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Madgwick.js
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Madgwick.js
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//= ====================================================================================================
// Based on MadgwickAHRS.c
//= ====================================================================================================
//
// Implementation of Madgwick's IMU and AHRS algorithms.
// See: http://www.x-io.co.uk/node/8#open_source_ahrs_and_imu_algorithms
//
//= ====================================================================================================
/**
* @typedef {Object} EulerAngles
* @property {number} heading - The direction of the object. Angle around Z-axis.
* @property {number} pitch - The forward/backward attitude of the object. Angle around Y-axis.
* @property {number} roll - The sideways angle of the object. Angle around X-axis.
*/
/* eslint-disable one-var-declaration-per-line */
'use strict';
/**
* The Madgwick algorithm. See: http://www.x-io.co.uk/open-source-imu-and-ahrs-algorithms/.
*
* @param {number} sampleInterval - The sample interval in milliseconds.
* @param {Object} options - The options.
*/
module.exports = function Madgwick(sampleInterval, options) {
//---------------------------------------------------------------------------------------------------
// Definitions
options = options || {};
const sampleFreq = 1000 / sampleInterval; // sample frequency in Hz
let beta = options.beta || 0.4; // 2 * proportional gain - lower numbers are smoother, but take longer to get to correct attitude.
let initalised = options.doInitialisation === true ? false : true;
//---------------------------------------------------------------------------------------------------
// Variable definitions
let q0 = 1.0,
q1 = 0.0,
q2 = 0.0,
q3 = 0.0; // quaternion of sensor frame relative to auxiliary frame
let recipSampleFreq = 1.0 / sampleFreq;
//= ===================================================================================================
// Functions
//---------------------------------------------------------------------------------------------------
// IMU algorithm update
/**
* @param {number} gx - gryo x
* @param {number} gy - gyro y
* @param {number} gz - gyro z
* @param {number} ax - accel x
* @param {number} ay - accel y
* @param {number} az - accel z
*/
function madgwickAHRSUpdateIMU(gx, gy, gz, ax, ay, az) {
let recipNorm;
let s0, s1, s2, s3;
let qDot1, qDot2, qDot3, qDot4;
let v2q0, v2q1, v2q2, v2q3, v4q0, v4q1, v4q2, v8q1, v8q2, q0q0, q1q1, q2q2, q3q3;
// Rate of change of quaternion from gyroscope
qDot1 = 0.5 * (-q1 * gx - q2 * gy - q3 * gz);
qDot2 = 0.5 * (q0 * gx + q2 * gz - q3 * gy);
qDot3 = 0.5 * (q0 * gy - q1 * gz + q3 * gx);
qDot4 = 0.5 * (q0 * gz + q1 * gy - q2 * gx);
// Compute feedback only if accelerometer measurement valid (avoids NaN in accelerometer normalisation)
if (!(ax === 0.0 && ay === 0.0 && az === 0.0)) {
// Normalise accelerometer measurement
recipNorm = (ax * ax + ay * ay + az * az) ** -0.5;
ax *= recipNorm;
ay *= recipNorm;
az *= recipNorm;
// Auxiliary variables to avoid repeated arithmetic
v2q0 = 2.0 * q0;
v2q1 = 2.0 * q1;
v2q2 = 2.0 * q2;
v2q3 = 2.0 * q3;
v4q0 = 4.0 * q0;
v4q1 = 4.0 * q1;
v4q2 = 4.0 * q2;
v8q1 = 8.0 * q1;
v8q2 = 8.0 * q2;
q0q0 = q0 * q0;
q1q1 = q1 * q1;
q2q2 = q2 * q2;
q3q3 = q3 * q3;
// Gradient decent algorithm corrective step
s0 = v4q0 * q2q2 + v2q2 * ax + v4q0 * q1q1 - v2q1 * ay;
s1 = v4q1 * q3q3 - v2q3 * ax + 4.0 * q0q0 * q1 - v2q0 * ay - v4q1 + v8q1 * q1q1 + v8q1 * q2q2 + v4q1 * az;
s2 = 4.0 * q0q0 * q2 + v2q0 * ax + v4q2 * q3q3 - v2q3 * ay - v4q2 + v8q2 * q1q1 + v8q2 * q2q2 + v4q2 * az;
s3 = 4.0 * q1q1 * q3 - v2q1 * ax + 4.0 * q2q2 * q3 - v2q2 * ay;
recipNorm = (s0 * s0 + s1 * s1 + s2 * s2 + s3 * s3) ** -0.5; // 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 * recipSampleFreq;
q1 += qDot2 * recipSampleFreq;
q2 += qDot3 * recipSampleFreq;
q3 += qDot4 * recipSampleFreq;
// Normalise quaternion
recipNorm = (q0 * q0 + q1 * q1 + q2 * q2 + q3 * q3) ** -0.5;
q0 *= recipNorm;
q1 *= recipNorm;
q2 *= recipNorm;
q3 *= recipNorm;
}
function cross_product(ax, ay, az, bx, by, bz) {
return {
x: ay * bz - az * by,
y: az * bx - ax * bz,
z: ax * by - ay * bx,
};
}
/**
* @param {number} ax - accel x
* @param {number} ay - accel y
* @param {number} az - accel z
* @param {number} mx - mag x
* @param {number} my - mag y
* @param {number} mz - mag z
* @returns {EulerAngles} - The Euler angles, in radians.
