Bicycle model

• Also called: single-track model
• Category: Foundations & spatial maths › Kinematics
• Tags: vehicle-dynamics, kinematics, autonomous-driving, mobile-robots, car-like-robots

A planar vehicle model that merges the two wheels of each axle into one wheel on the centerline, giving a simple model of a car-like robot's motion with steering and, in its dynamic form, tire forces.

Definition

The bicycle model, also called the single-track model, describes a four-wheeled vehicle moving in the plane by merging the left and right wheels of each axle into a single wheel on the vehicle's centerline. The result looks like a bicycle with a steerable front wheel and a fixed rear wheel, separated by the wheelbase. It ignores roll, pitch, suspension and the differences between left and right tires, but keeps the geometry that matters most for path following: a car-like vehicle must turn by steering, its turning radius depends on the steering angle and wheelbase, and it cannot move sideways at will.

The model comes in two common forms. The kinematic bicycle model assumes that each wheel rolls without lateral slip, so the velocity of each wheel points along its heading. Its state is position, heading and speed, its inputs are steering angle and acceleration, and with the rear axle as reference point the yaw rate is speed times the tangent of the steering angle divided by the wheelbase. It works well at low speeds and moderate lateral acceleration. The dynamic bicycle model adds the vehicle's mass, yaw inertia and the distances from the center of mass to each axle, and computes lateral tire forces from each axle's tire slip angle, through a linear cornering-stiffness model or a nonlinear tire model such as Pacejka's Magic Formula. It is needed when tires approach their grip limits, as in emergency maneuvers, racing or drifting.

The bicycle model sits between simpler and richer vehicle models. The unicycle model commands forward speed and yaw rate directly, with no steering geometry or minimum turning radius. The Dubins car is a planning abstraction: forward motion at constant speed with a bounded curvature, which is what a kinematic bicycle model becomes at constant speed with a steering limit. Four-wheel models with load transfer, suspension and individual tire forces are used when the bicycle model's lumping of left and right tires is too coarse.

Why it matters

Almost every autonomous car, Ackermann-steered mobile robot and small-scale racing platform uses some form of the bicycle model in its planner, controller or state estimator. It is simple enough to run inside model predictive control at high rates while still capturing steering limits and, in its dynamic form, tire saturation.

Key applications

• Model predictive control for autonomous cars and racing vehicles
• Path tracking controllers such as pure pursuit and Stanley-type steering laws
• Trajectory planning for Ackermann-steered mobile robots
• Vehicle state estimation, including sideslip estimation
• Drifting and limit-handling control

In practice

A team builds a model predictive controller for a small-scale autonomous race car. At low speed they use the kinematic bicycle model, with the wheelbase measured on the car and a steering limit taken from the servo's range, to predict the path over a short horizon. As lap speeds rise, the car starts to understeer in corners and the kinematic predictions drift, so they switch to a dynamic bicycle model. They identify mass, yaw inertia, axle distances and tire cornering stiffness from logged driving data, then check that predicted yaw rate and lateral acceleration match measurements from the IMU before tuning the controller.

Common misconceptions

• The bicycle model is only for bicycles — it is a model of four-wheeled, car-like vehicles in which each axle is lumped into one wheel
• The kinematic bicycle model holds at any speed — it assumes no tire slip and becomes inaccurate as lateral acceleration grows; the dynamic version with tire forces is needed near the grip limit
• The bicycle model is the same as the unicycle model — the unicycle commands yaw rate directly, while the bicycle model turns through steering geometry with a minimum turning radius

Who should know this

• Autonomous driving and vehicle control engineers
• Mobile robot developers working with Ackermann steering
• Motion planning researchers working on car-like robots
• Students and makers building small-scale autonomous race cars

Related technologies

• Ackermann Steering
• Pacejka Magic Formula
• Pure Pursuit
• Model Predictive Control
• Reeds-Shepp Curves

Related terms

• Unicycle Model
• Dubins Car
• Nonholonomic Constraint
• Tire slip angle
• Model Predictive Control

Related projects

• 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
• DART: C++ physics engine for articulated rigid body dynamics and robotics simulation
• MOLA: Modular C++/ROS 2 framework for LiDAR odometry, SLAM, and localization
• IKPy: Pure-Python inverse kinematics library that builds chains straight from URDF
• Vision Pilot: Autoware Foundation's open-source L2 ADAS stack from a single camera

Videos

• Self-Driving Cars - Lecture 5.2 (Vehicle Dynamics: Kinematic Bicycle Model): Tübingen Machine Learning
• Self-Driving Cars - Lecture 5.4 (Vehicle Dynamics: Dynamic Bicycle Model): Tübingen Machine Learning