Research Papers

Controlling the Movement of a TRR Spatial Chain With Coupled Six-Bar Function Generators for Biomimetic Motion

[+] Author and Article Information
Mark M. Plecnik

Robotics and Automation Laboratory,
Department of Mechanical
and Aerospace Engineering,
University of California,
Irvine, CA 92697
e-mail: mplecnik@uci.edu

J. Michael McCarthy

Robotics and Automation Laboratory,
Department of Mechanical
and Aerospace Engineering,
University of California,
Irvine, CA 92697
e-mail: jmmccart@uci.edu

1Corresponding author.

Manuscript received August 14, 2015; final manuscript received October 31, 2015; published online May 4, 2016. Assoc. Editor: Hai-Jun Su.

J. Mechanisms Robotics 8(5), 051005 (May 04, 2016) (10 pages) Paper No: JMR-15-1224; doi: 10.1115/1.4032105 History: Received August 14, 2015; Revised October 31, 2015

This paper describes a synthesis technique that constrains a spatial serial chain into a single degree-of-freedom mechanism using planar six-bar function generators. The synthesis process begins by specifying the target motion of a serial chain that is parameterized by time. The goal is to create a mechanism with a constant velocity rotary input that will achieve that motion. To do this, we solve the inverse kinematics equations to find functions of each serial joint angle with respect to time. Since a constant velocity input is desired, time is proportional to the angle of the input link, and each serial joint angle can be expressed as functions of the input angle. This poses a separate function generator problem to control each joint of the serial chain. Function generators are linkages that coordinate their input and output angles. Each function is synthesized using a technique that finds 11 position Stephenson II linkages, which are then packaged onto the serial chain. Using pulleys and the scaling capabilities of function generating linkages, the final device can be packaged compactly. We describe this synthesis procedure through the design of a biomimetic device for reproducing a flapping wing motion.

