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Lie-algebra Adaptive Tracking Control for Rigid Body Dynamics

Jiawei Tang, Shilei Li, Ling Shi

TL;DR

The paper tackles robust trajectory tracking for rigid bodies on $SE(3)$ under parameter uncertainty. It introduces a Lie-algebra–based adaptive tracking method by mapping group dynamics to the Lie algebra to obtain a linear error model that decouples unknown parameters from the state. It develops a discrete-time Lie-algebra adaptive optimal control framework that reconstructs inertia and mass from data via regularized least squares and computes an optimal policy by solving a discrete-time Riccati equation, with exploration noise to ensure persistence of excitation. Simulations on a fully actuated 3D rigid body demonstrate accurate tracking and real-time parameter estimation, with code made publicly available. This work advances geometry-aware adaptive control by combining Lie-group structure with data-driven parameter reconstruction for rigid-body systems.

Abstract

Adaptive tracking control for rigid body dynamics is of critical importance in control and robotics, particularly for addressing uncertainties or variations in system model parameters. However, most existing adaptive control methods are designed for systems with states in vector spaces, often neglecting the manifold constraints inherent to robotic systems. In this work, we propose a novel Lie-algebra-based adaptive control method that leverages the intrinsic relationship between the special Euclidean group and its associated Lie algebra. By transforming the state space from the group manifold to a vector space, we derive a linear error dynamics model that decouples model parameters from the system state. This formulation enables the development of an adaptive optimal control method that is both geometrically consistent and computationally efficient. Extensive simulations demonstrate the effectiveness and efficiency of the proposed method. We have made our source code publicly available to the community to support further research and collaboration.

Lie-algebra Adaptive Tracking Control for Rigid Body Dynamics

TL;DR

The paper tackles robust trajectory tracking for rigid bodies on under parameter uncertainty. It introduces a Lie-algebra–based adaptive tracking method by mapping group dynamics to the Lie algebra to obtain a linear error model that decouples unknown parameters from the state. It develops a discrete-time Lie-algebra adaptive optimal control framework that reconstructs inertia and mass from data via regularized least squares and computes an optimal policy by solving a discrete-time Riccati equation, with exploration noise to ensure persistence of excitation. Simulations on a fully actuated 3D rigid body demonstrate accurate tracking and real-time parameter estimation, with code made publicly available. This work advances geometry-aware adaptive control by combining Lie-group structure with data-driven parameter reconstruction for rigid-body systems.

Abstract

Adaptive tracking control for rigid body dynamics is of critical importance in control and robotics, particularly for addressing uncertainties or variations in system model parameters. However, most existing adaptive control methods are designed for systems with states in vector spaces, often neglecting the manifold constraints inherent to robotic systems. In this work, we propose a novel Lie-algebra-based adaptive control method that leverages the intrinsic relationship between the special Euclidean group and its associated Lie algebra. By transforming the state space from the group manifold to a vector space, we derive a linear error dynamics model that decouples model parameters from the system state. This formulation enables the development of an adaptive optimal control method that is both geometrically consistent and computationally efficient. Extensive simulations demonstrate the effectiveness and efficiency of the proposed method. We have made our source code publicly available to the community to support further research and collaboration.

Paper Structure

This paper contains 10 sections, 2 theorems, 36 equations, 3 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Special Euclidean Group $SE(n)$
  4. Rigid Body Dynamics
  5. $SE(3)$ Trajectory Tracking
  6. Main Results
  7. Lie-algebra Error Dynamics
  8. Lie-algebra Adaptive Control
  9. Simulations
  10. Conclusion

Key Result

Lemma 1

Given a feasible reference trajectory $\{X_d(t), \zeta_d(t), u_d(t)\}$ for rigid body rigid-body-dynamics to track, the error dynamics is as follows: Proof: Taking derivative of the pose error in full-error-state, we have Similarly, the twist error dynamics is as follows: Combining the pose error and twist error together, the overall error dynamics model is as follows: The proof is completed.

Figures (3)

  • Figure 1: Trajectory tracking comparison between initial model parameters and adaptive model parameters. The initial position is $0.400^{\top}$, the initial orientation is $I$, the initial angular velocity and linear velocity are $000^{\top}$. The dataset size for adaptive control is 1500 in this example.
  • Figure 2: Evolution of model parameters in terms of dataset size $N$.
  • Figure 3: Evolution of computation time in terms of dataset size.

Theorems & Definitions (4)

  • Lemma 1
  • Proposition 1
  • Remark 1
  • Remark 2