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3.3 Structure and Properties of the Robot Equation 141

The reader should verify that with these definitions, the dynamics may be
expressed as τ=Wϕ The matrix W is computed from measured joint positions,
and their velocities and accelerations.

Passivity and Conservation of Energy

The “Newtonian” form of the manipulator dynamics given in Table 3.3.1
obscures some important physical properties, which we should like to explore
here [Koditschek 1984], [Ortega and Spong 1988], [Johansson 1990], [Slotine
and Li 1987], [Slotine 1988]. Note that the dynamics can be written in terms
of the skew-symmetric matrix S(q, ) as

(3.3.63)

where friction and τ d are ignored. Now, with K the

whence (3.3.63) yields

(3.3.64)

or

(3.3.65)

Copyright © 2004 by Marcel Dekker, Inc.