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1. Analysis of the Rigid Motion of a Developable Conical Mechanism

   https://doi.org/10.1115/DETC2020-22643  

   Woodland, McKell; Hsiung, Michelle; Matheson, Erin L.; Safsten, C. Alex; Greenwood, Jacob; Halverson, Denise M.; Howell, Larry L. (January 2020, Proceedings of the 2020 International Design Engineering Conferences)

   null (Ed.)

   Abstract We demonstrate analytically that it is possible to construct a developable mechanism on a cone that has rigid motion. We solve for the paths of rigid motion and analyze the properties of this motion. In particular, we provide an analytical method for predicting the behavior of the mechanism with respect to the conical surface. Moreover, we observe that the conical developable mechanisms specified in this paper have motion paths that necessarily contain bifurcation points which lead to an unbounded array of motion paths in the parameterization plane. 

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2. Developable mechanisms on developable surfaces

   https://doi.org/10.1126/scirobotics.aau5171  

   Nelson, Todd G.; Zimmerman, Trent K.; Magleby, Spencer P.; Lang, Robert J.; Howell, Larry L. (February 2019, Science Robotics)

   The trend toward smaller mechanism footprints and volumes, while maintaining the ability to perform complex tasks, presents the opportunity for exploration of hypercompact mechanical systems integrated with curved surfaces. Developable surfaces are shapes that a flat sheet can take without tearing or stretching, and they represent a wide range of manufactured surfaces. This work introduces “developable mechanisms” as devices that emerge from or conform to developable surfaces. They are made possible by aligning hinge axes with developable surface ruling lines to enable mobility. Because rigid-link motion depends on the relative orientation of hinge axes and not link geometry, links can take the shape of the corresponding developable surface. Mechanisms are classified by their associated surface type, and these relationships are defined and demonstrated by example. Developable mechanisms show promise for meeting unfilled needs using systems not previously envisioned. 

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3. Designing Developable Mechanisms From Flat Patterns

   https://doi.org/10.1115/DETC2020-22445  

   Hyatt, Lance P.; Lytle, Amanda; Magleby, Spencer P.; Howell, Larry L. (January 2020, Proceedings of the 2020 International Design Engineering Conferences)

   null (Ed.)

   Abstract This paper presents tools and methods to design cylindrical and conical developable mechanisms from flat, planar patterns. Equations are presented that relate the link lengths and link angles of planar and spherical mechanisms to the dimensions in a flat configuration. These flat patterns can then be formed into curved, developable mechanisms. Guidelines are established to determine if a mechanism described by a flat pattern can exhibit intramobile or extramobile behavior. A developable mechanism can only potentially exhibit intramobile or extramobile behavior if none of the links extend beyond half of the flat pattern. The behavior of a mechanism can change depending on the location of the cut of the flat pattern. Different joint designs are discussed including lamina emergent torsional (LET) joints. Physical examples are presented. 

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4. Limits of Extramobile and Intramobile Motion of Cylindrical Developable Mechanisms

   https://doi.org/10.1115/DETC2020-22544  

   Butler, Jared; Greenwood, Jacob; Howell, Larry L.; Magleby, Spencer P. (January 2020, Proceedings of the 2020 International Design Engineering Conferences)

   null (Ed.)

   Abstract Mechanisms that can both deploy and perform desired tasks offer a multifunctional advantage over traditional mechanisms. Developable mechanisms (DMs) are devices capable of conforming to a predetermined developable surface and deploying from that surface to achieve specific motions. This paper presents new terminology that defines motion of developable mechanisms while interior and exterior to a developable surface. The limits of motion are identified using defined conditions, and it is shown that the more difficult of these conditions may be treated as a non-factor during the design of cylindrical developable mechanisms given certain assumptions. 

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5. Limits of Extramobile and Intramobile Motion of Cylindrical Developable Mechanisms

   https://doi.org/10.1115/1.4048833  

   Butler, Jared; Greenwood, Jacob; Howell, Larry L.; Magleby, Spencer (February 2021, Journal of Mechanisms and Robotics)

   null (Ed.)

   Abstract Mechanisms that can both deploy and provide motions to perform desired tasks offer a multifunctional advantage over traditional mechanisms. Developable mechanisms (DMs) are devices capable of conforming to a predetermined developable surface and deploying from that surface to achieve specific motions. This paper builds on the previously identified behaviors of extramobility and intramobility by introducing the terminology of extramobile and intramobile motions, which define the motion of developable mechanisms while interior and exterior to a developable surface. The limits of motion are identified using defined conditions. It is shown that the more difficult of these conditions to kinematically predict may be treated as a non-factor during the design of cylindrical developable mechanisms given certain assumptions. The impact of toggle positions for each case is discussed. Physical prototypes demonstrate the results. 

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