LibrePilot/flight/modules/AutoTune/autotune.c

/**
 ******************************************************************************
 * @addtogroup OpenPilotModules OpenPilot Modules
 * @{
 * @addtogroup StabilizationModule Stabilization Module
 * @brief Stabilization PID loops in an airframe type independent manner
 * @note This object updates the @ref ActuatorDesired "Actuator Desired" based on the
 * PID loops on the @ref AttitudeDesired "Attitude Desired" and @ref AttitudeState "Attitude State"
 * @{
 *
 * @file       AutoTune/autotune.c
 * @author     The LibrePilot Project, http://www.librepilot.org Copyright (C) 2016.
 *             dRonin, http://dRonin.org/, Copyright (C) 2015-2016
 *             Tau Labs, http://taulabs.org, Copyright (C) 2013-2014
 *             The OpenPilot Team, http://www.openpilot.org Copyright (C) 2012.
 * @brief      Automatic PID tuning module.
 *
 * @see        The GNU Public License (GPL) Version 3
 *
 *****************************************************************************/
/*
 * This program is free software; you can redistribute it and/or modify
 * it under the terms of the GNU General Public License as published by
 * the Free Software Foundation; either version 3 of the License, or
 * (at your option) any later version.
 *
 * This program is distributed in the hope that it will be useful, but
 * WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY
 * or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License
 * for more details.
 *
 * You should have received a copy of the GNU General Public License along
 * with this program; if not, write to the Free Software Foundation, Inc.,
 * 59 Temple Place, Suite 330, Boston, MA 02111-1307 USA
 */

#include "openpilot.h"
#include "pios.h"
#include "flightstatus.h"
#include "manualcontrolcommand.h"
#include "manualcontrolsettings.h"
#include "gyrosensor.h"
#include "actuatordesired.h"
#include "stabilizationdesired.h"
#include "stabilizationsettings.h"
#include "systemidentsettings.h"
#include "systemidentstate.h"
#include <pios_board_info.h>
#include "systemsettings.h"
#include "taskinfo.h"
#include "stabilization.h"
#include "hwsettings.h"
#include "stabilizationsettingsbank1.h"
#include "stabilizationsettingsbank2.h"
#include "stabilizationsettingsbank3.h"
#include "accessorydesired.h"

#if defined(PIOS_EXCLUDE_ADVANCED_FEATURES)
#define powapprox fastpow
#define expapprox fastexp
#else
#define powapprox powf
#define expapprox expf
#endif /* defined(PIOS_EXCLUDE_ADVANCED_FEATURES) */


// Private constants
#undef  STACK_SIZE_BYTES
// Nano locks up it seems in UAVObjSav() with 1340
// why did it lock up? 1540 now works (after a long initial delay) with 360 bytes left
#define STACK_SIZE_BYTES 1340
//#define TASK_PRIORITY PIOS_THREAD_PRIO_NORMAL
#define TASK_PRIORITY              (tskIDLE_PRIORITY + 1)

#define AF_NUMX 13
#define AF_NUMP 43

#if !defined(AT_QUEUE_NUMELEM)
#define AT_QUEUE_NUMELEM 18
#endif

#define MAX_PTS_PER_CYCLE            4 /* max gyro updates to process per loop see YIELD_MS and consider gyro rate */
#define INIT_TIME_DELAY_MS         100 /* delay to allow stab bank, etc. to be populated after flight mode switch change detection */
#define SYSTEMIDENT_TIME_DELAY_MS 2000 /* delay before starting systemident (shaking) flight mode */
#define INIT_TIME_DELAY2_MS       2500 /* delay before starting to capture data */
#define YIELD_MS                     2 /* delay this long between processing sessions see MAX_PTS_PER_CYCLE and consider gyro rate */

#define ROLL_BETA_LOW   1
#define PITCH_BETA_LOW  2
#define YAW_BETA_LOW    4
#define TAU_TOO_LONG    8
#define TAU_TOO_SHORT  16

#define SMOOTH_QUICK_DISABLED        0
#define SMOOTH_QUICK_ACCESSORY_BASE 10
#define SMOOTH_QUICK_TOGGLE_BASE    21


// Private types		<access gcs="readwrite" flight="readwrite"/>
enum AUTOTUNE_STATE { AT_INIT, AT_INIT_DELAY, AT_INIT_DELAY2, AT_START, AT_RUN, AT_FINISHED, AT_WAITING };

struct at_queued_data {
    float y[3];        /* Gyro measurements */
    float u[3];        /* Actuator desired */
    float throttle;    /* Throttle desired */

    uint32_t raw_time;    /* From PIOS_DELAY_GetRaw() */
};


// Private variables
static xTaskHandle taskHandle;
static bool moduleEnabled;
static xQueueHandle atQueue;
static volatile uint32_t atPointsSpilled;
static uint32_t throttleAccumulator;
static uint8_t rollMax, pitchMax;
static StabilizationBankManualRateData manualRate;
static float gX[AF_NUMX] = {0};
static float gP[AF_NUMP] = {0};
SystemIdentSettingsData systemIdentSettings;
SystemIdentStateData systemIdentState;
int8_t accessoryToUse;
int8_t flightModeSwitchTogglePosition;


// Private functions
static void AutoTuneTask(void *parameters);
static void AfPredict(float X[AF_NUMX], float P[AF_NUMP], const float u_in[3], const float gyro[3], const float dT_s, const float t_in);
static void AfInit(float X[AF_NUMX], float P[AF_NUMP]);
static uint8_t CheckSettingsRaw();
static uint8_t CheckSettings();
static void ComputeStabilizationAndSetPidsFromDampAndNoise(float damp, float noise);
static void ComputeStabilizationAndSetPids();
static void ProportionPidsSmoothToQuick(float min, float val, float max);
static void AtNewGyroData(UAVObjEvent * ev);
static void UpdateSystemIdentState(const float *X, const float *noise, float dT_s, uint32_t predicts, uint32_t spills, float hover_throttle);
static void UpdateStabilizationDesired(bool doingIdent);
static bool CheckFlightModeSwitchForPidRequest(uint8_t flightMode);
static void InitSystemIdent(bool loadDefaults);


/**
 * Initialise the module, called on startup
 * \returns 0 on success or -1 if initialisation failed
 */
int32_t AutoTuneInitialize(void)
{
    // Create a queue, connect to manual control command and flightstatus
#ifdef MODULE_AutoTune_BUILTIN
    moduleEnabled = true;
#else
    HwSettingsOptionalModulesData optionalModules;
    HwSettingsOptionalModulesGet(&optionalModules);
    if (optionalModules.AutoTune == HWSETTINGS_OPTIONALMODULES_ENABLED) {
        moduleEnabled = true;
    } else {
        moduleEnabled = false;
    }
#endif

    if (moduleEnabled) {
        SystemIdentSettingsInitialize();
        SystemIdentStateInitialize();
        atQueue = xQueueCreate(AT_QUEUE_NUMELEM, sizeof(struct at_queued_data));
        if (!atQueue) {
            moduleEnabled = false;
        }
    }

    return 0;
}


/**
 * Initialise the module, called on startup
 * \returns 0 on success or -1 if initialisation failed
 */
int32_t AutoTuneStart(void)
{
    // Start main task if it is enabled
    if (moduleEnabled) {
        //taskHandle = PIOS_Thread_Create(AutoTuneTask, "Autotune", STACK_SIZE_BYTES, NULL, TASK_PRIORITY);
        //TaskMonitorAdd(TASKINFO_RUNNING_AUTOTUNE, taskHandle);
        //PIOS_WDG_RegisterFlag(PIOS_WDG_AUTOTUNE);
        GyroSensorConnectCallback(AtNewGyroData);
        xTaskCreate(AutoTuneTask, "AutoTune", STACK_SIZE_BYTES / 4, NULL, TASK_PRIORITY, &taskHandle);
        PIOS_TASK_MONITOR_RegisterTask(TASKINFO_RUNNING_AUTOTUNE, taskHandle);
    }
    return 0;
}


MODULE_INITCALL(AutoTuneInitialize, AutoTuneStart);


