/**
******************************************************************************
*
* @file insgps.c
* @author The OpenPilot Team, http://www.openpilot.org Copyright (C) 2010.
* @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>
// constants/macros/typdefs
#define NUMX 13 // number of states, X is the state vector
#define NUMW 9 // 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
// 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
float F[NUMX][NUMX], G[NUMX][NUMW], 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], X[NUMX]; // covariance matrix and state vector
float Q[NUMW], R[NUMV]; // input noise and measurement noise variances
// ************* Exposed Functions ****************
// *************************************************
void INSGPSInit() //pretty much just a place holder for now
{
Be[0]=1; Be[1]=0; Be[2]=0; // local magnetic unit vector
P[0][0]=P[1][1]=P[2][2]=25; // initial position variance (m^2)
P[3][3]=P[4][4]=P[5][5]=5; // initial velocity variance (m/s)^2
P[6][6]=P[7][7]=P[8][8]=P[9][9]=1e-5; // initial quaternion variance
P[10][10]=P[11][11]=P[12][12]=1e-5; // initial gyro bias variance (rad/s)^2
X[0]=X[1]=X[2]=X[3]=X[4]=X[5]=0; // initial pos and vel (m)
X[6]=1; X[7]=X[8]=X[9]=0; // initial quaternion (level and North) (m/s)
X[10]=X[11]=X[12]=0; // initial gyro bias (rad/s)
Q[0]=Q[1]=Q[2]=50e-8; // gyro noise variance (rad/s)^2
Q[3]=Q[4]=Q[5]=0.01; // accelerometer noise variance (m/s^2)^2
Q[6]=Q[7]=Q[8]=2e-7; // gyro bias random walk variance (rad/s^2)^2
R[0]=R[1]=0.004; // High freq GPS horizontal position noise variance (m^2)
R[2]=0.036; // High freq GPS vertical position noise variance (m^2)
R[3]=R[4]=0.004; // High freq GPS horizontal velocity noise variance (m/s)^2
R[5]=0; // High freq GPS vertical velocity noise variance (m/s)^2
R[6]=R[7]=R[8]=0.005; // magnetometer unit vector noise variance
R[9]=1; // High freq altimeter noise variance (m^2)
}
void INSSetGyroBias(float gyro_bias[3])
{
X[10] = gyro_bias[0];
X[11] = gyro_bias[1];
X[12] = gyro_bias[2];
}
void INSSetAccelVar(float accel_var[3])
{
Q[3] = accel_var[0];
Q[4] = accel_var[1];
Q[5] = accel_var[2];
}
void INSSetGyroVar(float gyro_var[3])
{
Q[0] = gyro_var[0];
Q[1] = gyro_var[1];
Q[2] = gyro_var[2];
}
void INSSetMagVar(float scaled_mag_var[3])
{
R[6] = scaled_mag_var[0];
R[7] = scaled_mag_var[1];
R[8] = scaled_mag_var[2];
}
void INSSetMagNorth(float B[3])
{
Be[0] = B[0];
Be[1] = B[1];
Be[2] = B[2];
}
void INSPrediction(float gyro_data[3], 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(X,U,F,G);
RungeKutta(X,U,dT);
qmag=sqrt(X[6]*X[6] + X[7]*X[7] + X[8]*X[8] + X[9]*X[9]);
X[6] /= qmag; X[7] /= qmag; X[8] /= qmag; X[9] /= qmag;
