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Notebook[{
Cell[TextData[{
  "The ",
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      \(TraditionalForm\`Mathematica\^\[RegisteredTrademark]\)]],
  StyleBox[" ",
    FontSlant->"Italic"],
  "Journal"
}], "OnlineHeader"],

Cell[CellGroupData[{

Cell[TextData[StyleBox["Dynamic Rotation of Space Station Mir",
  FontSize->38]], "Title",
  TextAlignment->Left,
  TextJustification->0],

Cell[TextData[StyleBox["Michael Foale", "Text",
  FontSlant->"Plain"]], "Author",
  FontSlant->"Italic"],

Cell[TextData[StyleBox["NASA Johnson Space Center, Houston", "Text"]], \
"Address",
  FontSlant->"Italic"],

Cell["\<\
The Russian space station Mir is a solid body with unequal \
principal moments of inertia. Free body rotations initially begun about the \
middle principle axis, in general develop into rotations about the other pair \
of axes, in the course of free motion. This motion is problematic for a \
station that has lost attitude control, and that must maintain its solar \
arrays directed toward the sun to maintain electrical power. This paper uses \
methods of Euler to calculate the resultant motion of the Mir in inertial \
space, with given total angular momentum and specified initial angular \
velocities about the solid body axes. The results are used to illustrate the \
difficulty of setting up an initial free rotation suitable to provide \
constant illumination of the Mir solar arrays. \
\>", "Abstract"],

Cell[CellGroupData[{

Cell["Electric Power and Attitude Control on Station Mir", "SectionFirst"],

Cell["\<\
During the summer of 1997, the Russian space station Mir suffered a number of \
problems, resulting in the total loss of attitude control. A specific \
inertial attitude is essential for the Mir to keep its solar arrays \
illuminated by the sun. A rapid energy shortfall, and total loss of power to \
the station, can result if the arrays are not in a suitable orientation. The \
usual system of attitude control utilizes sets of momentum wheels, called \
gyrodynes, spun at high speeds, to provide reactive torques to keep the Mir \
inertially stable. When an attitude control failure occurs, the angular \
momentum stored in the gyrodynes is passed into the Mir body as a whole, as \
the gyrodynes slowly brake their rotation. So even if the Mir loses attitude \
control in a good orientation with no initial rotation rates, rates build up \
over the course of hours in all Mir axes, as the gyrodynes spin down.\
\>", "Text"],

Cell["\<\
The immediate problem presented to the crew of Mir was the establishment of a \
free body rotation of the Mir to provide sufficient solar illumination of the \
arrays. With some electrical energy restored, contact with the ground \
controllers can be established, and workarounds developed to replace the \
attitude control system box that had failed. The solution developed using the \
Soyuz transport spacecraft was very crude, but effective. The Soyuz docked to \
the Mir has a power system independent of the Mir, enabling it to be powered \
even when the station is not. The Soyuz reaction control jets were manually \
fired to first arrest the resultant rotation caused by the gyrodynes, and \
then subsequently to reorient Mir, finally followed by a spin of Mir. \
However, because of the tiny roll moment arm of the Soyuz jets about the Mir \
long axis, rotation about this axis could not be influenced, complicating the \
resultant rotation dynamics of Mir.\
\>", "Text"],

Cell["\<\
Much time was spent by the crew considering the effects of initial rotations \
on the resultant rotation dynamics of Mir. This paper presents the equations \
and visualization model that permit the correct rotation impulse strategy to \
be applied.\
\>", "Text"]
}, Closed]],

Cell[CellGroupData[{

Cell["Definition of the Euler Transformation Matrices", "Section"],

Cell[TextData[{
  "To transform motions about the Mir body axes into the inertial reference \
frame, define three Euler angles \[Phi], \[Theta], and \[Psi] as arguments of \
three rotation matrices. These matrices will be combined into one matrix to \
provide for a general multiaxis rotation. The Euler matrices ",
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          SubscriptBox[
            StyleBox["L",
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  " ",
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  ", and ",
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          StyleBox["L",
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  " rotate from the inertial unprimed frame ",
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  " to the primed frame ",
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  "is a vector in inertial space. First rotate about the inertial space axis \
",
  StyleBox["z",
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  " to get to the first primed frame."
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  "At this point ",
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  " axis (note, not ",
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  "Finally ",
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  " . Define ",
  Cell[BoxData[
      \(TraditionalForm\`L\_\[Psi]\[Theta]\[Phi]\)]],
  " to be the combined solid body rotation matrix ",
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  Cell[BoxData[
      \(TraditionalForm\`L\^T\)]],
  " be its inverse such that ",
  Cell[BoxData[
      \(TraditionalForm\`L\_\[Psi]\[Theta]\[Phi]\^T = \(L\_\[Phi]\^T\) \(L\_\
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Cell[BoxData[
    \(L\_\[Psi]\[Theta]\[Phi] = 
      L\_\[Psi] . L\_\[Theta] . L\_\[Phi]\)], "Input"],

