AP_NavEKF3_RngBcnFusion.cpp

#include "AP_NavEKF3.h"
#include "AP_NavEKF3_core.h"

/********************************************************
*                   FUSE MEASURED_DATA                  *
********************************************************/

// select fusion of range beacon measurements
void NavEKF3_core::SelectRngBcnFusion()
{
    // read range data from the sensor and check for new data in the buffer
    readRngBcnData();

    // Determine if we need to fuse range beacon data on this time step
    if (rngBcnDataToFuse) {
        if (PV_AidingMode == AID_ABSOLUTE) {
            if (!filterStatus.flags.using_gps && rngBcnAlignmentCompleted) {
                if (!bcnOriginEstInit) {
                    bcnOriginEstInit = true;
                    bcnPosOffsetNED.x = receiverPos.x - stateStruct.position.x;
                    bcnPosOffsetNED.y = receiverPos.y - stateStruct.position.y;
                }
                // If we aren't using GPS, then the beacons are used as the primary means of position reference
                FuseRngBcn();
            } else {
                // If we are using GPS, then GPS is the primary reference, but we continue to use the beacon data
                // to calculate an independant position that is used to update the beacon position offset if we need to
                // start using beacon data as the primary reference.
                FuseRngBcnStatic();
                // record that the beacon origin needs to be initialised
                bcnOriginEstInit = false;
            }
        } else {
            // If we aren't able to use the data in the main filter, use a simple 3-state filter to estimate position only
            FuseRngBcnStatic();
            // record that the beacon origin needs to be initialised
            bcnOriginEstInit = false;
        }
    }
}

void NavEKF3_core::FuseRngBcn()
{
    // declarations
    float pn;
    float pe;
    float pd;
    float bcn_pn;
    float bcn_pe;
    float bcn_pd;
    const float R_BCN = sq(MAX(rngBcnDataDelayed.rngErr , 0.1f));
    float rngPred;

    // health is set bad until test passed
    rngBcnHealth = false;

    if (activeHgtSource != HGT_SOURCE_BCN) {
        // calculate the vertical offset from EKF datum to beacon datum
        CalcRangeBeaconPosDownOffset(R_BCN, stateStruct.position, false);
    } else {
        bcnPosOffsetNED.z = 0.0f;
    }

    // copy required states to local variable names
    pn = stateStruct.position.x;
    pe = stateStruct.position.y;
    pd = stateStruct.position.z;
    bcn_pn = rngBcnDataDelayed.beacon_posNED.x;
    bcn_pe = rngBcnDataDelayed.beacon_posNED.y;
    bcn_pd = rngBcnDataDelayed.beacon_posNED.z + bcnPosOffsetNED.z;

    // predicted range
    Vector3f deltaPosNED = stateStruct.position - rngBcnDataDelayed.beacon_posNED;
    rngPred = deltaPosNED.length();

    // calculate measurement innovation
    innovRngBcn = rngPred - rngBcnDataDelayed.rng;

    // perform fusion of range measurement
    if (rngPred > 0.1f)
    {
        // calculate observation jacobians
        float H_BCN[24];
        memset(H_BCN, 0, sizeof(H_BCN));
        float t2 = bcn_pd-pd;
        float t3 = bcn_pe-pe;
        float t4 = bcn_pn-pn;
        float t5 = t2*t2;
        float t6 = t3*t3;
        float t7 = t4*t4;
        float t8 = t5+t6+t7;
        float t9 = 1.0f/sqrtf(t8);
        H_BCN[7] = -t4*t9;
        H_BCN[8] = -t3*t9;
        // If we are not using the beacons as a height reference, we pretend that the beacons
        // are at the same height as the flight vehicle when calculating the observation derivatives
        // and Kalman gains
        // TODO  - less hacky way of achieving this, preferably using an alternative derivation
        if (activeHgtSource != HGT_SOURCE_BCN) {
            t2 = 0.0f;
        }
        H_BCN[9] = -t2*t9;

