Noniterative Instantaneous Impact Point Prediction Algorithm for Launch Operations

Abstract

T HE instantaneous impact point (IIP) in launch operations is defined as the touchdown point of a rocket with an assumption of an immediate end of the propelled flight [1–3]. The IIP is very important information to judge whether the flight is normal or not, and it should be continuously monitored during the whole flight so that proper actions be taken in an emergency to protect human life and property. Therefore, the real-time prediction and monitoring of the IIP is one of the key safety requirements. The IIP prediction problem seeks for the intersection point between the Earth’s surface and the free-flight trajectory of a rocket subject to gravity and other disturbing forces. The current position and velocity vectors are given as inputs to the problem. A numerical integration of the dynamicmodel including all force elementswith an initial position and velocity could yield a very accurate answer to the problem with relative ease. But when the real-time calculation is considered as a constraint, the numerical integration that requires a lot of computational resources is not a feasible option in the majority of launch operations. There has been some research on developing the IIP prediction algorithm that provides the real-time solution within acceptable accuracy with limited computational resources. To calculate the IIP for shortto midrange rockets, simplified models in the localvertical–local-horizontal frame are frequently used [3–6]. These assume that the trajectory of a rocket is parabolic subject to the constant vertical downward gravitational acceleration and often add correction terms to compensate for the modeling errors. Montenbruck and Markgraf [6] developed a linearized IIP calculation algorithm and implemented it for the launch operation of the Maxus-5 sounding rocket. Their algorithm was based on a simple parabolic flight model, and its prediction accuracy was improved by considering the effects of the Earth curvature, the Coriolis force, and the gravity field variation. The Keplerian algorithm predicts the IIP using the equations describing the motion of a rocket under the inverse-square law of gravity. Regan and Anandakrishnan [4] and Zarchan [7] developed a series of formulations that can be used as the base of theKeplerian IIP prediction, such as the expressions for the flight angle and the flight time. Currently, an iterativeKeplerian algorithm is frequently used to predict the IIP for long-range space vehicles [8,9]. The iterative Keplerian algorithm can calculate the true IIP solution for the Keplerian motion of a rocket using an iterative procedure (e.g., successive substitution) to find parameters such as Kepler’s f and g series expansions, the eccentric anomaly of the impact point, and the surface radius at impact. Considering that the IIP prediction for an actual launch operation should be carried out in real time, the iterative procedure that has the risk of computational overload is not very desirable to be included in the prediction algorithm. Especially when the algorithm is implemented in an onboard computer that has limited computing resources but is responsible for multiple mission-critical tasks such as event sequencing, navigation, and attitude control, the potential damage associated with the risk is significantly increased. This engineering Note presents a Keplerian IIP prediction algorithm that does not require any iterative procedure and is thus remarkably faster than the iterative Keplerian algorithm. First, an exact and closed solution for the Keplerian IIP with the spherical Earth model is developed. Then, a noniterative procedure to compensate for the effect of the oblateness of the Earth is added. Finally, the validity of the proposed algorithm is proved through the performance comparison with the iterative algorithm in terms of computational efficiency and the prediction accuracy.