95
J(e)e + Tv(e,)  TA(t) + Tg(e) + Td(w(t),e) (6-1)
For a manipulator having N links, e(t), e(t) and *e(t) are the vectors of
length N defining angular positions, velocities, and accelerations of
the actuator joints. The matrix J(e) e R^ is the inertia matrix which
depends on the manipulator's configuration (i.e., the joint angles
e(t)). It can be shown that J(e) is positive definite for all 9 (see
[24]) and is thus always invertible. The inertia torque vector
Tv(0,0) e corresponds to dynamic torques caused by the velocities
of the manipulator's links. Denoting the j-th component of the intertia
torque vector as Tv(e,e). we have the following
J
TV(e,9), = 0'PJ(0)0 (6-2)
J
where P^(0) e R^ is a purely configuration dependent matrix referred
to in [28] as the intertia power modeling matrix. The term T^(t) e R^
is the control torque vector which is typically supplied to the actuator
joints by electric motors. The torques resulting from gravitational
loading are designated by T^ (0) e R^ which is a configuration
dependent vector. Finally, T^(w(t),e) e R^ is a torque vector resulting
from external uncontrollable forces. It is possible to write Td(w(t),
0) in the following form
Td(w(t),0) = D(e)w(t) (6-3)
where D(0) e R^xd depends only on the manipulator's configuration and
w(t) e R^ denotes the disturbance force vector.