*/
function eulerAnglesFromImuRad(ax, ay, az, mx, my, mz) {
const pitch = -Math.atan2(ax, Math.sqrt(ay * ay + az * az));
const tmp1 = cross_product(ax, ay, az, 1.0, 0.0, 0.0);
const tmp2 = cross_product(1.0, 0.0, 0.0, tmp1.x, tmp1.y, tmp1.z);
const roll = Math.atan2(tmp2.y, tmp2.z);
const cr = Math.cos(roll);
const sp = Math.sin(pitch);
const sr = Math.sin(roll);
const yh = my * cr - mz * sr;
const xh = mx * Math.cos(pitch) + my * sr * sp + mz * cr * sp;
const heading = -Math.atan2(yh, xh);
return {
heading,
pitch,
roll,
};
}
// Pinched from here: https://en.wikipedia.org/wiki/Conversion_between_quaternions_and_Euler_angles
function toQuaternion(eulerAngles) {
const cy = Math.cos(eulerAngles.heading * 0.5);
const sy = Math.sin(eulerAngles.heading * 0.5);
const cp = Math.cos(eulerAngles.pitch * 0.5);
const sp = Math.sin(eulerAngles.pitch * 0.5);
const cr = Math.cos(eulerAngles.roll * 0.5);
const sr = Math.sin(eulerAngles.roll * 0.5);
return {
w: cr * cp * cy + sr * sp * sy,
x: sr * cp * cy - cr * sp * sy,
y: cr * sp * cy + sr * cp * sy,
z: cr * cp * sy - sr * sp * cy,
};
}
/**
* Initalise the internal quaternion values. This function only needs to be
* called once at the beginning. The attitude will be set by the accelometer
* and the heading by the magnetometer.
*
* @param {number} ax - accel x
* @param {number} ay - accel y
* @param {number} az - accel z
* @param {number} mx - mag x
* @param {number} my - mag y
* @param {number} mz - mag z
*/
function init(ax, ay, az, mx, my, mz) {
const ea = eulerAnglesFromImuRad(ax, ay, az, mx, my, mz);
const iq = toQuaternion(ea);
// Normalise quaternion
const recipNorm = (iq.w * iq.w + iq.x * iq.x + iq.y * iq.y + iq.z * iq.z) ** -0.5;
q0 = iq.w * recipNorm;
q1 = iq.x * recipNorm;
q2 = iq.y * recipNorm;
q3 = iq.z * recipNorm;
initalised = true;
}
//---------------------------------------------------------------------------------------------------
// AHRS algorithm update
/**
* @param {number} gx - gryo x
* @param {number} gy - gyro y
* @param {number} gz - gyro z
* @param {number} ax - accel x
* @param {number} ay - accel y
* @param {number} az - accel z
* @param {number} mx - magetometer x
* @param {number} my - magetometer y
* @param {number} mz - magetometer z
* @param {number} deltaTimeSec
*/
function madgwickAHRSUpdate(gx, gy, gz, ax, ay, az, mx, my, mz, deltaTimeSec) {
recipSampleFreq = deltaTimeSec || recipSampleFreq;
if (!initalised) {
init(ax, ay, az, mx, my, mz);
}
let recipNorm;
let s0, s1, s2, s3;
let qDot1, qDot2, qDot3, qDot4;
let hx, hy;
let v2q0mx, v2q0my, v2q0mz, v2q1mx, v2bx, v2bz, v4bx, v4bz, v2q0, v2q1, v2q2, v2q3, v2q0q2, v2q2q3;
let q0q0, q0q1, q0q2, q0q3, q1q1, q1q2, q1q3, q2q2, q2q3, q3q3;
// Use IMU algorithm if magnetometer measurement invalid (avoids NaN in magnetometer normalisation)
if (mx === undefined || my === undefined || mz === undefined || (mx === 0 && my === 0 && mz === 0)) {
madgwickAHRSUpdateIMU(gx, gy, gz, ax, ay, az);
return;
}
// Rate of change of quaternion from gyroscope
qDot1 = 0.5 * (-q1 * gx - q2 * gy - q3 * gz);
qDot2 = 0.5 * (q0 * gx + q2 * gz - q3 * gy);
qDot3 = 0.5 * (q0 * gy - q1 * gz + q3 * gx);
qDot4 = 0.5 * (q0 * gz + q1 * gy - q2 * gx);
