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Geometric Impedance Control on SE(3) for Robotic Manipulators

Joohwan Seo, Nikhil Potu Surya Prakash, Alexander Rose, Jongeun Choi, Roberto Horowitz

TL;DR

This work addresses impedance control for robotic manipulators while respecting the geometry of the configuration space $SE(3)$. It introduces a differential geometric approach with a left-invariant error metric to derive geometrically consistent position and velocity error vectors, leading to a geometric impedance control law. A Lyapunov-based analysis proves asymptotic stability of the closed-loop system, and simulations with a UR5e demonstrate improved trajectory tracking over conventional, geometry-agnostic impedance controllers. The approach offers more robust interaction with unknown environments and highlights the benefits of enforcing SE(3) geometry in manipulation tasks, with future work extending to reinforcement learning and velocity-field task encodings on manifolds.

Abstract

After its introduction, impedance control has been utilized as a primary control scheme for robotic manipulation tasks that involve interaction with unknown environments. While impedance control has been extensively studied, the geometric structure of SE(3) for the robotic manipulator itself and its use in formulating a robotic task has not been adequately addressed. In this paper, we propose a differential geometric approach to impedance control. Given a left-invariant error metric in SE(3), the corresponding error vectors in position and velocity are first derived. We then propose the impedance control schemes that adequately account for the geometric structure of the manipulator in SE(3) based on a left-invariant potential function. The closed-loop stabilities for the proposed control schemes are verified using Lyapunov function-based analysis. The proposed control design clearly outperformed a conventional impedance control approach when tracking challenging trajectory profiles.

Geometric Impedance Control on SE(3) for Robotic Manipulators

TL;DR

This work addresses impedance control for robotic manipulators while respecting the geometry of the configuration space . It introduces a differential geometric approach with a left-invariant error metric to derive geometrically consistent position and velocity error vectors, leading to a geometric impedance control law. A Lyapunov-based analysis proves asymptotic stability of the closed-loop system, and simulations with a UR5e demonstrate improved trajectory tracking over conventional, geometry-agnostic impedance controllers. The approach offers more robust interaction with unknown environments and highlights the benefits of enforcing SE(3) geometry in manipulation tasks, with future work extending to reinforcement learning and velocity-field task encodings on manifolds.

Abstract

After its introduction, impedance control has been utilized as a primary control scheme for robotic manipulation tasks that involve interaction with unknown environments. While impedance control has been extensively studied, the geometric structure of SE(3) for the robotic manipulator itself and its use in formulating a robotic task has not been adequately addressed. In this paper, we propose a differential geometric approach to impedance control. Given a left-invariant error metric in SE(3), the corresponding error vectors in position and velocity are first derived. We then propose the impedance control schemes that adequately account for the geometric structure of the manipulator in SE(3) based on a left-invariant potential function. The closed-loop stabilities for the proposed control schemes are verified using Lyapunov function-based analysis. The proposed control design clearly outperformed a conventional impedance control approach when tracking challenging trajectory profiles.
Paper Structure (5 sections, 1 theorem, 24 equations, 2 figures, 1 table)

This paper contains 5 sections, 1 theorem, 24 equations, 2 figures, 1 table.

Table of Contents

  1. Simulation Result
  2. Conclusion and Future Works
  3. A derivation of geometric elastic force $f_g$ \ref{['eq:geometric_impedance']}
  4. A derivation of the time-derivative of the potential energy part \ref{['eq:time_derivative_potential']}
  5. Derivation of $\dot{f}_g$ \ref{['eq:f_g_dot']}

Key Result

Lemma 9

For $A \in \mathbb{R}^{3\times 3}$, $b \in \mathbb{R}^3$,

Figures (2)

  • Figure 1: The trajectory tracking results in $x$, $y$, and $z$ coordinates for the proposed and benchmark approach are plotted.
  • Figure 2: Trajectory results in a bird-eye-view for $t = 0 \sim 10 (\mathrm{s})$ with end-effector axes attached are plotted.

Theorems & Definitions (2)

  • Remark 8
  • Lemma 9