/// -*- tab-width: 4; Mode: C++; c-basic-offset: 4; indent-tabs-mode: nil -*- #include #if HAL_CPU_CLASS >= HAL_CPU_CLASS_150 // uncomment this to force the optimisation of this code, note that // this makes debugging harder #if CONFIG_HAL_BOARD == HAL_BOARD_AVR_SITL || CONFIG_HAL_BOARD == HAL_BOARD_LINUX #pragma GCC optimize("O0") #else #pragma GCC optimize("O3") #endif #include "AP_NavEKF.h" #include #include #include #include /* parameter defaults for different types of vehicle. The APM_BUILD_DIRECTORY is taken from the main vehicle directory name where the code is built. Note that this trick won't work for arduino builds on APM2, but NavEKF doesn't run on APM2, so that's OK */ #if APM_BUILD_TYPE(APM_BUILD_ArduCopter) // copter defaults #define VELNE_NOISE_DEFAULT 0.5f #define VELD_NOISE_DEFAULT 0.7f #define POSNE_NOISE_DEFAULT 0.5f #define ALT_NOISE_DEFAULT 1.0f #define MAG_NOISE_DEFAULT 0.05f #define GYRO_PNOISE_DEFAULT 0.015f #define ACC_PNOISE_DEFAULT 0.25f #define GBIAS_PNOISE_DEFAULT 1E-06f #define ABIAS_PNOISE_DEFAULT 0.0001f #define MAGE_PNOISE_DEFAULT 0.0003f #define MAGB_PNOISE_DEFAULT 0.0003f #define VEL_GATE_DEFAULT 2 #define POS_GATE_DEFAULT 10 #define HGT_GATE_DEFAULT 5 #define MAG_GATE_DEFAULT 3 #define MAG_CAL_DEFAULT 1 #define GLITCH_ACCEL_DEFAULT 150 #define GLITCH_RADIUS_DEFAULT 15 #elif APM_BUILD_TYPE(APM_BUILD_APMrover2) // rover defaults #define VELNE_NOISE_DEFAULT 0.5f #define VELD_NOISE_DEFAULT 0.7f #define POSNE_NOISE_DEFAULT 0.5f #define ALT_NOISE_DEFAULT 1.0f #define MAG_NOISE_DEFAULT 0.05f #define GYRO_PNOISE_DEFAULT 0.015f #define ACC_PNOISE_DEFAULT 0.25f #define GBIAS_PNOISE_DEFAULT 1E-06f #define ABIAS_PNOISE_DEFAULT 0.0002f #define MAGE_PNOISE_DEFAULT 0.0003f #define MAGB_PNOISE_DEFAULT 0.0003f #define VEL_GATE_DEFAULT 2 #define POS_GATE_DEFAULT 10 #define HGT_GATE_DEFAULT 5 #define MAG_GATE_DEFAULT 3 #define MAG_CAL_DEFAULT 1 #define GLITCH_ACCEL_DEFAULT 150 #define GLITCH_RADIUS_DEFAULT 15 #else // generic defaults (and for plane) #define VELNE_NOISE_DEFAULT 0.3f #define VELD_NOISE_DEFAULT 0.5f #define POSNE_NOISE_DEFAULT 0.5f #define ALT_NOISE_DEFAULT 0.5f #define MAG_NOISE_DEFAULT 0.05f #define GYRO_PNOISE_DEFAULT 0.015f #define ACC_PNOISE_DEFAULT 0.25f #define GBIAS_PNOISE_DEFAULT 1E-06f #define ABIAS_PNOISE_DEFAULT 0.0002f #define MAGE_PNOISE_DEFAULT 0.0003f #define MAGB_PNOISE_DEFAULT 0.0003f #define VEL_GATE_DEFAULT 2 #define POS_GATE_DEFAULT 10 #define HGT_GATE_DEFAULT 10 #define MAG_GATE_DEFAULT 3 #define MAG_CAL_DEFAULT 0 #define GLITCH_ACCEL_DEFAULT 150 #define GLITCH_RADIUS_DEFAULT 15 #endif // APM_BUILD_DIRECTORY extern const AP_HAL::HAL& hal; #define earthRate 0.000072921f // earth rotation rate (rad/sec) // maximum value for any element in the covariance matrix #define EKF_COVARIENCE_MAX 1.0e8f // when the wind estimation first starts with no airspeed sensor, // assume 3m/s to start #define STARTUP_WIND_SPEED 3.0f // Define tuning parameters const AP_Param::GroupInfo NavEKF::var_info[] PROGMEM = { // @Param: VELNE_NOISE // @DisplayName: GPS horizontal velocity measurement noise (m/s) // @Description: This is the RMS value of noise in the North and East GPS velocity measurements. Increasing it reduces the weighting on these measurements. // @Range: 0.05 - 5.0 // @Increment: 0.05 // @User: advanced AP_GROUPINFO("VELNE_NOISE", 0, NavEKF, _gpsHorizVelNoise, VELNE_NOISE_DEFAULT), // @Param: VELD_NOISE // @DisplayName: GPS vertical velocity measurement noise (m/s) // @Description: This is the RMS value of noise in the vertical GPS velocity measurement. Increasing it reduces the weighting on this measurement. // @Range: 0.05 - 5.0 // @Increment: 0.05 // @User: advanced AP_GROUPINFO("VELD_NOISE", 1, NavEKF, _gpsVertVelNoise, VELD_NOISE_DEFAULT), // @Param: POSNE_NOISE // @DisplayName: GPS horizontal position measurement noise (m) // @Description: This is the RMS value of noise in the GPS horizontal position measurements. Increasing it reduces the weighting on these measurements. // @Range: 0.1 - 10.0 // @Increment: 0.1 // @User: advanced AP_GROUPINFO("POSNE_NOISE", 2, NavEKF, _gpsHorizPosNoise, POSNE_NOISE_DEFAULT), // @Param: ALT_NOISE // @DisplayName: Altitude measurement noise (m) // @Description: This is the RMS value of noise in the altitude measurement. Increasing it reduces the weighting on this measurement. // @Range: 0.1 - 10.0 // @Increment: 0.1 // @User: advanced AP_GROUPINFO("ALT_NOISE", 3, NavEKF, _baroAltNoise, ALT_NOISE_DEFAULT), // @Param: MAG_NOISE // @DisplayName: Magnetometer measurement noise (Gauss) // @Description: This is the RMS value of noise in magnetometer measurements. Increasing it reduces the weighting on these measurements. // @Range: 0.01 - 0.5 // @Increment: 0.01 // @User: advanced AP_GROUPINFO("MAG_NOISE", 4, NavEKF, _magNoise, MAG_NOISE_DEFAULT), // @Param: EAS_NOISE // @DisplayName: Equivalent airspeed measurement noise (m/s) // @Description: This is the RMS value of noise in magnetometer measurements. Increasing it reduces the weighting on these measurements. // @Range: 0.5 - 5.0 // @Increment: 0.1 // @User: advanced AP_GROUPINFO("EAS_NOISE", 5, NavEKF, _easNoise, 1.4f), // @Param: WIND_PNOISE // @DisplayName: Wind velocity process noise (m/s^2) // @Description: This noise controls the growth of wind state error estimates. Increasing it makes wind estimation faster and noisier. // @Range: 0.01 - 1.0 // @Increment: 0.1 // @User: advanced AP_GROUPINFO("WIND_PNOISE", 6, NavEKF, _windVelProcessNoise, 0.1f), // @Param: WIND_PSCALE // @DisplayName: Height rate to wind procss noise scaler // @Description: Increasing this parameter increases how rapidly the wind states adapt when changing altitude, but does make wind speed estimation noiser. // @Range: 0.0 - 1.0 // @Increment: 0.1 // @User: advanced AP_GROUPINFO("WIND_PSCALE", 7, NavEKF, _wndVarHgtRateScale, 0.5f), // @Param: GYRO_PNOISE // @DisplayName: Rate gyro noise (rad/s) // @Description: This noise controls the growth of estimated error due to gyro measurement errors excluding bias. Increasing it makes the flter trust the gyro measurements less and other measurements more. // @Range: 0.001 - 0.05 // @Increment: 0.001 // @User: advanced AP_GROUPINFO("GYRO_PNOISE", 8, NavEKF, _gyrNoise, GYRO_PNOISE_DEFAULT), // @Param: ACC_PNOISE // @DisplayName: Accelerometer noise (m/s^2) // @Description: This noise controls the growth of estimated error due to accelerometer measurement errors excluding bias. Increasing it makes the flter trust the accelerometer measurements less and other measurements more. // @Range: 0.05 - 1.0 AP_Float _gpsNEVelVarAccScale; // scale factor applied to NE velocity measurement variance due to Vdot // @Increment: 0.01 // @User: advanced AP_GROUPINFO("ACC_PNOISE", 9, NavEKF, _accNoise, ACC_PNOISE_DEFAULT), // @Param: GBIAS_PNOISE // @DisplayName: Rate gyro bias process noise (rad/s) // @Description: This noise controls the growth of gyro bias state error estimates. Increasing it makes rate gyro bias estimation faster and noisier. // @Range: 0.0000001 - 0.00001 // @User: advanced AP_GROUPINFO("GBIAS_PNOISE", 10, NavEKF, _gyroBiasProcessNoise, GBIAS_PNOISE_DEFAULT), // @Param: ABIAS_PNOISE // @DisplayName: Accelerometer bias process noise (m/s^2) // @Description: This noise controls the growth of the vertical acelerometer bias state error estimate. Increasing it makes accelerometer bias estimation faster and noisier. // @Range: 0.00001 - 0.001 // @User: advanced AP_GROUPINFO("ABIAS_PNOISE", 11, NavEKF, _accelBiasProcessNoise, ABIAS_PNOISE_DEFAULT), // @Param: MAGE_PNOISE // @DisplayName: Earth magnetic field process noise (gauss/s) // @Description: This noise controls the growth of earth magnetic field state error estimates. Increasing it makes earth magnetic field bias estimation faster and noisier. // @Range: 0.0001 - 0.01 // @User: advanced AP_GROUPINFO("MAGE_PNOISE", 12, NavEKF, _magEarthProcessNoise, MAGE_PNOISE_DEFAULT), // @Param: MAGB_PNOISE // @DisplayName: Body magnetic field process noise (gauss/s) // @Description: This noise controls the growth of body magnetic field state error estimates. Increasing it makes compass offset estimation faster and noisier. // @Range: 0.0001 - 0.01 // @User: advanced AP_GROUPINFO("MAGB_PNOISE", 13, NavEKF, _magBodyProcessNoise, MAGB_PNOISE_DEFAULT), // @Param: VEL_DELAY // @DisplayName: GPS velocity measurement delay (msec) // @Description: This is the number of msec that the GPS velocity measurements lag behind the inertial measurements. // @Range: 0 - 500 // @Increment: 10 // @User: advanced AP_GROUPINFO("VEL_DELAY", 14, NavEKF, _msecVelDelay, 220), // @Param: POS_DELAY // @DisplayName: GPS position measurement delay (msec) // @Description: This is the number of msec that the GPS position measurements lag behind the inertial measurements. // @Range: 0 - 500 // @Increment: 10 // @User: advancedScale factor applied to horizontal position measurement variance due to manoeuvre acceleration AP_GROUPINFO("POS_DELAY", 15, NavEKF, _msecPosDelay, 220), // @Param: GPS_TYPE // @DisplayName: GPS velocity mode control // @Description: This parameter controls use of GPS velocity measurements : 0 = use 3D velocity, 1 = use 2D velocity, 2 = use no velocity // @Range: 0 - 3 // @Increment: 1 // @User: advanced AP_GROUPINFO("GPS_TYPE", 16, NavEKF, _fusionModeGPS, 0), // @Param: VEL_GATE // @DisplayName: GPS velocity measurement gate size // @Description: This parameter sets the number of standard deviations applied to the GPS velocity measurement innovation consistency check. Decreasing it makes it more likely that good measurements willbe rejected. Increasing it makes it more likely that bad measurements will be accepted. // @Range: 1 - 100 // @Increment: 1 // @User: advanced AP_GROUPINFO("VEL_GATE", 17, NavEKF, _gpsVelInnovGate, VEL_GATE_DEFAULT), // @Param: POS_GATE // @DisplayName: GPS position measurement gate size // @Description: This parameter sets the number of standard deviations applied to the GPS position measurement innovation consistency check. Decreasing it makes it more likely that good measurements will be rejected. Increasing it makes it more likely that bad measurements will be accepted. // @Range: 1 - 100 // @Increment: 1 // @User: advanced AP_GROUPINFO("POS_GATE", 18, NavEKF, _gpsPosInnovGate, POS_GATE_DEFAULT), // @Param: HGT_GATE // @DisplayName: Height measurement gate size // @Description: This parameter sets the number of standard deviations applied to the height measurement innovation consistency check. Decreasing it makes it more likely that good measurements will be rejected. Increasing it makes it more likely that bad measurements will be accepted. // @Range: 1 - 100 // @Increment: 1 // @User: advanced AP_GROUPINFO("HGT_GATE", 19, NavEKF, _hgtInnovGate, HGT_GATE_DEFAULT), // @Param: MAG_GATE // @DisplayName: Magnetometer measurement gate size // @Description: This parameter sets the number of standard deviations applied to the magnetometer measurement innovation consistency check. Decreasing it makes it more likely that good measurements will be rejected. Increasing it makes it more likely that bad measurements will be accepted. // @Range: 1 - 100 // @Increment: 1 // @User: advanced AP_GROUPINFO("MAG_GATE", 20, NavEKF, _magInnovGate, MAG_GATE_DEFAULT), // @Param: EAS_GATE // @DisplayName: Airspeed measurement gate size // @Description: This parameter sets the number of standard deviations applied to the airspeed measurement innovation consistency check. Decreasing it makes it more likely that good measurements will be rejected. Increasing it makes it more likely that bad measurements will be accepted. // @Range: 1 - 100 // @Increment: 1 // @User: advanced AP_GROUPINFO("EAS_GATE", 21, NavEKF, _tasInnovGate, 10), // @Param: MAG_CAL // @DisplayName: Magnetometer calibration mode // @Description: EKF_MAG_CAL = 0 enables calibration based on flying speed and altitude and is the default setting for Plane users. EKF_MAG_CAL = 1 enables calibration based on manoeuvre level and is the default setting for Copter and Rover users. EKF_MAG_CAL = 2 prevents magnetometer calibration regardless of flight condition and is recommended if in-flight magnetometer calibration is unreliable. // @Values: 0:Speed and Height,1:Acceleration,2:Never // @Increment: 1 // @User: advanced AP_GROUPINFO("MAG_CAL", 22, NavEKF, _magCal, MAG_CAL_DEFAULT), // @Param: GLITCH_ACCEL // @DisplayName: GPS glitch accel gate size (cm/s^2) // @Description: This parameter controls the maximum amount of difference in horizontal acceleration between the value predicted by the filter and the value measured by the GPS before the GPS position data is rejected. If this value is set too low, then valid GPS data will be regularly discarded, and the position accuracy will degrade. If this parameter is set too high, then large GPS glitches will cause large rapid changes in position. // @Range: 100 - 500 // @Increment: 50 // @User: advanced AP_GROUPINFO("GLITCH_ACCEL", 23, NavEKF, _gpsGlitchAccelMax, GLITCH_ACCEL_DEFAULT), // @Param: GLITCH_RAD // @DisplayName: GPS glitch radius gate size (m) // @Description: This parameter controls the maximum amount of difference in horizontal position (in m) between the value predicted by the filter and the value measured by the GPS before the long term glitch protection logic is activated and an offset is applied to the GPS measurement to compensate. Position steps smaller than this value will be temporarily ignored, but will then be accepted and the filter will move to the new position. Position steps larger than this value will be ignored initially, but the filter will then apply an offset to the GPS position measurement. // @Range: 10 - 50 // @Increment: 5 // @User: advanced AP_GROUPINFO("GLITCH_RAD", 24, NavEKF, _gpsGlitchRadiusMax, GLITCH_RADIUS_DEFAULT), AP_GROUPEND }; // constructor NavEKF::NavEKF(const AP_AHRS *ahrs, AP_Baro &baro) : _ahrs(ahrs), _baro(baro), state(*reinterpret_cast(&states)), covTimeStepMax(0.07f), // maximum time (sec) between covariance prediction updates covDelAngMax(0.05f), // maximum delta angle between covariance prediction updates TASmsecMax(200), // maximum allowed interval between airspeed measurement updates fuseMeNow(false), // forces airspeed and sythetic sideslip fusion to occur on the IMU frame that data arrives staticMode(true), // staticMode forces position and velocity fusion with zero values prevStaticMode(true), // staticMode from previous filter update yawAligned(false), // set true when heading or yaw angle has been aligned inhibitWindStates(true), // inhibit wind state updates on startup inhibitMagStates(true) // inhibit magnetometer state updates on startup #if CONFIG_HAL_BOARD == HAL_BOARD_PX4 || CONFIG_HAL_BOARD == HAL_BOARD_VRBRAIN ,_perf_UpdateFilter(perf_alloc(PC_ELAPSED, "EKF_UpdateFilter")), _perf_CovariancePrediction(perf_alloc(PC_ELAPSED, "EKF_CovariancePrediction")), _perf_FuseVelPosNED(perf_alloc(PC_ELAPSED, "EKF_FuseVelPosNED")), _perf_FuseMagnetometer(perf_alloc(PC_ELAPSED, "EKF_FuseMagnetometer")), _perf_FuseAirspeed(perf_alloc(PC_ELAPSED, "EKF_FuseAirspeed")), _perf_FuseSideslip(perf_alloc(PC_ELAPSED, "EKF_FuseSideslip")) #endif { AP_Param::setup_object_defaults(this, var_info); // Tuning parameters _gpsNEVelVarAccScale = 0.05f; // Scale factor applied to NE velocity measurement variance due to manoeuvre acceleration _gpsDVelVarAccScale = 0.07f; // Scale factor applied to vertical velocity measurement variance due to manoeuvre acceleration _gpsPosVarAccScale = 0.05f; // Scale factor applied to horizontal position measurement variance due to manoeuvre acceleration _msecHgtDelay = 60; // Height measurement delay (msec) _msecMagDelay = 40; // Magnetometer measurement delay (msec) _msecTasDelay = 240; // Airspeed measurement delay (msec) _gpsRetryTimeUseTAS = 20000; // GPS retry time with airspeed measurements (msec) _gpsRetryTimeNoTAS = 10000; // GPS retry time without airspeed measurements (msec) _hgtRetryTimeMode0 = 10000; // Height retry time with vertical velocity measurement (msec) _hgtRetryTimeMode12 = 5000; // Height retry time without vertical velocity measurement (msec) _magFailTimeLimit_ms = 10000; // number of msec before a magnetometer failing innovation consistency checks is declared failed (msec) _magVarRateScale = 0.05f; // scale factor applied to magnetometer variance due to angular rate _gyroBiasNoiseScaler = 2.0f; // scale factor applied to gyro bias state process noise when on ground _msecGpsAvg = 200; // average number of msec between GPS measurements _msecHgtAvg = 100; // average number of msec between height measurements _msecBetaAvg = 100; // average number of msec between synthetic sideslip measurements dtVelPos = 0.02; // number of seconds between position and velocity corrections. This should be a multiple of the imu update interval. // Misc initial conditions hgtRate = 0.0f; mag_state.q0 = 1; mag_state.DCM.identity(); IMU1_weighting = 0.5f; lastDivergeTime_ms = 0; memset(&faultStatus, 0, sizeof(faultStatus)); } // Check basic filter health metrics and return a consolidated health status bool NavEKF::healthy(void) const { if (!statesInitialised) { return false; } if (state.quat.is_nan()) { return false; } if (state.velocity.is_nan()) { return false; } if (filterDiverged || (hal.scheduler->millis() - lastDivergeTime_ms < 10000)) { return false; } // If measurements have failed innovation consistency checks for long enough to time-out // and force fusion then the nav solution can be conidered to be unhealthy // This will only be set as a transient if (posTimeout || velTimeout || hgtTimeout) { return false; } // all OK return true; } // return true if filter is dead-reckoning height bool NavEKF::HeightDrifting(void) const { // Set to true if height measurements are failing the innovation consistency check return !hgtHealth; } // return true if filter is dead-reckoning position bool NavEKF::PositionDrifting(void) const { // Set to true if position measurements are failing the innovation consistency check return !posHealth; } // resets position states to last GPS measurement or to zero if in static mode void NavEKF::ResetPosition(void) { if (staticMode) { states[7] = 0; states[8] = 0; } else if (_ahrs->get_gps().status() >= AP_GPS::GPS_OK_FIX_3D) { // read the GPS readGpsData(); // write to state vector states[7] = gpsPosNE.x + gpsPosGlitchOffsetNE.x; states[8] = gpsPosNE.y + gpsPosGlitchOffsetNE.y; } } // resets velocity states to last GPS measurement or to zero if in static mode void NavEKF::ResetVelocity(void) { if (staticMode) { state.velocity.zero(); state.vel1.zero(); state.vel2.zero(); } else if (_ahrs->get_gps().status() >= AP_GPS::GPS_OK_FIX_3D) { // read the GPS readGpsData(); // reset horizontal velocity states if (_fusionModeGPS <= 1) { states[4] = velNED[0]; // north velocity from blended accel data states[5] = velNED[1]; // east velocity from blended accel data states[23] = velNED[0]; // north velocity from IMU1 accel data states[24] = velNED[1]; // east velocity from IMU1 accel data states[27] = velNED[0]; // north velocity from IMU2 accel data states[28] = velNED[1]; // east velocity from IMU2 accel data } // reset vertical velocity states if (_fusionModeGPS <= 0) { states[6] = velNED[2]; // down velocity from blended accel data states[25] = velNED[2]; // down velocity from IMU1 accel data states[29] = velNED[2]; // down velocity from IMU2 accel data } } } // reset the vertical position state using the last height measurement void NavEKF::ResetHeight(void) { // read the altimeter readHgtData(); // write to the state vector states[9] = -hgtMea; // down position from blended accel data state.posD1 = -hgtMea; // down position from IMU1 accel data state.posD2 = -hgtMea; // down position from IMU2 accel data } // this function is used to initialise the filter whilst moving, using the AHRS DCM solution // it should NOT be used to re-initialise after a timeout as DCM will also be corrupted void NavEKF::InitialiseFilterDynamic(void) { // this forces healthy() to be false so that when we ask for ahrs // attitude we get the DCM attitude regardless of the state of AHRS_EKF_USE statesInitialised = false; // Set re-used variables to zero ZeroVariables(); // get initial time deltat between IMU measurements (sec) dtIMU = _ahrs->get_ins().get_delta_time(); // calculate initial orientation and earth magnetic field states Quaternion initQuat; initQuat = calcQuatAndFieldStates(_ahrs->roll, _ahrs->pitch); // write to state vector state.quat = initQuat; state.gyro_bias.zero(); state.accel_zbias1 = 0; state.accel_zbias2 = 0; state.wind_vel.zero(); ResetVelocity(); ResetPosition(); ResetHeight(); state.body_magfield = magBias; // set to true now that states have be initialised statesInitialised = true; // initialise the covariance matrix CovarianceInit(); // define Earth rotation vector in the NED navigation frame calcEarthRateNED(earthRateNED, _ahrs->get_home().lat); // initialise IMU pre-processing states readIMUData(); } // Initialise the states from accelerometer and magnetometer data (if present) // This method can only be used when the vehicle is static void NavEKF::InitialiseFilterBootstrap(void) { // set re-used variables to zero ZeroVariables(); // acceleration vector in XYZ body axes measured by the IMU (m/s^2) Vector3f initAccVec; // TODO we should average accel readings over several cycles initAccVec = _ahrs->get_ins().get_accel(); // read the magnetometer data readMagData(); // normalise the acceleration vector initAccVec.normalize(); // calculate initial pitch angle float pitch = asinf(initAccVec.x); // calculate initial roll angle float roll = -asinf(initAccVec.y / cosf(pitch)); // calculate initial orientation and earth magnetic field states Quaternion initQuat; initQuat = calcQuatAndFieldStates(roll, pitch); // read the GPS readGpsData(); // read the barometer readHgtData(); // check on ground status SetFlightAndFusionModes(); // write to state vector state.quat = initQuat; state.gyro_bias.zero(); state.accel_zbias1 = 0; state.accel_zbias2 = 0; state.wind_vel.zero(); state.body_magfield = magBias; // set to true now we have intialised the states statesInitialised = true; // initialise the covariance matrix CovarianceInit(); // define Earth rotation vector in the NED navigation frame calcEarthRateNED(earthRateNED, _ahrs->get_home().lat); // initialise IMU pre-processing states readIMUData(); } // Update Filter States - this should be called whenever new IMU data is available void NavEKF::UpdateFilter() { // don't run filter updates if states have not been initialised if (!statesInitialised) { return; } // start the timer used for load measurement perf_begin(_perf_UpdateFilter); // read IMU data and convert to delta angles and velocities readIMUData(); // detect if filter has diverged and do a dynamic reset using the DCM solution checkDivergence(); if (filterDiverged) { InitialiseFilterDynamic(); return; } // detect if the filter update has been delayed for too long if (dtIMU > 0.2f) { // we have stalled for too long - reset states ResetVelocity(); ResetPosition(); ResetHeight(); StoreStatesReset(); //Initialise IMU pre-processing states readIMUData(); // stop the timer used for load measurement perf_end(_perf_UpdateFilter); return; } // check if on ground SetFlightAndFusionModes(); // define rules used to set staticMode // staticMode enables ground operation without GPS by fusing zeros for position and height measurements if (static_mode_demanded()) { staticMode = true; } else { staticMode = false; } // check to see if static mode has changed and reset states if it has if (prevStaticMode != staticMode) { ResetVelocity(); ResetPosition(); ResetHeight(); StoreStatesReset(); calcQuatAndFieldStates(_ahrs->roll, _ahrs->pitch); prevStaticMode = staticMode; } // run the strapdown INS equations every IMU update UpdateStrapdownEquationsNED(); // store the predicted states for subsequent use by measurement fusion StoreStates(); // sum delta angles and time used by covariance prediction summedDelAng = summedDelAng + correctedDelAng; summedDelVel = summedDelVel + correctedDelVel1; dt += dtIMU; // perform a covariance prediction if the total delta angle has exceeded the limit // or the time limit will be exceeded at the next IMU update if (((dt >= (covTimeStepMax - dtIMU)) || (summedDelAng.length() > covDelAngMax))) { CovariancePrediction(); covPredStep = true; summedDelAng.zero(); summedDelVel.zero(); dt = 0.0; } else { covPredStep = false; } // Update states using GPS, altimeter, compass, airspeed and synthetic sideslip observations SelectVelPosFusion(); SelectMagFusion(); SelectTasFusion(); SelectBetaFusion(); // stop the timer used for load measurement perf_end(_perf_UpdateFilter); } // select fusion of velocity, position and height measurements void NavEKF::SelectVelPosFusion() { // calculate ratio of VelPos fusion to state prediction steps uint8_t velPosFuseStepRatio = floor(dtVelPos/dtIMU + 0.5f); // calculate the scale factor to be applied to GPS measurement variance to account for // the fact we repeat fusion of the same measurement to provide a smoother output gpsVarScaler = _msecGpsAvg/(1000.0f*dtVelPos); // calculate the scale factor to be applied to height measurement variance to account for // the fact we repeat fusion of the same measurement to provide a smoother output hgtVarScaler = _msecHgtAvg/(1000.0f*dtVelPos); // check for new data, specify which measurements should be used and check data for freshness if (!staticMode) { // check for and read new GPS data readGpsData(); // command fusion of GPS data and reset states as required if (newDataGps) { // reset data arrived flag newDataGps = false; // enable fusion fuseVelData = true; fusePosData = true; // reset the counter used to schedule updates so that we always fuse data on the frame GPS data arrives skipCounter = velPosFuseStepRatio; // If a long time since last GPS update, then reset position and velocity and reset stored state history uint32_t gpsRetryTimeout = useAirspeed() ? _gpsRetryTimeUseTAS : _gpsRetryTimeNoTAS; if (hal.scheduler->millis() - secondLastFixTime_ms > gpsRetryTimeout) { ResetPosition(); ResetVelocity(); StoreStatesReset(); } } else if (hal.scheduler->millis() - lastFixTime_ms > (uint32_t)(_msecGpsAvg + 40)) { // Timeout fusion of GPS data if stale. Needed because we repeatedly fuse the same // measurement until the next one arrives to provide a smoother output fuseVelData = false; fusePosData = false; } // command fusion of height data if (newDataHgt) { // reset data arrived flag newDataHgt = false; // enable fusion fuseHgtData = true; } else if (hal.scheduler->millis() - lastHgtTime_ms > (uint32_t)(_msecHgtAvg + 40)) { // timeout fusion of height data if stale. Needed because we repeatedly fuse the same // measurement until the next one arrives to provide a smoother output fuseHgtData = false; } } else { // in static mode use synthetic position measurements set to zero // only fuse synthetic measurements when rate of change of velocity is less than 0.5g to reduce attitude errors due to launch acceleration // do not use velocity fusion to reduce the effect of movement on attitude if (accNavMag < 4.9f) { fusePosData = true; } else { fusePosData = false; } fuseVelData = false; fuseHgtData = true; } // check for and read new height data readHgtData(); // perform fusion as commanded, and in accordance with specified time intervals if (fuseVelData || fusePosData || fuseHgtData) { // skip fusion as required to maintain ~dtVelPos time interval between corrections if (skipCounter >= velPosFuseStepRatio) { FuseVelPosNED(); // reset counter used to skip update frames skipCounter = 1; } else { // increment counter used to skip update frames skipCounter += 1; } } } // select fusion of magnetometer data void NavEKF::SelectMagFusion() { // check for and read new magnetometer measurements readMagData(); // If we are using the compass and the magnetometer has been unhealthy for too long we declare it failed if (magHealth) { lastHealthyMagTime_ms = hal.scheduler->millis(); } else { if ((hal.scheduler->millis() - lastHealthyMagTime_ms) > _magFailTimeLimit_ms && use_compass()) { magTimeout = true; } else { magTimeout = false; } } // determine if conditions are right to start a new fusion cycle bool dataReady = statesInitialised && use_compass() && newDataMag; if (dataReady) { MAGmsecPrev = IMUmsec; fuseMagData = true; } else { fuseMagData = false; } // call the function that performs fusion of magnetometer data FuseMagnetometer(); } // select fusion of true airspeed measurements void NavEKF::SelectTasFusion() { // get true airspeed measurement readAirSpdData(); // if the filter is initialised, wind states are not inhibited and we have data to fuse, then queue TAS fusion tasDataWaiting = (statesInitialised && !inhibitWindStates && (tasDataWaiting || newDataTas)); // if we have waited too long, set a timeout flag which will force fusion to occur bool timeout = ((IMUmsec - TASmsecPrev) >= TASmsecMax); // we don't fuse airspeed measurements if magnetometer fusion has been performed in the same frame, unless timed out or the fuseMeNow option is selected // this helps to spreasthe load associated with fusion of different measurements across multiple frames // setting fuseMeNow to true disables this load spreading feature if (tasDataWaiting && (!magFusePerformed || timeout || fuseMeNow)) { FuseAirspeed(); TASmsecPrev = IMUmsec; tasDataWaiting = false; } } // select fusion of synthetic sideslip measurements void NavEKF::SelectBetaFusion() { // Determine if synthetic sidelsip data should be fused // synthetic sidelip fusion only works for fixed wing aircraft and relies on the average sideslip being close to zero // it requires a stable wind estimate for best results and should not be used for aerobatic flight // we fuse synthetic sideslip measurements if: // we are a fly forward vehicle type AND NOT using a full range of sensors with healthy position // AND NOT on the ground AND enough time has lapsed since our last fusion // AND (we have not fused magnetometer data on this time step OR the immediate fusion flag is set) if (assume_zero_sideslip() && !(use_compass() && useAirspeed() && posHealth) && !inhibitWindStates && ((IMUmsec - BETAmsecPrev) >= _msecBetaAvg) && (!magFusePerformed || fuseMeNow)) { FuseSideslip(); BETAmsecPrev = IMUmsec; } } // update the quaternion, velocity and position states using IMU measurements void NavEKF::UpdateStrapdownEquationsNED() { Vector3f delVelNav; // delta velocity vector calculated using a blend of IMU1 and IMU2 data Vector3f delVelNav1; // delta velocity vector calculated using IMU1 data Vector3f delVelNav2; // delta velocity vector calculated using IMU2 data float rotationMag; // magnitude of rotation vector from previous to current time step float rotScaler; // scaling variable used to calculate delta quaternion from last to current time step Quaternion qUpdated; // quaternion at current time step after application of delta quaternion Quaternion deltaQuat; // quaternion from last to current time step const Vector3f gravityNED(0, 0, GRAVITY_MSS); // NED gravity vector m/s^2 // remove sensor bias errors correctedDelAng = dAngIMU - state.gyro_bias; correctedDelVel1 = dVelIMU1; correctedDelVel2 = dVelIMU2; correctedDelVel1.z -= state.accel_zbias1; correctedDelVel2.z -= state.accel_zbias2; // use weighted average of both IMU units for delta velocities correctedDelVel12 = correctedDelVel1 * IMU1_weighting + correctedDelVel2 * (1.0f - IMU1_weighting); // save current measurements prevDelAng = correctedDelAng; // apply corrections for earths rotation rate and coning errors // % * - and + operators have been overloaded correctedDelAng = correctedDelAng - prevTnb * earthRateNED*dtIMU + (prevDelAng % correctedDelAng) * 8.333333e-2f; // convert the rotation vector to its equivalent quaternion rotationMag = correctedDelAng.length(); if (rotationMag < 1e-12f) { deltaQuat[0] = 1; deltaQuat[1] = 0; deltaQuat[2] = 0; deltaQuat[3] = 0; } else { deltaQuat[0] = cosf(0.5f * rotationMag); rotScaler = (sinf(0.5f * rotationMag)) / rotationMag; deltaQuat[1] = correctedDelAng.x * rotScaler; deltaQuat[2] = correctedDelAng.y * rotScaler; deltaQuat[3] = correctedDelAng.z * rotScaler; } // update the quaternions by rotating from the previous attitude through // the delta angle rotation quaternion qUpdated[0] = states[0]*deltaQuat[0] - states[1]*deltaQuat[1] - states[2]*deltaQuat[2] - states[3]*deltaQuat[3]; qUpdated[1] = states[0]*deltaQuat[1] + states[1]*deltaQuat[0] + states[2]*deltaQuat[3] - states[3]*deltaQuat[2]; qUpdated[2] = states[0]*deltaQuat[2] + states[2]*deltaQuat[0] + states[3]*deltaQuat[1] - states[1]*deltaQuat[3]; qUpdated[3] = states[0]*deltaQuat[3] + states[3]*deltaQuat[0] + states[1]*deltaQuat[2] - states[2]*deltaQuat[1]; // normalise the quaternions and update the quaternion states qUpdated.normalize(); state.quat = qUpdated; // calculate the body to nav cosine matrix Quaternion q(states[0],states[1],states[2],states[3]); Matrix3f Tbn_temp; q.rotation_matrix(Tbn_temp); prevTnb = Tbn_temp.transposed(); // transform body delta velocities to delta velocities in the nav frame // * and + operators have been overloaded // blended IMU calc delVelNav = Tbn_temp*correctedDelVel12 + gravityNED*dtIMU; // single IMU calcs delVelNav1 = Tbn_temp*correctedDelVel1 + gravityNED*dtIMU; delVelNav2 = Tbn_temp*correctedDelVel2 + gravityNED*dtIMU; // calculate the rate of change of velocity (used for launch detect and other functions) velDotNED = delVelNav / dtIMU ; // apply a first order lowpass filter velDotNEDfilt = velDotNED * 0.05f + velDotNEDfilt * 0.95f; // calculate a magnitude of the filtered nav acceleration (required for GPS // variance estimation) accNavMag = velDotNEDfilt.length(); accNavMagHoriz = pythagorous2(velDotNEDfilt.x , velDotNEDfilt.y); // save velocity for use in trapezoidal intergration for position calcuation Vector3f lastVelocity = state.velocity; Vector3f lastVel1 = state.vel1; Vector3f lastVel2 = state.vel2; // sum delta velocities to get velocity state.velocity += delVelNav; state.vel1 += delVelNav1; state.vel2 += delVelNav2; // apply a trapezoidal integration to velocities to calculate position state.position += (state.velocity + lastVelocity) * (dtIMU*0.5f); state.posD1 += (state.vel1.z + lastVel1.z) * (dtIMU*0.5f); state.posD2 += (state.vel2.z + lastVel2.z) * (dtIMU*0.5f); // limit states to protect against divergence ConstrainStates(); } // calculate the predicted state covariance matrix void NavEKF::CovariancePrediction() { perf_begin(_perf_CovariancePrediction); float windVelSigma; // wind velocity 1-sigma process noise - m/s float dAngBiasSigma;// delta angle bias 1-sigma process noise - rad/s float dVelBiasSigma;// delta velocity bias 1-sigma process noise - m/s float magEarthSigma;// earth magnetic field 1-sigma process noise float magBodySigma; // body magnetic field 1-sigma process noise float daxCov; // X axis delta angle variance rad^2 float dayCov; // Y axis delta angle variance rad^2 float dazCov; // Z axis delta angle variance rad^2 float dvxCov; // X axis delta velocity variance (m/s)^2 float dvyCov; // Y axis delta velocity variance (m/s)^2 float dvzCov; // Z axis delta velocity variance (m/s)^2 float dvx; // X axis delta velocity (m/s) float dvy; // Y axis delta velocity (m/s) float dvz; // Z axis delta velocity (m/s) float dax; // X axis delta angle (rad) float day; // Y axis delta angle (rad) float daz; // Z axis delta angle (rad) float q0; // attitude quaternion float q1; // attitude quaternion float q2; // attitude quaternion float q3; // attitude quaternion float dax_b; // X axis delta angle measurement bias (rad) float day_b; // Y axis delta angle measurement bias (rad) float daz_b; // Z axis delta angle measurement bias (rad) float dvz_b; // Z axis delta velocity measurement bias (rad) // calculate covariance prediction process noise // use filtered height rate to increase wind process noise when climbing or descending // this allows for wind gradient effects. // filter height rate using a 10 second time constant filter float alpha = 0.1f * dt; hgtRate = hgtRate * (1.0f - alpha) - states[6] * alpha; // use filtered height rate to increase wind process noise when climbing or descending // this allows for wind gradient effects. if (!inhibitWindStates) { windVelSigma = dt * constrain_float(_windVelProcessNoise, 0.01f, 1.0f) * (1.0f + constrain_float(_wndVarHgtRateScale, 0.0f, 1.0f) * fabsf(hgtRate)); } else { windVelSigma = 0.0f; } dAngBiasSigma = dt * constrain_float(_gyroBiasProcessNoise, 1e-7f, 1e-5f); dVelBiasSigma = dt * constrain_float(_accelBiasProcessNoise, 1e-5f, 1e-3f); if (!inhibitMagStates) { magEarthSigma = dt * constrain_float(_magEarthProcessNoise, 1e-4f, 1e-2f); magBodySigma = dt * constrain_float(_magBodyProcessNoise, 1e-4f, 1e-2f); } else { magEarthSigma = 0.0f; magBodySigma = 0.0f; } for (uint8_t i= 0; i<=9; i++) processNoise[i] = 1.0e-9f; for (uint8_t i=10; i<=12; i++) processNoise[i] = dAngBiasSigma; // scale gyro bias noise when in static mode to allow for faster bias estimation for (uint8_t i=10; i<=12; i++) { processNoise[i] = dAngBiasSigma; if (staticMode) { processNoise[i] *= _gyroBiasNoiseScaler; } } processNoise[13] = dVelBiasSigma; for (uint8_t i=14; i<=15; i++) processNoise[i] = windVelSigma; for (uint8_t i=16; i<=18; i++) processNoise[i] = magEarthSigma; for (uint8_t i=19; i<=21; i++) processNoise[i] = magBodySigma; for (uint8_t i= 0; i<=21; i++) processNoise[i] = sq(processNoise[i]); // set variables used to calculate covariance growth dvx = summedDelVel.x; dvy = summedDelVel.y; dvz = summedDelVel.z; dax = summedDelAng.x; day = summedDelAng.y; daz = summedDelAng.z; q0 = states[0]; q1 = states[1]; q2 = states[2]; q3 = states[3]; dax_b = states[10]; day_b = states[11]; daz_b = states[12]; dvz_b = IMU1_weighting * states[13] + (1.0f - IMU1_weighting) * states[22]; _gyrNoise = constrain_float(_gyrNoise, 1e-3f, 5e-2f); daxCov = sq(dt*_gyrNoise); dayCov = sq(dt*_gyrNoise); dazCov = sq(dt*_gyrNoise); _accNoise = constrain_float(_accNoise, 5e-2f, 1.0f); dvxCov = sq(dt*_accNoise); dvyCov = sq(dt*_accNoise); dvzCov = sq(dt*_accNoise); // calculate the predicted covariance due to inertial sensor error propagation SF[0] = dvz - dvz_b; SF[1] = 2*q3*SF[0] + 2*dvx*q1 + 2*dvy*q2; SF[2] = 2*dvx*q3 - 2*q1*SF[0] + 2*dvy*q0; SF[3] = 2*q2*SF[0] + 2*dvx*q0 - 2*dvy*q3; SF[4] = day/2 - day_b/2; SF[5] = daz/2 - daz_b/2; SF[6] = dax/2 - dax_b/2; SF[7] = dax_b/2 - dax/2; SF[8] = daz_b/2 - daz/2; SF[9] = day_b/2 - day/2; SF[10] = 2*q0*SF[0]; SF[11] = q1/2; SF[12] = q2/2; SF[13] = q3/2; SF[14] = 2*dvy*q1; SG[0] = q0/2; SG[1] = sq(q3); SG[2] = sq(q2); SG[3] = sq(q1); SG[4] = sq(q0); SG[5] = 2*q2*q3; SG[6] = 2*q1*q3; SG[7] = 2*q1*q2; SQ[0] = dvzCov*(SG[5] - 2*q0*q1)*(SG[1] - SG[2] - SG[3] + SG[4]) - dvyCov*(SG[5] + 2*q0*q1)*(SG[1] - SG[2] + SG[3] - SG[4]) + dvxCov*(SG[6] - 2*q0*q2)*(SG[7] + 2*q0*q3); SQ[1] = dvzCov*(SG[6] + 2*q0*q2)*(SG[1] - SG[2] - SG[3] + SG[4]) - dvxCov*(SG[6] - 2*q0*q2)*(SG[1] + SG[2] - SG[3] - SG[4]) + dvyCov*(SG[5] + 2*q0*q1)*(SG[7] - 2*q0*q3); SQ[2] = dvzCov*(SG[5] - 2*q0*q1)*(SG[6] + 2*q0*q2) - dvyCov*(SG[7] - 2*q0*q3)*(SG[1] - SG[2] + SG[3] - SG[4]) - dvxCov*(SG[7] + 2*q0*q3)*(SG[1] + SG[2] - SG[3] - SG[4]); SQ[3] = (dayCov*q1*SG[0])/2 - (dazCov*q1*SG[0])/2 - (daxCov*q2*q3)/4; SQ[4] = (dazCov*q2*SG[0])/2 - (daxCov*q2*SG[0])/2 - (dayCov*q1*q3)/4; SQ[5] = (daxCov*q3*SG[0])/2 - (dayCov*q3*SG[0])/2 - (dazCov*q1*q2)/4; SQ[6] = (daxCov*q1*q2)/4 - (dazCov*q3*SG[0])/2 - (dayCov*q1*q2)/4; SQ[7] = (dazCov*q1*q3)/4 - (daxCov*q1*q3)/4 - (dayCov*q2*SG[0])/2; SQ[8] = (dayCov*q2*q3)/4 - (daxCov*q1*SG[0])/2 - (dazCov*q2*q3)/4; SQ[9] = sq(SG[0]); SQ[10] = sq(q1); SPP[0] = SF[10] + SF[14] - 2*dvx*q2; SPP[1] = 2*q2*SF[0] + 2*dvx*q0 - 2*dvy*q3; SPP[2] = 2*dvx*q3 - 2*q1*SF[0] + 2*dvy*q0; SPP[3] = 2*q0*q1 - 2*q2*q3; SPP[4] = 2*q0*q2 + 2*q1*q3; SPP[5] = sq(q0) - sq(q1) - sq(q2) + sq(q3); SPP[6] = SF[13]; SPP[7] = SF[12]; nextP[0][0] = P[0][0] + P[1][0]*SF[7] + P[2][0]*SF[9] + P[3][0]*SF[8] + P[10][0]*SF[11] + P[11][0]*SPP[7] + P[12][0]*SPP[6] + (daxCov*SQ[10])/4 + SF[7]*(P[0][1] + P[1][1]*SF[7] + P[2][1]*SF[9] + P[3][1]*SF[8] + P[10][1]*SF[11] + P[11][1]*SPP[7] + P[12][1]*SPP[6]) + SF[9]*(P[0][2] + P[1][2]*SF[7] + P[2][2]*SF[9] + P[3][2]*SF[8] + P[10][2]*SF[11] + P[11][2]*SPP[7] + P[12][2]*SPP[6]) + SF[8]*(P[0][3] + P[1][3]*SF[7] + P[2][3]*SF[9] + P[3][3]*SF[8] + P[10][3]*SF[11] + P[11][3]*SPP[7] + P[12][3]*SPP[6]) + SF[11]*(P[0][10] + P[1][10]*SF[7] + P[2][10]*SF[9] + P[3][10]*SF[8] + P[10][10]*SF[11] + P[11][10]*SPP[7] + P[12][10]*SPP[6]) + SPP[7]*(P[0][11] + P[1][11]*SF[7] + P[2][11]*SF[9] + P[3][11]*SF[8] + P[10][11]*SF[11] + P[11][11]*SPP[7] + P[12][11]*SPP[6]) + SPP[6]*(P[0][12] + P[1][12]*SF[7] + P[2][12]*SF[9] + P[3][12]*SF[8] + P[10][12]*SF[11] + P[11][12]*SPP[7] + P[12][12]*SPP[6]) + (dayCov*sq(q2))/4 + (dazCov*sq(q3))/4; nextP[0][1] = P[0][1] + SQ[8] + P[1][1]*SF[7] + P[2][1]*SF[9] + P[3][1]*SF[8] + P[10][1]*SF[11] + P[11][1]*SPP[7] + P[12][1]*SPP[6] + SF[6]*(P[0][0] + P[1][0]*SF[7] + P[2][0]*SF[9] + P[3][0]*SF[8] + P[10][0]*SF[11] + P[11][0]*SPP[7] + P[12][0]*SPP[6]) + SF[5]*(P[0][2] + P[1][2]*SF[7] + P[2][2]*SF[9] + P[3][2]*SF[8] + P[10][2]*SF[11] + P[11][2]*SPP[7] + P[12][2]*SPP[6]) + SF[9]*(P[0][3] + P[1][3]*SF[7] + P[2][3]*SF[9] + P[3][3]*SF[8] + P[10][3]*SF[11] + P[11][3]*SPP[7] + P[12][3]*SPP[6]) + SPP[6]*(P[0][11] + P[1][11]*SF[7] + P[2][11]*SF[9] + P[3][11]*SF[8] + P[10][11]*SF[11] + P[11][11]*SPP[7] + P[12][11]*SPP[6]) - SPP[7]*(P[0][12] + P[1][12]*SF[7] + P[2][12]*SF[9] + P[3][12]*SF[8] + P[10][12]*SF[11] + P[11][12]*SPP[7] + P[12][12]*SPP[6]) - (q0*(P[0][10] + P[1][10]*SF[7] + P[2][10]*SF[9] + P[3][10]*SF[8] + P[10][10]*SF[11] + P[11][10]*SPP[7] + P[12][10]*SPP[6]))/2; nextP[0][2] = P[0][2] + SQ[7] + P[1][2]*SF[7] + P[2][2]*SF[9] + P[3][2]*SF[8] + P[10][2]*SF[11] + P[11][2]*SPP[7] + P[12][2]*SPP[6] + SF[4]*(P[0][0] + P[1][0]*SF[7] + P[2][0]*SF[9] + P[3][0]*SF[8] + P[10][0]*SF[11] + P[11][0]*SPP[7] + P[12][0]*SPP[6]) + SF[8]*(P[0][1] + P[1][1]*SF[7] + P[2][1]*SF[9] + P[3][1]*SF[8] + P[10][1]*SF[11] + P[11][1]*SPP[7] + P[12][1]*SPP[6]) + SF[6]*(P[0][3] + P[1][3]*SF[7] + P[2][3]*SF[9] + P[3][3]*SF[8] + P[10][3]*SF[11] + P[11][3]*SPP[7] + P[12][3]*SPP[6]) + SF[11]*(P[0][12] + P[1][12]*SF[7] + P[2][12]*SF[9] + P[3][12]*SF[8] + P[10][12]*SF[11] + P[11][12]*SPP[7] + P[12][12]*SPP[6]) - SPP[6]*(P[0][10] + P[1][10]*SF[7] + P[2][10]*SF[9] + P[3][10]*SF[8] + P[10][10]*SF[11] + P[11][10]*SPP[7] + P[12][10]*SPP[6]) - (q0*(P[0][11] + P[1][11]*SF[7] + P[2][11]*SF[9] + P[3][11]*SF[8] + P[10][11]*SF[11] + P[11][11]*SPP[7] + P[12][11]*SPP[6]))/2; nextP[0][3] = P[0][3] + SQ[6] + P[1][3]*SF[7] + P[2][3]*SF[9] + P[3][3]*SF[8] + P[10][3]*SF[11] + P[11][3]*SPP[7] + P[12][3]*SPP[6] + SF[5]*(P[0][0] + P[1][0]*SF[7] + P[2][0]*SF[9] + P[3][0]*SF[8] + P[10][0]*SF[11] + P[11][0]*SPP[7] + P[12][0]*SPP[6]) + SF[4]*(P[0][1] + P[1][1]*SF[7] + P[2][1]*SF[9] + P[3][1]*SF[8] + P[10][1]*SF[11] + P[11][1]*SPP[7] + P[12][1]*SPP[6]) + SF[7]*(P[0][2] + P[1][2]*SF[7] + P[2][2]*SF[9] + P[3][2]*SF[8] + P[10][2]*SF[11] + P[11][2]*SPP[7] + P[12][2]*SPP[6]) - SF[11]*(P[0][11] + P[1][11]*SF[7] + P[2][11]*SF[9] + P[3][11]*SF[8] + P[10][11]*SF[11] + P[11][11]*SPP[7] + P[12][11]*SPP[6]) + SPP[7]*(P[0][10] + P[1][10]*SF[7] + P[2][10]*SF[9] + P[3][10]*SF[8] + P[10][10]*SF[11] + P[11][10]*SPP[7] + P[12][10]*SPP[6]) - (q0*(P[0][12] + P[1][12]*SF[7] + P[2][12]*SF[9] + P[3][12]*SF[8] + P[10][12]*SF[11] + P[11][12]*SPP[7] + P[12][12]*SPP[6]))/2; nextP[0][4] = P[0][4] + P[1][4]*SF[7] + P[2][4]*SF[9] + P[3][4]*SF[8] + P[10][4]*SF[11] + P[11][4]*SPP[7] + P[12][4]*SPP[6] + SF[3]*(P[0][0] + P[1][0]*SF[7] + P[2][0]*SF[9] + P[3][0]*SF[8] + P[10][0]*SF[11] + P[11][0]*SPP[7] + P[12][0]*SPP[6]) + SF[1]*(P[0][1] + P[1][1]*SF[7] + P[2][1]*SF[9] + P[3][1]*SF[8] + P[10][1]*SF[11] + P[11][1]*SPP[7] + P[12][1]*SPP[6]) + SPP[0]*(P[0][2] + P[1][2]*SF[7] + P[2][2]*SF[9] + P[3][2]*SF[8] + P[10][2]*SF[11] + P[11][2]*SPP[7] + P[12][2]*SPP[6]) - SPP[2]*(P[0][3] + P[1][3]*SF[7] + P[2][3]*SF[9] + P[3][3]*SF[8] + P[10][3]*SF[11] + P[11][3]*SPP[7] + P[12][3]*SPP[6]) - SPP[4]*(P[0][13] + P[1][13]*SF[7] + P[2][13]*SF[9] + P[3][13]*SF[8] + P[10][13]*SF[11] + P[11][13]*SPP[7] + P[12][13]*SPP[6]); nextP[0][5] = P[0][5] + P[1][5]*SF[7] + P[2][5]*SF[9] + P[3][5]*SF[8] + P[10][5]*SF[11] + P[11][5]*SPP[7] + P[12][5]*SPP[6] + SF[2]*(P[0][0] + P[1][0]*SF[7] + P[2][0]*SF[9] + P[3][0]*SF[8] + P[10][0]*SF[11] + P[11][0]*SPP[7] + P[12][0]*SPP[6]) + SF[1]*(P[0][2] + P[1][2]*SF[7] + P[2][2]*SF[9] + P[3][2]*SF[8] + P[10][2]*SF[11] + P[11][2]*SPP[7] + P[12][2]*SPP[6]) + SF[3]*(P[0][3] + P[1][3]*SF[7] + P[2][3]*SF[9] + P[3][3]*SF[8] + P[10][3]*SF[11] + P[11][3]*SPP[7] + P[12][3]*SPP[6]) - SPP[0]*(P[0][1] + P[1][1]*SF[7] + P[2][1]*SF[9] + P[3][1]*SF[8] + P[10][1]*SF[11] + P[11][1]*SPP[7] + P[12][1]*SPP[6]) + SPP[3]*(P[0][13] + P[1][13]*SF[7] + P[2][13]*SF[9] + P[3][13]*SF[8] + P[10][13]*SF[11] + P[11][13]*SPP[7] + P[12][13]*SPP[6]); nextP[0][6] = P[0][6] + P[1][6]*SF[7] + P[2][6]*SF[9] + P[3][6]*SF[8] + P[10][6]*SF[11] + P[11][6]*SPP[7] + P[12][6]*SPP[6] + SF[2]*(P[0][1] + P[1][1]*SF[7] + P[2][1]*SF[9] + P[3][1]*SF[8] + P[10][1]*SF[11] + P[11][1]*SPP[7] + P[12][1]*SPP[6]) + SF[1]*(P[0][3] + P[1][3]*SF[7] + P[2][3]*SF[9] + P[3][3]*SF[8] + P[10][3]*SF[11] + P[11][3]*SPP[7] + P[12][3]*SPP[6]) + SPP[0]*(P[0][0] + P[1][0]*SF[7] + P[2][0]*SF[9] + P[3][0]*SF[8] + P[10][0]*SF[11] + P[11][0]*SPP[7] + P[12][0]*SPP[6]) - SPP[1]*(P[0][2] + P[1][2]*SF[7] + P[2][2]*SF[9] + P[3][2]*SF[8] + P[10][2]*SF[11] + P[11][2]*SPP[7] + P[12][2]*SPP[6]) - (sq(q0) - sq(q1) - sq(q2) + sq(q3))*(P[0][13] + P[1][13]*SF[7] + P[2][13]*SF[9] + P[3][13]*SF[8] + P[10][13]*SF[11] + P[11][13]*SPP[7] + P[12][13]*SPP[6]); nextP[0][7] = P[0][7] + P[1][7]*SF[7] + P[2][7]*SF[9] + P[3][7]*SF[8] + P[10][7]*SF[11] + P[11][7]*SPP[7] + P[12][7]*SPP[6] + dt*(P[0][4] + P[1][4]*SF[7] + P[2][4]*SF[9] + P[3][4]*SF[8] + P[10][4]*SF[11] + P[11][4]*SPP[7] + P[12][4]*SPP[6]); nextP[0][8] = P[0][8] + P[1][8]*SF[7] + P[2][8]*SF[9] + P[3][8]*SF[8] + P[10][8]*SF[11] + P[11][8]*SPP[7] + P[12][8]*SPP[6] + dt*(P[0][5] + P[1][5]*SF[7] + P[2][5]*SF[9] + P[3][5]*SF[8] + P[10][5]*SF[11] + P[11][5]*SPP[7] + P[12][5]*SPP[6]); nextP[0][9] = P[0][9] + P[1][9]*SF[7] + P[2][9]*SF[9] + P[3][9]*SF[8] + P[10][9]*SF[11] + P[11][9]*SPP[7] + P[12][9]*SPP[6] + dt*(P[0][6] + P[1][6]*SF[7] + P[2][6]*SF[9] + P[3][6]*SF[8] + P[10][6]*SF[11] + P[11][6]*SPP[7] + P[12][6]*SPP[6]); nextP[0][10] = P[0][10] + P[1][10]*SF[7] + P[2][10]*SF[9] + P[3][10]*SF[8] + P[10][10]*SF[11] + P[11][10]*SPP[7] + P[12][10]*SPP[6]; nextP[0][11] = P[0][11] + P[1][11]*SF[7] + P[2][11]*SF[9] + P[3][11]*SF[8] + P[10][11]*SF[11] + P[11][11]*SPP[7] + P[12][11]*SPP[6]; nextP[0][12] = P[0][12] + P[1][12]*SF[7] + P[2][12]*SF[9] + P[3][12]*SF[8] + P[10][12]*SF[11] + P[11][12]*SPP[7] + P[12][12]*SPP[6]; nextP[0][13] = P[0][13] + P[1][13]*SF[7] + P[2][13]*SF[9] + P[3][13]*SF[8] + P[10][13]*SF[11] + P[11][13]*SPP[7] + P[12][13]*SPP[6]; nextP[0][14] = P[0][14] + P[1][14]*SF[7] + P[2][14]*SF[9] + P[3][14]*SF[8] + P[10][14]*SF[11] + P[11][14]*SPP[7] + P[12][14]*SPP[6]; nextP[0][15] = P[0][15] + P[1][15]*SF[7] + P[2][15]*SF[9] + P[3][15]*SF[8] + P[10][15]*SF[11] + P[11][15]*SPP[7] + P[12][15]*SPP[6]; nextP[0][16] = P[0][16] + P[1][16]*SF[7] + P[2][16]*SF[9] + P[3][16]*SF[8] + P[10][16]*SF[11] + P[11][16]*SPP[7] + P[12][16]*SPP[6]; nextP[0][17] = P[0][17] + P[1][17]*SF[7] + P[2][17]*SF[9] + P[3][17]*SF[8] + P[10][17]*SF[11] + P[11][17]*SPP[7] + P[12][17]*SPP[6]; nextP[0][18] = P[0][18] + P[1][18]*SF[7] + P[2][18]*SF[9] + P[3][18]*SF[8] + P[10][18]*SF[11] + P[11][18]*SPP[7] + P[12][18]*SPP[6]; nextP[0][19] = P[0][19] + P[1][19]*SF[7] + P[2][19]*SF[9] + P[3][19]*SF[8] + P[10][19]*SF[11] + P[11][19]*SPP[7] + P[12][19]*SPP[6]; nextP[0][20] = P[0][20] + P[1][20]*SF[7] + P[2][20]*SF[9] + P[3][20]*SF[8] + P[10][20]*SF[11] + P[11][20]*SPP[7] + P[12][20]*SPP[6]; nextP[0][21] = P[0][21] + P[1][21]*SF[7] + P[2][21]*SF[9] + P[3][21]*SF[8] + P[10][21]*SF[11] + P[11][21]*SPP[7] + P[12][21]*SPP[6]; nextP[1][0] = P[1][0] + SQ[8] + P[0][0]*SF[6] + P[2][0]*SF[5] + P[3][0]*SF[9] + P[11][0]*SPP[6] - P[12][0]*SPP[7] - (P[10][0]*q0)/2 + SF[7]*(P[1][1] + P[0][1]*SF[6] + P[2][1]*SF[5] + P[3][1]*SF[9] + P[11][1]*SPP[6] - P[12][1]*SPP[7] - (P[10][1]*q0)/2) + SF[9]*(P[1][2] + P[0][2]*SF[6] + P[2][2]*SF[5] + P[3][2]*SF[9] + P[11][2]*SPP[6] - P[12][2]*SPP[7] - (P[10][2]*q0)/2) + SF[8]*(P[1][3] + P[0][3]*SF[6] + P[2][3]*SF[5] + P[3][3]*SF[9] + P[11][3]*SPP[6] - P[12][3]*SPP[7] - (P[10][3]*q0)/2) + SF[11]*(P[1][10] + P[0][10]*SF[6] + P[2][10]*SF[5] + P[3][10]*SF[9] + P[11][10]*SPP[6] - P[12][10]*SPP[7] - (P[10][10]*q0)/2) + SPP[7]*(P[1][11] + P[0][11]*SF[6] + P[2][11]*SF[5] + P[3][11]*SF[9] + P[11][11]*SPP[6] - P[12][11]*SPP[7] - (P[10][11]*q0)/2) + SPP[6]*(P[1][12] + P[0][12]*SF[6] + P[2][12]*SF[5] + P[3][12]*SF[9] + P[11][12]*SPP[6] - P[12][12]*SPP[7] - (P[10][12]*q0)/2); nextP[1][1] = P[1][1] + P[0][1]*SF[6] + P[2][1]*SF[5] + P[3][1]*SF[9] + P[11][1]*SPP[6] - P[12][1]*SPP[7] + daxCov*SQ[9] - (P[10][1]*q0)/2 + SF[6]*(P[1][0] + P[0][0]*SF[6] + P[2][0]*SF[5] + P[3][0]*SF[9] + P[11][0]*SPP[6] - P[12][0]*SPP[7] - (P[10][0]*q0)/2) + SF[5]*(P[1][2] + P[0][2]*SF[6] + P[2][2]*SF[5] + P[3][2]*SF[9] + P[11][2]*SPP[6] - P[12][2]*SPP[7] - (P[10][2]*q0)/2) + SF[9]*(P[1][3] + P[0][3]*SF[6] + P[2][3]*SF[5] + P[3][3]*SF[9] + P[11][3]*SPP[6] - P[12][3]*SPP[7] - (P[10][3]*q0)/2) + SPP[6]*(P[1][11] + P[0][11]*SF[6] + P[2][11]*SF[5] + P[3][11]*SF[9] + P[11][11]*SPP[6] - P[12][11]*SPP[7] - (P[10][11]*q0)/2) - SPP[7]*(P[1][12] + P[0][12]*SF[6] + P[2][12]*SF[5] + P[3][12]*SF[9] + P[11][12]*SPP[6] - P[12][12]*SPP[7] - (P[10][12]*q0)/2) + (dayCov*sq(q3))/4 + (dazCov*sq(q2))/4 - (q0*(P[1][10] + P[0][10]*SF[6] + P[2][10]*SF[5] + P[3][10]*SF[9] + P[11][10]*SPP[6] - P[12][10]*SPP[7] - (P[10][10]*q0)/2))/2; nextP[1][2] = P[1][2] + SQ[5] + P[0][2]*SF[6] + P[2][2]*SF[5] + P[3][2]*SF[9] + P[11][2]*SPP[6] - P[12][2]*SPP[7] - (P[10][2]*q0)/2 + SF[4]*(P[1][0] + P[0][0]*SF[6] + P[2][0]*SF[5] + P[3][0]*SF[9] + P[11][0]*SPP[6] - P[12][0]*SPP[7] - (P[10][0]*q0)/2) + SF[8]*(P[1][1] + P[0][1]*SF[6] + P[2][1]*SF[5] + P[3][1]*SF[9] + P[11][1]*SPP[6] - P[12][1]*SPP[7] - (P[10][1]*q0)/2) + SF[6]*(P[1][3] + P[0][3]*SF[6] + P[2][3]*SF[5] + P[3][3]*SF[9] + P[11][3]*SPP[6] - P[12][3]*SPP[7] - (P[10][3]*q0)/2) + SF[11]*(P[1][12] + P[0][12]*SF[6] + P[2][12]*SF[5] + P[3][12]*SF[9] + P[11][12]*SPP[6] - P[12][12]*SPP[7] - (P[10][12]*q0)/2) - SPP[6]*(P[1][10] + P[0][10]*SF[6] + P[2][10]*SF[5] + P[3][10]*SF[9] + P[11][10]*SPP[6] - P[12][10]*SPP[7] - (P[10][10]*q0)/2) - (q0*(P[1][11] + P[0][11]*SF[6] + P[2][11]*SF[5] + P[3][11]*SF[9] + P[11][11]*SPP[6] - P[12][11]*SPP[7] - (P[10][11]*q0)/2))/2; nextP[1][3] = P[1][3] + SQ[4] + P[0][3]*SF[6] + P[2][3]*SF[5] + P[3][3]*SF[9] + P[11][3]*SPP[6] - P[12][3]*SPP[7] - (P[10][3]*q0)/2 + SF[5]*(P[1][0] + P[0][0]*SF[6] + P[2][0]*SF[5] + P[3][0]*SF[9] + P[11][0]*SPP[6] - P[12][0]*SPP[7] - (P[10][0]*q0)/2) + SF[4]*(P[1][1] + P[0][1]*SF[6] + P[2][1]*SF[5] + P[3][1]*SF[9] + P[11][1]*SPP[6] - P[12][1]*SPP[7] - (P[10][1]*q0)/2) + SF[7]*(P[1][2] + P[0][2]*SF[6] + P[2][2]*SF[5] + P[3][2]*SF[9] + P[11][2]*SPP[6] - P[12][2]*SPP[7] - (P[10][2]*q0)/2) - SF[11]*(P[1][11] + P[0][11]*SF[6] + P[2][11]*SF[5] + P[3][11]*SF[9] + P[11][11]*SPP[6] - P[12][11]*SPP[7] - (P[10][11]*q0)/2) + SPP[7]*(P[1][10] + P[0][10]*SF[6] + P[2][10]*SF[5] + P[3][10]*SF[9] + P[11][10]*SPP[6] - P[12][10]*SPP[7] - (P[10][10]*q0)/2) - (q0*(P[1][12] + P[0][12]*SF[6] + P[2][12]*SF[5] + P[3][12]*SF[9] + P[11][12]*SPP[6] - P[12][12]*SPP[7] - (P[10][12]*q0)/2))/2; nextP[1][4] = P[1][4] + P[0][4]*SF[6] + P[2][4]*SF[5] + P[3][4]*SF[9] + P[11][4]*SPP[6] - P[12][4]*SPP[7] - (P[10][4]*q0)/2 + SF[3]*(P[1][0] + P[0][0]*SF[6] + P[2][0]*SF[5] + P[3][0]*SF[9] + P[11][0]*SPP[6] - P[12][0]*SPP[7] - (P[10][0]*q0)/2) + SF[1]*(P[1][1] + P[0][1]*SF[6] + P[2][1]*SF[5] + P[3][1]*SF[9] + P[11][1]*SPP[6] - P[12][1]*SPP[7] - (P[10][1]*q0)/2) + SPP[0]*(P[1][2] + P[0][2]*SF[6] + P[2][2]*SF[5] + P[3][2]*SF[9] + P[11][2]*SPP[6] - P[12][2]*SPP[7] - (P[10][2]*q0)/2) - SPP[2]*(P[1][3] + P[0][3]*SF[6] + P[2][3]*SF[5] + P[3][3]*SF[9] + P[11][3]*SPP[6] - P[12][3]*SPP[7] - (P[10][3]*q0)/2) - SPP[4]*(P[1][13] + P[0][13]*SF[6] + P[2][13]*SF[5] + P[3][13]*SF[9] + P[11][13]*SPP[6] - P[12][13]*SPP[7] - (P[10][13]*q0)/2); nextP[1][5] = P[1][5] + P[0][5]*SF[6] + P[2][5]*SF[5] + P[3][5]*SF[9] + P[11][5]*SPP[6] - P[12][5]*SPP[7] - (P[10][5]*q0)/2 + SF[2]*(P[1][0] + P[0][0]*SF[6] + P[2][0]*SF[5] + P[3][0]*SF[9] + P[11][0]*SPP[6] - P[12][0]*SPP[7] - (P[10][0]*q0)/2) + SF[1]*(P[1][2] + P[0][2]*SF[6] + P[2][2]*SF[5] + P[3][2]*SF[9] + P[11][2]*SPP[6] - P[12][2]*SPP[7] - (P[10][2]*q0)/2) + SF[3]*(P[1][3] + P[0][3]*SF[6] + P[2][3]*SF[5] + P[3][3]*SF[9] + P[11][3]*SPP[6] - P[12][3]*SPP[7] - (P[10][3]*q0)/2) - SPP[0]*(P[1][1] + P[0][1]*SF[6] + P[2][1]*SF[5] + P[3][1]*SF[9] + P[11][1]*SPP[6] - P[12][1]*SPP[7] - (P[10][1]*q0)/2) + SPP[3]*(P[1][13] + P[0][13]*SF[6] + P[2][13]*SF[5] + P[3][13]*SF[9] + P[11][13]*SPP[6] - P[12][13]*SPP[7] - (P[10][13]*q0)/2); nextP[1][6] = P[1][6] + P[0][6]*SF[6] + P[2][6]*SF[5] + P[3][6]*SF[9] + P[11][6]*SPP[6] - P[12][6]*SPP[7] - (P[10][6]*q0)/2 + SF[2]*(P[1][1] + P[0][1]*SF[6] + P[2][1]*SF[5] + P[3][1]*SF[9] + P[11][1]*SPP[6] - P[12][1]*SPP[7] - (P[10][1]*q0)/2) + SF[1]*(P[1][3] + P[0][3]*SF[6] + P[2][3]*SF[5] + P[3][3]*SF[9] + P[11][3]*SPP[6] - P[12][3]*SPP[7] - (P[10][3]*q0)/2) + SPP[0]*(P[1][0] + P[0][0]*SF[6] + P[2][0]*SF[5] + P[3][0]*SF[9] + P[11][0]*SPP[6] - P[12][0]*SPP[7] - (P[10][0]*q0)/2) - SPP[1]*(P[1][2] + P[0][2]*SF[6] + P[2][2]*SF[5] + P[3][2]*SF[9] + P[11][2]*SPP[6] - P[12][2]*SPP[7] - (P[10][2]*q0)/2) - (sq(q0) - sq(q1) - sq(q2) + sq(q3))*(P[1][13] + P[0][13]*SF[6] + P[2][13]*SF[5] + P[3][13]*SF[9] + P[11][13]*SPP[6] - P[12][13]*SPP[7] - (P[10][13]*q0)/2); nextP[1][7] = P[1][7] + P[0][7]*SF[6] + P[2][7]*SF[5] + P[3][7]*SF[9] + P[11][7]*SPP[6] - P[12][7]*SPP[7] - (P[10][7]*q0)/2 + dt*(P[1][4] + P[0][4]*SF[6] + P[2][4]*SF[5] + P[3][4]*SF[9] + P[11][4]*SPP[6] - P[12][4]*SPP[7] - (P[10][4]*q0)/2); nextP[1][8] = P[1][8] + P[0][8]*SF[6] + P[2][8]*SF[5] + P[3][8]*SF[9] + P[11][8]*SPP[6] - P[12][8]*SPP[7] - (P[10][8]*q0)/2 + dt*(P[1][5] + P[0][5]*SF[6] + P[2][5]*SF[5] + P[3][5]*SF[9] + P[11][5]*SPP[6] - P[12][5]*SPP[7] - (P[10][5]*q0)/2); nextP[1][9] = P[1][9] + P[0][9]*SF[6] + P[2][9]*SF[5] + P[3][9]*SF[9] + P[11][9]*SPP[6] - P[12][9]*SPP[7] - (P[10][9]*q0)/2 + dt*(P[1][6] + P[0][6]*SF[6] + P[2][6]*SF[5] + P[3][6]*SF[9] + P[11][6]*SPP[6] - P[12][6]*SPP[7] - (P[10][6]*q0)/2); nextP[1][10] = P[1][10] + P[0][10]*SF[6] + P[2][10]*SF[5] + P[3][10]*SF[9] + P[11][10]*SPP[6] - P[12][10]*SPP[7] - (P[10][10]*q0)/2; nextP[1][11] = P[1][11] + P[0][11]*SF[6] + P[2][11]*SF[5] + P[3][11]*SF[9] + P[11][11]*SPP[6] - P[12][11]*SPP[7] - (P[10][11]*q0)/2; nextP[1][12] = P[1][12] + P[0][12]*SF[6] + P[2][12]*SF[5] + P[3][12]*SF[9] + P[11][12]*SPP[6] - P[12][12]*SPP[7] - (P[10][12]*q0)/2; nextP[1][13] = P[1][13] + P[0][13]*SF[6] + P[2][13]*SF[5] + P[3][13]*SF[9] + P[11][13]*SPP[6] - P[12][13]*SPP[7] - (P[10][13]*q0)/2; nextP[1][14] = P[1][14] + P[0][14]*SF[6] + P[2][14]*SF[5] + P[3][14]*SF[9] + P[11][14]*SPP[6] - P[12][14]*SPP[7] - (P[10][14]*q0)/2; nextP[1][15] = P[1][15] + P[0][15]*SF[6] + P[2][15]*SF[5] + P[3][15]*SF[9] + P[11][15]*SPP[6] - P[12][15]*SPP[7] - (P[10][15]*q0)/2; nextP[1][16] = P[1][16] + P[0][16]*SF[6] + P[2][16]*SF[5] + P[3][16]*SF[9] + P[11][16]*SPP[6] - P[12][16]*SPP[7] - (P[10][16]*q0)/2; nextP[1][17] = P[1][17] + P[0][17]*SF[6] + P[2][17]*SF[5] + P[3][17]*SF[9] + P[11][17]*SPP[6] - P[12][17]*SPP[7] - (P[10][17]*q0)/2; nextP[1][18] = P[1][18] + P[0][18]*SF[6] + P[2][18]*SF[5] + P[3][18]*SF[9] + P[11][18]*SPP[6] - P[12][18]*SPP[7] - (P[10][18]*q0)/2; nextP[1][19] = P[1][19] + P[0][19]*SF[6] + P[2][19]*SF[5] + P[3][19]*SF[9] + P[11][19]*SPP[6] - P[12][19]*SPP[7] - (P[10][19]*q0)/2; nextP[1][20] = P[1][20] + P[0][20]*SF[6] + P[2][20]*SF[5] + P[3][20]*SF[9] + P[11][20]*SPP[6] - P[12][20]*SPP[7] - (P[10][20]*q0)/2; nextP[1][21] = P[1][21] + P[0][21]*SF[6] + P[2][21]*SF[5] + P[3][21]*SF[9] + P[11][21]*SPP[6] - P[12][21]*SPP[7] - (P[10][21]*q0)/2; nextP[2][0] = P[2][0] + SQ[7] + P[0][0]*SF[4] + P[1][0]*SF[8] + P[3][0]*SF[6] + P[12][0]*SF[11] - P[10][0]*SPP[6] - (P[11][0]*q0)/2 + SF[7]*(P[2][1] + P[0][1]*SF[4] + P[1][1]*SF[8] + P[3][1]*SF[6] + P[12][1]*SF[11] - P[10][1]*SPP[6] - (P[11][1]*q0)/2) + SF[9]*(P[2][2] + P[0][2]*SF[4] + P[1][2]*SF[8] + P[3][2]*SF[6] + P[12][2]*SF[11] - P[10][2]*SPP[6] - (P[11][2]*q0)/2) + SF[8]*(P[2][3] + P[0][3]*SF[4] + P[1][3]*SF[8] + P[3][3]*SF[6] + P[12][3]*SF[11] - P[10][3]*SPP[6] - (P[11][3]*q0)/2) + SF[11]*(P[2][10] + P[0][10]*SF[4] + P[1][10]*SF[8] + P[3][10]*SF[6] + P[12][10]*SF[11] - P[10][10]*SPP[6] - (P[11][10]*q0)/2) + SPP[7]*(P[2][11] + P[0][11]*SF[4] + P[1][11]*SF[8] + P[3][11]*SF[6] + P[12][11]*SF[11] - P[10][11]*SPP[6] - (P[11][11]*q0)/2) + SPP[6]*(P[2][12] + P[0][12]*SF[4] + P[1][12]*SF[8] + P[3][12]*SF[6] + P[12][12]*SF[11] - P[10][12]*SPP[6] - (P[11][12]*q0)/2); nextP[2][1] = P[2][1] + SQ[5] + P[0][1]*SF[4] + P[1][1]*SF[8] + P[3][1]*SF[6] + P[12][1]*SF[11] - P[10][1]*SPP[6] - (P[11][1]*q0)/2 + SF[6]*(P[2][0] + P[0][0]*SF[4] + P[1][0]*SF[8] + P[3][0]*SF[6] + P[12][0]*SF[11] - P[10][0]*SPP[6] - (P[11][0]*q0)/2) + SF[5]*(P[2][2] + P[0][2]*SF[4] + P[1][2]*SF[8] + P[3][2]*SF[6] + P[12][2]*SF[11] - P[10][2]*SPP[6] - (P[11][2]*q0)/2) + SF[9]*(P[2][3] + P[0][3]*SF[4] + P[1][3]*SF[8] + P[3][3]*SF[6] + P[12][3]*SF[11] - P[10][3]*SPP[6] - (P[11][3]*q0)/2) + SPP[6]*(P[2][11] + P[0][11]*SF[4] + P[1][11]*SF[8] + P[3][11]*SF[6] + P[12][11]*SF[11] - P[10][11]*SPP[6] - (P[11][11]*q0)/2) - SPP[7]*(P[2][12] + P[0][12]*SF[4] + P[1][12]*SF[8] + P[3][12]*SF[6] + P[12][12]*SF[11] - P[10][12]*SPP[6] - (P[11][12]*q0)/2) - (q0*(P[2][10] + P[0][10]*SF[4] + P[1][10]*SF[8] + P[3][10]*SF[6] + P[12][10]*SF[11] - P[10][10]*SPP[6] - (P[11][10]*q0)/2))/2; nextP[2][2] = P[2][2] + P[0][2]*SF[4] + P[1][2]*SF[8] + P[3][2]*SF[6] + P[12][2]*SF[11] - P[10][2]*SPP[6] + dayCov*SQ[9] + (dazCov*SQ[10])/4 - (P[11][2]*q0)/2 + SF[4]*(P[2][0] + P[0][0]*SF[4] + P[1][0]*SF[8] + P[3][0]*SF[6] + P[12][0]*SF[11] - P[10][0]*SPP[6] - (P[11][0]*q0)/2) + SF[8]*(P[2][1] + P[0][1]*SF[4] + P[1][1]*SF[8] + P[3][1]*SF[6] + P[12][1]*SF[11] - P[10][1]*SPP[6] - (P[11][1]*q0)/2) + SF[6]*(P[2][3] + P[0][3]*SF[4] + P[1][3]*SF[8] + P[3][3]*SF[6] + P[12][3]*SF[11] - P[10][3]*SPP[6] - (P[11][3]*q0)/2) + SF[11]*(P[2][12] + P[0][12]*SF[4] + P[1][12]*SF[8] + P[3][12]*SF[6] + P[12][12]*SF[11] - P[10][12]*SPP[6] - (P[11][12]*q0)/2) - SPP[6]*(P[2][10] + P[0][10]*SF[4] + P[1][10]*SF[8] + P[3][10]*SF[6] + P[12][10]*SF[11] - P[10][10]*SPP[6] - (P[11][10]*q0)/2) + (daxCov*sq(q3))/4 - (q0*(P[2][11] + P[0][11]*SF[4] + P[1][11]*SF[8] + P[3][11]*SF[6] + P[12][11]*SF[11] - P[10][11]*SPP[6] - (P[11][11]*q0)/2))/2; nextP[2][3] = P[2][3] + SQ[3] + P[0][3]*SF[4] + P[1][3]*SF[8] + P[3][3]*SF[6] + P[12][3]*SF[11] - P[10][3]*SPP[6] - (P[11][3]*q0)/2 + SF[5]*(P[2][0] + P[0][0]*SF[4] + P[1][0]*SF[8] + P[3][0]*SF[6] + P[12][0]*SF[11] - P[10][0]*SPP[6] - (P[11][0]*q0)/2) + SF[4]*(P[2][1] + P[0][1]*SF[4] + P[1][1]*SF[8] + P[3][1]*SF[6] + P[12][1]*SF[11] - P[10][1]*SPP[6] - (P[11][1]*q0)/2) + SF[7]*(P[2][2] + P[0][2]*SF[4] + P[1][2]*SF[8] + P[3][2]*SF[6] + P[12][2]*SF[11] - P[10][2]*SPP[6] - (P[11][2]*q0)/2) - SF[11]*(P[2][11] + P[0][11]*SF[4] + P[1][11]*SF[8] + P[3][11]*SF[6] + P[12][11]*SF[11] - P[10][11]*SPP[6] - (P[11][11]*q0)/2) + SPP[7]*(P[2][10] + P[0][10]*SF[4] + P[1][10]*SF[8] + P[3][10]*SF[6] + P[12][10]*SF[11] - P[10][10]*SPP[6] - (P[11][10]*q0)/2) - (q0*(P[2][12] + P[0][12]*SF[4] + P[1][12]*SF[8] + P[3][12]*SF[6] + P[12][12]*SF[11] - P[10][12]*SPP[6] - (P[11][12]*q0)/2))/2; nextP[2][4] = P[2][4] + P[0][4]*SF[4] + P[1][4]*SF[8] + P[3][4]*SF[6] + P[12][4]*SF[11] - P[10][4]*SPP[6] - (P[11][4]*q0)/2 + SF[3]*(P[2][0] + P[0][0]*SF[4] + P[1][0]*SF[8] + P[3][0]*SF[6] + P[12][0]*SF[11] - P[10][0]*SPP[6] - (P[11][0]*q0)/2) + SF[1]*(P[2][1] + P[0][1]*SF[4] + P[1][1]*SF[8] + P[3][1]*SF[6] + P[12][1]*SF[11] - P[10][1]*SPP[6] - (P[11][1]*q0)/2) + SPP[0]*(P[2][2] + P[0][2]*SF[4] + P[1][2]*SF[8] + P[3][2]*SF[6] + P[12][2]*SF[11] - P[10][2]*SPP[6] - (P[11][2]*q0)/2) - SPP[2]*(P[2][3] + P[0][3]*SF[4] + P[1][3]*SF[8] + P[3][3]*SF[6] + P[12][3]*SF[11] - P[10][3]*SPP[6] - (P[11][3]*q0)/2) - SPP[4]*(P[2][13] + P[0][13]*SF[4] + P[1][13]*SF[8] + P[3][13]*SF[6] + P[12][13]*SF[11] - P[10][13]*SPP[6] - (P[11][13]*q0)/2); nextP[2][5] = P[2][5] + P[0][5]*SF[4] + P[1][5]*SF[8] + P[3][5]*SF[6] + P[12][5]*SF[11] - P[10][5]*SPP[6] - (P[11][5]*q0)/2 + SF[2]*(P[2][0] + P[0][0]*SF[4] + P[1][0]*SF[8] + P[3][0]*SF[6] + P[12][0]*SF[11] - P[10][0]*SPP[6] - (P[11][0]*q0)/2) + SF[1]*(P[2][2] + P[0][2]*SF[4] + P[1][2]*SF[8] + P[3][2]*SF[6] + P[12][2]*SF[11] - P[10][2]*SPP[6] - (P[11][2]*q0)/2) + SF[3]*(P[2][3] + P[0][3]*SF[4] + P[1][3]*SF[8] + P[3][3]*SF[6] + P[12][3]*SF[11] - P[10][3]*SPP[6] - (P[11][3]*q0)/2) - SPP[0]*(P[2][1] + P[0][1]*SF[4] + P[1][1]*SF[8] + P[3][1]*SF[6] + P[12][1]*SF[11] - P[10][1]*SPP[6] - (P[11][1]*q0)/2) + SPP[3]*(P[2][13] + P[0][13]*SF[4] + P[1][13]*SF[8] + P[3][13]*SF[6] + P[12][13]*SF[11] - P[10][13]*SPP[6] - (P[11][13]*q0)/2); nextP[2][6] = P[2][6] + P[0][6]*SF[4] + P[1][6]*SF[8] + P[3][6]*SF[6] + P[12][6]*SF[11] - P[10][6]*SPP[6] - (P[11][6]*q0)/2 + SF[2]*(P[2][1] + P[0][1]*SF[4] + P[1][1]*SF[8] + P[3][1]*SF[6] + P[12][1]*SF[11] - P[10][1]*SPP[6] - (P[11][1]*q0)/2) + SF[1]*(P[2][3] + P[0][3]*SF[4] + P[1][3]*SF[8] + P[3][3]*SF[6] + P[12][3]*SF[11] - P[10][3]*SPP[6] - (P[11][3]*q0)/2) + SPP[0]*(P[2][0] + P[0][0]*SF[4] + P[1][0]*SF[8] + P[3][0]*SF[6] + P[12][0]*SF[11] - P[10][0]*SPP[6] - (P[11][0]*q0)/2) - SPP[1]*(P[2][2] + P[0][2]*SF[4] + P[1][2]*SF[8] + P[3][2]*SF[6] + P[12][2]*SF[11] - P[10][2]*SPP[6] - (P[11][2]*q0)/2) - (sq(q0) - sq(q1) - sq(q2) + sq(q3))*(P[2][13] + P[0][13]*SF[4] + P[1][13]*SF[8] + P[3][13]*SF[6] + P[12][13]*SF[11] - P[10][13]*SPP[6] - (P[11][13]*q0)/2); nextP[2][7] = P[2][7] + P[0][7]*SF[4] + P[1][7]*SF[8] + P[3][7]*SF[6] + P[12][7]*SF[11] - P[10][7]*SPP[6] - (P[11][7]*q0)/2 + dt*(P[2][4] + P[0][4]*SF[4] + P[1][4]*SF[8] + P[3][4]*SF[6] + P[12][4]*SF[11] - P[10][4]*SPP[6] - (P[11][4]*q0)/2); nextP[2][8] = P[2][8] + P[0][8]*SF[4] + P[1][8]*SF[8] + P[3][8]*SF[6] + P[12][8]*SF[11] - P[10][8]*SPP[6] - (P[11][8]*q0)/2 + dt*(P[2][5] + P[0][5]*SF[4] + P[1][5]*SF[8] + P[3][5]*SF[6] + P[12][5]*SF[11] - P[10][5]*SPP[6] - (P[11][5]*q0)/2); nextP[2][9] = P[2][9] + P[0][9]*SF[4] + P[1][9]*SF[8] + P[3][9]*SF[6] + P[12][9]*SF[11] - P[10][9]*SPP[6] - (P[11][9]*q0)/2 + dt*(P[2][6] + P[0][6]*SF[4] + P[1][6]*SF[8] + P[3][6]*SF[6] + P[12][6]*SF[11] - P[10][6]*SPP[6] - (P[11][6]*q0)/2); nextP[2][10] = P[2][10] + P[0][10]*SF[4] + P[1][10]*SF[8] + P[3][10]*SF[6] + P[12][10]*SF[11] - P[10][10]*SPP[6] - (P[11][10]*q0)/2; nextP[2][11] = P[2][11] + P[0][11]*SF[4] + P[1][11]*SF[8] + P[3][11]*SF[6] + P[12][11]*SF[11] - P[10][11]*SPP[6] - (P[11][11]*q0)/2; nextP[2][12] = P[2][12] + P[0][12]*SF[4] + P[1][12]*SF[8] + P[3][12]*SF[6] + P[12][12]*SF[11] - P[10][12]*SPP[6] - (P[11][12]*q0)/2; nextP[2][13] = P[2][13] + P[0][13]*SF[4] + P[1][13]*SF[8] + P[3][13]*SF[6] + P[12][13]*SF[11] - P[10][13]*SPP[6] - (P[11][13]*q0)/2; nextP[2][14] = P[2][14] + P[0][14]*SF[4] + P[1][14]*SF[8] + P[3][14]*SF[6] + P[12][14]*SF[11] - P[10][14]*SPP[6] - (P[11][14]*q0)/2; nextP[2][15] = P[2][15] + P[0][15]*SF[4] + P[1][15]*SF[8] + P[3][15]*SF[6] + P[12][15]*SF[11] - P[10][15]*SPP[6] - (P[11][15]*q0)/2; nextP[2][16] = P[2][16] + P[0][16]*SF[4] + P[1][16]*SF[8] + P[3][16]*SF[6] + P[12][16]*SF[11] - P[10][16]*SPP[6] - (P[11][16]*q0)/2; nextP[2][17] = P[2][17] + P[0][17]*SF[4] + P[1][17]*SF[8] + P[3][17]*SF[6] + P[12][17]*SF[11] - P[10][17]*SPP[6] - (P[11][17]*q0)/2; nextP[2][18] = P[2][18] + P[0][18]*SF[4] + P[1][18]*SF[8] + P[3][18]*SF[6] + P[12][18]*SF[11] - P[10][18]*SPP[6] - (P[11][18]*q0)/2; nextP[2][19] = P[2][19] + P[0][19]*SF[4] + P[1][19]*SF[8] + P[3][19]*SF[6] + P[12][19]*SF[11] - P[10][19]*SPP[6] - (P[11][19]*q0)/2; nextP[2][20] = P[2][20] + P[0][20]*SF[4] + P[1][20]*SF[8] + P[3][20]*SF[6] + P[12][20]*SF[11] - P[10][20]*SPP[6] - (P[11][20]*q0)/2; nextP[2][21] = P[2][21] + P[0][21]*SF[4] + P[1][21]*SF[8] + P[3][21]*SF[6] + P[12][21]*SF[11] - P[10][21]*SPP[6] - (P[11][21]*q0)/2; nextP[3][0] = P[3][0] + SQ[6] + P[0][0]*SF[5] + P[1][0]*SF[4] + P[2][0]*SF[7] - P[11][0]*SF[11] + P[10][0]*SPP[7] - (P[12][0]*q0)/2 + SF[7]*(P[3][1] + P[0][1]*SF[5] + P[1][1]*SF[4] + P[2][1]*SF[7] - P[11][1]*SF[11] + P[10][1]*SPP[7] - (P[12][1]*q0)/2) + SF[9]*(P[3][2] + P[0][2]*SF[5] + P[1][2]*SF[4] + P[2][2]*SF[7] - P[11][2]*SF[11] + P[10][2]*SPP[7] - (P[12][2]*q0)/2) + SF[8]*(P[3][3] + P[0][3]*SF[5] + P[1][3]*SF[4] + P[2][3]*SF[7] - P[11][3]*SF[11] + P[10][3]*SPP[7] - (P[12][3]*q0)/2) + SF[11]*(P[3][10] + P[0][10]*SF[5] + P[1][10]*SF[4] + P[2][10]*SF[7] - P[11][10]*SF[11] + P[10][10]*SPP[7] - (P[12][10]*q0)/2) + SPP[7]*(P[3][11] + P[0][11]*SF[5] + P[1][11]*SF[4] + P[2][11]*SF[7] - P[11][11]*SF[11] + P[10][11]*SPP[7] - (P[12][11]*q0)/2) + SPP[6]*(P[3][12] + P[0][12]*SF[5] + P[1][12]*SF[4] + P[2][12]*SF[7] - P[11][12]*SF[11] + P[10][12]*SPP[7] - (P[12][12]*q0)/2); nextP[3][1] = P[3][1] + SQ[4] + P[0][1]*SF[5] + P[1][1]*SF[4] + P[2][1]*SF[7] - P[11][1]*SF[11] + P[10][1]*SPP[7] - (P[12][1]*q0)/2 + SF[6]*(P[3][0] + P[0][0]*SF[5] + P[1][0]*SF[4] + P[2][0]*SF[7] - P[11][0]*SF[11] + P[10][0]*SPP[7] - (P[12][0]*q0)/2) + SF[5]*(P[3][2] + P[0][2]*SF[5] + P[1][2]*SF[4] + P[2][2]*SF[7] - P[11][2]*SF[11] + P[10][2]*SPP[7] - (P[12][2]*q0)/2) + SF[9]*(P[3][3] + P[0][3]*SF[5] + P[1][3]*SF[4] + P[2][3]*SF[7] - P[11][3]*SF[11] + P[10][3]*SPP[7] - (P[12][3]*q0)/2) + SPP[6]*(P[3][11] + P[0][11]*SF[5] + P[1][11]*SF[4] + P[2][11]*SF[7] - P[11][11]*SF[11] + P[10][11]*SPP[7] - (P[12][11]*q0)/2) - SPP[7]*(P[3][12] + P[0][12]*SF[5] + P[1][12]*SF[4] + P[2][12]*SF[7] - P[11][12]*SF[11] + P[10][12]*SPP[7] - (P[12][12]*q0)/2) - (q0*(P[3][10] + P[0][10]*SF[5] + P[1][10]*SF[4] + P[2][10]*SF[7] - P[11][10]*SF[11] + P[10][10]*SPP[7] - (P[12][10]*q0)/2))/2; nextP[3][2] = P[3][2] + SQ[3] + P[0][2]*SF[5] + P[1][2]*SF[4] + P[2][2]*SF[7] - P[11][2]*SF[11] + P[10][2]*SPP[7] - (P[12][2]*q0)/2 + SF[4]*(P[3][0] + P[0][0]*SF[5] + P[1][0]*SF[4] + P[2][0]*SF[7] - P[11][0]*SF[11] + P[10][0]*SPP[7] - (P[12][0]*q0)/2) + SF[8]*(P[3][1] + P[0][1]*SF[5] + P[1][1]*SF[4] + P[2][1]*SF[7] - P[11][1]*SF[11] + P[10][1]*SPP[7] - (P[12][1]*q0)/2) + SF[6]*(P[3][3] + P[0][3]*SF[5] + P[1][3]*SF[4] + P[2][3]*SF[7] - P[11][3]*SF[11] + P[10][3]*SPP[7] - (P[12][3]*q0)/2) + SF[11]*(P[3][12] + P[0][12]*SF[5] + P[1][12]*SF[4] + P[2][12]*SF[7] - P[11][12]*SF[11] + P[10][12]*SPP[7] - (P[12][12]*q0)/2) - SPP[6]*(P[3][10] + P[0][10]*SF[5] + P[1][10]*SF[4] + P[2][10]*SF[7] - P[11][10]*SF[11] + P[10][10]*SPP[7] - (P[12][10]*q0)/2) - (q0*(P[3][11] + P[0][11]*SF[5] + P[1][11]*SF[4] + P[2][11]*SF[7] - P[11][11]*SF[11] + P[10][11]*SPP[7] - (P[12][11]*q0)/2))/2; nextP[3][3] = P[3][3] + P[0][3]*SF[5] + P[1][3]*SF[4] + P[2][3]*SF[7] - P[11][3]*SF[11] + P[10][3]*SPP[7] + (dayCov*SQ[10])/4 + dazCov*SQ[9] - (P[12][3]*q0)/2 + SF[5]*(P[3][0] + P[0][0]*SF[5] + P[1][0]*SF[4] + P[2][0]*SF[7] - P[11][0]*SF[11] + P[10][0]*SPP[7] - (P[12][0]*q0)/2) + SF[4]*(P[3][1] + P[0][1]*SF[5] + P[1][1]*SF[4] + P[2][1]*SF[7] - P[11][1]*SF[11] + P[10][1]*SPP[7] - (P[12][1]*q0)/2) + SF[7]*(P[3][2] + P[0][2]*SF[5] + P[1][2]*SF[4] + P[2][2]*SF[7] - P[11][2]*SF[11] + P[10][2]*SPP[7] - (P[12][2]*q0)/2) - SF[11]*(P[3][11] + P[0][11]*SF[5] + P[1][11]*SF[4] + P[2][11]*SF[7] - P[11][11]*SF[11] + P[10][11]*SPP[7] - (P[12][11]*q0)/2) + SPP[7]*(P[3][10] + P[0][10]*SF[5] + P[1][10]*SF[4] + P[2][10]*SF[7] - P[11][10]*SF[11] + P[10][10]*SPP[7] - (P[12][10]*q0)/2) + (daxCov*sq(q2))/4 - (q0*(P[3][12] + P[0][12]*SF[5] + P[1][12]*SF[4] + P[2][12]*SF[7] - P[11][12]*SF[11] + P[10][12]*SPP[7] - (P[12][12]*q0)/2))/2; nextP[3][4] = P[3][4] + P[0][4]*SF[5] + P[1][4]*SF[4] + P[2][4]*SF[7] - P[11][4]*SF[11] + P[10][4]*SPP[7] - (P[12][4]*q0)/2 + SF[3]*(P[3][0] + P[0][0]*SF[5] + P[1][0]*SF[4] + P[2][0]*SF[7] - P[11][0]*SF[11] + P[10][0]*SPP[7] - (P[12][0]*q0)/2) + SF[1]*(P[3][1] + P[0][1]*SF[5] + P[1][1]*SF[4] + P[2][1]*SF[7] - P[11][1]*SF[11] + P[10][1]*SPP[7] - (P[12][1]*q0)/2) + SPP[0]*(P[3][2] + P[0][2]*SF[5] + P[1][2]*SF[4] + P[2][2]*SF[7] - P[11][2]*SF[11] + P[10][2]*SPP[7] - (P[12][2]*q0)/2) - SPP[2]*(P[3][3] + P[0][3]*SF[5] + P[1][3]*SF[4] + P[2][3]*SF[7] - P[11][3]*SF[11] + P[10][3]*SPP[7] - (P[12][3]*q0)/2) - SPP[4]*(P[3][13] + P[0][13]*SF[5] + P[1][13]*SF[4] + P[2][13]*SF[7] - P[11][13]*SF[11] + P[10][13]*SPP[7] - (P[12][13]*q0)/2); nextP[3][5] = P[3][5] + P[0][5]*SF[5] + P[1][5]*SF[4] + P[2][5]*SF[7] - P[11][5]*SF[11] + P[10][5]*SPP[7] - (P[12][5]*q0)/2 + SF[2]*(P[3][0] + P[0][0]*SF[5] + P[1][0]*SF[4] + P[2][0]*SF[7] - P[11][0]*SF[11] + P[10][0]*SPP[7] - (P[12][0]*q0)/2) + SF[1]*(P[3][2] + P[0][2]*SF[5] + P[1][2]*SF[4] + P[2][2]*SF[7] - P[11][2]*SF[11] + P[10][2]*SPP[7] - (P[12][2]*q0)/2) + SF[3]*(P[3][3] + P[0][3]*SF[5] + P[1][3]*SF[4] + P[2][3]*SF[7] - P[11][3]*SF[11] + P[10][3]*SPP[7] - (P[12][3]*q0)/2) - SPP[0]*(P[3][1] + P[0][1]*SF[5] + P[1][1]*SF[4] + P[2][1]*SF[7] - P[11][1]*SF[11] + P[10][1]*SPP[7] - (P[12][1]*q0)/2) + SPP[3]*(P[3][13] + P[0][13]*SF[5] + P[1][13]*SF[4] + P[2][13]*SF[7] - P[11][13]*SF[11] + P[10][13]*SPP[7] - (P[12][13]*q0)/2); nextP[3][6] = P[3][6] + P[0][6]*SF[5] + P[1][6]*SF[4] + P[2][6]*SF[7] - P[11][6]*SF[11] + P[10][6]*SPP[7] - (P[12][6]*q0)/2 + SF[2]*(P[3][1] + P[0][1]*SF[5] + P[1][1]*SF[4] + P[2][1]*SF[7] - P[11][1]*SF[11] + P[10][1]*SPP[7] - (P[12][1]*q0)/2) + SF[1]*(P[3][3] + P[0][3]*SF[5] + P[1][3]*SF[4] + P[2][3]*SF[7] - P[11][3]*SF[11] + P[10][3]*SPP[7] - (P[12][3]*q0)/2) + SPP[0]*(P[3][0] + P[0][0]*SF[5] + P[1][0]*SF[4] + P[2][0]*SF[7] - P[11][0]*SF[11] + P[10][0]*SPP[7] - (P[12][0]*q0)/2) - SPP[1]*(P[3][2] + P[0][2]*SF[5] + P[1][2]*SF[4] + P[2][2]*SF[7] - P[11][2]*SF[11] + P[10][2]*SPP[7] - (P[12][2]*q0)/2) - (sq(q0) - sq(q1) - sq(q2) + sq(q3))*(P[3][13] + P[0][13]*SF[5] + P[1][13]*SF[4] + P[2][13]*SF[7] - P[11][13]*SF[11] + P[10][13]*SPP[7] - (P[12][13]*q0)/2); nextP[3][7] = P[3][7] + P[0][7]*SF[5] + P[1][7]*SF[4] + P[2][7]*SF[7] - P[11][7]*SF[11] + P[10][7]*SPP[7] - (P[12][7]*q0)/2 + dt*(P[3][4] + P[0][4]*SF[5] + P[1][4]*SF[4] + P[2][4]*SF[7] - P[11][4]*SF[11] + P[10][4]*SPP[7] - (P[12][4]*q0)/2); nextP[3][8] = P[3][8] + P[0][8]*SF[5] + P[1][8]*SF[4] + P[2][8]*SF[7] - P[11][8]*SF[11] + P[10][8]*SPP[7] - (P[12][8]*q0)/2 + dt*(P[3][5] + P[0][5]*SF[5] + P[1][5]*SF[4] + P[2][5]*SF[7] - P[11][5]*SF[11] + P[10][5]*SPP[7] - (P[12][5]*q0)/2); nextP[3][9] = P[3][9] + P[0][9]*SF[5] + P[1][9]*SF[4] + P[2][9]*SF[7] - P[11][9]*SF[11] + P[10][9]*SPP[7] - (P[12][9]*q0)/2 + dt*(P[3][6] + P[0][6]*SF[5] + P[1][6]*SF[4] + P[2][6]*SF[7] - P[11][6]*SF[11] + P[10][6]*SPP[7] - (P[12][6]*q0)/2); nextP[3][10] = P[3][10] + P[0][10]*SF[5] + P[1][10]*SF[4] + P[2][10]*SF[7] - P[11][10]*SF[11] + P[10][10]*SPP[7] - (P[12][10]*q0)/2; nextP[3][11] = P[3][11] + P[0][11]*SF[5] + P[1][11]*SF[4] + P[2][11]*SF[7] - P[11][11]*SF[11] + P[10][11]*SPP[7] - (P[12][11]*q0)/2; nextP[3][12] = P[3][12] + P[0][12]*SF[5] + P[1][12]*SF[4] + P[2][12]*SF[7] - P[11][12]*SF[11] + P[10][12]*SPP[7] - (P[12][12]*q0)/2; nextP[3][13] = P[3][13] + P[0][13]*SF[5] + P[1][13]*SF[4] + P[2][13]*SF[7] - P[11][13]*SF[11] + P[10][13]*SPP[7] - (P[12][13]*q0)/2; nextP[3][14] = P[3][14] + P[0][14]*SF[5] + P[1][14]*SF[4] + P[2][14]*SF[7] - P[11][14]*SF[11] + P[10][14]*SPP[7] - (P[12][14]*q0)/2; nextP[3][15] = P[3][15] + P[0][15]*SF[5] + P[1][15]*SF[4] + P[2][15]*SF[7] - P[11][15]*SF[11] + P[10][15]*SPP[7] - (P[12][15]*q0)/2; nextP[3][16] = P[3][16] + P[0][16]*SF[5] + P[1][16]*SF[4] + P[2][16]*SF[7] - P[11][16]*SF[11] + P[10][16]*SPP[7] - (P[12][16]*q0)/2; nextP[3][17] = P[3][17] + P[0][17]*SF[5] + P[1][17]*SF[4] + P[2][17]*SF[7] - P[11][17]*SF[11] + P[10][17]*SPP[7] - (P[12][17]*q0)/2; nextP[3][18] = P[3][18] + P[0][18]*SF[5] + P[1][18]*SF[4] + P[2][18]*SF[7] - P[11][18]*SF[11] + P[10][18]*SPP[7] - (P[12][18]*q0)/2; nextP[3][19] = P[3][19] + P[0][19]*SF[5] + P[1][19]*SF[4] + P[2][19]*SF[7] - P[11][19]*SF[11] + P[10][19]*SPP[7] - (P[12][19]*q0)/2; nextP[3][20] = P[3][20] + P[0][20]*SF[5] + P[1][20]*SF[4] + P[2][20]*SF[7] - P[11][20]*SF[11] + P[10][20]*SPP[7] - (P[12][20]*q0)/2; nextP[3][21] = P[3][21] + P[0][21]*SF[5] + P[1][21]*SF[4] + P[2][21]*SF[7] - P[11][21]*SF[11] + P[10][21]*SPP[7] - (P[12][21]*q0)/2; nextP[4][0] = P[4][0] + P[0][0]*SF[3] + P[1][0]*SF[1] + P[2][0]*SPP[0] - P[3][0]*SPP[2] - P[13][0]*SPP[4] + SF[7]*(P[4][1] + P[0][1]*SF[3] + P[1][1]*SF[1] + P[2][1]*SPP[0] - P[3][1]*SPP[2] - P[13][1]*SPP[4]) + SF[9]*(P[4][2] + P[0][2]*SF[3] + P[1][2]*SF[1] + P[2][2]*SPP[0] - P[3][2]*SPP[2] - P[13][2]*SPP[4]) + SF[8]*(P[4][3] + P[0][3]*SF[3] + P[1][3]*SF[1] + P[2][3]*SPP[0] - P[3][3]*SPP[2] - P[13][3]*SPP[4]) + SF[11]*(P[4][10] + P[0][10]*SF[3] + P[1][10]*SF[1] + P[2][10]*SPP[0] - P[3][10]*SPP[2] - P[13][10]*SPP[4]) + SPP[7]*(P[4][11] + P[0][11]*SF[3] + P[1][11]*SF[1] + P[2][11]*SPP[0] - P[3][11]*SPP[2] - P[13][11]*SPP[4]) + SPP[6]*(P[4][12] + P[0][12]*SF[3] + P[1][12]*SF[1] + P[2][12]*SPP[0] - P[3][12]*SPP[2] - P[13][12]*SPP[4]); nextP[4][1] = P[4][1] + P[0][1]*SF[3] + P[1][1]*SF[1] + P[2][1]*SPP[0] - P[3][1]*SPP[2] - P[13][1]*SPP[4] + SF[6]*(P[4][0] + P[0][0]*SF[3] + P[1][0]*SF[1] + P[2][0]*SPP[0] - P[3][0]*SPP[2] - P[13][0]*SPP[4]) + SF[5]*(P[4][2] + P[0][2]*SF[3] + P[1][2]*SF[1] + P[2][2]*SPP[0] - P[3][2]*SPP[2] - P[13][2]*SPP[4]) + SF[9]*(P[4][3] + P[0][3]*SF[3] + P[1][3]*SF[1] + P[2][3]*SPP[0] - P[3][3]*SPP[2] - P[13][3]*SPP[4]) + SPP[6]*(P[4][11] + P[0][11]*SF[3] + P[1][11]*SF[1] + P[2][11]*SPP[0] - P[3][11]*SPP[2] - P[13][11]*SPP[4]) - SPP[7]*(P[4][12] + P[0][12]*SF[3] + P[1][12]*SF[1] + P[2][12]*SPP[0] - P[3][12]*SPP[2] - P[13][12]*SPP[4]) - (q0*(P[4][10] + P[0][10]*SF[3] + P[1][10]*SF[1] + P[2][10]*SPP[0] - P[3][10]*SPP[2] - P[13][10]*SPP[4]))/2; nextP[4][2] = P[4][2] + P[0][2]*SF[3] + P[1][2]*SF[1] + P[2][2]*SPP[0] - P[3][2]*SPP[2] - P[13][2]*SPP[4] + SF[4]*(P[4][0] + P[0][0]*SF[3] + P[1][0]*SF[1] + P[2][0]*SPP[0] - P[3][0]*SPP[2] - P[13][0]*SPP[4]) + SF[8]*(P[4][1] + P[0][1]*SF[3] + P[1][1]*SF[1] + P[2][1]*SPP[0] - P[3][1]*SPP[2] - P[13][1]*SPP[4]) + SF[6]*(P[4][3] + P[0][3]*SF[3] + P[1][3]*SF[1] + P[2][3]*SPP[0] - P[3][3]*SPP[2] - P[13][3]*SPP[4]) + SF[11]*(P[4][12] + P[0][12]*SF[3] + P[1][12]*SF[1] + P[2][12]*SPP[0] - P[3][12]*SPP[2] - P[13][12]*SPP[4]) - SPP[6]*(P[4][10] + P[0][10]*SF[3] + P[1][10]*SF[1] + P[2][10]*SPP[0] - P[3][10]*SPP[2] - P[13][10]*SPP[4]) - (q0*(P[4][11] + P[0][11]*SF[3] + P[1][11]*SF[1] + P[2][11]*SPP[0] - P[3][11]*SPP[2] - P[13][11]*SPP[4]))/2; nextP[4][3] = P[4][3] + P[0][3]*SF[3] + P[1][3]*SF[1] + P[2][3]*SPP[0] - P[3][3]*SPP[2] - P[13][3]*SPP[4] + SF[5]*(P[4][0] + P[0][0]*SF[3] + P[1][0]*SF[1] + P[2][0]*SPP[0] - P[3][0]*SPP[2] - P[13][0]*SPP[4]) + SF[4]*(P[4][1] + P[0][1]*SF[3] + P[1][1]*SF[1] + P[2][1]*SPP[0] - P[3][1]*SPP[2] - P[13][1]*SPP[4]) + SF[7]*(P[4][2] + P[0][2]*SF[3] + P[1][2]*SF[1] + P[2][2]*SPP[0] - P[3][2]*SPP[2] - P[13][2]*SPP[4]) - SF[11]*(P[4][11] + P[0][11]*SF[3] + P[1][11]*SF[1] + P[2][11]*SPP[0] - P[3][11]*SPP[2] - P[13][11]*SPP[4]) + SPP[7]*(P[4][10] + P[0][10]*SF[3] + P[1][10]*SF[1] + P[2][10]*SPP[0] - P[3][10]*SPP[2] - P[13][10]*SPP[4]) - (q0*(P[4][12] + P[0][12]*SF[3] + P[1][12]*SF[1] + P[2][12]*SPP[0] - P[3][12]*SPP[2] - P[13][12]*SPP[4]))/2; nextP[4][4] = P[4][4] + P[0][4]*SF[3] + P[1][4]*SF[1] + P[2][4]*SPP[0] - P[3][4]*SPP[2] - P[13][4]*SPP[4] + dvyCov*sq(SG[7] - 2*q0*q3) + dvzCov*sq(SG[6] + 