Dynamic Rotation of Space Station Mir

Michael Foale, NASA Johnson Space Center, Houston

Abstract

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.

Electric Power and Attitude Control on Station Mir

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 looses 


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.







The immediate problem presented to the crew of Mir, is 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 


gyrodines, and then subsequently to reorient Mir, followed finally 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.  







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.

Definition of the Euler Transformation Matrices

To transform motions about the Mir body axes into the inertial reference 


frame,  define three Euler angles ϕ, θ, ψ as arguments of  


three rotation matrices. These matrices will be combined into one matrix to 


provide for a general multiaxis rotation. The Euler matrices (Subscript[L, ϕ]), (Subscript[L, θ]), (Subscript[L, ψ])  rotate from the inertial unprimed frame x to the primed frame x' use the form x'= (Subscript[L, ϕ])x where x is a vector in inertial space.  First rotate about the inertial space axis 


z, to get to the first primed frame..

(Subscript[L, ϕ])=(({{cos(ϕ), sin(ϕ), 0}, {-sin(ϕ), cos(ϕ), 0}, {0, 0, 1}}));

\(TextForm



    \`At\ this\ point\ \ x\&_ = \ 



      \(L\_ϕ\) x . \ Now\ we\ rotate\ about\ the\ primed\ frame\ y\ 



        axis, \ by\ θ\)

(Subscript[L, θ])=(({{cos(θ), 0, -sin(θ)}, {0, 1, 0}, {sin(θ), 0, cos(θ)}}));

At this point \(TraditionalForm\`x\&\(_\&_\) = \ \(L\_θ\) x\&_\) .  Now rotate about the double primed frame axis about the primed frame z 


axis (note, not x, since this is the Euler system)

(Subscript[L, ψ])=(({{cos(ψ), sin(ψ), 0}, {-sin(ψ), cos(ψ), 0}, {0, 0, 1}}));

Finally \(TraditionalForm



      \`x\&\(_\&\(_\&_\)\) = \ \(L\_ψ\) x\&\(_\&_\)\) . Define (Subscript[L, ψ])θϕ to be the combined solid body rotation matrix L =  (Subscript[L, ψ]) (Subscript[L, θ]) (Subscript[L, ϕ]) and (L^T) be its inverse such that \(TraditionalForm



      \`L\_ψθϕ\^T = 



        \(L\_ϕ\^T\) \(L\_θ\^T\) L\_ψ\^T\)

(Subscript[L, ψ])θϕ = (Subscript[L, ψ]) . (Subscript[L, θ]) . (Subscript[L, ϕ])

Definition of Inertia Tensor and Euler's Equations

The angular momentum (UnderBar[J]) of a collection of particles with momentum ((UnderBar[P])^( α)) for each particle α , with a position vector (((\(r\+_\)))^( α)) relative to some origin in inertial space is

(UnderBar[J]) = (Underscript[Σ, α])  ((UnderBar[r])^( α)) ⋀ ((UnderBar[P])^( α))

where ((UnderBar[P])^( α)) is given by ((UnderBar[P])^( α)) = (m^α) (UnderBar[Ω]) ^ ((UnderBar[r])^( α)) .  Expanding the summation, and using Einstein summation indices, we can 


define the inertia tensor (Subscript[I, i j])  that combines with the body angular momentum vector ω to give the 


net angular momentum (Subscript[J, i]).

\(\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 


\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 



          J\_i = \(\(Σ\+α\) 



                \(\(m\^α\)(



                  \(\(r\^2\)\&\((α)\)\) δ\_\(i\ j\) - 



                    \(r\_i\) r\_j)\) ω\_j = 



              \(I\_\(i\ j\)\) ω\_j\)\)   \(	\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 


\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 



        I\_\(i\ j\) = 



          \(Σ\+α\) 



            \(\(m\^α\)(



              \(\(r\^2\)\&\((α)\)\) δ\_\(i\ j\) - \(r\_i\) r\_j)



