Point Cloud Glue: constraining simulations using the procrustes transform

In physical simulation, it is frequently useful to define constraints between deformable objects, ensuring that one object follows another. Existing techniques for enforcing these constraints define the relationship between the objects using barycentric coordinates, a linear combination of vertices. While simple to implement and understand, barycentric coordinates have one important drawback: for stability, weights must be non-negative, which limits the types of constraints that can be defined. We introduce the Point Cloud Glue, which uses the nearest fit rigid rotation (the Procrustes transform) to the deformable object's particles. Our key contribution is to demonstrate that we can differentiate through this minimization in a numerically stable manner, allowing our method to be used in many constrained dynamics systems including those based on bindings/embeddings and those based on Lagrange multipliers. We demonstrate the flexibility of our method through several examples.