Spatial vector algebra

Pair angular and linear quantities#

A spatial motion vector combines angular and linear velocity into one six-dimensional object. A spatial force vector combines moment and force. These resemble a twist and a wrench, but spatial vector algebra also supplies the transforms, cross-product operators, and inertia quantities needed to use them consistently in rigid-body dynamics.

The two vector types transform differently when the reference frame or point changes. Keeping that distinction explicit lets an algorithm propagate motion outward through a kinematic tree and forces back toward the base without repeatedly expanding separate three-dimensional equations.

Build recursive robot dynamics algorithms#

Roy Featherstone's spatial-vector reference implementation includes separate motion and force cross-product operators, spatial coordinate transforms, spatial rigid-body inertia, and algorithms for forward, inverse, hybrid, and floating-base dynamics. The software accompanies his reference text on rigid-body dynamics algorithms.

For a humanoid, the same representation can organise the velocities, accelerations, inertias, and contact wrenches of a branching floating-base robot. It is a mathematical framework rather than one controller: an implementation can use it inside estimation, simulation, trajectory optimisation, or whole-body control.

Conventions must remain consistent#

Different libraries order the angular and linear components differently and may define transforms in opposite directions. Copying a six-vector or matrix between conventions without conversion can silently produce incorrect dynamics. Every quantity must also be expressed in a stated coordinate frame.

Spatial vector algebra does not determine contact forces, actuator limits, friction, or control objectives. It can make forward dynamics and inverse dynamics calculations compact, but the result is still only as accurate as the robot model, parameters, and external-force assumptions.