  To reliably model real robot characteristics, interval linear systems of
equations allow to describe families of problems that consider sets of values.
This allows to easily account for typical complexities such as sets of joint
states and design parameter uncertainties. Inner approximations of the
solutions to the interval linear systems can be used to describe the common
capabilities of a robotic manipulator corresponding to the considered sets of
values. In this work, several classes of problems are considered. For each
class, reliable and efficient polytope, n-cube, and n-ball inner approximations
are presented. The interval approaches usually proposed are inefficient because
they are too computationally heavy for certain applications, such as control.
We propose efficient new inner approximation theorems for the considered
classes of problems. This allows for usage with real-time applications as well
as rapid analysis of potential designs. Several applications are presented for
a redundant planar manipulator including locally evaluating the manipulator's
velocity, acceleration, and static force capabilities, and evaluating its
future acceleration capabilities over a given time horizon.
