# Complementary Dynamics

arXiv: https://arxiv.org/abs/2009.02462 · HTML: https://mlsanity.com/paper/2009.02462/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 5 Sep 2020.

## Built on by (most cited first, 8 of 8)

- [Learning Skeletal Articulations with Neural Blend Shapes](https://mlsanity.com/paper/2105.02451/) (2021)
- [High-order Differentiable Autoencoder for Nonlinear Model Reduction](https://mlsanity.com/paper/2102.11026/) (2021)
- [A Deep Emulator for Secondary Motion of 3D Characters](https://mlsanity.com/paper/2103.01261/) (2021)
- [Temporal Residual Jacobians For Rig-free Motion Transfer](https://mlsanity.com/paper/2407.14958/) (2024)
- [AnimaMimic: Imitating 3D Animation from Video Priors](https://mlsanity.com/paper/2512.14133/) (2025)
- [Analytically Integratable Zero-restlength Springs for Capturing Dynamic Modes unrepresented by Quasistatic Neural Networks](https://mlsanity.com/paper/2201.10122/) (2022)
- [From Rigging to Waving: 3D-Guided Diffusion for Natural Animation of Hand-Drawn Characters](https://mlsanity.com/paper/2509.06573/) (2025)
- [One-shot Compositional 3D Head Avatars with Deformable Hair](https://mlsanity.com/paper/2604.14782/) (2026)

## Abstract (arXiv)

We present a novel approach to enrich arbitrary rig animations with elastodynamic secondary effects. Unlike previous methods which pit rig displacements and physical forces as adversaries against each other, we advocate that physics should complement artists intentions. We propose optimizing for elastodynamic displacements in the subspace orthogonal to displacements that can be created by the rig. This ensures that the additional dynamic motions do not undo the rig animation. The complementary space is high dimensional, algebraically constructed without manual oversight, and capable of rich high-frequency dynamics. Unlike prior tracking methods, we do not require extra painted weights, segmentation into fixed and free regions or tracking clusters. Our method is agnostic to the physical model and plugs into non-linear FEM simulations, geometric as-rigid-as-possible energies, or mass-spring models. Our method does not require a particular type of rig and adds secondary effects to skeletal animations, cage-based deformations, wire deformers, motion capture data, and rigid-body simulations.