Slew Maneuver Constraints for Agile Flexible Spacecraft
Michael A. Marshall
∗
and Sergio Pellegrino
†
California Institute of Technology, Pasadena, California, 91125
Traditional spacecraft design paradigms rely on stiff bus structures with comparatively
flexible appendages. More recent trends, however, trade deployed stiffness for packaging
efficiency to stow apertures with larger areas inside existing launch vehicles. By leveraging
recent advances in materials and structures, these spacecraft may be up to several orders of
magnitude lighter and more flexible than the current state-of-the-art. Motivated by the goal
of achieving agility despite structural flexibility, this paper proposes a quantitative method
for determining structure-based performance limits for maneuvering flexible spacecraft. It
then uses a geometrically nonlinear flexible multibody dynamics model of a representative very
flexible spacecraft to verify this method. The results demonstrate that, contrary to common
assumptions, other constraints impose more restrictive limits on maneuverability than the
dynamics of the structure. In particular, it is shown that the available attitude control system
momentum and torque are often significantly more limiting than the compliance of the structure.
Consequently, these results suggest that there is an opportunity to design less-conservative,
higher-performance space systems that can either be maneuvered faster, assuming suitable
actuators are available, or built using lighter-weight, less-stiff architectures that move the
structure-based performance limits closer to those of the rest of the system.
I. Introduction
A
current
paradigm in spacecraft design trades deployed structural stiffness against packaging efficiency to build
higher-performing spacecraft with larger deployed apertures that can be stowed within existing launch vehicles.
Such spacecraft are currently envisioned for a variety of applications including astronomy [
1
]; planetary [
2
] and solar
system exploration [
3
]; space science [
4
]; communications, power transfer, and remote sensing [
5
]; and space solar
power [
6
]. Each application requires attitude slew maneuvers, i.e., maneuvers that change the spacecraft’s orientation.
Large-angle slew maneuvers in particular are commonly used, e.g., for reorienting sensors, antennas, and solar arrays.
Slew maneuvers are an overhead on a mission, meaning they are required for achieving the mission objectives, but
generally represent time lost from actively performing useful mission functions. For this reason, minimizing slew
times has important implications for space mission design. In particular, slewing faster leaves more time available for
executing a spacecraft’s intended mission.
Given the proliferation of applications for flexible spacecraft, a common question during mission concept development
and preliminary design pertains to how fast these spacecraft can be slewed. In some cases, a rapid slew capability can
even be a prerequisite for feasibility and/or viability of a particular mission concept. For example, in geostationary Earth
orbit (GEO), the space solar power satellites proposed by the Caltech Space Solar Power Project (SSPP) [
6
] require two
90-deg pitch-axis slews per day to maximize the energy delivered to the electrical grid [
7
]. In the SSPP concept, the
system efficiency decreases as the slew time increases [
8
]. All else being equal, slower slew maneuvers result in the
transmission of less energy, thereby increasing the cost of the electricity delivered to the grid. Thus, the slew time
directly impacts the system’s overall economic viability. More generally, as the development of increasingly large and
flexible spacecraft continues, so too does the importance of slew time as a design driver.
A common assumption about flexible spacecraft is that structural compliance limits how fast they can be slewed.
For highly compliant structures, very long slew times can make an otherwise promising mission concept infeasible. As
a result, it is important to demonstrate the feasibility of slewing large flexible spacecraft early in the design process.
However, to the authors’ knowledge, there is no standard framework for rigorously quantifying how fast flexible
spacecraft can be slewed. The most common heuristic states that the minimum slew time must be at least ten times the
∗
Graduate Research Assistant, Graduate Aerospace Laboratories, 1200 E. California Blvd., Mail Code 105-50. Member AIAA. Currently:
Guidance and Control Analyst, The Johns Hopkins University Applied Physics Laboratory, 11100 Johns Hopkins Road, Laurel, Maryland 20723.
michael.a.marshall@jhuapl.edu.
†
Joyce and Kent Kresa Professor of Aerospace and Civil Engineering, Graduate Aerospace Laboratories, 1200 E. California Blvd., Mail Code
105-50. AIAA Fellow. sergiop@caltech.edu.
1
structure’s lowest natural period. Such a heuristic is convenient but may lead to overly conservative, and in some cases,
prohibitively conservative spacecraft designs and mission scenarios.
Motivated by the goal of achieving agility despite structural flexibility, this paper proposes a framework for using
reduced-order models to predict minimum slew times for flexible spacecraft. The reduced-order models lead to simple
analytical and quasi-analytical slew time estimates, which in turn are useful for both requirements definition and for
establishing the feasibility of slewing flexible spacecraft during concept development and preliminary design.
Slew time verification then requires higher-fidelity analysis tools, such as geometrically nonlinear finite element
simulations. There are many academic examples in the literature that apply these types of simulations to flexible
spacecraft with simple structural geometries consisting of beams or plates; see e.g., [
9
,
10
]. These types of simulations
are also often used for modeling solar sails [
11
,
12
] and other advanced concepts [
13
]. However, they are by no means
standard for simulating the attitude dynamics of flexible spacecraft with complex structural geometries. To that end, this
paper uses geometrically nonlinear finite element simulations of a very flexible spacecraft with a complex structural
geometry both to verify slew time predictions and to promote the more widespread adoption of these types of simulations
in spacecraft engineering practice.
Spacecraft
Tile
Strip
Fig. 1 Caltech SSPP spacecraft structural architecture originally introduced in [6].
As a case study, this paper analyzes a representative problem based on the Caltech SSPP spacecraft structural
architecture [
6
] (see Fig. 1) during a 90-deg, nominally rest-to-rest pitch-axis slew maneuver. Such a maneuver is
representative of the maneuvers required to maximize the energy delivered to an electrical grid from an SSPP spacecraft
in GEO [
7
]. In the SSPP concept, the structural architecture is planar to facilitate packaging and deployment using a
kirigami-inspired folding scheme and is designed to be both modular and scalable, allowing the same basic components
(the photovoltaic-powered radio-frequency tiles and the structural strips) to be used for spacecraft designed for different
applications at different length scales. This paper specifically considers spacecraft with outermost strips that range from
5
m
to 50
m
long, corresponding to first-mode natural frequencies between approximately 1
Hz
and 1
mHz
. The SSPP
architecture and its derivatives (see e.g., [
14
]) are representative of a class of spacecraft structures referred to as bending
architectures [
15
], i.e., structural concepts that derive their load carrying capability from the bending stiffness of the
structural elements. With its approximately 100
g/m
2
areal density, the SSPP concept specifically occupies a middle
ground in terms of stiffness and areal density between membrane-based deployable structures concepts like solar sails
and more traditional spacecraft with deployable solar arrays and antennas.
