M10-T2 — SO(3) and Its Lie Algebra

Learning objectives

After this lesson, you should be able to:

• Write the hat map and show that [a]ₓb = a × b and that algebra elements are skew-symmetric matrices.
• Interpret ω = θn as a rotation vector and as an angular velocity, and read the Lie bracket as the cross product.
• Derive that the derivative of a rotation matrix satisfies Ṙ = [ω_world]ₓR = R[ω_body]ₓ.
• Explain how many parameters, and which, a small rotation needs.

Introduction and motivation

A gyroscope reports three numbers: the angular velocity of the body. To update an orientation matrix R the filter needs to turn those three numbers into a small rotation and multiply: R ← R exp([ω]ₓ Δt). That recipe is the Lie algebra of SO(3) in action. It also explains why a rotation error has three components rather than nine, and why attitude covariance is a 3 × 3 matrix even when the attitude is stored as a quaternion.

The intuition

Differentiate RᵀR = I along a path through the identity: Xᵀ + X = 0, so the tangent matrices are skew-symmetric, and a 3 × 3 skew-symmetric matrix has three free entries: a 3-vector. That vector points along the axis about which a small rotation happens, and its length is the rate (or the angle). Multiplying the skew matrix by a position vector gives the velocity of that point on a rotating body: exactly the familiar ω × r.

Geometric and physical interpretation

Geometric view. Near the identity, every rotation is exp([ω]ₓ) for a small vector ω: an angle ‖ω‖ about the axis ω/‖ω‖. The space of such ω is ℝ³, a flat 3-dimensional chart of the curved group. Physical view. For a rotating body, a point at position r has velocity ω × r = [ω]ₓ r. The angular velocity can be written in the world frame or the body frame; the two are related by ω_world = Rω_body.

The mathematics

[ω]ₓ

the skew-symmetric (“hat”) matrix of ω ∈ ℝ³; also written ω^ or hat(ω).

(·)∨

the inverse map (“vee”) from skew matrices back to vectors.

so(3)

the Lie algebra of SO(3): all 3 × 3 skew-symmetric matrices.

ω = θ n

a rotation vector: unit axis n, angle θ.

The hat map

[ω]ₓ = [ 0  −ω₃  ω₂ ; ω₃  0  −ω₁ ; −ω₂  ω₁  0 ],      [ω]ₓ v = ω × v

What it means. The matrix has the three entries of ω, with signs. Applying it to v gives the cross product, so the matrix encodes “rotate about ω”.

The Lie algebra and its bracket

so(3) = { [ω]ₓ : ω ∈ ℝ³ },        [ [a]ₓ, [b]ₓ ] = [a]ₓ[b]ₓ − [b]ₓ[a]ₓ = [ a × b ]ₓ

What it means. The set is closed under the commutator, which equals the hat of the cross product: the failure of rotations to commute at second order is measured by a × b.

Angular velocity

Ṙ = [ωw]ₓ R = R [ωb]ₓ,        ωw = R ωb

What it means. The velocity of the matrix R is an algebra element multiplied on the left (world angular velocity) or on the right (body angular velocity), the two forms being related by R.

so(3) = skew-symmetric 3×3 matrices ≅ ℝ³ via the hat map; [ω]ₓv = ω × v; bracket = cross product; Ṙ = [ω_w]ₓR = R[ω_b]ₓ.

Step-by-step derivation: why skew-symmetric, the bracket and Ṙ

1. Tangent matrices are skew. Let R(t) be a smooth path in SO(3) with R(0) = I and tangent X = Ṙ(0). Differentiating R(t)ᵀR(t) = I at t = 0 gives Xᵀ + X = 0: X is skew-symmetric.
2. Dimension. A 3 × 3 skew-symmetric matrix has zero diagonal and x₁₂ = −x₂₁ etc., so three independent entries: dim so(3) = 3 = dim SO(3), as the rotation counting of Module 7 required. Every skew matrix has the form [ω]ₓ.
3. Cross product. By expansion, [ω]ₓv = (−ω₃v₂ + ω₂v₃, ω₃v₁ − ω₁v₃, −ω₂v₁ + ω₁v₂) = ω × v, so it has the kinematic meaning v̇ = ω × v for a point rotating about the axis ω.
4. Bracket. Using the identity a × (b × v) − b × (a × v) = (a × b) × v (Jacobi identity for the cross product) we get ([a]ₓ[b]ₓ − [b]ₓ[a]ₓ)v = [a × b]ₓv for all v, hence [[a]ₓ, [b]ₓ] = [a × b]ₓ.
5. Powers of the hat matrix. For a unit axis n, [n]ₓ² = nnᵀ − I and [n]ₓ³ = −[n]ₓ. This closed pattern is what makes the exponential have a finite closed form (Rodrigues, next lesson).
6. Rate of rotation. A moving body with rotation R(t) has Ṙ Rᵀ skew-symmetric (differentiate RRᵀ = I: ṘRᵀ + RṘᵀ = 0). Define [ω_w]ₓ = ṘRᵀ, the world angular velocity, so Ṙ = [ω_w]ₓR. Likewise RᵀṘ is skew, defining the body angular velocity: Ṙ = R[ω_b]ₓ.
7. Relation between the two. From [ω_w]ₓR = R[ω_b]ₓ we get [ω_w]ₓ = R[ω_b]ₓRᵀ = [Rω_b]ₓ (rotation of a cross product), so ω_w = Rω_b. A gyroscope measures ω_b; to integrate it, multiply on the right: R(t + Δt) ≈ R(t) exp([ω_b Δt]ₓ).
8. Covariance of attitude. Small attitude errors are rotation vectors δ ∈ ℝ³, so an attitude covariance is a 3 × 3 matrix, even for a quaternion (4 numbers) or matrix (9 numbers) state: the covariance lives in the tangent space (Module 9 meets Module 10).

