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Prescribed Performance Control of Unknown Euler-Lagrange Systems Under Input Constraints

Ratnangshu Das, Pushpak Jagtap

TL;DR

The proposed approach enforces hard funnel constraints, meaning that the prescribed performance bounds must not be violated during operation, and derives feasibility conditions that guarantee the tracking error evolves within these predefined funnels while ensuring bounded control inputs.

Abstract

In this paper, we present a prescribed performance control framework for trajectory tracking in Euler-Lagrange systems with unknown dynamics and prescribed input constraints. The proposed approach enforces hard funnel constraints, meaning that the prescribed performance bounds must not be violated during operation. We derive feasibility conditions that guarantee the tracking error evolves within these predefined funnels while ensuring bounded control inputs. To handle situations where the feasibility conditions are not satisfied, we introduce two approximation-free control strategies: one that actively drives the error back toward the funnel and another that prioritizes safety by preventing further deviation. The effectiveness and robustness of the proposed method are demonstrated through simulation studies and hardware experiments, highlighting its suitability for real-world robotic systems operating under strict input limits.

Prescribed Performance Control of Unknown Euler-Lagrange Systems Under Input Constraints

TL;DR

The proposed approach enforces hard funnel constraints, meaning that the prescribed performance bounds must not be violated during operation, and derives feasibility conditions that guarantee the tracking error evolves within these predefined funnels while ensuring bounded control inputs.

Abstract

In this paper, we present a prescribed performance control framework for trajectory tracking in Euler-Lagrange systems with unknown dynamics and prescribed input constraints. The proposed approach enforces hard funnel constraints, meaning that the prescribed performance bounds must not be violated during operation. We derive feasibility conditions that guarantee the tracking error evolves within these predefined funnels while ensuring bounded control inputs. To handle situations where the feasibility conditions are not satisfied, we introduce two approximation-free control strategies: one that actively drives the error back toward the funnel and another that prioritizes safety by preventing further deviation. The effectiveness and robustness of the proposed method are demonstrated through simulation studies and hardware experiments, highlighting its suitability for real-world robotic systems operating under strict input limits.

Paper Structure

This paper contains 22 sections, 1 theorem, 34 equations, 7 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries and Problem Formulation
  3. Notations
  4. System Definition
  5. Problem Statement
  6. Controller Design and Theoretical Analysis
  7. Stage I
  8. Stage II
  9. Bounded Transformation Function
  10. Saturation Transformation Function
  11. Zeroing Transformation Function
  12. Feasibility Condition
  13. Stage I
  14. Stage II
  15. Tracking Error Analysis
  16. ...and 7 more sections

Key Result

Theorem 3.2

Consider the Euler–Lagrange system eqn:sysdyn satisfying Assumptions assum_d-assum_Md, and a reference trajectory under Assumption assum_xref. If the initial conditions satisfy $|e_x(0)| \!\prec\! p_x$ and $|e_v(0)| \!\prec\! p_v$, and the feasibility conditions eqn:feas1-eqn:feas2 hold, then the co

Figures (7)

  • Figure 1: (a) Omnidirectional mobile-robot. (b) FRANKA RESEARCH 3.
  • Figure 2: Bounded Transformation Functions.
  • Figure 3: Funnel constraints.
  • Figure 4: Simulation results for 2R manipulator. (a) Desired trajectory vs tracked trajectory, (b) evolution of errors within funnel boundaries, (c) sinusoidal disturbance with disturbance bounds, and (d) torque input with input bounds.
  • Figure 5: 7-DOF Franka Research 3 manipulator: tracking error constrained within funnels and torque input for (a) Nominal tracking, (b) Payload variation, and (c) External disturbances. https://drive.google.com/file/d/1O5uEoPiUZU9zdcU34QYLhEC3ImSzORug/view?usp=sharing.
  • ...and 2 more figures

Theorems & Definitions (5)

  • Remark 3.1
  • Theorem 3.2
  • Proof 3.3
  • Remark 3.4
  • Remark 3.5