1 //=====================================================================================================
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3 //=====================================================================================================
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5 // Implementation of Madgwick's IMU and AHRS algorithms.
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6 // See: http://www.x-io.co.uk/node/8#open_source_ahrs_and_imu_algorithms
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9 // 29/09/2011 SOH Madgwick Initial release
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10 // 02/10/2011 SOH Madgwick Optimised for reduced CPU load
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11 // 19/02/2012 SOH Madgwick Magnetometer measurement is normalised
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13 //=====================================================================================================
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15 //---------------------------------------------------------------------------------------------------
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18 #include "MadgwickAHRS.h"
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24 //---------------------------------------------------------------------------------------------------
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27 //#define sampleFreq 512.0f // sample frequency in Hz
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28 //#define sampleFreq 46.0f // sample frequency in Hz
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29 #define sampleFreq 85.0f // sample frequency in Hz
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30 #define betaDef 0.1f // 2 * proportional gain
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32 //---------------------------------------------------------------------------------------------------
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33 // Variable definitions
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35 volatile float beta = betaDef; // 2 * proportional gain (Kp)
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36 volatile float q0 = 1.0f, q1 = 0.0f, q2 = 0.0f, q3 = 0.0f; // quaternion of sensor frame relative to auxiliary frame
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39 //---------------------------------------------------------------------------------------------------
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40 // Function declarations
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42 float invSqrt(float x);
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44 //====================================================================================================
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47 //---------------------------------------------------------------------------------------------------
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48 // AHRS algorithm update
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50 void MadgwickAHRSupdate(float gx, float gy, float gz, float ax, float ay, float az, float mx, float my, float mz) {
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52 float s0, s1, s2, s3;
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53 float qDot1, qDot2, qDot3, qDot4;
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55 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;
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57 // Use IMU algorithm if magnetometer measurement invalid (avoids NaN in magnetometer normalisation)
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58 if((mx == 0.0f) && (my == 0.0f) && (mz == 0.0f)) {
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59 MadgwickAHRSupdateIMU(gx, gy, gz, ax, ay, az);
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63 // Rate of change of quaternion from gyroscope
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64 qDot1 = 0.5f * (-q1 * gx - q2 * gy - q3 * gz);
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65 qDot2 = 0.5f * (q0 * gx + q2 * gz - q3 * gy);
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66 qDot3 = 0.5f * (q0 * gy - q1 * gz + q3 * gx);
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67 qDot4 = 0.5f * (q0 * gz + q1 * gy - q2 * gx);
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69 // Compute feedback only if accelerometer measurement valid (avoids NaN in accelerometer normalisation)
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70 if(!((ax == 0.0f) && (ay == 0.0f) && (az == 0.0f))) {
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72 // Normalise accelerometer measurement
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73 recipNorm = invSqrt(ax * ax + ay * ay + az * az);
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78 // Normalise magnetometer measurement
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79 recipNorm = invSqrt(mx * mx + my * my + mz * mz);
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84 // Auxiliary variables to avoid repeated arithmetic
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85 _2q0mx = 2.0f * q0 * mx;
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86 _2q0my = 2.0f * q0 * my;
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87 _2q0mz = 2.0f * q0 * mz;
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88 _2q1mx = 2.0f * q1 * mx;
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93 _2q0q2 = 2.0f * q0 * q2;
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94 _2q2q3 = 2.0f * q2 * q3;
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106 // Reference direction of Earth's magnetic field
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107 hx = mx * q0q0 - _2q0my * q3 + _2q0mz * q2 + mx * q1q1 + _2q1 * my * q2 + _2q1 * mz * q3 - mx * q2q2 - mx * q3q3;
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108 hy = _2q0mx * q3 + my * q0q0 - _2q0mz * q1 + _2q1mx * q2 - my * q1q1 + my * q2q2 + _2q2 * mz * q3 - my * q3q3;
