Module spatial 

Spatial math: rotations, Lie groups, and spatial-algebra types.

• Quaternion — unit-quaternion rotations.
• SO2 / SO3 / SE2 / SE3 — 2D/3D rotation and rigid-transform Lie groups.
• Twist / Wrench — spatial velocity and force in [v; ω] / [force; torque] ordering.
• SpatialInertia — a body’s mass, centre of mass, and rotational inertia.
• FreeJointState — the pose and velocity of a body free to move in all six directions.

Structs

FreeJointState

Where a freely moving body is and how it is moving.

Quaternion

A quaternion w + x·i + y·j + z·k, stored scalar-first.

SE2

A 2D rigid-body transform: a rotation and a translation. The tangent is [vx, vy, ω].

SE3

A 3D rigid-body transform: a rotation and a translation. The tangent is [vx, vy, vz, ωx, ωy, ωz].

SO2

A 2D rotation, stored as a unit complex number (cosθ, sinθ). Composition is a complex product, so it takes no trigonometry. The group is abelian; exp/log are exact and need no fallback.

SO3

A 3D rotation. Wraps a unit Quaternion and carries the unit-rotation invariant. Composition uses the Hamilton product; call SO3::normalized to remove drift after long chains. The tangent is a rotation vector φ = θ·n̂ in radians, and the retract is R · exp(φ).

SpatialInertia

How a rigid body’s mass is spread out: how much there is, where it balances, and how hard it is to spin.

Twist

A spatial velocity (twist), stored linear-first in the crate-wide [v; ω] ordering.

Wrench

A spatial force (wrench), stored force-first in the [force; torque] ordering — reciprocal to a Twist, so the two line up component-for-component.