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3.3 Structure and Properties of the Robot Equation 125

3.3 Structure and Properties of the Robot Equation

In this section we investigate the detailed structure and properties of the
dynamical arm equations, for this structure should be reflected in the form of
the control law. The controller is simpler and more effective if the known
properties of the arm are incorporated in the design stage.

In reality, a robot arm is always affected by friction and disturbances. Therefore,
we shall generalize the arm model we have just derived by writing the
manipulator dynamics as

(3.3.1)

with q the joint variable n-vector and τ the n-vector of generalized forces.
M(q) is the inertia matrix, V(q, ) the Coriolis/centripetal vector, and G(q)
the gravity vector. We have added a friction term

(3.3.2)

with F v the coefficient matrix of viscous friction and F d a dynamic friction
term. Also added is a disturbance τ d, which could represent, for instance, any
inaccurately modeled dynamics.
Friction is not an easy term to model, and indeed, may be the most contrary
term to describe in the manipulator dynamics model. Some more discussion
on friction may be found in [Schilling 1990].
We shall sometimes write the arm dynamics as

(3.3.3)
where
(3.3.4)
represents nonlinear terms.
Let us examine the structure and properties of each of the terms in the
robot dynamics equation. This study will offer us a great deal of insight
which we use in deriving robot control schemes in subsequent chapters. A
summary of the properties we discover is given in Table 3.3.1, to which we
refer in the remainder of the book. As we develop each property, it will be
worthwhile to refer to Examples 3.2.1 to 3.2.3 in order to verify that the
properties indeed hold there. At the end of this section we illustrate in Example
3.3.1 several of the properties for a two-link planar elbow arm.

Copyright © 2004 by Marcel Dekker, Inc.