International Journal of Advanced Robotic Systems
ARTICLE
Multi-agent Orbit Design For Visual
PMeurlctie-a pgteionnt OErnbhit aDnecseigmn efonrt VPisuurapl oses
Perception Enhancement Purpose
Regular Paper
immediate
Hamidreza Nourzadeh1,* and John McInroy2
1 Rensselaer Polytechnic Institute, Troy, New York, USA
2 University of Wyoming, Laramie, Wyoming, USA
* Corresponding author E-mail: nourzh@rpi.edu
Received 27 Mar 2014; Accepted 23 Jul 2014
DOI: 10.5772/58894
∂ 2014 The Author(s). Licensee InTech. This is an open access article distributed under the terms of the Creative
Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use,
distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract This paper develops a robust optimization-based 1. Introduction
method to design orbits on which the sensory perception
of the desired physical quantities are maximized. It In many practical applications, there is an increasing
also demonstrates how to incorporate various constraints tendency to use a network of inexpensive agents with
imposed by many spacecraft missions, such as collision more sophisticated data-gathering equipment. This
avoidance, co-orbital configuration, altitude and frozen involves exploiting the coordinated operation of the
orbit constraints along with Sun-synchronous orbit agents to achieve more accurate measurements, and also
constraints. The paper specifically investigates designing makes the mission more robust to possible errors and
orbits for constrained visual sensor planning applications faults. Challenging problems in multi-sensor management
as its case study. For this purpose, the key elements to systems arise when the sensors operate in a dynamic
form an image in such vision systems are considered and environment, and they should be repeatedly moved to
effective factors are taken into account to define a metric perceive the desired physical quantities of non-stationary
for perception quality. The method employs a max-min targets. In such a multi-agent system, in order to leverage
model to ensure robustness against possible perturbations resources more efficiently, a meticulous plan of action is
and model uncertainties. While fulfilling the mission required to distribute tasks among the agents [1].
requirements, the algorithm devises orbits on which a
higher level collective observation quality for the desired In this paper, a special class of multi-agent systems to
sides of the targets is available. The simulation results acquire sensory measurements from physical phenomena
confirm the effectiveness of the proposed method for of interest is considered. Many orbit design problems
several scenarios involving low and medium Earth orbits belong to this class of multi-agent systems. In fact,
as well as a challenging space-based space surveillance one scenario that frequently happens in various space
program application. missions is the design of satellite trajectories on which
the perception qualities of certain physical quantities of
some targets are maximized. In this constrained path
Keywords Orbit Design, Perception Enhancement, Sensor planning problem, the orbit should also fulfill particular
Network, Multi-agent System requirements imposed by the mission. Depending upon
the system configuration and requirements, the design
problem could be quite elaborate, even for a small number
of agents and targets.
Hamidreza Nourzadeh and John McInroy: Multi- agent IOnrtb Ji tA Ddevs Rigonb ofot rS Vysistu, a2l0 P1e4r,c e1p1t:i1o6n1 E |n dhoain: c1e0m.5e7n7t 2P/u5r8p8o9s4e 1
There is an extensive literature on the subject of orbit Boain’s scheme [2] of SSO mission design to the
design for different science missions specifically oriented multi-agent case. (3) To handle possible perturbations and
towards special types of orbits [2–5]. In these studies, the uncertainties in the system, a max-min model is employed
main aim is to define the attributes of the mission, usually to bring robustness into the planning. (4) A perception
without providing information about multi-agent path model for the vision system is derived by considering
planning. In another interesting field of study, researchers key contributing factors affecting the quality of the
have addressed a coordinated trajectory design for images which encompasses a wide range of circumstances
multi-agent, multi-target systems in the context of occurring in space situational awareness applications. The
formation flying. The formation flying research field seeks model is particularly useful for spaced-based SSA (SBSSA)
to develop strategies to control the formation of a satellite applications, e.g., a spaced-based space surveillance
fleet in order to meet a global performance objective, and programwhosemain goal is to build a network of satellites
the focus of interest is to employ an adequate control equipped with powerful vision systems in order to detect
strategy to maintain the desired formation [6–10]. In many and track objects in Earth orbits.
applications, the ultimate goal is to effectively allocate each
sensor to a target at any point in time [11, 12]. However, This paper is organized as follows. Section 2 formulates
before performing the resource allocation process, there the coordinated trajectory design problem that determine
are fundamental questions that need to be addressed early the paths on which the quality of the received information
in the planning stage. The first and most important is maximized (in a sense). Section 3 discusses contributing
concerns how the agents (i.e., the leaders in a cluster elements in image formation and defines a quality metric
formation) should be placed or moved with respect to the for a space situational awareness (SSA) application by
targets during the operation to optimize that particular considering the dominating factors. Section 4 describes the
mission [13]. In other words, prior to the allocation motion equations of the objects and explains two sets of
problem, the agent trajectories that maximize the sensory parameters commonly used to uniquely identify an orbit.
perception of the desired physical quantities should be In Section 5, the constrained multi-agent, multi-target
determined by the planning scheme. trajectory optimization problem is constructed, and a
description is given of how to incorporate the constraints
In this article, the multi-agent orbit design procedure of different configurations that might be employed in
is formalized as a constrained parametric optimization a science mission. Brief remarks on choosing suitable
problem by taking into account different constraints that numerical algorithms to solve the optimization problem
might be involved in a science mission. In particular, the are given in Section 6. Simulation studies are included in
Sun-synchronous orbit (SSO) and frozen orbit constraints Section 7, and conclusions are drawn in the final section.
that are widely employed in practical application are
defined in terms of optimization parameters. The 2. Problem Formulation
developed method is not limited to any particular
sensor allocation scheme, and it increases the overall This section focuses on the problem of designing the
system performance since a higher level of perception best agent trajectories on which sensory perception is
quality of the targets is achievable on the designed maximized subject to some constraints imposed by the
orbits. Nourzadeh and McInroy [11] specifically system’s nature. Consider a general path planning
investigate the sensormanagement problem and propose a problem as illustrated in Figure 1. As depicted, two
non-myopic (multi-step) scheduling algorithm to address cube-shaped objects are moving on two different curved
the multi-agent, multi-target sensor allocation problem. paths (AB and CD). The objects passively or else actively
send out certain types of signals that can be perceived
To analyse the performance of the proposed method, at any point of the space with different quality levels.
