/** ****************************************************************************** * @addtogroup AHRS * @{ * @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. * @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 #include #include #include // 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 #pragma GCC optimize "O3" // Private functions void CovariancePrediction(float F[NUMX][NUMX], float G[NUMX][NUMW], float Q[NUMW], float dT, float P[NUMX][NUMX]); static 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); static void RungeKutta(float X[NUMX], float U[NUMU], float dT); static void StateEq(float X[NUMX], float U[NUMU], float Xdot[NUMX]); static void LinearizeFG(float X[NUMX], float U[NUMU], float F[NUMX][NUMX], float G[NUMX][NUMW]); static void MeasurementEq(float X[NUMX], float Be[3], float Y[NUMV]); static void LinearizeH(float X[NUMX], float Be[3], float H[NUMV][NUMX]); // Private variables // speed optimizations, describe matrix sparsity // derived from state equations in // LinearizeFG() and LinearizeH(): // // usage F: usage G: usage H: // -0123456789abc 012345678 0123456789abc // 0...X......... ......... X............ // 1....X........ ......... .X........... // 2.....X....... ......... ..X.......... // 3......XXXX... ...XXX... ...X......... // 4......XXXX... ...XXX... ....X........ // 5......XXXX... ...XXX... .....X....... // 6.....ooXXXXXX XXX...... ......XXXX... // 7.....oXoXXXXX XXX...... ......XXXX... // 8.....oXXoXXXX XXX...... ......XXXX... // 9.....oXXXoXXX XXX...... ..X.......... // a............. ......Xoo // b............. ......oXo // c............. ......ooX static int8_t FrowMin[NUMX] = { 3, 4, 5, 6, 6, 6, 5, 5, 5, 5, 13, 13, 13 }; static int8_t FrowMax[NUMX] = { 3, 4, 5, 9, 9, 9, 12, 12, 12, 12, -1, -1, -1 }; static int8_t GrowMin[NUMX] = { 9, 9, 9, 3, 3, 3, 0, 0, 0, 0, 6, 7, 8 }; static int8_t GrowMax[NUMX] = { -1, -1, -1, 5, 5, 5, 2, 2, 2, 2, 6, 7, 8 }; static int8_t HrowMin[NUMV] = { 0, 1, 2, 3, 4, 5, 6, 6, 6, 2 }; static int8_t HrowMax[NUMV] = { 0, 1, 2, 3, 4, 5, 9, 9, 9, 2 }; static struct EKFData { // linearized system matrices float F[NUMX][NUMX]; float G[NUMX][NUMW]; float H[NUMV][NUMX]; // local magnetic unit vector in NED frame float Be[3]; // covariance matrix and state vector float P[NUMX][NUMX]; float X[NUMX]; // input noise and measurement noise variances float Q[NUMW]; float R[NUMV]; } 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] = 0.0f; ekf.Be[2] = 0.0f; // 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.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-9f; // initial gyro bias variance (rad/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.Q[0] = ekf.Q[1] = ekf.Q[2] = 50e-4f; // gyro noise variance (rad/s)^2 ekf.Q[3] = ekf.Q[4] = ekf.Q[5] = 0.00001f; // accelerometer noise variance (m/s^2)^2 ekf.Q[6] = ekf.Q[7] = ekf.Q[8] = 2e-8f; // gyro bias random walk variance (rad/s^2)^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] = 100.0f; // 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] = .25f; // 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; } } void INSResetP(const float PDiag[NUMX]) { 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 INSGetVariance(float PDiag[NUMX]) { uint8_t i; // retrieve diagonal elements (aka state variance) if (PDiag != 0) { for (i = 0; i < NUMX; i++) { PDiag[i] = ekf.P[i][i]; } } } void INSSetState(const float pos[3], const float vel[3], const float q[4], const float gyro_bias[3], __attribute__((unused)) const float accel_bias[3]) { /* Note: accel_bias not used in 13 state INS */ 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]; } 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; // zero the first 6 rows and columns ekf.P[j][i] = 0; } } ekf.P[0][0] = ekf.P[1][1] = ekf.P[2][2] = 25; // initial position variance (m^2) ekf.P[3][3] = ekf.P[4][4] = ekf.P[5][5] = 5; // 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(const float PosVar[3], const float VelVar[3]) { 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]; } 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 