Pfaffian constraint

is the number of equations in a system of constraints.

Holonomic systems can always be written in Pfaffian constraint form.

are the n generalized coordinates that describe the system, and where

is the number of equations in a system of constraints, we can differentiate by the chain rule for each equation: By a simple substitution of nomenclature we arrive at: Consider a pendulum.

Because of how the motion of the weight is constrained by the arm, the velocity vector

of the weight must be perpendicular at all times to the position vector

Both position and velocity of the mass can be defined in terms of an

coordinate system: Simplifying the dot product yields: We multiply both sides by

This results in the Pfaffian form of the constraint equation: This Pfaffian form is useful, as we may integrate it to solve for the holonomic constraint equation of the system, if one exists.

In robot motion planning, a Pfaffian constraint is a set of k linearly independent constraints linear in velocity, i.e., of the form

One source of Pfaffian constraints is rolling without slipping in wheeled robots.