Spacecraft trajectory optimization is a near-infinite problem space with a wide variety of models and optimizers. As trajectory complexity increases, so too must the capabilities of modern optimizers. Common objective cost functions for these optimizers include the propellant utilized by the spacecraft and the time the spacecraft spends in flight. One effective method of minimizing these costs is the utilization of one or multiple gravity assists. Due to the phenomenon known as the Oberth effect, fuel burned at a high velocity results in a larger change in orbital energy than fuel burned at a low velocity. Since a spacecraft is flying fastest at the periapsis of its orbit, application of impulsive thrust at this closest approach is demonstrably capable of generating a greater change in orbital energy than at any other location in a trajectory. Harnessing this extra energy in order to lower relevant cost functions requires the modeling of these “powered flybys” or “powered gravity assists” (PGAs) within an interplanetary trajectory optimizer. This paper will discuss the use and modification of the Spacecraft Trajectory Optimization Suite, an optimizer built on evolutionary algorithms and the island model paradigm from the Parallel Global Multi-Objective Optimizer (PaGMO). This variant of STOpS enhances the STOpS library of tools with the capability of modeling and optimizing single and multiple powered gravity assist trajectories. Due to its functionality as a tool to optimize powered flybys, this variant of STOpS is named the Spacecraft Trajectory Optimization Suite - Flybys with Impulsive Thrust Engines (STOpS-FLITE). In three test scenarios, the PGA algorithm was able to converge to comparable or superior solutions to the unpowered gravity assist (uPGA) modeling used in previous STOpS versions, while providing extra options of trades between time of flight and propellant burned. Further, the PGA algorithm was able to find trajectories utilizing a PGA where uPGA trajectories were impossible due to limitations on time of flight and flyby altitude. Finally, STOpS-FLITE was able to converge to a uPGA trajectory when it was the most optimal solution, suggesting the algorithm does include and properly considers the uPGA case within its search space.

Solutions to trajectory optimization problems are carried out by the direct and indirect methods. Under broad heading of these methods, numerous algorithms such as collocation, direct, indirect and multiple shooting methods have been developed and reported in the literature. Each of these algorithms has certain advantages and limitations. For example, direct shooting technique is not suitable when the number of nonlinear programming variables is large. Indirect shooting method requires analytical derivatives of the control and co-states function and a poorly guessed initial condition can result in numerical unstable values of the adjoint variable. Multiple shooting techniques can alleviate some of these difficulties by breaking down the trajectory into several segments that help in reducing the non-linearity effects of early control on later parts of the trajectory. However, multiple shooting methods then have to handle more number of variables and constraints to satisfy the defects at the segment joints. The sie of the nonlinear programming problem in the collocation method is also large and proper locations of grid points are necessary to satisfy all the path constraints. Stochastic methods such as Genetic algorithms, on the other hand, also require large number of function evaluations before convergence. To overcome some of the limitations of the conventional methods, improved solution techniques are developed. Three improved methods are proposed for the solution of trajectory optimization problems. They are • a genetic algorithm employing dominance and diploidy concept. • a collocation method using chebyshev polynomials , and • a hybrid method that combines collocation and direct shooting technique A conventional binary-coded genetic algorithm uses a haploid chromosome, where a single string contains all the variable information in the coded from. A diploid, as the name suggests, uses pair of chromosomes to store the same characteristic feature. The diploid genetic algorithm uses a dominant map for decoding genotype into a stable, consistent phenotype. In dominance, one allele takes precedence over another. Diploidy and dominance helps in retaining the previous best solution discovered and shields them from harmful selection in a changing environment. Hence, diploid and dominance affect a king of long-term memory in the genetic algorithm. They allow alternate solutions to co-exist. One solution is expressed and the other is held in abeyance. In the improved diploid genetic algorithm, dominant and recessive genes are defined based on the fitness evaluation of each string. The genotype of fittest string is declared as the dominant map. The dominant map is dynamic in nature as it is replaced with a better individual in future generations. The concept of diploidy and dominance in the improved method mimics closer to the principles used in human genetics as compared to any such algorithms reported in the literature. It is observed that the improved diploid genetic algorithm is able to locate the optima for a given trajectory optimization problem with 10% lower computational time as compared to the haploid genetic algorithm. A parameter optimization problem arising from an optimal control problem where states and control are approximated by piecewise Chebyshev polynomials is well known. These polynomials are more accurate than the interpolating segments involving equal spaced data. In the collocation method involving Chebyshev polynomials, derivatives of two neighboring polynomials are matched with the dynamics at the nodal points. This leads to a large number of equality constraints in the optimization problem. In the improved method, derivative of the polynomial is also matched with the dynamics at the center of segments. Though is appears the problem size is merely increased, the additional computations improve the accuracy of the polynomial for a larger segment. The implicit integration step size is enhanced and overall size of the problem is brought down to one-fourth of the problem size defined with a conventional collocation method using Chebyshev polynomials. Hybrid method uses both collocation and direct shooting techniques. Advantages of both the methods are combined to give more synergy. Collocation method is used in the starting phase of the hybrid method. The disadvantage of standalone collocation method is that tuning of grid points is required to satisfy the path constraints. Nevertheless, collocation method does give a good guess required for the terminal phase of the hybrid method, which uses a direct shooting approach. Results show nearly 30% reduction in computation time for the hybrid approach as compared to a method in which direct shooting alone is used, for the same initial guess of control. The solutions obtained from the three improved methods are compared with an indirect method. The indirect method requires derivations of the control and adjoint equations, which are difficult and problem specific. Due to sensitivity of the costate variables, it is often difficult to find a solution through the indirect method. Nevertheless, these methods do provide an accurate result, which defines a benchmark for comparing the solutions obtained through the improved methods. Trajectory design and optimization of a RLV(Reusable Launch Vehicle) Demonstrator mission is considered as a test problem for evaluating the performance of the improved methods. The optimization problem is difficult than a conventional launch vehicle trajectory optimization problem because of the following two reasons. • aerodynamic lift forces in the RLV add one more dimension to the already complex launch vehicle optimization problem. • as RLV performs a sub orbital flight, the ascent phase trajectory influences the re-entry trajectory. Both the ascent and re-entry optimization problem of the RLV mission is addressed. It is observed that the hybrid method gives accurate results with least computational effort, as compared with other improved techniques for the trajectory optimization problem of RLV during its ascent flight. Hybrid method is then successfully used during the re-entry phase and in designing the feasible optimal trajectories under the dispersion conditions. Analytical solutions obtained from literature are used to compare the optimized trajectory during the re-entry phase. Trajectory optimization studies are also carried out for the off-nominal performances. Being a thrusting phase, the ascent trajectory is subjected to significant deviations, mainly arising out of solid booster performance dispersions. The performance index during rhe ascent phase is modified in a novel way for handling dispersions. It minimizes the state errors in a least square sense, defined at the burnout conditions ensure possibilities of safe re-entry trajectories. The optimal trajectories under dispersion conditions serve as a benchmark for validating the closed-loop guidance algorithm that is developed for the ascent phase flight. Finally, an on-line trajectory command-reshaping algorithm is developed which meets the flight objectives under the dispersion conditions. The guidance algorithm uses a pre-computed trajectory database along with some real-time measured parameters in generating the optimal steering profiles. The flight objectives are met under the dispersion conditions and the guidance generated steering profiles matches closely with the optimal trajectories.