Falling Spacecraft Trim Problem
Ron Graham
Submitted to Lockheed Martin Astronautics 01.2000

The characteristics of hazardous material release from an attached Radioisotope Thermoelectric Generator (RTG) impact with the Earth following a launch vehicle failure are dependent on impact velocity and orientation. These in turn are dependent on the initial conditions of the falling spacecraft at the time of failure, such as translation and rotation rates and spacecraft condition: Current source term models consider the answers to these questions as initial conditions only, and treat them as having assigned probabilities. In order for one to assess the actual likelihood of an impact at a given velocity and orientation, on a given surface and with the spacecraft in a given condition, a dynamic model is needed -- one that takes into consideration gravitational and aerodynamic effects on the falling body.

This report describes such a model. Here we consider a falling spacecraft (or stack) in two dimensions, with simulation results; and describe what steps are needed to extend the model to three dimensions.

Assumptions
Model Development

Gravitational acceleration has an almost negligible effect on the rotational motion of the body over a relatively short fall. Aerodynamic forces, on the other hand, affect rotational motion, as well as translational motion in both the X- and Y-directions. The extent of that effect depends on how much each face of the stack is "to the wind," and on whether the stack center of gravity (CG) is above or below each face's center of pressure (CP).

Trim Problem Parameters
Parameter Definition
vX stack velocity in direction of gravity
vY lateral stack velocity
g gravitational acceleration
Cp aerodynamic coefficient
L axial distance of CP from CG
I stack mass moment of inertia
M stack mass
AX projected area of stack normal to its axis
AY projected area of stack parallel to its axis
ρ atmospheric density
θ rotation of stack relative to an unmoving (inertial) coordinate system

The force acting on the body in the X-direction is as follows:

fX = M g - (ρ Cp / 2) (AX |cos θ| + AY |sin θ|) vX |vX|

And the force in the Y-direction is

fX = -(ρ Cp / 2) (AX |sin θ| + AY |cos θ|) vY |vY|

The aerodynamic drag forces take into account the fact that the same projected areas are acted on by air resistance regardless of the sign of the orientation. The moment about the CG is as follows:

τZ = (ρ Cp / 2) (AY vY cos θ |vY cos θ| + AX vX sin θ |vX sin θ|)

The equations of motion can be developed in these inertial coordinates as follows: for X-direction translation,

d/dt (vX) = fX / M

For Y-direction translation,

d/dt (vY) = fY / M

And for rotation of the stack about its CG,

d2/dt2 (θ) = τZ / I

The following are "generic" properties used for this set of equations:

Trim Problem Mass Properties
Parameter Value
Cp 0.6
L 0.3 m
I 2400 kg-m2
M 2900 kg
AX 180 m2
AY 480 m2

A Mathematica simulation was run for the following initial conditions: The simulation shows that convergence of downward velocity to terminal will take longer than the 75 seconds given - though a falling stack is highly unlikely to take that long to reach the surface. The stack tumbles for a cycle or two then wobbles for the remainder of the fall.

The simulation was also successfully run for rotational velocities of 4 and 7 rad/sec.

Extending To Three Dimensions

The problem is more complex in three dimensions primarily because of three-axis rotation of the stack. Depending on how the problem is set up, it's possible to have a singularity problem with rotational motion if the stack is tumbling. Such singularities are overcome with the use of quaternions, or an Eulerian development is appropriate if you have some idea in advance that at least one axis of the spacecraft will not tumble.

The following is an Eulerian development for three-axis rotational motion:

d/dt (ω)  =  Z d/dt (θ)  +  rot (z, θ) X d/dt (φ)  +  rot (z, θ) rot (x, φ) Y d/dt (ψ)

...where (φ, ψ, θ) are the Euler angles about the local X (roll), Y (yaw), and Z (pitch) axes, respectively. The order of rotation in converting from inertial to body axes is user-selectable, but when the above equation is inverted, the term sec φ appears -- this term will be singular if the roll angle goes through plus or minus π/2. We therefore must make the assumption that the stack does not tumble in roll in order for this development to be valid.

Boldface quantities are matrices and vectors.

The forces acting on the body are no longer decoupled, because we can't assume zero products of inertia in the matrix I.

Inverting the above form yields the following result:

d/dt (φ) = ωX cos θ + ωY sin θ

d/dt (ψ) = -ωX sin θ sec φ + ωY cos θ sec φ

d/dt (θ) = ωZ + ωX sin θ tan φ - ωY cos θ tan φ

From the Euler angles we construct the direction cosine matrix (DCM), defined for this order of rotation as

rot (z, θ) rot (x, φ) rot (y, ψ)

The direction cosine matrix is used to convert gravitational acceleration from inertial into body coordinates. In the two-dimensional development it was convenient to represent velocity in inertial coordinates -- but it's much more convenient to work in a body coordinate system for the three-dimensional case.

The equations of motion for the stack in three dimensions are as follows:

[fX, fY, fZ]T  =  M d/dt [vX, vY, vZ]T  -  M [DCM] [g 0 0]T  =  [LX, LY, LZ]T  X  [DCM] [(ρ / 2) CP vX |vX| 0 0]T

where [fX, fY, fZ]T  is the vector of forces acting on the vehicle due to drag; and

[τX, τY, τZ]T  =   I d/dt [ωX, ωY, ωZ]T  +  [ωX, ωY, ωZ]T  X  I [ωX, ωY, ωZ]T

where [τX, τY, τZ]T is the vector of moments acting on the vehicle due to drag.

References

"Final Safety Analysis Report for the Ulysses Mission: Volume II." GE Astro Space document 90SDS4203, March 1990.
"Cassini GPHS-RTG FSAR." Lockheed Martin, June 1997.
Graham, R. "ACTS Roll/Yaw Dynamics Math Model." NASA Lewis Research Center, Engineering Directorate internal report, 1988.
Kaplan, M. Modern Spacecraft Dynamics and Control. New York: John Wiley and Sons, 1976.
Wolfram, S. The Mathematica Book, 4th ed. New York: Cambridge Univ. Press, 1999.

[Engineer's Companion home] [Back to Ron's page]