2*q0*q2) + SF[3]*(P[4][0] + P[0][0]*SF[3] + P[1][0]*SF[1] + P[2][0]*SPP[0] - P[3][0]*SPP[2] - P[13][0]*SPP[4]) + SF[1]*(P[4][1] + P[0][1]*SF[3] + P[1][1]*SF[1] + P[2][1]*SPP[0] - P[3][1]*SPP[2] - P[13][1]*SPP[4]) + SPP[0]*(P[4][2] + P[0][2]*SF[3] + P[1][2]*SF[1] + P[2][2]*SPP[0] - P[3][2]*SPP[2] - P[13][2]*SPP[4]) - SPP[2]*(P[4][3] + P[0][3]*SF[3] + P[1][3]*SF[1] + P[2][3]*SPP[0] - P[3][3]*SPP[2] - P[13][3]*SPP[4]) - SPP[4]*(P[4][13] + P[0][13]*SF[3] + P[1][13]*SF[1] + P[2][13]*SPP[0] - P[3][13]*SPP[2] - P[13][13]*SPP[4]) + dvxCov*sq(SG[1] + SG[2] - SG[3] - SG[4]); nextP[4][5] = P[4][5] + SQ[2] + P[0][5]*SF[3] + P[1][5]*SF[1] + P[2][5]*SPP[0] - P[3][5]*SPP[2] - P[13][5]*SPP[4] + SF[2]*(P[4][0] + P[0][0]*SF[3] + P[1][0]*SF[1] + P[2][0]*SPP[0] - P[3][0]*SPP[2] - P[13][0]*SPP[4]) + SF[1]*(P[4][2] + P[0][2]*SF[3] + P[1][2]*SF[1] + P[2][2]*SPP[0] - P[3][2]*SPP[2] - P[13][2]*SPP[4]) + SF[3]*(P[4][3] + P[0][3]*SF[3] + P[1][3]*SF[1] + P[2][3]*SPP[0] - P[3][3]*SPP[2] - P[13][3]*SPP[4]) - SPP[0]*(P[4][1] + P[0][1]*SF[3] + P[1][1]*SF[1] + P[2][1]*SPP[0] - P[3][1]*SPP[2] - P[13][1]*SPP[4]) + SPP[3]*(P[4][13] + P[0][13]*SF[3] + P[1][13]*SF[1] + P[2][13]*SPP[0] - P[3][13]*SPP[2] - P[13][13]*SPP[4]); nextP[4][6] = P[4][6] + SQ[1] + P[0][6]*SF[3] + P[1][6]*SF[1] + P[2][6]*SPP[0] - P[3][6]*SPP[2] - P[13][6]*SPP[4] + SF[2]*(P[4][1] + P[0][1]*SF[3] + P[1][1]*SF[1] + P[2][1]*SPP[0] - P[3][1]*SPP[2] - P[13][1]*SPP[4]) + SF[1]*(P[4][3] + P[0][3]*SF[3] + P[1][3]*SF[1] + P[2][3]*SPP[0] - P[3][3]*SPP[2] - P[13][3]*SPP[4]) + SPP[0]*(P[4][0] + P[0][0]*SF[3] + P[1][0]*SF[1] + P[2][0]*SPP[0] - P[3][0]*SPP[2] - P[13][0]*SPP[4]) - SPP[1]*(P[4][2] + P[0][2]*SF[3] + P[1][2]*SF[1] + P[2][2]*SPP[0] - P[3][2]*SPP[2] - P[13][2]*SPP[4]) - (sq(q0) - sq(q1) - sq(q2) + sq(q3))*(P[4][13] + P[0][13]*SF[3] + P[1][13]*SF[1] + P[2][13]*SPP[0] - P[3][13]*SPP[2] - P[13][13]*SPP[4]); nextP[4][7] = P[4][7] + P[0][7]*SF[3] + P[1][7]*SF[1] + P[2][7]*SPP[0] - P[3][7]*SPP[2] - P[13][7]*SPP[4] + dt*(P[4][4] + P[0][4]*SF[3] + P[1][4]*SF[1] + P[2][4]*SPP[0] - P[3][4]*SPP[2] - P[13][4]*SPP[4]); nextP[4][8] = P[4][8] + P[0][8]*SF[3] + P[1][8]*SF[1] + P[2][8]*SPP[0] - P[3][8]*SPP[2] - P[13][8]*SPP[4] + dt*(P[4][5] + P[0][5]*SF[3] + P[1][5]*SF[1] + P[2][5]*SPP[0] - P[3][5]*SPP[2] - P[13][5]*SPP[4]); nextP[4][9] = P[4][9] + P[0][9]*SF[3] + P[1][9]*SF[1] + P[2][9]*SPP[0] - P[3][9]*SPP[2] - P[13][9]*SPP[4] + dt*(P[4][6] + P[0][6]*SF[3] + P[1][6]*SF[1] + P[2][6]*SPP[0] - P[3][6]*SPP[2] - P[13][6]*SPP[4]); nextP[4][10] = P[4][10] + P[0][10]*SF[3] + P[1][10]*SF[1] + P[2][10]*SPP[0] - P[3][10]*SPP[2] - P[13][10]*SPP[4]; nextP[4][11] = P[4][11] + P[0][11]*SF[3] + P[1][11]*SF[1] + P[2][11]*SPP[0] - P[3][11]*SPP[2] - P[13][11]*SPP[4]; nextP[4][12] = P[4][12] + P[0][12]*SF[3] + P[1][12]*SF[1] + P[2][12]*SPP[0] - P[3][12]*SPP[2] - P[13][12]*SPP[4]; nextP[4][13] = P[4][13] + P[0][13]*SF[3] + P[1][13]*SF[1] + P[2][13]*SPP[0] - P[3][13]*SPP[2] - P[13][13]*SPP[4]; nextP[4][14] = P[4][14] + P[0][14]*SF[3] + P[1][14]*SF[1] + P[2][14]*SPP[0] - P[3][14]*SPP[2] - P[13][14]*SPP[4]; nextP[4][15] = P[4][15] + P[0][15]*SF[3] + P[1][15]*SF[1] + P[2][15]*SPP[0] - P[3][15]*SPP[2] - P[13][15]*SPP[4]; nextP[4][16] = P[4][16] + P[0][16]*SF[3] + P[1][16]*SF[1] + P[2][16]*SPP[0] - P[3][16]*SPP[2] - P[13][16]*SPP[4]; nextP[4][17] = P[4][17] + P[0][17]*SF[3] + P[1][17]*SF[1] + P[2][17]*SPP[0] - P[3][17]*SPP[2] - P[13][17]*SPP[4]; nextP[4][18] = P[4][18] + P[0][18]*SF[3] + P[1][18]*SF[1] + P[2][18]*SPP[0] - P[3][18]*SPP[2] - P[13][18]*SPP[4]; nextP[4][19] = P[4][19] + P[0][19]*SF[3] + P[1][19]*SF[1] + P[2][19]*SPP[0] - P[3][19]*SPP[2] - P[13][19]*SPP[4]; nextP[4][20] = P[4][20] + P[0][20]*SF[3] + P[1][20]*SF[1] + P[2][20]*SPP[0] - P[3][20]*SPP[2] - P[13][20]*SPP[4]; nextP[4][21] = P[4][21] + P[0][21]*SF[3] + P[1][21]*SF[1] + P[2][21]*SPP[0] - P[3][21]*SPP[2] - P[13][21]*SPP[4]; nextP[5][0] = P[5][0] + P[0][0]*SF[2] + P[2][0]*SF[1] + P[3][0]*SF[3] - P[1][0]*SPP[0] + P[13][0]*SPP[3] + SF[7]*(P[5][1] + P[0][1]*SF[2] + P[2][1]*SF[1] + P[3][1]*SF[3] - P[1][1]*SPP[0] + P[13][1]*SPP[3]) + SF[9]*(P[5][2] + P[0][2]*SF[2] + P[2][2]*SF[1] + P[3][2]*SF[3] - P[1][2]*SPP[0] + P[13][2]*SPP[3]) + SF[8]*(P[5][3] + P[0][3]*SF[2] + P[2][3]*SF[1] + P[3][3]*SF[3] - P[1][3]*SPP[0] + P[13][3]*SPP[3]) + SF[11]*(P[5][10] + P[0][10]*SF[2] + P[2][10]*SF[1] + P[3][10]*SF[3] - P[1][10]*SPP[0] + P[13][10]*SPP[3]) + SPP[7]*(P[5][11] + P[0][11]*SF[2] + P[2][11]*SF[1] + P[3][11]*SF[3] - P[1][11]*SPP[0] + P[13][11]*SPP[3]) + SPP[6]*(P[5][12] + P[0][12]*SF[2] + P[2][12]*SF[1] + P[3][12]*SF[3] - P[1][12]*SPP[0] + P[13][12]*SPP[3]); nextP[5][1] = P[5][1] + P[0][1]*SF[2] + P[2][1]*SF[1] + P[3][1]*SF[3] - P[1][1]*SPP[0] + P[13][1]*SPP[3] + SF[6]*(P[5][0] + P[0][0]*SF[2] + P[2][0]*SF[1] + P[3][0]*SF[3] - P[1][0]*SPP[0] + P[13][0]*SPP[3]) + SF[5]*(P[5][2] + P[0][2]*SF[2] + P[2][2]*SF[1] + P[3][2]*SF[3] - P[1][2]*SPP[0] + P[13][2]*SPP[3]) + SF[9]*(P[5][3] + P[0][3]*SF[2] + P[2][3]*SF[1] + P[3][3]*SF[3] - P[1][3]*SPP[0] + P[13][3]*SPP[3]) + SPP[6]*(P[5][11] + P[0][11]*SF[2] + P[2][11]*SF[1] + P[3][11]*SF[3] - P[1][11]*SPP[0] + P[13][11]*SPP[3]) - SPP[7]*(P[5][12] + P[0][12]*SF[2] + P[2][12]*SF[1] + P[3][12]*SF[3] - P[1][12]*SPP[0] + P[13][12]*SPP[3]) - (q0*(P[5][10] + P[0][10]*SF[2] + P[2][10]*SF[1] + P[3][10]*SF[3] - P[1][10]*SPP[0] + P[13][10]*SPP[3]))/2; nextP[5][2] = P[5][2] + P[0][2]*SF[2] + P[2][2]*SF[1] + P[3][2]*SF[3] - P[1][2]*SPP[0] + P[13][2]*SPP[3] + SF[4]*(P[5][0] + P[0][0]*SF[2] + P[2][0]*SF[1] + P[3][0]*SF[3] - P[1][0]*SPP[0] + P[13][0]*SPP[3]) + SF[8]*(P[5][1] + P[0][1]*SF[2] + P[2][1]*SF[1] + P[3][1]*SF[3] - P[1][1]*SPP[0] + P[13][1]*SPP[3]) + SF[6]*(P[5][3] + P[0][3]*SF[2] + P[2][3]*SF[1] + P[3][3]*SF[3] - P[1][3]*SPP[0] + P[13][3]*SPP[3]) + SF[11]*(P[5][12] + P[0][12]*SF[2] + P[2][12]*SF[1] + P[3][12]*SF[3] - P[1][12]*SPP[0] + P[13][12]*SPP[3]) - SPP[6]*(P[5][10] + P[0][10]*SF[2] + P[2][10]*SF[1] + P[3][10]*SF[3] - P[1][10]*SPP[0] + P[13][10]*SPP[3]) - (q0*(P[5][11] + P[0][11]*SF[2] + P[2][11]*SF[1] + P[3][11]*SF[3] - P[1][11]*SPP[0] + P[13][11]*SPP[3]))/2; nextP[5][3] = P[5][3] + P[0][3]*SF[2] + P[2][3]*SF[1] + P[3][3]*SF[3] - P[1][3]*SPP[0] + P[13][3]*SPP[3] + SF[5]*(P[5][0] + P[0][0]*SF[2] + P[2][0]*SF[1] + P[3][0]*SF[3] - P[1][0]*SPP[0] + P[13][0]*SPP[3]) + SF[4]*(P[5][1] + P[0][1]*SF[2] + P[2][1]*SF[1] + P[3][1]*SF[3] - P[1][1]*SPP[0] + P[13][1]*SPP[3]) + SF[7]*(P[5][2] + P[0][2]*SF[2] + P[2][2]*SF[1] + P[3][2]*SF[3] - P[1][2]*SPP[0] + P[13][2]*SPP[3]) - SF[11]*(P[5][11] + P[0][11]*SF[2] + P[2][11]*SF[1] + P[3][11]*SF[3] - P[1][11]*SPP[0] + P[13][11]*SPP[3]) + SPP[7]*(P[5][10] + P[0][10]*SF[2] + P[2][10]*SF[1] + P[3][10]*SF[3] - P[1][10]*SPP[0] + P[13][10]*SPP[3]) - (q0*(P[5][12] + P[0][12]*SF[2] + P[2][12]*SF[1] + P[3][12]*SF[3] - P[1][12]*SPP[0] + P[13][12]*SPP[3]))/2; nextP[5][4] = P[5][4] + SQ[2] + P[0][4]*SF[2] + P[2][4]*SF[1] + P[3][4]*SF[3] - P[1][4]*SPP[0] + P[13][4]*SPP[3] + SF[3]*(P[5][0] + P[0][0]*SF[2] + P[2][0]*SF[1] + P[3][0]*SF[3] - P[1][0]*SPP[0] + P[13][0]*SPP[3]) + SF[1]*(P[5][1] + P[0][1]*SF[2] + P[2][1]*SF[1] + P[3][1]*SF[3] - P[1][1]*SPP[0] + P[13][1]*SPP[3]) + SPP[0]*(P[5][2] + P[0][2]*SF[2] + P[2][2]*SF[1] + P[3][2]*SF[3] - P[1][2]*SPP[0] + P[13][2]*SPP[3]) - SPP[2]*(P[5][3] + P[0][3]*SF[2] + P[2][3]*SF[1] + P[3][3]*SF[3] - P[1][3]*SPP[0] + P[13][3]*SPP[3]) - SPP[4]*(P[5][13] + P[0][13]*SF[2] + P[2][13]*SF[1] + P[3][13]*SF[3] - P[1][13]*SPP[0] + P[13][13]*SPP[3]); nextP[5][5] = P[5][5] + P[0][5]*SF[2] + P[2][5]*SF[1] + P[3][5]*SF[3] - P[1][5]*SPP[0] + P[13][5]*SPP[3] + dvxCov*sq(SG[7] + 2*q0*q3) + dvzCov*sq(SG[5] - 2*q0*q1) + SF[2]*(P[5][0] + P[0][0]*SF[2] + P[2][0]*SF[1] + P[3][0]*SF[3] - P[1][0]*SPP[0] + P[13][0]*SPP[3]) + SF[1]*(P[5][2] + P[0][2]*SF[2] + P[2][2]*SF[1] + P[3][2]*SF[3] - P[1][2]*SPP[0] + P[13][2]*SPP[3]) + SF[3]*(P[5][3] + P[0][3]*SF[2] + P[2][3]*SF[1] + P[3][3]*SF[3] - P[1][3]*SPP[0] + P[13][3]*SPP[3]) - SPP[0]*(P[5][1] + P[0][1]*SF[2] + P[2][1]*SF[1] + P[3][1]*SF[3] - P[1][1]*SPP[0] + P[13][1]*SPP[3]) + SPP[3]*(P[5][13] + P[0][13]*SF[2] + P[2][13]*SF[1] + P[3][13]*SF[3] - P[1][13]*SPP[0] + P[13][13]*SPP[3]) + dvyCov*sq(SG[1] - SG[2] + SG[3] - SG[4]); nextP[5][6] = P[5][6] + SQ[0] + P[0][6]*SF[2] + P[2][6]*SF[1] + P[3][6]*SF[3] - P[1][6]*SPP[0] + P[13][6]*SPP[3] + SF[2]*(P[5][1] + P[0][1]*SF[2] + P[2][1]*SF[1] + P[3][1]*SF[3] - P[1][1]*SPP[0] + P[13][1]*SPP[3]) + SF[1]*(P[5][3] + P[0][3]*SF[2] + P[2][3]*SF[1] + P[3][3]*SF[3] - P[1][3]*SPP[0] + P[13][3]*SPP[3]) + SPP[0]*(P[5][0] + P[0][0]*SF[2] + P[2][0]*SF[1] + P[3][0]*SF[3] - P[1][0]*SPP[0] + P[13][0]*SPP[3]) - SPP[1]*(P[5][2] + P[0][2]*SF[2] + P[2][2]*SF[1] + P[3][2]*SF[3] - P[1][2]*SPP[0] + P[13][2]*SPP[3]) - (sq(q0) - sq(q1) - sq(q2) + sq(q3))*(P[5][13] + P[0][13]*SF[2] + P[2][13]*SF[1] + P[3][13]*SF[3] - P[1][13]*SPP[0] + P[13][13]*SPP[3]); nextP[5][7] = P[5][7] + P[0][7]*SF[2] + P[2][7]*SF[1] + P[3][7]*SF[3] - P[1][7]*SPP[0] + P[13][7]*SPP[3] + dt*(P[5][4] + P[0][4]*SF[2] + P[2][4]*SF[1] + P[3][4]*SF[3] - P[1][4]*SPP[0] + P[13][4]*SPP[3]); nextP[5][8] = P[5][8] + P[0][8]*SF[2] + P[2][8]*SF[1] + P[3][8]*SF[3] - P[1][8]*SPP[0] + P[13][8]*SPP[3] + dt*(P[5][5] + P[0][5]*SF[2] + P[2][5]*SF[1] + P[3][5]*SF[3] - P[1][5]*SPP[0] + P[13][5]*SPP[3]); nextP[5][9] = P[5][9] + P[0][9]*SF[2] + P[2][9]*SF[1] + P[3][9]*SF[3] - P[1][9]*SPP[0] + P[13][9]*SPP[3] + dt*(P[5][6] + P[0][6]*SF[2] + P[2][6]*SF[1] + P[3][6]*SF[3] - P[1][6]*SPP[0] + P[13][6]*SPP[3]); nextP[5][10] = P[5][10] + P[0][10]*SF[2] + P[2][10]*SF[1] + P[3][10]*SF[3] - P[1][10]*SPP[0] + P[13][10]*SPP[3]; nextP[5][11] = P[5][11] + P[0][11]*SF[2] + P[2][11]*SF[1] + P[3][11]*SF[3] - P[1][11]*SPP[0] + P[13][11]*SPP[3]; nextP[5][12] = P[5][12] + P[0][12]*SF[2] + P[2][12]*SF[1] + P[3][12]*SF[3] - P[1][12]*SPP[0] + P[13][12]*SPP[3]; nextP[5][13] = P[5][13] + P[0][13]*SF[2] + P[2][13]*SF[1] + P[3][13]*SF[3] - P[1][13]*SPP[0] + P[13][13]*SPP[3]; nextP[5][14] = P[5][14] + P[0][14]*SF[2] + P[2][14]*SF[1] + P[3][14]*SF[3] - P[1][14]*SPP[0] + P[13][14]*SPP[3]; nextP[5][15] = P[5][15] + P[0][15]*SF[2] + P[2][15]*SF[1] + P[3][15]*SF[3] - P[1][15]*SPP[0] + P[13][15]*SPP[3]; nextP[5][16] = P[5][16] + P[0][16]*SF[2] + P[2][16]*SF[1] + P[3][16]*SF[3] - P[1][16]*SPP[0] + P[13][16]*SPP[3]; nextP[5][17] = P[5][17] + P[0][17]*SF[2] + P[2][17]*SF[1] + P[3][17]*SF[3] - P[1][17]*SPP[0] + P[13][17]*SPP[3]; nextP[5][18] = P[5][18] + P[0][18]*SF[2] + P[2][18]*SF[1] + P[3][18]*SF[3] - P[1][18]*SPP[0] + P[13][18]*SPP[3]; nextP[5][19] = P[5][19] + P[0][19]*SF[2] + P[2][19]*SF[1] + P[3][19]*SF[3] - P[1][19]*SPP[0] + P[13][19]*SPP[3]; nextP[5][20] = P[5][20] + P[0][20]*SF[2] + P[2][20]*SF[1] + P[3][20]*SF[3] - P[1][20]*SPP[0] + P[13][20]*SPP[3]; nextP[5][21] = P[5][21] + P[0][21]*SF[2] + P[2][21]*SF[1] + P[3][21]*SF[3] - P[1][21]*SPP[0] + P[13][21]*SPP[3]; nextP[6][0] = P[6][0] + P[1][0]*SF[2] + P[3][0]*SF[1] + P[0][0]*SPP[0] - P[2][0]*SPP[1] - P[13][0]*(sq(q0) - sq(q1) - sq(q2) + sq(q3)) + SF[7]*(P[6][1] + P[1][1]*SF[2] + P[3][1]*SF[1] + P[0][1]*SPP[0] - P[2][1]*SPP[1] - P[13][1]*(sq(q0) - sq(q1) - sq(q2) + sq(q3))) + SF[9]*(P[6][2] + P[1][2]*SF[2] + P[3][2]*SF[1] + P[0][2]*SPP[0] - P[2][2]*SPP[1] - P[13][2]*(sq(q0) - sq(q1) - sq(q2) + sq(q3))) + SF[8]*(P[6][3] + P[1][3]*SF[2] + P[3][3]*SF[1] + P[0][3]*SPP[0] - P[2][3]*SPP[1] - P[13][3]*(sq(q0) - sq(q1) - sq(q2) + sq(q3))) + SF[11]*(P[6][10] + P[1][10]*SF[2] + P[3][10]*SF[1] + P[0][10]*SPP[0] - P[2][10]*SPP[1] - P[13][10]*(sq(q0) - sq(q1) - sq(q2) + sq(q3))) + SPP[7]*(P[6][11] + P[1][11]*SF[2] + P[3][11]*SF[1] + P[0][11]*SPP[0] - P[2][11]*SPP[1] - P[13][11]*(sq(q0) - sq(q1) - sq(q2) + sq(q3))) + SPP[6]*(P[6][12] + P[1][12]*SF[2] + P[3][12]*SF[1] + P[0][12]*SPP[0] - P[2][12]*SPP[1] - P[13][12]*(sq(q0) - sq(q1) - sq(q2) + sq(q3))); nextP[6][1] = P[6][1] + P[1][1]*SF[2] + P[3][1]*SF[1] + P[0][1]*SPP[0] - P[2][1]*SPP[1] - P[13][1]*(sq(q0) - sq(q1) - sq(q2) + sq(q3)) + SF[6]*(P[6][0] + P[1][0]*SF[2] + P[3][0]*SF[1] + P[0][0]*SPP[0] - P[2][0]*SPP[1] - P[13][0]*(sq(q0) - sq(q1) - sq(q2) + sq(q3))) + SF[5]*(P[6][2] + P[1][2]*SF[2] + P[3][2]*SF[1] + P[0][2]*SPP[0] - P[2][2]*SPP[1] - P[13][2]*(sq(q0) - sq(q1) - sq(q2) + sq(q3))) + SF[9]*(P[6][3] + P[1][3]*SF[2] + P[3][3]*SF[1] + P[0][3]*SPP[0] - P[2][3]*SPP[1] - P[13][3]*(sq(q0) - sq(q1) - sq(q2) + sq(q3))) + SPP[6]*(P[6][11] + P[1][11]*SF[2] + P[3][11]*SF[1] + P[0][11]*SPP[0] - P[2][11]*SPP[1] - P[13][11]*(sq(q0) - sq(q1) - sq(q2) + sq(q3))) - SPP[7]*(P[6][12] + P[1][12]*SF[2] + P[3][12]*SF[1] + P[0][12]*SPP[0] - P[2][12]*SPP[1] - P[13][12]*(sq(q0) - sq(q1) - sq(q2) + sq(q3))) - (q0*(P[6][10] + P[1][10]*SF[2] + P[3][10]*SF[1] + P[0][10]*SPP[0] - P[2][10]*SPP[1] - P[13][10]*(sq(q0) - sq(q1) - sq(q2) + sq(q3))))/2; nextP[6][2] = P[6][2] + P[1][2]*SF[2] + P[3][2]*SF[1] + P[0][2]*SPP[0] - P[2][2]*SPP[1] - P[13][2]*(sq(q0) - sq(q1) - sq(q2) + sq(q3)) + SF[4]*(P[6][0] + P[1][0]*SF[2] + P[3][0]*SF[1] + P[0][0]*SPP[0] - P[2][0]*SPP[1] - P[13][0]*(sq(q0) - sq(q1) - sq(q2) + sq(q3))) + SF[8]*(P[6][1] + P[1][1]*SF[2] + P[3][1]*SF[1] + P[0][1]*SPP[0] - P[2][1]*SPP[1] - P[13][1]*(sq(q0) - sq(q1) - sq(q2) + sq(q3))) + SF[6]*(P[6][3] + P[1][3]*SF[2] + P[3][3]*SF[1] + P[0][3]*SPP[0] - P[2][3]*SPP[1] - P[13][3]*(sq(q0) - sq(q1) - sq(q2) + sq(q3))) + SF[11]*(P[6][12] + P[1][12]*SF[2] + P[3][12]*SF[1] + P[0][12]*SPP[0] - P[2][12]*SPP[1] - P[13][12]*(sq(q0) - sq(q1) - sq(q2) + sq(q3))) - SPP[6]*(P[6][10] + P[1][10]*SF[2] + P[3][10]*SF[1] + P[0][10]*SPP[0] - P[2][10]*SPP[1] - P[13][10]*(sq(q0) - sq(q1) - sq(q2) + sq(q3))) - (q0*(P[6][11] + P[1][11]*SF[2] + P[3][11]*SF[1] + P[0][11]*SPP[0] - P[2][11]*SPP[1] - P[13][11]*(sq(q0) - sq(q1) - sq(q2) + sq(q3))))/2; nextP[6][3] = P[6][3] + P[1][3]*SF[2] + P[3][3]*SF[1] + P[0][3]*SPP[0] - P[2][3]*SPP[1] - P[13][3]*(sq(q0) - sq(q1) - sq(q2) + sq(q3)) + SF[5]*(P[6][0] + P[1][0]*SF[2] + P[3][0]*SF[1] + P[0][0]*SPP[0] - P[2][0]*SPP[1] - P[13][0]*(sq(q0) - sq(q1) - sq(q2) + sq(q3))) + SF[4]*(P[6][1] + P[1][1]*SF[2] + P[3][1]*SF[1] + P[0][1]*SPP[0] - P[2][1]*SPP[1] - P[13][1]*(sq(q0) - sq(q1) - sq(q2) + sq(q3))) + SF[7]*(P[6][2] + P[1][2]*SF[2] + P[3][2]*SF[1] + P[0][2]*SPP[0] - P[2][2]*SPP[1] - P[13][2]*(sq(q0) - sq(q1) - sq(q2) + sq(q3))) - SF[11]*(P[6][11] + P[1][11]*SF[2] + P[3][11]*SF[1] + P[0][11]*SPP[0] - P[2][11]*SPP[1] - P[13][11]*(sq(q0) - sq(q1) - sq(q2) + sq(q3))) + SPP[7]*(P[6][10] + P[1][10]*SF[2] + P[3][10]*SF[1] + P[0][10]*SPP[0] - P[2][10]*SPP[1] - P[13][10]*(sq(q0) - sq(q1) - sq(q2) + sq(q3))) - (q0*(P[6][12] + P[1][12]*SF[2] + P[3][12]*SF[1] + P[0][12]*SPP[0] - P[2][12]*SPP[1] - P[13][12]*(sq(q0) - sq(q1) - sq(q2) + sq(q3))))/2; nextP[6][4] = P[6][4] + SQ[1] + P[1][4]*SF[2] + P[3][4]*SF[1] + P[0][4]*SPP[0] - P[2][4]*SPP[1] - P[13][4]*(sq(q0) - sq(q1) - sq(q2) + sq(q3)) + SF[3]*(P[6][0] + P[1][0]*SF[2] + P[3][0]*SF[1] + P[0][0]*SPP[0] - P[2][0]*SPP[1] - P[13][0]*(sq(q0) - sq(q1) - sq(q2) + sq(q3))) + SF[1]*(P[6][1] + P[1][1]*SF[2] + P[3][1]*SF[1] + P[0][1]*SPP[0] - P[2][1]*SPP[1] - P[13][1]*(sq(q0) - sq(q1) - sq(q2) + sq(q3))) + SPP[0]*(P[6][2] + P[1][2]*SF[2] + P[3][2]*SF[1] + P[0][2]*SPP[0] - P[2][2]*SPP[1] - P[13][2]*(sq(q0) - sq(q1) - sq(q2) + sq(q3))) - SPP[2]*(P[6][3] + P[1][3]*SF[2] + P[3][3]*SF[1] + P[0][3]*SPP[0] - P[2][3]*SPP[1] - P[13][3]*(sq(q0) - sq(q1) - sq(q2) + sq(q3))) - SPP[4]*(P[6][13] + P[1][13]*SF[2] + P[3][13]*SF[1] + P[0][13]*SPP[0] - P[2][13]*SPP[1] - P[13][13]*(sq(q0) - sq(q1) - sq(q2) + sq(q3))); nextP[6][5] = P[6][5] + SQ[0] + P[1][5]*SF[2] + P[3][5]*SF[1] + P[0][5]*SPP[0] - P[2][5]*SPP[1] - P[13][5]*(sq(q0) - sq(q1) - sq(q2) + sq(q3)) + SF[2]*(P[6][0] + P[1][0]*SF[2] + P[3][0]*SF[1] + P[0][0]*SPP[0] - P[2][0]*SPP[1] - P[13][0]*(sq(q0) - sq(q1) - sq(q2) + sq(q3))) + SF[1]*(P[6][2] + P[1][2]*SF[2] + P[3][2]*SF[1] + P[0][2]*SPP[0] - P[2][2]*SPP[1] - P[13][2]*(sq(q0) - sq(q1) - sq(q2) + sq(q3))) + SF[3]*(P[6][3] + P[1][3]*SF[2] + P[3][3]*SF[1] + P[0][3]*SPP[0] - P[2][3]*SPP[1] - P[13][3]*(sq(q0) - sq(q1) - sq(q2) + sq(q3))) - SPP[0]*(P[6][1] + P[1][1]*SF[2] + P[3][1]*SF[1] + P[0][1]*SPP[0] - P[2][1]*SPP[1] - P[13][1]*(sq(q0) - sq(q1) - sq(q2) + sq(q3))) + SPP[3]*(P[6][13] + P[1][13]*SF[2] + P[3][13]*SF[1] + P[0][13]*SPP[0] - P[2][13]*SPP[1] - P[13][13]*(sq(q0) - sq(q1) - sq(q2) + sq(q3))); nextP[6][6] = P[6][6] + P[1][6]*SF[2] + P[3][6]*SF[1] + P[0][6]*SPP[0] - P[2][6]*SPP[1] - P[13][6]*(sq(q0) - sq(q1) - sq(q2) + sq(q3)) + dvxCov*sq(SG[6] - 2*q0*q2) + dvyCov*sq(SG[5] + 2*q0*q1) - SPP[5]*(P[6][13] + P[1][13]*SF[2] + P[3][13]*SF[1] + P[0][13]*SPP[0] - P[2][13]*SPP[1] - P[13][13]*SPP[5]) + SF[2]*(P[6][1] + P[1][1]*SF[2] + P[3][1]*SF[1] + P[0][1]*SPP[0] - P[2][1]*SPP[1] - P[13][1]*(sq(q0) - sq(q1) - sq(q2) + sq(q3))) + SF[1]*(P[6][3] + P[1][3]*SF[2] + P[3][3]*SF[1] + P[0][3]*SPP[0] - P[2][3]*SPP[1] - P[13][3]*(sq(q0) - sq(q1) - sq(q2) + sq(q3))) + SPP[0]*(P[6][0] + P[1][0]*SF[2] + P[3][0]*SF[1] + P[0][0]*SPP[0] - P[2][0]*SPP[1] - P[13][0]*(sq(q0) - sq(q1) - sq(q2) + sq(q3))) - SPP[1]*(P[6][2] + P[1][2]*SF[2] + P[3][2]*SF[1] + P[0][2]*SPP[0] - P[2][2]*SPP[1] - P[13][2]*(sq(q0) - sq(q1) - sq(q2) + sq(q3))) + dvzCov*sq(SG[1] - SG[2] - SG[3] + SG[4]); nextP[6][7] = P[6][7] + P[1][7]*SF[2] + P[3][7]*SF[1] + P[0][7]*SPP[0] - P[2][7]*SPP[1] - P[13][7]*SPP[5] + dt*(P[6][4] + P[1][4]*SF[2] + P[3][4]*SF[1] + P[0][4]*SPP[0] - P[2][4]*SPP[1] - P[13][4]*SPP[5]); nextP[6][8] = P[6][8] + P[1][8]*SF[2] + P[3][8]*SF[1] + P[0][8]*SPP[0] - P[2][8]*SPP[1] - P[13][8]*SPP[5] + dt*(P[6][5] + P[1][5]*SF[2] + P[3][5]*SF[1] + P[0][5]*SPP[0] - P[2][5]*SPP[1] - P[13][5]*SPP[5]); nextP[6][9] = P[6][9] + P[1][9]*SF[2] + P[3][9]*SF[1] + P[0][9]*SPP[0] - P[2][9]*SPP[1] - P[13][9]*SPP[5] + dt*(P[6][6] + P[1][6]*SF[2] + P[3][6]*SF[1] + P[0][6]*SPP[0] - P[2][6]*SPP[1] - P[13][6]*SPP[5]); nextP[6][10] = P[6][10] + P[1][10]*SF[2] + P[3][10]*SF[1] + P[0][10]*SPP[0] - P[2][10]*SPP[1] - P[13][10]*SPP[5]; nextP[6][11] = P[6][11] + P[1][11]*SF[2] + P[3][11]*SF[1] + P[0][11]*SPP[0] - P[2][11]*SPP[1] - P[13][11]*SPP[5]; nextP[6][12] = P[6][12] + P[1][12]*SF[2] + P[3][12]*SF[1] + P[0][12]*SPP[0] - P[2][12]*SPP[1] - P[13][12]*SPP[5]; nextP[6][13] = P[6][13] + P[1][13]*SF[2] + P[3][13]*SF[1] + P[0][13]*SPP[0] - P[2][13]*SPP[1] - P[13][13]*SPP[5]; nextP[6][14] = P[6][14] + P[1][14]*SF[2] + P[3][14]*SF[1] + P[0][14]*SPP[0] - P[2][14]*SPP[1] - P[13][14]*SPP[5]; nextP[6][15] = P[6][15] + P[1][15]*SF[2] + P[3][15]*SF[1] + P[0][15]*SPP[0] - P[2][15]*SPP[1] - P[13][15]*SPP[5]; nextP[6][16] = P[6][16] + P[1][16]*SF[2] + P[3][16]*SF[1] + P[0][16]*SPP[0] - P[2][16]*SPP[1] - P[13][16]*SPP[5]; nextP[6][17] = P[6][17] + P[1][17]*SF[2] + P[3][17]*SF[1] + P[0][17]*SPP[0] - P[2][17]*SPP[1] - P[13][17]*SPP[5]; nextP[6][18] = P[6][18] + P[1][18]*SF[2] + P[3][18]*SF[1] + P[0][18]*SPP[0] - P[2][18]*SPP[1] - P[13][18]*SPP[5]; nextP[6][19] = P[6][19] + P[1][19]*SF[2] + P[3][19]*SF[1] + P[0][19]*SPP[0] - P[2][19]*SPP[1] - P[13][19]*SPP[5]; nextP[6][20] = P[6][20] + P[1][20]*SF[2] + P[3][20]*SF[1] + P[0][20]*SPP[0] - P[2][20]*SPP[1] - P[13][20]*SPP[5]; nextP[6][21] = P[6][21] + P[1][21]*SF[2] + P[3][21]*SF[1] + P[0][21]*SPP[0] - P[2][21]*SPP[1] - P[13][21]*SPP[5]; nextP[7][0] = P[7][0] + P[4][0]*dt + SF[7]*(P[7][1] + P[4][1]*dt) + SF[9]*(P[7][2] + P[4][2]*dt) + SF[8]*(P[7][3] + P[4][3]*dt) + SF[11]*(P[7][10] + P[4][10]*dt) + SPP[7]*(P[7][11] + P[4][11]*dt) + SPP[6]*(P[7][12] + P[4][12]*dt); nextP[7][1] = P[7][1] + P[4][1]*dt + SF[6]*(P[7][0] + P[4][0]*dt) + SF[5]*(P[7][2] + P[4][2]*dt) + SF[9]*(P[7][3] + P[4][3]*dt) + SPP[6]*(P[7][11] + P[4][11]*dt) - SPP[7]*(P[7][12] + P[4][12]*dt) - (q0*(P[7][10] + P[4][10]*dt))/2; nextP[7][2] = P[7][2] + P[4][2]*dt + SF[4]*(P[7][0] + P[4][0]*dt) + SF[8]*(P[7][1] + P[4][1]*dt) + SF[6]*(P[7][3] + P[4][3]*dt) + SF[11]*(P[7][12] + P[4][12]*dt) - SPP[6]*(P[7][10] + P[4][10]*dt) - (q0*(P[7][11] + P[4][11]*dt))/2; nextP[7][3] = P[7][3] + P[4][3]*dt + SF[5]*(P[7][0] + P[4][0]*dt) + SF[4]*(P[7][1] + P[4][1]*dt) + SF[7]*(P[7][2] + P[4][2]*dt) - SF[11]*(P[7][11] + P[4][11]*dt) + SPP[7]*(P[7][10] + P[4][10]*dt) - (q0*(P[7][12] + P[4][12]*dt))/2; nextP[7][4] = P[7][4] + P[4][4]*dt + SF[1]*(P[7][1] + P[4][1]*dt) + SF[3]*(P[7][0] + P[4][0]*dt) + SPP[0]*(P[7][2] + P[4][2]*dt) - SPP[2]*(P[7][3] + P[4][3]*dt) - SPP[4]*(P[7][13] + P[4][13]*dt); nextP[7][5] = P[7][5] + P[4][5]*dt + SF[2]*(P[7][0] + P[4][0]*dt) + SF[1]*(P[7][2] + P[4][2]*dt) + SF[3]*(P[7][3] + P[4][3]*dt) - SPP[0]*(P[7][1] + P[4][1]*dt) + SPP[3]*(P[7][13] + P[4][13]*dt); nextP[7][6] = P[7][6] + P[4][6]*dt + SF[2]*(P[7][1] + P[4][1]*dt) + SF[1]*(P[7][3] + P[4][3]*dt) + SPP[0]*(P[7][0] + P[4][0]*dt) - SPP[1]*(P[7][2] + P[4][2]*dt) - SPP[5]*(P[7][13] + P[4][13]*dt); nextP[7][7] = P[7][7] + P[4][7]*dt + dt*(P[7][4] + P[4][4]*dt); nextP[7][8] = P[7][8] + P[4][8]*dt + dt*(P[7][5] + P[4][5]*dt); nextP[7][9] = P[7][9] + P[4][9]*dt + dt*(P[7][6] + P[4][6]*dt); nextP[7][10] = P[7][10] + P[4][10]*dt; nextP[7][11] = P[7][11] + P[4][11]*dt; nextP[7][12] = P[7][12] + P[4][12]*dt; nextP[7][13] = P[7][13] + P[4][13]*dt; nextP[7][14] = P[7][14] + P[4][14]*dt; nextP[7][15] = P[7][15] + P[4][15]*dt; nextP[7][16] = P[7][16] + P[4][16]*dt; nextP[7][17] = P[7][17] + P[4][17]*dt; nextP[7][18] = P[7][18] + P[4][18]*dt; nextP[7][19] = P[7][19] + P[4][19]*dt; nextP[7][20] = P[7][20] + P[4][20]*dt; nextP[7][21] = P[7][21] + P[4][21]*dt; nextP[8][0] = P[8][0] + P[5][0]*dt + SF[7]*(P[8][1] + P[5][1]*dt) + SF[9]*(P[8][2] + P[5][2]*dt) + SF[8]*(P[8][3] + P[5][3]*dt) + SF[11]*(P[8][10] + P[5][10]*dt) + SPP[7]*(P[8][11] + P[5][11]*dt) + SPP[6]*(P[8][12] + P[5][12]*dt); nextP[8][1] = P[8][1] + P[5][1]*dt + SF[6]*(P[8][0] + P[5][0]*dt) + SF[5]*(P[8][2] + P[5][2]*dt) + SF[9]*(P[8][3] + P[5][3]*dt) + SPP[6]*(P[8][11] + P[5][11]*dt) - SPP[7]*(P[8][12] + P[5][12]*dt) - (q0*(P[8][10] + P[5][10]*dt))/2; nextP[8][2] = P[8][2] + P[5][2]*dt + SF[4]*(P[8][0] + P[5][0]*dt) + SF[8]*(P[8][1] + P[5][1]*dt) + SF[6]*(P[8][3] + P[5][3]*dt) + SF[11]*(P[8][12] + P[5][12]*dt) - SPP[6]*(P[8][10] + P[5][10]*dt) - (q0*(P[8][11] + P[5][11]*dt))/2; nextP[8][3] = P[8][3] + P[5][3]*dt + SF[5]*(P[8][0] + P[5][0]*dt) + SF[4]*(P[8][1] + P[5][1]*dt) + SF[7]*(P[8][2] + P[5][2]*dt) - SF[11]*(P[8][11] + P[5][11]*dt) + SPP[7]*(P[8][10] + P[5][10]*dt) - (q0*(P[8][12] + P[5][12]*dt))/2; nextP[8][4] = P[8][4] + P[5][4]*dt + SF[1]*(P[8][1] + P[5][1]*dt) + SF[3]*(P[8][0] + P[5][0]*dt) + SPP[0]*(P[8][2] + P[5][2]*dt) - SPP[2]*(P[8][3] + P[5][3]*dt) - SPP[4]*(P[8][13] + P[5][13]*dt); nextP[8][5] = P[8][5] + P[5][5]*dt + SF[2]*(P[8][0] + P[5][0]*dt) + SF[1]*(P[8][2] + P[5][2]*dt) + SF[3]*(P[8][3] + P[5][3]*dt) - SPP[0]*(P[8][1] + P[5][1]*dt) + SPP[3]*(P[8][13] + P[5][13]*dt); nextP[8][6] = P[8][6] + P[5][6]*dt + SF[2]*(P[8][1] + P[5][1]*dt) + SF[1]*(P[8][3] + P[5][3]*dt) + SPP[0]*(P[8][0] + P[5][0]*dt) - SPP[1]*(P[8][2] + P[5][2]*dt) - SPP[5]*(P[8][13] + P[5][13]*dt); nextP[8][7] = P[8][7] + P[5][7]*dt + dt*(P[8][4] + P[5][4]*dt); nextP[8][8] = P[8][8] + P[5][8]*dt + dt*(P[8][5] + P[5][5]*dt); nextP[8][9] = P[8][9] + P[5][9]*dt + dt*(P[8][6] + P[5][6]*dt); nextP[8][10] = P[8][10] + P[5][10]*dt; nextP[8][11] = P[8][11] + P[5][11]*dt; nextP[8][12] = P[8][12] + P[5][12]*dt; nextP[8][13] = P[8][13] + P[5][13]*dt; nextP[8][14] = P[8][14] + P[5][14]*dt; nextP[8][15] = P[8][15] + P[5][15]*dt; nextP[8][16] = P[8][16] + P[5][16]*dt; nextP[8][17] = P[8][17] + P[5][17]*dt; nextP[8][18] = P[8][18] + P[5][18]*dt; nextP[8][19] = P[8][19] + P[5][19]*dt; nextP[8][20] = P[8][20] + P[5][20]*dt; nextP[8][21] = P[8][21] + P[5][21]*dt; nextP[9][0] = P[9][0] + P[6][0]*dt + SF[7]*(P[9][1] + P[6][1]*dt) + SF[9]*(P[9][2] + P[6][2]*dt) + SF[8]*(P[9][3] + P[6][3]*dt) + SF[11]*(P[9][10] + P[6][10]*dt) + SPP[7]*(P[9][11] + P[6][11]*dt) + SPP[6]*(P[9][12] + P[6][12]*dt); nextP[9][1] = P[9][1] + P[6][1]*dt + SF[6]*(P[9][0] + P[6][0]*dt) + SF[5]*(P[9][2] + P[6][2]*dt) + SF[9]*(P[9][3] + P[6][3]*dt) + SPP[6]*(P[9][11] + P[6][11]*dt) - SPP[7]*(P[9][12] + P[6][12]*dt) - (q0*(P[9][10] + P[6][10]*dt))/2; nextP[9][2] = P[9][2] + P[6][2]*dt + SF[4]*(P[9][0] + P[6][0]*dt) + SF[8]*(P[9][1] + P[6][1]*dt) + SF[6]*(P[9][3] + P[6][3]*dt) + SF[11]*(P[9][12] + P[6][12]*dt) - SPP[6]*(P[9][10] + P[6][10]*dt) - (q0*(P[9][11] + P[6][11]*dt))/2; nextP[9][3] = P[9][3] + P[6][3]*dt + SF[5]*(P[9][0] + P[6][0]*dt) + SF[4]*(P[9][1] + P[6][1]*dt) + SF[7]*(P[9][2] + P[6][2]*dt) - SF[11]*(P[9][11] + P[6][11]*dt) + SPP[7]*(P[9][10] + P[6][10]*dt) - (q0*(P[9][12] + P[6][12]*dt))/2; nextP[9][4] = P[9][4] + P[6][4]*dt + SF[1]*(P[9][1] + P[6][1]*dt) + SF[3]*(P[9][0] + P[6][0]*dt) + SPP[0]*(P[9][2] + P[6][2]*dt) - SPP[2]*(P[9][3] + P[6][3]*dt) - SPP[4]*(P[9][13] + P[6][13]*dt); nextP[9][5] = P[9][5] + P[6][5]*dt + SF[2]*(P[9][0] + P[6][0]*dt) + SF[1]*(P[9][2] + P[6][2]*dt) + SF[3]*(P[9][3] + P[6][3]*dt) - SPP[0]*(P[9][1] + P[6][1]*dt) + SPP[3]*(P[9][13] + P[6][13]*dt); nextP[9][6] = P[9][6] + P[6][6]*dt + SF[2]*(P[9][1] + P[6][1]*dt) + SF[1]*(P[9][3] + P[6][3]*dt) + SPP[0]*(P[9][0] + P[6][0]*dt) - SPP[1]*(P[9][2] + P[6][2]*dt) - SPP[5]*(P[9][13] + P[6][13]*dt); nextP[9][7] = P[9][7] + P[6][7]*dt + dt*(P[9][4] + P[6][4]*dt); nextP[9][8] = P[9][8] + P[6][8]*dt + dt*(P[9][5] + P[6][5]*dt); nextP[9][9] = P[9][9] + P[6][9]*dt + dt*(P[9][6] + P[6][6]*dt); nextP[9][10] = P[9][10] + P[6][10]*dt; nextP[9][11] = P[9][11] + P[6][11]*dt; nextP[9][12] = P[9][12] + P[6][12]*dt; nextP[9][13] = P[9][13] + P[6][13]*dt; nextP[9][14] = P[9][14] + P[6][14]*dt; nextP[9][15] = P[9][15] + P[6][15]*dt; nextP[9][16] = P[9][16] + P[6][16]*dt; nextP[9][17] = P[9][17] + P[6][17]*dt; nextP[9][18] = P[9][18] + P[6][18]*dt; nextP[9][19] = P[9][19] + P[6][19]*dt; nextP[9][20] = P[9][20] + P[6][20]*dt; nextP[9][21] = P[9][21] + P[6][21]*dt; nextP[10][0] = P[10][0] + P[10][1]*SF[7] + P[10][2]*SF[9] + P[10][3]*SF[8] + P[10][10]*SF[11] + P[10][11]*SPP[7] + P[10][12]*SPP[6]; nextP[10][1] = P[10][1] + P[10][0]*SF[6] + P[10][2]*SF[5] + P[10][3]*SF[9] + P[10][11]*SPP[6] - P[10][12]*SPP[7] - (P[10][10]*q0)/2; nextP[10][2] = P[10][2] + P[10][0]*SF[4] + P[10][1]*SF[8] + P[10][3]*SF[6] + P[10][12]*SF[11] - P[10][10]*SPP[6] - (P[10][11]*q0)/2; nextP[10][3] = P[10][3] + P[10][0]*SF[5] + P[10][1]*SF[4] + P[10][2]*SF[7] - P[10][11]*SF[11] + P[10][10]*SPP[7] - (P[10][12]*q0)/2; nextP[10][4] = P[10][4] + P[10][1]*SF[1] + P[10][0]*SF[3] + P[10][2]*SPP[0] - P[10][3]*SPP[2] - P[10][13]*SPP[4]; nextP[10][5] = P[10][5] + P[10][0]*SF[2] + P[10][2]*SF[1] + P[10][3]*SF[3] - P[10][1]*SPP[0] + P[10][13]*SPP[3]; nextP[10][6] = P[10][6] + P[10][1]*SF[2] + P[10][3]*SF[1] + P[10][0]*SPP[0] - P[10][2]*SPP[1] - P[10][13]*SPP[5]; nextP[10][7] = P[10][7] + P[10][4]*dt; nextP[10][8] = P[10][8] + P[10][5]*dt; nextP[10][9] = P[10][9] + P[10][6]*dt; nextP[10][10] = P[10][10]; nextP[10][11] = P[10][11]; nextP[10][12] = P[10][12]; nextP[10][13] = P[10][13]; nextP[10][14] = P[10][14]; nextP[10][15] = P[10][15]; nextP[10][16] = P[10][16]; nextP[10][17] = P[10][17]; nextP[10][18] = P[10][18]; nextP[10][19] = P[10][19]; nextP[10][20] = P[10][20]; nextP[10][21] = P[10][21]; nextP[11][0] = P[11][0] + P[11][1]*SF[7] + P[11][2]*SF[9] + P[11][3]*SF[8] + P[11][10]*SF[11] + P[11][11]*SPP[7] + P[11][12]*SPP[6]; nextP[11][1] = P[11][1] + P[11][0]*SF[6] + P[11][2]*SF[5] + P[11][3]*SF[9] + P[11][11]*SPP[6] - P[11][12]*SPP[7] - (P[11][10]*q0)/2; nextP[11][2] = P[11][2] + P[11][0]*SF[4] + P[11][1]*SF[8] + P[11][3]*SF[6] + P[11][12]*SF[11] - P[11][10]*SPP[6] - (P[11][11]*q0)/2; nextP[11][3] = P[11][3] + P[11][0]*SF[5] + P[11][1]*SF[4] + P[11][2]*SF[7] - P[11][11]*SF[11] + P[11][10]*SPP[7] - (P[11][12]*q0)/2; nextP[11][4] = P[11][4] + P[11][1]*SF[1] + P[11][0]*SF[3] + P[11][2]*SPP[0] - P[11][3]*SPP[2] - P[11][13]*SPP[4]; nextP[11][5] = P[11][5] + P[11][0]*SF[2] + P[11][2]*SF[1] + P[11][3]*SF[3] - P[11][1]*SPP[0] + P[11][13]*SPP[3]; nextP[11][6] = P[11][6] + P[11][1]*SF[2] + P[11][3]*SF[1] + P[11][0]*SPP[0] - P[11][2]*SPP[1] - P[11][13]*SPP[5]; nextP[11][7] = P[11][7] + P[11][4]*dt; nextP[11][8] = P[11][8] + P[11][5]*dt; nextP[11][9] = P[11][9] + P[11][6]*dt; nextP[11][10] = P[11][10]; nextP[11][11] = P[11][11]; nextP[11][12] = P[11][12]; nextP[11][13] = P[11][13]; nextP[11][14] = P[11][14]; nextP[11][15] = P[11][15]; nextP[11][16] = P[11][16]; nextP[11][17] = P[11][17]; nextP[11][18] = P[11][18]; nextP[11][19] = P[11][19]; nextP[11][20] = P[11][20]; nextP[11][21] = P[11][21]; nextP[12][0] = P[12][0] + P[12][1]*SF[7] + P[12][2]*SF[9] + P[12][3]*SF[8] + P[12][10]*SF[11] + P[12][11]*SPP[7] + P[12][12]*SPP[6]; nextP[12][1] = P[12][1] + P[12][0]*SF[6] + P[12][2]*SF[5] + P[12][3]*SF[9] + P[12][11]*SPP[6] - P[12][12]*SPP[7] - (P[12][10]*q0)/2; nextP[12][2] = P[12][2] + P[12][0]*SF[4] + P[12][1]*SF[8] + P[12][3]*SF[6] + P[12][12]*SF[11] - P[12][10]*SPP[6] - (P[12][11]*q0)/2; nextP[12][3] = P[12][3] + P[12][0]*SF[5] + P[12][1]*SF[4] + P[12][2]*SF[7] - P[12][11]*SF[11] + P[12][10]*SPP[7] - (P[12][12]*q0)/2; nextP[12][4] = P[12][4] + P[12][1]*SF[1] + P[12][0]*SF[3] + P[12][2]*SPP[0] - P[12][3]*SPP[2] - P[12][13]*SPP[4]; nextP[12][5] = P[12][5] + P[12][0]*SF[2] + P[12][2]*SF[1] + P[12][3]*SF[3] - P[12][1]*SPP[0] + P[12][13]*SPP[3]; nextP[12][6] = P[12][6] + P[12][1]*SF[2] + P[12][3]*SF[1] + P[12][0]*SPP[0] - P[12][2]*SPP[1] - P[12][13]*SPP[5]; nextP[12][7] = P[12][7] + P[12][4]*dt; nextP[12][8] = P[12][8] + P[12][5]*dt; nextP[12][9] = P[12][9] + P[12][6]*dt; nextP[12][10] = P[12][10]; nextP[12][11] = P[12][11]; nextP[12][12] = P[12][12]; nextP[12][13] = P[12][13]; nextP[12][14] = P[12][14]; nextP[12][15] = P[12