}
template <typename TYPE, typename OTHER_TYPE>
static mat<TYPE, 3, 3> crossMatrix(const vec<TYPE, 3>& p, OTHER_TYPE diag) {
mat<TYPE, 3, 3> r;
r[0][0] = diag;
r[1][1] = diag;
r[2][2] = diag;
r[0][1] = p.z;
r[1][0] =-p.z;
r[0][2] =-p.y;
r[2][0] = p.y;
r[1][2] = p.x;
r[2][1] =-p.x;
return r;
}
template<typename TYPE, size_t SIZE>
class Covariance {
mat<TYPE, SIZE, SIZE> mSumXX;
vec<TYPE, SIZE> mSumX;
size_t mN;
public:
Covariance() : mSumXX(0.0f), mSumX(0.0f), mN(0) { }
void update(const vec<TYPE, SIZE>& x) {
mSumXX += x*transpose(x);
mSumX += x;
mN++;
}
mat<TYPE, SIZE, SIZE> operator()() const {
const float N = 1.0f / mN;
return mSumXX*N - (mSumX*transpose(mSumX))*(N*N);
}
void reset() {
mN = 0;
mSumXX = 0;
mSumX = 0;
}
size_t getCount() const {
return mN;
}
};
// -----------------------------------------------------------------------
Fusion::Fusion() {
Phi[0][1] = 0;
Phi[1][1] = 1;
Ba.x = 0;
Ba.y = 0;
Ba.z = 1;
Bm.x = 0;
Bm.y = 1;
Bm.z = 0;
x0 = 0;
x1 = 0;
init();
}
void Fusion::init() {
mInitState = 0;
mGyroRate = 0;
mCount[0] = 0;
mCount[1] = 0;
mCount[2] = 0;
mData = 0;
}
void Fusion::initFusion(const vec4_t& q, float dT)
{
// initial estimate: E{ x(t0) }
x0 = q;
x1 = 0;
// process noise covariance matrix: G.Q.Gt, with
//
// G = | -1 0 | Q = | q00 q10 |
// | 0 1 | | q01 q11 |
//
// q00 = sv^2.dt + 1/3.su^2.dt^3
// q10 = q01 = 1/2.su^2.dt^2
// q11 = su^2.dt
//
const float dT2 = dT*dT;
const float dT3 = dT2*dT;
// variance of integrated output at 1/dT Hz (random drift)
const float q00 = gyroVAR * dT + 0.33333f * biasVAR * dT3;
// variance of drift rate ramp
const float q11 = biasVAR * dT;
const float q10 = 0.5f * biasVAR * dT2;
const float q01 = q10;
GQGt[0][0] = q00; // rad^2
GQGt[1][0] = -q10;
GQGt[0][1] = -q01;
GQGt[1][1] = q11; // (rad/s)^2
// initial covariance: Var{ x(t0) }
// TODO: initialize P correctly
P = 0;
// it is unclear how to set the initial covariance. It does affect
// how quickly the fusion converges. Experimentally it would take
// about 10 seconds at 200 Hz to estimate the gyro-drift with an
// initial covariance of 0, and about a second with an initial covariance
// of about 1 deg/s.
const float covv = 0;
const float covu = 0.5f * (float(M_PI) / 180);
mat33_t& Pv = P[0][0];
Pv[0][0] = covv;
Pv[1][1] = covv;
Pv[2][2] = covv;
mat33_t& Pu = P[1][1];
Pu[0][0] = covu;
Pu[1][1] = covu;
Pu[2][2] = covu;
}
bool Fusion::hasEstimate() const {
return (mInitState == (MAG|ACC|GYRO));
}
bool Fusion::checkInitComplete(int what, const vec3_t& d, float dT) {
if (hasEstimate())
return true;
if (what == ACC) {
mData[0] += d * (1/length(d));
mCount[0]++;
mInitState |= ACC;
} else if (what == MAG) {
mData[1] += d * (1/length(d));
mCount[1]++;
mInitState |= MAG;
} else if (what == GYRO) {
mGyroRate = dT;
mData[2] += d*dT;
mCount[2]++;
if (mCount[2] == 64) {
// 64 samples is good enough to estimate the gyro drift and
// doesn't take too much time.
