一篇寫的非常好的解析:
IMU Data Fusing: Complementary, Kalman, and Mahony Filter
關於Madgwick算法的介紹:
macrodefs.h代碼(主要是宏定義):
#ifndef MACRODEFS_H
#define MACRODEFS_H
// CAN Frame Micro Defines
#define GYRO_X_Y 0x00000066
#define GYRO_Z_Temp 0x00000067
#define ACCEL_X_Y_Z 0x00000069
#define GYRO_FULL_32_BIT ((unsigned int)(0xFFFFFFFF))
#define ACCEL_FULL_16_BIT ((unsigned int)(0xFFFF))
#define GYRO_RATE_LSB 0.001
#define ACCEL_RATE_LSB 0.0001
#define TEMP_IMU_LSB 1
/* Blow macros are imported from http://mbed.org/cookbook/IMU and may not be used
*/
//Gravity at Earth's surface in m/s/s
#define g0 9.812865328
//Number of samples to average.
#define SAMPLES 4
//Number of samples to be averaged for a null bias calculation
//during calibration.
#define CALIBRATION_SAMPLES 128
//Convert from radians to degrees.
#define toDegrees(x) (x * 57.2957795)
//Convert from degrees to radians.
#define toRadians(x) (x * 0.01745329252)
//ITG-3200 sensitivity is 14.375 LSB/(degrees/sec).
#define GYROSCOPE_GAIN (1 / 14.375)
//Full scale resolution on the ADXL345 is 4mg/LSB.
#define ACCELEROMETER_GAIN (0.004 * g0)
//Sampling gyroscope at 200Hz.
#define GYRO_RATE 0.005
//Sampling accelerometer at 200Hz.
#define ACC_RATE 0.005
//Updating filter at 1/FILTER_RATE Hz.
#define FILTER_RATE 0.01
// Timer Interval, namely 1000*FILTER_RATE
#define TIMER_INTERVAL 10
// Sample Frames for Calibration
#define SAMPLE_FRAMES_FOR_CALIBRATION 10000
#endif // MACRODEFS_H
IMUfilter.h代碼:
/**
* @author Aaron Berk
*
* @section LICENSE
*
* Copyright (c) 2010 ARM Limited
*
* Permission is hereby granted, free of charge, to any person obtaining a copy
* of this software and associated documentation files (the "Software"), to deal
* in the Software without restriction, including without limitation the rights
* to use, copy, modify, merge, publish, distribute, sublicense, and/or sell
* copies of the Software, and to permit persons to whom the Software is
* furnished to do so, subject to the following conditions:
*
* The above copyright notice and this permission notice shall be included in
* all copies or substantial portions of the Software.
*
* THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR
* IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
* FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE
* AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER
* LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM,
* OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN
* THE SOFTWARE.
*
* @section DESCRIPTION
*
* IMU orientation filter developed by Sebastian Madgwick.
*
* Find more details about his paper here:
*
* http://code.google.com/p/imumargalgorithm30042010sohm/
*/
#ifndef IMU_FILTER_H
#define IMU_FILTER_H
/**
* Includes
*/
#include <math.h>
#include "macrodefs.h"
/**
* Defines
*/
#define PI 3.1415926536
/**
* IMU orientation filter.
*/
class IMUfilter {
public:
/**
* Constructor.
*
* Initializes filter variables.
*
* @param rate The rate at which the filter should be updated.
* @param gyroscopeMeasurementError The error of the gyroscope in degrees
* per second. This used to calculate a tuning constant for the filter.
* Try changing this value if there are jittery readings, or they change
* too much or too fast when rotating the IMU.陀螺儀誤差
*/
IMUfilter(double rate, double gyroscopeMeasurementError);
/**
* Update the filter variables.
*
* @param w_x X-axis gyroscope reading in rad/s.
* @param w_y Y-axis gyroscope reading in rad/s.
* @param w_z Z-axis gyroscope reading in rad/s.
* @param a_x X-axis accelerometer reading in m/s/s.
* @param a_y Y-axis accelerometer reading in m/s/s.
* @param a_z Z-axis accelerometer reading in m/s/s.
*/
void updateFilter(double w_x, double w_y, double w_z,
double a_x, double a_y, double a_z);
/**
* Compute the Euler angles based on the current filter data.
*/
void computeEuler(void);
/**
* Get the current roll.
*
* @return The current roll angle in radians.
*/
double getRoll(void);
/**
* Get the current roll.
*
* @return The current roll angle in degrees.
*/
double getRollInDegrees(void);
/**
* Get the current pitch.
*
* @return The current pitch angle in radians.
*/
double getPitch(void);
/**
* Get the current pitch.
*
* @return The current pitch angle in degrees.
*/
double getPitchInDegrees(void);
/**
* Get the current yaw.