Cell[BoxData[
    \({{\(-sin\^2\)\ \[Phi]\ \[Psi] + cos\^3\ \[Theta]\ \[Phi]\ \[Psi], 
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\[Psi]}, {\(-cos\)\ sin\ \[Phi]\ \[Psi] - 
          cos\^2\ sin\ \[Theta]\ \[Phi]\ \[Psi], 
        cos\^2\ \[Phi]\ \[Psi] - cos\ sin\^2\ \[Theta]\ \[Phi]\ \[Psi], 
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Cell[CellGroupData[{

Cell["\<\
Definition of Inertia Tensor and Euler\[CloseCurlyQuote]s Equations\
\>", "Section"],

Cell[TextData[{
  "The angular momentum ",
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  "where ",
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                    StyleBox["r",
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                    "_"], \(\(\ \)\(\[Alpha]\)\)]}]}]}], " "}], 
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  ". Expanding the summation, and using Einstein summation indices, we can \
define the inertia tensor ",
  Cell[BoxData[
      \(TraditionalForm\`I\_\(\(i\ j\)\(\ \ \)\)\)]],
  "that combines with the body angular momentum vector \[Omega] to give the",
  " ",
  "net angular momentum ",
  Cell[BoxData[
      \(TraditionalForm\`J\_i\)]],
  "."
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          "=", \(\(I\_\(i\ j\)\) \[Omega]\_j\)}]}], 
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    \(TraditionalForm\`\[Delta]\_\(i\ j\)\[ThickSpace]is\[ThickSpace]the\
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Cell[TextData[{
  "In an arbitrary body coordinate system, for an irregular body like the \
Mir, ",
  Cell[BoxData[
      \(TraditionalForm\`I\_\(i\ j\)\)]],
  " is not diagonal. However, a set of coordinate axes can be found in the \
solid body that correspond to the eigenvectors of ",
  Cell[BoxData[
      \(TraditionalForm\`I\_\(i\ j\)\)]],
  ", given by ",
  Cell[BoxData[
      \(TraditionalForm\`I  \[Omega]\+_ = \(\[Lambda]  \[Omega]\+_ = 
          J\+_\)\)]],
  ". The eigenvalues \[Lambda] of ",
  Cell[BoxData[
      \(TraditionalForm\`I\)]],
  " are the three principal moments of inertia ",
  Cell[BoxData[
      \(TraditionalForm\`\(\(A\)\(,\)\(\ \)\(B\)\(,\)\(\ \)\(C\)\(\ \)\)\)]],
  "about the three orthogonal eigenvectors defined as ",
  Cell[BoxData[
      \(TraditionalForm\`\(\[Omega]\_1\)\+_, \ \(\[Omega]\_2\)\+_, \(\[Omega]\
\_3\)\+_\)]],
  ". If the body has axes of symmetry, then the ",
  Cell[BoxData[
      \(TraditionalForm\`\[Omega]\+_\)]],
  "\[CloseCurlyQuote]s are in the same direction as the axes of symmetry. For \
arbitrary directions of ",
  Cell[BoxData[
      \(TraditionalForm\`J\+_\)]],
  ", the equation ",
  Cell[BoxData[
      \(TraditionalForm\`J\+_ = I  \[Omega]\+_\)]],
  " implies that ",
  Cell[BoxData[
      \(TraditionalForm\`\[Omega]\+_\)]],
  " will not be in the same direction as ",
  Cell[BoxData[
      \(TraditionalForm\`J\+_\)]],
  ". The coupling of the components ",
  Cell[BoxData[
      \(TraditionalForm\`\[Omega]\_1, \ \[Omega]\_2, \ \[Omega]\_3\)]],
  " to each other through the principle moments of inertia ",
  Cell[BoxData[
      \(TraditionalForm\`A, \ B, \ C\)]],
  " is given by the Euler equations, if no external forces are acting. The \
general Euler equations are"
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              B\ \(\[Omega]\& . \)\_2\), "=", \(\(0\)\(,\)\)},
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          }], TraditionalForm]], "DisplayFormula",
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Cell["\<\
The \[Omega]\[CloseCurlyQuote]s are evaluated in the body axes of \
the rotating frame. The total angular momentum is \
\>", "Text"],