CovariancePrediction(F,G,Q,dT,P);
// Update Nav solution structure
Nav.Pos[0] = X[0];
Nav.Pos[1] = X[1];
Nav.Pos[2] = X[2];
Nav.Vel[0] = X[3];
Nav.Vel[1] = X[4];
Nav.Vel[2] = X[5];
Nav.q[0] = X[6];
Nav.q[1] = X[7];
Nav.q[2] = X[8];
Nav.q[3] = X[9];
}
void MagCorrection(float mag_data[3])
{
float Z[10], Y[10];
float Bmag, qmag;
// magnetometer data in any units (use unit vector) and in body frame
Bmag = sqrt(mag_data[0]*mag_data[0] + mag_data[1]*mag_data[1] + mag_data[2]*mag_data[2]);
Z[6] = mag_data[0]/Bmag;
Z[7] = mag_data[1]/Bmag;
Z[8] = mag_data[2]/Bmag;
// EKF correction step
LinearizeH(X,Be,H);
MeasurementEq(X,Be,Y);
SerialUpdate(H,R,Z,Y,P,X,MagSensors);
qmag=sqrt(X[6]*X[6] + X[7]*X[7] + X[8]*X[8] + X[9]*X[9]);
X[6] /= qmag; X[7] /= qmag; X[8] /= qmag; X[9] /= qmag;
// Update Nav solution structure
Nav.Pos[0] = X[0];
Nav.Pos[1] = X[1];
Nav.Pos[2] = X[2];
Nav.Vel[0] = X[3];
Nav.Vel[1] = X[4];
Nav.Vel[2] = X[5];
Nav.q[0] = X[6];
Nav.q[1] = X[7];
Nav.q[2] = X[8];
Nav.q[3] = X[9];
}
void FullCorrection(float mag_data[3], float Pos[3], float Vel[3], float BaroAlt)
{
float Z[10], Y[10];
float Bmag, 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];
// magnetometer data in any units (use unit vector) and in body frame
Bmag = sqrt(mag_data[0]*mag_data[0] + mag_data[1]*mag_data[1] + mag_data[2]*mag_data[2]);
Z[6] = mag_data[0]/Bmag;
Z[7] = mag_data[1]/Bmag;
Z[8] = mag_data[2]/Bmag;
// barometric altimeter in meters and in local NED frame
Z[9] = BaroAlt;
// EKF correction step
LinearizeH(X,Be,H);
MeasurementEq(X,Be,Y);
SerialUpdate(H,R,Z,Y,P,X,FullSensors);
qmag=sqrt(X[6]*X[6] + X[7]*X[7] + X[8]*X[8] + X[9]*X[9]);
X[6] /= qmag; X[7] /= qmag; X[8] /= qmag; X[9] /= qmag;
// Update Nav solution structure
Nav.Pos[0] = X[0];
Nav.Pos[1] = X[1];
Nav.Pos[2] = X[2];
Nav.Vel[0] = X[3];
Nav.Vel[1] = X[4];
Nav.Vel[2] = X[5];
Nav.q[0] = X[6];
Nav.q[1] = X[7];
Nav.q[2] = X[8];
Nav.q[3] = X[9];
}
void GndSpeedAndMagCorrection(float Speed, float Heading, float mag_data[3])
{
float Z[10], Y[10];
float Bmag, qmag;
// Ground Speed in m/s and Heading in rad
Z[3] = Speed*cos((double)Heading);
Z[4] = Speed*sin((double)Heading);
// magnetometer data in any units (use unit vector) and in body frame
Bmag = sqrt(mag_data[0]*mag_data[0] + mag_data[1]*mag_data[1] + mag_data[2]*mag_data[2]);
Z[6] = mag_data[0]/Bmag;
Z[7] = mag_data[1]/Bmag;
Z[8] = mag_data[2]/Bmag;
// EKF correction step
LinearizeH(X,Be,H);
MeasurementEq(X,Be,Y);
SerialUpdate(H,R,Z,Y,P,X,GndSpeedAndMagSensors);
qmag=sqrt(X[6]*X[6] + X[7]*X[7] + X[8]*X[8] + X[9]*X[9]);
X[6] /= qmag; X[7] /= qmag; X[8] /= qmag; X[9] /= qmag;
// Update Nav solution structure
Nav.Pos[0] = X[0];
Nav.Pos[1] = X[1];
