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142 Robot Dynamics

This is a statement of the conservation of energy, with the right-hand side
representing the power input from the net external forces. The skew symmetry
of is nothing more than a statement that the fictitious forces
S(q, ) do no work. The work done by the external forces is given by

(3.3.66)

Recall at this point the passivity property of the robot arm from τ(t)to (Section
1.5), which merely states that the arm cannot create energy. From a controls
point of view, a passive system cannot go unstable. A problem with some
popular control schemes (e.g., standard computed torque, Section 3.4) is that
they destroy the passivity property, resulting in possible instability if the system
parameters are not exactly known or disturbances are present. Passivity-based
designs ensure that the closed-loop system is passive (see Section 4.3, the
references cited above, and [Anderson 1989]).
This analysis does not include the friction terms. A reasonable assumption
regardless of the form of f ( ) is that friction is dissipative, so that f i(x) lies in
the first and third quadrants only. This is equivalent to
(3.3.67)

Under this assumption, friction does not destroy the passivity of the
manipulator. It is then simple to modify a controller designed for (3.3.63) to
include the friction [Slotine 1988]. The dissipative nature of friction allows
one to increase the system’s bandwidth beyond classical limits.

3.4 State-Variable Representations and Feedback
Linearization

The robot arm dynamical equation in Table 3.3.1 is

(3.4.1)

n
with q(t) R the joint variable vector τ(t)and the control input. M(q) is the
inertia matrix, V(q, ) the Coriolis/centripetal vector, F ( ) the viscous friction,
v
F d ( ) the dynamic friction, G(q) the gravity, and τ d a disturbance. These
terms satisfy the properties shown in Table 3.3.1. We may also write the
dynamics as
(3.4.2)

with the nonlinear terms represented by

Copyright © 2004 by Marcel Dekker, Inc.