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Code Issues Releases Activity

LP-443 - Add INS14 to Sparky/Revo

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This commit is contained in:
Alessio Morale 2016-07-18 01:17:47 +02:00
parent c2cf44f8d8
commit ce87ce105e
5 changed files with 1011 additions and 23 deletions
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33
flight/libraries/inc/insgps.h
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@ -32,7 +32,8 @@
#ifndef INSGPS_H_
#define INSGPS_H_
#include "stdint.h"
#include <stdint.h>
#include <stdbool.h>
/**
* @addtogroup Constants
@ -54,22 +55,24 @@
// Exposed Function Prototypes
void INSGPSInit();
void INSStatePrediction(float gyro_data[3], float accel_data[3], float dT);
void INSStatePrediction(const float gyro_data[3], const float accel_data[3], float dT);
void INSCovariancePrediction(float dT);
void INSCorrection(float mag_data[3], float Pos[3], float Vel[3], float BaroAlt, uint16_t SensorsUsed);
void INSResetP(float PDiag[13]);
void INSGetP(float PDiag[13]);
void INSSetState(float pos[3], float vel[3], float q[4], float gyro_bias[3], float accel_bias[3]);
void INSCorrection(const float mag_data[3], const float Pos[3], const float Vel[3],
float BaroAlt, uint16_t SensorsUsed);
void INSResetP(const float PDiag[13]);
void INSGetVariance(float PDiag[13]);
void INSSetState(const float pos[3], const float vel[3], const float q[4], const float gyro_bias[3], const float accel_bias[3]);
void INSSetPosVelVar(float PosVar[3], float VelVar[3]);
void INSSetGyroBias(float gyro_bias[3]);
void INSSetAccelVar(float accel_var[3]);
void INSSetGyroVar(float gyro_var[3]);
void INSSetGyroBiasVar(float gyro_bias_var[3]);
void INSSetMagNorth(float B[3]);
void INSSetMagVar(float scaled_mag_var[3]);
void INSSetBaroVar(float baro_var);
void INSPosVelReset(float pos[3], float vel[3]);
void INSSetGyroBias(const float gyro_bias[3]);
void INSSetAccelVar(const float accel_var[3]);
void INSSetGyroVar(const float gyro_var[3]);
void INSSetGyroBiasVar(const float gyro_bias_var[3]);
void INSSetMagNorth(const float B[3]);
void INSSetMagVar(const float scaled_mag_var[3]);
void INSSetBaroVar(const float baro_var);
void INSSetArmed(bool armed);
void INSPosVelReset(const float pos[3], const float vel[3]);
void MagCorrection(float mag_data[3]);
void MagVelBaroCorrection(float mag_data[3], float Vel[3], float BaroAlt);

986
flight/libraries/insgps14state.c Normal file
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@ -0,0 +1,986 @@
/**
******************************************************************************
* @addtogroup Math
* @{
* @addtogroup INSGPS
* @{
* @brief INSGPS is a joint attitude and position estimation EKF
*
* @file insgps.c
* @author The OpenPilot Team, http://www.openpilot.org Copyright (C) 2010.
* Tau Labs, http://github.com/TauLabs Copyright (C) 2012-2013.
* The LibrePilot Project, http://www.librepilot.org Copyright (C) 2016.
* @brief An INS/GPS algorithm implemented with an EKF.
*
* @see The GNU Public License (GPL) Version 3
*
*****************************************************************************/
/*
* This program is free software; you can redistribute it and/or modify
* it under the terms of the GNU General Public License as published by
* the Free Software Foundation; either version 3 of the License, or
* (at your option) any later version.
*
* This program is distributed in the hope that it will be useful, but
* WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY
* or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License
* for more details.
*
* You should have received a copy of the GNU General Public License along
* with this program; if not, write to the Free Software Foundation, Inc.,
* 59 Temple Place, Suite 330, Boston, MA 02111-1307 USA
*/
#include "insgps.h"
#include <math.h>
#include <stdint.h>
#include <stdbool.h>
#include <pios_math.h>
#include <mathmisc.h>
#include <pios_constants.h>
// constants/macros/typdefs
#define NUMX 14 // number of states, X is the state vector
#define NUMW 10 // number of plant noise inputs, w is disturbance noise vector
#define NUMV 10 // number of measurements, v is the measurement noise vector
#define NUMU 6 // number of deterministic inputs, U is the input vector
#if defined(GENERAL_COV)
// This might trick people so I have a note here. There is a slower but bigger version of the
// code here but won't fit when debugging disabled (requires -Os)
#define COVARIANCE_PREDICTION_GENERAL
#endif
// Private functions
void CovariancePrediction(float F[NUMX][NUMX], float G[NUMX][NUMW],
float Q[NUMW], float dT, float P[NUMX][NUMX]);
void SerialUpdate(float H[NUMV][NUMX], float R[NUMV], float Z[NUMV],
float Y[NUMV], float P[NUMX][NUMX], float X[NUMX],
uint16_t SensorsUsed);
void RungeKutta(float X[NUMX], float U[NUMU], float dT);
void StateEq(float X[NUMX], float U[NUMU], float Xdot[NUMX]);
void LinearizeFG(float X[NUMX], float U[NUMU], float F[NUMX][NUMX],
float G[NUMX][NUMW]);
void MeasurementEq(float X[NUMX], float Be[3], float Y[NUMV]);
void LinearizeH(float X[NUMX], float Be[3], float H[NUMV][NUMX]);
// Private variables
static struct EKFData {
float F[NUMX][NUMX];
float G[NUMX][NUMW];
float H[NUMV][NUMX]; // linearized system matrices
// global to init to zero and maintain zero elements
float Be[3]; // local magnetic unit vector in NED frame
float P[NUMX][NUMX];
float X[NUMX]; // covariance matrix and state vector
float Q[NUMW];
float R[NUMV]; // input noise and measurement noise variances
float K[NUMX][NUMV]; // feedback gain matrix
} ekf;
// Global variables
struct NavStruct Nav;
// ************* Exposed Functions ****************
// *************************************************
uint16_t ins_get_num_states()
{
return NUMX;
}
void INSGPSInit() // pretty much just a place holder for now
{
ekf.Be[0] = 1.0f;
ekf.Be[1] = 0;
ekf.Be[2] = 0; // local magnetic unit vector
for (int i = 0; i < NUMX; i++) {
for (int j = 0; j < NUMX; j++) {
ekf.P[i][j] = 0.0f; // zero all terms
ekf.F[i][j] = 0.0f;
}
for (int j = 0; j < NUMW; j++) {
ekf.G[i][j] = 0.0f;
}
for (int j = 0; j < NUMV; j++) {
ekf.H[j][i] = 0.0f;
ekf.K[i][j] = 0.0f;
}
ekf.X[i] = 0.0f;
}
for (int i = 0; i < NUMW; i++) {
ekf.Q[i] = 0.0f;
}
for (int i = 0; i < NUMV; i++) {
ekf.R[i] = 0.0f;
}
ekf.P[0][0] = ekf.P[1][1] = ekf.P[2][2] = 25.0f; // initial position variance (m^2)
ekf.P[3][3] = ekf.P[4][4] = ekf.P[5][5] = 5.0f; // initial velocity variance (m/s)^2
ekf.P[6][6] = ekf.P[7][7] = ekf.P[8][8] = ekf.P[9][9] = 1e-5f; // initial quaternion variance
ekf.P[10][10] = ekf.P[11][11] = ekf.P[12][12] = 1e-6f; // initial gyro bias variance (rad/s)^2
ekf.P[13][13] = 1e-5f; // initial accel bias variance (deg/s)^2
ekf.X[0] = ekf.X[1] = ekf.X[2] = ekf.X[3] = ekf.X[4] = ekf.X[5] = 0.0f; // initial pos and vel (m)
ekf.X[6] = 1.0f;
ekf.X[7] = ekf.X[8] = ekf.X[9] = 0.0f; // initial quaternion (level and North) (m/s)
ekf.X[10] = ekf.X[11] = ekf.X[12] = 0.0f; // initial gyro bias (rad/s)
ekf.X[13] = 0.0f; // initial accel bias
ekf.Q[0] = ekf.Q[1] = ekf.Q[2] = 1e-5f; // gyro noise variance (rad/s)^2
ekf.Q[3] = ekf.Q[4] = ekf.Q[5] = 1e-5f; // accelerometer noise variance (m/s^2)^2
ekf.Q[6] = ekf.Q[7] = 1e-6f; // gyro x and y bias random walk variance (rad/s^2)^2
ekf.Q[8] = 1e-6f; // gyro z bias random walk variance (rad/s^2)^2
ekf.Q[9] = 5e-4f; // accel bias random walk variance (m/s^3)^2
ekf.R[0] = ekf.R[1] = 0.004f; // High freq GPS horizontal position noise variance (m^2)
ekf.R[2] = 0.036f; // High freq GPS vertical position noise variance (m^2)
ekf.R[3] = ekf.R[4] = 0.004f; // High freq GPS horizontal velocity noise variance (m/s)^2
ekf.R[5] = 0.004f; // High freq GPS vertical velocity noise variance (m/s)^2
ekf.R[6] = ekf.R[7] = ekf.R[8] = 0.005f; // magnetometer unit vector noise variance
ekf.R[9] = .05f; // High freq altimeter noise variance (m^2)
}
// ! Set the current flight state
void INSSetArmed(bool armed)
{
return;
// Speed up convergence of accel and gyro bias when not armed
if (armed) {
ekf.Q[9] = 1e-4f;
ekf.Q[8] = 2e-9f;
} else {
ekf.Q[9] = 1e-2f;
ekf.Q[8] = 2e-8f;
}
}
/**
* Get the current state estimate (null input skips that get)
* @param[out] pos The position in NED space (m)
* @param[out] vel The velocity in NED (m/s)
* @param[out] attitude Quaternion representation of attitude
* @param[out] gyros_bias Estimate of gyro bias (rad/s)
* @param[out] accel_bias Estiamte of the accel bias (m/s^2)
*/
void INSGetState(float *pos, float *vel, float *attitude, float *gyro_bias, float *accel_bias)
{
if (pos) {
pos[0] = ekf.X[0];
pos[1] = ekf.X[1];
pos[2] = ekf.X[2];
}
if (vel) {
vel[0] = ekf.X[3];
vel[1] = ekf.X[4];
vel[2] = ekf.X[5];
}
if (attitude) {
attitude[0] = ekf.X[6];
attitude[1] = ekf.X[7];
attitude[2] = ekf.X[8];
attitude[3] = ekf.X[9];
}
if (gyro_bias) {
gyro_bias[0] = ekf.X[10];
gyro_bias[1] = ekf.X[11];
gyro_bias[2] = ekf.X[12];
}
if (accel_bias) {
accel_bias[0] = 0.0f;
accel_bias[1] = 0.0f;
accel_bias[2] = ekf.X[13];
}
}
/**
* Get the variance, for visualizing the filter performance
* @param[out var_out The variances
*/
void INSGetVariance(float *var_out)
{
for (uint32_t i = 0; i < NUMX; i++) {
var_out[i] = ekf.P[i][i];
}
}
void INSResetP(const float *PDiag)
{
uint8_t i, j;
// if PDiag[i] nonzero then clear row and column and set diagonal element
for (i = 0; i < NUMX; i++) {
if (PDiag != 0) {
for (j = 0; j < NUMX; j++) {
ekf.P[i][j] = ekf.P[j][i] = 0.0f;
}
ekf.P[i][i] = PDiag[i];
}
}
}
void INSSetState(const float pos[3], const float vel[3], const float q[4], const float gyro_bias[3], const float accel_bias[3])
{
ekf.X[0] = pos[0];
ekf.X[1] = pos[1];
ekf.X[2] = pos[2];
ekf.X[3] = vel[0];
ekf.X[4] = vel[1];
ekf.X[5] = vel[2];
ekf.X[6] = q[0];
ekf.X[7] = q[1];
ekf.X[8] = q[2];
ekf.X[9] = q[3];
ekf.X[10] = gyro_bias[0];
ekf.X[11] = gyro_bias[1];
ekf.X[12] = gyro_bias[2];
ekf.X[13] = accel_bias[2];
}
void INSPosVelReset(const float pos[3], const float vel[3])
{
for (int i = 0; i < 6; i++) {
for (int j = i; j < NUMX; j++) {
ekf.P[i][j] = 0.0f; // zero the first 6 rows and columns
ekf.P[j][i] = 0.0f;
}
}
ekf.P[0][0] = ekf.P[1][1] = ekf.P[2][2] = 25.0f; // initial position variance (m^2)
ekf.P[3][3] = ekf.P[4][4] = ekf.P[5][5] = 5.0f; // initial velocity variance (m/s)^2
ekf.X[0] = pos[0];
ekf.X[1] = pos[1];
ekf.X[2] = pos[2];
ekf.X[3] = vel[0];
ekf.X[4] = vel[1];
ekf.X[5] = vel[2];
}
void INSSetPosVelVar(float PosVar[3], float VelVar[3])
// void INSSetPosVelVar(float PosVar, float VelVar, float VertPosVar)
{
ekf.R[0] = PosVar[0];
ekf.R[1] = PosVar[1];
ekf.R[2] = PosVar[2];
ekf.R[3] = VelVar[0];
ekf.R[4] = VelVar[1];
ekf.R[5] = VelVar[2]; // Don't change vertical velocity, not measured
}
void INSSetGyroBias(const float gyro_bias[3])
{
ekf.X[10] = gyro_bias[0];
ekf.X[11] = gyro_bias[1];
ekf.X[12] = gyro_bias[2];
}
void INSSetAccelBias(const float accel_bias[3])
{
ekf.X[13] = accel_bias[2];
}
void INSSetAccelVar(const float accel_var[3])
{
ekf.Q[3] = accel_var[0];
ekf.Q[4] = accel_var[1];
ekf.Q[5] = accel_var[2];
}
void INSSetGyroVar(const float gyro_var[3])
{
ekf.Q[0] = gyro_var[0];
ekf.Q[1] = gyro_var[1];
ekf.Q[2] = gyro_var[2];
}
void INSSetGyroBiasVar(const float gyro_bias_var[3])
{
ekf.Q[6] = gyro_bias_var[0];
ekf.Q[7] = gyro_bias_var[1];
ekf.Q[8] = gyro_bias_var[2];
}
void INSSetMagVar(const float scaled_mag_var[3])
{
ekf.R[6] = scaled_mag_var[0];
ekf.R[7] = scaled_mag_var[1];
ekf.R[8] = scaled_mag_var[2];
}
void INSSetBaroVar(const float baro_var)
{
ekf.R[9] = baro_var;
}
void INSSetMagNorth(const float B[3])
{
ekf.Be[0] = B[0];
ekf.Be[1] = B[1];
ekf.Be[2] = B[2];
}
void INSLimitBias()
{
// The Z accel bias should never wander too much. This helps ensure the filter
// remains stable.
