Nano-Satellite Swarm for SAR Applications: Design and ... - IEEE Xplore
I. INTRODUCTION
Nano-Satellite Swarm for SAR Applications: Design and Robust Scheduling
CHEE KHIANG PANG, Senior Member, IEEE AKASH KUMAR, Senior Member, IEEE CHER HIANG GOH, Senior Member, IEEE CAO VINH LE, Student Member, IEEE National University of Singapore
In this paper, the design and robust scheduling of a nano-satellite (nanosat) swarm is presented for synthetic aperture radar (SAR) applications. Based on the power budget and bandwidth limit of nanosats, the nanosats’ form factor and swarm size are chosen to ensure that requirements on ground resolution and signal-to-noise ratio are met. Next, an energy-efficient and robust scheduling considering stochastic failures is proposed using scenario optimization with convexification. The effectiveness of our proposed scheduling approach is verified with mathematical rigor as well as extensive simulation results on a realistic SAR application using strip and spot modes.
Manuscript received January 29, 2014; revised July 7, 2014, July 15, 2014; released for publication July 15, 2014. DOI. No. 10.1109/TAES.2014.140077. Refereeing of this contribution was handled by J. Saleh. This work is supported in part by Singapore MOE AcRF Tier 1 Grant No. R 263-000-A52-112. Authors’ addresses: Department of Electrical and Computer Engineering, National University of Singapore, 4 Engineering Drive 3, Singapore, 117583 Singapore, E-mail: ([email protected]). C 2015 IEEE 0018-9251/15/$26.00
With recent advancements in consumer electronics, the ability to operate co-orbital nano-satellites (nanosats) in swarm platforms to replace the more costly and bulkier satellites opens new opportunities and challenges for space industries [1, 2]. As compared to a single satellite, a nanosat swarm has many advantages, including synchronous measurements over a dispersed area and improvements in performance and survivability, which have been widely proposed for synthetic aperture radar (SAR) applications, such as Earth resource exploration, battlefield reconnaissance, natural disaster surveillance, and so on [3, 4]. In addition, smaller and lighter nanosats might be launched in multiples, requiring possibly cheaper dedicated launch vehicles [5]. They also allow for cheaper designs as well as ease of mass production. Nanosats might be included and removed from the swarm in an ad hoc manner, which provides easier in-orbit maintenance [6]. On the contrary, the low mass of nanosats restricts their functionalities, including restrictions on power budget, system reliability, radar bandwidth, and onboard computational capability [7]. As such, the power budget and bandwidth limit of nanosats must be carefully taken into account when designing a nanosat swarm. In addition, energy efficiency and reliability are highly essential scheduling objectives. Due to restrictive onboard computational capability, given nanosat specifications and mission details, it is crucial for a nanosat swarm to be properly designed and scheduled prior to launch. This is opposite to the traditional SAR applications performed by large satellites, where a built-in scheduler is able to receive the jobs submitted by users and perform scheduling online [8–10]. To the best of our knowledge, design of a nanosat swarm for SAR applications has been proposed only in [11], considering radar system design and revisit times of the nanosat constellation on different orbits. Early literature on satellite scheduling often focused on static scheduling; that is, a baseline schedule was created offline without anticipation of uncertainties. For example, a planning and scheduling method for fleets of Earth observing satellites was proposed using a greedy search algorithm [12]. Bianchessi et al. [13] considered a scheduling problem for a constellation of agile satellites based on Tabu search and column generation algorithms. The scheduling problem of agile Earth observing satellites was further studied in [14] using different scheduling algorithms, such as dynamic programming and constraint programming. Thirteen scheduling algorithms were compared and tested for the Earth observing satellites scheduling problem [15]. Finally, a two-phase scheduling method was also proposed with the consideration of task clustering for observing satellites [10]. Once propelled into orbits, there are many uncertainties that can immensely affect the satellites’
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baseline schedules. For instance, some members of a nanosat swarm may fail at uncertain temporal instants and hence impose additional workloads on the remaining members. Pemberton and Greenwald [16] identified the need for dynamic scheduling of imaging satellites in the presence of uncertain changes in the environment, desired jobs, and availability of resources. Wang et al. [8] proposed a heuristic dispatching rule for the dynamic scheduling problem of Earth observing satellites to deal with uncertain arrival of new jobs. A satellite mission scheduling problem was studied involving the scheduling of jobs to be performed by a satellite in which new job requests can arrive stochastically [9]. To the best of our knowledge, energy-efficient and robust