Other papers
AAS 04-100
In designing solar sail structures, structural engineers must
calculate structural loading and resulting deflections. Byproducts
of this analysis are the integrated external solar pressure forces
and moments on the mainsail and vanes, as well as mass properties.
This paper proposes a standard format and set of coordinate systems
for reporting forces, moments, and mass properties, as well as
other items relevant to performance, including packaging efficiencies
and life. Sail designers will report integrated external solar
pressure forces and moments on the mainsail and on the vanes (if
used), as well as mass properties. Forces and moments will be
non-dimensionalized into coefficients. These are tables of point
values, the independent variables being attitude relative to the
sun and solar distance. The user will interpolate between the
point values supplied by the sail designer. Separate solar propulsion
performance tables and mass properties will be reported for the
mainsail alone, and for various detached single vanes (if used).
Sail designers using gimbaled control mass will provide various
mast length designs. This will allow the user to mix and match
mainsails with control devices of various sizes. Combined forces,
moments, and mass properties for the assembled sailcraft will
be calculated and transformed by the user. Users can then insert
attitude or vane commands and propagate the resulting solar sail
trajectories and/or attitudes.
INTRODUCTION
First, reference coordinate systems are defined, followed
by discussions of factors which affect performance and how they
are to be represented, then coefficient definition. Finally, reporting
of life and packageability are discussed.
1 L’Garde, Inc., Tustin, CA
2 Jet Propulsion Laboratory, California Institute of
Technology, Pasadena, CA 91107
3 University of Michigan, Ann Arbor, MI
Copyright (c) 2004 by The American Astronomical Society
REFERENCE COORDINATE SYSTEMS
Sailcraft Body-Fixed Coordinate System “Sail”
As mass properties are reported in body-fixed systems, forces
and moments will be as well. The body-fixed coordinate system
“Sail” is illustrated below. The analogy is an aircraft
flying over the surface of the sun. The aircraft’s belly
is the reflective side. The origin is to be located by the sail
designer at some well-defined point in the geometric sail center.
Specification of Attitude Relative to the Sun
Two angles are to be used to identify the attitude of the Sail
system relative to the sun, “Sun Incidence” and “Flatspin”.
“Sun Incidence” is the angle between the sail normal
(Zsail) and the sail-sun line. This angle will have the greatest
effect on sail forces and moments.
“Flatspin” is rotation about the sail normal, Zsail
(note: NOT generally about the sail-sun line). For an idealized
flat plate sail, forces and moments would not change with Flatspin.
However, for a real sail, forces and moments will change somewhat
due to sail billow, particularly asymmetric sail billow, as well
as variations in relative beam bends.
“Top” is rotation about the sail-sun line. Beginning
with Zsail coincident with the sail-sun line, the full Euler sequence
is Top -> Sun Incidence -> Flatspin:
1. Top about Zsail, which is coincident with the sail-sun line
at this point
2. Sun Incidence about Ysail
3. Flatspin about Zsail, which is no longer coincident with the
sail-sun line
This is therefore a Yaw -> Pitch -> Yaw sequence.
Whatever the Sun Incidence and Flatspin angles are, the sail may
be rotated arbitrarily about the sail-sun line without changing
forces or moments, as expressed in the body-fixed Sail coordinate
system. Top is therefore not included in the reporting of forces
and moments.
Sun Incidence and Flatspin are believed to represent the propulsion
of a square sail in a revealing and sensible way. Propulsion is
a strong function of Sun Incidence, a weak function of Flatspin.
Sail billow asymmetries are discriminated by Flatspin.
While performance is reported vs Sun Incidence and Flatspin, the
user may prefer a different attitude description for interpolation
due to singularity problems or other reasons. The user will convert
Sun Incidence and Flatspin as necessary, but may require the sail
designer to report at specific intervals of Sun Incidence and
Flatspin in order not to oversample.
Note that Sun Incidence and Top are referenced to the body-fixed
coordinate system Sail. They differ from McInnes’s “Cone”
and “Clock” (Ref. 1), which are referenced to the net
solar force vector.
