Radians per Second to RPM

Because the whole middleware convention rests on one coherent unit system: metres, kilograms, seconds, radians. Once every message obeys that rule a planner can differentiate a position into a velocity, integrate an acceleration back into a position, and mix rotational with linear terms in a Jacobian without carrying conversion constants around. Radians are what keep that arithmetic clean — arc length equals radius multiplied by angle only when the angle is in radians, so a wheel's contact speed and an end-effector's tangential speed both come out right with no extra factor. A revolution count would serve perfectly well as a display unit but would poison every derivative in the stack, so it is deliberately kept out of the interfaces and left to documentation and purchasing.

Divide the ground speed by the wheel radius, never the diameter — the single most common slip in a first calculation, and it produces an answer exactly twice as large as it should be. A 130 mm wheel has a 65 mm radius, so 1 m/s becomes 15.4 rad/s. On a differential base the two wheels then diverge with the turn: adding and subtracting half the track width multiplied by the yaw rate gives the left and right speeds, and the wider the base, the harder a fast spin-in-place pushes the outer wheel. Use the loaded rolling radius rather than the moulded dimension if the tyre is soft, because a few millimetres of squash shows up directly as an odometry scale error.

Speed divides by the ratio and torque multiplies by it, less whatever the stage loses to friction. A 50:1 unit spun at 3 000 RPM on the input delivers 60 RPM at the flange, which is 6.28 rad/s. Two consequences matter while selecting. First, an encoder mounted on the rotor sees fifty times more counts per output revolution, which is why a modest encoder still resolves the joint finely. Second, reflected inertia scales with the square of the ratio, so a high reduction makes the load feel almost weightless to the motor while letting the rotor's own inertia dominate the dynamics. Belt or timing-pulley stages ahead of the gearbox fold into the same figure, so the ratio to use is the product of everything between rotor and flange.

No-load speed is the top corner of the torque–speed line: the shaft spinning with nothing attached, producing no useful torque at all. Rated speed is a point chosen inside that line where the winding can sit indefinitely without cooking, commonly around two-thirds to three-quarters of the no-load figure once a real load is applied. Sizing a joint against the no-load number is optimistic to the point of being wrong, because the moment the arm lifts anything the operating point slides down the line and the speed collapses with it. The honest approach is to take the flange speed the axis needs, multiply up through the ratio, and confirm the result sits comfortably below rated speed with the expected torque still available.

Several ceilings stack up and the lowest one wins. Continuous heating in the winding usually bites first, since an axis that spends its life accelerating and decelerating draws far more average current than a steady-speed test suggests. The gearbox contributes its own maximum permissible input speed, and exceeding it wears the flexible components long before anything sounds wrong. Structural and payload limits cap the tip speed the arm's stiffness can control without visible wobble. On top of that, machinery safety practice caps the speed a person may stand beside while teaching, a hard limit unrelated to what the hardware could do. The number that reaches the description file is whichever of those is smallest, and it is a specification rather than a measurement.