scheduling of a nanosat swarm considering stochastic failures has not been considered in state-of-the-art literature. In addition, the scheduling approaches of existing works are commonly proposed as a dispatching rule [8] or rescheduling [9]. While the former has no baseline schedule and jobs are dispatched online based on a predefined criterion, the latter has a baseline schedule to be revised multiple times online. These approaches require online computation and are not suitable for nanosats. Furthering our earlier work in [17], a design and robust scheduling framework of a swarm of identical nanosats is presented for SAR applications. Based on the power budget and bandwidth limit of nanosats, the nanosats’ form factor and swarm size are properly chosen to ensure minimum requirements on ground resolution and signal-to-noise ratio (SNR), which is defined as the ratio between radar-transmitted power versus antenna thermal noise. An energy-efficient and robust scheduling approach considering stochastic failures is then proposed based on scenario optimization (SO). SO finds a feasible solution that balances the energy-optimal schedules from individual scenarios of failures. Traditional SO approaches require one to assign occurrence probabilities of scenarios a priori [18]. Occurrence probabilities of scenarios in our problem depend on the Weibull reliability of nanosats, which is unknown until the scheduling problem is solved. As such, our idea is to formulate occurrence probabilities as decision variables of the scheduling problem and apply a convexification method to reduce its computational complexity. Our proposed convexified SO (CSO) is evaluated on SAR applications in Singapore using strip and spot modes. Its effectiveness is verified with mathematical rigor as well as extensive simulation results. The rest of this paper is organized as follows. Section II designs a nanosat swarm for SAR applications. Section III proposes CSO for energy-efficient and robust scheduling considering stochastic failures. Section IV presents simulation studies to compare CSO to two other scheduling approaches. Finally, Section V concludes this paper. 854
II. DESIGN OF NANOSAT SWARM BASED ON POWER BUDGET AND BANDWIDTH LIMIT
In this section, the design of a nanosat swarm for SAR applications is presented, which includes the following steps: 1) Provision of specifications of nanosats • Form factor • Power supplies, including battery pack and solar panels • Power budget • Bandwidth limit 2) Provision of mission details • Mission jobs • Choice of orbit • Required ground resolution and SNR 3) Proposal of nanosats’ form factor and swarm size • Translation of required ground resolution and SNR into minimum power and bandwidth requirements • Calculation of nanosats’ form factor and swarm size based on power and bandwidth limits of each nanosat SAR applications include a form of radar that uses relative motion between an antenna and its target region to provide distinctive long-term coherent signals. The received signals will be further processed for many purposes, such as imaging. A. Specifications of Nanosats
Nanosats are small satellites having wet mass between 1 and 10 kg [19]. The most common platform to build a nanosat is cubesat, which measures the satellites in units of 10 × 10 × 10 cm having a wet mass no more than 1 kg per cube [20]. As such, the form factor of nanosat is often denoted by the units of cube that it contains, for example, 1U, 2U, 3U, and so on. Nanosats are usually built between 1U and 5U, while some 6U cubesats having a wet mass below 10 kg are also classified as nanosats. Power supply of nanosats is provided by rechargeable battery packs that are tailored to fit in cubesat platforms. Ten percent of the space of nanosats is assumed to carry battery packs. For example, the lithium polymer battery is one of the most popular batteries used for cubesats with a typical energy density of 1.08 MJ/L. The power that can be supplied safely by battery packs is restricted due to thermal issues in removing the excess heat; for example, the maximum dissipating rate was reported as 15 W for a 2U cubesat [7]. In addition, it is assumed that battery charge/discharge efficiency is 100% and that the self-discharge rate is 0%. Recharging of battery packs is commonly done using solar panels. Multiple solar panels can be connected in parallel or series. Different materials can be used for solar panels. One approach is to use gallium arsenide thin film cells with an energy density of 40 W/m2 using multiple junction cells at high solar concentrations [21]. To ensure
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TABLE I Nanosat Specifications [7, 11, 21] Form Factor 1U 2U 3U 4U 5U
Dimensions (cm) 10 20 30 40 50
× × × × ×
10 10 10 10 10
× × × × ×
10 10 10 10 10
Battery Volume (L)
Density (MJ/L)
SAR Budget (MJ)
Solar Panel Area (m2 )
Density (W/m2 )
Charging Rate (W)
pmax (W)
f (MHz)
0.1 0.2 0.3 0.4 0.5
1.08 1.08 1.08 1.08 1.08
0.648 1.296 1.944 2.592 3.240
0.0225 0.0475 0.0750 0.1050 0.1350
40 40 40 40 40
0.9 1.9 3.0 4.2 5.4
7.5 15 22.5 30 37.5
10 20 30 40 50
Fig. 2. SAR mode: (a) strip mode and (b) spot mode. Fig. 1. Design of 3U cubesat with two deployed and one body-fixed solar panels.
