405 research outputs found

    Unscented Autoencoder

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    The Variational Autoencoder (VAE) is a seminal approach in deep generative modeling with latent variables. Interpreting its reconstruction process as a nonlinear transformation of samples from the latent posterior distribution, we apply the Unscented Transform (UT) -- a well-known distribution approximation used in the Unscented Kalman Filter (UKF) from the field of filtering. A finite set of statistics called sigma points, sampled deterministically, provides a more informative and lower-variance posterior representation than the ubiquitous noise-scaling of the reparameterization trick, while ensuring higher-quality reconstruction. We further boost the performance by replacing the Kullback-Leibler (KL) divergence with the Wasserstein distribution metric that allows for a sharper posterior. Inspired by the two components, we derive a novel, deterministic-sampling flavor of the VAE, the Unscented Autoencoder (UAE), trained purely with regularization-like terms on the per-sample posterior. We empirically show competitive performance in Fr\'echet Inception Distance (FID) scores over closely-related models, in addition to a lower training variance than the VAE

    Bayesian inference for dynamic pose estimation using directional statistics

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    The dynamic pose of an object, where the object can represent a spacecraft, aircraft, or mobile robot, among other possibilities, is defined to be the position, velocity, attitude, and angular velocity of the object. A new method to perform dynamic pose estimation is developed that leverages directional statistics and operates under the Bayesian estimation framework, as opposed to the minimum mean square error (MMSE) framework that conventional methods employ. No small attitude uncertainty assumption is necessary using this method, and, therefore, a more accurate estimate of the state can be obtained when the attitude uncertainty is large. Two new state densities, termed the Gauss-Bingham and Bingham-Gauss mixture (BGM) densities, are developed that probabilistically represent a state vector comprised of an attitude quaternion and other Euclidean states on their natural manifold, the unit hypercylinder. When the Euclidean states consist of position, velocity, and angular velocity, the state vector represents the dynamic pose. An uncertainty propagation scheme is developed for a Gauss-Bingham-distributed state vector, and two demonstrations of this uncertainty propagation scheme are presented that show its applicability to quantify the uncertainty in dynamic pose, especially when the attitude uncertainty becomes large. The BGM filter is developed, which is an approximate Bayesian filter in which the true temporal and measurement evolution of the BGM density, as quantified by the Chapman-Kolmogorov equation and Bayes\u27 rule, are approximated by a BGM density. The parameters of the approximating BGM density are found via integral approximation on a component-wise basis, which is shown to be the Kullback-Leibler divergence optimal parameters of each component. The BGM filter is then applied to three simulations in order to compare its performance to a multiplicative Kalman filter and demonstrate its efficacy in estimating dynamic pose. The BGM filter is shown to be more statistically consistent than the multiplicative Kalman filter when the attitude uncertainty is large --Abstract, page iii

    Numerical Fitting-based Likelihood Calculation to Speed up the Particle Filter

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    The likelihood calculation of a vast number of particles is the computational bottleneck for the particle filter in applications where the observation information is rich. For fast computing the likelihood of particles, a numerical fitting approach is proposed to construct the Likelihood Probability Density Function (Li-PDF) by using a comparably small number of so-called fulcrums. The likelihood of particles is thereby analytically inferred, explicitly or implicitly, based on the Li-PDF instead of directly computed by utilizing the observation, which can significantly reduce the computation and enables real time filtering. The proposed approach guarantees the estimation quality when an appropriate fitting function and properly distributed fulcrums are used. The details for construction of the fitting function and fulcrums are addressed respectively in detail. In particular, to deal with multivariate fitting, the nonparametric kernel density estimator is presented which is flexible and convenient for implicit Li-PDF implementation. Simulation comparison with a variety of existing approaches on a benchmark 1-dimensional model and multi-dimensional robot localization and visual tracking demonstrate the validity of our approach.Comment: 42 pages, 17 figures, 4 tables and 1 appendix. This paper is a draft/preprint of one paper submitted to the IEEE Transaction

