Formula & Calculator
Wheel Linear Speed from Angular Speed
Calculates the linear surface speed of a rotating wheel from its angular velocity and radius.
Interpretation
v = ω·r. Linear speed of a wheel equals angular speed times radius. Used to convert encoder readings to linear speeds.
Variables
| Symbol | Quantity | Unit |
|---|---|---|
| v | Linear (surface) speed of the wheel | m/s |
| omega | Angular velocity of the wheel | rad/s |
| r | Radius of the wheel | m |
What it means
For a wheel rolling without slipping, the linear speed at its contact point is equal to the product of its angular speed (ω) and its radius (r). This relation is used in odometry to compute the robot’s speed from wheel encoder counts (which give angular displacement). It is also used to convert motor speeds to wheel velocities. Understanding this formula is fundamental for mobile robot kinematics and for calibrating encoders.
Worked example
Wheel Linear Speed from Angular Speed – Two Detailed Examples
Real‑World| Parameter | Value |
|---|---|
| ω (rad/s) | 10 |
| r (m) | 0.05 |
| Parameter | Value |
|---|---|
| ω | 5 |
| r | 0.1 |
Common mistakes
- Wheel linear speed: v = ω·r – where ω is angular speed (rad/s) and r is wheel radius.
- Units: ω in rad/s, r in m → v in m/s.
- Radius: Use the effective rolling radius (may differ from nominal due to deformation).
- Slip: If there is slip, this formula gives the theoretical speed – actual may be lower.
- Sign: v is positive for forward motion – ω positive for forward rotation (right‑hand rule).
Applications
Wheel linear speed from angular speed, v = ω·r, relates the rotational speed of a wheel to its linear velocity. This is fundamental for wheeled robots and vehicles. Engineers use it to convert encoder counts (angular position) to distance travelled, to compute speed commands for motors, and to calibrate odometry. It is also used in designing traction control systems and in estimating vehicle speed from wheel sensors. By understanding this relation, they can accurately track motion and control wheeled platforms. This formula is a basic building block for mobile robotics and automotive engineering.
- Calibration of wheel odometry for mobile robots
- Conversion of motor RPM to vehicle speed
- Speed control of wheeled vehicles
- Design of traction and braking systems
- Educational understanding of rotational‑linear motion mapping
Frequently Asked Questions
The linear speed v of a point on the wheel's circumference is v = ω · r, where ω is the angular speed (rad/s) and r is the wheel radius (m). This assumes no slipping.
Using the wheel's diameter instead of its radius, which doubles the calculated linear speed. Always use radius.
ω in rad/s, r in meters, v in m/s. If ω is in RPM, convert to rad/s first: ω_rad/s = (RPM × 2π) / 60.
If there is a gear reduction, the wheel angular speed is the motor speed divided by the gear ratio. The linear speed then is v = (ω_motor / gear_ratio) · r.
A larger wheel diameter (and radius) results in a higher linear speed for the same angular speed. This is why larger wheels are used for higher speed robots.
ω = (Δcounts / CPR) × (2π / Δt), where CPR is counts per revolution (including quadrature multiplication if any).
Odometry, motor control, and speed estimation for mobile robots.
v (m/s) = (RPM × 2π × r) / 60.
Slip causes the actual linear speed to be less than the theoretical value (v = ω·r). The slip ratio is used to account for this in vehicle dynamics.
1 inch = 0.0254 m. If the radius is given in inches, multiply by 0.0254 to get meters.