Organizations: School of Computer Science and Engineering, Southeast University, Nanjing, China · Key Laboratory of New Generation Artificial Intelligence Technology and Its Interdisciplinary Applications (Southeast University), Ministry of Education, China
Morphology--control co-design jointly optimizes an agent's body structure and control policy as an integrated embodied system. However, existing methods typically model morphology design and control with separate networks coupled only indirectly through a shared task objective, limiting explicit high-level coordination. Inspired by natural genes that coordinate biological development, we introduce Morphogene, a compact latent blueprint that bridges an agent's body and brain. Through AdaConcat, Morphogene jointly conditions morphology and control generation at the limb level, allowing its variations to induce coordinated changes in both components. Building on this representation, we propose GeCode, which formulates co-design as exploration in the compact Morphogene space. Each Morphogene anchors a local design region in which nearby body--brain designs are explored, while performance-guided updates move these anchors toward promising regions for more efficient exploration of the broader design space. This process combines local refinement with global exploration while preserving body--brain compatibility. Extensive experiments across diverse 2D and 3D co-design tasks demonstrate that GeCode consistently outperforms existing state-of-the-art methods, achieving substantially faster convergence and higher final performance.
Figures & tables
Figure 1: (a) Traditional morphology–control co-design employs separate morphology-design and control networks coupled primarily through a shared task objective, resulting in weak body–brain coordination. (b) Inspired by biological genes, GeCode introduces Morphogene as a high-level latent blueprint that bridges body and brain. By propagating Morphogene variations to both morphology design and controller adaptation, GeCode drives their explicit coordination at the representation level while reformulating morphology search as exploration within a continuous latent space.
Figure 2: Overview of Morphogene space construction and conditioning. (a) An RAE encodes randomly generated morphologies into morphology codes, forming a continuous latent space in which Morphogenes serve as searchable anchors. (b) AdaConcat adapts an agent-level Morphogene into limb-specific conditions for the morphology-design and control networks.
Figure 3: Overview of GeCode. GeCode reformulates morphology–control co-design as exploration over the compact Morphogene space, using a small set of Morphogenes as searchable design anchors. At each co-design epoch, sampled anchors jointly condition morphology generation and controller adaptation, while stochastic rollouts guided by a proximity objective explore their local neighborhoods. Rollout performance subsequently drives Morphogene updates and adaptive resampling, propagating high-performing design information across the set and enabling global latent-space exploration.
Figure 4: Performance comparison of GeCode with state-of-the-art baselines, including StackelbergPPO ( Dai et al., 2026 ) , BodyGen ( Lu et al., 2025 ) , and Transform2Act ( Yuan et al., 2022 ) , and its ablations, GeCode-w/o Gene2Body and GeCode-w/o Gene2Brain, across twelve co-design environments. Shaded regions indicate the standard error over five random seeds.
Crawler
Stepper
Pusher
Cheetah
TerrainCrosser
Swimmer
w/ Direct Concat
14049.3 ± 533.4
3296.0 ± 560.1
4289.5 ± 989.3
18734.4 ± 523.1
7826.2 ± 798.0
1353.5 ± 27.4
w/ AdaConcat
16143.3 ± 583.0
3688.6 ± 271.0
6250.2 ± 757.9
19888.7 ± 944.6
9670.6 ± 1176.4
1422.5 ± 29.5
Glider-Regular
Glider-Medium
Glider-Hard
Walker-Regular
Walker-Medium
Walker-Hard
w/ Direct Concat
20998.1 ± 1002.0
20333.9 ± 884.1
18031.0 ± 1571.1
19197.0 ± 2119.3
19949.6 ± 856.5
20647.6 ± 851.3
w/ AdaConcat
23317.5 ± 627.9
22376.4 ± 809.9
19972.8 ± 332.1
24306.1 ± 1293.0
21973.3 ± 918.9
21614.6 ± 521.8
Table 1: Performance comparison of different Morphogene injection mechanisms. Results show mean episode rewards and standard errors over five random seeds.
Figure 5: (a) Representative morphologies designed by GeCode. (b) Visualization of limb-specific conditioning produced by AdaConcat. (c) Ablations on Morphogene updates and set size NG .
Figure 6: (a) Ablation study of structural constraints (i.e., Ltopo and Ltree ) in the latent space. We report topology-aware 1-NN consistency and the Spearman rank correlation between latent and tree-edit distances, denoted “1-NN ↑ ” and “Spearman ↑ ”, respectively. (b) Morphogene update trajectories over 1,000 co-design epochs. and denote the initial and final positions, respectively, while numerical labels indicate elimination epochs. See Figure 14 in App. E.4 for additional trajectories.
Figure 7: Computational efficiency of GeCode. (a) Performance comparison under a wall-clock time budget. (b) Detailed training costs, together with the resources required for GeCode to match each baseline’s best performance. Further results are provided in Table 7 and Table 8 in App. E.3 .
Appendix figures & tables3 assets
Supplementary material from the paper’s appendix.
