William Kingdon Clifford (1845-1879)

William Kingdon Clifford was an English mathematician and philosopher who worked extensively in many branches of pure mathematics and classical mechanics. Although he died young, he left a deep and long-lasting legacy, particularly in geometry. One of the main achievements that he is remembered for is his pioneering work on integrating Hamilton’s Elements of Quaternions with Grassmann’s Theory of Extension into a more general coherent corpus, now referred to eponymously as Clifford algebras. These geometric algebras are utilised in engineering mechanics (especially in robotics) as well as in mathematical physics (especially in quantum mechanics) for representing spatial relationships, motions, and dynamics within systems of particles and rigid bodies. Clifford’s study of geometric algebras in both Euclidean and non-Euclidean spaces led to his invention of the biquaternion, now used as an efficient representation for twists and wrenches in the same context as that of Ball’s Theory of Screws

Spencer Operator and Applications: From Continuum Mechanics to Mathematical physics

The Spencer operator, introduced by D.C. Spencer fifty years ago, is rarely used in mathematics today and, up to our knowledge, has never been used in engineering applications or mathematical physics. The main purpose of this paper, an extended version of a lecture at the second workshop on Differential Equations by Algebraic Methods (DEAM2, february 9-11, 2011, Linz, Austria) is to prove that the use of the Spencer operator constitutes the common secret of the three following famous books published about at the same time in the beginning of the last century, though they do not seem to have anything in common at first sight as they are successively dealing with elasticity theory, commutative algebra, electromagnetism and general relativity: (C) E. and F. COSSERAT: "Th\'eorie des Corps D\'eformables", Hermann, Paris, 1909. (M) F.S. MACAULAY: "The Algebraic Theory of Modular Systems", Cambridge University Press, 1916. (W) H. WEYL: "Space, Time, Matter", Springer, Berlin, 1918 (1922, 1958; Dover, 1952). Meanwhile, we shall point out the importance of (M) for studying control identifiability and of (C)+(W) for the group theoretical unification of finite elements in engineering sciences, recovering in a purely mathematical way well known field-matter coupling phenomena (piezzoelectricity, photoelasticity, streaming birefringence, viscosity, ...). As a byproduct and though disturbing it could be, we shall prove that these unavoidable new diferential and homological methods contradict the mathematical foundations of both engineering (continuum mechanics,electromagnetism) and mathematical (gauge theory, general relativity) physics.Comment: Though a few of the results presented are proved in the recent references provided, the way they are combined with others and patched together around the three books quoted is new. In view of the importance of the full paper, the present version is only a summary of the definitive version to appear later on. Finally, the reader must not forget that "each formula" appearing in this new general framework has been used explicitly or implicitly in (C), (M) and (W) for a mechanical, mathematical or physical purpos

Solving the nearest rotation matrix problem in three and four dimensions with applications in robotics

Aplicat embargament des de la data de defensa fins ei 31/5/2022Since the map from quaternions to rotation matrices is a 2-to-1 covering map, this map cannot be smoothly inverted. As a consequence, it is sometimes erroneously assumed that all inversions should necessarily contain singularities that arise in the form of quotients where the divisor can be arbitrarily small. This misconception was clarified when we found a new division-free conversion method. This result triggered the research work presented in this thesis. At first glance, the matrix to quaternion conversion does not seem to be a relevant problem. Actually, most researchers consider it as a well-solved problem whose revision is not likely to provide any new insight in any area of practical interest. Nevertheless, we show in this thesis how solving the nearest rotation matrix problem in Frobenius norm can be reduced to a matrix to quaternion conversion. Many problems, such as hand-eye calibration, camera pose estimation, location recognition, image stitching etc. require finding the nearest proper orthogonal matrix to a given matrix. Thus, the matrix to quaternion conversion becomes of paramount importance. While a rotation in 3D can be represented using a quaternion, a rotation in 4D can be represented using a double quaternion. As a consequence, the computation of the nearest rotation matrix in 4D, using our approach, essentially follow the same steps as in the 3D case. Although the 4D case might seem of theoretical interest only, we show in this thesis its practical relevance thanks to a little known mapping between 3D displacements and 4D rotations. In this thesis we focus our attention in obtaining closed-form solutions, in particular those that only require the four basic arithmetic operations because they can easily be implemented on microcomputers with limited computational resources. Moreover, closed-form methods are preferable for at least two reasons: they provide the most meaningful answer because they permit analyzing the influence of each variable on the result; and their computational cost, in terms of arithmetic operations, is fixed and assessable beforehand. We have actually derived closed-form methods specifically tailored for solving the hand-eye calibration and the pointcloud registration problems which outperform all previous approaches.Dado que la función que aplica a cada cuaternión su matrix de rotación correspondiente es 2 a 1, la inversa de esta función no es diferenciable en todo su dominio. Por consiguiente, a veces se asume erróneamente que todas las inversiones deben contener necesariamente singularidades que surgen en forma de cocientes donde el divisor puede ser arbitrariamente pequeño. Esta idea errónea se aclaró cuando encontramos un nuevo método de conversión sin división. Este resultado desencadenó el trabajo de investigación presentado en esta tesis. A primera vista, la conversión de matriz a cuaternión no parece un problema relevante. En realidad, la mayoría de los investigadores lo consideran un problema bien resuelto cuya revisión no es probable que proporcione nuevos resultados en ningún área de interés práctico. Sin embargo, mostramos en esta tesis cómo la resolución del problema de la matriz de rotación más cercana según la norma de Frobenius se puede reducir a una conversión de matriz a cuaternión. Muchos problemas, como el de la calibración mano-cámara, el de la estimación de la pose de una cámara, el de la identificación de una ubicación, el del solapamiento de imágenes, etc. requieren encontrar la matriz de rotación más cercana a una matriz dada. Por lo tanto, la conversión de matriz a cuaternión se vuelve de suma importancia. Mientras que una rotación en 3D se puede representar mediante un cuaternión, una rotación en 4D se puede representar mediante un cuaternión doble. Como consecuencia, el cálculo de la matriz de rotación más cercana en 4D, utilizando nuestro enfoque, sigue esencialmente los mismos pasos que en el caso 3D. Aunque el caso 4D pueda parecer de interés teórico únicamente, mostramos en esta tesis su relevancia práctica gracias a una función poco conocida que relaciona desplazamientos en 3D con rotaciones en 4D. En esta tesis nos centramos en la obtención de soluciones de forma cerrada, en particular aquellas que solo requieren las cuatro operaciones aritméticas básicas porque se pueden implementar fácilmente en microcomputadores con recursos computacionales limitados. Además, los métodos de forma cerrada son preferibles por al menos dos razones: proporcionan la respuesta más significativa porque permiten analizar la influencia de cada variable en el resultado; y su costo computacional, en términos de operaciones aritméticas, es fijo y evaluable de antemano. De hecho, hemos derivado nuevos métodos de forma cerrada diseñados específicamente para resolver el problema de la calibración mano-cámara y el del registro de nubes de puntos cuya eficiencia supera la de todos los métodos anteriores.Postprint (published version

