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Notebook[{

Cell[CellGroupData[{
Cell["Three Position Synthesis of Planar 4R chains", "Title"],

Cell["\<\
This notebook solves the constraint equation for a planar RR chain \
for three specified positions.
The design parameters are the location of the fixed G=(u,v) and moving pivot \
W=(\[Lambda], \[Mu]) pivot. 

We write the constraint equations ([D]W-G).([D]W-G)=R^2 for each of the \
specified positions.  Subtract the first from the remaining equations to \
eliminate R.  Then we choose values for G=(u,v) or W=(\[Lambda], \[Mu]) and \
solve for the remaining terms.

An analysis routine then examines the linkage to determine whether or not it \
can smoothly reach all three positions.
\
\>", "Text"],

Cell[BoxData[
    \(Quit[]\)], "Input"],

Cell[BoxData[
    \(<< Utilities`MemoryConserve`\)], "Input"],

Cell[CellGroupData[{

Cell["1. Specify Three Positions", "Section"],

Cell["\<\
A data file \"dataRR\" is read that lists the rotation angle, and \
x, y coordinates of the translation for each position.  The location of your \
file is listed within the quotes \"\".\
\>", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(data1 = {{0, 0, 0}, {22, \(-6\), 11}, {68, \(-17\), 13}}\)], "Input"],

Cell[BoxData[
    \({{0, 0, 0}, {22, \(-6\), 11}, {68, \(-17\), 13}}\)], "Output"]
}, Open  ]],

Cell["The number of specified positions is denoted \"npos\".", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(If[Length[data1] \[Equal] 3, \((Print["\<Three Positions\>"]; 
        npos = 3)\), \ 
      Print["\<Warning::Specify Three Positions\>"]]\)], "Input"],

Cell[BoxData[
    \("Three Positions"\)], "Print"],

Cell[BoxData[
    \(3\)], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell["Compute the Transformations", "Subsection"],

Cell[BoxData[
    \(Disp[t_, 
        d_] := {{Cos[t], \(-Sin[t]\), d[\([1]\)]}, \ {Sin[t], Cos[t], 
          d[\([2]\)]}, {0, 0, 1}}\)], "Input"],

Cell[BoxData[
    \(Zdisp[t_] := Disp[t, {0, 0}]\)], "Input"],

Cell[BoxData[
    \(Xdisp[t_] := Disp[0, {t, 0}]\)], "Input"],

Cell[BoxData[
    \(Ydisp[t_] := Disp[0, {0, t}]\)], "Input"],

Cell[CellGroupData[{

Cell[BoxData[
    \(DtoR = N[\[Pi]/180]\)], "Input"],

Cell[BoxData[
    \(0.017453292519943295`\)], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(position = 
      Table[Disp[
          DtoR*data1[\([i, 1]\)], {data1[\([i, 2]\)], 
            data1[\([i, 3]\)]}], {i, npos}]\)], "Input"],

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Cell[BoxData[
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Cell[BoxData[
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Cell[CellGroupData[{

Cell[BoxData[
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Cell[BoxData[
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        11.927183854566788`}, {\(-17.92718385456679`\), 
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Cell[BoxData[
    TagBox[\(\[SkeletonIndicator]  Graphics  \[SkeletonIndicator]\),
      False,
      Editable->False]], "Output"]
}, Open  ]],

Cell["Construct the relative transformations", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(relative = 
      Chop[Table[
          position[\([i]\)] . Inverse[position[\([1]\)]], \ {i, \ 
            npos}]]\)], "Input"],

Cell[BoxData[
    \({{{1, 0, 0}, {0, 1, 0}, {0, 0, 
          1}}, {{0.9271838545667874`, \(-0.374606593415912`\), \(-6.`\)}, \
{0.374606593415912`, 0.9271838545667874`, 11.`}, {0, 0, 
          1.`}}, {{0.37460659341591196`, \(-0.9271838545667874`\), \
\(-17.`\)}, {0.9271838545667874`, 0.37460659341591196`, 13.`}, {0, 0, 
          1.`}}}\)], "Output"]
}, Open  ]],

Cell[BoxData[
    \(relative >> "\<RelativeData\>"\)], "Input"],

Cell[BoxData[
    \(position >> "\<PositionData\>"\)], "Input"],

Cell[BoxData[
    \(data1 >> "\<ThreePosData\>"\)], "Input"]
}, Open  ]]
}, Open  ]],

Cell[CellGroupData[{

Cell["2. Design the Input and Output Cranks.", "Section"],

Cell["\<\
We construct the design equations which are bilinear in the \
coordinates of G and W..\
\>", "Text"],

Cell[BoxData[
    \(G = {u, v, \ 1}; \ W = {\[Lambda], \[Mu], \ 1};\)], "Input"],

Cell[CellGroupData[{

Cell[BoxData[
    \(Eqn = 
      Table[\((relative[\([i]\)] . W - G)\) . \((relative[\([i]\)] . W - 
                G)\) - R^2, {i, \ npos}]\)], "Input"],