              \)\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 



            δ\_\(i\ j\)\ \ \ \ is\ the\ identity\ matrix\)

In an arbitrary body coordinate system, for an irregular body like the 


Mir, (Subscript[I, i j]) is not diagonal.  However a set of coordinate axes can be found in the 


solid body that correspond to the eigen vectors of (Subscript[I, i j]), given by \(TraditionalForm



      \`I  ω\+_ = \(λ  ω\+_ = J\+_\)\).  The eigen values λ of I are the three principal moments of inertia A,  B,  C about the three orthogonal  eigen vectors defined as \(TraditionalForm\`\(ω\_1\)\+_, \ \(ω\_2\)\+_, 



      \(ω\_3\)\+_\).  If the body has axes of symmetry, then the \(TraditionalForm\`ω\+_\)'s are in the same direction as the axes of symmetry.  For arbitrary 


directions of \(TraditionalForm\`J\+_\) , the equation \(TraditionalForm\`J\+_ = I  ω\+_\) implies that \(TraditionalForm\`ω\+_\) will not be in the same direction as \(TraditionalForm\`J\+_\).  The coupling of the components (Subscript[ω, 1]),  (Subscript[ω, 2]),  (Subscript[ω, 3]) to each other through the principle moments of inertia A,  B,  C is given by the Euler equations, if no external forces are acting. The 


general Euler equations are .  
\(\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 


\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 



          \((C - B)\) \(ω\_2\) ω\_3\  + \ A  ω\& . \_1 = 



            0\)      
\(\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 


\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 



          \((A - C)\) \(ω\_1\) ω\_3\  + \ B  ω\& . \_2 = 



            0\ \ \ \ \ 





          



          \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 


\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 



          \((B - A)\) \(ω\_1\) ω\_2\  + \ C  ω\& . \_3 = 



            0\) 
   
The ω's are evaluated in the body axes of the rotating frame. The 


total angular momentum is 

              



      J = (Sqrt[(((A (Subscript[ω, 1])))^2) + (((B (Subscript[ω, 2])))^2) + 



            (((C (Subscript[ω, 3])))^2)])       and        \(TraditionalForm



      \`Ω\_1 = 



        \@\(J\^2 - \(C\^2\) n\^2 - \(B\^2\) Ω\_2\^2\)\/A\)

for initial values for (Subscript[ω, 1]), (Subscript[ω, 2]), (Subscript[ω, 3]) to be (Subscript[Ω, 1]),  (Subscript[Ω, 2]),  n respectively.  Obtaining solutions for the ω's in the fixed body 


frame, these must be related to the Euler angles ϕ, θ, ψ of 


the body to inertial transformation matrices (Subscript[L, ψ]), (Subscript[L, θ]), (Subscript[L, ϕ]).

(Subscript[L, ψ]).((Subscript[L, θ]).(({{0}, {0}, {ϕ'}}))+(({{0}, {θ'}, {0}})))+(({{0}, {0}, {ψ'}}))==(({{(Subscript[ω, 1])(t)}, {(Subscript[ω, 2])(t)}, {(Subscript[ω, 3])(t)}}))

These three additional equations can be solved simultaneously with the 


Euler equations to give solutions for the six parameters (Subscript[ω, 1]), (Subscript[ω, 2]), (Subscript[ω, 3]),  ϕ, θ, ψ as a function of time.

The solution of Euler's equations

Dynamic Equations

The Euler equations are

\(eqn1 = 



      A\ ∂\_t ω\_1[t] + 



          \((C - B)\)\ ω\_2[t]\ ω\_3[t] == 0; 




    eqn2 = B\ ∂\_t ω\_2[t] + 



          \((A - C)\)\ ω\_1[t]\ ω\_3[t] == 0; 




    eqn3 = C\ ∂\_t ω\_3[t] + 



          \((B - A)\) ω\_1[t] ω\_2[t] == 0; \)

The total angular momentum and initial conditions are

initialconditions = {t -> 0, 



        (Subscript[ω, 1])[t] -> (Subscript[Ω, 1]), 



        (Subscript[ω, 2])[t] -> (Subscript[Ω, 2]), (Subscript[ω, 3])[t] -> n}; 