/**
 * Module thread, should not return.
 */
static void AutoTuneTask(__attribute__((unused)) void *parameters)
{
    enum AUTOTUNE_STATE state = AT_INIT;
    uint32_t lastUpdateTime = 0; // initialization is only for compiler warning
    float noise[3] = {0};
    uint32_t lastTime = 0.0f;
    bool saveSiNeeded = false;
    bool savePidNeeded = false;

    // get max attitude / max rate
    // for use in generating Attitude mode commands from this module
    // note that the values could change when they change flight mode (and the associated bank)
    StabilizationBankRollMaxGet(&rollMax);
    StabilizationBankPitchMaxGet(&pitchMax);
    StabilizationBankManualRateGet(&manualRate);
    // correctly set accessoryToUse and flightModeSwitchTogglePosition
    // based on what is in SystemIdent
    // so that the user can use the PID smooth->quick slider in following flights
    InitSystemIdent(false);

    while (1) {
        static uint32_t updateCounter = 0;
        uint32_t diffTime;
        uint32_t measureTime = 60000;
        bool doingIdent = false;
        bool canSleep = true;
        FlightStatusData flightStatus;
        FlightStatusGet(&flightStatus);

        // I have never seen this module misbehave so not bothering making a watchdog
        //PIOS_WDG_UpdateFlag(PIOS_WDG_AUTOTUNE);

        if (flightStatus.Armed == FLIGHTSTATUS_ARMED_DISARMED) {
            if (saveSiNeeded) {
                saveSiNeeded = false;
                // Save SystemIdentSettings to permanent settings
                UAVObjSave(SystemIdentSettingsHandle(), 0);
            }
            if (savePidNeeded) {
                savePidNeeded = false;
                // Save PIDs to permanent settings
                switch (systemIdentSettings.DestinationPidBank) {
                case 1:
                    UAVObjSave(StabilizationSettingsBank1Handle(), 0);
                    break;
                case 2:
                    UAVObjSave(StabilizationSettingsBank2Handle(), 0);
                    break;
                case 3:
                    UAVObjSave(StabilizationSettingsBank3Handle(), 0);
                    break;
                }
            }
        }

        // if using flight mode switch quick toggle to "try smooth -> quick PIDs" is enabled
        // and user toggled into and back out of AutoTune
        // three times in the last two seconds
        // and the data gathering is complete
        // and the data gathered is good
        // note: CheckFlightModeSwitchForPidRequest(mode) only returns true if mode is not autotune
        if (flightModeSwitchTogglePosition!=-1 && CheckFlightModeSwitchForPidRequest(flightStatus.FlightMode)
            && systemIdentSettings.Complete && !CheckSettings()) {
            if (flightStatus.Armed == FLIGHTSTATUS_ARMED_ARMED) {
                // if user toggled while armed set PID's to next in sequence 2,3,4,0,1... or 1,2,0...
                // if smoothest is -100 and quickest is +100 this corresponds to 0,+50,+100,-100,-50... or 0,+100,-100
                ++flightModeSwitchTogglePosition;
                if (flightModeSwitchTogglePosition > systemIdentSettings.SmoothQuick - SMOOTH_QUICK_TOGGLE_BASE) {
                    flightModeSwitchTogglePosition = 0;
                }
            } else {
                // if they did it disarmed, then set PID's back to AutoTune default
                flightModeSwitchTogglePosition = (systemIdentSettings.SmoothQuick - SMOOTH_QUICK_TOGGLE_BASE) / 2;
            }
            ProportionPidsSmoothToQuick(0.0f,
                                        (float) flightModeSwitchTogglePosition,
                                        (float) (systemIdentSettings.SmoothQuick - SMOOTH_QUICK_TOGGLE_BASE));
            savePidNeeded = true;
        }

        // any time we are not in AutoTune mode:
        // - the user may be using the accessory0-3 knob/slider to request PID changes
        // - the state machine needs to be reset
        // - the local version of Attitude mode gets skipped
        if (flightStatus.FlightMode != FLIGHTSTATUS_FLIGHTMODE_AUTOTUNE) {
            // if accessory0-3 is configured as a PID changing slider/knob over the smooth to quick range
            // and FC is not currently running autotune
            // and accessory0-3 changed by at least 1/900 of full range
            // (don't bother checking to see if the requested accessory# is configured properly
            // if it isn't, the value will be 0 which is the center of [-1,1] anyway)
            if (accessoryToUse != -1 && systemIdentSettings.Complete && !CheckSettings()) {
                static AccessoryDesiredData accessoryValueOld = { 0.0f };
                AccessoryDesiredData accessoryValue;
                AccessoryDesiredInstGet(accessoryToUse, &accessoryValue);
                // if the accessory changed more than 1/900
                // (this test is intended to remove one unit jitter)
                if (fabsf(accessoryValueOld.AccessoryVal - accessoryValue.AccessoryVal) > (1.0f/900.0f)) {
                    accessoryValueOld = accessoryValue;
                    ProportionPidsSmoothToQuick(-1.0f, accessoryValue.AccessoryVal, 1.0f);
                    savePidNeeded = true;
                }
            }
            state = AT_INIT;
            vTaskDelay(50 / portTICK_RATE_MS);
            continue;
        }

        switch(state) {
        case AT_INIT:
            // beware that control comes here every time the user toggles the flight mode switch into AutoTune
            // and it isn't appropriate to reset the main state here
            // init must wait until after a delay has passed:
            // - to make sure they intended to stay in this mode
            // - to wait for the stab bank to get populated with the new bank info
            // This is a race.  It is possible that flightStatus.FlightMode has been changed,
            // but the stab bank hasn't been changed yet.
            state = AT_INIT_DELAY;
            lastUpdateTime = xTaskGetTickCount();
            break;

        case AT_INIT_DELAY:
            diffTime = xTaskGetTickCount() - lastUpdateTime;
            // after a small delay, get the stab bank values and SystemIdentSettings in case they changed
            // this is a very small delay, so fms toggle gets in here
            if (diffTime > INIT_TIME_DELAY_MS) {
                // do these here so the user has at most a 1/10th second
                // with controls that use the previous bank's rates
                StabilizationBankRollMaxGet(&rollMax);
                StabilizationBankPitchMaxGet(&pitchMax);
                StabilizationBankManualRateGet(&manualRate);
                // load SystemIdentSettings so that they can change it
                // and do smooth-quick on changed values
                InitSystemIdent(false);
                state = AT_INIT_DELAY2;
                lastUpdateTime = xTaskGetTickCount();
            }
            break;

        case AT_INIT_DELAY2:
            // delay for 2 seconds before actually starting the SystemIdent flight mode and AutoTune.
            // that allows the user to get his fingers on the sticks
            // and avoids starting the AutoTune if the user is toggling the flight mode switch
            // to select other PIDs on the "simulated Smooth Quick slider".
            diffTime = xTaskGetTickCount() - lastUpdateTime;
            // after 2 seconds start systemident flight mode
            if (diffTime > SYSTEMIDENT_TIME_DELAY_MS) {
                doingIdent = true;
                // after an additional .5 seconds start capturing data
                if (diffTime > INIT_TIME_DELAY2_MS) {
                    // Only start when armed and flying
                    if (flightStatus.Armed == FLIGHTSTATUS_ARMED_ARMED) {
                        // Reset save status
                        // save SI data even if partial or bad, aids in diagnostics
                        saveSiNeeded = true;
                        // don't save PIDs until data gathering is complete
                        // and the complete data has been sanity checked
                        savePidNeeded = false;
                        InitSystemIdent(true);
                        AfInit(gX, gP);
                        UpdateSystemIdentState(gX, NULL, 0.0f, 0, 0, 0.0f);
                        measureTime = (uint32_t)systemIdentSettings.TuningDuration * (uint32_t)1000;
                        state = AT_START;
                    }
                }
            }
            break;

        case AT_START:
            lastTime = PIOS_DELAY_GetRaw();
            doingIdent = true;
            /* Drain the queue of all current data */
            xQueueReset(atQueue);
            /* And reset the point spill counter */
            updateCounter = 0;
            atPointsSpilled = 0;
            throttleAccumulator = 0;
            state = AT_RUN;
            lastUpdateTime = xTaskGetTickCount();
            break;

