Adaptive saturated control for spacecraft rendezvous and docking under motion constraints

Abstract

This paper investigates the problem of position tracking and attitude synchronization for spacecraft rendezvous and docking with a tumbling target, in which kinematic couplings, unknown disturbances, motion constrains, and input saturation are considered. A novel artificial potential function (APF) is developed to encode both three-dimensional pose information and two-dimensional image information regarding path constraint and field of view constraint, respectively. By integrating the APF into the sliding mode technique, an adaptive control strategy is proposed to realize the arrival of the chaser at the docking port without any risk of collision or loss of target features. In addition, a linear anti-windup compensator is introduced to compensate for the input saturation effect. Uniformly ultimately bounded of the closed-loop system is guaranteed through Lyapunov analysis. A specific simulation scenario of docking with the defunct satellite Envisat is created to demonstrate the effectiveness and robustness of the proposed control strategy.

Introduction

Autonomous rendezvous and docking (ARD) is an important research direction in current aerospace technology. It has been widely used in on-orbit servicing, such as satellite salvage, repair, and refueling [1], [2], and debris removal [3], [4]. In order to accomplish the final attachment to the target, the chasing spacecraft needs to not only approach to but also synchronize with it, especially when the target is uncontrollable and tumbling, such as defunct satellite and space debris. In the process of rendezvous and synchronization, the relative translational and rotational motion between the chaser and target are highly nonlinear and coupled. Therefore, an integrated 6-degree-of-freedom (6-DOF) control is required to realize high-precision ARD, i.e., the spacecraft position and attitude should be considered concurrently in the controller design. For relative orbital motion, Clohessy and Wiltshire (C-W) presented a simplified model, assuming that the target is moving in a circular orbit [5], [6]. The C-W equations were generalized to elliptic orbits by Tschauner and Hempel (T-H) [7]. The C-W and T-H models are established with respect to the local-vertical-local-horizontal frame, which involves the target's orbital information. In addition, some researchers establish the dynamic model of translational motion in the target's body frame [8], [9], and line of sight frame [10], [11]. The relative attitude motion is typically described by quaternions [12], [13], rotation matrix [14], [15], and modified Rodrigues parameters [16], [17]. However, the spacecraft is considered to be a mass point in studies [8], [9], [10], [11], [12], [13], [14], [15], and the natural coupling characteristics of the spacecraft rotation-translation motion is neglected. In essence, the spacecraft relative motion is modeled and controlled separately. It is reported in [18] that the kinematic coupling effect of the rotation around the center of mass (COM) is significant in high-fidelity relative motion modeling. In [19], a coupled dynamic model is derived to represent the relative motion of docking ports on the chasing and target spacecraft, then a super twisting algorithm is proposed to realize the relative position tracking and attitude synchronization. Sun et al. [16], [17] established a novel mechanical model, describing the relative translational motion of the chasing spacecraft with respect to a fixed point along the direction of the target's docking port. In [16], a norm-estimation-based adaptive method is proposed to deal with the thrust misalignment, parametric uncertainties, external disturbance, and dynamic couplings. Then, the problem is further studied by introducing relative state constraints, actuator faults, and input saturation in [17].

In the final phase of rendezvous and docking, a safe approaching trajectory is the prerequisite to avoid accidental collisions with the target. Effective methods concerning the path constraint are largely available in the literature, one of which is based on optimal control theory. In [20], an adaptive Gaussian Pseudo-Spectral method is adopted to generate an optimal trajectory for safety in the ultra-close proximity mission. In [21], a sub-optimal control scheme is proposed for collision-free maneuvers of multiple spacecraft. In addition, model predictive control (MPC) is commonly used to deal with multi-constrained, multi-variable problems. In [22] and [23], different kinds of MPC strategies are investigated for respective spacecraft rendezvous missions under multiple engineering constraints. In recent years, the artificial potential function (APF) method has gained increased attention, because its clear physical meaning makes it convenient to interpret the motion constraints. One of the most commonly used path constraints is the final approach corridor, which can be defined according to different curves, such as cone [24], elliptic cissoid [25], and semicubical parabola [26]. Another type of path constraint is described by a forbidden zone surrounding the target. In [27], an artificial potential field is established by employing a cardioid-based curve to meet the collision-avoidance requirement. In [28], a bullet-like constrained zone and a constrained zone composed of multiple curves are designed to ensure flight safety. Although the above mentioned studies considered the path constraint in ARD, they regarded the target spacecraft as a mass point and neglected the kinematic coupling effect caused by target's tumbling.