if (ekf.X[13] > 0.1f) {
ekf.X[13] = 0.1f;
} else if (ekf.X[13] < -0.1f) {
ekf.X[13] = -0.1f;
}
// Make sure no gyro bias gets to more than 10 deg / s. This should be more than
// enough for well behaving sensors.
const float GYRO_BIAS_LIMIT = DEG2RAD(10);
for (int i = 10; i < 13; i++) {
if (ekf.X[i] < -GYRO_BIAS_LIMIT) {
ekf.X[i] = -GYRO_BIAS_LIMIT;
} else if (ekf.X[i] > GYRO_BIAS_LIMIT) {
ekf.X[i] = GYRO_BIAS_LIMIT;
}
}
}
void INSStatePrediction(const float gyro_data[3], const float accel_data[3], float dT)
{
float U[6];
float qmag;
// rate gyro inputs in units of rad/s
U[0] = gyro_data[0];
U[1] = gyro_data[1];
U[2] = gyro_data[2];
// accelerometer inputs in units of m/s
U[3] = accel_data[0];
U[4] = accel_data[1];
U[5] = accel_data[2];
// EKF prediction step
LinearizeFG(ekf.X, U, ekf.F, ekf.G);
RungeKutta(ekf.X, U, dT);
qmag = sqrtf(ekf.X[6] * ekf.X[6] + ekf.X[7] * ekf.X[7] + ekf.X[8] * ekf.X[8] + ekf.X[9] * ekf.X[9]);
ekf.X[6] /= qmag;
ekf.X[7] /= qmag;
ekf.X[8] /= qmag;
ekf.X[9] /= qmag;
// Update Nav solution structure
Nav.Pos[0] = ekf.X[0];
Nav.Pos[1] = ekf.X[1];
Nav.Pos[2] = ekf.X[2];
Nav.Vel[0] = ekf.X[3];
Nav.Vel[1] = ekf.X[4];
Nav.Vel[2] = ekf.X[5];
Nav.q[0] = ekf.X[6];
Nav.q[1] = ekf.X[7];
Nav.q[2] = ekf.X[8];
Nav.q[3] = ekf.X[9];
Nav.gyro_bias[0] = ekf.X[10];
Nav.gyro_bias[1] = ekf.X[11];
Nav.gyro_bias[2] = ekf.X[12];
Nav.gyro_bias[0] = 0.0f;
Nav.gyro_bias[1] = 0.0f;
Nav.gyro_bias[2] = ekf.X[13];
}
void INSCovariancePrediction(float dT)
{
CovariancePrediction(ekf.F, ekf.G, ekf.Q, dT, ekf.P);
}
void INSCorrection(const float mag_data[3], const float Pos[3], const float Vel[3],
float BaroAlt, uint16_t SensorsUsed)
{
float Z[10], Y[10];
float qmag;
// GPS Position in meters and in local NED frame
Z[0] = Pos[0];
Z[1] = Pos[1];
Z[2] = Pos[2];
// GPS Velocity in meters and in local NED frame
Z[3] = Vel[0];
Z[4] = Vel[1];
Z[5] = Vel[2];
if (SensorsUsed & MAG_SENSORS) {
// magnetometer data in any units (use unit vector) and in body frame
float Rbe_a[3][3];
float q0 = ekf.X[6];
float q1 = ekf.X[7];
float q2 = ekf.X[8];
float q3 = ekf.X[9];
float k1 = 1.0f / sqrtf(powf(q0 * q1 * 2.0f + q2 * q3 * 2.0f, 2.0f) + powf(q0 * q0 - q1 * q1 - q2 * q2 + q3 * q3, 2.0f));
float k2 = sqrtf(-powf(q0 * q2 * 2.0f - q1 * q3 * 2.0f, 2.0f) + 1.0f);
Rbe_a[0][0] = k2;
Rbe_a[0][1] = 0.0f;
Rbe_a[0][2] = q0 * q2 * -2.0f + q1 * q3 * 2.0f;
Rbe_a[1][0] = k1 * (q0 * q1 * 2.0f + q2 * q3 * 2.0f) * (q0 * q2 * 2.0f - q1 * q3 * 2.0f);
Rbe_a[1][1] = k1 * (q0 * q0 - q1 * q1 - q2 * q2 + q3 * q3);
Rbe_a[1][2] = k1 * sqrtf(-powf(q0 * q2 * 2.0f - q1 * q3 * 2.0f, 2.0f) + 1.0f) * (q0 * q1 * 2.0f + q2 * q3 * 2.0f);
Rbe_a[2][0] = k1 * (q0 * q2 * 2.0f - q1 * q3 * 2.0f) * (q0 * q0 - q1 * q1 - q2 * q2 + q3 * q3);
Rbe_a[2][1] = -k1 * (q0 * q1 * 2.0f + q2 * q3 * 2.0f);
Rbe_a[2][2] = k1 * k2 * (q0 * q0 - q1 * q1 - q2 * q2 + q3 * q3);
Z[6] = Rbe_a[0][0] * mag_data[0] + Rbe_a[1][0] * mag_data[1] + Rbe_a[2][0] * mag_data[2];
Z[7] = Rbe_a[0][1] * mag_data[0] + Rbe_a[1][1] * mag_data[1] + Rbe_a[2][1] * mag_data[2];
Z[8] = Rbe_a[0][2] * mag_data[0] + Rbe_a[1][2] * mag_data[1] + Rbe_a[2][2] * mag_data[2];
}
// barometric altimeter in meters and in local NED frame
Z[9] = BaroAlt;
// EKF correction step
LinearizeH(ekf.X, ekf.Be, ekf.H);
MeasurementEq(ekf.X, ekf.Be, Y);
SerialUpdate(ekf.H, ekf.R, Z, Y, ekf.P, ekf.X, SensorsUsed);
qmag = sqrtf(ekf.X[6] * ekf.X[6] + ekf.X[7] * ekf.X[7] + ekf.X[8] * ekf.X[8] + ekf.X[9] * ekf.X[9]);
ekf.X[6] /= qmag;
ekf.X[7] /= qmag;
ekf.X[8] /= qmag;
ekf.X[9] /= qmag;
INSLimitBias();
}
// ************* CovariancePrediction *************
// Does the prediction step of the Kalman filter for the covariance matrix
// Output, Pnew, overwrites P, the input covariance
// Pnew = (I+F*T)*P*(I+F*T)' + T^2*G*Q*G'
// Q is the discrete time covariance of process noise
// Q is vector of the diagonal for a square matrix with
// dimensions equal to the number of disturbance noise variables
// The General Method is very inefficient,not taking advantage of the sparse F and G
// The first Method is very specific to this implementation
// ************************************************
#ifdef COVARIANCE_PREDICTION_GENERAL
void CovariancePrediction(float F[NUMX][NUMX], float G[NUMX][NUMW],
float Q[NUMW], float dT, float P[NUMX][NUMX])
{
float Dummy[NUMX][NUMX], dTsq;
uint8_t i, j, k;
// Pnew = (I+F*T)*P*(I+F*T)' + T^2*G*Q*G' = T^2[(P/T + F*P)*(I/T + F') + G*Q*G')]
dTsq = dT * dT;
for (i = 0; i < NUMX; i++) { // Calculate Dummy = (P/T +F*P)
for (j = 0; j < NUMX; j++) {
Dummy[i][j] = P[i][j] / dT;
for (k = 0; k < NUMX; k++) {
Dummy[i][j] += F[i][k] * P[k][j];
}
}
}
for (i = 0; i < NUMX; i++) { // Calculate Pnew = Dummy/T + Dummy*F' + G*Qw*G'
for (j = i; j < NUMX; j++) { // Use symmetry, ie only find upper triangular
P[i][j] = Dummy[i][j] / dT;
for (k = 0; k < NUMX; k++) {
P[i][j] += Dummy[i][k] * F[j][k]; // P = Dummy/T + Dummy*F'
}
for (k = 0; k < NUMW; k++) {
P[i][j] += Q[k] * G[i][k] * G[j][k]; // P = Dummy/T + Dummy*F' + G*Q*G'
}
P[j][i] = P[i][j] = P[i][j] * dTsq; // Pnew = T^2*P and fill in lower triangular;
}
}
}
#else /* ifdef COVARIANCE_PREDICTION_GENERAL */
void CovariancePrediction(float F[NUMX][NUMX], float G[NUMX][NUMW],
float Q[NUMW], float dT, float P[NUMX][NUMX])
{
float D[NUMX][NUMX], T, Tsq;
uint8_t i, j;
// Pnew = (I+F*T)*P*(I+F*T)' + T^2*G*Q*G' = scalar expansion from symbolic manipulator
T = dT;
Tsq = dT * dT;
for (i = 0; i < NUMX; i++) { // Create a copy of the upper triangular of P
for (j = i; j < NUMX; j++) {
D[i][j] = P[i][j];
}
}
// Brute force calculation of the elements of P
P[0][0] = D[3][3] * Tsq + (2 * D[0][3]) * T + D[0][0];
P[0][1] = P[1][0] = D[3][4] * Tsq + (D[0][4] + D[1][3]) * T + D[0][1];
P[0][2] = P[2][0] = D[3][5] * Tsq + (D[0][5] + D[2][3]) * T + D[0][2];
P[0][3] = P[3][0] = (F[3][6] * D[3][6] + F[3][7] * D[3][7] + F[3][8] * D[3][8] + F[3][9] * D[3][9] + F[3][13] * D[3][13]) * Tsq + (D[3][3] + F[3][6] * D[0][6] + F[3][7] * D[0][7] + F[3][8] * D[0][8] + F[3][9] * D[0][9] + F[3][13] * D[0][13]) * T + D[0][3];
P[0][4] = P[4][0] = (F[4][6] * D[3][6] + F[4][7] * D[3][7] + F[4][8] * D[3][8] + F[4][9] * D[3][9] + F[4][13] * D[3][13]) * Tsq + (D[3][4] + F[4][6] * D[0][6] + F[4][7] * D[0][7] + F[4][8] * D[0][8] + F[4][9] * D[0][9] + F[4][13] * D[0][13]) * T + D[0][4];
P[0][5] = P[5][0] = (F[5][6] * D[3][6] + F[5][7] * D[3][7] + F[5][8] * D[3][8] + F[5][9] * D[3][9] + F[5][13] * D[3][13]) * Tsq + (D[3][5] + F[5][6] * D[0][6] + F[5][7] * D[0][7] + F[5][8] * D[0][8] + F[5][9] * D[0][9] + F[5][13] * D[0][13]) * T + D[0][5];
P[0][6] = P[6][0] = (F[6][7] * D[3][7] + F[6][8] * D[3][8] + F[6][9] * D[3][9] + F[6][10] * D[3][10] + F[6][11] * D[3][11] + F[6][12] * D[3][12]) * Tsq + (D[3][6] + F[6][7] * D[0][7] + F[6][8] * D[0][8] + F[6][9] * D[0][9] + F[6][10] * D[0][10] + F[6][11] * D[0][11] + F[6][12] * D[0][12]) * T + D[0][6];
P[0][7] = P[7][0] = (F[7][6] * D[3][6] + F[7][8] * D[3][8] + F[7][9] * D[3][9] + F[7][10] * D[3][10] + F[7][11] * D[3][11] + F[7][12] * D[3][12]) * Tsq + (D[3][7] + F[7][6] * D[0][6] + F[7][8] * D[0][8] + F[7][9] * D[0][9] + F[7][10] * D[0][10] + F[7][11] * D[0][11] + F[7][12] * D[0][12]) * T + D[0][7];
P[0][8] = P[8][0] = (F[8][6] * D[3][6] + F[8][7] * D[3][7] + F[8][9] * D[3][9] + F[8][10] * D[3][10] + F[8][11] * D[3][11] + F[8][12] * D[3][12]) * Tsq + (D[3][8] + F[8][6] * D[0][6] + F[8][7] * D[0][7] + F[8][9] * D[0][9] + F[8][10] * D[0][10] + F[8][11] * D[0][11] + F[8][12] * D[0][12]) * T + D[0][8];
P[0][9] = P[9][0] = (F[9][6] * D[3][6] + F[9][7] * D[3][7] + F[9][8] * D[3][8] + F[9][10] * D[3][10] + F[9][11] * D[3][11] + F[9][12] * D[3][12]) * Tsq + (D[3][9] + F[9][6] * D[0][6] + F[9][7] * D[0][7] + F[9][8] * D[0][8] + F[9][10] * D[0][10] + F[9][11] * D[0][11] + F[9][12] * D[0][12]) * T + D[0][9];
P[0][10] = P[10][0] = D[3][10] * T + D[0][10];
P[0][11] = P[11][0] = D[3][11] * T + D[0][11];
P[0][12] = P[12][0] = D[3][12] * T + D[0][12];
P[0][13] = P[13][0] = D[3][13] * T + D[0][13];
P[1][1] = D[4][4] * Tsq + (2 * D[1][4]) * T + D[1][1];
P[1][2] = P[2][1] = D[4][5] * Tsq + (D[1][5] + D[2][4]) * T + D[1][2];
P[1][3] = P[3][1] = (F[3][6] * D[4][6] + F[3][7] * D[4][7] + F[3][8] * D[4][8] + F[3][9] * D[4][9] + F[3][13] * D[4][13]) * Tsq + (D[3][4] + F[3][6] * D[1][6] + F[3][7] * D[1][7] + F[3][8] * D[1][8] + F[3][9] * D[1][9] + F[3][13] * D[1][13]) * T + D[1][3];
P[1][4] = P[4][1] = (F[4][6] * D[4][6] + F[4][7] * D[4][7] + F[4][8] * D[4][8] + F[4][9] * D[4][9] + F[4][13] * D[4][13]) * Tsq + (D[4][4] + F[4][6] * D[1][6] + F[4][7] * D[1][7] + F[4][8] * D[1][8] + F[4][9] * D[1][9] + F[4][13] * D[1][13]) * T + D[1][4];