        case AT_RUN:
            diffTime = xTaskGetTickCount() - lastUpdateTime;
            doingIdent = true;
            canSleep = false;
            // 4 gyro samples per cycle
            // 2ms cycle time
            // that is 500 gyro samples per second if it sleeps each time
            // actually less than 500 because it cycle time is processing time + 2ms
            for (int i=0; i<MAX_PTS_PER_CYCLE; i++) {
                struct at_queued_data pt;
                /* Grab an autotune point */
                if (xQueueReceive(atQueue, &pt, 0) != pdTRUE) {
                    /* We've drained the buffer fully */
                    canSleep = true;
                    break;
                }
                /* calculate time between successive points */
                float dT_s = PIOS_DELAY_DiffuS2(lastTime, pt.raw_time) * 1.0e-6f;
                /* This is for the first point, but
                 * also if we have extended drops */
                if (dT_s > 0.010f) {
                    dT_s = 0.010f;
                }
                lastTime = pt.raw_time;
                AfPredict(gX, gP, pt.u, pt.y, dT_s, pt.throttle);
                for (int j=0; j<3; ++j) {
                    const float NOISE_ALPHA = 0.9997f;  // 10 second time constant at 300 Hz
                    noise[j] = NOISE_ALPHA * noise[j] + (1-NOISE_ALPHA) * (pt.y[j] - gX[j]) * (pt.y[j] - gX[j]);
                }
                // This will work up to 8kHz with an 89% throttle position before overflow
                throttleAccumulator += 10000 * pt.throttle;
                // Update uavo every 256 cycles to avoid
                // telemetry spam
                if (((updateCounter++) & 0xff) == 0) {
                    float hover_throttle = ((float)(throttleAccumulator/updateCounter))/10000.0f;
                    UpdateSystemIdentState(gX, noise, dT_s, updateCounter, atPointsSpilled, hover_throttle);
                }
            }
            if (diffTime > measureTime) { // Move on to next state
                // permanent flag that AT is complete and PIDs can be calculated
                state = AT_FINISHED;
            }
            break;

        case AT_FINISHED:
            // data is automatically considered bad if FC was disarmed at the time AT completed
            if (flightStatus.Armed == FLIGHTSTATUS_ARMED_ARMED) {
                // always calculate and save PIDs if disabling sanity checks
                if (!CheckSettings()) {
                    ComputeStabilizationAndSetPids();
                    savePidNeeded = true;
                    // mark these results as good in the permanent settings so they can be used next flight too
                    systemIdentSettings.Complete = true;
                    // mark these results as good in the log settings so they can be viewed in playback
                    systemIdentState.Complete = true;
                }
                // always raise an alarm if sanity checks failed
                // even if disabling sanity checks
                // that way user can still see that they failed
                uint8_t failureBits = CheckSettingsRaw();
                if (failureBits) {
                    // raise a warning that includes failureBits to indicate what failed
                    ExtendedAlarmsSet(SYSTEMALARMS_ALARM_SYSTEMCONFIGURATION, SYSTEMALARMS_ALARM_WARNING,
                                      SYSTEMALARMS_EXTENDEDALARMSTATUS_AUTOTUNE, failureBits);
                }
            }
            float hover_throttle = ((float)(throttleAccumulator/updateCounter))/10000.0f;
            UpdateSystemIdentState(gX, noise, 0, updateCounter, atPointsSpilled, hover_throttle);
            SystemIdentSettingsSet(&systemIdentSettings);
            state = AT_WAITING;
            break;

        case AT_WAITING:
        default:
            // after tuning, wait here till user switches to another flight mode
            // or disarms
            break;
        }

        // fly in Attitude mode or in SystemIdent mode
        UpdateStabilizationDesired(doingIdent);

        if (canSleep) {
            vTaskDelay(YIELD_MS / portTICK_RATE_MS);
        }
    }
}


// gyro sensor callback
// get gyro data and actuatordesired into a packet
// and put it in the queue for later processing
static void AtNewGyroData(UAVObjEvent * ev) {
    static struct at_queued_data q_item;
    static bool last_sample_unpushed = false;
    GyroSensorData gyro;
    ActuatorDesiredData actuators;

    if (!ev || !ev->obj || ev->instId!=0 || ev->event!=EV_UPDATED) {
        return;
    }

    // object will at times change asynchronously so must copy data here, with locking
    // and do it as soon as possible
    GyroSensorGet(&gyro);
    ActuatorDesiredGet(&actuators);

    if (last_sample_unpushed) {
        /* Last time we were unable to queue up the gyro data.
         * Try again, last chance! */
        if (xQueueSend(atQueue, &q_item, 0) != pdTRUE) {
            atPointsSpilled++;
        }
    }

    q_item.raw_time = PIOS_DELAY_GetRaw();
    q_item.y[0] = gyro.x;
    q_item.y[1] = gyro.y;
    q_item.y[2] = gyro.z;
    q_item.u[0] = actuators.Roll;
    q_item.u[1] = actuators.Pitch;
    q_item.u[2] = actuators.Yaw;
    q_item.throttle = actuators.Thrust;

    if (xQueueSend(atQueue, &q_item, 0) != pdTRUE) {
        last_sample_unpushed = true;
    } else {
        last_sample_unpushed = false;
    }
}


// check for the user quickly toggling the flight mode switch
// into and out of AutoTune, 3 times
// that is a signal that the user wants to try the next PID settings
// on the scale from smooth to quick
// when it exceeds the quickest setting, it starts back at the smoothest setting
static bool CheckFlightModeSwitchForPidRequest(uint8_t flightMode) {
    static uint32_t lastUpdateTime;
    static uint8_t flightModePrev;
    static uint8_t counter;
    uint32_t updateTime;

    // only count transitions into and out of autotune
    if ((flightModePrev == FLIGHTSTATUS_FLIGHTMODE_AUTOTUNE) ^ (flightMode == FLIGHTSTATUS_FLIGHTMODE_AUTOTUNE)) {
        flightModePrev = flightMode;
        updateTime = xTaskGetTickCount();
        // if it has been over 2 seconds, reset the counter
        if (updateTime - lastUpdateTime > 2000) {
            counter = 0;
        }
        // if the counter is reset, start a new time period
        if (counter == 0) {
            lastUpdateTime = updateTime;
        }
        // if flight mode has toggled into autotune 3 times but is currently not autotune
        if (++counter >= 5 && flightMode != FLIGHTSTATUS_FLIGHTMODE_AUTOTUNE) {
            counter = 0;
            return true;
        }
    }

    return false;
}


// read SystemIdent uavos, update the local structures
// and set some flags based on the values
// it is used two ways:
// - on startup it reads settings so the user can reuse an old tune with smooth-quick
// - at tune time, it inits the state in preparation for tuning
static void InitSystemIdent(bool loadDefaults) {
    SystemIdentSettingsGet(&systemIdentSettings);
    uint8_t smoothQuick = systemIdentSettings.SmoothQuick;

    if (loadDefaults) {
        // get these 10.0 10.0 7.0 -4.0 from default values of SystemIdent (.Beta and .Tau)
        // so that if they are changed there (mainly for future code changes), they will be changed here too
        SystemIdentStateSetDefaults(SystemIdentStateHandle(), 0);
        SystemIdentStateGet(&systemIdentState);
        // Tau Beta and the Complete flag get default values
        // in preparation for running AutoTune
        systemIdentSettings.Tau = systemIdentState.Tau;
        memcpy(&systemIdentSettings.Beta, &systemIdentState.Beta, sizeof(SystemIdentSettingsBetaData));
        systemIdentSettings.Complete = systemIdentState.Complete;
    } else {
        // Tau Beta and the Complete flag get stored values
        // so the user can fly another battery to select and test PIDs with the slider/knob
        systemIdentState.Tau = systemIdentSettings.Tau;
        memcpy(&systemIdentState.Beta, &systemIdentSettings.Beta, sizeof(SystemIdentStateBetaData));
        systemIdentState.Complete = systemIdentSettings.Complete;
    }

