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Notebook[{

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

Cell["\<\
This notebook solves the constraint equation for a planar RR chain \
for as many as 5 specified positions.
The design parameters are the location of the fixed G=(u,v) and moving pivot \
W=(\[Lambda], \[Mu]) pivot. 
These parameters combine with the length R=|W-G| to define 5 unknowns.

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 for
\t2 positions:  we can choose values for 3 of the four parameters (u, v, \
\[Lambda], \[Mu]), and obtain 1 solution for the remaining parameter;
\t3 positions:  choose values for 2 of the four paramters to obtain 1 or 2 \
solutions;
\t4 positions:  choose values for 1 of these parameters to obtain 1 or 3 \
solutions;
\t5 positions:  all the parameters are variable and there are at most 4 real \
solutions.\
\>", "Text"],

Cell[CellGroupData[{

Cell["1. Specify 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 = 
      ReadList["\<jmmc:GeomDesign:dataRR\>", Number, 
        RecordLists \[Rule] True]\)], "Input"],

Cell[BoxData[
    \({{\(-70\), 10, 1}, {34, 3, 2}, {\(-10\), 3, 2}}\)], "Output"]
}, Open  ]],

Cell["\<\
The number of specified positions 2, 3, 4 or 5 is denoted \"npos\".\
\
\>", "Text"],

Cell[BoxData[
    \(If[Length[data1] < 2, 
      Print["\<Need More than One Position\>"]]\)], "Input"],

Cell[CellGroupData[{

Cell[BoxData[
    \(npos = If[Length[data1] < 6, Length[data1], \ 5]\)], "Input"],

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

Cell[CellGroupData[{

Cell["Functions for transformations", "Section"],

Cell[CellGroupData[{

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

Cell[BoxData[
    \(General::"spell1" \(\(:\)\(\ \)\) 
      "Possible spelling error: new symbol name \"\!\(Disp\)\" is similar to \
existing symbol \"\!\(Disk\)\"."\)], "Message"]
}, Open  ]],

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

Cell[CellGroupData[{

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

Cell[BoxData[
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Cell[CellGroupData[{

Cell[BoxData[
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Cell[BoxData[
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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"],

Cell[BoxData[
    \(General::"spell1" \(\(:\)\(\ \)\) 
      "Possible spelling error: new symbol name \"\!\(position\)\" is similar \
to existing symbol \"\!\(Position\)\"."\)], "Message"],

Cell[BoxData[
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          3}, {\(-0.17364817766693036`\), 0.984807753012208`, 2}, {0, 0, 
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Cell[CellGroupData[{

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

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Cell[BoxData[
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      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.`}}, {{\(-0.24192189559966773`\), \(-0.9702957262759966`\), 
          6.389514682272674`}, {0.9702957262759966`, \
\(-0.24192189559966767`\), \(-7.461035367160299`\)}, {0, 0, 
          1.`}}, {{0.5000000000000001`, \(-0.8660254037844386`\), \
\(-1.1339745962155634`\)}, {0.8660254037844386`, 
          0.5000000000000001`, \(-7.160254037844387`\)}, {0, 0, 
          1.`}}}\)], "Output"]
}, Open  ]]
}, Open  ]],

