For 6 pairs of legs (seen in many build-kits), with the phase difference of each successive pairs of legs just being Pi/3, you will get the following animation:

where I marked each pair by number and by F(front) and B(back).

The trajectory of one of the feet can be obtained like so:

pts = Subdivide[0, 2 Pi, 150];
trajectoriesdata = (FindPoints /@ pts)\[Transpose];
trajectoriesdata = trajectoriesdata[[-1]];
Graphics[{Arrow[trajectoriesdata], Point[trajectoriesdata]}, Axes -> True]

To get the angles when it touching we find the minima and maxima in X position:

ClearAll[XPosFeet,XPosFeet2]
XPosFeet[\[Theta]_?NumericQ]:=FindPoints[\[Theta]][[-1,1]]
XPosFeet2[\[Theta]_?NumericQ]:=FindPoints[\[Theta]][[7,1]]
min\[Theta]=\[Theta]/.Quiet@FindMaximum[XPosFeet[\[Theta]],{\[Theta],2,1,3}][[2]];
max\[Theta]=\[Theta]/.Quiet@FindMinimum[XPosFeet[\[Theta]],{\[Theta],4.5,3,6}][[2]];
{min\[Theta]2,max\[Theta]2}={\[Pi]/2+(\[Pi]/2-min\[Theta]),3\[Pi]/2+(3\[Pi]/2-max\[Theta])}

ClearAll[LeftOnGroundQ,RightOnGroundQ]
LeftOnGroundQ[\[Theta]_?NumericQ]:=!(min\[Theta]<Mod[\[Theta],2\[Pi]]<max\[Theta])
RightOnGroundQ[\[Theta]_?NumericQ]:=min\[Theta]2<Mod[\[Theta],2\[Pi]]<max\[Theta]2

Plot[XPosFeet[\[Theta]],{\[Theta],0,2\[Pi]},GridLines->{{min\[Theta],max\[Theta]},{}}]
Plot[XPosFeet2[\[Theta]],{\[Theta],0,2\[Pi]},GridLines->{{\[Pi]/2+(\[Pi]/2-min\[Theta]),3\[Pi]/2+(3\[Pi]/2-max\[Theta])},{}}]
Manipulate[Graphics[{Red,Line[FindLines[\[Theta]]],If[LeftOnGroundQ[\[Theta]],Disk[{-50,-100}],{}],If[RightOnGroundQ[\[Theta]],Disk[{50,-100}],{}]},PlotRange->{{-150,150},{-120,70}},ImageSize->800],{\[Theta],0,2\[Pi]}]

giving:

where the dot now indicates touching.

Now we can do some bigger code to generate a nice diagram:

mp=60;
n=6;
(*\[CurlyPhi]=Table[Mod[5\[Iota],n,1],{\[Iota],1,n}];*)
\[CurlyPhi]=Range[n];
Manipulate[
    Graphics3D[{Darker@Yellow,Table[
        {
        Line[ Map[Prepend[mp \[Iota]],FindLines[\[Theta]+\[CurlyPhi][[\[Iota]]] (2Pi/n)],{2}]],
        {Black,Text[\[Iota],{mp \[Iota],0,50}]}
        },
        {\[Iota],n}
    ]
    ,Black,Line[{{mp 1,0,0},{mp n,0,0}}]
    ,MapThread[Text[#1,{mp (n+1),#2,-100}]&,{{"F","B"},{-50,50}}]
    }
    ,
    Lighting->"Neutral",
    PlotRangePadding->Scaled[.1],
    PlotRange->{{-mp,(n+1)mp},{-150,150},{-150,150}},
    Boxed->False,
    ImageSize->700
    ]
,
{\[Theta],0,2\[Pi]}
]

separators={min\[Theta],max\[Theta],min\[Theta]2,max\[Theta]2};
separators=Table[separators-\[CurlyPhi][[\[Iota]]],{\[Iota],n}];
separators=Mod[Flatten[separators],2\[Pi]];
separators=separators//Prepend[0.0]//Append[2.0\[Pi]];
separators=Union[separators];
regions=Partition[separators,2,1];
expanded=Partition[Subdivide[0,2\[Pi],regions//Length],2,1];

ClearAll[MakeRegionPart]
MakeRegionPart[{\[Theta]start_,\[Theta]stop_},{start_,stop_}]:=Module[{eval,onground,ypos,xpos,pos,support},
    eval=(\[Theta]start+\[Theta]stop)/2.0;
    onground=Table[{LeftOnGroundQ[eval+\[CurlyPhi][[\[Iota]]]],RightOnGroundQ[eval+\[CurlyPhi][[\[Iota]]]]},{\[Iota],n}];
    xpos=Subdivide[start,stop,3][[2;;-2]];
    ypos=Subdivide[0.5,0,n+1][[2;;-2]];
    pos=Partition[Reverse/@Tuples[{ypos,xpos}],2];
    support=MapThread[If[#1,Disk[#2,0.02],{}]&,{onground,pos},2];
    (*Print[support];*)
    {
        Line[{{{\[Theta]start,0.75},{\[Theta]start,1.25}},{{\[Theta]stop,0.75},{\[Theta]stop,1.25}}}],
        Line[{{{start,0},{start,0.5}},{{stop,0},{stop,0.5}}}],
        {Dashed,Opacity[0.5],Line[{{{\[Theta]start,0.75},{start,0.5}},{{\[Theta]stop,0.75},{stop,0.5}}}]},
        support
    }
]

xpos1=Subdivide[Sequence@@expanded[[1]],3][[2;;-2]];
ypos1=Subdivide[0.5,0,n+1][[2;;-2]];

Graphics[{
    MapThread[MakeRegionPart,{regions,expanded}],
    Text[#,{#,1.35}]&/@Range[0,2\[Pi],\[Pi]/2],
    MapThread[Text[#1,{#2,-0.1}]&,{{"F","B"},xpos1}],
    MapThread[Text[#1,{-0.1,#2}]&,{Range[n],ypos1}]
}
]
(*
plts=Table[
Plot[{If[LeftOnGroundQ[\[Theta]+\[CurlyPhi][[\[Iota]]]],20-4\[Iota],Missing[]],If[RightOnGroundQ[\[Theta]+\[CurlyPhi][[\[Iota]]]],20-(4\[Iota]+1),Missing[]]},{\[Theta],0,2\[Pi]}]
,
{\[Iota],n}
];
Show[plts,PlotRange\[Rule]All,GridLines\[Rule]{separators,{}},AspectRatio->1/5,ImageSize\[Rule]1000]*)

Giving:

I attached my notebook so you can change the number of feet and the relative phases between the legs... Note that this notebook was not meant to be easily understood ;-)

Attachments:

@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

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:

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/

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

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

oops! one length is wrong! Well, easy to correct ;) Thanks for spotting!