# An Efficient Optimal Planning and Control Framework For Quadrupedal Locomotion

arXiv: https://arxiv.org/abs/1609.09861 · HTML: https://mlsanity.com/paper/1609.09861/

## Key facts

- Cited by 12 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 30 Sep 2016.

## Built on by (most cited first, 12 of 12)

- [Learning Humanoid Standing-up Control across Diverse Postures](https://mlsanity.com/paper/2502.08378/) (2025)
- [Articulated Object Interaction in Unknown Scenes with Whole-Body Mobile Manipulation](https://mlsanity.com/paper/2103.10534/) (2021)
- [CaT: Constraints as Terminations for Legged Locomotion Reinforcement Learning](https://mlsanity.com/paper/2403.18765/) (2024)
- [Imitation Learning from MPC for Quadrupedal Multi-Gait Control](https://mlsanity.com/paper/2103.14331/) (2021)
- [Text2Robot: Evolutionary Robot Design from Text Descriptions](https://mlsanity.com/paper/2406.19963/) (2024)
- [VAE-Loco: Versatile Quadruped Locomotion by Learning a Disentangled Gait Representation](https://mlsanity.com/paper/2205.01179/) (2022)
- [An Efficient Model-Based Approach on Learning Agile Motor Skills without Reinforcement](https://mlsanity.com/paper/2403.01962/) (2024)
- [Goal-Conditioned Terminal Value Estimation for Real-time and Multi-task Model Predictive Control](https://mlsanity.com/paper/2410.04929/) (2024)
- [Overcoming Slow Decision Frequencies in Continuous Control: Model-Based Sequence Reinforcement Learning for Model-Free Control](https://mlsanity.com/paper/2410.08979/) (2024)
- [Quadruped Robot Simulation Using Deep Reinforcement Learning -- A step towards locomotion policy](https://mlsanity.com/paper/2502.16401/) (2025)
- [Learning Natural and Robust Hexapod Locomotion over Complex Terrains via Motion Priors based on Deep Reinforcement Learning](https://mlsanity.com/paper/2511.03167/) (2025)
- [Energy-Efficient Quadruped Locomotion with Compliant Feet](https://mlsanity.com/paper/2605.14411/) (2026)

## Abstract (arXiv)

In this paper, we present an efficient Dynamic Programing framework for optimal planning and control of legged robots. First we formulate this problem as an optimal control problem for switched systems. Then we propose a multi--level optimization approach to find the optimal switching times and the optimal continuous control inputs. Through this scheme, the decomposed optimization can potentially be done more efficiently than the combined approach. Finally, we present a continuous-time constrained LQR algorithm which simultaneously optimizes the feedforward and feedback controller with O(n) time-complexity. In order to validate our approach, we show the performance of our framework on a quadrupedal robot. We choose the Center of Mass dynamics and the full kinematic formulation as the switched system model where the switching times as well as the contact forces and the joint velocities are optimized for different locomotion tasks such as gap crossing, walking and trotting.