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109 _2bx = sqrt(hx * hx + hy * hy);
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110 _2bz = -_2q0mx * q2 + _2q0my * q1 + mz * q0q0 + _2q1mx * q3 - mz * q1q1 + _2q2 * my * q3 - mz * q2q2 + mz * q3q3;
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111 _4bx = 2.0f * _2bx;
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112 _4bz = 2.0f * _2bz;
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114 // Gradient decent algorithm corrective step
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115 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);
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116 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);
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117 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);
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118 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);
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119 recipNorm = invSqrt(s0 * s0 + s1 * s1 + s2 * s2 + s3 * s3); // normalise step magnitude
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125 // Apply feedback step
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126 qDot1 -= beta * s0;
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127 qDot2 -= beta * s1;
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128 qDot3 -= beta * s2;
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129 qDot4 -= beta * s3;
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132 // Integrate rate of change of quaternion to yield quaternion
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133 q0 += qDot1 * (1.0f / sampleFreq);
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134 q1 += qDot2 * (1.0f / sampleFreq);
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135 q2 += qDot3 * (1.0f / sampleFreq);
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136 q3 += qDot4 * (1.0f / sampleFreq);
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138 // Normalise quaternion
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139 recipNorm = invSqrt(q0 * q0 + q1 * q1 + q2 * q2 + q3 * q3);
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146 //---------------------------------------------------------------------------------------------------
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147 // IMU algorithm update
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149 void MadgwickAHRSupdateIMU(float gx, float gy, float gz, float ax, float ay, float az) {
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151 float s0, s1, s2, s3;
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152 float qDot1, qDot2, qDot3, qDot4;
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153 float _2q0, _2q1, _2q2, _2q3, _4q0, _4q1, _4q2 ,_8q1, _8q2, q0q0, q1q1, q2q2, q3q3;
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155 // Rate of change of quaternion from gyroscope
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156 qDot1 = 0.5f * (-q1 * gx - q2 * gy - q3 * gz);
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157 qDot2 = 0.5f * (q0 * gx + q2 * gz - q3 * gy);
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158 qDot3 = 0.5f * (q0 * gy - q1 * gz + q3 * gx);
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159 qDot4 = 0.5f * (q0 * gz + q1 * gy - q2 * gx);
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162 // Compute feedback only if accelerometer measurement valid (avoids NaN in accelerometer normalisation)
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163 if(!((ax == 0.0f) && (ay == 0.0f) && (az == 0.0f))) {
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165 // Normalise accelerometer measurement
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166 recipNorm = invSqrt(ax * ax + ay * ay + az * az);
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171 // Auxiliary variables to avoid repeated arithmetic
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187 // Gradient decent algorithm corrective step
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188 s0 = _4q0 * q2q2 + _2q2 * ax + _4q0 * q1q1 - _2q1 * ay;
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189 s1 = _4q1 * q3q3 - _2q3 * ax + 4.0f * q0q0 * q1 - _2q0 * ay - _4q1 + _8q1 * q1q1 + _8q1 * q2q2 + _4q1 * az;
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190 s2 = 4.0f * q0q0 * q2 + _2q0 * ax + _4q2 * q3q3 - _2q3 * ay - _4q2 + _8q2 * q1q1 + _8q2 * q2q2 + _4q2 * az;
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191 s3 = 4.0f * q1q1 * q3 - _2q1 * ax + 4.0f * q2q2 * q3 - _2q2 * ay;
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192 recipNorm = invSqrt(s0 * s0 + s1 * s1 + s2 * s2 + s3 * s3); // normalise step magnitude
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202 // Apply feedback step
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203 qDot1 -= beta * s0;
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204 qDot2 -= beta * s1;
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205 qDot3 -= beta * s2;
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206 qDot4 -= beta * s3;
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209 // Integrate rate of change of quaternion to yield quaternion
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210 q0 += qDot1 * (1.0f / sampleFreq);
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211 q1 += qDot2 * (1.0f / sampleFreq);
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212 q2 += qDot3 * (1.0f / sampleFreq);
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213 q3 += qDot4 * (1.0f / sampleFreq);
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216 // Normalise quaternion
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217 recipNorm = invSqrt(q0 * q0 + q1 * q1 + q2 * q2 + q3 * q3);
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223 //printf("%+3.3f\t%+3.3f\t%+3.3f\r\n", q0, q1, q2);
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228 //---------------------------------------------------------------------------------------------------
