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Walking strandbeest dynamics

Sander Huisman

Sander Huisman, University of Twente

Posted 10 years ago

POSTED BY: Sander Huisman

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Anton Antonov

Anton Antonov, Accendo Data LLC

Posted 10 years ago

In the spirit of the original T. Jensen art I think it is a good idea to use XKCD styling of the graphics: Here is the code (using modifications of Mr.Wizard's answer at MSE): (********************************) ( Mr.Wizard's code for XKCD style) (*******************************) split[{a_, b_}] := If[a == b, {b}, With[{n = Ceiling[3 Norm[a - b]]}, Array[{n - #, #}/n &, n].{a, b}]] partition[{x_, y__}] := Partition[{x, x, y}, 2, 1] nudge[L : {a_, b_}, d_] := Mean@L + d Cross[a - b]; gap = {style__, x_BSplineCurve} :> {{White, AbsoluteThickness[10], x}, style, AbsoluteThickness[2], x}; wiggle[pts : {{_, _} ..}, d_: {-0.15, 0.15}] := ## &[#~nudge~RandomReal@d, #[[2]]] & /@ partition[Join @@ split /@ partition@pts] xkcdify[plot_Graphics] := Show[FullGraphics@plot, TextStyle -> {17, FontFamily -> "Humor Sans"}] /. Line[pts_] :> {AbsoluteThickness[2], BSplineCurve@wiggle@pts} // MapAt[# /. gap &, #, {1, 1}] & ( Modification ) xkcdify[plot : (Line \| Arrow)[_], d_: {-0.15, 0.15}] := Block[{funcHead = Head[plot], bf = BSplineCurve}, If[funcHead === Arrow, bf = Arrow@bf]; plot /. funcHead[pts_] :> {AbsoluteThickness[2], bf@(wiggle[#, d] &)@pts} // MapAt[# /. gap &, #, {1, 1}] & ]; (*****************************) ( Sander's code modification ) (*****************************) rescaleRule = {x_?NumericQ, y_?NumericQ} :> {Rescale[x, {-150, 150}, {-3, 3}], Rescale[y, {-120, 50}, {-2.5, 1}]}; FindLines[0.4] /. rescaleRule; trajectoriesdata = (FindPoints /@ Subdivide[0, 2 Pi, 100])\[Transpose]; trajectoriesdataXKCD = Map[xkcdify[Arrow[#], {-0.23, 0.23}] &, (trajectoriesdata /. rescaleRule)]; man = Manipulate[ Graphics[{Arrowheads[Large], trajectoriesdataXKCD, Thick, Red, Map[xkcdify[Line[#], {-0.1, 0.1}] &, FindLines[\[Theta]] /. rescaleRule]}, Frame -> True, PlotRange -> {{-3, 3}, {-3, 1}}, ImageSize -> 800], {\[Theta], 0, 2 \[Pi]}, AutorunSequencing -> {Automatic, 5}]

In the spirit of the original T. Jensen art I think it is a good idea to use XKCD styling of the graphics:

enter image description here

Here is the code (using modifications of Mr.Wizard's answer at MSE):

(*********************************)
(* Mr.Wizard's code for XKCD style*)
(*********************************)

split[{a_, b_}] := 
 If[a == b, {b}, 
  With[{n = Ceiling[3 Norm[a - b]]}, Array[{n - #, #}/n &, n].{a, b}]]

partition[{x_, y__}] := Partition[{x, x, y}, 2, 1]

nudge[L : {a_, b_}, d_] := Mean@L + d Cross[a - b];

gap = {style__, x_BSplineCurve} :> {{White, AbsoluteThickness[10], x}, style, 
    AbsoluteThickness[2], x};

wiggle[pts : {{_, _} ..}, 
  d_: {-0.15, 0.15}] := ## &[#~nudge~RandomReal@d, #[[2]]] & /@ 
  partition[Join @@ split /@ partition@pts]

xkcdify[plot_Graphics] := 
 Show[FullGraphics@plot, TextStyle -> {17, FontFamily -> "Humor Sans"}] /. 
   Line[pts_] :> {AbsoluteThickness[2], BSplineCurve@wiggle@pts} // 
  MapAt[# /. gap &, #, {1, 1}] &

(* Modification *)
xkcdify[plot : (Line | Arrow)[_], d_: {-0.15, 0.15}] :=
  Block[{funcHead = Head[plot], bf = BSplineCurve},
   If[funcHead === Arrow, bf = Arrow@*bf];
   plot /. funcHead[pts_] :> {AbsoluteThickness[2], bf@(wiggle[#, d] &)@pts} //
     MapAt[# /. gap &, #, {1, 1}] &
   ];

