Dynamics of the Caltech SSPP deployable structures:
structure–mechanism interaction and deployment envelope
Narravula Harshavardhan Reddy
∗
and Sergio Pellegrino
†
California Institute of Technology, Pasadena, CA, 91125
The Caltech Space Solar Power Project has been developing ultralight deployable space
structures consisting of thin-shell composite strips that support photovoltaic and RF elements.
These modular, square structures can potentially be scaled to tens of meters in size. This paper
studies the interaction between the deployment dynamics of the structure and the deployment
mechanism, both experimentally and numerically. Instead of considering a full structure, a
quadrant is considered to reduce the number of components and to better focus on the main
parameters that affect the deployment behavior. Outcomes of this research will not only benefit
the Caltech project but will also contribute to the design of future lightweight deployable space
structures that undergo unconstrained dynamic deployment.
Nomenclature
A
= in-plane stiffness matrix
D
= out-of-plane stiffness matrix
푑
= radial position of CG of cylinder with respect to hinge axis
퐹
= deploying force on the composite structure
퐹
푛
1
,퐹
푛
2
= total radial forces on pulleys
푔
= acceleration due to gravity
퐻
= height of the center of strip from the position of the cords
ℎ
= vertical displacement of weights
퐼
= rotation moment of inertia of pulley
퐼
ℎ
= rotation moment of inertia of deploying cylinder about hinge axis
M
= out-of-plane moments per length
푀
= mass of deploying cylinder
푀
1
,푀
2
= masses connected by cord over two pulleys
푀
푓
1
,푀
푓
2
= friction moments in pulleys
N
= in-plane forces per length
푅
= radius of pulley
푟
= inner radius of ball bearing
푇
1
,푇
2
,푇
= tensions in cord over two pulleys
푇
푒
= moment due to elastic torsion spring
푇
푓
= opposing moment due to friction in hinge
푥, 푦, 푧
= Cartesian coordinates
훽
= rotation of pulleys
휖
= mid-plane strain
휅
= mid-plane curvature
휇
1
,휇
2
= coefficients of friction relating moments and total radial forces
휃
= angle of rotation of deploying cylinder
휃
0
= angular position of CG of cylinder with respect to hinge axis
∗
Graduate Student, Graduate Aerospace Laboratories, MC 105-50. AIAA Student Member
†
Joyce and Kent Kresa Professor of Aerospace and Civil Engineering; Jet Propulsion Laboratory Senior Research Scientist; Co-Director,
Space-Based Solar Power Project, Graduate Aerospace Laboratories, MC 105-50. AIAA Fellow. E-mail: sergiop@caltech.edu
1
I. Introduction
U
nconstrained dynamic deployment
is driven by the release of the strain energy stored in a folded or coiled
structure. While the structure is simply released and allowed to deploy on its own, the deployment can be guided
by pulling the structure outward at specific points. Unconstrained deployment is less common in existing deployable
structures, however, it can potentially lead to simpler and lighter mechanical components, and thus allows for lighter and
more efficient deployable spacecraft. Studies of the fundamentals of uncontrolled deployment are relatively rare and in
this paper we attempt to fill this gap by studying the deployment of the ultralight deployable space structures developed
by the Caltech Space Solar Power Project (SSPP).
Figure 1 shows schematically the modular structural architecture adopted by the Caltech Space Solar Power Project.
It consists of bending-stiff trapezoidal strips bounded by foldable longerons with a TRAC cross-section [
1
] that are
connected by battens. The strips form structurally independent modules and are attached to diagonal cords suspended
between the tips of four diagonal booms and a central mechanism. Of the four identical quadrants in the full architecture,
the present study focuses on a single quadrant.
Fig. 1 Full architecture of the Caltech SSPP.
In particular, we are interested in studying how the motion of the structure during deployment is affected by the
presence of the deployment mechanism, and by the rate at which the outer ends of the cords are pulled outwards during
deployment. Metrics of interest are the time to fully deploy and the transverse motion of the structure during deployment,
which is linked to its deployment envelope. This latter metric is important as it indicates whether the deploying structure
clears any obstacles for a successful deployment that avoids damage or snagging against other components. Hence, in
this study, we use the extreme heights (normal to the plane of the fully deployed structure) reached by the structure
during deployment as a key metric.