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 mag_var[3]) { // scale variance down to unit vector float invBmag = invsqrtf(mag_var[0] * mag_var[0] + mag_var[1] * mag_var[1] + mag_var[2] * mag_var[2]); ekf.R[6] = mag_var[0] * invBmag; ekf.R[7] = mag_var[1] * invBmag; ekf.R[8] = mag_var[2] * invBmag; } void INSSetBaroVar(float baro_var) { ekf.R[9] = baro_var; } void INSSetMagNorth(const float B[3]) { float invmag = invsqrtf(B[0] * B[0] + B[1] * B[1] + B[2] * B[2]); ekf.Be[0] = B[0] * invmag; ekf.Be[1] = B[1] * invmag; ekf.Be[2] = B[2] * invmag; } void INSStatePrediction(const float gyro_data[3], const float accel_data[3], float dT) { float U[6]; float invqmag; // 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); invqmag = invsqrtf(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] *= invqmag; ekf.X[7] *= invqmag; ekf.X[8] *= invqmag; ekf.X[9] *= invqmag; // CovariancePrediction(ekf.F,ekf.G,ekf.Q,dT,ekf.P); // 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]; } void INSCovariancePrediction(float dT) { CovariancePrediction(ekf.F, ekf.G, ekf.Q, dT, ekf.P); } float zeros[3] = { 0, 0, 0 }; void MagCorrection(float mag_data[3]) { INSCorrection(mag_data, zeros, zeros, zeros[0], MAG_SENSORS); } void MagVelBaroCorrection(float mag_data[3], float Vel[3], float BaroAlt) { INSCorrection(mag_data, zeros, Vel, BaroAlt, MAG_SENSORS | HORIZ_SENSORS | VERT_SENSORS | BARO_SENSOR); } void GpsBaroCorrection(float Pos[3], float Vel[3], float BaroAlt) { INSCorrection(zeros, Pos, Vel, BaroAlt, HORIZ_SENSORS | VERT_SENSORS | BARO_SENSOR); } void FullCorrection(float mag_data[3], float Pos[3], float Vel[3], float BaroAlt) { INSCorrection(mag_data, Pos, Vel, BaroAlt, FULL_SENSORS); } void GpsMagCorrection(float mag_data[3], float Pos[3], float Vel[3]) { INSCorrection(mag_data, Pos, Vel, zeros[0], POS_SENSORS | HORIZ_SENSORS | MAG_SENSORS); } void VelBaroCorrection(float Vel[3], float BaroAlt) { INSCorrection(zeros, zeros, Vel, BaroAlt, HORIZ_SENSORS | VERT_SENSORS | BARO_SENSOR); } void INSCorrection(const float mag_data[3], const float Pos[3], const float Vel[3], const float BaroAlt, uint16_t SensorsUsed) { float Z[10] = { 0 }; float Y[10] = { 0 }; // 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 if (SensorsUsed & MAG_SENSORS) { // magnetometer data in any units (use unit vector) and in body frame float invBmag = invsqrtf(mag_data[0] * mag_data[0] + mag_data[1] * mag_data[1] + mag_data[2] * mag_data[2]); Z[6] = mag_data[0] * invBmag; Z[7] = mag_data[1] * invBmag; Z[8] = mag_data[2] * invBmag; } // 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); float invqmag = invsqrtf(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] *= invqmag; ekf.X[7] *= invqmag; ekf.X[8] *= invqmag; ekf.X[9] *= invqmag; // 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]; } // ************* 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 // ************************************************ void CovariancePrediction(float F[NUMX][NUMX], float G[NUMX][NUMW], float Q[NUMW], float dT, float P[NUMX][NUMX]) { // 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')] const float dT1 = 1.0f / dT; // multiplication is faster than division on fpu. const float dTsq = dT * dT; float Dummy[NUMX][NUMX]; int8_t i; int8_t k; for (i = 0; i < NUMX; i++) { // Calculate Dummy = (P/T +F*P) float *Firow = F[i]; float *Pirow = P[i]; float *Dirow = Dummy[i]; const int8_t Fistart = FrowMin[i]; const int8_t Fiend = FrowMax[i]; int8_t j; for (j = 0; j < NUMX; j++) { Dirow[j] = Pirow[j] * dT1; // Dummy = P / T ... } for (k = Fistart; k <= Fiend; k++) { for (j = 0; j < NUMX; j++) { Dirow[j] += Firow[k] * P[k][j]; // [] + F * P } } } for (i = 0; i < NUMX; i++) { // Calculate Pnew = (T^2) [Dummy/T + Dummy*F' + G*Qw*G'] float *Dirow = Dummy[i]; float *Girow = G[i]; float *Pirow = P[i]; const int8_t Gistart = GrowMin[i]; const int8_t Giend = GrowMax[i]; int8_t j; for (j = i; j < NUMX; j++) { // Use symmetry, ie only find upper triangular float Ptmp = Dirow[j] * dT1; // Pnew = Dummy / T ... const