MadgwickAHRS.cpp

00001 //=============================================================================================
00002 // MadgwickAHRS.c
00003 //=============================================================================================
00004 //
00005 // Implementation of Madgwick's IMU and AHRS algorithms.
00006 // See: http://www.x-io.co.uk/open-source-imu-and-ahrs-algorithms/
00007 //
00008 // From the x-io website "Open-source resources available on this website are
00009 // provided under the GNU General Public Licence unless an alternative licence
00010 // is provided in source."
00011 //
00012 // Date         Author          Notes
00013 // 29/09/2011   SOH Madgwick    Initial release
00014 // 02/10/2011   SOH Madgwick    Optimised for reduced CPU load
00015 // 19/02/2012   SOH Madgwick    Magnetometer measurement is normalised
00016 // 18/12/2016                   Added better fast inverse square root
00017 //
00018 //=============================================================================================
00019 
00020 //-------------------------------------------------------------------------------------------
00021 // Header files
00022 
00023 #include "MadgwickAHRS.h"
00024 #include <math.h>
00025 
00026 //-------------------------------------------------------------------------------------------
00027 // Definitions
00028 
00029 #define betaDef         0.25f            // 2 * proportional gain 0.1 - 0.5 - 5; 8
00030 
00031 
00032 //============================================================================================
00033 // Functions
00034 
00035 //-------------------------------------------------------------------------------------------
00036 // AHRS algorithm update
00037 
00038 Madgwick::Madgwick(float Ts) {
00039     beta = betaDef;
00040     q0 = 1.0f;
00041     q1 = 0.0f;
00042     q2 = 0.0f;
00043     q3 = 0.0f;
00044     invSampleFreq = Ts;
00045     anglesComputed = 0;
00046 }
00047 
00048 void Madgwick::update(float gx, float gy, float gz, float ax, float ay, float az, float mx, float my, float mz) {
00049     float recipNorm;
00050     float s0, s1, s2, s3;
00051     float qDot1, qDot2, qDot3, qDot4;
00052     float hx, hy;
00053     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;
00054 
00055     // Use IMU algorithm if magnetometer measurement invalid (avoids NaN in magnetometer normalisation)
00056     if((mx == 0.0f) && (my == 0.0f) && (mz == 0.0f)) {
00057         updateIMU(gx, gy, gz, ax, ay, az);
00058         return;
00059     }
00060 
00061     // Convert gyroscope degrees/sec to radians/sec
00062     gx *= 0.0174533f;
00063     gy *= 0.0174533f;
00064     gz *= 0.0174533f;
00065 
00066     // Rate of change of quaternion from gyroscope
00067     qDot1 = 0.5f * (-q1 * gx - q2 * gy - q3 * gz);
00068     qDot2 = 0.5f * (q0 * gx + q2 * gz - q3 * gy);
00069     qDot3 = 0.5f * (q0 * gy - q1 * gz + q3 * gx);
00070     qDot4 = 0.5f * (q0 * gz + q1 * gy - q2 * gx);
00071 
00072     // Compute feedback only if accelerometer measurement valid (avoids NaN in accelerometer normalisation)
00073     if(!((ax == 0.0f) && (ay == 0.0f) && (az == 0.0f))) {
00074 
00075         // Normalise accelerometer measurement
00076         recipNorm = invSqrt(ax * ax + ay * ay + az * az);
00077         ax *= recipNorm;
00078         ay *= recipNorm;
00079         az *= recipNorm;
00080 
00081         // Normalise magnetometer measurement
00082         recipNorm = invSqrt(mx * mx + my * my + mz * mz);