P[1][5] = P[5][1] = (F[5][6] * D[4][6] + F[5][7] * D[4][7] + F[5][8] * D[4][8] + F[5][9] * D[4][9] + F[5][13] * D[4][13]) * Tsq + (D[4][5] + F[5][6] * D[1][6] + F[5][7] * D[1][7] + F[5][8] * D[1][8] + F[5][9] * D[1][9] + F[5][13] * D[1][13]) * T + D[1][5];
P[1][6] = P[6][1] = (F[6][7] * D[4][7] + F[6][8] * D[4][8] + F[6][9] * D[4][9] + F[6][10] * D[4][10] + F[6][11] * D[4][11] + F[6][12] * D[4][12]) * Tsq + (D[4][6] + F[6][7] * D[1][7] + F[6][8] * D[1][8] + F[6][9] * D[1][9] + F[6][10] * D[1][10] + F[6][11] * D[1][11] + F[6][12] * D[1][12]) * T + D[1][6];
P[1][7] = P[7][1] = (F[7][6] * D[4][6] + F[7][8] * D[4][8] + F[7][9] * D[4][9] + F[7][10] * D[4][10] + F[7][11] * D[4][11] + F[7][12] * D[4][12]) * Tsq + (D[4][7] + F[7][6] * D[1][6] + F[7][8] * D[1][8] + F[7][9] * D[1][9] + F[7][10] * D[1][10] + F[7][11] * D[1][11] + F[7][12] * D[1][12]) * T + D[1][7];
P[1][8] = P[8][1] = (F[8][6] * D[4][6] + F[8][7] * D[4][7] + F[8][9] * D[4][9] + F[8][10] * D[4][10] + F[8][11] * D[4][11] + F[8][12] * D[4][12]) * Tsq + (D[4][8] + F[8][6] * D[1][6] + F[8][7] * D[1][7] + F[8][9] * D[1][9] + F[8][10] * D[1][10] + F[8][11] * D[1][11] + F[8][12] * D[1][12]) * T + D[1][8];
P[1][9] = P[9][1] = (F[9][6] * D[4][6] + F[9][7] * D[4][7] + F[9][8] * D[4][8] + F[9][10] * D[4][10] + F[9][11] * D[4][11] + F[9][12] * D[4][12]) * Tsq + (D[4][9] + F[9][6] * D[1][6] + F[9][7] * D[1][7] + F[9][8] * D[1][8] + F[9][10] * D[1][10] + F[9][11] * D[1][11] + F[9][12] * D[1][12]) * T + D[1][9];
P[1][10] = P[10][1] = D[4][10] * T + D[1][10];
P[1][11] = P[11][1] = D[4][11] * T + D[1][11];
P[1][12] = P[12][1] = D[4][12] * T + D[1][12];
P[1][13] = P[13][1] = D[4][13] * T + D[1][13];
P[2][2] = D[5][5] * Tsq + (2 * D[2][5]) * T + D[2][2];
P[2][3] = P[3][2] = (F[3][6] * D[5][6] + F[3][7] * D[5][7] + F[3][8] * D[5][8] + F[3][9] * D[5][9] + F[3][13] * D[5][13]) * Tsq + (D[3][5] + F[3][6] * D[2][6] + F[3][7] * D[2][7] + F[3][8] * D[2][8] + F[3][9] * D[2][9] + F[3][13] * D[2][13]) * T + D[2][3];
P[2][4] = P[4][2] = (F[4][6] * D[5][6] + F[4][7] * D[5][7] + F[4][8] * D[5][8] + F[4][9] * D[5][9] + F[4][13] * D[5][13]) * Tsq + (D[4][5] + F[4][6] * D[2][6] + F[4][7] * D[2][7] + F[4][8] * D[2][8] + F[4][9] * D[2][9] + F[4][13] * D[2][13]) * T + D[2][4];
P[2][5] = P[5][2] = (F[5][6] * D[5][6] + F[5][7] * D[5][7] + F[5][8] * D[5][8] + F[5][9] * D[5][9] + F[5][13] * D[5][13]) * Tsq + (D[5][5] + F[5][6] * D[2][6] + F[5][7] * D[2][7] + F[5][8] * D[2][8] + F[5][9] * D[2][9] + F[5][13] * D[2][13]) * T + D[2][5];
P[2][6] = P[6][2] = (F[6][7] * D[5][7] + F[6][8] * D[5][8] + F[6][9] * D[5][9] + F[6][10] * D[5][10] + F[6][11] * D[5][11] + F[6][12] * D[5][12]) * Tsq + (D[5][6] + F[6][7] * D[2][7] + F[6][8] * D[2][8] + F[6][9] * D[2][9] + F[6][10] * D[2][10] + F[6][11] * D[2][11] + F[6][12] * D[2][12]) * T + D[2][6];
P[2][7] = P[7][2] = (F[7][6] * D[5][6] + F[7][8] * D[5][8] + F[7][9] * D[5][9] + F[7][10] * D[5][10] + F[7][11] * D[5][11] + F[7][12] * D[5][12]) * Tsq + (D[5][7] + F[7][6] * D[2][6] + F[7][8] * D[2][8] + F[7][9] * D[2][9] + F[7][10] * D[2][10] + F[7][11] * D[2][11] + F[7][12] * D[2][12]) * T + D[2][7];
P[2][8] = P[8][2] = (F[8][6] * D[5][6] + F[8][7] * D[5][7] + F[8][9] * D[5][9] + F[8][10] * D[5][10] + F[8][11] * D[5][11] + F[8][12] * D[5][12]) * Tsq + (D[5][8] + F[8][6] * D[2][6] + F[8][7] * D[2][7] + F[8][9] * D[2][9] + F[8][10] * D[2][10] + F[8][11] * D[2][11] + F[8][12] * D[2][12]) * T + D[2][8];
P[2][9] = P[9][2] = (F[9][6] * D[5][6] + F[9][7] * D[5][7] + F[9][8] * D[5][8] + F[9][10] * D[5][10] + F[9][11] * D[5][11] + F[9][12] * D[5][12]) * Tsq + (D[5][9] + F[9][6] * D[2][6] + F[9][7] * D[2][7] + F[9][8] * D[2][8] + F[9][10] * D[2][10] + F[9][11] * D[2][11] + F[9][12] * D[2][12]) * T + D[2][9];
P[2][10] = P[10][2] = D[5][10] * T + D[2][10];
P[2][11] = P[11][2] = D[5][11] * T + D[2][11];
P[2][12] = P[12][2] = D[5][12] * T + D[2][12];
P[2][13] = P[13][2] = D[5][13] * T + D[2][13];
P[3][3] = (Q[3] * G[3][3] * G[3][3] + Q[4] * G[3][4] * G[3][4] + Q[5] * G[3][5] * G[3][5] + F[3][6] * (F[3][6] * D[6][6] + F[3][7] * D[6][7] + F[3][8] * D[6][8] + F[3][9] * D[6][9] + F[3][13] * D[6][13]) + F[3][7] * (F[3][6] * D[6][7] + F[3][7] * D[7][7] + F[3][8] * D[7][8] + F[3][9] * D[7][9] + F[3][13] * D[7][13]) + F[3][8] * (F[3][6] * D[6][8] + F[3][7] * D[7][8] + F[3][8] * D[8][8] + F[3][9] * D[8][9] + F[3][13] * D[8][13]) + F[3][9] * (F[3][6] * D[6][9] + F[3][7] * D[7][9] + F[3][8] * D[8][9] + F[3][9] * D[9][9] + F[3][13] * D[9][13]) + F[3][13] * (F[3][6] * D[6][13] + F[3][7] * D[7][13] + F[3][8] * D[8][13] + F[3][9] * D[9][13] + F[3][13] * D[13][13])) * Tsq + (2 * F[3][6] * D[3][6] + 2 * F[3][7] * D[3][7] + 2 * F[3][8] * D[3][8] + 2 * F[3][9] * D[3][9] + 2 * F[3][13] * D[3][13]) * T + D[3][3];
P[3][4] = P[4][3] = (F[4][6] * (F[3][6] * D[6][6] + F[3][7] * D[6][7] + F[3][8] * D[6][8] + F[3][9] * D[6][9] + F[3][13] * D[6][13]) + F[4][7] * (F[3][6] * D[6][7] + F[3][7] * D[7][7] + F[3][8] * D[7][8] + F[3][9] * D[7][9] + F[3][13] * D[7][13]) + F[4][8] * (F[3][6] * D[6][8] + F[3][7] * D[7][8] + F[3][8] * D[8][8] + F[3][9] * D[8][9] + F[3][13] * D[8][13]) + F[4][9] * (F[3][6] * D[6][9] + F[3][7] * D[7][9] + F[3][8] * D[8][9] + F[3][9] * D[9][9] + F[3][13] * D[9][13]) + F[4][13] * (F[3][6] * D[6][13] + F[3][7] * D[7][13] + F[3][8] * D[8][13] + F[3][9] * D[9][13] + F[3][13] * D[13][13]) + G[3][3] * G[4][3] * Q[3] + G[3][4] * G[4][4] * Q[4] + G[3][5] * G[4][5] * Q[5]) * Tsq + (F[3][6] * D[4][6] + F[4][6] * D[3][6] + F[3][7] * D[4][7] + F[4][7] * D[3][7] + F[3][8] * D[4][8] + F[4][8] * D[3][8] + F[3][9] * D[4][9] + F[4][9] * D[3][9] + F[3][13] * D[4][13] + F[4][13] * D[3][13]) * T + D[3][4];
P[3][5] = P[5][3] = (F[5][6] * (F[3][6] * D[6][6] + F[3][7] * D[6][7] + F[3][8] * D[6][8] + F[3][9] * D[6][9] + F[3][13] * D[6][13]) + F[5][7] * (F[3][6] * D[6][7] + F[3][7] * D[7][7] + F[3][8] * D[7][8] + F[3][9] * D[7][9] + F[3][13] * D[7][13]) + F[5][8] * (F[3][6] * D[6][8] + F[3][7] * D[7][8] + F[3][8] * D[8][8] + F[3][9] * D[8][9] + F[3][13] * D[8][13]) + F[5][9] * (F[3][6] * D[6][9] + F[3][7] * D[7][9] + F[3][8] * D[8][9] + F[3][9] * D[9][9] + F[3][13] * D[9][13]) + F[5][13] * (F[3][6] * D[6][13] + F[3][7] * D[7][13] + F[3][8] * D[8][13] + F[3][9] * D[9][13] + F[3][13] * D[13][13]) + G[3][3] * G[5][3] * Q[3] + G[3][4] * G[5][4] * Q[4] + G[3][5] * G[5][5] * Q[5]) * Tsq + (F[3][6] * D[5][6] + F[5][6] * D[3][6] + F[3][7] * D[5][7] + F[5][7] * D[3][7] + F[3][8] * D[5][8] + F[5][8] * D[3][8] + F[3][9] * D[5][9] + F[5][9] * D[3][9] + F[3][13] * D[5][13] + F[5][13] * D[3][13]) * T + D[3][5];
P[3][6] = P[6][3] = (F[6][7] * (F[3][6] * D[6][7] + F[3][7] * D[7][7] + F[3][8] * D[7][8] + F[3][9] * D[7][9] + F[3][13] * D[7][13]) + F[6][8] * (F[3][6] * D[6][8] + F[3][7] * D[7][8] + F[3][8] * D[8][8] + F[3][9] * D[8][9] + F[3][13] * D[8][13]) + F[6][9] * (F[3][6] * D[6][9] + F[3][7] * D[7][9] + F[3][8] * D[8][9] + F[3][9] * D[9][9] + F[3][13] * D[9][13]) + F[6][10] * (F[3][6] * D[6][10] + F[3][7] * D[7][10] + F[3][8] * D[8][10] + F[3][9] * D[9][10] + F[3][13] * D[10][13]) + F[6][11] * (F[3][6] * D[6][11] + F[3][7] * D[7][11] + F[3][8] * D[8][11] + F[3][9] * D[9][11] + F[3][13] * D[11][13]) + F[6][12] * (F[3][6] * D[6][12] + F[3][7] * D[7][12] + F[3][8] * D[8][12] + F[3][9] * D[9][12] + F[3][13] * D[12][13])) * Tsq + (F[3][6] * D[6][6] + F[3][7] * D[6][7] + F[6][7] * D[3][7] + F[3][8] * D[6][8] + F[6][8] * D[3][8] + F[3][9] * D[6][9] + F[6][9] * D[3][9] + F[6][10] * D[3][10] + F[6][11] * D[3][11] + F[6][12] * D[3][12] + F[3][13] * D[6][13]) * T + D[3][6];
P[3][7] = P[7][3] = (F[7][6] * (F[3][6] * D[6][6] + F[3][7] * D[6][7] + F[3][8] * D[6][8] + F[3][9] * D[6][9] + F[3][13] * D[6][13]) + F[7][8] * (F[3][6] * D[6][8] + F[3][7] * D[7][8] + F[3][8] * D[8][8] + F[3][9] * D[8][9] + F[3][13] * D[8][13]) + F[7][9] * (F[3][6] * D[6][9] + F[3][7] * D[7][9] + F[3][8] * D[8][9] + F[3][9] * D[9][9] + F[3][13] * D[9][13]) + F[7][10] * (F[3][6] * D[6][10] + F[3][7] * D[7][10] + F[3][8] * D[8][10] + F[3][9] * D[9][10] + F[3][13] * D[10][13]) + F[7][11] * (F[3][6] * D[6][11] + F[3][7] * D[7][11] + F[3][8] * D[8][11] + F[3][9] * D[9][11] + F[3][13] * D[11][13]) + F[7][12] * (F[3][6] * D[6][12] + F[3][7] * D[7][12] + F[3][8] * D[8][12] + F[3][9] * D[9][12] + F[3][13] * D[12][13])) * Tsq + (F[3][6] * D[6][7] + F[7][6] * D[3][6] + F[3][7] * D[7][7] + F[3][8] * D[7][8] + F[7][8] * D[3][8] + F[3][9] * D[7][9] + F[7][9] * D[3][9] + F[7][10] * D[3][10] + F[7][11] * D[3][11] + F[7][12] * D[3][12] + F[3][13] * D[7][13]) * T + D[3][7];
