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Robot Path Curvature

Curvature measures how sharply a path bends. It is simply 1 divided by the turning radius. A small radius means high curvature (sharp turn); a large radius means low curvature (gentle turn); infinite radius means zero curvature (straight line). For example, a path with a 0.5 m radius has curvature 2 m⁻¹. It is used in path-planning algorithms to smooth trajectories and in pure-pursuit controllers to follow curved paths.

Written by the Vorzep Team

Robotics Kinematics Calculator

⟹ Solve⟷κ, R

κ (curvature)

1/m

R (radius)

Please fix the errors above.

Solve for: κ R

Presets:

Curvature—

—

κ: —R: —

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Curvature Magnitude—

Low (< 0.5) Medium (0.5–2.0) High (2.0–5.0) Very High (> 5.0)

κ = 1 / R · Units: 1/m, m. Higher curvature means tighter turns.

Variables

Symbol	Quantity	Unit
kappa	Path curvature	1/m
R	Instantaneous turning radius	m

What it means

Curvature (κ) is a measure of how sharply a path bends. It is the inverse of the turning radius R. For a robot, curvature is related to the ratio of angular to linear velocities. Curvature is used in path planning (e.g., continuous curvature paths), in motion control to avoid jerky movements, and in analysing the geometry of trajectories. In differential drive robots, κ = ω/v. Understanding curvature helps in designing smooth and efficient paths for robots to follow.

Worked example

Robot Path Curvature – Two Detailed Examples

Real‑World

Scenario: A robot follows a circular path with turning radius R = 1 m. The curvature κ = 1 / R = 1 / 1 = 1 m⁻¹. Curvature is a measure of how sharply the path bends. A higher curvature means a tighter turn. The path planner uses curvature to generate trajectories that respect the robot's maximum turning capability, ensuring the robot can follow the planned path.

Parameter	Value
R (m)	1

1κ = 1 / 1 = 1 m⁻¹

Result 1 m⁻¹ ✓ Curvature

Scenario: A robot travels along a gentle curve with R = 10 m. The curvature is κ = 1 / 10 = 0.1 m⁻¹. This low curvature indicates a nearly straight path. The control engineer uses curvature to adjust the robot's steering angle and maintain smooth motion, which is particularly important for passenger comfort in autonomous vehicles.

Parameter	Value
R	10

1κ = 1 / 10 = 0.1 m⁻¹

Result 0.1 m⁻¹ ✓ Gentle curve

Insight: Curvature is the reciprocal of the radius of curvature. It is a key parameter in path planning for mobile robots and vehicles, influencing steering and speed control.

Common mistakes

Path curvature: κ = 1 / R – the reciprocal of the turning radius.
Units: κ in 1/m.
Sign: Positive for left turns, negative for right turns.
Curvature vs. radius: A straight line has κ = 0.
Used in: Path planning and control (e.g., pure pursuit).

Applications

Robot path curvature, κ = 1/R, is the reciprocal of the turning radius and is used in trajectory planning to describe how sharply a path bends. Engineers use curvature to define smooth, continuous paths for mobile robots and to ensure kinematic feasibility. In control, curvature is used in pure pursuit and Stanley controllers for autonomous navigation. Higher curvature corresponds to tighter turns, which may be limited by robot kinematics. By analysing curvature, they can avoid sharp turns that cause slipping or instability. This formula is essential for path generation and motion control in automated vehicles.

Trajectory smoothing and optimisation for autonomous vehicles
Controller design for path tracking (pure pursuit, Stanley)
Kinematic feasibility assessment of planned paths
Simulation of vehicle dynamics and stability
Race line optimisation for autonomous racing

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Published July 13, 2026
Last updated September 18, 2026