help calculating velocity up a ramp

C.R.C,

I'm trying to estimate the speed of a LEGO robot traveling as fast as possible up a ramp of a given slope.... here is what I have:

Given data:

The robot has these characteristics: mass: 0.55 kilograms ramp angle a = 20 degrees = 0.349 radians (there will be 2 other ramp angles as well, but this will show the math) wheel radius = 0.04445 meters gear ratio: 25:1 (there will be 2 other gear ratios as well, but this will show the math) 1 drive motor

The motor has these specs: RPM: 340 = 35.6 Radians per second Noload current: 0.009 Amps Stall Torque: 0.055 Newton-Meters Stall Current: 0.34 Amps

Here is the procedure I have tried thus far, but I get bogged down in the final Step...

Step 1: Calculate the power available at the wheel

Applying the gear ratio: RPM: 35.6 / 25 = 1.424 Radians per second Stall Torque: 0.055 * 25 = 1.375 Newton-Meters

Applying the wheel radius: Force at wheel = stall Torque / wheel radius = 1.375 / 0.04445 = 30.933 Newtons

Is that right?

From Inspiration to Implementation, page 217 I can calculate the theoretical maximum possible power:

Pm,max = 1/4 * (Stall Torque) * (radians per second) = 1/4 * (1.375 ) * (1.424) = 0.4895 Watts

So, Pm,max tells me how much power I have available at the wheel... 0.4895 Watts?

Step 2: Calculate how much force is needed on the ramp

Assuming constant velocity up the ramp, such that acceleration is ZERO, F = m * a = 0.55 * 0 = 0

So the net force must be zero, and the applied force "Framp" must be the sum of the frictional force "Ff" plus the force of the weight of the robot acting down "Fw"

Framp = Ff + Fw

Given that Ff = u * Fn where u = coefficient of friction, g = the normal gravitational force and (m*g) is just the weight of the robot, so that Ff = u * (m * g) * cos(ramp angle) = u * 0.55 * cos(20deg) = u * 0.55 * 0.9396 = u * 0.5167

Assuming the coefficient of friction u = 0.3 (from same book), then Ff = 0.3 * 0.5167 = 0.155

So we now have Framp = 0.155 + Fw

Given that Fw = m * g * sin(ramp angle) Fw = 0.55 * 9.8 * sin(20deg) = 0.55 * 9.8 * 0.3420 = 1.8434

So finally, the applied force is... Framp = 0.155 + 1.8434 = 1.9984

So, Framp tells me how much force is needed on a ramp of 20 degrees... 1.9984 (what are the units? Newtons or Newton-meters or Watts or kilograms?

Step 3: Calculate the speed up the ramp

Here is where I get uncertain. I need to apply these two formulas? 1) Power = Fapp * Velocity (Is Power for the available power.... 0.4895 Watts?)

I need to solve for Velocity, so rearranging them yields 1) Velocity = Power / Fapp 2) Velocity = (rotational velocity ) * wheelRadius

And here I am stuck, since I don't know: a) units for Framp b) is Power in equation 1 above the same as Pm,max? c) how to use equation 2 above...

Thanks for your help!