        // calculate Kalman gains
        float t10 = P[9][9]*t2*t9;
        float t11 = P[8][9]*t3*t9;
        float t12 = P[7][9]*t4*t9;
        float t13 = t10+t11+t12;
        float t14 = t2*t9*t13;
        float t15 = P[9][8]*t2*t9;
        float t16 = P[8][8]*t3*t9;
        float t17 = P[7][8]*t4*t9;
        float t18 = t15+t16+t17;
        float t19 = t3*t9*t18;
        float t20 = P[9][7]*t2*t9;
        float t21 = P[8][7]*t3*t9;
        float t22 = P[7][7]*t4*t9;
        float t23 = t20+t21+t22;
        float t24 = t4*t9*t23;
        varInnovRngBcn = R_BCN+t14+t19+t24;
        float t26;
        if (varInnovRngBcn >= R_BCN) {
            t26 = 1.0f/varInnovRngBcn;
            faultStatus.bad_rngbcn = false;
        } else {
            // the calculation is badly conditioned, so we cannot perform fusion on this step
            // we reset the covariance matrix and try again next measurement
            CovarianceInit();
            faultStatus.bad_rngbcn = true;
            return;
        }

        Kfusion[0] = -t26*(P[0][7]*t4*t9+P[0][8]*t3*t9+P[0][9]*t2*t9);
        Kfusion[1] = -t26*(P[1][7]*t4*t9+P[1][8]*t3*t9+P[1][9]*t2*t9);
        Kfusion[2] = -t26*(P[2][7]*t4*t9+P[2][8]*t3*t9+P[2][9]*t2*t9);
        Kfusion[3] = -t26*(P[3][7]*t4*t9+P[3][8]*t3*t9+P[3][9]*t2*t9);
        Kfusion[4] = -t26*(P[4][7]*t4*t9+P[4][8]*t3*t9+P[4][9]*t2*t9);
        Kfusion[5] = -t26*(P[5][7]*t4*t9+P[5][8]*t3*t9+P[5][9]*t2*t9);
        Kfusion[7] = -t26*(t22+P[7][8]*t3*t9+P[7][9]*t2*t9);
        Kfusion[8] = -t26*(t16+P[8][7]*t4*t9+P[8][9]*t2*t9);

        if (!inhibitDelAngBiasStates) {
            Kfusion[10] = -t26*(P[10][7]*t4*t9+P[10][8]*t3*t9+P[10][9]*t2*t9);
            Kfusion[11] = -t26*(P[11][7]*t4*t9+P[11][8]*t3*t9+P[11][9]*t2*t9);
            Kfusion[12] = -t26*(P[12][7]*t4*t9+P[12][8]*t3*t9+P[12][9]*t2*t9);
        } else {
            // zero indexes 10 to 12 = 3*4 bytes
            memset(&Kfusion[10], 0, 12);
        }

        if (!inhibitDelVelBiasStates) {
            Kfusion[13] = -t26*(P[13][7]*t4*t9+P[13][8]*t3*t9+P[13][9]*t2*t9);
            Kfusion[14] = -t26*(P[14][7]*t4*t9+P[14][8]*t3*t9+P[14][9]*t2*t9);
            Kfusion[15] = -t26*(P[15][7]*t4*t9+P[15][8]*t3*t9+P[15][9]*t2*t9);
        } else {
            // zero indexes 13 to 15 = 3*4 bytes
            memset(&Kfusion[13], 0, 12);
        }

        // only allow the range observations to modify the vertical states if we are using it as a height reference
        if (activeHgtSource == HGT_SOURCE_BCN) {
            Kfusion[6] = -t26*(P[6][7]*t4*t9+P[6][8]*t3*t9+P[6][9]*t2*t9);
            Kfusion[9] = -t26*(t10+P[9][7]*t4*t9+P[9][8]*t3*t9);
        } else {
            Kfusion[6] = 0.0f;
            Kfusion[9] = 0.0f;
        }

        if (!inhibitMagStates) {
            Kfusion[16] = -t26*(P[16][7]*t4*t9+P[16][8]*t3*t9+P[16][9]*t2*t9);
            Kfusion[17] = -t26*(P[17][7]*t4*t9+P[17][8]*t3*t9+P[17][9]*t2*t9);
            Kfusion[18] = -t26*(P[18][7]*t4*t9+P[18][8]*t3*t9+P[18][9]*t2*t9);
            Kfusion[19] = -t26*(P[19][7]*t4*t9+P[19][8]*t3*t9+P[19][9]*t2*t9);
            Kfusion[20] = -t26*(P[20][7]*t4*t9+P[20][8]*t3*t9+P[20][9]*t2*t9);
            Kfusion[21] = -t26*(P[21][7]*t4*t9+P[21][8]*t3*t9+P[21][9]*t2*t9);
        } else {
            // zero indexes 16 to 21 = 6*4 bytes
            memset(&Kfusion[16], 0, 24);
        }

