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a “safety” system must be very well designed to make sure it does not create a
false alarm and disrupt a perfectly good solution. CONTROL SYSTEMS 67
Speed of operation As with any robot control system, good performance is
always expected. The speed of operation is almost always one of the criteria. If the
step sizes are too small, it might take intolerably long to move to the proper solu-
tion. Choose the step size to optimize the robot’s behavior in terms of speed and
accuracy. Consider choosing the step size to best match the capability of the robot
to move and maneuver. If the match is close, the results will be better in the form
of smoother operation.
Now we need a bit of a reward for having slogged through so much “useful” math.
It’s time to dream a bit and talk about more esoteric matters that might not affect us
today or tomorrow but are important anyway.

Time

A little ways back in this book, we talked about the fact that the earth cannot be counted
on to be a stable reference point for our robot. As a practical point, it truly is stable
enough in every case I’ve ever seen, so I’m content not to worry about the earth.
But along comes Albert Einstein to throw us another curve ball (see Figure 2-31). It
turns out that we cannot count on time itself to be unvarying in our calculations.
However, if the robot is puttering around at a slow speed and stays away from black
holes, we can probably ignore the considerations that follow. If the robot will be mov-
ing at high speeds relative to the earth, then Einstein’s calculations come into play.
In the very early 1900s, Einstein came up with the special theory of relativity, which
holds that time does not always run at the same rate. If two bodies are moving with
respect to one another, they will experience time running at two different rates. The
effect does not become serious until the speeds are high. But even the astronauts cir-
cling the earth have to take relativisitic time into account or their orbital calculations
will be off. The following URLs show some of the calculations involved in the theory.
It was a Polish mathematician Minkowski who provided the math that eluded Einstein.
www.astro.ucla.edu/ wright/relatvty.htm
www.physics.syr.edu/courses/modules/LIGHTCONE/twins.html
Time varies roughly as 1/sqrt (1 (v/c) ), where v is the relative velocity of the
2
object and c is the speed of light. Using this formula, plugging in an orbital speed of