Education Lab / Robotics Theory Basics
Angular Velocity & Acceleration
Angular velocity tells us how quickly an angle changes. Angular acceleration tells us how quickly angular velocity changes. A joint can rotate quickly at constant speed while having zero angular acceleration.
Why it matters for a robot
A robot motion command specifies an angle and a time, but motor sizing also needs the speed and acceleration during that move. Our 90° joint study uses a motion profile to distinguish average speed from peak speed.
Core equations
ωavg = Δθ / Δt
αavg = Δω / Δt
θ is angle (rad), t is time (s), ω is angular velocity (rad/s), and α is angular acceleration (rad/s2). Δ means final minus initial. Convert degrees using radians = degrees × π/180. Convert rpm using rad/s = rpm × 2π/60.
A small worked example
A joint accelerates uniformly from rest to 30 rpm in 0.50 s. First convert the final speed: 30 × 2π/60 = π ≈ 3.14 rad/s. Then α = (3.14 − 0)/0.50 = 6.28 rad/s2. Because acceleration is constant during this ramp, the average acceleration equals the instantaneous acceleration throughout it.
The rising velocity line has a constant positive slope: angular acceleration.
Common mix-ups
Do not insert rpm or degrees directly into a radians-based torque calculation. Also, angle divided by total time gives average velocity; it does not give the peak velocity of an accelerating and decelerating move. The slope of a velocity graph is acceleration; its area is angle travelled.
Reference: OpenStax · Rotational Variables.
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Related concept: Motion Profiles →