Klann linkage

About this Book Klann linkage This wikibooks shall be a supplement to the Wikipedia article of the same name. On Wikipedia, there is a need to compromise between readability, details and amount of illustration. It is impossible, to give an overview to the general public and an in depth view for engineers diving deeper into the topic at the same time. So: Where the Wikipedia article brings a general overview, this book brings depth and details. This book will also talk about the relevant patent(s). This book will shall not replace studying the patent, but simplifying it by providing additional tables and illustrations. As the author of this book, I would like to cite larger sections of the patents. As the content of a patent may be copyrighted, this book will only cite small sections of the patent to be on the safe side. A big thanks has to go to the patent holder Joseph Klann for his website,https://web.archive.org/web/20170812223320/http://www.mechanicalspider.com/ which is a great reference and for uploading pictureshttps://commons.wikimedia.org/wiki/Special:Contributions/Jck56270 to Wikimedia Commons. Without the pictures from Joseph Klann on Commons and without his website, this book would not have been possible. This book is the partner book to Comparison of crank based leg mechanism. Parts To get started, we look at the parts of the Klann Linkage, starting with a leg unit. Leg Unit First we take a look at this illustration: File:Klann-Linkage-Nameing of the parts.svg In the patent, those parts are numbered as follow: Number Name 3 supportive Frame 5 first rocker arm 7 second rocker arm 6 spacer not shown in the picture above 13 crank 21 connecting rod 31 leg The supportive frame is of course simplified. The axes/joints are named: File:Klann-Linkage-Nameing of the axes.svg Number Name 9 first rocker arm axle 11 second rocker arm axle 15 crank shaft 27 elbow joint 29 crank 33 foot 35 knee joint/axle 37 hip joint a pair of legs Two leg units combine to form, as Mister Klann puts it,https://web.archive.org/web/20170728172050/http://www.mechanicalspider.com/concept.html a wheel replacement. The two units are one-half cycle out of phase (= 180 degree) to each other. a full device A full device consists of at least three pairs of legs, which means a total of six legs.https://www.google.com/patents/US6260862 Practical implementations might use eight legs for stability. (see also https://web.archive.org/web/20170812223320/http://www.mechanicalspider.com/ for designs using the Klann linkage) Geometry Now we take a look at the "Enable diagram of Klann linkage" from Mister Klann: Enable diagram of Klann linkage How to construct it is described in the patent and on the website of the patent holder. Our concern at the moment is how to interpret this drawing. Points ending with "x" belong to the fully extended leg. Points ending with "y" belong to the grounded gait position. Points ending without a letter are fix points. Points ending with "p" are auxiliary points used to construct the other points. The suffix "c" is used for circles, "s" for straight lines and "a" are angles. By "connecting the dots" without suffix, "x" suffix and "y" suffix, we can derive a first meaning from the drawing, showing us the frame and the leg in two positions. examples In the patent, there is are two tables with coordinates. Here we use those tables, but with the point description written next to it. Table 1 thumb|Figure 1 thumb|Animation Point X Y Description Fixpoints 9 1.366 1.366 first rocker arm axle 11 1.009 0.574 second rocker arm axle 15 1.599 0.750 crank shaft fully extended ground stride position 27X 0.741 0.750 elbow joint 29x 1.331 0.750 crank 33x 0.000 0.000 foot 35x 0.232 0.866 knee joint/axle 37x 0.866 1.500 hip joint grounded gait position 27Y 1.277 0.750 elbow joint 29y 1.867 0.750 crank 33y 1.000 0.000 foot 35y 0.768 0.866 knee joint/axle 37y 1.000 1.732 hip joint Auxiliary points 52p 0.500 0.866 56p 0.000 1.732 57p 1.000 1.732 62p 0.500 1.866 65p 0.500 -0.134 72p 0.500 0.901 74p 0.500 0.844 coordinate missing in the patent 75p 1.599 0.844 