Convergent Functions, Divergent Forms

We thank Reviewer  Rt4W for the constructive comments and feedback. We appreciate your recognition of our paper’s new compelling application for QD algorithms to morphology optimization, the comprehensive experimental results and a well-laid-out approach. Below, we address each concern you raised and clarify the perceived weaknesses.

1. It is unclear how random sampling of the morphology parameters is performed.

We apologize for the confusion and will include a detailed explanation in the Appendix for the camera-ready version. The process of sampling random morphologies in LOKI follows the procedure introduced in prior work, DERL [4] (see Table 1 in the Appendix of the arXiv version). During the population initialization phase, a new morphology is created by first sampling the total number of limbs to grow, followed by a series of mutation operations until the desired number of limbs is reached. These mutations include: growing or deleting limbs, mutating limb parameters, density, degrees of freedom (DoF), gear ratios, and joint angles. For each mutation, the parameters are uniformly sampled from predefined ranges specified in [4]. The key difference is that DERL [4] samples parameter values from discrete sets, while LOKI samples continuously within each range, enabling a denser and more comprehensive coverage of the morphology space. The table below presents the parameter ranges used to create our morphologies. The notation range(a, b) denotes a continuous range from minimum a to maximum b.

2. What exactly does Dynamic Local Search (DLS) refer to in this paper?

Stochastic Local Search (SLS) algorithms iteratively improve candidate solutions to an optimization problem based on an evaluation function. At each step, the current solution is typically replaced by a neighboring solution with a better evaluation value. Dynamic Local Search (DLS) is a subclass of these algorithms in which the evaluation function itself is dynamically adapted whenever a local optimum is encountered, allowing further improvements. In LOKI, we implement a variant of DLS by partially updating the “pool of elites” within each cluster: the two worst-performing morphologies are replaced by new candidates sampled from the same cluster. The evaluation policy is also updated after each step, co-evolving alongside the elite morphologies. This design follows the Dynamic Local Search framework described in Algorithm 54.1, Section 54.3.5 of reference [25] in the paper, from which LOKI draws its search mechanism.

3. It would be helpful to formally define cluster C_k as a set of morphologies or a set of latent embeddings. It can be clarified that the members of each cluster are fixed at the beginning of the search.

Thank you for the suggestions. We agree that the definition and static nature of the clusters could be stated more clearly. A cluster $C_k$​ is formally defined as a set of latent embeddings of morphologies, where the 500,000 randomly generated morphologies are partitioned into $C_1 \cup C_2 \cup \cdots \cup C_K$​. These clusters are constructed in advance using the VAE latent embeddings of a diverse, precomputed set of morphologies that densely cover the design space. The clusters remain fixed throughout the evolutionary process, and all candidate morphologies are sampled from within these predefined clusters.

4. Confusion about the coverage metric.

We apologize for the confusion and will clarify this in the revised manuscript. LOKI achieves 100% coverage because it explicitly maintains a dedicated policy for each of the 40 clusters. For fair comparison across variants, both the QD-score and coverage are always computed using the same set of 40 reference clusters, regardless of the number of clusters used during training (e.g., $N_c=10,20$). The coverage metric reflects how broadly the discovered solutions span this fixed 40-cluster space. For variants trained with fewer clusters, we assign their 100 final discovered morphologies to the fixed set of 40 reference clusters and compute coverage accordingly.

5. It would be helpful to formalize the QD problem statement used in this paper.

Thank you for the great suggestion. For the camera-ready version, we will include a Problem Formulation section and formally define the QD problem as applied in our work.

6. In Figure 1c, it is not clear whether red or blue indicates better performance. Perhaps a color bar would be useful?

The small heatmap in the figure indicates that red corresponds to higher reward. We will improve the figure in the camera-ready version.

7. I am unclear how each morphology is evaluated 8 times on average. Shouldn't this come out to 78,000*40/500,000=6.24 evaluations per morphology?

Thank you for identifying a miscalculation in our paper. We will revise the manuscript to correct this error.

8. In Appendix D Eq. , are $f_{max}$ and $f_{min}$ the max and min across all clusters of all runs, or only all clusters within a single run?

