title: Madgwick AHRS algorithm Android implementation
body: <p>I'm trying to implement Madgwick sensor fusion algorithm from  <a href="http://x-io.co.uk/open-source-imu-and-ahrs-algorithms/" rel="noreferrer">here</a>, to calculate orientation of my phone. It works relatively fine when I am moving, but it if the phone is steady on the table it starts to accumulate errors very quickly. I tried to play with sensor update rate and filter gain, but it didn't help </p>
  
  <p>I also tried to edit the code according to
  <a href="https://stackoverflow.com/questions/25902186/madgwick-sensor-fusion-on-lsm9ds0">this</a> solution, but it also didn't work.
  Is there a way how to fix this? Thanks </p>
  
  <p>MadgwickAHRS algorithm:</p>
  
  <pre><code> public class MadgwickAHRS
  {
      public float SamplePeriod ;
      public float Beta;
      public float[] Quaternion;
  
      public MadgwickAHRS(float samplePeriod)
      {
          SamplePeriod = samplePeriod;
          Beta = 1f;
          Quaternion = new float[] { 1f, 0f, 0f, 0f };
      }
  
      public MadgwickAHRS(float samplePeriod, float beta)
      {
          SamplePeriod = samplePeriod;
          Beta = beta;
          Quaternion = new float[] { 1f, 0f, 0f, 0f };
      }
  
      public void Update(float gx, float gy, float gz, float ax, float ay, float az, float mx, float my, float mz)
      {
          float q1 = Quaternion[0], q2 = Quaternion[1], q3 = Quaternion[2], q4 = Quaternion[3];   // short name local variable for readability
          float norm;
          float hx, hy, _2bx, _2bz;
          float s1, s2, s3, s4;
          float qDot1, qDot2, qDot3, qDot4;
  
          // Auxiliary variables to avoid repeated arithmetic
          float _2q1mx;
          float _2q1my;
          float _2q1mz;
          float _2q2mx;
          float _4bx;
          float _4bz;
          float _8bx;
          float _8bz;
          float _2q1 = 2f * q1;
          float _2q2 = 2f * q2;
          float _2q3 = 2f * q3;
          float _2q4 = 2f * q4;
          float _2q1q3 = 2f * q1 * q3;
          float _2q3q4 = 2f * q3 * q4;
          float q1q1 = q1 * q1;
          float q1q2 = q1 * q2;
          float q1q3 = q1 * q3;
          float q1q4 = q1 * q4;
          float q2q2 = q2 * q2;
          float q2q3 = q2 * q3;
          float q2q4 = q2 * q4;
          float q3q3 = q3 * q3;
          float q3q4 = q3 * q4;
          float q4q4 = q4 * q4;
  
          // Normalise accelerometer measurement
          norm = (float)Math.sqrt(ax * ax + ay * ay + az * az);
          if (norm == 0f) return; // handle NaN
          norm = 1 / norm;        // use reciprocal for division
          ax *= norm;
          ay *= norm;
          az *= norm;
  
          // Normalise magnetometer measurement
          norm = (float)Math.sqrt(mx * mx + my * my + mz * mz);
          if (norm == 0f) return; // handle NaN
          norm = 1 / norm;        // use reciprocal for division
          mx *= norm;
          my *= norm;
          mz *= norm;
  
          // Reference direction of Earth's magnetic field
          _2q1mx = 2f * q1 * mx;
          _2q1my = 2f * q1 * my;
          _2q1mz = 2f * q1 * mz;
          _2q2mx = 2f * q2 * mx;
          hx = mx * q1q1 - _2q1my * q4 + _2q1mz * q3 + mx * q2q2 + _2q2 * my * q3 + _2q2 * mz * q4 - mx * q3q3 - mx * q4q4;
          hy = _2q1mx * q4 + my * q1q1 - _2q1mz * q2 + _2q2mx * q3 - my * q2q2 + my * q3q3 + _2q3 * mz * q4 - my * q4q4;
          _2bx = (float)Math.sqrt(hx * hx + hy * hy);
          _2bz = -_2q1mx * q3 + _2q1my * q2 + mz * q1q1 + _2q2mx * q4 - mz * q2q2 + _2q3 * my * q4 - mz * q3q3 + mz * q4q4;
          _4bx = 2f * _2bx;
          _4bz = 2f * _2bz;
          _8bx = 2f * _4bx;
          _8bz = 2f * _4bz;
  