Example Placement in an Inertial Frame
Locating the sun-sail line in an inertial frame and specifying
a reference for Top is not necessary for sailcraft performance
specification, but an example is given for illustration. A suggested
inertial frame is the J2000 frame. The attitude of the sail-sun
line in this frame may typically be specified using “Right
Ascension” and “Declination.” In order to establish
a reference and sense for Top and to facilitate transformations,
a new coordinate system is introduced, the “Sun” system.
This system moves with the sail, its origin coincident with the
“Sail” system origin. Zsun always points toward the
J2000 origin, along the sail-sun line. Xsun is defined as parallel
to the J2000 x-y plane, and Ysun completes the orthogonal set.
The orientation of the Sun system relative to J2000 is defined
by the following Euler sequence:
“Flip” -90° about x -> -(“Right Ascension”
+ 90°) about y -> -“Declination” about x
The transformation matrix for a vector [V] between J2000 and Sun
systems is:
[V]Sun = [R]Su-J [V]J2000
RA = Right Ascension
D = Declination
R11 = cos(-RA-90°) |
R12 = -sin(-RA-90°) |
R13 = 0 |
c = cos()
s = sin()
TP = Top
SI = Sun Incidence
FS = Flatspin
R11 = cTPcSIcFS -sTPsFS |
R12 = sTPcSIcFS +cTPsFS |
R13 = -sSIcFS |
One may also determine the attitude of the vehicle relative
to the instantaneous velocity vector. Figure 7 illustrates the
Orbit Position Reference Frame, where the x-axis is parallel to
the sun-to-sailcraft line, the z-axis is parallel to the instantaneous
orbital angular velocity, and the y-axis completes the orthogonal
set. The attitude of both this system and the sail-fixed system
relative to the inertial system are known, so the attitude of
the sail relative to the Orbit Position Reference Frame is a simple
subtraction.
Specification of Beam Tips Relative to the “Sail”
System
If vanes are used, a position vector to each mainsail beam
tip will be reported, as well as the attitude of each mainsail
beam tip relative to the sailcraft body-fixed Sail system. The
Euler sequence for rotation of a Beam Tip system relative to the
Sail system is:
“Index” about z:
0° for the fore beam tip
90° for the starboard beam tip
180° for the aft beam tip
270° for the port beam tip
then:
“Bend” about y -> “Sway” about z ->
“Twist” about x
Positive sense is determined by the right-hand rule. The transformation
matrix for a vector [V] between Sail and Beam Tip systems is:
[V]Tip = [R]T-S [V]Sail
c = cos() I = Index S = Sway
s = sin() B = Bend T = Twist
R11 = cIcBcS -sIsS |
R12 = sIcBcS +cIsS |
R13 = -sBcS |
Specification of Vane Rotations Relative to the Beam Tip
If vanes are used, two types of 2-axis vane yoke designs are
considered. They are identified by the Vane Euler rotation sequence
relative to Beam Tip as either:
“Twirl” about x -> “Cant” about y
or,
“Cant” about y -> “Twirl” about x
Positive sense is determined by the right-hand rule.
It is assumed that yoke design will be such that there is no
significant offset between the Twirl and Cant axes. The Beam Tip
system origin is coincident with the Vane system origin. The mass
of the yoke and actuators is included in the mainsail.
The transformation matrix for a vector [V] between Beam Tip and
Vane systems is:
[V]Vane = [R]V-T [V]Tip
for Twirl about x -> Cant about y:
R11 = cos(Cant) |
R21 = 0 |
R31 = sin(Cant) |
R11 = cos(Cant) |
R12 = 0 |
R13 = -sin(Cant) |
Vane performance will be reported separately from mainsail
performance. Vane tables will actually specify Vane Sun Incidence
and Flatspin angles relative to the sun, which the user must calculate
from vehicle attitude, beam tip attitude, and user-desired Vane
Twirl & Cant. VaneSunIncidence is the angle between the vane
normal and the sail-sun line. The equation for VaneSunIncidence
(VSI) is developed using the dot product of a unit vector Zsun
along the sail-sun line and a unit vector Zvane:
Zsun • Zvane = cos(VSI) |Zsun| |Zvane|
VSI = acos(Zsunvanez)
The dot product will always return a [positive] included angle.