][15]; nextP[12][16] = P[12][16]; nextP[12][17] = P[12][17]; nextP[12][18] = P[12][18]; nextP[12][19] = P[12][19]; nextP[12][20] = P[12][20]; nextP[12][21] = P[12][21]; nextP[13][0] = P[13][0] + P[13][1]*SF[7] + P[13][2]*SF[9] + P[13][3]*SF[8] + P[13][10]*SF[11] + P[13][11]*SPP[7] + P[13][12]*SPP[6]; nextP[13][1] = P[13][1] + P[13][0]*SF[6] + P[13][2]*SF[5] + P[13][3]*SF[9] + P[13][11]*SPP[6] - P[13][12]*SPP[7] - (P[13][10]*q0)/2; nextP[13][2] = P[13][2] + P[13][0]*SF[4] + P[13][1]*SF[8] + P[13][3]*SF[6] + P[13][12]*SF[11] - P[13][10]*SPP[6] - (P[13][11]*q0)/2; nextP[13][3] = P[13][3] + P[13][0]*SF[5] + P[13][1]*SF[4] + P[13][2]*SF[7] - P[13][11]*SF[11] + P[13][10]*SPP[7] - (P[13][12]*q0)/2; nextP[13][4] = P[13][4] + P[13][1]*SF[1] + P[13][0]*SF[3] + P[13][2]*SPP[0] - P[13][3]*SPP[2] - P[13][13]*SPP[4]; nextP[13][5] = P[13][5] + P[13][0]*SF[2] + P[13][2]*SF[1] + P[13][3]*SF[3] - P[13][1]*SPP[0] + P[13][13]*SPP[3]; nextP[13][6] = P[13][6] + P[13][1]*SF[2] + P[13][3]*SF[1] + P[13][0]*SPP[0] - P[13][2]*SPP[1] - P[13][13]*SPP[5]; nextP[13][7] = P[13][7] + P[13][4]*dt; nextP[13][8] = P[13][8] + P[13][5]*dt; nextP[13][9] = P[13][9] + P[13][6]*dt; nextP[13][10] = P[13][10]; nextP[13][11] = P[13][11]; nextP[13][12] = P[13][12]; nextP[13][13] = P[13][13]; nextP[13][14] = P[13][14]; nextP[13][15] = P[13][15]; nextP[13][16] = P[13][16]; nextP[13][17] = P[13][17]; nextP[13][18] = P[13][18]; nextP[13][19] = P[13][19]; nextP[13][20] = P[13][20]; nextP[13][21] = P[13][21]; nextP[14][0] = P[14][0] + P[14][1]*SF[7] + P[14][2]*SF[9] + P[14][3]*SF[8] + P[14][10]*SF[11] + P[14][11]*SPP[7] + P[14][12]*SPP[6]; nextP[14][1] = P[14][1] + P[14][0]*SF[6] + P[14][2]*SF[5] + P[14][3]*SF[9] + P[14][11]*SPP[6] - P[14][12]*SPP[7] - (P[14][10]*q0)/2; nextP[14][2] = P[14][2] + P[14][0]*SF[4] + P[14][1]*SF[8] + P[14][3]*SF[6] + P[14][12]*SF[11] - P[14][10]*SPP[6] - (P[14][11]*q0)/2; nextP[14][3] = P[14][3] + P[14][0]*SF[5] + P[14][1]*SF[4] + P[14][2]*SF[7] - P[14][11]*SF[11] + P[14][10]*SPP[7] - (P[14][12]*q0)/2; nextP[14][4] = P[14][4] + P[14][1]*SF[1] + P[14][0]*SF[3] + P[14][2]*SPP[0] - P[14][3]*SPP[2] - P[14][13]*SPP[4]; nextP[14][5] = P[14][5] + P[14][0]*SF[2] + P[14][2]*SF[1] + P[14][3]*SF[3] - P[14][1]*SPP[0] + P[14][13]*SPP[3]; nextP[14][6] = P[14][6] + P[14][1]*SF[2] + P[14][3]*SF[1] + P[14][0]*SPP[0] - P[14][2]*SPP[1] - P[14][13]*SPP[5]; nextP[14][7] = P[14][7] + P[14][4]*dt; nextP[14][8] = P[14][8] + P[14][5]*dt; nextP[14][9] = P[14][9] + P[14][6]*dt; nextP[14][10] = P[14][10]; nextP[14][11] = P[14][11]; nextP[14][12] = P[14][12]; nextP[14][13] = P[14][13]; nextP[14][14] = P[14][14]; nextP[14][15] = P[14][15]; nextP[14][16] = P[14][16]; nextP[14][17] = P[14][17]; nextP[14][18] = P[14][18]; nextP[14][19] = P[14][19]; nextP[14][20] = P[14][20]; nextP[14][21] = P[14][21]; nextP[15][0] = P[15][0] + P[15][1]*SF[7] + P[15][2]*SF[9] + P[15][3]*SF[8] + P[15][10]*SF[11] + P[15][11]*SPP[7] + P[15][12]*SPP[6]; nextP[15][1] = P[15][1] + P[15][0]*SF[6] + P[15][2]*SF[5] + P[15][3]*SF[9] + P[15][11]*SPP[6] - P[15][12]*SPP[7] - (P[15][10]*q0)/2; nextP[15][2] = P[15][2] + P[15][0]*SF[4] + P[15][1]*SF[8] + P[15][3]*SF[6] + P[15][12]*SF[11] - P[15][10]*SPP[6] - (P[15][11]*q0)/2; nextP[15][3] = P[15][3] + P[15][0]*SF[5] + P[15][1]*SF[4] + P[15][2]*SF[7] - P[15][11]*SF[11] + P[15][10]*SPP[7] - (P[15][12]*q0)/2; nextP[15][4] = P[15][4] + P[15][1]*SF[1] + P[15][0]*SF[3] + P[15][2]*SPP[0] - P[15][3]*SPP[2] - P[15][13]*SPP[4]; nextP[15][5] = P[15][5] + P[15][0]*SF[2] + P[15][2]*SF[1] + P[15][3]*SF[3] - P[15][1]*SPP[0] + P[15][13]*SPP[3]; nextP[15][6] = P[15][6] + P[15][1]*SF[2] + P[15][3]*SF[1] + P[15][0]*SPP[0] - P[15][2]*SPP[1] - P[15][13]*SPP[5]; nextP[15][7] = P[15][7] + P[15][4]*dt; nextP[15][8] = P[15][8] + P[15][5]*dt; nextP[15][9] = P[15][9] + P[15][6]*dt; nextP[15][10] = P[15][10]; nextP[15][11] = P[15][11]; nextP[15][12] = P[15][12]; nextP[15][13] = P[15][13]; nextP[15][14] = P[15][14]; nextP[15][15] = P[15][15]; nextP[15][16] = P[15][16]; nextP[15][17] = P[15][17]; nextP[15][18] = P[15][18]; nextP[15][19] = P[15][19]; nextP[15][20] = P[15][20]; nextP[15][21] = P[15][21]; nextP[16][0] = P[16][0] + P[16][1]*SF[7] + P[16][2]*SF[9] + P[16][3]*SF[8] + P[16][10]*SF[11] + P[16][11]*SPP[7] + P[16][12]*SPP[6]; nextP[16][1] = P[16][1] + P[16][0]*SF[6] + P[16][2]*SF[5] + P[16][3]*SF[9] + P[16][11]*SPP[6] - P[16][12]*SPP[7] - (P[16][10]*q0)/2; nextP[16][2] = P[16][2] + P[16][0]*SF[4] + P[16][1]*SF[8] + P[16][3]*SF[6] + P[16][12]*SF[11] - P[16][10]*SPP[6] - (P[16][11]*q0)/2; nextP[16][3] = P[16][3] + P[16][0]*SF[5] + P[16][1]*SF[4] + P[16][2]*SF[7] - P[16][11]*SF[11] + P[16][10]*SPP[7] - (P[16][12]*q0)/2; nextP[16][4] = P[16][4] + P[16][1]*SF[1] + P[16][0]*SF[3] + P[16][2]*SPP[0] - P[16][3]*SPP[2] - P[16][13]*SPP[4]; nextP[16][5] = P[16][5] + P[16][0]*SF[2] + P[16][2]*SF[1] + P[16][3]*SF[3] - P[16][1]*SPP[0] + P[16][13]*SPP[3]; nextP[16][6] = P[16][6] + P[16][1]*SF[2] + P[16][3]*SF[1] + P[16][0]*SPP[0] - P[16][2]*SPP[1] - P[16][13]*SPP[5]; nextP[16][7] = P[16][7] + P[16][4]*dt; nextP[16][8] = P[16][8] + P[16][5]*dt; nextP[16][9] = P[16][9] + P[16][6]*dt; nextP[16][10] = P[16][10]; nextP[16][11] = P[16][11]; nextP[16][12] = P[16][12]; nextP[16][13] = P[16][13]; nextP[16][14] = P[16][14]; nextP[16][15] = P[16][15]; nextP[16][16] = P[16][16]; nextP[16][17] = P[16][17]; nextP[16][18] = P[16][18]; nextP[16][19] = P[16][19]; nextP[16][20] = P[16][20]; nextP[16][21] = P[16][21]; nextP[17][0] = P[17][0] + P[17][1]*SF[7] + P[17][2]*SF[9] + P[17][3]*SF[8] + P[17][10]*SF[11] + P[17][11]*SPP[7] + P[17][12]*SPP[6]; nextP[17][1] = P[17][1] + P[17][0]*SF[6] + P[17][2]*SF[5] + P[17][3]*SF[9] + P[17][11]*SPP[6] - P[17][12]*SPP[7] - (P[17][10]*q0)/2; nextP[17][2] = P[17][2] + P[17][0]*SF[4] + P[17][1]*SF[8] + P[17][3]*SF[6] + P[17][12]*SF[11] - P[17][10]*SPP[6] - (P[17][11]*q0)/2; nextP[17][3] = P[17][3] + P[17][0]*SF[5] + P[17][1]*SF[4] + P[17][2]*SF[7] - P[17][11]*SF[11] + P[17][10]*SPP[7] - (P[17][12]*q0)/2; nextP[17][4] = P[17][4] + P[17][1]*SF[1] + P[17][0]*SF[3] + P[17][2]*SPP[0] - P[17][3]*SPP[2] - P[17][13]*SPP[4]; nextP[17][5] = P[17][5] + P[17][0]*SF[2] + P[17][2]*SF[1] + P[17][3]*SF[3] - P[17][1]*SPP[0] + P[17][13]*SPP[3]; nextP[17][6] = P[17][6] + P[17][1]*SF[2] + P[17][3]*SF[1] + P[17][0]*SPP[0] - P[17][2]*SPP[1] - P[17][13]*SPP[5]; nextP[17][7] = P[17][7] + P[17][4]*dt; nextP[17][8] = P[17][8] + P[17][5]*dt; nextP[17][9] = P[17][9] + P[17][6]*dt; nextP[17][10] = P[17][10]; nextP[17][11] = P[17][11]; nextP[17][12] = P[17][12]; nextP[17][13] = P[17][13]; nextP[17][14] = P[17][14]; nextP[17][15] = P[17][15]; nextP[17][16] = P[17][16]; nextP[17][17] = P[17][17]; nextP[17][18] = P[17][18]; nextP[17][19] = P[17][19]; nextP[17][20] = P[17][20]; nextP[17][21] = P[17][21]; nextP[18][0] = P[18][0] + P[18][1]*SF[7] + P[18][2]*SF[9] + P[18][3]*SF[8] + P[18][10]*SF[11] + P[18][11]*SPP[7] + P[18][12]*SPP[6]; nextP[18][1] = P[18][1] + P[18][0]*SF[6] + P[18][2]*SF[5] + P[18][3]*SF[9] + P[18][11]*SPP[6] - P[18][12]*SPP[7] - (P[18][10]*q0)/2; nextP[18][2] = P[18][2] + P[18][0]*SF[4] + P[18][1]*SF[8] + P[18][3]*SF[6] + P[18][12]*SF[11] - P[18][10]*SPP[6] - (P[18][11]*q0)/2; nextP[18][3] = P[18][3] + P[18][0]*SF[5] + P[18][1]*SF[4] + P[18][2]*SF[7] - P[18][11]*SF[11] + P[18][10]*SPP[7] - (P[18][12]*q0)/2; nextP[18][4] = P[18][4] + P[18][1]*SF[1] + P[18][0]*SF[3] + P[18][2]*SPP[0] - P[18][3]*SPP[2] - P[18][13]*SPP[4]; nextP[18][5] = P[18][5] + P[18][0]*SF[2] + P[18][2]*SF[1] + P[18][3]*SF[3] - P[18][1]*SPP[0] + P[18][13]*SPP[3]; nextP[18][6] = P[18][6] + P[18][1]*SF[2] + P[18][3]*SF[1] + P[18][0]*SPP[0] - P[18][2]*SPP[1] - P[18][13]*SPP[5]; nextP[18][7] = P[18][7] + P[18][4]*dt; nextP[18][8] = P[18][8] + P[18][5]*dt; nextP[18][9] = P[18][9] + P[18][6]*dt; nextP[18][10] = P[18][10]; nextP[18][11] = P[18][11]; nextP[18][12] = P[18][12]; nextP[18][13] = P[18][13]; nextP[18][14] = P[18][14]; nextP[18][15] = P[18][15]; nextP[18][16] = P[18][16]; nextP[18][17] = P[18][17]; nextP[18][18] = P[18][18]; nextP[18][19] = P[18][19]; nextP[18][20] = P[18][20]; nextP[18][21] = P[18][21]; nextP[19][0] = P[19][0] + P[19][1]*SF[7] + P[19][2]*SF[9] + P[19][3]*SF[8] + P[19][10]*SF[11] + P[19][11]*SPP[7] + P[19][12]*SPP[6]; nextP[19][1] = P[19][1] + P[19][0]*SF[6] + P[19][2]*SF[5] + P[19][3]*SF[9] + P[19][11]*SPP[6] - P[19][12]*SPP[7] - (P[19][10]*q0)/2; nextP[19][2] = P[19][2] + P[19][0]*SF[4] + P[19][1]*SF[8] + P[19][3]*SF[6] + P[19][12]*SF[11] - P[19][10]*SPP[6] - (P[19][11]*q0)/2; nextP[19][3] = P[19][3] + P[19][0]*SF[5] + P[19][1]*SF[4] + P[19][2]*SF[7] - P[19][11]*SF[11] + P[19][10]*SPP[7] - (P[19][12]*q0)/2; nextP[19][4] = P[19][4] + P[19][1]*SF[1] + P[19][0]*SF[3] + P[19][2]*SPP[0] - P[19][3]*SPP[2] - P[19][13]*SPP[4]; nextP[19][5] = P[19][5] + P[19][0]*SF[2] + P[19][2]*SF[1] + P[19][3]*SF[3] - P[19][1]*SPP[0] + P[19][13]*SPP[3]; nextP[19][6] = P[19][6] + P[19][1]*SF[2] + P[19][3]*SF[1] + P[19][0]*SPP[0] - P[19][2]*SPP[1] - P[19][13]*SPP[5]; nextP[19][7] = P[19][7] + P[19][4]*dt; nextP[19][8] = P[19][8] + P[19][5]*dt; nextP[19][9] = P[19][9] + P[19][6]*dt; nextP[19][10] = P[19][10]; nextP[19][11] = P[19][11]; nextP[19][12] = P[19][12]; nextP[19][13] = P[19][13]; nextP[19][14] = P[19][14]; nextP[19][15] = P[19][15]; nextP[19][16] = P[19][16]; nextP[19][17] = P[19][17]; nextP[19][18] = P[19][18]; nextP[19][19] = P[19][19]; nextP[19][20] = P[19][20]; nextP[19][21] = P[19][21]; nextP[20][0] = P[20][0] + P[20][1]*SF[7] + P[20][2]*SF[9] + P[20][3]*SF[8] + P[20][10]*SF[11] + P[20][11]*SPP[7] + P[20][12]*SPP[6]; nextP[20][1] = P[20][1] + P[20][0]*SF[6] + P[20][2]*SF[5] + P[20][3]*SF[9] + P[20][11]*SPP[6] - P[20][12]*SPP[7] - (P[20][10]*q0)/2; nextP[20][2] = P[20][2] + P[20][0]*SF[4] + P[20][1]*SF[8] + P[20][3]*SF[6] + P[20][12]*SF[11] - P[20][10]*SPP[6] - (P[20][11]*q0)/2; nextP[20][3] = P[20][3] + P[20][0]*SF[5] + P[20][1]*SF[4] + P[20][2]*SF[7] - P[20][11]*SF[11] + P[20][10]*SPP[7] - (P[20][12]*q0)/2; nextP[20][4] = P[20][4] + P[20][1]*SF[1] + P[20][0]*SF[3] + P[20][2]*SPP[0] - P[20][3]*SPP[2] - P[20][13]*SPP[4]; nextP[20][5] = P[20][5] + P[20][0]*SF[2] + P[20][2]*SF[1] + P[20][3]*SF[3] - P[20][1]*SPP[0] + P[20][13]*SPP[3]; nextP[20][6] = P[20][6] + P[20][1]*SF[2] + P[20][3]*SF[1] + P[20][0]*SPP[0] - P[20][2]*SPP[1] - P[20][13]*SPP[5]; nextP[20][7] = P[20][7] + P[20][4]*dt; nextP[20][8] = P[20][8] + P[20][5]*dt; nextP[20][9] = P[20][9] + P[20][6]*dt; nextP[20][10] = P[20][10]; nextP[20][11] = P[20][11]; nextP[20][12] = P[20][12]; nextP[20][13] = P[20][13]; nextP[20][14] = P[20][14]; nextP[20][15] = P[20][15]; nextP[20][16] = P[20][16]; nextP[20][17] = P[20][17]; nextP[20][18] = P[20][18]; nextP[20][19] = P[20][19]; nextP[20][20] = P[20][20]; nextP[20][21] = P[20][21]; nextP[21][0] = P[21][0] + P[21][1]*SF[7] + P[21][2]*SF[9] + P[21][3]*SF[8] + P[21][10]*SF[11] + P[21][11]*SPP[7] + P[21][12]*SPP[6]; nextP[21][1] = P[21][1] + P[21][0]*SF[6] + P[21][2]*SF[5] + P[21][3]*SF[9] + P[21][11]*SPP[6] - P[21][12]*SPP[7] - (P[21][10]*q0)/2; nextP[21][2] = P[21][2] + P[21][0]*SF[4] + P[21][1]*SF[8] + P[21][3]*SF[6] + P[21][12]*SF[11] - P[21][10]*SPP[6] - (P[21][11]*q0)/2; nextP[21][3] = P[21][3] + P[21][0]*SF[5] + P[21][1]*SF[4] + P[21][2]*SF[7] - P[21][11]*SF[11] + P[21][10]*SPP[7] - (P[21][12]*q0)/2; nextP[21][4] = P[21][4] + P[21][1]*SF[1] + P[21][0]*SF[3] + P[21][2]*SPP[0] - P[21][3]*SPP[2] - P[21][13]*SPP[4]; nextP[21][5] = P[21][5] + P[21][0]*SF[2] + P[21][2]*SF[1] + P[21][3]*SF[3] - P[21][1]*SPP[0] + P[21][13]*SPP[3]; nextP[21][6] = P[21][6] + P[21][1]*SF[2] + P[21][3]*SF[1] + P[21][0]*SPP[0] - P[21][2]*SPP[1] - P[21][13]*SPP[5]; nextP[21][7] = P[21][7] + P[21][4]*dt; nextP[21][8] = P[21][8] + P[21][5]*dt; nextP[21][9] = P[21][9] + P[21][6]*dt; nextP[21][10] = P[21][10]; nextP[21][11] = P[21][11]; nextP[21][12] = P[21][12]; nextP[21][13] = P[21][13]; nextP[21][14] = P[21][14]; nextP[21][15] = P[21][15]; nextP[21][16] = P[21][16]; nextP[21][17] = P[21][17]; nextP[21][18] = P[21][18]; nextP[21][19] = P[21][19]; nextP[21][20] = P[21][20]; nextP[21][21] = P[21][21]; // add the general state process noise variances for (uint8_t i=0; i<= 21; i++) { nextP[i][i] = nextP[i][i] + processNoise[i]; } // if the total position variance exceeds 1E6 (1000m), then stop covariance // growth by setting the predicted to the previous values // This prevent an ill conditioned matrix from occurring for long periods // without GPS if ((P[7][7] + P[8][8]) > 1e6f) { for (uint8_t i=7; i<=8; i++) { for (uint8_t j=0; j<=21; j++) { nextP[i][j] = P[i][j]; nextP[j][i] = P[j][i]; } } } // copy covariances to output and fix numerical errors CopyAndFixCovariances(); // constrain diagonals to prevent ill-conditioning ConstrainVariances(); perf_end(_perf_CovariancePrediction); } // fuse selected position, velocity and height measurements, checking dat for consistency // provide a static mode that allows maintenance of the attitude reference without GPS provided the vehicle is not accelerating // check innovation consistency of velocity states calculated using IMU1 and IMU2 and calculate the optimal weighting of accel data void NavEKF::FuseVelPosNED() { // start performance timer perf_begin(_perf_FuseVelPosNED); // health is set bad until test passed velHealth = false; posHealth = false; hgtHealth = false; // declare variables used to check measurement errors Vector3f velInnov; Vector3f velInnov1; Vector3f velInnov2; Vector2 posInnov; float hgtInnov = 0; // declare variables used to control access to arrays bool fuseData[6] = {false,false,false,false,false,false}; uint8_t stateIndex; uint8_t obsIndex; // declare variables used by state and covariance update calculations float NEvelErr; float DvelErr; float posErr; Vector6 R_OBS; Vector6 observation; float SK; // perform sequential fusion of GPS measurements. This assumes that the // errors in the different velocity and position components are // uncorrelated which is not true, however in the absence of covariance // data from the GPS receiver it is the only assumption we can make // so we might as well take advantage of the computational efficiencies // associated with sequential fusion if (fuseVelData || fusePosData || fuseHgtData) { // if static mode is active use the current states to calculate the predicted // measurement rather than use states from a previous time. We need to do this // because there may be no stored states due to lack of real measurements. // in static mode, only position and height fusion is used if (staticMode) { statesAtPosTime = state; statesAtHgtTime = state; } // set the GPS data timeout depending on whether airspeed data is present uint32_t gpsRetryTime; if (useAirspeed()) gpsRetryTime = _gpsRetryTimeUseTAS; else gpsRetryTime = _gpsRetryTimeNoTAS; // form the observation vector and zero velocity and horizontal position observations if in static mode if (!staticMode) { for (uint8_t i=0; i<=2; i++) observation[i] = velNED[i]; observation[3] = gpsPosNE.x + gpsPosGlitchOffsetNE.x; observation[4] = gpsPosNE.y + gpsPosGlitchOffsetNE.y; } else { for (uint8_t i=0; i<=4; i++) observation[i] = 0.0f; } observation[5] = -hgtMea; // calculate additional error in GPS velocity caused by manoeuvring NEvelErr = _gpsNEVelVarAccScale * accNavMag; DvelErr = _gpsDVelVarAccScale * accNavMag; // calculate additional error in GPS position caused by manoeuvring posErr = _gpsPosVarAccScale * accNavMag; // estimate the GPS Velocity, GPS horiz position and height measurement variances. R_OBS[0] = gpsVarScaler*(sq(constrain_float(_gpsHorizVelNoise, 0.05f, 5.0f)) + sq(NEvelErr)); R_OBS[1] = R_OBS[0]; R_OBS[2] = gpsVarScaler*(sq(constrain_float(_gpsVertVelNoise, 0.05f, 5.0f)) + sq(DvelErr)); R_OBS[3] = gpsVarScaler*(sq(constrain_float(_gpsHorizPosNoise, 0.1f, 10.0f)) + sq(posErr)); R_OBS[4] = R_OBS[3]; R_OBS[5] = hgtVarScaler*sq(constrain_float(_baroAltNoise, 0.1f, 10.0f)); // if vertical GPS velocity data is being used, check to see if the GPS vertical velocity and barometer // innovations have the same sign and are outside limits. If so, then it is likely aliasing is affecting // the accelerometers and we should disable the GPS and barometer innovation consistency checks. bool badIMUdata = false; if (_fusionModeGPS == 0 && fuseVelData && fuseHgtData) { // calculate innovations for height and vertical GPS vel measurements float hgtErr = statesAtVelTime.position.z - observation[5]; float velDErr = statesAtVelTime.velocity.z - observation[2]; // check if they are the same sign and both more than 3-sigma out of bounds if ((hgtErr*velDErr > 0.0f) && (sq(hgtErr) > 9.0f * (P[9][9] + R_OBS[5])) && (sq(velDErr) > 9.0f * (P[6][6] + R_OBS[2]))) { badIMUdata = true; } else { badIMUdata = false; } } // calculate innovations and check GPS data validity using an innovation consistency check // test position measurements if (fusePosData) { // test horizontal position measurements posInnov[0] = statesAtPosTime.position.x - observation[3]; posInnov[1] = statesAtPosTime.position.y - observation[4]; varInnovVelPos[3] = P[7][7] + R_OBS[3]; varInnovVelPos[4] = P[8][8] + R_OBS[4]; // apply an innovation consistency threshold test, but don't fail if bad IMU data // calculate max valid position innovation squared based on a maximum horizontal inertial nav accel error and GPS noise parameter // max inertial nav error is scaled with horizontal g to allow for increased errors when manoeuvring float accelScale = (1.0f + 0.1f * accNavMagHoriz); float maxPosInnov2 = sq(_gpsPosInnovGate * _gpsHorizPosNoise + 0.005f * accelScale * float(_gpsGlitchAccelMax) * sq(0.001f * float(hal.scheduler->millis() - posFailTime))); posTestRatio = (sq(posInnov[0]) + sq(posInnov[1])) / maxPosInnov2; posHealth = ((posTestRatio < 1.0f) || badIMUdata); // declare a timeout condition if we have been too long without data posTimeout = ((hal.scheduler->millis() - posFailTime) > gpsRetryTime); // use position data if healthy, timed out, or in static mode if (posHealth || posTimeout || staticMode) { posHealth = true; posFailTime = hal.scheduler->millis(); // if timed out or outside the specified glitch radius, increment the offset applied to GPS data to compensate for large GPS position jumps if (posTimeout || (maxPosInnov2 > sq(float(_gpsGlitchRadiusMax)))) { gpsPosGlitchOffsetNE.x += posInnov[0]; gpsPosGlitchOffsetNE.y += posInnov[1]; // limit the radius of the offset to 100m and decay the offset to zero radially decayGpsOffset(); // reset the position to the current GPS position which will include the glitch correction offset ResetPosition(); // don't fuse data on this time step fusePosData = false; } } else { posHealth = false; } } // test velocity measurements if (fuseVelData) { // test velocity measurements uint8_t imax = 2; if (_fusionModeGPS == 1) { imax = 1; } float K1 = 0; // innovation to error ratio for IMU1 float K2 = 0; // innovation to error ratio for IMU2 float innovVelSumSq = 0; // sum of squares of velocity innovations float varVelSum = 0; // sum of velocity innovation variances for (uint8_t i = 0; i<=imax; i++) { // velocity states start at index 4 stateIndex = i + 4; // calculate innovations using blended and single IMU predicted states velInnov[i] = statesAtVelTime.velocity[i] - observation[i]; // blended velInnov1[i] = statesAtVelTime.vel1[i] - observation[i]; // IMU1 velInnov2[i] = statesAtVelTime.vel2[i] - observation[i]; // IMU2 // calculate innovation variance varInnovVelPos[i] = P[stateIndex][stateIndex] + R_OBS[i]; // calculate error weightings for singloe IMU velocity states using // observation error to normalise float R_hgt; if (i == 2) { R_hgt = sq(constrain_float(_gpsVertVelNoise, 0.05f, 5.0f)); } else { R_hgt = sq(constrain_float(_gpsHorizVelNoise, 0.05f, 5.0f)); } K1 += R_hgt / (R_hgt + sq(velInnov1[i])); K2 += R_hgt / (R_hgt + sq(velInnov2[i])); // sum the innovation and innovation variances innovVelSumSq += sq(velInnov[i]); varVelSum += varInnovVelPos[i]; } // calculate weighting used by fuseVelPosNED to do IMU accel data blending // this is used to detect and compensate for aliasing errors with the accelerometers // provide for a first order lowpass filter to reduce noise on the weighting if required IMU1_weighting = 1.0f * (K1 / (K1 + K2)) + 0.0f * IMU1_weighting; // filter currently inactive // apply an innovation consistency threshold test, but don't fail if bad IMU data // calculate the test ratio velTestRatio = innovVelSumSq / (varVelSum * sq(_gpsVelInnovGate)); // fail if the ratio is greater than 1 velHealth = ((velTestRatio < 1.0f) || badIMUdata); // declare a timeout if we have not fused velocity data for too long velTimeout = (hal.scheduler->millis() - velFailTime) > gpsRetryTime; // if data is healthy or in static mode we fuse it if (velHealth || staticMode) { velHealth = true; velFailTime = hal.scheduler->millis(); } else if (velTimeout && !posHealth) { // if data is not healthy and timed out and position is unhealthy we reset the velocity, but do not fuse data on this time step ResetVelocity(); StoreStatesReset(); fuseVelData = false; } else { // if data is unhealthy and position is healthy, we do not fuse it velHealth = false; } } // test height measurements if (fuseHgtData) { // set the height data timeout depending on whether vertical velocity data is being used uint32_t hgtRetryTime; if (_fusionModeGPS == 0) hgtRetryTime = _hgtRetryTimeMode0; else hgtRetryTime = _hgtRetryTimeMode12; // calculate height innovations hgtInnov = statesAtHgtTime.position.z - observation[5]; varInnovVelPos[5] = P[9][9] + R_OBS[5]; // calculate the innovation consistency test ratio hgtTestRatio = sq(hgtInnov) / (sq(_hgtInnovGate) * varInnovVelPos[5]); // fail if the ratio is > 1, but don't fail if bad IMU data hgtHealth = ((hgtTestRatio < 1.0f) || badIMUdata); hgtTimeout = (hal.scheduler->millis() - hgtFailTime) > hgtRetryTime; // Fuse height data if healthy or timed out or in static mode if (hgtHealth || hgtTimeout || staticMode) { hgtHealth = true; hgtFailTime = hal.scheduler->millis(); // if timed out, reset the height, but do not fuse data on this time step if (hgtTimeout) { ResetHeight(); StoreStatesReset(); fuseHgtData = false; } } else { hgtHealth = false; } } // set range for sequential fusion of velocity and position measurements depending on which data is available and its health if (fuseVelData && _fusionModeGPS == 0 && velHealth && !staticMode) { fuseData[0] = true; fuseData[1] = true; fuseData[2] = true; } if (fuseVelData && _fusionModeGPS == 1 && velHealth && !staticMode) { fuseData[0] = true; fuseData[1] = true; } if ((fusePosData && _fusionModeGPS <= 2 && posHealth) || staticMode) { fuseData[3] = true; fuseData[4] = true; } if ((fuseHgtData && hgtHealth) || staticMode) { fuseData[5] = true; } // fuse measurements sequentially for (obsIndex=0; obsIndex<=5; obsIndex++) { if (fuseData[obsIndex]) { stateIndex = 4 + obsIndex; // calculate the measurement innovation, using states from a different time coordinate if fusing height data if (obsIndex <= 2) { innovVelPos[obsIndex] = statesAtVelTime.velocity[obsIndex] - observation[obsIndex]; } else if (obsIndex == 3 || obsIndex == 4) { innovVelPos[obsIndex] = statesAtPosTime.position[obsIndex-3] - observation[obsIndex]; } else { innovVelPos[obsIndex] = statesAtHgtTime.position[obsIndex-3] - observation[obsIndex]; } // calculate the Kalman gain and calculate innovation variances varInnovVelPos[obsIndex] = P[stateIndex][stateIndex] + R_OBS[obsIndex]; SK = 1.0f/varInnovVelPos[obsIndex]; for (uint8_t i= 0; i<=12; i++) { Kfusion[i] = P[i][stateIndex]*SK; } // Only height observations are used to update z accel bias estimate // Protect Kalman gain from ill-conditioning // Don't update Z accel bias if off-level by greater than 60 degrees to avoid scale factor error effects if (obsIndex == 5 && prevTnb.c.z > 0.5f) { Kfusion[13] = constrain_float(P[13][stateIndex]*SK,-1.0f,0.0f); } else { Kfusion[13] = 0.0f; } // inhibit wind state estimation by setting Kalman gains to zero if (!inhibitWindStates) { Kfusion[14] = P[14][stateIndex]*SK; Kfusion[15] = P[15][stateIndex]*SK; } else { Kfusion[14] = 0.0f; Kfusion[15] = 0.0f; } // inhibit magnetic field state estimation by setting Kalman gains to zero if (!inhibitMagStates) { for (uint8_t i = 16; i<=21; i++) { Kfusion[i] = P[i][stateIndex]*SK; } } else { for (uint8_t i = 16; i<=21; i++) { Kfusion[i] = 0.0f; } } // Set the Kalman gain values for the single IMU states Kfusion[22] = Kfusion[13]; // IMU2 Z accel bias Kfusion[26] = Kfusion[9]; // IMU1 posD Kfusion[30] = Kfusion[9]; // IMU2 posD for (uint8_t i = 0; i<=2; i++) { Kfusion[i+23] = Kfusion[i+4]; // IMU1 velNED Kfusion[i+27] = Kfusion[i+4]; // IMU2 velNED } // Correct states that have been predicted using single (not blended) IMU data if (obsIndex == 5){ // Calculate height measurement innovations using single IMU states float hgtInnov1 = statesAtHgtTime.posD1 - observation[obsIndex]; float hgtInnov2 = statesAtHgtTime.posD2 - observation[obsIndex]; // Correct single IMU prediction states using height measurement, limiting rate of change of bias to 0.02 m/s3 float correctionLimit = 0.02f * dtIMU *dtVelPos; states[13] = states[13] - constrain_float(Kfusion[13] * hgtInnov1, -correctionLimit, correctionLimit); // IMU1 Z accel bias states[22] = states[22] - constrain_float(Kfusion[22] * hgtInnov2, -correctionLimit, correctionLimit); // IMU2 Z accel bias for (uint8_t i = 23; i<=26; i++) { states[i] = states[i] - Kfusion[i] * hgtInnov1; // IMU1 velNED,posD } for (uint8_t i = 27; i<=30; i++) { states[i] = states[i] - Kfusion[i] * hgtInnov2; // IMU2 velNED,posD } } else if (obsIndex == 0 || obsIndex == 1 || obsIndex == 2) { // Correct single IMU prediction states using velocity measurements for (uint8_t i = 23; i<=26; i++) { states[i] = states[i] - Kfusion[i] * velInnov1[obsIndex]; // IMU1 velNED,posD } for (uint8_t i = 27; i<=30; i++) { states[i] = states[i] - Kfusion[i] * velInnov2[obsIndex]; // IMU2 velNED,posD } } // calculate state corrections and re-normalise the quaternions for blended IMU data predicted states for (uint8_t i = 0; i<=21; i++) { states[i] = states[i] - Kfusion[i] * innovVelPos[obsIndex]; } state.quat.normalize(); // update the covariance - take advantage of direct observation of a single state at index = stateIndex to reduce computations // this is a