MiniSumo Velocity and Momentum

Momentum is stored energy related to the velocity of movement (kinetic energy) and mass. In this section we consider how it can be optimised and used to advantage.

It's usual to have the output speed of a motor/gearbox stated in Revolutions Per Minute (rpm), or in the case of servo drives time to turn through a number of degrees. We want to translate this rotational speed into a velocity for our 'bot, velocity is use as it includes a directional component that speed does not. In our calculations we'll quote Velocity in meters per second, often shortened to m/s or ms-1.

Revs to Velocity With a typical gearbox output speeds of 65 rpm and a 40mm diameter wheel we can calculate velocity, translating to standard units we'll consider seconds and metres. We calculate the circumference of our wheel from:-

      π x D     giving us 3.14 x 0.04m = 0.125m

      65 rpm or 1.08 rev per second gives us 0.135 m/s
      calculated from 1.08 revs per sec x 0.125 m

If this wheel was driven by a servo drive it may have an output speed quoted as a time per number of degrees, for example 0.19sec/60 deg (Hitec HS-311). Here we want to multiply up to get a full second and find how many revolutions are made.

      degrees / seconds = 60 / 0.19 = 315.8 degrees per second

With 360 degree per revolution we get to rev per second

      315.8 / 360 = 0.88 revs per second     or     0.11 m/s given the 40 mm wheel example.

Later we might want to look at minisumo performance and refine these velocities, but for the purpose of this comparison we're taking a simplified model. Note, with a larger wheel we can increase the velocity. If you reconsider the servo drive with a 50 mm wheel, the circumference becomes 0.157 m and the velocity increases to 0.138 m/s

Having calculated a velocity for our 'bot we want to look what force this might generate. Earlier we said that the 'bot has stored kenitic energy based upon it's mass and the velocity of travel. The force this can generate is proportional to the rate at which we give up this energy and is described by:-

      Force = Mass x Acceleration       F = MA

Acceleration is a rate of change in velocity over time, giving us metres per second per second (m/s2 or ms-2).

      Acceleration = (Start Velocity - Final Velocity) / time

We have the starting velocity from our earlier calculations and will make some assumptions about our final velocity and the time period. It can been send that the longer the time period the smaller the acceleration (or deceleration), similarly the smaller the difference from start to final velocity the smaller the acceleration. While we are making some assumptions for our 'bots this doesn't really matter too much. We are using some known values for starting velocity and mass, these calculations give us a simple tool to compare expected performance.

Using the example of the 65 rpm gear motor with 40 mm wheel we had a velocity of 0.135 m/s, we are making the assumption that our minisumo impact on another and continues to push forward at a quarter of it's original velocity. The impact is estimated at taking 0.1 seconds.

      Acceleration = (0.135 m/s - 0.034 m/s) / 0.1s = 1.01 m/s2

      Force = 0.5 kg x 1.01 m/s2 = 0.505 kg.m/s2     or     0.505N

We can see that this only works for an initial impact, with a continuous slow push there is little gain from any small change in velocity. Considering our calculation, based upon friction, for our push at the dohyo surface of 2.38N we have gained another 20% push by virtue of the speed we had at time of impact. We can see that an increase in speed will give us a greater advantage. There is practical limit, that is the velocity we can reach before impacting upon the opponent.

The factors effecting any practical velocity are time before impact and acceleration. The first will be made by assumption, the second will be based upon the characteristics of the drive and any limiting factor the supporting electronics might have in drive control.

Comments and discussion on this page are welcome within the mechanical section of the MiniSumo Forum.

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