The returned angle will further be limited to the luff limit of
the vane (Zsunvanez will be positive). The lack of
negative values of VSI is OK as long as VSI and Vane Flatspin
(VFS) are not used to derive attitude of the vane relative to
the Beam Tip, only for coefficient lookup.
To express Zsun in the Vane coordinate system, transform from
Sun to Sail to Tip to Vane. The appropriate matrix must be chosen
for Twirl -> Cant vs. Cant -> Twirl:
[Zsun]vane = [R]V-T [R]T-S
[R]S-Su [0,0,1]
VSI = acos({[R]V-T [R]T-S [R]S-Su
[0,0,1]}z)
In order to find VaneFlatspin (VFS), take the dot product between
the projection of Zsun onto the Vane x-y plane and a unit vector
along the negative Vane X-axis (the negative axis because VSI
as calculated above will always be positive):
Zsunvane x-y • -Xvane = cos(VFS) |Zsunvane
x-y| |-Xvane|
VFS = SIGN(Zsunvaney) acos{-Zsunvanex ÷
(Zsunvanex2 + Zsunvaney2)}
[Zsun]vane = [R]V-T [R]T-S [R]S-Su
[0,0,1]
SIGN(Zsunvaney) = Zsunvaney ÷ |Zsunvaney|
The assignment of sense is necessary because the dot product
will only return an angle between 0° and +180°. Note that
this assignment again relies on VSI being positive, as calculated
above.
As mentioned earlier, if VSI 0° (the vane is ~normal to
the vane-sun line), the user should skip these equations and return
“N/A” (or equivalent) for VFS. Otherwise, the dot product
may have an x/0 error.
Specification of Gimbaled Mass Relative to the “Sail”
System
If a gimbaled mass such as an antenna is used, the mass of
the gimbal and actuators will be included in the mainsail. The
gimbaled mass itself will be reported separately from the mainsail
mass, and will be assumed constant (no significant expendables).
Several such gimbaled masses may be included, as with an instrument
boom pointing toward the sun, together with an antenna pointing
toward the ground.
There are two categories of gimbaled masses, those which can be
pointed for control purposes, and those which cannot. The payload
is generally divided into sail-fixed mass and control-gimbaled
mass, both of which can be modified by the user.
The position vector from the Sail system origin to the gimbal
will be reported in the Sail system. The position of the gimbaled
mass center, as well as its moments of inertia, will be reported
in a gimbaled mass-fixed frame. This frame will have its origin
at the gimbal, its Z-axis pointing generally along positive or
negative Zsail (to be indicated by the sail designer). The position
of such a frame relative to Sail will be described using the convention
of “Azimuth” and “Elevation,” with “Phase”
describing any final rotation about the Zaxis of the gimbaled
mass. The positive sense and reference for these angles will be
described by the sail designer.
FACTORS AFFECTING
PERFORMANCE
Shape Effects Due to Solar Distance and Centralized Mass
The closer the sail is to the sun, the more the propulsive load
transfer to the centralized payload and bus, therefore the more
the beams bend. This changes sail shape somewhat, so solar distance
must be specified in the tables of coefficients. Also, closer
distance to the sun will amplify the effects of solar disc diameter
and limb darkening.
Sail designs will differ for different minimum design solar approaches.
The minimum solar loading (maximum solar distance) a sail can
be structurally designed to handle is of course at 1.02 AU (Earth
Aphelion), where the sail is deployed and has to bear load. Sails
designed to fly to Mercury at ~0.3 AU, for example, will need
stronger (and therefore heavier) structures in order to preserve
structural safety factor, as well as more emissive backside sail
coatings. These will affect propulsion and deflected shape across
all solar distances. Minimum solar approach is a design variable,
like sail area or assumed packaging volume available, as opposed
to one which may vary over a trajectory, or just using larger
vanes or a heavier payload with the same sail, thus will be presented
as an entirely different set of tables. Structural safety factor
at the design minimum solar approach will also be reported for
fair comparisons.
