[Paper Review] Tracking Control of Fully-actuated Mechanical port-Hamiltonian Systems using Sliding Manifolds and Contraction

Rodolfo Reyes‐Báez, Arjan van der Schaft+1 more|arXiv (Cornell University)|Nov 22, 2016

Control and Stability of Dynamical Systems3 citations

TL;DR

This paper proposes a novel trajectory tracking controller for fully-actuated port-Hamiltonian systems by combining sliding manifold control with contraction-based design. The method ensures exponential convergence to a desired reference trajectory by rendering a sliding manifold attractive through partial contraction of the error system, with theoretical guarantees via differential Lyapunov functions and Finsler geometry.

ABSTRACT

In this paper, we propose a novel trajectory tracking controller for fully-actuated mechanical port-Hamiltonian (pH) systems, which is based on recent advances in contraction-based control theory. Our proposed controller renders a desired sliding manifold (where the reference trajectory lies) attractive by making the corresponding error system partially contracting. Finally, we present numerical simulation results where a SCARA robot is commanded by our proposed tracking control law.

Motivation & Objective

• Address the challenge of trajectory tracking in fully-actuated mechanical port-Hamiltonian systems where time-varying references disrupt passivity and dissipativity properties.
• Overcome limitations of standard passivity-based control in time-varying output regulation by preserving the pH structure in the error dynamics.
• Develop a control framework that ensures attractivity of a sliding manifold containing the reference trajectory using contraction theory.
• Provide a physically interpretable control law with incremental stability guarantees through a Riemannian distance metric induced by a contraction measure.
• Demonstrate the effectiveness of the controller on a 3-DOF SCARA robot with exponential convergence and smooth control effort.

Proposed method

• Define a sliding manifold where the reference trajectory lies, using a generalized coordinate transformation to express the error dynamics.
• Construct a virtual system in the transformed coordinates to analyze incremental stability via a differential Lyapunov function.
• Apply partial contraction theory to ensure the error system contracts exponentially with rate $\beta$, using a symmetric positive definite matrix $\boldsymbol{\Theta}$ and a Riemannian metric $\overline{V}(\vec{x}_v, \delta\vec{x}_v) = \frac{1}{2}\delta\vec{x}_v^T \boldsymbol{\Theta}^T \vec{P}(\tilde{\vec{x}}) \boldsymbol{\Theta} \delta\vec{x}_v$.
• Use the Finsler structure induced by the contraction measure to define a Riemannian distance $d(\vec{x}, \vec{x}_d) < e^{-\beta t}$, ensuring exponential convergence to the desired trajectory.

Experimental results

Research questions

• RQ1 How can trajectory tracking be achieved in fully-actuated port-Hamiltonian systems while preserving the system's structural properties under time-varying references?
• RQ2 Can contraction theory be effectively combined with sliding manifold control to ensure exponential convergence and robustness in pH systems?
• RQ3 What conditions ensure that the error dynamics are partially contracting, and how does this lead to incremental stability and attractivity of the sliding manifold?
• RQ4 How does the proposed controller maintain physical interpretability through energy-based analysis despite time-varying references?
• RQ5 What is the performance of the controller in terms of convergence speed, control effort, and robustness in a realistic mechanical system like a SCARA robot?

Key findings

• The proposed controller ensures exponential convergence of the state error to zero with a rate bounded by $\beta$, as shown by the contraction measure $d(\vec{x}, \vec{x}_d) < e^{-\beta t}$.
• The sliding manifold is made attractive through a feedback control law that induces partial contraction in the error system, guaranteeing asymptotic stability.
• Numerical simulations on a 3-DOF SCARA robot show all error variables ($\tilde{\vec{q}}$, $\tilde{\vec{p}}$, and the sliding variable) converge to zero exponentially after transients.
• The contraction measure exhibits an initial overshoot but shrinks over time, confirming the incremental stability of the virtual system.
• The Hamiltonian of the closed-loop system converges to the desired Hamiltonian value, indicating energy-based consistency.
• Control effort is smooth after a large transient, which is attributed to the initial compensation required by the high-gain terms in the control law (27).