## Rectilinear Six-Bar (Candy Coating Linkage)

Jeff Glabe designed this six-bar linkage to move through six task positions while maintaining a parallel orientation.  This required the calculation of 55,000 linkages to find 26 that work.  This one has the additional feature that it is operated by a rotating crank (the red link).  The video is a collaboration of Jeff Glabe and Benjamin Liu.

This spatial six-bar linkage provides a compact flapping wing mechanism for a hovering micro air vehicle that controls both swing and pitch movements.  It is the result of a collaboration between Benjamin Liu and Peter Wang.

## Flapping Wing Prototype

This is the manufacturing prototype for the flapping wing mechanism designed by Peter Wang and modeled by Benjamin Liu. The wing is 5cm in length to match the dimensions of a hummingbird.

Here is a link to the youku.com video for our Chinese colleagues: Flapping Wing Prototype.

Youku Flapping Wing

## RPR-2SS Soil Conditioning Valve

This is an animation by Benjamin Liu of a spatial RPR-2SS six-bar linkage designed by Peter Wang for use on tunnel boring machines. This new device controls a valve that cleans and closes the soil conditioning port when the flow of soil lubricant is stopped. Soil conditioning provides lubricant to manage the cutting operation. This self-cleaning valve stops clogging of the port which is a common problem.

## Flapping Wing Mechanism

Flapping wing mechanism

Benjamin Liu prepared this animation of the flapping wing mechanism designed by Peter Wang. It controls both the swing and the pitch of the wing to improve aerodynamics in hovering flight.

This is the Youtube version of the animation.

The youku version for our Chinese colleagues is available at the link: Hummingbird Linkage.

Flapping wing mechanism on youku

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,