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This paper investigates a robust guaranteed cost tracking control problem for thrust-limited spacecraft rendezvous in near-circular orbits. Relative motion model is established based on the two-body problem with noncircularity of the target orbit described as a parameter uncertainty. A guaranteed cost tracking controller with input saturation is designed via a linear matrix inequality (LMI) method, and sufficient conditions for the existence of the robust tracking controller are derived, which is more concise and less conservative compared with the previous works. Numerical examples are provided for both time-invariant and time-variant reference signals to illustrate the effectiveness of the proposed control scheme when applied to the terminal rendezvous and other astronautic missions with scheduled states signal.

Autonomous rendezvous is a key operational technology in astronautic missions that involve more than one spacecraft, such as crew exchange, spacecraft assembly, maintenance, and monitoring. Generally speaking, a complete rendezvous mission can be divided into several specific phases: launch, phasing, far range rendezvous, close range rendezvous, docking, and departure [

Absolute navigation and relative navigation are the two major navigation methods used in a rendezvous mission. Varying with the distance between two spacecrafts, navigation methods for each phase are varied. For the launch and phasing phases where the spacecrafts are not able to detect each other, absolute navigation is in operation, while for the last four phases, as the distance between vehicles is short enough, it is common to switch to the relative navigation mode. According to the source of the navigation data, different types of controlled plants and control methods are required. The rest of this paper mainly focuses on the rendezvous problem based on the relative navigation method, and some works relying on absolute navigation are given in [

Researchers used to establish the rendezvous model on the basis of the two-body problem. Differed by the methods used in linearizing the equations of motion, rendezvous in circular, near-circular, and elliptical orbits are the three main branches that have been studied by precursors. Among these branches, rendezvous in circular and elliptical orbits seemed more attractive. Clohessy and Wiltshire [

Tracking control is often employed in realizing systems that are able to track reference signals. Gao and Chen [

In order to achieve some advanced aeronautic and astronautic tasks where trajectories and velocities are required specifically, such as obstruction and detection avoidances, tracking control is often applied on the vehicles to track the preplanned reference signals. Liao et al. [

Motivated by the above discussions, this paper designs a guaranteed cost tracking controller for thrust-limited rendezvous in near-circular orbits via a LMI method. Based on the works of Melton [

The remainder of this paper is organized as follows. Section

In this section, a relative motion model that describes the rendezvous process in near-circular target orbits is established based on the two-body problem. Then the rendezvous control problem is converted into a robust tracking control problem, and multiple requirements on the controller are raised.

Suppose that a target vehicle is on a near-circular orbit with a chase vehicle adjacent. It is assumed that these two spacecrafts are only influenced by a central gravitational source. To illustrate this rendezvous system, a Cartesian coordinate system is defined with the origin fixed at the centroid of the target vehicle, the

Cartesian coordinate system for spacecraft rendezvous.

When a chase vehicle experiences an additional perturbing force

In order to utilize linear control theory, model (

Let

According to the conversions between the orbit parameters and Kepler’s time equation

Adopting (

Assume that reference state vector

According to relative motion model (

Based on controlled plant (

The tracking error

The control inputs along each axis

During the rendezvous, the quadratic cost function defined in (

In this section, a robust guaranteed cost tracking controller with input saturation for autonomous rendezvous is designed via a LMI method, and a convex optimization problem for solving the controller is presented at the end of this section.

First of all, a lemma needed by the subsequent derivation is given. Proofs and applications for this lemma can be found in [

Given matrices

For each moment, the reference signal

Consider the closed-loop system (

Consider the Lyapunov function

When matrices

Consider the situation where reference signal

Substituting (

In order to design a controller with input saturation, squaring both sides of (

Based on Schur complements, inequalities (

From (

In [

In this section, two examples are presented to demonstrate the effectiveness and advantages of the control scheme presented above. The first example is for the situation in which the reference signal is time invariant, while the other example is for the situation where the reference signal is time variant, and both of the simulations are carried out in a two-body system.

Firstly, a simulated scene of spacecraft rendezvous is set. Consider a pair of adjacent spacecrafts. The target vehicle is circling on a low earth orbit with semimajor axis

Then, in both examples, the matrices in (

Suppose that the chase vehicle starts at a point which is 3000 m, −4000 m, and 20 m from the target vehicle along the

With the time-invariant reference signal

Relative distances between two spacecrafts along each axis.

Propulsive thrusts of the chase vehicle along each axis.

Rendezvous trajectory of the chase vehicle.

From Figures

In order to study the controller’s performance in tracking a time-variant reference signal, a straight-line rendezvous trajectory is planned in advance. The initial state vector is set to ^{2}, which is less than 0.25 m/s^{2}, the maximum acceleration that can be generated by the chase vehicle. Referring to the initial state

To demonstrate the maximum tolerant tracking error’s impacts on the rendezvous system, two groups of parameters are adopted and compared. For the precise group, we set the tolerant error to

With reference signals in (

Relative position and velocity between two spacecrafts along the

Propulsive thrusts of the chase vehicle along the

Figure

This paper has discussed a guaranteed cost tracking control problem for spacecraft rendezvous with an upper bound on thrust. A relative motion model with parameter uncertainty for rendezvous in near-circular orbits has been established. Via a LMI approach, an integrated, concise, and less conservative control scheme has been proposed. Then the controller has been demonstrated by two numerical examples with time-invariant and time-variant reference signals. The results show that our control scheme is effective for the terminal phase of rendezvous with all purposed requirements met, and, due to its strong ability in tracking reference signal, this control method should also be valid for flying by, departure, and other potential missions with planned path.

The authors declare that there is no conflict of interests regarding the publication of this paper.

This work is partially supported by the National Natural Science Foundation of China (Grant no. 61104101), the China Postdoctoral Science Foundation Funded Project (Grant no. 2011M500058), the Special Chinese National Postdoctoral Science Foundation (Grant no. 2012T50356), the Heilongjiang Postdoctoral Fund (Grant no. LBH-Z11144), the open fund of national defense key discipline laboratory of Micro-Spacecraft Technology (Grant no. HIT.KLOF.MST.2012006), and Research Fund for the Doctoral Program of Higher Education of China (Grant no. 20112302120011), and we wish to thank Robert G. Melton at Pennsylvania State University for his suggestion on this paper.