<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>
| Identifer | oai:union.ndltd.org:purdue.edu/oai:figshare.com:article/23528169 |
| Date | 15 June 2023 |
| Creators | Sean M. Nolan (5930771) |
| Source Sets | Purdue University |
| Detected Language | English |
| Type | Text, Thesis |
| Rights | CC BY 4.0 |
| Relation | https://figshare.com/articles/thesis/Practical_Numerical_Trajectory_Optimization_via_Indirect_Methods/23528169 |
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