Solving Optimal Continuous Thrust Rendezvous Problems With Generating Functions

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Near-optimal continuous control for spacecraft collision avoidance maneuvers via generating functions

Abstract

This paper presents a suboptimal continuous control algorithm that would enable an active spacecraft to avoid collision with inactive space objects. To prevent collision, a penalty function whose value soars as a spacecraft is about to collide with other objects is inserted into the cost functional of an optimal control problem. Then, a two-point boundary value problem for a Hamiltonian system is constructed, and is solved with the generating functions. This algorithm can be coded step-by-step to obtain suboptimal feedback continuous control laws as truncated power series with no initial guess or iteration. This advantage over direct optimizations, however, requires moderate efforts to develop higher-order generating functions and update the penalty function parameters heuristically. In the illustrative examples, the above process allows an active satellite to smoothly circumvent other space objects or forbidden regions. The overall process is also useful for obtaining an appropriate initial guess for the direct optimization approaches in case of numerically sensitive problems because of the definiteness in its solution procedure.

Introduction

The process of trajectory design and maneuvering for preventing collision with other space objects is becoming increasingly important in space missions because of the growing amount of space debris and number of spacecraft in orbit. In addition, because the concept of operating multiple small spacecraft in close proximity has drawn more attention as a result of its efficiency and flexibility, researchers are actively studying the development of a collision avoidance algorithm. Such an algorithm is essential for operating multiple spacecraft because the collision of multiple spacecraft can cause irreversible damage to the entire mission, which can lead to the possibility of mission failure.

Algorithms developed for an active spacecraft to avoid collision with other objects often adapt direct optimizations. Richards et al. [1] introduced a method to obtain optimal trajectories that ensure collision avoidance by employing mixed-integer linear programming. Scharf et al. [2] developed a reactive collision avoidance algorithm based on model predictive control and direct optimization methods. Sultan et al. [3] developed a gradient-based algorithm to design suboptimal collision-free trajectories for the reconfiguration of multiple spacecraft on a large scale. Wu et al. [4] provided an optimal open-loop solution for collision avoidance maneuvers by applying the Legendre pseudospectral optimization method. These studies can assign the so-called forbidden regions around obstacles explicitly as inequality constraints, but the solution procedure generally requires an initial guess and iterative process.

While the above mentioned works mainly applied the inequality constraint explicitly, others have also implemented the inequality constraint implicitly. Slater et al. [5] studied an impulsive control strategy to reduce the collision probability while minimizing fuel expenditure. Since Leitmann and Skowronski [6] presented that the avoidance strategy using Lyapunov-type function guarantees a collision-free motion, the collision avoidance algorithms using the gradient of Lyapunov-type function have been studied steadily [7], [8], [9]. These techniques were extended to the path planning algorithms to realize autonomous multiple spacecraft assembly without collision by using an artificial potential function [10], [11], [12]. Although the approaches using Lyapunov-type function could implement collision avoidance simply and robustly, the optimality of fuel expenditure was not considered in the control/guidance law. Umehara and McInnes [13] and Epenoy [14] obtained suboptimal collision-free trajectories by using the penalty function and indirect shooting method. They formulated inequality-constrained problems for collision avoidance maneuvers as unconstrained problems by introducing a penalty function into the performance index of fuel consumption. However, the indirect method still needs an initial guess and iteration like the direct optimizations.

This study presents another approach to developing a suboptimal collision-free trajectory and continuous feedback control law by extending the study conducted by Lee et al. [15]. As in the case of the studies by Umehara and McInnes [13] and Epenoy [14], a penalty function is augmented into the performance index to indirectly designate the forbidden region around prescribed trajectories of free space objects. Then, the collision avoidance problem is constructed as an optimal feedback control (OFC) problem. Then, the algorithm based on generating functions provides collision-free trajectories and series-based continuous control laws in feedback form [16], [17]. The overall procedure does not involve any guesses or iteration for initial adjoints unlike typical gradient-based shooting methods, and is not affected by the complexity of the dynamics or trajectories of the space objects to be detoured. Based on this definitive solution procedure, our approach is combined with a direct optimization method by using the collision-free solution obtained from our approach as an initial guess for the direct optimization method. These advantages and practicality are obtained at the expense of moderate efforts to develop an appropriate generating function and to empirically update the design parameters of the penalty function.