    // default to disable PID changing with flight mode switch and accessory0-3
    accessoryToUse = -1;
    flightModeSwitchTogglePosition = -1;
    systemIdentSettings.SmoothQuick = SMOOTH_QUICK_DISABLED;
    switch (smoothQuick) {
    case SMOOTH_QUICK_ACCESSORY_BASE+0: // use accessory0
    case SMOOTH_QUICK_ACCESSORY_BASE+1: // use accessory1
    case SMOOTH_QUICK_ACCESSORY_BASE+2: // use accessory2
    case SMOOTH_QUICK_ACCESSORY_BASE+3: // use accessory3
        accessoryToUse = smoothQuick - SMOOTH_QUICK_ACCESSORY_BASE;
        systemIdentSettings.SmoothQuick = smoothQuick;
        break;
    case SMOOTH_QUICK_TOGGLE_BASE+2: // use flight mode switch toggle with 3 points
    case SMOOTH_QUICK_TOGGLE_BASE+4: // use flight mode switch toggle with 5 points
        // first test PID is in the middle of the smooth -> quick range
        flightModeSwitchTogglePosition = (smoothQuick - SMOOTH_QUICK_TOGGLE_BASE) / 2;
        systemIdentSettings.SmoothQuick = smoothQuick;
        break;
    }
}


// update the gain and delay with current calculated value
// these are stored in the settings for use with next battery
// and also in the state for logging purposes
static void UpdateSystemIdentState(const float *X, const float *noise,
    float dT_s, uint32_t predicts, uint32_t spills, float hover_throttle) {
    systemIdentState.Beta.Roll    = X[6];
    systemIdentState.Beta.Pitch   = X[7];
    systemIdentState.Beta.Yaw     = X[8];
    systemIdentState.Bias.Roll    = X[10];
    systemIdentState.Bias.Pitch   = X[11];
    systemIdentState.Bias.Yaw     = X[12];
    systemIdentState.Tau          = X[9];
    // 'settings' beta and tau have same value as state versions
    // the state version produces a GCS log
    // the settings version is remembered after power off/on
    systemIdentSettings.Tau       = systemIdentState.Tau;
    memcpy(&systemIdentSettings.Beta, &systemIdentState.Beta, sizeof(SystemIdentSettingsBetaData));

    if (noise) {
        systemIdentState.Noise.Roll  = noise[0];
        systemIdentState.Noise.Pitch = noise[1];
        systemIdentState.Noise.Yaw   = noise[2];
    }
    systemIdentState.Period = dT_s * 1000.0f;

    systemIdentState.NumAfPredicts = predicts;
    systemIdentState.NumSpilledPts = spills;

    systemIdentState.HoverThrottle = hover_throttle;

    SystemIdentStateSet(&systemIdentState);
}


// when running AutoTune mode, this bypasses manualcontrol.c / stabilizedhandler.c
// to control exactly when the multicopter should be in Attitude mode vs. SystemIdent mode
static void UpdateStabilizationDesired(bool doingIdent) {
    StabilizationDesiredData stabDesired;
    ManualControlCommandData manualControlCommand;
    ManualControlCommandGet(&manualControlCommand);

    stabDesired.Roll = manualControlCommand.Roll * rollMax;
    stabDesired.Pitch = manualControlCommand.Pitch * pitchMax;
    stabDesired.Yaw = manualControlCommand.Yaw * manualRate.Yaw;
    stabDesired.Thrust = manualControlCommand.Thrust;

    if (doingIdent) {
        stabDesired.StabilizationMode.Roll  = STABILIZATIONDESIRED_STABILIZATIONMODE_SYSTEMIDENT;
        stabDesired.StabilizationMode.Pitch = STABILIZATIONDESIRED_STABILIZATIONMODE_SYSTEMIDENT;
        stabDesired.StabilizationMode.Yaw   = STABILIZATIONDESIRED_STABILIZATIONMODE_SYSTEMIDENT;
    } else {
        stabDesired.StabilizationMode.Roll  = STABILIZATIONDESIRED_STABILIZATIONMODE_ATTITUDE;
        stabDesired.StabilizationMode.Pitch = STABILIZATIONDESIRED_STABILIZATIONMODE_ATTITUDE;
        stabDesired.StabilizationMode.Yaw   = STABILIZATIONDESIRED_STABILIZATIONMODE_RATE;
    }
    stabDesired.StabilizationMode.Thrust    = STABILIZATIONDESIRED_STABILIZATIONMODE_MANUAL;

    StabilizationDesiredSet(&stabDesired);
}


// check the completed autotune state (mainly gain and delay)
// to see if it is reasonable
// return a bit mask of errors detected
static uint8_t CheckSettingsRaw()
{
    uint8_t retVal = 0;

    // Check the axis gains
    // Extreme values: Your roll or pitch gain was lower than expected. This will result in large PID values.
    if (systemIdentState.Beta.Roll < 6) {
        retVal |= ROLL_BETA_LOW;
    }
    if (systemIdentState.Beta.Pitch < 6) {
        retVal |= PITCH_BETA_LOW;
    }
    if (systemIdentState.Beta.Yaw < 4) {
        retVal |= YAW_BETA_LOW;
    }
    // Check the response speed
    // Extreme values: Your estimated response speed (tau) is slower than normal. This will result in large PID values.
    if (expf(systemIdentState.Tau) > 0.1f) {
        retVal |= TAU_TOO_LONG;
    }
    // Extreme values: Your estimated response speed (tau) is faster than normal. This will result in large PID values.
    else if (expf(systemIdentState.Tau) < 0.008f) {
        retVal |= TAU_TOO_SHORT;
    }

    return retVal;
}


// check the completed autotune state (mainly gain and delay)
// to see if it is reasonable
// override bad yaw values if configured that way
// return a bit mask of errors detected
static uint8_t CheckSettings()
{
    uint8_t retVal = CheckSettingsRaw();

    if (systemIdentSettings.DisableSanityChecks) {
    	retVal = 0;
    } else if  (systemIdentSettings.CalculateYaw != SYSTEMIDENTSETTINGS_CALCULATEYAW_TRUE) {
        retVal &= ~YAW_BETA_LOW;
    }

    return retVal;
}


// given Tau(delay) and Beta(gain) from the tune (and user selection of smooth to quick) calculate the PIDs
// this code came from dRonin GCS and uses double precision math
// most of the doubles could be replaced with floats
static void ComputeStabilizationAndSetPidsFromDampAndNoise(float dampRate, float noiseRate)
{
    StabilizationSettingsBank1Data stabSettingsBank;

    switch (systemIdentSettings.DestinationPidBank) {
    case 1:
        StabilizationSettingsBank1Get((void *)&stabSettingsBank);
        break;
    case 2:
        StabilizationSettingsBank2Get((void *)&stabSettingsBank);
        break;
    case 3:
        StabilizationSettingsBank3Get((void *)&stabSettingsBank);
        break;
    }

    // These three parameters define the desired response properties
    // - rate scale in the fraction of the natural speed of the system
    //   to strive for.
    // - damp is the amount of damping in the system. higher values
    //   make oscillations less likely
    // - ghf is the amount of high frequency gain and limits the influence
    //   of noise
    const double ghf  = (double) noiseRate / 1000.0d;
    const double damp = (double) dampRate  / 100.0d;

    double tau = exp(systemIdentState.Tau);
    double exp_beta_roll_times_ghf = exp(systemIdentState.Beta.Roll)*ghf;
    double exp_beta_pitch_times_ghf = exp(systemIdentState.Beta.Pitch)*ghf;

    double wn = 1.0d/tau;
    double tau_d = 0.0d;
    for (int i = 0; i < 30; i++) {
        double tau_d_roll = (2.0d*damp*tau*wn - 1.0d)/(4.0d*tau*damp*damp*wn*wn - 2.0d*damp*wn - tau*wn*wn + exp_beta_roll_times_ghf);
        double tau_d_pitch = (2.0d*damp*tau*wn - 1.0d)/(4.0d*tau*damp*damp*wn*wn - 2.0d*damp*wn - tau*wn*wn + exp_beta_pitch_times_ghf);
        // Select the slowest filter property
        tau_d = (tau_d_roll > tau_d_pitch) ? tau_d_roll : tau_d_pitch;
        wn = (tau + tau_d) / (tau*tau_d) / (2.0d * damp + 2.0d);
    }

