Learning Purposeful Co-designs of Bio-inspired Quadrupedal Robots

Fabio Boekel; Jiatao Ding; Cosimo Della Santina

cs.RONov 2, 2026

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

Abstract

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 ( $l_{1}^{n}$ , $l_{2}^{n}$ , $l_{3}^{n}$ ) using scaling factors ( $\lambda_{1}$ , $\lambda_{2}$ , $\lambda_{3}$ ). 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 $\pi_{u}$ is trained in simulation across a broad design space $\mathcal{D}$ using RL. In Step 2 , a BO-based selector identifies the optimal design $d^{*}_{k}$ for a specific task $k$ by evaluating energy performance. For real-world deployment, the policy maps observations $\bm{o}_{t}$ to actions $\bm{a}_{t}$ while incorporating the feedforward spring torque $\bm{\tau}_{t}^{\text{ff}}$ , which helps to bridge the sim-to-real gap.

Panel of Figure 3 — 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.

Panel of Figure 4 — Fig. 4: Task-driven co-design selects energy-efficient configurations. (A) The four designs: $d^{n}_{1}$ , $d^{n}_{2}$ , $d^{*}_{1}$ , and $d^{*}_{2}$ , where $d^{n}_{1}$ and $d^{n}_{2}$ share the same link parameters. The radar chart shows their five design parameters. Design colours: purple ( $d^{n}_{1}$ ), blue ( $d^{n}_{2}$ ), green ( $d^{*}_{1}$ ), and yellow ( $d^{*}_{2}$ ). (B) Relative $CoT$ changes compared with $d^{n}_{1}$ 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.

Panel of Figure 5 — 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, $d^{n}_{1}$ ; 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.

Panel of Figure 6 — 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.

Appendix figures & tables6 assets

Supplementary material from the paper’s appendix.

Appendix

Parameter	Value
Number of Timesteps	$150,000,000$
Episode Length	$1000$
Normalize Observations	True
Unroll Length	$20$
Number of Minibatches	$32$
Number of Updates per Batch	$4$

Appendix

TABLE I: PPO training hyperparameters.

Parameter	Value
Velocity Command Bounds
Linear Velocity X	[ $-0.6$ , $0.6$ ]
Linear Velocity Y	[ $-0.6$ , $0.8$ ]
Angular Velocity Yaw	[ $-1.0$ , $1.0$ ]
Action Space Bounds
Lower Bounds	[ $-1.0$ , $0.5$ , $-1.75$ ]

Appendix

TABLE II: Environment-configuration hyperparameters.

Reward Component	Expression
Tracking Reward Functions
Tracking Linear Velocity	$\exp\left(-\frac{(v_{\text{des},x}-v_{x})^{2}+(v_{\text{des},y}-v_{y})^{2}}{\sigma}\right)$
Tracking Angular Velocity	$\exp\left(-\frac{(\omega_{\text{des}}-\omega_{z})^{2}}{\sigma}\right)$
Regularization Reward Functions
Linear Velocity along Z	$v_{z}^{2}$
Angular Velocity in XY	$\omega_{x}^{2}+\omega_{y}^{2}$

Appendix

TABLE III: Reward functions.

Symbol	Description
$\bm{q}$	Joint position vector for all actuators.
$\bm{q}^{\text{default}}$	Default joint position vector for all actuators.
$\dot{q}_{i}$	Angular velocity of the $i$ -th actuator.
$\tau_{i}$	Torque applied to the $i$ -th actuator.
$a_{i}$	Action command for the $i$ -th actuator.
$a_{i}^{-1}$	Previous action command for the $i$ -th actuator.

Appendix

TABLE IV: Symbols used in the reward functions.

Reward Component	Weight
Tracking Reward Weights
Tracking Linear Velocity	$1.5$
Tracking Angular Velocity	$0.8$
Tracking Sigma	$0.25$
Regularization Reward Weights
Linear Velocity along Z	$-2.0$

Appendix

TABLE V: Reward weights.

Parameter	Range	Unit	Notes
$\lambda_{1}$	[ $0.7$ , $1.3$ ]	[-]	Factor
$\lambda_{2}$	[ $0.7$ , $1.3$ ]	[-]	Factor
$\lambda_{3}$	[ $0.7$ , $1.3$ ]	[-]	Factor
$\diameter$	[ $0$ , $50$ ]	[mm]	-
$o$	[ $0$ , $10$ ]	[mm]	-
Friction	[ $1.0$ , $2.0$ ]	[-]	Multiplicative factor

Appendix

TABLE VI: Variable ranges for domain randomization.

Robot Morphology-Control Co-Design

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Efficient Morphology-Control Co-Design via Stackelberg Proximal Policy Optimization

Yanning Dai, Yuhui Wang, Dylan R. Ashley, Jürgen Schmidhuber

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

Robot Morphology-Control Co-Design Bilevel Optimization Proximal Policy Optimization

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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.

Abstract

Robot Morphology-Control Co-Design

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