# Combining Trajectory Optimization, Supervised Machine Learning, and Model Structure for Mitigating the Curse of Dimensionality in the Control of Bipedal Robots

arXiv: https://arxiv.org/abs/1711.02223 · HTML: https://mlsanity.com/paper/1711.02223/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 6 Nov 2017.

## Built on by (most cited first, 5 of 5)

- [GLiDE: Generalizable Quadrupedal Locomotion in Diverse Environments with a Centroidal Model](https://mlsanity.com/paper/2104.09771/) (2021)
- [A Unified and General Humanoid Whole-Body Controller for Versatile Locomotion](https://mlsanity.com/paper/2502.03206/) (2025)
- [Learning Linear Policies for Robust Bipedal Locomotion on Terrains with Varying Slopes](https://mlsanity.com/paper/2104.01662/) (2021)
- [MASH: Cooperative-Heterogeneous Multi-Agent Reinforcement Learning for Single Humanoid Robot Locomotion](https://mlsanity.com/paper/2508.10423/) (2025)
- [GCImOpt: Learning efficient goal-conditioned policies by imitating optimal trajectories](https://mlsanity.com/paper/2604.22724/) (2026)

## Abstract (arXiv)

To overcome the obstructions imposed by high-dimensional bipedal models, we embed a stable walking motion in an attractive low-dimensional surface of the system's state space. The process begins with trajectory optimization to design an open-loop periodic walking motion of the high-dimensional model and then adding to this solution, a carefully selected set of additional open-loop trajectories of the model that steer toward the nominal motion. A drawback of trajectories is that they provide little information on how to respond to a disturbance. To address this shortcoming, Supervised Machine Learning is used to extract a low-dimensional state-variable realization of the open-loop trajectories. The periodic orbit is now an attractor of the low-dimensional state-variable model but is not attractive in the full-order system. We then use the special structure of mechanical models associated with bipedal robots to embed the low-dimensional model in the original model in such a manner that the desired walking motions are locally exponentially stable. The design procedure is first developed for ordinary differential equations and illustrated on a simple model. The methods are subsequently extended to a class of hybrid models and then realized experimentally on an Atrias-series 3D bipedal robot.