The rotational dynamics of space station Mir
By Michael Foale
NASA Johnson Space Center, Houston
Updated by Paul Abbott
By Michael Foale
NASA Johnson Space Center, Houston
Updated by Paul Abbott
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
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 gyrodynes, 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
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 , , rotate from the inertial unprimed frame to the primed frame use the form =x where is a vector in inertial space. First rotate about the inertial space axis , to get to the first primed frame,
L
ϕ
L
θ
L
ψ
x
′
x
′
x
L
ϕ
x
z
L
ϕ
cos(ϕ) | sin(ϕ) | 0 |
-sin(ϕ) | cos(ϕ) | 0 |
0 | 0 | 1 |
At this point =x. Now we rotate about the primed frame axis, by ,
x
L
ϕ
y
θ
L
θ
cos(θ) | 0 | -sin(θ) |
0 | 1 | 0 |
sin(θ) | 0 | cos(θ) |
At this point = . Now rotate about the double primed frame axis about the primed frame axis (note, not , since this is the Euler system),
_
x
L
θ
x
z
x
L
ψ
cos(ψ) | sin(ψ) | 0 |
-sin(ψ) | cos(ψ) | 0 |
0 | 0 | 1 |
Finally = . Define to be the combined solid body rotation matrix :
_
_
x
L
ψ
_
x
L
ψθϕ
L=
L
ψ
L
θ
L
ϕ
L
ψθϕ
L
ψ
L
θ
L
ϕ
cos(θ)cos(ψ)cos(ϕ)-sin(ψ)sin(ϕ) | cos(θ)cos(ψ)sin(ϕ)+sin(ψ)cos(ϕ) | sin(θ)(-cos(ψ)) |
-cos(θ)sin(ψ)cos(ϕ)-cos(ψ)sin(ϕ) | cos(ψ)cos(ϕ)-cos(θ)sin(ψ)sin(ϕ) | sin(θ)sin(ψ) |
sin(θ)cos(ϕ) | sin(θ)sin(ϕ) | cos(θ) |
L
ψθϕ
True
and be its inverse such that =:
T
L
T
L
ψθϕ
T
L
ϕ
T
L
θ
T
L
ψ
T
L
ψθϕ
-1
L
ψθϕ
True
Definition of Inertia Tensor and Euler's Equations
Definition of Inertia Tensor and Euler's Equations
The angular momentum of a collection of particles with momentum for each particle α , with a position vector relative to some origin in inertial space is
J
α
P
α
()
r
J
Σ
α
α
r
α
P
where is given by =^. Expanding the summation, and using Einstein summation indices, we can define the inertia tensor that combines with the body angular momentum vector ω to give the net angular momentum .
α
P
α
P
α
m
Ω
α
r
I
ij
J
i
J
i
Σ
α
α
m
(α)
2
r
δ
i,j
r
i
r
j
ω
j
I
ij
ω
j
I
ij
Σ
α
α
m
(α)
2
r
δ
i,j
r
i
r
j
I
ij
ω
j
In an arbitrary body coordinate system, for an irregular body like the Mir, is not diagonal. However a set of coordinate axes can be found in the solid body that correspond to the eigenvectors of , given by . The eigenvalues λ of are the three principal moments of inertia about the three orthogonal eigenvectors defined as ,,. If the body has axes of symmetry, then the 's are in the same direction as the axes of symmetry. For arbitrary directions of , the equation =I implies that will not be in the same direction as . The coupling of the components ,, to each other through the principle moments of inertia is given by the Euler equations, if no external forces are acting. The general Euler equations are
I
ij
I
ij
I=λ=
ω
ω
J
I
A,B,C
ω
1
ω
2
ω
3
ω
J
J
ω
ω
J
ω
1
ω
2
ω
3
A,B,C
(C-B)+A0(A-C)+B=0(B-A)+C=0
ω
2
ω
3
ω
1
ω
1
ω
3
ω
2
ω
1
ω
2
ω
3
The ω's are evaluated in the body axes of the rotating frame. The total angular momentum is
J=++
2
(A)
ω
1
2
(B)
ω
2
2
(C)
ω
3
and
Ω
1
2
J
2
C
2
n
2
B
2
Ω
2
A
for initial values for ,, to be ,,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 ,,.