We appreciate the reviewer’s clarification. $f_{max}$ and $f_{min}$ are computed across all clusters from all runs, not just within a single run. We will revise the manuscript to explicitly clarify this to avoid any potential confusion for readers.

9. In Figure 7, it seems the term LOKI (10 clusters) refers to the top 10 clusters rather than LOKI with N_c=10. I think it would be more clear to say LOKI (top 10 clusters).

We will revise the figure label in the camera-ready version to “LOKI (top 10 clusters)” for clarity.

10. Limitations and positive/negative societal impacts.

We appreciate the reviewer’s suggestion. We will include a section on societal impact in the appendix and highlight key points in the camera-ready version. Positive impacts: Our work aims to challenge prevailing assumptions about optimal robot morphology, potentially reshaping how robotic design is approached. By promoting diversity and functionality beyond conventional forms, LOKI encourages exploration of unconventional yet effective morphologies. We believe this contributes toward a more inclusive and biologically inspired understanding of embodiment, potentially serving as a bridge between AI, robotics, and the natural sciences. Negative impacts: A potential concern is that the ability to autonomously generate a large number of novel and capable morphologies could be misused in contexts that prioritize performance over safety. For instance, this approach could enable rapid prototyping of morphologies for autonomous systems without adequate human oversight, increasing the risk of deploying untested designs in sensitive environments.

We hope we have fully addressed the reviewer’s concerns and would greatly appreciate it if the rating could be raised. Thank you.

We thank Reviewer  oRP8, for your constructive feedback. We are encouraged that you recognized the strengths of our framework such as computational gains and generalization to novel tasks. Below, we address the limitations and recommendations you highlighted:

1. It is not clear how the VAE -based latent space affects performance.

We address this through an ablation study presented in Appendix A, where we evaluate the impact of the VAE-based latent space by comparing it to a variant of LOKI that uses clustering over raw morphology parameters instead (LOKI w/ raw-param-cluster). Clustering in the raw parameter space fails to capture structural and behavioral similarities between morphologies, resulting in poor generalization of the multi-design policies trained within each cluster. In contrast, the VAE latent space captures meaningful structure by learning a smooth, low-dimensional manifold of the morphology space. As a result, LOKI with raw-parameter clustering performs significantly worse in terms of overall performance, morphological diversity, and downstream task generalization.

2. Underexplored relationship between number of clusters and performance.

In our LOKI framework, the number of clusters is treated as a hyperparameter, similar to the number of cells in MAP-Elites-based methods [22, 23]. Our experiments show that using 40 clusters yields better performance than using 10 or 20. A larger number of clusters enables finer-grained distinctions between the clusters but also increases the computational budget due to the need for additional policy training. As such, this hyperparameter should be tuned based on the available computational resources. LOKI’s performance is generally robust to the number of clusters, as long as there are enough to avoid overly coarse groupings. For example, using only 10 clusters often results in clusters aligning with broad structural features such as limb count, limiting specialization.

3. Pros and cons of dynamic or adaptive clustering instead of initial clustering.

Thank you for the insightful question. It is true that in the current version of LOKI, clustering is performed only once at the beginning using a VAE-based latent space, and the clusters remain fixed throughout training. LOKI frames co-design as a search problem by constructing a large, predefined pool of morphologies, generated via repeated random mutations to densely and exhaustively cover the design space. Rather than generating new morphologies during optimization, the algorithm samples from these predefined clusters. While we have not explored dynamic or adaptive clustering during training, we agree that incorporating task-aware or behavior-driven re-clustering is a promising direction. As the policies are being trained, periodically updating clusters based on evolving behavioral or functional similarities could lead to more meaningful groupings and finer specialization. This could be particularly beneficial in continual or lifelong learning scenarios, where task distributions and optimal behaviors may shift over time. However, this approach presents several challenges. Re-clustering and reassigning morphologies during training may introduce significant computational overhead and could require resetting or transferring policies, potentially disrupting convergence of cluster-specific controllers and destabilizing training. Despite these challenges, we consider dynamic clustering a compelling direction for future work.

4. Explain whether generalization comes from the evolutionary processes or simply because of the variety of a rich set of initial morphologies.

