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Cell[CellGroupData[{
Cell["Synthesis of Spherical 4R chains", "Title",
  FontColor->RGBColor[0, 0, 1]],

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

We write the constraint equations ([D]W.G=cos\[Rho]  for each of the \
specified positions.  Then, we subtract the first from the remaining \
equations to eliminate cos\[Rho].  Finally, we choose values for G=(u, v, w) \
or W=(\[Lambda], \[Mu], \[Nu]) and solve for the remaining components.

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

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

Cell["Specify Three Orientations", "Section",
  FontColor->RGBColor[1, 0, 0]],

Cell["\<\
We specify the positions in terms of the latitude \[Theta] and \
longitude \[Phi] of the z-axis and a roll \[Psi] about this axis.\
\>", "Text"],

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Cell[BoxData[
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\>"],
  ImageRangeCache->{{{0, 287}, {205.062, 0}} -> {-0.130697, 0.072077, \
0.00393973, 0.00393973}}],

Cell[BoxData[
    TagBox[\(\[SkeletonIndicator]  Graphics3D  \[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, 0.9999999999999999`, 0}, {0, 0, 
          1.`}}, {{0.9295238176541111`, \(-0.3334114327522509`\), 
          0.15755090899077157`}, {0.34975325424928766`, 
          0.9324970084943001`, \(-0.09012208547972216`\)}, \
{\(-0.11686801767703722`\), 0.13887456807951715`, 
          0.983389912895677`}}, {{0.32017743858068526`, \
\(-0.5595828460305189`\), 0.7644301447825667`}, {0.3651960701677784`, 
          0.8174480566277952`, 
          0.4454329411364317`}, {\(-0.8741385691970825`\), 
          0.13654930663985534`, 0.46608588124749295`}}}\)], "Output"]
}, Open  ]]
}, Open  ]]
}, Open  ]],

Cell[CellGroupData[{

Cell["Design the Input and Output Cranks.", "Section",
  FontColor->RGBColor[1, 0, 0]],

Cell[CellGroupData[{

Cell[BoxData[
    \(Id = {{1, 0, 0}, \ {0, 1, 0}, {0, 0, 1}}\)], "Input"],

Cell[BoxData[
    \({{1, 0, 0}, {0, 1, 0}, {0, 0, 1}}\)], "Output"]
}, Open  ]],

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

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

Cell[CellGroupData[{

Cell[BoxData[
    \(Eqn = 
      Table[Dot[relative[\([i]\)] . W, G] - cos\[Rho], {i, \ 
          npos}]\)], "Input"],

Cell[BoxData[
    \({\(-cos\[Rho]\) + 1.`\ u\ \[Lambda] + 0.9999999999999999`\ v\ \[Mu] + 
        1.`\ w\ \[Nu], \(-cos\[Rho]\) + 
        v\ \((0.34975325424928766`\ \[Lambda] + 0.9324970084943001`\ \[Mu] - 
              0.09012208547972216`\ \[Nu])\) + 
        u\ \((0.9295238176541111`\ \[Lambda] - 0.3334114327522509`\ \[Mu] + 
              0.15755090899077157`\ \[Nu])\) + 
        w\ \((\(-0.11686801767703722`\)\ \[Lambda] + 
              0.13887456807951715`\ \[Mu] + 
              0.983389912895677`\ \[Nu])\), \(-cos\[Rho]\) + 
        v\ \((0.3651960701677784`\ \[Lambda] + 0.8174480566277952`\ \[Mu] + 
              0.4454329411364317`\ \[Nu])\) + 
        w\ \((\(-0.8741385691970825`\)\ \[Lambda] + 
              0.13654930663985534`\ \[Mu] + 0.46608588124749295`\ \[Nu])\) + 
        u\ \((0.32017743858068526`\ \[Lambda] - 0.5595828460305189`\ \[Mu] + 
              0.7644301447825667`\ \[Nu])\)}\)], "Output"]
}, Open  ]],

Cell[CellGroupData[{

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

Cell[BoxData[
    \({0, \(-0.07047618234588893`\)\ u\ \[Lambda] + 
        0.34975325424928766`\ v\ \[Lambda] - 
        0.11686801767703722`\ w\ \[Lambda] - 0.3334114327522509`\ u\ \[Mu] - 
        0.06750299150569983`\ v\ \[Mu] + 0.13887456807951715`\ w\ \[Mu] + 
        0.15755090899077157`\ u\ \[Nu] - 0.09012208547972216`\ v\ \[Nu] - 
        0.01661008710432299`\ w\ \[Nu], \(-0.6798225614193147`\)\ u\ \
\[Lambda] + 0.3651960701677784`\ v\ \[Lambda] - 
        0.8741385691970825`\ w\ \[Lambda] - 0.5595828460305189`\ u\ \[Mu] - 
        0.1825519433722047`\ v\ \[Mu] + 0.13654930663985534`\ w\ \[Mu] + 
        0.7644301447825667`\ u\ \[Nu] + 0.4454329411364317`\ v\ \[Nu] - 
        0.533914118752507`\ w\ \[Nu]}\)], "Output"]
}, Open  ]],