float *Fjrow = F[j]; int8_t Fjstart = FrowMin[j]; int8_t Fjend = FrowMax[j]; k = Fjstart; while (k <= Fjend - 3) { Ptmp += Dirow[k] * Fjrow[k]; // [] + Dummy*F' ... Ptmp += Dirow[k + 1] * Fjrow[k + 1]; Ptmp += Dirow[k + 2] * Fjrow[k + 2]; Ptmp += Dirow[k + 3] * Fjrow[k + 3]; k += 4; } while (k <= Fjend) { Ptmp += Dirow[k] * Fjrow[k]; k++; } float *Gjrow = G[j]; const int8_t Gjstart = MAX(Gistart, GrowMin[j]); const int8_t Gjend = MIN(Giend, GrowMax[j]); k = Gjstart; while (k <= Gjend - 2) { Ptmp += Q[k] * Girow[k] * Gjrow[k]; // [] + G*Q*G' ... Ptmp += Q[k + 1] * Girow[k + 1] * Gjrow[k + 1]; Ptmp += Q[k + 2] * Girow[k + 2] * Gjrow[k + 2]; k += 3; } if (k <= Gjend) { Ptmp += Q[k] * Girow[k] * Gjrow[k]; if (k <= Gjend - 1) { Ptmp += Q[k + 1] * Girow[k + 1] * Gjrow[k + 1]; } } P[j][i] = Pirow[j] = Ptmp * dTsq; // [] * (T^2) } } } // ************* 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; float Km[NUMX]; 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 = HrowMin[m]; k <= HrowMax[m]; k++) { for (j = 0; j < NUMX; j++) { // Find Hp = H*P HP[j] += H[m][k] * P[k][j]; } } HPHR = R[m]; // Find HPHR = H*P*H' + R for (k = HrowMin[m]; k <= HrowMax[m]; k++) { HPHR += HP[k] * H[m][k]; } float invHPHR = 1.0f / HPHR; for (k = 0; k < NUMX; k++) { Km[k] = HP[k] * invHPHR; // 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] - Km[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] + Km[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) { const float dT2 = dT / 2.0f; float 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]) * (1.0f / 6.0f); } } // ************* 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 // ************************************************ static 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 + 2.0f * (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 * (q2 * q3 - q0 * q1) * az; Xdot[5] = 2.0f * (q1 * q3 - q0 * q2) * ax + 2 * (q2 * q3 + q0 * q1) * ay + (q0 * q0 - q1 * q1 - q2 * q2 + q3 * q3) * az + 9.81f; // 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; } 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] = 1.0f; // 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 - 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 / 2.0f; 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 - NO BIAS STATES ON ACCELS - S0 REPEAT FOR G HERE G[3][3] = -q0 * q0 - q1 * q1 + q2 * q2 + q3 * q3; G[3][4] = 2.0f * (-q1 * q2 + q0 * q3); G[3][5] = -2.0f * (q1 * q3 + q0 * q2); G[4][3] = -2.0f * (q1 * q2 + q0 * q3); G[4][4] = -q0 * q0 + q1 * q1 - q2 * q2 + q3 * q3; G[4][5] = 2.0f * (-q2 * q3 + q0 * q1); G[5][3] = 2.0f * (-q1 * q3 + q0 * q2); G[5][4] = -2.0f * (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; // dwbias = random walk noise G[10][6] = G[11][7] = G[12][8] = 1.0f; // 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] + 2.0f * (q1 * q2 + q0 * q3) * Be[1] + 2.0f * (q1 * q3 - q0 * q2) * Be[2]; Y[7] = 2.0f * (q1 * q2 - q0 * q3) * Be[0] + (q0 * q0 - q1 * q1 + q2 * q2 - q3 * q3) * Be[1] + 2.0f * (q2 * q3 + q0 * q1) * Be[2]; Y[8] = 2.0f * (q1 * q3 + q0 * q2) * Be[0] + 2.0f * (q2 * q3 - q0 * q1) * Be[1] + (q0 * q0 - q1 * q1 - q2 * q2 + q3 * q3) * Be[2]; // Alt = -Pz Y[9] = -1.0f * 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] = 1.0f; // dV/dV=I; H[3][3] = H[4][4] = H[5][5] = 1.0f; // dBb/dq H[6][6] = 2.0f * (q0 * Be[0] + q3 * Be[1] - q2 * Be[2]); H[6][7] = 2.0f * (q1 * Be[0] + q2 * Be[1] + q3 * Be[2]); H[6][8] = 2.0f * (-q2 * Be[0] + q1 * Be[1] - q0 * Be[2]); H[6][9] = 2.0f * (-q3 * Be[0] + q0 * Be[1] + q1 * Be[2]); H[7][6] = 2.0f * (-q3 * Be[0] + q0 * Be[1] + q1 * Be[2]); H[7][7] = 2.0f * (q2 * Be[0] - q1 * Be[1] + q0 * Be[2]); H[7][8] = 2.0f * (q1 * Be[0] + q2 * Be[1] + q3 * Be[2]); H[7][9] = 2.0f * (-q0 * Be[0] - q3 * Be[1] + q2 * Be[2]); H[8][6] = 2.0f * (q2 * Be[0] - q1 * Be[1] + q0 * Be[2]); H[8][7] = 2.0f * (q3 * Be[0] - q0 * Be[1] - q1 * Be[2]); H[8][8] = 2.0f * (q0 * Be[0] + q3 * Be[1] - q2 * Be[2]); H[8][9] = 2.0f * (q1 * Be[0] + q2 * Be[1] + q3 * Be[2]); // dAlt/dPz = -1 H[9][2] = -1.0f; } /** * @} * @} */