On the other hand, spacecraft rendezvous and docking requires continuous and accurate information of the relative position and attitude, which is mainly obtained by vision-based measurement. Therefore, the spacecraft motion must be constrained to keep the target pattern within the field of view (FOV) of optical sensors (e.g., CCD camera). This problem is addressed in [29], [30], [31], [32], where the FOV constraint is defined by utilizing the camera's boresight and a line-of-sight angle. However, this kind of constraint is not strict and cannot cover all of the feature elements on the target pattern. The relative navigation may fail as a result of losing some critical visual features. In this paper, we design an image-based field of view (IFOV) constraint, which defines the potential field in the two-dimensional (2D) image space based on each feature element. The original idea of introducing image information into the control system comes from the image-based visual servoing in the robotic area [33]. Image-based path planning strategies using potential field can be found in [34], [35]. Compared with the velocity-based control of the robotic manipulator, the spacecraft control with the consideration of nonlinear relative dynamics is more challenging.

Although the aforementioned APF methods have been extensively studied and proven to be effective in spacecraft rendezvous and docking under different motion constraints, the input saturation problem is not taken into account. However, this problem usually occurs in practical control systems, especially in the vicinity of motion constraints, where the APF method generates an unexpected excessive control effort, which may exceed the actuator limits and lead to instability of the system. Therefore, the input saturation effect should be considered simultaneously in the controller design. In [36], a linear anti-windup compensator system is introduced to deal with the input saturation problem in spacecraft proximity maneuvers. This kind of first-order linear auxiliary system is simple but effective to eliminate the input saturation. In [37], a finite-time controller combining input saturation and safe constraint is proposed for spacecraft rendezvous and docking. However, the proposed compensator may lead to a singularity problem. In [38], Guo et al. developed an integral sliding mode surface by using the saturation function. The outputs of the controller are proven to be bounded and will not exceed the actuator limits. These studies did not systematically take into account motion constrains, especially the visibility of the target features.

The spacecraft autonomous rendezvous and docking is essentially a nonlinear system subjected to kinematic couplings, unknown disturbances, and multiple constrains, including path constraint, IFOV constraint, and input saturation. The main contributions of this paper are summarized as follows: (i) Motivated by the idea of image-based path planning in [34], we propose an IFOV constraint to ensure the visibility of the target and introduce image features in the controller design. A potential field is developed using pixel coordinates of the image features to interpret the IFOV constraint. (ii) An APF-based sliding-mode controller is designed subjected to both 3D Cartesian space and 2D image space constraints. The interaction between different coordinate frames is described by Jacobian matrices. (iii) Compared with the APF methods in [24], [25], [26], [27], [28], [29], [30], [31], [32], the kinematic coupling of the off-COM point (i.e., docking port) is explicitly considered in the controller design, and the input saturation effect is handled by a linear anti-windup compensator. It is proven that the ARD mission can be accomplished, while restraining the spacecraft in the pre-analyzed approach corridor and keeping the target pattern within the FOV of the camera.

The rest of the paper is organized as follows. Section 2 presents the definition of coordinate frames and mathematical description of the coupled dynamic model during the spacecraft rendezvous and docking mission. Then, we detailed describe the motion constraints and input saturation considered in this work. Section 3 provides the APF-based controller design and rigorous analysis of the closed-loop system stability. Numerical simulations of the spacecraft behavior, during the relative position tracking and attitude synchronization under multiple constraints, are conducted in Section 4. Finally, a concluding discussion is presented in Section 5.