Multistage rocket preliminary design and trajectory optimization using a multidisciplinary approach

Abstract

A procedure for rocket preliminary design was developed using a multidisciplinary coupled approach that simultaneously finds the optimal design and trajectory parameters for a given representative insertion in orbit launch mission. Given the nature of the performance metrics and design space, and the distinct design and trajectory problems, heuristic methods were used in a multilevel design optimization architecture. For the design, a continuous genetic algorithm able to perform parallel optimization was developed and benchmarked. The results were obtained with mass and sizing models, required to estimate the rocket structure, and created using historical data regression. For the trajectory, once defined its assumptions, the optimality equations are deduced and the optimal values are found using a particle swarm optimization. The multidisciplinary optimization procedure was demonstrated by designing a small launch vehicle and comparing it to a state-of-the-art existing rocket. Promising results were obtained in both design and trajectory optimization, with the imposed constraints adequately handled and the optimal rocket preliminary design with appropriate optimal trajectory found with an affordable computational cost.

Appendix

1.1 A benchmark of developed GA algorithm

The population initialization was defined using a maximin Latin hypercube method, maximizing the smallest distance between any two design points, spreading them evenly over the entire design region (Husslage et al. 2006). The parents are selected through tournament, followed by an uniform crossover and a Gaussian mutation, creating the children for the next generation. The GA parameters were tuned using the Ackley function as objective function, to a crossover rate p_c = 0.75 and mutation rate $p_m=0.5e^{-0.025gen_k}$, where $gen_k$ is the generation number. The chosen step-size for the Gaussian mutation is $\lambda =1.0e^{-0.075gen_k}$.

The developed GA was benchmark against the proved DEAP’s GA (Fortin et al. 2012) and PyGMO’s GA (Biscani and Izzo 2019), for the set of test functions listed in Table 18.

Table 18 Optimization benchmark functions

The benchmark results are summarized in Table 19, where the three algorithms used uniform crossover, tournament selection and Gaussian mutation throughout the generations. For a better comparison of the three algorithms, the crossover rate and the mutation rate chosen were p_c = 0.75 and p_m = 0.2, respectively, as suggested by Eiben et al. (1999) and Haupt and Haupt (2004). However, it is not possible to guarantee that DEAP and PyGMO do not internally change the parameters set at the beginning of the optimization to ease the search of the global minima. Therefore, to the best of the authors’ knowledge, the most fair comparison was performed. The population size (Popul.) and the maximum number of generations (Gener.) were adjusted for each function.

Table 19 Benchmark results of implemented GA

As seen in Table 19, our developed GA algorithm exhibits a significantly better solution accuracy than DEAP and PyGMO, believed to be due to the better control of the Gaussian standard deviation required for the individual mutation. However, our GA presently requires a slightly higher computational time (up to 100% increase when compared to DEAP and to 460% increase when compared to PyGMO), but faster times can be achieved in future through the compilation of our implemented GA code. Additionally, our GA presents an easy-to-use interface, easily enabling the combined optimization of discrete and continuous variables. Observing these results, we confidently used our developed algorithm in the MDO rocket design procedure presented in this work.