Arrgggg Stupid Fisher Price Motors

[Paul Copioli]

Wysiwyg,

I just to make sure you understand the relationship between weight and the force needed to propel your robot.

F=ma right? Right. However, the largest determination in how your robot will behave is friction in your drivetrain. The actual equation I use is:
F(at wheels) = m(130lbs or 59 kg) * a + Ff(friction)

An easy way to determine Ff is to take your motors off, but leave the transmissions and pull the robot at a constant speed with a spring scale. I usually add 10% and use that.

F=Tout / Rwheel and Tout = Tmotor * GR * eff

Tout is the torque output of your drivetrain
Rwheel is your wheel radius (not diameter)
Tmotor is motor torque
GR is your gear ratio (usually > 1)
eff is your drivetrain efficiency (0.70 < eff < 0.95)

In DC motors there exists a speed-Torque relationship as follows:

Tmotor = K*Smotor + Tstall

K is the slope of the Torque - Speed curve of the motor (negative)
Smotor is the motor speed
Tstall is the stall torque of the motor
Please watch your units!!

Putting it all together:

(K*Smotor+Tstall) * GR * eff / Rwheel = M * a + Ff


'a' is the time rate of change of speed of the robot, Vout and if you take small time increments the formula for a at any instance of time 'i" is:

a_i = (Vout_i - Vout_i-1)/(t_i - t_i-1)
to shorten the notation we will use a=dVout/dt

Sout is the ROTATIONAL speed of the wheels and is related to Vout using the following equation:

Vout = Sout * Rwheel,

so the equation is a = Rwheel *dSout/dt

Remember that Smotor is related to Sout by the relationship:

Smotor = Sout * GR

Again, putting it all together:

(K*Sout*GR+Tstall)*GR*eff/Rwheel = M*Rwheel*dSout/dt +Ff

You can solve this for Sout at any instance in time, i, and put it into a spreadsheet formula and get robot speed vs. time for various wheel and gear ratio combinations. The final spreadsheet equation using 'i' as the time right now and 'i-1' as the previous time (use a time step between 0.01 and 0.1 seconds) time step is notated as 'dt':

Sout_i = (Tstall*GR*eff*dt + M*Rwheel*Sout_i-1 - Ff*dt) / BIG Y

BIG Y = M*Rwheel - (K*GR^2*eff*dt/Rwheel)

The equation seems long, but it is pretty straightforward. If you use a spreadsheet and use initial conditions at t=0 if Sout=0, you can solve for Sout at each time increment and can figure out how long it will take you to get to max speed.

I haven't checked my notes, but this looks right. I will double check tonight.

I hope this helps.

Paul