The task of "relative placement" is to predict the placement of one object in relation to another, e.g. placing a mug onto a mug rack. Through explicit object-centric geometric reasoning, recent methods for relative placement have made tremendous progress towards data-efficient learning for robot manipulation while generalizing to unseen task variations. However, they have yet to represent deformable transformations, despite the ubiquity of non-rigid bodies in real world settings. As a first step towards bridging this gap, we propose ``cross-displacement" - an extension of the principles of relative placement to geometric relationships between deformable objects - and present a novel vision-based method to learn cross-displacement through dense diffusion. To this end, we demonstrate our method's ability to generalize to unseen object instances, out-of-distribution scene configurations, and multimodal goals on multiple highly deformable tasks (both in simulation and in the real world) beyond the scope of prior works. Supplementary information and videos can be found at https://sites.google.com/view/tax3d-corl-2024 .

Relative placement tasks are an important category of tasks in which one object needs to be placed in a desired pose relative to another object. Previous work has shown success in learning relative placement tasks from just a small number of demonstrations when using relational reasoning networks with geometric inductive biases. However, such methods cannot flexibly represent multimodal tasks, like a mug hanging on any of n racks. We propose a method that incorporates additional properties that enable learning multimodal relative placement solutions, while retaining the provably translation-invariant and relational properties of prior work. We show that our method is able to learn precise relative placement tasks with only 10-20 multimodal demonstrations with no human annotations across a diverse set of objects within a category.