mInitState |= GYRO;
}
}
if (mInitState == (MAG|ACC|GYRO)) {
// Average all the values we collected so far
mData[0] *= 1.0f/mCount[0];
mData[1] *= 1.0f/mCount[1];
mData[2] *= 1.0f/mCount[2];
// calculate the MRPs from the data collection, this gives us
// a rough estimate of our initial state
mat33_t R;
vec3_t up(mData[0]);
vec3_t east(cross_product(mData[1], up));
east *= 1/length(east);
vec3_t north(cross_product(up, east));
R << east << north << up;
const vec4_t q = matrixToQuat(R);
initFusion(q, mGyroRate);
}
return false;
}
void Fusion::handleGyro(const vec3_t& w, float dT) {
if (!checkInitComplete(GYRO, w, dT))
return;
predict(w, dT);
}
status_t Fusion::handleAcc(const vec3_t& a) {
// ignore acceleration data if we're close to free-fall
if (length_squared(a) < FREE_FALL_THRESHOLD_SQ) {
return BAD_VALUE;
}
if (!checkInitComplete(ACC, a))
return BAD_VALUE;
const float l = 1/length(a);
update(a*l, Ba, accSTDEV*l);
return NO_ERROR;
}
status_t Fusion::handleMag(const vec3_t& m) {
// the geomagnetic-field should be between 30uT and 60uT
// reject if too large to avoid spurious magnetic sources
const float magFieldSq = length_squared(m);
if (magFieldSq > MAX_VALID_MAGNETIC_FIELD_SQ) {
return BAD_VALUE;
} else if (magFieldSq < MIN_VALID_MAGNETIC_FIELD_SQ) {
// Also reject if too small since we will get ill-defined (zero mag)
// cross-products below
return BAD_VALUE;
}
if (!checkInitComplete(MAG, m))
return BAD_VALUE;
// Orthogonalize the magnetic field to the gravity field, mapping it into
// tangent to Earth.
const vec3_t up( getRotationMatrix() * Ba );
const vec3_t east( cross_product(m, up) );
// If the m and up vectors align, the cross product magnitude will
// approach 0.
// Reject this case as well to avoid div by zero problems and
// ill-conditioning below.
if (length_squared(east) < MIN_VALID_CROSS_PRODUCT_MAG_SQ) {
return BAD_VALUE;
}
// If we have created an orthogonal magnetic field successfully,
// then pass it in as the update.
vec3_t north( cross_product(up, east) );
const float l = 1 / length(north);
north *= l;
update(north, Bm, magSTDEV*l);
return NO_ERROR;
}
void Fusion::checkState() {
// P needs to stay positive semidefinite or the fusion diverges. When we
// detect divergence, we reset the fusion.
// TODO(braun): Instead, find the reason for the divergence and fix it.