    Jrule = 



      JTotal -> 



        (Sqrt[(((A (Subscript[ω, 1])[t]))^2) + (((B (Subscript[ω, 2])[t]))^2) + 



            (((C (Subscript[ω, 3])[t]))^2)]);

omega1rule = 



      (Subscript[Ω, 1]) -> 



        ((Sqrt[(J^2) - (C^2) (n^2) - (B^2) (Subscript[Ω, 2]^2)])/A);

The Euler angles ϕ, θ, ψ using the transformation 


matrices  (Subscript[L, ψ]), (Subscript[L, θ]), (Subscript[L, ϕ]) give the equations

eqn4=Sin[ψ[t]] θ'[t]-Cos[ψ[t]] Sin[θ[t]] ϕ'[t]==(Subscript[ω, 1])[t];
eqn5=Cos[ψ[t]] θ'[t]+Sin[θ[t]] Sin[ψ[t]] ϕ'[t]==(Subscript[ω, 2])[t];
eqn6=Cos[θ[t]] ϕ'[t]+ψ'[t]==(Subscript[ω, 3])[t];

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 


z-axis.  The Euler angles (Subscript[ψ, 0]),  (Subscript[θ, 0]),  (Subscript[ϕ, 0])   at t = 0 are given by the transformation

(Subscript[L, ψ])θϕ.(({{0}, {0}, {J}}))==(({{A (Subscript[Ω, 1])}, {B (Subscript[Ω, 2])}, {C n}}))

Solve 2 equations for 2 initial angles, the angle ϕ being 


indeterminate.

Solve[{J Sin[(Subscript[ψ, 0])] Sin[(Subscript[θ, 0])] == B(Subscript[Ω, 2]), 



          J Cos[(Subscript[θ, 0])] == C n }, {(Subscript[θ, 0]), (Subscript[ψ, 0])}] /. 



      initialconditions

The first solution in the list will be used, using a small initial (Subscript[Ω, 2]) with positive value, so that there is correspondence with the analytic 


solution found elsewhere for the case of the symmetric top when A = B ≠ C.

anglerule = {



        (Subscript[ψ, 0]) -> 



          -ArcSin[((B (Subscript[Ω, 2]))/(Sqrt[(J^2) - (C^2) (n^2)]))], 



        (Subscript[θ, 0]) -> -ArcCos[((C n)/J)]};

Initial Conditions

The task is to solve the six parameters (Subscript[ω, 1]), (Subscript[ω, 2]), (Subscript[ω, 3]),  ϕ, θ, ψ as a function of time, starting with initial 


values for each of the parameters, using the three principle moments of 


inertia 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's  moments of inertia cannot be known at any one 


time precisely, because of the uncertain arrangement of internal stores 


through out the Mir, the basic fixed hardware results in moments of inertia 


whose ratios can be characterized by setting A  = 1,  



      B ≃ 1.5,  C ≃ 1.2 . The initial values for (Subscript[ω, 1]), (Subscript[ω, 2]), (Subscript[ω, 3]) are denoted to be (Subscript[Ω, 1]),  (Subscript[Ω, 2]), 



      and n respectively.  The rate about the C axis is denoted as n, 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 (Subscript[Ω, 1]) about the A axis,  



      and (Subscript[Ω, 2]) about the B  axis as small as possible.  However, because of the lack of Soyuz control 


moment about the A axis (Soyuz is docked along this axis), (Subscript[Ω, 1]) could not be controlled, or reduced from the value developed during 


gyrodyne braking, so here (Subscript[Ω, 1]) is simply calculated from initial values of J and n, and (Subscript[Ω, 2]).  Note that we are using dimensionless quantities, and that our time 


parameter t should be scaled roughly by 20 to get the physical time in 


minutes. 

Tmax is the total time of integration. Define a very small real 


number ϵ to avoid causing difficulties in NDSolve. List our Mir physical properties as a single list, called dynamics.

