Convert quaternion to euler rotations
Here’s a short, self contained c++ program for demonstrating conversion of quaternion rotations to euler rotations based on various rotation sequences:
// COMPILE: g++ -o quat2EulerTest quat2EulerTest.cpp
#include <iostream>
#include <cmath>
using namespace std;
///////////////////////////////
// Quaternion struct
// Simple incomplete quaternion struct for demo purpose
///////////////////////////////
struct Quaternion{
Quaternion():x(0), y(0), z(0), w(1){};
Quaternion(double x, double y, double z, double w):x(x), y(y), z(z), w(w){};
void normalize(){
double norm = std::sqrt(x*x + y*y + z*z + w*w);
x /= norm;
y /= norm;
z /= norm;
w /= norm;
}
double norm(){
return std::sqrt(x*x + y*y + z*z + w*w);
}
double x;
double y;
double z;
double w;
};
///////////////////////////////
// Quaternion to Euler
///////////////////////////////
enum RotSeq{zyx, zyz, zxy, zxz, yxz, yxy, yzx, yzy, xyz, xyx, xzy,xzx};
void twoaxisrot(double r11, double r12, double r21, double r31, double r32, double res[]){
res[0] = atan2( r11, r12 );
res[1] = acos ( r21 );
res[2] = atan2( r31, r32 );
}
void threeaxisrot(double r11, double r12, double r21, double r31, double r32, double res[]){
res[0] = atan2( r31, r32 );
res[1] = asin ( r21 );
res[2] = atan2( r11, r12 );
}
// note:
// return values of res[] depends on rotSeq.
// i.e.
// for rotSeq zyx,
// x = res[0], y = res[1], z = res[2]
// for rotSeq xyz
// z = res[0], y = res[1], x = res[2]
// ...
void quaternion2Euler(const Quaternion& q, double res[], RotSeq rotSeq)
{
switch(rotSeq){
case zyx:
threeaxisrot( 2*(q.x*q.y + q.w*q.z),
q.w*q.w + q.x*q.x - q.y*q.y - q.z*q.z,
-2*(q.x*q.z - q.w*q.y),
2*(q.y*q.z + q.w*q.x),
q.w*q.w - q.x*q.x - q.y*q.y + q.z*q.z,
res);
break;
case zyz:
twoaxisrot( 2*(q.y*q.z - q.w*q.x),
2*(q.x*q.z + q.w*q.y),
q.w*q.w - q.x*q.x - q.y*q.y + q.z*q.z,
2*(q.y*q.z + q.w*q.x),
-2*(q.x*q.z - q.w*q.y),
res);
break;
case zxy:
threeaxisrot( -2*(q.x*q.y - q.w*q.z),
q.w*q.w - q.x*q.x + q.y*q.y - q.z*q.z,
2*(q.y*q.z + q.w*q.x),
-2*(q.x*q.z - q.w*q.y),
q.w*q.w - q.x*q.x - q.y*q.y + q.z*q.z,
res);
break;
case zxz:
twoaxisrot( 2*(q.x*q.z + q.w*q.y),
-2*(q.y*q.z - q.w*q.x),
q.w*q.w - q.x*q.x - q.y*q.y + q.z*q.z,
2*(q.x*q.z - q.w*q.y),
2*(q.y*q.z + q.w*q.x),
res);
break;
case yxz:
threeaxisrot( 2*(q.x*q.z + q.w*q.y),
q.w*q.w - q.x*q.x - q.y*q.y + q.z*q.z,
-2*(q.y*q.z - q.w*q.x),
2*(q.x*q.y + q.w*q.z),
q.w*q.w - q.x*q.x + q.y*q.y - q.z*q.z,
res);
break;
case yxy:
twoaxisrot( 2*(q.x*q.y - q.w*q.z),
2*(q.y*q.z + q.w*q.x),
q.w*q.w - q.x*q.x + q.y*q.y - q.z*q.z,
2*(q.x*q.y + q.w*q.z),
-2*(q.y*q.z - q.w*q.x),
res);
break;
case yzx:
threeaxisrot( -2*(q.x*q.z - q.w*q.y),
q.w*q.w + q.x*q.x - q.y*q.y - q.z*q.z,
2*(q.x*q.y + q.w*q.z),
-2*(q.y*q.z - q.w*q.x),
q.w*q.w - q.x*q.x + q.y*q.y - q.z*q.z,
res);
break;
case yzy:
twoaxisrot( 2*(q.y*q.z + q.w*q.x),
-2*(q.x*q.y - q.w*q.z),
q.w*q.w - q.x*q.x + q.y*q.y - q.z*q.z,
2*(q.y*q.z - q.w*q.x),
2*(q.x*q.y + q.w*q.z),
res);
break;
case xyz:
threeaxisrot( -2*(q.y*q.z - q.w*q.x),
q.w*q.w - q.x*q.x - q.y*q.y + q.z*q.z,
2*(q.x*q.z + q.w*q.y),
