5
Kinematics, Polynomials, and Computers—A Brief History J. Michael McCarthy Department of Mechanical and Aerospace Engineering University of California, Irvine Irvine, CA 92697 JMR Editorial – February 2011 As we move into the adolescent years of the 21st century, allow me to discuss where research in mechanisms and robotics has been as a prelude to considering where it is going. Polynomials: Mechanisms have been characterized by the curves that they trace since the time of Archimedes [1]. In the 1800’s, F. Reuleaux, A. B. W. Kennedy and L. Burmester formalized this by applying the descriptive geometry of Gaspard Monge to the analysis and synthesis of machines [2]. James Watt invented a straight-line linkage to convert the linear expansion of steam into the ro- tation of the great beam making the steam engine practical (Fig. 1), and captured the imagination of the mathematician P. L. Chebyshev, who introduced the mathematical analysis and synthesis of linkages. 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 link- age which generates a pure linear movement from a rotating link [3]. Influenced by Sylvester, A. B. Kempe developed a method for designing a linkage that traces a given algebraic curve [4] that even now inspires research at the intersection of geometry [5, 6] and computation. (a) An engraving of Watt’s steam en- gine (R. Stuart 1824). (b) Illustration showing the operation Watt’s linkage. Figure 1: Watt’s linkage transforms the rotational motion of the great beam into the linear motion of the cylinder In the mid 1950’s, J. Denavit and R. S. Hartenberg introduced a matrix formulation of the loop equations of a mechanism to obtain polynomials that defined its movement [7]. During a speech

Kinematics, Polynomials, and Computers|A Brief Historysynthetica.eng.uci.edu/mechanicaldesign101/JMR-Editorial-Feb2011... · Kinematics, Polynomials, and Computers|A Brief History

  • Upload
    dodieu

  • View
    231

  • Download
    0

Embed Size (px)

Citation preview

Page 1: Kinematics, Polynomials, and Computers|A Brief Historysynthetica.eng.uci.edu/mechanicaldesign101/JMR-Editorial-Feb2011... · Kinematics, Polynomials, and Computers|A Brief History

Kinematics, Polynomials, and Computers—A Brief History

J. Michael McCarthyDepartment of Mechanical and Aerospace EngineeringUniversity of California, IrvineIrvine, CA 92697

JMR Editorial – February 2011

As we move into the adolescent years of the 21st century, allow me to discuss where research inmechanisms and robotics has been as a prelude to considering where it is going.

Polynomials: Mechanisms have been characterized by the curves that they trace since the timeof Archimedes [1]. In the 1800’s, F. Reuleaux, A. B. W. Kennedy and L. Burmester formalized thisby applying the descriptive geometry of Gaspard Monge to the analysis and synthesis of machines[2]. James Watt invented a straight-line linkage to convert the linear expansion of steam into the ro-tation of the great beam making the steam engine practical (Fig. 1), and captured the imaginationof the mathematician P. L. Chebyshev, who introduced the mathematical analysis and synthesisof linkages. About the same time, J. J. Sylvester, who introduced the Sylvester resultant for thesolution of polynomial equations, went on to lecture about the importance of the Peaucellier link-age which generates a pure linear movement from a rotating link [3]. Influenced by Sylvester,A. B. Kempe developed a method for designing a linkage that traces a given algebraic curve [4] thateven now inspires research at the intersection of geometry [5, 6] and computation.

(a) An engraving of Watt’s steam en-gine (R. Stuart 1824).

(b) Illustration showing the operation Watt’s linkage.

