Animations of linkage movement.

Prototype Butterfly Linkage

Manufacturing Prototype for the Butterfly Linkage

Prototype Butterfly Linkage

Prototype 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

Youku Prototype Butterfly

Whale Linkage

Bezier Linkages

Whale Linkage

Whale Linkage

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

Whale on Youku

Butterfly drawing mechanism

Design of Drawing Mechanisms

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.

Butterfly curve

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

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,

Link dimensions

Link dimensions

and the initial angles,

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

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

Table of Butterfly coefficients

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

Table of Butterfly link dimensions

Table of Butterfly link dimensions

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

Butterfly drawing mechanism

Mechanisms and Robotics Conference

2016 Mechanisms and Robotics Conference

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.

 

MotionGen Horse

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 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:

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

 

Elliptic curve drawing

Mechanical computation and algebraic curves

Animation.  Here is Yang Liu’s animation of the mechanical system that draws an elliptic cubic curve.

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,

Algebraic curve

 Algebraic curve

where ui and vi are integer exponents, and ci 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.

RR serial chain

 Figure 1. The joints of an RR serial chain are coordinated to guide the point P along a specified curve.

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

RR forward kinematics

 RR forward kinematics

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 formula

 Kempe formula

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

Kempe link angles

 Kempe link angles

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 Psij 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,

Elliptic cubic curve

 Elliptic cubic curve

Let the RR chain have link lengths L1=L2=1, and obtain Kempe’s formula for this curve,

Elliptic cubic formula

 Elliptic cubic formula

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.

Elliptic curve drawing

 Figure 2. The Kempe’s serial chain and the mechanical computations that constrain the RR serial chain to draw an elliptic cubic curve.

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.

Trifolium using 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

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

SIAM News: Biologically inspired linkage design

SIAM News

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.

Heart Linkage

Heart Trajectory

Heart Linkage

Heart Linkage

This mechanical system was designed by Yang Liu to trace the shape of a heart. The work is inspired by the mechanical 32-element harmonic synthesizer described by Dayton Miller in the 1916 article in the Journal of the Franklin Institute.