\(tmax = 35; 
ϵ = 0.0000000001; 




    dynamics = {A -> 1, \ B -> 1.5, C -> 1.2, J -> 2.41, 



        Ω\_2 -> ϵ, n -> 2}; \)

Check the consistency of our parameters, by finding the minimum value of J for n,  assuming   (Subscript[Ω, 1]) =  0.

JTotal /. Jrule /. initialconditions /. 



        (Subscript[Ω, 1]) -> 0 /. dynamics

This should always be less than the J value specified in dynamics, to ensure a non-imaginary value for (Subscript[Ω, 1]).

\(\(
\(\(Ω\_1 /. anglerule\) /. omega1rule\) /. dynamics\ 



    \)\)

Numerical Solutions

Using the parameters given, the Euler equations are solved simultaneously 


with the 3 equations for the Euler angles, applying our geometric constraint 


anglerule that \(TraditionalForm\`J\+_\) be directed along the inertial Z axis.

\(\(soln = \ 



      NDSolve[\(\({eqn1, eqn2, eqn3, eqn4, eqn5, eqn6, \ 



                ω\_1[0] == Ω\_1, 



                ω\_2[0] == Ω\_2, ω\_3[0] == n, \ 



                ϕ[0] == ϵ, θ[0] == θ\_0, 



                ψ[0] == ψ\_0} /. anglerule\) /. omega1rule\) /. 



          dynamics\ , 




        		{ω\_1, ω\_2, ω\_3, ϕ, θ, 



          ψ}, {t, 0, tmax}, MaxSteps -> 40000]; \)\)

To understand our solution from the point of the crew members on board 


Mir, plot the angular rates (Subscript[ω, 1]), (Subscript[ω, 2]), (Subscript[ω, 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 C axis, after t = 9 the direction of rotation is reversed, and oscillates with a period of 


roughly t = 35 .  The spin about the A axis, initially small, grows periodically, but never reverses.

\(\(Plot[



      Evaluate[{ω\_1[t], ω\_2[t], ω\_3[t]} /. soln], 



      \ {t, \ 0, \ tmax}, 
	



      PlotStyle -> {{}, {Dashing[{ .02}]}, {Dashing[{ .04}]}}, 



      PlotRange -> All]; \)\)

Solid curve is (Subscript[ω, 1]), Short dashed curve is (Subscript[ω, 2]), Long dashed curve is (Subscript[ω, 3])

No physical intuition is readily apparent when the Euler angles are 


plotted, but they determine the behaviour of the physical transformation 


matrix (Subscript[L, ψ])θϕ used to do the animation of the Mir model.  It should be noted that the 


roughly linear growth of ϕ  with time corresponds to the continuous spin 


of the Mir complex about the inertial Z axis, along which the total angular 


momentum was constrained as an initial condition using anglerule.

\(\(Plot[Evaluate[{ϕ[t], θ[t], ψ[t]} /. soln], 



      \ {t, \ 0, \ tmax}, 
	



      PlotStyle -> {{}, {Dashing[{ .02}]}, {Dashing[{ .04}]}}, 



      PlotRange -> All]; \)\)

Solid curve is ϕ, Short dashed curve is θ, Long dashed curve is ψ

Three Dimensional Graphical Depiction of Mir Rotation

A Graphical Model of the Mir

<< Graphics`Shapes`

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 behaviour of the function RotationMatrix3D, in the package Graphics`Shapes`, so that the Euler transformation matrices defined above are used to 


perform the rotations as a function of ϕ,θ,ψ rather than the 


matrix built into the package, which uses a different Euler sequence.  To 


carryout the transformation from the body axes to inertial axes, the 


transformation (Subscript[L, ψ])θ(ϕ^T) is used.

