text: //=============================================================================================
  // Madgwick.c
  //=============================================================================================
  //
  // Implementation of Madgwick's IMU and AHRS algorithms.
  // See: http://www.x-io.co.uk/open-source-imu-and-ahrs-algorithms/
  //
  // From the x-io website "Open-source resources available on this website are
  // provided under the GNU General Public Licence unless an alternative licence
  // is provided in source."
  //
  // Date			Author          Notes
  // 29/09/2011	SOH Madgwick    Initial release
  // 02/10/2011	SOH Madgwick	Optimised for reduced CPU load
  // 19/02/2012	SOH Madgwick	Magnetometer measurement is normalised
  //
  //=============================================================================================
  
  //-------------------------------------------------------------------------------------------
  // Header files
  
  #include "Madgwick.h"
  
  //-------------------------------------------------------------------------------------------
  // Definitions
  
  #define sampleFreqDef   512.0f          // sample frequency in Hz
  #define betaDef         0.1f            // 2 * proportional gain
  
  
  //============================================================================================
  // Functions
  
  //-------------------------------------------------------------------------------------------
  // AHRS algorithm update
  
  Madgwick::Madgwick() {
  	beta = betaDef;
  	q0 = 1.0f;
  	q1 = 0.0f;
  	q2 = 0.0f;
  	q3 = 0.0f;
  	invSampleFreq = 1.0f / sampleFreqDef;
  	anglesComputed = 0;
  }
  
  void Madgwick::update(float gx, float gy, float gz, float ax, float ay, float az, float mx, float my, float mz) {
  	float recipNorm;
  	float s0, s1, s2, s3;
  	float qDot1, qDot2, qDot3, qDot4;
  	float hx, hy;
  	float _2q0mx, _2q0my, _2q0mz, _2q1mx, _2bx, _2bz, _4bx, _4bz, _2q0, _2q1, _2q2, _2q3, _2q0q2, _2q2q3, q0q0, q0q1, q0q2, q0q3, q1q1, q1q2, q1q3, q2q2, q2q3, q3q3;
  
  	// Use IMU algorithm if magnetometer measurement invalid (avoids NaN in magnetometer normalisation)
  	if((mx == 0.0f) && (my == 0.0f) && (mz == 0.0f)) {
  		update(gx, gy, gz, ax, ay, az);
  		return;
  	}
  
  	// Convert gyroscope degrees/sec to radians/sec
  	//gx *= 0.0174533f;
  	//gy *= 0.0174533f;
  	//gz *= 0.0174533f;
  
  	// Rate of change of quaternion from gyroscope
  	qDot1 = 0.5f * (-q1 * gx - q2 * gy - q3 * gz);
  	qDot2 = 0.5f * ( q0 * gx + q2 * gz - q3 * gy);
  	qDot3 = 0.5f * ( q0 * gy - q1 * gz + q3 * gx);
  	qDot4 = 0.5f * ( q0 * gz + q1 * gy - q2 * gx);
  
  	// Compute feedback only if accelerometer measurement valid (avoids NaN in accelerometer normalisation)
  	if(!((ax == 0.0f) && (ay == 0.0f) && (az == 0.0f))) {
  
  		// Normalise accelerometer measurement
  		recipNorm = invSqrt(ax * ax + ay * ay + az * az);
  		ax *= recipNorm;
  		ay *= recipNorm;
  		az *= recipNorm;
  
  		// Normalise magnetometer measurement
  		recipNorm = invSqrt(mx * mx + my * my + mz * mz);
  		mx *= recipNorm;
  		my *= recipNorm;
  		mz *= recipNorm;
  
  		// Auxiliary variables to avoid repeated arithmetic
  		_2q0mx = 2.0f * q0 * mx;
  		_2q0my = 2.0f * q0 * my;
  		_2q0mz = 2.0f * q0 * mz;
  		_2q1mx = 2.0f * q1 * mx;
  		_2q0 = 2.0f * q0;
  		_2q1 = 2.0f * q1;
  		_2q2 = 2.0f * q2;
  		_2q3 = 2.0f * q3;
  		_2q0q2 = 2.0f * q0 * q2;
  		_2q2q3 = 2.0f * q2 * q3;
  		q0q0 = q0 * q0;
  		q0q1 = q0 * q1;
  		q0q2 = q0 * q2;
  		q0q3 = q0 * q3;
  		q1q1 = q1 * q1;
  		q1q2 = q1 * q2;
  		q1q3 = q1 * q3;
  		q2q2 = q2 * q2;
  		q2q3 = q2 * q3;
  		q3q3 = q3 * q3;
  