Figure 1: Watt’s linkage transforms the rotational motion of the great beam into the linear motionof the cylinder

In the mid 1950’s, J. Denavit and R. S. Hartenberg introduced a matrix formulation of the loopequations of a mechanism to obtain polynomials that defined its movement [7]. During a speech

Page 2: Kinematics, Polynomials, and Computers|A Brief Historysynthetica.eng.uci.edu/mechanicaldesign101/JMR-Editorial-Feb2011... · Kinematics, Polynomials, and Computers|A Brief History

in 1972, F. Freudenstein famously used the phrase “Mount Everest of kinematics” to describe thesolution of these polynomials for the 7R spatial linkage [8]. In this context “solution” is not a singleroot but an algorithm that yields all of the roots of the polynomial system, which in turn definesall of the configurations of the linkage for a given input.

It was immediately recognized that the 7R analysis problem was equivalent to solving the inversekinematics for a general robot manipulator to obtain the configurations that are available to pickup an object. By the end of the 1970’s, J. Duffy [9] had formulated an efficient set of equations forthis problem, but it was not until the late 1980’s when the degree 16 polynomial that yields the 16robot configurations was obtained by H. Y. Lee and C. G. Liang [10].

By the mid-1990’s, computer algebra and sparse resultant techniques were the most advancedtools for formulating and solving increasingly complex arrays of polynomials obtained in the studyof mechanisms and robotics systems [11, 12]. In 1996, M. Husty used computer algebra to reduceeight quadratic equations in eight soma coordinates that locate the end-effector of a general six-legged Stewart platform to a degree 40 polynomial [13], which allowed the calculation of the 40configurations of the system.

Computers: In 1959 F. Freudenstein and G. N. Sandor [14] used the newly developed digitalcomputer and the loop equations of a linkage to determine its dimensions, initiating the computer-aided design of mechanisms. Within two decades the computer solution of the equations introducedby Denavit and Hartenberg was integral to the analysis of complex machine systems [15, 16] andthe control of robot manipulators [17].

R. E. Kaufman [18, 19] combined the computer’s ability to rapidly compute the roots of poly-nomial equations with a graphical display to unite Freudenstein’s techniques with the geometricalmethods of Reuleaux and Burmester to form KINSYN, an interactive computer graphics systemfor mechanism design (Fig. 2). This was followed by A. G. Erdman’s LINCAGES system [20, 21]and K. J. Waldron’s RECSYN system [22], which combined sophisticated computer graphics andpolynomial solvers to implement Burmester’s strategy for linkage synthesis. Computerized linkagesynthesis was extended to spherical linkages [23, 24] and spatial linkages [25] by the turn of the21st century.

Figure 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

Page 3: Kinematics, Polynomials, and Computers|A Brief Historysynthetica.eng.uci.edu/mechanicaldesign101/JMR-Editorial-Feb2011... · Kinematics, Polynomials, and Computers|A Brief History

finding a four-bar linkage that traces a curve through nine specified points lead Freudenstein andB. Roth [26, 27] to develop a unique solution strategy, now called numerical continuation. Theystarted with a set of polynomials with a known solution, which was then deformed slightly and thesolution updated numerically. Iterating this “parameter-perturbation procedure,” they obtained asequence of polynomials and solutions that converged to the target polynomials and the desiredsolution. While this yielded the first solutions to the nine-point problem, their heuristic deformationprocedure could not find all of the solutions.

By the 1980’s theoretical advances in numerical continuation yielded algorithms that couldreliably and efficiently find all solutions to small sets of polynomial equations [28, 29]. L. W. Tsaiand A. P. Morgan [30] applied the polynomial continuation routine SYMPOL to the eight quadraticpolynomials of the inverse kinematics problem for a general manipulator and obtained the 16 rootsin four minutes. In the early 1990’s, C. W. Wampler and A. P. Morgan [31] revisited Freudensteinand Roth’s nine-point synthesis problem to obtain 1442 solutions, demonstrating that polynomialcontinuation algorithms had come of age.

A few years later M. Raghavan and Roth [32, 33] included polynomial continuation with resul-tant elimination among the strategies to obtain complete solutions to kinematics problems. In fact,Raghavan [34] used polynomial continuation to obtain 40 configurations for the general Stewartplatform anticipating Husty’s degree 40 polynomial.

