T2 Two and Three Position Synthesis

Four-bar Linkage Synthesis, Two and Three Task Positions

These notes for the Stanford University course ME 322 Kinematic Synthesis of Mechanisms introduce the synthesis of a four-bar linkages that guide the coupler link through two and three specified task positions.

More notes on Kinematic Synthesis    Also see my book Kinematic Synthesis of Mechanisms: a project based approach

December 12, 2017.  These notes have been revised to introduce Sandor and Erdman’s formulas for linkage synthesis which solve for the input and output cranks.  This provides a nice pairing with the related equations that are solved for the fixed and moving pivots, directly.

T2 Two and Three Position Synthesis revised v2



T1 Four-bar Linkage Analysis

Four-bar Linkage Analysis Notes

These notes have been prepared for the Stanford University graduate course ME 322 Kinematic Synthesis of Mechanisms.  This first set details the position and velocity analysis of a four-bar linkage. 

Update:  December 12, 2017.  These notes have been revised to represent rotation matrices using boldfaced letters.  Remarkably, because 2×2 matrices commute this allows these matrices to be replaced by complex exponentials and the coordinate vectors to be replaced by complex numbers and the derivations and calculations do not change.

T1 Four-bar Linkage Analysis revised

Superluminal Chapter 4

David finds that surpassing Siggy and Kwan to get to Satellite 9 may have benefits he did not expect. 



Superluminal Chapter 4
Chapter 1 Busted

Superluminal Chapter 3

More of R. W. Frost’s story following David Minor’s uneasy relationship with MitsukoTek Corporation as he is drawn into helping their wayward Satellite 9.

(Read by scrolling the image or by using the page down and up buttons at the bottom.)

Superluminal Classic 1012 C3
Hummingbird Spatial Six-bar

Hummingbird Spatial Six-bar Linkage

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.


Superluminal Chapters 1-2

Superluminal, the new book by R. W. Frost, follows David Minor’s experiences with MitsukoTek Corporation’s troublesome Satellite 9.  It is the year 2035 and a new technology has swept the globe…

(Read by scrolling the image or by using the page down and up buttons at the bottom.)

Superluminal Chapter 1-2
Type synthesis

Type Synthesis, Assur Groups, and Rigidity

These lecture notes introduce students to research on classifying linkage systems which involves a variety of ideas such as Assur groups, Baranov trusses, and Rigidity theory.  Organizing the increasing complexity of machine systems to guide inventors has attracted researchers for literally generations and has generally been called “Type synthesis.”  The goal is to provide a systematic way to explore the types of linkage systems that are available to address a design need.

Type Synthesis
Flapping Wing Prototype

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

Youku Flapping Wing

Soil Conditioning Port Valve

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.

Beijing Walking Robot

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

Beijing Design Research