numerically optimised implementation of standard equation P = (I - K*H)*P; for (uint8_t i= 0; i<=21; i++) { for (uint8_t j= 0; j<=21; j++) { KHP[i][j] = Kfusion[i] * P[stateIndex][j]; } } for (uint8_t i= 0; i<=21; i++) { for (uint8_t j= 0; j<=21; j++) { P[i][j] = P[i][j] - KHP[i][j]; } } } } } // force the covariance matrix to me symmetrical and limit the variances to prevent ill-condiioning. ForceSymmetry(); ConstrainVariances(); // stop performance timer perf_end(_perf_FuseVelPosNED); } // fuse magnetometer measurements and apply innovation consistency checks // fuse each axis on consecutive time steps to spread computional load void NavEKF::FuseMagnetometer() { // start performance timer perf_begin(_perf_FuseMagnetometer); // declarations ftype &q0 = mag_state.q0; ftype &q1 = mag_state.q1; ftype &q2 = mag_state.q2; ftype &q3 = mag_state.q3; ftype &magN = mag_state.magN; ftype &magE = mag_state.magE; ftype &magD = mag_state.magD; ftype &magXbias = mag_state.magXbias; ftype &magYbias = mag_state.magYbias; ftype &magZbias = mag_state.magZbias; uint8_t &obsIndex = mag_state.obsIndex; Matrix3f &DCM = mag_state.DCM; Vector3f &MagPred = mag_state.MagPred; ftype &R_MAG = mag_state.R_MAG; ftype *SH_MAG = &mag_state.SH_MAG[0]; Vector22 H_MAG; Vector6 SK_MX; Vector6 SK_MY; Vector6 SK_MZ; // perform sequential fusion of magnetometer measurements. // this assumes that the errors in the different components are // uncorrelated which is not true, however in the absence of covariance // data fit is the only assumption we can make // so we might as well take advantage of the computational efficiencies // associated with sequential fusion if (fuseMagData || obsIndex == 1 || obsIndex == 2) { // calculate observation jacobians and Kalman gains if (fuseMagData) { // copy required states to local variable names q0 = statesAtMagMeasTime.quat[0]; q1 = statesAtMagMeasTime.quat[1]; q2 = statesAtMagMeasTime.quat[2]; q3 = statesAtMagMeasTime.quat[3]; magN = statesAtMagMeasTime.earth_magfield[0]; magE = statesAtMagMeasTime.earth_magfield[1]; magD = statesAtMagMeasTime.earth_magfield[2]; magXbias = statesAtMagMeasTime.body_magfield[0]; magYbias = statesAtMagMeasTime.body_magfield[1]; magZbias = statesAtMagMeasTime.body_magfield[2]; // rotate predicted earth components into body axes and calculate // predicted measurements DCM[0][0] = q0*q0 + q1*q1 - q2*q2 - q3*q3; DCM[0][1] = 2*(q1*q2 + q0*q3); DCM[0][2] = 2*(q1*q3-q0*q2); DCM[1][0] = 2*(q1*q2 - q0*q3); DCM[1][1] = q0*q0 - q1*q1 + q2*q2 - q3*q3; DCM[1][2] = 2*(q2*q3 + q0*q1); DCM[2][0] = 2*(q1*q3 + q0*q2); DCM[2][1] = 2*(q2*q3 - q0*q1); DCM[2][2] = q0*q0 - q1*q1 - q2*q2 + q3*q3; MagPred[0] = DCM[0][0]*magN + DCM[0][1]*magE + DCM[0][2]*magD + magXbias; MagPred[1] = DCM[1][0]*magN + DCM[1][1]*magE + DCM[1][2]*magD + magYbias; MagPred[2] = DCM[2][0]*magN + DCM[2][1]*magE + DCM[2][2]*magD + magZbias; // scale magnetometer observation error with total angular rate R_MAG = sq(constrain_float(_magNoise, 0.01f, 0.5f)) + sq(_magVarRateScale*dAngIMU.length() / dtIMU); // calculate observation jacobians SH_MAG[0] = 2*magD*q3 + 2*magE*q2 + 2*magN*q1; SH_MAG[1] = 2*magD*q0 - 2*magE*q1 + 2*magN*q2; SH_MAG[2] = 2*magD*q1 + 2*magE*q0 - 2*magN*q3; SH_MAG[3] = sq(q3); SH_MAG[4] = sq(q2); SH_MAG[5] = sq(q1); SH_MAG[6] = sq(q0); SH_MAG[7] = 2*magN*q0; SH_MAG[8] = 2*magE*q3; for (uint8_t i=0; i<=21; i++) H_MAG[i] = 0; H_MAG[0] = SH_MAG[7] + SH_MAG[8] - 2*magD*q2; H_MAG[1] = SH_MAG[0]; H_MAG[2] = 2*magE*q1 - 2*magD*q0 - 2*magN*q2; H_MAG[3] = SH_MAG[2]; H_MAG[16] = SH_MAG[5] - SH_MAG[4] - SH_MAG[3] + SH_MAG[6]; H_MAG[17] = 2*q0*q3 + 2*q1*q2; H_MAG[18] = 2*q1*q3 - 2*q0*q2; H_MAG[19] = 1; // calculate Kalman gain float temp = (P[19][19] + R_MAG + P[1][19]*SH_MAG[0] + P[3][19]*SH_MAG[2] - P[16][19]*(SH_MAG[3] + SH_MAG[4] - SH_MAG[5] - SH_MAG[6]) - (2*magD*q0 - 2*magE*q1 + 2*magN*q2)*(P[19][2] + P[1][2]*SH_MAG[0] + P[3][2]*SH_MAG[2] - P[16][2]*(SH_MAG[3] + SH_MAG[4] - SH_MAG[5] - SH_MAG[6]) + P[17][2]*(2*q0*q3 + 2*q1*q2) - P[18][2]*(2*q0*q2 - 2*q1*q3) - P[2][2]*(2*magD*q0 - 2*magE*q1 + 2*magN*q2) + P[0][2]*(SH_MAG[7] + SH_MAG[8] - 2*magD*q2)) + (SH_MAG[7] + SH_MAG[8] - 2*magD*q2)*(P[19][0] + P[1][0]*SH_MAG[0] + P[3][0]*SH_MAG[2] - P[16][0]*(SH_MAG[3] + SH_MAG[4] - SH_MAG[5] - SH_MAG[6]) + P[17][0]*(2*q0*q3 + 2*q1*q2) - P[18][0]*(2*q0*q2 - 2*q1*q3) - P[2][0]*(2*magD*q0 - 2*magE*q1 + 2*magN*q2) + P[0][0]*(SH_MAG[7] + SH_MAG[8] - 2*magD*q2)) + SH_MAG[0]*(P[19][1] + P[1][1]*SH_MAG[0] + P[3][1]*SH_MAG[2] - P[16][1]*(SH_MAG[3] + SH_MAG[4] - SH_MAG[5] - SH_MAG[6]) + P[17][1]*(2*q0*q3 + 2*q1*q2) - P[18][1]*(2*q0*q2 - 2*q1*q3) - P[2][1]*(2*magD*q0 - 2*magE*q1 + 2*magN*q2) + P[0][1]*(SH_MAG[7] + SH_MAG[8] - 2*magD*q2)) + SH_MAG[2]*(P[19][3] + P[1][3]*SH_MAG[0] + P[3][3]*SH_MAG[2] - P[16][3]*(SH_MAG[3] + SH_MAG[4] - SH_MAG[5] - SH_MAG[6]) + P[17][3]*(2*q0*q3 + 2*q1*q2) - P[18][3]*(2*q0*q2 - 2*q1*q3) - P[2][3]*(2*magD*q0 - 2*magE*q1 + 2*magN*q2) + P[0][3]*(SH_MAG[7] + SH_MAG[8] - 2*magD*q2)) - (SH_MAG[3] + SH_MAG[4] - SH_MAG[5] - SH_MAG[6])*(P[19][16] + P[1][16]*SH_MAG[0] + P[3][16]*SH_MAG[2] - P[16][16]*(SH_MAG[3] + SH_MAG[4] - SH_MAG[5] - SH_MAG[6]) + P[17][16]*(2*q0*q3 + 2*q1*q2) - P[18][16]*(2*q0*q2 - 2*q1*q3) - P[2][16]*(2*magD*q0 - 2*magE*q1 + 2*magN*q2) + P[0][16]*(SH_MAG[7] + SH_MAG[8] - 2*magD*q2)) + P[17][19]*(2*q0*q3 + 2*q1*q2) - P[18][19]*(2*q0*q2 - 2*q1*q3) - P[2][19]*(2*magD*q0 - 2*magE*q1 + 2*magN*q2) + (2*q0*q3 + 2*q1*q2)*(P[19][17] + P[1][17]*SH_MAG[0] + P[3][17]*SH_MAG[2] - P[16][17]*(SH_MAG[3] + SH_MAG[4] - SH_MAG[5] - SH_MAG[6]) + P[17][17]*(2*q0*q3 + 2*q1*q2) - P[18][17]*(2*q0*q2 - 2*q1*q3) - P[2][17]*(2*magD*q0 - 2*magE*q1 + 2*magN*q2) + P[0][17]*(SH_MAG[7] + SH_MAG[8] - 2*magD*q2)) - (2*q0*q2 - 2*q1*q3)*(P[19][18] + P[1][18]*SH_MAG[0] + P[3][18]*SH_MAG[2] - P[16][18]*(SH_MAG[3] + SH_MAG[4] - SH_MAG[5] - SH_MAG[6]) + P[17][18]*(2*q0*q3 + 2*q1*q2) - P[18][18]*(2*q0*q2 - 2*q1*q3) - P[2][18]*(2*magD*q0 - 2*magE*q1 + 2*magN*q2) + P[0][18]*(SH_MAG[7] + SH_MAG[8] - 2*magD*q2)) + P[0][19]*(SH_MAG[7] + SH_MAG[8] - 2*magD*q2)); if (temp >= R_MAG) { SK_MX[0] = 1.0f / temp; faultStatus.bad_xmag = false; } else { // the calculation is badly conditioned, so we cannot perform fusion on this step // we increase the state variances and try again next time P[19][19] += 0.1f*R_MAG; obsIndex = 1; faultStatus.bad_xmag = true; return; } SK_MX[1] = SH_MAG[3] + SH_MAG[4] - SH_MAG[5] - SH_MAG[6]; SK_MX[2] = 2*magD*q0 - 2*magE*q1 + 2*magN*q2; SK_MX[3] = SH_MAG[7] + SH_MAG[8] - 2*magD*q2; SK_MX[4] = 2*q0*q2 - 2*q1*q3; SK_MX[5] = 2*q0*q3 + 2*q1*q2; Kfusion[0] = SK_MX[0]*(P[0][19] + P[0][1]*SH_MAG[0] + P[0][3]*SH_MAG[2] + P[0][0]*SK_MX[3] - P[0][2]*SK_MX[2] - P[0][16]*SK_MX[1] + P[0][17]*SK_MX[5] - P[0][18]*SK_MX[4]); Kfusion[1] = SK_MX[0]*(P[1][19] + P[1][1]*SH_MAG[0] + P[1][3]*SH_MAG[2] + P[1][0]*SK_MX[3] - P[1][2]*SK_MX[2] - P[1][16]*SK_MX[1] + P[1][17]*SK_MX[5] - P[1][18]*SK_MX[4]); Kfusion[2] = SK_MX[0]*(P[2][19] + P[2][1]*SH_MAG[0] + P[2][3]*SH_MAG[2] + P[2][0]*SK_MX[3] - P[2][2]*SK_MX[2] - P[2][16]*SK_MX[1] + P[2][17]*SK_MX[5] - P[2][18]*SK_MX[4]); Kfusion[3] = SK_MX[0]*(P[3][19] + P[3][1]*SH_MAG[0] + P[3][3]*SH_MAG[2] + P[3][0]*SK_MX[3] - P[3][2]*SK_MX[2] - P[3][16]*SK_MX[1] + P[3][17]*SK_MX[5] - P[3][18]*SK_MX[4]); Kfusion[4] = SK_MX[0]*(P[4][19] + P[4][1]*SH_MAG[0] + P[4][3]*SH_MAG[2] + P[4][0]*SK_MX[3] - P[4][2]*SK_MX[2] - P[4][16]*SK_MX[1] + P[4][17]*SK_MX[5] - P[4][18]*SK_MX[4]); Kfusion[5] = SK_MX[0]*(P[5][19] + P[5][1]*SH_MAG[0] + P[5][3]*SH_MAG[2] + P[5][0]*SK_MX[3] - P[5][2]*SK_MX[2] - P[5][16]*SK_MX[1] + P[5][17]*SK_MX[5] - P[5][18]*SK_MX[4]); Kfusion[6] = SK_MX[0]*(P[6][19] + P[6][1]*SH_MAG[0] + P[6][3]*SH_MAG[2] + P[6][0]*SK_MX[3] - P[6][2]*SK_MX[2] - P[6][16]*SK_MX[1] + P[6][17]*SK_MX[5] - P[6][18]*SK_MX[4]); Kfusion[7] = SK_MX[0]*(P[7][19] + P[7][1]*SH_MAG[0] + P[7][3]*SH_MAG[2] + P[7][0]*SK_MX[3] - P[7][2]*SK_MX[2] - P[7][16]*SK_MX[1] + P[7][17]*SK_MX[5] - P[7][18]*SK_MX[4]); Kfusion[8] = SK_MX[0]*(P[8][19] + P[8][1]*SH_MAG[0] + P[8][3]*SH_MAG[2] + P[8][0]*SK_MX[3] - P[8][2]*SK_MX[2] - P[8][16]*SK_MX[1] + P[8][17]*SK_MX[5] - P[8][18]*SK_MX[4]); Kfusion[9] = SK_MX[0]*(P[9][19] + P[9][1]*SH_MAG[0] + P[9][3]*SH_MAG[2] + P[9][0]*SK_MX[3] - P[9][2]*SK_MX[2] - P[9][16]*SK_MX[1] + P[9][17]*SK_MX[5] - P[9][18]*SK_MX[4]); Kfusion[10] = SK_MX[0]*(P[10][19] + P[10][1]*SH_MAG[0] + P[10][3]*SH_MAG[2] + P[10][0]*SK_MX[3] - P[10][2]*SK_MX[2] - P[10][16]*SK_MX[1] + P[10][17]*SK_MX[5] - P[10][18]*SK_MX[4]); Kfusion[11] = SK_MX[0]*(P[11][19] + P[11][1]*SH_MAG[0] + P[11][3]*SH_MAG[2] + P[11][0]*SK_MX[3] - P[11][2]*SK_MX[2] - P[11][16]*SK_MX[1] + P[11][17]*SK_MX[5] - P[11][18]*SK_MX[4]); Kfusion[12] = SK_MX[0]*(P[12][19] + P[12][1]*SH_MAG[0] + P[12][3]*SH_MAG[2] + P[12][0]*SK_MX[3] - P[12][2]*SK_MX[2] - P[12][16]*SK_MX[1] + P[12][17]*SK_MX[5] - P[12][18]*SK_MX[4]); // this term has been zeroed to improve stability of the Z accel bias Kfusion[13] = 0.0f;//SK_MX[0]*(P[13][19] + P[13][1]*SH_MAG[0] + P[13][3]*SH_MAG[2] + P[13][0]*SK_MX[3] - P[13][2]*SK_MX[2] - P[13][16]*SK_MX[1] + P[13][17]*SK_MX[5] - P[13][18]*SK_MX[4]); // zero Kalman gains to inhibit wind state estimation if (!inhibitWindStates) { Kfusion[14] = SK_MX[0]*(P[14][19] + P[14][1]*SH_MAG[0] + P[14][3]*SH_MAG[2] + P[14][0]*SK_MX[3] - P[14][2]*SK_MX[2] - P[14][16]*SK_MX[1] + P[14][17]*SK_MX[5] - P[14][18]*SK_MX[4]); Kfusion[15] = SK_MX[0]*(P[15][19] + P[15][1]*SH_MAG[0] + P[15][3]*SH_MAG[2] + P[15][0]*SK_MX[3] - P[15][2]*SK_MX[2] - P[15][16]*SK_MX[1] + P[15][17]*SK_MX[5] - P[15][18]*SK_MX[4]); } else { Kfusion[14] = 0.0; Kfusion[15] = 0.0; } // zero Kalman gains to inhibit magnetic field state estimation if (!inhibitMagStates) { Kfusion[16] = SK_MX[0]*(P[16][19] + P[16][1]*SH_MAG[0] + P[16][3]*SH_MAG[2] + P[16][0]*SK_MX[3] - P[16][2]*SK_MX[2] - P[16][16]*SK_MX[1] + P[16][17]*SK_MX[5] - P[16][18]*SK_MX[4]); Kfusion[17] = SK_MX[0]*(P[17][19] + P[17][1]*SH_MAG[0] + P[17][3]*SH_MAG[2] + P[17][0]*SK_MX[3] - P[17][2]*SK_MX[2] - P[17][16]*SK_MX[1] + P[17][17]*SK_MX[5] - P[17][18]*SK_MX[4]); Kfusion[18] = SK_MX[0]*(P[18][19] + P[18][1]*SH_MAG[0] + P[18][3]*SH_MAG[2] + P[18][0]*SK_MX[3] - P[18][2]*SK_MX[2] - P[18][16]*SK_MX[1] + P[18][17]*SK_MX[5] - P[18][18]*SK_MX[4]); Kfusion[19] = SK_MX[0]*(P[19][19] + P[19][1]*SH_MAG[0] + P[19][3]*SH_MAG[2] + P[19][0]*SK_MX[3] - P[19][2]*SK_MX[2] - P[19][16]*SK_MX[1] + P[19][17]*SK_MX[5] - P[19][18]*SK_MX[4]); Kfusion[20] = SK_MX[0]*(P[20][19] + P[20][1]*SH_MAG[0] + P[20][3]*SH_MAG[2] + P[20][0]*SK_MX[3] - P[20][2]*SK_MX[2] - P[20][16]*SK_MX[1] + P[20][17]*SK_MX[5] - P[20][18]*SK_MX[4]); Kfusion[21] = SK_MX[0]*(P[21][19] + P[21][1]*SH_MAG[0] + P[21][3]*SH_MAG[2] + P[21][0]*SK_MX[3] - P[21][2]*SK_MX[2] - P[21][16]*SK_MX[1] + P[21][17]*SK_MX[5] - P[21][18]*SK_MX[4]); } else { for (uint8_t i=16; i<=21; i++) { Kfusion[i] = 0.0f; } } // calculate the observation innovation variance varInnovMag[0] = 1.0f/SK_MX[0]; // reset the observation index to 0 (we start by fusing the X measurement) obsIndex = 0; // set flags to indicate to other processes that fusion has been performed and is required on the next frame // this can be used by other fusion processes to avoid fusing on the same frame as this expensive step magFusePerformed = true; magFuseRequired = true; } else if (obsIndex == 1) // we are now fusing the Y measurement { // calculate observation jacobians for (uint8_t i=0; i<=21; i++) H_MAG[i] = 0; H_MAG[0] = SH_MAG[2]; H_MAG[1] = SH_MAG[1]; H_MAG[2] = SH_MAG[0]; H_MAG[3] = 2*magD*q2 - SH_MAG[8] - SH_MAG[7]; H_MAG[16] = 2*q1*q2 - 2*q0*q3; H_MAG[17] = SH_MAG[4] - SH_MAG[3] - SH_MAG[5] + SH_MAG[6]; H_MAG[18] = 2*q0*q1 + 2*q2*q3; H_MAG[20] = 1; // calculate Kalman gain float temp = (P[20][20] + R_MAG + P[0][20]*SH_MAG[2] + P[1][20]*SH_MAG[1] + P[2][20]*SH_MAG[0] - P[17][20]*(SH_MAG[3] - SH_MAG[4] + SH_MAG[5] - SH_MAG[6]) - (2*q0*q3 - 2*q1*q2)*(P[20][16] + P[0][16]*SH_MAG[2] + P[1][16]*SH_MAG[1] + P[2][16]*SH_MAG[0] - P[17][16]*(SH_MAG[3] - SH_MAG[4] + SH_MAG[5] - SH_MAG[6]) - P[16][16]*(2*q0*q3 - 2*q1*q2) + P[18][16]*(2*q0*q1 + 2*q2*q3) - P[3][16]*(SH_MAG[7] + SH_MAG[8] - 2*magD*q2)) + (2*q0*q1 + 2*q2*q3)*(P[20][18] + P[0][18]*SH_MAG[2] + P[1][18]*SH_MAG[1] + P[2][18]*SH_MAG[0] - P[17][18]*(SH_MAG[3] - SH_MAG[4] + SH_MAG[5] - SH_MAG[6]) - P[16][18]*(2*q0*q3 - 2*q1*q2) + P[18][18]*(2*q0*q1 + 2*q2*q3) - P[3][18]*(SH_MAG[7] + SH_MAG[8] - 2*magD*q2)) - (SH_MAG[7] + SH_MAG[8] - 2*magD*q2)*(P[20][3] + P[0][3]*SH_MAG[2] + P[1][3]*SH_MAG[1] + P[2][3]*SH_MAG[0] - P[17][3]*(SH_MAG[3] - SH_MAG[4] + SH_MAG[5] - SH_MAG[6]) - P[16][3]*(2*q0*q3 - 2*q1*q2) + P[18][3]*(2*q0*q1 + 2*q2*q3) - P[3][3]*(SH_MAG[7] + SH_MAG[8] - 2*magD*q2)) - P[16][20]*(2*q0*q3 - 2*q1*q2) + P[18][20]*(2*q0*q1 + 2*q2*q3) + SH_MAG[2]*(P[20][0] + P[0][0]*SH_MAG[2] + P[1][0]*SH_MAG[1] + P[2][0]*SH_MAG[0] - P[17][0]*(SH_MAG[3] - SH_MAG[4] + SH_MAG[5] - SH_MAG[6]) - P[16][0]*(2*q0*q3 - 2*q1*q2) + P[18][0]*(2*q0*q1 + 2*q2*q3) - P[3][0]*(SH_MAG[7] + SH_MAG[8] - 2*magD*q2)) + SH_MAG[1]*(P[20][1] + P[0][1]*SH_MAG[2] + P[1][1]*SH_MAG[1] + P[2][1]*SH_MAG[0] - P[17][1]*(SH_MAG[3] - SH_MAG[4] + SH_MAG[5] - SH_MAG[6]) - P[16][1]*(2*q0*q3 - 2*q1*q2) + P[18][1]*(2*q0*q1 + 2*q2*q3) - P[3][1]*(SH_MAG[7] + SH_MAG[8] - 2*magD*q2)) + SH_MAG[0]*(P[20][2] + P[0][2]*SH_MAG[2] + P[1][2]*SH_MAG[1] + P[2][2]*SH_MAG[0] - P[17][2]*(SH_MAG[3] - SH_MAG[4] + SH_MAG[5] - SH_MAG[6]) - P[16][2]*(2*q0*q3 - 2*q1*q2) + P[18][2]*(2*q0*q1 + 2*q2*q3) - P[3][2]*(SH_MAG[7] + SH_MAG[8] - 2*magD*q2)) - (SH_MAG[3] - SH_MAG[4] + SH_MAG[5] - SH_MAG[6])*(P[20][17] + P[0][17]*SH_MAG[2] + P[1][17]*SH_MAG[1] + P[2][17]*SH_MAG[0] - P[17][17]*(SH_MAG[3] - SH_MAG[4] + SH_MAG[5] - SH_MAG[6]) - P[16][17]*(2*q0*q3 - 2*q1*q2) + P[18][17]*(2*q0*q1 + 2*q2*q3) - P[3][17]*(SH_MAG[7] + SH_MAG[8] - 2*magD*q2)) - P[3][20]*(SH_MAG[7] + SH_MAG[8] - 2*magD*q2)); if (temp >= R_MAG) { SK_MY[0] = 1.0f / temp; faultStatus.bad_ymag = false; } else { // the calculation is badly conditioned, so we cannot perform fusion on this step // we increase the state variances and try again next time P[20][20] += 0.1f*R_MAG; obsIndex = 2; faultStatus.bad_ymag = true; return; } SK_MY[1] = SH_MAG[3] - SH_MAG[4] + SH_MAG[5] - SH_MAG[6]; SK_MY[2] = SH_MAG[7] + SH_MAG[8] - 2*magD*q2; SK_MY[3] = 2*q0*q3 - 2*q1*q2; SK_MY[4] = 2*q0*q1 + 2*q2*q3; Kfusion[0] = SK_MY[0]*(P[0][20] + P[0][0]*SH_MAG[2] + P[0][1]*SH_MAG[1] + P[0][2]*SH_MAG[0] - P[0][3]*SK_MY[2] - P[0][17]*SK_MY[1] - P[0][16]*SK_MY[3] + P[0][18]*SK_MY[4]); Kfusion[1] = SK_MY[0]*(P[1][20] + P[1][0]*SH_MAG[2] + P[1][1]*SH_MAG[1] + P[1][2]*SH_MAG[0] - P[1][3]*SK_MY[2] - P[1][17]*SK_MY[1] - P[1][16]*SK_MY[3] + P[1][18]*SK_MY[4]); Kfusion[2] = SK_MY[0]*(P[2][20] + P[2][0]*SH_MAG[2] + P[2][1]*SH_MAG[1] + P[2][2]*SH_MAG[0] - P[2][3]*SK_MY[2] - P[2][17]*SK_MY[1] - P[2][16]*SK_MY[3] + P[2][18]*SK_MY[4]); Kfusion[3] = SK_MY[0]*(P[3][20] + P[3][0]*SH_MAG[2] + P[3][1]*SH_MAG[1] + P[3][2]*SH_MAG[0] - P[3][3]*SK_MY[2] - P[3][17]*SK_MY[1] - P[3][16]*SK_MY[3] + P[3][18]*SK_MY[4]); Kfusion[4] = SK_MY[0]*(P[4][20] + P[4][0]*SH_MAG[2] + P[4][1]*SH_MAG[1] + P[4][2]*SH_MAG[0] - P[4][3]*SK_MY[2] - P[4][17]*SK_MY[1] - P[4][16]*SK_MY[3] + P[4][18]*SK_MY[4]); Kfusion[5] = SK_MY[0]*(P[5][20] + P[5][0]*SH_MAG[2] + P[5][1]*SH_MAG[1] + P[5][2]*SH_MAG[0] - P[5][3]*SK_MY[2] - P[5][17]*SK_MY[1] - P[5][16]*SK_MY[3] + P[5][18]*SK_MY[4]); Kfusion[6] = SK_MY[0]*(P[6][20] + P[6][0]*SH_MAG[2] + P[6][1]*SH_MAG[1] + P[6][2]*SH_MAG[0] - P[6][3]*SK_MY[2] - P[6][17]*SK_MY[1] - P[6][16]*SK_MY[3] + P[6][18]*SK_MY[4]); Kfusion[7] = SK_MY[0]*(P[7][20] + P[7][0]*SH_MAG[2] + P[7][1]*SH_MAG[1] + P[7][2]*SH_MAG[0] - P[7][3]*SK_MY[2] - P[7][17]*SK_MY[1] - P[7][16]*SK_MY[3] + P[7][18]*SK_MY[4]); Kfusion[8] = SK_MY[0]*(P[8][20] + P[8][0]*SH_MAG[2] + P[8][1]*SH_MAG[1] + P[8][2]*SH_MAG[0] - P[8][3]*SK_MY[2] - P[8][17]*SK_MY[1] - P[8][16]*SK_MY[3] + P[8][18]*SK_MY[4]); Kfusion[9] = SK_MY[0]*(P[9][20] + P[9][0]*SH_MAG[2] + P[9][1]*SH_MAG[1] + P[9][2]*SH_MAG[0] - P[9][3]*SK_MY[2] - P[9][17]*SK_MY[1] - P[9][16]*SK_MY[3] + P[9][18]*SK_MY[4]); Kfusion[10] = SK_MY[0]*(P[10][20] + P[10][0]*SH_MAG[2] + P[10][1]*SH_MAG[1] + P[10][2]*SH_MAG[0] - P[10][3]*SK_MY[2] - P[10][17]*SK_MY[1] - P[10][16]*SK_MY[3] + P[10][18]*SK_MY[4]); Kfusion[11] = SK_MY[0]*(P[11][20] + P[11][0]*SH_MAG[2] + P[11][1]*SH_MAG[1] + P[11][2]*SH_MAG[0] - P[11][3]*SK_MY[2] - P[11][17]*SK_MY[1] - P[11][16]*SK_MY[3] + P[11][18]*SK_MY[4]); Kfusion[12] = SK_MY[0]*(P[12][20] + P[12][0]*SH_MAG[2] + P[12][1]*SH_MAG[1] + P[12][2]*SH_MAG[0] - P[12][3]*SK_MY[2] - P[12][17]*SK_MY[1] - P[12][16]*SK_MY[3] + P[12][18]*SK_MY[4]); // this term has been zeroed to improve stability of the Z accel bias Kfusion[13] = 0.0f;//SK_MY[0]*(P[13][20] + P[13][0]*SH_MAG[2] + P[13][1]*SH_MAG[1] + P[13][2]*SH_MAG[0] - P[13][3]*SK_MY[2] - P[13][17]*SK_MY[1] - P[13][16]*SK_MY[3] + P[13][18]*SK_MY[4]); // zero Kalman gains to inhibit wind state estimation if (!inhibitWindStates) { Kfusion[14] = SK_MY[0]*(P[14][20] + P[14][0]*SH_MAG[2] + P[14][1]*SH_MAG[1] + P[14][2]*SH_MAG[0] - P[14][3]*SK_MY[2] - P[14][17]*SK_MY[1] - P[14][16]*SK_MY[3] + P[14][18]*SK_MY[4]); Kfusion[15] = SK_MY[0]*(P[15][20] + P[15][0]*SH_MAG[2] + P[15][1]*SH_MAG[1] + P[15][2]*SH_MAG[0] - P[15][3]*SK_MY[2] - P[15][17]*SK_MY[1] - P[15][16]*SK_MY[3] + P[15][18]*SK_MY[4]); } else { Kfusion[14] = 0.0; Kfusion[15] = 0.0; } // zero Kalman gains to inhibit magnetic field state estimation if (!inhibitMagStates) { Kfusion[16] = SK_MY[0]*(P[16][20] + P[16][0]*SH_MAG[2] + P[16][1]*SH_MAG[1] + P[16][2]*SH_MAG[0] - P[16][3]*SK_MY[2] - P[16][17]*SK_MY[1] - P[16][16]*SK_MY[3] + P[16][18]*SK_MY[4]); Kfusion[17] = SK_MY[0]*(P[17][20] + P[17][0]*SH_MAG[2] + P[17][1]*SH_MAG[1] + P[17][2]*SH_MAG[0] - P[17][3]*SK_MY[2] - P[17][17]*SK_MY[1] - P[17][16]*SK_MY[3] + P[17][18]*SK_MY[4]); Kfusion[18] = SK_MY[0]*(P[18][20] + P[18][0]*SH_MAG[2] + P[18][1]*SH_MAG[1] + P[18][2]*SH_MAG[0] - P[18][3]*SK_MY[2] - P[18][17]*SK_MY[1] - P[18][16]*SK_MY[3] + P[18][18]*SK_MY[4]); Kfusion[19] = SK_MY[0]*(P[19][20] + P[19][0]*SH_MAG[2] + P[19][1]*SH_MAG[1] + P[19][2]*SH_MAG[0] - P[19][3]*SK_MY[2] - P[19][17]*SK_MY[1] - P[19][16]*SK_MY[3] + P[19][18]*SK_MY[4]); Kfusion[20] = SK_MY[0]*(P[20][20] + P[20][0]*SH_MAG[2] + P[20][1]*SH_MAG[1] + P[20][2]*SH_MAG[0] - P[20][3]*SK_MY[2] - P[20][17]*SK_MY[1] - P[20][16]*SK_MY[3] + P[20][18]*SK_MY[4]); Kfusion[21] = SK_MY[0]*(P[21][20] + P[21][0]*SH_MAG[2] + P[21][1]*SH_MAG[1] + P[21][2]*SH_MAG[0] - P[21][3]*SK_MY[2] - P[21][17]*SK_MY[1] - P[21][16]*SK_MY[3] + P[21][18]*SK_MY[4]); } else { for (uint8_t i=16; i<=21; i++) { Kfusion[i] = 0.0f; } } // calculate the observation innovation variance varInnovMag[1] = 1.0f/SK_MY[0]; // set flags to indicate to other processes that fusion has been performede and is required on the next frame // this can be used by other fusion processes to avoid fusing on the same frame as this expensive step magFusePerformed = true; magFuseRequired = true; } else if (obsIndex == 2) // we are now fusing the Z measurement { // calculate observation jacobians for (uint8_t i=0; i<=21; i++) H_MAG[i] = 0; H_MAG[0] = SH_MAG[1]; H_MAG[1] = 2*magN*q3 - 2*magE*q0 - 2*magD*q1; H_MAG[2] = SH_MAG[7] + SH_MAG[8] - 2*magD*q2; H_MAG[3] = SH_MAG[0]; H_MAG[16] = 2*q0*q2 + 2*q1*q3; H_MAG[17] = 2*q2*q3 - 2*q0*q1; H_MAG[18] = SH_MAG[3] - SH_MAG[4] - SH_MAG[5] + SH_MAG[6]; H_MAG[21] = 1; // calculate Kalman gain float temp = (P[21][21] + R_MAG + P[0][21]*SH_MAG[1] + P[3][21]*SH_MAG[0] + P[18][21]*(SH_MAG[3] - SH_MAG[4] - SH_MAG[5] + SH_MAG[6]) - (2*magD*q1 + 2*magE*q0 - 2*magN*q3)*(P[21][1] + P[0][1]*SH_MAG[1] + P[3][1]*SH_MAG[0] + P[18][1]*(SH_MAG[3] - SH_MAG[4] - SH_MAG[5] + SH_MAG[6]) + P[16][1]*(2*q0*q2 + 2*q1*q3) - P[17][1]*(2*q0*q1 - 2*q2*q3) - P[1][1]*(2*magD*q1 + 2*magE*q0 - 2*magN*q3) + P[2][1]*(SH_MAG[7] + SH_MAG[8] - 2*magD*q2)) + (SH_MAG[7] + SH_MAG[8] - 2*magD*q2)*(P[21][2] + P[0][2]*SH_MAG[1] + P[3][2]*SH_MAG[0] + P[18][2]*(SH_MAG[3] - SH_MAG[4] - SH_MAG[5] + SH_MAG[6]) + P[16][2]*(2*q0*q2 + 2*q1*q3) - P[17][2]*(2*q0*q1 - 2*q2*q3) - P[1][2]*(2*magD*q1 + 2*magE*q0 - 2*magN*q3) + P[2][2]*(SH_MAG[7] + SH_MAG[8] - 2*magD*q2)) + SH_MAG[1]*(P[21][0] + P[0][0]*SH_MAG[1] + P[3][0]*SH_MAG[0] + P[18][0]*(SH_MAG[3] - SH_MAG[4] - SH_MAG[5] + SH_MAG[6]) + P[16][0]*(2*q0*q2 + 2*q1*q3) - P[17][0]*(2*q0*q1 - 2*q2*q3) - P[1][0]*(2*magD*q1 + 2*magE*q0 - 2*magN*q3) + P[2][0]*(SH_MAG[7] + SH_MAG[8] - 2*magD*q2)) + SH_MAG[0]*(P[21][3] + P[0][3]*SH_MAG[1] + P[3][3]*SH_MAG[0] + P[18][3]*(SH_MAG[3] - SH_MAG[4] - SH_MAG[5] + SH_MAG[6]) + P[16][3]*(2*q0*q2 + 2*q1*q3) - P[17][3]*(2*q0*q1 - 2*q2*q3) - P[1][3]*(2*magD*q1 + 2*magE*q0 - 2*magN*q3) + P[2][3]*(SH_MAG[7] + SH_MAG[8] - 2*magD*q2)) + (SH_MAG[3] - SH_MAG[4] - SH_MAG[5] + SH_MAG[6])*(P[21][18] + P[0][18]*SH_MAG[1] + P[3][18]*SH_MAG[0] + P[18][18]*(SH_MAG[3] - SH_MAG[4] - SH_MAG[5] + SH_MAG[6]) + P[16][18]*(2*q0*q2 + 2*q1*q3) - P[17][18]*(2*q0*q1 - 2*q2*q3) - P[1][18]*(2*magD*q1 + 2*magE*q0 - 2*magN*q3) + P[2][18]*(SH_MAG[7] + SH_MAG[8] - 2*magD*q2)) + P[16][21]*(2*q0*q2 + 2*q1*q3) - P[17][21]*(2*q0*q1 - 2*q2*q3) - P[1][21]*(2*magD*q1 + 2*magE*q0 - 2*magN*q3) + (2*q0*q2 + 2*q1*q3)*(P[21][16] + P[0][16]*SH_MAG[1] + P[3][16]*SH_MAG[0] + P[18][16]*(SH_MAG[3] - SH_MAG[4] - SH_MAG[5] + SH_MAG[6]) + P[16][16]*(2*q0*q2 + 2*q1*q3) - P[17][16]*(2*q0*q1 - 2*q2*q3) - P[1][16]*(2*magD*q1 + 2*magE*q0 - 2*magN*q3) + P[2][16]*(SH_MAG[7] + SH_MAG[8] - 2*magD*q2)) - (2*q0*q1 - 2*q2*q3)*(P[21][17] + P[0][17]*SH_MAG[1] + P[3][17]*SH_MAG[0] + P[18][17]*(SH_MAG[3] - SH_MAG[4] - SH_MAG[5] + SH_MAG[6]) + P[16][17]*(2*q0*q2 + 2*q1*q3) - P[17][17]*(2*q0*q1 - 2*q2*q3) - P[1][17]*(2*magD*q1 + 2*magE*q0 - 2*magN*q3) + P[2][17]*(SH_MAG[7] + SH_MAG[8] - 2*magD*q2)) + P[2][21]*(SH_MAG[7] + SH_MAG[8] - 2*magD*q2)); if (temp >= R_MAG) { SK_MZ[0] = 1.0f / temp; faultStatus.bad_zmag = false; } else { // the calculation is badly conditioned, so we cannot perform fusion on this step // we increase the state variances and try again next time P[21][21] += 0.1f*R_MAG; obsIndex = 3; faultStatus.bad_zmag = true; return; } SK_MZ[1] = SH_MAG[3] - SH_MAG[4] - SH_MAG[5] + SH_MAG[6]; SK_MZ[2] = 2*magD*q1 + 2*magE*q0 - 2*magN*q3; SK_MZ[3] = SH_MAG[7] + SH_MAG[8] - 2*magD*q2; SK_MZ[4] = 2*q0*q1 - 2*q2*q3; SK_MZ[5] = 2*q0*q2 + 2*q1*q3; Kfusion[0] = SK_MZ[0]*(P[0][21] + P[0][0]*SH_MAG[1] + P[0][3]*SH_MAG[0] - P[0][1]*SK_MZ[2] + P[0][2]*SK_MZ[3] + P[0][18]*SK_MZ[1] + P[0][16]*SK_MZ[5] - P[0][17]*SK_MZ[4]); Kfusion[1] = SK_MZ[0]*(P[1][21] + P[1][0]*SH_MAG[1] + P[1][3]*SH_MAG[0] - P[1][1]*SK_MZ[2] + P[1][2]*SK_MZ[3] + P[1][18]*SK_MZ[1] + P[1][16]*SK_MZ[5] - P[1][17]*SK_MZ[4]); Kfusion[2] = SK_MZ[0]*(P[2][21] + P[2][0]*SH_MAG[1] + P[2][3]*SH_MAG[0] - P[2][1]*SK_MZ[2] + P[2][2]*SK_MZ[3] + P[2][18]*SK_MZ[1] + P[2][16]*SK_MZ[5] - P[2][17]*SK_MZ[4]); Kfusion[3] = SK_MZ[0]*(P[3][21] + P[3][0]*SH_MAG[1] + P[3][3]*SH_MAG[0] - P[3][1]*SK_MZ[2] + P[3][2]*SK_MZ[3] + P[3][18]*SK_MZ[1] + P[3][16]*SK_MZ[5] - P[3][17]*SK_MZ[4]); Kfusion[4] = SK_MZ[0]*(P[4][21] + P[4][0]*SH_MAG[1] + P[4][3]*SH_MAG[0] - P[4][1]*SK_MZ[2] + P[4][2]*SK_MZ[3] + P[4][18]*SK_MZ[1] + P[4][16]*SK_MZ[5] - P[4][17]*SK_MZ[4]); Kfusion[5] = SK_MZ[0]*(P[5][21] + P[5][0]*SH_MAG[1] + P[5][3]*SH_MAG[0] - P[5][1]*SK_MZ[2] + P[5][2]*SK_MZ[3] + P[5][18]*SK_MZ[1] + P[5][16]*SK_MZ[5] - P[5][17]*SK_MZ[4]); Kfusion[6] = SK_MZ[0]*(P[6][21] + P[6][0]*SH_MAG[1] + P[6][3]*SH_MAG[0] - P[6][1]*SK_MZ[2] + P[6][2]*SK_MZ[3] + P[6][18]*SK_MZ[1] + P[6][16]*SK_MZ[5] - P[6][17]*SK_MZ[4]); Kfusion[7] = SK_MZ[0]*(P[7][21] + P[7][0]*SH_MAG[1] + P[7][3]*SH_MAG[0] - P[7][1]*SK_MZ[2] + P[7][2]*SK_MZ[3] + P[7][18]*SK_MZ[1] + P[7][16]*SK_MZ[5] - P[7][17]*SK_MZ[4]); Kfusion[8] = SK_MZ[0]*(P[8][21] + P[8][0]*SH_MAG[1] + P[8][3]*SH_MAG[0] - P[8][1]*SK_MZ[2] + P[8][2]*SK_MZ[3] + P[8][18]*SK_MZ[1] + P[8][16]*SK_MZ[5] - P[8][17]*SK_MZ[4]); Kfusion[9] = SK_MZ[0]*(P[9][21] + P[9][0]*SH_MAG[1] + P[9][3]*SH_MAG[0] - P[9][1]*SK_MZ[2] + P[9][2]*SK_MZ[3] + P[9][18]*SK_MZ[1] + P[9][16]*SK_MZ[5] - P[9][17]*SK_MZ[4]); Kfusion[10] = SK_MZ[0]*(P[10][21] + P[10][0]*SH_MAG[1] + P[10][3]*SH_MAG[0] - P[10][1]*SK_MZ[2] + P[10][2]*SK_MZ[3] + P[10][18]*SK_MZ[1] + P[10][16]*SK_MZ[5] - P[10][17]*SK_MZ[4]); Kfusion[11] = SK_MZ[0]*(P[11][21] + P[11][0]*SH_MAG[1] + P[11][3]*SH_MAG[0] - P[11][1]*SK_MZ[2] + P[11][2]*SK_MZ[3] + P[11][18]*SK_MZ[1] + P[11][16]*SK_MZ[5] - P[11][17]*SK_MZ[4]); Kfusion[12] = SK_MZ[0]*(P[12][21] + P[12][0]*SH_MAG[1] + P[12][3]*SH_MAG[0] - P[12][1]*SK_MZ[2] + P[12][2]*SK_MZ[3] + P[12][18]*SK_MZ[1] + P[12][16]*SK_MZ[5] - P[12][17]*SK_MZ[4]); // this term has been zeroed to improve stability of the Z accel bias Kfusion[13] = 0.0f;//SK_MZ[0]*(P[13][21] + P[13][0]*SH_MAG[1] + P[13][3]*SH_MAG[0] - P[13][1]*SK_MZ[2] + P[13][2]*SK_MZ[3] + P[13][18]*SK_MZ[1] + P[13][16]*SK_MZ[5] - P[13][17]*SK_MZ[4]); // zero Kalman gains to inhibit wind state estimation if (!inhibitWindStates) { Kfusion[14] = SK_MZ[0]*(P[14][21] + P[14][0]*SH_MAG[1] + P[14][3]*SH_MAG[0] - P[14][1]*SK_MZ[2] + P[14][2]*SK_MZ[3] + P[14][18]*SK_MZ[1] + P[14][16]*SK_MZ[5] - P[14][17]*SK_MZ[4]); Kfusion[15] = SK_MZ[0]*(P[15][21] + P[15][0]*SH_MAG[1] + P[15][3]*SH_MAG[0] - P[15][1]*SK_MZ[2] + P[15][2]*SK_MZ[3] + P[15][18]*SK_MZ[1] + P[15][16]*SK_MZ[5] - P[15][17]*SK_MZ[4]); } else { Kfusion[14] = 0.0; Kfusion[15] = 0.0; } // zero Kalman gains to inhibit magnetic field state estimation if (!inhibitMagStates) { Kfusion[16] = SK_MZ[0]*(P[16][21] + P[16][0]*SH_MAG[1] + P[16][3]*SH_MAG[0] - P[16][1]*SK_MZ[2] + P[16][2]*SK_MZ[3] + P[16][18]*SK_MZ[1] + P[16][16]*SK_MZ[5] - P[16][17]*SK_MZ[4]); Kfusion[17] = SK_MZ[0]*(P[17][21] + P[17][0]*SH_MAG[1] + P[17][3]*SH_MAG[0] - P[17][1]*SK_MZ[2] + P[17][2]*SK_MZ[3] + P[17][18]*SK_MZ[1] + P[17][16]*SK_MZ[5] - P[17][17]*SK_MZ[4]); Kfusion[18] = SK_MZ[0]*(P[18][21] + P[18][0]*SH_MAG[1] + P[18][3]*SH_MAG[0] - P[18][1]*SK_MZ[2] + P[18][2]*SK_MZ[3] + P[18][18]*SK_MZ[1] + P[18][16]*SK_MZ[5] - P[18][17]*SK_MZ[4]); Kfusion[19] = SK_MZ[0]*(P[19][21] + P[19][0]*SH_MAG[1] + P[19][3]*SH_MAG[0] - P[19][1]*SK_MZ[2] + P[19][2]*SK_MZ[3] + P[19][18]*SK_MZ[1] + P[19][16]*SK_MZ[5] - P[19][17]*SK_MZ[4]); Kfusion[20] = SK_MZ[0]*(P[20][21] + P[20][0]*SH_MAG[1] + P[20][3]*SH_MAG[0] - P[20][1]*SK_MZ[2] + P[20][2]*SK_MZ[3] + P[20][18]*SK_MZ[1] + P[20][16]*SK_MZ[5] - P[20][17]*SK_MZ[4]); Kfusion[21] = SK_MZ[0]*(P[21][21] + P[21][0]*SH_MAG[1] + P[21][3]*SH_MAG[0] - P[21][1]*SK_MZ[2] + P[21][2]*SK_MZ[3] + P[21][18]*SK_MZ[1] + P[21][16]*SK_MZ[5] - P[21][17]*SK_MZ[4]); } else { for (uint8_t i=16; i<=21; i++) { Kfusion[i] = 0.0f; } } // calculate the observation innovation variance varInnovMag[2] = 1.0f/SK_MZ[0]; // set flags to indicate to other processes that fusion has been performede and is required on the next frame // this can be used by other fusion processes to avoid fusing on the same frame as this expensive step magFusePerformed = true; magFuseRequired = false; } // calculate the measurement innovation innovMag[obsIndex] = MagPred[obsIndex] - magData[obsIndex]; // calculate the innovation test ratio magTestRatio[obsIndex] = sq(innovMag[obsIndex]) / (sq(_magInnovGate) * varInnovMag[obsIndex]); // check the last values from all components and set magnetometer health accordingly magHealth = (magTestRatio[0] < 1.0f && magTestRatio[1] < 1.0f && magTestRatio[2] < 1.0f); // Don't fuse unless all componenets pass. The exception is if the bad health has timed out and we are not a fly forward vehicle // In this case we might as well try using the magnetometer, but with a reduced weighting if (magHealth || ((magTestRatio[obsIndex] < 1.0f) && !assume_zero_sideslip() && magTimeout)) { // correct the state vector for (uint8_t j= 0; j<=21; j++) { // If we are forced to use a bad compass, we reduce the weighting by a factor of 4 if (!magHealth) { Kfusion[j] *= 0.25f; } states[j] = states[j] - Kfusion[j] * innovMag[obsIndex]; } // normalise the quaternion states state.quat.normalize(); // correct the covariance P = (I - K*H)*P // take advantage of the empty columns in KH to reduce the // number of operations for (uint8_t i = 0; i<=21; i++) { for (uint8_t j = 0; j<=3; j++) { KH[i][j] = Kfusion[i] * H_MAG[j]; } for (uint8_t j = 4; j<=15; j++) { KH[i][j] = 0.0f; } if (!inhibitMagStates) { for (uint8_t j = 16; j<=21; j++) { KH[i][j] = Kfusion[i] * H_MAG[j]; } } else { for (uint8_t j = 16; j<=21; j++) { KH[i][j] = 0.0f; } } } for (uint8_t i = 0; i<=21; i++) { for (uint8_t j = 0; j<=21; j++) { KHP[i][j] = 0; for (uint8_t k = 0; k<=3; k++) { KHP[i][j] = KHP[i][j] + KH[i][k] * P[k][j]; } if (!inhibitMagStates) { for (uint8_t k = 16; k<=21; k++) { KHP[i][j] = KHP[i][j] + KH[i][k] * P[k][j]; } } } } for (uint8_t i = 0; i<=21; i++) { for (uint8_t j = 0; j<=21; j++) { P[i][j] = P[i][j] - KHP[i][j]; } } } obsIndex = obsIndex + 1; } else { // set flags to indicate to other processes that fusion has not been performed and is not required on the next time step magFusePerformed = false; magFuseRequired = false; } // force the covariance matrix to be symmetrical and limit the variances to prevent // ill-condiioning. ForceSymmetry(); ConstrainVariances(); // stop performance timer perf_end(_perf_FuseMagnetometer); } // fuse true airspeed measurements void NavEKF::FuseAirspeed() { // start performance timer perf_begin(_perf_FuseAirspeed); // declarations float vn; float ve; float vd; float vwn; float vwe; float EAS2TAS = _ahrs->get_EAS2TAS(); const float R_TAS = sq(constrain_float(_easNoise, 0.5f, 5.0f) * constrain_float(EAS2TAS, 0.9f, 10.0f)); Vector3f SH_TAS; float SK_TAS; Vector22 H_TAS; float VtasPred; // copy required states to local variable names vn = statesAtVtasMeasTime.velocity.x; ve = statesAtVtasMeasTime.velocity.y; vd = statesAtVtasMeasTime.velocity.z; vwn = statesAtVtasMeasTime.wind_vel.x; vwe = statesAtVtasMeasTime.wind_vel.y; // calculate the predicted airspeed VtasPred = pythagorous3((ve - vwe) , (vn - vwn) , vd); // perform fusion of True Airspeed measurement if (VtasPred > 1.0f) { // calculate observation jacobians SH_TAS[0] = 1.0f/VtasPred; SH_TAS[1] = (SH_TAS[0]*(2*ve - 2*vwe))/2; SH_TAS[2] = (SH_TAS[0]*(2*vn - 2*vwn))/2; for (uint8_t i=0; i<=21; i++) H_TAS[i] = 0.0f; H_TAS[4] = SH_TAS[2]; H_TAS[5] = SH_TAS[1]; H_TAS[6] = vd*SH_TAS[0]; H_TAS[14] = -SH_TAS[2]; H_TAS[15] = -SH_TAS[1]; // calculate Kalman gains float temp = (R_TAS + SH_TAS[2]*(P[4][4]*SH_TAS[2] + P[5][4]*SH_TAS[1] - P[14][4]*SH_TAS[2] - P[15][4]*SH_TAS[1] + P[6][4]*vd*SH_TAS[0]) + SH_TAS[1]*(P[4][5]*SH_TAS[2] + P[5][5]*SH_TAS[1] - P[14][5]*SH_TAS[2] - P[15][5]*SH_TAS[1] + P[6][5]*vd*SH_TAS[0]) - SH_TAS[2]*(P[4][14]*SH_TAS[2] + P[5][14]*SH_TAS[1] - P[14][14]*SH_TAS[2] - P[15][14]*SH_TAS[1] + P[6][14]*vd*SH_TAS[0]) - SH_TAS[1]*(P[4][15]*SH_TAS[2] + P[5][15]*SH_TAS[1] - P[14][15]*SH_TAS[2] - P[15][15]*SH_TAS[1] + P[6][15]*vd*SH_TAS[0]) + vd*SH_TAS[0]*(P[4][6]*SH_TAS[2] + P[5][6]*SH_TAS[1] - P[14][6]*SH_TAS[2] - P[15][6]*SH_TAS[1] + P[6][6]*vd*SH_TAS[0])); if (temp >= R_TAS) { SK_TAS = 1.0f / temp; faultStatus.bad_airspeed = false; } else { // the calculation is badly conditioned, so we cannot perform fusion on this step // we increase the wind state variances and try again next time P[14][14] += 0.05f*R_TAS; P[15][15] += 0.05f*R_TAS; faultStatus.bad_airspeed = true; return; } Kfusion[0] = SK_TAS*(P[0][4]*SH_TAS[2] - P[0][14]*SH_TAS[2] + P[0][5]*SH_TAS[1] - P[0][15]*SH_TAS[1] + P[0][6]*vd*SH_TAS[0]); Kfusion[1] = SK_TAS*(P[1][4]*SH_TAS[2] - P[1][14]*SH_TAS[2] + P[1][5]*SH_TAS[1] - P[1][15]*SH_TAS[1] + P[1][6]*vd*SH_TAS[0]); Kfusion[2] = SK_TAS*(P[2][4]*SH_TAS[2] - P[2][14]*SH_TAS[2] + P[2][5]*SH_TAS[1] - P[2][15]*SH_TAS[1] + P[2][6]*vd*SH_TAS[0]); Kfusion[3] = SK_TAS*(P[3][4]*SH_TAS[2] - P[3][14]*SH_TAS[2] + P[3][5]*SH_TAS[1] - P[3][15]*SH_TAS[1] + P[3][6]*vd*SH_TAS[0]); Kfusion[4] = SK_TAS*(P[4][4]*SH_TAS[2] - P[4][14]*SH_TAS[2] + P[4][5]*SH_TAS[1] - P[4][15]*SH_TAS[1] + P[4][6]*vd*SH_TAS[0]); Kfusion[5] = SK_TAS*(P[5][4]*SH_TAS[2] - P[5][14]*SH_TAS[2] + P[5][5]*SH_TAS[1] - P[5][15]*SH_TAS[1] + P[5][6]*vd*SH_TAS[0]); Kfusion[6] = SK_TAS*(P[6][4]*SH_TAS[2] - P[6][14]*SH_TAS[2] + P[6][5]*SH_TAS[1] - P[6][15]*SH_TAS[1] + P[6][6]*vd*SH_TAS[0]); Kfusion[7] = SK_TAS*(P[7][4]*SH_TAS[2] - P[7][14]*SH_TAS[2] + P[7][5]*SH_TAS[1] - P[7][15]*SH_TAS[1] + P[7][6]*vd*SH_TAS[0]); Kfusion[8] = SK_TAS*(P[8][4]*SH_TAS[2] - P[8][14]*SH_TAS[2] + P[8][5]*SH_TAS[1] - P[8][15]*SH_TAS[1] + P[8][6]*vd*SH_TAS[0]); Kfusion[9] = SK_TAS*(P[9][4]*SH_TAS[2] - P[9][14]*SH_TAS[2] + P[9][5]*SH_TAS[1] - P[9][15]*SH_TAS[1] + P[9][6]*vd*SH_TAS[0]); Kfusion[10] = SK_TAS*(P[10][4]*SH_TAS[2] - P[10][14]*SH_TAS[2] + P[10][5]*SH_TAS[1] - P[10][15]*SH_TAS[1] + P[10][6]*vd*SH_TAS[0]); Kfusion[11] = SK_TAS*(P[11][4]*SH_TAS[2] - P[11][14]*SH_TAS[2] + P[11][5]*SH_TAS[1] - P[11][15]*SH_TAS[1] + P[11][6]*vd*SH_TAS[0]); Kfusion[12] = SK_TAS*(P[12][4]*SH_TAS[2] - P[12][14]*SH_TAS[2] + P[12][5]*SH_TAS[1] - P[12][15]*SH_TAS[1] + P[12][6]*vd*SH_TAS[0]); // this term has been zeroed to improve stability of the Z accel bias Kfusion[13] = 0.0f;//SK_TAS*(P[13][4]*SH_TAS[2] - P[13][14]*SH_TAS[2] + P[13][5]*SH_TAS[1] - P[13][15]*SH_TAS[1] + P[13][6]*vd*SH_TAS[0]); Kfusion[14] = SK_TAS*(P[14][4]*SH_TAS[2] - P[14][14]*SH_TAS[2] + P[14][5]*SH_TAS[1] - P[14][15]*SH_TAS[1] + P[14][6]*vd*SH_TAS[0]); Kfusion[15] = SK_TAS*(P[15][4]*SH_TAS[2] - P[15][14]*SH_TAS[2] + P[15][5]*SH_TAS[1] - P[15][15]*SH_TAS[1] + P[15][6]*vd*SH_TAS[0]); // zero Kalman gains to inhibit magnetic field state estimation if (!inhibitMagStates) { Kfusion[16] = SK_TAS*(P[16][4]*SH_TAS[2] - P[16][14]*SH_TAS[2] + P[16][5]*SH_TAS[1] - P[16][15]*SH_TAS[1] + P[16][6]*vd*SH_TAS[0]); Kfusion[17] = SK_TAS*(P[17][4]*SH_TAS[2] - P[17][14]*SH_TAS[2] + P[17][5]*SH_TAS[1] - P[17][15]*SH_TAS[1] + P[17][6]*vd*SH_TAS[0]); Kfusion[18] = SK_TAS*(P[18][4]*SH_TAS[2] - P[18][14]*SH_TAS[2] + P[18][5]*SH_TAS[1] - P[18][15]*SH_TAS[1] + P[18][6]*vd*SH_TAS[0]); Kfusion[19] = SK_TAS*(P[19][4]*SH_TAS[2] - P[19][14]*SH_TAS[2] + P[19][5]*SH_TAS[1] - P[19][15]*SH_TAS[1] + P[19][6]*vd*SH_TAS[0]); Kfusion[20] = SK_TAS*(P[20][4]*SH_TAS[2] - P[20][14]*SH_TAS[2] + P[20][5]*SH_TAS[1] - P[20][15]*SH_TAS[1] + P[20][6]*vd*SH_TAS[0]); Kfusion[21] = SK_TAS*(P[21][4]*SH_TAS[2] - P[21][14]*SH_TAS[2] + P[21][5]*SH_TAS[1] - P[21][15]*SH_TAS[1] + P[21][6]*vd*SH_TAS[0]); } else { for (uint8_t i=16; i<=21; i++) { Kfusion[i] = 0.0f; } } // calculate measurement innovation variance varInnovVtas = 1.0f/SK_TAS; // calculate measurement innovation innovVtas = VtasPred - VtasMeas; // calculate the innovation consistency test ratio tasTestRatio = sq(innovVtas) / (sq(_tasInnovGate) * varInnovVtas); // test the ratio before fusing data if (tasTestRatio < 1.0f) { // correct the state vector for (uint8_t j=0; j<=21; j++) { states[j] = states[j] - Kfusion[j] * innovVtas; } Quaternion q(states[0], states[1], states[2], states[3]); q.normalize(); for (uint8_t i = 0; i<=3; i++) { states[i] = q[i]; } // correct the covariance P = (I - K*H)*P // take advantage of the empty columns in H to reduce the number of operations for (uint8_t i = 0; i<=21; i++) { for (uint8_t j = 0; j<=3; j++) KH[i][j] = 0.0; for (uint8_t j = 4; j<=6; j++) { KH[i][j] = Kfusion[i] * H_TAS[j]; } for (uint8_t j = 7; j<=13; j++) KH[i][j] = 0.0; for (uint8_t j = 14; j<=15; j++) { KH[i][j] = Kfusion[i] * H_TAS[j]; } for (uint8_t j = 16; j<=21; j++) KH[i][j] = 0.0; } for (uint8_t i = 0; i<=21; i++) { for (uint8_t j = 0; j<=21; j++) { KHP[i][j] = 0; for (uint8_t k = 4; k<=6; k++) { KHP[i][j] = KHP[i][j] + KH[i][k] * P[k][j]; } for (uint8_t k = 14; k<=15; k++) { KHP[i][j] = KHP[i][j] + KH[i][k] * P[k][j]; } } } for (uint8_t i = 0; i<=21; i++) { for (uint8_t j = 0; j<=21; j++) { P[i][j] = P[i][j] - KHP[i][j]; } } } } // force the covariance matrix to me symmetrical and limit the variances to prevent ill-condiioning. ForceSymmetry(); ConstrainVariances(); // stop performance timer perf_end(_perf_FuseAirspeed); } // fuse sythetic sideslip measurement of zero void NavEKF::FuseSideslip() { // start performance timer perf_begin(_perf_FuseSideslip); // declarations float q0; float q1; float q2; float q3; float vn; float ve; float vd; float vwn; float vwe; const float R_BETA = 0.03f; // assume a sideslip angle RMS of ~10 deg float SH_BETA[13]; float SK_BETA[8]; Vector3f vel_rel_wind; Vector22 H_BETA; float innovBeta; // copy required states to local variable names q0 = states[0]; q1 = states[1]; q2 = states[2]; q3 = states[3]; vn = states[4]; ve = states[5]; vd = states[6]; vwn = states[14]; vwe = states[15]; // calculate predicted wind relative velocity in NED vel_rel_wind.x = vn - vwn; vel_rel_wind.y = ve - vwe; vel_rel_wind.z = vd; // rotate into body axes vel_rel_wind = prevTnb * vel_rel_wind; // perform fusion of assumed sideslip = 0 if (vel_rel_wind.x > 5.0f) { // Calculate observation jacobians SH_BETA[0] = (vn - vwn)*(sq(q0) + sq(q1) - sq(q2) - sq(q3)) - vd*(2*q0*q2 - 2*q1*q3) + (ve - vwe)*(2*q0*q3 + 2*q1*q2); if (fabsf(SH_BETA[0]) <= 1e-9f) { faultStatus.bad_sideslip = true; return; } else { faultStatus.bad_sideslip = false; } SH_BETA[1] = (ve - vwe)*(sq(q0) - sq(q1) + sq(q2) - sq(q3)) + vd*(2*q0*q1 + 2*q2*q3) - (vn - vwn)*(2*q0*q3 - 2*q1*q2); SH_BETA[2] = vn - vwn; SH_BETA[3] = ve - vwe; SH_BETA[4] = 1/sq(SH_BETA[0]); SH_BETA[5] = 1/SH_BETA[0]; SH_BETA[6] = SH_BETA[5]*(sq(q0) - sq(q1) + sq(q2) - sq(q3)); SH_BETA[7] = sq(q0) + sq(q1) - sq(q2) - sq(q3); SH_BETA[8] = 2*q0*SH_BETA[3] - 2*q3*SH_BETA[2] + 2*q1*vd; SH_BETA[9] = 2*q0*SH_BETA[2] + 2*q3*SH_BETA[3] - 2*q2*vd; SH_BETA[10] = 2*q2*SH_BETA[2] - 2*q1*SH_BETA[3] + 2*q0*vd; SH_BETA[11] = 2*q1*SH_BETA[2] + 2*q2*SH_BETA[3] + 2*q3*vd; SH_BETA[12] = 2*q0*q3; for (uint8_t i=0; i<=21; i++) { H_BETA[i] = 0.0f; } H_BETA[0] = SH_BETA[5]*SH_BETA[8] - SH_BETA[1]*SH_BETA[4]*SH_BETA[9]; H_BETA[1] = SH_BETA[5]*SH_BETA[10] - SH_BETA[1]*SH_BETA[4]*SH_BETA[11]; H_BETA[2] = SH_BETA[5]*SH_BETA[11] + SH_BETA[1]*SH_BETA[4]*SH_BETA[10]; H_BETA[3] = - SH_BETA[5]*SH_BETA[9] - SH_BETA[1]*SH_BETA[4]*SH_BETA[8]; H_BETA[4] = - SH_BETA[5]*(SH_BETA[12] - 2*q1*q2) - SH_BETA[1]*SH_BETA[4]*SH_BETA[7]; H_BETA[5] = SH_BETA[6] - SH_BETA[1]*SH_BETA[4]*(SH_BETA[12] + 2*q1*q2); H_BETA[6] = SH_BETA[5]*(2*q0*q1 + 2*q2*q3) + SH_BETA[1]*SH_BETA[4]*(2*q0*q2 - 2*q1*q3); H_BETA[14] = SH_BETA[5]*(SH_BETA[12] - 2*q1*q2) + SH_BETA[1]*SH_BETA[4]*SH_BETA[7]; H_BETA[15] = SH_BETA[1]*SH_BETA[4]*(SH_BETA[12] + 2*q1*q2) - SH_BETA[6]; // Calculate Kalman gains float temp = (R_BETA - (SH_BETA[5]*(SH_BETA[12] - 2*q1*q2) + SH_BETA[1]*SH_BETA[4]*SH_BETA[7])*(P[14][4]*(SH_BETA[5]*(SH_BETA[12] - 2*q1*q2) + SH_BETA[1]*SH_BETA[4]*SH_BETA[7]) - P[4][4]*(SH_BETA[5]*(SH_BETA[12] - 2*q1*q2) + SH_BETA[1]*SH_BETA[4]*SH_BETA[7]) + P[5][4]*(SH_BETA[6] - SH_BETA[1]*SH_BETA[4]*(SH_BETA[12] + 2*q1*q2)) - P[15][4]*(SH_BETA[6] - SH_BETA[1]*SH_BETA[4]*(SH_BETA[12] + 2*q1*q2)) + P[0][4]*(SH_BETA[5]*SH_BETA[8] - SH_BETA[1]*SH_BETA[4]*SH_BETA[9]) + P[1][4]*(SH_BETA[5]*SH_BETA[10] - SH_BETA[1]*SH_BETA[4]*SH_BETA[11]) + P[2][4]*(SH_BETA[5]*SH_BETA[11] + SH_BETA[1]*SH_BETA[4]*SH_BETA[10]) - P[3][4]*(SH_BETA[5]*SH_BETA[9] + SH_BETA[1]*SH_BETA[4]*SH_BETA[8]) + P[6][4]*(SH_BETA[5]*(2*q0*q1 + 2*q2*q3) + SH_BETA[1]*SH_BETA[4]*(2*q0*q2 - 2*q1*q3))) + (SH_BETA[5]*(SH_BETA[12] - 2*q1*q2) + SH_BETA[1]*SH_BETA[4]*SH_BETA[7])*(P[14][14]*(SH_BETA[5]*(SH_BETA[12] - 2*q1*q2) + SH_BETA[1]*SH_BETA[4]*SH_BETA[7]) - P[4][14]*(SH_BETA[5]*(SH_BETA[12] - 2*q1*q2) + SH_BETA[1]*SH_BETA[4]*SH_BETA[7]) + P[5][14]*(SH_BETA[6] - SH_BETA[1]*SH_BETA[4]*(SH_BETA[12] + 2*q1*q2)) - P[15][14]*(SH_BETA[6] - SH_BETA[1]*SH_BETA[4]*(SH_BETA[12] + 2*q1*q2)) + P[0][14]*(SH_BETA[5]*SH_BETA[8] - SH_BETA[1]*SH_BETA[4]*SH_BETA[9]) + P[1][14]*(SH_BETA[5]*SH_BETA[10] - SH_BETA[1]*SH_BETA[4]*SH_BETA[11]) + P[2][14]*(SH_BETA[5]*SH_BETA[11] + SH_BETA[1]*SH_BETA[4]*SH_BETA[10]) - P[3][14]*(SH_BETA[5]*SH_BETA[9] + SH_BETA[1]*SH_BETA[4]*SH_BETA[8]) + P[6][14]*(SH_BETA[5]*(2*q0*q1 + 2*q2*q3) + SH_BETA[1]*SH_BETA[4]*(2*q0*q2 - 2*q1*q3))) + (SH_BETA[6] - SH_BETA[1]*SH_BETA[4]*(SH_BETA[12] + 2*q1*q2))*(P[14][5]*(SH_BETA[5]*(SH_BETA[12] - 2*q1*q2) + SH_BETA[1]*SH_BETA[4]*SH_BETA[7]) - P[4][5]*(SH_BETA[5]*(SH_BETA[12] - 2*q1*q2) + SH_BETA[1]*SH_BETA[4]*SH_BETA[7]) + P[5][5]*(SH_BETA[6] - SH_BETA[1]*SH_BETA[4]*(SH_BETA[12] + 2*q1*q2)) - P[15][5]*(SH_BETA[6] - SH_BETA[1]*SH_BETA[4]*(SH_BETA[12] + 2*q1*q2)) + P[0][5]*(SH_BETA[5]*SH_BETA[8] - SH_BETA[1]*SH_BETA[4]*SH_BETA[9]) + P[1][5]*(SH_BETA[5]*SH_BETA[10] - SH_BETA[1]*SH_BETA[4]*SH_BETA[11]) + P[2][5]*(SH_BETA[5]*SH_BETA[11] + SH_BETA[1]*SH_BETA[4]*SH_BETA[10]) - P[3][5]*(SH_BETA[5]*SH_BETA[9] + SH_BETA[1]*SH_BETA[4]*SH_BETA[8]) + P[6][5]*(SH_BETA[5]*(2*q0*q1 + 2*q2*q3) + SH_BETA[1]*SH_BETA[4]*(2*q0*q2 - 2*q1*q3))) - (SH_BETA[6] - SH_BETA[1]*SH_BETA[4]*(SH_BETA[12] + 2*q1*q2))*(P[14][15]*(SH_BETA[5]*(SH_BETA[12] - 2*q1*q2) + SH_BETA[1]*SH_BETA[4]*SH_BETA[7]) - P[4][15]*(SH_BETA[5]*(SH_BETA[12] - 2*q1*q2) + SH_BETA[1]*SH_BETA[4]*SH_BETA[7]) + P[5][15]*(SH_BETA[6] - SH_BETA[1]*SH_BETA[4]*(SH_BETA[12] + 2*q1*q2)) - P[15][15]*(SH_BETA[6] - SH_BETA[1]*SH_BETA[4]*(SH_BETA[12] + 2*q1*q2)) + P[0][15]*(SH_BETA[5]*SH_BETA[8] - SH_BETA[1]*SH_BETA[4]*SH_BETA[9]) + P[1][15]*(SH_BETA[5]*SH_BETA[10] - SH_BETA[1]*SH_BETA[4]*SH_BETA[11]) + P[2][15]*(SH_BETA[5]*SH_BETA[11] + SH_BETA[1]*SH_BETA[4]*SH_BETA[10]) - P[3][15]*(SH_BETA[5]*SH_BETA[9] + SH_BETA[1]*SH_BETA[4]*SH_BETA[8]) + P[6][15]*(SH_BETA[5]*(2*q0*q1 + 2*q2*q3) + SH_BETA[1]*SH_BETA[4]*(2*q0*q2 - 2*q1*q3))) + (SH_BETA[5]*SH_BETA[8] - SH_BETA[1]*SH_BETA[4]*SH_BETA[9])*(P[14][0]*(SH_BETA[5]*(SH_BETA[12] - 2*q1*q2) + SH_BETA[1]*SH_BETA[4]*SH_BETA[7]) - P[4][0]*(SH_BETA[5]*(SH_BETA[12] - 2*q1*q2) + SH_BETA[1]*SH_BETA[4]*SH_BETA[7]) + P[5][0]*(SH_BETA[6] - SH_BETA[1]*SH_BETA[4]*(SH_BETA[12] + 2*q1*q2)) - P[15][0]*(SH_BETA[6] - SH_BETA[1]*SH_BETA[4]*(SH_BETA[12] + 2*q1*q2)) + P[0][0]*(SH_BETA[5]*SH_BETA[8] - SH_BETA[1]*SH_BETA[4]*SH_BETA[9]) + P[1][0]*(SH_BETA[5]*SH_BETA[10] - SH_BETA[1]*SH_BETA[4]*SH_BETA[11]) + P[2][0]*(SH_BETA[5]*SH_BETA[11] + SH_BETA[1]*SH_BETA[4]*SH_BETA[10]) - P[3][0]*(SH_BETA[5]*SH_BETA[9] + SH_BETA[1]*SH_BETA[4]*SH_BETA[8]) + P[6][0]*(SH_BETA[5]*(2*q0*q1 + 2*q2*q3) + SH_BETA[1]*SH_BETA[4]*(2*q0*q2 - 2*q1*q3))) + (SH_BETA[5]*SH_BETA[10] - SH_BETA[1]*SH_BETA[4]*SH_BETA[11])*(P[14][1]*(SH_BETA[5]*(SH_BETA[12] - 2*q1*q2) + SH_BETA[1]*SH_BETA[4]*SH_BETA[7]) - P[4][1]*(SH_BETA[5]*(SH_BETA[12] - 2*q1*q2) + SH_BETA[1]*SH_BETA[4]*SH_BETA[7]) + P[5][1]*(SH_BETA[6] - SH_BETA[1]*SH_BETA[4]*(SH_BETA[12] + 2*q1*q2)) - P[15][1]*(SH_BETA[6] - SH_BETA[1]*SH_BETA[4]*(SH_BETA[12] + 2*q1*q2)) + P[0][1]*(SH_BETA[5]*SH_BETA[8] - SH_BETA[1]*SH_BETA[4]*SH_BETA[9]) + P[1][1]*(SH_BETA[5]*SH_BETA[10] - SH_BETA[1]*SH_BETA[4]*SH_BETA[11]) + P[2][1]*(SH_BETA[5]*SH_BETA[11] + SH_BETA[1]*SH_BETA[4]*SH_BETA[10]) - P[3][1]*(SH_BETA[5]*SH_BETA[9] + SH_BETA[1]*SH_BETA[4]*SH_BETA[8]) + P[6][1]*(SH_BETA[5]*(2*q0*q1 + 2*q2*q3) + SH_BETA[1]*SH_BETA[4]*(2*q0*q2 - 2*q1*q3))) + (SH_BETA[5]*SH_BETA[11] + SH_BETA[1]*SH_BETA[4]*SH_BETA[10])*(P[14][2]*(SH_BETA[5]*(SH_BETA[12] - 2*q1*q2) + SH_BETA[1]*SH_BETA[4]*SH_BETA[7]) - P[4][2]*(SH_BETA[5]*(SH_BETA[12] - 2*q1*q2) + SH_BETA[1]*SH_BETA[4]*SH_BETA[7]) + P[5][2]*(SH_BETA[6] - SH_BETA[1]*SH_BETA[4]*(SH_BETA[12] + 2*q1*q2)) - P[15][2]*(SH_BETA[6] - SH_BETA[1]*SH_BETA[4]*(SH_BETA[12] + 2*q1*q2)) + P[0][2]*(SH_BETA[5]*SH_BETA[8] - SH_BETA[1]*SH_BETA[4]*SH_BETA[9]) + P[1][2]*(SH_BETA[5]*SH_BETA[10] - SH_BETA[1]*SH_BETA[4]*SH_BETA[11]) + P[2][2]*(SH_BETA[5]*SH_BETA[11] + SH_BETA[1]*SH_BETA[4]*SH_BETA[10]) - P[3][2]*(SH_BETA[5]*SH_BETA[9] + SH_BETA[1]*SH_BETA[4]*SH_BETA[8]) + P[6][2]*(SH_BETA[5]*(2*q0*q1 + 2*q2*q3) + SH_BETA[1]*SH_BETA[4]*(2*q0*q2 - 2*q1*q3))) - (SH_BETA[5]*SH_BETA[9] + SH_BETA[1]*SH_BETA[4]*SH_BETA[8])*(P[14][3]*(SH_BETA[5]*(SH_BETA[12] - 2*q1*q2) + SH_BETA[1]*SH_BETA[4]*SH_BETA[7]) - P[4][3]*(SH_BETA[5]*(SH_BETA[12] - 2*q1*q2) + SH_BETA[1]*SH_BETA[4]*SH_BETA[7]) + P[5][3]*(SH_BETA[6] - SH_BETA[1]*SH_BETA[4]*(SH_BETA[12] + 2*q1*q2)) - P[15][3]*(SH_BETA[6] - SH_BETA[1]*SH_BETA[4]*(SH_BETA[12] + 2*q1*q2)) + P[0][3]*(SH_BETA[5]*SH_BETA[8] - SH_BETA[1]*SH_BETA[4]*SH_BETA[9]) + P[1][3]*(SH_BETA[5]*SH_BETA[10] - SH_BETA[1]*SH_BETA[4]*SH_BETA[11]) + P[2][3]*(SH_BETA[5]*SH_BETA[11] + SH_BETA[1]*SH_BETA[4]*SH_BETA[10]) - P[3][3]*(SH_BETA[5]*SH_BETA[9] + SH_BETA[1]*SH_BETA[4]*SH_BETA[8]) + P[6][3]*(SH_BETA[5]*(2*q0*q1 + 2*q2*q3) + SH_BETA[1]*SH_BETA[4]*(2*q0*q2 - 2*q1*q3))) + (SH_BETA[5]*(2*q0*q1 + 2*q2*q3) + SH_BETA[1]*SH_BETA[4]*(2*q0*q2 - 2*q1*q3))*(P[14][6]*(SH_BETA[5]*(SH_BETA[12] - 2*q1*q2) + SH_BETA[1]*SH_BETA[4]*SH_BETA[7]) - P[4][6]*(SH_BETA[5]*(SH_BETA[12] - 2*q1*q2) + SH_BETA[1]*SH_BETA[4]*SH_BETA[7]) + P[5][6]*(SH_BETA[6] - SH_BETA[1]*SH_BETA[4]*(SH_BETA[12] + 2*q1*q2)) - P[15][6]*(SH_BETA[6] - SH_BETA[1]*SH_BETA[4]*(SH_BETA[12] + 2*q1*q2)) + P[0][6]*(SH_BETA[5]*SH_BETA[8] - SH_BETA[1]*SH_BETA[4]*SH_BETA[9]) + P[1][6]*(SH_BETA[5]*SH_BETA[10] - SH_BETA[1]*SH_BETA[4]*SH_BETA[11]) + P[2][6]*(SH_BETA[5]*SH_BETA[11] + SH_BETA[1]*SH_BETA[4]*SH_BETA[10]) - P[3][6]*(SH_BETA[5]*SH_BETA[9] + SH_BETA[1]*SH_BETA[4]*SH_BETA[8]) + P[6][6]*(SH_BETA[5]*(2*q0*q1 + 2*q2*q3) + SH_BETA[1]*SH_BETA[4]*(2*q0*q2 - 2*q1*q3)))); if (temp >= R_BETA) { SK_BETA[0] = 1.0f / temp; faultStatus.bad_sideslip = false; } else { // the calculation is badly conditioned, so we cannot perform fusion on this step faultStatus.bad_sideslip = true; return; } SK_BETA[1] = SH_BETA[5]*(SH_BETA[12] - 2*q1*q2) + SH_BETA[1]*SH_BETA[4]*SH_BETA[7]; SK_BETA[2] = SH_BETA[6] - SH_BETA[1]*SH_BETA[4]*(SH_BETA[12] + 2*q1*q2); SK_BETA[3] = SH_BETA[5]*(2*q0*q1 + 2*q2*q3) + SH_BETA[1]*SH_BETA[4]*(2*q0*q2 - 2*q1*q3); SK_BETA[4] = SH_BETA[5]*SH_BETA[10] - SH_BETA[1]*SH_BETA[4]*SH_BETA[11]; SK_BETA[5] = SH_BETA[5]*SH_BETA[8] - SH_BETA[1]*SH_BETA[4]*SH_BETA[9]; SK_BETA[6] = SH_BETA[5]*SH_BETA[11] + SH_BETA[1]*SH_BETA[4]*SH_BETA[10]; SK_BETA[7] = SH_BETA[5]*SH_BETA[9] + SH_BETA[1]*SH_BETA[4]*SH_BETA[8]; Kfusion[0] = SK_BETA[0]*(P[0][0]*SK_BETA[5] + P[0][1]*SK_BETA[4] - P[0][4]*SK_BETA[1] + P[0][5]*SK_BETA[2] + P[0][2]*SK_BETA[6] + P[0][6]*SK_BETA[3] - P[0][3]*SK_BETA[7] + P[0][14]*SK_BETA[1] - P[0][15]*SK_BETA[2]); Kfusion[1] = SK_BETA[0]*(P[1][0]*SK_BETA[5] + P[1][1]*SK_BETA[4] - P[1][4]*SK_BETA[1] + P[1][5]*SK_BETA[2] + P[1][2]*SK_BETA[6] + P[1][6]*SK_BETA[3] - P[1][3]*SK_BETA[7] + P[1][14]*SK_BETA[1] - P[1][15]*SK_BETA[2]); Kfusion[2] = SK_BETA[0]*(P[2][0]*SK_BETA[5] + P[2][1]*SK_BETA[4] - P[2][4]*SK_BETA[1] + P[2][5]*SK_BETA[2] + P[2][2]*SK_BETA[6] + P[2][6]*SK_BETA[3] - P[2][3]*SK_BETA[7] + P[2][14]*SK_BETA[1] - P[2][15]*SK_BETA[2]); Kfusion[3] = SK_BETA[0]*(P[3][0]*SK_BETA[5] + P[3][1]*SK_BETA[4] - P[3][4]*SK_BETA[1] + P[3][5]*SK_BETA[2] + P[3][2]*SK_BETA[6] + P[3][6]*SK_BETA[3] - P[3][3]*SK_BETA[7] + P[3][14]*SK_BETA[1] - P[3][15]*SK_BETA[2]); Kfusion[4] = SK_BETA[0]*(P[4][0]*SK_BETA[5] + P[4][1]*SK_BETA[4] - P[4][4]*SK_BETA[1] + P[4][5]*SK_BETA[2] + P[4][2]*SK_BETA[6] + P[4][6]*SK_BETA[3] - P[4][3]*SK_BETA[7] + P[4][14]*SK_BETA[1] - P[4][15]*SK_BETA[2]); Kfusion[5] = SK_BETA[0]*(P[5][0]*SK_BETA[5] + P[5][1]*SK_BETA[4] - P[5][4]*SK_BETA[1] + P[5][5]*SK_BETA[2] + P[5][2]*SK_BETA[6] + P[5][6]*SK_BETA[3] - P[5][3]*SK_BETA[7] + P[5][14]*SK_BETA[1] - P[5][15]*SK_BETA[2]); Kfusion[6] = SK_BETA[0]*(P[6][0]*SK_BETA[5] + P[6][1]*SK_BETA[4] - P[6][4]*SK_BETA[1] + P[6][5]*SK_BETA[2] + P[6][2]*SK_BETA[6] + P[6][6]*SK_BETA[3] - P[6][3]*SK_BETA[7] + P[6][14]*SK_BETA[1] - P[6][15]*SK_BETA[2]); Kfusion[7] = SK_BETA[0]*(P[7][0]*SK_BETA[5] + P[7][1]*SK_BETA[4] - P[7][4]*SK_BETA[1] + P[7][5]*SK_BETA[2] + P[7][2]*SK_BETA[6] + P[7][6]*SK_BETA[3] - P[7][3]*SK_BETA[7] + P[7][14]*SK_BETA[1] - P[7][15]*SK_BETA[2]); Kfusion[8] = SK_BETA[0]*(P[8][0]*SK_BETA[5] + P[8][1]*SK_BETA[4] - P[8][4]*SK_BETA[1] + P[8][5]*SK_BETA[2] + P[8][2]*SK_BETA[6] + P[8][6]*SK_BETA[3] - P[8][3]*SK_BETA[7] + P[8][14]*SK_BETA[1] - P[8][15]*SK_BETA[2]); Kfusion[9] = SK_BETA[0]*(P[9][0]*SK_BETA[5] + P[9][1]*SK_BETA[4] - P[9][4]*SK_BETA[1] + P[9][5]*SK_BETA[2] + P[9][2]*SK_BETA[6] + P[9][6]*SK_BETA[3] - P[9][3]*SK_BETA[7] + P[9][14]*SK_BETA[1] - P[9][15]*SK_BETA[2]); Kfusion[10] = SK_BETA[0]*(P[10][0]*SK_BETA[5] + P[10][1]*SK_BETA[4] - P[10][4]*SK_BETA[1] + P[10][5]*SK_BETA[2] + P[10][2]*SK_BETA[6] + P[10][6]*SK_BETA[3] - P[10][3]*SK_BETA[7] + P[10][14]*SK_BETA[1] - P[10][15]*SK_BETA[2]); Kfusion[11] = SK_BETA[0]*(P[11][0]*SK_BETA[5] + P[11][1]*SK_BETA[4] - P[11][4]*SK_BETA[1] + P[11][5]*SK_BETA[2] + P[11][2]*SK_BETA[6] + P[11][6]*SK_BETA[3] - P[11][3]*SK_BETA[7] + P[11][14]*SK_BETA[1] - P[11][15]*SK_BETA[2]); Kfusion[12] = SK_BETA[0]*(P[12][0]*SK_BETA[5] + P[12][1]*SK_BETA[4] - P[12][4]*SK_BETA[1] + P[12][5]*SK_BETA[2] + P[12][2]*SK_BETA[6] + P[12][6]*SK_BETA[3] - P[12][3]*SK_BETA[7] + P[12][14]*SK_BETA[1] - P[12][15]*SK_BETA[2]); // this term has been zeroed to improve stability of the Z accel bias Kfusion[13] = 0.0f;//SK_BETA[0]*(P[13][0]*SK_BETA[5] + P[13][1]*SK_BETA[4] - P[13][4]*SK_BETA[1] + P[13][5]*SK_BETA[2] + P[13][2]*SK_BETA[6] + P[13][6]*SK_BETA[3] - P[13][3]*SK_BETA[7] + P[13][14]*SK_BETA[1] - P[13][15]*SK_BETA[2]); Kfusion[14] = SK_BETA[0]*(P[14][0]*SK_BETA[5] + P[14][1]*SK_BETA[4] - P[14][4]*SK_BETA[1] + P[14][5]*SK_BETA[2] + P[14][2]*SK_BETA[6] + P[14][6]*SK_BETA[3] - P[14][3]*SK_BETA[7] + P[14][14]*SK_BETA[1] - P[14][15]*SK_BETA[2]); Kfusion[15] = SK_BETA[0]*(P[15][0]*SK_BETA[5] + P[15][1]*SK_BETA[4] - P[15][4]*SK_BETA[1] + P[15][5]*SK_BETA[2] + P[15][2]*SK_BETA[6] + P[15][6]*SK_BETA[3] - P[15][3]*SK_BETA[7] + P[15][14]*SK_BETA[1] - P[15][15]*SK_BETA[2]); // zero Kalman gains to inhibit magnetic field state estimation if (!inhibitMagStates) { Kfusion[16] = SK_BETA[0]*(P[16][0]*SK_BETA[5] + P[16][1]*SK_BETA[4] - P[16][4]*SK_BETA[1] + P[16][5]*SK_BETA[2] + P[16][2]*SK_BETA[6] + P[16][6]*SK_BETA[3] - P[16][3]*SK_BETA[7] + P[16][14]*SK_BETA[1] - P[16][15]*SK_BETA[2]); Kfusion[17] = SK_BETA[0]*(P[17][0]*SK_BETA[5] + P[17][1]*SK_BETA[4] - P[17][4]*SK_BETA[1] + P[17][5]*SK_BETA[2] + P[17][2]*SK_BETA[6] + P[17][6]*SK_BETA[3] - P[17][3]*SK_BETA[7] + P[17][14]*SK_BETA[1] - P[17][15]*SK_BETA[2]); Kfusion[18] = SK_BETA[0]*(P[18][0]*SK_BETA[5] + P[18][1]*SK_BETA[4] - P[18][4]*SK_BETA[1] + P[18][5]*SK_BETA[2] + P[18][2]*SK_BETA[6] + P[18][6]*SK_BETA[3] - P[18][3]*SK_BETA[7] + P[18][14]*SK_BETA[1] - P[18][15]*SK_BETA[2]); Kfusion[19] = SK_BETA[0]*(P[19][0]*SK_BETA[5] + P[19][1]*SK_BETA[4] - P[19][4]*SK_BETA[1] + P[19][5]*SK_BETA[2] + P[19][2]*SK_BETA[6] + P[19][6]*SK_BETA[3] - P[19][3]*SK_BETA[7] + P[19][14]*SK_BETA[1] - P[19][15]*SK_BETA[2]); Kfusion[20] = SK_BETA[0]*(P[20][0]*SK_BETA[5] + P[20][1]*SK_BETA[4] - P[20][4]*SK_BETA[1] + P[20][5]*SK_BETA[2] + P[20][2]*SK_BETA[6] + P[20][6]*SK_BETA[3] - P[20][3]*SK_BETA[7] + P[20][14]*SK_BETA[1] - P[20][15]*SK_BETA[2]); Kfusion[21] = SK_BETA[0]*(P[21][0]*SK_BETA[5] + P[21][1]*SK_BETA[4] - P[21][4]*SK_BETA[1] + P[21][5]*SK_BETA[2] + P[21][2]*SK_BETA[6] + P[21][6]*SK_BETA[3] - P[21][3]*SK_BETA[7] + P[21][14]*SK_BETA[1] - P[21][15]*SK_BETA[2]); } else { for (uint8_t i=16; i<=21; i++) { Kfusion[i] = 0.0f; } } // calculate predicted sideslip angle and innovation using small angle approximation innovBeta = vel_rel_wind.y / vel_rel_wind.x; // reject measurement if greater than 3-sigma inconsistency if (innovBeta > 0.5f) { return; } // correct the state vector for (uint8_t j=0; j<=21; j++) { states[j] = states[j] - Kfusion[j] * innovBeta; } Quaternion q(states[0], states[1], states[2], states[3]); q.normalize(); for (uint8_t i = 0; i<=3; i++) { states[i] = q[i]; } // correct the covariance P = (I - K*H)*P // take advantage of the empty columns in H to reduce the // number of operations for (uint8_t i = 0; i<=21; i++) { for (uint8_t j = 0; j<=6; j++) { KH[i][j] = Kfusion[i] * H_BETA[j]; } for (uint8_t j = 7; j<=13; j++) KH[i][j] = 0.0; for (uint8_t j = 14; j<=15; j++) { KH[i][j] = Kfusion[i] * H_BETA[j]; } for (uint8_t j = 16; j<=21; j++) KH[i][j] = 0.0; } for (uint8_t i = 0; i<=21; i++) { for (uint8_t j = 0; j<=21; j++) { KHP[i][j] = 0; for (uint8_t k = 0; k<=6; k++) { KHP[i][j] = KHP[i][j] + KH[i][k] * P[k][j]; } for (uint8_t k = 14; k<=15; k++) { KHP[i][j] = KHP[i][j] + KH[i][k] * P[k][j]; } } } for (uint8_t i = 0; i<=21; i++) { for (uint8_t j = 0; j<=21; j++) { P[i][j] = P[i][j] - KHP[i][j]; } } } // force the covariance matrix to me symmetrical and limit the variances to prevent ill-condiioning. ForceSymmetry(); ConstrainVariances(); // stop the performance timer perf_end(_perf_FuseSideslip); } // zero specified range of rows in the state covariance matrix void NavEKF::zeroRows(Matrix22 &covMat, uint8_t first, uint8_t last) { uint8_t row; for (row=first; row<=last; row++) { memset(&covMat[row][0], 