(******************************)
(* Sander's code modification *)
(******************************)

rescaleRule = {x_?NumericQ, y_?NumericQ} :> {Rescale[x, {-150, 150}, {-3, 3}],
    Rescale[y, {-120, 50}, {-2.5, 1}]}; FindLines[0.4] /. rescaleRule;

trajectoriesdata = (FindPoints /@ Subdivide[0, 2 Pi, 100])\[Transpose];
trajectoriesdataXKCD = 
  Map[xkcdify[Arrow[#], {-0.23, 0.23}] &, (trajectoriesdata /. 
     rescaleRule)];
man = Manipulate[
  Graphics[{Arrowheads[Large], trajectoriesdataXKCD, Thick, Red, 
    Map[xkcdify[Line[#], {-0.1, 0.1}] &, 
     FindLines[\[Theta]] /. rescaleRule]}, Frame -> True, 
   PlotRange -> {{-3, 3}, {-3, 1}}, ImageSize -> 800], {\[Theta], 0, 
   2 \[Pi]}, AutorunSequencing -> {Automatic, 5}]

POSTED BY: Anton Antonov

Sander Huisman

Sander Huisman, University of Twente

Posted 10 years ago

Attachments:

POSTED BY: Sander Huisman

Vitaliy Kaurov

Vitaliy Kaurov, WOLFRAM Research

Posted 10 years ago

@Sander Huisman amazing post! I also would like to point to two demonstrations on the subject, see below. @Marco Thiel, indeed an interesting exciting idea about gait pattern. Sander, do you think we really have to adapt to 12, - is due to instability or inefficiency of this strandbeest with 4 legs? Perhaps it could walk with 4? Jansen Walker by Contributed by Karl Scherer, Additional contributions: Theo Jansen A Theo Jansen Walking Linkage by Sándor Kabai

POSTED BY: Vitaliy Kaurov

Robert Jacobson

Robert Jacobson, Roger Williams University

Posted 7 years ago

This work may be known to those in this thread, but a research group at Disney developed a nice program to compute the gearing mechanisms required to approximate a motion curve specified by the user: Stelian Coros, Bernhard Thomaszewski, Gioacchino Noris, Shinjiro Sueda, Moira Forberg, Robert W. Sumner, Wojciech Matusik, and Bernd Bickel. 2013. Computational design of mechanical characters. ACM Trans. Graph. 32, 4, Article 83 (July 2013), 12 pages. DOI: https://doi.org/10.1145/2461912.2461953 A video explaining their paper: https://www.youtube.com/watch?v=DfznnKUwywQ. The project webpage: https://www.disneyresearch.com/publication/computational-design-of-mechanical-characters/. I seem to recall they released a computer program as well, but now I can't find it. The same lab (but mostly different researchers) designed a similar system for kinetic wire characters: Hongyi Xu, Espen Knoop, Stelian Coros, and Moritz Bächer. 2018. Bend-it: design and fabrication of kinetic wire characters. ACM Trans. Graph. 37, 6, Article 239 (December 2018), 15 pages. DOI: https://doi.org/10.1145/3272127.3275089 Video: https://www.youtube.com/watch?v=4X9ORR-z_tY Project webpage: https://www.disneyresearch.com/publication/bend-it/

POSTED BY: Robert Jacobson

Erik Mahieu

Posted 10 years ago

Here are the links to some demonstrations about walking machine mechanisms I made some time ago: Chebyshev Walking Machine Walking Mechanism Using a Klann Linkage These are in fact all variations of coupler curves of the 4-bar linkage and like mechanisms. Here is a link to my demonstration about this: Coupler Curves of a Four-Bar Linkage In fact, almost any curve can be generated by some mechanical linkage. See more of these demonstrations: Polar Curve Generator with Five-Bar Linkage Designs from Mechanical Linkages Back in 1951, Hrones and Nelson made an comprehensive atlas of about 10,000 coupler curves that can be obtained by changing the parameters. This was (and is) used to construct a mechanism adapted to generate a desired curve with a specific position and speed variation along it. I am preparing another demonstration that can generate all curves as explored in the Hrones and Nelson atlas. This gives an idea of the curves/ speed combinations that can be generated. The walking machine mechanisms are part of these. Just find a parameter (bar dimension) combination with a sufficient straight line/constant speed section. Coupler Curve Atlas: Analysis of the four-bar linkage

POSTED BY: Erik Mahieu

Sander Huisman

Sander Huisman, University of Twente

Posted 10 years ago

Here a bit nicer with small figures: And as an animation: Notebook as attachment. (can now be ran by just executing the entire notebook). Attachments:

POSTED BY: Sander Huisman

Marco Thiel

Marco Thiel, University of Aberdeen - Dept. of Physics/Mathematics

Posted 10 years ago

Hi Sander, very nice indeed! I haven't had the opportunity to run any of this yet - I am on my phone, but I wonder whether it would be interesting to look at the gait pattern with some group theory. Ian Stewart has described such an approach here. Thanks for posting! Marco

