Answer Happy

QUESTION:
You are working for a startup robotics company designing a smalldifferential-drive mobile robot, and your job is to choose themotors and gearing. A diff-drive robot has two wheels, each drivendirectly by its own motor, as well as a caster wheel or two forbalance. Your design specifications say that the robot should becapable of continuously climbing a 20◦ slope at 20 cm/s. Tosimplify the problem, assume that the mass of the whole robot,including motor amplifiers, motors, and gearing, will be 2 kg,regardless of
the motors and gearing you choose. Further assume that the robotmust overcome a viscous damping force of (10 Ns/m)×v when it movesforward at a constant velocity v,
regardless of the slope. The radius of the wheels has alreadybeen chosen to be 4 cm, and you can assume they never slip. If youneed to make other assumptions to complete the problem, clearlystate them.
You will choose among the 15 V motors in Table 25.1, as well asgearheads with G = 1, 10, 20, 50, or 100. Assume the gearingefficiency η for G = 1 is 100%, and for the others,
75%. (Do not combine gearheads! You get to use only one.)
A). Provide a list of all combinations ofmotor and gearhead that satisfy the specifications, and explainyour reasoning. (There are 20 possible combinations: four motorsand five gearheads.) “Satisfy the specifications” means that themotor and gearhead can provide at least what is required by thespecifications. Remember that each motor only needs to provide halfof the total force needed, since there are two wheels.
B). To optimize your design, you decide to usethe motor with the lowest power rating, since it is the leastexpensive. You also decide to use the lowest gear ratio that workswith this motor. (Even though we are not modeling it, a lower gearratio likely means
higher efficiency, less backlash, less mass in a smallerpackage, a higher top-end speed (though lower top-end torque), andlower cost.) Which motor and gearing do you choose?
C). Instead of optimizing the cost, youdecide to optimize the power efficiency—the motor and gearingcombination that uses the least electrical power when climbing upthe 20◦ slope at a constant 20 cm/s. This is in recognition thatbattery life is very important to your customers. Which motor andgearhead do you
choose?
Table 25.1: Motors to choose from 10 15 Assigned power rating Nominal voltage No load speed No load current Nominal speed Max continuous torque Max continuous current Stall torque Stall current Max efficiency Terminal resistance Terminal inductance Torque constant Speed constant Mechanical time constant Rotor inertia Max permissible speed Cost W V rpm mA rpm mNm mA mNm mA % Ohm mH mNm/A rpm/V ms gcm² rpm USD 3 15 13,400 36.8 4980 21.8 10,400 3920 2.31 28.2 259 10.5 1030 65 14.6 0.486 10.2 932 7.51 1010 131 4570 0.541 16,000 87 3.28 0.353 28.6 334 4.23 10.6 5500 88 228 20 15 9660 60.8 8430 20.5 1500 225 15,800 82 0.952 0.088 14.3 670 4.87 10.4 14,000 236 90 15 7180 247 6500 73.1 4000 929 47,800 83 0.314 0.085 19.4 491 5.65 68.1 12,000 239 Note that sometimes the "Assigned power rating" is different from the mechanical power output at the nominal operating point, for manufacturer-specific reasons. The meanings of the other terms in the table are unambiguous.