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136 Robot Dynamics

Properties of the Disturbance Term. The arm Equation (3.3.1) has a disturbance
term τ d which could represent inaccurately modeled dynamics, and so on. We
shall assume that it is bounded so that
(3.3.61)

where d is a scalar constant that may be computed for a given arm and ||·|| is
any suitable norm.
Linearity in the Parameters

The robot dynamical equation enjoys one last property that will be of great
use to us in Chapter 5. Namely, it is linear in the parameters, a property first
exploited in [Craig 1988] in adaptive control. This is important, since some
or all of the parameters may be unknown; thus the dynamics are linear in the
unknown terms.

This property may be expressed as

(3.3.62)

with ϕ the parameter vector and W(q, , ) a matrix of robot functions
depending on the joint variables, joint velocities, and joint accelerations.
This matrix may be computed for any given robot arm and so is known.
See Example 3.3.1. Note that the disturbance τ d is not included in this
equation.

EXAMPLE 3.3–1: Structure and Bounds for Two-Link Planar
Elbow Arm
The dynamics of a two-link planar arm are given in Example 3.2.2. We
should now like to compute the structural matrices defined in this section,
as well as the bounds needed in Table 3.3.1. The friction bounds are
straightforward, so we do not mention them here. The dynamical matrices
are

Copyright © 2004 by Marcel Dekker, Inc.