*
* @return The current yaw angle in radians.
*/
double getYaw(void);
/**
* Get the current yaw.
*
* @return The current yaw angle in degrees.
*/
double getYawInDegrees(void);
/**
* Reset the filter.
*/
void reset(void);
private:
int firstUpdate;
//Quaternion orientation of earth frame relative to auxiliary frame.
double AEq_1;
double AEq_2;
double AEq_3;
double AEq_4;
//Estimated orientation quaternion elements with initial conditions.
double SEq_1;
double SEq_2;
double SEq_3;
double SEq_4;
//Sampling period
double deltat;
//Gyroscope measurement error (in degrees per second).
double gyroMeasError;
//Compute beta (filter tuning constant..
double beta;
double phi;
double theta;
double psi;
};
#endif /* IMU_FILTER_H */
IMUfilter.cpp代碼:
/**
* @author Aaron Berk
*
* @section LICENSE
*
* Copyright (c) 2010 ARM Limited
*
* Permission is hereby granted, free of charge, to any person obtaining a copy
* of this software and associated documentation files (the "Software"), to deal
* in the Software without restriction, including without limitation the rights
* to use, copy, modify, merge, publish, distribute, sublicense, and/or sell
* copies of the Software, and to permit persons to whom the Software is
* furnished to do so, subject to the following conditions:
*
* The above copyright notice and this permission notice shall be included in
* all copies or substantial portions of the Software.
*
* THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR
* IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
* FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE
* AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER
* LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM,
* OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN
* THE SOFTWARE.
*
* @section DESCRIPTION
*
* IMU orientation filter developed by Sebastian Madgwick.
*
* Find more details about his paper here:
*
* http://code.google.com/p/imumargalgorithm30042010sohm/
*/
/**
* Includes
*/
#include "IMUfilter.h"
IMUfilter::IMUfilter(double rate, double gyroscopeMeasurementError){
firstUpdate = 0;
//Quaternion orientation of earth frame relative to auxiliary frame.
AEq_1 = 1;
AEq_2 = 0;
AEq_3 = 0;
AEq_4 = 0;
//Estimated orientation quaternion elements with initial conditions.
SEq_1 = 1;
SEq_2 = 0;
SEq_3 = 0;
SEq_4 = 0;
//Sampling period (typical value is ~0.1s).
deltat = rate;
//Gyroscope measurement error (in degrees per second).
gyroMeasError = gyroscopeMeasurementError;
//Compute beta.
beta = sqrt(3.0 / 4.0) * (PI * (gyroMeasError / 180.0));
}
void IMUfilter::updateFilter(double w_x, double w_y, double w_z, double a_x, double a_y, double a_z) {
//Local system variables.
//Vector norm.
double norm;
//Quaternion rate from gyroscope elements.
double SEqDot_omega_1;
double SEqDot_omega_2;
double SEqDot_omega_3;
double SEqDot_omega_4;
//Objective function elements.
double f_1;
double f_2;
double f_3;
//Objective function Jacobian elements.
double J_11or24;
double J_12or23;
double J_13or22;
double J_14or21;
double J_32;
double J_33;
//Objective function gradient elements.
double nablaf_1;
double nablaf_2;
double nablaf_3;
double nablaf_4;
//Auxiliary variables to avoid reapeated calcualtions.
double halfSEq_1 = 0.5 * SEq_1;
double halfSEq_2 = 0.5 * SEq_2;
double halfSEq_3 = 0.5 * SEq_3;
double halfSEq_4 = 0.5 * SEq_4;
double twoSEq_1 = 2.0 * SEq_1;
double twoSEq_2 = 2.0 * SEq_2;
double twoSEq_3 = 2.0 * SEq_3;
//Compute the quaternion rate measured by gyroscopes.
SEqDot_omega_1 = -halfSEq_2 * w_x - halfSEq_3 * w_y - halfSEq_4 * w_z;
SEqDot_omega_2 = halfSEq_1 * w_x + halfSEq_3 * w_z - halfSEq_4 * w_y;
SEqDot_omega_3 = halfSEq_1 * w_y - halfSEq_2 * w_z + halfSEq_4 * w_x;
SEqDot_omega_4 = halfSEq_1 * w_z + halfSEq_2 * w_y - halfSEq_3 * w_x;
//Normalise the accelerometer measurement.
norm = sqrt(a_x * a_x + a_y * a_y + a_z * a_z);
a_x /= norm;
a_y /= norm;
a_z /= norm;
//Compute the objective function and Jacobian.
f_1 = twoSEq_2 * SEq_4 - twoSEq_1 * SEq_3 - a_x;
f_2 = twoSEq_1 * SEq_2 + twoSEq_3 * SEq_4 - a_y;
f_3 = 1.0 - twoSEq_2 * SEq_2 - twoSEq_3 * SEq_3 - a_z;
//J_11 negated in matrix multiplication.
