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Real-Time Linear MPC for Quadrotors on SE(3): An Analytical Koopman-based Realization

Santosh M. Rajkumar, Chengyu Yang, Yuliang Gu, Sheng Cheng, Naira Hovakimyan, Debdipta Goswami

TL;DR

This work tackles real-time trajectory tracking for quadrotors on SE(3) by deriving an analytical Koopman embedding that yields a finite-dimensional, lifted LPV representation while preserving the full input dimension. Building on this, the authors formulate KQ-LMPC, a convex QP-based MPC that enforces both state and input constraints through the lifted dynamics and an LPV structure, supported by controllability and ISPS guarantees. They prove controllability of the lifted finite-dimensional model, derive stability bounds under disturbances and truncation, and validate the approach through numerical simulations and hardware experiments that show substantial real-time feasibility with tracking performance competitive to NMPC. The combination of data-free Koopman embedding, LPV-MPC formulation, and experimental validation demonstrates a practical path toward high-performance, real-time quadrotor control with rigorous theoretical underpinnings.

Abstract

This letter presents an analytical linear parameter-varying (LPV) representation of quadrotor dynamics utilizing Koopman theory, facilitating computationally efficient linear model predictive control (LMPC) for real-time trajectory tracking. By leveraging carefully designed Koopman observables, the proposed approach enables a compact lifted-space evolution that mitigates the curse of dimensionality while preserving the nonlinear characteristics of the system. Although model predictive control (MPC) is a powerful strategy for quadrotor control, it faces a trade-off between the high computational cost of nonlinear MPC (NMPC) and the reduced accuracy of LMPC. To address this gap, we introduce KQ-LMPC (Koopman Quasilinear LPV MPC), which leverages the Koopman-lifted LPV formulation to enforce constraints, ensure lower computational burden and real-time feasibility, and deliver tracking performance comparable to NMPC. Experimental validation confirms the effectiveness of the framework in reasonably agile flight. To the best of our knowledge, this is the first experimentally validated LMPC for quadrotors that employs analytically derived Koopman observables without requiring training data.

Real-Time Linear MPC for Quadrotors on SE(3): An Analytical Koopman-based Realization

TL;DR

This work tackles real-time trajectory tracking for quadrotors on SE(3) by deriving an analytical Koopman embedding that yields a finite-dimensional, lifted LPV representation while preserving the full input dimension. Building on this, the authors formulate KQ-LMPC, a convex QP-based MPC that enforces both state and input constraints through the lifted dynamics and an LPV structure, supported by controllability and ISPS guarantees. They prove controllability of the lifted finite-dimensional model, derive stability bounds under disturbances and truncation, and validate the approach through numerical simulations and hardware experiments that show substantial real-time feasibility with tracking performance competitive to NMPC. The combination of data-free Koopman embedding, LPV-MPC formulation, and experimental validation demonstrates a practical path toward high-performance, real-time quadrotor control with rigorous theoretical underpinnings.

Abstract

This letter presents an analytical linear parameter-varying (LPV) representation of quadrotor dynamics utilizing Koopman theory, facilitating computationally efficient linear model predictive control (LMPC) for real-time trajectory tracking. By leveraging carefully designed Koopman observables, the proposed approach enables a compact lifted-space evolution that mitigates the curse of dimensionality while preserving the nonlinear characteristics of the system. Although model predictive control (MPC) is a powerful strategy for quadrotor control, it faces a trade-off between the high computational cost of nonlinear MPC (NMPC) and the reduced accuracy of LMPC. To address this gap, we introduce KQ-LMPC (Koopman Quasilinear LPV MPC), which leverages the Koopman-lifted LPV formulation to enforce constraints, ensure lower computational burden and real-time feasibility, and deliver tracking performance comparable to NMPC. Experimental validation confirms the effectiveness of the framework in reasonably agile flight. To the best of our knowledge, this is the first experimentally validated LMPC for quadrotors that employs analytically derived Koopman observables without requiring training data.
Paper Structure (22 sections, 8 theorems, 53 equations, 3 figures, 1 table, 2 algorithms)

This paper contains 22 sections, 8 theorems, 53 equations, 3 figures, 1 table, 2 algorithms.

Table of Contents

  1. Introduction
  2. Terminology
  3. Preliminaries
  4. Quadrotor Dynamics
  5. Koopman Operator Theory
  6. Analytical Koopman Embedding for Quadrotor Dynamics
  7. Invariant Observables and Lifted Quasilinear Realization
  8. Finite-Dimensional Representation and Controllability
  9. Controller Synthesis
  10. Proposed KQ-LMPC Scheme
  11. Closed-loop Stability and Robustness of KQ-LMPC
  12. Numerical and Experimental Results
  13. Approximation Error in Lifted Formulation
  14. Numerical Simulation Results
  15. Experimental Evaluation
  16. ...and 7 more sections

Key Result

Lemma 1

Under Assumption assum:control_limits, $\mathcal{F}\triangleq \overline{\mathrm{span}\{\varphi(\cdot)\}}$, i.e., the closure of the finite linear combinations of the components of $\varphi$ is a Hilbert space with respect to $L^2$ norm, and hence it is also a Banach space.

Figures (3)

  • Figure 1: (a) Quadrotor's coordinate frames, (b) The quadrotor used in the experiment.
  • Figure 2: Approximation errors for a period of 10 sec with different lifted-state dimension.
  • Figure 3: Experimental validation: trajectory tracking performance of KQ-LMPC (proposed) and NMPC (baseline). The values reported above are the tracking RMSEs ($\mathcal{E}_s$) in meters.

Theorems & Definitions (13)

  • Remark 1
  • Lemma 1
  • Theorem 1
  • Theorem 2
  • Remark 2
  • Remark 3
  • Proposition 1
  • Proposition 2
  • Lemma 2
  • Lemma 3
  • ...and 3 more