a special spaced-based space situational awareness At a particular point in time, depending on the system
application is chosen. In this application, the main configuration, specific quality fields are formed for any
objective is to determine agents’ trajectories on which perceptive physical aspects of the objects. In fact, when
the observation quality of some resident space objects objects move on their paths, a time-dependent vector or
(RSOs) is maximized. Although the orbit design procedure scalar field is formed for any of these physical quantities.
is performed in the context of inspection quality, the
proposed formulation is quite general, and any type of
physical quantity can be incorporated and optimized. To
deal with large-scale cases, some guidelines are proposed
to choose an adequate numerical optimization algorithm.
The contributions of this paper are: (1) The proposed
formulation allows us to solve more general multi-agent
orbit design problems seeking to optimize the sensory
perception of multiple attributes of targets. This extends
the existing single scenario optimization schemes. (2)
Figure 1. Path-planning to maximize sensory reception
It also provides a way of incorporating many prevailing
configurations in spacecraft missions, such as collision
avoidance, co-orbital configuration, altitude and frozen In the case of a scalar field, given the state of a targeti j
orbit constraints as well as the SSO. The latter extends i( XR) and of a sensor j( XS), and effective physical factors,
the total achievable quality along the sensor trajectories
2 Int J Adv Robot Syst, 2014, 11:161 | doi: 10.5772/58894
(e.g., EF and GH) in the space is the sum of values of the As mentioned above, two prime requirements to construct
perceptions at all points on the curve. It can be expressed the optimization problem 4 are the perception function,
by Equation 1: Φ, and the trajectories parameter vector, p. In the
next sections, these elements are defined for a visual
· · · measurement SSA application. In this scenario, it is∫t i = 1, , n2 assumed that all the agents are equipped with passive
Φ (iX jl R(t), XS(t))dt, j = 1, · · · ,m (1) visual sensors and that the physical quantities that are
t1 l = 1, · · · , L maximized along the trajectories are inspection qualities
of the different sides of the objects. Furthermore, each
where Φl is a function that describes the perception value trajectory is parametrized by orbital elements parameters.
of quantity l sensed by sensor j from object i at any point in While the following section elaborates the calculation
time. The symbols m,n and L denote the number of agents, process of the observation quality given the states of the
targets and physical quantities, respectively. objects and the cameras in this scenario, Section 4 describes
the motion dynamics of agents and targets as well as the
In the discrete case, the perception quality of target i trajectories parametrization process.
obtained by all the sensors can be written as Equation (2).
For i = 1, · · · , n and l = 1, · · · , L, (2) gives a perception
quality matrix J n×Lsum ∈ R . The (i, l) element of J 3. Optical Imaging Quality Calculationsum
gives the lth quality attribute of the ith object along the There are many vision systems with different sensors
trajectories: and means of data acquisition, and the process of sensor
m N planning is highly dependent on the way in which
J i jsum = ∑ ∑ Φl( XR[k], XS[k]). (2) information is gathered in that specific application. In
j=1 k=1 some vision systems, only passive sensors are used while
both passive and active sensors are used in some other
In this paper, it is presumed that each trajectory can be machine vision systems. For simplicity, suppose that all
parametrized with a set of parameters. Concretely, jX the vision systems in our application are equipped withS
moves on the trajectory j that is parametrized by p ∈ passive sensors (although the methods developed herej
npj where np is the number of parameters required to also apply to active sensing). To compute the qualities ofR j
m the resulting images, all the effective factors for forming an
∑ npj image should be considered, i.e., the intrinsic parameters
represent trajectory j. Let p ∈ R j=1 be a vector consisting of the sensors and environmental factors. There are a lot
of all the requisite parameters to describe the sensors’ of techniques for automatically determining image quality
trajectories (Equation (3)): [14–18].
m In this section, the preliminary requirements to compute
p = ⊕ pj. (3) the observation quality in an optical imaging system will
j=1
be demonstrated in detail by investigating all the effective
Here, ⊕ is the vertical concatenation symbol. aspects in forming images and defining a quality metric
based upon them.
It is worthwhile noting that the relation between the
parameters pj and the trajectory j does not have to be 3.1 Effective Factors on Image Quality
explicit, and even a heuristic path planning algorithm that
generates a trajectory j based on a tuning parameter pj can The output of an optical imaging system depends
be utilized here. on several physical factors, i.e., lighting, atmospheric
attenuation, light diffraction, occlusion, imaging sensor
Depending upon the application and how to maximize resolution and sensitivity, electronics parts and output
the Jsum matrix, various possible optimization strategies devices (Figure 2). The modulation transfer function
can be considered to acquire different sets of sensors’ (MTF) and the contrast transfer function (CTF) are two
trajectories. In this article, to deal with worst cases, prevalent techniques for assessing the performance of a
a max-min optimization model is employed to design vision system. In fact, the MTF enables the quantifying
trajectories which enforce planning redundancy, andmake of the resolving ability of each component of the system
the result robust to the probable uncertainty sources in in different spatial frequencies. The cut-off frequency of
the system. Therefore, the optimization formulation MTFs can be used to determine the performance of each
determines the sensors’ trajectories that maximize the component [19, 20].
minimum achievable perception for all objects (the
minimum element of Jsum) subject to some constraints In SSA applications, only some physical aspects have a
imposed by the system (Equation (4)). Figure 1 portrays significant influence on the output image. As such, in
the case when each agent’s trajectory is confined to planes subsequent sections, only the illumination, light fraction,
z = 0 and x = 0: occlusion and side observation quality will be taken into
account.