While our initial set of 500k morphologies densely covers the design space, these morphologies are generated through repeated random mutations and are not expected to be inherently optimal for locomotion or manipulation tasks. In fact, most of them are challenging to train for simple locomotion. To illustrate this, we include a “Random” baseline in Table 2 and Figure 6, which consists of 100 morphologies randomly sampled from the predefined set. The poor performance of the “Random” baseline confirms that simply sampling from this pool is insufficient, indicating that the observed generalization results from the targeted optimization performed by LOKI—not merely from the diversity introduced by the initial clustering. The generalization of our morphologies stems from identifying a diverse set of high-performing designs, each exhibiting optimal behaviors within their respective niche. The downstream tasks require different skills in agility, stability, and manipulation. LOKI discovers such optimal morphologies within each cluster. These top-performing agents display different locomotion behaviors, enabling them to excel in various downstream tasks.

5. How could LOKI be adapted to situations where there are morphological constraints such as real-world applications? Would that affect the training of the VAE?

LOKI can be adapted to scenarios with morphological constraints by applying those constraints to the initial pool of randomly sampled designs. The VAE learns to represent and reconstruct the morphology space it is trained on, as long as the constrained design space remains sufficiently dense and well-structured. However, if the constraints result in a highly non-convex or sparse design space, the learned latent space may no longer support smooth interpolations that remain within the feasible region. In such cases, it may be beneficial to replace the VAE with a more expressive latent model that can better capture the structure of constrained morphology spaces. This improved latent representation can then be used for clustering.

6. Lack of non-evolutionary baselines. But how does LOKI compare with gradient-descent based approaches that could be used on the same benchmarks? I.e., Metamorph or One Policy to Control Them All: Shared Modular Policies for Agent-Agnostic Control?

The two mentioned approaches, Metamorph and One Policy to Control Them All, propose global policies that generalize across a wide range of agent morphologies. However, they are not co-design methods and thus are not direct baselines for comparison. LOKI, by contrast, focuses on discovering high-performing morphologies rather than solely maximizing policy performance. In fact, we adopt the policy architecture from Metamorph [63] to train multi-agent policies within each cluster.

We would be grateful if, after reviewing our response and finding your concerns fully addressed, you would consider raising our score.

We would like to thank Reviewer  T5wK, for the thoughtful feedback and suggestions. We are pleased that you recognize how LOKI creatively integrates different techniques to improve efficiency and diversity in robot morphology optimization. Below we address each of the raised concerns.

1. Would incorporating a behavioral or performance-aware loss component into the VAE training improve the latent representation for clustering?

We agree with the reviewer that morphological similarity does not necessarily imply behavioral similarity, and that our VAE’s latent space, trained solely with a reconstruction loss, primarily captures structural features. Incorporating a behavior-aware loss could indeed enhance the latent representation. However, behavioral information is not available a priori at the clustering stage, as it requires training an optimal policy for each design. That said, in practice, we find that structural features serve as a reasonable proxy for behavioral similarity. As a result, our clustering tends to group morphologies with similar behaviors—for example, spinners and crawlers naturally fall into separate clusters.

2. Sensitivity to the Number of Clusters and Relation to Population-based Training.

In our LOKI framework, the number of clusters is treated as a hyperparameter, similar to the number of cells in MAP-Elites-based methods [22, 23]. Our experiments show that using 40 clusters yields better performance than using 10 or 20. A larger number of clusters allows for finer-grained morphological distinctions but also increases computational cost, as more policies need to be trained. Therefore, this hyperparameter should be selected based on the available computational resources. LOKI’s performance is generally robust to the number of clusters, as long as there are enough to avoid overly coarse groupings. For example, using only 10 clusters often results in clusters aligning with broad structural features such as limb count, limiting specialization. Each per-cluster evolutionary process resembles population-based training (PBT). Similar to PBT, a population of initial morphologies is sampled from the cluster and periodically evaluated. Underperforming agents are replaced with new morphologies from the cluster. Also similar to PBT, morphologies are evaluated continuously in parallel (simultaneous with policy training), rather than sequentially or only at the end.

3. Insufficient literature survey and rationale for using QD-based methods as primary baselines rather than Brain-body co-design approaches.

