Distributed optimal control allocation for 6-dof spacecraft with redundant thrusters

Abstract

Control allocation for 6-dof spacecraft maneuvers is considered. Retired spacecraft, whose own control system is inoperable, can be reused by installing multiple cellular thruster modules to provide thrusts for translational and rotational control. Such cellular thruster modules collaborate with their neighbors in a network topology to calculate an optimal distribution matrix for thrust allocation. A pseudo-inverse optimal thrust allocation scheme is extended to a distributed iteration algorithm to enable this optimal problem to be solved by cellular thruster modules in a distributed manner. Subsequently, a norm-based distributed pseudo-inverse optimal thrust allocation scheme is developed to solve the same optimal thrust allocation problem. The syntheses of two distributed optimal thrust allocation schemes are discussed. Numerical simulations are demonstrated in the end to show the effectiveness of two proposed distributed thrust allocation schemes. Performance comparisons between two schemes are analyzed as well.

Introduction

Maintenance of retired spacecraft has attracted more research on on-orbit servicing technology. Such retired spacecraft usually have some still functional components and are valuable to be reused. To proceed the maintenance process, the primary stage is that using servicing space robots to approximate and capture the retired spacecraft [1]. This stage has high requirements for the control performance of servicing space robots. It not only needs dedicated design of mechanisms [2], but also requires a powerful control system in space robots, particularly when the retired spacecraft has already been tumbling [3]. To facilitate it, the concept of cellular space modules has been put forward for the reuse of retired spacecraft in some space missions, such as “Phoenix project” [4]. This project planned to employ multiple cellularized space modules (or called “Satlet”) aggregated to a single retired spacecraft to fulfill the functions which were completed by subsystems in conventional monolithic spacecraft [4]. This cellularization idea was inspired by biological cells and intended to degenerate the conventional monolithic spacecraft to decentralized small modules. These modules can communicate and interact with each other through wireless links and aggregate a distributed network system to reuse the retired spacecraft. The iBOSS (Intelligent Building Blocks for On-Orbit Satellite Servicing) project proposed “plug-and-play” interfaces for the transmission of power, data and thermal among cellular space modules [5].

Rendezvous and proximity operations play dominant roles in some on-orbit servicing missions, where the translation and rotation of spacecraft as well as their mutual coupling effects should be considered concurrently. In traditional space research, spacecraft translation and rotation were studied separately and their couplings were usually omitted in dynamic modeling and controller design, which would bring about poor precision and efficiency issues in control performance [6]. To accommodate such issues, the coupled 6-dof relative motion dynamics are important to be analyzed to establish powerful control scheme for the stability of spacecraft [7], [8], [9], [10], [11], [12]. The 6-dof dynamic model can be simplified as the combination of classical Euler's equations and relative motion equations (like Hill's equations) to design control algorithm in complicated fashions [13], [14], [15]. As the extension of traditional quaternions, dual quaternions had been widely used to describe 6-dof spacecraft motion. The dual-quaternion-based sliding mode control for an integrated 6-dof spacecraft motion was discussed in [16]. Using the dual-quaternion formulation, various kinds of control algorithms have been exploited, such as adaptive control with inertia identification [17], and finite-time output feedback control [18]. The way to design dual-quaternion-based controller could always be inspired by quaternion-based controllers because of the similar mathematical properties. The unwinding problem of quaternions, however, is also inherited by the dual quaternions [19]. The special Euclidean group, or SE(3), is an alternative to dual quaternions to model the 6-dof dynamics [20]. An observer-based robust adaptive control scheme was proposed in [21] using SE(3) formulation with the consideration of actuator saturation and misalignment.