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Plecnik, M. , and McCarthy, J. M. , 2015, “ Computational Design of Stephenson II Six-Bar Function Generators for 11 Accuracy Points,” ASME J. Mech. Rob., 8(1), p. 011017. [CrossRef]
Roth, B. , 1967, “ The Kinematics of Motion Through Finitely Separated Positions,” ASME J. Appl. Mech., 34(3), pp. 591–598. [CrossRef]
Roth, B. , 1967, “ Finite-Position Theory Applied to Mechanism Synthesis,” ASME J. Appl. Mech., 34(3), pp. 599–605. [CrossRef]
Sandor, G. N. , 1968, “ Principles of a General Quaternion-Operator Method of Spatial Kinematic Synthesis,” ASME J. Appl. Mech., 35(1), pp. 40–46. [CrossRef]
Suh, C. H. , 1968, “ Design of Space Mechanisms for Rigid Body Guidance,” ASME J. Manuf. Sci. Eng., 90(3), pp. 499–506.
Tsai, L. W. , and Roth, B. , 1972, “ Design of Dyads With Helical, Cylindrical, Spherical, Revolute and Prismatic Joints,” Mech. Mach. Theory, 7(1), pp. 85–102. [CrossRef]
Sandor, G. N. , Weng, T. , and Xu, Y. , 1988, “ The Synthesis of Spatial Motion Generators With Prismatic, Revolute and Cylindric Pairs Without Branching Defect,” Mech. Mach. Theory, 23(4), pp. 269–274. [CrossRef]
Innocenti, C. , 1995, “ Polynomial Solution of the Spatial Burmester Problem,” ASME J. Mech. Des., 117(1), pp. 64–68. [CrossRef]
Liao, Q. , and McCarthy, J. M. , 2001, “ On the Seven Position Synthesis of a 5-SS Platform Linkage,” ASME J. Mech. Des., 123(1), pp. 74–79. [CrossRef]
Plecnik, M. , and McCarthy, J. M. , 2012, “ Design of a 5-SS Spatial Steering Linkage,” ASME Paper No. DETC2012-71405.
Murray, A. P. , and McCarthy, J. M. , 1999, “ Burmester Lines of Spatial Five Position Synthesis From the Analysis of a 3-CPC Platform,” ASME J. Mech. Des., 121(1), pp. 45–49. [CrossRef]
Larochelle, P. , 2012, “ Synthesis of Spatial CC Dyads and 4C Mechanisms for Pick & Place Tasks With Guiding Locations,” Latest Advances in Robot Kinematics, Springer, Dordrecht, pp. 437–444.
Mavroidis, C. , Lee, E. , and Alam, M. , 2001, “ A New Polynomial Solution to the Geometric Design Problem of Spatial RR Robot Manipulators Using the Denavit and Hartenberg Parameters,” ASME J. Mech. Des., 123(1), pp. 58–67. [CrossRef]
Lee, E. , Mavroidis, C. , and Merlet, J. P. , 2002, “ Five precision points synthesis of spatial RRR manipulators using interval analysis,” ASME Paper No. DETC2002/MECH-34272.
Lee, E. , and Mavroidis, C. , 2002, “ Solving the Geometric Design Problem of Spatial 3R Robot Manipulators Using Polynomial Homotopy Continuation,” ASME J. Mech. Des., 124(4), pp. 652–661. [CrossRef]
Lee, E. , and Mavroidis, C. , 2004, “ Geometric Design of 3r Robot Manipulators for Reaching Four End-Effector Spatial Poses,” Int. J. Rob. Res., 23(3), pp. 247–254. [CrossRef]
Su, H. , Wampler, C. W. , and McCarthy, J. M. , 2004, “ Geometric Design of Cylindric PRS Serial Chains,” ASME J. Mech. Des., 126(2), pp. 269–277. [CrossRef]
Su, H. , and McCarthy, J. M. , 2005, “ The Synthesis of an RPS Serial Chain to Reach a Given Set of Task Positions,” Mech. Mach. Theory, 40(7), pp. 757–775. [CrossRef]
Perez-Gracia, A. , 2011, “ Synthesis of Spatial RPRP Closed Linkages for a Given Screw System,” ASME J. Mech. Rob., 3(2), p. 021009. [CrossRef]
Batbold, B. , Yihun, Y. , Wolper, J. S. , and Perez-Gracia, A. , 2014, “ Exact Workspace Synthesis for RCCR Linkages,” Computational Kinematics, Springer, Dordrecht, pp. 349–357.
Perez, A. , and McCarthy, J. M. , 2004, “ Dual Quaternion Synthesis of Constrained Robotic Systems,” ASME J. Mech. Des., 126(3), pp. 425–435. [CrossRef]
Perez, A. , Su, H. , and McCarthy, J. M. , 2004, “ Synthetica 2.0: Software for the Synthesis of Constrained Serial Chains,” ASME Paper No. DETC2004-57524.
Perez-Gracia, A. , and McCarthy, J. M. , 2006, “ Kinematic Synthesis of Spatial Serial Chains Using Clifford Algebra Exponentials,” Proc. Inst. Mech. Eng., Part C, 220(7), pp. 953–968. [CrossRef]
Simo-Serra, E. , and Perez-Gracia, A. , 2014, “ Kinematic Synthesis Using Tree Topologies,” Mech. Mach. Theory, 72, pp. 94–113. [CrossRef]
Soh, G. S. , 2014, “ Rigid Body Guidance of Human Gait as Constrained TRS Serial Chain,” ASME Paper No. DETC2014-34881.
Svoboda, A. , 1948, Computing Mechanisms and Linkages, McGraw-Hill, New York.
McLarnan, C. W. , 1963, “ Synthesis of Six-Link Plane Mechanisms by Numerical Analysis,” J. Eng. Ind., 85(1), pp. 5–10. [CrossRef]
Dhingra, A. K. , Cheng, J. C. , and Kohli, D. , 1994, “ Synthesis of Six-Link, Slider-Crank and Four-Link Mechanisms for Function, Path and Motion Generation Using Homotopy With m-Homogenization,” ASME J. Mech. Des., 116(4), pp. 1122–1131. [CrossRef]
Luo, Z. , and Dai, J. S. , 2007, “ Patterned Bootstrap: A New Method That Gives Efficiency for Some Precision Position Synthesis Problems,” ASME J. Mech. Des., 129(2), pp. 173–183. [CrossRef]
Banala, S. K. , and Agrawal, S. K. , 2005, “ Design and Optimization of a Mechanism for Out-Of-Plane Insect Winglike Motion With Twist,” ASME J. Mech. Des., 127(4), pp. 841–844. [CrossRef]
McDonald, M. , and Agrawal, S. K. , 2010, “ Design of a Bio-Inspired Spherical Four-Bar Mechanism for Flapping-Wing Micro Air-Vehicle Applications,” ASME J. Mech. Rob., 2(2), p. 021012. [CrossRef]
Sreetharan, P. S. , Whitney, J. P. , Strauss, M. D. , and Wood, R. J. , 2012, “ Monolithic Fabrication of Millimeter-Scale Machines,” J. Micromech. Microeng., 22(5), p. 055027. [CrossRef]
Teoh, Z. E. , and Wood, R. J. , 2013, “ A Flapping-Wing Microrobot With a Differential Angle-of-Attack Mechanism,” 2013 IEEE International Conference, Robotics and Automation (ICRA), Karlsruhe, Germany, May 6–10, pp. 1381–1388.
Keennon, M. , Andryukov, A. , Klingebiel, K. , and Won, H. , 2014, “ Air Vehicle Flight Mechanism and Control Method for Non-Sinusoidal Wing Flapping,” U.S. patent 2014/0158821 A1.
Haas, F. , and Wootton, R. J. , 1996, “ Two Basic Mechanisms in Insect Wing Folding,” Proc. R. Soc. London, Ser. B, 263(1377), pp. 1651–1658. [CrossRef]
Tobalske, B. , and Dial, K. , 1996, “ Flight Kinematics of Black-Billed Magpies and Pigeons Over a Wide Range of Speeds,” J. Exp. Biol., 199(2), pp. 263–280. [PubMed]
Tobalske, B. W. , Olson, N. E. , and Dial, K. P. , 1997, “ Flight Style of the Black-Billed Magpie: Variation in Wing Kinematics, Neuromuscular Control, and Muscle Composition,” J. Exp. Zool., 279(4), pp. 313–329. [CrossRef] [PubMed]
Gan, D. M. , Liao, Q. Z. , Dai, J. S. , Wei, S. M. , and Qiao, S. G. , 2008, “ Dual Quaternion Based Inverse Kinematics of the General Spatial 7R Mechanism,” Proc. IMechE, Part C, 222(8), pp. 1593–1598. [CrossRef]
McCarthy, J. M. , and Soh, G. S. , 2010, Geometric Design of Linkages, 2nd ed., Springer, New York.
Greenberg, M. D. , 1988, Advanced Engineering Mathematics, Prentice-Hall, Upper Saddle River, NJ.
Proctor, N. S. , 1993, Manual of Ornithology: Avian Structure & Function, Yale University Press, New Haven, CT.
Bates, D. J. , Hauenstein, J. D. , Sommese, A. J. , and Wampler, C. W. , 2006, “ Bertini: Software for Numerical Algebraic Geometry,” available at www.bertini.nd.edu
Bates, D. J. , Hauenstein, J. D. , Sommese, A. J. , and Wampler, C. W. , 2013, Numerically Solving Polynomial Systems With Bertini, SIAM Press, Philadelphia, PA.
Dijksman, E. A. , 1976, Motion Geometry of Mechanisms, Cambridge University Press, London, UK.