    // Set the real pole position. The first pole is quite slow, which
    // prevents the integral being too snappy and driving too much
    // overshoot.
    const double a = ((tau+tau_d) / tau / tau_d - 2.0d * damp * wn) / 20.0d;
    const double b = ((tau+tau_d) / tau / tau_d - 2.0d * damp * wn - a);

    // Calculate the gain for the outer loop by approximating the
    // inner loop as a single order lpf. Set the outer loop to be
    // critically damped;
    const double zeta_o = 1.3d;
    const double kp_o = 1.0d / 4.0d / (zeta_o * zeta_o) / (1.0d/wn);

    // dRonin simply uses default PID settings for yaw
    float kpMax = 0.0f;
    for (int i = 0; i < ((systemIdentSettings.CalculateYaw != SYSTEMIDENTSETTINGS_CALCULATEYAW_FALSE) ? 3 : 2); i++) {
        double beta = exp(SystemIdentStateBetaToArray(systemIdentState.Beta)[i]);
        double ki = a * b * wn * wn * tau * tau_d / beta;
        double kp = tau * tau_d * ((a+b)*wn*wn + 2.0d*a*b*damp*wn) / beta - ki*tau_d;
        double kd = (tau * tau_d * (a*b + wn*wn + (a+b)*2.0d*damp*wn) - 1.0d) / beta - kp * tau_d;

        if (i<2) {
            if (kpMax < (float) kp) {
            	kpMax = (float) kp;
            }
        } else {
            // use the ratio with the largest roll/pitch kp to limit yaw kp to a reasonable value
            // use largest roll/pitch kp because it is the axis most slowed by rotational inertia
            // and yaw is also slowed maximally by rotational inertia
            // note that kp, ki, kd are all proportional in beta
            // so reducing them all proportionally is the same as changing beta
            float min = 0.0f;
            float max = 0.0f;
            switch (systemIdentSettings.YawPIDRatioFunction) {
            case SYSTEMIDENTSETTINGS_YAWPIDRATIOFUNCTION_DISABLE:
                max = 1000.0f;
                min = 0.0f;
                break;
            case SYSTEMIDENTSETTINGS_YAWPIDRATIOFUNCTION_LIMIT:
                max = kpMax * systemIdentSettings.YawToRollPitchPIDRatioMax;
                min = kpMax * systemIdentSettings.YawToRollPitchPIDRatioMin;
                break;
            }
            float ratio = 1.0f;
            if (min > 0.0f && (float) kp < min) {
            	ratio = (float) kp / min;
            } else if (max > 0.0f && (float) kp > max) {
            	ratio = (float) kp / max;
            }
         	kp /= (double) ratio;
           	ki /= (double) ratio;
           	kd /= (double) ratio;
        }

        switch(i) {
        case 0: // Roll
            stabSettingsBank.RollRatePID.Kp = kp;
            stabSettingsBank.RollRatePID.Ki = ki;
            stabSettingsBank.RollRatePID.Kd = kd;
            stabSettingsBank.RollPI.Kp = kp_o;
            stabSettingsBank.RollPI.Ki = 0;
            break;
        case 1: // Pitch
            stabSettingsBank.PitchRatePID.Kp = kp;
            stabSettingsBank.PitchRatePID.Ki = ki;
            stabSettingsBank.PitchRatePID.Kd = kd;
            stabSettingsBank.PitchPI.Kp = kp_o;
            stabSettingsBank.PitchPI.Ki = 0;
            break;
        case 2: // Yaw
            stabSettingsBank.YawRatePID.Kp = kp;
            stabSettingsBank.YawRatePID.Ki = ki;
            stabSettingsBank.YawRatePID.Kd = kd;
            stabSettingsBank.YawPI.Kp = kp_o;
            stabSettingsBank.YawPI.Ki = 0;
            break;
        }
    }

    // Librepilot might do something with this some time
    // stabSettingsBank.DerivativeCutoff = 1.0d / (2.0d*M_PI*tau_d);

    // Save PIDs to UAVO RAM (not permanently yet)
    switch (systemIdentSettings.DestinationPidBank) {
    case 1:
        StabilizationSettingsBank1Set((void *)&stabSettingsBank);
        break;
    case 2:
        StabilizationSettingsBank2Set((void *)&stabSettingsBank);
        break;
    case 3:
        StabilizationSettingsBank3Set((void *)&stabSettingsBank);
        break;
    }
}


// calculate PIDs using default smooth-quick settings
static void ComputeStabilizationAndSetPids()
{
    ComputeStabilizationAndSetPidsFromDampAndNoise(systemIdentSettings.DampRate, systemIdentSettings.NoiseRate);
}


// scale the damp and the noise to generate PIDs according to how a slider or other user specified ratio is set
//
// when val is half way between min and max, it generates the default PIDs
// when val is min, it generates the smoothest configured PIDs
// when val is max, it generates the quickest configured PIDs
//
// when val is between min and (min+max)/2, it scales val over the range [min, (min+max)/2] to generate PIDs between smoothest and default
// when val is between (min+max)/2 and max, it scales val over the range [(min+max)/2, max] to generate PIDs between default and quickest
//
// this is done piecewise because we are not guaranteed that default-min == max-default
// but we are given that [smoothDamp,smoothNoise] [defaultDamp,defaultNoise] [quickDamp,quickNoise] are all good parameterizations
static void ProportionPidsSmoothToQuick(float min, float val, float max)
{
    float ratio, damp, noise;

    // translate from range [min, max] to range [0, max-min]
    // takes care of min < 0 too
    val -= min;
    max -= min;
    ratio = val / max;

    if (ratio <= 0.5f) {
        // scale ratio in [0,0.5] to produce PIDs in [smoothest,default]
        ratio *= 2.0f;
        damp  = (systemIdentSettings.DampMax  * (1.0f - ratio)) + (systemIdentSettings.DampRate  * ratio);
        noise = (systemIdentSettings.NoiseMin * (1.0f - ratio)) + (systemIdentSettings.NoiseRate * ratio);
    } else {
        // scale ratio in [0.5,1.0] to produce PIDs in [default,quickest]
        ratio = (ratio - 0.5f) * 2.0f;
        damp  = (systemIdentSettings.DampRate  * (1.0f - ratio)) + (systemIdentSettings.DampMin  * ratio);
        noise = (systemIdentSettings.NoiseRate * (1.0f - ratio)) + (systemIdentSettings.NoiseMax * ratio);
    }

    ComputeStabilizationAndSetPidsFromDampAndNoise(damp, noise);
}


/**
 * Prediction step for EKF on control inputs to quad that
 * learns the system properties
 * @param X the current state estimate which is updated in place
 * @param P the current covariance matrix, updated in place
 * @param[in] the current control inputs (roll, pitch, yaw)
 * @param[in] the gyro measurements
 */
__attribute__((always_inline)) static inline void AfPredict(float X[AF_NUMX], float P[AF_NUMP], const float u_in[3], const float gyro[3], const float dT_s, const float t_in)
{
    const float Ts = dT_s;
    const float Tsq = Ts * Ts;
    const float Tsq3 = Tsq * Ts;
    const float Tsq4 = Tsq * Tsq;

    // for convenience and clarity code below uses the named versions of
    // the state variables
    float w1 = X[0];           // roll rate estimate
    float w2 = X[1];           // pitch rate estimate
    float w3 = X[2];           // yaw rate estimate
    float u1 = X[3];           // scaled roll torque
    float u2 = X[4];           // scaled pitch torque
    float u3 = X[5];           // scaled yaw torque
    const float e_b1 = expapprox(X[6]);   // roll torque scale
    const float b1 = X[6];
    const float e_b2 = expapprox(X[7]);   // pitch torque scale
    const float b2 = X[7];
    const float e_b3 = expapprox(X[8]);   // yaw torque scale
    const float b3 = X[8];
    const float e_tau = expapprox(X[9]); // time response of the motors
    const float tau = X[9];
    const float bias1 = X[10];       // bias in the roll torque
    const float bias2 = X[11];       // bias in the pitch torque
    const float bias3 = X[12];       // bias in the yaw torque

