Page 47 - Anatomy of a Robot

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32 CHAPTER TWO
grease in the bearings still has friction. The decrease in speed is somewhat lin-
ear in time. Friction is proportional to velocity and has a force of
F B v
where B is the coefficient of friction, and v is the velocity. This makes intuitive
sense. When you rub your hands together, you have to work harder to rub faster.
The friction grows hotter the faster you go. The force increases and the energy
mounts up faster.
Friction comes in disguised forms. We often think of friction as something
dragging over a surface. Often, elements will have their own internal friction.
A motor will coast to a stop by itself. Springs will heat up as they bounce and
will slowly stop bouncing by themselves. If the coefficient of friction is not
specified inside a system, you can often determine it empirically. The quick
way to do so is to compute the instantaneous deceleration of a mass and com-
pare the two forces:
F m a for the mass
F B v for the friction, so
B m a/v
This technique works for rotational, linear, or spring-type movements.
So now we have to pick a mechanical model of the robot in order to make a mathe-
matical model for it. We will pick an arbitrary model that will probably be different than
our robot’s actual mechanics. However, once we learn how to analyze and manipulate
this arbitrary model, it will be second nature for us to extend our knowledge to other
models. Most systems, even unusual nonlinear ones with spasmodic motions, can be
treated similarly to the model we will study. The math is close to the same. We are look-
ing at what is called a second-order system, so called because the forces are based upon
three different representations of positions (as represented in terms of calculus):

Position The position, x, of a mass. For springs, the force is proportional to x.
Velocity v, the rate of change of position, x, of a mass, the first derivative of x.
In calculus, this is called the first derivative of x with respect to time (v dx/dt).
In everyday terms, we think of it as miles per hour. The force of friction is pro-
portional to dx/dt.
Acceleration a is the rate of change of velocity, the first derivative of v, the sec-
ond derivative of position x. In calculus, a dv/dt or, when written in terms of x,
2
2
a d x/dt .
In a simple system where the acceleration is a constant (such as gravity acting on a
falling object near the surface of the earth):