        if (!inhibitWindStates) {
            Kfusion[22] = -t26*(P[22][7]*t4*t9+P[22][8]*t3*t9+P[22][9]*t2*t9);
            Kfusion[23] = -t26*(P[23][7]*t4*t9+P[23][8]*t3*t9+P[23][9]*t2*t9);
        } else {
            // zero indexes 22 to 23 = 2*4 bytes
            memset(&Kfusion[22], 0, 8);
        }

        // Calculate innovation using the selected offset value
        Vector3f delta = stateStruct.position - rngBcnDataDelayed.beacon_posNED;
        innovRngBcn = delta.length() - rngBcnDataDelayed.rng;

        // calculate the innovation consistency test ratio
        rngBcnTestRatio = sq(innovRngBcn) / (sq(MAX(0.01f * (float)frontend->_rngBcnInnovGate, 1.0f)) * varInnovRngBcn);

        // fail if the ratio is > 1, but don't fail if bad IMU data
        rngBcnHealth = ((rngBcnTestRatio < 1.0f) || badIMUdata);

        // test the ratio before fusing data
        if (rngBcnHealth) {

            // restart the counter
            lastRngBcnPassTime_ms = imuSampleTime_ms;

            // correct the covariance P = (I - K*H)*P
            // take advantage of the empty columns in KH to reduce the
            // number of operations
            for (unsigned i = 0; i<=stateIndexLim; i++) {
                for (unsigned j = 0; j<=6; j++) {
                    KH[i][j] = 0.0f;
                }
                for (unsigned j = 7; j<=9; j++) {
                    KH[i][j] = Kfusion[i] * H_BCN[j];
                }
                for (unsigned j = 10; j<=23; j++) {
                    KH[i][j] = 0.0f;
                }
            }
            for (unsigned j = 0; j<=stateIndexLim; j++) {
                for (unsigned i = 0; i<=stateIndexLim; i++) {
                    ftype res = 0;
                    res += KH[i][7] * P[7][j];
                    res += KH[i][8] * P[8][j];
                    res += KH[i][9] * P[9][j];
                    KHP[i][j] = res;
                }
            }
            // Check that we are not going to drive any variances negative and skip the update if so
            bool healthyFusion = true;
            for (uint8_t i= 0; i<=stateIndexLim; i++) {
                if (KHP[i][i] > P[i][i]) {
                    healthyFusion = false;
                }
            }
            if (healthyFusion) {
                // update the covariance matrix
                for (uint8_t i= 0; i<=stateIndexLim; i++) {
                    for (uint8_t j= 0; j<=stateIndexLim; j++) {
                        P[i][j] = P[i][j] - KHP[i][j];
                    }
                }

                // force the covariance matrix to be symmetrical and limit the variances to prevent ill-conditioning.
                ForceSymmetry();
                ConstrainVariances();

                // correct the state vector
                for (uint8_t j= 0; j<=stateIndexLim; j++) {
                    statesArray[j] = statesArray[j] - Kfusion[j] * innovRngBcn;
                }

                // record healthy fusion
                faultStatus.bad_rngbcn = false;

            } else {
                // record bad fusion
                faultStatus.bad_rngbcn = true;

            }
        }

        // Update the fusion report
        rngBcnFusionReport[rngBcnDataDelayed.beacon_ID].beaconPosNED = rngBcnDataDelayed.beacon_posNED;
        rngBcnFusionReport[rngBcnDataDelayed.beacon_ID].innov = innovRngBcn;
        rngBcnFusionReport[rngBcnDataDelayed.beacon_ID].innovVar = varInnovRngBcn;
        rngBcnFusionReport[rngBcnDataDelayed.beacon_ID].rng = rngBcnDataDelayed.rng;
        rngBcnFusionReport[rngBcnDataDelayed.beacon_ID].testRatio = rngBcnTestRatio;
    }
}

