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3.2 Lagrange-Euler Dynamics 113

with n a torque and fa force. The torque n and force f may be provided by
either motors or hydraulic actuators. We discuss the dynamics of actuators in
Section 3.6.
To determine the arm dynamics, we must now compute the quantities
required for the Lagrange equation.

a. Kinetic and Potential Energy
The total kinetic energy due to the angular motion and the linear motion
is

(3)
and the potential energy is

(4)

b. Lagrange’s Equation

The Lagrangian is
(5)

Now we obtain

(6)

(7)

(8)

Therefore, (3.2.14) shows that the arm dynamical equations are

(9)

(10)

This is a set of coupled nonlinear differential equations which describe the
motion q(t)=[θ(t) r(t)] given the control input torque n(t) and force f(t). We
T

Copyright © 2004 by Marcel Dekker, Inc.