<p>Numerical trajectory optimization is helpful not only for mission planning but also design</p> <p>space exploration and quantifying vehicle performance. Direct methods for solving the opti-</p> <p>mal control problems, which first discretize the problem before applying necessary conditions</p> <p>of optimality, dominate the field of trajectory optimization because they are easier for the</p> <p>user to set up and are less reliant on a forming a good initial guess. On the other hand,</p> <p>many consider indirect methods, which apply the necessary conditions of optimality prior to</p> <p>discretization, too difficult to use for practical applications. Indirect methods though provide</p> <p>very high quality solutions, easily accessible sensitivity information, and faster convergence</p> <p>given a sufficiently good guess. Those strengths make indirect methods especially well-suited</p> <p>for generating large data sets for system analysis and worth revisiting.</p> <p>Recent advancements in the application of indirect methods have already mitigated many</p> <p>of the often cited issues. Automatic derivation of the necessary conditions with computer</p> <p>algebra systems have eliminated the manual step which was time-intensive and error-prone.</p> <p>Furthermore, regularization techniques have reduced problems which traditionally needed</p> <p>many phases and complex staging, like those with inequality path constraints, to a signifi-</p> <p>cantly easier to handle single arc. Finally, continuation methods can circumvent the small</p> <p>radius of convergence of indirect methods by gradually changing the problem and use previ-</p> <p>ously found solutions for guesses.</p> <p>The new optimal control problem solver Giuseppe incorporates and builds upon these</p> <p>advancements to make indirect methods more accessible and easily used. It seeks to enable</p> <p>greater research and creative approaches to problem solving by being more flexible and</p> <p>extensible than previous solvers. The solver accomplishes this by implementing a modular</p> <p>design with well-defined internal interfaces. Moreover, it allows the user easy access to and</p> <p>manipulation of component objects and functions to be use in the way best suited to solve</p> <p>a problem.</p> <p>A new technique simplifies and automates what was the predominate roadblock to using</p> <p>continuation, the generation of an initial guess for the seed solution. Reliable generation of</p> <p>a guess sufficient for convergence still usually required advanced knowledge optimal contrtheory or sometimes incorporation of an entirely separate optimization method. With the</p> <p>new method, a user only needs to supply initial states, a control profile, and a time-span</p> <p>over which to integrate. The guess generator then produces a guess for the “primal” problem</p> <p>through propagation of the initial value problem. It then estimates the “dual” (adjoint)</p> <p>variables by the Gauss-Newton method for solving the nonlinear least-squares problem. The</p> <p>decoupled approach prevents poorly guessed dual variables from altering the relatively easily</p> <p>guess primal variables. As a result, this method is simpler to use, faster to iterate, and much</p> <p>more reliable than previous guess generation techniques.</p> <p>Leveraging the continuation process also allows for greater insight into the solution space</p> <p>as there is only a small marginal cost to producing an additional nearby solutions. As a</p> <p>result, a user can quickly generate large families of solutions by sweeping parameters and</p> <p>modifying constraints. These families provide much greater insight in the general problem</p> <p>and underlying system than is obtainable with singular point solutions. One can extend</p> <p>these analyses to high-dimensional spaces through construction of compound continuation</p> <p>strategies expressible by directed trees.</p> <p>Lastly, a study into common convergence explicates their causes and recommends mitiga-</p> <p>tion strategies. In this area, the continuation process also serves an important role. Adaptive</p> <p>step-size routines usually suffice to handle common sensitivity issues and scaling constraints</p> <p>is simpler and out-performs scaling parameters directly. Issues arise when a cost functional</p> <p>becomes insensitive to the control, which a small control cost mitigates. The best perfor-</p> <p>mance of the solver requires proper sizing of the smoothing parameters used in regularization</p> <p>methods. An asymptotic increase in the magnitude of adjoint variables indicate approaching</p> <p>a feasibility boundary of the solution space.</p> <p>These techniques for indirect methods greatly facilitate their use and enable the gen-</p> <p>eration of large libraries of high-quality optimal trajectories for complex problems. In the</p> <p>future, these libraries can give a detailed account of vehicle performance throughout its flight</p> <p>envelope, feed higher-level system analyses, or inform real-time control applications.</p>