Understanding the difference between two rotations is crucial in various fields, from robotics and computer graphics to aerospace engineering. When dealing with rotations represented in axis-angle form, quantifying the “error” or difference often boils down to finding the smallest angle that one rotation needs to turn to align with the other. This article delves into methods to Compare Two Angles of rotation, leveraging rotation matrices and quaternions for a comprehensive comparison.
Comparing Rotations using Rotation Matrices
One approach to compare two rotations, denoted as $r_1$ and $r_2$ in axis-angle representation, involves converting them into rotation matrices. This conversion can be achieved using the Rodrigues’ rotation formula:
$$ R(r) = I + frac{sin(|r|)}{|r|}S(r) + frac{1-cos(|r|)}{|r|^2}S(r)^2, $$
where $I$ is the 3×3 identity matrix, $r = begin{bmatrix}r_1 & r_2 & r_3end{bmatrix}^top$ is the axis-angle vector, and $S(r)$ is the skew-symmetric matrix:
$$ S(r) = begin{bmatrix} 0 & -r_3 & r_2 r_3 & 0 & -r_1 -r_2 & r_1 & 0 end{bmatrix}. $$
To find the relative rotation between $r_1$ and $r_2$, we can compute the error rotation matrix $R_e$. Since the inverse of a rotation matrix is its transpose, and transposing $R(r)$ is equivalent to $R(-r)$, the relative rotation can be expressed as:
$$ R_e = R(r_1)R(-r_2). $$
The angle $theta$ representing the difference between the two rotations can then be extracted from the trace of the error rotation matrix $R_e$. This angle, which lies in the interval $[0, pi]$, is given by:
$$ theta = cos^{-1}left(frac{text{tr}(R_e) – 1}{2}right). $$
This method provides a geometrically meaningful measure of the difference between two rotations by calculating the angle of the relative rotation.
Comparing Rotations using Quaternions
Alternatively, we can compare two rotations using unit quaternions. The axis-angle representation $r$ can be converted to a quaternion $q(r)$ using the scalar-vector representation:
$$ q(r) = left(cosleft(frac{|r|}{2}right), sinleft(frac{|r|}{2}right)frac{r}{|r|}right). $$
Similar to rotation matrices, we can “divide” quaternions to find the relative rotation. The inverse of a unit quaternion is obtained by negating its vector part. When multiplying quaternions, the Hamilton product is used. To directly obtain the angle difference $theta$, we can focus on the scalar part of the resulting quaternion product. The angle $theta$ in the interval $[0, pi]$ can be calculated as:
$$ theta = 2cos^{-1}left(left|cosleft(frac{|r_1|}{2}right) cosleft(frac{|r_2|}{2}right) + sinleft(frac{|r_1|}{2}right)sinleft(frac{|r_2|}{2}right)frac{r_1^top r_2}{|r_1||r_2|}right|right). $$
This formula offers a potentially faster computational approach compared to the rotation matrix method, especially when only the angle difference is required.
Conclusion
Both the rotation matrix method and the quaternion method provide equivalent results for comparing two angles of rotation, up to machine precision. While the rotation matrix approach offers a more direct geometric interpretation through the trace of the relative rotation matrix, the quaternion method can be computationally more efficient. Choosing between these methods often depends on the specific application and computational resources available. Understanding these methods is essential for accurately quantifying and comparing rotational differences in various engineering and scientific disciplines.