Payload mass will also affect sail shape and therefore forces
and moments. This is because the heavier the centralized mass,
the less distributed the load will be and the more “cantilevered”
the beams will be, thus deflecting more and altering sail shape.
A user may desire to increase or decrease payload mass, therefore
the sail designer must report the payload mass assumed in structural
shape calculations, along with the accuracy that a user can expect
by varying the payload mass a certain percentage. The mainsail
mass properties will not include any payload mass. The user will
select his own payload mass based on mission needs, and calculate
the combined mass properties of the sailcraft himself. A user
may request a sail designer recalculate shape and report new coefficients
using his specific payload mass, in order to improve accuracy.
The position vector to the payload center of mass will be given,
and will be assumed constant with user variations in payload mass.
If part or all of the payload is control-gimbaled, the gimbaled
portion will be described separately, in the gimbaled mass frame.
In the current format, it is assumed that from a mass standpoint
no significant expendables or ballast are used, and all parasitic
masses that could be jettisoned have already been released. All
masses are constant.
Effect of Vanes or Gimbaled Masses on the Mainsail Shape
Vane solar, inertial, gravitational, and dynamic actuation
loads affect mainsail beam tip deflections, therefore mainsail
shape and propulsion. Gimbaled masses also apply forces and moments
to the mainsail, and their actuators apply torques.
All sails with the center of mass forward of the center of pressure
experience some passive stability about the two in-plane axes
due to the solar “drag” force of an imperfectly reflecting
sail, and sail billow imparts more stability due to the effective
shuttlecock angle. Vanes may also be canted anti-sunward for added
strong passive stabilization, and vane neutral angles can be reset
to stabilize about an axis not in the sail plane. Vane or control
mass angles at which there is no net moment about a certain axis
on the vehicle for a given vehicle attitude are called “trim”
settings for that axis. Note that net vane or control mass forces
on the mainsail will still exist with the vehicle trimmed. Internal
moments may also exist, affecting shape. Various combinations
of vane angles can achieve trim, with different internal moments.
The sail designer will report separate solar propulsion performance
tables for the mainsail alone, and for various detached single
vanes (if used). Sail designers using gimbaled control mass will
provide various mast length designs. Mainsail, vane, and gimbal
mass properties are reported separately. A certain vane or gimbaled
control mass will be assumed when the sail designer calculates
mainsail shape (therefore propulsion), but the solar, inertial,
and gravitational forces on the vanes or control masses themselves
will not be included in reported mainsail propulsion. The vane
or control-gimbaled payload mass assumed in mainsail shape calculation
will be reported, along with the error associated with using a
vane a certain percentage larger or a control-gimbaled payload
mass a certain percentage heavier or a longer control mast. This
will allow the user to mix and match mainsails with control devices
of various sizes. Combined forces, moments, and mass properties
for the assembled sailcraft will be calculated and transformed
by the user. Once a user has determined an appropriate vane or
control mass size for his particular GN&C design and mission,
he may ask the sail designer to recalculate mainsail shape and
report new coefficients using these exact control sizes, in order
to improve accuracy. This iteration may reoccur several times.
In calculating mainsail shape, the sail designer will not include
any torques due to actuation, nor any actuator rates. The max
allowable reaction torque will be supplied. Also, when a sail
designer calculates sail shape at an off-trim attitude, he will
assume zero inertial rotation rates. The sail will be undergoing
rotational acceleration due to the net moment, but at zero angular
velocity, steady state. Note that the sail is always undergoing
translational accelerations, due to the imbalance and angles between
the solar gravity and solar pressure. The coefficients will thus
be static; transient dynamic response of the [flexible] sail will
not be represented. If a user wishes to investigate a spinning
sail, he may ask the sail designer to recalculate sail shape assuming
a constant nonzero rotation rate about some axis, and report a
new set of coefficients.