P[3][8] = P[8][3] = (F[8][6] * (F[3][6] * D[6][6] + F[3][7] * D[6][7] + F[3][8] * D[6][8] + F[3][9] * D[6][9] + F[3][13] * D[6][13]) + F[8][7] * (F[3][6] * D[6][7] + F[3][7] * D[7][7] + F[3][8] * D[7][8] + F[3][9] * D[7][9] + F[3][13] * D[7][13]) + F[8][9] * (F[3][6] * D[6][9] + F[3][7] * D[7][9] + F[3][8] * D[8][9] + F[3][9] * D[9][9] + F[3][13] * D[9][13]) + F[8][10] * (F[3][6] * D[6][10] + F[3][7] * D[7][10] + F[3][8] * D[8][10] + F[3][9] * D[9][10] + F[3][13] * D[10][13]) + F[8][11] * (F[3][6] * D[6][11] + F[3][7] * D[7][11] + F[3][8] * D[8][11] + F[3][9] * D[9][11] + F[3][13] * D[11][13]) + F[8][12] * (F[3][6] * D[6][12] + F[3][7] * D[7][12] + F[3][8] * D[8][12] + F[3][9] * D[9][12] + F[3][13] * D[12][13])) * Tsq + (F[3][6] * D[6][8] + F[3][7] * D[7][8] + F[8][6] * D[3][6] + F[8][7] * D[3][7] + F[3][8] * D[8][8] + F[3][9] * D[8][9] + F[8][9] * D[3][9] + F[8][10] * D[3][10] + F[8][11] * D[3][11] + F[8][12] * D[3][12] + F[3][13] * D[8][13]) * T + D[3][8];
P[3][9] = P[9][3] = (F[9][6] * (F[3][6] * D[6][6] + F[3][7] * D[6][7] + F[3][8] * D[6][8] + F[3][9] * D[6][9] + F[3][13] * D[6][13]) + F[9][7] * (F[3][6] * D[6][7] + F[3][7] * D[7][7] + F[3][8] * D[7][8] + F[3][9] * D[7][9] + F[3][13] * D[7][13]) + F[9][8] * (F[3][6] * D[6][8] + F[3][7] * D[7][8] + F[3][8] * D[8][8] + F[3][9] * D[8][9] + F[3][13] * D[8][13]) + F[9][10] * (F[3][6] * D[6][10] + F[3][7] * D[7][10] + F[3][8] * D[8][10] + F[3][9] * D[9][10] + F[3][13] * D[10][13]) + F[9][11] * (F[3][6] * D[6][11] + F[3][7] * D[7][11] + F[3][8] * D[8][11] + F[3][9] * D[9][11] + F[3][13] * D[11][13]) + F[9][12] * (F[3][6] * D[6][12] + F[3][7] * D[7][12] + F[3][8] * D[8][12] + F[3][9] * D[9][12] + F[3][13] * D[12][13])) * Tsq + (F[9][6] * D[3][6] + F[9][7] * D[3][7] + F[9][8] * D[3][8] + F[3][6] * D[6][9] + F[3][7] * D[7][9] + F[3][8] * D[8][9] + F[3][9] * D[9][9] + F[9][10] * D[3][10] + F[9][11] * D[3][11] + F[9][12] * D[3][12] + F[3][13] * D[9][13]) * T + D[3][9];
P[3][10] = P[10][3] = (F[3][6] * D[6][10] + F[3][7] * D[7][10] + F[3][8] * D[8][10] + F[3][9] * D[9][10] + F[3][13] * D[10][13]) * T + D[3][10];
P[3][11] = P[11][3] = (F[3][6] * D[6][11] + F[3][7] * D[7][11] + F[3][8] * D[8][11] + F[3][9] * D[9][11] + F[3][13] * D[11][13]) * T + D[3][11];
P[3][12] = P[12][3] = (F[3][6] * D[6][12] + F[3][7] * D[7][12] + F[3][8] * D[8][12] + F[3][9] * D[9][12] + F[3][13] * D[12][13]) * T + D[3][12];
P[3][13] = P[13][3] = (F[3][6] * D[6][13] + F[3][7] * D[7][13] + F[3][8] * D[8][13] + F[3][9] * D[9][13] + F[3][13] * D[13][13]) * T + D[3][13];
P[4][4] = (Q[3] * G[4][3] * G[4][3] + Q[4] * G[4][4] * G[4][4] + Q[5] * G[4][5] * G[4][5] + F[4][6] * (F[4][6] * D[6][6] + F[4][7] * D[6][7] + F[4][8] * D[6][8] + F[4][9] * D[6][9] + F[4][13] * D[6][13]) + F[4][7] * (F[4][6] * D[6][7] + F[4][7] * D[7][7] + F[4][8] * D[7][8] + F[4][9] * D[7][9] + F[4][13] * D[7][13]) + F[4][8] * (F[4][6] * D[6][8] + F[4][7] * D[7][8] + F[4][8] * D[8][8] + F[4][9] * D[8][9] + F[4][13] * D[8][13]) + F[4][9] * (F[4][6] * D[6][9] + F[4][7] * D[7][9] + F[4][8] * D[8][9] + F[4][9] * D[9][9] + F[4][13] * D[9][13]) + F[4][13] * (F[4][6] * D[6][13] + F[4][7] * D[7][13] + F[4][8] * D[8][13] + F[4][9] * D[9][13] + F[4][13] * D[13][13])) * Tsq + (2 * F[4][6] * D[4][6] + 2 * F[4][7] * D[4][7] + 2 * F[4][8] * D[4][8] + 2 * F[4][9] * D[4][9] + 2 * F[4][13] * D[4][13]) * T + D[4][4];
P[4][5] = P[5][4] = (F[5][6] * (F[4][6] * D[6][6] + F[4][7] * D[6][7] + F[4][8] * D[6][8] + F[4][9] * D[6][9] + F[4][13] * D[6][13]) + F[5][7] * (F[4][6] * D[6][7] + F[4][7] * D[7][7] + F[4][8] * D[7][8] + F[4][9] * D[7][9] + F[4][13] * D[7][13]) + F[5][8] * (F[4][6] * D[6][8] + F[4][7] * D[7][8] + F[4][8] * D[8][8] + F[4][9] * D[8][9] + F[4][13] * D[8][13]) + F[5][9] * (F[4][6] * D[6][9] + F[4][7] * D[7][9] + F[4][8] * D[8][9] + F[4][9] * D[9][9] + F[4][13] * D[9][13]) + F[5][13] * (F[4][6] * D[6][13] + F[4][7] * D[7][13] + F[4][8] * D[8][13] + F[4][9] * D[9][13] + F[4][13] * D[13][13]) + G[4][3] * G[5][3] * Q[3] + G[4][4] * G[5][4] * Q[4] + G[4][5] * G[5][5] * Q[5]) * Tsq + (F[4][6] * D[5][6] + F[5][6] * D[4][6] + F[4][7] * D[5][7] + F[5][7] * D[4][7] + F[4][8] * D[5][8] + F[5][8] * D[4][8] + F[4][9] * D[5][9] + F[5][9] * D[4][9] + F[4][13] * D[5][13] + F[5][13] * D[4][13]) * T + D[4][5];
P[4][6] = P[6][4] = (F[6][7] * (F[4][6] * D[6][7] + F[4][7] * D[7][7] + F[4][8] * D[7][8] + F[4][9] * D[7][9] + F[4][13] * D[7][13]) + F[6][8] * (F[4][6] * D[6][8] + F[4][7] * D[7][8] + F[4][8] * D[8][8] + F[4][9] * D[8][9] + F[4][13] * D[8][13]) + F[6][9] * (F[4][6] * D[6][9] + F[4][7] * D[7][9] + F[4][8] * D[8][9] + F[4][9] * D[9][9] + F[4][13] * D[9][13]) + F[6][10] * (F[4][6] * D[6][10] + F[4][7] * D[7][10] + F[4][8] * D[8][10] + F[4][9] * D[9][10] + F[4][13] * D[10][13]) + F[6][11] * (F[4][6] * D[6][11] + F[4][7] * D[7][11] + F[4][8] * D[8][11] + F[4][9] * D[9][11] + F[4][13] * D[11][13]) + F[6][12] * (F[4][6] * D[6][12] + F[4][7] * D[7][12] + F[4][8] * D[8][12] + F[4][9] * D[9][12] + F[4][13] * D[12][13])) * Tsq + (F[4][6] * D[6][6] + F[4][7] * D[6][7] + F[6][7] * D[4][7] + F[4][8] * D[6][8] + F[6][8] * D[4][8] + F[4][9] * D[6][9] + F[6][9] * D[4][9] + F[6][10] * D[4][10] + F[6][11] * D[4][11] + F[6][12] * D[4][12] + F[4][13] * D[6][13]) * T + D[4][6];
P[4][7] = P[7][4] = (F[7][6] * (F[4][6] * D[6][6] + F[4][7] * D[6][7] + F[4][8] * D[6][8] + F[4][9] * D[6][9] + F[4][13] * D[6][13]) + F[7][8] * (F[4][6] * D[6][8] + F[4][7] * D[7][8] + F[4][8] * D[8][8] + F[4][9] * D[8][9] + F[4][13] * D[8][13]) + F[7][9] * (F[4][6] * D[6][9] + F[4][7] * D[7][9] + F[4][8] * D[8][9] + F[4][9] * D[9][9] + F[4][13] * D[9][13]) + F[7][10] * (F[4][6] * D[6][10] + F[4][7] * D[7][10] + F[4][8] * D[8][10] + F[4][9] * D[9][10] + F[4][13] * D[10][13]) + F[7][11] * (F[4][6] * D[6][11] + F[4][7] * D[7][11] + F[4][8] * D[8][11] + F[4][9] * D[9][11] + F[4][13] * D[11][13]) + F[7][12] * (F[4][6] * D[6][12] + F[4][7] * D[7][12] + F[4][8] * D[8][12] + F[4][9] * D[9][12] + F[4][13] * D[12][13])) * Tsq + (F[4][6] * D[6][7] + F[7][6] * D[4][6] + F[4][7] * D[7][7] + F[4][8] * D[7][8] + F[7][8] * D[4][8] + F[4][9] * D[7][9] + F[7][9] * D[4][9] + F[7][10] * D[4][10] + F[7][11] * D[4][11] + F[7][12] * D[4][12] + F[4][13] * D[7][13]) * T + D[4][7];
P[4][8] = P[8][4] = (F[8][6] * (F[4][6] * D[6][6] + F[4][7] * D[6][7] + F[4][8] * D[6][8] + F[4][9] * D[6][9] + F[4][13] * D[6][13]) + F[8][7] * (F[4][6] * D[6][7] + F[4][7] * D[7][7] + F[4][8] * D[7][8] + F[4][9] * D[7][9] + F[4][13] * D[7][13]) + F[8][9] * (F[4][6] * D[6][9] + F[4][7] * D[7][9] + F[4][8] * D[8][9] + F[4][9] * D[9][9] + F[4][13] * D[9][13]) + F[8][10] * (F[4][6] * D[6][10] + F[4][7] * D[7][10] + F[4][8] * D[8][10] + F[4][9] * D[9][10] + F[4][13] * D[10][13]) + F[8][11] * (F[4][6] * D[6][11] + F[4][7] * D[7][11] + F[4][8] * D[8][11] + F[4][9] * D[9][11] + F[4][13] * D[11][13]) + F[8][12] * (F[4][6] * D[6][12] + F[4][7] * D[7][12] + F[4][8] * D[8][12] + F[4][9] * D[9][12] + F[4][13] * D[12][13])) * Tsq + (F[4][6] * D[6][8] + F[4][7] * D[7][8] + F[8][6] * D[4][6] + F[8][7] * D[4][7] + F[4][8] * D[8][8] + F[4][9] * D[8][9] + F[8][9] * D[4][9] + F[8][10] * D[4][10] + F[8][11] * D[4][11] + F[8][12] * D[4][12] + F[4][13] * D[8][13]) * T + D[4][8];
P[4][9] = P[9][4] = (F[9][6] * (F[4][6] * D[6][6] + F[4][7] * D[6][7] + F[4][8] * D[6][8] + F[4][9] * D[6][9] + F[4][13] * D[6][13]) + F[9][7] * (F[4][6] * D[6][7] + F[4][7] * D[7][7] + F[4][8] * D[7][8] + F[4][9] * D[7][9] + F[4][13] * D[7][13]) + F[9][8] * (F[4][6] * D[6][8] + F[4][7] * D[7][8] + F[4][8] * D[8][8] + F[4][9] * D[8][9] + F[4][13] * D[8][13]) + F[9][10] * (F[4][6] * D[6][10] + F[4][7] * D[7][10] + F[4][8] * D[8][10] + F[4][9] * D[9][10] + F[4][13] * D[10][13]) + F[9][11] * (F[4][6] * D[6][11] + F[4][7] * D[7][11] + F[4][8] * D[8][11] + F[4][9] * D[9][11] + F[4][13] * D[11][13]) + F[9][12] * (F[4][6] * D[6][12] + F[4][7] * D[7][12] + F[4][8] * D[8][12] + F[4][9] * D[9][12] + F[4][13] * D[12][13])) * Tsq + (F[9][6] * D[4][6] + F[9][7] * D[4][7] + F[9][8] * D[4][8] + F[4][6] * D[6][9] + F[4][7] * D[7][9] + F[4][8] * D[8][9] + F[4][9] * D[9][9] + F[9][10] * D[4][10] + F[9][11] * D[4][11] + F[9][12] * D[4][12] + F[4][13] * D[9][13]) * T + D[4][9];
P[4][10] = P[10][4] = (F[4][6] * D[6][10] + F[4][7] * D[7][10] + F[4][8] * D[8][10] + F[4][9] * D[9][10] + F[4][13] * D[10][13]) * T + D[4][10];
P[4][11] = P[11][4] = (F[4][6] * D[6][11] + F[4][7] * D[7][11] + F[4][8] * D[8][11] + F[4][9] * D[9][11] + F[4][13] * D[11][13]) * T + D[4][11];