/*
Use range beacon measurements to calculate a static position using a 3-state EKF algorithm.
Algorithm based on the following:
https://github.com/priseborough/InertialNav/blob/master/derivations/range_beacon.m
*/
void NavEKF3_core::FuseRngBcnStatic()
{
    // get the estimated range measurement variance
    const float R_RNG = sq(MAX(rngBcnDataDelayed.rngErr , 0.1f));

    /*
    The first thing to do is to check if we have started the alignment and if not, initialise the
    states and covariance to a first guess. To do this iterate through the available beacons and then
    initialise the initial position to the mean beacon position. The initial position uncertainty
    is set to the mean range measurement.
    */
    if (!rngBcnAlignmentStarted) {
        if (rngBcnDataDelayed.beacon_ID != lastBeaconIndex) {
            rngBcnPosSum += rngBcnDataDelayed.beacon_posNED;
            lastBeaconIndex = rngBcnDataDelayed.beacon_ID;
            rngSum += rngBcnDataDelayed.rng;
            numBcnMeas++;

            // capture the beacon vertical spread
            if (rngBcnDataDelayed.beacon_posNED.z > maxBcnPosD) {
                maxBcnPosD = rngBcnDataDelayed.beacon_posNED.z;
            } else if(rngBcnDataDelayed.beacon_posNED.z < minBcnPosD) {
                minBcnPosD = rngBcnDataDelayed.beacon_posNED.z;
            }
        }
        if (numBcnMeas >= 100) {
            rngBcnAlignmentStarted = true;
            float tempVar = 1.0f / (float)numBcnMeas;
            // initialise the receiver position to the centre of the beacons and at zero height
            receiverPos.x = rngBcnPosSum.x * tempVar;
            receiverPos.y = rngBcnPosSum.y * tempVar;
            receiverPos.z = 0.0f;
            receiverPosCov[2][2] = receiverPosCov[1][1] = receiverPosCov[0][0] = rngSum * tempVar;
            lastBeaconIndex  = 0;
            numBcnMeas = 0;
            rngBcnPosSum.zero();
            rngSum = 0.0f;
        }
    }

    if (rngBcnAlignmentStarted) {
        numBcnMeas++;

        if (numBcnMeas >= 100) {
            // 100 observations is enough for a stable estimate under most conditions
            // TODO monitor stability of the position estimate
            rngBcnAlignmentCompleted = true;

        }

        if (rngBcnAlignmentCompleted) {
            if (activeHgtSource != HGT_SOURCE_BCN) {
                // We are using a different height reference for the main EKF so need to estimate a vertical
                // position offset to be applied to the beacon system that minimises the range innovations
                // The position estimate should be stable after 100 iterations so we use a simple dual
                // hypothesis 1-state EKF to estimate the offset
                Vector3f refPosNED;
                refPosNED.x = receiverPos.x;
                refPosNED.y = receiverPos.y;
                refPosNED.z = stateStruct.position.z;
                CalcRangeBeaconPosDownOffset(R_RNG, refPosNED, true);

            } else {
                // we are using the beacons as the primary height source, so don't modify their vertical position
                bcnPosOffsetNED.z = 0.0f;

            }
        } else {
            if (activeHgtSource != HGT_SOURCE_BCN) {
                // The position estimate is not yet stable so we cannot run the 1-state EKF to estimate
                // beacon system vertical position offset. Instead we initialise the dual hypothesis offset states
                // using the beacon vertical position, vertical position estimate relative to the beacon origin
                // and the main EKF vertical position

                // Calculate the mid vertical position of all beacons
                float bcnMidPosD = 0.5f * (minBcnPosD + maxBcnPosD);

                // calculate the delta to the estimated receiver position
                float delta = receiverPos.z - bcnMidPosD;

                // calcuate the two hypothesis for our vertical position
                float receverPosDownMax;
                float receverPosDownMin;
                if (delta >= 0.0f) {
                    receverPosDownMax = receiverPos.z;
                    receverPosDownMin = receiverPos.z - 2.0f * delta;
                } else {
                    receverPosDownMax = receiverPos.z - 2.0f * delta;
                    receverPosDownMin = receiverPos.z;
                }

                bcnPosDownOffsetMax = stateStruct.position.z - receverPosDownMin;
                bcnPosDownOffsetMin = stateStruct.position.z - receverPosDownMax;
            } else {
                // We are using the beacons as the primary height reference, so don't modify their vertical position
                bcnPosOffsetNED.z = 0.0f;
            }
        }

