Rotation matrices in two and three dimensions

Position is only half of a pose. A gripper at the right place but facing the wrong way cannot pick anything up, and a camera that knows where it is but not which way it looks cannot locate what it sees. Orientation is harder to describe than position because rotations do not add like vectors: turning 90 degrees about x and then 90 degrees about y leaves an object in a different attitude from doing the same turns in the opposite order. The rotation matrix is the representation that handles this correctly. It is the workhorse of robot kinematics, the form used inside almost every library, and the foundation for the transforms, Denavit–Hartenberg parameters and Jacobians later in the course.

Rotation in the plane

Start in two dimensions. Suppose frame B is rotated by an angle \( \theta \) counter-clockwise relative to frame A, with the same origin. The unit x-axis of B, written in A's coordinates, is \( (\cos\theta, \sin\theta) \), and B's y-axis is \( (-\sin\theta, \cos\theta) \). Stacking these two columns gives the 2-D rotation matrix:

This matrix does two jobs, and keeping them distinct is the key to using rotations without confusion. First, it describes orientation: its columns are frame B's axes expressed in frame A. Second, it changes coordinates: multiplying a point's coordinates in B by \( {}^{A}R_{B} \) gives the same point's coordinates in A, \( {}^{A}\mathbf{p} = {}^{A}R_{B}\,{}^{B}\mathbf{p} \). The same matrix can also be read as an operator that rotates a vector by \( \theta \) within a single frame. All three readings are correct; trouble arises only when we switch between them without noticing.

Rotation matrices in three dimensions

In three dimensions the idea is the same. The rotation matrix \( {}^{A}R_{B} \) is a 3 × 3 matrix whose columns are the unit vectors of frame B's x, y and z axes, each expressed in frame A:

Each entry is a direction cosine, the cosine of the angle between one axis of B and one axis of A, because the dot product of two unit vectors is the cosine of the angle between them. That is why rotation matrices are also called direction cosine matrices, especially in aerospace. Reading a rotation matrix is a skill worth practicing: a column of (0, 0, −1) means that axis of B points straight down along A's negative z-axis; a column of (0.707, 0.707, 0) means it points diagonally between A's x and y axes.

The three elementary rotations

Rotations about the coordinate axes are the building blocks of all others. A rotation by \( \theta \) about x leaves the x-axis fixed and turns y and z in their plane; similarly for y and z. Writing \( c = \cos\theta \) and \( s = \sin\theta \):

Look closely at the sign pattern in \( R_y \): the negative sine sits below the diagonal, unlike the other two. This is not a typo. It follows from the right-hand rule, because the cyclic order of axes is x to y to z to x, so rotating about y turns z toward x. Many hand-derived kinematics errors trace back to getting this one sign wrong, so it is worth verifying with a quick test: \( R_y(90^\circ) \) should send the z-axis \( (0, 0, 1) \) to the x-axis \( (1, 0, 0) \).

Properties of rotation matrices

Not every 3 × 3 matrix is a rotation. Because the columns are perpendicular unit vectors, a rotation matrix is orthonormal: \( R^\mathsf{T} R = I \). Because the frames are both right-handed, its determinant is +1 rather than −1, which would indicate a reflection. The set of all such matrices is called the special orthogonal group in three dimensions, SO(3):

These two conditions have useful consequences. The inverse of a rotation is simply its transpose, \( R^{-1} = R^\mathsf{T} \), which is far cheaper to compute than a general matrix inverse and numerically exact. Rotations preserve lengths and angles, so a rotated vector has the same norm, and the dot product of two rotated vectors equals the dot product of the originals. The product of two rotations is another rotation. Although a rotation matrix has nine entries, the orthonormality conditions impose six constraints, leaving three free parameters, which matches the three rotational degrees of freedom of a rigid body.

Composing rotations

Suppose frame B is rotated relative to A, and frame C is rotated relative to B. The orientation of C relative to A is the product, with the frame labels canceling:

To use this, we need to know how each rotation was specified. There are two ways to describe a sequence of rotations, and they give products in opposite orders. If each rotation is about an axis of the current, moving frame, called intrinsic or body-fixed rotations, we multiply on the right: first rotation leftmost. If each rotation is about an axis of the original, fixed frame, called extrinsic or space-fixed rotations, we multiply on the left: first rotation rightmost. A useful slogan is: moving frame, multiply right; fixed frame, multiply left.

