## Design of Linkages to Draw Curves, GRASPLab Seminar

On May 8, 2018, I was pleased to give a seminar at the University of Pennsylvania GRASPLab:  McCarthy Seminar.

They also videotaped my lecture.  Here it is:

## Design Research in China: Beijing

This video shows the excellent research in design at universities in Beijing.  This is the final of five videos highlighting design research across China taken during a visit in September 2016.

For our colleagues in China, here is a link to a YouKu version.

Beijing Design Research

## 2017 ASME Mechanisms and Robotics Conference

The program for the 2017 ASME Mechanisms and Robotics Conference is now on-line. You can access it at the link: ASME MR Conference

The papers from the UCI Robotics and Automation Lab are:

2017 MR Conf Program

## Design Research in China: Changzhou

Our trip through China concluded at a conference and workshop at Changzhou University. This video highlights the Changzhou conference on innovation in Robotics and Intelligent Manufacturing, the beauty of the city of Changzhou, and a rainy night in Shanghai.

Our colleagues in China can see this video on YouKu at: Design Research in China: Changzhou

Changzhou YouKu

## Design Research in China: Dalian

Design Research Dalian

This is the third of five videos highlighting design research across China.  This captures the beauty of Dalian, a city on the Yellow sea, and the excellent research in precision machine design by colleagues and their students at Dalian University of Technology.

Here is a link to this video on Youku for our colleagues in China:
http://v.youku.com/v_show/id_XMTg5MDg4MzI0NA==.html

Youku Dalian

## Design Research in China: Xi’an

Design Research Xian

This video of our visit to Xi’an captures the beauty of the city and its surroundings, as well as the personality of the excellent professors and students at Xidian University.

For our colleagues in China, here is a link to a Youku version of this video: http://v.youku.com/v_show/id_XMTg2MzQ5NDI4MA==.html

Youku Xian Video

## Prototype of the Trifolium Mechanism

Trifolium prototype

Yang Liu and Peter Yang designed and built this physical prototype of our Trifolium mechanism.  It is fabricated from ABS using the Stratasys Fortus system in UCI’s Institute for Design and Manufacturing Innovation.

Our Chinese colleagues can view this video on Youku at the link: http://v.youku.com/v_show/id_XMTg2MzQ2MjE5Mg==.html

Youku Trifolium Prototype

## Design Research in China: Tianjin

Design Research Tianjin

Chris McCarthy filmed and edited this video of our visit to Tianjin, which showcases the design research in mechanisms and robotics at Tianjin University and captures the energy and beauty of the city and its people.

For our colleagues in China this video is available on youku.com: http://v.youku.com/v_show/id_XMTgzMTIyMTg1Ng==.html

Youku Design Tianjin

## Manufacturing Prototype for the Butterfly Linkage

This animation is taken from Yang Liu’s detailed design drawings for the manufacturing prototype of the Butterfly Linkage. The component parts are to be constructed by additive manufacturing.

This animation includes the music of Explosions in the Sky:

For our colleagues in China this animation is available through youku.com: http://v.youku.com/v_show/id_XMTgzMzA5ODQyOA==.html

Youku Prototype Butterfly

## Design of Drawing Mechanisms

Mechanical systems that draw trigonometric curves provide a versatile way to draw complex curves. Yang Liu designed this serial chain consisting of 14 links coupled by a belt drive to draw the Butterfly curve.

The Butterfly curve is an example of a trigonometric plane curve, and our study of Kempe’s design of linkages to draw algebraic curves has lead us to a way to design serial chains that draw these curves.

Here is how it is done.

Butterfly curve

Trigonometric curves. A trigonometric plane curve is a parametrized curve with coordinate functions, P = (x, y), that are finite Fourier series,

Trigonmetric equation

where ak , bk , ck and dk are real coefficients and theta ranges from 0 to $2\pi$.

A large number of well-known curves have this form, such as Limacon of Pascal, the Cardioid, Trifolium, Hypocycloid and Lissajous figures.

A coupled serial chain. Without going into too much detail, it is possible to use the coefficients ak , bk, ck and dk to define the link dimensions,

and the initial angles,

Initial angles

These parameters allow us to redefine the trigonometric equations of the curve as the coordinate equations of the links of a serial chain,

Serial chain equations

This equation identifies the curve as the end of a serial chain consisting of a sequence of links L1, M1, L2, M2 and so on, such that the Lk links rotate counter clockwise and the Mk links rotate clockwise both at the rate $k\theta$.

The initial angles define the configuration of the serial chain when  $\theta=0$.

The Butterfly drawing mechanism. The trigonometric curve of the Butterfly linkage is defined by the coefficients in the following table,

Table of Butterfly coefficients

These coefficients are used to calculate the dimensions and initial configuration of the serial chain listed in the following table,