This paper is organized as follows: Sec. II describes the canonical model for flexible spacecraft attitude dynamics.
Sec. III uses the Craig-Bampton method [
16
] to develop reduced-order modal models from flexible spacecraft finite
element models that are analogous to this canonical model. Sec. IV proposes a framework for predicting slew times for
flexible spacecraft. Sec. V applies the tools from Secs. III and IV to predict the minimum slew times for a flexible
spacecraft based on the Caltech SSPP spacecraft structural architecture. It subsequently verifies these predictions using
geometrically nonlinear simulations of a high-fidelity finite element model. The paper concludes with a discussion of
the results and their implications in Sec. VI.
2
II. Canonical Flexible Spacecraft Model
The classical approach for attitude control system (ACS) analysis and design reduces complex flexible spacecraft
dynamics into three decoupled, single-axis modal models, one for rotation about each axis (roll, pitch, and yaw) [
17
,
18
].
Each model includes a single rigid body mode and one or more dynamically significant elastic modes. In particular,
preliminary analysis and design often rely on single-mode models, i.e., single-axis modal models with a single retained
elastic mode. This is the simplest structural dynamic model that includes both rigid body and flexible modes, and hence,
is taken as the canonical model for flexible spacecraft dynamics. The canonical model takes the form of the unrestrained
spring-mass-damper system with two degrees of freedom (DOFs) depicted in Fig. 2.
Fig. 2 The canonical model of a flexible spacecraft is a floating spring-mass-damper system with two DOFs.
The equations of motion for the canonical model are
"
푚
1
0
0
푚
2
# "
¥
푥
1
¥
푥
2
#
+
"
푐
−
푐
−
푐 푐
# "
¤
푥
1
¤
푥
2
#
+
"
푘
−
푘
−
푘 푘
# "
푥
1
푥
2
#
=
"
푢
1
0
#
(1)
where
푚
1
denotes the mass of the spacecraft “bus” with position
푥
1
,
푚
2
is the mass of the flexible “appendage” with
position
푥
2
,
푘
is the spring stiffness,
푐
is the viscous damping coefficient,
푢
1
is the control input on
푚
1
, and dot
notation denotes differentiation with respect to time
푡
. In practice,
푥
1
is the bus orientation,
푥
2
is the modal coordinate
corresponding to the dominant flexible mode (which is not necessarily the lowest-frequency mode), and
푢
1
is the attitude
control torque. The remaining parameters are related to the rigid and flexible body properties of the spacecraft. Sec. III
shows how to reduce arbitrary finite element models into single-axis modal models and, by doing so, derives expressions
for these parameters. Importantly, even though the focus of this paper is on attitude slew maneuvers, Eq.
(1)
applies for
either translational or rotational motion. Hence, the parameters and variables in Eq.
(1)
are to be interpreted in the
generalized sense; e.g.,
푚
1
and
푚
2
are generalized masses that can represent either masses or moments of inertia.
The classical ACS analysis and design approach treats flexibility as a disturbance acting on the spacecraft bus. Thus,
the parameter of interest for ACS design and analysis is the influence of
푚
2
on
푚
1
, not the motion of
푚
2
itself. To
eliminate the motion of
푚
2
, the standard approach is to rewrite Eq.
(1)
in the Laplace domain and evaluate the transfer
function from
푢
1
to
푥
1
. Taking the Laplace transform of Eq. (1) with zero initial conditions gives
푚
1
푠
2
푋
1
(
푠
)+
푐푠
(
푋
1
(
푠
)−
푋
2
(
푠
)
)
+
푘
(
푋
1
(
푠
)−
푋
2
(
푠
)
)
=
푈
1
(
푠
)
(2)
푚
2
푠
2
푋
2
(
푠
)+
푐푠
(
푋
2
(
푠
)−
푋
1
(
푠
)
)
+
푘
(
푋
2
(
푠
)−
푋
1
(
푠
)
)
=
0
(3)
where
푋
1
(
푠
)
=
L(
푥
1
(
푡
))
,
푋
2
(
푠
)
=
L(
푥
2
(
푡
))
,
푈
1
(
푠
)
=
L(
푢
1
(
푡
))
, and
L(·)
denotes the Laplace transform that converts a
function of time
푡
to a function of the complex frequency
푠
. Solving Eqs. (2) and (3) for
푋
1
(
푠
)/
푈
1
(
푠
)
, taking a partial
fraction expansion, and simplifying yields
푋
1
(
푠
)
푈
′
1
(
푠
)
=
1
푠
2
+
푚
2
/
푚
1
푠
2
+
2
(
1
+
푚
2
/
푚
1
)
휁휔
푛
푠
+
(
1
+
푚
2
/
푚
1
)
휔
2
푛
(4)
where
휔
푛
=
√︁
푘
/
푚
2
is the fixed-base natural frequency,
휁
=
푐
/
2
√
푘푚
2
is the fixed-base damping ratio (fraction
of critical damping), and
푢
′
1
is the acceleration input to the system, i.e.,
푢
1
=
(
푚
1
+
푚
2
)
푢
′
1
[equivalently,
푈
1
(
푠
)
=
(
푚
1
+
푚
2
)
푈
′
1
(
푠
)
].
Equation
(4)
consists of two terms, the rigid body translation of
푚
1
and a perturbation due to the motion of
푚
2
, i.e.,
due to flexibility. To make this more explicit, let
푋
1
(
푠
)
=
푋
1
,푟
(
푠
)+
푋
1
, 푓
(
푠
)
where the subscripts
푟
and
푓
denote the
3
rigid body and flexible terms, with corresponding transfer functions
푋
1
,푟
(
푠
)
푈
′
1
(
푠
)
=
1
푠
2
(5)
푋
1
, 푓
(
푠
)
푈
′
1
(
푠
)
=
푚
2
/
푚
1
푠
2
+
2
(
1
+
푚
2
/
푚
1
)
휁휔
푛
푠
+
(
1
+
푚
2
/
푚
1
)
휔
2
푛
(6)
Taking the inverse Laplace transforms of Eqs. (5) and (6) then gives
¥
푥
1
,푟
=
푢
′
1
(7)
¥
푥
1
, 푓
+
2
1
+
푚
2
푚
1
휁휔
푛
¤
푥
1
, 푓
+
1
+
푚
2
푚
1
휔
2
푛
푥
1
, 푓
=
푚
2
푚
1
푢
′
1
(8)
From Eq.
