Many robotics problems require the computation and/or knowledge of 3D orientations of co-ordinate frames and co-ordinate transformations. This post will deal with some relevant background and an elegant way of describing 3D orientation, quaternions.
Given a valid rotation matrix, one can recover Euler angles by simple algebra. Hence, Euler angles and rotation matrices are related - you have to use matrices to use Euler angles; and every rotation matrix can be resolved to Euler angles (which in some cases may not be unique, as discussed below).
Rotation Matrix: The most basic approach
A rotation matrix is a 3 X 3 matrix that represents the unit vectors of the rotated frame in the global frame. Suppose 0 is the global frame and 1 is the rotated frame. The rotation matrix that represents 1 in 0 is
$$R_1^0 = [u_x \quad u_y \quad u_z]$$
Where $u_x$, $u_y$ and $u_z$ are column vectors, the three unit vectors of frame 1 as they are seen in frame 0.
Basic rotation matrices are ones that represent rotations about just X, or Y, or Z axes.
$$R_{X,\theta} = \begin{bmatrix}1&0&0\\1&\cos{\theta}&-\sin{\theta}\\0&sin{\theta}&\cos{\theta}\end{bmatrix}$$
$$R_{Y,\theta} = \begin{bmatrix}\cos{\theta}&0&\sin{\theta}\\0&1&0\\-\sin{\theta}&0&\cos{\theta}\end{bmatrix}$$
$$R_{Z,\theta} = \begin{bmatrix}\cos{\theta}&-\sin{\theta}&0\\\sin{\theta}&\cos{\theta}&0\\0&0&1\end{bmatrix}$$
Basic rotation matrices are ones that represent rotations about just X, or Y, or Z axes.
$$R_{X,\theta} = \begin{bmatrix}1&0&0\\1&\cos{\theta}&-\sin{\theta}\\0&sin{\theta}&\cos{\theta}\end{bmatrix}$$
$$R_{Y,\theta} = \begin{bmatrix}\cos{\theta}&0&\sin{\theta}\\0&1&0\\-\sin{\theta}&0&\cos{\theta}\end{bmatrix}$$
$$R_{Z,\theta} = \begin{bmatrix}\cos{\theta}&-\sin{\theta}&0\\\sin{\theta}&\cos{\theta}&0\\0&0&1\end{bmatrix}$$
Rotating a vector using rotation matrices
A vector $v^1$ in frame 1 is transformed the frame 0 like so-
$$v^0 = R_1^0 v^1$$
Remember that these vectors are free vectors and can be placed anywhere; they just indicate direction. If a free vector is placed starting at the origin of a co-ordinate frame and ending at a point, it also represents that point in that frame.
Composition of rotations
Composition of rotations is the combining of rotations. If rotations are represented as matrices, they are composed by post-multiplication. For example, if frame 0 is the global frame, frame 1 is obtained from frame 0 by rotation $R_1^0$ and frame 2 is obtained from frame 1 by rotation $R_2^1$, the composite rotation between frames 0 and 2 is,
$$R_2^0 = R_1^0 R_2^1$$
Hence a vector $v^2$ in frame 2 looks thus in frame 0-
$$\begin{align*}v^0&= R_2^0 v^2\\&= R_1^0 R_2^1 v^2\end{align*}$$
Note also that if each successive frame is obtained by rotating with respect to the global frame only (instead of the previous frame), the rotation matrices have to be pre-multiplied instead of post-multiplied. Unless mentioned otherwise, rotations are usually with respect to the previous frame.
Note also that if each successive frame is obtained by rotating with respect to the global frame only (instead of the previous frame), the rotation matrices have to be pre-multiplied instead of post-multiplied. Unless mentioned otherwise, rotations are usually with respect to the previous frame.
Euler angles
Euler angles are three numbers that can be used to represent a 3D rotation. These numbers represent the rotations about the certain axes of the successive frames. It is important to know the order of rotations by which the Euler angles are obtained. For example, if frame 1 is represented with respect to frame 0 by Euler angles $\phi$, $theta$ and $psi$ and the order is $ZYZ$, you first rotate frame 0 by $\phi$ along the Z axis, the resulting frame by $\theta$ along the Y axis and the resulting frame by $\psi$ along the Z axis to get frame 1.
Hence
$$R_{euler} = R_{Z,\phi} R_{Y,\theta} R_{Z,\psi}$$
Given a valid rotation matrix, one can recover Euler angles by simple algebra. Hence, Euler angles and rotation matrices are related - you have to use matrices to use Euler angles; and every rotation matrix can be resolved to Euler angles (which in some cases may not be unique, as discussed below).
Quaternions
It can be proved that any 3D orientation can be represented in the axis-angle form - an axis and an angle of rotation about that axis. Quaternions represent 3D orientations by 4 numbers - 3 for an axis and one for the angle about that axis. However, they are not simply axis-angle. A quaternion that represents an orientation is of the form
$$q = [\cos{\left(\psi/2\right)} \quad \sin{\left(\psi/2\right)}[a_x \quad a_y \quad a_z]]$$
where $[a_x \quad a_y \quad a_z]$ and $\psi$ is the angle of rotation about that axis.
You will observe that the norm of this quaternion is 1. Only quaternions with a unit norm represent an orientation.
The mathematical and algebraic properties of quaternions are explained in great detail here.
Rotating a vector using quaternion orientation
To rotate a vector $v$ using a quaternion $q$, we construct what is called a vector quaternion from the vector by appending a 0 to its top. This quadruple is now considered a quaternion and 'sandwich-multiplied' with $q$ to get a rotated vector quaternion, whose first element will always be 0. This is stripped off to get the rotated vector.
$$\begin{align*}v_q&=[0 \quad v]\\\Rightarrow v_q^{\prime}&=q v_q q^{-1}\\\Rightarrow v^{\prime}&=\text{last 3 elements of }v_q^{\prime}\end{align*}$$
Composing quaternion rotations
Looking at the way quaternions rotate vectors, the resultant rotation of two quaternion rotations $q_1$ and $q_2$ is $q_2q_1$
Why quaternions?
Why do we use quaternions when rotation matrices make the math so intuitive or when Euler angles can represent the same rotation using just three parameters instead of 4?
- When we compose rotations using multiplication of 3x3 rotation matrices, the result will no longer be quite orthonormal due to numerical and precision problems. Hence periodically one has to normalize the rotation matrix. However it is difficult to find the nearest orthonormal matrix to one that is not quite orthonormal. In contrast, quaternions can be normalized quite easily - just divide the quaternion by its norm.
- Quaternions can deal with gimbal lock, which is a major drawback of Euler angles (and hence rotation matrices). In short, if a rotation is expressed as a composition of three rotations about three perpendicular axes, and if the second rotation is zero, the other two angles cannot be recovered exactly from the composite matrix. They can only be recovered as their sum.