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229 // IMU algorithm update
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237 void Mad_f32_init()
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239 f_q0 = f32_from_double((double)1.0);
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240 f_q1 = f32_from_double((double)0.0);
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241 f_q2 = f32_from_double((double)0.0);
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242 f_q3 = f32_from_double((double)0.0);
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245 void MadgwickAHRSupdateIMU_f32(float gx, float gy, float gz, float ax, float ay, float az) {
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247 f32 f_s0, f_s1, f_s2, f_s3;
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248 f32 f_qDot1, f_qDot2, f_qDot3, f_qDot4;
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249 f32 f__2q0, f__2q1, f__2q2, f__2q3, f__4q0, f__4q1, f__4q2 ,f__8q1, f__8q2, f_q0q0, f_q1q1, f_q2q2, f_q3q3;
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251 f32 f_gx, f_gy, f_gz;
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252 f32 f_ax, f_ay, f_az;
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258 f_gx = f32_from_double((double)gx);
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259 f_gy = f32_from_double((double)gy);
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260 f_gz = f32_from_double((double)gz);
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262 f_ax = f32_from_double((double)ax);
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263 f_ay = f32_from_double((double)ay);
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264 f_az = f32_from_double((double)az);
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266 f_beta = f32_from_double((double)beta);
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267 f_sampleFreq = f32_from_double((double)sampleFreq);
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270 // Rate of change of quaternion from gyroscope
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272 qDot1 = 0.5f * (-q1 * gx - q2 * gy - q3 * gz);
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273 qDot2 = 0.5f * (q0 * gx + q2 * gz - q3 * gy);
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274 qDot3 = 0.5f * (q0 * gy - q1 * gz + q3 * gx);
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275 qDot4 = 0.5f * (q0 * gz + q1 * gy - q2 * gx);
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278 f_qDot1 = f32_mul(f32_sub(f32_sub(f32_neg(f32_mul(f_gx, f_q1)), f32_mul(f_gy, f_q2)), f32_mul(f_gz, f_q3)), f32_from_double(0.5));
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279 f_qDot2 = f32_mul(f32_sub(f32_add(f32_mul(f_gx, f_q0), f32_mul(f_gz, f_q2)), f32_mul(f_gy, f_q3)), f32_from_double(0.5));
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280 f_qDot3 = f32_mul(f32_add(f32_mul(f_gx, f_q3), f32_sub(f32_mul(f_gy, f_q0), f32_mul(f_gz, f_q1))), f32_from_double(0.5));
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281 f_qDot4 = f32_mul(f32_sub(f32_add(f32_mul(f_gy, f_q1), f32_mul(f_gz, f_q0)), f32_mul(f_gx, f_q2)), f32_from_double(0.5));
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286 // Compute feedback only if accelerometer measurement valid (avoids NaN in accelerometer normalisation)
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287 if(!((ax == 0.0f) && (ay == 0.0f) && (az == 0.0f))) {
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289 // Normalise accelerometer measurement
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290 //recipNorm = invSqrt(ax * ax + ay * ay + az * az);
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291 f_recipNorm = f32_inv(f32_sqrt((f32_add(f32_mul(f_ax, f_ax), f32_add(f32_mul(f_ay, f_ay), f32_mul(f_az, f_az))))));
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298 f_ax = f32_mul(f_ax, f_recipNorm);
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299 f_ay = f32_mul(f_ay, f_recipNorm);
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300 f_az = f32_mul(f_az, f_recipNorm);
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302 // Auxiliary variables to avoid repeated arithmetic
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320 f__2q0 = f32_mul(f32_from_double(2.0f), f_q0);
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321 f__2q1 = f32_mul(f32_from_double(2.0f), f_q1);
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322 f__2q2 = f32_mul(f32_from_double(2.0f), f_q2);
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323 f__2q3 = f32_mul(f32_from_double(2.0f), f_q3);
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324 f__4q0 = f32_mul(f32_from_double(4.0f), f_q0);
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325 f__4q1 = f32_mul(f32_from_double(4.0f), f_q1);
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326 f__4q2 = f32_mul(f32_from_double(4.0f), f_q2);
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327 f__8q1 = f32_mul(f32_from_double(8.0f), f_q1);
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328 f__8q2 = f32_mul(f32_from_double(8.0f), f_q2);
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329 f_q0q0 = f32_mul(f_q0, f_q0);
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330 f_q1q1 = f32_mul(f_q1, f_q1);
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331 f_q2q2 = f32_mul(f_q2, f_q2);
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332 f_q3q3 = f32_mul(f_q3, f_q3);
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335 // Gradient decent algorithm corrective step
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338 s0 = _4q0 * q2q2 + _2q2 * ax + _4q0 * q1q1 - _2q1 * ay;
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339 s1 = _4q1 * q3q3 - _2q3 * ax + 4.0f * q0q0 * q1 - _2q0 * ay - _4q1 + _8q1 * q1q1 + _8q1 * q2q2 + _4q1 * az;
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340 s2 = 4.0f * q0q0 * q2 + _2q0 * ax + _4q2 * q3q3 - _2q3 * ay - _4q2 + _8q2 * q1q1 + _8q2 * q2q2 + _4q2 * az;
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341 s3 = 4.0f * q1q1 * q3 - _2q1 * ax + 4.0f * q2q2 * q3 - _2q2 * ay;
