NUMERICAL VALIDATION AND PARAMETER ANALYSIS OF A FIVE-PHASE PITCH PROGRAM FOR LAUNCHING SPACECRAFT INTO CIRCULAR ORBITS
Abstract
The aim of this study is the systematic numerical validation of a semi-analytical five-phase pitch program model (FPPM), which is critically important for optimizing the trajectory of the active flight phase of launch vehicles (LV) performing orbital insertion into circular orbits. The validation is carried out by applying the model to numerical configurations of launch vehicles from two fundamentally different classes: Falcon 9 (characterized by high thrust-to-weight ratio and using LOX/RP-1) and JAXA H3 (distinguished by high specific impulse and using LOX/LH2). The ultimate goal is to establish and quantitatively analyze the dependence of optimal pitch control parameters on the altitude of the target circular orbit. The proposed FPPM parameterizes the angular profile of LV motion on the active segment, ensuring continuity of angular velocity and angular acceleration. The model is governed by a set of five key variables (t1, k2, k3, t4, t5), where the dimensionless coefficient defines the relative duration of the phase with constant angular velocity. Optimization was conducted via numerical integration of the full system of equations of motion of the LV center of mass using a grid search method in a multidimensional parameter space. The objective function minimized a composite error metric representing deviations from the final orbital constraints (prescribed orbital altitude and circular velocity ). The numerical simulations yielded sets of quasi-optimal FPPM parameters for orbital altitudes of 400–700 km for Falcon 9 and 1000–1200 km for H3. A stable and universal trend was identified: an increase in target orbital altitude correlates with an increase in the total insertion time and a significant growth of the parameter . Increasing directly leads to a reduction of the peak angular velocity , indicating the necessity of generating a flatter (more shallow) trajectory profile to effectively minimize gravity losses. The resulting quantitative dependencies demonstrate strong universality, as identical tendencies were observed for launch vehicles with fundamentally different dynamics. The obtained results may serve as reliable heuristic guidelines for rapid preliminary synthesis of ascent trajectories for a wide class of launch vehicles.
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