Cell[BoxData[
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Cell[TextData[{
  "for initial values for ",
  Cell[BoxData[
      \(TraditionalForm\`\[Omega]\_1, \[Omega]\_2, \[Omega]\_3\)]],
  " to be ",
  Cell[BoxData[
      \(TraditionalForm\`\[CapitalOmega]\_1, \ \[CapitalOmega]\_2, \ n\)]],
  " respectively. Obtaining solutions for the \[Omega]\[CloseCurlyQuote]s in \
the fixed body frame, these must be related to the Euler angles \[Phi], \
\[Theta], and \[Psi] of the body to inertial transformation matrices ",
  Cell[BoxData[
      \(TraditionalForm\`L\_\[Psi], L\_\[Theta], L\_\[Phi]\)]],
  ". "
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Cell[CellGroupData[{

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      "==", \({{t\ \[Omega]\_1}, {t\ \[Omega]\_2}, {t\ \[Omega]\_3}}\)}]], \
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Cell[TextData[{
  "These three additional equations can be solved simultaneously with the \
Euler equations to give solutions for the six parameters ",
  Cell[BoxData[
      \(TraditionalForm\`\[Omega]\_1, \[Omega]\_2, \[Omega]\_3\)]],
  ", \[Phi], \[Theta], and \[Psi] as a function of time."
}], "Text"]
}, Closed]],

Cell[CellGroupData[{

Cell["\<\
The Solution of Euler\[CloseCurlyQuote]s Equations \
\>", "Section"],

Cell[CellGroupData[{

Cell["Dynamic Equations ", "SubsectionNoSpace"],

Cell["The Euler equations are ", "Text"],

Cell[BoxData[{
    \(\(eqn1 = 
        A\ \[PartialD]\_t \[Omega]\_1[t] + \((C - B)\)\ \[Omega]\_2[
                t]\ \[Omega]\_3[t] == 0;\)\), "\n", 
    \(\(eqn2 = 
        B\ \[PartialD]\_t \[Omega]\_2[t] + \((A - C)\)\ \[Omega]\_1[
                t]\ \[Omega]\_3[t] == 0;\)\), "\n", 
    \(\(eqn3 = 
        C\ \[PartialD]\_t \[Omega]\_3[t] + \((B - A)\) \[Omega]\_1[
                t] \[Omega]\_2[t] == 0;\)\)}], "InputOnly"],

Cell["The total angular momentum and initial conditions are", "Text"],

Cell[BoxData[{
    \(\(initialconditions = {t \[Rule] 
            0, \[Omega]\_1[t] \[Rule] \[CapitalOmega]\_1, \[Omega]\_2[
              t] \[Rule] \[CapitalOmega]\_2, \[Omega]\_3[t] \[Rule] 
            n};\)\), "\n", 
    \(\(Jrule = 
        JTotal \[Rule] \@\(\((A\ \[Omega]\_1[t])\)\^2 + \((B\ \[Omega]\_2[t])\
\)\^2 + \((C\ \[Omega]\_3[t])\)\^2\);\)\)}], "InputOnly"],

Cell[BoxData[
    \(\(omega1rule = \[CapitalOmega]\_1 \[Rule] \@\(J\^2 - C\^2\ n\^2 - B\^2\ \
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Cell[TextData[{
  "The Euler angles \[Phi], \[Theta], and \[Psi] using the transformation",
  " ",
  "matrices ",
  Cell[BoxData[
      \(TraditionalForm\`\(\(\ \)\(L\_\[Psi], L\_\[Theta], L\_\[Phi]\)\)\)]],
  " give the equations"
}], "Text"],

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            RowBox[{\(Sin[\[Psi][t]]\), " ", 
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                  MultilineFunction->None], "[", "t", "]"}]}], "-", 
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          "==", \(\[Omega]\_1[t]\)}]}], ";"}], "\n", 
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          RowBox[{
            RowBox[{\(Cos[\[Psi][t]]\), " ", 
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          "==", \(\[Omega]\_2[t]\)}]}], ";"}], "\n", 
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          RowBox[{
            RowBox[{\(Cos[\[Theta][t]]\), " ", 
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                  MultilineFunction->None], "[", "t", "]"}]}], "+", 
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                MultilineFunction->None], "[", "t", "]"}]}], 
          "==", \(\[Omega]\_3[t]\)}]}], ";"}]}], "InputOnly"],

