Servo - September 2011

www.servomagazine.com/index.php?/magazine/article/september2011_MrRoboto

The force we need to apply () must be greater than the force of friction plus the force of weight (due to gravity) to move. The force due to gravity is shown as this formula:

θ sin mg Fw =

which means the product of our mass and acceleration due to gravity times the sin of the angle from the perpendicular to the ground is the force due to gravity that needs to be overcome. As obvious as this formula may seem (that was a joke, really), it may need some explanation.

We all get the “heavy things are hard to move” part, but what is that sin theta thing all about? It’s easy, really. Push something on level ground, grunt a bit, and off it goes, right? Well, now push it up your driveway at a 20 degree angle. Suddenly it isn’t so easy to push, right? Now, for the next bit, friction — the other half of our simple equation above. This is the formula for the force to overcome friction:

θ µ cos mg Ff =

The first term, mu, is the coefficient of friction — static friction (more on that later.) You have seen “mg” before; it is weight — the cosine is of the angle off of the horizontal plane. What this all comes down to is that friction is highest on flat ground and zero when falling straight down; that makes sense. There are two kinds of friction: static friction, which is the friction to overcome to get moving; and dynamic friction — that which needs to be overcome while you are in motion. The latter is far less than the former. Even a wheel has a surface area in contact with the ground; it isn’t just a single point, it has width and length. Until that wheel starts to turn, it is basically just skidding on the ground.

Once the wheel starts turning though, everything changes and the friction goes way down. Push anything with wheels. Did you notice that you really had to heave to get it going? Once it is moving, it is much easier to keep moving. That is partially friction; the other part is Newtonian physics — momentum; objects in motion tend to want to stay in motion.

Back to the point. The coefficient of friction is rarely given, but can be measured. If you are interested, it is the ratio between the resistive friction over the normal friction. Normal friction is basically the force holding things together, mg, or weight. To get the resistive friction, get a fish scale and attach it to the object and pull. That force will look like weight since this is a scale; take that value. Simple.

Now, we have the final formula for measuring the force needed to move your robot lawn mower:

θ µθ cos sin mg mg Fapp + =

I like to work in metric units, so g = 9.8m/s2, therefore, I use mass in kilograms. Finally, we need power. This formula is: which means power is force times velocity. For that to be useful, we need this formula for motors:

r v rv ω ω =⇒ =/

Rotational velocity is velocity divided by the radius of the arm from the center of the motor. The second formula just puts it into the form we want for the power formula above.

Now for units. These are the units to work with when using these formulas to get the power you need for your motors:

Term Power Force Mass Angle Rotational Velocity

Unit Watts (W) Newton Meters (Nm) Kilograms (kg) Radians Radians/Sec

Pick the angle to match the inclines that will most likely be the worst case scenarios in your yard. Pick a coefficient of friction to be something like a car tire, which I’ve found to be about . 9 to 1.0 for a rubber tire on concrete (to be conservative). If you choose 1.0, obviously that term falls out. Just plug in your numbers.

How do you know the power of your motors? Sadly, no one ever gives you that. If you get a motor brand new from a manufacturer, you might be able to get the maximum torque and maximum rotational velocity, which gives you the power by this formula:

ω T Pm =

The motor power is the torque times the rotational velocity; this gives you the power at any chosen torque and velocity, but not the maximum power. The maximum torque in a DC motor is at zero rotational velocity, which is no power. The maximum rotational velocity is where the motor torque is at its minimum. To get a motor’s maximum power, it needs to be where the torque is at 1/2 and the velocity is at 1/2, which gives this formula:

max max max 4

1 ωT P=

Torque is force and (here) is shown as Nm; rotational velocity in is radians/second. I don’t know the English units; I’m more comfortable in metric. Torque is the angular force that a motor can deliver at some distance from the shaft. If your motor could lift 1 kg on a pulley with a one meter radius, that would be one Newton meter. There are 2p radians in a full circle.