**Animation.** This mechanical system designed by Yang Liu is a serial chain consisting of 14 links that are coupled by a belt drive so that its end-point draws the Butterfly curve as the base rotates.

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.

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

where a_{k} , b_{k} , c_{k} and d_{k} are real coefficients and theta ranges from 0 to .

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 a_{k} , b_{k}, c_{k} and d_{k} to define the link dimensions,

and the 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,

This equation identifies the curve as the end of a serial chain consisting of a sequence of links links L_{1}, M_{1}, L_{2}, M_{2} and so on, such that the L_{k} links rotate counter clockwise and the M_{k} links rotate clockwise both at the rate .

The initial angles define the configuration of the serial chain when .

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

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.

]]>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.

You can find out more about each of the symposia at these links:

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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 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:

MotionGen is available as a free download at both Google Play- and Apple’s iTunes-Stores.

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It is an interesting challenge to design a mechanical computer to draw an algebraic curve. Mathematicians Michael Kapovich and John Millson have shown this is always possible using linkages, see Alex Kobel’s example below.

Here is how it is done.

**Algebraic curve**. An algebraic curve is the set of points **P**=(x, y) with coordinates that satisfy an algebraic equation,

where u_{i} and v_{i} are integer exponents, and c_{i} are real coefficients.

**Forward kinematics**. In order to draw this curve, we use an RR serial chain shown in Figure 1. All that is needed is to coordinate the angles of this chain so its end-point **P** traces the curve.

The relationship between the joint angles and the coordinates **P**=(x, y) is given by the forward kinematics equations of this RR chain,

**Kempe’s formula**. In his 1876 paper, “On a general method of describing plane curvesof nth degree by linkwork,” A. B. Kempe substituted this into the equation of the curve and simplified the result to obtain,

Kempe interpreted this formula to be the x-coordinate of an N link serial chain, where each link has the length A_{j} and is positioned at the angle,

The end-point of Kempe’s serial chain maintains the value x=C, which means it only moves along the y-direction.

**Mechanical computation**. Now connect cable drives to the two joints of the RR chain so they can be used as inputs to N mechanical computers that calculate the Psi_{j} angular values. The output of the mechanical computers are connected by cable drives to the joints of Kempe’s serial chain.

The operations needed in the mechanical computers are simply multiplication and addition. Multiplication of an angle by a given factor is achieved by using a cable drive with the ratio of the pulley sizes set to the specified factor. The sum of two angular values is achieved using a bevel gear differential.

The result is a mechanical system that constrains the angles of an RR chain so it draws the specified algebraic curve within the workspace of RR serial chain.

**Elliptic cubic curve**. As an example, consider the elliptic cubic curve,

Let the RR chain have link lengths L_{1}=L_{2}=1, and obtain Kempe’s formula for this curve,

This equation has 12 terms, so Kempe’s serial chain has 12 links. The 12 mechanical computers connecting the angles of Kempes serial chain to the driving joints for the RR chain, include six additions and eight multiplications, Figure 2.

Rather than use differentials and cable drives, Kempe used linkages to perform the mechanical computations and couple the results. The outcome is a lot of links as can be seen in this animation provided by Alex Kobel.

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

]]>**Add two bars to a four-bar to get a six-bar.** A four-bar linkage, familiar to all mechanical designers, has an input lever connected by a rod to an output lever, Figure 2 (top). Add two more bars and you have a six-bar linkage. Unfortunately, the standard way to add those bars yields sequence of two four-bar linkages, Figure 2 (bottom), which is not really new. There are other ways to add these two bars but they are beyond the state-of-the-art.

**Other ways to add two bars.** The two additional bars can be added to a four-bar linkage by attaching one end to the connecting rod and the other end to the output link. The result is a stack of two four-bar linkages, known as the Watt I six-bar linkage, Figure 3(a).