practical limits for launch, solar panels of nanosats must not be very large. In our design, two deployable solar panels and one body-fixed solar panel are given for each nanosat, as shown in Fig. 1. Let pmax be the maximum power that can be supplied to a nanosat. pmax is determined based on two factors. First, the safe power that can be supplied by battery packs must not be exceeded. Second, nanosats must be fully recharged by solar panels every orbital cycle. Herein, the power budget of nanosats is planned such that 60% of energy capacity is used for SAR operations and 40% for other processes. SAR operations and recharging of battery packs spend up to 5% and 45% of orbital cycle under sunlight, respectively, while other processes are carried out during 50% of orbital cycle in darkness. Our plan is realistic, as SAR operations need a higher power budget to ensure good ground resolution and SNR. It is also obvious to plan SAR operations and the recharging of battery packs by solar panels when the nanosat swarm is under sunlight. Planning 5% of orbital cycle for SAR operations is sufficient, as the nanosat swarm is able to cover a strip length of more than 2000 km. It is assumed that the orbits are circular by nature. Let f be the bandwidth of SAR. One of the common antennas used in nanosats is the patch antenna. The satellite must be facing the Earth during transmission. Two or more antennas may be required, which can take up a significant amount of space in the satellite. They also have limited bandwidth compared to other types of antennas. A typical value for the antenna bandwidth of 3U cubesats is f = 30 MHz and is assumed to be scalable by form factors [11]. Patch antennas do not require deployment, as they are embedded in the nanosats. Based on the above information, we can calculate nanosat specifications with different form factors, as reported in Table I. PANG ET AL.: NANO-SATELLITE SWARM FOR SAR APPLICATIONS
B. Mission Details
SAR jobs are commonly executed under three modes, namely, strip, spot, and scan. Strip mode maintains a fixed pointing direction of the radar antenna broadside to the platform track, as shown in Fig. 2a. A strip map is formed in width by the swath of the SAR and follows the length contour of the nanosat’s ground track. On the contrary, spot mode is used for obtaining high-resolution signals by steering the radar beam to keep the target within the beam for a longer time and thus form a longer synthetic aperture, as shown in Fig. 2b. Scan mode is omitted, as it is too energy consuming for implementation on nanosats. Satellite orbits are usually classified by orbital altitude (low, medium, and high Earth orbits) and inclination (polar, sun-synchronous, and equatorial orbits). Low Earth orbit includes altitudes up to 2000 km, medium Earth orbit ranges in altitude from 2000 to 35 786 km, and high Earth orbit includes altitudes above 35 786 km. For SAR applications in countries near the Equator, such as Singapore, it is worth noting that these countries are usually covered by equatorial cloud bands. Typically, a band of clouds girdles the Equator. This band of persistent clouds is called the Intertropical Convergence Zone, the place where the easterly trade winds in the northern and southern hemispheres meet. Sun-synchronous orbits with a long revisit time have difficulty seeing the ground during their pass since a large percentage of the ground is covered by clouds. As such, high observation frequency is crucial for SAR applications, and equatorial low Earth orbit should be selected. Mission details also provide user requirements on ground resolution and SNR of SAR jobs. Ground resolution is defined as the minimum distance on the ground at which two object points can be imaged separately. 855
