Page 45 - Anatomy of a Robot

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30 CHAPTER TWO
An ideal control system would follow the step input function and produce the same
step output function. The robot would instantly move to the new position and stop on a
dime with no steady state error. We’ve already seen how the robot will have a steady state
error (not fully reaching the desired final position). The truth is, the robot cannot move
instantly and it cannot stop on a dime. The control system in the robot sees the step input,
delays a bit for reaction time, finally starts moving, and tries to stop near the final posi-
tion. The response is going to be imperfect no matter which way we slice the pie.
So before we look at how control systems really behave, we’re going to have to stop
and do some math. After that, we’ll have to tools to see the following:
How the design of the control system determines how the robot will react
How to characterize the robot’s performance in a few parameters
How to know which design parameters to alter based on the robot’s performance
How to get optimum performance from the robot

To get the tools we need to analyze and manipulate the performance of the robot,
we’re going to pick a mathematical model for the robot and derive some of the equa-
tions. We’re going to skip the easier models of robot behavior and go straight to a
slightly more complex case. We are going to use math and physics that might be beyond
the casual reader’s abilities, but we will return to a usable, intuitive model of what’s
going on. We will start with physics, calculus, Laplace transforms, and algebra to arrive
at usable results. Once we have that math in front of us, we will explore the tools it
affords us.
First, we need a way to look at the parts of the robot and assign numbers to the move-
ments we observe. This can be done in a couple of ways:
Energy evaluation One way to analyze dynamic movement is by looking at
everything in terms of energy: where it’s stored and how it’s used. We are not going
to use this technique any further in this book, but it’s worth mentioning the alter-
nate technique. Energy is stored in multiple places in a robot, certainly in the bat-
teries, but it is also temporarily stored in other places
Springs (potential energy) A good mathematical description of springs will
be provided a little further along in this book. As a spring is compressed, the
energy E in the spring is

E 0.5 K x 2
where x is the compression distance. Note this equation only works for smaller
values of x because an overly compressed spring becomes nonlinear and runs
out of springiness. K is the spring constant; a bigger, stronger spring has a
larger K value.