{
Maximize min(Jsum)p (4)
subject to : Constraints
Hamidreza Nourzadeh and John McInroy: Multi- agent Orbit Design for Visual Perception Enhancement Purpose 3
Electronics /
Scene illumination Atmosphere Optics Detector array Output DeviceCommunication i.e., diffuse, Lambertian and specular surfaces. Specular
LCD surfaces behave like a mirror and reflect light into a
CRT lobe of the specular direction, and so the reflected light
is highly dependent on the illumination direction, while
CRT Projector
for Lambertian and diffuse surfaces such as cotton cloth,
Printer matte paper and matte paint, the radiance leaving the
i(x, y) hatm (x, y) hoptics (x, y) h h (x, y)det (x, y) elec hOutDev (x, y) surface does not have any meaningful correlation with
Input scene O(x, y)
(Optical flux) Atmosphere Optics Detector Electronics
Output
Device the illumination direction, and the bidirectional reflectance
distribution function (BRDF) of these surfaces is constant
Figure 2. Contributing elements on the image formation and [22]. Since the outer surfaces of man-made satellites are
quality highly reflective, the angle of the light with respect to the
viewing axis is extremely critical. Let αlit be the angle
between the viewing axis and incoming sunlight (Figure
3.1.1 Light diffraction 4). When αlit 0, the image contains much specular glare.
On the other hand, if αlit > 90◦, the satellite will be back-lit
In optics, diffraction is categorized into two different by the Sun, meaning that it will produce a silhouette with
classes, i.e., Fraunhofer diffraction (or far-field diffraction) fewer details. α = 45◦ will provide the best illumination
and Fresnel diffraction (or near-field diffraction). level. A suitable function that describes the illumination
Fraunhofer diffraction occurs when waves of a far-field effect and which lies between zero and one can be written
distance are passed through an aperture or slit. This as Equation (7), where qlit is the minimum value that the
happens when the Fresnel number F 1 and the parallel illumination quality takes. Figure 3 depicts the values
rays approximation is applicable. which qlit takes on as αlit changes from zero to 360 degrees
in polar coordinates when q = 0.2:
Because of the Fraunhofer diffraction phenomenon, lit
even an optical lens with perfect quality has limited {
performance and the MTF of each lens expresses this q + (1− q )sin2lit lit (2α πlit) αlit < 2
limitation in a spatial frequency. In fact, light coming qlit = qlit αlit ≥ π
(7)
2
from a point light source diffracts and forms an airy disk
pattern. Using this pattern, Rayleigh proposed a criterion
to find an optical resolution of the lens. The criterion states Figure 4 illustrates the illumination quality of an
that two points with a angular separation equal to the observation for three RSOs which can be observed by two
angular radius of the airy disk can be resolved. Equation 5 observer satellites (O1, O2 ). Observer 2 inspects RSO
relates the angular resolution to a wavelength of light ( ) 1, and the angle α
12k
lit , the angle between sunlight andθ λ
and the diameter of a lens aperture D [21]: the observation direction, is around 45
◦. Therefore, the
illumination quality will be around 1. Observer 1 has two
options to inspect: either RSO 1 or RSO 3. The image
taken of RSO 3 is of better quality in terms of illumination
λ 21k ◦
sinθ = 1.22 (5) (αlit ≈ 45 ), while the RSO 1 image will have a silhouetteD problem, since α11k > 90◦lit .
The image-resolving quality for a system with a fixed
aperture diameter D and a fixed wavelength λ can be 3.1.3 Occlusion and Sunlight Quality
expressed by Equation (6), where δ is the distance between
the object and the lens. Since an occluded object cannot be inspected by the
observer, the resulting image quality for that object should
be considered to be zero. On the other hand, if there
D Effect of the light angle on the image quality
qres = (6) 90 11.22λδ 120 60
0.8
0.6
Thus, by increasing the object distance, the resolving 150 30
0.4
ability decreases because of the diffraction.
0.2
180 0
3.1.2 Illumination
The relationship between the reflected light (radiance)
and the incoming illumination depends on the direction 210 330
from which the light arrives as well as the shape and
type of surface. Absorption, transmission, scattering or 240 300
a combination of these effects applies to the incoming 270
light when it strikes the surface. In other words,
the intensity and the colour reflected depends on the Figure 3. Illumination quality of a specular surface for different
illumination and reflection angle and the surface material. values of the observation angle
Surfaces can be categorized into three different groups,
4 Int J Adv Robot Syst, 2014, 11:161 | doi: 10.5772/58894
is no obstacle between the object and the observer, the D 2
possibility of having a perfect image of that object exists D 3 V R O
(Equation 8):
D 4 D1
{ D 5
qlos = 0 occlusion (8) D 6qlos = 1 no occlusion
Figure 5. Assigned normals to specify the sides of interest and
In order to observe the object, an acceptable level of the computation of the sides observation quality (inner product)
illumination is required. For instance, in satellite imaging, 3.2 Single Quality Metric
sunlit images are more desirable while dark images
obtained from shadow areas behind the Earth are not In order to form a single number reflecting the overall
as useful as sunlit images. To incorporate the sunlight observation quality of an object i inspected by an observer j
quality factor, let qlum = 1 when the object is in sunlight. during a sample k along a face l, all the calculated qualities
Otherwise, qlum is a small positive number(Equation 9): should be combined. Since the overall quality varies from
application to application, this can be done by multiplying
{ weighted qualities. This method allows us to control
q = 1 lit image the participation of all the constituent parts in a singlelum
q 1 dark image (9) quality metric. Let qijkl be the product of all the mentionedlum
qualities (Equation (11)):
3.1.4 Sides Observation Quality
Φ iX k , jX k q qijk ijk ijk ijk ijklIn visual sensor planning, the vast majority of studies l( R[ ] j[ ]) = ijkl = resqlosqlumqangleqview (11)
are mainly concerned with finding the vision system
parameters to inspect every side of an object with a
minimum number of observations [23]. One way to do this For different values of i, j, k and l, qijkl forms a 4D
n×m×N×L
is to assign an outward facing unit normal for each side of observation quality array Q ∈ R+ .