    // inputs to the system (roll, pitch, yaw)
    const float u1_in = 4*t_in*u_in[0];
    const float u2_in = 4*t_in*u_in[1];
    const float u3_in = 4*t_in*u_in[2];

    // measurements from gyro
    const float gyro_x = gyro[0];
    const float gyro_y = gyro[1];
    const float gyro_z = gyro[2];

    // update named variables because we want to use predicted
    // values below
    w1 = X[0] = w1 - Ts*bias1*e_b1 + Ts*u1*e_b1;
    w2 = X[1] = w2 - Ts*bias2*e_b2 + Ts*u2*e_b2;
    w3 = X[2] = w3 - Ts*bias3*e_b3 + Ts*u3*e_b3;
    u1 = X[3] = (Ts*u1_in)/(Ts + e_tau) + (u1*e_tau)/(Ts + e_tau);
    u2 = X[4] = (Ts*u2_in)/(Ts + e_tau) + (u2*e_tau)/(Ts + e_tau);
    u3 = X[5] = (Ts*u3_in)/(Ts + e_tau) + (u3*e_tau)/(Ts + e_tau);
    // X[6] to X[12] unchanged

    /**** filter parameters ****/
    const float q_w = 1e-3f;
    const float q_ud = 1e-3f;
    const float q_B = 1e-6f;
    const float q_tau = 1e-6f;
    const float q_bias = 1e-19f;
    const float s_a = 150.0f; // expected gyro measurment noise

    const float Q[AF_NUMX] = {q_w, q_w, q_w, q_ud, q_ud, q_ud, q_B, q_B, q_B, q_tau, q_bias, q_bias, q_bias};

    float D[AF_NUMP];
    for (uint32_t i = 0; i < AF_NUMP; i++)
        D[i] = P[i];

    const float e_tau2    = e_tau * e_tau;
    const float e_tau3    = e_tau * e_tau2;
    const float e_tau4    = e_tau2 * e_tau2;
    const float Ts_e_tau2 = (Ts + e_tau) * (Ts + e_tau);
    const float Ts_e_tau4 = Ts_e_tau2 * Ts_e_tau2;

