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Homogeneous Proportional-Integral-Derivative Controller in Mobile Robotic Manipulators

Luis Luna, Isaac Chairez, Andrey Polyakov

TL;DR

This paper advances control of mobile robotic manipulators by introducing a homogeneous PID (hPID) framework that leverages weighted dilations and canonical homogeneous norms to achieve faster, more robust tracking of coordinated base-and-arm motion. A rigorous stability analysis shows that, for small values of the homogeneity parameter $\mu$, the hPID inherits global asymptotic stability from a well-tuned linear PID, with a Lyapunov function built from the canonical homogeneous norm. The authors apply the method to a six-DoF arm on a mobile base, deriving a decentralized eight-controller hPID implementation and validating it through hardware-in-the-loop experiments; results indicate substantial reductions in control energy and smoother actuation, with only marginal differences in tracking error. Overall, the work provides a scalable, analytically grounded control framework for next-generation mobile manipulation systems in both structured and unstructured environments, with clear benefits in efficiency and robustness over traditional PID schemes.

Abstract

Mobile robotic manipulators (MRMs), which integrate mobility and manipulation capabilities, present significant control challenges due to their nonlinear dynamics, underactuation, and coupling between the base and manipulator subsystems. This paper proposes a novel homogeneous Proportional-Integral-Derivative (hPID) control strategy tailored for MRMs to achieve robust and coordinated motion control. Unlike classical PID controllers, the hPID controller leverages the mathematical framework of homogeneous control theory to systematically enhance the stability and convergence properties of the closed-loop system, even in the presence of dynamic uncertainties and external disturbances involved into a system in a homogeneous way. A homogeneous PID structure is designed, ensuring improved convergence of tracking errors through a graded homogeneity approach that generalizes traditional PID gains to nonlinear, state-dependent functions. Stability analysis is conducted using Lyapunov-based methods, demonstrating that the hPID controller guarantees global asymptotic stability and finite-time convergence under mild assumptions. Experimental results on a representative MRM model validate the effectiveness of the hPID controller in achieving high-precision trajectory tracking for both the mobile base and manipulator arm, outperforming conventional linear PID controllers in terms of response time, steady-state error, and robustness to model uncertainties. This research contributes a scalable and analytically grounded control framework for enhancing the autonomy and reliability of next-generation mobile manipulation systems in structured and unstructured environments.

Homogeneous Proportional-Integral-Derivative Controller in Mobile Robotic Manipulators

TL;DR

This paper advances control of mobile robotic manipulators by introducing a homogeneous PID (hPID) framework that leverages weighted dilations and canonical homogeneous norms to achieve faster, more robust tracking of coordinated base-and-arm motion. A rigorous stability analysis shows that, for small values of the homogeneity parameter , the hPID inherits global asymptotic stability from a well-tuned linear PID, with a Lyapunov function built from the canonical homogeneous norm. The authors apply the method to a six-DoF arm on a mobile base, deriving a decentralized eight-controller hPID implementation and validating it through hardware-in-the-loop experiments; results indicate substantial reductions in control energy and smoother actuation, with only marginal differences in tracking error. Overall, the work provides a scalable, analytically grounded control framework for next-generation mobile manipulation systems in both structured and unstructured environments, with clear benefits in efficiency and robustness over traditional PID schemes.

Abstract

Mobile robotic manipulators (MRMs), which integrate mobility and manipulation capabilities, present significant control challenges due to their nonlinear dynamics, underactuation, and coupling between the base and manipulator subsystems. This paper proposes a novel homogeneous Proportional-Integral-Derivative (hPID) control strategy tailored for MRMs to achieve robust and coordinated motion control. Unlike classical PID controllers, the hPID controller leverages the mathematical framework of homogeneous control theory to systematically enhance the stability and convergence properties of the closed-loop system, even in the presence of dynamic uncertainties and external disturbances involved into a system in a homogeneous way. A homogeneous PID structure is designed, ensuring improved convergence of tracking errors through a graded homogeneity approach that generalizes traditional PID gains to nonlinear, state-dependent functions. Stability analysis is conducted using Lyapunov-based methods, demonstrating that the hPID controller guarantees global asymptotic stability and finite-time convergence under mild assumptions. Experimental results on a representative MRM model validate the effectiveness of the hPID controller in achieving high-precision trajectory tracking for both the mobile base and manipulator arm, outperforming conventional linear PID controllers in terms of response time, steady-state error, and robustness to model uncertainties. This research contributes a scalable and analytically grounded control framework for enhancing the autonomy and reliability of next-generation mobile manipulation systems in structured and unstructured environments.

Paper Structure

This paper contains 15 sections, 1 theorem, 59 equations, 7 figures, 1 table.

Table of Contents

  1. Introduction
  2. Mathematical preliminaries on homogeneity
  3. Linear Dilation
  4. Homogeneous norm
  5. Homogeneous systems
  6. Homogeneous PID control
  7. Design of hPID
  8. Stability of hPID
  9. Application of hPID control to robotic manipulators
  10. The selected mobile manipulator
  11. Mechanical configuration of the mobile manipulator
  12. Electrical instrumentation of the mobile manipulator
  13. Implementation of hPID on the mobile manipulator
  14. Experimental assessment of HPID on the mobile manipulator
  15. Conclusion

Key Result

Theorem 1

Let a linear dilation $\mathbf{d}$ in $\mathbb{R}^2$ be defined by the formula $\mathbf{d}(s)=\mathrm{diag} (e^{(1-\mu)s},e^s)$, $s\in \mathbb{R}$. Let $\|\cdot\|_{\mathbf{d}}$ be an arbitrary $\mathbf{d}$-homogeneous norm in $\mathbb{R}^2$. If the system eq:model with the linear PID eq:PID is globa

Figures (7)

  • Figure 1: General view of the design of the mobile robotic manipulator.
  • Figure 2: Simplified diagram of electrical instrumentation for actuating/measuring the activity of each joint in the mobile manipulator.
  • Figure 3: Comparison of the temporal evolution of trajectories for all joints in the arm section of the mobile manipulator produced with the implementation of the hPID (red lines) and PID (blue lines).
  • Figure 4: Comparison of the temporal evolution of control functions for all joints in the arm section of the mobile manipulator produced with the implementation of the hPID (red lines) and PID (blue lines).
  • Figure 5: Comparison of the temporal evolution of trajectory tracking errors for all joints in the arm section of the mobile manipulator produced with the implementation of the hPID (red lines) and PID (blue lines).
  • ...and 2 more figures

Theorems & Definitions (1)

  • Theorem 1