/**************************************************************************** * * Copyright (c) 2015 Estimation and Control Library (ECL). All rights reserved. * * Redistribution and use in source and binary forms, with or without * modification, are permitted provided that the following conditions * are met: * * 1. Redistributions of source code must retain the above copyright * notice, this list of conditions and the following disclaimer. * 2. Redistributions in binary form must reproduce the above copyright * notice, this list of conditions and the following disclaimer in * the documentation and/or other materials provided with the * distribution. * 3. Neither the name ECL nor the names of its contributors may be * used to endorse or promote products derived from this software * without specific prior written permission. * * THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS * "AS IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT * LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS * FOR A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE * COPYRIGHT OWNER OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, * INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, * BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS * OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED * AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT * LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN * ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE * POSSIBILITY OF SUCH DAMAGE. * ****************************************************************************/ /** * @file ekf.cpp * Core functions for ekf attitude and position estimator. * * @author Roman Bast * @author Paul Riseborough */ #include "ekf.h" #include #include bool Ekf::init(uint64_t timestamp) { bool ret = initialise_interface(timestamp); _state.vel.setZero(); _state.pos.setZero(); _state.gyro_bias.setZero(); _state.accel_bias.setZero(); _state.mag_I.setZero(); _state.mag_B.setZero(); _state.wind_vel.setZero(); _state.quat_nominal.setZero(); _state.quat_nominal(0) = 1.0f; _output_new.vel.setZero(); _output_new.pos.setZero(); _output_new.quat_nominal.setZero(); _output_new.quat_nominal(0) = 1.0f; _delta_angle_corr.setZero(); _imu_down_sampled.delta_ang.setZero(); _imu_down_sampled.delta_vel.setZero(); _imu_down_sampled.delta_ang_dt = 0.0f; _imu_down_sampled.delta_vel_dt = 0.0f; _imu_down_sampled.time_us = timestamp; _q_down_sampled(0) = 1.0f; _q_down_sampled(1) = 0.0f; _q_down_sampled(2) = 0.0f; _q_down_sampled(3) = 0.0f; _imu_updated = false; _NED_origin_initialised = false; _gps_speed_valid = false; _filter_initialised = false; _terrain_initialised = false; _sin_tilt_rng = sinf(_params.rng_sens_pitch); _cos_tilt_rng = cosf(_params.rng_sens_pitch); _control_status.value = 0; _control_status_prev.value = 0; _dt_ekf_avg = FILTER_UPDATE_PERIOD_S; _fault_status.value = 0; _innov_check_fail_status.value = 0; _accel_mag_filt = 0.0f; _ang_rate_mag_filt = 0.0f; _prev_dvel_bias_var.zero(); return ret; } bool Ekf::update() { bool updated = false; if (!_filter_initialised) { _filter_initialised = initialiseFilter(); if (!_filter_initialised) { return false; } } // Only run the filter if IMU data in the buffer has been updated if (_imu_updated) { // perform state and covariance prediction for the main filter predictState(); predictCovariance(); // control fusion of observation data controlFusionModes(); // run a separate filter for terrain estimation runTerrainEstimator(); updated = true; } // the output observer always runs // Use full rate IMU data at the current time horizon calculateOutputStates(); return updated; } bool Ekf::initialiseFilter() { // Keep accumulating measurements until we have a minimum of 10 samples for the required sensors // Sum the IMU delta angle measurements const imuSample &imu_init = _imu_buffer.get_newest(); _delVel_sum += imu_init.delta_vel; // Sum the magnetometer measurements if (_mag_buffer.pop_first_older_than(_imu_sample_delayed.time_us, &_mag_sample_delayed)) { if ((_mag_counter == 0) && (_mag_sample_delayed.time_us != 