# Eduardo José Bayro-CorrochanoCenter for Research and Advanced Studies Campus Guadalajara · ELectrical Engineering and Computer Science

Eduardo José Bayro-Corrochano

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275

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## Publications

Publications (275)

The principal objective of the paper is to show the importance of the Hamiltonian in control theory. Instead of using the Lagrangian formulation of electromechanical or robotic systems, our work is focused on robot dynamics by its Hamiltonian. Using the iterative Newton–Euler, we generate the local Hamiltonians and the derivative of the moments at...

Interpolating trajectories of points and geometric entities is an important problem for kinematics. To describe these trajectories, several algorithms have been proposed using matrices, quaternions, dual-quaternions, and the Study quadric; the last one allows the embedding of motors as 8D vectors into projective space P7, where the interpolation of...

This paper presents a set of generalized iterative algorithms to find the inverse position kinematics of n-degree-of-freedom kinematic chains with revolute joints. As a first approach, an iterative algorithm is developed using the gradient descent method in Quaternion Algebra to find both the inverse position and velocity kinematics solution in red...

This paper presents a novel multi-stage perception system for collision avoidance in mobile robots. In the here considered scenario, a mobile robot stands in a workspace with a set of potential targets to reach or interact with. When a human partner appears gesturing to the target, the robot must plan a collision-free trajectory to reach the goal....

Geometric Algebra (GA) has proven to be an advanced language for mathematics, physics, computer science, and engineering. This review presents a comprehensive study of works on Quaternion Algebra and GA applications in computer science and engineering from 1995 to 2020. After a brief introduction of GA, the applications of GA are reviewed across ma...

Unmanned autonomous vehicles, especially multi-copters, are becoming nowadays ubiquitous. Its popularity is due to its relative maneuverability in civil field for performing a wide range of applications, for example, monitoring roads or areas at risk, remote surveillance, inspection of power lines, etc. However, some of these applications require m...

This chapter presents the formulation of robot manipulator kinematics within the geometric algebra framework. In this algebraic system, the 3D Euclidean motion of points, lines, and planes can be advantageously represented using the algebra of motors. The computational complexity of direct and indirect kinematics and other problems concerning robot...

In this chapter, we will discuss the programming issues to compute in the geometric algebra framework. We will explain the technicalities for the programming which you have to take into account to generate a sound source code. At the end, we will discuss the use of specialized hardware as FPGA and Nvidia CUDA to improve the efficiency of the code p...

In this chapter, we utilize the conformal geometric algebra for the development of concepts and computer algorithms in the domain of robot vision. We present an interesting application of fussy logic and conformal geometric algebra for grasping using the Barret Hand. We present real-time algorithms for a real scenario of perception, approach, and a...

We have learned that readers of the work of Hestenes and Sobzyk [1, Chap. 8] and a late article of Doran et al. [2] section IV may have difficulties to understand the subject and practitioners have difficulties to try the equations in certain applications. For this reason, this chapter reviews concepts and equations most of them introduced by Heste...

This chapter presents the computing of the dynamic model, the generation of trajectories using quadratic programming with geometric constraints, and nonlinear control for robot manipulators using the geometric algebra framework.

In medical image analysis, the availability of 3D models is of great interest to physicians because it allows them to have a better understanding of the situation, and such models are relatively easy to build. However, sometimes and in special situations (such as surgical procedures), some structures (such as the brain or tumors) suffer a (nonrigid...

In this chapter, we will discuss the advantages for geometric computing that geometric algebra offers for solving problems and developing algorithms in the fields of artificial intelligence, robotics, and intelligent machines acting within the perception and action cycle. We begin with a short tour of the history of mathematics to find the roots of...

This chapter is dedicated to the estimation of 3D Euclidean transformation using motor algebra. Two illustrations of estimation procedures are given: the first uses a batch approach for the estimation of the unknown 3D transformation between the coordinate reference systems of a robot neck, or arm, and of a digital camera. This problem is called th...