(8)
, the perturbation due to flexibility (i.e., the flexible dynamics) can be modeled as a damped harmonic
oscillator with increased natural frequency
휔
푛
√︁
1
+
푚
2
/
푚
1
and damping ratio
휁
√︁
1
+
푚
2
/
푚
1
relative to the fixed-base
case. The shifted natural frequency
휔
푛
√︁
1
+
푚
2
/
푚
1
is the free-free natural frequency of Eq. (4).
Classical approaches for flexible spacecraft ACS analysis and design are usually predicated on minimizing the
magnitude of any disturbances induced by flexibility, i.e., by making the magnitude of
푥
1
, 푓
and its derivatives “small”.
This entails moving the system sufficiently “slowly” to prevent significant excitation of the flexible mode(s). With this in
mind, a standard practice is to require that the closed-loop ACS bandwidth is at least an order of magnitude below the
free-free natural frequency
휔
푛
√︁
1
+
푚
2
/
푚
1
[
17
].
∗
In this case, the ACS reacts at least an order of magnitude slower
than the natural time scale of the system’s dynamics. Using this approach, it is often possible to neglect flexibility in
ACS design and instead simply design a control system for the rigid body motion, as is done, e.g., in [19].
A similar philosophy is usually adopted for designing slew maneuvers. A nominally rest-to-rest slew maneuver for
a rigid spacecraft, i.e., a spacecraft that can be modeled as a rigid body, leads to residual structural vibrations for a
flexible one. In light of Eq.
(4)
, the spacecraft bus perceives these vibrations as angular position and velocity errors, the
magnitudes of which often appear in ACS pointing error budgets (see e.g., [
20
]) and are a proxy for pointing stability
and jitter. Here, jitter refers to the classical definition of unwanted mechanical vibrations, as opposed to more nuanced
definitions typically used for space-borne optical systems [
21
,
22
]. A flexible spacecraft ACS with its closed-loop
bandwidth set an order of magnitude below its lowest flexible-mode frequency is incapable of rejecting jitter [
17
]. For
these reasons, minimizing jitter is imperative for pointing accuracy and stability.
A common heuristic for minimizing jitter states that the slew maneuver duration
푇
must be at least an order of
magnitude longer than the natural period
푇
푛
=
2
휋
/
휔
푛
. However, such a requirement is shown to be misguided in
Secs. IV and V. In particular, “slow” is relative, and depends on both the “shape” of the forcing applied to the system
and the ratio
푇
/
푇
푛
. With this in mind, this paper instead proposes using quantitative requirements on the residual (i.e.,
post-slew) amplitude of
푥
1
, 푓
and its derivatives (specifically, on
¤
푥
1
, 푓
) to calculate feasible slew times. For a given
spacecraft and slew maneuver, specifying a requirement on the residual amplitude of
푥
1
, 푓
or any of its derivatives
indirectly specifies a requirement on the minimum slew time. Hedgepeth [
23
] uses similar arguments to determine
a first-mode natural frequency requirement for slewing flexible spacecraft, although his approach underpredicts the
amplitudes of
푥
1
, 푓
and its derivatives; for additional details, see [8].
III. Derivation of Single-Axis Modal Models
Practical applications of the canonical flexible spacecraft model require relationships between the parameters
푚
1
,
푚
2
,
휁
,
휔
푛
, and a flexible spacecraft finite element model. To that end, this section uses the Craig-Bampton
method [
16
] to rigorously and systematically derive single-axis modal models analogous to the canonical model from
unrestrained (free-free) finite element models. In doing so, it derives a fully coupled 6-DOF generalization of the
transfer function from
푢
1
to
푥
1
[Eq.
(4)
] and shows that the correct set of vibration modes for ACS analysis and design
are the Craig-Bampton method’s fixed-interface normal modes. It also discusses methods for identifying the most
dynamically significant mode(s) and special considerations for symmetric structures. The Craig-Bampton method
generalizes the notion of a “bus” with a flexible “appendage” to arbitrarily complex flexible spacecraft.
∗
In practice, this depends on the spacing of the structural modes. For a system with a few well-separated modes, it is possible to achieve higher
bandwidth linear control systems by filtering the structural modes (see e.g., [
18
] and the references therein). However, this becomes difficult, if not
impossible for large space structures with many closely spaced modes (see e.g., [
19
]), in which case the aforementioned requirement on closed-loop
bandwidth becomes imperative.
4
A. 6-DOF Transfer Function
The derivation of the 6-DOF generalization of the transfer function from
푢
1
to
푥
1
[Eq.
(4)
] starts from the standard
equation of motion for a free-free linear finite element model:
M
¥
x
+
C
¤
x
+
Kx
=
Bu
(9)
Here,
x
∈
R
푛
contains the nodal displacement DOFs,
u
∈
R
푚
contains the external forces and moments,
M
∈
R
푛
×
푛
is the
symmetric positive definite mass matrix,
C
∈
R
푛
×
푛
is the symmetric positive semi-definite damping matrix,
K
∈
R
푛
×
푛
is the symmetric positive semi-definite stiffness matrix, and
B
∈
R
푛
×
푚
maps the external forces and moments to the
nodal DOFs. In general, each node has six DOFs, three translations and three rotations, from which it follows that
Eq.
(9)
admits six rigid body modes. The damping model (e.g., Rayleigh or modal) determines the rank deficiency of
C
;
the number of rigid body modes corresponds to the rank deficiency of
K
.
For the Craig-Bampton method [
16
], Eq.
(9)
is partitioned into
푛
퐼
interior (
퐼
) and
푛
퐵
boundary (
퐵
) coordinates, as
follows:
"
M
퐼퐼
M
퐼퐵
M
퐵퐼
M
퐵퐵
# "
¥
x
퐼
¥
x
퐵
#
+
"
C
퐼퐼
C
퐼퐵
C
퐵퐼
C
퐵퐵
# "
¤
x
퐼
¤
x
퐵
#
+
"
K
퐼퐼
K
퐼퐵
K
퐵퐼
K
퐵퐵
# "
x
퐼
x
퐵
#
=
"
0
푛
퐼
×
1
u
퐵
#
(10)
where
0
푛
퐼
×
1
∈
R
푛
퐼
is a zero vector and
푛
=
푛
퐼
+
푛
퐵
. Typically, the
퐵
-set contains DOFs either shared with adjacent
components (when the Craig-Bampton substructure is a component of a larger structural dynamic model) or loaded by
external forces or moments; the remaining DOFs belong to the
퐼
-set [
16
]. For a flexible spacecraft, the
퐵
-set coordinates
are the six rigid body DOFs of the bus, and hence, correspond to
푥
1
from the canonical model [Eq.
