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?)

        2) angular Speed (a.k.a. rotational velocity)  = Velocity / wheelRadius
               (Is rotational velocity just the RPMs at the wheel?)


    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!