Time-Constrained Model Predictive Control for Autonomous Satellite Rendezvous, Proximity Operations, and Docking

TL;DR

The paper addresses autonomous rendezvous, proximity operations, and docking (ARPOD) between a deputy and an uncontrolled chief under strict onboard computation limits. It proposes a time-constrained Model Predictive Control approach that solves a finite-horizon optimization with a cap on iterations, enabling real-time execution on a space-grade processor. The 6 DOF ARPOD model combines translational Clohessy-Wiltshire dynamics with coupled attitudinal dynamics, using a quaternion-based attitude representation and error definitions to drive docking. Contributions include a full 6 DOF ARPOD model under time-constrained MPC, hardware-in-the-loop validation on a space-grade processor, and empirical docking demonstrations under timing constraints. Results indicate successful docking in simulations and favorable timing performance, validating the practicality of onboard, nonlinear MPC for autonomous spacecraft rendezvous in constrained compute environments.

Abstract

This paper presents a time-constrained model predictive control strategy for the six degree-of-freedom autonomous rendezvous, proximity, operations and docking problem between a controllable "deputy" satellite and an uncontrolled "chief" satellite. The objective is to achieve a docking configuration defined by both the translational and attitudinal states of the deputy relative to the chief, whose dynamics are respectively governed by both the Clohessy-Wiltshire equations and Euler's second law of motion. The proposed control strategy explicitly addresses computational time constraints that are common to state-of-the-art space vehicles. Thus, a time-constrained model predictive control strategy is implemented on a space-grade processor. Although suboptimal with regards to energy consumption when compared to conventional optimal RPO trajectories, it is empirically demonstrated via numerical simulations that the deputy spacecraft still achieves a successful docking configuration while subject to computational time constraints.

Figures (20)

• Figure 1: The chief (with rotating orbit-fixed frame $\mathcal{O}$) and a deputy (with body-fixed frame $\mathcal{D}$) are orbiting about the Earth with inertial frame $\mathcal{E}$. The dashed lines are the closed orbital trajectories of both spacecraft. The red solid line depicts the rendezvous trajectory from the deputy to the chief as seen in the inertial frame.
• Figure 2: Open loop optimal control strategies compute the control sequence $u$ offline prior to the mission by solving an Optimal Control Problem which takes into account the initial state $x_0$, system dynamics $f_d$, and other constraints such as input limits $\bar{u},\underline{u}$. The control sequence is applied to the Dynamic System where $u(t) = u(i)$ for all $t \in [i\Delta,i\Delta+\Delta]$, for all $i=0,\dots,N-1$, and $\Delta>0$ is the sampling time which results in the state sequence $\pmb{x}$. Open loop control strategies may not be able to account for perturbations, such as atmospheric drag in the ARPOD problem, since the Optimal Control Problem is solved once before the mission and assumes perfect state information.
• Figure 3: Closed loop feedback control laws utilize state feedback from the Dynamic System to drive the state of the system to a desired state, for example the docking configuration in the ARPOD Problem. In this figure, the state $x(t)$ is fed back to the Static Control Law $u$ along with the Reference Trajectory $r(t)$ to compute the next control input. Typically, the static control law is designed to ensure stability of the system and does not consider a performance index for optimality or actuator limits.
• Figure 4: At each time step $k$ the state of the Dynamic System $x_0$ is provided to the MPC Controller. Then, the MPC controller computes the optimal control sequence $u^*(k)$. Then, the first element of the sequence is applied to the Dynamic System where $u(t) = u^*(k)$ for all $t \in [k\Delta ,k \Delta + \Delta]$ and $\Delta > 0$ is the sampling time. This process is repeated until the prediction horizon $N$ is met. Conventional MPC can account for nonlinear dynamics, input limits, and perturbations since the optimal control problem is solved at each time step $k$. However, it can be computationally expensive and not able to be solved at a fast enough timescale to be implemented online. Thus, computational time constraints should be addressed in the proposed control strategy for the ARPOD problem.
• Figure 5: At each time step $k$ the state of the Dynamic System $x_0$ is provided to the TC-MPC Controller. Then, the TC-MPC controller computes a number of algorithmic iterations $j_k$ to compute the possibly sub-optimal control sequence $\tilde{u}(k)$. Then, the first element of the sequence is applied to the Dynamic System where $u(t) = \tilde{u}(k)$ for all $t \in [k\Delta ,k \Delta + \Delta]$ and $\Delta > 0$ is the sampling time.. This process is repeated until the prediction horizon or a stopping condition is met.