ω
1
ω
2
ω
3
Ω
1
Ω
2
L
ψ
L
θ
L
ϕ
L
ψ
L
θ
0 |
0 |
′ ϕ |
0 |
′ θ |
0 |
0 |
0 |
′ ψ |
ω 1 |
ω 2 |
ω 3 |
′ θ ′ ϕ |
′ θ ′ ϕ |
cos(θ) ′ ϕ ′ ψ |
ω 1 |
ω 2 |
ω 3 |
These three additional equations can be solved simultaneously with the Euler equations to give solutions for the six parameters ,,, ϕ, θ, ψ as a function of time.
ω
1
ω
2
ω
3
The Solution of Euler's Equations
The Solution of Euler's Equations
Dynamic Equations
Dynamic Equations
The Euler equations:
eqn1=+(-ℬ)(t)(t)0;
∂(t)
ω
1
∂t
ω
2
ω
3
eqn2=ℬ+(-)(t)(t)0;
∂(t)
ω
2
∂t
ω
1
ω
3
eqn3=+(ℬ-)(t)(t)0;
∂(t)
ω
3
∂t
ω
1
ω
2
The total angular momentum and initial conditions:
initialconditions={t0,(t),(t),(t)n};
ω
1
Ω
1
ω
2
Ω
2
ω
3
rule=Total++;
2
((t))
ω
1
2
(ℬ(t))
ω
2
2
((t))
ω
3
omega1rule=;
Ω
1
--+
2
ℬ
2
Ω
2
2
2
n
2
The Euler angles ϕ, θ, ψ using the transformation matrices ,, give the equations
L
ψ
L
θ
L
ϕ
L
ψ
L
θ
0 |
0 |
′ ϕ |
0 |
′ θ |
0 |
0 |
0 |
′ ψ |
ω 1 |
ω 2 |
ω 3 |
′ θ ′ ϕ |
′ θ ′ ϕ |
cos(θ) ′ ϕ ′ ψ |
ω 1 |
ω 2 |
ω 3 |
Explicitly:
eqn4=(t)sin(ψ(t))-sin(θ(t))cos(ψ(t))(t)(t);eqn5=(t)cos(ψ(t))+sin(θ(t))sin(ψ(t))(t)(t);eqn6=cos(θ(t))(t)+(t)(t);
′
θ
′
ϕ
ω
1
′
θ
′
ϕ
ω
2
′
ϕ
′
ψ
ω
3
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 -axis.
z
The Euler angles ,, at are given by the transformation:
ψ
0
θ
0
ϕ
0
t=0
L
ψθϕ
0 |
0 |
|
Ω 1 |
ℬ Ω 2 |
n |
-sin(θ)cos(ψ) |
sin(θ)sin(ψ) |
cos(θ) |
Ω 1 |
Ω 2 |
n |
Solve 2 equations for 2 initial angles, the angle ϕ being indeterminate:
Solve[{sin()sin()ℬ,cos()n},{,}]/.initialconditions
θ
0
ψ
0
Ω
2
θ
0
θ
0
ψ
0
,,,,,,,
θ
0
ψ
0
θ
0
ψ
0
θ
0
ψ
0
θ
0
ψ
0
Simplify these solutions:
Initial Conditions
Initial Conditions
Define the total time of integration:
List our Mir physical properties as a single list:
Numerical Solutions
Numerical Solutions
Three Dimensional Graphical Depiction of Mir Rotation
Three Dimensional Graphical Depiction of Mir Rotation
A Graphical Model of the Mir
A Graphical Model of the Mir
The Mir model includes a depiction of the solar arrays, critical in their arrangement toward the sun.
Transform and rescale the model so that the center of mass of the Mir is roughly located at the central sphere, representing the node:
Animated 3D Rotation of Mir
Animated 3D Rotation of Mir
Biography
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.