The overall discussion in the rest of the paper begins with the formulation of an optimal collision-free control problem with penalty function. Then, the generating functions are discussed briefly as a main tool of solving two-point-boundary-value problem for a Hamiltonian system. This approach can provide collision-free trajectories and yield a series-based OFC law. The solutions from the proposed method are validated, and are compared with those obtained using a direct optimization method. Additional analysis on the inherent singularity in our approach and its circumvention is also presented. Finally, conclusions are drawn.

Section snippets

Formulation of fuel-optimal collision avoidance problem

Consider a typical fuel-optimal control problem:

Problem 1 Fuel-optimal trajectory and continuous control

Minimize the quadratic performance index $J=J_{\mathit{fuel}}=\frac{1}{2}\underset{t_0}{\overset{t_f}{\int }}{\mathit{u}}^T(t)\mathit{u}(t)dt$ subject to general nonlinear equations of motion in an affine form with rendezvous-type boundary conditions $\dot{\mathit{x}}(t)=\mathit{f}(\mathit{x}(t),t)+\mathit{g}(\mathit{x}(t),t)\mathit{u}(t),\mathit{x}(t_0)={\mathit{x}}_0,\mathit{x}(t_f)={\mathit{x}}_f$ where x and u are the state and control vectors of an active spacecraft, respectively; t is a general time index; and $t_0$ and $t_f$ are the initial and final epochs, respectively.

According to the appropriate rocket equation, the

Solutions using generating functions

The solution procedure of the generating function approach for Problem 2 is started by constructing a two-point boundary value problem (TPBVP) for a Hamiltonian system [21]. Then, the generating functions interpret the TPBVP as a canonical transformation between two Hamiltonian systems and provide the adjoint as a function of states (refer to [16], [17], [22], [23] for details).

The canonical transformation can be defined by the following principal kinds of generating functions: $\begin{array}{cc}\hfill F_1& =F_1(\mathit{x},{\mathit{x}}_f,t),F_2=F_{}\hfill \end{array}$

Collision avoidance maneuvers in proximity operation

To demonstrate the proposed approach, optimal collision-free transfer problems are solved by our approach for a spacecraft subject to the nonlinear equations of relative motion with respect to a nominal circular orbit of the Earth's central gravitational field. Fig. 1 shows the direction of the axes of the earth-centered inertial (ECI) frame and relative motion frame. Eq. (12) represents the nonlinear equations of relative motion normalized by the orbital radius and the frequency of the chief

Combination with direct optimization method

Although the proposed approach has the advantage that the solution procedure is related to neither an initial guess nor an iterative process, it has a limitation in that it is a suboptimal approach, because of the indirectly/empirically assigned forbidden region. In contrast, the direct optimization method is an optimal approach that can assign the forbidden region directly/exactly, but choosing an appropriate initial guess might be crucial to obtain the best (or even a feasible) solution as a

Singularity analysis

In the procedure for obtaining the generating function, the technical issue of singularity needs to be resolved. When ${\mathit{r}}_O$ is equivalent to the nominal value of the Taylor expansion of the Hamiltonian, a singularity occurs because the coefficients in the Taylor series of the penalty function defined in Eq. (4) approach infinity. However, this mathematical singularity can be simply circumvented by defining an alternative penalty function $P^{⁎}$ with ${\mathit{r}}_O^{⁎}$ , which is assigned relatively close to the

Conclusion

In this paper, we have proposed an alternative approach based on generating functions for developing suboptimal collision-free trajectories and continuous control laws in a feedback form. For this purpose, a penalty function that increases sharply as an active spacecraft approaches obstacles is augmented into the performance index, which is minimized to develop a collision-free trajectory while reducing fuel expenditure. The proposed approach has the advantage that it does not require any

Conflict of interest statement

The authors claim no conflict of interest.

Acknowledgements

This work was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Science, ICT Future Planning (NRF-2015R1A1A1A05001063) and by the Yonsei University Future-leading Research Initiative of 2014 (2014-22-0109).

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