interest, i.e., let n be the lthilk outward facing unit normal
on an object i at a sample time k. To determine whether 4. Orbital Elements and Motion Equations
a specific side of an object is viewed, let vijk be the unit
vector pointing from object i to observer j at sample time 4.1 Motion Dynamics
k. Thus, the observation quality of side l of object i can be
computed as Equation (10): Assuming a uniform, spherical Earth, Equation (12)
represents the motion equation of the target/object on its
orbit:
{
nT v nT
qilkl = ilk ijk ilk
vijk > 0
view (10) µ0 nTilkvijk ≤ 0 ̈r = − r3r + d (12)
Figure 5 depicts a RSOwith some assigned sides of interest where r = [x y z]T is the relative position of the
(Dl). According to Equation (10), (D1) and (D2) have a target with respect to the centre of the Earth. The vector
positive quality and the inner product of v and the other d = [dx dy dz]T denotes exogenous forces, including
side normals are less and equal to zero. This means that an the attitude control system (ACS) actions and modelled
observer in this direction can partially inspect two sides. or unmodelled disturbing forces affecting the target, such
as solar pressure, atmospheric drag, etc. The Earth
Sun light direction gravitational constant µ 398600.4418 Km
3s−2, and r =‖
α lit12k r ‖2, together represent the Euclidean distance of vectorr.O2
Letting the state vector X = [r ̇r]T and the velocity vector
v = ̇r = [ẋ ẏ ż]T , Equation (12) can be rewritten as
RSO2 a set of first-order ordinary differential equations (ODEs)
consisting of only the orbital state vector (Equation (13)):
αlit11k RSO1
αlit O21k 1 [ ]
r
Ẋ = f (X, d) = − r (13)µ r3 + d
RSO3
Figure 4. Determining illumination quality in a SSA application The discretized motion equation can be numerically
solved by choosing an adequate ODE solver (Equation
(14)):
Hamidreza Nourzadeh and John McInroy: Multi- agent Orbit Design for Visual Perception Enhancement Purpose 5
X[k+ 1] = F(X[k], d[k]), X[k] X(kTs). (14) maximize f (p)p
where Ts denotes the sampling rate. subject to gi(p) ≤ 0, i = 1, · · · , ng
(15)
hj(p) = 0, j = 1, · · · , nh
4.2 Orbital Elements
where ng and nh refer to the number of inequality and
When the ACS actions and perturbing forces cancel each equality constraints, respectively:
other (i.e., perfect control), the d will be a zero vector and
the target trajectory forms an elliptic orbit. This orbit can m m
be uniquely determined by knowing the orbital elements p = ⊕ pj = ⊕ [aj, ej,Ωj,ωj, ιj, νj]
T . (16)
j=1 j=1
at an epoch (time t). There are several sets of orbital
parameters that result in the same motion dynamics. In The subscript j denotes the number assigned to the agent
classical two-body systems, Keplerian orbital elements that moves in orbit with the orbital parameter pj, and
and the orbital state vector X comprise two commonly f (p) = min(Jsum(p)).
used sets of orbital elements. The orbital state vector
contains two physical quantities, the position and velocity
of the moving target at an epoch written in the reference 5.1 Co-orbital Configuration
frame.
In many surveillance missions, some agents will be
On the other hand, the Keplerian orbital elements are moving on the same orbit. In this case, all the orbital
directly related to the shape of the resultant orbit and elements of these agents are the same except for ν, which
the position of the target on the path. They consist of can be freely chosen. To enforce this constraint for agents j1
six parameters: the semi-major axis (a) is the sum of and j2 sharing the same orbit, the following linear equality
the periapsis (perigee) and apoapsis (apogee) distances constraint should be added to the optimization problem:
divided by two; eccentricity (e) controls the shape of the T
elliptic orbit and specifies by how much the orbit deviates A p = 0 (17)
from a circular orbit; the inclination ι is the angle between
the equatorial plane and the plane containing the elliptical where AT is a 5× 6m matrix consisting of m horizontally
orbit (elliptical plane); the longitude of the ascending node concatenated 5 × 6 block matrices. All of these
(Ω) is the angle between the ascending node vector and the sub-matrices are a zero matrix except for jth and jth, which
direction of the vernal equinox; the argument of perigee 1 2are replaced by As and −As, respectively. Mathematically,
(ω) orients the elliptical orbit in the elliptical plane, and matrix A can be constructed by the following equations:
it is the angle between the right ascending node direction
and the Perigee’s direction; and finally the true anomaly
(ν) at epoch t represents the geometric angle of the target T
in the orbital plane (Figure 6). m As j = j1
A = ⊕ −ATs j = j2 , As = [I5×5 , 05×1]
Since an orbital state vector can be converted j=1
06×5 otherwise
into Keplerian orbital parameters by a nonlinear .
transformation at any epoch [24], throughout this
paper, any of these representations might be employed
wherever it is more convenient to do so. 5.2 Uniform Distribution
Another configuration that might be considered in orbit
design would be to uniformly distribute those agents
which have a common orbit (trailing formation). Let
O be the number of distinct observers’ orbits and let
mo denote the number of satellites in the orbit o. In
this case, the true anomaly difference of the initial states
of two adjacent satellites in orbit o should be 360
◦
m . Ito
is worthwhile to mention that only in an unperturbed
circular orbit will the difference be maintained during
motion. Figure 7 illustrates one possible configuration
Figure 6. Keplerian and orbital state elements when three observer satellites are evenly distributed with
a constant true anomaly distance.
