Memory Clustering Using Persistent Homology for Multimodality- and Discontinuity-Sensitive Learning of Optimal Control Warm-Starts

Shooting methods are an efficient approach to solving nonlinear optimal control problems. As they use local optimization, they exhibit favorable convergence when initialized with a good warm-start but may not converge at all if provided with a poor initial guess. Recent work has focused on providing an initial guess from a learned model trained on samples generated during an offline exploration of the problem space. However, in practice the solutions contain discontinuities introduced by system dynamics or the environment. Additionally, in many cases multiple equally suitable, i.e., multi-modal, solutions exist to solve a problem. Classic learning approaches smooth across the boundary of these discontinuities and thus generalize poorly. In this work, we apply tools from algebraic topology to extract information on the underlying structure of the solution space. In particular, we introduce a method based on persistent homology to automatically cluster the dataset of precomputed solutions to obtain different candidate initial guesses. We then train a Mixture-of-Experts within each cluster to predict state and control trajectories to warm-start the optimal control solver and provide a comparison with modality-agnostic learning. We demonstrate our method on a cart-pole toy problem and a quadrotor avoiding obstacles, and show that clustering samples based on inherent structure improves the warm-start quality.Comment: 12 pages, 10 figures, accepted as a regular paper in IEEE Transactions on Robotics (T-RO). Supplementary video: https://youtu.be/lUULTWCFxY8 Code: https://github.com/wxmerkt/topological_memory_clustering The first two authors contributed equall

Exploiting structures of trajectory optimization for efficient optimal motion planning

Trajectory optimization is an important tool for optimal motion planning due to its flexibility in cost design, capability to handle complex constraints, and optimality certification. It has been widely used in robotic applications such as autonomous vehicles, unmanned aerial vehicles, humanoid robots, and highly agile robots. However, practical robotic applications often possess nonlinear dynamics and non-convex constraints and cost functions, which makes the trajectory optimization problem usually difficult to be efficiently solved to global optimum. The long computation time, possibility of non-convergence, and existence of local optima impose significant challenges to applying trajectory optimization in reactive tasks with requirements of real-time replanning. In this thesis, two structures of optimization problems are exploited to significantly improve the efficiency, i.e. computation time, reliability, i.e. success rate, and optimality, i.e. quality of the solution. The first structure is the existence of a convex sub-problem, i.e. the problem becomes convex if a subset of optimization variables is fixed and removed from the optimization. This structure exists in a wide variety of problems, especially where decomposition of spatial and temporal variables may result in convex sub-problem. A bilevel optimization framework is proposed that optimizes the subset and its complement hierarchically where the upper level optimizes the subset with convex constraints and the lower level uses convex optimization to solve its complement. The key is to use the solution of the lower level problem to compute analytic gradients for the upper-level problem. The bilevel framework is reliable due to its convex lower problem, efficient due to its simple upper problem, and yields better solutions than alternatives, although the existing requirement of convex sub-problem is generally too strict for many applications. The second structure is the local continuity of the argmin function for parametric optimization problems which map from problem parameters to the corresponding optimal solutions. The argmin function can be approximated from data which is collected offline by sampling the problem parameters and solving them to optima. Three approaches are proposed to learn the argmin from data each suited best for distinct applications. The nearest-neighbor optimal control searches problems with similar parameters and uses their solution to initialize nonlinear optimization. For problems with globally continuous argmin, neural networks can be used to learn from data and a few steps of convex optimization can further improve their predictions. As for problems with discontinuous argmin, mixture of experts (MoE) models are used. The MoE contains several experts and a classifier and is trained by splitting the data first according to discontinuity of argmin and then training each expert independently. Both empirical $k$-Means and theoretical topological data analysis approaches are explored for discontinuity identification and finding suitable data splits. Both methods result in data splits that help train MoE models that outperform the discontinuity-agnostic learning pipeline using standard neural networks. The trajectory learning approach is efficient since it only requires model evaluation to compute a trajectory, reliable since the MoE model is accurate after correctly handling discontinuity, and optimal since the data are collected offline and solved to optimal. Moreover, this local continuity structure is less restrictive and exists for a wide range of non-degenerate problems. The exploitation of these two structures helps build an efficient optimal motion planner with high reliability

Detection of presence, position and correct positioning of components on a PCB by image processing

Un recorrido por el funcionamiento del procesamiento de imagen y sus diferentes procesos, entrando en diferentes funcionamiento matemáticos de los mismos. Además, voy un poco más allá entrando en procesos adyacentes no imprescindibles del proceso de procesar una imagen, como puede ser el tratamiento que se hace del color o cuando se realiza un realce de una imagen durante el procesamiento de esta. Por último, se cuenta el funcionamiento de las redes neuronales, tanto como funciona matemática y lógicamente una neurona por si sola y un esquema de la misma. Añadiendo como se realizaría una red neuronal para el procesamiento de imagen en Python.Departamento de Organización de Empresas y Comercialización e Investigación de MercadosGrado en Ingeniería en Electrónica Industrial y Automátic

Learning Interaction Primitives for Biomechanical Prediction

abstract: This dissertation is focused on developing an algorithm to provide current state estimation and future state predictions for biomechanical human walking features. The goal is to develop a system which is capable of evaluating the current action a subject is taking while walking and then use this to predict the future states of biomechanical features. This work focuses on the exploration and analysis of Interaction Primitives (Amor er al, 2014) and their relevance to biomechanical prediction for human walking. Built on the framework of Probabilistic Movement Primitives, Interaction Primitives utilize an EKF SLAM algorithm to localize and map a distribution over the weights of a set of basis functions. The prediction properties of Bayesian Interaction Primitives were utilized to predict real-time foot forces from a 9 degrees of freedom IMUs mounted to a subjects tibias. This method shows that real-time human biomechanical features can be predicted and have a promising link to real-time controls applications.Dissertation/ThesisMasters Thesis Electrical Engineering 201

Adapting and Mitigating to Climate Change: Balancing the Choice under Uncertainty

Nowadays, as stressed by important strategic documents like for instance the 2009 EU White Paper on Adaptation or the recent 2009 “Copenhagen Accord”, it is amply recognized that both mitigation and adaptation strategies are necessary to combat climate change. This paper enriches the rapidly expanding literature trying to devise normative indications on the optimal combination of the two introducing the role of catastrophic and spatial uncertainty related to climate change damages. Applying a modified version of the Nordhaus’ Regional Dynamic Integrated Model of Climate and the Economy it is shown that in both cases uncertainty works in the direction to make mitigation a more attractive strategy than adaptation. When catastrophic uncertainty is concerned mitigation becomes relatively more important as, by curbing emissions, it helps to reduce temperature increase and hence the probability of the occurrence of the event. Adaptation on the contrary has no impact on this. It is also shown that optimal mitigation responses are much less sensitive than adaptation responses to spatial uncertainty. Mitigation responds to global damages, while adaptation to local damages. The first, being aggregated, change less than the second in the presence of spatial uncertainty as higher expected losses in some regions are compensated by lower expected losses in other. Accordingly, mitigation changes less than adaptation. Thus if it cannot be really claimed that spatial uncertainty increases the weight of mitigation respect to that of adaptation, however its presence makes mitigation a “safer” or more robust strategy to a policy decision maker than adaptation.Climate Change, Mitigation, Adaptation, Uncertainty, Integrated Assessment Model