          /** Gradient decent algorithm corrective step old
          s1 = -_2q3 * (2f * q2q4 - _2q1q3 - ax) + _2q2 * (2f * q1q2 + _2q3q4 - ay) - _2bz * q3 * (_2bx * (0.5f - q3q3 - q4q4) + _2bz * (q2q4 - q1q3) - mx) + (-_2bx * q4 + _2bz * q2) * (_2bx * (q2q3 - q1q4) + _2bz * (q1q2 + q3q4) - my) + _2bx * q3 * (_2bx * (q1q3 + q2q4) + _2bz * (0.5f - q2q2 - q3q3) - mz);
          s2 = _2q4 * (2f * q2q4 - _2q1q3 - ax) + _2q1 * (2f * q1q2 + _2q3q4 - ay) - 4f * q2 * (1 - 2f * q2q2 - 2f * q3q3 - az) + _2bz * q4 * (_2bx * (0.5f - q3q3 - q4q4) + _2bz * (q2q4 - q1q3) - mx) + (_2bx * q3 + _2bz * q1) * (_2bx * (q2q3 - q1q4) + _2bz * (q1q2 + q3q4) - my) + (_2bx * q4 - _4bz * q2) * (_2bx * (q1q3 + q2q4) + _2bz * (0.5f - q2q2 - q3q3) - mz);
          s3 = -_2q1 * (2f * q2q4 - _2q1q3 - ax) + _2q4 * (2f * q1q2 + _2q3q4 - ay) - 4f * q3 * (1 - 2f * q2q2 - 2f * q3q3 - az) + (-_4bx * q3 - _2bz * q1) * (_2bx * (0.5f - q3q3 - q4q4) + _2bz * (q2q4 - q1q3) - mx) + (_2bx * q2 + _2bz * q4) * (_2bx * (q2q3 - q1q4) + _2bz * (q1q2 + q3q4) - my) + (_2bx * q1 - _4bz * q3) * (_2bx * (q1q3 + q2q4) + _2bz * (0.5f - q2q2 - q3q3) - mz);
          s4 = _2q2 * (2f * q2q4 - _2q1q3 - ax) + _2q3 * (2f * q1q2 + _2q3q4 - ay) + (-_4bx * q4 + _2bz * q2) * (_2bx * (0.5f - q3q3 - q4q4) + _2bz * (q2q4 - q1q3) - mx) + (-_2bx * q1 + _2bz * q3) * (_2bx * (q2q3 - q1q4) + _2bz * (q1q2 + q3q4) - my) + _2bx * q2 * (_2bx * (q1q3 + q2q4) + _2bz * (0.5f - q2q2 - q3q3) - mz);
          norm = 1f / (float)Math.sqrt(s1 * s1 + s2 * s2 + s3 * s3 + s4 * s4);    // normalise step magnitude
          */
  