Cell[TextData[{
  "For display purposes, the initial Euler angles will be chosen so that \
output plots of the motion have the net angular momentum directed along the \
",
  StyleBox["z ",
    FontSlant->"Italic"],
  "axis. The Euler angles ",
  Cell[BoxData[
      \(TraditionalForm\`\[Psi]\_0, \ \[Theta]\_0, \ \[Phi]\_\(\(0\)\(\ \ \
\)\)\)]],
  " at ",
  Cell[BoxData[
      \(TraditionalForm\`t = 0\)]],
  " are given by the transformation "
}], "Text"],

Cell[CellGroupData[{

Cell[BoxData[
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        RowBox[{"(", GridBox[{
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            {\(A\ \[CapitalOmega]\_1\)},
            {\(B\ \[CapitalOmega]\_2\)},
            {\(C\ n\)}
            }], ")"}]}]], "Input"],

Cell[BoxData[
    \({{\(-cos\)\ J\ sin\ \[Theta]\ \[Psi]}, {J\ sin\^2\ \[Theta]\ \[Psi]}, \
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n}}\)], "Output"]
}, Open  ]],

Cell["\<\
Solve two equations for two initial angles, the angle \[Phi] being \
indeterminate.\
\>", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(Solve[{J\ Sin[\[Psi]\_0]\ Sin[\[Theta]\_0] == B\[CapitalOmega]\_2, 
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      initialconditions\)], "Input"],

Cell[BoxData[
    \({{\[Psi]\_0 \[Rule] \(-ArcSin[
              B\[CapitalOmega]\_2\/\@\(J\^2 - C\^2\ n\^2\)]\), \[Theta]\_0 \
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              B\[CapitalOmega]\_2\/\@\(J\^2 - C\^2\ n\^2\)]\), \[Theta]\_0 \
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          ArcSin[B\[CapitalOmega]\_2\/\@\(J\^2 - C\^2\ n\^2\)], \[Theta]\_0 \
\[Rule] \(-ArcCos[\(C\ n\)\/J]\)}, {\[Psi]\_0 \[Rule] 
          ArcSin[B\[CapitalOmega]\_2\/\@\(J\^2 - C\^2\ n\^2\)], \[Theta]\_0 \
\[Rule] ArcCos[\(C\ n\)\/J]}}\)], "Output"]
}, Open  ]],

Cell[TextData[{
  "The first solution in the list will be used, using a small initial ",
  Cell[BoxData[
      \(TraditionalForm\`\[CapitalOmega]\_2\)]],
  " with positive value, so that there is correspondence with the analytic",
  " ",
  "solution found elsewhere for the case of the symmetric top when ",
  Cell[BoxData[
      \(TraditionalForm\`A = B \[NotEqual] C\)]],
  "."
}], "Text"],

Cell[BoxData[
    \(\(anglerule = {\[Psi]\_0 \[Rule] \(-ArcSin[\(B\ \
\[CapitalOmega]\_2\)\/\@\(J\^2 - C\^2\ n\^2\)]\), \[Theta]\_0 \[Rule] \
\(-ArcCos[\(C\ n\)\/J]\)};\)\)], "InputOnly"]
}, Open  ]],

