Flatness-based hypersonic reentry guidance of a lifting-body vehicle

Abstract

This paper deals with the design of a guidance algorithm for the hypersonic phase of a lifting-body vehicle. The guidance strategy is based on a particular kind of nonlinear dynamic inversion, the so-called flatness approach. The main advantage of this approach is that the longitudinal guidance law is in-flight self-adaptive to any feasible hypersonic trajectory and can be written in analytical form with a small set of design parameters. Therefore, the required on-board computational resources are limited, and a reduced off-line design effort is needed for the change of vehicle parameters. Moreover, the closed-loop longitudinal guidance commands are computed on-board in a coupled way without relying on an explicit deceleration profile. Consequently, the approach leads to an efficient management of the degree of freedom associated with the angle-of-attack. PID controllers are then designed based on the longitudinal flat model in order to circumvent uncertainties and parameters dispersions. The crossrange is controlled by a series of bank reversals determined by an azimuth error deadband. The robustness and performance of the proposed guidance law are assessed by performing Monte Carlo runs with various sets of dispersions.

Introduction

In late 1998, the successful flight of the Atmospheric Reentry Demonstrator (ARD) was a milestone for Europe in the design and operation of civil space transportation vehicles that can return to Earth, whether carrying payloads or people (Pignié, Clar, Ferreira, Bouaziz, & Caillaud, 1996). For the first time, Europe flew a complete space mission, launching a vehicle into space and recovering it safely. The hypersonic reentry guidance scheme initially developed for the ARD (Vernis & Ferreira, 1999), and retained for the ARES-H winged-body demonstrator (Atmospheric Re-entry Experimental Spaceplane-Hypersonic) (Vernis & Ferreira, 2003), derives from the U.S. Space Shuttle entry guidance, as described by Harpold and Graves (1978). This scheme proceeds by a decoupling of the in-plane motion, for the downrange control, and the out-of-plane motion, for the crossrange control, the management of the aerodynamic forces leading to the typical Mach 2 gate being achieved by the vertical $L/D$ modulation. A neighboring linear guidance law that is scheduled on the current velocity is used onboard and can be computed using look-up tables. When the vehicle has a weak banking authority, a mixed modulation of both the bank angle and the angle-of-attack may be used. In such cases, the angle-of-attack modulation is introduced in a simple way in order to improve the tracking of the Drag–Velocity (D–V) profile, essentially during the roll reversals. At the same time, the out-of-plane guidance is essentially ensured by the classical roll reversal technique. The guidance law has to cope with design constraints such as the maximum g-load, for the integrity of the onboard equipment and the maximum heat flux and heat load for the structural integrity of the vehicle during the hypersonic phase. For most hypersonic entry guidance schemes, the mission scenario is designed with sufficient margins left on these parameters so that these requirements correspond only to passive path constraints.

Following the works of Harpold and Graves (1978), several hypersonic guidance schemes for reentry vehicles based on nonlinear control theory have been proposed. In Mease and Kremer (1994) and Lu (1997), guidance laws based on feedback linearization were introduced in order to manage the drag modulation, the bank angle being considered as the only control for the translational dynamics. While these techniques provide an expanded region of stability as compared with that used on the Space Shuttle, the main difficulty is to accurately estimate onboard the drag and also the associated acceleration. In Bharadwaj, Rao, and Mease (1998), a linear control design method is applied to the approximately linearized dynamics to achieve bounded-input/bounded-output tracking globally in the absence of control saturation. This scheme makes use of an approximate feedback linearization technique in order to compute the bank angle and the angle-of-attack in a coupled way. Even if the previous nonlinear guidance laws (Bharadwaj et al., 1998, Lu, 1997, Mease and Kremer, 1994) do not rely on a precomputed set of neighboring trajectories onboard, they are still slightly more computationally intensive than the linear guidance law proposed in Harpold and Graves (1978). Therefore, an alternative guidance law is proposed in Bikdash, Sartor, and Hofaimar (1999) that can track a family of reference trajectories. Sugeno approximators are used to perform multi-dimensional interpolation of a set of predefined neighboring hypersonic trajectories. However, a significant off-line design effort is needed to train the approximators from the set of reference trajectories.

For the purposes of improving the performance of the Shuttle-like hypersonic guidance schemes, one objective of the nonlinear control community was to make use of exact linearization techniques (the so-called flatness approach) so as to efficiently manage the nonlinear coupling terms existing between the in-plane and out-of-plane motions. Initial attempts in this direction were made in Neckel, Talbot, and Petit (2003) and Carson, Epstein, MacMynowski, and Murray (2006), but the authors concluded that the full nonlinear model of the vehicle is not input-to-state linearizable. In this paper, the vehicle longitudinal dynamics are inverted in a geocentric frame using altitude and curvilinear abscissa as linearizing outputs. Thus, the proposed longitudinal guidance law does not rely explicitly on a predefined D–V profile. While the proposed approach shows some similarities with the approximate feedback linearization technique introduced in Bharadwaj et al. (1998), it has the advantage of being based on an exact input-to-state linearization, thus avoiding the potential instability of the dynamics that are made unobservable. This approach appears to be an appealing alternative to other Shuttle-like guidance schemes since the longitudinal guidance law is in-flight self-adaptive to any feasible hypersonic trajectory and can be written formally with a small set of design parameters. Therefore, onboard computational resources are limited, and a reduced off-line design effort is needed for the change of vehicle parameters. In this way, the absolute bank angle and the angle-of-attack are computed onboard in a coupled way, offering an efficient management of the degree of freedom associated with the angle-of-attack modulation, as long as this modulation is compliant with the thermal flux constraint. Then, PID controllers are designed based on the longitudinal flat model in order to circumvent model uncertainties and atmospheric dispersions and also take account of several navigation scenarios. As the same time, the lateral guidance is ensured using the well-known roll reversal technique. The robustness and performance of the guidance law are assessed by performing Monte Carlo runs with a dedicated nonlinear simulation tool, considering various sets of dispersions.

The remainder of this paper is organized as follows. Flatness theory and its application to trajectory tracking are briefly reviewed in the next section. An application of these concepts is presented in Section 3 for the hypersonic reentry of a lifting-body. A flat model corresponding to the simplified longitudinal dynamics of the vehicle is computed and a linear feedback is designed. Finally, numerical results are provided to demonstrate the performance and robustness of the flatness-based guidance scheme.