(1)
]. The
퐼
-set
coordinates (or the corresponding modal coordinates) are then analogous to
푥
2
in Eq.
(1)
. With the
퐵
-set coordinates
defined in this way,
K
퐼퐼
is the full-rank stiffness matrix corresponding to fixed (clamped) boundary conditions at the bus.
u
퐵
then contains the forces (e.g., due to thrusters) and moments (e.g., due to the ACS) acting on the bus. Equation
(10)
is simply a permutation of the rows and columns of Eq. (9).
Following Sec. II, the immediate goal is to derive the transfer function
H
(
푠
)
that relates a force or moment on
퐵
to the corresponding translations and rotations, i.e., to find
H
(
푠
)
=
G
−
1
(
푠
)
such that
X
퐵
(
푠
)
=
H
(
푠
)
U
퐵
(
푠
)
where
X
퐵
(
푠
)
=
L(
x
퐵
(
푡
))
and
U
퐵
(
푠
)
=
L(
u
퐵
(
푡
))
.
H
(
푠
)
is subsequently simplified for the special case of a single-axis slew to
obtain an expression analogous to Eq.
(4)
. The derivation of
G
(
푠
)
closely follows the procedure in [
24
, p. 187–190] for
the undamped sinusoidal (steady-state) transfer function
G
(
푗휔
)
(referred to as “mechanical impedance” in [
24
]) where
푗
2
=
−
1
and
휔
is the frequency of the harmonic forcing.
The derivation of
G
(
푠
)
requires taking the Laplace transform of Eq.
(10)
(again with zero initial conditions), from
which it follows that
푠
2
"
M
퐼퐼
M
퐼퐵
M
퐵퐼
M
퐵퐵
#
+
푠
"
C
퐼퐼
C
퐼퐵
C
퐵퐼
C
퐵퐵
#
+
"
K
퐼퐼
K
퐼퐵
K
퐵퐼
K
퐵퐵
#! "
X
퐼
(
푠
)
X
퐵
(
푠
)
#
=
"
0
푛
퐼
×
1
U
퐵
(
푠
)
#
(11)
where
X
퐼
(
푠
)
=
L(
x
퐼
(
푡
))
. Solving the first equation in Eq.
(11)
for
X
퐼
(
푠
)
and substituting this result into the second
equation then yields
G
(
푠
)
=
푠
2
M
퐵퐵
+
푠
C
퐵퐵
+
K
퐵퐵
−
Z
퐵퐼
(
푠
)
Z
−
1
퐼퐼
(
푠
)
Z
퐼퐵
(
푠
)
(12)
where
Z
푘푙
(
푠
)
=
푠
2
M
푘푙
+
푠
C
푘푙
+
K
푘푙
. Equation
(12)
obscures the modal properties of the structure, and hence, is
rewritten explicitly in terms of mode shapes and natural frequencies next. Truncating the resulting modal expansion
yields a reduced-order modal model.
The fixed-interface normal modes, i.e., the eigenmodes corresponding to fixed (clamped) boundary DOFs, are the
solutions to the following generalized eigenproblem [16]:
K
퐼퐼
흓
푖
=
휔
2
푖
M
퐼퐼
흓
푖
(13)
Each fixed-interface normal mode
휙
푖
(for
푖
=
1
, ..., 푛
퐼
) is orthogonal to
M
퐼퐼
and normalized such that
흓
푇
푖
M
퐼퐼
흓
푗
=
훿
푖푗
.
†
The
푛
퐼
solutions to Eq. (13) can equivalently be written in the form
K
퐼퐼
횽
=
M
퐼퐼
횽훀
2
(14)
†
훿
푖푗
is the Kronecker delta symbol defined such that
훿
푖푗
=
1
for
푖
=
푗
and
훿
푖푗
=
0
otherwise.
5
where
횽
=
흓
1
,...,
흓
푛
퐼
is the matrix of generalized eigenvectors and
훀
2
=
diag
휔
2
1
,...,휔
2
푛
퐼
. Since each
흓
푖
is
orthogonal with respect to
M
퐼퐼
,
횽
푇
M
퐼퐼
횽
=
I
푛
퐼
×
푛
퐼
(15)
횽
푇
K
퐼퐼
횽
=
훀
2
(16)
where
I
푛
퐼
×
푛
퐼
∈
R
푛
퐼
×
푛
퐼
is an identity matrix. Subsequent developments additionally assume that
횽
푇
C
퐼퐼
횽
=
2
Z
훀
(17)
where
Z
=
diag
휁
1
,..., 휁
푛
퐼
is the matrix of modal damping coefficients and
휁
푖
≥
0
for all
푖
=
1
, ..., 푛
퐼
.
The
푛
퐼
generalized eigenvectors are linearly independent, i.e.,
횽
is invertible. Hence, rearranging Eqs.
(15)
–
(17)
results in the following identities:
M
퐼퐼
=
횽
−
푇
횽
−
1
(18)
K
퐼퐼
=
횽
−
푇
훀
2
횽
−
1
(19)
C
퐼퐼
=
2
횽
−
푇
Z
훀횽
−
1
(20)
Inverting
M
퐼퐼
and
K
퐼퐼
then yields the following modal expansions [24, p. 187–190]:
M
−
1
퐼퐼
=
횽횽
푇
=
푛
퐼
∑︁
푖
=
1
흓
푖
흓
푇
푖
(21)
K
−
1
퐼퐼
=
횽훀
−
2
횽
푇
=
푛
퐼
∑︁
푖
=
1
흓
푖
흓
푇
푖
휔
2
푖
(22)
from which it readily follows that
K
−
1
퐼퐼
M
퐼퐼
K
−
1
퐼퐼
=
횽훀
−
4
횽
푇
=
푛
퐼
∑︁
푖
=
1
흓
푖
흓
푇
푖
휔
4
푖
(23)
By analogy with Eq. (23),
K
−
1
퐼퐼
C
퐼퐼
K
−
1
퐼퐼
=
횽훀
−
2
(
2
Z
훀
)
훀
−
2
횽
푇
=
푛
퐼
∑︁
푖
=
1
흓
푖
(
2
휁
푖
휔
푖
)
흓
푇
푖
휔
4
푖
(24)
Substituting Eqs. (18)–(20) into
Z
−
1
퐼퐼
(
푠
)
then yields
Z
−
1
퐼퐼
(
푠
)
=
횽
푠
2
I
푛
퐼
×
푛
퐼
+
2
푠
Z
훀
+
훀
2
−
1
횽
푇
(25)
which is equivalent to the following modal expansion:
Z
−
1
퐼퐼
(
푠
)
=
푛
퐼
∑︁
푖
=
1
흓
푖
흓
푇
푖
푠
2
+
2
휁
푖
휔
푖
푠
+
휔
2
푖
(26)
Further simplifications require the identity [24, p. 189]
1
푠
2
+
휔
2
푖
=
1
휔
2
푖
−
푠
2
휔
4
푖
+
푠
4
휔
4
푖
푠
2
+
휔
2
푖
(27)
and the substitution
푠
2
→
푠
2
+
2
휁
푖
휔
푖
푠
. Using Eq.