Nav.Pos[2] = X[2];
Nav.Vel[0] = X[3];
Nav.Vel[1] = X[4];
Nav.Vel[2] = X[5];
Nav.q[0] = X[6];
Nav.q[1] = X[7];
Nav.q[2] = X[8];
Nav.q[3] = X[9];
}
// ************* 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
// Could be much more efficient using the sparse, block structure of F and G
// ************************************************
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;
}
}
// ************* 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], K[NUMX], HPHR, Error;
uint8_t i,j,k,m;
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;
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++) K[k] = 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] - K[i]*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] + K[i]*Error;
}
}
}
// ************* 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, 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*K2[i]+2*K3[i]+K4[i])/6;
}
// ************* Model Specific Stuff ***************************
// *** StateEq, MeasurementEq, LinerizeFG, and LinearizeH ********
//
// State Variables = [Pos Vel Quaternion GyroBias NO-AccelBias]
// Deterministic Inputs = [AngularVel Accel]
// Disturbance Noise = [GyroNoise AccelNoise GyroRandomWalkNoise NO-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]){
float ax, ay, az, wx, wy, wz, q0, q1, q2, q3;
// ax=U[3]-X[13]; ay=U[4]-X[14]; az=U[5]-X[15]; // subtract the biases on accels
ax=U[3]; ay=U[4]; az=U[5]; // NO BIAS STATES ON ACCELS
wx=U[0]-X[10]; wy=U[1]-X[11]; wz=U[2]-X[12]; // subtract the biases on gyros
q0=X[6]; q1=X[7]; q2=X[8]; 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*(q1*q2-q0*q3)*ay + 2*(q1*q3+q0*q2)*az;
Xdot[4]=2*(q1*q2+q0*q3)*ax + (q0*q0-q1*q1+q2*q2-q3*q3)*ay + 2*(q2*q3-q0*q1)*az;
Xdot[5]=2*(q1*q3-q0*q2)*ax + 2*(q2*q3+q0*q1)*ay + (q0*q0-q1*q1-q2*q2+q3*q3)*az + 9.81;
// qdot = Q*w
Xdot[6] = (-q1*wx-q2*wy-q3*wz)/2;
Xdot[7] = (q0*wx-q3*wy+q2*wz)/2;
Xdot[8] = (q3*wx+q0*wy-q1*wz)/2;
Xdot[9] = (-q2*wx+q1*wy+q0*wz)/2;
// best guess is that bias stays constant
Xdot[10]=Xdot[11]=Xdot[12]=0;
}
void LinearizeFG(float X[NUMX],float U[NUMU], float F[NUMX][NUMX], float G[NUMX][NUMW]){
float ax, ay, az, wx, wy, wz, q0, q1, q2, q3;
// ax=U[3]-X[13]; ay=U[4]-X[14]; az=U[5]-X[15]; // subtract the biases on accels
ax=U[3]; ay=U[4]; az=U[5]; // NO BIAS STATES ON ACCELS
wx=U[0]-X[10]; wy=U[1]-X[11]; wz=U[2]-X[12]; // subtract the biases on gyros
q0=X[6]; q1=X[7]; q2=X[8]; q3=X[9];
// Pdot = V
F[0][3]=F[1][4]=F[2][5]=1;
// dVdot/dq
F[3][6]=2*(q0*ax-q3*ay+q2*az); F[3][7]=2*(q1*ax+q2*ay+q3*az); F[3][8]=2*(-q2*ax+q1*ay+q0*az); F[3][9]=2*(-q3*ax-q0*ay+q1*az);
