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1. Biased Kernel Density Estimators for Chance Constrained Optimal Control Problems

   https://doi.org/10.23919/ACC45564.2020.9148040  

   Keil, Rachel E.; Miller, Alexander; Kumar, Mrinal; Rao, Anil V. (July 2020, 2020 American Control Conference)

   null (Ed.)

   A method is developed for transforming chance constrained optimization problems to a form numerically solvable. The transformation is accomplished by reformulating the chance constraints as nonlinear constraints using a method that combines the previously developed Split-Bernstein approximation and kernel density estimator (KDE) methods. The Split-Bernstein approximation in a particular form is a biased kernel density estimator. The bias of this kernel leads to a nonlinear approximation that does not violate the bounds of the original chance constraint. The method of applying biased KDEs to reformulate chance constraints as nonlinear constraints transforms the chance constrained optimization problem to a deterministic optimization problems that retains key properties of the chance constrained optimization problem and can be solved numerically. This method can be applied to chance constrained optimal control problems. As a result, the Split-Bernstein and Gaussian kernels are applied to a chance constrained optimal control problem and the results are compared. 

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2. Data-Driven Chance Constrained Control using Kernel Distribution Embeddings

   Thorpe, Adam; and Oishi, Meeko (May 2021, Proceedings of The 4th Annual Learning for Dynamics and Control Conference)

   We present a data-driven algorithm for efficiently computing stochastic control policies for general joint chance constrained optimal control problems. Our approach leverages the theory of kernel distribution embeddings, which allows representing expectation operators as inner products in a reproducing kernel Hilbert space. This framework enables approximately reformulating the original problem using a dataset of observed trajectories from the system without imposing prior assumptions on the parameterization of the system dynamics or the structure of the uncertainty. By optimizing over a finite subset of stochastic open-loop control trajectories, we relax the original problem to a linear program over the control parameters that can be efficiently solved using standard convex optimization techniques. We demonstrate our proposed approach in simulation on a system with nonlinear non-Markovian dynamics navigating in a cluttered environment. 

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3. A Robust Optimal Guidance Strategy for Mars Entry

   Palmer, Emily M.; Rao, Anil V. (August 2022, 2022 AAS/AIAA Astrodynamics Specialist Conference)

   A robust optimal guidance strategy is proposed. The guidance strategy is designed to reduce the possibility of violations in inequality path constraints in the presence of modeling errors and perturbations. The guidance strategy solves a constrained nonlinear optimal control problem at the start of every guidance cycle. In order to reduce the possibility of path constraint violations, the objective functional for the optimal control problem is modified at the start of a guidance cycle if it is found that the solution lies within a user-specified threshold of a path constraint limit. The modified objective functional is designed such that it maximizes the margin in the solution relative to the path constraint limit that could potentially be violated in the future. The method is validated on a path-constrained Mars entry problem where the reference model and the perturbed model differ in their atmospheric density. It is found for the example studied that the approach significantly improves the path constraint margin and maintains feasibility relative to a guidance approach that maintains the original objective functional for each guidance update. 

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4. Backup Plan Constrained Model Predictive Control with Guaranteed Stability

   https://doi.org/10.2514/1.G007627  

   Tao, Ran; Kim, Hunmin; Yoon, Hyung-Jin; Wan, Wenbin; Hovakimyan, Naira; Sha, Lui; Voulgaris, Petros (October 2023, Journal of Guidance, Control, and Dynamics)

   Safe control designs for robotic systems remain challenging because of the difficulties of explicitly solving optimal control with nonlinear dynamics perturbed by stochastic noise. However, recent technological advances in computing devices enable online optimization or sampling-based methods to solve control problems. For example, Control Barrier Functions (CBFs) have been proposed to numerically solve convex optimization problems that ensure the control input to stay in the safe set. Model Predictive Path Integral (MPPI) control uses forward sampling of stochastic differential equations to solve optimal control problems online. Both control algorithms are widely used for nonlinear systems because they avoid calculating the derivatives of the nonlinear dynamic functions. In this paper, we use Stochastic Control Barrier Functions (SCBFs) constraints to limit sample regions in the samplingbased algorithm, ensuring safety in a probabilistic sense and improving sample efficiency with a stochastic differential equation. We also show that our algorithm needs fewer samples than the original MPPI algorithm does by providing a sampling complexity analysis. 

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5. Structure Detection Method for Solving State-Variable Inequality Path Constrained Optimal Control Problems

   Byczkowski, Cale A.; Rao, Anil V. (January 2023, American Astronautical Society)

   A structure detection method is developed for solving state-variable inequality path con- strained optimal control problems. The method obtains estimates of activation and deactiva- tion times of active state-variable inequality path constraints (SVICs), and subsequently al- lows for the times to be included as decision variables in the optimization process. Once the identification step is completed, the method partitions the problem into a multiple-domain formulation consisting of constrained and unconstrained domains. Within each domain, Legendre-Gauss-Radau (LGR) orthogonal direct collocation is used to transcribe the infinite- dimensional optimal control problem into a finite-dimensional nonlinear programming (NLP) problem. Within constrained domains, the corresponding time derivative of the active SVICs that are explicit in the control are enforced as equality path constraints, and at the beginning of the constrained domains, the necessary tangency conditions are enforced. The accuracy of the proposed method is demonstrated on a well-known optimal control problem where the analytical solution contains a state constrained arc. 

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