Research Papers

The Spherical Equivalent of Bresse's Circles: The Case of Crossed Double-Crank Linkages

[+] Author and Article Information
Giorgio Figliolini

Department of Civil and Mechanical Engineering,
University of Cassino and Southern Lazio,
Via G. Di Biasio 43,
Cassino (Fr) 03043, Italy
e-mail: figliolini@unicas.it

Jorge Angeles

Department of Mechanical Engineering & CIM,
McGill University,
817 Sherbrooke Street W.,
Montréal, QC H3A 2K6, Canada
e-mail: angeles@cim.mcgill.ca

Manuscript received March 5, 2016; final manuscript received December 9, 2016; published online January 12, 2017. Assoc. Editor: Jian S. Dai.

J. Mechanisms Robotics 9(1), 011014 (Jan 12, 2017) (11 pages) Paper No: JMR-16-1056; doi: 10.1115/1.4035504 History: Received March 05, 2016; Revised December 09, 2016

The subject of Bresse's circles is classical in the kinematics of planar mechanisms. These are the loci of the coupler points that exhibit either zero normal or zero tangential acceleration. Described in this paper is the construction of the spherical equivalent of Bresse's circles, which take the form of an inflexion spherical cubic and a Thales ellipse, respectively. An algorithm is developed to produce these loci for the case of the spherical antiparallelogram. As a byproduct, the corresponding polodes and their evolutes are obtained.

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Hirschhorn, J. , 1962, Kinematics and Dynamics of Plane Mechanisms, McGraw-Hill, New York.
Hunt, K. H. , 1978, Kinematic Geometry of Mechanisms, Clarendon Press, Oxford, UK.
Bottema, O. , and Roth, B. , 1979, Theoretical Kinematics, North Holland Publishing, Amsterdam, The Netherlands.
Di Benedetto, A. , and Pennestrì, E. , 1993, Introduction to the Kinematics of Mechanisms, Vol. 1–3, Casa Editrice Ambrosiana, Milan, Italy.
Chiang, C. H. , 2000, Kinematics of Planar Mechanisms, Krieger Publishing Company, Malabar, FL.
Meyer zur Capellen, W. , and Dittrich, G. , 1966, “ Note on the Determination of Accelerations in Plane Kinematics,” J. Mech, 1(3), pp. 315–319. [CrossRef]
Gilmore, B. J. , and Cipra, R. J. , 1983, “ An Analytical Method for Computing the Instant Centers, Centrodes, Inflection Circles, and Centers of Curvature of the Centrodes by Successively Grounding Each Link,” J. Mech., Transm., Autom., 105(3), pp. 407–414. [CrossRef]
Figliolini, G. , Conte, M. , and Rea, P. , 2012, “ Algebraic Algorithm for the Kinematic Analysis of Slider-Crank/Rocker Mechanisms,” ASME J. Mech. Rob., 4(1), p. 011003. [CrossRef]
Chiang, C. H. , 2000, Kinematics of Spherical Mechanisms, Krieger Publishing Company, Malabar, FL.
Chiang, C. H. , 1992, “ Spherical Kinematics in Contrast to Planar Kinematics,” Mech. Mach. Theory, 27(3), pp. 243–250. [CrossRef]
McCarthy, J. M. , and Ravani, B. , 1986, “ Differential Kinematics of Spherical and Spatial Motions Using Kinematic Mapping,” J. Appl. Mech., 53(1), pp. 15–22. [CrossRef]
Veldkamp, G. R. , 1967, “ An Approach to Spherical Kinematics Using Tools Suggested by Plane Kinematics,” J. Mech., 2(4), pp. 437–450. [CrossRef]
Bisshopp, K. E. , 1969, “ Note on Spherical Motion,” J. Mech., 4(2), pp. 159–166. [CrossRef]
Kamphuis, H. J. , 1969, “ Application of Spherical Instantaneous Kinematics to the Spherical Slider-Crank Mechanism,” J. Mech., 4(1), pp. 43–56. [CrossRef]
Fu, T.-T. , and Chang, C. H. , 1994, “ Simulating a Given Spherical Motion by the Polode Method,” Mech. Mach. Theory, 29(2), pp. 237–249. [CrossRef]
Sodhi, R. , and Shoup, T. E. , 1982, “ Axodes for the Four-Revolute Spherical Mechanism,” Mech. Mach, Theory, 17(3), pp. 173–178. [CrossRef]
Meyer zur Capellen, W. , and Dittrich, G. , 1966, “ The Instantaneous Distribution of Acceleration of a Spherically Moving System,” J. Mech., 1(1), pp. 23–42. [CrossRef]
Bottema, O. , 1965, “ Acceleration Axes in Spherical Kinematics,” J. Eng. Ind., 87(2), pp. 150–153. [CrossRef]
Bresse, A. , 1853, “ Mémoire sur un Théorème Nouveau Concernant les Mouvements Plans et l'Application de la Cinématique à la Détermination des Rayons de Courbure,” J. Éc. Polytech., Paris, 20, pp. 89–115.
Koetsier, T. , 1986, “ From Kinematically Generated Curves to Instantaneous Invariants: Episodes in the History of Instantaneous Planar Kinematics,” Mech. Mach. Theory, 21(6), pp. 489–498. [CrossRef]
Shoup, T. E. , 1991, “ Centrodes of the Slider-Crank Mechanism,” 8th IFToMM World Congress on the Theory of Machines and Mechanisms, Prague, Vol. 1, pp. 59–62.
Figliolini, G. , Rea, P. , and Angeles, J. , 2015, “ The Synthesis of the Axodes of RCCC Linkages,” ASME J. Mech. Rob., 8(2), p. 021011. [CrossRef]
Figliolini, G. , and Angeles, J. , 2006, “ The Synthesis of the Pitch Surfaces of Internal and External Skew-Gears and Their Racks,” ASME J. Mech. Des., 128(4), pp. 794–802. [CrossRef]
Figliolini, G. , and Angeles, J. , 2011, “ Synthesis of the Pitch Cones of N-Lobed Elliptical Bevel Gears,” ASME J. Mech. Des., 133(3), p. 031002. [CrossRef]
Figliolini, G. , Stachel, H. , and Angeles, J. , 2015, “ The Role of the Orthogonal Helicoid in the Generation of the Tooth Flanks of Involute-Gear Pairs With Skew Axes,” ASME J. Mech. Rob., 7(1), p. 011003. [CrossRef]
Schaaf, J. A. , and Yang, A.-T. , 1992, “ Kinematic Geometry of Spherical Evolutes,” ASME J. Mech. Des., 114(1), pp. 109–116. [CrossRef]
Dirnböck, H. , 1999, “ Absolute Polarity on the Sphere; Conics; Loxodrome, Tractrix,” Math. Commun., 4(2), pp. 225–240.
Serret, P. J. , 1860, Théorie Nouvelle Géométrique et Mécanique des Lignes a Double Courbure, Mallet-Bachelier, Paris, pp. 56–59.
Garnier, R. , 1949, Cours de Cinématique, Tome II, Chap. VI, Mouvement Sphérique, Gauthier-Villars, Paris, pp. 86–114.
Ting, K.-L. , and Bunduwongse, R. , 1991, “ Unified Spherical Curvature Theory of Point-, Plane-, and Circle-Paths,” ASME J. Mech. Des., 113(2), pp. 142–149. [CrossRef]
Hirschhorn, J. , 1989, “ Path Curvature in Three-Dimensional Constrained Motion of a Rigid Body,” Mech. Mach. Theory, 24(2), pp. 73–81. [CrossRef]
Stachel, H. , 2015, “ Strophoids, a Family of Cubic Curves With Remarkable Properties,” J. Ind. Des. Eng. Graph., 10(ICEGD), pp. 65–72.
Özçelik, Z. , and Şaka, Z. , 2010, “ Ball and Burmester Points in Spherical Kinematics and Their Special Cases,” Forsch Ingenierwes, 74(2), pp. 111–122. [CrossRef]
Pottmann, H. , 1985, “ Zur Konstruktion der Sphärischen Wendekurve,” Mech. Mach. Theory, 20(1), pp. 77–79. [CrossRef]
Tipparthi, H. , and Larochelle, P. , 2011, “ Orientation Order Analysis of Spherical Four-Bar Mechanisms,” ASME J. Mech. Rob., 3(4), p. 044501. [CrossRef]
Al-Widyan, K. , and Angeles, J. , 2014, “ The Synthesis of Spherical Motion Generators in the Presence of an Incomplete Set of Attitudes,” ASME J. Mech. Rob., 6(3), p. 031008. [CrossRef]