0)) { // initialise the counter when we start getting data from the buffer _mag_counter = 1; } else if ((_mag_counter != 0) && (_mag_sample_delayed.time_us != 0)) { // increment the sample count and apply a LPF to the measurement _mag_counter ++; // don't start using data until we can be certain all bad initial data has been flushed if (_mag_counter == (uint8_t)(_obs_buffer_length + 1)) { // initialise filter states _mag_filt_state = _mag_sample_delayed.mag; } else if (_mag_counter > (uint8_t)(_obs_buffer_length + 1)) { // noise filter the data _mag_filt_state = _mag_filt_state * 0.9f + _mag_sample_delayed.mag * 0.1f; } } } // Count the number of external vision measurements received if (_ext_vision_buffer.pop_first_older_than(_imu_sample_delayed.time_us, &_ev_sample_delayed)) { if ((_ev_counter == 0) && (_ev_sample_delayed.time_us != 0)) { // initialise the counter _ev_counter = 1; // set the height fusion mode to use external vision data when we start getting valid data from the buffer if (_primary_hgt_source == VDIST_SENSOR_EV) { _control_status.flags.baro_hgt = false; _control_status.flags.gps_hgt = false; _control_status.flags.rng_hgt = false; _control_status.flags.ev_hgt = true; } } else if ((_ev_counter != 0) && (_ev_sample_delayed.time_us != 0)) { // increment the sample count _ev_counter ++; } } // set the default height source from the adjustable parameter if (_hgt_counter == 0) { _primary_hgt_source = _params.vdist_sensor_type; } // accumulate enough height measurements to be confident in the quality of the data // we use baro height initially and switch to GPS/range/EV finder later when it passes checks. if (_baro_buffer.pop_first_older_than(_imu_sample_delayed.time_us, &_baro_sample_delayed)) { if ((_hgt_counter == 0) && (_baro_sample_delayed.time_us != 0)) { // initialise the counter and height fusion method when we start getting data from the buffer setControlBaroHeight(); _hgt_counter = 1; } else if ((_hgt_counter != 0) && (_baro_sample_delayed.time_us != 0)) { // increment the sample count and apply a LPF to the measurement _hgt_counter ++; // don't start using data until we can be certain all bad initial data has been flushed if (_hgt_counter == (uint8_t)(_obs_buffer_length + 1)) { // initialise filter states _baro_hgt_offset = _baro_sample_delayed.hgt; } else if (_hgt_counter > (uint8_t)(_obs_buffer_length + 1)) { // noise filter the data _baro_hgt_offset = 0.9f * _baro_hgt_offset + 0.1f * _baro_sample_delayed.hgt; } } } // check to see if we have enough measurements and return false if not bool hgt_count_fail = _hgt_counter <= 2u * _obs_buffer_length; bool mag_count_fail = _mag_counter <= 2u * _obs_buffer_length; if (hgt_count_fail || mag_count_fail) { return false; } else { // reset variables that are shared with post alignment GPS checks _gps_drift_velD = 0.0f; _gps_alt_ref = 0.0f; // Zero all of the states _state.vel.setZero(); _state.pos.setZero(); _state.gyro_bias.setZero(); _state.accel_bias.setZero(); _state.mag_I.setZero(); _state.mag_B.setZero(); _state.wind_vel.setZero(); // get initial roll and pitch estimate from delta velocity vector, assuming vehicle is static float pitch = 0.0f; float roll = 0.0f; if (_delVel_sum.norm() > 0.001f) { _delVel_sum.normalize(); pitch = asinf(_delVel_sum(0)); roll = atan2f(-_delVel_sum(1), -_delVel_sum(2)); } else { return false; } // calculate initial tilt alignment Eulerf euler_init(roll, pitch, 0.0f); _state.quat_nominal = Quatf(euler_init); // update transformation matrix from body to world frame _R_to_earth = Dcmf(_state.quat_nominal); // calculate the initial magnetic field and yaw alignment _control_status.flags.yaw_align = resetMagHeading(_mag_filt_state, false, false); // initialise the state covariance matrix initialiseCovariance(); // update the yaw angle variance using the variance of the measurement if (_control_status.flags.ev_yaw) { // using error estimate from external vision data TODO: this is never true increaseQuatYawErrVariance(sq(fmaxf(_ev_sample_delayed.angErr, 1.0e-2f))); } else if (_params.mag_fusion_type <= MAG_FUSE_TYPE_AUTOFW) { // using magnetic heading