It is the belief that imaginary numbers appeared for the first time around 1540 when the mathematicians Tartaglia and Cardano represented real roots of a cubic equation in terms of conjugated complex numbers. A Norwegian surveyor, Caspar Wessel, was in 1798 the first one to represent complex numbers by points on a plane with its vertical axis imagi...

The study of the kinematics and dynamics of robot mechanisms has employed different frameworks, such as vector calculus, quaternion algebra, or linear algebra; the last is used most often. However, in these frameworks, handling the kinematics and dynamics involving only points and lines is very complicated. In previous chapter, the motor algebra wa...

In this chapter, we show the treatment of a variety of tasks of medical robotics handled using a powerful, non-redundant coefficient geometric language. This chapter is based on our previous works [1, 2]. You will see how we can treat the representation and modeling using geometric primitives like points, lines, and spheres. The screw and motors ar...

In general, when the sensors are mounted on a robot arm, one can use the hand–eye calibration algorithm to calibrate them. In this chapter, we present the calibration of an endoscopic camera with respect to a tracking system and the case of a mobile robot for that one has to calibrate the robot’s sensors with respect to the robot’s global coordinat...

This chapter presents the geometric algebra framework for dealing with 3D kinematics. The reader will see the usefulness of this mathematical approach for applications in computer vision and kinematics. We start with an introduction to 4D geometric algebra for 3D kinematics. Then we reformulate, using 3D and 4D geometric algebras, the classic model...

In this chapter, we present the localized control policy which is possible due to the computing of the local dynamic model at each robot joint computed using the recursive Newton–Euler algorithm. We compute the local Hamiltonians at each joint and derive their localized controllers as well. In the experimental part, we compare the performance of PD...

Biological creatures are able to perform complex tasks, due to the capacity of the brain to store information and to adapt its neuro connections as necessary, and this is known as synaptic plasticity [1]. The neuroplasticity was investigated and later used in Artificial Neural Networks (ANN), where these ANN were called the third generation of neur...

This chapter presents applications of body–eye calibration algorithms using motors of the conformal geometric algebra. A scan-matching algorithm, based on such algorithm, aligns the scans by representing the scan points as lines. We show then a path-following procedure that also uses the conformal geometric algebra techniques to estimate the geomet...

In this work the dynamic model and the nonlinear control for a multi-copter have been developed using the geometric algebra framework specifically using the motor algebra \(G^+_{3,0,1}\). The kinematics for the aircraft model and the dynamics based on Newton-Euler formalism are presented. Block-control technique is applied to the multi-copter model...

This work introduces a modern and intuitive geometric language to support and to enhance the handling of different tasks in medical robot vision. By reformulating screw theory (generalization of quaternions) in the conformal geometric algebra framework, we address the hand eye calibration, 3D model registration using Kinect, interpolation, haptics,...

The goal of Geometric Algebra Applications Vol. II: Robot Modeling and Control is to present a unified mathematical treatment of diverse problems in the general domain of robotics and associated fields using Clifford, or geometric algebra. By treating a wide spectrum of problems in a common language, this Volume II offers both new insights and new...

In this work, the H∞ control for mechanical systems and its application in Robotics is discussed. The controller is designed in discrete time and it is synthesized for mechanical systems that are modeled by means of the Euler–Lagrange formulation. Making use of the discrete Hamilton–Jacobi–Isaacs equation the control law is derived. The discrete co...

In medical image analysis, the availability of 3D models is of great interest to physicians because it allows them to have a better understanding of the situation, and such models are relatively easy to build. However, sometimes and in special situations (such as surgical procedures), some structures (such as the brain or tumors) suffer a (non-rigi...

In this chapter, we present a series of experiments in order to demonstrate the capabilities of geometric neural networks. We show cases of learning of a high nonlinear mapping and prediction. In the second part experiments of multiclass classification, object recognition, and robot trajectories interpolation using CSVM are included.

It appears that for biological creatures, the external world may be internalized in terms of intrinsic geometric representations. We can formalize the relationships between the physical signals of external objects and the internal signals of a biological creature by using extrinsic vectors to represent those signals coming from the world and intrin...

