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Adaptive Modular Geometric Control of Robotic Manipulators

Mahdi Hejrati, Amir Hossein Barjini, Gokhan Alcan, Jouni Mattila

TL;DR

This paper proposes an adaptive modular geometric control framework for robotic manipulators that decomposes the overall manipulator dynamics into individual modules, enabling the design of local geometric control laws at the module level to address parametric uncertainties.

Abstract

This paper proposes an adaptive modular geometric control framework for robotic manipulators. The proposed methodology decomposes the overall manipulator dynamics into individual modules, enabling the design of local geometric control laws at the module level. To address parametric uncertainties, a geometric adaptive law is incorporated into the control structure. The adaptation mechanism updates only the spatial inertia parameters using a single adaptation gain for the entire system, while guaranteeing physically consistent and drift-free parameter estimates. Numerical simulations are provided to validate the effectiveness of the proposed approach in comparison to the existing modular and geometric methods.

Adaptive Modular Geometric Control of Robotic Manipulators

TL;DR

This paper proposes an adaptive modular geometric control framework for robotic manipulators that decomposes the overall manipulator dynamics into individual modules, enabling the design of local geometric control laws at the module level to address parametric uncertainties.

Abstract

This paper proposes an adaptive modular geometric control framework for robotic manipulators. The proposed methodology decomposes the overall manipulator dynamics into individual modules, enabling the design of local geometric control laws at the module level. To address parametric uncertainties, a geometric adaptive law is incorporated into the control structure. The adaptation mechanism updates only the spatial inertia parameters using a single adaptation gain for the entire system, while guaranteeing physically consistent and drift-free parameter estimates. Numerical simulations are provided to validate the effectiveness of the proposed approach in comparison to the existing modular and geometric methods.
Paper Structure (10 sections, 4 theorems, 74 equations, 7 figures)

This paper contains 10 sections, 4 theorems, 74 equations, 7 figures.

Table of Contents

  1. INTRODUCTION
  2. Mathematical Preliminaries
  3. Notation and Geometric Preliminaries
  4. Manipulator Kinematics
  5. Manipulator Dynamics
  6. Virtual Stability
  7. Modular Geometric Control
  8. Adaptive Modular Geometric Control
  9. Results
  10. Conclusion

Key Result

Theorem 1

Consider a complex robotic system which is decomposed into the subsystems, as shown in Fig. fig:interconnected, with the rigid body and joint dynamics of (Fi) and (Ji), respectively. Each subsystem is virtually stable in the sense of Definition VS under the modular geometric control (Fri) and (Jri).

Figures (7)

  • Figure 1: Interconnected multi-rigid body system decomposed into modules of rigid body and joints.
  • Figure 2: Geometric illustration of the SPD manifold $\mathscr{P}(4)$ with the true inertia $L_i$, estimate $\hat{L}_i$, and nominal model $L_i^0$. The curves indicates the Riemannian geodesic between the points on the manifold.
  • Figure 3: Simulation comparison of the different controllers based on position tracking performance. Error convergence in (a) x-y plane, (b) y-z plane, and (c) x-z plane.
  • Figure 4: Time history of the error convergence. Position error in (a) x direction, (b) y direction, (c) z direction, and (d) orientation error.
  • Figure 5: Comparison of AMGC and MGC under parametric uncertainties. Error convergence in (a) x-y plane, (b) y-z plane, and (c) x-z plane.
  • ...and 2 more figures

Theorems & Definitions (12)

  • Definition 1: Configuration Spaces and Matrix Manifolds
  • Definition 2: Inner Products and Riemannian Metrics
  • Definition 3
  • Definition 4
  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Lemma 1
  • proof
  • ...and 2 more