We appreciate the reviewer’s suggestion and agree that recent brain-body co-design research deserves clearer discussion in our manuscript. Most of the recent co-design methods [A,B,C,D,E,F] train a transferable universal controller and they focus on task-specific optimization of morphology and control with no additional objective for diversity and generalization to downstream tasks. LOKI introduces a cluster-based Quality-Diversity (QD) framework that co-evolves both morphology and control within behaviorally similar clusters. Unlike methods that aim to train a single universal controller across all morphologies, LOKI leverages structural similarity within each cluster to train per-cluster policies. This is particularly important in high-dimensional and exponentially large design spaces such as UNIMAL, where learning a single controller is especially challenging [G]. Furthermore, this clustering enables localized competition within each group, helping discover morphologies that are optimal within their respective behavioral niches. While the reviewer suggests that co-design approaches are more appropriate baselines for LOKI, we argue that Quality-Diversity (QD) methods are actually a better fit for comparison. Most co-design methods focus on optimizing morphologies for a specific task and aim to identify one or a few highly performant designs. These approaches typically lack mechanisms to promote morphological diversity. In contrast, QD methods are explicitly designed to discover a diverse set of high-performing solutions across multiple behavioral niches—an objective that aligns closely with the goals of the LOKI framework. LOKI is designed to efficiently discover a diverse population of agents capable of generalizing to a wide range of downstream tasks. While the agents it discovers may not be globally optimal for the locomotion task used during training, they exhibit niche-optimal behaviors that are more transferable to other domains, such as manipulation. As such, comparing LOKI to co-design methods that prioritize task-specific optimality alone would not yield a fair or meaningful evaluation. Among co-design methods, DERL [4] is the most relevant baseline, as it also produces a population of 100 morphologies and evaluates their transferability on similar downstream tasks.

• [A]: Luke Strgar et al. Accelerated co-design of robots through morphological pretraining. arXiv preprint arXiv:2502.10862, 2025.

• [B]: Yuxing Wang et al. PreCo: Enhancing Generalization in Co-Design of Modular Soft Robots via Brain-Body Pre-Training. In the ​​proceedings of The 7th Conference on Robot Learning, PMLR 229:478-498, 2023.

• [C]: Muhan Li et al. Generating freeform endoskeletal robots. In International conference on learning representations, 2025.

• [D]: Yuxing Wang et al. Curriculum-based Co-design of Morphology and Control of Voxel-based Soft Robots. In International conference on learning representations, 2023.

• [E]: Haofei Lu et al. Bodygen: Advancing towards efficient embodiment co-design. In The Thirteenth International Conference on Learning Representations, 2025.

• [F]: Ye Yuan et al. Transform2act: Learning a transform-and-control policy for efficient agent design. arXiv preprint arXiv:2110.03659, 2021.

• [G]: Agrim Gupta et al. Metamorph: Learning universal controllers with transformers. arXiv preprint arXiv:2203.11931, 2022.

4. Visualization of Morphological Diversity.

Thank you for the suggestion. We agree that additional qualitative results would help better illustrate the diversity of the discovered morphologies. To address this, we have provided videos showcasing the morphologies and locomotion behaviors of our 100 discovered designs on the supplementary website uploaded in the supplementary material (index.html). Due to space limitations in the main paper, we were unable to include more visual examples. We will include more visualizations in the Appendix of the paper for the camera-ready version.

We hope we have fully addressed the reviewer’s concerns and would greatly appreciate it if the rating could be reconsidered and raised. Thank you.

We thank Reviewer  uzg9 for the constructive feedback. We appreciate your recognition of our paper’s strong motivation and promising results on a variety of tasks. Below, we address each concern you raised and clarify the perceived weaknesses.

1. The claim of co-design between morphology and control is questionable.

Although the reviewer is correct that our morphologies are predefined and we sample from this set during the evolutionary process, we argue that LOKI still qualifies as a co-design method. Co-design refers to the joint optimization of morphology and control for a given task, typically framed as a bi-level optimization problem: the outer loop searches over morphologies (often using evolutionary strategies), while the inner loop trains a control policy for each morphology to assess its performance [4]. Most existing methods alternate between these two loops. To simplify the optimization, some approaches integrate morphology refinement directly into the reinforcement learning process, enabling joint optimization in a single loop [11, 13]. In LOKI, we frame co-design as a search problem over a large, predefined space of morphologies. This space is generated through extensive random mutations to densely cover the design landscape. Instead of evolving new morphologies online, our design optimization loop samples from this rich, precomputed set. While the implementation differs from traditional co-design pipelines, it still falls under the broader definition of co-design.