Cell[CellGroupData[{

Cell["Initial Conditions", "Subsection"],

Cell[TextData[{
  "The task is to solve the six parameters ",
  Cell[BoxData[
      \(TraditionalForm\`\[Omega]\_1, \[Omega]\_2, \[Omega]\_3\)]],
  ", \[Phi], \[Theta], and \[Psi] as a function of time, starting with \
initial values for each of the parameters, using the three principle moments \
of inertia ",
  Cell[BoxData[
      \(TraditionalForm\`A, \ B, \ C\)]],
  " for the Mir station. The principal axes of inertia of Mir roughly \
coincide with the module axes originating from the center section, known as \
the node. Although the Mir\[CloseCurlyQuote]s moments of inertia cannot be \
known at any one time precisely because of the uncertain arrangement of \
internal stores throughout the Mir, the basic fixed hardware results in \
moments of inertia whose ratios can be characterized by setting ",
  Cell[BoxData[
      \(TraditionalForm\`A \[InvisiblePrefixScriptBase] = 1, \ 
      B \[TildeEqual] 1.5, \ C \[TildeEqual] 1.2\)]],
  ". The initial values for ",
  Cell[BoxData[
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  " are denoted to be ",
  Cell[BoxData[
      \(TraditionalForm\`\[CapitalOmega]\_1, \ \[CapitalOmega]\_2, 
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  " respectively. The rate about the ",
  Cell[BoxData[
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  " axis is denoted as ",
  StyleBox["n",
    FontSlant->"Italic"],
  ", distinguishing it from the other two axes, because it was about this \
axis that a spin was attempted to be established manually, using the Soyuz \
spacecraft. The crew attempted to begin the spin with ",
  Cell[BoxData[
      FormBox[
        RowBox[{\(\[CapitalOmega]\_1\), Cell[""]}], TraditionalForm]]],
  "about the ",
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    FontSlant->"Italic"],
  " axis and ",
  Cell[BoxData[
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  "about the ",
  Cell[BoxData[
      FormBox[
        RowBox[{"B", 
          StyleBox[
            RowBox[{
              StyleBox[" ",
                FontWeight->"Plain"], " "}]]}], TraditionalForm]]],
  "axis as small as possible. However, because of the lack of Soyuz control \
moment about the ",
  Cell[BoxData[
      \(TraditionalForm\`A\)]],
  " axis (Soyuz is docked along this axis), ",
  Cell[BoxData[
      \(TraditionalForm\`\[CapitalOmega]\_1\)]],
  " could not be controlled, or reduced from the value developed during \
gyrodyne braking, so here ",
  Cell[BoxData[
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  " is simply calculated from initial values of ",
  StyleBox["J",
    FontSlant->"Italic"],
  " and ",
  StyleBox["n",
    FontSlant->"Italic"],
  ", and ",
  Cell[BoxData[
      \(TraditionalForm\`\[CapitalOmega]\_2\)]],
  ". Note that we are using dimensionless quantities, and that our time \
parameter ",
  StyleBox["t",
    FontSlant->"Italic"],
  " should be scaled roughly by 20 to get the physical time in minutes. "
}], "Text"],

Cell[TextData[{
  StyleBox["tmax", "InlineInput"],
  " is the total time of integration. Define a very small real number \
\[Epsilon] to avoid causing difficulties in ",
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    FontFamily->"Courier New",
    FontWeight->"Bold"],
  ". List the Mir physical properties as a single list, called ",
  StyleBox["dynamics",
    FontFamily->"Courier New",
    FontWeight->"Bold"],
  "."
}], "Text"],

Cell[BoxData[{
    \(\(tmax = 35;\)\), "\n", 
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          J -> 2.41, \[CapitalOmega]\_2 -> \[Epsilon], 
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  "Check the consistency of our parameters, by finding the minimum value of \
",
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  " for ",
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Cell[BoxData[
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Cell[BoxData[
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Cell[TextData[{
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  " value specified in ",
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  ", to ensure a non-imaginary value for ",
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  "."
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Cell[BoxData[
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Cell[BoxData[
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Cell[CellGroupData[{

Cell["Numerical Solutions", "Subsection"],

Cell[TextData[{
  "Using the parameters given, the Euler equations are solved simultaneously \
with the three equations for the Euler angles, applying our geometric \
constraint ",
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    FontFamily->"Courier New",
    FontWeight->"Bold"],
  " that ",
  Cell[BoxData[
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  " be directed along the inertial ",
  StyleBox["z",
    FontSlant->"Italic"],
  " axis."
}], "Text"],

Cell[BoxData[
    \(\(soln = \ 
        NDSolve[\(\({eqn1, eqn2, eqn3, eqn4, eqn5, 
                  eqn6, \ \[Omega]\_1[0] == \[CapitalOmega]\_1, \[Omega]\_2[
                      0] == \[CapitalOmega]\_2, \[Omega]\_3[0] == 
                    n, \ \[Phi][0] == \[Epsilon], \[Theta][
                      0] == \[Theta]\_0, \[Psi][0] == \[Psi]\_0} /. 
                anglerule\) /. omega1rule\) /. 
            dynamics\ , \n\t\t{\[Omega]\_1, \[Omega]\_2, \[Omega]\_3, \[Phi], \
\[Theta], \[Psi]}, {t, 0, tmax}, MaxSteps -> 40000];\)\)], "InputOnly"],