Cell[BoxData[
    \({\(-R\^2\) + \((\(-u\) + \[Lambda])\)\^2 + \((\(-v\) + \[Mu])\)\^2, \
\(\(0.`\)\(\[InvisibleSpace]\)\) - 
        R\^2 + \((\(-6.`\) - u + 0.9271838545667874`\ \[Lambda] - \
0.374606593415912`\ \[Mu])\)\^2 + \((\(\(11.`\)\(\[InvisibleSpace]\)\) - v + \
0.374606593415912`\ \[Lambda] + 0.9271838545667874`\ \[Mu])\)\^2, \(\(0.`\)\(\
\[InvisibleSpace]\)\) - 
        R\^2 + \((\(-17.`\) - u + 0.37460659341591196`\ \[Lambda] - \
0.9271838545667874`\ \[Mu])\)\^2 + \((\(\(13.`\)\(\[InvisibleSpace]\)\) - v + \
0.9271838545667874`\ \[Lambda] + 0.37460659341591196`\ \[Mu])\)\^2}\)], \
"Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(Design = 
      Table[Chop[Expand[Eqn[\([i]\)] - Eqn[\([1]\)]]], {i, npos}]\)], "Input"],

Cell[BoxData[
    \({0, \(\(157.`\)\(\[InvisibleSpace]\)\) + 12.`\ u - 22.`\ v - 
        2.884861199651384`\ \[Lambda] + 0.14563229086642515`\ u\ \[Lambda] - 
        0.749213186831824`\ v\ \[Lambda] + 24.893323921460265`\ \[Mu] + 
        0.749213186831824`\ u\ \[Mu] + 
        0.14563229086642515`\ v\ \[Mu], \(\(458.`\)\(\[InvisibleSpace]\)\) + 
        34.`\ u - 26.`\ v + 11.370156042595468`\ \[Lambda] + 
        1.250786813168176`\ u\ \[Lambda] - 
        1.8543677091335748`\ v\ \[Lambda] + 41.264022484084485`\ \[Mu] + 
        1.8543677091335748`\ u\ \[Mu] + 
        1.250786813168176`\ v\ \[Mu]}\)], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell["Specify Parameters of the Fixed and Moving Pivots.", "Subsection"],

Cell["\<\
We assume that we can give values arbitrarly to any of the four \
parameters of G=(u,v) and W=(\[Lambda],\[Mu]) and solve for the remaining \
parameters.\
\>", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(InputCrank = {\(-15\), 0, \[Lambda], \[Mu]}\)], "Input"],

Cell[BoxData[
    \({\(-15\), 0, \[Lambda], \[Mu]}\)], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(OutputCrank = {5, 1, \ \[Lambda], \ \[Mu]}\)], "Input"],

Cell[BoxData[
    \({5, 1, \[Lambda], \[Mu]}\)], "Output"]
}, Open  ]],

Cell["\<\
This function assembles the design equations \"System\" and calls \
the solver:\
\>", "Text"],

Cell[BoxData[
    \(Compute[
        datapv_] := \((deseqns = 
          Table[Design[\([i + 1]\)] /. {u \[Rule] datapv[\([1]\)], 
                v \[Rule] datapv[\([2]\)], \[Lambda] \[Rule] 
                  datapv[\([3]\)], \[Mu] \[Rule] datapv[\([4]\)]}, {i, 
              npos - 1}]; NSolve[deseqns])\)\)], "Input"],

Cell[CellGroupData[{

Cell[BoxData[
    \(incrank = Flatten[InputCrank /. Compute[InputCrank]]\)], "Input"],

Cell[BoxData[
    \({\(-15\), 
      0, \(-12.23340699382514`\), \(-2.8571956875574873`\)}\)], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(outcrank = Flatten[OutputCrank /. Compute[OutputCrank]]\)], "Input"],

Cell[BoxData[
    \({5, 1, \(-11.962204093248042`\), \(-7.981967927707509`\)}\)], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(pO = {incrank[\([1]\)], \ incrank[\([2]\)]}\)], "Input"],

Cell[BoxData[
    \({\(-15\), 0}\)], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(pA = {incrank[\([3]\)], \ incrank[\([4]\)]}\)], "Input"],

Cell[BoxData[
    \({\(-12.23340699382514`\), \(-2.8571956875574873`\)}\)], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(pC = {outcrank[\([1]\)], outcrank[\([2]\)]}\)], "Input"],

Cell[BoxData[
    \({5, 1}\)], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(pB = {outcrank[\([3]\)], outcrank[\([4]\)]}\)], "Input"],

Cell[BoxData[
    \({\(-11.962204093248042`\), \(-7.981967927707509`\)}\)], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(fbar = 
      Graphics[{Line[{pO, pA, \ pB, \ pC}], \ Disk[pO,  .2], \ 
          Circle[pC,  .2]}, Axes \[Rule] True, 
        AspectRatio \[Rule] Automatic]\)], "Input"],

Cell[BoxData[
    TagBox[\(\[SkeletonIndicator]  Graphics  \[SkeletonIndicator]\),
      False,
      Editable->False]], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(Show[fbar, frames]\)], "Input"],

Cell[GraphicsData["PostScript", "\<\
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