A caution on notation. Texts write the hat as ω^, [ω]ₓ or S(ω) and the inverse as vee. Check the sign convention of the cross-product matrix (some define the skew matrix of the transpose); the safe test is [ω]ₓv = ω × v.

A worked example: hats, brackets and angular velocity

• Hat. ω = (1, 2, 3): [ω]ₓ = [[0, −3, 2], [3, 0, −1], [−2, 1, 0]]; its transpose is its negative ✓.
• Cross product by matrix. b = (−1, 0, 2): [ω]ₓb = (−3·0 + 2·2, 3·(−1) − 1·2, −2·(−1) + 0) = (4, −5, 2) = ω × b ✓.
• Cube of the hat. θ² = 14 and [ω]ₓ³ = −14 [ω]ₓ (the pattern K³ = −θ²K).
• Bracket. a = (1, 0, 0), b = (0, 1, 0): [a]ₓ[b]ₓ − [b]ₓ[a]ₓ = [a × b]ₓ = [e₃]ₓ, the matrix with entries ∓1 in the upper left: the commutator of rotations about x and y is a rotation about z.
• A rotation vector. ω = (0, 0, π/2) gives exp([ω]ₓ) = Rz(90°); ω = (0.3, −0.2, 0.5) has angle ‖ω‖ = 0.616 rad (35.3°) and exp([ω]ₓ) = [[0.860, −0.498, −0.115], [0.440, 0.835, −0.330], [0.260, 0.233, 0.937]].
• Angular velocity. For R = Rz(90°) and a world angular velocity ω_w = (1, 0, 0), the body-frame rate is ω_b = Rᵀω_w = (0, −1, 0), and [ω_w]ₓR = R[ω_b]ₓ ✓ (both give the same Ṙ).
• Gyro integration. A gyro reads ω_b = (0, 0, 1) rad/s for 0.1 s: ΔR = exp([ω_b Δt]ₓ) = Rz(5.73°) and R ← R ΔR (post-multiplication, body rates).

Robotics application

UAV attitude estimation. Gyro integration, the attitude error state and its 3 × 3 covariance all use so(3); see attitude representation and rotation matrices, quaternion-based orientation and the error-state Kalman filter.

SLAM. A relative rotation between scans is measured, and its error expressed as a rotation vector δ with a 3 × 3 covariance; see rotation matrices, quaternions and poses and pose graphs.

IMU preintegration. Preintegrated rotation increments are products of exp([ω Δt]ₓ) over many samples; see IMU preintegration.

Manipulators and mobile robots. Angular velocities of links, wheel-based yaw rates and the angular part of the Jacobian (Module 8) are algebra elements; the angular rows of the 6 × n Jacobian are the world angular velocity ω_w as a function of joint rates.

Engineering implications and limitations

Practical rules

• Know the frame of your angular velocity. A gyro gives body rates: use R exp(ω_b Δt). World rates give exp(ω_w Δt) R. Mixing them is a common attitude bug.
• Test the hat sign. Check [ω]ₓv = ω × v with a numeric example in your code base.
• Use 3-vectors for errors and covariances. Do not carry a 4 × 4 covariance for a unit quaternion or a 9 × 9 for a matrix: keep 3 × 3 in the tangent space.
• Re-orthonormalise after products. Even though exp gives exact rotations in exact arithmetic, repeated products drift in floating point.
• Use closed forms with small-angle guards. The Rodrigues coefficients sin θ/θ and (1 − cos θ)/θ² need series expansions near θ = 0 (next lessons).

Limits

The rotation vector is a chart: it is unique and smooth only for ‖ω‖ < π. At exactly π the axis can be flipped (n and −n give the same rotation), and beyond π the vector wraps. Tangent-space algorithms must stay in the neighbourhood of the current estimate.

Common misconceptions

Engineering takeaways

1. so(3) = skew-symmetric 3 × 3 matrices ≅ ℝ³; [ω]ₓv = ω × v.
2. A rotation vector ω = θn encodes axis and angle; exp([ω]ₓ) is the rotation.
3. The Lie bracket on so(3) is the cross product: [[a]ₓ, [b]ₓ] = [a × b]ₓ.
4. Ṙ = [ω_w]ₓR = R[ω_b]ₓ with ω_w = Rω_b; gyro integration is R ← R exp([ω_b Δt]ₓ).
5. Attitude errors and covariances are three-dimensional, living in the tangent space.

References and further reading

Primary papers and authoritative documentation.

• Solà, Deray and Atchuthan, A micro Lie theory for state estimation in robotics: A micro Lie theory for state estimation in robotics: so(3), rotation vectors and the hat operator
• Solà, Quaternion kinematics for the error-state Kalman filter: Quaternion kinematics for the error-state Kalman filter: rotation vectors and angular velocity
• Modern Robotics (Lynch and Park), free preprint: chapter 3.2.3, angular velocity and exponential coordinates
• Barfoot, State Estimation for Robotics (author-hosted first edition, free to read): chapter 7, rotation Lie algebra

Knowledge check

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What’s next

Rigid motions combine rotation and translation, and their velocities, twists, combine angular and linear velocity. The next lesson builds the Lie algebra of SE(3) and shows that a constant twist traces a screw motion.

Related content

Nothing below is required to finish this lesson.

• Before
  M10-T1: Introduction to Lie Groups and Lie Algebras

  Say what a Lie group and its Lie algebra are, using rotations in the plane as the first example.

• Before
  M7-T4: SO(3)

  State what makes a matrix a rotation (orthogonal, determinant +1) and why such matrices form a group.