F[4][6]=2*(q3*ax+q0*ay-q1*az); F[4][7]=2*(q2*ax-q1*ay-q0*az); F[4][8]=2*(q1*ax+q2*ay+q3*az); F[4][9]=2*(q0*ax-q3*ay+q2*az);
F[5][6]=2*(-q2*ax+q1*ay+q0*az); F[5][7]=2*(q3*ax+q0*ay-q1*az); F[5][8]=2*(-q0*ax+q3*ay-q2*az); F[5][9]=2*(q1*ax+q2*ay+q3*az);
// dVdot/dabias & dVdot/dna - NO BIAS STATES ON ACCELS - S0 REPEAT FOR G BELOW
// F[3][13]=G[3][3]=-q0*q0-q1*q1+q2*q2+q3*q3; F[3][14]=G[3][4]=2*(-q1*q2+q0*q3); F[3][15]=G[3][5]=-2*(q1*q3+q0*q2);
// F[4][13]=G[4][3]=-2*(q1*q2+q0*q3); F[4][14]=G[4][4]=-q0*q0+q1*q1-q2*q2+q3*q3; F[4][15]=G[4][5]=2*(-q2*q3+q0*q1);
// F[5][13]=G[5][3]=2*(-q1*q3+q0*q2); F[5][14]=G[5][4]=-2*(q2*q3+q0*q1); F[5][15]=G[5][5]=-q0*q0+q1*q1+q2*q2-q3*q3;
// dqdot/dq
F[6][6]=0; F[6][7]=-wx/2; F[6][8]=-wy/2; F[6][9]=-wz/2;
F[7][6]=wx/2; F[7][7]=0; F[7][8]=wz/2; F[7][9]=-wy/2;
F[8][6]=wy/2; F[8][7]=-wz/2; F[8][8]=0; F[8][9]=wx/2;
F[9][6]=wz/2; F[9][7]=wy/2; F[9][8]=-wx/2; F[9][9]=0;
// dqdot/dwbias
F[6][10]=q1/2; F[6][11]=q2/2; F[6][12]=q3/2;
F[7][10]=-q0/2; F[7][11]=q3/2; F[7][12]=-q2/2;
F[8][10]=-q3/2; F[8][11]=-q0/2; F[8][12]=q1/2;
F[9][10]=q2/2; F[9][11]=-q1/2; F[9][12]=-q0/2;
// dVdot/dna - NO BIAS STATES ON ACCELS - S0 REPEAT FOR G HERE
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; G[6][1]=q2/2; G[6][2]=q3/2;
G[7][0]=-q0/2; G[7][1]=q3/2; G[7][2]=-q2/2;
G[8][0]=-q3/2; G[8][1]=-q0/2; G[8][2]=q1/2;
G[9][0]=q2/2; G[9][1]=-q1/2; G[9][2]=-q0/2;
// dwbias = random walk noise
G[10][6]=G[11][7]=G[12][8]=1;
// dabias = random walk noise
// G[13][9]=G[14][10]=G[15][11]=1; // NO BIAS STATES ON ACCELS
}
void MeasurementEq(float X[NUMX], float Be[3], float Y[NUMV]){
float q0, q1, q2, q3;
q0=X[6]; q1=X[7]; q2=X[8]; 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];
// Bb=Rbe*Be
Y[6]=(q0*q0+q1*q1-q2*q2-q3*q3)*Be[0] + 2*(q1*q2+q0*q3)*Be[1] + 2*(q1*q3-q0*q2)*Be[2];
Y[7]=2*(q1*q2-q0*q3)*Be[0] + (q0*q0-q1*q1+q2*q2-q3*q3)*Be[1] + 2*(q2*q3+q0*q1)*Be[2];
Y[8]=2*(q1*q3+q0*q2)*Be[0] + 2*(q2*q3-q0*q1)*Be[1] + (q0*q0-q1*q1-q2*q2+q3*q3)*Be[2];
// Alt = -Pz
Y[9] = -X[2];
}
void LinearizeH(float X[NUMX], float Be[3], float H[NUMV][NUMX]){
float q0, q1, q2, q3;
q0=X[6]; q1=X[7]; q2=X[8]; q3=X[9];
// dP/dP=I;
H[0][0]=H[1][1]=H[2][2]=1;
// dV/dV=I;
H[3][3]=H[4][4]=H[5][5]=1;
// dBb/dq
H[6][6]=2*(q0*Be[0]+q3*Be[1]-q2*Be[2]); H[6][7]=2*(q1*Be[0]+q2*Be[1]+q3*Be[2]); H[6][8]=2*(-q2*Be[0]+q1*Be[1]-q0*Be[2]); H[6][9]=2*(-q3*Be[0]+q0*Be[1]+q1*Be[2]);
H[7][6]=2*(-q3*Be[0]+q0*Be[1]+q1*Be[2]); H[7][7]=2*(q2*Be[0]-q1*Be[1]+q0*Be[2]); H[7][8]=2*(q1*Be[0]+q2*Be[1]+q3*Be[2]); H[7][9]=2*(-q0*Be[0]-q3*Be[1]+q2*Be[2]);
H[8][6]=2*(q2*Be[0]-q1*Be[1]+q0*Be[2]); H[8][7]=2*(q3*Be[0]-q0*Be[1]-q1*Be[2]); H[8][8]=2*(q0*Be[0]+q3*Be[1]-q2*Be[2]); H[8][9]=2*(q1*Be[0]+q2*Be[1]+q3*Be[2]);
// dAlt/dPz = -1
H[9][2]=-1;
}