# Learning-based Design and Control for Quadrupedal Robots with Parallel-Elastic Actuators

arXiv: https://arxiv.org/abs/2301.03509 · HTML: https://mlsanity.com/paper/2301.03509/

## Key facts

- Cited by 11 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 9 Jan 2023.

## Built on by (most cited first, 11 of 11)

- [Learning Robust Autonomous Navigation and Locomotion for Wheeled-Legged Robots](https://mlsanity.com/paper/2405.01792/) (2024)
- [ZSL-RPPO: Zero-Shot Learning for Quadrupedal Locomotion in Challenging Terrains using Recurrent Proximal Policy Optimization](https://mlsanity.com/paper/2403.01928/) (2024)
- [Embodied Co-Design for Rapidly Evolving Agents: Taxonomy, Frontiers, and Challenges](https://mlsanity.com/paper/2512.04770/) (2025)
- [Toward Humanoid Brain-Body Co-design: Joint Optimization of Control and Morphology for Fall Recovery](https://mlsanity.com/paper/2510.22336/) (2025)
- [Efficient and Versatile Quadrupedal Skating: Optimal Co-design via Reinforcement Learning and Bayesian Optimization](https://mlsanity.com/paper/2603.18408/) (2026)
- [TurboMPC: Fast, Scalable, and Differentiable Model Predictive Control on the GPU](https://mlsanity.com/paper/2606.24039/) (2026)
- [Latent Diffeomorphic Co-Design of End-Effectors for Deformable and Fragile Object Manipulation](https://mlsanity.com/paper/2602.17921/) (2026)
- [Explosive Jumping with Rigid and Articulated Soft Quadrupeds via Example Guided Reinforcement Learning](https://mlsanity.com/paper/2503.16197/) (2025)
- [RobotDesignGPT: Automated Robot Design Synthesis using Vision Language Models](https://mlsanity.com/paper/2601.11801/) (2026)
- [Physical Imitation Learning: Distilling Control Policies into Passive Elasticity](https://mlsanity.com/paper/2604.00611/) (2026)
- [Model-Based Reinforcement Learning Exploits Passive Body Dynamics for High-Performance Biped Robot Locomotion](https://mlsanity.com/paper/2604.14565/) (2026)

## Abstract (arXiv)

Parallel-elastic joints can improve the efficiency and strength of robots by assisting the actuators with additional torques. For these benefits to be realized, a spring needs to be carefully designed. However, designing robots is an iterative and tedious process, often relying on intuition and heuristics. We introduce a design optimization framework that allows us to co-optimize a parallel elastic knee joint and locomotion controller for quadrupedal robots with minimal human intuition. We design a parallel elastic joint and optimize its parameters with respect to the efficiency in a model-free fashion. In the first step, we train a design-conditioned policy using model-free Reinforcement Learning, capable of controlling the quadruped in the predefined range of design parameters. Afterwards, we use Bayesian Optimization to find the best design using the policy. We use this framework to optimize the parallel-elastic spring parameters for the knee of our quadrupedal robot ANYmal together with the optimal controller. We evaluate the optimized design and controller in real-world experiments over various terrains. Our results show that the new system improves the torque-square efficiency of the robot by 33% compared to the baseline and reduces maximum joint torque by 30% without compromising tracking performance. The improved design resulted in 11% longer operation time on flat terrain.