(27)
to expand the denominator in Eq.
(26)
and simplifying with
Eqs. (22)–(24) then gives
Z
−
1
퐼퐼
(
푠
)
=
K
−
1
퐼퐼
−
푠
2
K
−
1
퐼퐼
M
퐼퐼
K
−
1
퐼퐼
−
푠
K
−
1
퐼퐼
C
퐼퐼
K
−
1
퐼퐼
+
푛
퐼
∑︁
푖
=
1
흓
푖
푠
2
+
2
휁
푖
휔
푖
푠
2
흓
푇
푖
휔
4
푖
푠
2
+
2
휁
푖
휔
푖
푠
+
휔
2
푖
(28)
6
Substituting Eq. (28) into Eq. (12) and simplifying ultimately results in the following expression for
G
(
푠
)
:
G
(
푠
)
=
푠
2
M
∗
퐵퐵
+
푠
C
∗
퐵퐵
+
K
∗
퐵퐵
−
푠
2
푛
퐼
∑︁
푖
=
1
(
e
푖
+
푠
f
푖
)(
e
푖
+
푠
f
푖
)
푇
휔
4
푖
푠
2
+
2
휁
푖
휔
푖
푠
+
휔
2
푖
(29)
where
e
푖
=
2
휁
푖
휔
푖
K
퐵퐼
−
휔
2
푖
C
퐵퐼
흓
푖
and
f
푖
=
K
퐵퐼
−
휔
2
푖
M
퐵퐼
흓
푖
are modal vectors and
M
∗
퐵퐵
=
M
퐵퐵
−
M
퐵퐼
K
−
1
퐼퐼
K
퐼퐵
−
K
퐵퐼
K
−
1
퐼퐼
M
퐼퐵
+
K
퐵퐼
K
−
1
퐼퐼
M
퐼퐼
K
−
1
퐼퐼
K
퐼퐵
(30)
K
∗
퐵퐵
=
K
퐵퐵
−
K
퐵퐼
K
−
1
퐼퐼
K
퐼퐵
(31)
C
∗
퐵퐵
=
C
퐵퐵
−
C
퐵퐼
K
−
1
퐼퐼
K
퐼퐵
−
K
퐵퐼
K
−
1
퐼퐼
C
퐼퐵
+
K
퐵퐼
K
−
1
퐼퐼
C
퐼퐼
K
−
1
퐼퐼
K
퐼퐵
(32)
The derivation of Eq.
(29)
is conceptually straightforward, but the details are involved and are left to [
8
]. Equation
(29)
emphasizes that a force or moment on
퐵
induces both rigid body and elastic motions. In practice, it is advantageous to
precompute
e
푖
and
f
푖
to avoid unnecessary calculations of computationally expensive matrix-vector products during
repeated evaluations of Eq. (29).
If the
푛
퐵
boundary DOFs fully restrain the structure’s
푛
퐵
rigid body modes, then
M
∗
퐵퐵
is the rigid body mass
matrix of the unrestrained and undeformed structure (evaluated with respect to
퐵
) and
K
∗
퐵퐵
is the projection of the
unrestrained stiffness matrix
K
onto the rigid body modes, i.e.,
K
∗
퐵퐵
=
0
푛
퐵
×
푛
퐵
(otherwise,
K
∗
퐵퐵
is singular but non-zero).
The properties of
C
∗
퐵퐵
depend on the damping model. A particularly convenient choice for the damping model is
stiffness-proportional damping, in which case
C
=
(
2
휁
1
/
휔
1
)
K
where
휔
1
and
휁
1
are respectively the first-mode natural
frequency and damping ratio. With stiffness-proportional damping, higher-frequency modes are more heavily damped;
specifically,
휁
푖
=
휁
1
(
휔
푖
/
휔
1
)
for
푖
=
1
, ..., 푛
퐼
. Thus,
C
∗
퐵퐵
=
0
푛
퐵
×
푛
퐵
and
e
푖
=
0
푛
퐵
×
1
. Together, these assumptions
reduce Eq. (29) to
G
(
푠
)
=
푠
2
M
∗
퐵퐵
−
푠
4
푛
퐼
∑︁
푖
=
1
f
푖
f
푇
푖
휔
4
푖
푠
2
+
2
휁
푖
휔
푖
푠
+
휔
2
푖
(33)
Equation (33) is equivalent to Eq. (8.21) in [24] when
푠
=
푗휔
and
휁
푖
=
0
for
푖
=
1
, ..., 푛
퐼
.
The modal participation vector
f
푖
is related to the dynamic reaction of the
푖
th mode on
퐵
due to an external force or
moment on
퐵
[
24
, p. 187–190]. The corresponding modal mass matrix
M
푖
=
f
푖
f
푇
푖
/
휔
4
푖
determines the flexible body
accelerations required to dynamically react an external force or moment on
퐵
and how those accelerations are distributed
among the eigenmodes. The total modal mass matrix is
푛
퐼
∑︁
푖
=
1
M
푖
=
K
퐵퐼
K
−
1
퐼퐼
M
퐼퐼
K
−
1
퐼퐼
K
퐼퐵
−
K
퐵퐼
K
−
1
퐼퐼
M
퐼퐵
−
M
퐵퐼
K
−
1
퐼퐼
K
퐼퐵
+
M
퐵퐼
M
−
1
퐼퐼
M
퐼퐵
(34)
which is derived from
M
푖
=
f
푖
f
푇
푖
/
휔
4
푖
and Eqs.
(21)
–
(23)
. Equation
(34)
is only a function of the finite element mass and
stiffness matrices, i.e., it is independent of the computed eigenmodes.