  		// Reference direction of Earth's magnetic field
  		hx = mx * q0q0 - _2q0my * q3 + _2q0mz * q2 + mx * q1q1 + _2q1 * my * q2 + _2q1 * mz * q3 - mx * q2q2 - mx * q3q3;
  		hy = _2q0mx * q3 + my * q0q0 - _2q0mz * q1 + _2q1mx * q2 - my * q1q1 + my * q2q2 + _2q2 * mz * q3 - my * q3q3;
  		_2bx = sqrtf(hx * hx + hy * hy);
  		_2bz = -_2q0mx * q2 + _2q0my * q1 + mz * q0q0 + _2q1mx * q3 - mz * q1q1 + _2q2 * my * q3 - mz * q2q2 + mz * q3q3;
  		_4bx = 2.0f * _2bx;
  		_4bz = 2.0f * _2bz;
  
  		// Gradient decent algorithm corrective step
  		s0 = -_2q2 * (2.0f * q1q3 - _2q0q2 - ax) + _2q1 * (2.0f * q0q1 + _2q2q3 - ay) - _2bz * q2 * (_2bx * (0.5f - q2q2 - q3q3) + _2bz * (q1q3 - q0q2) - mx) + (-_2bx * q3 + _2bz * q1) * (_2bx * (q1q2 - q0q3) + _2bz * (q0q1 + q2q3) - my) + _2bx * q2 * (_2bx * (q0q2 + q1q3) + _2bz * (0.5f - q1q1 - q2q2) - mz);
  		s1 = _2q3 * (2.0f * q1q3 - _2q0q2 - ax) + _2q0 * (2.0f * q0q1 + _2q2q3 - ay) - 4.0f * q1 * (1 - 2.0f * q1q1 - 2.0f * q2q2 - az) + _2bz * q3 * (_2bx * (0.5f - q2q2 - q3q3) + _2bz * (q1q3 - q0q2) - mx) + (_2bx * q2 + _2bz * q0) * (_2bx * (q1q2 - q0q3) + _2bz * (q0q1 + q2q3) - my) + (_2bx * q3 - _4bz * q1) * (_2bx * (q0q2 + q1q3) + _2bz * (0.5f - q1q1 - q2q2) - mz);
  		s2 = -_2q0 * (2.0f * q1q3 - _2q0q2 - ax) + _2q3 * (2.0f * q0q1 + _2q2q3 - ay) - 4.0f * q2 * (1 - 2.0f * q1q1 - 2.0f * q2q2 - az) + (-_4bx * q2 - _2bz * q0) * (_2bx * (0.5f - q2q2 - q3q3) + _2bz * (q1q3 - q0q2) - mx) + (_2bx * q1 + _2bz * q3) * (_2bx * (q1q2 - q0q3) + _2bz * (q0q1 + q2q3) - my) + (_2bx * q0 - _4bz * q2) * (_2bx * (q0q2 + q1q3) + _2bz * (0.5f - q1q1 - q2q2) - mz);
  		s3 = _2q1 * (2.0f * q1q3 - _2q0q2 - ax) + _2q2 * (2.0f * q0q1 + _2q2q3 - ay) + (-_4bx * q3 + _2bz * q1) * (_2bx * (0.5f - q2q2 - q3q3) + _2bz * (q1q3 - q0q2) - mx) + (-_2bx * q0 + _2bz * q2) * (_2bx * (q1q2 - q0q3) + _2bz * (q0q1 + q2q3) - my) + _2bx * q1 * (_2bx * (q0q2 + q1q3) + _2bz * (0.5f - q1q1 - q2q2) - mz);
  		recipNorm = invSqrt(s0 * s0 + s1 * s1 + s2 * s2 + s3 * s3); // normalise step magnitude
  		s0 *= recipNorm;
  		s1 *= recipNorm;
  		s2 *= recipNorm;
  		s3 *= recipNorm;
  
  		// Apply feedback step
  		qDot1 -= beta * s0;
  		qDot2 -= beta * s1;
  		qDot3 -= beta * s2;
  		qDot4 -= beta * s3;
  	}
  
  	// Integrate rate of change of quaternion to yield quaternion
  	q0 += qDot1 * invSampleFreq;
  	q1 += qDot2 * invSampleFreq;
  	q2 += qDot3 * invSampleFreq;
  	q3 += qDot4 * invSampleFreq;
  
  	// Normalise quaternion
  	recipNorm = invSqrt(q0 * q0 + q1 * q1 + q2 * q2 + q3 * q3);
  	q0 *= recipNorm;
  	q1 *= recipNorm;
  	q2 *= recipNorm;
  	q3 *= recipNorm;
  	anglesComputed = 0;
  }
  