tuning parameter increaseQuatYawErrVariance(sq(fmaxf(_params.mag_heading_noise, 1.0e-2f))); } if (_control_status.flags.rng_hgt) { // if we are using the range finder as the primary source, then calculate the baro height at origin so we can use baro as a backup // so it can be used as a backup ad set the initial height using the range finder const baroSample &baro_newest = _baro_buffer.get_newest(); _baro_hgt_offset = baro_newest.hgt; _state.pos(2) = -math::max(_rng_filt_state * _R_rng_to_earth_2_2, _params.rng_gnd_clearance); ECL_INFO_TIMESTAMPED("EKF using range finder height - commencing alignment"); } else if (_control_status.flags.ev_hgt) { // if we are using external vision data for height, then the vertical position state needs to be reset // because the initialisation position is not the zero datum resetHeight(); } // try to initialise the terrain estimator _terrain_initialised = initHagl(); // reset the essential fusion timeout counters _time_last_hgt_fuse = _time_last_imu; _time_last_pos_fuse = _time_last_imu; _time_last_delpos_fuse = _time_last_imu; _time_last_vel_fuse = _time_last_imu; _time_last_hagl_fuse = _time_last_imu; _time_last_of_fuse = _time_last_imu; // reset the output predictor state history to match the EKF initial values alignOutputFilter(); return true; } } void Ekf::predictState() { if (!_earth_rate_initialised) { if (_NED_origin_initialised) { calcEarthRateNED(_earth_rate_NED, (float)_pos_ref.lat_rad); _earth_rate_initialised = true; } } // apply imu bias corrections Vector3f corrected_delta_ang = _imu_sample_delayed.delta_ang - _state.gyro_bias; Vector3f corrected_delta_vel = _imu_sample_delayed.delta_vel - _state.accel_bias; // correct delta angles for earth rotation rate corrected_delta_ang -= -_R_to_earth.transpose() * _earth_rate_NED * _imu_sample_delayed.delta_ang_dt; // convert the delta angle to a delta quaternion Quatf dq; dq.from_axis_angle(corrected_delta_ang); // rotate the previous quaternion by the delta quaternion using a quaternion multiplication _state.quat_nominal = _state.quat_nominal * dq; // quaternions must be normalised whenever they are modified _state.quat_nominal.normalize(); // save the previous value of velocity so we can use trapzoidal integration Vector3f vel_last = _state.vel; // update transformation matrix from body to world frame _R_to_earth = Dcmf(_state.quat_nominal); // Calculate an earth frame delta velocity Vector3f corrected_delta_vel_ef = _R_to_earth * corrected_delta_vel; // calculate a filtered horizontal acceleration with a 1 sec time constant // this are used for manoeuvre detection elsewhere float alpha = 1.0f - _imu_sample_delayed.delta_vel_dt; _accel_lpf_NE(0) = _accel_lpf_NE(0) * alpha + corrected_delta_vel_ef(0); _accel_lpf_NE(1) = _accel_lpf_NE(1) * alpha + corrected_delta_vel_ef(1); // calculate the increment in velocity using the current orientation _state.vel += corrected_delta_vel_ef; // compensate for acceleration due to gravity _state.vel(2) += CONSTANTS_ONE_G * _imu_sample_delayed.delta_vel_dt; // predict position states via trapezoidal integration of velocity _state.pos += (vel_last + _state.vel) * _imu_sample_delayed.delta_vel_dt * 0.5f; constrainStates(); // calculate an average filter update time float input = 0.5f * (_imu_sample_delayed.delta_vel_dt + _imu_sample_delayed.delta_ang_dt); // filter and limit input between -50% and +100% of nominal value input = math::constrain(input, 0.5f * FILTER_UPDATE_PERIOD_S, 2.0f * FILTER_UPDATE_PERIOD_S); _dt_ekf_avg = 0.99f * _dt_ekf_avg + 0.01f * input; } bool Ekf::collect_imu(const imuSample &imu) { // accumulate and downsample IMU data across a period FILTER_UPDATE_PERIOD_MS long // copy imu data to local variables _imu_sample_new = imu; // accumulate the time deltas _imu_down_sampled.delta_ang_dt += imu.delta_ang_dt; _imu_down_sampled.delta_vel_dt += imu.delta_vel_dt; // use a quaternion to accumulate delta angle data // this quaternion represents the rotation from the start to end of the accumulation period Quatf delta_q; delta_q.rotate(imu.delta_ang); _q_down_sampled = _q_down_sampled * delta_q; _q_down_sampled.normalize(); // rotate the accumulated delta velocity data forward each time so it is always in the updated rotation frame Dcmf delta_R(delta_q.inversed()); _imu_down_sampled.delta_vel = delta_R * _imu_down_sampled.delta_vel; // accumulate the most recent delta velocity data at the updated rotation frame // assume effective sample time is halfway between the previous and current rotation frame _imu_down_sampled.delta_vel += (imu.delta_vel + delta_R * imu.delta_vel) * 0.5f; // if the target time delta between filter prediction steps has been exceeded // write the accumulated IMU data to the ring buffer const float target_dt = FILTER_UPDATE_PERIOD_S; if (_imu_down_sampled.delta_ang_dt >= target_dt - _imu_collection_time_adj) { // accumulate the amount of time to advance the IMU collection time so that we meet the // average EKF update rate requirement _imu_collection_time_adj += 0.01f * (_imu_down_sampled.delta_ang_dt - target_dt); _imu_collection_time_adj = math::constrain(_imu_collection_time_adj, -0.5f * target_dt, 0.5f * target_dt); imuSample imu_sample_new; imu_sample_new.delta_ang = _q_down_sampled.to_axis_angle(); imu_sample_new.delta_vel = _imu_down_sampled.delta_vel; imu_sample_new.delta_ang_dt = _imu_down_sampled.delta_ang_dt; imu_sample_new.delta_vel_dt = _imu_down_sampled.delta_vel_dt; imu_sample_new.time_us = imu.time_us; _imu_buffer.push(imu_sample_new); // get the oldest data from the buffer _imu_sample_delayed = _imu_buffer.get_oldest(); // calculate the minimum interval between observations required to guarantee no loss of data // this will occur if data is overwritten before its time stamp falls behind the fusion time horizon _min_obs_interval_us = (imu_sample_new.time_us - _imu_sample_delayed.time_us) / (_obs_buffer_length - 1); // reset _imu_down_sampled.delta_ang.setZero(); _imu_down_sampled.delta_vel.setZero(); _imu_down_sampled.delta_ang_dt = 0.0f; _imu_down_sampled.delta_vel_dt = 0.0f; _q_down_sampled(0) = 1.0f; _q_down_sampled(1) = _q_down_sampled(2) = _q_down_sampled(3) = 0.0f; _imu_updated = true; } else { _imu_updated = false; } return _imu_updated; } /* * Implement a strapdown INS algorithm using the latest IMU data at the current time horizon. * Buffer the INS states and calculate the difference with the EKF states at the delayed fusion time horizon. * Calculate delta angle, delta velocity and velocity corrections from the differences and apply them at the * current time horizon so that the INS states track the EKF states at the delayed fusion time horizon. * The inspiration for using a complementary filter to correct for time delays in the EKF * is based on the work by A Khosravian: * “Recursive Attitude Estimation in the Presence of Multi-rate and Multi-delay Vector Measurements” * A Khosravian, J Trumpf, R Mahony, T Hamel, Australian National University */ void Ekf::calculateOutputStates() { // Use full rate IMU data at the current time horizon const imuSample &imu = _imu_sample_new; // correct delta angles for bias offsets const float dt_scale_correction = _dt_imu_avg / _dt_ekf_avg; // Apply corrections to the delta angle required to track the quaternion states at the EKF fusion time horizon const Vector3f delta_angle{imu.delta_ang - _state.gyro_bias * dt_scale_correction + _delta_angle_corr}; // calculate a yaw change about the earth frame vertical const float spin_del_ang_D = _R_to_earth_now(2, 0) * delta_angle(0) + _R_to_earth_now(2, 1) * delta_angle(1) + _R_to_earth_now(2, 2) * delta_angle(2); _yaw_delta_ef += spin_del_ang_D; // Calculate filtered yaw rate to be used by the magnetometer fusion type selection logic // Note fixed coefficients are used to save operations. The exact time constant is not important. _yaw_rate_lpf_ef = 0.95f * _yaw_rate_lpf_ef + 0.05f * spin_del_ang_D / imu.delta_ang_dt; // convert the delta angle to an equivalent delta quaternions Quatf dq; dq.from_axis_angle(delta_angle); // rotate the previous INS quaternion by the delta quaternions _output_new.time_us = imu.time_us; _output_new.quat_nominal = _output_new.quat_nominal * dq; // the quaternions must always be normalised