A closed-form expression for the transfer function
H
(
푠
)
=
G
−
1
(
푠
)
can be derived using the Woodbury matrix
identity [
25
], or equivalently, repeated applications of the Sherman-Morrison formula [
25
] (see also [
26
]). The latter
is useful for developing reduced-order models with only a handful of retained modes because it results in explicit
relationships for the modal interactions.
Using the Woodbury matrix identity to invert
G
(
푠
)
requires rewriting the modal expansion in Eq.
(33)
as a matrix
product. To do this, let
횲
(
푠
)
=
diag
푠
2
+
2
휁
1
휔
1
푠
+
휔
2
1
,..., 푠
2
+
2
휁
푛
퐼
휔
푛
퐼
푠
+
휔
2
푛
퐼
and
F
=
f
1
/
휔
2
1
,...,
f
푛
퐼
/
휔
2
푛
퐼
, from
which it follows that
G
(
푠
)
can be written as
G
(
푠
)
=
푠
2
M
∗
퐵퐵
−
푠
4
F
횲
−
1
(
푠
)
F
푇
(35)
Applying the Woodbury matrix identity then results in the following exact expression for
H
(
푠
)
=
G
−
1
(
푠
)
:
H
(
푠
)
=
1
푠
2
M
∗−
1
퐵퐵
+
M
∗−
1
퐵퐵
F
횲
(
푠
)−
푠
2
F
푇
M
∗−
1
퐵퐵
F
−
1
F
푇
M
∗−
1
퐵퐵
(36)
This is the 6-DOF generalization of the transfer function from
푢
1
to
푥
1
from Sec. II [Eq.
(4)
]. Equation
(36)
is
subsequently specialized for the case of a single-axis, single-mode model in Sec. III.C.
7
B. Mode Selection Criteria
Following [
24
, p. 191–192], Eq.
(33)
shows how to select “dominant” eigenmodes for general reduced-order models,
i.e., which modes to retain in a truncated modal expansion. Specifically,
f
푖
measures the modal participation of mode
푖
;
the larger the magnitude of
f
푖
, the larger the dynamic reaction on
퐵
, the larger the modal mass, and the more dominant
the mode. For these reasons, [
24
] suggests using the magnitude of the
푖
th term in the modal expansion from Eq.
(33)
to
rank modes from most dominant to least dominant:
푝
푖
=
||
f
푖
/
휔
2
푖
||
2
2
=
tr
M
푖
(37)
Here,
||·||
2
denotes the Euclidean norm. Larger values of
푝
푖
correspond to more dominant modes. The
휔
4
푖
in the
denominator of Eq.
(37)
implies that dominant modes tend to be lower-frequency modes, but in general, the dominant
mode is not necessarily the lowest-frequency mode.
Equation
(37)
is ill-defined for structures with both translational and rotational DOFs. To remedy this, the coordinate
partition from [
27
] is used to develop a modified mode selection criterion. Specifically,
f
푖
is partitioned into translational
(
푇
) and rotational (
푅
) DOFs such that
f
푇
푖
=
f
푇
푖,푇
,
f
푇
푖,푅
. The modes are then sorted according to the following criterion:
푞
푖
=
1
2
©
«
||
f
푖,푇
/
휔
2
푖
||
2
2
푛
퐼
∑︁
푖
=
1
||
f
푖,푇
/
휔
2
푖
||
2
2
+
||
f
푖,푅
/
휔
2
푖
||
2
2
푛
퐼
∑︁
푖
=
1
||
f
푖,푅
/
휔
2
푖
||
2
2
ª
®
®
®
®
®
¬
(38)
where
f
푖,푇
f
푇
푖,푇
/
휔
4
푖
and
f
푖,푅
f
푇
푖,푅
/
휔
4
푖
are the translational and rotational blocks of the modal mass matrix
M
푖
, the
summations in the denominators are evaluated using Eq.
(34)
, and again larger values correspond to more dominant
modes. Equation
(38)
is the average of the
푖
th mode’s translational modal mass (normalized by the total modal mass)
and rotational modal inertia (normalized by the total modal inertia). Equivalently, it is a normalized measure of the
푖
th
mode’s dynamic reaction on
퐵
. For models with either translational DOFs or rotational DOFs, but not both, Eq.
(38)
reduces to a normalized version of Eq.
(37)
. Importantly, both Eqs.
(37)
and
(38)
, and hence, the resulting mode
sortings, are invariant to reference frame transformations.
Equation
(38)
is equivalent to the effective interface mass introduced by Kammer and Triller [
27
]. Specifically,
M
푖
is equivalent to their matrix
[
M
푖
]
. This is straightforward to show by expanding
[
M
푖
]
, substituting Eq.
(22)
, and using
the orthogonality of
횽
[Eq.
(15)
]. With collocated sensors and actuators, effective interface mass, and by extension,
Eq.
(38)
, also measures the relative controllability and observability of each fixed-interface mode [
28
]. The higher the
value of Eq.
(38)
, the more controllable and observable the mode. Moreover, effective interface mass is closely related
to the balanced singular values [
29
] from the balanced truncation method of model reduction [
30
]. For undamped free
vibrations, balanced truncation yields normal vibration modes [31].
For a single-axis modal model, the dominant mode is the mode with the maximum modal mass or inertia per
axis, i.e., the maximum absolute value in the corresponding DOF in
f
푖
. In general, different modes are dominant for
translational and rotational motions about different axes.
Special considerations are required for symmetric structures. These structures have natural frequencies (eigenvalues)
with multiplicities greater than one; the actual multiplicity of a given eigenvalue depends on a structure’s specific
symmetries [
32
]. Due to the limitations of floating point computations and the accumulation of round-off errors, it is
often difficult to distinguish between symmetric modes with repeated eigenvalues and merely closely spaced modes.
Conveniently, the same criteria used to sort modes can also be used to identify repeated eigenvalues. In particular,
both the magnitude of the modal participation vector
f
푖
and the trace of the modal mass matrix
M
푖
are invariant to the
operations of a symmetry group. Hence, these quantities are both invariant for symmetric modes, meaning symmetric
modes have the same values (to within close numerical tolerances) of both Eq.
(37)
and Eq.
(38)
. Closely spaced modes,
on the other hand, typically have distinct values of both Eq.
(37)
and Eq.
(38)
. This has important implications for the
development of reduced-order models for symmetric structures, as discussed further in Sec. III.D.