J_11or24 = twoSEq_3;
J_12or23 = 2 * SEq_4;
//J_12 negated in matrix multiplication
J_13or22 = twoSEq_1;
J_14or21 = twoSEq_2;
//Negated in matrix multiplication.
J_32 = 2 * J_14or21;
//Negated in matrix multiplication.
J_33 = 2 * J_11or24;
//Compute the gradient (matrix multiplication).
nablaf_1 = J_14or21 * f_2 - J_11or24 * f_1;
nablaf_2 = J_12or23 * f_1 + J_13or22 * f_2 - J_32 * f_3;
nablaf_3 = J_12or23 * f_2 - J_33 * f_3 - J_13or22 * f_1;
nablaf_4 = J_14or21 * f_1 + J_11or24 * f_2;
//Normalise the gradient.
norm = sqrt(nablaf_1 * nablaf_1 + nablaf_2 * nablaf_2 + nablaf_3 * nablaf_3 + nablaf_4 * nablaf_4);
nablaf_1 /= norm;
nablaf_2 /= norm;
nablaf_3 /= norm;
nablaf_4 /= norm;
//Compute then integrate the estimated quaternion rate.
SEq_1 += (SEqDot_omega_1 - (beta * nablaf_1)) * deltat;
SEq_2 += (SEqDot_omega_2 - (beta * nablaf_2)) * deltat;
SEq_3 += (SEqDot_omega_3 - (beta * nablaf_3)) * deltat;
SEq_4 += (SEqDot_omega_4 - (beta * nablaf_4)) * deltat;
//Normalise quaternion
norm = sqrt(SEq_1 * SEq_1 + SEq_2 * SEq_2 + SEq_3 * SEq_3 + SEq_4 * SEq_4);
SEq_1 /= norm;
SEq_2 /= norm;
SEq_3 /= norm;
SEq_4 /= norm;
if (firstUpdate == 0) {
//Store orientation of auxiliary frame.
AEq_1 = SEq_1;
AEq_2 = SEq_2;
AEq_3 = SEq_3;
AEq_4 = SEq_4;
firstUpdate = 1;
}
}
void IMUfilter::computeEuler(void){
//Quaternion describing orientation of sensor relative to earth.
double ESq_1, ESq_2, ESq_3, ESq_4;
//Quaternion describing orientation of sensor relative to auxiliary frame.
double ASq_1, ASq_2, ASq_3, ASq_4;
//Compute the quaternion conjugate.
ESq_1 = SEq_1;
ESq_2 = -SEq_2;
ESq_3 = -SEq_3;
ESq_4 = -SEq_4;
//Compute the quaternion product.
ASq_1 = ESq_1 * AEq_1 - ESq_2 * AEq_2 - ESq_3 * AEq_3 - ESq_4 * AEq_4;
ASq_2 = ESq_1 * AEq_2 + ESq_2 * AEq_1 + ESq_3 * AEq_4 - ESq_4 * AEq_3;
ASq_3 = ESq_1 * AEq_3 - ESq_2 * AEq_4 + ESq_3 * AEq_1 + ESq_4 * AEq_2;
ASq_4 = ESq_1 * AEq_4 + ESq_2 * AEq_3 - ESq_3 * AEq_2 + ESq_4 * AEq_1;
//Compute the Euler angles from the quaternion.
phi = atan2(2 * ASq_3 * ASq_4 - 2 * ASq_1 * ASq_2, 2 * ASq_1 * ASq_1 + 2 * ASq_4 * ASq_4 - 1);
theta = asin(2 * ASq_2 * ASq_3 - 2 * ASq_1 * ASq_3);
psi = atan2(2 * ASq_2 * ASq_3 - 2 * ASq_1 * ASq_4, 2 * ASq_1 * ASq_1 + 2 * ASq_2 * ASq_2 - 1);
}
double IMUfilter::getRoll(void){
return phi;
}
double IMUfilter::getPitch(void){
return theta;
}
double IMUfilter::getYaw(void){
return psi;
}
// Degrees
double IMUfilter::getRollInDegrees(void){
return toDegrees(phi);
}
double IMUfilter::getPitchInDegrees(void){
return toDegrees(theta);
}
double IMUfilter::getYawInDegrees(void){
return toDegrees(psi);
}
void IMUfilter::reset(void) {
firstUpdate = 0;
//Quaternion orientation of earth frame relative to auxiliary frame.
AEq_1 = 1;
AEq_2 = 0;
AEq_3 = 0;
AEq_4 = 0;
//Estimated orientation quaternion elements with initial conditions.
SEq_1 = 1;
SEq_2 = 0;
SEq_3 = 0;
SEq_4 = 0;
}