0, sizeof(covMat[0][0])*22); } } // zero specified range of columns in the state covariance matrix void NavEKF::zeroCols(Matrix22 &covMat, uint8_t first, uint8_t last) { uint8_t row; for (row=0; row<=21; row++) { memset(&covMat[row][first], 0, sizeof(covMat[0][0])*(1+last-first)); } } // store states in a history array along with time stamp void NavEKF::StoreStates() { // Don't need to store states more often than every 10 msec if (hal.scheduler->millis() - lastStateStoreTime_ms >= 10) { lastStateStoreTime_ms = hal.scheduler->millis(); if (storeIndex > 49) { storeIndex = 0; } storedStates[storeIndex] = state; statetimeStamp[storeIndex] = lastStateStoreTime_ms; storeIndex = storeIndex + 1; } } // reset the stored state history and store the current state void NavEKF::StoreStatesReset() { // clear stored state history memset(&storedStates[0], 0, sizeof(storedStates)); memset(&statetimeStamp[0], 0, sizeof(statetimeStamp)); // store current state vector in first column storeIndex = 0; storedStates[storeIndex] = state; statetimeStamp[storeIndex] = hal.scheduler->millis(); storeIndex = storeIndex + 1; } // recall state vector stored at closest time to the one specified by msec void NavEKF::RecallStates(state_elements &statesForFusion, uint32_t msec) { uint32_t timeDelta; uint32_t bestTimeDelta = 200; uint8_t bestStoreIndex = 0; for (uint8_t i=0; i<=49; i++) { timeDelta = msec - statetimeStamp[i]; if (timeDelta < bestTimeDelta) { bestStoreIndex = i; bestTimeDelta = timeDelta; } } if (bestTimeDelta < 200) // only output stored state if < 200 msec retrieval error { statesForFusion = storedStates[bestStoreIndex]; } else // otherwise output current state { statesForFusion = state; } } // calculate nav to body quaternions from body to nav rotation matrix void NavEKF::quat2Tbn(Matrix3f &Tbn, const Quaternion &quat) const { // Calculate the body to nav cosine matrix quat.rotation_matrix(Tbn); } // return the Euler roll, pitch and yaw angle in radians void NavEKF::getEulerAngles(Vector3f &euler) const { Quaternion q(states[0], states[1], states[2], states[3]); q.to_euler(&euler.x, &euler.y, &euler.z); euler = euler - _ahrs->get_trim(); } // return NED velocity in m/s void NavEKF::getVelNED(Vector3f &vel) const { vel.x = states[4]; vel.y = states[5]; vel.z = states[6]; } // return the last calculated NED position relative to the reference point (m). // return false if no position is available bool NavEKF::getPosNED(Vector3f &pos) const { pos.x = states[7]; pos.y = states[8]; pos.z = states[9]; return true; } // return body axis gyro bias estimates in rad/sec void NavEKF::getGyroBias(Vector3f &gyroBias) const { if (dtIMU == 0) { gyroBias.zero(); return; } gyroBias.x = states[10] / dtIMU; gyroBias.y = states[11] / dtIMU; gyroBias.z = states[12] / dtIMU; } // return weighting of first IMU in blending function and the individual Z-accel bias estimates in m/s^2 void NavEKF::getAccelBias(Vector3f &accelBias) const { accelBias.x = IMU1_weighting; if (dtIMU == 0) { accelBias.y = 0; accelBias.z = 0; } else { accelBias.y = states[22] / dtIMU; accelBias.z = states[13] / dtIMU; } } // return the NED wind speed estimates in m/s (positive is air moving in the direction of the axis) void NavEKF::getWind(Vector3f &wind) const { wind.x = states[14]; wind.y = states[15]; wind.z = 0.0f; // currently don't estimate this } // return earth magnetic field estimates in measurement units / 1000 void NavEKF::getMagNED(Vector3f &magNED) const { magNED.x = states[16]*1000.0f; magNED.y = states[17]*1000.0f; magNED.z = states[18]*1000.0f; } // return body magnetic field estimates in measurement units / 1000 void NavEKF::getMagXYZ(Vector3f &magXYZ) const { magXYZ.x = states[19]*1000.0f; magXYZ.y = states[20]*1000.0f; magXYZ.z = states[21]*1000.0f; } // return the last calculated latitude, longitude and height bool NavEKF::getLLH(struct Location &loc) const { loc.lat = _ahrs->get_home().lat; loc.lng = _ahrs->get_home().lng; loc.alt = _ahrs->get_home().alt - states[9]*100; location_offset(loc, states[7], states[8]); return true; } // calculate whether the flight vehicle is on the ground or flying from height, airspeed and GPS speed void NavEKF::SetFlightAndFusionModes() { const AP_Airspeed *airspeed = _ahrs->get_airspeed(); uint8_t highAirSpd = (airspeed && airspeed->use() && airspeed->get_airspeed() * airspeed->get_EAS2TAS() > 8.0f); float gndSpdSq = sq(velNED[0]) + sq(velNED[1]); uint8_t highGndSpdStage1 = (uint8_t)(gndSpdSq > 9.0f); uint8_t highGndSpdStage2 = (uint8_t)(gndSpdSq > 36.0f); uint8_t highGndSpdStage3 = (uint8_t)(gndSpdSq > 81.0f); uint8_t largeHgt = (uint8_t)(fabsf(hgtMea) > 15.0f); uint8_t inAirSum = highAirSpd + highGndSpdStage1 + highGndSpdStage2 + highGndSpdStage3 + largeHgt; // if magnetometer calibration mode 1 is selected, use a manoeuvre threshold test // otherwise use a height and velocity test if ((_magCal == 1) && (accNavMagHoriz > 0.5f) && !static_mode_demanded() && use_compass()) { onGround = false; } else { // detect on-ground to in-air transition // if we are already on the ground then 3 or more out of 5 criteria are required // if we are in the air then only 2 or more are required // this prevents rapid tansitions if ((onGround && (inAirSum >= 3)) || (!onGround && (inAirSum >= 2))) { onGround = false; } else { onGround = true; } // force a yaw alignment if exiting onGround without a compass or if compass is timed out and we are a fly forward vehicle if (!onGround && prevOnGround && (!use_compass() || (magTimeout && assume_zero_sideslip()))) { alignYawGPS(); } // If we are flying a fly-forward type vehicle without an airspeed sensor and exiting onGround // we set the wind velocity to the reciprocal of the velocity vector and scale states so that the // wind speed is equal to the 6m/s. This prevents gains being too high at the start // of flight if launching into a headwind until the first turn when the EKF can form a wind speed // estimate if (!onGround && prevOnGround && !useAirspeed() && assume_zero_sideslip()) { setWindVelStates(); } } // store current on-ground status for next time prevOnGround = onGround; // If we are on ground, or in static mode, or don't have the right vehicle and sensing to estimate wind, inhibit wind states inhibitWindStates = ((!useAirspeed() && !assume_zero_sideslip()) || onGround || staticMode); // If magnetometer calibration mode is turned off by the user or we are on ground or in static mode, then inhibit magnetometer states inhibitMagStates = ((_magCal == 2) || !use_compass() || onGround || staticMode); } // initialise the covariance matrix void NavEKF::CovarianceInit() { // zero the matrix for (uint8_t i=1; i<=21; i++) { for (uint8_t j=0; j<=21; j++) { P[i][j] = 0.0f; } } // quaternions - TODO better maths for initial quaternion covariances that uses roll, pitch and yaw P[0][0] = 1.0e-9f; P[1][1] = 0.25f*sq(radians(1.0f)); P[2][2] = 0.25f*sq(radians(1.0f)); P[3][3] = 0.25f*sq(radians(1.0f)); // velocities P[4][4] = sq(0.7f); P[5][5] = P[4][4]; P[6][6] = sq(0.7f); // positions P[7][7] = sq(15.0f); P[8][8] = P[7][7]; P[9][9] = sq(5.0f); // delta angle biases P[10][10] = sq(radians(0.1f * dtIMU)); P[11][11] = P[10][10]; P[12][12] = P[10][10]; // Z delta velocity bias P[13][13] = 0.0f; // wind velocities P[14][14] = 0.0f; P[15][15] = P[14][14]; // earth magnetic field P[16][16] = 0.0f; P[17][17] = P[16][16]; P[18][18] = P[16][16]; // body magnetic field P[19][19] = 0.0f; P[20][20] = P[19][19]; P[21][21] = P[19][19]; } // force symmetry on the covariance matrix to prevent ill-conditioning void NavEKF::ForceSymmetry() { for (uint8_t i=1; i<=21; i++) { for (uint8_t j=0; j<=i-1; j++) { if (fabsf(P[i][j]) > EKF_COVARIENCE_MAX || fabsf(P[j][i]) > EKF_COVARIENCE_MAX) { // set the filter status as diverged and re-initialise the filter filterDiverged = true; faultStatus.diverged = true; lastDivergeTime_ms = hal.scheduler->millis(); InitialiseFilterDynamic(); return; } float temp = 0.5f*(P[i][j] + P[j][i]); P[i][j] = temp; P[j][i] = temp; } } } // copy covariances across from covariance prediction calculation and fix numerical errors void NavEKF::CopyAndFixCovariances() { // copy predicted variances for (uint8_t i=0; i<=21; i++) { P[i][i] = nextP[i][i]; } // copy predicted covariances and force symmetry for (uint8_t i=1; i<=21; i++) { for (uint8_t j=0; j<=i-1; j++) { P[i][j] = 0.5f*(nextP[i][j] + nextP[j][i]); P[j][i] = P[i][j]; } } } // constrain variances (diagonal terms) in the state covariance matrix to prevent ill-conditioning void NavEKF::ConstrainVariances() { for (uint8_t i=0; i<=3; i++) P[i][i] = constrain_float(P[i][i],0.0f,1.0f); // quaternions for (uint8_t i=4; i<=6; i++) P[i][i] = constrain_float(P[i][i],0.0f,1.0e3f); // velocities for (uint8_t i=7; i<=9; i++) P[i][i] = constrain_float(P[i][i],0.0f,1.0e6f); // positions for (uint8_t i=10; i<=12; i++) P[i][i] = constrain_float(P[i][i],0.0f,sq(0.175f * dtIMU)); // delta angle biases P[13][13] = constrain_float(P[13][13],0.0f,sq(10.0f * dtIMU)); // delta velocity bias for (uint8_t i=14; i<=15; i++) P[i][i] = constrain_float(P[i][i],0.0f,1.0e3f); // earth magnetic field for (uint8_t i=16; i<=21; i++) P[i][i] = constrain_float(P[i][i],0.0f,1.0f); // body magnetic field } // constrain states to prevent ill-conditioning void NavEKF::ConstrainStates() { // quaternions are limited between +-1 for (uint8_t i=0; i<=3; i++) states[i] = constrain_float(states[i],-1.0f,1.0f); // velocity limit 500 m/sec (could set this based on some multiple of max airspeed * EAS2TAS) for (uint8_t i=4; i<=6; i++) states[i] = constrain_float(states[i],-5.0e2f,5.0e2f); // position limit 1000 km - TODO apply circular limit for (uint8_t i=7; i<=8; i++) states[i] = constrain_float(states[i],-1.0e6f,1.0e6f); // height limit covers home alt on everest through to home alt at SL and ballon drop states[9] = constrain_float(states[9],-4.0e4f,1.0e4f); // gyro bias limit ~6 deg/sec (this needs to be set based on manufacturers specs) for (uint8_t i=10; i<=12; i++) states[i] = constrain_float(states[i],-0.1f*dtIMU,0.1f*dtIMU); // Z accel bias limit 1.0 m/s^2 (this needs to be finalised from test data) states[13] = constrain_float(states[13],-1.0f*dtIMU,1.0f*dtIMU); states[22] = constrain_float(states[22],-1.0f*dtIMU,1.0f*dtIMU); // wind velocity limit 100 m/s (could be based on some multiple of max airspeed * EAS2TAS) - TODO apply circular limit for (uint8_t i=14; i<=15; i++) states[i] = constrain_float(states[i],-100.0f,100.0f); // earth magnetic field limit for (uint8_t i=16; i<=18; i++) states[i] = constrain_float(states[i],-1.0f,1.0f); // body magnetic field limit for (uint8_t i=19; i<=21; i++) states[i] = constrain_float(states[i],-0.5f,0.5f); } // update IMU delta angle and delta velocity measurements void NavEKF::readIMUData() { Vector3f angRate; // angular rate vector in XYZ body axes measured by the IMU (rad/s) Vector3f accel1; // acceleration vector in XYZ body axes measured by IMU1 (m/s^2) Vector3f accel2; // acceleration vector in XYZ body axes measured by IMU2 (m/s^2) // get the time the IMU data was read IMUmsec = hal.scheduler->millis(); // limit IMU delta time to prevent numerical problems elsewhere dtIMU = constrain_float(_ahrs->get_ins().get_delta_time(), 0.001f, 1.0f); // get accels and gyro data from dual sensors if healthy if (_ahrs->get_ins().get_accel_health(0) && _ahrs->get_ins().get_accel_health(1)) { accel1 = _ahrs->get_ins().get_accel(0); accel2 = _ahrs->get_ins().get_accel(1); } else { accel1 = _ahrs->get_ins().get_accel(); accel2 = accel1; } // average the available gyro sensors angRate.zero(); uint8_t gyro_count = 0; for (uint8_t i = 0; i<_ahrs->get_ins().get_gyro_count(); i++) { if (_ahrs->get_ins().get_gyro_health(i)) { angRate += _ahrs->get_ins().get_gyro(i); gyro_count++; } } if (gyro_count != 0) { angRate /= gyro_count; } // trapezoidal integration dAngIMU = (angRate + lastAngRate) * dtIMU * 0.5f; lastAngRate = angRate; dVelIMU1 = (accel1 + lastAccel1) * dtIMU * 0.5f; lastAccel1 = accel1; dVelIMU2 = (accel2 + lastAccel2) * dtIMU * 0.5f; lastAccel2 = accel2; } // check for new valid GPS data and update stored measurement if available void NavEKF::readGpsData() { // check for new GPS data if ((_ahrs->get_gps().last_message_time_ms() != lastFixTime_ms) && (_ahrs->get_gps().status() >= AP_GPS::GPS_OK_FIX_3D)) { // store fix time from previous read secondLastFixTime_ms = lastFixTime_ms; // get current fix time lastFixTime_ms = _ahrs->get_gps().last_message_time_ms(); // set flag that lets other functions know that new GPS data has arrived newDataGps = true; // get state vectors that were stored at the time that is closest to when the the GPS measurement // time after accounting for measurement delays RecallStates(statesAtVelTime, (IMUmsec - constrain_int16(_msecVelDelay, 0, 500))); RecallStates(statesAtPosTime, (IMUmsec - constrain_int16(_msecPosDelay, 0, 500))); // read the NED velocity from the GPS velNED = _ahrs->get_gps().velocity(); // Check if GPS can output vertical velocity and set GPS fusion mode accordingly if (!_ahrs->get_gps().have_vertical_velocity()) { // vertical velocity should not be fused if (_fusionModeGPS == 0) { _fusionModeGPS = 1; } } // read latitutde and longitude from GPS and convert to NE position const struct Location &gpsloc = _ahrs->get_gps().location(); gpsPosNE = location_diff(_ahrs->get_home(), gpsloc); // decay and limit the position offset which is applied to NE position wherever it is used throughout code to allow GPS position jumps to be accommodated gradually decayGpsOffset(); } } // check for new altitude measurement data and update stored measurement if available void NavEKF::readHgtData() { // check to see if baro measurement has changed so we know if a new measurement has arrived if (_baro.get_last_update() != lastHgtMeasTime) { // time stamp used to check for new measurement lastHgtMeasTime = _baro.get_last_update(); // time stamp used to check for timeout lastHgtTime_ms = hal.scheduler->millis(); // get measurement and set flag to let other functions know new data has arrived hgtMea = _baro.get_altitude(); newDataHgt = true; // get states that wer stored at the time closest to the measurement time, taking measurement delay into account RecallStates(statesAtHgtTime, (IMUmsec - _msecHgtDelay)); } else { newDataHgt = false; } } // check for new magnetometer data and update store measurements if available void NavEKF::readMagData() { if (use_compass() && _ahrs->get_compass()->last_update != lastMagUpdate) { // store time of last measurement update lastMagUpdate = _ahrs->get_compass()->last_update; // read compass data and assign to bias and uncorrected measurement // body fixed magnetic bias is opposite sign to APM compass offsets // we scale compass data to improve numerical conditioning magBias = -_ahrs->get_compass()->get_offsets() * 0.001f; magData = _ahrs->get_compass()->get_field() * 0.001f + magBias; // get states stored at time closest to measurement time after allowance for measurement delay RecallStates(statesAtMagMeasTime, (IMUmsec - _msecMagDelay)); // let other processes know that new compass data has arrived newDataMag = true; } else { newDataMag = false; } } // check for new airspeed data and update stored measurements if available void NavEKF::readAirSpdData() { // if airspeed reading is valid and is set by the user to be used and has been updated then // we take a new reading, convert from EAS to TAS and set the flag letting other functions // know a new measurement is available const AP_Airspeed *aspeed = _ahrs->get_airspeed(); if (aspeed && aspeed->use() && aspeed->last_update_ms() != lastAirspeedUpdate) { VtasMeas = aspeed->get_airspeed() * aspeed->get_EAS2TAS(); lastAirspeedUpdate = aspeed->last_update_ms(); newDataTas = true; RecallStates(statesAtVtasMeasTime, (IMUmsec - _msecTasDelay)); } else { newDataTas = false; } } // calculate the NED earth spin vector in rad/sec void NavEKF::calcEarthRateNED(Vector3f &omega, int32_t latitude) const { float lat_rad = radians(latitude*1.0e-7f); omega.x = earthRate*cosf(lat_rad); omega.y = 0; omega.z = -earthRate*sinf(lat_rad); } // initialise the earth magnetic field states using declination, suppled roll/pitch // and magnetometer measurements and return initial attitude quaternion // if no magnetometer data, do not update amgentic field states and assume zero yaw angle Quaternion NavEKF::calcQuatAndFieldStates(float roll, float pitch) { // declare local variables required to calculate initial orientation and magnetic field float yaw; Matrix3f Tbn; Vector3f initMagNED; Quaternion initQuat; if (use_compass()) { // calculate rotation matrix from body to NED frame Tbn.from_euler(roll, pitch, 0.0f); // read the magnetometer data readMagData(); // rotate the magnetic field into NED axes initMagNED = Tbn*(magData - magBias); // calculate heading of mag field rel to body heading float magHeading = atan2f(initMagNED.y, initMagNED.x); // get the magnetic declination float magDecAng = use_compass() ? _ahrs->get_compass()->get_declination() : 0; // calculate yaw angle rel to true north yaw = magDecAng - magHeading; yawAligned = true; // calculate initial filter quaternion states initQuat.from_euler(roll, pitch, yaw); // calculate initial Tbn matrix and rotate Mag measurements into NED // to set initial NED magnetic field states initQuat.rotation_matrix(Tbn); initMagNED = Tbn * (magData - magBias); // write to earth magnetic field state vector state.earth_magfield = initMagNED; } else { initQuat.from_euler(roll, pitch, 0.0f); yawAligned = false; } // return attitude quaternion return initQuat; } // this function is used to do a forced alignment of the yaw angle to aligwith the horizontal velocity // vector from GPS. It is used to align the yaw angle after launch or takeoff without a magnetometer. void NavEKF::alignYawGPS() { if ((sq(velNED[0]) + sq(velNED[1])) > 16.0f) { float roll; float pitch; float oldYaw; float newYaw; float yawErr; // get quaternion from existing filter states and calculate roll, pitch and yaw angles Quaternion initQuat; Quaternion newQuat; for (uint8_t i=0; i<=3; i++) initQuat[i] = states[i]; initQuat.to_euler(&roll, &pitch, &oldYaw); // calculate yaw angle from GPS velocity newYaw = atan2f(velNED[1],velNED[0]); // modify yaw angle using GPS ground course if more than 45 degrees away or if not previously aligned yawErr = fabsf(newYaw - oldYaw); if (((yawErr > 0.7854f) && (yawErr < 5.4978f)) || !yawAligned) { // calculate new filter quaternion states from Euler angles newQuat.from_euler(roll, pitch, newYaw); for (uint8_t i=0; i<=3; i++) states[i] = newQuat[i]; // the yaw angle is now aligned so update its status yawAligned = true; // set the velocity states if (_fusionModeGPS < 2) { states[4] = velNED[0]; states[5] = velNED[1]; } // reinitialise the quaternion, velocity and position covariances // zero the matrix entries zeroRows(P,0,9); zeroCols(P,0,9); // quaternions - TODO maths that sets them based on different roll, yaw and pitch uncertainties P[0][0] = 1.0e-9f; P[1][1] = 0.25f*sq(radians(1.0f)); P[2][2] = 0.25f*sq(radians(1.0f)); P[3][3] = 0.25f*sq(radians(1.0f)); // velocities - we could have a big error coming out of static mode due to GPS lag P[4][4] = 400.0f; P[5][5] = P[4][4]; P[6][6] = sq(0.7f); // positions - we could have a big error coming out of static mode due to GPS lag P[7][7] = 400.0f; P[8][8] = P[7][7]; P[9][9] = sq(5.0f); } } } // This function is used to do a forced alignment of the wind velocity // states so that they are set to the reciprocal of the ground speed // and scaled to STARTUP_WIND_SPEED m/s. This is used when launching a // fly-forward vehicle without an airspeed sensor on the assumption // that launch will be into wind and STARTUP_WIND_SPEED is // representative of typical launch wind void NavEKF::setWindVelStates() { float gndSpd = sqrtf(sq(states[4]) + sq(states[5])); if (gndSpd > 4.0f) { // set the wind states to be the reciprocal of the velocity and scale float scaleFactor = STARTUP_WIND_SPEED / gndSpd; states[14] = - states[4] * scaleFactor; states[15] = - states[5] * scaleFactor; // reinitialise the wind state covariances zeroRows(P,14,15); zeroCols(P,14,15); P[14][14] = 64.0f; P[15][15] = P[14][14]; } } // return the transformation matrix from XYZ (body) to NED axes void NavEKF::getRotationBodyToNED(Matrix3f &mat) const { Vector3f trim = _ahrs->get_trim(); Quaternion q(states[0], states[1], states[2], states[3]); q.rotation_matrix(mat); mat.rotateXYinv(trim); } // return the innovations for the NED Pos, NED Vel, XYZ Mag and Vtas measurements void NavEKF::getInnovations(Vector3f &velInnov, Vector3f &posInnov, Vector3f &magInnov, float &tasInnov) const { velInnov.x = innovVelPos[0]; velInnov.y = innovVelPos[1]; velInnov.z = innovVelPos[2]; posInnov.x = innovVelPos[3]; posInnov.y = innovVelPos[4]; posInnov.z = innovVelPos[5]; magInnov.x = 1e3f*innovMag[0]; // Convert back to sensor units magInnov.y = 1e3f*innovMag[1]; // Convert back to sensor units magInnov.z = 1e3f*innovMag[2]; // Convert back to sensor units tasInnov = innovVtas; } // return the innovation consistency test ratios for the velocity, position, magnetometer and true airspeed measurements // this indicates the amount of margin available when tuning the various error traps // also return the current offsets applied to the GPS position measurements void NavEKF::getVariances(float &velVar, float &posVar, float &hgtVar, Vector3f &magVar, float &tasVar, Vector2f &offset) const { velVar = sqrtf(velTestRatio); posVar = sqrtf(posTestRatio); hgtVar = sqrtf(hgtTestRatio); magVar.x = sqrtf(magTestRatio.x); magVar.y = sqrtf(magTestRatio.y); magVar.z = sqrtf(magTestRatio.z); tasVar = sqrtf(tasTestRatio); offset = gpsPosGlitchOffsetNE; } // zero stored variables - this needs to be called before a full filter initialisation void NavEKF::ZeroVariables() { velTimeout = false; posTimeout = false; hgtTimeout = false; filterDiverged = false; lastStateStoreTime_ms = 0; lastFixTime_ms = 0; secondLastFixTime_ms = 0; lastMagUpdate = 0; lastAirspeedUpdate = 0; velFailTime = 0; posFailTime = 0; hgtFailTime = 0; storeIndex = 0; TASmsecPrev = 0; BETAmsecPrev = 0; MAGmsecPrev = 0; HGTmsecPrev = 0; lastMagUpdate = 0; lastAirspeedUpdate = 0; lastHgtMeasTime = 0; dtIMU = 0; dt = 0; hgtMea = 0; storeIndex = 0; lastGyroBias.zero(); prevDelAng.zero(); lastAngRate.zero(); lastAccel1.zero(); lastAccel2.zero(); velDotNEDfilt.zero(); summedDelAng.zero(); summedDelVel.zero(); velNED.zero(); gpsPosGlitchOffsetNE.zero(); gpsPosNE.zero(); prevTnb.zero(); memset(&P[0][0], 0, sizeof(P)); memset(&nextP[0][0], 0, sizeof(nextP)); memset(&processNoise[0], 0, sizeof(processNoise)); memset(&storedStates[0], 0, sizeof(storedStates)); memset(&statetimeStamp[0], 0, sizeof(statetimeStamp)); } // return true if we should use the airspeed sensor bool NavEKF::useAirspeed(void) const { if (_ahrs->get_airspeed() == NULL) { return false; } return _ahrs->get_airspeed()->use(); } // return true if the vehicle code has requested use of static mode // in static mode, position and height are constrained to zero, allowing an attitude // reference to be initialised and maintained when on the ground and without GPS lock bool NavEKF::static_mode_demanded(void) const { return !_ahrs->get_armed() || !_ahrs->get_correct_centrifugal(); } // return true if we should use the compass bool NavEKF::use_compass(void) const { return _ahrs->get_compass() && _ahrs->get_compass()->use_for_yaw(); } // decay GPS horizontal position offset to close to zero at a rate of 1 m/s // limit radius to a maximum of 100m // apply glitch offset to GPS measurements void NavEKF::decayGpsOffset() { float lapsedTime = 0.001f*float(hal.scheduler->millis() - lastDecayTime_ms); lastDecayTime_ms = hal.scheduler->millis(); float offsetRadius = pythagorous2(gpsPosGlitchOffsetNE.x, gpsPosGlitchOffsetNE.y); // decay radius if larger than velocity of 1.0 multiplied by lapsed time (plus a margin to prevent divide by zero) if (offsetRadius > (lapsedTime + 0.1f)) { // calculate scale factor to be applied to both offset components float scaleFactor = constrain_float((offsetRadius - lapsedTime), 0.0f, 100.0f) / offsetRadius; gpsPosGlitchOffsetNE.x *= scaleFactor; gpsPosGlitchOffsetNE.y *= scaleFactor; } } /* should we assume zero sideslip? */ bool NavEKF::assume_zero_sideslip(void) const { // we don't assume zero sideslip for ground vehicles as EKF could // be quite sensitive to a rapid spin of the ground vehicle if // traction is lost return _ahrs->get_fly_forward() && _ahrs->get_vehicle_class() != AHRS_VEHICLE_GROUND; } // Check for filter divergence void NavEKF::checkDivergence() { // If filter is diverging, then fail for 10 seconds // delay checking to allow bias estimate to settle after reset // filter divergence is detected by looking for rapid changes in gyro bias Vector3f tempVec = state.gyro_bias - lastGyroBias; float tempLength = tempVec.length(); if (tempLength != 0.0f) { float temp = constrain_float((P[10][10] + P[11][11] + P[12][12]),1e-12f,1e-8f); scaledDeltaGyrBiasLgth = (5e-7f / temp) * tempVec.length() / dtIMU; } bool divergenceDetected = (scaledDeltaGyrBiasLgth > 1.0f); lastGyroBias = state.gyro_bias; if (hal.scheduler->millis() - lastDivergeTime_ms > 10000) { if (divergenceDetected) { filterDiverged = true; faultStatus.diverged = true; lastDivergeTime_ms = hal.scheduler->millis(); } else { filterDiverged = false; } } } /* return the filter fault status as a bitmasked integer 0 = filter divergence detected via gyro bias growth 1 = filter divergence detected by large covariances 2 = badly conditioned X magnetometer fusion 3 = badly conditioned Y magnetometer fusion 4 = badly conditioned Z magnetometer fusion 5 = badly conditioned airspeed fusion 6 = badly conditioned synthetic sideslip fusion 7 = unassigned return normalised delta gyro bias length used for divergence test */ void NavEKF::getFilterFaults(uint8_t &faults, float &deltaGyroBias) const { faults = (faultStatus.diverged<<0 | faultStatus.large_covarience<<1 | faultStatus.bad_xmag<<2 | faultStatus.bad_ymag<<3 | faultStatus.bad_zmag<<4 | faultStatus.bad_airspeed<<5 | faultStatus.bad_sideslip<<6); deltaGyroBias = scaledDeltaGyrBiasLgth; } #endif // HAL_CPU_CLASS