Mathematics, Systems and Robotics Seminar 

After introducing the notion of algebraic surface, some ideas related to the classification of complex algebraic surfaces will be explained.

The usual problem in multi-view reconstruction consists of estimating the 3D shape of a rigid object from 2D images taken from diferent viewpoints. Of course several other variants of the problem have been tackled, particularly when dealing with non-rigid objects. This seminar will be about reconstructing from 2d images an unknown 2d object isometrically embedded in 3D space.

The best way to imagine it is to think of an image sequence taken of a waving flag. Althought the 2D object (the flag) is the same, at each time instant its 3D embedding changes due to the waving effect.

The intent is to present some preliminary results recently obtained, but the problem is not yet completely solved.

Many applications require a computer representation of a two-dimensional (2D) shape, usually described by a set of 2D points. The challenge of this representation is that it must not only capture the characteristics of the shape but also be invariant to relevant transformations. Invariance to geometric transformations such as translation, rotation and scale, has received attention in the past, usually under the assumption that the points are previously labeled, i.e., that the shape is characterized by an ordered set of landmarks. However, in many practical scenarios the points describing the shape are obtained from an automatic process, e.g., edge detection, thus without natural ordering. Obviously, the combinatorial problem of computing the correspondences between the points of two shapes in the presence of geometrical distortions becomes a quagmire when the number of points is large. Within our framework, a 2D shape is mapped to an analytic function on the complex plane, leading to what we call its analytic signature (ANSIG), circumventing the combinatorial search. We further show how easy it is to factor out geometric transformations when comparing shapes using the ANSIG representation. We illustrate these ANSIG capabilities for shape-based image classification, particularly in automatic trademark retrieval.

To a given hybrid system we associate its hybrid manifold and the corresponding flow on it. This enables us to study hybrid systems as (possibly non-smooth) dynamical systems.

We will talk about the Mather problem in a generalized setting and how this setting is motivated by a optimization problem. We start defining the appropriated setting, next we will use the duality to establish the existence of Mather measures. Finally we use viscosity solution to understand the support of such measures.

Planning is the process of computing a sequence of actions that fulfill a given task as well as possible. It is a crucial part of any intelligent agent; human, robot or software agent alike. In real-world planning an agent has to deal with several sources of uncertainty. First of all, the agent might be uncertain regarding the exact consequence of executing a particular action. Furthermore, the agent's sensors may be noisy or provide only a limited view of the environment. Partially observable Markov decision processes (POMDPs) provide a mathematical framework for acting optimally in such partially observable and stochastic environments. In this talk, we will introduce POMDPs and give an overview of available optimal and sub-optimal solution methods.

We address the problem of stabilizing systems that evolve on the Special Euclidean Group $\mathrm{SE}(3)$. The proposed solution consists of an output-feedback controller that guarantees almost global asymptotic stability (GAS) of the desired equilibrium point, in the sense that the equilibrium point is stable and we have convergence for all initial conditions except for those in a set of measure zero. The output vector is formed by the position coordinates, expressed in the body frame, of a collection of landmarks fixed in the environment. As such, the method can be used to perform a variety of robotic tasks such as positioning a manipulator’s end-effector with respect to an object to be grasped or landing an UAV over a predefined target using onboard sensor measurements. The resulting closed-loop system exhibits the following properties:

1. the position and orientation subsystems are decoupled,
2. the position error is globally exponentially stable, and
3. the orientation error is almost globally exponentially stable.

Results are also provided that allow one to select landmark configurations so as to control how the position and orientation of the rigid body converge to their final equilibrium values. Building on these results, a vision-based controller is proposed that not only guarantees almost GAS of the target configuration, but also ensures the positive invariance of a subset of $\mathrm{SE}(3)$, specifically defined to enforce feature visibility throughout the closed-loop trajectories of the camera.

Blood coagulation is one of the important defense mechanisms preventing the loss of blood following a vascular injury. When the endothelium is damaged a complex physiological process called hemostasis is set into action: the blood vessel diameter is diminished slowing bleeding, blood platelets get activated and a complex sequence of chemical reactions occurs, leading to the formation of a fibrin clot (thrombus) localized at the site of vessel wall damage. The process of platelet activation and blood coagulation is quite complex and not yet completely understood. A number of researchers have attempted to tackle this challenging problem. Recently a phenomenological comprehensive model for clot formation and lysis in flowing blood that extends existing models to integrate biochemical, physiologic and rheological factors, has been developed. The aim of this talk is to present a numerical method and results for this model. Three-dimensional numerical simulations are obtained in a straight vessel for a shear-thinning non-Newtonian blood flow model coupled with a system of advection-reaction diffusion equations, using a finite volume semi-discretization in space, on structured grids, and a multistage scheme for time integration. These are preliminary results to validate the model and the numerical code.