The composite structure deploys dynamically due to its own stored strain energy. The deployment mechanism is
designed to hold the structure in its folded configuration, to avoid premature deployment. Structural stability after the
structure is fully deployed is provided by the diagonal cords, which are held under tension by the diagonal booms.
Various components (such as the cords, deploying cylinder, pulleys) of the mechanism have an influence on the
deployment dynamics. Two major factors affecting the deployment dynamics are as follows.
1)
The deploying cylinder assists in maintaining a safe radius of curvature in the elastic folds and keeps the structure
from deploying in stowed configuration. Once released, the cylinder rotates about the hinge axis and remains in
contact with the structure for a significant duration thus imposing kinematic constraints on the structure.
2)
The suspension cords carry the deploying force and dictate the speed of deployment. Since the three strips are
assembled using the cords, the cords also impose kinematic constraints on the structure.
In this study, we investigate these two factors using experiments and finite element simulations.
II. Experimental setup
The only way to truly verify the fidelity of numerical simulations is to compare them with experiments. The
composite structure of interest and the deployment mechanism that was designed to study its deployment are described
below.
2
A. Structure
Fig. 2 Geometry of structure.
The composite structure used in this study comprises three strips (Figure 2). Each strip consists of two
bending-stiff longerons of TRAC cross-section ([
1
,
2
]) and have the composite layup shown in Figure 3. The
composite layup in the flanges of a longeron is
[±
45
퐺퐹푃푊
/
0
퐶퐹
/±
45
퐺퐹푃푊
]
, and that in the web region is
[±
45
퐺퐹푃푊
/
0
퐶퐹
/±
45
3
,퐺퐹푃푊
/
0
퐶퐹
/±
45
퐺퐹푃푊
]
, where CF represents a thin ply with unidirectional Pyrofil MR
70 12P Carbon fibers made by Mitsubishi Chemical (30 gsm). GFPW represents plain weave scrim glass (25 gsm). Both
plies are impregnated with North Thin Ply Technology’s ThinPreg 415 resin. The longerons are fabricated in-house
using the techniques laid out in [
3
]. A strip is made torsion-stiff by connecting its two longerons at specific locations by
carbon fiber rods called battens. The battens, made of carbon fibers, have a rectangular cross-section with a width of 3
mm and thickness 0.6 mm. Placement of the battens in a strip and the placement of strips in relation to one another was
adopted from [5]. The battens are bonded to the web regions of the longerons with epoxy.
The three strips are connected via cords. To support a strip with the cord, two 3D printed sleeves called strip-cord
connectors are rigidly bonded to the leftmost and rightmost battens (Figure 2). The cord is bonded, using epoxy, to the
outermost strip-cord connectors of every strip. These connectors are labeled as C12, C14, C16, C22, C24, and C26 in
Figure 6.
105
o
105
o
13.6 mm
13.6 mm
8 mm
web
flange
flange
fl
ange
fl
ange
web
0
o
CF, 30 μm thick
+
45
o
GFPW, 25 μm thick
_
Fig. 3 Cross-section of a TRAC longeron. Each flange has the composition
[±
45
퐺퐹푃푊
/
0
퐶퐹
/±
45
퐺퐹푃푊
]
and the
web
[±
45
퐺퐹푃푊
/
0
퐶퐹
/±
45
3
,퐺퐹푃푊
/
0
퐶퐹
/±
45
퐺퐹푃푊
]
.
B. The deployment mechanism
Inspired by the designs described in [
4
,
5
], we designed a simplified deployment mechanism that incorporates
only the essential components (Fig. 5) to trigger the release of the structure and study its deployment, but does not
provide other key functions of the full flight mechanism. Also, instead of using constant-force retractors attached to the
ends of the diagonal cords, for the purpose of verifying the numerical simulations, hanging masses were used in our
3
Table 1 Mass properties of the composite structure.
Areal density of flange
1
.
196
×
10
−
4
g
/
mm
2
Areal density of web
2
.
908
×
10
−
4
g
/
mm
2
Density of batten
1
.