          s1 = -_2q3 * (2f * (q2q4 -q1q3) - ax) + _2q2 * (2f *(q1q2 + q3q4) - ay) + -_4bz * q3 * (_4bx * (0.5f -q3q3 - q4q4) + _4bz * (q2q4 - q1q3) - mx) + (-_4bx * q4 + _4bz * q2) * (_4bx * (q2q3 - q1q4) + _4bz * (q1q2 + q3q4) - my) + _4bx * q3 * (_4bx * (q1q3 + q2q4) + _4bz * (0.5f - q2q2 - q3q3) - mz);
          s2 = (float)(_2q4 * (2f *(q2q4 - q1q3) - ax) + _2q1 * (2f *(q1q2 + q3q4) - ay) - 4f * q2 * (2  *(0.5 - q2q2 - q3q3) - az) + _4bz * q4 * (_4bx * (0.5f - q3q3 - q4q4) + _4bz * (q2q4 - q1q3) - mx) + (_4bx * q3 + _4bz * q1) * (_4bx * (q2q3 - q1q4) + _4bz * (q1q2 + q3q4) - my) + (_4bx * q4 - _8bz * q2) * (_4bx * (q1q3 + q2q4) + _4bz * (0.5f - q2q2 - q3q3) - mz));
          s3 = (float)(-_2q1 * (2f * (q2q4 -q1q3) - ax) + _2q4 * (2f * (q1q2 + q3q4) - ay) +( -4f *q3) * (2 * (0.5 - q2q2 - q3q3) - az) + (-_8bx * q3 - _4bz * q1) * (_4bx * (0.5f - q3q3 - q4q4) + _4bz * (q2q4 - q1q3) - mx) + (_4bx * q2 + _4bz * q4) * (_4bx * (q2q3 - q1q4) + _4bz * (q1q2 + q3q4) - my) + (_4bx * q1 - _8bz * q3) * (_4bx * (q1q3 + q2q4) + _4bz * (0.5f - q2q2 - q3q3) - mz));
          s4 = _2q2 * (2f *(q2q4 - q1q3) - ax) + _2q3 * (2f *(q1q2 + q3q4) - ay) + (-_8bx * q4 + _4bz * q2) * (_4bx * (0.5f - q3q3- q4q4) + _4bz * (q2q4 - q1q3) - mx) + (-_4bx * q1 + _4bz * q3) * (_4bx * (q2q3 - q1q4) + _4bz * (q1q2 + q3q4) - my) + (_4bx * q2) * (_4bx * (q1q3 + q2q4) + _4bz * (0.5f - q2q2 - q3q3) - mz);
          norm = 1f / (float)Math.sqrt(s1 * s1 + s2 * s2 + s3 * s3 + s4 * s4);
  
          s1 *= norm;
          s2 *= norm;
          s3 *= norm;
          s4 *= norm;
  
          // Compute rate of change of quaternion
          qDot1 = 0.5f * (-q2 * gx - q3 * gy - q4 * gz) - Beta * s1;
          qDot2 = 0.5f * (q1 * gx + q3 * gz - q4 * gy) - Beta * s2;
          qDot3 = 0.5f * (q1 * gy - q2 * gz + q4 * gx) - Beta * s3;
          qDot4 = 0.5f * (q1 * gz + q2 * gy - q3 * gx) - Beta * s4;
  
          // Integrate to yield quaternion
          q1 += qDot1 * SamplePeriod;
          q2 += qDot2 * SamplePeriod;
          q3 += qDot3 * SamplePeriod;
          q4 += qDot4 * SamplePeriod;
          norm = 1f / (float)Math.sqrt(q1 * q1 + q2 * q2 + q3 * q3 + q4 * q4);    // normalise quaternion
          Quaternion[0] = q1 * norm;
          Quaternion[1] = q2 * norm;
          Quaternion[2] = q3 * norm;
          Quaternion[3] = q4 * norm;
  
      }
  
      public double[] computeAngles()
      {
          double roll = Math.atan2(Quaternion[0] * Quaternion[1] + Quaternion[2] * Quaternion[3], 0.5f - Quaternion[1] * Quaternion[1] - Quaternion[2] * Quaternion[2]);
          double pitch = Math.asin(-2.0f * (Quaternion[1] * Quaternion[3] - Quaternion[0] * Quaternion[2]));
          double yaw = Math.atan2(Quaternion[1] * Quaternion[2] + Quaternion[0] * Quaternion[3], 0.5f - Quaternion[2] * Quaternion[2] - Quaternion[3] * Quaternion[3]);
          return new double[] { roll,pitch,yaw };
  
      }
  }
  </code></pre>
  
  <p>Charts of Input data: 
  <a href="https://i.stack.imgur.com/B41Gb.png" rel="noreferrer">Input data</a></p>