We have learned that readers of the work of D. Hestenes and G. Sobzyk (Hestenes and Sobczyk (1984). Clifford Algebra to Geometric Calculus: A Unified Language for Mathematics and Physics.) [138] Chap. 8 and a late article of Ch. Doran, D. Hestenes and F. Sommen (Doran, Hestenes, Sommen and Van Acker (1993). Journal of Mathematical Physics, 34(8), p...

This chapter gives a detailed outline of geometric algebra and explains the related traditional algebras in common use by mathematicians, physicists, computer scientists, and engineers.

The geometric algebra of a 3D Euclidean space \(G_{3,0,0}\) has a point basis and the motor algebra \(G_{3,0,1}^+\) a line basis. In the latter, the lines are expressed in terms of Plücker coordinates and the points and planes in terms of bivectors. The reader can find a comparison of representations of points, lines, and planes using vector calcul...

This chapter presents the theory and use of the Clifford Fourier transforms and Clifford wavelet transforms. We will show that using the mathematical system of the geometric algebra, it is possible to develop different kinds of Clifford Fourier and wavelet transforms which are very useful for image filtering, pattern recognition, feature detection,...

The geometric algebra of a 3D Euclidean space \(G_{3,0,0}\) has a point basis and the motor algebra \(G_{3,0,1}\) a line basis. In the latter geometric algebra, the lines expressed in terms of Plücker coordinates can be used to represent points and planes as well. The reader can find a comparison of representations of points, lines, and planes usin...

This chapter presents a mathematical approach based on geometric algebra for the computation of problems in computer vision. We will show that geometric algebra is a well-founded and elegant language for expressing and implementing those aspects of linear algebra and projective geometry that are useful for computer vision. Since geometric algebra o...

This chapter gives a detailed outline of differentiation, linear, and multilinear functions, eigenblades, and tensors formulated in geometric algebra and explains the related operators and transformations in common use by mathematicians, physicists, computer scientists, and engineers.

Clifford algebras were created and classified by William K. Clifford (1878–1882) Clifford (Proc. London Math Soc, 4:381–395, 1873, [59]), Clifford (Am J Math, 1:350–358, 1878, [57], Clifford (In Mathematical Papers, Macmillan, London, [58], when he presented a new multiplication rule for vectors in Grassmann’s exterior algebra Open image in new win...

We have learned that readers of the chapter on geometriccalculus of the book (Hestenes and Sobczyk (1984). Clifford Algebra to Geometric Calculus: A Unified Language for Mathematics and Physics.) [138] may have difficulties to understand the subject and practitioners have difficulties to try the equations in certain applications. For this reason, t...

It is the believe that imaginary numbers appeared for the first time around 1540 when the mathematicians Tartaglia and Cardano represented real roots of a cubic equation in terms of conjugated complex numbers. A Norwegian surveyor, Caspar Wessel, was in 1798 the first one to represent complex numbers by points on a plane with its vertical axis imag...

This chapter first presents Lie operators for key points detection working in the affine plane. This approach is stimulated by certain evidence of the human visual system; therefore, these Lie filters appear to be very useful for implementing in near future of an humanoid vision system.

This chapter will demonstrate that geometric algebra provides a simple mechanism for unifying current approaches in the computation and application of projective invariants using n-uncalibrated cameras. First, we describe Pascal’s theorem as a type of projective invariant, and then the theorem is applied for computing camera-intrinsic parameters. T...

In this chapter, we will discuss the programming issues to compute in the geometric algebra framework. We will explain the technicalities for the programming which you have to take into account to generate a sound source code. At the end, we will discuss the use of specialized hardware as FPGA and NVidia CUDA to improve the efficiency of the code p...

The Volume I is devoted to geometric algebra for computer vision, graphics, and machine learning within the broad scope of cybernetics. The Vol II handles the theme geometric algebra for robotics and control. The Vol III presents geometric algebra for integral transforms for science and engineering. As a matter of fact, these topics are fundamental...