61
×
10
−
3
g
/
mm
3
Total mass of 6 longerons
45
.
4g
Total mass of 18 battens
12
.
4g
Total mass of 12 strip-cord connectors
7
.
5g
experiments. The masses can be more easily tailored in the experiment, and are easier to quantify compared to other
sources of deployment force such as retractor springs.
The deployment mechanism shown in Figure 4 is built around a rigid cylinder labelled central shaft. Two annular
plates made of acrylic are placed at the top and bottom of this shaft to mimic the mechanism in [
4
,
5
]. The central shaft
supports the quick release mechanism at the very top and is connected to the hinge and diagonal frames at the bottom.
One end of each of the two cords is rigidly attached to the central shaft. The metallic frames placed along the diagonals
of the quadrant support pulleys mounted on vertical columns at the outer ends. The central shaft, deploying cylinder,
and the hinge components are made of aluminum while the top and bottom plates are made of acrylic.
lift to deploy
quick-release link
deploying
cylinder
hinge with
torsion spring
cord connection
central shaft
top plate
bottom plate
reflective
marker
Fig. 4 Schematic of the deployment mechanism in stowed configuration. The middle portion of the folded
structure lies in between the central shaft and deploying cylinder. One end of each cord is attached to the central
shaft at the cord connection point.
C. Motion capture
In order to verify the numerical simulations of the deployment, the motion of the structure was recorded using six
infrared motion capture cameras, set a frame rate of
180
fps, marketed as
Prime 41
by
NaturalPoint Inc
. These cameras
track the reflective markers attached to the structure at chosen locations. Flat reflective stickers were attached at all the
longeron-batten intersections. A reflective sticker was placed also on the webs of the longerons, in the middle of two
longeron-batten intersections. These flat markers are shown as white dots in Figure 6. In addition, to track the motion of
the strip-cord connectors, reflective spheres 6
.
4 mm in diameter were glued to the connectors.
4
(a)
(b)
Fig. 5 Deployment mechanism and structure in (a) stowed and (b) deployed configurations. Strip 1 of the
structure is shown in orange, Strip 2 in blue, and Strip 3 in green.
Fig. 6 Reflective stickers are glued to the structure at the intersections of longerons and battens, and on the web
regions of the TRAC longerons between in between battens. Reflective spheres attached to strip-cord connectors
are labeled
C
.
5
III. Characterization of composite laminates and deployment mechanism
To develop high-fidelity simulations of the dynamic deployment, it is essential to quantify the constitutive behavior
of the composites, tensile behavior of the cords, and friction in the hinge and pulleys (shown in Fig. 5). The quantities
measured will be used in the dynamic simulations presented in Section IV.
A. Material properties
The elastic stiffness of the flange and web laminates (Figure 3) was modeled with the generalized stiffness matrices
푨
,
푫
for symmetric laminates [6],
{
푵
푴
}
=
[
푨
0
0
푫
]{
흐
휿
}
,
(1)
where
푵
and
푴
are the in-plane forces and out-of-plane moments per unit length,
흐
and
휿
are the mid-plane strains and
curvatures, respectively;
푨
is the in-plane stiffness matrix, and
푫
is the bending stiffness matrix.
To fold a longeron, the flanges are first flattened inducing a bending strain in the transverse direction (along the
circumference of the circular arc, direction
2
) followed by bending the flanges and web in longitudinal direction
(direction
1
). Hence, the elements of the stiffness matrices
푫
contributing significantly to the stored strain energy due to
bending are
퐷
web
11
,
퐷
flange
11
and
퐷
flange
22
.
We conducted four-point bending tests on the flange and web laminates to experimentally measure the stiffness
values
퐷
11
and
퐷
22
, and tensile tests in longitudinal direction to measure the compliance values
푎
11
and
푎
21
, where the
compliance matrix
풂
=
푨
−
1
. The missing elements of
푫
and
풂
for both flange and web laminates were taken from an
earlier study [
3
] conducted on similar laminates. The non-zero stiffness values used in the numerical simulations are
listed in Table 2.
Table 2 Generalized stiffness values for flange and web laminates.
