M2-T3 — Rotation Matrices, Quaternions, and Pose Representation

Learning objectives

After this lesson, you should be able to:

• Explain gimbal lock and show it with a concrete Euler-angle example.
• Rotate a vector and compose rotations with a unit quaternion.
• Convert between axis-angle, quaternion and matrix and state the sign and double-cover subtleties.
• Choose a representation for storage, composition, interpolation and optimisation.

The intuition

A rotation has three degrees of freedom, and that is the root of all the trouble. You cannot map three numbers onto the set of rotations without a seam somewhere: some orientations have no clean three-number description, or two descriptions that mean the same thing. The different representations choose where to put the seam.

The mathematics

Euler angles and gimbal lock

Euler angles describe a rotation as three turns about chosen axes in order, for example yaw about z, then pitch about y, then roll about x (the common “ZYX” aerospace sequence): R = Rz(ψ) Ry(θ) Rx(φ). When the pitch is ±90°, the first and third axes line up, and only the sum or difference of roll and yaw is determined. One degree of freedom is lost: this is gimbal lock. Near it, a small change of orientation needs a huge change of the angles, and numerical results become unstable.

Unit quaternions

A quaternion q = (w, x, y, z) = w + xi + yj + zk with unit length represents a rotation by angle θ about unit axis n as

w

the scalar part, cos(θ/2). Some libraries (ROS geometry_msgs, Eigen’s coefficient order) store it last as (x, y, z, w).

(x, y, z)

the vector part: the axis scaled by sin(θ/2).

q*

the conjugate (w, −x, −y, −z): for a unit quaternion this is the inverse.

⊗

the Hamilton product of two quaternions: composition of rotations, with the same order as matrix products.

The rotation of a vector v is q ⊗ (0, v) ⊗ q*. Composing two rotations is q₁ ⊗ q₂. Because q and −q describe the same rotation (a double cover), equality tests must allow for the sign, and interpolation must pick the shorter path by flipping the sign if q₁·q₂ < 0.

Interpolation and averaging

Spherical linear interpolation (slerp) moves along the great arc at constant angular speed: q(t) = sin((1−t)Ω)/sinΩ · q₁ + sin(tΩ)/sinΩ · q₂, where cosΩ = q₁·q₂. Interpolating Euler angles, by contrast, does not give a constant-speed turn and can pass through gimbal lock.

Conventions that bite

• Component order: (w, x, y, z) in Eigen’s constructor and in most papers; (x, y, z, w) in ROS messages and Eigen’s raw coefficient array.
• Hamilton versus JPL: the two quaternion algebras differ in the sign of the ij = ±k rule and therefore in multiplication order and in what the quaternion means (frame-to-world or world-to-frame). Never mix them in one system. Solà (2017) documents both.
• Euler sequence and frame: ZYX body-fixed is not the same as XYZ fixed-axis. State it where the angles are stored.

A worked example: quaternion and gimbal lock

1. A 90° turn about z. θ = 90°, n = (0, 0, 1). q = (cos 45°, 0, 0, sin 45°) = (0.7071, 0, 0, 0.7071).
2. Rotate the vector (1, 0, 0). The answer must be (0, 1, 0) because x turns into y. Using the matrix equivalent from the previous lesson, Rz(90°)(1, 0, 0) = (0, 1, 0). The quaternion formula gives the same by q ⊗ (0, 1, 0, 0) ⊗ q*, which evaluates to (0, 0, 1, 0), that is, the vector (0, 1, 0).
3. Compose two such turns. q ⊗ q = (w² − z², 0, 0, 2wz) = (0.5 − 0.5, 0, 0, 2 · 0.7071 · 0.7071) = (0, 0, 0, 1): a 180° turn about z, since cos 90° = 0 and sin 90° = 1. ✓
4. The double cover. −q = (−0.7071, 0, 0, −0.7071) rotates every vector exactly as q does. A test that compares quaternions elementwise would call them different; a test that compares |q₁·q₂| ≈ 1 would not.
5. Gimbal lock. With ZYX Euler angles, take pitch θ = 90°. Then R = Rz(ψ) Ry(90°) Rx(φ), and one can check that R depends only on ψ − φ. The pairs (ψ, φ) = (30°, 0°) and (60°, 30°) give the same rotation. A system that stores Euler angles cannot tell them apart, and a controller that differentiates them sees a spike.

The quaternion for (30°, 0°) and (60°, 30°) with pitch 90° are identical to numerical precision, which is exactly what “one degree of freedom lost” looks like in a quaternion: no problem is visible, because the quaternion has no seam there.

Engineering implications and limitations

How to choose

Engineering rules

• Normalise a quaternion after each product or integration step. A drift of 1e-3 in length slowly stretches rotated vectors.
• Convert Euler angles at the edge of the system and never inside the estimator or the optimiser.
• Write one test per convention: a known rotation about each axis, compared in all representations. The 90° turn about z above is a good first one.
• Always compare orientations by the angle of the relative rotation, 2 acos(|q₁·q₂|), not by differences of components.

Limits

Quaternions fix the singularity, but they cost a constraint (unit length) and a sign ambiguity. They are also not natural for covariance: uncertainty about orientation is described as a small three-vector perturbation in the tangent space, not as a four-by-four covariance of the quaternion (Module 4 and 6).

Common misconceptions

Engineering takeaways

1. Every representation of a 3D rotation has a seam; choose by where you work and what you do.
2. Euler angles lose a degree of freedom at ±90° pitch (gimbal lock): use them only at the human boundary.
3. A unit quaternion rotates by q ⊗ v ⊗ q*, composes by multiplication, and has the double cover q ≡ −q.
4. Conventions differ: component order, Hamilton versus JPL, Euler sequence. State and test them.
5. Optimise with small rotation-vector increments; store and exchange with quaternions or matrices.

References and further reading

Primary papers and authoritative documentation.

• Diebel, J. (2006). Representing Attitude: Euler Angles, Unit Quaternions, and Rotation Vectors. Stanford University technical report.
• Solà, J. (2017). Quaternion kinematics for the error-state Kalman filter. arXiv:1711.02508: Hamilton and JPL conventions side by side.
• Shuster, M. D. (1993). A survey of attitude representations. The Journal of the Astronautical Sciences 41(4).
• Kuipers, J. B. (1999). Quaternions and Rotation Sequences. Princeton University Press.
• ROS REP 103 (Standard Units of Measure and Coordinate Conventions) and Eigen documentation, Geometry module: quaternion storage order.

Knowledge check

Three conceptual questions. Write an answer, then reveal the explanation. Your answers stay in your browser.

What’s next

With frames and rotations settled, the next lessons deal with what a robot does not know: probability and uncertainty. Start with Gaussians and covariance.

Related content

Nothing below is required to finish this lesson. The last 3 entries are lessons of other courses on this site that teach the background or go further.

• Before
  M2-T2: 3D Rotations and Rigid-Body Transformations

  Extend to SO(3) and SE(3): composition order, non-commutativity and the exponential map.

• Lab
  M2-L2: Diagnose a Pose-Transformation Error

  A pose chain gives a wrong result: find the composition-order, inverse or angle-wrapping error among 3D and 2D variants.

• Linear Algebra
  Euler Angles and Quaternions

  The three parameterisations side by side, with their singularities.

• Linear Algebra
  SO(3) and Its Lie Algebra

  Rotation vectors and skew-symmetric matrices: the minimal, singularity-free way to express a small rotation.

• Linear Algebra
  The Exponential Map

  How a rotation vector becomes a rotation matrix, used whenever an optimiser updates an orientation.