Safe Control Design for Quadruped Locomotion in Unstructured Environments using Linear Transfer Operators

Author

• Sriram Sundar Krishnamoorthy Shankara Narayanan, Clemson UniversityFollow

Date of Award

5-2026

Document Type

Dissertation

Degree Name

Doctor of Philosophy (PhD)

Department

Mechanical Engineering

Committee Chair/Advisor

Umesh Vaidya

Committee Member

Venkat Krovi

Committee Member

Ardalan Vahidi

Committee Member

Ge Lv

Abstract

Deploying quadruped robots in unstructured, obstacle-rich environments requires control and planning methods that remain safe and reliable despite complex terrain geometry, limited sensing, and inevitable modeling errors. This thesis develops operator-theoretic tools for safe control design of robotic systems using linear transfer operators, with a focus on quadruped locomotion in unstructured environments. The central goal is to develop a unified operator-theoretic framework for safe control design based on the Perron–Frobenius (P–F) and Koopman operators. In particular, the thesis leverages \emph{density functions} to develop safe navigation frameworks in the dual space of densities. In the operator-theoretic perspective, the P–F operator governs how density is transported by the closed-loop dynamics. Further, the thesis develops data-driven models based on \emph{Koopman operator} theory by learning lifted linear predictors from measurement data. By embedding nonlinear dynamics into a higher-dimensional feature space, these predictors capture key nonlinear effects that are not well represented by nominal templates in real-world conditions. When used within a predictive control setting, the lifted linear structure yields convex Koopman-based MPC formulations that can be solved in real time.

First, the thesis formulates a safe navigation framework based on density functions. Safety and convergence are certified through operator-theoretic conditions posed in the dual space of densities. Navigation density functions encode an occupancy interpretation of safety. Unsafe regions are assigned zero density, and the induced closed-loop behavior avoids the unsafe sets while converging to a target set in an almost-everywhere sense. The thesis provides analytical constructions of navigation density functions for complex environments, including time-varying settings, where the densities explicitly encode obstacle sets and target sets. These density functions can also be updated online in local-observation settings using only locally sensed obstacle information. This density-based viewpoint yields constructive safety certificates that accommodate arbitrary obstacle geometries and can be extended to high-dimensional systems. Further, they can be readily incorporated as safety constraints in optimization-based control synthesis. We demonstrate this framework through density-informed motion planning pipelines for safe quadruped navigation in static environments and unstructured off-road settings.

Second, the thesis investigates data-driven model identification and predictive control using Koopman operators. We use finite-dimensional approximation methods to learn linear predictors in a lifted space directly from measurement data. These predictors capture dominant nonlinear effects while preserving a structure that enables convex model predictive control (MPC) formulations. For quadruped locomotion, the thesis introduces the Residual Koopman MPC (RK-MPC) framework, which combines a nominal template model with a compact, data-driven residual predictor learned in Koopman lifted coordinates. This residual correction systematically compensates for model mismatch caused by contact variability and terrain disturbances, while the resulting predictor remains linear in the lifted space. Furthermore, it can be easily integrated into MPC formulations as a linear constraint. RK-MPC is validated in simulation and on a Unitree Go1 quadruped through extensive hardware experiments on off-road terrains, including grass and snow, demonstrating reliable and robust locomotion under disturbances.

Overall, this thesis develops operator-theoretic tools that support safe and reliable real-time quadruped autonomy in unstructured, obstacle-rich environments.

Comments

Google Scholar URL: https://scholar.google.com/citations?user=-kMTFjwAAAAJ&hl=en

Recommended Citation

Krishnamoorthy Shankara Narayanan, Sriram Sundar, "Safe Control Design for Quadruped Locomotion in Unstructured Environments using Linear Transfer Operators" (2026). All Dissertations. 4216.
https://open.clemson.edu/all_dissertations/4216

Author ORCID Identifier

https://orcid.org/0000-0002-3163-9275