To incorporate this scheme in the optimization problem,
5. Optimization Formulation consider a subset of agents’ indices that share the same
The problem of maximizing the minimum observation trajectory (orbit o) {j1, j2, · · · , jmo}. Assuming different
quality for all sides of the targets with respect to the types of satellites with different characteristics, there will
sensor’s trajectories parameters can be written as Equation be (mo − 1)! ways to arrange these mo distinct objects
(15): in orbit o. To find the optimal arrangement, one should
solve the optimization problem for all possible orders
6 Int J Adv Robot Syst, 2014, 11:161 | doi: 10.5772/58894
O1 For instance, regardless of the fuel required to retain
Perigee
ν1 orbital speeds, depending upon the material technology, a120 o
satellite can withstand a limited amount of heating caused
by atmospheric resistance. Therefore, for any SSAmission,
O3 there is a minimum operational altitude associated with a
satellite. This restriction can be included in the trajectories
120 o optimization problem by adding the following nonlinear
constraint:
aj(1− ej) > RE + cj, j = 1, · · · ,m (20)
O2
where RE = 6371 Km is the Earth’s radius and cj denotes
Figure 7. Spreading satellites in an orbit the minimum orbital altitude for an agent j.
and choose the one that has the best resultant cost Depending upon the altitude, an orbiting satellite receives
function. This method is impractical when dealing with a different kinds of radiation (such as cosmic rays, van Allen
large-scale system, and to avoid this cumbersome process, radiation, solar flares, etc.) which can cause damage to the
an approximate approach should be taken into account. satellite equipment. To avoid such hazardous exposure,
One heuristic way that results in satisfactory suboptimal the mission designer can choose different strategies based
solutions can be obtained by assuming identical agents on any constraints and key requirements. For a satellite
and solving the corresponding optimization problem. in low Earth orbit (LEO), one way to stay away from
Next, check the performance of all possible arrangements a region of intense radiation is by keeping the altitude
with the resulting trajectories and choose the best of the satellite lower than the Allen radiation inner belt
configuration. altitude, which is around 1,000 Km. Another way is to use
expensive radiation-tolerant components in the satellite
Assuming identical agents, the true anomaly of the first building process to make sure that the equipment operates
agent (ν) in orbit o(agent j1) should vary between 0 and properly throughout the mission.
360 ◦
m . In addition, the following constraints should beo
added to the optimization problem for adjacent agents in Therefore, for a space mission, an upper bound associated
orbit o (Figure 7). with the altitude might be defined. The upper bound
constraint can be expressed by Equation (21):
360 ◦ aj(1− ej) < RE + c̄j, j = 1, · · · ,m (21)
νj2 − ν j1
= m
o− 360 ◦ Here, c̄j denotes the agent j’s maximum orbital altitude.νj3 νj2 = mo
. (18).
. 360 ◦ 5.5 Specify Orbit Typeνjm − ν =o jmo−1 mo
A satellite is classified into various categories according
to its orbital altitude, inclination, eccentricity and period,
5.3 Collision Avoidance Constraint etc. Practically every constraint regarding the shape of
To avoid collisions between agents and moving objects the orbit expressed by orbital parameters can be included
around the Earth, the optimization problem should in the proposed trajectory optimization problem. For
guarantee that all the sensors retain a minimum safe instance, to design an orbit for agent j within LEO, its
distance from other moving objects during the mission. altitude should vary within the 0-2,000 km range, i.e.,
This can be incorporated in the optimization problem (0 < aj < RE + 2000). Similarly, the motion of an agent
(Equation (15)) by heavily penalizing the cost function j moving in a polar orbit can be planned by adding the
whenever the minimum distance (d) between agents and equality constraint ιj = 90
◦ to the problem.
any orbiting objects in all time instances is less than an
acceptable threshold tr (Equation (19)): 5.6 Sun-synchronous Orbit
{
≥ In science missions, one of the most widely-used typesf (p J) = sum(p) d tr (19) of orbits is the SSO. The SSO is a near-polar and0 d < tr almost-circular geocentric orbit whose nodal precession
rate (Ω̇) is equal to the Earth’s mean rotation rate around
5.4 Altitude Constraints the Sun. Geometrically, a SSO approximately orients in
such away that the angle between the orbital plane and the
Satellite altitude is an important parameter that relates vector from the Sun to the Earth remains the same during
to many mission requirements. In the mission design the mission. Therefore, the illumination angle of the
process, many factors, such as launch constraints, type of ground trackwill be constantlymaintained throughout the
satellite, desired perception resolution, mission lifetime, mission. Besides this interesting characteristic, a SSO has
ground track repeatability and mission expenses directly other orbital properties that make it highly desirable for
affect the altitude of the satellite. various applications [2]. In the subsequent subsections, the
SSO orbital parameters’ constraints imposed by a common
Hamidreza Nourzadeh and John McInroy: Multi- agent Orbit Design for Visual Perception Enhancement Purpose 7
scientific mission are briefly discussed, and it is shown
how the mission requirements greatly restrict the feasible
space of the optimization parameters.
5.6.1 Precession Rate Constraint
Due to the out-of-plane gravitational force caused by the
Earth’s equatorial bulge, an orbit plane gyroscopically
precesses. The corresponding nodal precession rate Ω̇
can be operatively computed by Equation (22). To have
a SSO, Ω̇ should be equal to the Earth’s mean orbital
rate around the Sun, i.e., Ω̇ = 365365.242199degree/day =
1.991063802746144× 10−7rad/sec: Figure 9. The locus of the frozen SSO parameters within the LEO
( ) √
Ω̇ − 3
2
J̃ RE µ= 2 2 a(1−e2 a3 cos(ι) (22))
where J̃2 = 1.08263 × 10−3 denotes the Earth’s
dimensionless zonal harmonic coefficient.
Figure 8 illustrates the surface on which the precession
rate condition is satisfied for a SSO within the operational
region of a LEO where RE + 300Km ≤ a ≤ RE + 2000Km.
Although every point on this surface meets the precession
rate condition, a relatively small portion of the surface
contains the admissible orbits’ parameters - for instance,
most points on the surface do not meet the minimum
altitude requirement. In Figure 8, the area surrounded
by black lines contains the points satisfying the minimum
altitude constraint a(1− e) > RE + 300Km.