The first section presents a non-iterative algorithm that combines the power of expression of geometric algebra with the robustness of Tensor Voting to find the correspondences between two sets of 3D points with an underlying rigid transformation. In addition, we present experiments of the conformal geometric algebra voting scheme using synthetic a...

In geometric algebra, there exist specific operators named versors to model rotations, translations, and dilations, and are called rotors, translators and dilators respectively. In general, a versor \({\varvec{G}}\) is a multivector which can be expressed as the geometric product of non-singular vectors $$\begin{aligned} G = \pm {\varvec{v}}_1 {\va...

This chapter presents the geometric algebra framework for dealing with 3D kinematics. The reader will see the usefulness of this mathematical approach for applications in computer vision and kinematics. We start with an introduction to 4D geometric algebra for 3D kinematics. Then we reformulate, using 3D and 4D geometric algebras, the classic model...

In this work, by reformulating screw theory (generalization of quaternions) in the conformal geometric algebra framework, we address the interpolation, virtual reality, graphics engineering, haptics. We derive intuitive geometric equations to handle surface operations like in kidney surgery. The interpolation can handle the interpolation and dilati...

In this work we propose a robust controller to do tracking using the sub-optimal H∞technique with the approach of differential game theory. The problem is solved in two steps using the Block Control technique. The controller is designed in discrete time and it is synthesized for electromechanical systems which are modeled by means of the Euler-Lagr...

This paper presents an implementation of the conformal voting scheme using a reconfigurable hardware approach in the frame of conformal geometric algebra G
<sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">3,1</sub>
. This algorithm is able to extract geometric entities, such as circles and lines from ed...

This paper describes a novel method for solving the inverse kinematics of a humanoid robot leg anthropomorphically configured with 6 degrees of freedom using conformal geometric algebra. We have used different geometric entities such as lines, planes, and spheres in order to achieve the desired position and orientation of the body and the foot, ind...

In this work, we present a novel architecture in a field programmable gate array (FPGA) for accelerating image processing algorithms based on conformal geometric algebra (CGA). This implementation specifically accelerates the execution of the Conformal Geometric Algebra Voting Scheme for detection of circles and lines in images. All geometric opera...

This work presents a parallelization method for the Clifford support vector machines, based in two characteristics of the Gaussian Kernel. The pure real-valued result and its commutativity allows us to separate the multivector data in its defining subspaces. These subspaces are independent from each other, so we can solve the problem using parallel...

This paper presents a novel algorithm to stabilize a bipedal robot while performing dynamic walking. The problem is divided into the formulation of translational and rotational stabilizers. This work is focused in the translational stabilizer, thus, time-dependent functions based in the zero-moment point stability criterion are established to defin...

Object tracking and manipulation is an important process for many applications in robotics and computer vision. A novel 3D pose estimation of objects using reflectionally symmetry formulated in conformal geometric algebra (CGA) is proposed in this work. The synthesis of the kinematical model for robots using the CGA approach is described. Real time...

This work presents a new type of Spike Neural Networks (SNN) developed in the quaternion algebra framework. This new neural structure based on SNN is developed using the quaternion algebra. The training algorithm was extended adjusting the weights according to the quaternion multiplication rule, which allows accurate results with a decreased networ...

Controlling the pose of a manipulator involves finding the correct configuration of the robot's elements to move the end effector to a desired position and orientation. In order to find the geometric relationships between the elements of a robot manipulator, it is necessary to define the kinematics of the robot. We present a synthesis of the kinema...

This paper presents the design of radial basis function geometric bioinspired networks and their applications. Until now, the design of neural networks has been inspired by the biological models of neural networks but mostly using vector calculus and linear algebra. However, these designs have never shown the role of geometric computing. The questi...

Traditional methods for geometric entities resort to the Hough transform and tensor voting schemes for detect lines and circles. In this work, the authors extend these approaches using representations in terms of k-vectors of the Conformal Geometric Algebra. Of interest is the detection of lines and circles in images, and planes, circles, and spher...