Conclusion: The goal of this survey is to show that in the past century our ability to analyze anddesign mechanisms and robotic systems of increasing complexity has depended on our ability toderive and solve the associated increasingly complex polynomial systems. From this, we can expectthat advances in computer algebra and numerical continuation for the derivation and solution ofthe even more complex polynomial systems will advance research in mechanisms and robotics.

References

[1] T. Koetsier, “From Kinematically Generated Curves to Instantaneous Invariants: Episodes inthe History of Instantaneous Planar Kinematics,” Mechanism and Machine Theory, 21(6):489-498, 1986.

[2] F. C. Moon, “History of the Dynamics of Machines and Mechanisms from Leonardo to Tim-oshenko,” International Symposium on History of Machines and Mechanisms, (H. S. Yan andM. Ceccarelli, eds.), 2009. doi: 10.1007/978-1-4020-9485-9-1

[3] T. Koetsier, “A Contribution to the History of Kinematics–II,” Mechanism and Machine The-ory, 18(1):43-48, 1983.

[4] A. B. Kempe, “On a general method of describing plane curves of the nth degree by linkwork,”Proceedings of the London Mathematical Society, VII:213-216, 1976.

[5] D. Jordan and M. Steiner, “Configuration Spaces of Mechanical Linkages,” Discrete and Com-putational Geometry, 22:297-315, 1999.

[6] R. Connelly and E. D. Demaine, “Geometry and Topology of Polygonal Linkages,” Chapter9, Handbook of discrete and computational geometry, (J. E. Goodman and J. O’Rourke, eds.),CRC Press, 2004.

[7] J. Denavit and R. S. Hartenberg, “A kinematic notation for lower-pair mechanisms based onmatrices,” ASME Journal of Applied Mechanics, 22:215-221, 1955.

Page 4: Kinematics, Polynomials, and Computers|A Brief Historysynthetica.eng.uci.edu/mechanicaldesign101/JMR-Editorial-Feb2011... · Kinematics, Polynomials, and Computers|A Brief History

[8] F. Freudenstein, “Kinematics: Past, Present and Future,” Mechanisms and Machine Theory,8:151-160, 1973

[9] J. Duffy, The Analysis of Mechanisms and Robot Manipulators, John Wiley, New York, 419pp.,1980.

[10] H. Y. Lee, and C. G. Liang, “Displacement Analysis of the General Spatial 7-Link 7R Mech-anisms,” Mechanism and Machine Theory, 23(2):219-226, 1988.

[11] J. Canny and I. Emiris, “An efficient algorithm for the sparse matrix resultant,” AppliedAlgebra, Algebraic Algorithms and Error Correcting Codes: Lecture notes in Computer Science,673:89-104, 1993. doi: 10.1007/3-540-56686-4-36

[12] J. Neilsen and B. Roth, “Elimination Methods for Spatial Synthesis,” Computational Kine-matics, (eds. J. P. Merlet and B. Ravani), Vol. 40 Solid Mechanics and Its Applications, pp.51-62, 1995.

[13] M. L. Husty, “An Algorithm for Solving the Direct Kinematics of General Stewart-GoughPlatforms,” Mechanism and Machine Theory, 31(4):365-380, 1996.

[14] F. Freudenstein and G. N. Sandor, “Synthesis of Path Generating Mechanisms by Means ofa Programmed Digital Computer,” ASME Journal of Engineering for Industry, 81:159-168,1959.

[15] P. N. Sheth and J. J. Uicker, “IMP (Integrated Mechanisms Program), A Computer-AidedDesign Analysis system for Mechanisms and Linkages,” ASME Journal of Engineering forIndustry, 94:454-464, 1972.

[16] C. H. Suh and C. W. Radcliffe, Kinematics and Mechanism Design, John Wiley, pp:458, 1978.

[17] R. P. Paul, Robot Manipulators: Mathematics, Programming and Control, MIT Press, 1981.

