Differential Drive Robot Angular Velocity

If ω = 0 the wheels must be equal (v_R = v_L), because the wheelbase L is a strictly positive physical quantity. Dividing by zero gives no information about L. The relation L = (v_R − v_L)/ω is mathematically singular whenever the numerator is zero and the denominator is zero, and physically the wheelbase simply cannot be inferred from wheel velocities when the robot is going straight. In practice this is never an issue: the wheelbase is a design parameter that you measure once with a ruler, not a quantity to be solved for from kinematic data.

The standard robotics convention is that positive ω corresponds to counter-clockwise rotation (a left turn), when viewed from above with the x-axis forward and the y-axis to the left. Under this convention, positive ω happens when the right wheel is faster than the left (v_R > v_L). Some textbooks and some robot firmware use the opposite convention (positive = clockwise), so always check the documentation of the specific system you are working with. The magnitude is unaffected by the convention — only the interpretation of the sign changes.

The turn radius R is the radius of the circular arc traced by the robot's axle midpoint. It is given by R = v/ω, where v is the body velocity and ω is the yaw rate. When ω = 0 (straight line) the radius is infinite. When v = 0 (spin in place) the radius is zero. The curvature κ = 1/R = ω/v tells you how tightly the path bends per unit length of travel. Together, (v, ω) and (R, κ) are two equivalent parameterisations of the same local motion — the choice between them is a matter of convenience. For path following, R and κ are often more natural; for motor control, (v, ω) is often more natural.

Yes — the wheelbase is central to the entire kinematic model. It appears in the inverse relation v_R = v + L·ω/2 and v_L = v − L·ω/2, which converts a desired (v, ω) into wheel speeds. It also determines the physical turn radius achievable at a given speed: for a fixed wheel-speed limit, smaller wheelbases allow tighter turns. And it influences stability: large wheelbases resist yaw rotation, which is one reason long vehicles feel more stable but turn less sharply. In the pure angular-velocity relation ω = Δv / L, the wheelbase is the only geometric parameter that appears, and it acts as a simple scaling factor on the wheel-speed difference.

The maximum spin-in-place yaw rate is set by the maximum wheel velocity magnitude the motors can sustain, divided by half the wheelbase: ω_max = 2·v_max/L, where v_max is the maximum forward speed of one wheel and the other wheel runs at −v_max. For a 0.30 m wheelbase and wheels that can do 0.50 m/s, that gives ω_max = 2×0.50 / 0.30 = 3.33 rad/s, or about 191 °/s — the robot can complete a full rotation in under 2 seconds. Small indoor robots with 0.15 m wheelbases can spin twice as fast, which is why they feel so agile.

Use the inverse relations: v_R = v + L·ω/2 and v_L = v − L·ω/2. Given a desired straight-line speed v and a desired yaw rate ω, these two equations give the wheel speeds that will produce them. Because the relations are linear and unconstrained (any (v, ω) pair produces a unique wheel-speed pair, and vice versa), there is no solution space to search — the calculation is direct. This is what a differential-drive motor controller does on every control cycle: takes the commanded (v, ω), computes the wheel speeds, and commands each motor individually.

Not directly in the ω formula — the relation is expressed in linear wheel velocities (m/s), not angular wheel rotations. However, in a real robot the motor commands are angular velocities (rad/s at the motor shaft), which are converted to linear wheel velocities by multiplying by the wheel radius r: v_wheel = ω_motor · r. So the wheel radius does enter the complete system, but it acts on both wheels symmetrically and does not change the fundamental relation ω_body = (v_R − v_L)/L. When specifying motors, larger wheels give higher top speed but lower torque; the wheelbase L remains the only geometric parameter that scales the yaw-rate response.