C. Nanosat Swarm Design K Let us denote by R = rj j =1 and V = {vi }Q i=1 the nanosat swarm and the set of mission jobs, respectively. The required bandwidth of job vi at a desired ground resolution under strip mode and spot mode is [22]
fimin =
c , 2ψi sin (90◦ − ηi )
(1)
where ψ i denotes desired ground resolution of vi and c is the speed of light. ηi denotes the steering angle of the radar beam of vi . When ηi = 0◦ , the nanosat is said to be operating in the strip mode; otherwise, it is operating in the spot mode. The required power of vi at a desired ground resolution and SNR is calculated by [22] pimin =
SNRi × (4π)3 H 2 kB F Ta ψi , G 2 λ2 σ
(2)
where SNRi represents the desired SNR of vi . kB , Ta , and H denote Boltzmann constant, antenna noise temperature, and orbital altitude, respectively. G is antenna gain, and λ expresses radar beam wavelength. σ and F denote radar target cross section and amplifier stage noise factor, respectively. We assume that long-time coherent integration is used during processing of SAR signals. For SAR applications using small satellite systems, as both transmitting antenna bandwidth and signal power are generally small, long-time coherent integration is crucial to improve output SNR and ground resolution [23]. When an ideal integrator is used (100% efficiency) and noise samples from different nanosats are uncorrelated, we can essentially consider the output of the coherent integrator from np input pulses as a single pulse whose SNR is improved by the same factor [24]. As such, the bandwidth and power requirements when using a nanosat swarm can be formulated by pij ≥ pimin , (3) rj ∈R
fj ≥ fimin ,
(4)
rj ∈R
where pij and fj denote power of rj to perform vi and bandwidth of rj , respectively. fj can be obtained based on nanosat specifications given in Table I, while pij will be explained later in Section III-A. To the best of our knowledge, other novel SAR signal processing techniques for nanosat swarms have not been proposed in state-of-the-art literature. Nevertheless, (3) and (4) can always be modified accordingly, and our proposed design and robust scheduling framework can be updated systematically with regard to different SAR signal processing techniques. 856
As the nanosat swarm must satisfy fimin and pimin for all vi ∈ V , let us define the following: Q min = max pimin i=1 , (5) pmax Q min = max fimin i=1 . fmax
(6) min
Based on Table I, the minimum swarm size K is computed for each form factor by min min K min = arg min kpmax ≥ pmax ∧ kf ≥ fmax ,
(7)
k
where ∧ denotes logical conjunction. Equation (7) indicates that the power and bandwidth to execute a certain task by a nanosat will not exceed pmax and f, respectively, as long as the swarm size is kept greater than or equal to Kmin . In essence, pmax is the minimum safe power and the level to which the nanosat must be fully recharged during the orbital period. Using our proposed design methodology, a design table for nanosat swarms with different form factors based on bandwidth and power requirements can be calculated, as shown in Table II. III. ENERGY-EFFICIENT AND ROBUST SCHEDULING CONSIDERING STOCHASTIC FAILURES
In this section, standard notations are used. R, R∗+ , and R+ are used to denote a set of real numbers, a set of nonnegative real numbers, and a set of strictly positive real numbers, respectively. Z is used to denote a set of integer numbers. N and N∗ denote a set of natural numbers and a set of nonzero natural numbers, respectively. [a, b] is used to denote a range or interval defined by {a ≤ x ≤ b}. A. Problem Formulation min min Given fmax and pmax , the mission manager first refers to Table II for references on the nanosats’ form factor and Kmin . Depending on the number of failures that the mission manager wants the nanosat swarm to tolerate, the number of nanosats K is determined by
K = K min + N,
(8)