It would be impractical and unnecessary for a sail designer to
calculate mainsail shape at all possible combinations of vehicle
attitude and vane or control mass angles. As a minimum, sail designers
will determine mainsail shape at each attitude with the sail trimmed,
and report the vane or gimbaled mass angles used to achieve trim
at each sail attitude. The assumed yoke design (Twirl->Cant
vs Cant->Twirl) for a vane must also be given. Users may later
request accurate coefficients with the vanes at different angles
that still achieve trim. An example would be if a GN&C designer
desires greater passive stability, he would ask for mainsail coefficients
to include the shape effect of all 4 vanes canted further anti-sunward.
Users may also request accurate coefficients in an off-trim condition,
such as after an attitude perturbation that drove the vehicle
off-trim, or with vanes or control mass rotated to effect a maneuver.
Factors Affecting Vane Shape
The vane will bend and billow as a structure as well, so major
parameters affecting its shape must be reported. Solar distance
will affect loading and bend, and is therefore reported. Also,
just as the mainsail shape is affected by heavier centralized
masses, so the vane is affected by what mainsail it is attached
to. Something more than mass is required, however, as the sail
is propulsive. The lightness factor of the sailcraft assumed in
the vane structural analysis will be reported in the vane performance
table.
Momentum-Producing Biases
Sail shape will vary from ideal due to manufacturing errors, deployment
effects, and other factors. These effects, along with asymmetric
shape effects already represented in the coefficients, will give
rise to bias moments on the sailcraft. These will generally be
a function of solar distance and attitude, as they ultimately
result from solar pressure. They need to be trimmed out, or the
resulting momentum dumped. It is not possible to passively trim
Top disturbances, as there is no inertial reference about the
sun-sail line. Active stabilization must be used. It is therefore
important for the sail designer to determine how much bias moment
might be developed, especially about the sail-sun line.
Two approaches are taken. The sail designer will supply estimates
of estimated average Top bias moment vs solar distance. He will
also postulate defects in the sail due to manufacturing, deployment,
and the like, and generate entirely new tables of coefficients
for the corrupted sailcraft.
Reflectivity Degradation
There is some belief that the reflectivity of certain aluminized
sail materials may degrade with extended exposure. Two approaches
are again taken. The sail designer will provide estimates or data
(if available) on the total reflectivity of his sail material
vs. accumulated dose. The user will apply this degradation to
zero-dose coefficients in a bulk manner. The sail designer will
also supply a set of tables, one for each total dose, with the
correct reflectivity used in coefficient calculation.
Rigid Body Damping
It is believed that structural damping will result in some rigid
body damping of the sailcraft. Sail designers will supply estimates
of rigid body damping coefficients.
Attitude & Angle Limitations and Fault Tolerance
Sails will generally have Sun Incidence limits for safe operation.
One practical limit is that at which the max camber line of
an aft sail quadrant becomes shadowed, the “luff limit,”
which will be reported. Tighter limits may be specified by the
sail designer as well for other reasons.
Vanes and gimbals will generally have practical limits, as well
as mechanical stops put in place to prevent oversteering. These
will be reported, along with any “watchdogs” implemented
for safe operation, such as hardwire logic observing vane angles
and vehicle attitudes and rates. Sail designers should also postulate
and report credible faults affecting attitude control.
NON-DIMENSIONALIZED COEFFICIENTS OF FORCE
AND MOMENT
A force coefficient is non-dimensionalized as follows:
force coefficient “Cf” = F/[PA]
= force / [solar pressure at distance used in calculation X nominal
projected area]
solar pressure = insolation (W/m2) / speed of light
(m/s)
“nominal” projected sail area is held as a constant
The idea is that these coefficients could be applied within
a range about the solar distance used to come up with the beam
bend, without affecting results much. Also, different people use
different values for solar insolation, so the format tries to
take out sensitivity to this.
Mainsail moments are given about the “Sail” system origin;
vane moments are about the “Vane” system origin; a gimbaled
mass moment about its system. A moment coefficient is non-dimensionalized
as follows:
moment coefficient “Cm” = M/[PAL]
= moment / [solar pressure X nominal area X reference length]
The reference length will be the square root of the nominal projected
area
The tables will also include a position vector to the mainsail,
vane, and gimbaled mass centers of mass, as well as the moments
of inertia about the “Sail” origin for the mainsail;
about the “Vane” origin for the vane, about a gimbaled
mass’s own system. Payload centers of mass are described
separately.
