C++ 如何从旋转矩阵计算角度
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How to calculate the angle from rotation matrix
提问by N.J
I am using two image of the single object the object is roated certain degree from its first image.
我正在使用单个对象的两个图像，该对象从它的第一个图像旋转了一定程度。
I have calculated the POSE of each image and converted the rotational vector to Matrix using Rodergues(). Now how do I calculate and see how much it is rotated from its first position?
我已经计算了每个图像的姿势并使用 Rodergues() 将旋转向量转换为矩阵。现在我如何计算并查看它从第一个位置旋转了多少？
I have tried many ways but answers was no were close
我尝试了很多方法，但答案是否定的
EDIT: My camera is fixed only the object is moving.
编辑：我的相机是固定的，只有物体在移动。
回答by Krish
We can get Euler angles from rotation matrix using following formula.
我们可以使用以下公式从旋转矩阵中获得欧拉角。
Given a 3×3 rotation matrix
给定一个 3×3 的旋转矩阵
The 3 Euler angles are
3个欧拉角是
Here atan2 is the same arc tangent function, with quadrant checking, you typically find in C or Matlab.
这里 atan2 是相同的反正切函数，带有象限检查，您通常在 C 或 Matlab 中找到。
Note: Care must be taken if the angle around the y-axis is exactly +/-90°. In that case all elements in the first column and last row, except the one in the lower corner, which is either 1 or -1, will be 0 (cos(1)=0). One solution would be to fix the rotation around the x-axis at 180° and compute the angle around the z-axis from: atan2(r_12, -r_22).
注意：如果绕 y 轴的角度正好是 +/-90°，则必须小心。在这种情况下，第一列和最后一行中的所有元素，除了下角的元素，即 1 或 -1，都将为 0 (cos(1)=0)。一种解决方案是将绕 x 轴的旋转固定为 180°，并根据以下公式计算绕 z 轴的角度：atan2(r_12, -r_22)。
See also https://www.geometrictools.com/Documentation/EulerAngles.pdf, which includes implementations for six different orders of Euler angles.
另请参阅https://www.geometrictools.com/Documentation/EulerAngles.pdf，其中包括六种不同欧拉角阶数的实现。
回答by Beta
If Ris the (3x3) rotation matrix, then the angle of rotation will be acos((tr(R)-1)/2), where tr(R) is the trace of the matrix (i.e. the sum of the diagonal elements).
如果R是 (3x3) 旋转矩阵，则旋转角度将为 acos((tr( R)-1)/2)，其中 tr( R) 是矩阵的迹（即对角线元素的总和）。
That is what you asked for; I estimate a 90% chance that it is not what you want.
这就是你所要求的；我估计有 90% 的机会不是你想要的。
回答by Curnane
For your reference, this code computes the Euler angles in MATLAB:
供您参考，此代码在 MATLAB 中计算欧拉角：
function Eul = RotMat2Euler(R)
if R(1,3) == 1 | R(1,3) == -1
%special case
E3 = 0; %set arbitrarily
dlta = atan2(R(1,2),R(1,3));
if R(1,3) == -1
E2 = pi/2;
E1 = E3 + dlta;
else
E2 = -pi/2;
E1 = -E3 + dlta;
end
else
E2 = - asin(R(1,3));
E1 = atan2(R(2,3)/cos(E2), R(3,3)/cos(E2));
E3 = atan2(R(1,2)/cos(E2), R(1,1)/cos(E2));
end
Eul = [E1 E2 E3];
Code provided by Graham Taylor, Geoff Hinton and Sam Roweis. For more information, see here
代码由 Graham Taylor、Geoff Hinton 和 Sam Roweis 提供。有关更多信息，请参阅此处
回答by Francesco Callari
Let R1c and R2c be the 2 rotation matrices you have computed. These express the rotations from the object in poses 1 and 2 respectively to the camera frame (hence the second c suffix). The rotation matrix you want is from pose 1 to pose 2, i.e. R12. To compute it you must rotate, in your mind, the object from pose_1-to-camera, then from the camera-to-pose_2. The latter rotation is the inverse of the pose_2-to-camera espressed by R2c, hence:
让 R1c 和 R2c 是您计算的 2 个旋转矩阵。它们分别表示从姿势 1 和姿势 2 中的对象到相机帧的旋转（因此是第二个 c 后缀）。你想要的旋转矩阵是从pose 1到pose 2，即R12。要计算它，您必须在脑海中旋转物体，从pose_1 到相机，然后从相机到pose_2。后一个旋转是由 R2c 压缩的pose_2-to-camera的逆，因此：
R12 = R1c * inv(R2c)
R12 = R1c * inv(R2c)
From matrix R12 you can then compute the angle and axis of rotation using Rodiguez's formula.
然后，您可以从矩阵 R12 使用 Rodiguez 公式计算角度和旋转轴。