퐴
11
(N
/
mm)
퐴
12
(N
/
mm)
퐴
22
(N
/
mm)
퐴
33
(N
/
mm)
퐷
11
(Nmm)
퐷
12
(Nmm)
퐷
22
(Nmm)
퐷
33
(Nmm)
Flange
6218.5
637.38
1078.55
736.5
0.76
0.48
0.59
0.46
Web
11476.1
1112.8
2291.54
1727.4
39.61
4.32
10.42
4.93
B. Behavior of the cord
Cords are the primary members carrying the deploying force in addition to holding various strips of the structure
together (Figures 2 and 5). Knowing the tensile behavior of the cord helps accurately estimate the transfer of forces
along the cord in numerical simulations. In addition, deformed lengths of the various segments of a cord decide the
positions of the strips relative to one another (see Figure 14).
Cords used in this study are 8 strand weave fishing lines with a load rating of
50
lb, marketed as J-Braid by Daiwa.
To measure the elastic behavior (reaction force vs extension) of these cords, we conducted tensile tests on a sample
using Instron machine. Extension was measured using two laser extensometers (LE-01 and LE-05 from Electronics
Instrument Research), and the reaction force in the cord was measured using a 500 N load cell. The resulting data
(Figure 7) was given as a direct input (in the form of a table) to the simulations.
C. Characterization of the hinge
Deployment of the structure begins the instant the deploying cylinder is released. Initial dynamics of the structure
depends on the rotational speed of the cylinder. This speed in turn is governed by the stiffness of the torsion springs and
friction in the hinge (see Figure 4). Therefore, quantifying the elastic and friction contributions to the moment at the
hinge axis is essential to accurately simulate the initial dynamics of the deployment. Elastic stiffness of the torsion
springs was obtained from the data sheets, and was verified experimentally. To measure the friction moment at the hinge
axis, we released only the cylinder, without any structure, and tracked its motion using motion capture cameras (Figure
4). The measured rotation of the cylinder as a function of time,
휃
푒푥푝
(
푡
)
, will be used in the following analysis to obtain
a constant value for friction moment in the hinge.
6
Fig. 7 Response of the cord to applied tensile strain. Measurements from three repetitions of the tensile test are
presented.
The equation of motion of the cylinder at a rotation angle
휃
is
푀푔푑
sin
(
휃
+
휃
0
)+
푇
푒
−
푇
푓
=
퐼
ℎ
휃,
(2)
with the initial conditions
휃
(
0
)
=
0 and
휃
=
0
,
(3)
where
푀
is the measured mass of the cylinder,
푔
is the acceleration due to gravity,
푇
푒
is the elastic moment due to the
torsion spring in the hinge,
푇
푓
is the opposing friction moment,
퐼
ℎ
is the rotational moment of inertia about the hinge
axis,
푑
is the distance of the center of gravity from the hinge axis,
휃
0
is the angular position of center of gravity from the
hinge axis with respect to the longer edge of the cylinder.
The hinge has two torsion springs with measured stiffness of
54
.
1 Nmm
/
rad
each. Since each spring was
pre-compressed by 180
표
, the elastic moment is given by
푇
푒
=
108
.
2
(
휋
−
휃
)
Nmm
.
(4)
The location of the center of gravity of the cylinder was obtained from the CAD model and was verified with
experiments. Moment of inertia
퐼
ℎ
was obtained from the CAD model.
The equation of motion 2 has two unknowns, constant
푇
푓
and
휃
(
푡
)
, and needs to be solved iteratively. For a given
value of friction moment
푇
푓
, Equation 2 can be integrated numerically to obtain the rotation as a function of time,
휃
푎푛
(
푡
)
. The value of
푇
푓
that minimizes the difference between the experimental measurement and the analytical result,
퐽
=
norm
(
휃
푒푥푝
−
휃
푎푛
)
,
(5)
was searched for using the unconstrained optimization solver
fminunc
in MATLAB.
The aforementioned experiment was repeated four times and four corresponding values for friction moment were
obtained. The average value
푇
푓
=
137
.
9 Nmm was used in the numerical simulations that follow.
D. Coefficient of friction of the pulleys
Deployment of the cord ends is driven by gravity. The deploying force is carried by the cords which go over pulleys.
