Robot Turning Radius Calculator
Solve the kinematic turning relation R = v / ω for any of its three unknowns — turning radius, linear velocity, or angular velocity — with step-by-step math, a live path schematic, and companion motion metrics.
Turning radius couples speed and yaw rate
For a robot following a circular path at constant speed, the turning radius is simply the forward velocity divided by the angular velocity. This is the kinematic definition of curvature: a tighter turn (smaller R) requires either lower speed or higher yaw rate. All path-following controllers work within this constraint — you cannot simultaneously demand high speed and a tight radius without exceeding the robot's yaw-rate capability.
Yaw rate sets the turning limit
Every robot has a maximum yaw rate ω_max set by its steering geometry or by the differential-drive wheel speed limits. At speed v, the tightest possible turn is R_min = v / ω_max. To turn more tightly, you must slow down. This trade-off is why high-speed vehicles have large turning radii and why tight indoor robots move slowly. The product v · ω is the lateral acceleration v²/R, which is also limited by friction and comfort.
Differential drive reaches zero radius
A differential-drive robot with two independently driven wheels can turn in place: by driving the wheels in opposite directions, ω is nonzero while v = 0, giving R = 0. This is the tightest possible turn, but it costs energy and causes wheel slip and odometry drift. Ackermann-steered vehicles have a mechanical minimum radius set by the steering angle limit: R_min = L / tan(δ_max), where L is the wheelbase and δ_max is the maximum steering angle.
Lateral acceleration limits real turns
Although the kinematic relation R = v / ω has no force in it, real turns are limited by lateral acceleration a_lat = v²/R = v · ω. Tyres lose grip and passengers feel uncomfortable above about 4–6 m/s² for cars and 0.5–2 m/s² for service robots carrying payloads. This dynamical limit is what ultimately sets the maximum comfortable speed for a given radius, and it is why highway curves are designed with much larger radii than city intersections.
How to Use This Calculator
The turning relation links three quantities: R, v, and ω. Pick which one you want the tool to solve, then fill in the other two.
1. Choose what to solve for
Pick R, v, or ω from the toggle. The calculator rearranges R = v / ω automatically for whichever you choose.
2. Use realistic presets
Presets cover typical values: radius from 0.15 m (small indoor robot) to 20 m (highway), velocity from 0.2 m/s (slow robot) to 15 m/s (car), and angular velocity from 0.1 rad/s to 3 rad/s.
3. Match units carefully
Use SI units throughout: R in m, v in m/s, ω in rad/s. If you have ω in degrees per second, divide by 57.296. If you have ω in rpm, multiply by 0.10472. The hint under each field shows the value in alternative units.
4. Read the result and verdict
The result card shows the solved value plus equivalent units. A colour-coded verdict classifies the radius from very tight (indoor robot) through moderate (service robot) to wide (highway).
5. Check the schematic and charts
The schematic shows the circular path, the robot, the velocity tangent, and the angular velocity. The chart plots turning radius against velocity for three yaw rates.
6. Switch to Advanced for depth
Advanced mode adds a chart of R vs. ω, a sensitivity check on a +1% input change, and companion metrics including lateral acceleration, turn time, and path circumference.
What This Tool Offers
Built around the kinematic turning relation R = v / ω, the fundamental link between speed, yaw rate, and path curvature.
Three-way solving
Solve for turning radius, linear velocity, or angular velocity. The calculator rearranges the formula for whichever variable you choose.
Clean path schematic
A top-view circular path with the robot, velocity tangent, angular velocity, and radius dimension, all scaled with magnitude.
Turn-class verdict
Each result is classified from very tight (indoor robot) through moderate (service robot) to wide (highway), with a coloured badge.
Multi-unit hints
Each field shows the value converted to related units — cm and ft for radius; km/h and ft/s for velocity; °/s and rpm for angular velocity.
Dual comparison charts
R vs. v for three yaw rates, plus an advanced R vs. ω chart for three speeds.
Reference tables
Typical turning radii by vehicle and robot type, a unit converter, and a full variable reference table.
Reference Tables & Conversions
An angular velocity converter, a turning-radius lookup grid, and typical values.
Angular velocity unit converter
Turning Radius Lookup at ω = 0.75 rad/s
Computed from R = v / ω. Values in metres.
| v (m/s) | ω = 0.1 rad/s | ω = 0.3 rad/s | ω = 0.75 rad/s | ω = 1.5 rad/s | ω = 3.0 rad/s |
|---|
Typical Turning Radii by Vehicle and Robot Type
Representative values at normal operating speed.
