ekf.cpp

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/**
 * @file ekf.cpp
 * Core functions for ekf attitude and position estimator.
 *
 * @author Roman Bast <bapstroman@gmail.com>
 * @author Paul Riseborough <p_riseborough@live.com.au>
 */

#include "ekf.h"

#include <ecl.h>
#include <mathlib/mathlib.h>

bool Ekf::init(uint64_t timestamp)
{
    bool ret = initialise_interface(timestamp);
    reset(timestamp);

    return ret;
}

void Ekf::reset(uint64_t timestamp)
{
    resetStatesAndCovariances();

    _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();
}

void Ekf::resetStatesAndCovariances()
{
    resetStates();
    initialiseCovariance();
}

void Ekf::resetStates()
{
    _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;
}

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)) {
                _mag_lpf.reset(_mag_sample_delayed.mag);

            } else if (_mag_counter > (uint8_t)(_obs_buffer_length + 1)) {
                _mag_lpf.update(_mag_sample_delayed.mag);
            }
        }
    }

    // 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_lpf.getState(), 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_3D) {
            // 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 &current_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();
}