00083         mx *= recipNorm;
00084         my *= recipNorm;
00085         mz *= recipNorm;
00086 
00087         // Auxiliary variables to avoid repeated arithmetic
00088         _2q0mx = 2.0f * q0 * mx;
00089         _2q0my = 2.0f * q0 * my;
00090         _2q0mz = 2.0f * q0 * mz;
00091         _2q1mx = 2.0f * q1 * mx;
00092         _2q0 = 2.0f * q0;
00093         _2q1 = 2.0f * q1;
00094         _2q2 = 2.0f * q2;
00095         _2q3 = 2.0f * q3;
00096         _2q0q2 = 2.0f * q0 * q2;
00097         _2q2q3 = 2.0f * q2 * q3;
00098         q0q0 = q0 * q0;
00099         q0q1 = q0 * q1;
00100         q0q2 = q0 * q2;
00101         q0q3 = q0 * q3;
00102         q1q1 = q1 * q1;
00103         q1q2 = q1 * q2;
00104         q1q3 = q1 * q3;
00105         q2q2 = q2 * q2;
00106         q2q3 = q2 * q3;
00107         q3q3 = q3 * q3;
00108 
00109         // Reference direction of Earth's magnetic field
00110         hx = mx * q0q0 - _2q0my * q3 + _2q0mz * q2 + mx * q1q1 + _2q1 * my * q2 + _2q1 * mz * q3 - mx * q2q2 - mx * q3q3;
00111         hy = _2q0mx * q3 + my * q0q0 - _2q0mz * q1 + _2q1mx * q2 - my * q1q1 + my * q2q2 + _2q2 * mz * q3 - my * q3q3;
00112         _2bx = sqrtf(hx * hx + hy * hy);
00113         _2bz = -_2q0mx * q2 + _2q0my * q1 + mz * q0q0 + _2q1mx * q3 - mz * q1q1 + _2q2 * my * q3 - mz * q2q2 + mz * q3q3;
00114         _4bx = 2.0f * _2bx;
00115         _4bz = 2.0f * _2bz;
00116 
00117         // Gradient decent algorithm corrective step
00118         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);
00119         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);
00120         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);
00121         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);
00122         recipNorm = invSqrt(s0 * s0 + s1 * s1 + s2 * s2 + s3 * s3); // normalise step magnitude
00123         s0 *= recipNorm;
00124         s1 *= recipNorm;
00125         s2 *= recipNorm;
00126         s3 *= recipNorm;
00127 
00128         // Apply feedback step
00129         qDot1 -= beta * s0;
00130         qDot2 -= beta * s1;
00131         qDot3 -= beta * s2;
00132         qDot4 -= beta * s3;
00133     }
00134 
00135     // Integrate rate of change of quaternion to yield quaternion
00136     q0 += qDot1 * invSampleFreq;
00137     q1 += qDot2 * invSampleFreq;
00138     q2 += qDot3 * invSampleFreq;
00139     q3 += qDot4 * invSampleFreq;
00140 
00141     // Normalise quaternion
00142     recipNorm = invSqrt(q0 * q0 + q1 * q1 + q2 * q2 + q3 * q3);
00143     q0 *= recipNorm;
00144     q1 *= recipNorm;
00145     q2 *= recipNorm;
00146     q3 *= recipNorm;
00147     anglesComputed = 0;
00148 }
00149 
00150 //-------------------------------------------------------------------------------------------
00151 // IMU algorithm update
00152 
00153 void Madgwick::updateIMU(float gx, float gy, float gz, float ax, float ay, float az) {
00154     float recipNorm;
00155     float s0, s1, s2, s3;
00156     float qDot1, qDot2, qDot3, qDot4;
00157     float _2q0, _2q1, _2q2, _2q3, _4q0, _4q1, _4q2 ,_8q1, _8q2, q0q0, q1q1, q2q2, q3q3;
00158 
00159     // Convert gyroscope degrees/sec to radians/sec
00160     gx *= 0.0174533f;
00161     gy *= 0.0174533f;
00162     gz *= 0.0174533f;
00163 
00164     // Rate of change of quaternion from gyroscope
00165     qDot1 = 0.5f * (-q1 * gx - q2 * gy - q3 * gz);
00166     qDot2 = 0.5f * (q0 * gx + q2 * gz - q3 * gy);
