Klann linkage

About this Book

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, which is a great reference and for uploading pictures 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:

In the patent, those parts are numbered as follow:

Number	Name
3	supportive Frame
5	first rocker arm
7	second rocker arm
6	spacer
13	crank
21	connecting rod
31	leg

The supportive frame is of course simplified.

The axes/joints are named:

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, 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.

• 

Practical implementations might use eight legs for stability.

• 
  Lobsterbot - a robot with a total of eight legs
• 
  Mondospider - A robot with a total of eight legs

(see also https://web.archive.org/web/20170812223320/http://www.mechanicalspider.com/ [3] for designs using the Klann linkage)

Geometry

Now we take a look at the "Enable diagram of Klann linkage" from Mister Klann:

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

Figure 1

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
75p	1.599	0.844
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

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
75p	17.607	13.925
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 website 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 2 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:

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:

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:

Further reading

• http://www.mechanicalspider.com/