coordinate missing in the patent 78p 1.599 1.018 80p 1.022 0.894 Table 1a (the table in the Wikipedia article is a cut down version of this table) Table 2 thumb thumb|Animation Point X Y Description Fixpoints 9 17.818 16.076 first rocker arm axle 11 12.101 10.186 second rocker arm axle 15 17.607 11.807 crank shaft fully extended ground stride position 27X 9.125 11.807 elbow joint 29X 14.631 11.807 crank 33x 0.000 0.000 foot 35x 3.024 13.099 knee joint/axle 37x 11.119 19.200 hip joint grounded gait position 27Y 15.077 11.807 elbow joint 29Y 20.583 11.807 crank 33y 12.000 0.000 foot 35y 8.976 13.099 knee joint/axle 37y 13.578 22.130 hip joint Auxiliary points 52p 6.000 13.992 56p 0.000 24.384 57p 12.000 24.384 62p 6.000 25.992 65p 6.000 1.992 72p 6.000 13.491 74p 6.000 13.925 coordinate missing in the patent 75p 17.607 13.925 coordinate missing in the patent 78p 17.607 14.783 80p 12.236 13.572 Table 2a (this table is omitted in the wikipedia article) Input Variables The geometric construction according to U.S. Patent 6,260,862 or more specifically the patent holders websitehttp://www.mechanicalspider.com/enable.html has six input variables. We will consider the stride length as an input as well (here input 0), as table 2 uses bigger stride length. (notice that the same effect can be achieved by scaling all points and lengths proportionally) An other thing to keep in mind is, that not all combinations of inputs result in a working linkage. Table 1 The input variables for figure 17 (and therefore table 1) according U.S. Patent 6,260,862 are as follow: the values on the patent holders website for figure 2http://www.mechanicalspider.com/enable.html are identical, but rounded Input Value Reference Description Input 0 1.0000 units length of line 50s (distance between 33x and 33y) length of the stride Input 1 0.8660 units height of point 52p above line 50s Input 2 1.0000 units Radius of circle 53c (circle 53c centred at 52p) Input 3 60 degree Input 4 -90 degree Location of point 65p on circle above horizontal Input 5 1.0991 units Distance from point 74p to point 75p Input 6 0.5895 units distance of between points 29x and 27x Table 1b Notice that this set of values has some special properties: point 33x and 33y are both on circle 53c (which is not necessary as we see in table 2 from the patent) line 67s and 51n overlap (which is caused by the -90 degree angle as input 4) Therefore: Do not over generalize the drawing. (if your colour blind: sorry for the heavy colour coding, but given the complexity of the drawing, I could not avoid to colour code the drawing) Table 2 The input values for table 2 aren´t given in the patent on or the patent holders website, but can be reconstructed from table 2. Input Value Reference Description Input 0 12.000 units length of line 50s (distance between 33x and 33y) length of the stride Input 1 13.992 units height of point 52p above line 50s Input 2 12.000 units Radius of circle 53c (circle 53c centered at 52p) Input 3 80 degree Input 4 -90 degree Location of point 65p on circle above horizontal Input 5 11.607 units Distance from point 74p to point 75p Input 6 5.506 units distance of between points 29x and 27x Table 2b Notice that this set of values has some special properties: line 67s and 51n overlap (which is caused by the -90 degree angle as input 4) Therefore: Do not over generalize the drawing. Inputs In this drawing, you see how the input variables influence the linkage: 300px Foot An important part for a complete robot is the foot. Inspiration for the foot can be found in the patent, Mondo spider Design and Theo Jansens Strandbeest Design. Detailed analysis - walking Table 1 Presuming that the fasted foot defines the speed of the whole robot, we can animate a Robot based on the values in table 1: 480px 480px It is a bit tricky to realize in the animation, but obvious in the graph: The robot comes to an almost stand still during each crank revolution. Table 2 with the same assumption, we can also animate the robot according to table 2: 480px Further reading http://www.mechanicalspider.com/ References Mechanical engineering 25%