Due to friction in the pulleys, the tension force available to deploy the structure is smaller than the force applied at the
loose end of a cord.
The miniature pulleys made of acrylic are mounted over ball bearings. We aim to approximate the friction moment
in this assembly as a linear function of the total radial force acting on it. This linear behavior will then be simulated
using
hinge
connector element in Abaqus [7].
The experimental setup consists of two identical pulleys at the same horizontal level (Figure 9). The cord (of the
same material used in the deployment experiments) running over the two pulleys is connected to two masses. When
released, the heavier mass moves downward and the lighter mass upward. The reflective spheres attached to two masses
were tracked using OptiTrack motion capture cameras at a speed of 200 fps.
7
(a)
θ
θ0
d
Mg
T
e
T
f
(b)
Fig. 8 (a) Schematic of the deploying cylinder in the absence of the structure. (b) Rotation of the cylinder
with time. Friction moment
푻
풇
=
137
.
9 Nmm
was used for the curves from numerical integration and Abaqus
simulation. Cylinder locks at a rotation of 109
풐
in the simulation.
Applying Newton’s second law of motion at masses
푀
1
and
푀
2
,
푇
1
=
푀
1
(
푔
+
ℎ
)
,
(6)
푇
2
=
푀
2
(
푔
−
ℎ
)
,
(7)
where
푇
1
and
푇
2
are the tension forces in the cord at
푀
1
and
푀
2
, respectively,
푔
is the acceleration due to gravity, and
ℎ
is the acceleration of the masses.
The equations of rotational motion of the two pulleys are
푇푅
−
푇
1
푅
−
푀
푓
1
=
퐼
훽,
(8)
푇
2
푅
−
푇푅
−
푀
푓
2
=
퐼
훽,
(9)
where
푇
is the tension in the portion of the cord between the two pulleys,
푅
and
퐼
are the radius and moment of inertia
of the pulleys, respectively,
푀
푓
1
and
푀
푓
2
are the friction moments in the left and right pulleys, respectively. When
there is no slip between the cord and a pulley, rotation
훽
=
ℎ
/
푅
.
The vertical displacement
ℎ
of the masses we measured is a quadratic function of time, so the acceleration
ℎ
is a
constant and is known. Therefore, tension forces
푇
1
and
푇
2
in Equation 6 are known. The moment of inertia
퐼
was
previously measured in [
?
]tobe
1
.
9
×
10
−
8
kgm
2
. To solve the three unknowns
푇
,
푀
푓
1
, and
푀
푓
2
in Equation 8, we
used the optimization solver
fmincon
in MATLAB.
Assuming that the friction moment in a pulley is directly proportional to the total radial force at the pulley bearing,
푀
푓
1
=
휇
1
푟퐹
푛
1
,
(10)
푀
푓
2
=
휇
2
푟퐹
푛
2
,
(11)
where
휇
1
and
휇
2
are the coefficients of friction in the pulleys,
푟
=
1
.
59 mm
is the radius of the bore, and
퐹
푛
1
and
퐹
푛
2
are the total reaction forces at the pulleys.
The force resultants on the pulleys are
퐹
푛
1
=
√
푇
2
1
+
푇
2
,
(12)
퐹
푛
2
=
√
푇
2
2
+
푇
2
.
(13)
This experiment was conducted with 21 different combinations of masses (Figure 10) with 2 to 4 repetitions for each
combination. Using linear fits, the coefficients of friction were found to be 0.0323 and 0.0326 for the two pulleys.
8
M
f1
R
r
M
f2
R
r
M
1
M
2
T
1
T
2
T
2
T
1
TT
h
Pulley 1
Pulley 2
M1g
M2g
F
n1
F
n2
h
Fig. 9 Schematic of the experiment to measure the friction coefficients of the pulleys.
Fig. 10 Friction moments in the pulleys in relation to the total radial forces acting at the centers. In pulley 1,
coefficient of friction,
흁
1
=
0
.
0322
, and in pulley 2,
흁
2
=
0
.
0326
. The mean value
흁
=
0
.
0324
is used in the finite
element analyses.
9