[18] R. E. Kaufman and W. G. Maurer, ”Interactive Linkage Synthesis on a Small Computer”,ACM National Conference, Aug.3-5, 1971.

[19] A. J. Rubel and R. E. Kaufman, 1977, “KINSYN III: A New Human-Engineered Systemfor Interactive Computer-aided Design of Planar Linkages,” ASME Transactions, Journal ofEngineering for Industry, May.

[20] A. G. Erdman and J. Gustafson, 1977, “LINCAGES-A Linkage Interactive Computer Analysisand Graphically Enhanced Synthesis Package,” ASME Paper No. 77-DTC-5, Chicago, Illinois.

[21] L. Hunt, A. G. Erdman, and D. R. Riley, 1981, “MicroLINCAGES: Microcomputer Synthe-sis and Analysis of Planar Linkages,” Proceedings of the Seventh OSU Applied MechanismsConference, Dec.

[22] J. C. Chuang, R. T. Strong, and K. J. Waldron, “Implementation of Solution RectificationTechniques in an Interactive Linkage Synthesis Program,” ASME Journal of Mechanical De-sign, 103:657-664.

[23] D.A. Ruth and J. M. McCarthy, “SphinxPC: An Implementation of Four Position Synthesisfor Planar and Spherical Linkages,” Proceedings of the ASME Design Engineering TechnicalConferences, Sacramento, CA, Sept. 14-17, 1997.

Page 5: Kinematics, Polynomials, and Computers|A Brief Historysynthetica.eng.uci.edu/mechanicaldesign101/JMR-Editorial-Feb2011... · Kinematics, Polynomials, and Computers|A Brief History

[24] T. J. Furlong, J. M. Vance, and P. M. Larochelle, 1999, “Spherical Mechanism Synthesis inVirtual Reality,” ASME Journal of Mechanical Design, 121:515.

[25] Q Liao and J M McCarthy, “On the Seven Position Synthesis of a 5-SS Platform Linkage,”ASME Journal of Mechanical Design, 123:74-79, 2001.

[26] B. Roth and F. Freudenstein, “Synthesis of path-generating mechanisms by numerical meth-ods,” ASME Journal of Engineering for Industry, 85B-3:298306, 1963.

[27] F. Freudenstein and B. Roth, “Numerical solution of systems of nonlinear equations,” Journalof ACM, 10(4):550-556, 1963.

[28] L. T. Watson, “A Globally Convergent Algorithm for Computing Fixed Points of C2 Maps,”Journal of Applied Mathematics and Computation, 5:297-311, 1979.

[29] A. P. Morgan, “A Homotopy for Solving Polynomial Systems,” Journal of Applied Mathematicsand Computation, 18:87-92, 1986.

[30] L. W. Tsai and A. P. Morgan, “Solving the Kinematics of the Most General Six- and Five-degree-of-freedom Manipulators by Continuation Methods,” Journal of Mechanisms, Trans-missions, and Automation in Design, 107:189-200, 1985.

[31] C. W. Wampler and A. P. Morgan, “Complete Solution fo the Nine-Point Path Synthe-sis Problem for Four-Bar Linkages,” Journal of Mechanical Design, 114(1):153-159, 1992.doi:10.1115/1.2916909

[32] M. Raghavan and B. Roth, “Inverse kinematics of the general 6R manipulator and relatedlinkages,” Journal of Mechanical Design, 115(3):502-508, 1993. doi:10.1115/1.2919218

[33] M. Raghavan and B. Roth, “Solving polynomial systems for kinematic analysis and synthesisof mechanisms and robot manipulators,” Journal of Mechanical Design, 117(B):71-79, 1995.doi:10.1115/1.2836473

[34] M. Raghavan, “The Stewart Platform of General Geometry has 40 Configurations,” Journalof Mechanical Design, 115(2):277-282, 1993. doi:10.1115/1.2919188

Correction: This version was revised to correct an error in the references [28] and [29].