Cell[TextData[{
  "To understand our solution from the point of view of the crew members on \
board Mir, plot the angular rates ",
  Cell[BoxData[
      \(\[Omega]\_1, \[Omega]\_2, \[Omega]\_3\)]],
  " as observed about the three principle axes of Mir. These can be measured \
directly by the crew, utilizing the stars as an inertial reference. It is \
apparent that the motion is that of an irregular body, with most angular \
momentum about its middle axis of inertia, causing an unstable exchange of \
angular momentum between the principle axes. Although the Mir starts spinning \
initially about the ",
  Cell[BoxData[
      \(TraditionalForm\`C\)]],
  " axis, after",
  Cell[BoxData[
      \(TraditionalForm\`\(\(\ \)\(t = 9\)\)\)]],
  " the direction of rotation is reversed, and oscillates with a period of \
roughly ",
  Cell[BoxData[
      \(TraditionalForm\`t = 35\)]],
  ". The spin about the ",
  Cell[BoxData[
      \(TraditionalForm\`A\)]],
  " axis, initially small, grows periodically, but never reverses."
}], "Text"],

Cell[CellGroupData[{

Cell[BoxData[
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        PlotStyle -> {{}, {Dashing[{ .02}]}, {Dashing[{ .04}]}}, 
        PlotRange -> All];\)\)], "Input"],

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}, Open  ]],

Cell[TextData[{
  "Solid curve is ",
  Cell[BoxData[
      \(TraditionalForm\`\[Phi]\)]],
  ", short dashed curve is ",
  Cell[BoxData[
      \(TraditionalForm\`\[Theta]\)]],
  ", long dashed curve is ",
  Cell[BoxData[
      \(TraditionalForm\`\[Psi]\)]]
}], "Caption"]
}, Open  ]]
}, Closed]],

Cell[CellGroupData[{

Cell["Three-Dimensional Graphical Depiction of Mir Rotation", "Section"],

Cell[CellGroupData[{

Cell["A Graphical Model of the Mir", "SubsectionNoSpace"],

Cell[BoxData[
    \(<< Graphics`Shapes`\)], "InputOnly"],

Cell[TextData[{
  "A Mir model is constructed, using polygons, and then rotated repeatedly to \
create an animation of the Mir rotation in inertial space. However, it is \
necessary to modify the behavior of the function ",
  StyleBox["RotationMatrix3D", "InlineInput",
    FontFamily->"Courier New",
    FontWeight->"Bold"],
  ", in the package ",
  StyleBox["Graphics`Shapes`", "InlineInput",
    FontFamily->"Courier New",
    FontWeight->"Bold"],
  ", so that the Euler transformation matrices defined above are used to \
perform the rotations as a function of \[Phi], \[Theta], and \[Psi] rather \
than the matrix built into the package, which uses a different Euler \
sequence. To carry out the transformation from the body axes to the inertial \
axes, the transformation ",
  Cell[BoxData[
      \(TraditionalForm\`L\_\[Psi]\[Theta]\[Phi]\^T\)]],
  " is used."
}], "Text"],

Cell[BoxData[{
    \(\(Unprotect[RotationMatrix3D];\)\), "\n", 
    \(\(Clear[RotationMatrix3D];\)\), "\n", 
    \(RotationMatrix3D[\[Phi]_, \[Theta]_, \[Psi]_] := \n\t{{Cos[\[Theta]]\ \
Cos[\[Phi]]\ Cos[\[Psi]] - 
            Sin[\[Phi]]\ Sin[\[Psi]], \(-Cos[\[Psi]]\)\ Sin[\[Phi]] - 
            Cos[\[Theta]]\ Cos[\[Phi]]\ Sin[\[Psi]], 
          Cos[\[Phi]]\ Sin[\[Theta]]}, {Cos[\[Theta]]\ Cos[\[Psi]]\ \
Sin[\[Phi]] + Cos[\[Phi]]\ Sin[\[Psi]], 
          Cos[\[Phi]]\ Cos[\[Psi]] - Cos[\[Theta]]\ Sin[\[Phi]]\ Sin[\[Psi]], 
          Sin[\[Theta]]\ Sin[\[Phi]]}, {\(-Cos[\[Psi]]\)\ Sin[\[Theta]], 
          Sin[\[Theta]]\ Sin[\[Psi]], Cos[\[Theta]]}}\), "\n", 
    \(\(Protect[RotationMatrix3D];\)\)}], "InputOnly"],

