A hybrid dynamical system approach to the impulsive control of spacecraft rendezvous (extended version)

Abstract

This paper introduces a hybrid dynamical system methodology for managing impulsive control in spacecraft rendezvous and proximity operations under the Hill-Clohessy-Wiltshire model. We address the control design problem by isolating the out-of-plane from the in-plane dynamics and present a feedback control law for each of them. This law is based on a Lyapunov function tailored to each of the dynamics, capable of addressing thruster saturation and also a minimum impulse bit. These Lyapunov functions were found by reformulating the system's dynamics into coordinates that more intuitively represent their physical behavior. The effectiveness of our control laws is then shown through numerical simulation. This is an extended version of an ECC24 article of the same name, which includes the proofs omitted for lack of space.

Figures (4)

• Figure 1: Local-Vertical, Local-Horizontal (LVLH) frame.
• Figure 2: Simulation of system \ref{['eq:closed_loop_z']} with $\tau_z^M=0.01$ (left) and $\tau_z^M=0.25$ (right). The figure shows the evolution of the state variables $r_z,v_z$ (top), the magnitude of the control input (middle) and the instants of impulses (bottom).
• Figure 3: Evolution in the original coordinates $r_x,v_x/n$ together with the transformed state $2\beta/3$ from \ref{['def:tildexy']} (top plot), as well as the original coordinates $r_y,v_y/n$ together with the transformed state $\alpha$ (second plot). Magnitude of the control input (third plot) and instants of impulses (bottom plot).
• Figure 4: Evolution of the state variables $x,y/n$ (top) i.e. in the transformed coordinates $\zeta$ of \ref{['def:tildexy']}, and the resulting trajectories of the original coordinates ($r_x,r_y)$ (bottom).

Theorems & Definitions (11)

• Remark 1
• Theorem 1
• proof
• Theorem 2
• proof
• Remark 2
• Remark 3
• Theorem 3
• proof
• Theorem 4