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3.3 Structure and Properties of the Robot Equation 137

The selection of a suitable norm in Table 3.3.1 is not always straightforward.
In the control algorithms to be developed in subsequent chapters, we prove
suitable performance in terms of some norm, which can often be any norm
desired. For implementation of the controller, a specific norm must be selected
and the bounds evaluated. This choice often depends simply on which norm
makes it possible to evaluate the bounds in the table. For instance, choosing
the 2—norm for vectors requires the evaluation of the maximum singular
value of M(q), a very difficult task.

Selecting the ∞—norm for vectors means determining at each sampling time
the element [of V(q(t), (t)) for instance] with the largest magnitude. This
requires decision logic, and the norm may not be continuous. Therefore, let
us use the 1—norm in this example. The corresponding matrix induced norm
is then the maximum absolute column sum (Chapter 2).

a. Bounds on the Intertia Matrix

The evaluation of µ 1, and µ 2 amounts to the determination of the minimum
and maximum eigenvalues of M(q) over all q. This is not an easy affair and
requires the solution of some quadratic equations, although it can be carried
out without too much trouble using software such as Mathematica or Maple.
Thus, let us find m 1 and m 2.

The induced 1—norm for M(q) is the maximum absolute column sum. In
determining bounds for this norm, it is important to consider the range of
allowed motion of the joint angles. To illustrate, suppose that θ 1, and θ 2 are
limited by ±π/2. Then the 1—norm is always given in terms of column 1 as

Copyright © 2004 by Marcel Dekker, Inc.