Cell["\<\
The Mir graphical model is limited in its fidelity, but includes a depiction \
of the solar arrays, critical in their arrangement toward the sun.\
\>", "Text"],

Cell[BoxData[{
    \(\ \ \(solararray\  = 
        Polygon[{{\(-1.8\),  .4, 0}, {1.8,  .4, 0}, {1.8, \(- .4\), 
              0}, {\(-1.8\), \(- .4\), 0}}];\)\), "\n", 
    \(\t\(mir = \ 
        Graphics3D[{Thickness[
              0.03], \n\t\t\tLine[{{\(-1\), 0, 0}, {5, 0, 0}}], \t
            Thickness[0.02], \n\t\t\tLine[{{0, \(-2.2\), 0}, \ {0, 4, 0}}], 
            Thickness[0.01], \n\t\t\tLine[{{0, 0, \(-2\)}, \ {0, 0, 3}}], \ 
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              RotateShape[Cylinder[ .7,  .9, 10], 0, \[Pi]/2, 0], {1.9, 0, 
                0}], \n\t\t\tTranslateShape[
              RotateShape[Cylinder[ .7,  .4, 10], 0, \[Pi]/2, 0], {3.5, 0, 
                0}], \n\t\t\tTranslateShape[
              RotateShape[Cylinder[ .5,  .7, 10], 0, 0, 0], \ {0, 0, 
                1.5}], \n\t\t\tTranslateShape[
              RotateShape[Cylinder[ .5,  .7, 10], 0, 0, 0], \ {0, 
                0, \(-1.5\)}], \n\t\t\tTranslateShape[
              RotateShape[Cylinder[ .5, 1, 10], \[Pi]/2, \[Pi]/2, 0], \ {0, 
                2, 0}], \n\t\t\tTranslateShape[
              RotateShape[Cylinder[ .5,  .7, 10], \[Pi]/2, \[Pi]/2, 
                0], \ {0, \(-1.5\), 0}], \n\t\t\tTranslateShape[
              RotateShape[Cone[ .3, 1, 10], 0, \[Pi]/2, 0], {\(-1.5\), 0, 
                0}], \n\t\t\tTranslateShape[
              RotateShape[Cone[ .3, 1, 10], 0, \(-\[Pi]\)/2, 0], {5, 0, 
                0}], \n\t\t\tTranslateShape[
              RotateShape[solararray, 0, \[Pi]/6, \[Pi]/2], {3.5, \(-2.6\), 
                0}], \n\t\t\tTranslateShape[
              RotateShape[solararray, 0, \[Pi]/6, \[Pi]/2], {3.5, 2.6, 
                0}], \n\t\t\tTranslateShape[
              RotateShape[\n\t\t\t\t\ \ \ AffineShape[
                  solararray, { .7,  .9, 1}], 
                0, \[Pi]/6, \[Pi]/2], {1.8, \(-2.1\), 
                0}], \n\t\t\tTranslateShape[
              RotateShape[\n\t\t\t\t\ \ AffineShape[
                  solararray, { .7,  .9, 1}], 0, \[Pi]/6, \[Pi]/2], {1.8, 
                2.1, 0}], \n\t\t\tTranslateShape[
              RotateShape[solararray, \(-\[Pi]\)/2, \[Pi]/4, 
                0], {0, \(-1.9\), \(-3.3\)}], \n\t\t\tTranslateShape[
              RotateShape[solararray, \[Pi]/2, \[Pi]/4, 0], {0, 
                1.9, \(-3.3\)}]}];\)\)}], "InputOnly"],

Cell[CellGroupData[{

Cell[BoxData[
    \(\(\(\ \)\(Show[mir, 
        Graphics3D[{Text[
              StyleForm["\<A\>", 
                FontFamily -> \ "\<Arial\>", \n\t\t\t\ \ \ \ \ \ FontWeight -> \
"\<Bold\>", \ FontSize -> 20], {7, 0, 0}], \n\t\t\ \ \ \ \ \ \ \ Text[
              StyleForm["\<B\>", FontFamily -> \ "\<Arial\>", \ 
                FontWeight -> "\<Bold\>", \n\t\t\t\ \ \ \ \ \ FontSize -> 
                  20], {0, 6, 0}], \ 
            Text[StyleForm["\<C\>", 
                FontFamily -> \ "\<Arial\>", \n\t\t\t\ \ \ \ \ \ FontWeight -> \
"\<Bold\>", FontSize -> 20], {0, 0, 4}], \ 
            Text["\<Soyuz\>", {\(-5\), 0, 
                0}]}], \n\ \ \ PlotRange -> {{\(-7\), 7}, {\(-7\), 
              7}, {\(-7\), 7}}, \n\ \ \ BoxRatios -> {1, 1, 1}, 
        Axes -> True, \ AxesLabel -> {"\<X\>", "\<Y\>", \ "\<Z\>"}, \n\t
        ViewPoint -> {1.864, \ \(-2.603\), \ 1.096}];\)\)\)], "Input"],

