shaft. Torque is measured in units of force and distance, for
example in-lbs. What does this mean? If you have a torque
of 6 in-lbs, you can expect 6 lbs of force, one inch from the
shaft of the motor. In our case, our drive wheels are six
inches in diameter. This means that the force is applied
three inches from the drive motor. We could expect 2 lbs of
force to be applied by the wheel to the ground. As you will
see, many of these values come in a variety of units. Some
popular measurements of torque are ft-lb (foot-pounds), oz-in (ounce-inches), and Nm (Newton-meters). Some units will
be in English units and some will be in metric units. Values
will need to be converted to metric before plugging them
into the equations. In most cases, these resulting answers
will need to be converted to English units in order to
determine what motor to order from most suppliers.
Torque = Force x Distance
Finally, velocity is the speed at which our robot will
move up the incline and acceleration represents how fast
our robot will reach the desired velocity.
How does acceleration relate to speed (velocity)?
Now that we understand the forces acting on our
robot, we can begin the process of sizing the drive motors.
To determine what size motors we need for our robot,
we will need to define the following:
Velocity = Acceleration Time + initial velocity 2
Weight of the robot: w = 25 lbs
Maximum Speed: 60 ft/min v = 1 ft/s
Maximum incline to climbs: θ= 10 degrees
Reach maximum speed in two seconds: a = .254 m/s
Drive wheels will be six inches in diameter: r = 3 in
From our free body diagram, we will focus on the
forces working in parallel to the inclined surface. Let’s also
assume that our robot will start from rest and need to
accelerate up the incline to full speed.
fw = the force pushing against the wheel
fg = the force pulling robot down incline due to gravity
Torque : T = fw r
In physics, all forces must balance which gives the
ΣForces = 0
If your robot is moving at a constant speed, the
sumation of all forces will equal zero.
ΣForces = ftotal = fw - fg = 0
To properly size our motor, we will focus on the
situation where the robot is accelerating from rest to full
speed. This is where you want to size your motors to be
large enough to get the job done. The torque required to
get your robot moving can be much greater than keeping it
in motion. In this case, the sumation of the forces acting on
our robot will equal the total mass multiplied by
acceleration. I usually accelerate my robot to full speed in
one second or less in the calculations.
ΣForces = ftotal = fw - fg = Ma
fw= Ma + fg
T/r = Ma + Msinθ
T = M(a + gsinθ)r
We must convert weight to mass in metric units:
M = 25 lbs ((1 kg)/( 2. 2 lbs)) = 11. 36 kg
Next, we convert radius from three inches to meters.
T = ( 11. 36 kg) .254 m + 9. 8 m sin( 10) .0762 m = 1.69 Nm
r = 3 in 2. 54 cm 100 m = .0762 m in cm
(() ) s2 s2
Most of the motors I have worked with define torque
in in-lbs or oz-in. We will convert to in-lbs:
Tft-lbs = 1.69 Nm .225 lb 100 cm 1 in = 14.97 in-lb ( ( ) ) 1 N 1 m 2. 54 cm
This is the total torque required to drive the robot. Since
we are using two drive motors, we can divide this in half.
Tper motor = 6. 5 in - lb
Next, we will determine how fast in rpms the motor
will need to turn.
Rev = Velocity ft ) ( Min min
Finally, to determine how much power the motors are
required to supply, the following equation should be used:
P = T ω ω = angular velocity
Angular velocity is measured in radians per second.
One Revolution = 2 π radians
( 2 π Drive wheel radius) 1 ft = 38 Rev/Min ) ( 12 in
ω = 38 Rev 2πRad 1 min sec = 3.98 rad/sec Min 1 Rev 60
P = 1.69 Nm 3.98 rad / sec = 6. 72 watts
Tips For Selecting DC Motors
SERVO 01.2010 35