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120 Robot Dynamics

with q the joint-variable vector and τ the generalized force/torque vector. In
this subsection we derive the dynamics for a general robot manipulator. They
will be of this same form.
To obtain the general robot arm dynamical equation, we determine the
arm kinetic and potential energies, then the Lagrangian, and then substitute
into Lagrange’s Equation (3.2.14) to obtain the final result [Paul 1981, Lee
et al. 1983, Asada and Slotine 1986, Spong and Vidyasagar 1989].
i
Arm Kinetic Energy. Given a point on link i with coordinates of r with respect
to frame i attached to that link, the base coordinates of the point are
(3.2.17)

where T i is the 4×4 homogeneous transformation defined in Appendix A.
Note that T i is a function of joint variables q 1 , q 2 ,…, q i . Consequently, the
velocity of the point in base coordinates is

(3.2.18)

Since ∂T i /∂q j =0, j>i, we may replace the upper summation limit by n, the
number of links. The 4×4 matrices ∂T i /∂q j may be computed if the arm matrices
T i are known.

i
The kinetic energy of an infinitesimal mass dm at r that has a velocity of
v=[v x v y v z] is
T

(3.2.19)

Thus the total kinetic energy for link i is given by

Substituting for dK i from (3.2.19), we may move the integral inside the
summations. Then, defining the 4×4 pseudo-inertia matrix for link i as

(3.2.20)

Copyright © 2004 by Marcel Dekker, Inc.