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3.2 Lagrange-Euler Dynamics 111

2
I=mr . (3.2.8)
Therefore,
(3.2.9)
The potential energy of a mass m at a height h in a gravitational field with
constant g is given by

P=mgh. (3.2.10)

The origin, corresponding to zero potential energy, may be selected arbitrarily
since only differences in potential energy are meaningful in terms of physical
forces.
The momentum of a mass m moving with velocity v is given by

p=mv. (3.2.11)
The angular momentum of a mass m with respect to an origin from which
the mass has distance r is

Pang=r×p. (3.2.12)

The torque or moment of a force F with respect to the same origin is defined
to be
N=r×F. (3.2.13)

Lagrange’s Equations of Motion
Lagrange’s equation of motion for a conservative system are given by [Marion
1965]

(3.2.14)

where q is an n-vector of generalized coordinates q i, is an n-vector of
generalized forces i, and the Lagrangian is the difference between the kinetic
and potential energies

L=K-P. (3.2.15)

In our usage, q will be the joint-variable vector, consisting of joint angles θ i;
(in degrees or radians) and joint offsets d i (in meters). Then τ is a vector that
has components n i of torque (newton-meters) corresponding to the joint
angles, and f i of force (newtons) corresponding to the joint offsets. Note that
we denote the scalar components of τ by lowercase letters.

Copyright © 2004 by Marcel Dekker, Inc.