C. Reduction to Canonical Model
For the case when only the
푖
th mode is retained in the modal expansion, i.e., the
푖
th mode is the dominant mode,
Eq. (36) reduces to
H
(
푠
)
=
1
푠
2
M
∗−
1
퐵퐵
+
M
∗−
1
퐵퐵
M
푖
M
∗−
1
퐵퐵
1
−
tr
M
∗−
1
퐵퐵
M
푖
푠
2
+
2
휁
푖
휔
푖
푠
+
휔
2
푖
(39)
8
which uses the definition of the modal mass matrix
M
푖
=
f
푖
f
푇
푖
/
휔
4
푖
and the identity
f
푇
푖
M
∗−
1
퐵퐵
f
푖
/
휔
4
푖
=
tr
M
∗−
1
퐵퐵
M
푖
. In
turn, the relationship between
X
퐵
(
푠
)
and
U
′
퐵
(
푠
)
takes the form
X
퐵
(
푠
)
=
1
푠
2
I
푛
퐵
×
푛
퐵
+
M
∗−
1
퐵퐵
M
푖
1
−
tr
M
∗−
1
퐵퐵
M
푖
푠
2
+
2
휁
푖
휔
푖
푠
+
휔
2
푖
!
U
′
퐵
(
푠
)
(40)
where
U
퐵
(
푠
)
=
M
∗
퐵퐵
U
′
퐵
(
푠
)
. Aside from the modal truncation, Eqs. (39) and (40) are both exact.
From here, several assumptions are required to reduce Eq.
(40)
to a form analogous to the canonical model [Eq.
(4)
].
Such a model relates the response of a generalized displacement
푋
퐵,푗
(
푠
)
to the corresponding scalar control input
푈
′
퐵,푗
(
푠
)
for some
푗
=
1
, ..., 푛
퐵
. For this reason,
U
′
퐵
(
푠
)
is restricted to a generalized acceleration input about a single
axis, i.e.,
U
′
(
푠
)
=
e
푗
푈
′
퐵,푗
(
푠
)
where
e
푗
is a standard unit basis vector in
R
푛
퐵
. Similarly, the generalized displacement
X
퐵
(
푠
)
is restricted to a translation or rotation about a single axis by left-multiplying both sides of Eq.
(40)
by
e
푇
푗
, i.e.,
푋
퐵,푗
(
푠
)
=
e
푇
푗
X
퐵
(
푠
)
. Additional assumptions decouple the
퐵
-set coordinates. In particular, it is assumed that the
퐵
-set
coordinates correspond to a node located at the structure’s undeformed center of mass and that the global finite element
reference frame coincides with principal inertial axes. These assumptions diagonalize
M
∗
퐵퐵
. It follows that
푋
퐵,푗
(
푠
)
푈
′
퐵,푗
(
푠
)
=
1
푠
2
+
푀
푖,푗푗
/
푀
∗
퐵퐵,푗푗
1
−
tr
M
∗−
1
퐵퐵
M
푖
푠
2
+
2
휁
푖
휔
푖
푠
+
휔
2
푖
(41)
where the subscript
푗 푗
denotes the
푗
th main diagonal entry of the corresponding matrix. Lastly, it is assumed that
tr
M
∗−
1
퐵퐵
M
푖
≈
푀
푖,푗푗
/
푀
∗
퐵퐵,푗푗
, i.e., the matrix
M
∗−
1
퐵퐵
M
푖
has a single dominant main diagonal term. With this assumption,
Eq. (41) simplifies to
푋
퐵,푗
(
푠
)
푈
′
퐵,푗
(
푠
)
=
1
푠
2
+
휂
푗푗
푠
2
+
2
1
+
휂
푗푗
휁
푖
휔
푖
푠
+
1
+
휂
푗푗
휔
2
푖
(42)
with mass ratio
휂
푗푗
=
푀
푖,푗푗
푀
∗
퐵퐵,푗푗
−
푀
푖,푗푗
(43)
Comparing Eqs.
(42)
and
(4)
reveals that
푚
1
=
푀
∗
퐵퐵,푗푗
−
푀
푖,푗푗
, i.e., the
difference
between the rigid body mass and the
modal mass, not the rigid body mass itself;
푚
2
=
푀
푖,푗푗
;
휁
=
휁
푖
; and
휔
푛
=
휔
푖
. Thus, the parameters in the canonical
model are related to the rigid body mass and the dominant mode’s modal mass, damping ratio, and natural frequency
(where again “mass” is to be interpreted in the generalized sense of either translational or rotational inertia)
D. Canonical Model for Symmetric Structures
The derivation of single-axis modal models requires special considerations when the dominant eigenmode corresponds
to a repeated eigenvalue. This is particularly important for flexible spacecraft with symmetric structural architectures.
The analysis that follows specifically considers 4-fold symmetric structures, a class of structures that includes solar
sails and the Caltech SSPP spacecraft [
6
], but the approach and conclusions readily generalize to structures with other
symmetries. 4-fold symmetric structures have eigenvalues with multiplicities of either one or two [32].
When the dominant eigenmode corresponds to a repeated eigenvalue, Eq. (42) underpredicts the magnitude of the
elastic disturbance on the spacecraft bus. Instead of truncating the modal expansion after the dominant mode, the correct
truncation includes both the dominant mode (assumed to be the
푖
th mode with natural frequency
휔
푖
) and the associated
symmetric mode (assumed to be the
(
푖
+
1
)
th mode with natural frequency
휔
푖
+
1
=
휔
푖
). In this case,
f
푖
=
Tf
푖
+
1
where
T
is an orthogonal matrix corresponding to a symmetry operation [
32
]. Assuming both modes share the same damping
ratio
휁
푖
=
휁
푖
+
1
, then the truncated form of Eq. (36) is
H
(
푠
)
=
1
푠
2
M
∗−
1
퐵퐵
+
M
∗−
1
퐵퐵
(
M
푖
+
M
푖
+
1
)
M
∗−
1
퐵퐵
1
−
tr
M
∗−
1
퐵퐵
M
푖
푠
2
+
2
휁
푖
휔
푖
푠
+
휔
2
푖
(44)
the derivation of which uses
f
푖
=
Tf
푖
+
1
; the invariance of the rigid body mass matrix
M
∗
퐵퐵
to symmetry operations, i.e.,
M
∗
퐵퐵
=
T
푇
M
∗
퐵퐵
T
; the definition of the modal mass matrix
M
푖
=
f
푖
f
푇
푖
/
휔
4
푖
; and the identity
f
푇
푖
M
∗−
1
퐵퐵
f
푖
/
휔
4
푖
=
tr
M
∗−
1
퐵퐵
M
푖
.
The derivation of Eq.