There is a wide range of applications for unmanned aerial vehicles that requires the capability of having several and robust flight controllers available. This paper presents the main framework of a multi-mode flight control system for a quadrotor based on the super twisting control algorithm. The design stages for the four flight control modes enc...

The understanding of scenes is a key aspect of computer vision. Edge detection helps us to understand more about the scene structure since the edges mark a clear distinction for a transition from one region with similar properties to another one. When the edges are obtained from changes in orientation, we can use them to find key planes and describ...

DNA microarrays is a technology that can be used to diagnose cancer and other diseases. To automate the analysis of such data, pattern recognition and machine learning algorithms can be applied. However, the curse of dimensionality is unavoidable: very few samples to train, and many attributes in each sample. As the predictive accuracy of supervise...

In this paper, we introduce a novel geometric voting scheme that extends previous algorithms, like Hough transform and tensor voting, in order to tackle perceptual organization problems. Our approach is grounded in three methodologies: representation of information using Conformal Geometric Algebra, a local voting process, which introduce global pe...

This paper presents the Quaternion Support Vector Machines for classification as a generalization of the real- and complex- valued Support Vector Machines. In this framework we handle the design of kernels involving the Clifford or quaternion product. The QSVM allows to change the metric involved in the quaternion product. The application section s...

The pose tracking problem for the 5 DOF (degrees of freedom) arm of a humanoid robot is studied. The kinematic and dynamic models of the manipulator are obtained using the conformal geometric algebra framework. Then, using the obtained models, the well known super-twisting algorithm, is used to design a controller in terms of the conformal geometri...

The fundamental purpose of this paper is to present a stabilizing control for rolling ball (ball-plate system), which is developed using the framework of the theory of high order control variation in combination with integrator backstepping, and sliding mode control. Under this scheme, control laws are proposed to solve the problem of stabilization...

It has been shown that lines and edges are important for biological visual systems and this information can be described in terms of symmetric relations (even and odd) which permits a compact data representation. In order to define symmetry, we need two basic concepts, an object definition and transforms definitions. The main aim of this work is th...

A controller, based on sliding mode control, is proposed for the n-link robotic manipulator pose tracking problem. The point pair (a geometric entity expressed in geometric algebra) is used to represent position and orientation of the end-effector of a manipulator. This permits us to express the direct and differential kinematics of the endeffector...

This paper presents different methods, some based on geometric algebra, for ultrasound probe tracking in endoscopic images, 3D allocation of the ultrasound probe, ultrasound image segmentation (to extract objects like tumors), and 3D reconstruction of the surface defined by a set of points. The tracking of the ultrasound probe in endoscopic images...

This book constitutes the refereed proceedings of the 19th Iberoamerican Congress on Pattern Recognition, CIARP 2014, held in Puerto Vallarta, Jalisco, Mexico, in November 2014. The 115 papers presented were carefully reviewed and selected from 160 submissions. The papers are organized in topical sections on image coding, processing and analysis; s...

Complex and hyper-complex valued filtering play a substantial role in signal processing, especially to obtain local features in the frequency and phase domain. In the case of 1D signals, the analytic signal is typically computed using the Hilbert transform. Such complex representation allows us to compute the phase and magnitude of the signal. For...

This work presents a new method to apply the Hough Transform to 2D and 3D cloud points using the conformal geometric algebra framework. The objective is to detect geometric entities, with the use of simple parametric equations and the properties of the geometric algebra. We show with real images and RGB-D data that this new method is very useful to...

In this work we address the topic of image processing using an atomic function (AF) in a representation of quaternionic algebra. Our approach is based on the most important AF, the up (x) function. The main reason to use the atomic function up (x) is that this function can express analytically multiple operations commonly used in image processing s...

Controlling walking biped robots is a challenging problem due to its complex and uncertain dynamics. In order to tackle this, we propose a sliding mode controller based on a dynamic model which was obtained using the conformal geometric algebra approach (CGA). The CGA framework permits us to use lines, points, and other geometric entities, to obtai...

