歐拉角轉旋轉矩陣公式：
旋轉矩陣轉歐拉角公式：
旋轉矩陣轉四元數公式，其中1+r11+r22+r33>0：
四元數轉旋轉矩陣公式，q0^2+q1^2+q2^2+q3^2=1：
歐拉角轉四元數公式：
四元數轉歐拉角公式：
matlab代碼如下：
1 clear all; 2 close all; 3 clc; 4
5 %歐拉角 6 x = 0.5; 7 y = 0.6; 8 z = 0.7; 9 Ang1 = [x y z]; 10
11 %歐拉角轉旋轉矩陣 12 Rx = [1 0 0; 13 0 cos(x) -sin(x); 14 0 sin(x) cos(x)]; 15 Ry = [cos(y) 0 sin(y); 16 0 1 0; 17 -sin(y) 0 cos(y)]; 18 Rz = [cos(z) -sin(z) 0; 19 sin(z) cos(z) 0; 20 0 0 1]; 21 R = Rz*Ry*Rx; 22 R1 = R; 23
24 %旋轉矩陣轉歐拉角 25 x = atan2(R(3,2),R(3,3)); 26 y = atan2(-R(3,1),sqrt(R(3,2)^2+R(3,3)^2)); 27 z = atan2(R(2,1),R(1,1)); 28 Ang2 = [x y z]; 29
30 %旋轉矩陣轉四元數 31 t=sqrt(1+R(1,1)+R(2,2)+R(3,3))/2; 32 q=[t (R(3,2)-R(2,3))/(4*t) (R(1,3)-R(3,1))/(4*t) (R(2,1)-R(1,2))/(4*t)]; 33 Q1 = q; 34
35 %四元數轉旋轉矩陣 36 R=[ 2*q(1).^2-1+2*q(2)^2 2*(q(2)*q(3)-q(1)*q(4)) 2*(q(2)*q(4)+q(1)*q(3)); 37 2*(q(2)*q(3)+q(1)*q(4)) 2*q(1)^2-1+2*q(3)^2 2*(q(3)*q(4)-q(1)*q(2)); 38 2*(q(2)*q(4)-q(1)*q(3)) 2*(q(3)*q(4)+q(1)*q(2)) 2*q(1)^2-1+2*q(4)^2]; 39 R2 = R; 40
41 %歐拉角轉四元數 42 q = [cos(x/2)*cos(y/2)*cos(z/2) + sin(x/2)*sin(y/2)*sin(z/2) ... 43 sin(x/2)*cos(y/2)*cos(z/2) - cos(x/2)*sin(y/2)*sin(z/2) ... 44 cos(x/2)*sin(y/2)*cos(z/2) + sin(x/2)*cos(y/2)*sin(z/2) ... 45 cos(x/2)*cos(y/2)*sin(z/2) - sin(x/2)*sin(y/2)*cos(z/2)]; 46 Q2 = q; 47
48 %四元數轉歐拉角 49 x = atan2(2*(q(1)*q(2)+q(3)*q(4)),1 - 2*(q(2)^2+q(3)^2)); 50 y = asin(2*(q(1)*q(3) - q(2)*q(4))); 51 z = atan2(2*(q(1)*q(4)+q(2)*q(3)),1 - 2*(q(3)^2+q(4)^2)); 52 Ang3 = [x y z]; 53
54 Ang1 55 Ang2 56 Ang3 57
58 R1 59 R2 60
61 Q1 62 Q2