where N is the maximum number of failures under consideration. Note that a new orbit insertion must be carried out before the occurrence of (N + 1)th failure in order to prevent mission failure, namely, N ≤ K. It is assumed that the nanosats have full information of failures; that is, all failures can be detected by the entire nanosat swarm. This is independent and regardless of previous failures. The mission manager’s next task is to plan the power levels for nanosats according to N. In the case of failures, the functional nanosats have to increase their powers to predefined levels in order for incomplete jobs to be executed correctly at desired ground resolution and SNR. While being designed identically, nanosats are subjected to a practical range of variation in power consumption, denoted by [−ε%, + ε%], because of the lumped and undecoupled errors throughout many
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TABLE II Nanosat Swarm Design Based on Bandwidth and Power Requirements of Mission Jobs Min Swarm Size (Kmin ) Max Bandwidth min ) (MHz) (fmax
Signal-to-Noise Ratio (SNR) (dB)
Max Power min ) (W) (pmax
1U
2U
3U
4U
5U
5 5 5 10 10 10 15 15 15 20 20 20 25 25 25 30 30 30 35 35 35 40 40 40 45 45 45 50 50
35 40 45 35 40 45 35 40 45 30 35 40 30 35 40 30 35 40 30 35 40 30 35 40 30 35 40 30 35
32.11 101.54 321.08 64.22 203.07 642.16 96.33 304.6 963.23 40.62 128.44 406.14 50.77 160.54 507.67 60.92 192.65 609.2 71.08 224.76 710.74 81.23 256.87 812.27 91.38 288.97 913.8 101.54 321.08
5 14 43 9 28 86 13 41 129 6 18 55 7 22 68 9 26 82 10 30 95 11 35 109 13 39 122 14 43
3 7 22 5 14 43 7 21 65 3 9 28 4 11 34 5 13 41 5 15 48 6 18 55 7 20 61 7 22
2 5 15 3 10 29 5 14 43 2 6 19 3 8 23 3 9 28 4 10 32 4 12 37 5 13 41 5 15
2 4 11 3 7 22 4 11 33 2 5 14 2 6 17 3 7 21 3 8 24 3 9 28 4 10 31 4 11
1 3 9 2 6 18 3 9 26 2 4 11 2 5 14 2 6 17 2 6 19 3 7 22 3 8 25 3 9
production phases. The value of ε depends on production methods of nanosats: ε = 2.5, 5, and 7.5 for job production, batch production, and mass production, respectively [25]. pi and pn are used to denote the nominal powers that the mission manager sets for vi in the case of zero failure and for all jobs in the case of n failures, respectively. The actual realizations of pi and pn for rj are pij and pjn , respectively, with variations defined by pij ∈ [pi (1 − ε%), pi (1 + ε%)] ,
(9)
pjn ∈ pn (1 − ε%) , p n (1 + ε%) ,
(10)
where {pij , pjn } ∈ R+ , n ∈ N+ , and n ≤ N. Let di ∈ R+ denote the processing time of vi and t e ∈ R+ denote the end time of the mission. Based on mission details, di and t e are known prior to launch. Decision variables are denoted by xij . If rj is assigned to vi , xij = 1; otherwise, xij = 0. B. Weibull Reliability Analysis
In general, reliability of a device can be defined by the probability to execute its intended functions. Reliability of satellite components and systems are commonly modeled PANG ET AL.: NANO-SATELLITE SWARM FOR SAR APPLICATIONS
by the Weibull distribution [26, 27]. The Weibull probability density function of rj is ⎧
β−1 y β − α ⎪ y ⎨ β j e y ≥ 0, bj (y; αj , β) = αj αj (11) ⎪ ⎩ 0 y < 0, where β ∈ R+ is the shape parameter and αj ∈ R+ is the scale parameter. The reliability function of rj is then given by
β ∞ − αt j bj (y; αj , β)dy = e , (12) Rej (t) = t
for t ≥ 0, and Rej (t) = 1 for t < 0. In state-of-the-art literature, several authors have recently considered the costs of machine usage; that is, a machine with more workload assigned degrades faster and incurs more costs [28–30]. Analogously, a dynamic Weibull reliability of rj is proposed herein such that the costs of usage of nanosats are included. The usage of rj is sj = xij , (13) vi ∈V
and α is modified by α j (sj ), where α j (·) is a strictly monotonic decreasing function such that αj : N → R+ . 857
For simplicity, α j (sj ) is chosen by αj sj =
δ1 , sj + δ2
(14)
where {δ1 , δ2 } ∈ R+ . β, δ1 , δ2 can be determined based on offline testing or the commercial failure database of satellites, such as SpaceTraks [26].