Cell[GraphicsData["PostScript", "\<\
%!
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%%AspectRatio: .97402 
MathPictureStart
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Cell[TextData[{
  "The three principle axes ",
  Cell[BoxData[
      \(TraditionalForm\`A, \ B, \ C\)]],
  " are shown, and the Soyuz spacecraft is represented as a slender cone on \
the ",
  Cell[BoxData[
      \(TraditionalForm\`\(-A\)\)]],
  " axis. The center of mass of the Mir is roughly located at the central \
sphere, representing the node. It should be noted that the moment arm of the \
Soyuz reaction control jets at the end of the arm from the center of mass is \
small compared to the dimensions of the Mir. Effectively, only rotations \
about the ",
  Cell[BoxData[
      \(TraditionalForm\`B\)]],
  " and ",
  Cell[BoxData[
      \(TraditionalForm\`C\)]],
  " axes are achievable without excessive usage of Soyuz fuel. The \
orientation of the Mir is roughly that desired at the start of a manual \
rotation, if the sun is located on the +",
  StyleBox["z",
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  " axis. The critical solar arrays are mounted perpendicular to the base \
block of Mir, which are the two cylinders along the ",
  Cell[BoxData[
      \(TraditionalForm\`A\)]],
  " axis. The arrays are able to slowly rotate about their long axes to be as \
perpendicular as possible to the line of sight toward the sun. "
}], "Text"]
}, Open  ]],

Cell[CellGroupData[{

Cell["Animated 3D Rotation of Mir", "Subsection"],

Cell["\<\
The Mir model is animated using the calculated interpolation functions for \
\[Phi], \[Theta], and \[Psi].\
\>", "Text"],

Cell[CellGroupData[{

Cell[BoxData[{
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        i\ \((tend - tstart)\)/nframes + 
          tstart; \n\ \ \ \ \ \ \ phi\  = \ \[Phi][s] /. First[soln]; 
      theta\  = \ \[Theta][s] /. First[soln]; 
      psi\  = \ \[Psi][s] /. First[soln]; \n\ \ \ \ \ \ \ Show[
        RotateShape[mir, phi, theta, 
          psi], \n\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ PlotRange -> \
{{\(-6\), 6}, {\(-6\), 6}, {\(-6\), 
              6}}, \n\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ BoxRatios -> {1, 
            1, 1}, \ Boxed -> 
          False, \n\t\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ ViewPoint -> {1.886, \ \
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.45723 .55343 m .40582 .60209 L .4207 .59371 L p .669 .971 .805 r
F P
0 g
s
.4207 .59371 m .47217 .54516 L .45723 .55343 L p .669 .971 .805 r
F P
0 g
s
.5161 .60479 m .46319 .65178 L .44857 .66062 L p .62 .947 .76 r
F P
0 g
s
.40179 .62018 m .40582 .60209 L p .40655 .60162 L .122 0 .312 r
F P
0 g
s
.41939 .60367 m .40179 .62018 L p .40655 .60162 L .122 0 .312 r
F P
0 g
s
.40179 .62018 m .45344 .57172 L .46248 .59309 L p .434 .374 .691 r
F P
0 g
s
.29986 .61822 m .41433 .60465 L .42859 .60513 L p .138 0 0 r
F P
0 g
s
.46248 .59309 m .4104 .64108 L .40179 .62018 L p .434 .374 .691 r
F P
0 g
s
.42859 .60513 m .43969 .60551 L p .43412 .60602 L .138 0 0 r
F P
0 g
s
.43412 .60602 m .29986 .61822 L .42859 .60513 L .138 0 0 r
F
.45344 .57172 m .41939 .60367 L p .40655 .60162 L .122 0 .312 r
F P
0 g
s
.40582 .60209 m .45723 .55343 L .45344 .57172 L p .122 0 .312 r
F P
0 g
s
.53311 .55356 m .59037