P[4][12] = P[12][4] = (F[4][6] * D[6][12] + F[4][7] * D[7][12] + F[4][8] * D[8][12] + F[4][9] * D[9][12] + F[4][13] * D[12][13]) * T + D[4][12];
P[4][13] = P[13][4] = (F[4][6] * D[6][13] + F[4][7] * D[7][13] + F[4][8] * D[8][13] + F[4][9] * D[9][13] + F[4][13] * D[13][13]) * T + D[4][13];
P[5][5] = (Q[3] * G[5][3] * G[5][3] + Q[4] * G[5][4] * G[5][4] + Q[5] * G[5][5] * G[5][5] + F[5][6] * (F[5][6] * D[6][6] + F[5][7] * D[6][7] + F[5][8] * D[6][8] + F[5][9] * D[6][9] + F[5][13] * D[6][13]) + F[5][7] * (F[5][6] * D[6][7] + F[5][7] * D[7][7] + F[5][8] * D[7][8] + F[5][9] * D[7][9] + F[5][13] * D[7][13]) + F[5][8] * (F[5][6] * D[6][8] + F[5][7] * D[7][8] + F[5][8] * D[8][8] + F[5][9] * D[8][9] + F[5][13] * D[8][13]) + F[5][9] * (F[5][6] * D[6][9] + F[5][7] * D[7][9] + F[5][8] * D[8][9] + F[5][9] * D[9][9] + F[5][13] * D[9][13]) + F[5][13] * (F[5][6] * D[6][13] + F[5][7] * D[7][13] + F[5][8] * D[8][13] + F[5][9] * D[9][13] + F[5][13] * D[13][13])) * Tsq + (2 * F[5][6] * D[5][6] + 2 * F[5][7] * D[5][7] + 2 * F[5][8] * D[5][8] + 2 * F[5][9] * D[5][9] + 2 * F[5][13] * D[5][13]) * T + D[5][5];
P[5][6] = P[6][5] = (F[6][7] * (F[5][6] * D[6][7] + F[5][7] * D[7][7] + F[5][8] * D[7][8] + F[5][9] * D[7][9] + F[5][13] * D[7][13]) + F[6][8] * (F[5][6] * D[6][8] + F[5][7] * D[7][8] + F[5][8] * D[8][8] + F[5][9] * D[8][9] + F[5][13] * D[8][13]) + F[6][9] * (F[5][6] * D[6][9] + F[5][7] * D[7][9] + F[5][8] * D[8][9] + F[5][9] * D[9][9] + F[5][13] * D[9][13]) + F[6][10] * (F[5][6] * D[6][10] + F[5][7] * D[7][10] + F[5][8] * D[8][10] + F[5][9] * D[9][10] + F[5][13] * D[10][13]) + F[6][11] * (F[5][6] * D[6][11] + F[5][7] * D[7][11] + F[5][8] * D[8][11] + F[5][9] * D[9][11] + F[5][13] * D[11][13]) + F[6][12] * (F[5][6] * D[6][12] + F[5][7] * D[7][12] + F[5][8] * D[8][12] + F[5][9] * D[9][12] + F[5][13] * D[12][13])) * Tsq + (F[5][6] * D[6][6] + F[5][7] * D[6][7] + F[6][7] * D[5][7] + F[5][8] * D[6][8] + F[6][8] * D[5][8] + F[5][9] * D[6][9] + F[6][9] * D[5][9] + F[6][10] * D[5][10] + F[6][11] * D[5][11] + F[6][12] * D[5][12] + F[5][13] * D[6][13]) * T + D[5][6];
P[5][7] = P[7][5] = (F[7][6] * (F[5][6] * D[6][6] + F[5][7] * D[6][7] + F[5][8] * D[6][8] + F[5][9] * D[6][9] + F[5][13] * D[6][13]) + F[7][8] * (F[5][6] * D[6][8] + F[5][7] * D[7][8] + F[5][8] * D[8][8] + F[5][9] * D[8][9] + F[5][13] * D[8][13]) + F[7][9] * (F[5][6] * D[6][9] + F[5][7] * D[7][9] + F[5][8] * D[8][9] + F[5][9] * D[9][9] + F[5][13] * D[9][13]) + F[7][10] * (F[5][6] * D[6][10] + F[5][7] * D[7][10] + F[5][8] * D[8][10] + F[5][9] * D[9][10] + F[5][13] * D[10][13]) + F[7][11] * (F[5][6] * D[6][11] + F[5][7] * D[7][11] + F[5][8] * D[8][11] + F[5][9] * D[9][11] + F[5][13] * D[11][13]) + F[7][12] * (F[5][6] * D[6][12] + F[5][7] * D[7][12] + F[5][8] * D[8][12] + F[5][9] * D[9][12] + F[5][13] * D[12][13])) * Tsq + (F[5][6] * D[6][7] + F[7][6] * D[5][6] + F[5][7] * D[7][7] + F[5][8] * D[7][8] + F[7][8] * D[5][8] + F[5][9] * D[7][9] + F[7][9] * D[5][9] + F[7][10] * D[5][10] + F[7][11] * D[5][11] + F[7][12] * D[5][12] + F[5][13] * D[7][13]) * T + D[5][7];
P[5][8] = P[8][5] = (F[8][6] * (F[5][6] * D[6][6] + F[5][7] * D[6][7] + F[5][8] * D[6][8] + F[5][9] * D[6][9] + F[5][13] * D[6][13]) + F[8][7] * (F[5][6] * D[6][7] + F[5][7] * D[7][7] + F[5][8] * D[7][8] + F[5][9] * D[7][9] + F[5][13] * D[7][13]) + F[8][9] * (F[5][6] * D[6][9] + F[5][7] * D[7][9] + F[5][8] * D[8][9] + F[5][9] * D[9][9] + F[5][13] * D[9][13]) + F[8][10] * (F[5][6] * D[6][10] + F[5][7] * D[7][10] + F[5][8] * D[8][10] + F[5][9] * D[9][10] + F[5][13] * D[10][13]) + F[8][11] * (F[5][6] * D[6][11] + F[5][7] * D[7][11] + F[5][8] * D[8][11] + F[5][9] * D[9][11] + F[5][13] * D[11][13]) + F[8][12] * (F[5][6] * D[6][12] + F[5][7] * D[7][12] + F[5][8] * D[8][12] + F[5][9] * D[9][12] + F[5][13] * D[12][13])) * Tsq + (F[5][6] * D[6][8] + F[5][7] * D[7][8] + F[8][6] * D[5][6] + F[8][7] * D[5][7] + F[5][8] * D[8][8] + F[5][9] * D[8][9] + F[8][9] * D[5][9] + F[8][10] * D[5][10] + F[8][11] * D[5][11] + F[8][12] * D[5][12] + F[5][13] * D[8][13]) * T + D[5][8];
P[5][9] = P[9][5] = (F[9][6] * (F[5][6] * D[6][6] + F[5][7] * D[6][7] + F[5][8] * D[6][8] + F[5][9] * D[6][9] + F[5][13] * D[6][13]) + F[9][7] * (F[5][6] * D[6][7] + F[5][7] * D[7][7] + F[5][8] * D[7][8] + F[5][9] * D[7][9] + F[5][13] * D[7][13]) + F[9][8] * (F[5][6] * D[6][8] + F[5][7] * D[7][8] + F[5][8] * D[8][8] + F[5][9] * D[8][9] + F[5][13] * D[8][13]) + F[9][10] * (F[5][6] * D[6][10] + F[5][7] * D[7][10] + F[5][8] * D[8][10] + F[5][9] * D[9][10] + F[5][13] * D[10][13]) + F[9][11] * (F[5][6] * D[6][11] + F[5][7] * D[7][11] + F[5][8] * D[8][11] + F[5][9] * D[9][11] + F[5][13] * D[11][13]) + F[9][12] * (F[5][6] * D[6][12] + F[5][7] * D[7][12] + F[5][8] * D[8][12] + F[5][9] * D[9][12] + F[5][13] * D[12][13])) * Tsq + (F[9][6] * D[5][6] + F[9][7] * D[5][7] + F[9][8] * D[5][8] + F[5][6] * D[6][9] + F[5][7] * D[7][9] + F[5][8] * D[8][9] + F[5][9] * D[9][9] + F[9][10] * D[5][10] + F[9][11] * D[5][11] + F[9][12] * D[5][12] + F[5][13] * D[9][13]) * T + D[5][9];
P[5][10] = P[10][5] = (F[5][6] * D[6][10] + F[5][7] * D[7][10] + F[5][8] * D[8][10] + F[5][9] * D[9][10] + F[5][13] * D[10][13]) * T + D[5][10];
P[5][11] = P[11][5] = (F[5][6] * D[6][11] + F[5][7] * D[7][11] + F[5][8] * D[8][11] + F[5][9] * D[9][11] + F[5][13] * D[11][13]) * T + D[5][11];
P[5][12] = P[12][5] = (F[5][6] * D[6][12] + F[5][7] * D[7][12] + F[5][8] * D[8][12] + F[5][9] * D[9][12] + F[5][13] * D[12][13]) * T + D[5][12];
P[5][13] = P[13][5] = (F[5][6] * D[6][13] + F[5][7] * D[7][13] + F[5][8] * D[8][13] + F[5][9] * D[9][13] + F[5][13] * D[13][13]) * T + D[5][13];
P[6][6] = (Q[0] * G[6][0] * G[6][0] + Q[1] * G[6][1] * G[6][1] + Q[2] * G[6][2] * G[6][2] + F[6][7] * (F[6][7] * D[7][7] + F[6][8] * D[7][8] + F[6][9] * D[7][9] + F[6][10] * D[7][10] + F[6][11] * D[7][11] + F[6][12] * D[7][12]) + F[6][8] * (F[6][7] * D[7][8] + F[6][8] * D[8][8] + F[6][9] * D[8][9] + F[6][10] * D[8][10] + F[6][11] * D[8][11] + F[6][12] * D[8][12]) + F[6][9] * (F[6][7] * D[7][9] + F[6][8] * D[8][9] + F[6][9] * D[9][9] + F[6][10] * D[9][10] + F[6][11] * D[9][11] + F[6][12] * D[9][12]) + F[6][10] * (F[6][7] * D[7][10] + F[6][8] * D[8][10] + F[6][9] * D[9][10] + F[6][10] * D[10][10] + F[6][11] * D[10][11] + F[6][12] * D[10][12]) + F[6][11] * (F[6][7] * D[7][11] + F[6][8] * D[8][11] + F[6][9] * D[9][11] + F[6][10] * D[10][11] + F[6][11] * D[11][11] + F[6][12] * D[11][12]) + F[6][12] * (F[6][7] * D[7][12] + F[6][8] * D[8][12] + F[6][9] * D[9][12] + F[6][10] * D[10][12] + F[6][11] * D[11][12] + F[6][12] * D[12][12])) * Tsq + (2 * F[6][7] * D[6][7] + 2 * F[6][8] * D[6][8] + 2 * F[6][9] * D[6][9] + 2 * F[6][10] * D[6][10] + 2 * F[6][11] * D[6][11] + 2 * F[6][12] * D[6][12]) * T + D[6][6];
P[6][7] = P[7][6] = (F[7][6] * (F[6][7] * D[6][7] + F[6][8] * D[6][8] + F[6][9] * D[6][9] + F[6][10] * D[6][10] + F[6][11] * D[6][11] + F[6][12] * D[6][12]) + F[7][8] * (F[6][7] * D[7][8] + F[6][8] * D[8][8] + F[6][9] * D[8][9] + F[6][10] * D[8][10] + F[6][11] * D[8][11] + F[6][12] * D[8][12]) + F[7][9] * (F[6][7] * D[7][9] + F[6][8] * D[8][9] + F[6][9] * D[9][9] + F[6][10] * D[9][10] + F[6][11] * D[9][11] + F[6][12] * D[9][12]) + F[7][10] * (F[6][7] * D[7][10] + F[6][8] * D[8][10] + F[6][9] * D[9][10] + F[6][10] * D[10][10] + F[6][11] * D[10][11] + F[6][12] * D[10][12]) + F[7][11] * (F[6][7] * D[7][11] + F[6][8] * D[8][11] + F[6][9] * D[9][11] + F[6][10] * D[10][11] + F[6][11] * D[11][11] + F[6][12] * D[11][12]) + F[7][12] * (F[6][7] * D[7][12] + F[6][8] * D[8][12] + F[6][9] * D[9][12] + F[6][10] * D[10][12] + F[6][11] * D[11][12] + F[6][12] * D[12][12]) + G[6][0] * G[7][0] * Q[0] + G[6][1] * G[7][1] * Q[1] + G[6][2] * G[7][2] * Q[2]) * Tsq + (F[7][6] * D[6][6] + F[6][7] * D[7][7] + F[6][8] * D[7][8] + F[7][8] * D[6][8] + F[6][9] * D[7][9] + F[7][9] * D[6][9] + F[6][10] * D[7][10] + F[7][10] * D[6][10] + F[6][11] * D[7][11] + F[7][11] * D[6][11] + F[6][12] * D[7][12] + F[7][12] * D[6][12]) * T + D[6][7];
P[6][8] = P[8][6] = (F[8][6] * (F[6][7] * D[6][7] + F[6][8] * D[6][8] + F[6][9] * D[6][9] + F[6][10] * D[6][10] + F[6][11] * D[6][11] + F[6][12] * D[6][12]) + F[8][7] * (F[6][7] * D[7][7] + F[6][8] * D[7][8] + F[6][9] * D[7][9] + F[6][10] * D[7][10] + F[6][11] * D[7][11] + F[6][12] * D[7][12]) + F[8][9] * (F[6][7] * D[7][9] + F[6][8] * D[8][9] + F[6][9] * D[9][9] + F[6][10] * D[9][10] + F[6][11] * D[9][11] + F[6][12] * D[9][12]) + F[8][10] * (F[6][7] * D[7][10] + F[6][8] * D[8][10] + F[6][9] * D[9][10] + F[6][10] * D[10][10] + F[6][11] * D[10][11] + F[6][12] * D[10][12]) + F[8][11] * (F[6][7] * D[7][11] + F[6][8] * D[8][11] + F[6][9] * D[9][11] + F[6][10] * D[10][11] + F[6][11] * D[11][11] + F[6][12] * D[11][12]) + F[8][12] * (F[6][7] * D[7][12] + F[6][8] * D[8][12] + F[6][9] * D[9][12] + F[6][10] * D[10][12] + F[6][11] * D[11][12] + F[6][12] * D[12][12]) + G[6][0] * G[8][0] * Q[0] + G[6][1] * G[8][1] * Q[1] + G[6][2] * G[8][2] * Q[2]) * Tsq + (F[6][7] * D[7][8] + F[8][6] * D[6][6] + F[8][7] * D[6][7] + F[6][8] * D[8][8] + F[6][9] * D[8][9] + F[8][9] * D[6][9] + F[6][10] * D[8][10] + F[8][10] * D[6][10] + F[6][11] * D[8][11] + F[8][11] * D[6][11] + F[6][12] * D[8][12] + F[8][12] * D[6][12]) * T + D[6][8];