Thrusters are quite common actuators for spacecraft steering. Compared with other types of actuator, such as reaction wheels, all-thruster actuation systems are simple to be implemented in 6-dof motion control [22], [23]. In particular, when thrust vectors produced by propulsion systems do not cross through the mass center of spacecraft, torques can be generated together with thrusts and their corresponding moment arms [24]. This causes the dynamic coupling between translation and rotation of spacecraft [13]. It is obvious that at least six thrusters are needed to accomplish 6-dof spacecraft rendezvous and proximity operations. However, thruster redundancy is a quite common practice for spacecraft maneuvering, particularly for large-mass monolithic spacecraft. Using a redundant thruster configuration needs a control allocation algorithm to apportion the desired forces and torques into multiple thrusters to produce desired control commands. The control allocation problem has been investigated substantially in [25]. There are fruitful results about control allocation, such as pseudo-inverse allocation [26], dynamic control allocation [27], [28], [29], programming control allocation [30], [31], etc. For a 6-dof spacecraft with thruster-only actuation, a Lyapunov-based method was proposed in [13] for thrust allocation and thruster selection. An optimal combination table was proposed in [8] to choose specific thrusters and calculate the firing time intervals based on linear programming. A simplex method has been implemented to have an optimal thruster management function [32]. A pre-computing distributed matrix for thruster control allocation was formulated in [33], [34] using min-max optimization. This method is a fusion of the selection matrix method [8] and the process of Linear Programming [35], which only needs to calculate a single constant distribution matrix. Based on the concept of cellularized space robots, Chang et al. [36] proposed a control allocation algorithm for the cellular space modules which contain corresponding actuators to produce torques for 3-dof attitude takeover control in a distributed manner. The scheme proposed in [36] was actually inspired by the idea on the estimation of the truth values over sensor networks [37], which has been exploited extensively in distributed computing of multi-agent systems. This kind of multi-agent-based algorithm can also be used to solve the distributed optimization problems for resource allocation in a decentralized manner [38]. Fault-tolerant attitude tracking control problem for over-actuated spacecraft was addressed in Ref. [39] when the actuator failure happens, and the problem was tackled with an on-line control allocation scheme and a novel fault-tolerant control algorithm. A robust control allocation was proposed in Ref. [40] to deal with spacecraft attitude tracking control when an actuator has fault but fault detection and diagnosis scheme is imprecise. An application of direct force control was presented with a novel closed-loop control allocation method for force and moment increment distribution [41]. A control allocation scheme with a fuzzy controller was presented in Ref. [42] to solve the attitude control problem in a Cubesat-based space debris removal mission which is an over-actuated space system. A quadratic-programming-based control allocation was used by Ref. [43] to generate control commands for thrusters from the control signals calculated by a novel dynamic-programming-based optimal method. In Ref. [44], the authors proposed a dynamic near-optimal control allocator augmented with penalty terms for spacecraft control. However, all of the above 6-dof control allocation algorithms are centralized approaches. When the spacecraft is equipped with multiple redundant cellular space modules for motion control, a scheme is needed for control allocation in a distributed pattern. The distributed control allocation scheme for spacecraft 6-dof translation and rotation control using cellular thruster modules is still not available. The dynamic coupling of forces and torques induced by thruster-only actuation configuration poses a challenge to obtain the optimal thrust allocation matrix in a distributed way. To tackle this challenge, this paper develops the distributed optimal thrust allocation algorithm for 6-dof spacecraft maneuvers with multiple cellular thruster modules which consist of thrusters and communication subsystems. Each cellular thruster module could interact with its neighbors within a certain network topology to yield an optimal thrust allocation matrix using the proposed distributed algorithm.

The rest of this work is structured as follows. Some preliminaries about spacecraft 6-dof dynamics using SE(3) are reviewed in Section 2. Section 3 is the main result of this paper. In Section 3.2, a pseudo-inverse optimal thrust allocation is implemented in a distributed way which enables all cellular thruster modules to calculate an optimal thrust allocation matrix jointly. Based on this scheme, a norm-based distributed pseudo-inverse optimal thrust allocation scheme is developed in Section 3.3. In Section 4, numerical simulations are conducted to show the effectiveness of the two proposed distributed thrust allocation algorithms. Moreover, comparisons between the two schemes are also carried out to manifest the advantages of the norm-based scheme. Conclusions are given in Section 5.