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Fig. 1

Jacques Antoine Charles Bresse

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Fig. 2

Paper and original drawings of the Bresse circles1

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Fig. 3

Kinematic sketch of the spherical four-bar linkage A0ABB0 along with its instant centers of rotation P and R

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Fig. 4

Fixed and moving reference frames

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Fig. 5

Conical fixed axodes (elliptic cones) for a spherical four-bar linkage with a = b = 80 deg and c = f = 40 deg: (a) antiparallelogram mechanism and (b) double-crank mechanism

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Fig. 6

Fixed and moving axodes for an antiparallelogram mechanism with a = b = 80 deg and c = f = 40 deg: (a) ϕ = 0 deg, (b) ϕ = 30 deg, and (c) ϕ = 90 deg

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Fig. 7

Fixed and moving axodes for an antiparallelogram mechanism with a = b = 60 deg and c = f = 45 deg: (a) ϕ = 0 deg, (b) ϕ = 30 deg, and (c) ϕ = 90 deg

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Fig. 8

Elliptic cone (axode), spherical ellipse (polode), and its evolute curve for a = b = 80 deg and c = f = 70 deg

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Fig. 9

Canonical frame centered at P and composed by the poletangent great circle and polenormal great circle

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Fig. 10

Inflection surface (IS) for Θ=0.4 and inflection spherical cubic for Θ=0.505 : (a) axonometric view of the IS, (b) front-view (xz-plane), (c) side-view (yz-plane), (d) top-view (xy-plane), (e) intersection circles of IS with planes parallel to the xy-plane, and (f) inflection spherical cubic as intersection of the IS with the unit sphere of the spherical motion

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Fig. 11

Inflection spherical cubic and Thales ellipse, along with both polodes and their evolutes: (a) and (b) for a = 67 deg, f = 43 deg, ω = 2 rad/s; (c) and (d) for a = 75 deg, f = 70 deg, ω = 1 rad/s; (e) and (f) for a = 73 deg, f = 71 deg, ω = 1 rad/s




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