  //-------------------------------------------------------------------------------------------
  // IMU algorithm update
  
  void Madgwick::update(float gx, float gy, float gz, float ax, float ay, float az) {
  	float recipNorm;
  	float s0, s1, s2, s3;
  	float qDot1 = 0, qDot2 = 0, qDot3 = 0, qDot4 = 0;
  	float _2q0, _2q1, _2q2, _2q3, _4q0, _4q1, _4q2 ,_8q1, _8q2, q0q0, q1q1, q2q2, q3q3;
  
  	// Convert gyroscope degrees/sec to radians/sec
  	//gx *= 0.0174533f;
  	//gy *= 0.0174533f;
  	//gz *= 0.0174533f;
  
  	bool hasGyro  = !((gx == 0.0f) && (gy == 0.0f) && (gz == 0.0f));
  	bool hasAccel = !((ax == 0.0f) && (ay == 0.0f) && (az == 0.0f));
  
  	// Rate of change of quaternion from gyroscope
  	if(hasGyro) {
  		qDot1 = 0.5f * (-q1 * gx - q2 * gy - q3 * gz);
  		qDot2 = 0.5f * ( q0 * gx + q2 * gz - q3 * gy);
  		qDot3 = 0.5f * ( q0 * gy - q1 * gz + q3 * gx);
  		qDot4 = 0.5f * ( q0 * gz + q1 * gy - q2 * gx);
  	}
  
  	// Compute feedback only if accelerometer measurement valid (avoids NaN in accelerometer normalisation)
  	if(hasAccel) {
  		// Normalise accelerometer measurement
  		recipNorm = invSqrt(ax * ax + ay * ay + az * az);
  		ax *= recipNorm;
  		ay *= recipNorm;
  		az *= recipNorm;
  
  		// Auxiliary variables to avoid repeated arithmetic
  		_2q0 = 2.0f * q0;
  		_2q1 = 2.0f * q1;
  		_2q2 = 2.0f * q2;
  		_2q3 = 2.0f * q3;
  		_4q0 = 4.0f * q0;
  		_4q1 = 4.0f * q1;
  		_4q2 = 4.0f * q2;
  		_8q1 = 8.0f * q1;
  		_8q2 = 8.0f * q2;
  		q0q0 = q0 * q0;
  		q1q1 = q1 * q1;
  		q2q2 = q2 * q2;
  		q3q3 = q3 * q3;
  
  		// Gradient decent algorithm corrective step
  		s0 = _4q0 * q2q2 + _2q2 * ax + _4q0 * q1q1 - _2q1 * ay;
  		s1 = _4q1 * q3q3 - _2q3 * ax + 4.0f * q0q0 * q1 - _2q0 * ay - _4q1 + _8q1 * q1q1 + _8q1 * q2q2 + _4q1 * az;
  		s2 = 4.0f * q0q0 * q2 + _2q0 * ax + _4q2 * q3q3 - _2q3 * ay - _4q2 + _8q2 * q1q1 + _8q2 * q2q2 + _4q2 * az;
  		s3 = 4.0f * q1q1 * q3 - _2q1 * ax + 4.0f * q2q2 * q3 - _2q2 * ay;
  		recipNorm = invSqrt(s0 * s0 + s1 * s1 + s2 * s2 + s3 * s3); // normalise step magnitude
  		s0 *= recipNorm;
  		s1 *= recipNorm;
  		s2 *= recipNorm;
  		s3 *= recipNorm;
  
  		// Apply feedback step
  		qDot1 -= beta * s0;
  		qDot2 -= beta * s1;
  		qDot3 -= beta * s2;
  		qDot4 -= beta * s3;
  	}
  
  	// Integrate rate of change of quaternion to yield quaternion
  	q0 += qDot1 * invSampleFreq;
  	q1 += qDot2 * invSampleFreq;
  	q2 += qDot3 * invSampleFreq;
  	q3 += qDot4 * invSampleFreq;
  
  	// Normalise quaternion
  	recipNorm = invSqrt(q0 * q0 + q1 * q1 + q2 * q2 + q3 * q3);
  	q0 *= recipNorm;
  	q1 *= recipNorm;
  	q2 *= recipNorm;
  	q3 *= recipNorm;
  	anglesComputed = 0;
  }
  
  //-------------------------------------------------------------------------------------------
  
  void Madgwick::computeAngles()
  {
  	roll = atan2f(q0*q1 + q2*q3, 0.5f - q1*q1 - q2*q2);
  	pitch = asinf(-2.0f * (q1*q3 - q0*q2));
  	yaw = atan2f(q1*q2 + q0*q3, 0.5f - q2*q2 - q3*q3);
  	anglesComputed = 1;
  }
  