2. The theoretical novelty and contribution are limited.

LOKI’s strength lies in introducing a compute-efficient framework for co-designing a diverse set of high-performing morphologies that generalize well to a variety of unseen downstream tasks. While the individual components are not novel, the overall framework effectively addresses key challenges in evolution-based co-design methods, such as computational efficiency and the diversity of final solutions.

3. What is the advantage of using VAE + k-means over recent deep learning-based clustering methods?

We agree with the reviewer that the VAE is not explicitly trained for clustering. However, its latent space is designed to capture meaningful structure in the morphology space by learning a smooth, low-dimensional manifold. Applying k-means to the VAE embeddings offers a simple yet effective way to partition this space based on topological similarity. While more advanced deep clustering methods could potentially improve performance, their exploration is beyond the scope of this work. Our focus is on the overall framework rather than the specific choice of clustering algorithm. In fact, the effectiveness of LOKI despite using a simple clustering method underscores the robustness of our approach.

4. The argument for diversity in addressing small local changes and avoiding local minima is unconvincing.

The key to LOKI’s broader exploration lies in how the initial set of morphologies is generated and utilized. Instead of relying on small, incremental mutations from current solutions, as is common in traditional evolutionary approaches, LOKI samples from a large, diverse pool of precomputed morphologies that densely cover the design space. This pool is created by repeatedly applying random mutations to produce a wide variety of designs. Dynamic Local Search (DLS) is then applied within each individual cluster to identify the best-performing morphologies. This sampling strategy allows for non-local jumps across the morphology space, helping to escape local minima that incremental mutations often fail to overcome. Furthermore, standard evolutionary methods with global competition tend to favor a single dominant behavior (e.g., faster learning), which can reduce behavioral diversity. Quality-Diversity (QD) methods address this by maintaining a repertoire of diverse solutions and localizing competition [20], allowing the algorithm to discover morphologies that are optimal within their own behavioral niches (e.g., spinning may lead to faster locomotion but requires more training to master). Similarly, LOKI restricts competition within clusters rather than across the entire population (e.g., spinners compete only with other spinners), preserving behavioral diversity across different regions of the morphology space.

5. The claim of efficiency is not well justified. The proposed method requires training a separate policy for each morphology cluster.

While LOKI requires training a separate policy for each cluster, it is still significantly more sample-efficient than prior evolution-based and Quality-Diversity approaches, which typically train a separate policy for each individual morphology to evaluate its fitness. In contrast, LOKI evaluates 500,000 morphologies using only 40 policies. Table 1 presents efficiency metrics compared to the most closely related baseline, DERL [4], highlighting LOKI’s advantage. Moreover, maintaining separate policies across clusters serves a purpose beyond efficiency: it localizes competition within each cluster, allowing the discovery of morphologies that are optimal within distinct behavioral niches. Co-design methods based on curriculum learning typically operate within more restricted and structured design spaces, such as modular voxel-based soft robots [A], and have been shown to converge to local morphological optima [B]. In contrast, LOKI is designed as an efficient evolutionary framework capable of navigating exponentially large morphology spaces (e.g., UNIMAL) while preserving diversity and avoiding premature convergence to local optima. Although LOKI trains a separate policy for each cluster, this approach enables scalable and diverse policy learning across a wide range of morphologies, without relying on restrictive assumptions or handcrafted curricula.

• [A] Yuxing Wang et al. Curriculum-based Co-design of Morphology and Control of Voxel-based Soft Robots. In International conference on learning representations, 2023.

• [B] Yuxing Wang et al. PreCo: Enhancing Generalization in Co-Design of Modular Soft Robots via Brain-Body Pre-Training. In the ​​proceedings of The 7th Conference on Robot Learning, PMLR 229:478-498, 2023.

We hope to address all the reviewer's concerns, and would be grateful if after reviewing our response you would consider raising your score.