// Compute feedback only if accelerometer measurement valid (avoids NaN in accelerometer normalisation)
if (!(ax === 0.0 && ay === 0.0 && az === 0.0)) {
// Normalise accelerometer measurement
recipNorm = (ax * ax + ay * ay + az * az) ** -0.5;
ax *= recipNorm;
ay *= recipNorm;
az *= recipNorm;
// Normalise magnetometer measurement
recipNorm = (mx * mx + my * my + mz * mz) ** -0.5;
mx *= recipNorm;
my *= recipNorm;
mz *= recipNorm;
// Auxiliary variables to avoid repeated arithmetic
v2q0mx = 2.0 * q0 * mx;
v2q0my = 2.0 * q0 * my;
v2q0mz = 2.0 * q0 * mz;
v2q1mx = 2.0 * q1 * mx;
v2q0 = 2.0 * q0;
v2q1 = 2.0 * q1;
v2q2 = 2.0 * q2;
v2q3 = 2.0 * q3;
v2q0q2 = 2.0 * q0 * q2;
v2q2q3 = 2.0 * 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 - v2q0my * q3 + v2q0mz * q2 + mx * q1q1 + v2q1 * my * q2 + v2q1 * mz * q3 - mx * q2q2 - mx * q3q3;
hy = v2q0mx * q3 + my * q0q0 - v2q0mz * q1 + v2q1mx * q2 - my * q1q1 + my * q2q2 + v2q2 * mz * q3 - my * q3q3;
v2bx = Math.sqrt(hx * hx + hy * hy);
v2bz = -v2q0mx * q2 + v2q0my * q1 + mz * q0q0 + v2q1mx * q3 - mz * q1q1 + v2q2 * my * q3 - mz * q2q2 + mz * q3q3;
v4bx = 2.0 * v2bx;
v4bz = 2.0 * v2bz;
// Gradient decent algorithm corrective step
s0 =
-v2q2 * (2.0 * q1q3 - v2q0q2 - ax) +
v2q1 * (2.0 * q0q1 + v2q2q3 - ay) -
v2bz * q2 * (v2bx * (0.5 - q2q2 - q3q3) + v2bz * (q1q3 - q0q2) - mx) +
(-v2bx * q3 + v2bz * q1) * (v2bx * (q1q2 - q0q3) + v2bz * (q0q1 + q2q3) - my) +
v2bx * q2 * (v2bx * (q0q2 + q1q3) + v2bz * (0.5 - q1q1 - q2q2) - mz);
s1 =
v2q3 * (2.0 * q1q3 - v2q0q2 - ax) +
v2q0 * (2.0 * q0q1 + v2q2q3 - ay) -
4.0 * q1 * (1 - 2.0 * q1q1 - 2.0 * q2q2 - az) +
v2bz * q3 * (v2bx * (0.5 - q2q2 - q3q3) + v2bz * (q1q3 - q0q2) - mx) +
(v2bx * q2 + v2bz * q0) * (v2bx * (q1q2 - q0q3) + v2bz * (q0q1 + q2q3) - my) +
(v2bx * q3 - v4bz * q1) * (v2bx * (q0q2 + q1q3) + v2bz * (0.5 - q1q1 - q2q2) - mz);
s2 =
-v2q0 * (2.0 * q1q3 - v2q0q2 - ax) +
v2q3 * (2.0 * q0q1 + v2q2q3 - ay) -
4.0 * q2 * (1 - 2.0 * q1q1 - 2.0 * q2q2 - az) +
(-v4bx * q2 - v2bz * q0) * (v2bx * (0.5 - q2q2 - q3q3) + v2bz * (q1q3 - q0q2) - mx) +
(v2bx * q1 + v2bz * q3) * (v2bx * (q1q2 - q0q3) + v2bz * (q0q1 + q2q3) - my) +
(v2bx * q0 - v4bz * q2) * (v2bx * (q0q2 + q1q3) + v2bz * (0.5 - q1q1 - q2q2) - mz);
s3 =
v2q1 * (2.0 * q1q3 - v2q0q2 - ax) +
v2q2 * (2.0 * q0q1 + v2q2q3 - ay) +
(-v4bx * q3 + v2bz * q1) * (v2bx * (0.5 - q2q2 - q3q3) + v2bz * (q1q3 - q0q2) - mx) +
(-v2bx * q0 + v2bz * q2) * (v2bx * (q1q2 - q0q3) + v2bz * (q0q1 + q2q3) - my) +
v2bx * q1 * (v2bx * (q0q2 + q1q3) + v2bz * (0.5 - q1q1 - q2q2) - mz);
recipNorm = (s0 * s0 + s1 * s1 + s2 * s2 + s3 * s3) ** -0.5; // 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 * recipSampleFreq;
q1 += qDot2 * recipSampleFreq;
q2 += qDot3 * recipSampleFreq;
q3 += qDot4 * recipSampleFreq;
// Normalise quaternion
recipNorm = (q0 * q0 + q1 * q1 + q2 * q2 + q3 * q3) ** -0.5;
q0 *= recipNorm;
q1 *= recipNorm;
q2 *= recipNorm;
q3 *= recipNorm;
}
return {
update: madgwickAHRSUpdate,
init,
getQuaternion() {
return {
w: q0,
x: q1,
y: q2,
z: q3,
};
},
};
//= ===================================================================================================
// END OF CODE
//= ===================================================================================================
};