Mccarthy - 21st Century Kinematics

21st Century Kinematics — read online for free the complete book (whole text) full work

Below is the text of the book, divided by pages. System saving the place of the last page read, allows you to conveniently read the book "21st Century Kinematics" online for free, without having to search again every time where you left off. Put a bookmark, and you can go to the page where you finished reading at any time.

Light

Font size:

↓

↑

Georgia Tahoma Arial Verdana

Reset

Interval:

↓

↑

Bookmark:

Make

1234567...74

J. Michael McCarthy (ed.) 21st Century Kinematics 2013 The 2012 NSF Workshop 10.1007/978-1-4471-4510-3_1 Springer-Verlag London 2013

1. Polynomials, Computers, and Kinematics for the 21st Century

J. Michael McCarthy 1

(1)

Robotics and Automation Laboratory, University of California, Irvine, USA

J. Michael McCarthy

Email:

Abstract

A review of the history of kinematics and machine theory shows a direct connection between the ability to solve polynomial systems using algebraic and numerical techniques and the advancement of the analysis and synthesis of machine systems including robots. Research challenges in kinematic synthesis, compliant mechanisms and cable and tensegrity systems show an ever increasing need for the solutions of complex polynomial systems.

1.1 Introduction

It is a regular practice to organize a session to discuss research directions in the theory and practice, and such a panel was organized at the 2010 ASME Mechanisms and Robotics Conference of the International Design Engineering Technical Conferences. Unlike the usual format which has senior researchers advise their younger colleagues of recent successes and current challenges, the session in 2010 included our younger colleagues who took the opportunity to make their views known.

What our young colleagues had to say opened the eyes of many of us and motivated the organization of an National Science Foundation Workshop on 21st Century Kinematics held in conjunction with the 2012 Mechanisms and Robotics Conference. Our younger colleagues conveyed their excitement regarding many new and different opportunities for research built on theoretical and applied kinematics. However they expressed a clear frustration that the foundational material that was readily available to an earlier generation has not been taught for decades, and therefore seems lost to our younger colleagues and their students.

It took two years to organize, but the 2012 NSF Workshop is our response to this concern. However, rather than revive the lessons and notes from the past, we have sought to place that theory in a context that serves research goals in this new century.

1.2 The Recent Past

In order to consider the needs for the 21st century, it seems reasonable to consider the historical relationship between kinematics, mathematics and machine design. The following is taken from Editorials published in the February 2011 and May 2011 issues of the ASME Journal of Mechanisms and Robotics.

1.2.1 Polynomials and Kinematics

Mechanisms have been characterized by the curves that they trace since the time of Archimedes [].

James Watts invention of a straight-line linkage to convert the linear expansion of steam into the rotation of the great beam making the steam engine practical (Fig. ) captured the imagination of the mathematician P.L. Chebyshev, who introduced the mathematical analysis and synthesis of linkages.

Fig. 1.1

An engraving of Watts steam engine (R. Stuart 1824)

About the same time, J.J. Sylvester, who introduced the Sylvester resultant for the solution of polynomial equations, went on to lecture about the importance of the Peaucellier linkage which generates a pure linear movement from a rotating link [] and computation.

In the mid 1950s, J. Denavit and R.S. Hartenberg introduced a matrix formulation of the loop equations of a mechanism to obtain polynomials that define- its movement []. In this context solution is not a single root but an algorithm that yields all of the roots of the polynomial system, which in turn defines all of the configurations of the linkage for a given input.

It was immediately recognized that the 7R analysis problem was equivalent to solving the inverse kinematics for a general robot manipulator to obtain the configurations that are available to pick up an object. By the end of the 1970s, J. Duffy [].

By the mid-1990s, computer algebra and sparse resultant techniques were the most advanced tools for formulating and solving increasingly complex arrays of polynomials obtained in the study of mechanisms and robotics systems [], which allowed the calculation of the 40 configurations of the system.

1.2.2 Computers and Kinematics

In 1959 F. Freudenstein and G.N. Sandor [].