#if 0
    // covariance propagation - D is stored copy of covariance
    P[0] = D[0] + Q[0] + 2*Ts*e_b1*(
        D[3] - D[28] - D[9]*bias1 + D[9]*u1
    ) + Tsq*(e_b1*e_b1)*(
        D[4] - 2*D[29] + D[32] - 2*D[10]*bias1 + 2*D[30]*bias1 + 2*D[10]*u1
        - 2*D[30]*u1 + D[11]*(bias1*bias1) + D[11]*(u1*u1) - 2*D[11]*bias1*u1
    );
    P[1] = D[1] + Q[1] + 2*Ts*e_b2*(
        D[5] - D[33] - D[12]*bias2 + D[12]*u2
    ) + Tsq*(e_b2*e_b2)*(
        D[6] - 2*D[34] + D[37] - 2*D[13]*bias2 + 2*D[35]*bias2 + 2*D[13]*u2
        - 2*D[35]*u2 + D[14]*(bias2*bias2) + D[14]*(u2*u2) - 2*D[14]*bias2*u2
    );
    P[2] = D[2] + Q[2] + 2*Ts*e_b3*(
        D[7] - D[38] - D[15]*bias3 + D[15]*u3
    ) + Tsq*(e_b3*e_b3)*(
        D[8] - 2*D[39] + D[42] - 2*D[16]*bias3 + 2*D[40]*bias3 + 2*D[16]*u3
        - 2*D[40]*u3 + D[17]*(bias3*bias3) + D[17]*(u3*u3) - 2*D[17]*bias3*u3
    );
    P[3] = (
        D[3]*(
            e_tau2 + Ts*e_tau
        ) + Ts*e_b1*e_tau2*(
            D[4] - D[29]
        ) + Tsq*e_b1*e_tau*(
            D[4] - D[29]
        ) + D[18]*Ts*e_tau*(
            u1 - u1_in
        ) + D[10]*e_b1*(
            u1*(
                Ts*e_tau2 + Tsq*e_tau
            ) - bias1*(
                Ts*e_tau2 + Tsq*e_tau
            )
        ) + D[21]*Tsq*e_b1*e_tau*(
            u1 - u1_in
        ) + D[31]*Tsq*e_b1*e_tau*(
            u1_in - u1
        ) + D[24]*Tsq*e_b1*e_tau*(
            u1*(
                u1 - bias1
            ) + u1_in*(
                bias1 - u1
            )
        )
    ) / Ts_e_tau2;
    P[4] = (
        Q[3]*Tsq4 + e_tau4*(
            D[4] + Q[3]
        ) + 2*Ts*e_tau3*(
            D[4] + 2*Q[3]
        ) + 4*Q[3]*Tsq3*e_tau + Tsq*e_tau2*(
            D[4] + 6*Q[3] + u1*(
                D[27]*u1 + 2*D[21]
            ) + u1_in*(
                D[27]*u1_in - 2*D[21]
            )
        ) + 2*D[21]*Ts*e_tau3*(
            u1 - u1_in
        ) - 2*D[27]*Tsq*u1*u1_in*e_tau2
    ) / Ts_e_tau4;
    P[5] = (
        D[5]*(
            e_tau2 + Ts*e_tau
        ) + Ts*e_b2*e_tau2*(
            D[6] - D[34]
        ) + Tsq*e_b2*e_tau*(
            D[6] - D[34]
        ) + D[19]*Ts*e_tau*(
            u2 - u2_in
        ) + D[13]*e_b2*(
            u2*(
                Ts*e_tau2 + Tsq*e_tau
            ) - bias2*(
                Ts*e_tau2 + Tsq*e_tau
            )
        ) + D[22]*Tsq*e_b2*e_tau*(
            u2 - u2_in
        ) + D[36]*Tsq*e_b2*e_tau*(
            u2_in - u2
        ) + D[25]*Tsq*e_b2*e_tau*(
            u2*(
                u2 - bias2
            ) + u2_in*(
                bias2 - u2
            )
        )
    ) / Ts_e_tau2;
    P[6] = (
        Q[4]*Tsq4 + e_tau4*(
            D[6] + Q[4]
        ) + 2*Ts*e_tau3*(
            D[6] + 2*Q[4]
        ) + 4*Q[4]*Tsq3*e_tau + Tsq*e_tau2*(
            D[6] + 6*Q[4] + u2*(
                D[27]*u2 + 2*D[22]
            ) + u2_in*(
                D[27]*u2_in - 2*D[22]
            )
        ) + 2*D[22]*Ts*e_tau3*(
            u2 - u2_in
        ) - 2*D[27]*Tsq*u2*u2_in*e_tau2
    ) / Ts_e_tau4;
    P[7] = (
        D[7]*(
            e_tau2 + Ts*e_tau
        ) + Ts*e_b3*e_tau2*(
            D[8] - D[39]
        ) + Tsq*e_b3*e_tau*(
            D[8] - D[39]
        ) + D[20]*Ts*e_tau*(
            u3 - u3_in
        ) + D[16]*e_b3*(
            u3*(
                Ts*e_tau2 + Tsq*e_tau
            ) - bias3*(
                Ts*e_tau2 + Tsq*e_tau
            )
        ) + D[23]*Tsq*e_b3*e_tau*(
            u3 - u3_in
        ) + D[41]*Tsq*e_b3*e_tau*(
            u3_in - u3
        ) + D[26]*Tsq*e_b3*e_tau*(
            u3*(
                u3 - bias3
            ) + u3_in*(
                bias3 - u3
            )
        )
    ) / Ts_e_tau2;
    P[8] = (
        Q[5]*Tsq4 + e_tau4*(
            D[8] + Q[5]
        ) + 2*Ts*e_tau3*(
            D[8] + 2*Q[5]
        ) + 4*Q[5]*Tsq3*e_tau + Tsq*e_tau2*(
            D[8] + 6*Q[5] + u3*(
                D[27]*u3 + 2*D[23]
            ) + u3_in*(
                D[27]*u3_in - 2*D[23]
            )
        ) + 2*D[23]*Ts*e_tau3*(
            u3 - u3_in
        ) - 2*D[27]*Tsq*u3*u3_in*e_tau2
    ) / Ts_e_tau4;
#endif
    // covariance propagation - D is stored copy of covariance
    P[0] = D[0] + Q[0] + 2*Ts*e_b1*(D[3] - D[28] - D[9]*bias1 + D[9]*u1)
           + Tsq*(e_b1*e_b1)*(D[4] - 2*D[29] + D[32] - 2*D[10]*bias1 + 2*D[30]*bias1 + 2*D[10]*u1 - 2*D[30]*u1
                  + D[11]*(bias1*bias1) + D[11]*(u1*u1) - 2*D[11]*bias1*u1);
    P[1] = D[1] + Q[1] + 2*Ts*e_b2*(D[5] - D[33] - D[12]*bias2 + D[12]*u2)
           + Tsq*(e_b2*e_b2)*(D[6] - 2*D[34] + D[37] - 2*D[13]*bias2 + 2*D[35]*bias2 + 2*D[13]*u2 - 2*D[35]*u2
                  + D[14]*(bias2*bias2) + D[14]*(u2*u2) - 2*D[14]*bias2*u2);
    P[2] = D[2] + Q[2] + 2*Ts*e_b3*(D[7] - D[38] - D[15]*bias3 + D[15]*u3)
           + Tsq*(e_b3*e_b3)*(D[8] - 2*D[39] + D[42] - 2*D[16]*bias3 + 2*D[40]*bias3 + 2*D[16]*u3 - 2*D[40]*u3
                  + D[17]*(bias3*bias3) + D[17]*(u3*u3) - 2*D[17]*bias3*u3);
    P[3] = (D[3]*(e_tau2 + Ts*e_tau) + Ts*e_b1*e_tau2*(D[4] - D[29]) + Tsq*e_b1*e_tau*(D[4] - D[29])
            + D[18]*Ts*e_tau*(u1 - u1_in) + D[10]*e_b1*(u1*(Ts*e_tau2 + Tsq*e_tau) - bias1*(Ts*e_tau2 + Tsq*e_tau))
            + D[21]*Tsq*e_b1*e_tau*(u1 - u1_in) + D[31]*Tsq*e_b1*e_tau*(u1_in - u1)
            + D[24]*Tsq*e_b1*e_tau*(u1*(u1 - bias1) + u1_in*(bias1 - u1)))/Ts_e_tau2;
    P[4] = (Q[3]*Tsq4 + e_tau4*(D[4] + Q[3]) + 2*Ts*e_tau3*(D[4] + 2*Q[3]) + 4*Q[3]*Tsq3*e_tau
            + Tsq*e_tau2*(D[4] + 6*Q[3] + u1*(D[27]*u1 + 2*D[21]) + u1_in*(D[27]*u1_in - 2*D[21]))
            + 2*D[21]*Ts*e_tau3*(u1 - u1_in) - 2*D[27]*Tsq*u1*u1_in*e_tau2)/Ts_e_tau4;
    P[5] = (D[5]*(e_tau2 + Ts*e_tau) + Ts*e_b2*e_tau2*(D[6] - D[34])
            + Tsq*e_b2*e_tau*(D[6] - D[34]) + D[19]*Ts*e_tau*(u2 - u2_in)
            + D[13]*e_b2*(u2*(Ts*e_tau2 + Tsq*e_tau) - bias2*(Ts*e_tau2 + Tsq*e_tau))
            + D[22]*Tsq*e_b2*e_tau*(u2 - u2_in) + D[36]*Tsq*e_b2*e_tau*(u2_in - u2)
            + D[25]*Tsq*e_b2*e_tau*(u2*(u2 - bias2) + u2_in*(bias2 - u2)))/Ts_e_tau2;
    P[6] = (Q[4]*Tsq4 + e_tau4*(D[6] + Q[4]) + 2*Ts*e_tau3*(D[6] + 2*Q[4]) + 4*Q[4]*Tsq3*e_tau
            + Tsq*e_tau2*(D[6] + 6*Q[4] + u2*(D[27]*u2 + 2*D[22]) + u2_in*(D[27]*u2_in - 2*D[22]))
            + 2*D[22]*Ts*e_tau3*(u2 - u2_in) - 2*D[27]*Tsq*u2*u2_in*e_tau2)/Ts_e_tau4;
    P[7] = (D[7]*(e_tau2 + Ts*e_tau) + Ts*e_b3*e_tau2*(D[8] - D[39])
            + Tsq*e_b3*e_tau*(D[8] - D[39]) + D[20]*Ts*e_tau*(u3 - u3_in)
            + D[16]*e_b3*(u3*(Ts*e_tau2 + Tsq*e_tau) - bias3*(Ts*e_tau2 + Tsq*e_tau))
            + D[23]*Tsq*e_b3*e_tau*(u3 - u3_in) + D[41]*Tsq*e_b3*e_tau*(u3_in - u3)
            + D[26]*Tsq*e_b3*e_tau*(u3*(u3 - bias3) + u3_in*(bias3 - u3)))/Ts_e_tau2;
    P[8] = (Q[5]*Tsq4 + e_tau4*(D[8] + Q[5]) + 2*Ts*e_tau3*(D[8] + 2*Q[5]) + 4*Q[5]*Tsq3*e_tau
            + Tsq*e_tau2*(D[8] + 6*Q[5] + u3*(D[27]*u3 + 2*D[23]) + u3_in*(D[27]*u3_in - 2*D[23]))
            + 2*D[23]*Ts*e_tau3*(u3 - u3_in) - 2*D[27]*Tsq*u3*u3_in*e_tau2)/Ts_e_tau4;
    P[9] = D[9] - Ts*e_b1*(D[30] - D[10] + D[11]*(bias1 - u1));
    P[10] = (D[10]*(Ts + e_tau) + D[24]*Ts*(u1 - u1_in))*(e_tau/Ts_e_tau2);
    P[11] = D[11] + Q[6];
    P[12] = D[12] - Ts*e_b2*(D[35] - D[13] + D[14]*(bias2 - u2));
    P[13] = (D[13]*(Ts + e_tau) + D[25]*Ts*(u2 - u2_in))*(e_tau/Ts_e_tau2);
    P[14] = D[14] + Q[7];
    P[15] = D[15] - Ts*e_b3*(D[40] - D[16] + D[17]*(bias3 - u3));
    P[16] = (D[16]*(Ts + e_tau) + D[26]*Ts*(u3 - u3_in))*(e_tau/Ts_e_tau2);
    P[17] = D[17] + Q[8];
    P[18] = D[18] - Ts*e_b1*(D[31] - D[21] + D[24]*(bias1 - u1));
    P[19] = D[19] - Ts*e_b2*(D[36] - D[22] + D[25]*(bias2 - u2));
    P[20] = D[20] - Ts*e_b3*(D[41] - D[23] + D[26]*(bias3 - u3));
    P[21] = (D[21]*(Ts + e_tau) + D[27]*Ts*(u1 - u1_in))*(e_tau/Ts_e_tau2);
    P[22] = (D[22]*(Ts + e_tau) + D[27]*Ts*(u2 - u2_in))*(e_tau/Ts_e_tau2);
    P[23] = (D[23]*(Ts + e_tau) + D[27]*Ts*(u3 - u3_in))*(e_tau/Ts_e_tau2);
    P[24] = D[24];
    P[25] = D[25];
    P[26] = D[26];
    P[27] = D[27] + Q[9];
    P[28] = D[28] - Ts*e_b1*(D[32] - D[29] + D[30]*(bias1 - u1));
    P[29] = (D[29]*(Ts + e_tau) + D[31]*Ts*(u1 - u1_in))*(e_tau/Ts_e_tau2);
    P[30] = D[30];
    P[31] = D[31];
    P[32] = D[32] + Q[10];
    P[33] = D[33] - Ts*e_b2*(D[37] - D[34] + D[35]*(bias2 - u2));
    P[34] = (D[34]*(Ts + e_tau) + D[36]*Ts*(u2 - u2_in))*(e_tau/Ts_e_tau2);
    P[35] = D[35];
    P[36] = D[36];
    P[37] = D[37] + Q[11];
    P[38] = D[38] - Ts*e_b3*(D[42] - D[39] + D[40]*(bias3 - u3));
    P[39] = (D[39]*(Ts + e_tau) + D[41]*Ts*(u3 - u3_in))*(e_tau/Ts_e_tau2);
    P[40] = D[40];
    P[41] = D[41];
    P[42] = D[42] + Q[12];