-2*(q.x*q.y - q.w*q.z),
q.w*q.w + q.x*q.x - q.y*q.y - q.z*q.z,
res);
break;
case xyx:
twoaxisrot( 2*(q.x*q.y + q.w*q.z),
-2*(q.x*q.z - q.w*q.y),
q.w*q.w + q.x*q.x - q.y*q.y - q.z*q.z,
2*(q.x*q.y - q.w*q.z),
2*(q.x*q.z + q.w*q.y),
res);
break;
case xzy:
threeaxisrot( 2*(q.y*q.z + q.w*q.x),
q.w*q.w - q.x*q.x + q.y*q.y - q.z*q.z,
-2*(q.x*q.y - q.w*q.z),
2*(q.x*q.z + q.w*q.y),
q.w*q.w + q.x*q.x - q.y*q.y - q.z*q.z,
res);
break;
case xzx:
twoaxisrot( 2*(q.x*q.z - q.w*q.y),
2*(q.x*q.y + q.w*q.z),
q.w*q.w + q.x*q.x - q.y*q.y - q.z*q.z,
2*(q.x*q.z + q.w*q.y),
-2*(q.x*q.y - q.w*q.z),
res);
break;
default:
std::cout << "Unknown rotation sequence" << std::endl;
break;
}
}
///////////////////////////////
// Helper functions
///////////////////////////////
Quaternion operator*(Quaternion& q1, Quaternion& q2){
Quaternion q;
q.w = q1.w*q2.w - q1.x*q2.x - q1.y*q2.y - q1.z*q2.z;
q.x = q1.w*q2.x + q1.x*q2.w + q1.y*q2.z - q1.z*q2.y;
q.y = q1.w*q2.y - q1.x*q2.z + q1.y*q2.w + q1.z*q2.x;
q.z = q1.w*q2.z + q1.x*q2.y - q1.y*q2.x + q1.z*q2.w;
return q;
}
ostream& operator <<(std::ostream& stream, const Quaternion& q) {
cout << q.w << " "<< showpos << q.x << "i " << q.y << "j " << q.z << "k";
cout << noshowpos;
}
double rad2deg(double rad){
return rad*180.0/M_PI;
}
///////////////////////////////
// Main
///////////////////////////////
int main(){
Quaternion q; // x,y,z,w
Quaternion qx45(sin(M_PI/8), 0,0, cos(M_PI/8) );
Quaternion qy45(0, sin(M_PI/8), 0, cos(M_PI/8));
Quaternion qz45(0, 0, sin(M_PI/8), cos(M_PI/8));
Quaternion qx90(sin(M_PI/4), 0,0, cos(M_PI/4) );
Quaternion qy90(0, sin(M_PI/4), 0, cos(M_PI/4));
Quaternion qz90(0, 0, sin(M_PI/4), cos(M_PI/4));
double res[3];
q = qz45*qx45;
q.normalize();
quaternion2Euler(q, res, zyx);
cout << "Rotation sequence: X->Y->Z" << endl;
cout << "x45 -> z45" << endl;
cout << "q: " << q << endl;
cout << "z: " << rad2deg(res[0]) << " y: " << rad2deg(res[1]) << " x: " << rad2deg(res[2]) << endl << endl;
q = qz90*qx90;
q.normalize();
quaternion2Euler(q, res, zyx);
cout << "Rotation sequence: X->Y->Z" << endl;
cout << "x90 -> z90" << endl;
cout << "q: " << q << endl;
cout << "x: " << rad2deg(res[0]) << " y: " << rad2deg(res[1]) << " z: " << rad2deg(res[2]) << endl << endl;
q = qx90*qz90;
q.normalize();
quaternion2Euler(q, res, xyz);
cout << "Rotation sequence: Z->Y->X" << endl;
cout << "z90 -> x90" << endl;
cout << "q: " << q << endl;
cout << "z: " << rad2deg(res[0]) << " y: " << rad2deg(res[1]) << " x: " << rad2deg(res[2]) << endl;
q = qx90*qz45;
q.normalize();
quaternion2Euler(q, res, xyz);
cout << "Rotation sequence: Z->Y->X" << endl;
cout << "z45 -> x90" << endl;
cout << "q: " << q << endl;
cout << "z: " << rad2deg(res[0]) << " y: " << rad2deg(res[1]) << " x: " << rad2deg(res[2]) << endl << endl;
q = qx90*qz45;
q.normalize();
quaternion2Euler(q, res, zyx);
cout << "Rotation sequence: X->Y->Z" << endl;
cout << "z45 -> x90" << endl;
cout << "q: " << q << endl;
cout << "x: " << rad2deg(res[0]) << " y: " << rad2deg(res[1]) << " z: " << rad2deg(res[2]) << endl;
}
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