Page 54 - Anatomy of a Robot

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Multiplying the top and bottom by v brings us to the two roots of the quadratic:
2
s 1d>v2 11d 12 1>2 CONTROL SYSTEMS 39
Now we perform the inverse Laplace transform using the tables (which are not repli-
cated herein). For the cases where d is less than 1, we have what’s called an under-
damped system that responds much like the overshoot chart. In this case, the Laplace
tables show the basic solution to be of the form

2 sin 1v 11 d 2 t c22

x1t2 1 c1 1e 1 d v t2 2 0.5
where c1 and c2 are to be determined by initial conditions. To find the initial condi-
tions, we look at the equations for the rest state of x and dx/dt. This gives us two equa-
tions in two unknowns and leads to the final solution, which is

x1t2 1 1e 1 d v t2 2 0.5 2 0.5
2 sin 1v 11 d 2 t acos 1d22>11 d 2
This is the final solution and was used to generate the charts earlier. This equation
represents a unit step function starting from x 0 at time 0 and settling at the value of
x 1 after the transients settle out. This can be seen in the behavior of the individual
functions in the solution. The exponential function e (-d v t) dies off over time as t goes
to infinity. The larger the damping, the faster it does so. The function

2 0.5
sin 1v 11 d 2 t . . .2
oscillates and provides the ringing.

Designing the Control System

Well, we’ve come through the math gauntlet and come up with a closed solution of how
the model system behaves. Now, how do we make this useable? Remember our goals;
we are going to answer the following:

How the design of the control system determines how the robot will react
How to characterize the robot’s performance and which design parameters to alter
How to alter the robot’s design parameters
How to get optimum performance from the robot

Let’s tackle the first goal.