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Research Papers

# Computational Design of Stephenson II Six-Bar Function Generators for 11 Accuracy Points

[+] Author and Article Information
Mark M. Plecnik

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

J. Michael McCarthy

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

Manuscript received March 9, 2015; final manuscript received July 8, 2015; published online August 18, 2015. Assoc. Editor: James Schmiedeler.

J. Mechanisms Robotics 8(1), 011017 (Aug 18, 2015) (9 pages) Paper No: JMR-15-1055; doi: 10.1115/1.4031124 History: Received March 09, 2015; Revised July 08, 2015

## Abstract

This paper presents a design methodology for Stephenson II six-bar function generators that coordinate 11 input and output angles. A complex number formulation of the loop equations yields 70 quadratic equations in 70 unknowns, which is reduced to a system of ten eighth degree polynomial equations of total degree $810=1.07×109$. These equations have a monomial structure which yields a multihomogeneous degree of 264,241,152. A sequence of polynomial homotopies was used to solve these equations and obtain 1,521,037 nonsingular solutions. Contained in these solutions are linkage design candidates which are evaluated to identify cognates, and then analyzed to determine their input–output angles in each assembly. The result is a set of feasible linkage designs that reach the required accuracy points in a single assembly. As an example, three Stephenson II function generators are designed, which provide the input–output functions for the hip, knee, and ankle of a humanoid walking gait.

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Topics: Linkages , Design , Generators , Knee

## References

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## Figures

Fig. 1

(a) A Stephenson II linkage in its reference configuration and (b) a Stephenson II linkage displaced from its reference configuration drawn with dotted lines

Fig. 2

(a) Link vectors of the Stephenson II linkage and (b) link vectors in the jth displaced configuration

Fig. 3

Three Stephenson II cognate function generators shown in a single overconstrained mechanism. The three cognate mechanisms are comprised of pivots {A,B,C,D,G,H,F}, {A,B,C,D,G′,H′,F′}, and {A,B,C,F′,G″,H″,F}.

Fig. 4

The coupler curve traced by C as part of the four-bar DGHF is the same for two additional cognate linkages DG′H′F′ and F′G″H″F

Fig. 5

A humanoid leg is modeled as a planar 3R chain where l1=18.7 and l2=14

Fig. 6

The specified functions for the hip, knee, and ankle joints: (a) hip joint function ΔyA=fA(t), (b) knee joint function ΔyB=fB(t), and (c) ankle joint function ΔyC=fC(t). These functions are plotted alongside the functions generated by the mechanisms shown in Fig. 7.

Fig. 7

The hip, knee, and ankle Stephenson II six-bar function generators that generate the desired functions: (a) hip joint ΔyA=fA(t) function generator that reaches 10 accuracy points, (b) knee joint ΔyB=fB(t) function generator that reaches 11 accuracy points, and (c) ankle joint ΔyC=fC(t) function generator that reaches 11 accuracy points

Fig. 8

Solid model of the hip, knee, and ankle function generators integrated into a humanoid walker. The schematics in Fig. 7 overlay their physical embodiments for each function generator.

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