Nonholonomic Constraint
- Also called: non-holonomic constraint
- Category: Foundations & spatial maths › Kinematics
- Tags: kinematics, mobile-robots, motion-planning, wheeled-robots, control-theory
A constraint on a system's velocities that cannot be integrated into a constraint on its configuration, such as a wheel that cannot slide sideways but can still reach any position.
Definition
A nonholonomic constraint restricts the directions in which a system can move at each instant without restricting which configurations it can eventually reach. Holonomic constraints can be written as equations on the configuration itself, such as a fixed distance between two points, and they remove dimensions from the reachable configuration space. Nonholonomic constraints are written on velocities, typically in the Pfaffian form A(q) q̇ = 0, and cannot be integrated into such equations. The classic example is a wheel rolling without slipping: its velocity must point along its heading, yet by combining forward motion and turning, it can reach any position and orientation on the plane.
Wheeled mobile robots are the most common case. A differential-drive robot or a car cannot move sideways instantaneously, so parallel parking requires a sequence of forward and backward arcs. Other sources include steerable wheels on wheeled-legged robots, ball-and-plate rolling, and conservation of angular momentum in free-floating space robots, which constrains velocities in a way that cannot be integrated. Whether a system with such constraints can reach all configurations is a controllability question, answered by checking whether Lie brackets of the allowed motion directions span the configuration space (Chow's theorem).
Nonholonomic constraints have practical consequences for both planning and control. Straight-line paths in configuration space are often not feasible, so planners use steering functions such as Dubins or Reeds-Shepp curves, lattice primitives, or trajectory optimization with the constraint included. In control, Brockett's 1983 result shows that many nonholonomic systems, including the unicycle, cannot be stabilized to a point with a smooth, time-invariant state feedback law; practical controllers instead track a moving reference trajectory, use time-varying or discontinuous feedback, or control a point ahead of the wheel axle.
Why it matters
Most wheeled robots are nonholonomic, and ignoring this produces plans the robot cannot follow and controllers that fail near the goal. The concept also explains why omnidirectional bases and holonomic drives are chosen when sideways motion matters.
Key applications
- Path planning for cars, differential-drive robots and tractor-trailers
- Trajectory tracking control of wheeled mobile robots
- Motion optimization for wheeled-legged robots
- Parking and maneuvering in tight spaces
- Attitude control of free-floating space robots
In practice
A team building an indoor delivery robot with a differential drive finds that their global planner, written for a point robot, produces paths with sharp corners and sideways steps in narrow doorways. They switch the local planner to sample velocity commands that respect the drive's constraint, such as forward speed and turn rate, and add a final approach behavior that comes to the docking station along a curve instead of trying to correct small lateral errors on the spot. On a wheeled-legged quadruped, a similar constraint is added to the whole-body optimizer: each wheel's ground velocity must point along its steering angle, so the solver never plans lateral wheel slip.
Common misconceptions
- A nonholonomic robot cannot reach some positions — it can usually reach every configuration, just not along every path
- Nonholonomic means the robot has fewer degrees of freedom — the configuration space keeps its dimension; only instantaneous velocities are restricted
- Any velocity limit is a nonholonomic constraint — speed and acceleration bounds are inequality limits, not non-integrable equality constraints on direction
- Standard feedback control can park a car-like robot at a point — Brockett's condition rules out smooth time-invariant feedback for such systems, so other strategies are needed
Who should know this
- Mobile robot planning and control engineers
- Autonomous vehicle motion planning teams
- Wheeled-legged and field robot developers
- Robotics students studying kinematics and control
Related technologies
- Dubins Paths
- Reeds-Shepp Curves
- State Lattice Planners
- Pure Pursuit
- Mecanum Wheels
Related terms
- Unicycle Model
- Configuration Space
- Omnidirectional Drive
- Kinematics
- Trajectory Planning
- Underactuated system
Related projects
- OMPL: Sampling-based motion planning library powering MoveIt and countless robotics applications
- MoveIt 2: Motion planning framework for ROS 2 robot arms, with kinematics and collision checking
- PyRoki: Modular Python toolkit for kinematic optimization across CPU, GPU, and TPU via JAX
- mink: Python inverse kinematics library built on MuJoCo supporting 20+ robot platforms
- Ruckig: Real-time jerk-constrained time-optimal trajectory generation for robots and machines
- DART: C++ physics engine for articulated rigid body dynamics and robotics simulation
Videos
- Holonomic vs. Nonholonomic Constraints for Robots | Fundamentals of Robotics | Lesson 4: Mecharithm - Robotics and Mechatronics
- Nonholonomic Constraints & Lagrange Multipliers Explained | ESM 5314 Lecture 22: Dr. Shane Ross