00167     qDot3 = 0.5f * (q0 * gy - q1 * gz + q3 * gx);
00168     qDot4 = 0.5f * (q0 * gz + q1 * gy - q2 * gx);
00169 
00170     // Compute feedback only if accelerometer measurement valid (avoids NaN in accelerometer normalisation)
00171     if(!((ax == 0.0f) && (ay == 0.0f) && (az == 0.0f))) {
00172 
00173         // Normalise accelerometer measurement
00174         recipNorm = invSqrt(ax * ax + ay * ay + az * az);
00175         ax *= recipNorm;
00176         ay *= recipNorm;
00177         az *= recipNorm;
00178 
00179         // Auxiliary variables to avoid repeated arithmetic
00180         _2q0 = 2.0f * q0;
00181         _2q1 = 2.0f * q1;
00182         _2q2 = 2.0f * q2;
00183         _2q3 = 2.0f * q3;
00184         _4q0 = 4.0f * q0;
00185         _4q1 = 4.0f * q1;
00186         _4q2 = 4.0f * q2;
00187         _8q1 = 8.0f * q1;
00188         _8q2 = 8.0f * q2;
00189         q0q0 = q0 * q0;
00190         q1q1 = q1 * q1;
00191         q2q2 = q2 * q2;
00192         q3q3 = q3 * q3;
00193 
00194         // Gradient decent algorithm corrective step
00195         s0 = _4q0 * q2q2 + _2q2 * ax + _4q0 * q1q1 - _2q1 * ay;
00196         s1 = _4q1 * q3q3 - _2q3 * ax + 4.0f * q0q0 * q1 - _2q0 * ay - _4q1 + _8q1 * q1q1 + _8q1 * q2q2 + _4q1 * az;
00197         s2 = 4.0f * q0q0 * q2 + _2q0 * ax + _4q2 * q3q3 - _2q3 * ay - _4q2 + _8q2 * q1q1 + _8q2 * q2q2 + _4q2 * az;
00198         s3 = 4.0f * q1q1 * q3 - _2q1 * ax + 4.0f * q2q2 * q3 - _2q2 * ay;
00199         recipNorm = invSqrt(s0 * s0 + s1 * s1 + s2 * s2 + s3 * s3); // normalise step magnitude
00200         s0 *= recipNorm;
00201         s1 *= recipNorm;
00202         s2 *= recipNorm;
00203         s3 *= recipNorm;
00204 
00205         // Apply feedback step
00206         qDot1 -= beta * s0;
00207         qDot2 -= beta * s1;
00208         qDot3 -= beta * s2;
00209         qDot4 -= beta * s3;
00210     }
00211 
00212     // Integrate rate of change of quaternion to yield quaternion
00213     q0 += qDot1 * invSampleFreq;
00214     q1 += qDot2 * invSampleFreq;
00215     q2 += qDot3 * invSampleFreq;
00216     q3 += qDot4 * invSampleFreq;
00217 
00218     // Normalise quaternion
00219     recipNorm = invSqrt(q0 * q0 + q1 * q1 + q2 * q2 + q3 * q3);
00220     q0 *= recipNorm;
00221     q1 *= recipNorm;
00222     q2 *= recipNorm;
00223     q3 *= recipNorm;
00224     anglesComputed = 0;
00225 }
00226 
00227 //-------------------------------------------------------------------------------------------
00228 // Fast inverse square-root
00229 // See: http://en.wikipedia.org/wiki/Fast_inverse_square_root
00230 
00231 /*float Madgwick::invSqrt(float x) {
00232     float halfx = 0.5f * x;
00233     float y = x;
00234     long i = *(long*)&y;
00235     i = 0x5f3759df - (i>>1);
00236     y = *(float*)&i;
00237     y = y * (1.5f - (halfx * y * y));
00238     y = y * (1.5f - (halfx * y * y));
00239     return y;
00240 } */
00241 
00242 float Madgwick::invSqrt(float x){
00243    unsigned int i = 0x5F1F1412 - (*(unsigned int*)&x >> 1);
00244    float tmp = *(float*)&i;
00245    return tmp * (1.69000231f - 0.714158168f * x * tmp * tmp);
00246 }
00247 
00248 //-------------------------------------------------------------------------------------------
00249 
00250 void Madgwick::computeAngles()
00251 {
00252     roll = atan2f(q0*q1 + q2*q3, 0.5f - q1*q1 - q2*q2);
00253     pitch = asinf(-2.0f * (q1*q3 - q0*q2));
00254     yaw = atan2f(q1*q2 + q0*q3, 0.5f - q2*q2 - q3*q3);
00255     anglesComputed = 1;
00256 }