Hybrid Near-Optimal Atmospheric Guidance for Aeroassisted Orbit Transfer

In this paper, a hybrid methodology is applied to the problem of determining a near-optimal control which approximately minimizes energy loss for a class of aeroassisted orbit transfer maneuvers. The result is an ap- proximate solution incorporating aspects of both an analytical zero-order solution from regular perturbation and a numerical solution from collocation. This approach is used to compute both open-loop and closed-loop near- optimal controls for a particular vehicle. Results from several approximations are quantitatively compared with an exact optimal solution. N orbit plane change is one of the most fuel-intensive ma- neuvers that can be required of an orbiting space vehicle. It has been shown that an aeroassisted orbit transfer consumes signifi- cantly less fuel than a similar maneuver executed with thrusters only. This advantage is only realizable, however, when optimal control laws are developed that take full advantage of the available aerody- namic forces. Because the dynamics governing this type of maneu- ver are highly nonlinear, an analytical solution to the optimal control problem is impossible. Furthermore, an exact numerical solution is too computationally demanding for real-time implementation on- board the orbiting vehicle. Approximation techniques that allow near-optimal guidance laws to be determined either analytically or with a minimum of computational effort are, therefore, highly de- sirable for this application. In this paper, a method is considered that combines elements of both approaches. An analytical approx- imation that holds for the atmospheric phase of flight is obtained from the zero-order solution to a regular perturbation problem. This solution is then augmented numerically by a technique borrowed from the method of collocation, in which the trajectory is divided into finite elements. In recent efforts, numerous approximate methods have been applied to the task of determining a near-optimal control for aeroassisted orbit transfer. These approximations have addressed the atmospheric portion of the maneuver, assuming that the effects of the orbital dynamics are small.1"3 Keplerian effects are treated by introducing an approximation of Loh's term. Some approxima- tions of this type have included assuming that Loh's term is con- stant or piecewise constant.2 A second alternative is to account for Keplerian effects through a regular perturbation expansion of the control solution.1 A related approach to the problem addresses both the orbital and atmospheric stages of the trajectory by invoking sin- gular perturbation theory and matched asymptotic expansions.4 It has been shown in Ref. 4 that the problem in question is, in fact, a singularly perturbed problem and that variations in Loh's term can be accounted for by properly matching a zero-order outer solution with the zero-order solution used in Ref. 1. That is, the zero-order solution of Ref. 1 serves as the inner solution in a matched asymp- totic expansion. The paper begins with a review of the dynamics of aeroas- sisted orbit transfer and the associated optimal control problem. A summary of the problem formulation as presented in Ref. 1 is included in the interests of clarity and completeness. The prob- lem definition is followed by a description of its analytical zero- order solution. An overview of the method of collocation is then presented as an entirely numerical methodology for this prob- lem. Next, the hybrid methodology incorporating elements of both solutions is described. This method was originally developed in Ref. 5 and applied to the problem of launch vehicle guidance. Finally, numerical results are presented for the maneuverable re- search re-entry vehicle (MRRV) which was featured in Ref. 3. These results are discussed in detail with recommendations for further analysis. Aeroassisted Orbit Transfer Problem The aeroassisted orbit transfer maneuver may be defined as an optimal control problem. This problem is characterized by highly nonlinear coupled differential constraints and is not amenable to an analytical solution. The equations may, however, be transformed and then used under certain simplifying assumptions to determine a near-optimal control which is valid for the atmospheric portion of the maneuver. In this section, the equations of motion which govern this type of maneuver are presented, and a scheme for modifying the equations is reviewed. The associated optimal con- trol problem is then defined, along with the general form of its solution.