Appendix
Figure 8: Overview of Task-Specific Initial Morphologies. Initial Type 1 is a three-dimensional morphology comprising four limbs. Initial Type 2 comprises two limbs connected by one joint in the x – y plane. Initial Type 3 comprises two limbs connected by one joint in the x – z plane. Initial Type 4 comprises three limbs connected by two joints in the x – z plane. These morphologies serve as the starting points for sequential morphology design.
Figure 9: Overview of the Benchmark Environments. Crawler , Stepper , and Pusher are three-dimensional (3D) tasks, whereas the remaining environments are two-dimensional (2D), operating in either the x – z or x – y plane. The benchmarks span diverse morphology depths, symmetries, limb configurations, terrains, and task objectives, enabling a comprehensive evaluation of morphology–control co-design across locomotion and manipulation settings.
Table 2: Hyperparameters of GeCode adopted in all experiments.
Morphology-control co-design concerns the coupled optimization of an agent's body structure and control policy. This problem exhibits a bi-level structure, where the control dynamically adapts to the morphology to maximize performance. Existing methods typically neglect the control's adaptation dynamics by adopting a single-level formulation that treats the control policy as fixed when optimizing morphology. This can lead to inefficient optimization, as morphology updates may be misaligned with control adaptation. In this paper, we revisit the co-design problem from a game-theoretic perspective, modeling the intrinsic coupling between morphology and control as a novel variant of a Stackelberg game. We propose Stackelberg Proximal Policy Optimization (Stackelberg PPO), which explicitly incorporates the control's adaptation dynamics into morphology optimization. By modeling this intrinsic coupling, our method aligns morphology updates with control adaptation, thereby stabilizing training and improving learning efficiency. Experiments across diverse co-design tasks demonstrate that Stackelberg PPO outperforms standard PPO in both stability and final performance, opening the way for dramatically more efficient robotics designs.
Yanning Dai, Yuhui Wang, Dylan R. Ashley +1
Center of Excellence for Generative AI, King Abdullah University of Science and Technology (KAUST), Thuwal, Saudi Arabia. · Dalle Molle Institute for Artificial Intelligence Research (IDSIA), Lugano, Switzerland. · Università della Svizzera italiana (USI), Lugano, Switzerland. +1
The intelligent behavior of robots does not emerge solely from control systems, but from the tight coupling between body and brain, a principle known as embodied intelligence. Designing soft robots that leverage this interaction remains a significant challenge, particularly when morphology and control require simultaneous optimization. A significant obstacle in this co-design process is that morphological evolution can disrupt learned control strategies, making it difficult to reuse or adapt existing knowledge. We address this by develop a Graph Neural Network-based approach for the co-design of morphology and controller. Each robot is represented as a graph, with a graph attention network (GAT) encoding node features and a pooled representation passed through a multilayer perceptron (MLP) head to produce actuator commands or value estimates. During evolution, inheritance follows a topology-consistent mapping: shared GAT layers are reused, MLP hidden layers are transferred intact, matched actuator outputs are copied, and unmatched ones are randomly initialized and fine-tuned. This morphology-aware policy class lets the controller adapt when the body mutates. On the benchmark, our GAT-based approach achieves higher final fitness and stronger adaptability to morphological variations compared to traditional MLP-only co-design methods. These results indicate that graph-structured policies provide a more effective interface between evolving morphologies and control for embodied intelligence.
Robot co-design via bi-level optimization couples within-lifetime controller learning for fitness evaluation with cross-generational morphological evolution. Prior work has established that well-adapted morphology facilitates faster control learning, a property termed morphological intelligence. Yet how control learning reciprocally shapes morphological evolution remains unexplored. This paper examines both directions for a holistic account of brain-body interplay. We first show that morphological contributions to control learning decouple into two orthogonal dimensions. We formalize the convergence speed as morphological intelligence and identify the performance ceiling as a complementary quantity termed true potential. A concise functional relation is then established to jointly characterize both quantities from individual learning curves, which, when aggregated at the population level, capture evolutionary profiles. Through extensive experiments on simulated voxel-based soft robots, we reveal that premature fitness evaluation systematically underestimates true potential and biases selection towards fast learners. This restricts design space exploration, compromising both optimization efficiency and morphological diversity. Notably, the widely recognized morphological Baldwin effect emerges as an artifact of this bias rather than a general evolutionary tendency. We therefore propose AdaControl, which monitors disproportionate selection for morphological intelligence during evolution and allocates minimally sufficient control learning for unbiased fitness evaluation. With AdaControl, a simple genetic algorithm rivals state-of-the-art generative-model-based co-design methods in discovering diverse high-performing designs while cutting computation by up to 80% versus exhaustive control.
Junru Song, Yang Yang, Yaqing Xu +5
Shanghai Jiao Tong University, Shanghai 200240, China · Intelligent Game and Decision Laboratory, Beijing 100048, China · Renmin University of China, Beijing 100872, China +1