holonomic vs nonholonomic constraints

Holonomic and nonholonomic are two kinds of rules that limit how a robot can move, and the difference is whether the rule can be boiled down to a relationship among positions or only ties down the velocities — how the robot may move from one instant to the next. A holonomic constraint is one you can express purely in terms of where the parts are: it locks certain positions together, so it actually shrinks the set of places the robot can ever reach. A nonholonomic constraint is subtler — it restricts the robot's instantaneous motion without forbidding any destination, so the robot can still reach anywhere, just never by heading straight there.

A shopping cart is the perfect picture. There is no spot in the store the cart cannot reach, so nothing rules out any final position — yet the cart cannot slide directly sideways; at every instant it is only allowed to roll along the way its wheels point. To move sideways into a gap you have to shuffle forward and back, wiggling the wheels, until you have crept over. That ban on sideways sliding is a nonholonomic constraint: a limit on instantaneous velocity that can never be rewritten as a rule about position alone, which is exactly what makes it nonholonomic. A car and a differential-drive robot live under the same rule, which is why parallel parking takes a careful little dance.

Why bother with the distinction? Because it tells you how hard path planning will be, and what counts as a holonomic robot. A holonomic robot — like an omnidirectional drive, which is free of any such velocity restriction and can scoot in any direction at any instant — can be steered straight to its goal along the most obvious line, so planning its motion is easy. A nonholonomic robot can reach the same goal but must obey its moment-to-moment limits along the way, so its planner has to thread together a sequence of allowed wiggles and curves. Recognizing which kind of constraint a robot faces is the first thing an engineer checks before deciding how to make it move.