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**Survey of numerical methods for trajectory optimization.**
*(English)*
Zbl 1158.49303

Introduction: It is not surprising that the development of numerical methods for trajectory optimization have closely paralleled the exploration of space and the development of the digital computer. Space exploration provided the impetus by presenting scientists and engineers with challenging technical problems.The digital computer provided the tool for solving these new problems.The goal of this paper is to review the state of the art in the field loosely referred to as trajectory optimization.

Presenting a survey of a field as diverse as trajectory optimization is a daunting task. Perhaps the most difficult issue is restricting the scope of the survey to permit a meaningful discussion within a limited amount of space. To achieve this goal, I made a conscious decision to focus on the two types of methods most widely used today, namely, direct and indirect. I begin the discussion with a brief review of the underlying mathematics in both direct and indirect methods. I then discuss the complications that occur when path and boundary constraints are imposed on the problem description. Finally, I describe unresolved issues that are the subject of ongoing research.

A number of recurrent themes appear throughout the paper. First, the aforementioned direct vs. indirect is introduced as a means of categorizing an approach. Unfortunately, not every technique falls neatly into one category or another. I will attempt to describe the benefits and deficiencies in both approaches and then suggest that the techniques may ultimately merge. Second, I shall attempt to discriminate between method vs implementation. A numerical method is usually described by mathematical equations and/or algorithmic logic. Computational results are achieved by implementing the algorithm as software, e.g., Fortran code. A second level of implementation may involve translating a (preflight) scientific software implementation into an (on board) hardware implementation. In general, method and implementation are not the same, and I shall try to emphasize that fact. Third, I shall focus the discussion on algorithms instead of physical models. The definition of a trajectory problem necessarily entails a definition of the dynamic environment such as gravitational, propulsion, and aerodynamic forces. Thus it is common to use the same algorithm, with different physical models, to solve different problems. Conversely, different algorithms may be applied to the same physical models (with entirely different results).

Finally, I shall attempt to focus on general rather than special purpose methods. A great deal of research has been directed toward solving specific problems. Carefully specialized techniques can either be very effective or very ad hoc. Unfortunately, what works well for a launch vehicle guidance problem may be totally inappropriate for a low-thrust orbit transfer.

Presenting a survey of a field as diverse as trajectory optimization is a daunting task. Perhaps the most difficult issue is restricting the scope of the survey to permit a meaningful discussion within a limited amount of space. To achieve this goal, I made a conscious decision to focus on the two types of methods most widely used today, namely, direct and indirect. I begin the discussion with a brief review of the underlying mathematics in both direct and indirect methods. I then discuss the complications that occur when path and boundary constraints are imposed on the problem description. Finally, I describe unresolved issues that are the subject of ongoing research.

A number of recurrent themes appear throughout the paper. First, the aforementioned direct vs. indirect is introduced as a means of categorizing an approach. Unfortunately, not every technique falls neatly into one category or another. I will attempt to describe the benefits and deficiencies in both approaches and then suggest that the techniques may ultimately merge. Second, I shall attempt to discriminate between method vs implementation. A numerical method is usually described by mathematical equations and/or algorithmic logic. Computational results are achieved by implementing the algorithm as software, e.g., Fortran code. A second level of implementation may involve translating a (preflight) scientific software implementation into an (on board) hardware implementation. In general, method and implementation are not the same, and I shall try to emphasize that fact. Third, I shall focus the discussion on algorithms instead of physical models. The definition of a trajectory problem necessarily entails a definition of the dynamic environment such as gravitational, propulsion, and aerodynamic forces. Thus it is common to use the same algorithm, with different physical models, to solve different problems. Conversely, different algorithms may be applied to the same physical models (with entirely different results).

Finally, I shall attempt to focus on general rather than special purpose methods. A great deal of research has been directed toward solving specific problems. Carefully specialized techniques can either be very effective or very ad hoc. Unfortunately, what works well for a launch vehicle guidance problem may be totally inappropriate for a low-thrust orbit transfer.