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342 recipNorm = invSqrt(s0 * s0 + s1 * s1 + s2 * s2 + s3 * s3); // normalise step magnitude
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345 f_s0 = f32_sub(f32_add(f32_mul(f__2q2, f_ax), f32_add(f32_mul(f__4q0, f_q1q1), f32_mul(f__4q0, f_q2q2))), f32_mul(f__2q1, f_ay));
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346 f_s1 = f32_add(f32_mul(f__4q1, f_az), f32_add(f32_mul(f__8q1, f_q1q1), f32_add(f32_mul(f__8q1, f_q2q2), f32_sub(f32_sub(f32_add(f32_mul(f_q0q0, f32_mul(f_q1, f32_from_double(4.0f))), f32_sub(f32_mul(f__4q1, f_q3q3), f32_mul(f__2q3, f_ax))), f32_mul(f__2q0, f_ay)), f__4q1))));
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347 f_s2 = f32_add(f32_mul(f__4q2, f_az), f32_add(f32_mul(f__8q2, f_q1q1), f32_add(f32_mul(f__8q2, f_q2q2), f32_sub(f32_sub(f32_add(f32_mul(f__2q0, f_ax), f32_add(f32_mul(f__4q2, f_q3q3), f32_mul(f_q0q0, f32_mul(f_q2, f32_from_double(4.0))))), f32_mul(f__2q3, f_ay)), f__4q2))));
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348 f_s3 = f32_sub(f32_add(f32_mul(f_q2q2, f32_mul(f_q3, f32_from_double(4.0))), f32_sub(f32_mul(f_q1q1, f32_mul(f_q3, f32_from_double(4.0))), f32_mul(f__2q1, f_ax))), f32_mul(f__2q2, f_ay));
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349 f_recipNorm = f32_inv(f32_sqrt(f32_add(f32_mul(f_s0, f_s0), f32_add(f32_mul(f_s1, f_s1), f32_add(f32_mul(f_s2, f_s2), f32_mul(f_s3, f_s3))))));
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358 f_s0 = f32_mul(f_s0, f_recipNorm);
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359 f_s1 = f32_mul(f_s1, f_recipNorm);
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360 f_s2 = f32_mul(f_s2, f_recipNorm);
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361 f_s3 = f32_mul(f_s3, f_recipNorm);
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364 // Apply feedback step
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367 qDot1 -= beta * s0;
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368 qDot2 -= beta * s1;
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369 qDot3 -= beta * s2;
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370 qDot4 -= beta * s3;
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373 f_qDot1 = f32_sub(f_qDot1, f32_mul(f_beta, f_s0));
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374 f_qDot2 = f32_sub(f_qDot2, f32_mul(f_beta, f_s1));
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375 f_qDot3 = f32_sub(f_qDot3, f32_mul(f_beta, f_s2));
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376 f_qDot4 = f32_sub(f_qDot4, f32_mul(f_beta, f_s3));
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380 // Integrate rate of change of quaternion to yield quaternion
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383 q0 += qDot1 * (1.0f / sampleFreq);
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384 q1 += qDot2 * (1.0f / sampleFreq);
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385 q2 += qDot3 * (1.0f / sampleFreq);
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386 q3 += qDot4 * (1.0f / sampleFreq);
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389 f_q0 = f32_add(f_q0, f32_mul(f_qDot1, f32_inv(f_sampleFreq)));
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390 f_q1 = f32_add(f_q1, f32_mul(f_qDot2, f32_inv(f_sampleFreq)));
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391 f_q2 = f32_add(f_q2, f32_mul(f_qDot3, f32_inv(f_sampleFreq)));
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392 f_q3 = f32_add(f_q3, f32_mul(f_qDot4, f32_inv(f_sampleFreq)));
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394 // Normalise quaternion
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395 //recipNorm = invSqrt(q0 * q0 + q1 * q1 + q2 * q2 + q3 * q3);
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397 f_recipNorm = f32_inv(f32_sqrt(f32_add(f32_mul(f_q0, f_q0), f32_add(f32_mul(f_q1, f_q1), f32_add(f32_mul(f_q2, f_q2), f32_mul(f_q3, f_q3))))));
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406 f_q0 = f32_mul(f_q0, f_recipNorm);
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407 f_q1 = f32_mul(f_q1, f_recipNorm);
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408 f_q2 = f32_mul(f_q2, f_recipNorm);
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409 f_q3 = f32_mul(f_q3, f_recipNorm);
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412 q0 = f32_to_double(f_q0);
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413 q1 = f32_to_double(f_q1);
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414 q2 = f32_to_double(f_q2);
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415 q3 = f32_to_double(f_q3);
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417 //printf("%+3.3f\t%+3.3f\t%+3.3f\r\n", q0, q1, q2);
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422 //---------------------------------------------------------------------------------------------------
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423 // Fast inverse square-root
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424 // See: http://en.wikipedia.org/wiki/Fast_inverse_square_root
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426 float invSqrt(float x) {
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427 float halfx = 0.5f * x;
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429 long i = *(long*)&y;
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430 i = 0x5f3759df - (i>>1);
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432 y = y * (1.5f - (halfx * y * y));
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436 float invSqrt(float x) {
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437 return 1.0f / sqrtf(x);
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439 //====================================================================================================
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441 //====================================================================================================
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