(44)
additionally assumes that
f
푇
푖
f
푖
+
1
=
0
(which reflects that the symmetry operation for a 4-fold
symmetric structure is a rotation by 90 deg) and that
M
∗
퐵퐵
is diagonal with equal inertias about the axes of symmetry.
9
From here, the reduction to the canonical model [Eq.
(4)
] mirrors the derivation from Sec. III.C with the caveat
that it is now assumed that the matrix
M
∗−
1
퐵퐵
M
푖
contains two main diagonal terms of similar magnitudes. Thus,
tr
M
∗−
1
퐵퐵
M
푖
≈
푀
푖,푗푗
/
푀
∗
퐵퐵,푗푗
+
푀
푖,푘푘
/
푀
∗
퐵퐵,푘푘
, which due to symmetry is equivalent to
(
푀
푖,푗푗
+
푀
푖
+
1
,푗푗
)/
푀
∗
퐵퐵,푗푗
.
Since the
푖
th mode is the dominant mode,
푀
푖,푗푗
> 푀
푖
+
1
,푗푗
. With this assumption, the transfer function from
푈
′
퐵,푗
(
푠
)
to
푋
퐵,푗
(
푠
)
takes the canonical form
푋
퐵,푗
(
푠
)
푈
′
퐵,푗
(
푠
)
=
1
푠
2
+
휂
푗푗
푠
2
+
2
1
+
휂
푗푗
휁
푖
휔
푖
푠
+
1
+
휂
푗푗
휔
2
푖
(45)
where the mass ratio is now given by
휂
푗푗
=
푀
푖,푗푗
+
푀
푖
+
1
,푗푗
푀
∗
퐵퐵,푗푗
−
푀
푖,푗푗
+
푀
푖
+
1
,푗푗
(46)
Equivalently,
휂
푗푗
=
1
+
푀
푖
+
1
,푗푗
푀
푖,푗푗
푀
푖,푗푗
푀
∗
퐵퐵,푗푗
−
푀
푖,푗푗
+
푀
푖
+
1
,푗푗
(47)
which shows that the elastic disturbance due to the
(
푖
+
1
)
th mode is proportional to the ratio between its modal mass
푀
푖
+
1
,푗푗
and the dominant mode’s modal mass
푀
푖,푗푗
.
IV. Structure-Based Slew Maneuver Requirements
The canonical flexible spacecraft model from Sec. II (or an analogous reduced-order model from Sec. III) provides a
useful tool for developing slew maneuver requirements, i.e., requirements on the ratio
푇
/
푇
푛
between the slew maneuver
duration
푇
and the fixed-base natural period
푇
푛
. To that end, this section first uses a bang-bang slew maneuver to
demonstrate that settling time is a poor metric for deriving flexible spacecraft slew maneuver requirements. This
motivates the use of a metric based on the amplitude of the residual disturbance due to the flexible dynamics instead.
A smooth slew maneuver then highlights how tailoring the “shape” of the slew profile can decrease the excitation of
the flexible mode relative to the baseline bang-bang case. The latter is a fairly well-known result in general (see e.g.,
[
33
–
36
]) and is the premise underlying the use of input shaping [
37
,
38
] for reducing residual vibrations, but is less
well-used in the definition of flexible spacecraft slew maneuver requirements.
A. Reference Slew Maneuvers
Two reference slew maneuvers are considered in this paper: a bang-bang slew and a smooth polynomial slew, both
of which are nominally rest-to-rest maneuvers through a generalized displacement
Δ
푥
in time
푇
. Their accelerations,
velocities, and displacements are depicted in Fig. 3. It is emphasized that these maneuvers are not necessarily appropriate
for implementation on actual flight systems. Rather, they are merely intended to illustrate some of the important design
considerations associated with slewing flexible spacecraft.
A bang-bang slew is the time-optimal, rest-to-rest, single-axis reorientation maneuver for a rigid body with angular
acceleration (torque) constraints [39]. Each “bang” is a step acceleration input of magnitude
¥
푥
max
=
4
Δ
푥
푇
2
(48)
and duration
푇
/
2
, as depicted in Fig. 3a. For a rigid body, the first “bang” linearly accelerates the system from rest to a
peak velocity of
¤
푥
max
=
¥
푥
max
푇
2
=
2
Δ
푥
푇
(49)
at time
푡
=
푇
/
2
. The second “bang” then linearly decelerates the system back to rest; see Fig. 3b. The constant
magnitude accelerations yield the quadratic variation in the generalized displacement
푥
(
푡
)
shown in Fig. 3c.
Comparing the bang-bang and smooth polynomial slew maneuvers emphasizes that both the slew profile and the
ratio
푇
/
푇
푛
between the slew maneuver duration
푇
and the natural period
푇
푛
determine the disturbance due to the flexible
dynamics. Following [
33
], the smooth slew maneuver considered here is based on a higher-order (in this case, 7th-order)
polynomial for the generalized displacement
푥
(
푡
)
:
푥
(
푡
)
Δ
푥
=
−
20
푡
푇
7
+
70
푡
푇
6
−
84
푡
푇
5
+
35
푡
푇
4
(50)
10
(a)
(b)
(c)
Fig. 3 Comparison of rest-to-rest bang-bang and polynomial slew profiles for rigid spacecraft: (a) accelerations,
(b) velocities, and (c) displacements.
This is the lowest-order polynomial that can simultaneous satisfy boundary conditions on
푥
(
푡
)
and its velocity,
acceleration, and jerk (the time derivative of the acceleration). The polynomial coefficients in Eq.
(50)
correspond to a
rest-to-rest slew through a generalized displacement
Δ
푥
in time
푇
with zero-velocity, zero-acceleration, and zero-jerk
boundary conditions. The zero-jerk boundary conditions reduce jerk by flattening the acceleration curve in the vicinities
of the start and end points. Reducing jerk usually reduces the amplitude of the disturbance due to the flexible dynamics
[
33
–
36
]. Additionally, momentum control systems are usually jerk-limited [
36
], meaning hardware limitations may
also mandate the use of low-jerk slew trajectories. The polynomial slew smoothly accelerates and then decelerates a
nominally rigid spacecraft from rest-to-rest.
Compared to the baseline bang-bang slew, the polynomial slew requires higher peak accelerations to achieve the
same total displacement in the same time. In this case, the peak acceleration is
¥
푥
max
=
84
√
5
Δ
푥
25
푇
2
(51)
which occurs at times
푡
=
(
5
∓
√
5
)
푇
/
10
. The peak acceleration is approximately 1.9 times higher than the peak
acceleration for the bang-bang slew and leads to proportionally higher peak structural loads. Larger accelerations also
11