LinkD: AutoRegressive Diffusion Model for Mechanical Linkage Synthesis

TL;DR

This work tackles planar 1-DoF mechanical linkage synthesis by casting it as autoregressive graph generation conditioned on target trajectories. It combines a causal transformer with a DDPM to sequentially generate topology and geometry, leveraging the dyadic compositionality of linkages and a node-level retry mechanism to maintain kinematic validity. Empirical results show high generation success rates and reasonable curve fidelity, with substantial topological diversity across valid solutions. The approach offers scalable inverse design for complex mechanisms and suggests avenues to further improve conditioning robustness and design exploration. Practically, it enables fast, diverse generation of feasible linkages that realize a wide range of trajectories, with potential impact on robotic design and mechanism synthesis workflows.

Abstract

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.

Figures (8)

• Figure 1: Overview of the proposed framework. (a) The input curve is divided into smaller sub-curves, each encoded using (b) a PointNet encoder with curve centers serving as positional encodings. The resulting embeddings are processed by (c) a Transformer encoder with self-attention. The learned “[CLS]” token is combined with the mean sub-curve embedding to form a conditioning vector for an autoregressive causal (d) Transformer decoder, which operates on (e) the graph representation of the mechanism. (f) A DDPM head with a U-Net backbone further refines the Transformer output to regenerate the final graph structure.
• Figure 2: Graph representation of mechanical linkage. The mechanism graph is represented as an $N \times 24$ matrix, where $N$ is the number of nodes. The first entry in each 24-dimensional feature vector encodes node validity, distinguishing valid nodes from padding. The second dimension specifies the node type, followed by the normalized nodal positions in the $x$ and $y$ directions. The remaining entries correspond to the lower-triangular half of the adjacency matrix, representing pairwise node connections. Each subgraph derived from this representation constitutes a valid mechanism instance.
• Figure 3: Transformer backbone. The transformer backbone encodes each node using categorical validity embeddings, discrete type embeddings, continuous learned Fourier positional embeddings, and discrete adjacency embeddings. These features are fused through a mixer MLP and concatenated with the validity embeddings to form nodal representations. A diffusion-inspired transformer (DiT) structure is then applied, where each block performs layer normalization, FiLM-style scale–shift modulation, and causal multi-head attention followed by a feed-forward network. The entire model is conditioned on the target curve trajectory, which modulates the transformer layers to guide node generation toward matching the desired motion pattern.
• Figure 4: Qualitative results of mechanism generation conditioned on target curves. Each row shows a different example with the complete mechanism visualization (left) and curve comparison (right). Dashed lines represent the conditioning curves, while solid green lines show the curves traced by the generated mechanism's end effector. The model successfully generates mechanically valid linkages across diverse curve geometries (elongated ellipses, semi-circular arcs, and circular paths) with close but not perfect alignment between conditioning and generated curves, reflecting the stochastic nature of conditional DDPM generation.
• Figure 5: Evaluation of Generation Success Rates Across Retry Strategies. Comparison of one-shot generation with graph-level and node-level retry mechanisms across three independent experimental runs. Graph-level retry improves success rate from 29.4% to 96.7%, while node-level retry achieves 99.6% success rate.