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Nussbaum Function Based Approach for Tracking Control of Robot Manipulators

Hamed Rahimi Nohooji, Holger Voos

TL;DR

This work tackles tracking control for robotic manipulators with unknown control direction and dynamics by integrating a Nussbaum function $N(\zeta)$ into a PID-like framework. The proposed controller, with adaptive gains and neural-network-based weight estimation, is analyzed via Lyapunov theory to guarantee bounded signals and convergence of the tracking error to a small neighborhood, while reducing tuning complexity. A Nussbaum function $N(\zeta)$ is used to handle unknown direction, and update laws for $\hat{\psi}$ and $\zeta$ ensure stability; a two-link planar manipulator confirms practical effectiveness through numerical simulations. The approach yields a simple, low-complexity control strategy with automatic gain adjustment that maintains robust tracking under uncertain actuator direction, offering a feasible path toward real-time implementation in uncertain robotic systems.

Abstract

This paper introduces a novel Nussbaum function-based PID control for robotic manipulators. The integration of the Nussbaum function into the PID framework provides a solution with a simple structure that effectively tackles the challenge of unknown control directions. Stability is achieved through a combination of neural network-based estimation and Lyapunov analysis, facilitating automatic gain adjustment without the need for system dynamics. Our approach offers a gain determination with minimum parameter requirements, significantly reducing the complexity and enhancing the efficiency of robotic manipulator control. The paper guarantees that all signals within the closed-loop system remain bounded. Lastly, numerical simulations validate the theoretical framework, confirming the effectiveness of the proposed control strategy in enhancing robotic manipulator control.

Nussbaum Function Based Approach for Tracking Control of Robot Manipulators

TL;DR

This work tackles tracking control for robotic manipulators with unknown control direction and dynamics by integrating a Nussbaum function into a PID-like framework. The proposed controller, with adaptive gains and neural-network-based weight estimation, is analyzed via Lyapunov theory to guarantee bounded signals and convergence of the tracking error to a small neighborhood, while reducing tuning complexity. A Nussbaum function is used to handle unknown direction, and update laws for and ensure stability; a two-link planar manipulator confirms practical effectiveness through numerical simulations. The approach yields a simple, low-complexity control strategy with automatic gain adjustment that maintains robust tracking under uncertain actuator direction, offering a feasible path toward real-time implementation in uncertain robotic systems.

Abstract

This paper introduces a novel Nussbaum function-based PID control for robotic manipulators. The integration of the Nussbaum function into the PID framework provides a solution with a simple structure that effectively tackles the challenge of unknown control directions. Stability is achieved through a combination of neural network-based estimation and Lyapunov analysis, facilitating automatic gain adjustment without the need for system dynamics. Our approach offers a gain determination with minimum parameter requirements, significantly reducing the complexity and enhancing the efficiency of robotic manipulator control. The paper guarantees that all signals within the closed-loop system remain bounded. Lastly, numerical simulations validate the theoretical framework, confirming the effectiveness of the proposed control strategy in enhancing robotic manipulator control.
Paper Structure (7 sections, 3 theorems, 22 equations, 5 figures)

This paper contains 7 sections, 3 theorems, 22 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Problem Formulation And Priliminaries
  3. Problem Formulation
  4. Priliminaries
  5. Nussbaum Function-based Control Design and Stability Analysis
  6. Numerical Simulation
  7. Conclusion

Key Result

Lemma 1

chen2021tracking Given the intermediate variable $\Psi(t)$ as defined in eq:E, if $\Psi(t) \to 0$ as $t \to \infty$, then the tracking errors $e(t)$ and $\dot{e}(t)$, and their integrals are bounded and converge to zero over time.

Figures (5)

  • Figure 1: Desired and actual trajectories of joint positions
  • Figure 2: Desired and actual trajectories of joint velocities
  • Figure 3: Trajectories of the filtered error $\Psi(t)$
  • Figure 4: Norms of neural networks weignts
  • Figure 5: Trajectories of control input

Theorems & Definitions (5)

  • Remark 1
  • Lemma 1
  • Definition 1: Nussbaum Function
  • Lemma 2
  • Theorem 1