Cell[CellGroupData[{

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

Cell["\<\
We can select arbitrary values to the coordinates of either G=(u, \
v, w) or W=(\[Lambda], \[Mu], \[Nu]).  Use latitude and longitude \
coordinates:\
\>", "Text"],

Cell[BoxData[
    \(LongLatV[long_, lat_] := {Cos[N[Pi/180]*lat]*Sin[N[Pi/180]*long], 
        Sin[N[Pi/180]*lat], 
        Cos[N[Pi/180]*lat]*Cos[N[Pi/180]*long]}\)], "Input"],

Cell[CellGroupData[{

Cell[BoxData[
    \(outFixed = LongLatV[30, \(-30\)]\)], "Input"],

Cell[BoxData[
    \({0.43301270189221935`, \(-0.5`\), 0.7500000000000001`}\)], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(inFixed = LongLatV[\(-30\), \ \(-10\)]\)], "Input"],

Cell[BoxData[
    \({\(-0.492403876506104`\), \(-0.17364817766693036`\), 
      0.8528685319524433`}\)], "Output"]
}, Open  ]],

Cell[CellGroupData[{

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

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

Cell[CellGroupData[{

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

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

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

Cell[BoxData[
    \(Compute[f_, \ 
        m_] := \((DesEqns = 
          Table[Design[\([i + 1]\)] /. {u \[Rule] f[\([1]\)], 
                v \[Rule] f[\([2]\)], 
                w \[Rule] f[\([3]\)], \[Lambda] \[Rule] 
                  m[\([1]\)], \[Mu] \[Rule] m[\([2]\)], \ \[Nu] \[Rule] 
                  m[\([3]\)]}, {i, npos - 1}]; NSolve[DesEqns])\)\)], "Input"],

Cell[CellGroupData[{

Cell[BoxData[
    \(incrank = 
      Flatten[{inFixed, inMoving} /. Compute[inFixed, inMoving]]\)], "Input"],

Cell[BoxData[
    \({\(-0.492403876506104`\), \(-0.17364817766693036`\), 
      0.8528685319524433`, \(-2.7266797076624893`\), \
\(-0.9059697828696482`\), 1}\)], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(outcrank = 
      Flatten[{outFixed, outMoving} /. 
          Compute[outFixed, outMoving]]\)], "Input"],

Cell[BoxData[
    \({0.43301270189221935`, \(-0.5`\), 0.7500000000000001`, 
      0.9803486747854139`, \(-28.846230713514746`\), 1}\)], "Output"]
}, Open  ]]
}, Open  ]],

Cell[CellGroupData[{

Cell["Draw the linkage", "Subsection"],

Cell[CellGroupData[{

Cell[BoxData[
    \(sO = 
      UnitV[{incrank[\([1]\)], \ incrank[\([2]\)], 
          incrank[\([3]\)]}]\)], "Input"],

Cell[BoxData[
    \({\(-0.492403876506104`\), \(-0.17364817766693036`\), 
      0.8528685319524433`}\)], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(sA = 
      UnitV[{incrank[\([4]\)], \ incrank[\([5]\)], 
          incrank[\([6]\)]}]\)], "Input"],

Cell[BoxData[
    \({\(-0.896257275778341`\), \(-0.2977914887657769`\), 
      0.3286991402986157`}\)], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(sC = 
      UnitV[{outcrank[\([1]\)], outcrank[\([2]\)], 
          outcrank[\([3]\)]}]\)], "Input"],

Cell[BoxData[
    \({0.43301270189221935`, \(-0.5`\), 0.7500000000000001`}\)], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(sB = 
      UnitV[{outcrank[\([4]\)], outcrank[\([5]\)], 
          outcrank[\([6]\)]}]\)], "Input"],

Cell[BoxData[
    \({0.033945352941164395`, \(-0.9988236917919516`\), 
      0.03462579571354509`}\)], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(inScale =  .75\)], "Input"],

Cell[BoxData[
    \(0.75`\)], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(outScale = 1.25\)], "Input"],

Cell[BoxData[
    \(1.25`\)], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(SpLinkage = 
      Graphics3D[{Thickness[ .015], Line[{Origin, sO}], Line[{Origin, sC}], \ 
          Line[{Origin, outScale*sA}], Line[{Origin, outScale*sB}], 
          Polygon[{Origin, \ inScale*sO, inScale*sA}], 
          Polygon[{inScale*sC, sC, \ sB, \ inScale*sB}], \ 
          Polygon[{sA, \ outScale*sA, \ outScale*sB, \ sB}], PointSize[ .05], 
          Point[sA], Point[outScale*sA], Point[sB], Point[outScale*sB]\ }, \ 
        PlotRange \[Rule] {{\(-1.25\), 1.25}, {\(-1.25\), 1.25}, {\(-1.25\), 
              1.25}}, AspectRatio \[Rule] Automatic]\)], "Input"],

Cell[BoxData[
    TagBox[\(\[SkeletonIndicator]  Graphics3D  \[SkeletonIndicator]\),
      False,
      Editable->False]], "Output"]
}, Open  ]],

Cell[CellGroupData[{

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

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*)




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