P[6][9] = P[9][6] = (F[9][6] * (F[6][7] * D[6][7] + F[6][8] * D[6][8] + F[6][9] * D[6][9] + F[6][10] * D[6][10] + F[6][11] * D[6][11] + F[6][12] * D[6][12]) + F[9][7] * (F[6][7] * D[7][7] + F[6][8] * D[7][8] + F[6][9] * D[7][9] + F[6][10] * D[7][10] + F[6][11] * D[7][11] + F[6][12] * D[7][12]) + F[9][8] * (F[6][7] * D[7][8] + F[6][8] * D[8][8] + F[6][9] * D[8][9] + F[6][10] * D[8][10] + F[6][11] * D[8][11] + F[6][12] * D[8][12]) + F[9][10] * (F[6][7] * D[7][10] + F[6][8] * D[8][10] + F[6][9] * D[9][10] + F[6][10] * D[10][10] + F[6][11] * D[10][11] + F[6][12] * D[10][12]) + F[9][11] * (F[6][7] * D[7][11] + F[6][8] * D[8][11] + F[6][9] * D[9][11] + F[6][10] * D[10][11] + F[6][11] * D[11][11] + F[6][12] * D[11][12]) + F[9][12] * (F[6][7] * D[7][12] + F[6][8] * D[8][12] + F[6][9] * D[9][12] + F[6][10] * D[10][12] + F[6][11] * D[11][12] + F[6][12] * D[12][12]) + G[6][0] * G[9][0] * Q[0] + G[6][1] * G[9][1] * Q[1] + G[6][2] * G[9][2] * Q[2]) * Tsq + (F[9][6] * D[6][6] + F[9][7] * D[6][7] + F[9][8] * D[6][8] + F[6][7] * D[7][9] + F[6][8] * D[8][9] + F[6][9] * D[9][9] + F[6][10] * D[9][10] + F[9][10] * D[6][10] + F[6][11] * D[9][11] + F[9][11] * D[6][11] + F[6][12] * D[9][12] + F[9][12] * D[6][12]) * T + D[6][9];
P[6][10] = P[10][6] = (F[6][7] * D[7][10] + F[6][8] * D[8][10] + F[6][9] * D[9][10] + F[6][10] * D[10][10] + F[6][11] * D[10][11] + F[6][12] * D[10][12]) * T + D[6][10];
P[6][11] = P[11][6] = (F[6][7] * D[7][11] + F[6][8] * D[8][11] + F[6][9] * D[9][11] + F[6][10] * D[10][11] + F[6][11] * D[11][11] + F[6][12] * D[11][12]) * T + D[6][11];
P[6][12] = P[12][6] = (F[6][7] * D[7][12] + F[6][8] * D[8][12] + F[6][9] * D[9][12] + F[6][10] * D[10][12] + F[6][11] * D[11][12] + F[6][12] * D[12][12]) * T + D[6][12];
P[6][13] = P[13][6] = (F[6][7] * D[7][13] + F[6][8] * D[8][13] + F[6][9] * D[9][13] + F[6][10] * D[10][13] + F[6][11] * D[11][13] + F[6][12] * D[12][13]) * T + D[6][13];
P[7][7] = (Q[0] * G[7][0] * G[7][0] + Q[1] * G[7][1] * G[7][1] + Q[2] * G[7][2] * G[7][2] + F[7][6] * (F[7][6] * D[6][6] + F[7][8] * D[6][8] + F[7][9] * D[6][9] + F[7][10] * D[6][10] + F[7][11] * D[6][11] + F[7][12] * D[6][12]) + F[7][8] * (F[7][6] * D[6][8] + F[7][8] * D[8][8] + F[7][9] * D[8][9] + F[7][10] * D[8][10] + F[7][11] * D[8][11] + F[7][12] * D[8][12]) + F[7][9] * (F[7][6] * D[6][9] + F[7][8] * D[8][9] + F[7][9] * D[9][9] + F[7][10] * D[9][10] + F[7][11] * D[9][11] + F[7][12] * D[9][12]) + F[7][10] * (F[7][6] * D[6][10] + F[7][8] * D[8][10] + F[7][9] * D[9][10] + F[7][10] * D[10][10] + F[7][11] * D[10][11] + F[7][12] * D[10][12]) + F[7][11] * (F[7][6] * D[6][11] + F[7][8] * D[8][11] + F[7][9] * D[9][11] + F[7][10] * D[10][11] + F[7][11] * D[11][11] + F[7][12] * D[11][12]) + F[7][12] * (F[7][6] * D[6][12] + F[7][8] * D[8][12] + F[7][9] * D[9][12] + F[7][10] * D[10][12] + F[7][11] * D[11][12] + F[7][12] * D[12][12])) * Tsq + (2 * F[7][6] * D[6][7] + 2 * F[7][8] * D[7][8] + 2 * F[7][9] * D[7][9] + 2 * F[7][10] * D[7][10] + 2 * F[7][11] * D[7][11] + 2 * F[7][12] * D[7][12]) * T + D[7][7];
P[7][8] = P[8][7] = (F[8][6] * (F[7][6] * D[6][6] + F[7][8] * D[6][8] + F[7][9] * D[6][9] + F[7][10] * D[6][10] + F[7][11] * D[6][11] + F[7][12] * D[6][12]) + F[8][7] * (F[7][6] * D[6][7] + F[7][8] * D[7][8] + F[7][9] * D[7][9] + F[7][10] * D[7][10] + F[7][11] * D[7][11] + F[7][12] * D[7][12]) + F[8][9] * (F[7][6] * D[6][9] + F[7][8] * D[8][9] + F[7][9] * D[9][9] + F[7][10] * D[9][10] + F[7][11] * D[9][11] + F[7][12] * D[9][12]) + F[8][10] * (F[7][6] * D[6][10] + F[7][8] * D[8][10] + F[7][9] * D[9][10] + F[7][10] * D[10][10] + F[7][11] * D[10][11] + F[7][12] * D[10][12]) + F[8][11] * (F[7][6] * D[6][11] + F[7][8] * D[8][11] + F[7][9] * D[9][11] + F[7][10] * D[10][11] + F[7][11] * D[11][11] + F[7][12] * D[11][12]) + F[8][12] * (F[7][6] * D[6][12] + F[7][8] * D[8][12] + F[7][9] * D[9][12] + F[7][10] * D[10][12] + F[7][11] * D[11][12] + F[7][12] * D[12][12]) + G[7][0] * G[8][0] * Q[0] + G[7][1] * G[8][1] * Q[1] + G[7][2] * G[8][2] * Q[2]) * Tsq + (F[7][6] * D[6][8] + F[8][6] * D[6][7] + F[8][7] * D[7][7] + F[7][8] * D[8][8] + F[7][9] * D[8][9] + F[8][9] * D[7][9] + F[7][10] * D[8][10] + F[8][10] * D[7][10] + F[7][11] * D[8][11] + F[8][11] * D[7][11] + F[7][12] * D[8][12] + F[8][12] * D[7][12]) * T + D[7][8];
P[7][9] = P[9][7] = (F[9][6] * (F[7][6] * D[6][6] + F[7][8] * D[6][8] + F[7][9] * D[6][9] + F[7][10] * D[6][10] + F[7][11] * D[6][11] + F[7][12] * D[6][12]) + F[9][7] * (F[7][6] * D[6][7] + F[7][8] * D[7][8] + F[7][9] * D[7][9] + F[7][10] * D[7][10] + F[7][11] * D[7][11] + F[7][12] * D[7][12]) + F[9][8] * (F[7][6] * D[6][8] + F[7][8] * D[8][8] + F[7][9] * D[8][9] + F[7][10] * D[8][10] + F[7][11] * D[8][11] + F[7][12] * D[8][12]) + F[9][10] * (F[7][6] * D[6][10] + F[7][8] * D[8][10] + F[7][9] * D[9][10] + F[7][10] * D[10][10] + F[7][11] * D[10][11] + F[7][12] * D[10][12]) + F[9][11] * (F[7][6] * D[6][11] + F[7][8] * D[8][11] + F[7][9] * D[9][11] + F[7][10] * D[10][11] + F[7][11] * D[11][11] + F[7][12] * D[11][12]) + F[9][12] * (F[7][6] * D[6][12] + F[7][8] * D[8][12] + F[7][9] * D[9][12] + F[7][10] * D[10][12] + F[7][11] * D[11][12] + F[7][12] * D[12][12]) + G[7][0] * G[9][0] * Q[0] + G[7][1] * G[9][1] * Q[1] + G[7][2] * G[9][2] * Q[2]) * Tsq + (F[9][6] * D[6][7] + F[9][7] * D[7][7] + F[9][8] * D[7][8] + F[7][6] * D[6][9] + F[7][8] * D[8][9] + F[7][9] * D[9][9] + F[7][10] * D[9][10] + F[9][10] * D[7][10] + F[7][11] * D[9][11] + F[9][11] * D[7][11] + F[7][12] * D[9][12] + F[9][12] * D[7][12]) * T + D[7][9];
P[7][10] = P[10][7] = (F[7][6] * D[6][10] + F[7][8] * D[8][10] + F[7][9] * D[9][10] + F[7][10] * D[10][10] + F[7][11] * D[10][11] + F[7][12] * D[10][12]) * T + D[7][10];
P[7][11] = P[11][7] = (F[7][6] * D[6][11] + F[7][8] * D[8][11] + F[7][9] * D[9][11] + F[7][10] * D[10][11] + F[7][11] * D[11][11] + F[7][12] * D[11][12]) * T + D[7][11];
P[7][12] = P[12][7] = (F[7][6] * D[6][12] + F[7][8] * D[8][12] + F[7][9] * D[9][12] + F[7][10] * D[10][12] + F[7][11] * D[11][12] + F[7][12] * D[12][12]) * T + D[7][12];
P[7][13] = P[13][7] = (F[7][6] * D[6][13] + F[7][8] * D[8][13] + F[7][9] * D[9][13] + F[7][10] * D[10][13] + F[7][11] * D[11][13] + F[7][12] * D[12][13]) * T + D[7][13];
P[8][8] = (Q[0] * G[8][0] * G[8][0] + Q[1] * G[8][1] * G[8][1] + Q[2] * G[8][2] * G[8][2] + F[8][6] * (F[8][6] * D[6][6] + F[8][7] * D[6][7] + F[8][9] * D[6][9] + F[8][10] * D[6][10] + F[8][11] * D[6][11] + F[8][12] * D[6][12]) + F[8][7] * (F[8][6] * D[6][7] + F[8][7] * D[7][7] + F[8][9] * D[7][9] + F[8][10] * D[7][10] + F[8][11] * D[7][11] + F[8][12] * D[7][12]) + F[8][9] * (F[8][6] * D[6][9] + F[8][7] * D[7][9] + F[8][9] * D[9][9] + F[8][10] * D[9][10] + F[8][11] * D[9][11] + F[8][12] * D[9][12]) + F[8][10] * (F[8][6] * D[6][10] + F[8][7] * D[7][10] + F[8][9] * D[9][10] + F[8][10] * D[10][10] + F[8][11] * D[10][11] + F[8][12] * D[10][12]) + F[8][11] * (F[8][6] * D[6][11] + F[8][7] * D[7][11] + F[8][9] * D[9][11] + F[8][10] * D[10][11] + F[8][11] * D[11][11] + F[8][12] * D[11][12]) + F[8][12] * (F[8][6] * D[6][12] + F[8][7] * D[7][12] + F[8][9] * D[9][12] + F[8][10] * D[10][12] + F[8][11] * D[11][12] + F[8][12] * D[12][12])) * Tsq + (2 * F[8][6] * D[6][8] + 2 * F[8][7] * D[7][8] + 2 * F[8][9] * D[8][9] + 2 * F[8][10] * D[8][10] + 2 * F[8][11] * D[8][11] + 2 * F[8][12] * D[8][12]) * T + D[8][8];
P[8][9] = P[9][8] = (F[9][6] * (F[8][6] * D[6][6] + F[8][7] * D[6][7] + F[8][9] * D[6][9] + F[8][10] * D[6][10] + F[8][11] * D[6][11] + F[8][12] * D[6][12]) + F[9][7] * (F[8][6] * D[6][7] + F[8][7] * D[7][7] + F[8][9] * D[7][9] + F[8][10] * D[7][10] + F[8][11] * D[7][11] + F[8][12] * D[7][12]) + F[9][8] * (F[8][6] * D[6][8] + F[8][7] * D[7][8] + F[8][9] * D[8][9] + F[8][10] * D[8][10] + F[8][11] * D[8][11] + F[8][12] * D[8][12]) + F[9][10] * (F[8][6] * D[6][10] + F[8][7] * D[7][10] + F[8][9] * D[9][10] + F[8][10] * D[10][10] + F[8][11] * D[10][11] + F[8][12] * D[10][12]) + F[9][11] * (F[8][6] * D[6][11] + F[8][7] * D[7][11] + F[8][9] * D[9][11] + F[8][10] * D[10][11] + F[8][11] * D[11][11] + F[8][12] * D[11][12]) + F[9][12] * (F[8][6] * D[6][12] + F[8][7] * D[7][12] + F[8][9] * D[9][12] + F[8][10] * D[10][12] + F[8][11] * D[11][12] + F[8][12] * D[12][12]) + G[8][0] * G[9][0] * Q[0] + G[8][1] * G[9][1] * Q[1] + G[8][2] * G[9][2] * Q[2]) * Tsq + (F[9][6] * D[6][8] + F[9][7] * D[7][8] + F[9][8] * D[8][8] + F[8][6] * D[6][9] + F[8][7] * D[7][9] + F[8][9] * D[9][9] + F[8][10] * D[9][10] + F[9][10] * D[8][10] + F[8][11] * D[9][11] + F[9][11] * D[8][11] + F[8][12] * D[9][12] + F[9][12] * D[8][12]) * T + D[8][9];