POSTED BY: Marco Thiel

Henrik Schachner

Henrik Schachner, Radiation Therapy Center, Weilheim, Germany

Posted 10 years ago

Hi Sander, thank you very much for sharing this nice code - I already had a lot of fun playing around with it! I have just a little remark to make: It seems to me that in your function `FindPoints` there is a typo; it should read (see comments, c.f. line "d"): FindPoints[\[Theta]_] := Module[{p1, p2, p3, p4, p5, p6, p7, p8, p9, p10, p11, p12, p13, p14, p15, p16}, p1 = {0, 0}; p4 = {38, -7.8}; p11 = {-38, -7.8}; p2 = 15 {Cos[\[Theta]], Sin[\[Theta]]}; p3 = FindPoint[p2, p4, 50, 41.5, Left]; p6 = FindPoint[p2, p4, 61.9, 39.3, Right]; p5 = FindPoint[p3, p4, 55.8, 40.1(41.5), Left]; p8 = FindPoint[p5, p6, 39.4, 36.7, Left]; p7 = FindPoint[p6, p8, 49, 65.7, Right]; p10 = FindPoint[p2, p11, 50, 41.5, Right]; p13 = FindPoint[p2, p11, 61.9, 39.3, Left]; p12 = FindPoint[p10, p11, 55.8, 40.1(41.5), Right]; p14 = FindPoint[p12, p13, 39.4, 36.7, Right]; p15 = FindPoint[p13, p14, 49, 65.7, Left]; {p1, p2, p3, p4, p5, p6, p7, p8, p10, p11, p12, p13, p14, p15}] Then the resulting trajectory of the foot motion (along the ground) looks better: Best regards -- Henrik

Hi Sander,

thank you very much for sharing this nice code - I already had a lot of fun playing around with it! I have just a little remark to make: It seems to me that in your function FindPoints there is a typo; it should read (see comments, c.f. line "d"):

FindPoints[\[Theta]_] := 
 Module[{p1, p2, p3, p4, p5, p6, p7, p8, p9, p10, p11, p12, p13, p14, 
   p15, p16},
  p1 = {0, 0};
  p4 = {38, -7.8};
  p11 = {-38, -7.8};
  p2 = 15 {Cos[\[Theta]], Sin[\[Theta]]};
  p3 = FindPoint[p2, p4, 50, 41.5, Left];
  p6 = FindPoint[p2, p4, 61.9, 39.3, Right];
  p5 = FindPoint[p3, p4, 55.8, 40.1(*41.5*), Left];
  p8 = FindPoint[p5, p6, 39.4, 36.7, Left];
  p7 = FindPoint[p6, p8, 49, 65.7, Right];
  p10 = FindPoint[p2, p11, 50, 41.5, Right];
  p13 = FindPoint[p2, p11, 61.9, 39.3, Left];
  p12 = FindPoint[p10, p11, 55.8, 40.1(*41.5*), Right];
  p14 = FindPoint[p12, p13, 39.4, 36.7, Right];
  p15 = FindPoint[p13, p14, 49, 65.7, Left];
  {p1, p2, p3, p4, p5, p6, p7, p8, p10, p11, p12, p13, p14, p15}]

Then the resulting trajectory of the foot motion (along the ground) looks better:

enter image description here

Best regards -- Henrik

POSTED BY: Henrik Schachner

Sander Huisman

Sander Huisman, University of Twente

Posted 10 years ago

Indeed an interesting idea, let me see what I can do! These diagrams are made for 2 and 4 legs, i have to adapt it two e.g. 12 legs...

POSTED BY: Sander Huisman

Sander Huisman

Sander Huisman, University of Twente

Posted 10 years ago

Theo Jansen, the strandbeest inventor, actually also used evolutionary algorithms on an Atari (!!) that made 1500 perturbation of a known state, then keep 150, make for each of them 10 perturbations and keep 150, and do this several times. It took many months on this Atari to run!

POSTED BY: Sander Huisman

Vitaliy Kaurov

Vitaliy Kaurov, WOLFRAM Research

Posted 10 years ago

An interesting variation with genetic algorithms used to improve the linkage. I could not find more detailed info though. I think it was 3D printed and is highly maneuvering. The City Beest by Hans Jørgen Grimstad

POSTED BY: Vitaliy Kaurov

EDITORIAL BOARD

EDITORIAL BOARD, WOLFRAM

Posted 10 years ago

- another post of yours has been selected for the Staff Picks group, congratulations ! We are happy to see you at the tops of the "Featured Contributor" board. Thank you for your wonderful contributions, and please keep them coming!