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3.2 Lagrange-Euler Dynamics 123

Since ∂T i /∂q j =0 for j>i, we may write this more efficiently as

(3.2.31)

Equation (3.2.29) is what we have been seeking; it provides a convenient
expression for the arm kinetic energy in terms of known quantities and the
joint variables q. Since m jk=m kj, the inertia matrix M(q) is symmetric. Since
the kinetic energy is positive, vanishing only when the generalized velocity
equal zero, the inertia matrix M(q) is also positive definite. Note that the
kinetic energy depends on q and .

Arm Potential Energy. If link i has a mass m i and a center of gravity
expressed in the coordinates of its frame i, the potential energy of the link is
given by

(3.2.32)

where the gravity vector is expressed in base coordinates as

(3.2.33)

If the arm is level, at sea level, and the base z-axis is directed vertically
upward, then
(3.2.34)

2
with units of m/s .
The total arm potential energy, therefore, is

(3.2.35)

Note that P depends only on the joint variables q, not on the joint velocities
.
Noting that is the last column of the link i pseudo-inertia matrix T i,
we may write

(3.2.36)

T
with e 4 the last column of the 4×4 identity matrix (i.e., e 4=[0 0 0 1] ). Lagrange’s
Equation. The arm Lagrangian is

Copyright © 2004 by Marcel Dekker, Inc.