Degrees of freedom

Counting independent motion#

A free rigid body in three-dimensional space has six degrees of freedom: three for position and three for orientation. A rigid body restricted to a plane has three. Modern Robotics derives these counts by subtracting independent geometric constraints from the available point coordinates.

A revolute joint leaves one relative rotation between two links. A prismatic joint leaves one relative translation. A spherical joint allows three relative rotational freedoms.

Constraints change the count#

For an open serial chain, independent joint coordinates provide a convenient count. Closed loops require more care because joint motions constrain one another. The mechanism-counting lesson shows that a simple mobility formula can give the wrong answer when its assumed constraints are dependent.

A joint's finite travel range restricts its allowed values but does not remove its degree of freedom throughout the interior of that range.

State what the count includes#

A fixed-base arm and the same arm on a floating base have different configuration counts. MIT's modeling notes explain how fixing a base removes its floating freedoms. When comparing humanoids, establish whether the reported count includes the base, hands, and coupled mechanisms rather than treating every published number as directly comparable.