Modified Newton’s method in the leapfrog method for mobile robot path planning

The problem of determining an optimal trajectory for an autonomous mobile robot in an environment with obstacles is considered. The Leapfrog approach is used to solve the ensuing system of equations derived from the first-order optimality conditions of the Pontryagin’s Minimum Principle. A comparison is made between a case in which the classical Newton Method and the Modified Newton Method are used in the shooting method for solving the two-point boundary value problem in the inner loop of the Leapfrog algorithm. It can be observed that with this modification there is an improvement in the convergence rate of the Leapfrog algorithm in general.http://www.springer.comseries/111562019-03-20hj2018Mathematics and Applied Mathematic

Convergence analysis of leapfrog for geodesics

Geodesics are of fundamental interest in mathematics, physics, computer science, and many other subjects. The so-called leapfrog algorithm was proposed in [L. Noakes, J. Aust. Math. Soc., 65 (1998), pp. 37-50] (but not named there as such) to find geodesics joining two given points x0 and x1 on a path-connected complete Riemannian manifold. The basic idea is to choose some junctions between x0 and x1 that can be joined by geodesics locally and then adjust these junctions. It was proved that the sequence of piecewise geodesics { k}k ≥ 1 generated by this algorithm converges to a geodesic joining x0 and x1. The present paper investigates leapfrog\u27s convergence rate i,n of ith junction depending on the manifold M. A relationship is found with the maximal root n of a polynomial of degree n-3, where n (n \u3e 3) is the number of geodesic segments. That is, the minimal i,n is upper bounded by n(1 + c+), where c+ is a sufficiently small positive constant depending on the curvature of the manifold M. Moreover, we show that n increases as n increases. These results are illustrated by implementing leapfrog on two Riemannian manifolds: the unit 2-sphere and the manifold of all 2 × 2 symmetric positive definite matrices

Learning Generalized Relational Heuristic Networks for Model-Agnostic Planning

Computing goal-directed behavior (sequential decision-making, or planning) is essential to designing efficient AI systems. Due to the computational complexity of planning, current approaches rely primarily upon hand-coded symbolic domain models and hand-coded heuristic-function generators for efficiency. Learned heuristics for such problems have been of limited utility as they are difficult to apply to problems with objects and object quantities that are significantly different from those in the training data. This paper develops a new approach for learning generalized heuristics in the absence of symbolic domain models using deep neural networks that utilize an input predicate vocabulary but are agnostic to object names and quantities. It uses an abstract state representation to facilitate data efficient, generalizable learning. Empirical evaluation on a range of benchmark domains show that in contrast to prior approaches, generalized heuristics computed by this method can be transferred easily to problems with different objects and with object quantities much larger than those in the training data.Comment: Submitted to NIPS 2020, 11 pages, 3 figure

Focal Spot, Winter 2005/2006

https://digitalcommons.wustl.edu/focal_spot_archives/1101/thumbnail.jp

Inference Algorithms and Sensorimotor Representations in Brains and Machines

Animals function in a 3D world in which survival depends on robust, well-controlled actions. Historically, researchers in Artificial Intelligence (AI) and neuroscience have explored sensory and motor systems independently. There is a growing body of literature in AI and neuroscience to suggest that they actually work in tandem. While there has been a great deal of work on vision and audition as sensory modalities in these fields, one could argue that a more fundamental modality in biology is haptics, or the sense of touch. In this thesis, we will look at building computational models that integrate tactile sensing with other sensory modalities to perform manipulation-like tasks in robots and discrimination tasks in mice. We will also explore the problem of inference through the lens of Markov Chain Monte Carlo methods (MCMC). We elaborate on the ideas discussed in this thesis in the introduction presented in Chapter 1. A challenging problem one often faces when applying probabilistic mathematical models to the study of sensory-motor systems and other problems involving learning of inference is sampling. Hamiltonian Markov Chain Monte Carlo (HMC) algorithms can efficiently draw representative samples from complex probabilistic models. Most MCMC methods rely on detailed balance to ensure that we can sample from the correct distribution. This constraint can be relaxed in discrete state spaces such as those employed by HMC type methods. In Chapter 2, we study HMC methods without detailed balance to explore faster convergence. Markov jump processes are stochastic processes on discrete state space but continuous in time. In Chapter 3, we use Markov Jump Processes to simulate waiting times along with generalized detailed balance. This waiting time ,we show, helps generate samples faster. Most MCMC methods are plagued by slow simulation times on discrete computing systems. In Chapter 4, we explore HMC in analog circuits where the problem of generating samples from a distribution is mapped to the problem of sampling charge in a capacitor.The second half of this dissertation focuses on the role of haptics in perception and action. Manipulation is a fundamental problem for artificial and biological agents. High dimensional actuators (say, fingers, trunks,etc) are really hard to control. In Chapter 5, we present an approach to learn to actuate dexterous manipulators to grasp objects in simulation. Haptics as a sensory modality is critical to many manipulation tasks. Employing haptics in high dimensional dextrous actuators is challenging. In Chapter 6, we explore how intrinsic curiosity and haptics can be used to learn exploration strategies for discrimination of objects with dextrous hands. A key component to make tactile sensing a possibility is the availability of cheap, efficient, scalable hardware. Chapter 7 presents results for tactile servoing using a physical gelsight sensor. Traditional neuroscience texts delineate sensory and motor systems as two independent systems yet recent results suggest that this may not be entirely complete. That is, there is evidence to suggest that the representations in the cortex is more distributed than is accepted. Finally in Chapter 8, we explore building a computational model of spiking neural data collected from both the barrel and motor cortices during free and active whisking. These works help towards understanding sensorimotor representations in the context of haptics and high dimensional controls. We conclude with a discussion on future directions in Chapter 9