P[8][10] = P[10][8] = (F[8][6] * D[6][10] + F[8][7] * D[7][10] + F[8][9] * D[9][10] + F[8][10] * D[10][10] + F[8][11] * D[10][11] + F[8][12] * D[10][12]) * T + D[8][10];
P[8][11] = P[11][8] = (F[8][6] * D[6][11] + F[8][7] * D[7][11] + F[8][9] * D[9][11] + F[8][10] * D[10][11] + F[8][11] * D[11][11] + F[8][12] * D[11][12]) * T + D[8][11];
P[8][12] = P[12][8] = (F[8][6] * D[6][12] + F[8][7] * D[7][12] + F[8][9] * D[9][12] + F[8][10] * D[10][12] + F[8][11] * D[11][12] + F[8][12] * D[12][12]) * T + D[8][12];
P[8][13] = P[13][8] = (F[8][6] * D[6][13] + F[8][7] * D[7][13] + F[8][9] * D[9][13] + F[8][10] * D[10][13] + F[8][11] * D[11][13] + F[8][12] * D[12][13]) * T + D[8][13];
P[9][9] = (Q[0] * G[9][0] * G[9][0] + Q[1] * G[9][1] * G[9][1] + Q[2] * G[9][2] * G[9][2] + F[9][6] * (F[9][6] * D[6][6] + F[9][7] * D[6][7] + F[9][8] * D[6][8] + F[9][10] * D[6][10] + F[9][11] * D[6][11] + F[9][12] * D[6][12]) + F[9][7] * (F[9][6] * D[6][7] + F[9][7] * D[7][7] + F[9][8] * D[7][8] + F[9][10] * D[7][10] + F[9][11] * D[7][11] + F[9][12] * D[7][12]) + F[9][8] * (F[9][6] * D[6][8] + F[9][7] * D[7][8] + F[9][8] * D[8][8] + F[9][10] * D[8][10] + F[9][11] * D[8][11] + F[9][12] * D[8][12]) + F[9][10] * (F[9][6] * D[6][10] + F[9][7] * D[7][10] + F[9][8] * D[8][10] + F[9][10] * D[10][10] + F[9][11] * D[10][11] + F[9][12] * D[10][12]) + F[9][11] * (F[9][6] * D[6][11] + F[9][7] * D[7][11] + F[9][8] * D[8][11] + F[9][10] * D[10][11] + F[9][11] * D[11][11] + F[9][12] * D[11][12]) + F[9][12] * (F[9][6] * D[6][12] + F[9][7] * D[7][12] + F[9][8] * D[8][12] + F[9][10] * D[10][12] + F[9][11] * D[11][12] + F[9][12] * D[12][12])) * Tsq + (2 * F[9][6] * D[6][9] + 2 * F[9][7] * D[7][9] + 2 * F[9][8] * D[8][9] + 2 * F[9][10] * D[9][10] + 2 * F[9][11] * D[9][11] + 2 * F[9][12] * D[9][12]) * T + D[9][9];
P[9][10] = P[10][9] = (F[9][6] * D[6][10] + F[9][7] * D[7][10] + F[9][8] * D[8][10] + F[9][10] * D[10][10] + F[9][11] * D[10][11] + F[9][12] * D[10][12]) * T + D[9][10];
P[9][11] = P[11][9] = (F[9][6] * D[6][11] + F[9][7] * D[7][11] + F[9][8] * D[8][11] + F[9][10] * D[10][11] + F[9][11] * D[11][11] + F[9][12] * D[11][12]) * T + D[9][11];
P[9][12] = P[12][9] = (F[9][6] * D[6][12] + F[9][7] * D[7][12] + F[9][8] * D[8][12] + F[9][10] * D[10][12] + F[9][11] * D[11][12] + F[9][12] * D[12][12]) * T + D[9][12];
P[9][13] = P[13][9] = (F[9][6] * D[6][13] + F[9][7] * D[7][13] + F[9][8] * D[8][13] + F[9][10] * D[10][13] + F[9][11] * D[11][13] + F[9][12] * D[12][13]) * T + D[9][13];
P[10][10] = Q[6] * Tsq + D[10][10];
P[10][11] = P[11][10] = D[10][11];
P[10][12] = P[12][10] = D[10][12];
P[10][13] = P[13][10] = D[10][13];
P[11][11] = Q[7] * Tsq + D[11][11];
P[11][12] = P[12][11] = D[11][12];
P[11][13] = P[13][11] = D[11][13];
P[12][12] = Q[8] * Tsq + D[12][12];
P[12][13] = P[13][12] = D[12][13];
P[13][13] = Q[9] * Tsq + D[13][13];
}
#endif /* ifdef COVARIANCE_PREDICTION_GENERAL */
// ************* SerialUpdate *******************
// Does the update step of the Kalman filter for the covariance and estimate
// Outputs are Xnew & Pnew, and are written over P and X
// Z is actual measurement, Y is predicted measurement
// Xnew = X + K*(Z-Y), Pnew=(I-K*H)*P,
// where K=P*H'*inv[H*P*H'+R]
// NOTE the algorithm assumes R (measurement covariance matrix) is diagonal
// i.e. the measurment noises are uncorrelated.
// It therefore uses a serial update that requires no matrix inversion by
// processing the measurements one at a time.
// Algorithm - see Grewal and Andrews, "Kalman Filtering,2nd Ed" p.121 & p.253
// - or see Simon, "Optimal State Estimation," 1st Ed, p.150
// The SensorsUsed variable is a bitwise mask indicating which sensors
// should be used in the update.
// ************************************************
void SerialUpdate(float H[NUMV][NUMX], float R[NUMV], float Z[NUMV],
float Y[NUMV], float P[NUMX][NUMX], float X[NUMX],
uint16_t SensorsUsed)
{
float HP[NUMX], HPHR, Error;
uint8_t i, j, k, m;
// Iterate through all the possible measurements and apply the
// appropriate corrections
for (m = 0; m < NUMV; m++) {
if (SensorsUsed & (0x01 << m)) { // use this sensor for update
for (j = 0; j < NUMX; j++) { // Find Hp = H*P
HP[j] = 0.0f;
for (k = 0; k < NUMX; k++) {
HP[j] += H[m][k] * P[k][j];
}
}
HPHR = R[m]; // Find HPHR = H*P*H' + R
for (k = 0; k < NUMX; k++) {
HPHR += HP[k] * H[m][k];
}
for (k = 0; k < NUMX; k++) {
ekf.K[k][m] = HP[k] / HPHR; // find K = HP/HPHR
}
for (i = 0; i < NUMX; i++) { // Find P(m)= P(m-1) + K*HP
for (j = i; j < NUMX; j++) {
P[i][j] = P[j][i] =
P[i][j] - ekf.K[i][m] * HP[j];
}
}
Error = Z[m] - Y[m];
for (i = 0; i < NUMX; i++) { // Find X(m)= X(m-1) + K*Error
X[i] = X[i] + ekf.K[i][m] * Error;
}
}
}
INSLimitBias();
}
// ************* RungeKutta **********************
// Does a 4th order Runge Kutta numerical integration step
// Output, Xnew, is written over X
// NOTE the algorithm assumes time invariant state equations and
// constant inputs over integration step
// ************************************************
void RungeKutta(float X[NUMX], float U[NUMU], float dT)
{
float dT2 =
dT / 2.0f, K1[NUMX], K2[NUMX], K3[NUMX], K4[NUMX], Xlast[NUMX];
uint8_t i;
for (i = 0; i < NUMX; i++) {
Xlast[i] = X[i]; // make a working copy
}
StateEq(X, U, K1); // k1 = f(x,u)
for (i = 0; i < NUMX; i++) {
X[i] = Xlast[i] + dT2 * K1[i];
}
StateEq(X, U, K2); // k2 = f(x+0.5*dT*k1,u)
for (i = 0; i < NUMX; i++) {
X[i] = Xlast[i] + dT2 * K2[i];
}
StateEq(X, U, K3); // k3 = f(x+0.5*dT*k2,u)
for (i = 0; i < NUMX; i++) {
X[i] = Xlast[i] + dT * K3[i];
}
StateEq(X, U, K4); // k4 = f(x+dT*k3,u)
// Xnew = X + dT*(k1+2*k2+2*k3+k4)/6
for (i = 0; i < NUMX; i++) {
X[i] =
Xlast[i] + dT * (K1[i] + 2.0f * K2[i] + 2.0f * K3[i] +
K4[i]) / 6.0f;
}
}
// ************* Model Specific Stuff ***************************
// *** StateEq, MeasurementEq, LinerizeFG, and LinearizeH ********
//
// State Variables = [Pos Vel Quaternion GyroBias AccelBias]
// Deterministic Inputs = [AngularVel Accel]
// Disturbance Noise = [GyroNoise AccelNoise GyroRandomWalkNoise AccelRandomWalkNoise]
//
// Measurement Variables = [Pos Vel BodyFrameMagField Altimeter]
// Inputs to Measurement = [EarthFrameMagField]
//
// Notes: Pos and Vel in earth frame
// AngularVel and Accel in body frame
// MagFields are unit vectors
// Xdot is output of StateEq()
// F and G are outputs of LinearizeFG(), all elements not set should be zero
// y is output of OutputEq()
// H is output of LinearizeH(), all elements not set should be zero
// ************************************************
void StateEq(float X[NUMX], float U[NUMU], float Xdot[NUMX])
{
const float wx = U[0] - X[10];
const float wy = U[1] - X[11];
const float wz = U[2] - X[12]; // subtract the biases on gyros
const float ax = U[3];
const float ay = U[4];
const float az = U[5] - X[13]; // subtract the biases on accels
const float q0 = X[6];
const float q1 = X[7];
const float q2 = X[8];
const float q3 = X[9];
// Pdot = V
Xdot[0] = X[3];
Xdot[1] = X[4];
Xdot[2] = X[5];
// Vdot = Reb*a
Xdot[3] =
(q0 * q0 + q1 * q1 - q2 * q2 - q3 * q3) * ax + 2.0f * (q1 * q2 -
q0 * q3) *
ay + 2.0f * (q1 * q3 + q0 * q2) * az;
Xdot[4] =
2.0f * (q1 * q2 + q0 * q3) * ax + (q0 * q0 - q1 * q1 + q2 * q2 -
q3 * q3) * ay + 2.0f * (q2 * q3 -
q0 * q1) *
az;
Xdot[5] =
2.0f * (q1 * q3 - q0 * q2) * ax + 2.0f * (q2 * q3 + q0 * q1) * ay +
(q0 * q0 - q1 * q1 - q2 * q2 + q3 * q3) * az + PIOS_CONST_MKS_GRAV_ACCEL_F;
// qdot = Q*w
Xdot[6] = (-q1 * wx - q2 * wy - q3 * wz) / 2.0f;
Xdot[7] = (q0 * wx - q3 * wy + q2 * wz) / 2.0f;
Xdot[8] = (q3 * wx + q0 * wy - q1 * wz) / 2.0f;
Xdot[9] = (-q2 * wx + q1 * wy + q0 * wz) / 2.0f;
// best guess is that bias stays constant
Xdot[10] = Xdot[11] = Xdot[12] = 0;
// For accels to make sure things stay stable, assume bias always walks weakly
// towards zero for the horizontal axis. This prevents drifting around an
// unobservable manifold of possible attitudes and gyro biases. The z-axis
// we assume no drift becaues this is teh one we want to estimate most accurately.