        // Add some process noise to the states at each time step
        for (uint8_t i= 0; i<=2; i++) {
            receiverPosCov[i][i] += 0.1f;
        }

        // calculate the observation jacobian
        float t2 = rngBcnDataDelayed.beacon_posNED.z - receiverPos.z + bcnPosOffsetNED.z;
        float t3 = rngBcnDataDelayed.beacon_posNED.y - receiverPos.y;
        float t4 = rngBcnDataDelayed.beacon_posNED.x - receiverPos.x;
        float t5 = t2*t2;
        float t6 = t3*t3;
        float t7 = t4*t4;
        float t8 = t5+t6+t7;
        if (t8 < 0.1f) {
            // calculation will be badly conditioned
            return;
        }
        float t9 = 1.0f/sqrtf(t8);
        float t10 = rngBcnDataDelayed.beacon_posNED.x*2.0f;
        float t15 = receiverPos.x*2.0f;
        float t11 = t10-t15;
        float t12 = rngBcnDataDelayed.beacon_posNED.y*2.0f;
        float t14 = receiverPos.y*2.0f;
        float t13 = t12-t14;
        float t16 = rngBcnDataDelayed.beacon_posNED.z*2.0f;
        float t18 = receiverPos.z*2.0f;
        float t17 = t16-t18;
        float H_RNG[3];
        H_RNG[0] = -t9*t11*0.5f;
        H_RNG[1] = -t9*t13*0.5f;
        H_RNG[2] = -t9*t17*0.5f;

        // calculate the Kalman gains
        float t19 = receiverPosCov[0][0]*t9*t11*0.5f;
        float t20 = receiverPosCov[1][1]*t9*t13*0.5f;
        float t21 = receiverPosCov[0][1]*t9*t11*0.5f;
        float t22 = receiverPosCov[2][1]*t9*t17*0.5f;
        float t23 = t20+t21+t22;
        float t24 = t9*t13*t23*0.5f;
        float t25 = receiverPosCov[1][2]*t9*t13*0.5f;
        float t26 = receiverPosCov[0][2]*t9*t11*0.5f;
        float t27 = receiverPosCov[2][2]*t9*t17*0.5f;
        float t28 = t25+t26+t27;
        float t29 = t9*t17*t28*0.5f;
        float t30 = receiverPosCov[1][0]*t9*t13*0.5f;
        float t31 = receiverPosCov[2][0]*t9*t17*0.5f;
        float t32 = t19+t30+t31;
        float t33 = t9*t11*t32*0.5f;
        varInnovRngBcn = R_RNG+t24+t29+t33;
        float t35 = 1.0f/varInnovRngBcn;
        float K_RNG[3];
        K_RNG[0] = -t35*(t19+receiverPosCov[0][1]*t9*t13*0.5f+receiverPosCov[0][2]*t9*t17*0.5f);
        K_RNG[1] = -t35*(t20+receiverPosCov[1][0]*t9*t11*0.5f+receiverPosCov[1][2]*t9*t17*0.5f);
        K_RNG[2] = -t35*(t27+receiverPosCov[2][0]*t9*t11*0.5f+receiverPosCov[2][1]*t9*t13*0.5f);

        // calculate range measurement innovation
        Vector3f deltaPosNED = receiverPos - rngBcnDataDelayed.beacon_posNED;
        deltaPosNED.z -= bcnPosOffsetNED.z;
        innovRngBcn = deltaPosNED.length() - rngBcnDataDelayed.rng;

        // calculate the innovation consistency test ratio
        rngBcnTestRatio = sq(innovRngBcn) / (sq(MAX(0.01f * (float)frontend->_rngBcnInnovGate, 1.0f)) * varInnovRngBcn);

        // fail if the ratio is > 1, but don't fail if bad IMU data
        rngBcnHealth = ((rngBcnTestRatio < 1.0f) || badIMUdata || !rngBcnAlignmentCompleted);

        // test the ratio before fusing data
        if (rngBcnHealth) {

            // update the state
            receiverPos.x -= K_RNG[0] * innovRngBcn;
            receiverPos.y -= K_RNG[1] * innovRngBcn;
            receiverPos.z -= K_RNG[2] * innovRngBcn;