    Moment Estimation Using a Marginalized Transform

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    We present a method for estimating mean and covariance of a transformed Gaussian random variable. The method is based on evaluations of the transforming function and resembles the unscented transform and Gauss-Hermite integration in that respect. The information provided by the evaluations is used in a Bayesian framework to form a posterior description of the parameters in a model of the transforming function. Estimates are then derived by marginalizing these parameters from the analytical expression of the mean and covariance. An estimation algorithm, based on the assumption that the transforming function can be described using Hermite polynomials, is presented and applied to the non-linear filtering problem. The resulting marginalized transform (MT) estimator is compared to the cubature rule, the unscented transform and the divided difference estimator. The evaluations show that the presented method performs better than these methods, more specifically in estimating the covariance matrix. Contrary to the unscented transform, the resulting approximation of the covariance matrix is guaranteed to be positive-semidefinite

    Towards Efficient MPPI Trajectory Generation with Unscented Guidance: U-MPPI Control Strategy

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    The classical Model Predictive Path Integral (MPPI) control framework lacks reliable safety guarantees since it relies on a risk-neutral trajectory evaluation technique, which can present challenges for safety-critical applications such as autonomous driving. Additionally, if the majority of MPPI sampled trajectories concentrate in high-cost regions, it may generate an infeasible control sequence. To address this challenge, we propose the U-MPPI control strategy, a novel methodology that can effectively manage system uncertainties while integrating a more efficient trajectory sampling strategy. The core concept is to leverage the Unscented Transform (UT) to propagate not only the mean but also the covariance of the system dynamics, going beyond the traditional MPPI method. As a result, it introduces a novel and more efficient trajectory sampling strategy, significantly enhancing state-space exploration and ultimately reducing the risk of being trapped in local minima. Furthermore, by leveraging the uncertainty information provided by UT, we incorporate a risk-sensitive cost function that explicitly accounts for risk or uncertainty throughout the trajectory evaluation process, resulting in a more resilient control system capable of handling uncertain conditions. By conducting extensive simulations of 2D aggressive autonomous navigation in both known and unknown cluttered environments, we verify the efficiency and robustness of our proposed U-MPPI control strategy compared to the baseline MPPI. We further validate the practicality of U-MPPI through real-world demonstrations in unknown cluttered environments, showcasing its superior ability to incorporate both the UT and local costmap into the optimization problem without introducing additional complexity.Comment: This paper has 15 pages, 10 figures, 4 table

    Development of a Solution for Start-up Optimization of a Thermal Power Plant

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    This thesis covers optimizing the first phase of the start-up of a thermal power plant using Nonlinear Model Predictive Control (NMPC) and state estimation using an Unscented Kalman Filter (UKF). The start-up has been optimized in regards to time and fuel usage. The thesis is done as a joint project between Vattenfall and Modelon. Both NMPC and UKF are nonlinear methods and require a model of the power plant. The model used in this thesis has been developed in the language Modelica in a previous master thesis and has been extended and improved upon during this thesis. The optimization and simulation of the model required by the NMPC and UKF was done within the framework of JModelica.org. Another, more detailed, model of the power plant, developed by Vattenfall, was originally planned to be used as the process to be controlled. State estimation using the UKF has been successful, with a maximum mean absolute error of 0.7 % when estimating the states of the detailed model in a reference startup. When using the NMPC to control the optimization model itself, the simulated start-up time is 70 minutes faster compared to a reference start-up using the detailed model. This is more than half the time of the first phase of the start-up. The total firing power, which relates to the fuel amount, is also considerably less, with the optimized value being about 40 % of that in the reference soft start with the detailed model. Due to difficulties in initializing the detailed model, it was not possible to run it online together with the NMPC and UKF. Running the NMPC and UKF together on the optimization model worked, but the NMPC failed to find an optimal trajectory 8 out of 10 iterations. The conclusion is that the start-up has potential for optimization, but requires more robust models to work with

    Nonlinear Bayesian Estimation with Compactly Supported Wavelets

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    Bayesian estimation for nonlinear systems is still a challenging problem, as in general the type of the true probability density changes and the complexity increases over time. Hence, approximations of the occurring equations and/or of the underlying probability density functions are inevitable. In this paper, we propose an approximation of the conditional densities by wavelet expansions. This kind of representation allows a sparse set of characterizing coefficients, especially for smooth or piecewise smooth density functions. Besides its good approximation properties, fast algorithms operating on sparse vectors are applicable and thus, a good trade-off between approximation quality and run-time can be achieved. Moreover, due to its highly generic nature, it can be applied to a large class of nonlinear systems with a high modeling accuracy. In particular, the noise acting upon the system can be modeled by an arbitrary probability distribution and can influence the system in any way
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