Convex Optimization of Launch Vehicle Ascent Trajectory with Heat-Flux and Splash-Down Constraints

Abstract

This paper presents a convex programming approach to the optimization of a multistage launch vehicle ascent trajectory, from the liftoff to the payload injection into the target orbit, taking into account multiple nonconvex constraints, such as the maximum heat flux after fairing jettisoning and the splash-down of the burned-out stages. Lossless and successive convexification are employed to convert the problem into a sequence of convex subproblems. Virtual controls and buffer zones are included to ensure the recursive feasibility of the process and a state-of-the-art method for updating the reference solution is implemented to filter out undesired phenomena that may hinder convergence. A hp pseudospectral discretization scheme is used to accurately capture the complex ascent and return dynamics with a limited computational effort. The convergence properties, computational efficiency, and robustness of the algorithm are discussed on the basis of numerical results. The ascent of the VEGA launch vehicle toward a polar orbit is used as case study to discuss the interaction between the heat flux and splash-down constraints. Finally, a sensitivity analysis of the launch vehicle carrying capacity to different splash-down locations is presented.

Advances in Trajectory Optimization for Space Vehicle Control

“…The target orbit is prescribed by the vector function ψ in (73g), which is the same as in (70d) and specifies some or all of the target orbital elements. (Benedikter et al, 2020) chose boundary conditions based on the radius and inclination of a circular target orbit. A final nuance is that, if the rocket is assumed to be a two-stage vehicle, a stage separation event must be scheduled at a pre-determined time t 1 via (73h).…”

“…A final nuance is that, if the rocket is assumed to be a two-stage vehicle, a stage separation event must be scheduled at a pre-determined time t 1 via (73h). At the separation instant, the mass variable experiences a discontinuous decrease that amounts to the dry weight of the first stage (Benedikter et al, 2019a(Benedikter et al, , 2020. A related constraint for stage separation requires θ(t 1 ) = 0 in order to reduce lateral load (Zhengxiang et al, 2018).…”

“…A related constraint for stage separation requires θ(t 1 ) = 0 in order to reduce lateral load (Zhengxiang et al, 2018). Furthermore, the splashdown location of a burnt-out separated stage can also be constrained (Benedikter et al, 2020).…”

“…This choice makes drag a linear function of the control, while the constraint u 2 1 = u 2 is relaxed to u 2 1 ≤ u 2 via lossless convexification. Similarly, (Benedikter et al, 2019a(Benedikter et al, , 2020 chose thrust direction as input and losslessly convexified the unit norm constraint on the thrust direction to a convex inequality. (Cheng et al, 2017) considered a 3D problem with AoA and bank angle as control inputs, and applied lossless convexification to a constraint of the form u 2 1 + u 2 2 + u 2 3 = 1.…”