### Posts

Here is an example of our Bezier curve drawing linkages. The first draws Yang Liu’s first name in cursive. Rather than show the linkages all at once, they are separated so it is easier to see the curves that they draw. Also shown here is the linkage that draws the a cursive version of the Chinese character “long” which means dragon. If you compare this to previous work, I hope you see that the curves we can draw are more complicated while the linkages are becoming simpler.

Here are animations of a linkage system that writes a cursive Yang, and one that writes the cursive Chinese character “long” which means dragon.

For our colleagues in China, here are the Youku versions:

Cursive Yang Youku

Cursive Dragon Youku

Yang Liu has developed Bezier linkages that trace trigonometric Bezier curves.  In this video he shows that he can design these linkages to draw his name in cursive.

This is a demonstration of the colloquial description of Kempe’s Universality Theorem, which says “there is a linkage that signs your name,” described by William Thurston.  See the article by Joseph Malkevitch.

Our Chinese colleagues can see this video on YouKu at the link:

Yang Signature YouKu

## 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

Recent research on the design of linkages by Yang Liu has resulted in “Bezier linkages” that can be used to draw arbitrary Bezier curves. The trick is to use trigonometric Bezier curves. This whale consists of four Bezier segments and is drawn by four Bezier linkage elements.

This youtube version includes music by Explosions in the Sky:

A youku.com version for colleagues in China can be seen here:
http://v.youku.com/v_show/id_XMTgzMjU4NTQwMA==.html

Whale on Youku

## 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,

The result is a serial chain consisting of 14 links that are coordinated to move together as the base rotates. The end-point of the chain draws the Butterfly curve.

Butterfly drawing mechanism

## 2016 Mechanisms and Robotics Conference

Symposia organized for the 2016 Mechanisms and Robotics Conference

The 2016 Mechanisms and Robotics conference is part of International Design Engineering Technical Conferences organized by ASME International in Charlotte, North Caroline, August 22-24.

Plenary speaker Bernard Roth is the Academic Director of Stanford University’s d.school and the author of the Achievement Habit.

For some reason, ASME has broken these links to the 2016 IDETC conference, but you can find out more about each of the symposia at the conference overview link: 2016 ASME Mechanism and Robotics Conference Overview.  Then select the Expand all Symposia Link to see the sessions and a list of papers.

## Motion Gen Linkage Design App

MotionGen is a planar four-bar linkage simulation and synthesis app that helps users synthesize planar four-bar linkages by assembling two of the planar RR-, RP- and PR-dyad types, (R refers to a revolute or hinged joint and P refers to a prismatic or sliding joint).

The input task is a planar motion given as a set of discrete positions and orientations and the app computes type and dimensions of synthesized planar four-bar linkages, where their coupler interpolates through the given poses either exactly or approximately while minimizing an algebraic fitting error.  The algorithm implemented in the app extracts the geometric constraints (circular, fixed-line or line-tangent-to-a-circle) implicit in a given motion and matches them with corresponding mechanical dyad types enumerated earlier.  In the process, the dimensions of the dyads are also computed. By picking two dyads at a time, a planar four-bar linkage is formed. Due to the degree of polynomial system created in the solution, up to a total of six four-bar linkages can be computed for a given motion.

MotionGen display

MotionGen also lets users simulate planar four-bar linkages by assembling the constraints of planar dyads on a blank- or image-overlaid screen. This constraint-based simulation approach mirrors the synthesis approach and allows users to input simple geometric features (circles and lines) for assembly and animation. As an example of the Simulation capabilities of the app, Figure shows a walking robot driven by two sets of planar four-bar linkages where the foot approximately traces a trajectory of walking motion. The users can input two dyads on top of an imported image of a robot or machine to verify the motion and make interactive changes to the trajectories.

Anurag Purwar describes MotionGen and its applications in this video:

## Trifolium using contra-parallelograms

Trifolium Contra-parallelograms

Yang Liu shows that a simple linkage can draw the trifolium curve.

For comparison here is the Kempe linkage that draws a trifolium curve obtained by Alexander Kobel.

## Fourier Curve Tracing

Fourier Curve Tracing

This animation by Yang Liu is inspired by the mechanical Fourier synthesizer described by Dayton Miller, see A 32-element harmonic synthesizer.

This mechanical system combines the terms of a Fourier approximation of the batman curve found on Wolfram.com. The video below shows this device draws the batman curve.

## SIAM News: Biologically inspired linkage design

SIAM News

This article by Jon Hauenstein with me for SIAM News (Society for Industrial and Applied Mathematics) describes research by Mark Plecnik in the computer-aided design of linkages to provide mechanical movement of a bird’s wing. Here is Mark’s video of the his wing flapping mechanism.