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1. A Monte Carlo Analysis of Contingency Optimal Guidance for Mars Entry

   Palmer, Emily M.; Rao, Anil V. (January 2023, American Astronautical Society)

   A Monte Carlo analysis of a contingency optimal guidance strategy is conducted. The guidance strategy is applied to a Mars Entry problem in which it is assumed that the surface level atmospheric density is a random variable. First, a nominal guidance strategy is employed such that the optimal control problem is re-solved at constant guidance cycles. When the trajectory lies within a particular distance from a path constraint boundary, the nominal guidance strategy is replaced with a contingency guidance strategy, where the contingency guidance strategy attempts to prevent a violation in the the relevant path constraint. The contingency guidance strategy utilizes the reference optimal control problem formulation, but modifies the objective functional to maximize the margin between the path constraint limit and path constraint function value. The ability of the contingency guidance strategy to prevent violations in the path constraints is assessed via a Monte Carlo simulation. 

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2. 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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3. Desensitized Optimal Guidance Using Adaptive Radau Collocation

   Winkler, K L; Rao, A V (December 2024, IEEE Conference on Decision and Control)

   Abstract—An optimal guidance method is developed that reduces sensitivity to parametric uncertainties in the dynamic model. The method combines a previously developed method for guidance and control using adaptive Legendre-Gauss-Radau (LGR) collocation and a previously developed approach for desensitized optimal control. Guidance updates are performed such that the desensitized optimal control problem is re-solved on the remaining horizon at the start of each guidance cycle. The effectiveness of the method is demonstrated on a simple example using Monte Carlo simulation. The application of the method results in a smaller final state error distribution when compared to desensitized optimal control without guidance as well as a previously developed method for optimal guidance and control. 

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4. Adaptive Risk Sensitive Path Integral for Model Predictive Control via Reinforcement Learning

   https://doi.org/10.1109/MED59994.2023.10185876  

   Yoon, Hyung-Jin; Tao, Chuyuan; Kim, Hunmin; Hovakimyan, Naira; Voulgaris, Petros (June 2023, editerranean Conference on Control and Automation (MED))

   We propose a reinforcement learning framework where an agent uses an internal nominal model for stochastic model predictive control (MPC) while compensating for a disturbance. Our work builds on the existing risk-aware optimal control with stochastic differential equations (SDEs) that aims to deal with such disturbance. However, the risk sensitivity and the noise strength of the nominal SDE in the riskaware optimal control are often heuristically chosen. In the proposed framework, the risk-taking policy determines the behavior of the MPC to be risk-seeking (exploration) or riskaverse (exploitation). Specifcally, we employ the risk-aware path integral control that can be implemented as a Monte-Carlo (MC) sampling with fast parallel simulations using a GPU. The MC sampling implementations of the MPC have been successful in robotic applications due to their real-time computation capability. The proposed framework that adapts the noise model and the risk sensitivity outperforms the standard model predictive path integ 

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5. Method for solving chance constrained optimal control problems using biased kernel density estimators

   https://doi.org/10.1002/oca.2675  

   Keil, Rachel_E; T Miller, Alexander; Kumar, Mrinal; Rao, Anil_V (October 2020, Optimal Control Applications and Methods)

   Summary A method is developed to numerically solve chance constrained optimal control problems. The chance constraints are reformulated as nonlinear constraints that retain the probability properties of the original constraint. The reformulation transforms the chance constrained optimal control problem into a deterministic optimal control problem that can be solved numerically. The new method developed in this paper approximates the chance constraints using Markov Chain Monte Carlo sampling and kernel density estimators whose kernels have integral functions that bound the indicator function. The nonlinear constraints resulting from the application of kernel density estimators are designed with bounds that do not violate the bounds of the original chance constraint. The method is tested on a nontrivial chance constrained modification of a soft lunar landing optimal control problem and the results are compared with results obtained using a conservative deterministic formulation of the optimal control problem. Additionally, the method is tested on a complex chance constrained unmanned aerial vehicle problem. The results show that this new method can be used to reliably solve chance constrained optimal control problems. 

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