Figure 10. Observation quality field evolution in time
Figure 8. Surface of all the SSO within a LEO that satisfy the
precession rate condition
5.6.2 Frozen Orbit Constraint
Owing to the perturbing forces caused by the oblateness
of the Earth, the satellite nominal trajectory based on the
selected orbital elements does not remain fixed, instead
varying as a function of time. In a SSO orbit, this fact Figure 11. The best obtained suboptimal solution of Equation
is exploited to achieve the gyroscopic precession of the (25)
orbital plane. On the other hand, perturbing forces also
adversely affect the eccentricity, e, and argument of the
perigee, ω, of the orbit. Perturbation theory states that (Equation (23)). Theoretically, a satellite in such an orbit
systematic choices of the orbital parameters can minimize has a minimum propellant usage requirement during a
the drift from the selected initial values. In a frozen SSO long-term mission:
orbit, the parameters e, ι and ω are picked in such a way
that the secular perturbations of J2 and J
{
3 cancel out, ω̃ = 90 or 270 degrees
and parameters only undergo a relatively small periodic ẽ = − J2 sin(ι) (23)
perturbation with a period equal to the orbit period J3 2a(1−e2)
8 Int J Adv Robot Syst, 2014, 11:161 | doi: 10.5772/58894
where ω̃ and ẽ are the mean value of the desired frozen barrier parameters. Recently, successful attempts have
orbit parameters. been made to develop more robust IPM-type algorithms
[25].
Figure 9 depicts the locus of the all the SSO parameters
that satisfy the frozen orbit constraint. As highlighted, In this paper, since the trajectory design problem
a very small part of the precession rate surface results formulation is block-structured in terms of optimization
in the minimum perturbed orbital parameters set. This parameters, to treat large-scale optimization problems,
has a direct practical implication for the optimization IPM-type solvers are employed, and for small-scale
algorithm performance by drastically decreasing the size problems, SQP methods are used.
of the feasible sets.
7. Simulations
6. Numerical Optimization Algorithm
To overcome obstacles in ground-based SSA in detecting
This section considers the problem of choosing an and tracking space objects, SBSSA has begun to build
adequate optimization strategy to solve the trajectory a network of satellites equipped with different sensors.
design problem governed by Equation (15). The To fulfil this aim, on 26 September 2010, Minotaur IV
optimization algorithm selection is the most important performed its first orbital launch of the SBSS system. The
part of solving a practical optimization problem. This main objective of the SBSS program is to search, detect and
process is performed by considering different factors that track objects in Earth orbits, especially geosynchronous
can affect the overall performance. The performance orbit (GEO) objects. The SBSS program is about to build
of an algorithm is commonly assessed according to a larger constellation of observer satellites to cover wider
its convergence speed, solution optimality, robustness areas of space. With the development of technology, it
to perturbation and uncertainty, required resources, is expected that there will be more SBSSA missions in
computational complexity and implementation difficulty, the near future. The simulation part of this paper is
etc. Quantifying all the effective factors is extremely intended to exploit the proposed multi-agent trajectory
difficult and occasionally impossible for an optimization design method to plan agents’ movement in the context
method without any knowledge about the problem itself. of SBSSA.
Equation (15) is a deterministic constrained nonlinear This section consists of two parts. In the first subsection, an
optimization problem with continuous variables. For illustrative simulation study for a fairly small-scale system
such a large-scale system with several targets and agents, with five RSOs and two agents is presented. In this system,
the idea of utilizing global optimization methods seems all the targets and agents are orbiting within a LEO and
contrived. In fact, current known global approaches in the equatorial plane. In section 7.2, multi-agent orbit
offer successful performance in small- and medium-sized design is performed for more general scenarios involving
problems, as the solution space size exponentially unperturbed a SSO.
increases with respect to the number of decision variables.
Therefore, in this paper, a local optimization algorithm is To perform the numerical integration of the motion
employed to tackle (15). dynamics of all the objects (Equation (13)), the
fourth-order Runge-Kutta (RK4) method is employed
In practice, variations of sequential quadratic with an adequate step size. Since the chosen method to
programming (SQP) and the interior point method solve the trajectory optimization problem results in the
(IPM) are commonly-used algorithms to solve large-scale largest function value in some feasible neighbourhood
general constrained optimization problems. In each (i.e., the local optimum solution), to achieve a decent
iteration, a SQP method generates admissible steps sub-optimal solution for each problem the optimization
towards a local minimum by solving a quadratic model algorithm will be run from different initial parameters.
of the objective function subject to a linearized version
of the problem constraints. SQP-type algorithms have 7.1 Illustrative Case Study
been quite successful, especially in dealing with nonlinear
constraints, and has shown more robustness in relation In this case study, five RSOs are moving in two elliptic
to the badly-scaled optimization problem than IPM orbits in the equatorial plane with the following Keplerian
algorithms. elements at time t = 0. The objective is to plan two agents’
Like SQPs , IPM-type algorithms are also well-known
a e ι Ω ω ν
for their superior performance in solving nonlinear Orbit 1 8033 0.126 0 0 68.02 247, 62, 237
constrained problems. IPM algorithm’s promising Orbit 2 7898 0.057 0 0 225.2 281, 271
performance in solving linear problems has motivated
experts to utilize its key ideas, i.e., primal-dual steps, to
devise powerful nonlinear optimization algorithms. IPMs motion so that the visual information gathered from two
often outperform SQPs in dealing with large-scale faces of the targets is maximized. The simulation duration
applications, especially when the system has a is equal to five-times the period of the slowest orbit(t2 −
block-structure and sparse representation. On the √ a3
other hand, IPMs have shown a lack of robustness in t1 = 5× 2π µ = 35830.7 Sec), and the sunlight direction
relation to initial point selection, problem scaling and is assumed to be from the positive sides of the x-axis to its
Hamidreza Nourzadeh and John McInroy: Multi- agent Orbit Design for Visual Perception Enhancement Purpose 9
Figure 12. Optimal trajectory and sides quality pattern of all RSOs for both design scenarios. (Co-orbital case (Top) and two distinct
agents’ trajectories design case (Bottom)).