Rotating vectors versus changing frames

A common source of confusion is the difference between rotating an object and re-expressing a fixed object in a rotated frame. Suppose a vector \( \mathbf{v} \) is fixed in space and we rotate our frame by \( \theta \) about z. The vector's new coordinates are obtained with \( R_z(\theta)^\mathsf{T} = R_z(-\theta) \), because from the turned frame's point of view the vector appears to have rotated the other way. If instead we rotate the vector itself by \( \theta \) within a fixed frame, its new coordinates are \( R_z(\theta)\mathbf{v} \). Both are valid operations; the frame-label notation tells us which is which. \( {}^{A}\mathbf{v} = {}^{A}R_{B}\,{}^{B}\mathbf{v} \) is a change of frame for the same vector; \( {}^{B}\mathbf{v} = ({}^{A}R_{B})^\mathsf{T}\,{}^{A}\mathbf{v} \) goes the other way.

Numerical care

Rotation matrices computed from floating-point numbers drift slightly away from orthonormality as they are multiplied repeatedly, for example when integrating a gyroscope's readings over many time steps. The columns gradually lose unit length and perpendicularity, and the matrix starts to scale and shear vectors as well as rotate them. The standard fix is to re-orthonormalize from time to time, either by a Gram–Schmidt step on the columns or, more robustly, by replacing R with the nearest rotation matrix from its singular value decomposition, \( R \leftarrow UV^\mathsf{T} \) where \( R = U\Sigma V^\mathsf{T} \). Checking \( \lVert R^\mathsf{T}R - I \rVert \) is a cheap health test.

import numpy as np
def Rx(t): c, s = np.cos(t), np.sin(t); return np.array([[1,0,0],[0,c,-s],[0,s,c]])
def Rz(t): c, s = np.cos(t), np.sin(t); return np.array([[c,-s,0],[s,c,0],[0,0,1]])
np.set_printoptions(precision=3, suppress=True)
R = Rx(np.radians(30)) @ Rz(np.radians(45))
print("R =\n", R)
print("R^T R = I ?", np.allclose(R.T @ R, np.eye(3)), " det =", round(np.linalg.det(R), 6))

# drift: compose many small noisy steps, then repair with the SVD
rng = np.random.default_rng(1)
Q = np.eye(3)
for _ in range(10000):
    Q = Q @ (Rz(0.001) + rng.normal(0, 1e-6, (3, 3)))
print("error before repair:", f"{np.linalg.norm(Q.T @ Q - np.eye(3)):.1e}")
U, _, Vt = np.linalg.svd(Q); Q = U @ Vt
print("error after repair: ", f"{np.linalg.norm(Q.T @ Q - np.eye(3)):.1e}")

R =
 [[ 0.707 -0.707  0.   ]
 [ 0.612  0.612 -0.5  ]
 [ 0.354  0.354  0.866]]
R^T R = I ? True  det = 1.0
error before repair: 5.0e-04
error after repair:  1.7e-15

Rotation matrices in practice

Rotation matrices are ideal for computation: composing rotations is a matrix product, rotating a vector is a matrix-vector product, inverting is a transpose, and there are no special cases or singularities. Their drawbacks are redundancy, nine numbers for three degrees of freedom, and opacity: few people can look at nine direction cosines and picture the orientation. For storage, communication and human interfaces, robot software therefore uses more compact representations, Euler angles, axis-angle and quaternions, which are the subject of the next lesson. Internally, almost every robotics library converts to rotation matrices or quaternions to do the arithmetic.

A useful mental check when reading any rotation matrix is to look at each column in turn and ask where that axis of the rotated frame points. For a camera looking straight down at a table from above, with the image's x to the right, the matrix relative to a world frame with z up has a third column of (0, 0, −1), because the camera's optical z-axis points down. If a calibration routine returns a third column with a positive z component, something is upside down.