Bounding the Locus of the Center of Mass for a Part with Shape Variation

The shape and center of mass of a part are crucial parameters to algorithms for planning
automated manufacturing tasks. As industrial parts are generally manufactured to tolerances,
the shape is subject to variations, which, in turn, also cause variations in the location of the
center of mass. Planning algorithms should take into account both types of variation to
prevent failure when the resulting plans are applied to manufactured incarnations of a model
part.
We study the relation between variation in part shape and variation in the location of the
center of mass for a part with uniform mass distribution. We consider a general model for
shape variation that only assumes that every valid instance contains a shape PI while it is
contained in another shape PE. We characterize the worst-case displacement of the center of
mass in a given direction in terms of PI and PE. The characterization allows us to determine
an adequate polytopic approximation of the locus of the center of mass. We also show that
the worst-case displacement is small if PI is convex and fat and the distance between the
boundary of PE and PI is bounded.