Another way is to connect one end of the two bars to the input lever and the other to output lever. This can be done in two ways, either on top of or beneath the four-bar linkage. When added on top, the result is a five-bar loop stacked on the four-bar linkage, known as the Stephenson I six-bar, Figure 3(b). When added beneath, the four-bar linkage is stacked on a five-bar loop, which is a Stephenson II sixbar, Figure 3 (c).

A systematic procedure for design of these alternative six-bar linkages is simply not available to mechanical designers.

**Solving the loop equations for four-bar and six-bar linkages.** There is a simple, though mathematical, reason why four-bar linkages are easy to design and six-bar linkages that are more than a sequence of two four-bar linkages are very hard to design.

Almost sixty years ago in 1954, Ferdinand Freudenstein showed that if we define the movement required of a four-bar linkage, then its dimensions can be computed from its loop equations, which in modern form are give by,

He also showed that the solution of these equations is equivalent to finding the roots of a fourth degree polynomial, which is easy to do.

Soon afterward researchers obtained two sets of loop equations for six-bar linkages and showed that for a required movement, the solution of its loop equations,

is equivalent to finding the roots of a polynomial system of degree, d=264×10^6. A complete solution was only recently achieved after 300 hours of computation on a high performance computer cluster.

This stunning increase in complexity gets worse for eight-bar linkages obtained by adding two bars to a six-bar linkage, which yields three sets of loop equations. In this case calculating the dimensions of the linkage involves the solution of a polynomial system estimated to be of degree, d=10^15, which is massively beyond capabilities of even theoretical polynomial solvers.

**A four-bar with additional design parameters.** Once we understand the structure of the design equations for four-bar, six-bar and even eight-bar linkages, it is possible to take a different approach to the problem of calculating a design from a movement requirement.

A four-bar linkage has eight design variables, which are the four pairs of coordinates that define its hinged joints. Similarly, a six-bar linkage has seven hinged joints or 14 design parameters, and an eight-bar linkage has 10 joints or 20 design parameters. It is possible to consider a six-bar and even an eight-bar linkage as a four-bar linkage with extra design variables. The question then becomes how to use these extra design variables to improve the performance of the linkage system.

**Six-bar linkages provide simple and effective movement.** But is there ever a situation where the complexity of a six-bar linkage is preferred over a four-bar linkage that provides the same movement? Of course there is, but let’s let design engineers describe their experience.

**Søren Matthesen**, design engineer for Vendlet Aps, which makes automated equipment for beds, studied the use of gears, guide rails and four-bar linkages and wound up using a six-bar linkage for his application. He states,

“What really mattered to me was the fact that the six-bar linkage enabled me to solve my design task unlike the four-bar linkage. This is a huge advantage in functionality. Compared to my initial solutions, the six-bar linkage system ends up being more simple and stable to produce, use and maintain.”

**Mike Sutherland**, design engineer, Zennen Engineering, designed a six-bar linkage to fold the rear wheel of a new full-scale folding bicycle, Figure 4. He states,

“The six-bar rearstay folding linkage provided very tight package; that besides being convenient for transportation, also has something of a ‘Transformer’ attraction about it…”

**The simple answer** is that a six-bar linkage designed to achieve the same movement as a four-bar linkage will have free design parameters that allow optimization of other features important to the designer, and this definitely justifies the increase in complexity. And it may be an invention ready for a U. S. Patent.

This video shows the initial testing of UCI’s entry for the 2016 SAE Mini-Baja competition.

You can follow the UCI race team at the web links:

- webpage: http://sites.uci.edu/uciracing/
- instagram: https://www.instagram.com/uciracing/
- facebook: https://www.facebook.com/uciracing/
- linkedin: https://www.linkedin.com/groups/7467197

The 2016 UCI Energy Invitational will be May 21, 2016 on UCI’s Campus. The course will be along East Peltason Drive between Los Trancos and Bison.

The video below shows the 2015 Energy Invitational on UCI’s Campus:

This is the latest from Yang Liu. It is a mechanical synthesis of the Fourier expansion of the batman logo found on Wolfram.com. The video below shows the operation of this harmonic synthesis. ]]>

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.

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