g(xij ; γ ) consists of two costs, namely, the energy cost of scenario ω0 , = πω0 xij pij di , (23) vi ∈V rj ∈R
and the energy cost of other scenarios, =
C. SO
A set of scenarios is constructed, each of which represents a distinctive combination of failures. Let ω denote a scenario and R,n denote a superset of n-combination sets of R. For example, R,2 = {{r1 , r2 }, {r1 , r3 }, {r2 , r3 }} for R = {r1 , r2 , r3 }. Let ω ⊂ , where
denotes the set of possible failure scenarios; then clearly
=
N
R,n ∪ ω0 ,
(15)
n=1
where ω0 denotes the scenario of zero failure. Each scenario denotes the case when the vehicles remain operational until end of mission. As such, ω0 will be the null set for the scenario where there is no failure. If there is a failure, ω will be used according to different scenarios. As discussed in Section I, occurrence probabilities of scenarios in our problem depend on the reliability of nanosats, which in turn depends on the usages of nanosats. It is worth noting that usages of nanosats are unknown until the scheduling problem is solved. As such, our idea is to formulate occurrence probabilities as decision variables. Let us introduce an additional decision variable γ ∈ R and 0 ≤ γ ≤ 1 to denote the worst reliability of all nanosats during the mission. Let π ω denote the occurrence probability of scenario ω; then π ω is K−n (1 − γ )n ∀ω ⊂ R,n , γ (16) πω = ω = ω0 . γK Thus, our proposed SO problem is Min g xij ; γ = + s.t., Rej t e ; sj > γ , ∀rj ∈ R,
(18)
xij fj ≥
fimin , ∀vi
∈ V , ∀ω ⊂ ,
(19)
rj ∈R\ω
xij pij ≥ pimin , ∀vi ∈ V ,
∀ω∈
PROOF If all possible scenarios are considered, we have N = K,
≥
(25)
and from (15), we have
πω = γ K +
∀ω∈
N
γ K−n (1 − γ )n
n=1 ∀ω⊂ R,n
= γK +
N
N! γ K−n (1 − γ )n . (26) n!(N − n)!
Then, combination of (25) and (26) yields
πω = γ K +
N n=1
K! γ K−n (1 − γ )n n!(K − n)!
= (γ + 1 − γ )K = 1.
(27)
D. Convexity and Convexification
pimin , ∀vi
∈ V , ∀ω ⊂ R,n , ∀n,
(21)
rj ∈R\ω
0 ≤ γ ≤ 1, xij ∈ {0, 1} .
(22)
(18) indicates the worst reliability of all nanosats throughout the mission. Equation (19) specifies the minimum bandwidth requirements, while (20) and (21) indicate the minimum power requirements. Finally, the bounds of decision variables are given in (22). 858
PROPOSITION If all possible scenarios are considered, πω = 1 holds. then
(20)
rj ∈R
xij pjn
(24)
We now wish to prove the properness for our formulation of the occurrence probabilities of scenarios in (16). It is essential to show that if all possible scenarios are considered, then probability mass functions of occurrence probabilities are summed up to one regardless of the true value of γ . This means that the scenario in which all nanosats are failed must also be taken into account. It is worth noting that we have used the occurrence probability of a scenario as a decision variable for scenario-based optimization. This is possible as long as the occurrence probabilities are independent, mutually exclusive, and sum up to one. These properties are mathematically justified in the following proposition.
n=1
(17)
πω xij pjn di .
