Page 64 - Anatomy of a Robot

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CONTROL SYSTEMS 49
altering d changes just the overshoot, with minimal changes to the rise time. Altering v
changes just the ringing frequency with minimal changes to the overshoot. Here’s how
to alter v and d:
Altering v
We know that 1/v m/K.
2
v (K/m) 0.5
To change v, change K or m or both. We can change K by putting a different
spring in. A stiffer spring has a higher value of K. We can change m by alter-
ing the mass of the robot.
Beware!
We know that 2 d/v B/K.
If we change v or K, then we must change B if we want to hold d constant.
Altering d
We know that 2 d/v B/K.
Given v is held constant, in order to change d , alter B if possible. Only alter K
if we must.
Beware!
2
We know that 1/v m/K.
If we change K, then change m to hold v constant.
Most of us are familiar with a particular way of altering d. Many older or used
cars will exhibit a very bouncy suspension. When driven over a bumpy road, the
car will bounce along and be difficult to control. The wheels will often leave the
ground as the car bounces. Most experienced drivers will realize that the car
needs new shock absorbers. But what exactly is happening here? The mass m of
the car is not changing. The springs (spring constant K), installed at the factory
near each wheel, have not changed. The shock absorbers have simply worn out.
The shock absorbers look like tubes, about the size of a toddler’s baseball bat, and
are generally found inside the coil spring of each wheel. These shock absorbers
are filled with a viscuous fluid and provide a resistance to motion as the tires
bounce over potholes. They exhibit a fluid friction coefficient of B. Unfortunately,
the shock absorbers can develop internal leaks and the value of B decreases.
When this happens, the overshoot of the second-order system becomes too great,
and the wheels start to leave the ground. Replacing the shocks restores the orig-
inal value of B and brings the overshoot back to the design levels. Bigger cars
have more mass, bigger springs, and generally have larger shocks.
Here is a PDF file and a web site dealing with the management of shock:
www.lordmed.com/docs/ia_CATALOG.pdf
Let’s tackle the fourth and final goal.