Learning Purposeful Co-designs of Bio-inspired Quadrupedal Robots
Authors: Fabio Boekel, Jiatao Ding, Cosimo Della Santina
Organizations: Cognitive Robotics, Delft University of Technology, Building 34, Mekelweg 2, 2628 CD Delft, Netherlands · AI Core, Beijing Zhongguancun Academy, Beijing, 100094, China · Institute of Robotics and Mechatronics, German Aerospace Center (DLR), 82234 Wessling, Germany
Treating morphology and control as separate design problems has constrained the efficiency and adaptability of quadrupedal robots, while current co-design approaches are often constrained by restricted design spaces. We overcome this limit by introducing Dogger, a bio-inspired quadruped with modular three-segment legs, four-bar linkages and adjustable passive knee compliance. A two-stage framework combining reinforcement learning and Bayesian optimization is developed: the morphology-conditioned control policy is trained across designs and then held fixed during task-specific design selection. The resulting designs and policies deployed zero-shot reduce the cost of transport by 18.6% and 35.7% for inspection and payload-carrying tasks. Comparisons demonstrate that full geometry-compliance co-design outperforms fixed and restricted design scopes. Cross-task tests further reveal distinct task-specific morphologies with shared mechanical trade-offs. These results provide hardware-level evidence that expanding the physical design space alongside learned control is essential to realize the full potential of task-driven morphology-control co-design in legged robotics.
Figures & tables
Fig. 1: Design of the bio-inspired leg. The lengths of the leg’s links are scaled from their nominal values ( l1n , l2n , l3n ) using scaling factors ( λ1 , λ2 , λ3 ). The spring wheel’s diameter ( \diameter ) adjusts the effective rotational stiffness, while the spring offset ( o ) specifies the engagement angle. The middle inset shows the clamps used to set o . The right panel shows three spring-wheel configurations, with \diameter decreasing from left to right. The wire extending upward is connected to the spring, which is not shown here.
Fig. 2: Two-stage optimization for morphology-control co-design. In Step 1 , a universal control policy πu is trained in simulation across a broad design space D using RL. In Step 2 , a BO-based selector identifies the optimal design dk∗ for a specific task k by evaluating energy performance. For real-world deployment, the policy maps observations ot to actions at while incorporating the feedforward spring torque τtff , which helps to bridge the sim-to-real gap.
Fig. 3: Snapshots from indoor experiments. For task 1, the robot walks forward at 0.5 m/s. For task 2, the robot walks sideways to the left at 0.25 m/s. Consecutive frames are separated by 0.12 s. All four selected designs were tested on hardware; only a subset is shown because of space constraints.
Fig. 4: Task-driven co-design selects energy-efficient configurations. (A) The four designs: d1n , d2n , d1∗ , and d2∗ , where d1n and d2n share the same link parameters. The radar chart shows their five design parameters. Design colours: purple ( d1n ), blue ( d2n ), green ( d1∗ ), and yellow ( d2∗ ). (B) Relative CoT changes compared with d1n for task 1 (left) and task 2 (right). Solid bars represent simulation and hatched bars represent hardware experiments; error bars show standard deviations. (C) Torque profiles of the front-right knee motor during task 1.
Fig. 5: Design-space ablation for tasks 1 and 2. The left and right columns correspond to tasks 1 and 2, respectively. The top row shows the selected design parameters for the fixed, link-length-only, spring-only and full link-length–spring design scopes. The bottom row shows the relative CoT changes compared with the fixed configuration, d1n ; error bars indicate standard deviations. In (B), purple, green and yellow bars represent link-length-only, spring-only and full geometry–compliance co-design, respectively.
Fig. 6: Comparison with alternative co-design baselines. The top row shows the physical configurations selected by CPG+CMA-ES, privileged RL+BO and our method. The bottom row shows the mean CoT normalized to that of our method for each task; error bars indicate standard deviations.
Fig. 7: Robust locomotion in outdoor environments. Two distinct robot configurations are deployed across unstructured terrains; the configuration shown in the bottom two rows has a longer shank.
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
Co-design is a high-dimensional search problem in the robot morphology and control design space. Efficient search requires exploiting the structure shaped by their interaction. To understand this structure, we analyze the landscapes of soft locomotion and manipulation tasks. We identify three patterns consistent across regions of their co-design spaces: 1) Within a region, quality varies along a low-dimensional manifold, with minimal variation orthogonal to it, reducing the effective search space dimensionality. 2) In higher-quality regions, the variance in quality is spread across more dimensions, necessitating search to expand dimensionality as quality improves. 3) In higher-quality regions, quality varies along joint morphology-control dimensions, requiring search along them. Using these insights, we devise an efficient co-design algorithm that yields 36% better co-designs than state-of-the-art baselines. We examine their exploration patterns and show that these baselines required an order of magnitude more function evaluations to find co-designs of comparable quality. Finally, we ablate our algorithm to verify that exploiting the identified structure was the key to efficient co-design.
Apoorv Vaish, Oliver Brock
Robotics and Biology Laboratory, Technische Universität Berlin · Science of Intelligence (SCIoI), Cluster of Excellence, Berlin · Robotics Institute Germany (RIG)
In this paper, we present a hardware-control co-design approach that enables efficient and versatile roller skating on quadrupedal robots equipped with passive wheels. Passive-wheel skating reduces leg inertia and improves energy efficiency, particularly at high speeds. However, the absence of direct wheel actuation tightly couples mechanical design and control. To unlock the full potential of this modality, we formulate a bilevel optimization framework: an upper-level Bayesian Optimization searches the mechanical design space, while a lower-level Reinforcement Learning trains a motor control policy for each candidate design. The resulting design-policy pairs not only outperform human-engineered baselines, but also exhibit versatile behaviors such as hockey stop (rapid braking by turning sideways to maximize friction) and self-aligning motion (automatic reorientation to improve energy efficiency in the direction of travel), offering the first system-level study of dynamic skating motion on quadrupedal robots.
Hanwen Wang, Zhenlong Fang, Josiah Hanna +1
Hanwen Wang, Zhenlong Fang, and Josiah Hanna are with the University of Wisconsin - Madison. · Xiaobin Xiong was with the University of Wisconsin - Madison, and now with the Shanghai Innovation Institute (SII).