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Chapter 10 Controllers for automation 247

result, leading to either a critical or an overdamped system; therefore, z, has to be greater
than or equal to one. However, if the system is considerably overdamped, the response
to a change in the demand may be so poor as to make the system useless. An additional
constraint that should be considered is the relationship of the undamped natural
frequency to the vibrational characteristics of the mechanical structure. It is normally
recommended that a servo loop’s undamped natural frequency is no more than half that
of the mechanical structure’s resonant frequency. The derivation of the natural
frequency for a rotating shaft is discussed in Section 3.6. While the detailed derivation for
a manipulator system is far more complex, it can be shown that for a single joint the
natural frequency is given by,
s ffiffiffiffiffiffiffiffi
K s
u n ¼ (10.10)
I tot
where K s is the effective stiffness of a joint, I tot is the total inertia of the system. As
discussed in Chapter 3, Power Transmission and Sizing, a torque resulting from the wind
up of the shaft, K s q m (t), opposes the inertial torque of the motor, so that,
I tot a m ðtÞþ K s q m ðtÞ¼ 0 (10.11)
where a m is the motor’s acceleration.
A further critical factor in the consideration of a servo system is the steady state error,
e ss , which should be as close to zero as possible in a robotic or a machine-tool appli-
cation. The error within the system, e(t), can be defined as the difference between the
actual and the demanded position,

d
eðtÞ¼ q ðtÞ q L ðtÞ (10.12)
L
d
For a step input of magnitude X, that is, q ðtÞ¼ X, and if the disturbance input is
L
unknown, then the steady state error of the system can be determined by the use of the
final-value theorem. This theorem states that the steady-state error, is given by,
e ss ¼ limeðtÞ¼ limsEðsÞ (10.13)
t/N s/0
Using the overall transfer function, the steady-state error for a step input (X/s) can be
determined as,

2
½s R a I tot þ sðR a B þ K t K v Þ X=s þ nR a T d ðsÞ
e ss ¼ lim (10.14)
s/0 2
s R I tot þ sðR a B þ K e K t þ K t K v Þþ K t K p
which simplifies to,

R a T D ðsÞ
e ss ¼ lim 2 (10.15)
s/0 s R a I tot þ sðR a B þ K e K t þ K t K v Þþ K t K p
This shows that the steady-state positional error for a step input in a second order
system is a function of the external disturbance. In a fuller analysis, the disturbance
torque can be determined if it is the result of gravity loading and centrifugal forces;
however other disturbances, such as friction, are difficult to analyse. If the determination