\(Unprotect[RotationMatrix3D]; 
Clear[RotationMatrix3D]; 




    RotationMatrix3D[ϕ_, θ_, ψ_] := 




      	{{Cos[θ]\ Cos[ϕ]\ Cos[ψ] - Sin[ϕ]\ Sin[ψ], 



          \(-Cos[ψ]\)\ Sin[ϕ] - 



            Cos[θ]\ Cos[ϕ]\ Sin[ψ], 



          Cos[ϕ]\ Sin[θ]}, {



          Cos[θ]\ Cos[ψ]\ Sin[ϕ] + Cos[ϕ]\ Sin[ψ], 



          Cos[ϕ]\ Cos[ψ] - Cos[θ]\ Sin[ϕ]\ Sin[ψ], 



          Sin[θ]\ Sin[ϕ]}, {\(-Cos[ψ]\)\ Sin[θ], 



          Sin[θ]\ Sin[ψ], Cos[θ]}}\)Protect[RotationMatrix3D];

The Mir graphical model is limited in its fidelity, but includes a depiction 


of the solar arrays, critical in their arrangement toward the sun.

\(\(\ \ 



    solararray\  = 



      Polygon[{{\(-1.8\),  .4, 0}, {1.8,  .4, 0}, {1.8, \(- .4\), 0}, {



            \(-1.8\), \(- .4\), 0}}]; 
	



    mir = \ Graphics3D[{Thickness[0.03], 
			



          Line[{{\(-1\), 0, 0}, {5, 0, 0}}], 	Thickness[0.02], 
			



          Line[{{0, \(-2.2\), 0}, \ {0, 4, 0}}], Thickness[0.01], 
			



          Line[{{0, 0, \(-2\)}, \ {0, 0, 3}}], \ Sphere[ .5, 10, 7], 
			



          TranslateShape[



            RotateShape[Cylinder[ .7,  .9, 10], 0, π/2, 0], {1.9, 0, 0}], 



          
			TranslateShape[



            RotateShape[Cylinder[ .7,  .4, 10], 0, π/2, 0], {3.5, 0, 0}], 



          
			TranslateShape[



            RotateShape[Cylinder[ .5,  .7, 10], 0, 0, 0], \ {0, 0, 1.5}], 




          			TranslateShape[RotateShape[Cylinder[ .5,  .7, 10], 0, 0, 0], 



            \ {0, 0, \(-1.5\)}], 
			



          TranslateShape[



            RotateShape[Cylinder[ .5, 1, 10], π/2, π/2, 0], 



            \ {0, 2, 0}], 
			



          TranslateShape[



            RotateShape[Cylinder[ .5,  .7, 10], π/2, π/2, 0], 



            \ {0, \(-1.5\), 0}], 
			



          TranslateShape[



            RotateShape[Cone[ .3, 1, 10], 0, π/2, 0], {\(-1.5\), 0, 0}], 



          
			TranslateShape[



            RotateShape[Cone[ .3, 1, 10], 0, \(-π\)/2, 0], {5, 0, 0}], 




          			TranslateShape[



            RotateShape[solararray, 0, π/6, π/2], {3.5, \(-2.6\), 



              0}], 
			



          TranslateShape[



            RotateShape[solararray, 0, π/6, π/2], {3.5, 2.6, 0}], 




          			TranslateShape[



            RotateShape[
				\ \ \ 



              AffineShape[solararray, { .7,  .9, 1}], 0, π/6, π/2], {



              1.8, \(-2.1\), 0}], 
			



          TranslateShape[



            RotateShape[
				\ \ AffineShape[solararray, { .7,  .9, 1}], 



              0, π/6, π/2], {1.8, 2.1, 0}], 
			



          TranslateShape[



            RotateShape[solararray, \(-π\)/2, π/4, 0], {0, \(-1.9\), 



              \(-3.3\)}], 
			



          TranslateShape[



            RotateShape[solararray, π/2, π/4, 0], {0, 1.9, 



              \(-3.3\)}]}]; \)\)