Generalised Bayesian matrix factorisation models

Factor analysis and related models for probabilistic matrix factorisation are of central importance to the unsupervised analysis of data, with a colourful history more than a century long. Probabilistic models for matrix factorisation allow us to explore the underlying structure in data, and have relevance in a vast number of application areas including collaborative filtering, source separation, missing data imputation, gene expression analysis, information retrieval, computational finance and computer vision, amongst others. This thesis develops generalisations of matrix factorisation models that advance our understanding and enhance the applicability of this important class of models. The generalisation of models for matrix factorisation focuses on three concerns: widening the applicability of latent variable models to the diverse types of data that are currently available; considering alternative structural forms in the underlying representations that are inferred; and including higher order data structures into the matrix factorisation framework. These three issues reflect the reality of modern data analysis and we develop new models that allow for a principled exploration and use of data in these settings. We place emphasis on Bayesian approaches to learning and the advantages that come with the Bayesian methodology. Our port of departure is a generalisation of latent variable models to members of the exponential family of distributions. This generalisation allows for the analysis of data that may be real-valued, binary, counts, non-negative or a heterogeneous set of these data types. The model unifies various existing models and constructs for unsupervised settings, the complementary framework to the generalised linear models in regression. Moving to structural considerations, we develop Bayesian methods for learning sparse latent representations. We define ideas of weakly and strongly sparse vectors and investigate the classes of prior distributions that give rise to these forms of sparsity, namely the scale-mixture of Gaussians and the spike-and-slab distribution. Based on these sparsity favouring priors, we develop and compare methods for sparse matrix factorisation and present the first comparison of these sparse learning approaches. As a second structural consideration, we develop models with the ability to generate correlated binary vectors. Moment-matching is used to allow binary data with specified correlation to be generated, based on dichotomisation of the Gaussian distribution. We then develop a novel and simple method for binary PCA based on Gaussian dichotomisation. The third generalisation considers the extension of matrix factorisation models to multi-dimensional arrays of data that are increasingly prevalent. We develop the first Bayesian model for non-negative tensor factorisation and explore the relationship between this model and the previously described models for matrix factorisation.Supported by a Commonwealth Scholarship awarded by the Commonwealth Scholarship and Fellowship Programme (CSFP) [Award number ZACS-2207-363] Supported by award from the National Research Foundation, South Africa (NRF) [Award number SFH2007072200001

Viability of Numerical Full-Wave Techniques in Telecommunication Channel Modelling

In telecommunication channel modelling the wavelength is small compared to the physical features of interest, therefore deterministic ray tracing techniques provide solutions that are more efficient, faster and still within time constraints than current numerical full-wave techniques. Solving fundamental Maxwell's equations is at the core of computational electrodynamics and best suited for modelling electrical field interactions with physical objects where characteristic dimensions of a computing domain is on the order of a few wavelengths in size. However, extreme communication speeds, wireless access points closer to the user and smaller pico and femto cells will require increased accuracy in predicting and planning wireless signals, testing the accuracy limits of the ray tracing methods. The increased computing capabilities and the demand for better characterization of communication channels that span smaller geographical areas make numerical full-wave techniques attractive alternative even for larger problems. The paper surveys ways of overcoming excessive time requirements of numerical full-wave techniques while providing acceptable channel modelling accuracy for the smallest radio cells and possibly wider. We identify several research paths that could lead to improved channel modelling, including numerical algorithm adaptations for large-scale problems, alternative finite-difference approaches, such as meshless methods, and dedicated parallel hardware, possibly as a realization of a dataflow machine