In this talk we will survey some new problems in control theory: relaxation of optimal control problems, homogenization, and discuss some recent related problems studied by P.L. Lions: mean field games.

A notória TLEBS (reforma terminológica no ensino de português ) é condenada ou defendida como pouco, nada ou pouco 'científica'. Referem-se ciências da filosofia, ciências da linguagem, ciências da religião, ciências sociais, ciências políticas, ciências económicas, ciências jurídicas (muitas.), etc., etc.

Formam-se comissões "científicas" para avaliação de funcionários, adornam-se as instituições de ensino com indispensáveis conselhos "científicos".

Os conteúdos dos livros (não importa a sua natureza) são mais ou menos "cientificamente" correctos. A Fundação da Ciência e da Tecnologia do Ministério da Ciência e Ensino Superior português patrocina um prémio para a melhor tradução científica e concede-o, em 2007, à tradução da "Ciência Nova" do filósofo Giambattista Vico!!! Não deixa de ser irónico o episódio sabendo-se que, por princípio, Vico negava a possibilidade da ciência da natureza. Temos enfim "ciência" aqui, ali e acolá, em todo o sítio ciência. E, de facto, não se pode dizer que estes usos estejam incorrectos. A scientia para os latinos podia referir-se ao saber, à erudição, à doutrina, ao conhecimento, à inteligência, à habilidade, à aptidão. Sciens non faciam significava não fazer de propósito, não ter intenção. Hoje ainda continuamos a dizer "estou ciente disso", e fala-se, compreensivelmente, da "ciência do bem e do mal", por exemplo. Podemos falar, em bom português, da ciência do barbeiro, do serralheiro, do canalizador mas suspeitamos que um "cientista" social, para só falar dessa espécie, não apreciaria esta associação às classes laborais. Assim, o termo cientista pode ser usado quase como se fosse um título nobiliárquico. Para além da questão semântica, no entanto, levantam-se outras questões mais importantes, complexas e menos inocentes. Um problema constante é o da demarcação da ciência de outro tipo de actividades. é quase um falso problema devido à sua simplicidade. É certo que na base da ciência está uma crença, como foi realçado principalmente pelos importantes filósofos Hume e Berkeley, a da existência dum mundo exterior ao do pensamento do homem, acessível pelos sentidos humanos e de que não se pode logicamente demonstrar a existência. No entanto, como disse o grande matemático e famoso filósofo Alfred North Whitehead os únicos seres humanos que não assumem a existência desse mundo exterior como realidade estão num asilo de "lunáticos". Ou, poderíamos acrescentar, em certas faculdades. Depois, a ciência desenvolve-se obrigatoriamente em dois planos: o teórico, o pensamento, e a realidade que é observada ou representada pelas experiências e que, como sublinharam sóbrios pensadores como Charles Peirce ou Albert Einstein, é simplesmente "aquilo" que é independente do que pode pensar dele um indivíduo "qualquer". Em ciência tem de haver uma correspondência obrigatória, necessária entre a teoria, o pensamento, e a realidade. Este é o problema mais complexo da verificação, da validação ou da falseação. Tudo isto se foi tornando perfeitamente claro principalmente no século XIX mas arranca mais claramente desde o século XVI com Kepler, Galileu, Newton, Huyghems e colaboradores e discípulos. Só no século XIX em 1833 o reverendo William Whewell, por solicitação do poeta Coleridge propõe a palavra cientista para o indivíduo que contribui para a ciência. Quanto ao que era ciência julgou-se dispensável a sua definição, tão clara se julgava a sua natureza e exigências. Não se contava, no entanto, com as ideologias, as divagações e os delírios provocados pela quimera da originalidade. Em particular, as ideologias funcionam muitas vezes como agentes ancilosantes da actividade intelectual disciplinada que a ciência requer. As consequências podem ser mais do que lamentáveis. Dão-se exemplos.

This talk addresses a possible application of Rational Trigonometry to Robotics, more specifically, to Robot Kinematics. Rational Trigonometry is a relatively new mathematical framework for trigonometric analysis, yet to be widely accepted. It replaces many classic trigonometric concepts with new ones and has as its main concepts quadrance (square of distance) and spread (square of the sine of an angle). A new model for direct kinematics is described, introducing new concepts such as Reference Frame Matrices, Structure Matrices or Sensor-actuator data conversion matrices. Series and parallel robotic structures are discussed and exemplified.

