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108 Robot Dynamics

differential equation. In Section 3.4 we show several ways to convert this
formulation to a state-variable description. The state-variable description is a
first-order vector differential equation that is extremely useful for developing
many arm control schemes. Feedback linearization techniques and Hamiltonian
mechanics are used in this section.
The robot arm dynamics in Section 3.2 are given in joint-space coordinates.
In Section 3.5 we show a very general approach to obtaining the arm
dynamical description in any desired coordinates, including Cartesian or
workspace coordinates and the coordinates of a camera frame or reference.
In Section 3.6 we analyze the electrical or hydraulic actuators that perform
the work required to move the links of a robot arm. It is shown how to
incorporate dynamical models for the actuators into the arm dynamics to
provide a complete dynamical description of the arm-plus-actuator system.
This finally leaves us in a position to move on to the next chapters, where
robot manipulator control design is discussed.

3.2 Lagrange-Euler Dynamics

For control design purposes, it is necessary to have a mathematical model
that reveals the dynamical behavior of a system. Therefore, in this section we
derive the dynamical equations of motion for a robot manipulator. Our
approach is to derive the kinetic and potential energy of the manipulator and
then use Lagrange’s equations of motion.
In this section we ignore the dynamics of the electric or hydraulic motors
that drive the robot arm; actuator dynamics is covered in Section 3.6.
Force, Inertia, and Energy

Let us review some basic concepts from physics that will enable us to better
understand the arm dynamics [Marion 1965]. In this subsection we use boldface
to denote vectors and normal type to denote their magnitudes.
The centripetal force of a mass m orbiting a point at a radius r and angular
velocity ω is given by

(3.2.1)

See Figure 3.2.1. The linear velocity is given by
v=w×r, (3.2.2)

which in this case means simply that v=ωr.

Copyright © 2004 by Marcel Dekker, Inc.