Analytic Development of a Reference Trajectory for Skip Entry

Abstract

F OR entry vehicles with relatively low lift-to-drag ratios (L=D), a known strategy for achieving long downrange is to allow the vehicle to skip out of the atmosphere [1]. During this high-altitude and low-drag skip phase of the entry, the vehicle can dramatically increase its range. This Note is exclusively focused on entries from supercircular velocities that require an exoatmospheric phase to reach the landing site. Various methods have been suggested and implemented for atmospheric guidance of such trajectories. Most fall into one of three broad categories [2,3]: numeric predictor– correctors [4–9], analytic predictor–correctors [10,11], and reference-following controllers [12–16]. This Note presents a method, primarily intended for use under the latter category, that generates analytically a reference drag profile for the first entry portion of a skip entry when the exit conditions (and the initial conditions) are known. The analytic generation of such a reference profile has not been attempted before. In thisway, thisNote intends to contribute to the effort of developing a yet-unaccomplished complete analytic solution to the skip-entry guidance problem. Note that a complete analytic solution to the skip-entry problem should determine the conditions at exit (range to go, velocity V, and flightpath angle ) that render the entry conditions for the final phase. The central difficulty in analytically determining the exit conditions lies in the limitations that various approximations and linearization assumptions have been found to have when estimating the range flown at low-drag altitudes [2,17–19]. The analytic determination of the exit conditions remains an elusive problem whose resolution is not intended in this Note. The analytic development of a drag reference profile for a subcircular entry is the basis for the Space Shuttle Orbiter guidance logic [20]. This idea is based on the fact that the range to be flown during entry is a unique function of the drag acceleration maintained throughout the flight. This range is predictable using analytic techniques for simple geometric drag acceleration functions of the relative velocity (quadratic, linear, and constant, in the case of the orbiter), provided the local flight-path angle is small, which is the case at high speeds. Flight throughout the entry corridor can be achieved by linking these geometric functions together in a series. It is conceivable to divide the first entry in a skip trajectory in segments with linear and/or quadratic drag functions, as is the case in the space shuttle entry guidance; however, this approach will not be pursued in this Note. It is proposed in this Note to express the drag reference of the complete first entry as a polynomial function of the velocity, with degree higher than two. In addition, the generic method proposed to obtain the drag reference profilewill be further simplified by thinking of the drag as the probability density function of the velocity or, conversely, by thinking of the velocity as the distribution function of the drag. With this notion, it will be shown that the reference drag profile can be generated by solving a system of linear algebraic equations. For completeness, the drag profiles generated with this method will be tracked through the implementation of the feedback linearization method of differential geometric control as a guidance law with the error dynamics of a second-order homogeneous equation in the formof a damped oscillator [21].Although this approachwasfirst proposed as a revisited version of the Space Shuttle Orbiter entry guidance to demonstrate the commonality of both guidance laws, it has never been used to fly a skiplike entry trajectory, where the drag profile for the first entry does not fit a quadratic polynomial of the velocity. A number of different approaches to skip-entry guidance for the Orion Crew Exploration Vehicle (CEV) spacecraft have been under evaluation at the Flight Mechanics and Trajectory Design Branch at NASA’s Johnson Space Center [22]. A guidance algorithm called NSEG [19] (Numerical Skip Entry Guidance), which combines features of the original Apollo Guidance algorithm [23,24] with a numerical scheme for computing a real-time long-range skip trajectory, was found to provide very reliable means of meeting the skip-entry range requirement. Out of a comprehensive set of 60,000 skip-entry cases (20 nominal and 59,980 dispersed) that have been simulated for the CEV using NSEG, the 20 nominal cases will be used as test cases for this Note. As explained before, obtaining the skip-out exit conditions from the knowledge of the landing site and entry interface is not the objective of this Note; therefore, the initial and final conditions for testing purposes will be those pertaining to the 20 nominal trajectories. The 20 cases are subdivided into four groups. Each group is composed of five trajectories that have a common entry interface (EI in Fig. 1), but different target landing sites in the western continental United States and differentL=D (0.3, 0.33, and 0.35).