Grahic Jump Location
Fig. 1

This figure illustrates the wing gait of an accelerating black-billed magpie. (Reprinted with permission from Tobalske and Dial [36]. Copyright 1996 by the Company of Biologists Ltd.)

Grahic Jump Location
Fig. 2

The links of a TRR spatial chain correlate to the bones of a bird's wing. (Magpie image is reprinted with permission from Tobalske et al. [37]. Copyright 1997 by John Wiley and Sons.)

Grahic Jump Location
Fig. 3

A smooth trajectory of the wrist point is produced by fitting Fourier series to the discrete joint angle functions for ψA, ψB, ψC

Grahic Jump Location
Fig. 4

The joint angle functions for (a) ψA, (b) ψB, (c) ψC, and (d) ψD. The discrete data curves were fitted with Fourier series functions. Eleven task points were chosen from the Fourier functions to synthesize a mechanism.

Grahic Jump Location
Fig. 5

The addition of sprung joints creates a compliant hand wing design. Hard stops prevent dorsal flexure of the hand wing. The new segment lengths are l4 = 0.375, l5 = 2.125, l6 = 2.5, and l7 = 2.6.

Grahic Jump Location
Fig. 6

A Stephenson II function generator in (a) the initial configuration and (b) the jth configuration

Grahic Jump Location
Fig. 7

The linkages selected to mechanize the joint angle trajectories of the magpie wing. The linkage that generates (a) ψA=fA(ϕ) passes through seven task positions, (b) ψB=fB(ϕ) passes through nine task positions, (c) ψC=fC(ϕ) passes through eight task positions, and (d) ψD=fD(ϕ) passes through nine task positions.

Grahic Jump Location
Fig. 8

Diagram of power transmission from the motor, through the function generators, to the joints

Grahic Jump Location
Fig. 9

A prototype design for the function generator controlled magpie wing

Grahic Jump Location
Fig. 10

The wingtip path (black) of the mechanized wing alongside the desired path (gray) interpreted from the data recorded in Fig. 1




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