nonholonomic

Nonholonomic refers to a class of mechanical systems and constraints defined by velocity restrictions that cannot be integrated into constraints on the configuration coordinates alone. In contrast, holonomic constraints depend only on the positions (and possibly time) and can be expressed as equations f(q,t)=0. Nonholonomic constraints are typically expressed in Pfaffian form as A(q) qdot = b(q), meaning they restrict the allowable instantaneous motions rather than directly limiting the configurations themselves.

A characteristic feature is that nonholonomic constraints are not generally integrable to constraints on the configuration variables. As a result, the set of configurations reachable from a given state through admissible motions depends on the path taken, not simply on the current position. This leads to systems whose instantaneous motion directions form a restricted distribution of possible velocities at each configuration.

Common examples include rolling without slipping: a car cannot move directly sideways, a bicycle’s wheel rolls without lateral slip, and a rolling disk on a plane. These constraints reduce the instantaneous degrees of freedom without reducing the configuration space by a fixed equation on q alone.

In control theory and robotics, nonholonomic systems pose unique challenges for motion planning and controllability. Even though they seem underactuated locally, they can often reach a large set of configurations through sequences of maneuvers. The Chow–Rashevskii (or Chow) theorem provides conditions under which such systems are controllable, based on the Lie algebra generated by constraint vector fields.

Nonholonomic considerations influence modeling, simulation, and path planning across robotics, vehicle dynamics, and related fields.