Optimization Problem of the Launch Vehicle Pitch Angle during Spacecraft Insertion into Circular Orbits

Purpose. The aim of the study is to formulate and analytically pose the task of constructing the pitch program of a launch vehicle, ensuring the insertion of a spacecraft into a circular orbit, considering constraints on the motion parameters and requirements for the final state. Design / Method / Approach. An analytical model for the pitch program is proposed, consisting of five phases: vertical ascent, angular acceleration, constant angular velocity motion, deceleration, and a final fixed-angle phase. Dimensionless coefficients are introduced to characterize the relative durations of active angular maneuver phases. Findings. Analytical expressions were derived for the pitch angle, angular velocity, and acceleration for each control phase, ensuring continuity and physical realizability. Expressions for the resulting orbital altitude and control effort provide a basis for multi-criteria optimization. Theoretical Implications. The developed formalization enables analytical optimization of the pitch program without a full vehicle motion model, aiding understanding of the link between control structure and orbital injection conditions. Practical Implications. The proposed model can be used in preliminary design stages for initial synthesis of the launch vehicle pitch program, enabling rapid assessment of trajectory controllability and compliance with specified orbital parameters, avoiding full-scale simulation. Originality / Value. Unlike approaches based on full numerical integration, this study presents a simplified, analytically manageable pitch program model, allowing efficient optimization considering physical constraints and final conditions. Research Limitations / Future Research. Limitations involve the simplified analytical model with fixed control phases. Future work involves numerical optimization (using gradient-based, heuristic, or global methods) to precisely determine optimal pitch program parameters. Article Type. Methodological paper.