Unicycle Model
- Also called: unicycle kinematic model, unicycle kinematics
- Category: Foundations & spatial maths › Kinematics
- Tags: kinematics, mobile-robots, wheeled-robots, control, navigation
A kinematic model of a wheeled robot as a single wheel with planar position and heading, driven by a forward speed and a turn rate and unable to move sideways.
Definition
The unicycle model describes a wheeled robot by its position (x, y) on the plane and its heading θ, with two inputs: a forward speed v and a turn rate ω. The equations are ẋ = v cos θ, ẏ = v sin θ and θ̇ = ω. Combining them gives ẋ sin θ − ẏ cos θ = 0, the nonholonomic constraint that the robot cannot move sideways. The model is purely kinematic: it ignores mass, wheel slip and motor dynamics and assumes the commanded speeds are achieved.
A differential-drive robot maps directly onto the model. With wheel radius r, axle length L and wheel speeds ω_R and ω_L, the forward speed is r(ω_R + ω_L)/2 and the turn rate is r(ω_R − ω_L)/L, so controllers designed for the unicycle can be converted into wheel commands. Car-like robots are usually described by the related bicycle model, which adds a steering angle and wheelbase and so limits the turning radius. In multi-wheeled and wheeled-legged systems, each steerable wheel can be treated as its own unicycle, described by its contact position and steering angle, with the body motion coupling the wheels.
The model is the standard starting point for mobile robot control and planning. Its simplicity makes it differentially flat, with the planar position as flat output, so smooth paths in x and y can be turned into feasible speed and turn-rate commands. Common controllers include pure pursuit, Lyapunov-based posture regulation, and feedback linearization of a point placed ahead of the wheel axle, which avoids the singularity of controlling the axle center itself. Its limits show up at higher speeds and on loose ground, where wheel slip, lateral forces and actuator dynamics make the kinematic predictions inaccurate.
Why it matters
Almost every wheeled mobile robot is first modeled, planned for and controlled as a unicycle. Understanding the model and its assumptions is the basis for writing odometry, local planners and tracking controllers for such robots.
Key applications
- Trajectory tracking control of differential-drive robots
- Local planners and velocity sampling in navigation stacks
- Wheel odometry for mobile bases
- Modeling steerable wheels on wheeled-legged robots
- Teaching and benchmarking nonlinear control methods
In practice
A developer writing a controller for a small differential-drive robot represents it as a unicycle. The navigation stack outputs a path, and a pure pursuit controller picks a lookahead point and computes the turn rate needed to reach it at a chosen forward speed. Those two commands are converted into left and right wheel speeds using the measured wheel radius and axle length, and the same formulas, run in reverse on encoder readings, give the robot's wheel odometry. When the robot is moved to a carpeted floor, the developer sees heading errors grow and recalibrates the effective axle length, since slip makes the real turning behavior differ from the nominal geometry.
Common misconceptions
- The unicycle model only applies to one-wheeled robots — it describes any planar vehicle that moves forward and turns but cannot move sideways, including differential-drive bases
- The model captures a robot's dynamics — it is kinematic and ignores mass, slip and motor limits
- The unicycle and bicycle models are interchangeable — the bicycle model adds a minimum turning radius, while an ideal unicycle can turn in place
- A simple state-feedback law can stabilize a unicycle to any pose — its nonholonomic constraint rules out smooth time-invariant feedback for point stabilization
Who should know this
- Mobile robot software and controls engineers
- Navigation stack developers
- Wheeled-legged robot researchers
- Robotics students learning kinematic modeling
Related technologies
- Differential Drive
- Bicycle Model
- Pure Pursuit
- Nav2
- Dynamic Window Approach
Related terms
- Nonholonomic Constraint
- Differential Flatness
- Feedback Linearization
- Odometry
- Omnidirectional Drive
- Model Predictive Control
Related projects
- Nav2: ROS 2 navigation stack with path planning and obstacle avoidance for mobile robots
- DimOS: Python robotics runtime with agents, SLAM and teleop, no ROS required
- Drake: MIT robotics toolbox for model-based dynamics, optimization, and control system design
- ROS Motion Planning: One repo implementing 40+ classic and learning-based robot path planners as ROS plugins
- MRPT: Comprehensive C++ toolkit for mobile robotics, SLAM, and sensor-based navigation
- MoveIt 2: Motion planning framework for ROS 2 robot arms, with kinematics and collision checking