Given accurate points, the user will develop means to interpolate
between. An example of what this might look like can be demonstrated
using a polar plot of 1 AU flatsail thrust vs. thrust “cone”
(Ref. 1) angle. Polar plots are commonly used to specify performance
of racing sailboats, whose speed vary with course.
Estimated Error
Estimates of the error between reported coefficient values
and what will actually be experienced in space will be supplied
by the sail designer. This will allow a user to design and utilize
smartly. For example, a GN&C designer will be able to decide
the most reliable, perhaps not the highest performing, scheme
for thrust vectoring.
The designer will also identify the analysis models used to generate
the tables, under “modeling accuracy.” For example,
early tables may assume the sun is a point source, or they may
assume a sail billow which is independent of attitude or solar
distance. Any data relied on from ground test or in-space calibration
should be identified in this section as well.
LIFE REPORTING
A user must know how long he can operate the sail. Sail designers
will report factors that limit life. One of these is the use of
expendables, such as to dump momentum, or the use of primary batteries.
If so used, the approximate average expenditure rate must be reported,
as a function of whatever it may be strongly dependent on, such
as solar distance. Note that expendable mass is assumed insignificant
and not accounted for from a mass standpoint, but its effect on
life is modeled.
Beam and sail materials can also limit life. Material life is
generally a function of accumulated particulate and UV doses,
which the user will calculate for the desired trajectory. Maximum
accumulated doses before failure vs AU will be given in the tables.
Any information known about combined effects should also be given.
Sailcraft avionics contractors will supply estimates of component
life.
LAUNCH PROPULSION REQUIRED (PACKAGING PERFORMANCE)
The cost of an Ariane µASAP launch is currently $3M
- $4M, perhaps another $1M for an Earth escape kick motor. Another
commonly proposed launcher for solar sails is the Delta II, at
least an order of magnitude higher in cost at $40M - $50M. Discussion
of the propulsive performance of a sail cannot go without an evaluation
of the propulsion needed to launch it, which comes down to packaging
volume, as gossamer structures will generally be volume-limited,
rather than mass-limited for launch.
This will only come into play in the trajectory as initial conditions
from the launcher, but packaging performance should always be
reported with any performance numbers for fair comparisons between
sails, as well as to clearly identify a specific sail design.
Packaging accommodations do actually affect the sailcraft design
itself. Other cost drivers will be reported separately to interested
program managers.
The driver in sailcraft packaging generally turns out to be the
length and cross-section area of the packaged beams. A large deployed
cross-section is necessary for structural stiffness, but can consume
much of the packaging volume if not collapsed somehow.
The following packaging performance parameters will be reported:
Assumed Launch Slot and Lift Capability
Earth Escape Kick Motor (if any), and Kick Capability
Payload & Bus Packaging and Mass Limitations
Sailcraft Packageability (Deployed Sail Area ÷ Stowed Volume
of Sail + Beams)
Packaged Volumetric Density of Sail + Beams
Beam Length Contraction Ratio (= deployed length ÷ packaged
length)
Packaged Beam Volume % of Deployed (= packaged beam volume ÷
deployed)
Beam Effective Cross-Section Area Ratio (= [1/Volume Percent]
÷ Length Ratio)
Launch vehicle utilization, sailcraft packageability, and sailcraft
mass performance are related as:
Volumetric Density (g/cc) X 1,000,000 = Packageability (m2/m3)
X Areal Density (g/m2)
CONCLUSION
We hope that this specification allows accurate and sensible
representation of the predicted performance of gossamer solar
sail structures. This is intended to be an open specification,
for which this is the first version. Please feel free to contact
the authors with suggested changes or questions for clarification.
REFERENCES
1) McInnes, C.R., Solar Sailing Technology, Dynamics and
Mission Applications,
Springer-Praxis, London, UK, 1st Ed.