Mechanisms: Kinematic Analysis and Applications in Robotics

By Jaime Gallardo-Alvarado and José Gallardo-Razo

Theory of mechanisms is an applied science of mechanics that studies the relationship between geometry, mobility, topology, and relative motion between rigid bodies connected by geometric forms. Recently, knowledge in kinematics and mechanisms has considerably increased, causing a renovation in the methods of kinematic analysis. With the progress of the algebras of kinematics and the mathematical methods used in the optimal solution of polynomial equations, it has become possible to formulate and elegantly solve problems.

Mechanisms: Kinematic Analysis and Applications in Robotics provides an updated approach to kinematic analysis methods and a review of the mobility criteria most used in planar and spatial mechanisms. Applications in the kinematic analysis of robot manipulators complement the material presented in the book, growing in importance when one recognizes that kinematics is a basic area in the control and modeling of robot manipulators.

• Presents an organized review of general mathematical methods and classical concepts of the theory of mechanisms
• Introduces methods approaching time derivatives of arbitrary vectors employing general approaches based on the vector angular velocity concept introduced by Kane and Levinson
• Proposes a strategic approach not only in acceleration analysis but also to jerk analysis in an easy to understand and systematic way
• Explains kinematic analysis of serial and parallel manipulators by means of the theory of screws

• Language: English
• Publisher: Academic Press
• Release date: Jun 18, 2022
• ISBN: 9780323953474

Jaime Gallardo-Alvarado

Jaime Gallardo-Alvarado received his Ph.D. degree from the Tecnológico Nacional de México in La Laguna. He is a member of the National System of Researchers of Mexico, and is currently a professor in the Department of Mechanical Engineering at the Tecnológico Nacional de México in Celaya. His areas of interest include kinematics and dynamics of rigid bodies, Lie algebras, screw theory, and robot kinematics. Dr. Gallardo-Alvarado is author of the book “Kinematic Analysis of Parallel Manipulators by Algebraic Screw Theory” and has published more than 70 research papers in scientific journals.

Book preview

Preface

Jaime Gallardo-Alvarado; José Gallardo-Razo     

Theory of Mechanisms is an applied science of mechanics that studies the relationship between geometry, mobility (degrees of freedom), topology, and relative motion between rigid bodies connected by geometric forms. During the last decades, knowledge in the areas of kinematics and mechanisms has been increased considerably, causing a whole renovation of the methods of kinematic analysis. Simultaneously, with the progress of the algebras of the kinematics and the mathematical methods used in the optimal solution of polynomial equations, it has been possible to formulate and elegantly solve problems that at the time were considered by Freudenstein, better known as the father of mechanisms, as the Everest of modern kinematics. The main objective of this book is to provide an updated approach to kinematic analysis methods as well as a review of the mobility criteria most used in planar and spatial mechanisms. Applications in the kinematic analysis of robot manipulators complement the material presented in the book. The material exposed in the book can be useful for undergraduate and graduate students, professors, researchers, and professionals in the areas of Mechanical Engineering, Mechatronics, Electrical, Robotics, Aeronautics, and related careers. The material has been organized in 7 parts. Part 1 introduces the subject of the book while Part 2 focuses on the study of systems of polynomial equations and their solution applying numerical and algebraic methods. Polynomial equations frequently emerge from the displacement analysis of complex kinematic chains. The inclusion of mathematical procedures such as the algebraic method, Sylvester's dialytic method of elimination and homotopy principles, are subjects that undoubtedly allow the formulation and resolution of the displacement analysis of mechanisms in an accurate and legible way. Furthermore, these methods allow solving the spatial position analysis of robot manipulators by minimizing, and even avoiding in some cases, conventional numerical methods. The study of time derivatives of arbitrary vectors involving various rotating reference frames, an essential topic in higher order analyzes, completes Part 2. Part 3 addresses the classic theme of any book on mechanism theory: the geometry of motion. In this part, in addition to studying the constitutive elements of the mechanisms, a review is made of conventional criteria in the calculation of the mobility of mechanisms, such as the Grübler criterion, the Kutzbach-Grübler-Chebyshev formula and the Kutzbach-Grübler-Malyshev criterion. The applications and exceptions to these mobility criteria are illustrated with applications in planar and spatial manipulators. In Part 4 the displacement analysis is divided into closed and open kinematic chains. In the displacement analysis of closed kinematic chains, algebraic geometry is used with the intention of obtaining closed-form solutions or, if this is not possible given the complexity of the mechanism, numerical methods such as the Newton-Raphson technique and the Newton-homotopy method are employed. On the other hand, in open kinematic chains, homogeneous coordinate transformation matrices are used, which are precursors of the Denavit-Hartenberg representation. In Part 5 the subject of velocity is approached by applying vectorial, graphical, and analytical methods in the solution of the velocity analysis of mechanisms. In particular, the derivation of the input-output equation of velocity by the analytical method makes it possible to clearly elucidate the possible singular configurations of the mechanisms. Following the trend of Part 5, Part 6 provides the acceleration analysis using vectorial and analytical methods. The jerk analysis using the vectorial method is included in Part 6. However, the graphical method is excluded from the acceleration analysis due to its impractical application in higher order kinematic analyses. Lastly, Part 7 reviews the fundamentals of screw theory and its applications in the kinematic analysis of both serial and parallel manipulators up to the jerk level.