In this talk we recall a little bit of the history of the Calculus of Variations and present some results concerning the shape of energy-minimizers under some classical constrains.

We study a countable class of real-valued functions inductively defined from a basic set of trivial functions by composition, solving first-order differential equations and the taking of infinite limits. This class is the analytical counterpart of Kleene's partial recursive functions. By counting the number of nested limits required to define a function, this class can be stratified by a potentially infinite hierarchy - a hierarchy of infinite limits. In the first meaningful level of the hierarchy we have the extensions of classical primitive recursive functions. In the next level we find partial recursive functions, and in the following level we find the solution to the halting problem. We use methods from numerical analysis to show that the hierarchy does not collapse, concluding that the taking of infinite limits can always produce new functions from functions in the previous levels of the hierarchy.

Continuous-time marked point processes appear in many areas of science and engineering including queuing theory, seismology, neuroscience and finance. In numerous applications, these point processes are unobserved but actually drive an observation process. Here, we are interested in optimal sequential Bayesian estimation of such partially observed point processes. This class of filtering problems is non-standard as there is typically no underlying Markov structure and the likelihood function relating the observations to the point process has a complex form. Hence, except in very specific cases it is impossible to solve them in closed-form. Using this problem, I present Sequential Monte Carlo samplers which have been recently proposed as a generalization of Sequential Monte Carlo (a.k.a Particle Filters) and an alternative to MCMC to perform sequential estimation. We develop an original trans-dimensional Sequential Monte Carlo method to address this class of problems. An application to partially observed queues is presented.

Reinforcement learning is a framework used to address the general problem of an agent faced with a sequential decision problem and using evaluative feedback as a performance measure. In this talk I will briefly review this topic of research. I will describe the standard framework of Markov decision processes and some classical solution methods (dynamic programming methods and stochastic approximation methods). I will then go through some more recent results and describe some important open issues in this area.

We consider the problem of inferring the structure of a network from co-occurrence data; observations that indicate which nodes occur in a signaling pathway but do not directly reveal node order within the pathway. This problem is motivated by network inference problems arising in computational biology and communication systems, in which it is difficult or impossible to obtain precise time ordering information. Without order information, every permutation of the activated nodes leads to a different feasible solution, resulting in combinatorial explosion of the feasible set. However, physical principles underlying most networked systems suggest that not all feasible solutions are equally likely. Intuitively, nodes which co-occur more frequently are probably more closely connected. Building on this intuition, we model path co-activations as randomly shuffled samples of a random walk on the network. We derive a computationally efficient network inference algorithm and, via novel concentration inequalities for importance sampling estimators, prove that a polynomial complexity Monte Carlo version of the algorithm converges with high probability.

We overview some of the tools available today for the study of dynamical systems generated by retarded functional differential equations or partial differential equations. For this we recall basic results of the theory of ordinary differential equations. To exemplify in infinite dimensions we consider the flow generated by the heat equation.

Visual resolution of humans and animals is very different from the common constant resolution images we are used to deal with in photographs, television and computer files. In biological systems, the view field is observed in a non-uniform way, in order to optimize the allocation of neuronal resources to the animal's tasks. For instance, insects require a very enlarged view field (almost 360º) to be able to navigate in complex environments, but their visual resolution is limited. Humans instead, require a much larger resolution in order to recognize and manipulate successfuly very distinct objects. However this high resolution cannot be present in all the view field, otherwise humans would require an extremely large brain to process all the data. Therefore, the view field is more limited than insects, and resolution decays toward the periphery. In this talk we will overview two visual geometries inspired in the biological systems and applied to robotic systems. In one hand, omnidirectional vision, combining cameras and convex mirrors, tries to emulate insect vision and is used for navigation of autonomous robots. Foveal vision emulates the human retinas and is applied to the control of an humanoid robot. An interesting engineering application combines omnidirectional and foveal vision to acquire real-time panoramas of the environment.

Pseudogroups, inverse semigroups, and groupoids provide generalizations of the mathematical concept of a group that are useful for describing local notions of symmetry, and they are used extensively in differential geometry and operator algebra theory. (Groupoids in particular have already been addressed in this seminar series.) In this talk I shall present a brief introduction to these concepts, taking as a starting point an elementary look at the semigroup of all the binary relations on a fixed set, through which the talk ties up to yet another class of algebraic structures known as quantales. Along the way I shall try to hint at possible connections to application areas such as the control theory of hybrid systems.