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154 Robot Dynamics

Independent Joint Dynamics. In many commercial robot arms the gear
ratios r i are very small, providing a large torque advantage in the actuator/
link coupling. This has important ramifications that greatly simplify the design
of robot arm controllers.
To explore this, let us write the complete dynamics by components as

(3.6.7)

where B≡diag{B i } and d i is a disturbance given by

(3.6.8)

with m ij the off-diagonal elements of M’, V jki the tensor components of
the friction of the ith link, and G i the ith gravity component.
This equations reveals that if r i is small, the arm dynamics are
approximately given by n decoupled second-order equations with constant
coefficients. The dynamical effects of joint coupling and gravity appear only
as disturbance terms multiplied by . That is, robot controls design is virtually
the problem of simply controlling the actuator dynamics.
Unfortunately, modern high-performance tasks make the Coriolis and
centripetal terms large, so that d i is not small. Moreover, modern high-
performance arms have near-unity gear ratios (e.g., direct drive arms), so
that the nonlinearities must be taken into account in any conscientious controls
design.
Third-Order Arm-Plus-Actuator Dynamics

An alternative model of the complete robot arm is sometimes used in controls
design [Tarn et al. 1991]. It is a third-order differential equation that should
be used when the motor armature inductance is not negligible.
When the armature inductances L i are not negligible, instead of (3.6.2) we
must use the armature-controlled do motor equations

(3.6.9)

(3.6.10)

n
with I R the vector of armature currents,

Copyright © 2004 by Marcel Dekker, Inc.