after modification _output_new.quat_nominal.normalize(); // calculate the rotation matrix from body to earth frame _R_to_earth_now = Dcmf(_output_new.quat_nominal); // correct delta velocity for bias offsets const Vector3f delta_vel{imu.delta_vel - _state.accel_bias * dt_scale_correction}; // rotate the delta velocity to earth frame Vector3f delta_vel_NED{_R_to_earth_now * delta_vel}; // correct for measured acceleration due to gravity delta_vel_NED(2) += CONSTANTS_ONE_G * imu.delta_vel_dt; // calculate the earth frame velocity derivatives if (imu.delta_vel_dt > 1e-4f) { _vel_deriv_ned = delta_vel_NED * (1.0f / imu.delta_vel_dt); } // save the previous velocity so we can use trapezoidal integration const Vector3f vel_last{_output_new.vel}; // increment the INS velocity states by the measurement plus corrections // do the same for vertical state used by alternative correction algorithm _output_new.vel += delta_vel_NED; _output_vert_new.vel_d += delta_vel_NED(2); // use trapezoidal integration to calculate the INS position states // do the same for vertical state used by alternative correction algorithm const Vector3f delta_pos_NED = (_output_new.vel + vel_last) * (imu.delta_vel_dt * 0.5f); _output_new.pos += delta_pos_NED; _output_vert_new.vel_d_integ += delta_pos_NED(2); // accumulate the time for each update _output_vert_new.dt += imu.delta_vel_dt; // correct velocity for IMU offset if (imu.delta_ang_dt > 1e-4f) { // calculate the average angular rate across the last IMU update const Vector3f ang_rate = imu.delta_ang * (1.0f / imu.delta_ang_dt); // calculate the velocity of the IMU relative to the body origin const Vector3f vel_imu_rel_body = cross_product(ang_rate, _params.imu_pos_body); // rotate the relative velocity into earth frame _vel_imu_rel_body_ned = _R_to_earth_now * vel_imu_rel_body; } // store the INS states in a ring buffer with the same length and time coordinates as the IMU data buffer if (_imu_updated) { _output_buffer.push(_output_new); _output_vert_buffer.push(_output_vert_new); // get the oldest INS state data from the ring buffer // this data will be at the EKF fusion time horizon _output_sample_delayed = _output_buffer.get_oldest(); _output_vert_delayed = _output_vert_buffer.get_oldest(); // calculate the quaternion delta between the INS and EKF quaternions at the EKF fusion time horizon Quatf quat_inv = _state.quat_nominal.inversed(); Quatf q_error = quat_inv * _output_sample_delayed.quat_nominal; q_error.normalize(); // convert the quaternion delta to a delta angle float scalar; if (q_error(0) >= 0.0f) { scalar = -2.0f; } else { scalar = 2.0f; } const Vector3f delta_ang_error{scalar * q_error(1), scalar * q_error(2), scalar * q_error(3)}; // calculate a gain that provides tight tracking of the estimator attitude states and // adjust for changes in time delay to maintain consistent damping ratio of ~0.7 const float time_delay = fmaxf((imu.time_us - _imu_sample_delayed.time_us) * 1e-6f, _dt_imu_avg); const float att_gain = 0.5f * _dt_imu_avg / time_delay; // calculate a corrrection to the delta angle // that will cause the INS to track the EKF quaternions _delta_angle_corr = delta_ang_error * att_gain; // calculate velocity and position tracking errors const Vector3f vel_err{_state.vel - _output_sample_delayed.vel}; const Vector3f pos_err{_state.pos - _output_sample_delayed.pos}; // collect magnitude tracking error for diagnostics _output_tracking_error[0] = delta_ang_error.norm(); _output_tracking_error[1] = vel_err.norm(); _output_tracking_error[2] = pos_err.norm(); /* * Loop through the output filter state history and apply the corrections to the velocity and position states. * This method is too expensive to use for the attitude states due to the quaternion operations required * but because it eliminates the time delay in the 'correction loop' it allows higher tracking gains * to be used and reduces tracking error relative to EKF states. */ // Complementary filter gains const float vel_gain = _dt_ekf_avg / math::constrain(_params.vel_Tau, _dt_ekf_avg, 10.0f); const float pos_gain = _dt_ekf_avg / math::constrain(_params.pos_Tau, _dt_ekf_avg, 10.0f); { /* * Calculate a correction to be applied to vel_d that casues vel_d_integ to track the EKF * down position state at the fusion time horizon using an alternative algorithm to what * is used for the vel and pos state tracking. The algorithm applies a correction to the vel_d * state history and propagates vel_d_integ forward in time using the corrected vel_d history. * This provides an alternative vertical velocity output that is closer to the first derivative * of the position but does degrade tracking relative to the EKF state. */ // calculate down velocity and position tracking errors const float vel_d_err = (_state.vel(2) - _output_vert_delayed.vel_d); const float pos_d_err = (_state.pos(2) - _output_vert_delayed.vel_d_integ); // calculate a velocity correction that will be applied to the output state history // using a PD feedback tuned to a 5% overshoot const float vel_d_correction = pos_d_err * pos_gain + vel_d_err * pos_gain * 1.1f; /* * Calculate corrections to be applied to vel and pos output state history. * The vel and pos state history are corrected individually so they track the EKF states at * the fusion time horizon. This option provides the most accurate tracking of EKF states. */ // loop through the vertical output filter state history starting at the oldest and apply the corrections to the // vel_d states and propagate vel_d_integ forward using the corrected vel_d uint8_t index = _output_vert_buffer.get_oldest_index(); const uint8_t size = _output_vert_buffer.get_length(); for (uint8_t counter = 0; counter < (size - 1); counter++) { const uint8_t index_next = (index + 1) % size; outputVert ¤t_state = _output_vert_buffer[index]; outputVert &next_state = _output_vert_buffer[index_next]; // correct the velocity if (counter == 0) { current_state.vel_d += vel_d_correction; } next_state.vel_d += vel_d_correction; // position is propagated forward using the corrected velocity and a trapezoidal integrator next_state.vel_d_integ = current_state.vel_d_integ + (current_state.vel_d + next_state.vel_d) * 0.5f * next_state.dt; // advance the index index = (index + 1) % size; } // update output state to corrected values _output_vert_new = _output_vert_buffer.get_newest(); // reset time delta to zero for the next accumulation of full rate IMU data _output_vert_new.dt = 0.0f; } { /* * Calculate corrections to be applied to vel and pos output state history. * The vel and pos state history are corrected individually so they track the EKF states at * the fusion time horizon. This option provides the most accurate tracking of EKF states. */ // calculate a velocity correction that will be applied to the output state history _vel_err_integ += vel_err; const Vector3f vel_correction = vel_err * vel_gain + _vel_err_integ * sq(vel_gain) * 0.1f; // calculate a position correction that will be applied to the output state history _pos_err_integ += pos_err; const Vector3f pos_correction = pos_err * pos_gain + _pos_err_integ * sq(pos_gain) * 0.1f; // loop through the output filter state history and apply the corrections to the velocity and position states for (uint8_t index = 0; index < _output_buffer.get_length(); index++) { // a constant velocity correction is applied _output_buffer[index].vel += vel_correction; // a constant position correction is applied _output_buffer[index].pos += pos_correction; } // update output state to corrected values _output_new = _output_buffer.get_newest(); } } } /* * Predict the previous quaternion output state forward using the latest IMU delta angle data. */ Quatf Ekf::calculate_quaternion() const { // Correct delta angle data for bias errors using bias state estimates from the EKF and also apply // corrections required to track the EKF quaternion states const Vector3f delta_angle{_imu_sample_new.delta_ang - _state.gyro_bias * (_dt_imu_avg / _dt_ekf_avg) + _delta_angle_corr}; // increment the quaternions using the corrected delta angle vector // the quaternions must always be normalised after modification return Quatf{_output_new.quat_nominal * AxisAnglef{delta_angle}}.unit(); }