Large Scale Constrained Trajectory Optimization Using Indirect Methods

State-of-the-art direct and indirect methods face significant challenges when solving large scale constrained trajectory optimization problems. Two main challenges when using indirect methods to solve such problems are difficulties in handling path inequality constraints, and the exponential increase in computation time as the number of states and constraints in problem increases. The latter challenge affects both direct and indirect methods. A methodology called the Integrated Control Regularization Method (ICRM) is developed for incorporating path constraints into optimal control problems when using indirect methods. ICRM removes the need for multiple constrained and unconstrained arcs and converts constrained optimal control problems into two-point boundary value problems. Furthermore, it also addresses the issue of transcendental control law equations by re-formulating the problem so that it can be solved by existing numerical solvers for two-point boundary value problems (TPBVP). The capabilities of ICRM are demonstrated by using it to solve some representative constrained trajectory optimization problems as well as a five vehicle problem with path constraints. Regularizing path constraints using ICRM represents a first step towards obtaining high quality solutions for highly constrained trajectory optimization problems which would generally be considered practically impossible to solve using indirect or direct methods. The Quasilinear Chebyshev Picard Iteration (QCPI) method builds on prior work and uses Chebyshev Polynomial series and the Picard Iteration combined with the Modified Quasi-linearization Algorithm. The method is developed specifically to utilize parallel computational resources for solving large TPBVPs. The capabilities of the numerical method are validated by solving some representative nonlinear optimal control problems. The performance of QCPI is benchmarked against single shooting and parallel shooting methods using a multi-vehicle optimal control problem. The results demonstrate that QCPI is capable of leveraging parallel computing architectures and can greatly benefit from implementation on highly parallel architectures such as GPUs. The capabilities of ICRM and QCPI are explored further using a five-vehicle constrained optimal control problem. The scenario models a co-operative, simultaneous engagement of two targets by five vehicles. The problem involves 3DOF dynamic models, control constraints for each vehicle and a no-fly zone path constraint. Trade studies are conducted by varying different parameters in the problem to demonstrate smooth transition between constrained and unconstrained arcs. Such transitions would be highly impractical to study using existing indirect methods. The study serves as a demonstration of the capabilities of ICRM and QCPI for solving large-scale trajectory optimization methods. An open source, indirect trajectory optimization framework is developed with the goal of being a viable contender to state-of-the-art direct solvers such as GPOPS and DIDO. The framework, named beluga, leverages ICRM and QCPI along with traditional indirect optimal control theory. In its current form, as illustrated by the various examples in this dissertation, it has made significant advances in automating the use of indirect methods for trajectory optimization. Following on the path of popular and widely used scientific software projects such as SciPy [1] and Numpy [2], beluga is released under the permissive MIT license [3]. Being an open source project allows the community to contribute freely to the framework, further expanding its capabilities and allow faster integration of new advances to the state-of-the-art

A radial basis function method for solving optimal control problems.

This work presents two direct methods based on the radial basis function (RBF) interpolation and arbitrary discretization for solving continuous-time optimal control problems: RBF Collocation Method and RBF-Galerkin Method. Both methods take advantage of choosing any global RBF as the interpolant function and any arbitrary points (meshless or on a mesh) as the discretization points. The first approach is called the RBF collocation method, in which states and controls are parameterized using a global RBF, and constraints are satisfied at arbitrary discrete nodes (collocation points) to convert the continuous-time optimal control problem to a nonlinear programming (NLP) problem. The resulted NLP is quite sparse and can be efficiently solved by well-developed sparse solvers. The second proposed method is a hybrid approach combining RBF interpolation with Galerkin error projection for solving optimal control problems. The proposed solution, called the RBF-Galerkin method, applies a Galerkin projection to the residuals of the optimal control problem that make them orthogonal to every member of the RBF basis functions. Also, RBF-Galerkin costate mapping theorem will be developed describing an exact equivalency between the Karush–Kuhn–Tucker (KKT) conditions of the NLP problem resulted from the RBF-Galerkin method and discretized form of the first-order necessary conditions of the optimal control problem, if a set of conditions holds. Several examples are provided to verify the feasibility and viability of the RBF method and the RBF-Galerkin approach as means of finding accurate solutions to general optimal control problems. Then, the RBF-Galerkin method is applied to a very important drug dosing application: anemia management in chronic kidney disease. A multiple receding horizon control (MRHC) approach based on the RBF-Galerkin method is developed for individualized dosing of an anemia drug for hemodialysis patients. Simulation results are compared with a population-oriented clinical protocol as well as an individual-based control method for anemia management to investigate the efficacy of the proposed method

Scaling and Balancing for High-Performance Computation of Optimal Controls

The article of record may be found at ttps://doi.org/10.2514/1.G003382It is well known that proper scaling can increase the efficiency of computational problems. In this paper, we define and show that a balancing technique can substantially improve the computational efficiency of optimal-control algorithms. We also show that noncanonical scaling and balancing procedures may be used quite effectively to reduce the computational difficulty of some hard problems. These results have been used successfully for several flight and field operations at NASA and the U.S. Department of Defense. A surprising aspect of our analysis shows that it may be inadvisable to use autoscaling procedures employed in some software packages. The new results are agnostic to the specifics of the computational method; hence, they can be used to enhance the utility of any existing algorithm or software

Simulation-Based Sailboat Trajectory Optimization using On-Board Heterogeneous Computers