Finally, the authors are sure about the usefulness of the material reported in this book. The theory is easy to follow and can be applied in the solution of related problems of any standard book of mechanisms as well as in the kinematic analysis of robot manipulators.

Part 1: Introduction

Outline

Chapter 1. Overview of kinematics and its algebras

Chapter 2. Overview of mechanisms and robot manipulators

Chapter 1: Overview of kinematics and its algebras

Abstract

Understanding the motion of objects has been a major factor in the survival and development of civilization. With the advent of astronomy and with it the documentation in stone monuments of the changing position of celestial bodies, people ceased to be nomadic and formed sedentary communities in which knowledge of the sciences and the arts flourished splendidly. With the passage of time, the demand for precise quantification of motion gave birth to kinematics. With more than three centuries of formal study, kinematics by its own merits has earned its own place in mechanics. On the other hand, its applications in the design process of spatial robot manipulators demand efficient mathematical methods that allow solving real labyrinths of equations which inevitably arise from the mathematical models associated with the problems of analysis and synthesis of mechanisms, among others. This chapter presents an overview of the history of kinematics and its algebras.

Keywords

quaternion; Lie algebra; motor algebra; screw algebra; Euclidean geometry

1.1 Preamble

Kinematics is the branch of mechanics that studies the geometry of the motion of solid objects without considering the causes that originate it, i.e., the forces that cause the motion are ignored. To this end, it resorts to the use of velocities and accelerations, both linear and angular, which describe how the position of an object changes with time. The fundamental elements of kinematics are space, time and a body or particle in motion. In classical mechanics, space is absolute and independent of the existence of material objects. Thus, physical space is represented by a Euclidean space in which all physical phenomena occur in such a way that the laws of physics are fulfilled without exception in all regions of space. Similarly, in classical mechanics time is absolute and independent of the existence of material objects and the occurrence of physical phenomena. The framework of classical mechanics was developed by Newton and therefore frequently it is referred as Newtonian mechanics. Depending on the moving object, kinematics is classified into rigid body kinematics and particle kinematics. The first being the more complex. Kinematics is linked to the instantaneous change in the position of moving objects. Velocity indicates the instantaneous change in displacement while acceleration indicates the instantaneous change in velocity. Kinematics does not end with the study of acceleration and thus extends indefinitely to the so-called higher order analysis. For example, the jerk describes the ratio of acceleration to time used and is of great utility in a variety of problems ranging from understanding surface deterioration in cams, smoothing of trajectory generation, the study of human body movement, to the characterization of singularities in robotic manipulators. As any discipline of mechanics, kinematics is studied under the rigorous optics of mathematics. Throughout history, various algebras have been developed with the purpose of elucidating the rules of nature that govern the motion of both particles and rigid bodies. The Aristotelian vision of motion has been modified and adapted to reality thanks to algebraic structures that with mathematical arguments have solidified or refuted the various philosophies of kinematics. Thus, the algebras of kinematics have become the mathematical tools with which scholars of the subject have been able to decipher the geometry of motion and the instantaneous displacement of rigid bodies. Kinematics has found its own place in mechanics as a discipline that is part of a wide variety of engineering branches. Despite hundreds of years of formal study, the development and refinement of methods of kinematic analysis will be a recurring topic of study for both scientists and engineers whenever it is necessary to move objects in controlled spaces and respecting the physical environment.

1.2 Origins of kinematics

Astronomy and movement are concepts that are closely linked given the importance of quantifying and documenting, for various purposes, the change of position of celestial bodies. Nomads are prehistoric peoples whose survival was based mainly on the hunting of animals, for which they constantly traveled chasing animals in search of sustenance, guided mainly by instinct. With the stabilization of the climate and the advent of agriculture, nomadic peoples drastically changed their lifestyle and founded sedentary cultures in which the understanding of the cyclical behavior of nature became a priority activity. In that sense, the study of the motion of celestial bodies has been a recurring theme that has obsessed researchers of all times due to the need to quantify the passage and effect of time in nature. For example, the effect of the Moon's movement on the Earth, especially in the oceans and in the regulation of the cycle of life on our planet, is an example of why the subject has captivated the attention of practically all cultures. The construction of stone monuments with the purpose of administering the time according to the change of position of some celestial bodies and thus to follow the change of the seasons, is another representative example.