| Type | Typical v (m/s) | Typical ω (rad/s) | Turning radius R (m) | Notes |
|---|---|---|---|---|
| Differential-drive indoor robot | 0.2 – 0.5 | 0.5 – 2.0 | 0.1 – 1.0 | Can turn in place (R = 0) |
| Ackermann indoor robot | 0.3 – 1.0 | 0.2 – 0.8 | 0.5 – 3.0 | Mechanical minimum radius |
| Service / delivery robot | 0.5 – 1.5 | 0.3 – 1.0 | 1.0 – 4.0 | Outdoor sidewalks |
| Warehouse AGV | 0.5 – 2.0 | 0.2 – 1.0 | 1.0 – 5.0 | Follows marked paths |
| Forklift (industrial) | 1.0 – 3.0 | 0.2 – 0.6 | 2.0 – 6.0 | Rear-wheel steering, tight |
| Passenger car (city turn) | 5 – 10 | 0.5 – 1.0 | 5 – 15 | Intersection turn |
| Passenger car (highway curve) | 25 – 30 | 0.01 – 0.03 | 200 – 1000 | Lane change / gentle curve |
| Articulated bus | 5 – 10 | 0.1 – 0.3 | 20 – 50 | Large swept path |
| Train (track curve) | 20 – 40 | 0.005 – 0.02 | 500 – 4000 | Rail geometry |
Variable Reference
| Symbol | Meaning | Typical range | Units |
|---|---|---|---|
| R | Turning radius | 0.01 – 10⁴ | m |
| v | Linear velocity | 0.01 – 50 | m/s |
| ω | Angular velocity (yaw rate) | 10⁻³ – 20 | rad/s |
| a_lat | Lateral acceleration | — | v²/R = v·ω (m/s²) |
| T | Period of the circular path | — | 2π/ω (s) |
| L | Wheelbase (Ackermann) | — | m; R = L/tan(δ) |
What Does R = v / ω Measure?
The turning radius relation gives the radius of the circular path followed by a robot or vehicle moving at constant forward speed v while rotating at yaw rate ω. It is the kinematic definition of path curvature and the starting point for all path planning, trajectory generation, and steering controller design. The relation is exact for steady circular motion and is a good approximation for slowly varying turns.
Turning Radius Formula
The defining relationship is:
Rearranged to solve for the other two unknowns:
ω = v / R
The period of the circular path, the lateral acceleration, and the path length are useful companions:
a_lat = v² / R = v · ω
C = 2π · R
For an Ackermann-steered vehicle, the radius is set by the wheelbase L and steering angle δ:
For a differential-drive robot with wheel separation b, the radius is set by the wheel velocities:
Turning Variables
| Symbol | Meaning | Notes |
|---|---|---|
| R | Turning radius | Radius of the circular path traced by the reference point |
| v | Linear velocity | Forward speed along the path |
| ω | Angular velocity | Yaw rate; rate of change of heading |
| a_lat | Lateral acceleration | v²/R; limited by friction and comfort |
| T | Period | Time for one complete circle; 2π/ω |
| δ | Steering angle (Ackermann) | Angle of the front wheels relative to the body |
| L | Wheelbase | Distance between front and rear axles |
How Each Variable Affects Turning Radius
The turning radius is directly proportional to speed and inversely proportional to yaw rate. Doubling speed doubles the radius at the same yaw rate; doubling yaw rate halves the radius at the same speed. This is why a robot moving twice as fast needs twice the space to make a turn of the same yaw rate, or needs twice the yaw rate to keep the same radius. The relation is the kinematic backbone of every path-following controller: to follow a given curvature κ = 1/R at speed v, the controller must command yaw rate ω = v · κ.
In practice, the achievable radius is limited by three factors. First, the robot's maximum yaw rate ω_max caps the curvature at any speed: R_min = v / ω_max. Second, the lateral acceleration limit a_lat,max caps the product v · ω: R_min = v² / a_lat,max. Third, for Ackermann vehicles, the steering angle limit δ_max sets a geometric minimum radius R_geo = L / tan(δ_max) that cannot be reduced by slowing down. The tightest achievable turn is the largest of these three limits, and the designer must balance speed, yaw authority, and steering geometry to meet the mission requirements.
Turning Radius Calculation Examples
R = 1.5 / 0.75 = 2.000 m
v = 2 × 0.75 = 1.5 m/s
ω = 1.5 / 2 = 0.75 rad/s
a_lat = v²/R = 1²/0.5 = 2 m/s²
R = 25 / 0.025 = 1000 m
Frequently Asked Questions
They are reciprocals: curvature κ = 1/R. A small radius corresponds to high curvature (a tight turn), and a large radius corresponds to low curvature (a gentle turn). The curvature formulation is often more convenient in path planning because it is well-behaved as R → ∞ (a straight line has κ = 0, while R would be infinite). The kinematic relation can be written either way: ω = v · κ or R = v / ω. Both express the same physical constraint between speed, yaw rate, and path shape.