Xdot[13] = 0.0f;
}
/**
* Linearize the state equations around the current state estimate.
* @param[in] X the current state estimate
* @param[in] U the control inputs
* @param[out] F the linearized natural dynamics
* @param[out] G the linearized influence of disturbance model
*
* so the prediction of the next state is
* Xdot = F * X + G * U
* where X is the current state and U is the current input
*
* For reference the state order (in F) is pos, vel, attitude, gyro bias, accel bias
* and the input order is gyro, bias
*/
void LinearizeFG(float X[NUMX], float U[NUMU], float F[NUMX][NUMX],
float G[NUMX][NUMW])
{
const float wx = U[0] - X[10];
const float wy = U[1] - X[11];
const float wz = U[2] - X[12]; // subtract the biases on gyros
const float ax = U[3];
const float ay = U[4];
const float az = U[5] - X[13]; // subtract the biases on accels
const float q0 = X[6];
const float q1 = X[7];
const float q2 = X[8];
const float q3 = X[9];
// Pdot = V
F[0][3] = F[1][4] = F[2][5] = 1.0f;
// dVdot/dq
F[3][6] = 2.0f * (q0 * ax - q3 * ay + q2 * az);
F[3][7] = 2.0f * (q1 * ax + q2 * ay + q3 * az);
F[3][8] = 2.0f * (-q2 * ax + q1 * ay + q0 * az);
F[3][9] = 2.0f * (-q3 * ax - q0 * ay + q1 * az);
F[4][6] = 2.0f * (q3 * ax + q0 * ay - q1 * az);
F[4][7] = 2.0f * (q2 * ax - q1 * ay - q0 * az);
F[4][8] = 2.0f * (q1 * ax + q2 * ay + q3 * az);
F[4][9] = 2.0f * (q0 * ax - q3 * ay + q2 * az);
F[5][6] = 2.0f * (-q2 * ax + q1 * ay + q0 * az);
F[5][7] = 2.0f * (q3 * ax + q0 * ay - q1 * az);
F[5][8] = 2.0f * (-q0 * ax + q3 * ay - q2 * az);
F[5][9] = 2.0f * (q1 * ax + q2 * ay + q3 * az);
// dVdot/dabias & dVdot/dna - the equations for how the accel input and accel bias influence velocity are the same
F[3][13] = G[3][5] = -2.0f * (q1 * q3 + q0 * q2);
F[4][13] = G[4][5] = 2.0f * (-q2 * q3 + q0 * q1);
F[5][13] = G[5][5] = -q0 * q0 + q1 * q1 + q2 * q2 - q3 * q3;
// dqdot/dq
F[6][6] = 0;
F[6][7] = -wx / 2.0f;
F[6][8] = -wy / 2.0f;
F[6][9] = -wz / 2.0f;
F[7][6] = wx / 2.0f;
F[7][7] = 0;
F[7][8] = wz / 2.0f;
F[7][9] = -wy / 2.0f;
F[8][6] = wy / 2.0f;
F[8][7] = -wz / 2.0f;
F[8][8] = 0;
F[8][9] = wx / 2.0f;
F[9][6] = wz / 2.0f;
F[9][7] = wy / 2.0f;
F[9][8] = -wx / 2.0f;
F[9][9] = 0;
// dqdot/dwbias
F[6][10] = q1 / 2.0f;
F[6][11] = q2 / 2.0f;
F[6][12] = q3 / 2.0f;
F[7][10] = -q0 / 2.0f;
F[7][11] = q3 / 2.0f;
F[7][12] = -q2 / 2.0f;
F[8][10] = -q3 / 2.0f;
F[8][11] = -q0 / 2.0f;
F[8][12] = q1 / 2.0f;
F[9][10] = q2 / 2.0f;
F[9][11] = -q1 / 2.0f;
F[9][12] = -q0 / 2.0f;
// dVdot/dna
G[3][3] = -q0 * q0 - q1 * q1 + q2 * q2 + q3 * q3; G[3][4] = 2 * (-q1 * q2 + q0 * q3); G[3][5] = -2 * (q1 * q3 + q0 * q2);
G[4][3] = -2 * (q1 * q2 + q0 * q3); G[4][4] = -q0 * q0 + q1 * q1 - q2 * q2 + q3 * q3; G[4][5] = 2 * (-q2 * q3 + q0 * q1);
G[5][3] = 2 * (-q1 * q3 + q0 * q2); G[5][4] = -2 * (q2 * q3 + q0 * q1); G[5][5] = -q0 * q0 + q1 * q1 + q2 * q2 - q3 * q3;
// dqdot/dnw
G[6][0] = q1 / 2.0f;
G[6][1] = q2 / 2.0f;
G[6][2] = q3 / 2.0f;
G[7][0] = -q0 / 2.0f;
G[7][1] = q3 / 2.0f;
G[7][2] = -q2 / 2.0f;
G[8][0] = -q3 / 2.0f;
G[8][1] = -q0 / 2.0f;
G[8][2] = q1 / 2.0f;
G[9][0] = q2 / 2.0f;
G[9][1] = -q1 / 2.0f;
G[9][2] = -q0 / 2.0f;
}
/**
* Predicts the measurements from the current state. Note
* that this is very similar to @ref LinearizeH except this
* directly computes the outputs instead of a matrix that
* you transform the state by
*/
void MeasurementEq(float X[NUMX], float Be[3], float Y[NUMV])
{
const float q0 = X[6];
const float q1 = X[7];
const float q2 = X[8];
const float q3 = X[9];
// first six outputs are P and V
Y[0] = X[0];
Y[1] = X[1];
Y[2] = X[2];
Y[3] = X[3];
Y[4] = X[4];
Y[5] = X[5];
// Rotate Be by only the yaw heading
const float a1 = 2 * q0 * q3 + 2 * q1 * q2;
const float a2 = q0 * q0 + q1 * q1 - q2 * q2 - q3 * q3;
const float r = sqrtf(a1 * a1 + a2 * a2);
const float cP = a2 / r;
const float sP = a1 / r;
Y[6] = Be[0] * cP + Be[1] * sP;
Y[7] = -Be[0] * sP + Be[1] * cP;
Y[8] = 0; // don't care
// Alt = -Pz
Y[9] = X[2] * -1.0f;
}
/**
* Linearize the measurement around the current state estiamte
* so the predicted measurements are
* Z = H * X
*/
void LinearizeH(float X[NUMX], float Be[3], float H[NUMV][NUMX])
{
const float q0 = X[6];
const float q1 = X[7];
const float q2 = X[8];
const float q3 = X[9];
// dP/dP=I; (expect position to measure the position)
H[0][0] = H[1][1] = H[2][2] = 1.0f;
// dV/dV=I; (expect velocity to measure the velocity)
H[3][3] = H[4][4] = H[5][5] = 1.0f;
// dBb/dq (expected magnetometer readings in the horizontal plane)
// these equations were generated by Rhb(q)*Be which is the matrix that
// rotates the earth magnetic field into the horizontal plane, and then
// taking the partial derivative wrt each term in q. Maniuplated in
// matlab symbolic toolbox
const float Be_0 = Be[0];
const float Be_1 = Be[1];
const float a1 = q0 * q3 * 2.0f + q1 * q2 * 2.0f;
const float a1s = a1 * a1;
const float a2 = q0 * q0 + q1 * q1 - q2 * q2 - q3 * q3;
const float a2s = a2 * a2;
const float a3 = 1.0f / powf(a1s + a2s, 3.0f / 2.0f) * (1.0f / 2.0f);
const float k1 = 1.0f / sqrtf(a1s + a2s);
const float k3 = a3 * a2;
const float k4 = a2 * 4.0f;
const float k5 = a1 * 4.0f;
const float k6 = a3 * a1;
H[6][6] = Be_0 * q0 * k1 * 2.0f + Be_1 * q3 * k1 * 2.0f - Be_0 * (q0 * k4 + q3 * k5) * k3 - Be_1 * (q0 * k4 + q3 * k5) * k6;
H[6][7] = Be_0 * q1 * k1 * 2.0f + Be_1 * q2 * k1 * 2.0f - Be_0 * (q1 * k4 + q2 * k5) * k3 - Be_1 * (q1 * k4 + q2 * k5) * k6;
H[6][8] = Be_0 * q2 * k1 * -2.0f + Be_1 * q1 * k1 * 2.0f + Be_0 * (q2 * k4 - q1 * k5) * k3 + Be_1 * (q2 * k4 - q1 * k5) * k6;
H[6][9] = Be_1 * q0 * k1 * 2.0f - Be_0 * q3 * k1 * 2.0f + Be_0 * (q3 * k4 - q0 * k5) * k3 + Be_1 * (q3 * k4 - q0 * k5) * k6;
H[7][6] = Be_1 * q0 * k1 * 2.0f - Be_0 * q3 * k1 * 2.0f - Be_1 * (q0 * k4 + q3 * k5) * k3 + Be_0 * (q0 * k4 + q3 * k5) * k6;
H[7][7] = Be_0 * q2 * k1 * -2.0f + Be_1 * q1 * k1 * 2.0f - Be_1 * (q1 * k4 + q2 * k5) * k3 + Be_0 * (q1 * k4 + q2 * k5) * k6;
H[7][8] = Be_0 * q1 * k1 * -2.0f - Be_1 * q2 * k1 * 2.0f + Be_1 * (q2 * k4 - q1 * k5) * k3 - Be_0 * (q2 * k4 - q1 * k5) * k6;
H[7][9] = Be_0 * q0 * k1 * -2.0f - Be_1 * q3 * k1 * 2.0f + Be_1 * (q3 * k4 - q0 * k5) * k3 - Be_0 * (q3 * k4 - q0 * k5) * k6;
H[8][6] = 0.0f;
H[8][7] = 0.0f;
H[8][9] = 0.0f;
// dAlt/dPz = -1 (expected baro readings)
H[9][2] = -1.0f;
}
/**
* @}
* @}
*/

11
flight/modules/StateEstimation/filterekf.c
View File

@ -202,8 +202,8 @@ static filterResult filter(stateFilter *self, stateEstimation *state)
float dT;
uint16_t sensors = 0;
this->work.updated |= state->updated;
this->work.updated |= state->updated;
// check magnetometer alarm, discard any magnetometer readings if not OK
// during initialization phase (but let them through afterwards)
SystemAlarmsAlarmData alarms;
@ -241,10 +241,9 @@ static filterResult filter(stateFilter *self, stateEstimation *state)
// Reset the INS algorithm
INSGPSInit();
// variance is measured in mGaus, but internally the EKF works with a normalized vector. Scale down by Be^2
float Be2 = this->homeLocation.Be[0] * this->homeLocation.Be[0] + this->homeLocation.Be[1] * this->homeLocation.Be[1] + this->homeLocation.Be[2] * this->homeLocation.Be[2];
INSSetMagVar((float[3]) { this->ekfConfiguration.R.MagX / Be2,
this->ekfConfiguration.R.MagY / Be2,
this->ekfConfiguration.R.MagZ / Be2 }
INSSetMagVar((float[3]) { this->ekfConfiguration.R.MagX,
this->ekfConfiguration.R.MagY,
this->ekfConfiguration.R.MagZ }
);
INSSetAccelVar((float[3]) { this->ekfConfiguration.Q.AccelX,
this->ekfConfiguration.Q.AccelY,
@ -421,7 +420,7 @@ static filterResult filter(stateFilter *self, stateEstimation *state)
EKFStateVarianceData vardata;
EKFStateVarianceGet(&vardata);
INSGetP(EKFStateVariancePToArray(vardata.P));
INSGetVariance(EKFStateVariancePToArray(vardata.P));
EKFStateVarianceSet(&vardata);
int t;
for (t = 0; t < EKFSTATEVARIANCE_P_NUMELEM; t++) {

2
flight/targets/boards/revolution/firmware/Makefile
View File

@ -95,7 +95,7 @@ ifndef TESTAPP
SRC += $(FLIGHTLIB)/paths.c
SRC += $(FLIGHTLIB)/plans.c
SRC += $(FLIGHTLIB)/WorldMagModel.c
SRC += $(FLIGHTLIB)/insgps13state.c
SRC += $(FLIGHTLIB)/insgps14state.c
SRC += $(FLIGHTLIB)/auxmagsupport.c
SRC += $(FLIGHTLIB)/lednotification.c
SRC += $(FLIGHTLIB)/sha1.c

2
flight/targets/boards/sparky2/firmware/Makefile
View File

@ -95,7 +95,7 @@ ifndef TESTAPP
SRC += $(FLIGHTLIB)/paths.c
SRC += $(FLIGHTLIB)/plans.c
SRC += $(FLIGHTLIB)/WorldMagModel.c
SRC += $(FLIGHTLIB)/insgps13state.c
SRC += $(FLIGHTLIB)/insgps14state.c
SRC += $(FLIGHTLIB)/auxmagsupport.c
SRC += $(FLIGHTLIB)/lednotification.c
SRC += $(FLIGHTLIB)/sha1.c