R.E. Kaufman [] by the turn of the 21st century.

Fig. 1.2

Roger Kaufman using interactive computer graphics for linkage synthesis at MIT in 1970

The pursuit of solutions to the design equations for the particularly challenging problem of finding a four-bar linkage that traces a curve through nine specified points lead Freudenstein and B. Roth [] to develop a unique solution strategy, now called numerical continuation. They started with a set of polynomials with a known solution, which was then deformed slightly and the solution updated numerically. Iterating this parameter-perturbation procedure, they obtained a sequence of polynomials and solutions that converged to the target polynomials and the desired solution. While this yielded the first solutions to the nine-point problem, their heuristic deformation procedure could not find all of the solutions.

By the 1980s theoretical advances in numerical continuation yielded algorithms that could reliably and efficiently find all solutions to small sets of polynomial equations [] revisited Freudenstein and Roths nine-point synthesis problem to obtain 1442 solutions, demonstrating that polynomial continuation algorithms had come of age.

A few years later M. Raghavan and Roth [] used polynomial continuation to obtain 40 configurations for the general Stewart platform anticipating Hustys degree 40 polynomial.

1.3 Recent Results

The past century shows that our ability to analyze and design mechanisms and robotic systems of increasing complexity has depended on our ability to derive and solve the associated increasingly complex polynomial systems. From this, we can expect that advances in computer algebra and numerical continuation for the derivation and solution of the even more complex polynomial systems will advance research in mechanisms and robotics.

Now examining recent research, we can identify three research trends that we can expect to persist into the future. They are the analysis and synthesis of (i) spatial mechanisms and robotic systems, (ii) compliant linkage systems, and (iii) tensegrity and cable-driven systems. In each case, we find that researchers are formulating and solving polynomial systems of total degrees that dwarf those associated with major kinematics problems of the previous century.

1.3.1 Analysis and Synthesis of Machine Systems

Early in the 2000s, E. Lee and D. Mavroidis [].

In 2004, H. Su and colleagues reported that a parallel version of the POLSYS code [] obtained 42,625 solutions to the RRS design problem consisting of 11 polynomials in 11 unknowns for a total degree of 4,194,304. The computation time was 42 minutes on 128 nodes of the San Diego Supercomputer Centers Blue Horizon system.

A. Perez used dual quaternion coordinates to formulate the design equations for general spatial serial chains and obtained 126 synthesis equations in 126 unknowns for a spatial 5R chain that reaches a set of 21 task positions [). This set of equations has been solved numerically to verify that they are correct, but so far a complete solution has not been achieved.

Next page

1234567...74

Light

Font size:

↓

↑

Georgia Tahoma Arial Verdana

Reset

Interval:

↓

↑

Bookmark:

Make

Similar books «21st Century Kinematics»

Look at similar books to 21st Century Kinematics. We have selected literature similar in name and meaning in the hope of providing readers with more options to find new, interesting, not yet read works.

Josephine McCarthy

The Work Of The Hierophant

Stephanie Smith Budhai

Teaching the 4Cs with Technology: How do I use 21st century tools to teach 21st century skills?

Cormac McCarthy

Blood Meridian: Mccarthy Cormac

Josephine McCarthy

Tarot Skills for the 21st Century

Pete McCarthy

The Road to McCarthy

L. Egon Schiele

Nudes for the 21st Century: Photography and commentary (21st Century Nudes Photography Book 1)

Snyder

Quicklet on the Road by Cormac Mccarthy

McCarthy Cormac

Cormac McCarthys house: reading McCarthy without walls

Dominique Paul ChevallierJean Lerbet

Multi-Body Kinematics and Dynamics with Lie Groups

Erin McCarthy [McCarthy

It’s a Ghost’s Life

McCarthy

Jam On: The Craft of Canning Fruit

Ron Krannich

The Best Jobs For the 21st Century, Third Edition

Discussion, reviews of the book 21st Century Kinematics and just readers' own opinions. Leave your comments, write what you think about the work, its meaning or the main characters. Specify what exactly you liked and what you didn't like, and why you think so.