Convergent Functions, Divergent Forms

TL;DR

LOKI addresses the high cost and limited transferability of joint morphology–controller optimization by clustering morphologies in a learned latent space and training cluster-specific convergent policies that reuse behaviors. It replaces mutation with Dynamic Local Search to foster divergent forms within clusters, enabling broad exploration with significantly fewer retrainings. The approach combines a Transformer-based VAE for clustering, multi-design Transformer policies, and DLS-based co-evolution, achieving strong quality-diversity metrics and superior transfer to unseen tasks while reducing compute. This framework enables scalable, transferable co-design across large morphology spaces like UNIMAL, with meaningful implications for efficient embodied AI development.

Abstract

We introduce LOKI, a compute-efficient framework for co-designing morphologies and control policies that generalize across unseen tasks. Inspired by biological adaptation -- where animals quickly adjust to morphological changes -- our method overcomes the inefficiencies of traditional evolutionary and quality-diversity algorithms. We propose learning convergent functions: shared control policies trained across clusters of morphologically similar designs in a learned latent space, drastically reducing the training cost per design. Simultaneously, we promote divergent forms by replacing mutation with dynamic local search, enabling broader exploration and preventing premature convergence. The policy reuse allows us to explore 780$\times$ more designs using 78% fewer simulation steps and 40% less compute per design. Local competition paired with a broader search results in a diverse set of high-performing final morphologies. Using the UNIMAL design space and a flat-terrain locomotion task, LOKI discovers a rich variety of designs -- ranging from quadrupeds to crabs, bipedals, and spinners -- far more diverse than those produced by prior work. These morphologies also transfer better to unseen downstream tasks in agility, stability, and manipulation domains (e.g., 2$\times$ higher reward on bump and push box incline tasks). Overall, our approach produces designs that are both diverse and adaptable, with substantially greater sample efficiency than existing co-design methods. (Project website: https://loki-codesign.github.io/)

Figures (9)

• Figure 1: We introduce Loki, a compute-efficient co-design framework that discovers diverse, high-performing robot morphologies (divergent forms) using shared control policies (convergent functions) and dynamic local search instead of mutation. (A) The design space is clustered in a learned latent space so that morphologies within each cluster share structural similarities and exhibit similar behaviors. (B) A shared control policy is trained within each cluster on a dynamic pool of elite morphologies. (C) Morphologies co-evolve with the shared policy as elites are iteratively refined through dynamic local search.
• Figure 2: Lokidiscovers high-performing morphologies with diverse locomotion behaviors that effectively generalize across various unseen tasks.
• Figure 3: Clustering using raw morphology parameters fails to capture the topological similarity. The 3 designs are from the same cluster. Indices show the DFS traversal.
• Figure 4: Coevolution of morphologies with multi-design policies.Left: Training rewards for each policy. Right: Mean pairwise distance of morphologies in the elite pool of each cluster.
• Figure 5: Loki evolves significantly more sparse solutions (measured by the average distance to $k$-nearest neighbors). This is due to using Dynamic Local Search for sampling designs, rather than mutations. Higher sparseness indicates more diversity. (Statistical significance was assessed using independent samples t-tests; $\text{*}P<0.005$; $\text{**}P<10^{-11}$; $\text{***}P<10^{-16}$).