This paper describes a method for biped walking pattern generation by using discrete Optimal Preview Integral Sliding Mode Control (OPISMC) of the zero-moment point (ZMP). First, the 3D-linear inverted pendulum mode (3D-LIPM) model used to model biped dynamics is presented. The optimal preview controller that uses future ZMP reference information i...

Several methods (discrete and continuous) for surface reconstruction have been proposed over the past years. Convex hull is one of them, which is the minimal convex envelope for a set of points X in a real vector space V. We present a method to 3D surface reconstruction which refines the convex hull by means of a peeling process with an adaptive ra...

The use of haptic interfaces in surgery could provide the surgeon useful sensing information about the patient tissues. Our goal in this work, is to use the haptic interface to obtain some sample points on the surface of an object or organ tissue in medical applications. This elasticity information feeds an artificial neural network. The output of...

Atomic Functions are widely used in different applications in image processing, pattern recognition, computational physics and also in the digital interpretation of signal measurements. In 1D signals, is usual to compute the phase and the magnitude of a signal using the analytic signal (the signal and its Hilbert transform using complex numbers). H...

The main goal of this work is to develop a geometric neural network which can be used as an interface between sensors and robot mechanisms. For this goal we have developed two new geometric network called Spherical Radial Basis Function Network and Spherical General Regression Network using the conformal geometric algebra framework. The motivation...

The main goal of this work is to develop a geometric neural network which can be used as an interface between sensors and robot mechanisms. For this goal we have developed a new geometric network called Spherical Radial Basis Function Network using the conformal geometric algebra framework. The motivation to use circles or spheres as activation fun...

In this paper we present a new hypercomplex-valued model of recurrent neural network which is based on the Geometric Radial Basis (RBF) and Elman Network Models. This model is useful to recognize temporal sequences of geometric entities using geometric algebra. Our model combines features from the Elman recurrent neural network and geometric RBF ne...

In this paper we present the design of a device to guide the visually impaired person who normally uses a cane. We propose a non-invasive device that will help blind and visually impaired people to navigate. The system uses stereoscopic vision, a RGB-D sensor and an IMU to process images and to compute the distances from obstacles relative to camer...

The main goal of this work is to develop a geometric neural network which can be used as an interface between sensors and robot mechanisms. For this goal we have developed a new geometric network called Spherical General Regression Network using the conformal geometric algebra framework. The motivation to use circles or spheres as activation functi...

This chapter presents a novel quaternion wavelet transform using the atomic function. Atomic functions are compactly supported and infinitely differentiable solutions of differential equations. Consequently, they constitute a promising kernel for wavelet multi-resolution analysis. This work makes use of the phase concept for the detection of key fe...

Image denoising is probably one of the most studied problems in the image processing community. Recently a new paradigm on non-local denoising was introduced. The non-local means method proposed by Buades, Morel and Coll computes the denoised image as ...

This paper presents the application of 2D and 3D Hough Transforms together with conformal geometric algebra to build 3D geometric maps using the geometric entities of lines and planes. Among several existing techniques for robot self-localization, a new approach is proposed for map matching in the Hough domain. The geometric Hough representation is...

In this work, the problem of nonlinear regulation of an underactuated system is treated by means of sliding mode continuous control actions combined with block control technique. The sliding mode state feedback output regulator based on the super twisting algorithm, is applied to the Pendubot system. The transformation of the original system to reg...

This work presents the problem of nonlinear regulation of an underactuated system that is treated by means of sliding mode control actions combined with block control technique. The state feedback output regulator based on sliding mode control, is applied to the Pendubot system. The transformation of the original system to regular form and then blo...

The authors use a new algorithm to compute the forward Dynamics of n degree of freedom serial kinematic chains, which is less complex to handle than the classical approaches. This algorithm was created rewriting the Lagrange equation in terms of lines and points in the framework of conformal geometric algebra, which allows us to have a new equation...

A methodology for 3D modeling of virtualized reality objects using neural computing is presented. In this paper the objects are represented in virtualized reality and their 3D data are acquired by one of three acquisition systems: endoneurosonographic equipment (ENS), stereo vision system and non-contact 3D digitizer. These objects are modeled by o...