Cell[CellGroupData[{

Cell["2. Construct the Constraint Equations.", "Section"],

Cell["\<\
We subtract eq[1] because sometimes position 1 is closer to \
identity. The n-1 equations e[i] we obtain are bilinear in the coordinates of \
b and p.\
\>", "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[
    \({\(\(0.`\)\(\[InvisibleSpace]\)\) - 
        R\^2 + \((\(-u\) + 1.`\ \[Lambda])\)\^2 + \((\(-v\) + 1.`\ \
\[Mu])\)\^2, \(\(0.`\)\(\[InvisibleSpace]\)\) - 
        R\^2 + \((\(\(6.389514682272674`\)\(\[InvisibleSpace]\)\) - u - \
0.24192189559966773`\ \[Lambda] - 0.9702957262759966`\ \[Mu])\)\^2 + \
\((\(-7.461035367160299`\) - v + 0.9702957262759966`\ \[Lambda] - \
0.24192189559966767`\ \[Mu])\)\^2, \(\(0.`\)\(\[InvisibleSpace]\)\) - 
        R\^2 + \((\(-1.1339745962155634`\) - u + 0.5000000000000001`\ \
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+ 0.8660254037844386`\ \[Lambda] + 0.5000000000000001`\ \[Mu])\)\^2}\)], \
"Output"]
}, Open  ]],

Cell[CellGroupData[{

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

Cell[BoxData[
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        1.9405914525519932`\ v\ \[Lambda] - 8.78946194005465`\ \[Mu] + 
        1.9405914525519932`\ u\ \[Mu] + 
        2.483843791199335`\ v\ \[Mu], \(\(52.555136271329104`\)\(\
\[InvisibleSpace]\)\) + 2.267949192431127`\ u + 14.320508075688775`\ v - 
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        1.7320508075688772`\ u\ \[Mu] + 
        0.9999999999999998`\ v\ \[Mu]}\)], "Output"]
}, Open  ]]
}, Open  ]],

Cell[CellGroupData[{

Cell["3.  Specify Parameters of the Fixed and Moving Pivots.", "Section"],

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[
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Cell[BoxData[
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}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
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Cell[BoxData[
    \({3, \(-1\), \[Lambda], \[Mu]}\)], "Output"]
}, Open  ]],

Cell["\<\
Test data to make sure the correct number of parameters are \
specified\
\>", "Text"],

Cell[BoxData[
    \(dataQ[datapv_] := \((test = Table[NumberQ[datapv[\([i]\)]], {i, 4}]; 
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          Print["\<Data and Equations Match\>"], \ 
          Print["\<Warning::Data and Equations do NOT Match\>"]])\)\)], \
"Input"],

Cell[CellGroupData[{

Cell[BoxData[
    \(dataQ[InputCrank]\)], "Input"],

Cell[BoxData[
    \("Data and Equations Match"\)], "Print"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(dataQ[OutputCrank]\)], "Input"],

Cell[BoxData[
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}, Open  ]]
}, Open  ]],

Cell[CellGroupData[{

Cell["4.  Solve the System of Equations", "Section"],

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, 
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Cell[CellGroupData[{

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

Cell[BoxData[
    \({{1, \(-1\), 1.682429225715445`, 5.001200778707891`}}\)], "Output"]
}, Open  ]],

Cell[CellGroupData[{

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

Cell[BoxData[
    \({{3, \(-1\), 5.0807145247639935`, 0.30863951317098104`}}\)], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(nin = Length[incranks]\)], "Input"],

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

Cell[CellGroupData[{

Cell[BoxData[
    \(nout = Length[outcranks]\)], "Input"],

Cell[BoxData[
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}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
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      Table[{incranks[\([i, 1]\)], \ incranks[\([i, 2]\)]}, {i, 
          nin}]\)], "Input"],

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

Cell[CellGroupData[{

Cell[BoxData[
    \(pA = 
      Table[{incranks[\([i, 3]\)], \ incranks[\([i, 4]\)]}, {i, 
          nin}]\)], "Input"],

Cell[BoxData[
    \({{1.682429225715445`, 5.001200778707891`}}\)], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(pC = 
      Table[{outcranks[\([i, 1]\)], outcranks[\([i, 2]\)]}, {i, 
          nout}]\)], "Input"],

Cell[BoxData[
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}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
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      Table[{outcranks[\([i, 3]\)], outcranks[\([i, 4]\)]}, {i, 
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Cell[BoxData[
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}, Open  ]],

Cell[CellGroupData[{

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

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*)




(***********************************************************************
End of Mathematica Notebook file.
***********************************************************************)