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Sliding Mode Control and Subspace Stabilization Methodology for the Orbital Stabilization of Periodic Trajectories

Maksim Surov, Leonid Freidovich

TL;DR

This paper addresses orbital stabilization of periodic trajectories in underactuated mechanical systems by combining partial feedback linearization, transverse linearization, Floquet theory, and sliding-mode control to steer trajectories into a stable subspace; this avoids costly periodic LQR computations and enhances robustness to disturbances. The method introduces a transverse dynamics framework with a stable Floquet subspace, and a sliding-mode controller that drives the system into that subspace while mapping back to a nonlinear input for the original system. Experimental validation on the Butterfly robot demonstrates feasibility and highlights practical considerations such as parasitic dynamics, actuator limits, and chattering. The work suggests natural extensions to higher-dimensional systems and raises questions about applicability when actuator count is reduced further.

Abstract

This paper presents a combined sliding-mode control and subspace stabilization methodology for orbital stabilization of periodic trajectories in underactuated mechanical systems with one degree of underactuation. The approach starts with partial feedback linearization and stabilization. Then, transverse linearization along the reference orbit is computed, resulting in a periodic linear time-varying system with a stable subspace. Sliding-mode control drives trajectories toward this subspace. The proposed design avoids solving computationally intensive periodic LQR problems and improves robustness to matched disturbances. The methodology is validated through experiments on the Butterfly robot.

Sliding Mode Control and Subspace Stabilization Methodology for the Orbital Stabilization of Periodic Trajectories

TL;DR

This paper addresses orbital stabilization of periodic trajectories in underactuated mechanical systems by combining partial feedback linearization, transverse linearization, Floquet theory, and sliding-mode control to steer trajectories into a stable subspace; this avoids costly periodic LQR computations and enhances robustness to disturbances. The method introduces a transverse dynamics framework with a stable Floquet subspace, and a sliding-mode controller that drives the system into that subspace while mapping back to a nonlinear input for the original system. Experimental validation on the Butterfly robot demonstrates feasibility and highlights practical considerations such as parasitic dynamics, actuator limits, and chattering. The work suggests natural extensions to higher-dimensional systems and raises questions about applicability when actuator count is reduced further.

Abstract

This paper presents a combined sliding-mode control and subspace stabilization methodology for orbital stabilization of periodic trajectories in underactuated mechanical systems with one degree of underactuation. The approach starts with partial feedback linearization and stabilization. Then, transverse linearization along the reference orbit is computed, resulting in a periodic linear time-varying system with a stable subspace. Sliding-mode control drives trajectories toward this subspace. The proposed design avoids solving computationally intensive periodic LQR problems and improves robustness to matched disturbances. The methodology is validated through experiments on the Butterfly robot.

Paper Structure

This paper contains 16 sections, 4 theorems, 65 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Problem formulation
  3. Orbital Stabilization Approach
  4. Feedback Transformation for VHC Stabilization
  5. Transverse Dynamics
  6. Stable Floquet Subspace of Transverse Linearization
  7. Sliding-Mode Control for Transverse Linearization
  8. Control Law for Nonlinear System
  9. Butterfly Robot Example
  10. Periodic Trajectory
  11. Transverse Coordinates
  12. Feedback Design
  13. Concluding Remarks
  14. Proof of Proposition \ref{['prop:stability-of-y']}
  15. Proof of Proposition \ref{['prop:stable-subspace']}
  16. ...and 1 more sections

Key Result

Proposition 1

Consider the system obtained from eq:dynamics-normal-form by setting $w=0$ and feedback transformation parameters $\nu_{1},\nu_{2}>0$. Define Then there exist a positive definite matrix $P\in\mathbb{R}^{2\times2}$ and a constant $\alpha>0$ such that the quadratic form decays exponentially along all solutions of the unforced system within $U_{\varepsilon}$, that is,

Figures (5)

  • Figure 1: Kinematics of the Butterfly Robot. Definition of the generalized coordinates $q \equiv (\vartheta, \varphi)$.
  • Figure 2: Phase portrait of the reduced dynamics; projections of the reference trajectory onto the phase planes; control signal along the reference trajectory.
  • Figure 3: Components of the vector $n\left(\tau\right)$ defining the stable subspace $\mathcal{S}_{\tau}$, and the coefficient $b(\tau) \equiv n^{\top}\!\left(\tau\right)B\left(\tau\right)$.
  • Figure 4: Transient behavior of the transverse coordinates $\xi$, sliding variable $s$, coefficient $b$, and control signal $u$. Regions where $b$ is close to zero -- and thus the sliding variable dynamics are uncontrollable -- are highlighted in yellow.
  • Figure 5: Projection of the closed-loop system trajectory, measured in a real experiment, onto the phase-space planes.

Theorems & Definitions (5)

  • Proposition 1
  • Proposition 2
  • Proposition 3
  • Theorem 1
  • proof