            // calculate the covariance correction
            for (unsigned i = 0; i<=2; i++) {
                for (unsigned j = 0; j<=2; j++) {
                    KH[i][j] = K_RNG[i] * H_RNG[j];
                }
            }
            for (unsigned j = 0; j<=2; j++) {
                for (unsigned i = 0; i<=2; i++) {
                    ftype res = 0;
                    res += KH[i][0] * receiverPosCov[0][j];
                    res += KH[i][1] * receiverPosCov[1][j];
                    res += KH[i][2] * receiverPosCov[2][j];
                    KHP[i][j] = res;
                }
            }

            // prevent negative variances
            for (uint8_t i= 0; i<=2; i++) {
                if (receiverPosCov[i][i] < 0.0f) {
                    receiverPosCov[i][i] = 0.0f;
                    KHP[i][i] = 0.0f;
                } else if (KHP[i][i] > receiverPosCov[i][i]) {
                    KHP[i][i] = receiverPosCov[i][i];
                }
            }

            // apply the covariance correction
            for (uint8_t i= 0; i<=2; i++) {
                for (uint8_t j= 0; j<=2; j++) {
                    receiverPosCov[i][j] -= KHP[i][j];
                }
            }

            // ensure the covariance matrix is symmetric
            for (uint8_t i=1; i<=2; i++) {
                for (uint8_t j=0; j<=i-1; j++) {
                    float temp = 0.5f*(receiverPosCov[i][j] + receiverPosCov[j][i]);
                    receiverPosCov[i][j] = temp;
                    receiverPosCov[j][i] = temp;
                }
            }

        }

        if (numBcnMeas >= 100) {
            // 100 observations is enough for a stable estimate under most conditions
            // TODO monitor stability of the position estimate
            rngBcnAlignmentCompleted = true;
        }
        // Update the fusion report
        rngBcnFusionReport[rngBcnDataDelayed.beacon_ID].beaconPosNED = rngBcnDataDelayed.beacon_posNED;
        rngBcnFusionReport[rngBcnDataDelayed.beacon_ID].innov = innovRngBcn;
        rngBcnFusionReport[rngBcnDataDelayed.beacon_ID].innovVar = varInnovRngBcn;
        rngBcnFusionReport[rngBcnDataDelayed.beacon_ID].rng = rngBcnDataDelayed.rng;
        rngBcnFusionReport[rngBcnDataDelayed.beacon_ID].testRatio = rngBcnTestRatio;
    }
}

/*
Run a single state Kalman filter to estimate the vertical position offset of the range beacon constellation
Calculate using a high and low hypothesis and select the hypothesis with the lowest innovation sequence
*/
void NavEKF3_core::CalcRangeBeaconPosDownOffset(float obsVar, Vector3f &vehiclePosNED, bool aligning)
{
    // Handle height offsets between the primary height source and the range beacons by estimating
    // the beacon systems global vertical position offset using a single state Kalman filter
    // The estimated offset is used to correct the beacon height when calculating innovations
    // A high and low estimate is calculated to handle the ambiguity in height associated with beacon positions that are co-planar
    // The main filter then uses the offset with the smaller innovations

    float innov;    // range measurement innovation (m)
    float innovVar; // range measurement innovation variance (m^2)
    float gain;     // Kalman gain
    float obsDeriv; // derivative of observation relative to state

    const float stateNoiseVar = 0.1f; // State process noise variance
    const float filtAlpha = 0.1f; // LPF constant
    const float innovGateWidth = 5.0f; // width of innovation consistency check gate in std

    // estimate upper value for offset

    // calculate observation derivative
    float t2 = rngBcnDataDelayed.beacon_posNED.z - vehiclePosNED.z + bcnPosDownOffsetMax;
    float t3 = rngBcnDataDelayed.beacon_posNED.y - vehiclePosNED.y;
    float t4 = rngBcnDataDelayed.beacon_posNED.x - vehiclePosNED.x;
    float t5 = t2*t2;
    float t6 = t3*t3;
    float t7 = t4*t4;
    float t8 = t5+t6+t7;
    float t9;
    if (t8 > 0.1f) {
        t9 = 1.0f/sqrtf(t8);
        obsDeriv = t2*t9;