Yes, if it is a differential-drive, skid-steer, or omni-directional robot. In a differential-drive robot, the two wheels can be driven in opposite directions, producing yaw motion with zero forward speed: v = 0 and ω ≠ 0, so R = 0. This is called turning in place. Ackermann-steered vehicles cannot do this because the front wheels have a finite maximum steering angle, giving a geometric minimum radius R_min = L/tan(δ_max). In practice, turning in place causes wheel slip, tyre scrub, and odometry drift, so it is used sparingly on robots with soft or high-friction wheels.
For an Ackermann-steered vehicle (the standard car geometry), the turning radius is R = L / tan(δ), where L is the wheelbase and δ is the steering angle of the front wheels. A larger steering angle gives a tighter turn (smaller R), and a longer wheelbase gives a wider turn (larger R). The maximum steering angle δ_max sets the geometric minimum radius R_min = L/tan(δ_max), which is the tightest turn the vehicle can make at any speed. This is why small cars with short wheelbases can manoeuvre in tighter spaces than long trucks or buses.
Although the kinematic relation R = v / ω has no force in it, real turns require a lateral force to change the direction of motion. That force comes from tyre friction (or, for a robot on a smooth floor, from wheel friction), and it produces a lateral acceleration a_lat = v²/R = v · ω. If a_lat exceeds the friction limit (about μg, where μ is the friction coefficient and g is gravity), the vehicle skids. For passenger comfort, a_lat is usually limited to 2–4 m/s² for cars, and 0.5–2 m/s² for service robots carrying open containers. The lateral acceleration limit is what sets the maximum comfortable speed for a given radius.
1 rad/s = 57.2958 °/s = 9.5493 rpm = 0.15915 rev/s. To convert rad/s to °/s, multiply by 57.2958. To convert rad/s to rpm, multiply by 9.5493. To convert °/s to rad/s, multiply by 0.017453. To convert rpm to rad/s, multiply by 0.10472. The angular velocity converter card above performs all these conversions in one step. All of the formulas in this calculator use rad/s as the internal unit, since rad/s is the SI unit of angular velocity.
Yes. The turning radius is defined with respect to a specific reference point on the robot — usually the centre of the rear axle for Ackermann vehicles, or the midpoint between the drive wheels for differential-drive robots. If you choose a different reference point, you get a different radius (or, for an off-centre point, a path that is not a circle at all). The kinematic relation R = v / ω applies to the reference point whose velocity v and yaw rate ω are quoted. In controller design, this is why the reference point must be specified: a controller that regulates the path of the rear axle is different from one that regulates the path of the front bumper.
A typical passenger car has a minimum turning radius (measured to the outer front wheel) of about 5–6 m, corresponding to a turning circle diameter of 10–12 m. Compact cars can reach 4.5–5 m, and large SUVs and trucks are typically 6–7 m. The minimum radius is set by the wheelbase and the maximum steering angle: R_min = L/tan(δ_max). Electric vehicles with independent wheel motors and four-wheel steering can achieve much tighter turns, sometimes under 4 m, by steering the rear wheels in the opposite direction to the front at low speed. The turning circle is a standard published specification for every production vehicle.
Variables
| Symbol | Quantity | Unit |
|---|---|---|
| R | Turning radius | m |
| v | Linear (forward) velocity | m/s |
| omega | Angular (turning) velocity | rad/s |
What it means
The turning radius R is the radius of the circle traced by the robot’s centre when moving with linear velocity v and angular velocity ω. For a differential drive robot, it is determined by the ratio of speeds. This parameter is used in path planning to generate arcs, in obstacle avoidance, and in control to ensure smooth turns. A small R means a tight turn. Understanding the turning radius helps in designing robot motion to navigate confined spaces and to avoid collisions.
Worked example
Robot Turning Radius – Two Detailed Examples
Real‑World| Parameter | Value |
|---|---|
| v (m/s) | 1 |
| ω (rad/s) | 2 |
| Parameter | Value |
|---|---|
| v | 3 |
| ω | 1 |
Common mistakes
- Turning radius: R = v / ω – the radius of curvature of the robot’s path.
- Sign: R is positive for left turns (if ω>0) – direction matters.
- Units: v in m/s, ω in rad/s → R in m.
- Infinite radius: If ω = 0, the robot goes straight (R → ∞).
- Instantaneous centre of curvature: The robot rotates about the ICC located at distance R from the centre.
Applications
Robot turning radius, R = v/ω, determines the radius of the circular path followed by a differential drive robot given its linear and angular velocities. This is used in trajectory planning and control to assess manoeuvrability and to generate smooth paths. Engineers use it to check if the robot can make a required turn within available space, to avoid obstacles, and to compute path curvature. By adjusting the speed ratio, they can control the turn radius. This relation is also used in autonomous driving algorithms. Understanding turning radius is essential for mobile robot navigation and for designing safe and efficient paths.
- Path planning and obstacle avoidance
- Manoeuvrability assessment for mobile robots
- Trajectory generation for curved paths
- Autonomous parking and navigation in tight spaces
- Simulation of wheeled vehicle dynamics
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