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Desired Impedance Allocation for Robotic Systems

Mahdi Hejrati, Jouni Mattila

TL;DR

This work addresses the limitation of Virtual Decomposition Control (VDC) to first-order impedance by introducing a second-order impedance allocation that includes inertia. By defining a pseudo-impedance dynamics with a sliding surface and linking end-effector acceleration to joint commands, the authors realize true second-order impedance within VDC while preserving modularity. A rigorous stability analysis extends the VDC framework to the second-order regime, and experiments on a 7-DoF haptic exoskeleton show improved tracking, contact performance, and a 70% increase in maximum renderable stiffness (Z-width) without additional control effort. The approach enables deliberate shaping of dynamic responses during contact, with practical implications for high-fidelity interaction in pHRI and teleoperation settings.

Abstract

Virtual Decomposition Control (VDC) has emerged as a powerful modular framework for real-world robotic control, particularly in contact-rich tasks. Despite its widespread use, VDC has been fundamentally limited to first-order impedance allocation, inherently neglecting the desired inertia due to the mathematical complexity of second-order behavior allocation. However, inertia is crucial, not only for shaping dynamic responses during contact phases, but also for enabling smooth acceleration and deceleration in trajectory tracking. Motivated by the growing demand for high-fidelity interaction control, this work introduces, for the first time in the VDC framework, a method to realize second-order impedance behavior. By redefining the required end-effector velocity and introducing a required acceleration and a pseudo-impedance term, we achieve second-order impedance control while preserving the modularity of VDC. Rigorous stability analysis confirms the robustness of the proposed controller. Experimental validation on a 7-degree-of-freedom haptic exoskeleton demonstrates superior tracking and contact performance compared to first-order methods. Notably, incorporating inertia enables stable interaction with environments up to 70% stiffer, highlighting the effectiveness of the approach in real-world contact-rich scenarios.

Desired Impedance Allocation for Robotic Systems

TL;DR

This work addresses the limitation of Virtual Decomposition Control (VDC) to first-order impedance by introducing a second-order impedance allocation that includes inertia. By defining a pseudo-impedance dynamics with a sliding surface and linking end-effector acceleration to joint commands, the authors realize true second-order impedance within VDC while preserving modularity. A rigorous stability analysis extends the VDC framework to the second-order regime, and experiments on a 7-DoF haptic exoskeleton show improved tracking, contact performance, and a 70% increase in maximum renderable stiffness (Z-width) without additional control effort. The approach enables deliberate shaping of dynamic responses during contact, with practical implications for high-fidelity interaction in pHRI and teleoperation settings.

Abstract

Virtual Decomposition Control (VDC) has emerged as a powerful modular framework for real-world robotic control, particularly in contact-rich tasks. Despite its widespread use, VDC has been fundamentally limited to first-order impedance allocation, inherently neglecting the desired inertia due to the mathematical complexity of second-order behavior allocation. However, inertia is crucial, not only for shaping dynamic responses during contact phases, but also for enabling smooth acceleration and deceleration in trajectory tracking. Motivated by the growing demand for high-fidelity interaction control, this work introduces, for the first time in the VDC framework, a method to realize second-order impedance behavior. By redefining the required end-effector velocity and introducing a required acceleration and a pseudo-impedance term, we achieve second-order impedance control while preserving the modularity of VDC. Rigorous stability analysis confirms the robustness of the proposed controller. Experimental validation on a 7-degree-of-freedom haptic exoskeleton demonstrates superior tracking and contact performance compared to first-order methods. Notably, incorporating inertia enables stable interaction with environments up to 70% stiffer, highlighting the effectiveness of the approach in real-world contact-rich scenarios.
Paper Structure (11 sections, 2 theorems, 56 equations, 9 figures)

This paper contains 11 sections, 2 theorems, 56 equations, 9 figures.

Table of Contents

  1. INTRODUCTION
  2. Mathematical Preliminaries
  3. Virtual Decomposition Control Approach
  4. Natural Adaptation Law
  5. Control Action
  6. Virtual Stability
  7. Second-order Impedance Allocation
  8. Proposed method
  9. Virtual Stability Analysis
  10. Experimental Results
  11. Conclusion

Key Result

Theorem 1

For a complex system decomposed into multiple subsystems, if each decomposed subsystem satisfies the virtual stability condition in Definition Def 1, then the overall system is stable.

Figures (9)

  • Figure 1: Interconnected multi-body system
  • Figure 2: Implementation setup with attached VDC frames: 7-DoF Haptic exoskeleton communicates with the driver. The driver transmits data to host PC through EtherCAT.
  • Figure 3: In-contact experimental results of FOI with $t_f = 5s$.
  • Figure 4: In-contact experimental results of proposed method with $t_f = 5s$.
  • Figure 5: In-contact experimental results of FOI with $t_f = 3s$.
  • ...and 4 more figures

Theorems & Definitions (9)

  • Remark 1
  • Remark 2
  • Definition 1
  • Theorem 1
  • Theorem 2
  • Proof 1
  • Remark 3
  • Remark 4
  • Remark 5