        // Calculate innovation
        innov = sqrtf(t8) - rngBcnDataDelayed.rng;

        // covariance prediction
        bcnPosOffsetMaxVar += stateNoiseVar;

        // calculate the innovation variance
        innovVar = obsDeriv * bcnPosOffsetMaxVar * obsDeriv + obsVar;
        innovVar = MAX(innovVar, obsVar);

        // calculate the Kalman gain
        gain = (bcnPosOffsetMaxVar * obsDeriv) / innovVar;

        // calculate a filtered state change magnitude to be used to select between the high or low offset
        float stateChange = innov * gain;
        maxOffsetStateChangeFilt = (1.0f - filtAlpha) * maxOffsetStateChangeFilt + fminf(fabsf(filtAlpha * stateChange) , 1.0f);

        // Reject range innovation spikes using a 5-sigma threshold unless aligning
        if ((sq(innov) < sq(innovGateWidth) * innovVar) || aligning) {

            // state update
            bcnPosDownOffsetMax -= stateChange;

            // covariance update
            bcnPosOffsetMaxVar -= gain * obsDeriv * bcnPosOffsetMaxVar;
            bcnPosOffsetMaxVar = MAX(bcnPosOffsetMaxVar, 0.0f);
        }
    }

    // estimate lower value for offset

    // calculate observation derivative
    t2 = rngBcnDataDelayed.beacon_posNED.z - vehiclePosNED.z + bcnPosDownOffsetMin;
    t5 = t2*t2;
    t8 = t5+t6+t7;
    if (t8 > 0.1f) {
        t9 = 1.0f/sqrtf(t8);
        obsDeriv = t2*t9;

        // Calculate innovation
        innov = sqrtf(t8) - rngBcnDataDelayed.rng;

        // covariance prediction
        bcnPosOffsetMinVar += stateNoiseVar;

        // calculate the innovation variance
        innovVar = obsDeriv * bcnPosOffsetMinVar * obsDeriv + obsVar;
        innovVar = MAX(innovVar, obsVar);

        // calculate the Kalman gain
        gain = (bcnPosOffsetMinVar * obsDeriv) / innovVar;

        // calculate a filtered state change magnitude to be used to select between the high or low offset
        float stateChange = innov * gain;
        minOffsetStateChangeFilt = (1.0f - filtAlpha) * minOffsetStateChangeFilt + fminf(fabsf(filtAlpha * stateChange) , 1.0f);

        // Reject range innovation spikes using a 5-sigma threshold unless aligning
        if ((sq(innov) < sq(innovGateWidth) * innovVar) || aligning) {

            // state update
            bcnPosDownOffsetMin -= stateChange;

            // covariance update
            bcnPosOffsetMinVar -= gain * obsDeriv * bcnPosOffsetMinVar;
            bcnPosOffsetMinVar = MAX(bcnPosOffsetMinVar, 0.0f);
        }
    }

    // calculate the mid vertical position of all beacons
    float bcnMidPosD = 0.5f * (minBcnPosD + maxBcnPosD);

    // ensure the two beacon vertical offset hypothesis place the mid point of the beacons below and above the flight vehicle
    bcnPosDownOffsetMax = MAX(bcnPosDownOffsetMax, vehiclePosNED.z - bcnMidPosD + 0.5f);
    bcnPosDownOffsetMin  = MIN(bcnPosDownOffsetMin,  vehiclePosNED.z - bcnMidPosD - 0.5f);

    // calculate the innovation for the main filter using the offset that is most stable
    // apply hysteresis to prevent rapid switching
    if (!usingMinHypothesis && (minOffsetStateChangeFilt < (0.8f * maxOffsetStateChangeFilt))) {
        usingMinHypothesis = true;
    } else if (usingMinHypothesis && (maxOffsetStateChangeFilt < (0.8f * minOffsetStateChangeFilt))) {
        usingMinHypothesis = false;
    }
    if (usingMinHypothesis) {
        bcnPosOffsetNED.z = bcnPosDownOffsetMin;
    } else {
        bcnPosOffsetNED.z = bcnPosDownOffsetMax;
    }

    // apply the vertical offset to the beacon positions
    rngBcnDataDelayed.beacon_posNED.z += bcnPosOffsetNED.z;
}