if (!isPositiveSemidefinite(P[0][0], SYMMETRY_TOLERANCE) ||
!isPositiveSemidefinite(P[1][1], SYMMETRY_TOLERANCE)) {
ALOGW("Sensor fusion diverged; resetting state.");
P = 0;
}
}
vec4_t Fusion::getAttitude() const {
return x0;
}
vec3_t Fusion::getBias() const {
return x1;
}
mat33_t Fusion::getRotationMatrix() const {
return quatToMatrix(x0);
}
mat34_t Fusion::getF(const vec4_t& q) {
mat34_t F;
// This is used to compute the derivative of q
// F = | [q.xyz]x |
// | -q.xyz |
F[0].x = q.w; F[1].x =-q.z; F[2].x = q.y;
F[0].y = q.z; F[1].y = q.w; F[2].y =-q.x;
F[0].z =-q.y; F[1].z = q.x; F[2].z = q.w;
F[0].w =-q.x; F[1].w =-q.y; F[2].w =-q.z;
return F;
}
void Fusion::predict(const vec3_t& w, float dT) {
const vec4_t q = x0;
const vec3_t b = x1;
const vec3_t we = w - b;
// q(k+1) = O(we)*q(k)
// --------------------
//
// O(w) = | cos(0.5*||w||*dT)*I33 - [psi]x psi |
// | -psi' cos(0.5*||w||*dT) |
//
// psi = sin(0.5*||w||*dT)*w / ||w||
//
//
// P(k+1) = Phi(k)*P(k)*Phi(k)' + G*Q(k)*G'
// ----------------------------------------
//
// G = | -I33 0 |
// | 0 I33 |
//
// Phi = | Phi00 Phi10 |
// | 0 1 |
//
// Phi00 = I33
// - [w]x * sin(||w||*dt)/||w||
// + [w]x^2 * (1-cos(||w||*dT))/||w||^2
//
// Phi10 = [w]x * (1 - cos(||w||*dt))/||w||^2
// - [w]x^2 * (||w||*dT - sin(||w||*dt))/||w||^3
// - I33*dT
const mat33_t I33(1);
const mat33_t I33dT(dT);
const mat33_t wx(crossMatrix(we, 0));
const mat33_t wx2(wx*wx);
const float lwedT = length(we)*dT;
const float hlwedT = 0.5f*lwedT;
const float ilwe = 1/length(we);
const float k0 = (1-cosf(lwedT))*(ilwe*ilwe);
const float k1 = sinf(lwedT);
const float k2 = cosf(hlwedT);
const vec3_t psi(sinf(hlwedT)*ilwe*we);
const mat33_t O33(crossMatrix(-psi, k2));
mat44_t O;
O[0].xyz = O33[0]; O[0].w = -psi.x;
O[1].xyz = O33[1]; O[1].w = -psi.y;
O[2].xyz = O33[2]; O[2].w = -psi.z;
O[3].xyz = psi; O[3].w = k2;
Phi[0][0] = I33 - wx*(k1*ilwe) + wx2*k0;
Phi[1][0] = wx*k0 - I33dT - wx2*(ilwe*ilwe*ilwe)*(lwedT-k1);
x0 = O*q;
if (x0.w < 0)
x0 = -x0;
P = Phi*P*transpose(Phi) + GQGt;
checkState();
}
void Fusion::update(const vec3_t& z, const vec3_t& Bi, float sigma) {
vec4_t q(x0);
// measured vector in body space: h(p) = A(p)*Bi
const mat33_t A(quatToMatrix(q));
const vec3_t Bb(A*Bi);
// Sensitivity matrix H = dh(p)/dp
// H = [ L 0 ]
const mat33_t L(crossMatrix(Bb, 0));
// gain...
// K = P*Ht / [H*P*Ht + R]
vec<mat33_t, 2> K;
const mat33_t R(sigma*sigma);
const mat33_t S(scaleCovariance(L, P[0][0]) + R);
const mat33_t Si(invert(S));
const mat33_t LtSi(transpose(L)*Si);
K[0] = P[0][0] * LtSi;
K[1] = transpose(P[1][0])*LtSi;
// update...
// P = (I-K*H) * P
// P -= K*H*P
// | K0 | * | L 0 | * P = | K0*L 0 | * | P00 P10 | = | K0*L*P00 K0*L*P10 |
// | K1 | | K1*L 0 | | P01 P11 | | K1*L*P00 K1*L*P10 |
// Note: the Joseph form is numerically more stable and given by:
// P = (I-KH) * P * (I-KH)' + K*R*R'
const mat33_t K0L(K[0] * L);
const mat33_t K1L(K[1] * L);
P[0][0] -= K0L*P[0][0];
P[1][1] -= K1L*P[1][0];
P[1][0] -= K0L*P[1][0];
P[0][1] = transpose(P[1][0]);
const vec3_t e(z - Bb);
const vec3_t dq(K[0]*e);
const vec3_t db(K[1]*e);
q += getF(q)*(0.5f*dq);
x0 = normalize_quat(q);
x1 += db;
checkState();
}
// -----------------------------------------------------------------------
}; // namespace android