Semi-global Exponential Stability for Dual Quaternion Based Rigid-Body Tracking Control

TL;DR

The paper tackles joint 6-DOF rigid-body tracking using dual quaternions and proves Semi-Global Exponential Stability (SGES) for attitude and position errors under a novel nonlinear feedback law. It derives a Lyapunov-based SGES result with explicit bounds that hold without restrictive gain assumptions and shows local ISS robustness to time-varying disturbances. A Control Barrier Function (CBF) framework is integrated to enforce safety and motion constraints, yielding safe input updates via a quadratic program. The approach is validated across realistic aerospace scenarios, including MarCO, Apollo transposition/docking, Starship maneuvers, Martian-rock collision avoidance, and Dragon 2 docking with the ISS, highlighting benefits in convergence speed, fuel efficiency, and safety guarantees. Overall, the work advances rigorous stability and safety guarantees for 6-DOF spacecraft control using dual-quaternion representations, with practical implications for autonomous rendezvous and docking operations.

Abstract

Semi-Global Exponential Stability (SGES) is proved for the combined attitude and position rigid body motion tracking problem, which was previously only known to be asymptotically stable. Dual quaternions are used to jointly represent the rotational and translation tracking error dynamics of the rigid body. A novel nonlinear feedback tracking controller is proposed and a Lyapunov based analysis is provided to prove the semi-global exponential stability of the closed-loop dynamics. Our analysis does not place any restrictions on the reference trajectory or the feedback gains. This stronger SGES result aids in further analyzing the robustness of the rigid body system by establishing Input-to-State Stability (ISS) in the presence of time-varying additive and bounded external disturbances. Motivated by the fact that in many aerospace applications, stringent adherence to safety constraints such as approach path and input constraints is critical for overall mission success, we present a framework for safe control of spacecraft that combines the proposed feedback controller with Control Barrier Functions. Numerical simulations are provided to verify the SGES and ISS results and also showcase the efficacy of the proposed nonlinear feedback controller in several non-trivial scenarios including the Mars Cube One (MarCO) mission, Apollo transposition and docking problem, Starship flip maneuver, collision avoidance of spherical robots, and the rendezvous of SpaceX Dragon 2 with the International Space Station.

Figures (12)

• Figure 1: The figure illustrates the main steps involved in proving Theorem \ref{['thm:es']}. We begin by choosing a candidate Lyapunov function $V$, as demonstrated in Fig. (1a), and select a suitably large independent parameter $c$. This selection ensures that the function $V$ will be radially unbounded, i.e., $V\in\mathcal{KR}$. Next, we leverage Lemma \ref{['lemma:inequality_new']} in conjunction with dual quaternion algebra to derive an upper limit for $\dot{N}$ and $S$ (Fig. (1b)). With the established condition that $S<0$, we then determine an independent parameter $0<\beta(R)< 1$, which is used to establish an upper boundary for $\dot{V}$, as depicted in Fig. (1c). In our final step, as illustrated in Fig. (1d), we use the previously obtained upper bounds for $\dot{N}$ and $S$ to derive an upper limit for $\dot{V}$. This, in turn, permits us to establish the semi-global exponential stability, thus completing the proof of Theorem \ref{['thm:es']}.
• Figure 2: Schematic diagram of trajectories under the proposed feedback controller \ref{['eqn:feedback_control_law']} converging to a ball of radius equal or at most $\psi d_m$
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Theorems & Definitions (7)

• Remark 1
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• Remark 2