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Log-linear Dynamic Inversion for Thrusting Spacecraft on SE2(3)

Micah K. Condie, Abigaile E. Woodbury, Li-Yu Lin, Kartik A. Pant, Mike Walker, James Goppert

TL;DR

Powered spacecraft proximity operations are traditionally modeled via local linearizations that ignore Lie-group structure. The paper embeds disturbance-free dynamics in SE2(3) as a group-affine system, yielding exact log-linear tracking-error dynamics in Lie algebra, enabling direct use of linear tools for global analysis. A simple numerical example confirms that nonlinear SE2(3) error trajectories coincide with the linear Lie-algebra predictions, validating the approach. This foundational structure paves the way for invariant controllers, observer design, and convex safety certification in docking and rendezvous, with future work addressing disturbances and robustness through LMIs and related methods.

Abstract

We show that the dynamics of a thrusting spacecraft can be embedded in the Lie group SE2(3) in a form that is group-affine with application of a feed-forward control law. This structure implies that the configuration-tracking error evolves exactly linearly in the associated Lie algebra coordinates (log-linear dynamics), rather than arising from a local linearization of the nonlinear system. As a result, a broad class of linear analysis and synthesis tools becomes directly applicable to powered spacecraft motion on SE2(3). A simple numerical example confirms that the error predicted by the linear Lie-algebra dynamics matches the error computed from the full nonlinear system, illustrating the exact log-linear behavior. This foundational property opens a path toward rigorous tools for satellite docking, autonomous rendezvous and proximity operations, robust controller design, and convex safety certification-capabilities that are difficult to achieve with classical local linearizations such as Tschauner-Hempel/Yamanaka-Ankersen (TH/YA).

Log-linear Dynamic Inversion for Thrusting Spacecraft on SE2(3)

TL;DR

Powered spacecraft proximity operations are traditionally modeled via local linearizations that ignore Lie-group structure. The paper embeds disturbance-free dynamics in SE2(3) as a group-affine system, yielding exact log-linear tracking-error dynamics in Lie algebra, enabling direct use of linear tools for global analysis. A simple numerical example confirms that nonlinear SE2(3) error trajectories coincide with the linear Lie-algebra predictions, validating the approach. This foundational structure paves the way for invariant controllers, observer design, and convex safety certification in docking and rendezvous, with future work addressing disturbances and robustness through LMIs and related methods.

Abstract

We show that the dynamics of a thrusting spacecraft can be embedded in the Lie group SE2(3) in a form that is group-affine with application of a feed-forward control law. This structure implies that the configuration-tracking error evolves exactly linearly in the associated Lie algebra coordinates (log-linear dynamics), rather than arising from a local linearization of the nonlinear system. As a result, a broad class of linear analysis and synthesis tools becomes directly applicable to powered spacecraft motion on SE2(3). A simple numerical example confirms that the error predicted by the linear Lie-algebra dynamics matches the error computed from the full nonlinear system, illustrating the exact log-linear behavior. This foundational property opens a path toward rigorous tools for satellite docking, autonomous rendezvous and proximity operations, robust controller design, and convex safety certification-capabilities that are difficult to achieve with classical local linearizations such as Tschauner-Hempel/Yamanaka-Ankersen (TH/YA).

Paper Structure

This paper contains 10 sections, 1 theorem, 42 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Background and Problem Statement
  3. SE$_2(3)$ Lie Group Representation
  4. Problem Statement
  5. Mixed-Invariant Dynamics and Error Formulation
  6. Equations of Motion of Spacecraft as Mixed-Invariant System
  7. Other notation
  8. Simulation Results
  9. Conclusion
  10. Normalizer of the $C$ Matrix

Key Result

Theorem III.1

For the mixed-invariant spacecraft dynamics on $\mathrm{SE}_2(3)$, the left-invariant log–error $\xi = \operatorname{Log}(\bar{X}^{-1}X)^\vee$ satisfies Choosing the feedforward input which further simplifies to, results in $\tilde{m}$ dropping from the error dynamics, yielding, The log–error dynamics are then exactly linear in the Lie algebra. If the reference acceleration and rotation term $

Figures (2)

  • Figure 1: Validation of exact log-linear error dynamics for a thrusting spacecraft in a near-geostationary orbit. The reference and actual trajectories are propagated for $500\,\text{s}$ under a constant body-frame acceleration of $10~\text{m/s}^2$ along the body $x$–axis and a constant body-rate of $0.01~\text{rad/s}$ about the same axis. The plots show the components of the right-invariant Lie-algebra error $\xi = [\xi_p^\top,\xi_v^\top,\xi_R^\top]^\top$: solid lines are obtained by mapping the nonlinear SE$_2(3)$ error through the logarithm, and dashed lines are obtained by integrating the linear system $\dot{\xi} = (-\operatorname{ad}_{\bar{n}} + A_C)\xi$. The near-perfect overlap demonstrates that the tracking error evolves exactly linearly in the Lie-algebra coordinates.
  • Figure 2: Residual between the nonlinear log-error and the solution of the exact log-linear system. The error $e_\xi(t) = \xi_{\text{nl}}(t) - \xi(t)$ is computed componentwise for all nine Lie-algebra coordinates during the $500\,\text{s}$ maneuver. The residual remains at machine-level numerical precision, demonstrating that the nonlinear SE$_2(3)$ tracking error and the log-linear error dynamics coincide exactly, consistent with the group-affine structure of the spacecraft dynamics.

Theorems & Definitions (5)

  • Remark 1: Coordinate Ordering and Physical Meaning
  • Theorem III.1: Log-Linear Error Dynamics
  • proof
  • Remark 2: Structure of the Log-Linearized Dynamics
  • Remark 3: Role of the $C$ matrix