Approximating displacements in R^3 by rotations in R^4 and its application to pointcloud registration

© 2022 IEEE. Personal use of this material is permitted. Permission from IEEE must be obtained for all other uses, in any current or future media, including reprinting /republishing this material for advertising or promotional purposes, creating new collective works, for resale or redistribution to servers or lists, or reuse of any copyrighted component of this work in other worksNo proper norm exists to measure the distance between two object poses essentially because a general pose is defined by a rotation and a translation, and thus it involves magnitudes with different units. As a means to solve this dimensional-inhomogeneity problem, the concept of characteristic length has been put forward in the area of kinematics. The idea consists in scaling translations according to this characteristic length and then approximating the corresponding displacement defining the object pose in R^3 by a rotation in R^4, for which a norm exists. This paper sheds new light on this kind of approximations which permits simplifying optimization problem whose cost functions involve translations and rotations simultaneously. A good example of this kind of problems is the pointcloud registration problem in which the optimal rotation and translation between two sets of corresponding 3D point data, so that they are aligned/registered, have to be found. As a result, a simple closed-form formula for solving this problem is presented which is shown to be an attractive alternative to the previous approaches.Peer ReviewedPostprint (author's final draft

Development and Characterization of Velocity Workspaces for the Human Knee.

The knee joint is the most complex joint in the human body. A complete understanding of the physical behavior of the joint is essential for the prevention of injury and efficient treatment of infirmities of the knee. A kinematic model of the human knee including bone surfaces and four major ligaments was studied using techniques pioneered in robotic workspace analysis. The objective of this work was to develop and test methods for determining displacement and velocity workspaces for the model and investigate these workspaces. Data were collected from several sources using magnetic resonance imaging (MRI) and computed tomography (CT). Geometric data, including surface representations and ligament lengths and insertions, were extracted from the images to construct the kinematic model. Fixed orientation displacement workspaces for the tibia relative to the femur were computed using ANSI C programs and visualized using commercial personal computer graphics packages. Interpreting the constraints at a point on the fixed orientation displacement workspace, a corresponding velocity workspace was computed based on extended screw theory, implemented using MATLAB(TM), and visually interpreted by depicting basis elements. With the available data and immediate application of the displacement workspace analysis to clinical settings, fixed orientation displacement workspaces were found to hold the most promise. Significant findings of the velocity workspace analysis include the characterization of the velocity workspaces depending on the interaction of the underlying two-systems of the constraint set, an indication of the contributions from passive constraints to force closure of the joint, computational means to find potentially harmful motions within the model, and realistic motions predicted from solely geometric constraints. Geometric algebra was also investigated as an alternative method of representing the underlying mathematics of the computations with promising results. Recommendations for improving and continuing the research may be divided into three areas: the evolution of the knee model to allow a representation for cartilage and the menisci to be used in the workspace analysis, the integration of kinematic data with the workspace analysis, and the development of in vivo data collection methods to foster validation of the techniques outlined in this dissertation

Improvement of Functional Performance of Spatial Parallel Manipulators Using Mechanisms of Variable Structure

International audienceA procedure for the increase of singularity-free zones in the workspace of spatial parallel manipulators is presented in this paper. The procedure is based on the control of the pressure angles in the joints of the manipulator. The zones, which cannot be reached by the manipulator, are detected. For increase of the reachable workspace of the manipulator the legs of variable structure are proposed. The design of the optimal structure of the spatial parallel manipulator 3-RPS is illustrated by a numerical simulation