Exo-Atmospheric Trajectory Optimization for Small Satellite Launch Vehicles: A Robust Pseudo-Spectral Approach

Exo-atmospheric trajectory optimization for small satellite launch vehicles faces a critical challenge: state variable dispersions from atmospheric flight uncertainties can compromise orbital insertion accuracy. This paper presents a computationally efficient pseudo-spectral optimization framework for India’s Small Satellite Launch Vehicle (SSLV), enabling rapid trajectory re-optimization to accommodate varying atmospheric conditions and expand effective launch windows. The methodology employs Legendre-Gauss-Lobatto (LGL) collocation to transform nonlinear differential equations into algebraic constraints, achieving spectral accuracy with minimal computational overhead. Pitch steering angle and inter-stage coasting time serve as control variables, optimized via Broyden’s quasi-Newton method. Monte Carlo analysis with 300 samples under 7% (3Formula: see text) dispersions in altitude, velocity, and flight path angle demonstrates 100% convergence success, with terminal errors well within the Velocity Trimming Module’s 300 m/s correction capability (apogee/perigee errors Formula: see text15 km). Mesh refinement studies confirm spectral convergence with 40 collocation points achieving sub-kilometer accuracy. The framework’s 1.34-second average solve time-validated across worst-case scenarios-enables ground-based pre-launch trajectory regeneration within seconds, effectively expanding usable launch windows and reducing costly delays. A boundary condition-based initial guess strategy eliminates complex preprocessing, demonstrating the method’s robustness and operational practicality. The open-loop reference trajectories, uploaded to onboard guidance systems pre-flight, establish practical viability for responsive launch operations supporting commercial small satellite deployment.