\(\(\ Show[mir, 



      Graphics3D[{



          Text[StyleForm["A", FontFamily -> \ "Arial", 




              			\ \ \ \ \ \ FontWeight -> "Bold", \ FontSize -> 20], {



              7, 0, 0}], 
		\ \ \ \ \ \ \ \ 



          Text[StyleForm["B", FontFamily -> \ "Arial", \ 



              FontWeight -> "Bold", 
			\ \ \ \ \ \ FontSize -> 20], {



              0, 6, 0}], \ 



          Text[StyleForm["C", FontFamily -> \ "Arial", 




              			\ \ \ \ \ \ FontWeight -> "Bold", FontSize -> 20], {



              0, 0, 4}], \ Text["Soyuz", {\(-4\), 0, 0}]}], 
\ \ \ 



      PlotRange -> {{\(-7\), 7}, {\(-7\), 7}, {\(-7\), 7}}, 
\ \ \ 



      BoxRatios -> {1, 1, 1}, Axes -> True, \ 



      AxesLabel -> {"X", "Y", \ "Z"}, 
	



      ViewPoint -> {1.864, \ \(-2.603\), \ 1.096}]\)\)

The three priciple axes A,  B,  C are shown, and the Soyuz space craft is represented as a slender cone on 


the -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 B and 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 +Z axis.  The critical solar arrays 


are mounted perpendicular to the Base Block of the Mir, which are the two 


cylinders along the 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.

Animated 3D Rotation of Mir

The Mir model is animated using the calculated interpolation functions for 


ϕ,θ,ψ .

\(nframes\  = \ 30; tstart = 0; tend = 17; 




    Do[s = i\ \((tend - tstart)\)/nframes + tstart; 
\ \ \ \ \ \ \ 



      phi\  = \ ϕ[s] /. First[soln]; 



      theta\  = \ θ[s] /. First[soln]; 



      psi\  = \ ψ[s] /. First[soln]; 
\ \ \ \ \ \ \ 



      Show[RotateShape[mir, phi, theta, psi], 




        \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 



        PlotRange -> {{\(-6\), 6}, {\(-6\), 6}, {\(-6\), 6}}, 




        \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ BoxRatios -> {1, 1, 1}, \ 



        Boxed -> False, 
	\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 



        ViewPoint -> {1.886, \ \(-2.634\), \ 0.975}]
\ , {i, 0, nframes}]



      \)

It is apparent that although the Mir, with an initial spin about the C 


axis, is able to present the plane of the solar arrays roughly perpendicular 


to the sun on the Z axis, the condition does not hold for long.  After 


roughly t ≃ 9 , or about 180 minutes, the A axis is roughly along the Z axis, toward the sun, rotating about the C axis so that the solar arrays cannot properly face the sun.  However, 


this condition is also fairly short lived, and after another 180 minutes the 


Mir is effectively  oriented opposite to the starting condition, but with the 


A  axis again roughly perpendicular to the sun, as can be the solar arrays, 


if they have rotated their active surface toward the sun.  

Analysis of 


the Euler equations, and their solutions, shows that in general, pure 


rotations about the principle axis whose moment is the middle value of the 


three moments A,  B,  C ,  (in this case C) are unstable and always result in the behaviour shown 


above.  Pure rotations initiated about axes with the lowest or highest 


moments of inertia are stable.  This suggests that an attempt to spin about 


the B axis using the Soyuz would result in a stable spin.  This is true for 


Mir, and can be demonstrated using the method above,  but unfortunately the 


solar arrays in that case cannot be oriented continuously with their planes 


perpendicular to the spin axis, and therefore toward the sun, due to the Mir 


geometry.

Although clearly not an optimum situation, the rotation 


described about the C axis was the only realizable option available.  Fortunately, it 


nonetheless provided enough electrical energy for the most essential Mir life 


support systems to be powered, including communication with the ground.  This 


permitted a stable situation in which repairs could be made, and Mir attitude 


control could eventually be recovered.

Biography

The author is presently at the National Aeronautics and Space Administration 


(NASA) Johnson Space Center, Houston, Texas and is a  NASA astronaut.  The 


author was a crew member on board space station Mir from May until October of 


1997.