Figure 13. Optimization progress in the co-orbital scenario Figure 14. Optimization progress in the scenario with two
distinct orbits
negative side, i.e., [−1 0 0]T . The threshold for the collision be determined by solving the optimization problem, i.e.,
avoidance constraint is considered as tr = 3Km. p = [a1, e1,Ω1,ω1, ι1, ν1, a2, e2,Ω T2,ω2, ι2, ν2] . The p
vector will be limited to the upper and lower p̄ and p
Figure 10 depicts how the observation quality field vectors in Equation 24. Since the systems are small-scale
changes in five time instances as the system evolves. in these scenarios, a SQP-type algorithm is employed
As shown, the quality field varies in a highly nonlinear to numerically solve the corresponding optimization
manner. When the Sun shines on one side of the Earth, it problem.
casts a shadow on the opposite side. To provide a basis for
comparison, two different scenarios are considered. In the
p̄ = [8033, 0.15, 360, 360, 10, 360, 8033, 0.15, 360, 360, 10, 360]T
first situation, it is assumed that the agents are co-orbital p 7898.3, 0, 0, 0, 0, 0, 7898.3, 0, 0, 0, 0, 0 T (24)= [ ]
and uniformly distributed on an equatorial orbit, while
in the second case, two different equatorial orbits are
designed. For both cases, 12 unknown variables should
10 Int J Adv Robot Syst, 2014, 11:161 | doi: 10.5772/58894
7.1.1 Co-orbital Case
Equation (25) is the nonlinear optimization problem
corresponding to the first scenario.
maximize min( f (p))
p
subject to :
p ≤ p ≤ p̄
aj(1− e j
) > RE + 300, j = 1, 2
Ω1 = 0 → [01×2 1 01×11]p = 0
Ω2 = 0 → [01×8 1 01×3]p = 0
(25)
ι1 = 0 → [01×4 1 01×7]p = 0 ι2 = 0 → [01×10 1 0]p = 0 A
T p = [I5×5 0
5×1
− I5×5 05×1]p = 05×1
ν1 ≤ 180 → [05×1 1 05×1 0]p ≤ 180
ν2 − ν1 = 180 → [05×1 − 1 05×1 1]p = 180
Figure 15. RSOs arrangement in a typical SBSSA application Assuming the minimum operational altitude for both
agents to be 300 Km, the altitude constraints are aj(1 −
Histogram of optimization results initialized by different orbit configurations
14 ej) > RE + 300 for j = 1, 2. Equality constraints
Best Cost Function= 0.633 Ω12 1 = Ω2 = ι1 = ι2 = 0 force the orbits to be in the
equatorial plane. Constraint AT p = 0 makes the two
10
agents co-orbital, and constraints ν1 ≤ 180 and ν2 − ν1 =
8 180 result in equally-spaced agents in the orbit:
6
4
7.1.2 Agents with Different Orbits
2
00.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 The second scenario formulation is like Equation (25) with
Cost Function min(Jsum) only the six first constraints to make sure that the resultant
The best obtained sub-optimal solution from 200 orbits satisfy the boundary conditions, minimum altitudeFigure 16.
optimization runs for the SSO design case constraints and position in the equatorial plane. Both
optimization problems are run from several feasible initial
Evolution of the Cost Functioin parameters to obtain satisfactory sub-optimal solutions.
Figure 11 describes how the best trajectory achieved
0.7 evolved from a feasible initial solution in the co-orbital
0.6330 case. As can be seen, the agents are uniformly distributed
0.6
along the trajectories in all iterations.
0.5
Figures 13 and 14 illustrate how the scaled optimization
0.4 variables and the minimum observation quality of all the
targets’ sides converge for each scenario. As highlighted,
0.3 after convergence, the optimization cost function in the
co-orbital case is 90.449, which is clearly less than the
0.2
1 2 3 4 5 6 7 8 9
Iterations corresponding obtained observation quality along two
other trajectories (146.826). This indicates that, in systems
Figure 17. The cost function evolution of the best obtained in which agents have more degrees of freedom, a higher
solution in the SSO design scenario perception level can be achieved.
Figure 12 depicts the sum of the quality pattern for each
side of the targets, as well as the initial and optimized
trajectories for both design scenarios. The graph shows
how the initial trajectories are altered and developed to be
well-positioned in areas with a higher perception level on
both sides.
7.2 Frozen SSO Design
In this part, the multi-agent orbit design procedure is
performed for larger SBSSA systems. In the first part,
because of the desirable characteristics of the SSO reported
for the SBSS program [26], Sun-synchronous frozen orbits
are found for a SBSSA system with five agents which
Figure 18. The optimized frozen SSOs for a multi-agent system must characterize 60 RSOs. In this scenario, 51 RSOs are
Hamidreza Nourzadeh and John McInroy: Multi- agent Orbit Design for Visual Perception Enhancement Purpose 11
Cost Function min(Jsum)
travelling on a GEO and the rest are moving within a LEO [3] E. Hernandez, J. Bolivar, and Q. Wang.
(Figure 15). Geosynchronous transfer orbit design: A practical
approach. In Modeling, Simulation and Applied
In this design problem, the Keplerian parameters of the Optimization (ICMSAO), 2011 4th International
desired orbits should satisfy the SSO and frozen orbit Conference on, pages 1–4, April 2011.
constraints (Equation (22) and (23)). Furthermore, the [4] Zhanxia Zhu, Jianping Yuan, and Lili Zheng.
lower and upper band for altitude are assumed to be cj = Shape-based method for non-keplerian orbit design.
300Km and c̄j = 1000Km, the orbital inclination lies in In 12th Biennial International Conference on Engineering,
the [96.5, 102.5] interval and the argument of the perigee Construction, and Operations in Challenging
ωj = 90. Thus, the optimization problem can be written as Environments; and Fourth NASA/ARO/ASCE Workshop
Equation (26): on Granular Materials in Lunar and Martian Exploration:
Earth and Space 2010, pages 1934–1940, Honolulu,
Hawaii, United States, March 2010.
maximize min( f (p)) p [5] H.-C. Cho and S.-Y. Park. Analytic solution for
subject to : p ≤ p ≤ p̄ fuel-optimal reconfiguration in relative motion.
aj(1− ej) > RE + 300, j = 1, · · · , 5 Journal of Optimization Theory and Applications, aj(1−(ej) < RE)+
2√
1000, j = 1, · · · , 5 141(3):495–512, 2009.
