Biologically-inspired Linkage Design: Computing Form From Function

1 Newsjournal of the Society for Industrial and Applied Mathematics S. |. a.ITL|'lE|l. |S sinews.siam. 0rg Volume 48/ Number 8 October 2015 Biologically-inspired Linkage Design Computing Form From Function By Jonathan D. Hauenstein and J. Michael McCarthy For several decades, engineers have been designing robots resembling the gen- eral structure of a human, called humanoid robots. In response to the nuclear disaster at Fukushima, Japan in 2011, the DARPA Robotics Challenge was created to accelerate the development of robots that could perform tasks in areas too dangerous for humans. A total of 23 teams participated in the ﬁnals of this challenge in June 2015, which was won, along with a $2 million prize, by the human- oid DRC-HUBO (derived from “HUmanoid roBOt”) from Team KAIST (Daejeon, Republic of Korea). Coincidentally, the National Institute for Mathematical Sciences (NIMS), also in Daejeon, hosted the SIAM Conference on Applied Algebraic Geometry in August 2015. Although this Daejeon com- monality is fortuitous, there is a long and storied history of the relationship between algebraic geometry and mechanical engi- neering, particularly in the area of kinemat- ics, which studies the motion of systems such as a robotic arm. The arm of the DRC-HUBO robot con- sists of four linkages coupled together: a hand linkage, a wrist linkage, an elbow linkage, and a shoulder linkage. In contrast to Rube Goldberg machines, biologically- inspired linkages should be minimalist based on the function required and thus follow an “0ccarn’s razor” mentality. That is, the form should be derived to yield the function- ality required in a simple way, as unnecessar- ily complicated designs increase the chances of failure. In addition to mimicking human motion, linkages may also be inspired by other organ- isms. One such example is the elegance of a bird flapping its wings. The develop- ment of this linkage [2], which received the A.T. Yang Memorial Award in Theoretical Kinematics at the ASME 2015 International Design Engineering Technical Conferences, provides an excellent case study of the symbiotic relationship between mechanism design and numerical algebraic geometry, the area of algebraic geometry that designs and implements numerical algorithms for solving and manipulating solution sets to sys- tems of polynomial equations. Numerically Figure 1. Four coordinated six-bar linkages designed to take a constant rotational input and transform it to angular outputs that match the wing movement of a bird. Solving Polynomial Systems with Bertini, published by SIAM [1], provides an over- view of numerical algebraic geometry with a focus on using the software Bertini. This bird—ﬂapping linkage, as shown in Figure I and in the video (http://y2u. be/7aXmze9Ynis), was designed to take a constant rotational input from a motor. From this functionality, the general form of the linkage began to take shape, in this case, four coordinated six—bar linkages. Figure 2 (on page 3) shows a schematic drawing of a special kind of six-bar linkage used in the bird—ﬂapping linkage, called a Stephenson II six-bar linkage. Working with a family of four coordi- nated six—bar linkages, the next step for the See Linkage on page 3 Is There _a Role for Mathematics ; i 5 ll ~':