Practical Numerical Trajectory Optimization Via Indirect Methods

Sean Matthew Nolan, Purdue University


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




Grant, Purdue University.

Subject Area

Aerospace engineering|Mathematics|Systems science

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