# LinkD: AutoRegressive Diffusion Model for Mechanical Linkage Synthesis

arXiv: https://arxiv.org/abs/2601.04054 · HTML: https://mlsanity.com/paper/2601.04054/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Placed in Diffusion and flow policies for robotics, one of 14 directions in Diffusion models.
- Builds on 18 of its 56 references that are in the map.
- First posted on arXiv 7 Jan 2026.

## Placed in

- Diffusion models / [Diffusion and flow policies for robotics](https://mlsanity.com/diffusion/diffusion-and-flow-policies-for-robotics/)

## Builds on (most cited first, 18 of 18)

- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [Proximal Policy Optimization Algorithms](https://mlsanity.com/paper/1707.06347/) (2017)
- [Denoising Diffusion Implicit Models](https://mlsanity.com/paper/2010.02502/) (2020)
- [PointNet: Deep Learning on Point Sets for 3D Classification and Segmentation](https://mlsanity.com/paper/1612.00593/) (2016)
- [A Comprehensive Survey on Graph Neural Networks](https://mlsanity.com/paper/1901.00596/) (2019)
- [FiLM: Visual Reasoning with a General Conditioning Layer](https://mlsanity.com/paper/1709.07871/) (2017)
- [Graph Neural Networks: A Review of Methods and Applications](https://mlsanity.com/paper/1812.08434/) (2018)
- [Point-BERT: Pre-training 3D Point Cloud Transformers with Masked Point Modeling](https://mlsanity.com/paper/2111.14819/) (2021)
- [Autoregressive Image Generation without Vector Quantization](https://mlsanity.com/paper/2406.11838/) (2024)
- [GraphEBM: Molecular Graph Generation with Energy-Based Models](https://mlsanity.com/paper/2102.00546/) (2021)
- [CktGNN: Circuit Graph Neural Network for Electronic Design Automation](https://mlsanity.com/paper/2308.16406/) (2023)
- [Text2PDE: Latent Diffusion Models for Accessible Physics Simulation](https://mlsanity.com/paper/2410.01153/) (2024)
- [SculptDiff: Learning Robotic Clay Sculpting from Humans with Goal Conditioned Diffusion Policy](https://mlsanity.com/paper/2403.10401/) (2024)
- [LInK: Learning Joint Representations of Design and Performance Spaces through Contrastive Learning for Mechanism Synthesis](https://mlsanity.com/paper/2405.20592/) (2024)
- [Airfoil Diffusion: Denoising Diffusion Model For Conditional Airfoil Generation](https://mlsanity.com/paper/2408.15898/) (2024)
- [Joint Search of Optimal Topology and Trajectory for Planar Linkages](https://mlsanity.com/paper/2109.03392/) (2021)
- [Molecular Graph Generation by Decomposition and Reassembling](https://mlsanity.com/paper/2302.00587/) (2022)
- [MechaFormer: Sequence Learning for Kinematic Mechanism Design Automation](https://mlsanity.com/paper/2508.09005/) (2025)

## Abstract (arXiv)

Designing mechanical linkages to achieve target end-effector trajectories presents a fundamental challenge due to the intricate coupling between continuous node placements, discrete topological configurations, and nonlinear kinematic constraints. The highly nonlinear motion-to-configuration relationship means small perturbations in joint positions drastically alter trajectories, while the combinatorially expanding design space renders conventional optimization and heuristic methods computationally intractable. We introduce an autoregressive diffusion framework that exploits the dyadic nature of linkage assembly by representing mechanisms as sequentially constructed graphs, where nodes correspond to joints and edges to rigid links. Our approach combines a causal transformer with a Denoising Diffusion Probabilistic Model (DDPM), both conditioned on target trajectories encoded via a transformer encoder. The causal transformer autoregressively predicts discrete topology node-by-node, while the DDPM refines each node's spatial coordinates and edge connectivity to previously generated nodes. This sequential generation enables adaptive trial-and-error synthesis where problematic nodes exhibiting kinematic locking or collisions can be selectively regenerated, allowing autonomous correction of degenerate configurations during design. Our graph-based, data-driven methodology surpasses traditional optimization approaches, enabling scalable inverse design that generalizes to mechanisms with arbitrary node counts. We demonstrate successful synthesis of linkage systems containing up to 20 nodes with extensibility to N-node architectures. This work advances autoregressive graph generation methodologies and computational kinematic synthesis, establishing new paradigms for scalable inverse design of complex mechanical systems.