The movement of the Sun and the Moon with the purpose of perfecting the calendars were motives of deep astronomical studies by civilizations that occupied the Middle East in the antiquity among which stand out the Assyrians, Sumerians, Akkadians, and mainly the Babylonians. For example, those cultures used to designate the beginning of each month according to the day following the day of the new moon. The Babylonians were the first to calculate the beginning of each month without resorting to repetitive and rudimentary observations. Thus, it is in ancient Mesopotamia that, thanks to the contributions of the Babylonian civilization, astronomy with a deep religious meaning was born. The relevance of Babylonian astronomy did not pass unnoticed by the Greeks. The Greek civilization, developed mainly in the northeastern end of the Mediterranean Sea, detonated with creativity and knowledge the evolution of architecture, arts, and sciences in all their aspects. Deep and well-documented studies on astronomy are a clear proof of the magnificence of Greek culture.

Beyond the Mediterranean Sea and the Atlantic Ocean, pre-Columbian cultures in the Americas endeavored to unravel the mystery of the cosmos through the eyes of astronomy. In the Inca culture, astronomy was embedded as a surprisingly accurate tool that allowed the calendarization of agricultural work and the succession of solstices and equinoxes. On the other hand, the Mayan culture was a civilization that flourished in Mesoamerica and integrated astronomy into its style life. Although Mayan astronomical knowledge was reserved for the priestly class, the people respected it and conducted themselves according to the predictions made by their astronomers. The Maya made exact calculations of the synodic periods of Mercury, Venus, Mars, Jupiter, and Saturn. The calculation of the periods of the Moon, the Sun, and stars such as the Pleiades were also achievements of the Mayas and thus confirmed

the splendor of their civilization. Like the Mayan culture, in the Aztec culture astronomy was also studied from a religious perspective. The Aztecs built monuments that, in addition to predicting eclipses, allowed them to measure the revolutions of the Sun, the Moon, and some planets. The Aztec calendar is a jewel of astronomy with which the Aztecs described the movement of the celestial bodies and the different cycles with which time was measured. The Aztec calendar is a monolithic symbol of the conception of time and the worldview of the period. Contrary to Greek culture, despite the colossal achievements of pre-Columbian cultures, there is no material evidence that they made contributions in the field of kinematics. The precision with which various events of celestial bodies are depicted on these stone monuments is matched by the precision of some of the instruments used in contemporary astronomy. Moreover, pre-Columbian calendars do not require the necessary adjustments used in the Gregorian calendar.

Since its beginnings, the practice of astronomy has been so universal and transcendental that detailed studies of the motion of the celestial bodies have been found to have the greatest influence on the social and political behavior of human in all those parts on the Earth where they have inhabited. It is not strange that many of the most significant discoveries in kinematics are contributions of renowned astronomers such as Galileo and Newton. Although Chinese culture is said to have been the first to divide the sky into constellations, Greek astronomers and philosophers, influenced by scientists from other ancient cultures mainly from India and Babylon, were the first to make relevant and well-documented contributions to kinematics. Democritus (460 BC-370 BC), the creator of atomicism, postulated that motion is a real fact resulting from the combined effect of forces and inertias. Despite the correctness of this postulate, Democritus' idea contrasted with the believe of the Eleates, who conceived motion as a phenomenon and not as a reality. The Eleates were part of a philosophical current of ancient Greece that held that sensible things are in essence a single immutable substance [1]. Aristarchus of Samos (310-230 B.C.) proposed a theory in which it was affirmed that the stars of the Universe moved around our Sun. By the time a disturbing idea, since in those days it was taken without doubt that the Earth was the center of creation. The heliocentric model of Aristarchus of Samos did not pass unnoticed at the time, nor was Aristarchus of Samos was condemned for his controversial theory. On the contrary, the seminal work of Aristarchus of Samos was included in the work of Archimedes (287-212 B.C.) entitled Archimedes Syracusani arenarius et Dimensio Circuli, a work in which Archimedes attempts to determine the precise upper limit of grains required to fill the universe. A sign that Byzantine discussions have been around forever. Ptolemy, who lived and worked most of his life in the legendary library of Alexandria, was the author of the Almagest. In this formidable work can be found the Hipparchus' star catalog, which describes the positions of more than 800 stars in 48 constellations. In addition, it includes a geocentric model of the solar system that was valid for more than 1200 years in