Page 72 - Anatomy of a Robot

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CONTROL SYSTEMS 57
A CAUTION
So far, we’ve been talking about robot control systems in a very abstract way. The equa-
tions show very nicely that our mathematics will cleanly control the position of our
robot in a very predictable manner. Further, we can smugly make minor parametric
changes in the equation and our robot will blissfully change his ways to suit our best
hopes for his behavior.
Well, it’s very easy to get lost in such a mathematically perfect world. Those of us who
have had kids are well acquainted with a higher law than math called Murphy’s Law. Visit
www.murphys-laws.com for the surprising history of Murphy’s Law on the variants
thereof that apply to technology. I had long suspected that such wisdom would be bibli-
cal in its origin, but it came into being in 1949.
Murphy’s Law, as commonly quoted, states “Anything that can go wrong will go
wrong.” All along, we have been plotting and scheming to build and control a second-
order control system. We’ve got that pretty well down. The trouble is our model will
never exactly fit the real-world robot we’re building. We have a mathematical control
system that will control a single variable, such as our robot’s position, to ever-exacting
precision. However, this will not be the only requirement we will have to satisfy. We have
ignored other unstated requirements along the way. To satisfy these other requirements,
we may have to change the behavior of our simple control system, or we may have to put
in even more controls. The following section on multivariable control systems speaks to
this issue somewhat. Here’s a few other requirements that are liable to crop up:
Speed Great, we’ve designed our position control system so our robot will move
to where it belongs. But what about speed traps? Velocity is the first derivative of
position. In the parlance of the variables we have been using, v dx/dt. We really
haven’t worried about speed at all so far. Clearly, it is partially related to the rise time
of the position variable. The quicker the control system can react to changes in posi-
tion, the faster it is likely to go. But there will be various restrictions on speed:
Safety Sometimes it’s just not safe to have a robot moving around at higher
speeds.
Power Sometimes it’s wasteful to go too fast. Some motors and actuators are
not as efficient at top speed.
Maneuvering Some robots don’t corner well. It can be advisable to slow
down on the curves.
Acceleration Fine, we’ve designed our velocity control system so our robot will
not speed or be a hazard. But how fast can we punch the accelerator? Acceleration
is the first derivative of velocity and the second derivative of position. In the parl-
ance of the variables we have been using,

2
A dv>dt d x>dt 2