  Cloth manipulation is a category of deformable object manipulation of great
interest to the robotics community, from applications of automated
laundry-folding and home organizing and cleaning to textiles and flexible
manufacturing. Despite the desire for automated cloth manipulation, the
thin-shell dynamics and under-actuation nature of cloth present significant
challenges for robots to effectively interact with them. Many recent works omit
explicit modeling in favor of learning-based methods that may yield control
policies directly. However, these methods require large training sets that must
be collected and curated. In this regard, we create a framework for
differentiable modeling of cloth dynamics leveraging an Extended Position-based
Dynamics (XPBD) algorithm. Together with the desired control objective,
physics-aware regularization terms are designed for better results, including
trajectory smoothness and elastic potential energy. In addition, safety
constraints, such as avoiding obstacles, can be specified using signed distance
functions (SDFs). We formulate the cloth manipulation task with safety
constraints as a constrained optimization problem, which can be effectively
solved by mainstream gradient-based optimizers thanks to the end-to-end
differentiability of our framework. Finally, we assess the proposed framework
for manipulation tasks with various safety thresholds and demonstrate the
feasibility of result trajectories on a surgical robot. The effects of the
regularization terms are analyzed in an additional ablation study.
