  We present a sampling-based approach to reasoning about the caging-based
manipulation of rigid and a simplified class of deformable 3D objects subject
to energy constraints. Towards this end, we propose the notion of soft fixtures
extending earlier work on energy-bounded caging to include a broader set of
energy function constraints and settings, such as gravitational and elastic
potential energy of 3D deformable objects. Previous methods focused on
establishing provably correct algorithms to compute lower bounds or
analytically exact estimates of escape energy for a very restricted class of
known objects with low-dimensional C-spaces, such as planar polygons. We
instead propose a practical sampling-based approach that is applicable in
higher-dimensional C-spaces but only produces a sequence of upper-bound
estimates that, however, appear to converge rapidly to actual escape energy. We
present 8 simulation experiments demonstrating the applicability of our
approach to various complex quasi-static manipulation scenarios. Quantitative
results indicate the effectiveness of our approach in providing upper-bound
estimates for escape energy in quasi-static manipulation scenarios. Two
real-world experiments also show that the computed normalized escape energy
estimates appear to correlate strongly with the probability of escape of an
object under randomized pose perturbation.