A dynamic programming-based algorithm adapted to on-board heterogeneouscomputers for simulation-based trajectory optimization was studied inthe context of high-performance sailing. The algorithm can efficiently utilizeall OpenCL-capable devices, starting the computation (if necessary, in singleprecision)on a GPU and finalizing it (if necessary, in double-precision) withthe use of a CPU. The serial and parallel versions of the algorithm are presentedin detail. Possible extensions of the basic algorithm are also described. Theexperimental results show that contemporary heterogeneous on-board/mobilecomputers can be treated as micro HPC platforms. They offer high performance(the OpenCL-capable GPU was found to accelerate the optimization routine 41fold) while remaining energy and cost efficient. The simulation-based approachhas the potential to give very accurate results, as the mathematical model uponwhich the simulator is based may be as complex as required. The black-box representedperformance measure and the use of OpenCL make the presentedapproach applicable to many trajectory optimization problems

Optimization Techniques for Long Term Active Debris Removal Mission Design Applications

Debris in Low Earth Orbit (LEO) poses a great risk to humanity’s access to space in the coming years. Seemingly the only way to prevent a full scale catastrophe rendering LEO unusable is to begin a series of active debris removal (ADR) missions. An approach for designing a series of active debris removal missions is presented in this thesis. This approach has multiple steps and begins with a large list of high risk, high mass debris objects. The first stage of the planning process entails the use of clustering algorithms to partition a large catalogue of orbits into groups of user defined sizes. It is shown that this operation is tantamount to histogram analysis in non-Euclidean spaces. The second stage of the approach then solves a dynamic traveling salesman problem to compute an order of visitation through each cluster. Lastly, a novel trajectory optimization technique is presented, using Chebyshev polynomials to approximate the dynamics of the system (rather than the states, which is typical). This technique is then used to further optimize the solution of the dynamic traveling salesman problem in order to arrive at a locally optimal solution to each multi-rendezvous problem

Optimal Sizing and Control of Hybrid Rocket Vehicles

In the present work, a genetic algorithm is used to optimize a hybrid rocket engine in order to minimize the propellant required for a specific mission. In a hybrid rocket engine, the mass flow rate of the oxidizer can be throttled to enhance the performance of the rocket. First, an analysis of the internal ballistics and the ascent trajectory has been carried out for different mass flow rates of the oxidizer as a function of time, for a fixed amount of oxidizer, in order to study the effect of throttling. Two equivalent problems are considered: in the first problem the amount of propellant is fixed, and we are seeking the oxidizer mass flow rate as the function of time such as to maximize the altitude. In the second problem, we obtain the mass flow rate of the oxidizer as a function of time in order to minimize the propellant required to reach a specific altitude. A genetic algorithm is used to find the best mass flow rate of the oxidizer. The optimization is carried out for two different regression rate laws, one depending only on the oxidizer mass flux rate and the other one depending on the mass flux rate of the oxidizer and the fuel. The results obtained in both cases are similar and show that the mass flow rate of the oxidizer should be maximized up to about one-third of the burn time and then decreased gradually. Using this mass flow rate of the oxidizer, we obtain the best initial oxidizer to fuel ratio in order to perform an optimal sizing of the rocket

Overview of System Identification with Focus on Inverse Modeling: Literature Review

The intention behind this literature review is to obtain knowledge about the current status in the field of system identification with special focus put on the inverse modelling step. There the parameters for a model are to be determined by taking data obtained from the true system into account. The application in mind is located in geophysics, especially oil reservoir engineering, so special focus is put on methods which are relevant for system identification problems that arise in that context. Nonetheless the review should be interesting for everybody who works on system identification problems.--- Die Intention des Literaturreviews ist eine Übersicht über den Bereich der Systemidentifikation, im speziellen den Bereich der inversen Modellierung, zu erhalten. In diesem Schritt werden Parameter für ein Modell durch Konditionierung auf gemessene Daten eines realen Systems bestimmt. Das Anwendungsgebiet ist im Bereich der Geophysik, im speziellen Erdöl-Reservoirs, angesiedelt. Daher werden besonders die dort genutzten Methoden betrachtet

Optimal Control Theory for Undergraduates

Dynamic optimization is widely used in financial economics, macroeconomics and resource economics. This is accounting for some tension between the undergraduate and graduate teaching of economics because most undergraduate programs still concentrate on static economic analysis. This paper shows how, with the help of the Microsoft Excel Solver tool, the principles of dynamic economics can be taught to students with minimal knowledge of calculus. As it is assumed that the reader has no prior knowledge of optimal control theory, some attention is paid to the main concepts of dynamic optimization.Optimal Control Theory, Economic Education, Microsoft Excel

Continuous Control Artificial Potential Function Methods and Optimal Control

Artificial potential function methods (APFMs) are a class of computationally inexpensive control methods for driving a system to a desired goal while avoiding obstacles. Although APFMs have been applied successfully to a wide range systems since the late 1980s, these control methods do have notable drawbacks. The general suboptimality of APFM results is one of these drawbacks, which is due to the fact that APFMs contain no cost function in their formulation. This thesis first develops a new continuous control APFM for fully actuated systems called the Velocity Artificial Potential Function (VAPF) Method, which causes the system velocity to converge to the negative gradient of an artificial potential function. Then, methods for increasing APFM optimality are studied. First, an investigation is undertaken to determine if placing an APFM into an optimal control framework is a practical way of addressing the suboptimality of APFMs. While effective at increasing optimality of APFM results, this approach proves to be too computationally expensive to be practical. Finally, the Adaptive Artificial Potential Function developed by Munoz is studied and implemented via the VAPF Method. This approach produce results with higher optimality than traditional APFMs but negligibly greater computational expense