    /********* this is the update part of the equation ***********/
    float S[3] = {P[0] + s_a, P[1] + s_a, P[2] + s_a};
    X[0] = w1 + P[0]*((gyro_x - w1)/S[0]);
    X[1] = w2 + P[1]*((gyro_y - w2)/S[1]);
    X[2] = w3 + P[2]*((gyro_z - w3)/S[2]);
    X[3] = u1 + P[3]*((gyro_x - w1)/S[0]);
    X[4] = u2 + P[5]*((gyro_y - w2)/S[1]);
    X[5] = u3 + P[7]*((gyro_z - w3)/S[2]);
    X[6] = b1 + P[9]*((gyro_x - w1)/S[0]);
    X[7] = b2 + P[12]*((gyro_y - w2)/S[1]);
    X[8] = b3 + P[15]*((gyro_z - w3)/S[2]);
    X[9] = tau + P[18]*((gyro_x - w1)/S[0]) + P[19]*((gyro_y - w2)/S[1]) + P[20]*((gyro_z - w3)/S[2]);
    X[10] = bias1 + P[28]*((gyro_x - w1)/S[0]);
    X[11] = bias2 + P[33]*((gyro_y - w2)/S[1]);
    X[12] = bias3 + P[38]*((gyro_z - w3)/S[2]);

    // update the duplicate cache
    for (uint32_t i = 0; i < AF_NUMP; i++)
        D[i] = P[i];

    // This is an approximation that removes some cross axis uncertainty but
    // substantially reduces the number of calculations
    P[0] = -D[0]*(D[0]/S[0] - 1);
    P[1] = -D[1]*(D[1]/S[1] - 1);
    P[2] = -D[2]*(D[2]/S[2] - 1);
    P[3] = -D[3]*(D[0]/S[0] - 1);
    P[4] = D[4] - D[3]*D[3]/S[0];
    P[5] = -D[5]*(D[1]/S[1] - 1);
    P[6] = D[6] - D[5]*D[5]/S[1];
    P[7] = -D[7]*(D[2]/S[2] - 1);
    P[8] = D[8] - D[7]*D[7]/S[2];
    P[9] = -D[9]*(D[0]/S[0] - 1);
    P[10] = D[10] - D[3]*(D[9]/S[0]);
    P[11] = D[11] - D[9]*(D[9]/S[0]);
    P[12] = -D[12]*(D[1]/S[1] - 1);
    P[13] = D[13] - D[5]*(D[12]/S[1]);
    P[14] = D[14] - D[12]*(D[12]/S[1]);
    P[15] = -D[15]*(D[2]/S[2] - 1);
    P[16] = D[16] - D[7]*(D[15]/S[2]);
    P[17] = D[17] - D[15]*(D[15]/S[2]);
    P[18] = -D[18]*(D[0]/S[0] - 1);
    P[19] = -D[19]*(D[1]/S[1] - 1);
    P[20] = -D[20]*(D[2]/S[2] - 1);
    P[21] = D[21] - D[3]*(D[18]/S[0]);
    P[22] = D[22] - D[5]*(D[19]/S[1]);
    P[23] = D[23] - D[7]*(D[20]/S[2]);
    P[24] = D[24] - D[9]*(D[18]/S[0]);
    P[25] = D[25] - D[12]*(D[19]/S[1]);
    P[26] = D[26] - D[15]*(D[20]/S[2]);
    P[27] = D[27] - D[18]*(D[18]/S[0]) - D[19]*(D[19]/S[1]) - D[20]*(D[20]/S[2]);
    P[28] = -D[28]*(D[0]/S[0] - 1);
    P[29] = D[29] - D[3]*(D[28]/S[0]);
    P[30] = D[30] - D[9]*(D[28]/S[0]);
    P[31] = D[31] - D[18]*(D[28]/S[0]);
    P[32] = D[32] - D[28]*(D[28]/S[0]);
    P[33] = -D[33]*(D[1]/S[1] - 1);
    P[34] = D[34] - D[5]*(D[33]/S[1]);
    P[35] = D[35] - D[12]*(D[33]/S[1]);
    P[36] = D[36] - D[19]*(D[33]/S[1]);
    P[37] = D[37] - D[33]*(D[33]/S[1]);
    P[38] = -D[38]*(D[2]/S[2] - 1);
    P[39] = D[39] - D[7]*(D[38]/S[2]);
    P[40] = D[40] - D[15]*(D[38]/S[2]);
    P[41] = D[41] - D[20]*(D[38]/S[2]);
    P[42] = D[42] - D[38]*(D[38]/S[2]);

    // apply limits to some of the state variables
    if (X[9] > -1.5f)
        X[9] = -1.5f;
    else if (X[9] < -5.5f)        /* 4ms */
        X[9] = -5.5f;
    if (X[10] > 0.5f)
        X[10] = 0.5f;
    else if (X[10] < -0.5f)
        X[10] = -0.5f;
    if (X[11] > 0.5f)
        X[11] = 0.5f;
    else if (X[11] < -0.5f)
        X[11] = -0.5f;
    if (X[12] > 0.5f)
        X[12] = 0.5f;
    else if (X[12] < -0.5f)
        X[12] = -0.5f;
}


/**
 * Initialize the state variable and covariance matrix
 * for the system identification EKF
 */
static void AfInit(float X[AF_NUMX], float P[AF_NUMP])
{
    static const float qInit[AF_NUMX] = {
        1.0f, 1.0f, 1.0f,
        1.0f, 1.0f, 1.0f,
        0.05f, 0.05f, 0.005f,
        0.05f,
        0.05f, 0.05f, 0.05f
    };

    // X[0] = X[1] = X[2] = 0.0f;    // assume no rotation
    // X[3] = X[4] = X[5] = 0.0f;    // and no net torque
    // X[6] = X[7]        = 10.0f;   // medium amount of strength
    // X[8]               = 7.0f;    // yaw strength
    // X[9] = -4.0f;                 // and 50 (18?) ms time scale
    // X[10] = X[11] = X[12] = 0.0f; // zero bias

    memset(X, 0, AF_NUMX*sizeof(X[0]));
    // get these 10.0 10.0 7.0 -4.0 from default values of SystemIdent (.Beta and .Tau)
    // so that if they are changed there (mainly for future code changes), they will be changed here too
    //SystemIdentSetDefaults(SystemIdentHandle(), 0);
    //SystemIdentBetaArrayGet(&X[6]);
    memcpy(&X[6], &systemIdentState.Beta, sizeof(systemIdentState.Beta));
    //SystemIdentTauGet(&X[9]);
    X[9] = systemIdentState.Tau;

    // P initialization
    memset(P, 0, AF_NUMP*sizeof(P[0]));
    P[0] = qInit[0];
    P[1] = qInit[1];
    P[2] = qInit[2];
    P[4] = qInit[3];
    P[6] = qInit[4];
    P[8] = qInit[5];
    P[11] = qInit[6];
    P[14] = qInit[7];
    P[17] = qInit[8];
    P[27] = qInit[9];
    P[32] = qInit[10];
    P[37] = qInit[11];
    P[42] = qInit[12];
}

/**
 * @}
 * @}
 */