(26)
− 3 R µ [6] Michael James Tillerson. Orbit Design Convex E J̃ cos(ι ) = 1.99× 10
−7
2 2 a 2j(1−ej ) a3 jj Optimization Technique. PhD thesis, Massachusetts
j = 1, · · · , 5 Institute of Technology. ej = −
J2 sin(ιj)
J , j = 1, · · · , 5 3 2aj(1−e2j ) [7] Gokhan Inalhan, Michael Tillerson, and Jonathan P.
ωj = 90, j = 1, · · · , 5 How. Relative dynamics and control of spacecraft
formations in eccentric orbits. AIAA Journal of
Guidance, Control, and Dynamics (0731-5090, 25:48–59,
Figure 16 and 17 report the procedure to achieve a 2002.
sup-optimal solution for the optimization problem in [8] M. Tillerson and J. How. Formation flying control in
Equation (26). Figure 16 depicts 200 runs of the eccentric orbits. In Proceedings of the AIAA Guidance,
optimization from different initial parameters. The Navigation, and Control Conference, Montreal.
best minimum inspection quality for the desired sides [9] T. Vladimirova, C.P. Bridges, J.R. Paul, S.A. Malik,
of RSOs is 0.633, and Figure 17 indicates how the and M.N. Sweeting. Space-based wireless sensor
initial objective function is evolved in each optimization networks: Design issues. In Aerospace Conference, 2010
iteration. Figure 18 illustrates the designed trajectories on IEEE, pages 1–14, March 2010.
which the minimum perception quality is maximized, and [10] Y. Kim, M. Mesbahi, and F.Y. Hadaegh.
it corresponds to the best sub-optimal solution acquired. Multiple-spacecraft reconfiguration through collision
avoidance, bouncing, and stalemate. Journal of
8. Conclusion Optimization Theory and Applications, 122(2):323–343,
2004.
In this paper, a systematic means of optimizing trajectories [11] Hamidreza Nourzadeh and John E. McInroy.
for a sensor network system for perception enhancement Planning the visual measurement of n moving
purposes is presented. This has a major impact on the objects by m moving cameras, given their relative
performance of the resource allocation stage in any sensor trajectories. In IEEE Multi-Conference on Systems and
management system. Specifically, the proposed method is Control, September 2011.
of interest for large-scale systems characterized by highly [12] H. Nourzadeh and J.E. McInroy. Robust visual
dynamic behaviour because of its robustness in relation measurement planning in multi-robot systems. In
to possible perturbations and model uncertainties. The Automation Science and Engineering (CASE), 2013 IEEE
case study results for SBSSA applications indicate the International Conference on, pages 176–182, 2013.
effectiveness of the method in improving the received [13] H. Nourzadeh and J.E. McInroy. Integrated planning
perception qualities along the designed trajectories where of constraint sensor management and patrolling. In
the agents cooperatively meet the mission requirements, Automation Science and Engineering (CASE), 2013 IEEE
such that less effort is needed to serve the main purpose. International Conference on, pages 837–843, 2013.
[14] J.M. Irvine. National imagery interpretability rating
9. References scales (niirs): Overview and methodology. In
Proceedings of the International Society for Optical
[1] T. Vladimirova, C.P. Bridges, G. Prassinos, Xiaofeng Engineering (SPIE), volume 3128, pages 93–103, July
Wu, K. Sidibeh, D.J. Barnhart, A.-H. Jallad, J.R. 1997.
Paul, V. Lappas, A. Baker, K. Maynard, and [15] J.M. Irvine. National imagery intelligence rating scale
R. Magness. Characterising wireless sensor motes for (NIIRS). In R.G. Driggers, editor, The Encyclopedia of
space applications. In Adaptive Hardware and Systems, Optical Engineering. Marcel Dekker, 2003.
2007. AHS 2007. Second NASA/ESA Conference on, [16] J.C. Leachtenauer. National imagery interpretability
pages 43–50, Aug. 2007. rating scales: Overview and product description. In
[2] R. J. Boain. A-B-Cs of sun-synchronous orbit mission Proceedings of the American Society of Photogrammetry
design. In AAS/AIAA Space Flight Mechanics Meeting, and Remote Sensing Annual Meetings, April 1996.
Maui, HI; United States, 2004.
12 Int J Adv Robot Syst, 2014, 11:161 | doi: 10.5772/58894
[17] J. C. Leachtenauer, W. Malila, J. M. Irvine, L. Colburn, [22] David A. Forsyth and Jean Ponce. Computer Vision:
and N. Salvaggio. General image-quality equation: A Modern Approach, pages 9–11. Prentice Hall, US ed
GIQE. Applied Optics, 36:8322–8328, 1997. edition, August 2002.
[18] L.A. Maver, C.D. Erdman, and K. Riehl. Imagery [23] Shengyong Chen, Y. F. Li, Jianwei Zhang, and
interpretability rating scales. In Digest of Technical Wanliang Wang. Active Sensor Planning for Multiview
Papers, International Symposium Society for Information Vision Tasks, pages 4–5. Springer, 1 edition, 2008.
Display, volume XXVI, pages 117–120, 1995. [24] Howard Curtis. Orbital Mechanics: For Engineering
[19] Glenn D. Boreman. Modulation Transfer Function in Students, pages 158–161. Butterworth-Heinemann,
Optical and ElectroOptical Systems, pages 31–42. SPIE January 2005.
Publications, July 2001. [25] J. Nocedal and S.J. Wright. Numerical Optimization,
[20] Hamidreza Nourzadeh. Multi-agent sensor pages 352–417. Springer Series in Operations
management in the presence of uncertainty. PhD Research. Springer, 1999.
thesis, University of Wyoming, Laramie, WY, USA, [26] Su Zengli Hou Yuzhuo, Huang Xuexiang. A
2013. satellite orbit design method for space-based space
[21] E. Hecht. Optics, pages 347–472. Pearson education. surveillance. In European Space Surveillance Conference,
Addison-Wesley, 2002. June 7-9 2011.
Hamidreza Nourzadeh and John McInroy: Multi- agent Orbit Design for Visual Perception Enhancement Purpose 13