n=1 ω⊂ R,n vi ∈V rj ∈R\ω
∀ω∈
N
In the existing literature, one needs to know and fix γ a priori before optimization to convexify the problem for a solution using linear programming. Unfortunately, once propelled into space, γ will not be available, and we have chosen the Weibull distribution commonly used in the reliability literature to estimate the remaining useful life of the nanosat. This is a commonly used technique when determining remaining useful life in realistic engineering systems where the distributions are estimated using
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run-to-failure tests. A similar analysis and approach to model the reliability of satellites (and their subsystems) before launching is also discussed in Castet and Saleh [26] and Chen et al. [27]. Our SO problem consists of two nonlinear terms, namely, g(xij ; γ ) and Rej (t e ; s j ). While it is straightforward to show that Rej (t e ; sj ) is convex, the convexity of g(xij ; γ ) requires further analysis. We have the following result.
g(xij ; γ ) is nonconvex, as it contains some products of decision variables in the form ργ n x (the subscripts of xij are dropped for brevity), where ρ ∈ R. First, γ can be approximated by a set of discrete values {γk }M k=1 , γk ∈ R, and 0 ≤ γ k ≤ 1 such that
for any THEOREM 1 g(xij ; γ ) is nonconvex on RQ+1 + {K, Q, N} ∈ N.
x can be represented by {x1 , x2 } = {0, 1} such that
PROOF Proof by induction on K, N, and Q, respectively. BASIS K = 2, N = 1, Q = 1. Rewrite g(xij ; γ ) by g1,1,2 . By expanding (17), we have g1,1,2 = γ (x11 p11 d1 + x12 p12 d1 ) + γ (1 − γ ) x11 p11 d1 + x12 p21 d1 .
(28)
Let H(g) = ∇ xx f(x) be the Hessian of multivariate function g(x). Then ⎡ ⎤ 0 0 h1 0 h2 ⎦ , H g1,1,2 = ⎣ 0 (29) h1 h2 h3
γ =
M
ξi γk and
k=1
x = ζ1 x1 +ζ2 x2 and ζ1 +ζ2 = 1.
x = eX and γ = e ,
(40)
⇔ Min ρen+X
(41)
M
ξk ln (γk ),
(42)
k=1
h2 = 2γp12 d1 + (1 − 2γ ) p21 d1 ,
(31)
k=1
(32)
H(g1,1,2 ) has three eigenvalues: (33) (34)
ξk = 1,
(43)
X = ζ1 ln (x1 ) +ζ2 ln (x2 ) ,
(44)
ζ1 +ζ2 = 1,
(45)
{ξk , ζk } ∈ {0, 1} .
(46)
If ρ < 0, then ργ n x is convexified by introducing additional variables X and such that 1
(35)
1
x = X 1+n and γ = 1+n ,
(47)
and then the following approximate convex representation is obtained:
INDUCTIVE STEP Assume that g1,1,k is nonconvex for all k > 2. We have g1,1,k+1 = g1,1,k + γ (x1k+1 p1k+1 d1 ) + γ (1 − γ ) x1k+1 p1k+1 d1 = g1,1,k + g ∗ .
(39)
Min ργ n x
M
1 1 λ 2 = h3 − 4h21 + 4h1 h2 + h23 , 2 2 1 1 4h21 + 4h1 h2 + h23 . λ 3 = h3 + 2 2 As λ2 < 0 and λ3 > 0, g1,1,2 is nonconvex.
(38)
and then the following approximate convex representation is obtained:
(30)
λ1 = 0,
(37)
If ρ > 0, then ργ n x is convexified by introducing additional variables X and such that
h1 = 2γp11 d1 + (1 − 2γ ) p11 d1 ,
h3 = 2x11 p11 − p11 d1 + 2x12 p12 − p21 d1 .
ξk = 1.
k=1
s.t., =
where
M
Min ργ n x n
(48) 1
⇔ Min ρ 1+n X 1+n (36)
The projection of g∗ on the domain space of g1,1,k is null. As such, g∗ does not convexify g1,1,k , which implies that g1,1,k+1 is nonconvex. Analogous inductions can be applied for K and N in order to complete the proof.
s.t., =
M
ξk γk1+n ,
(49)
(50)
k=1 M
ξk = 1,
(51)
k=1
To convexify g(xij ; γ ), the following results are needed [31]:
X = ζ1 x11+n +ζ2 x21+n ,
(52)
LEMMA 1 ρer1 x1 +r1 x2 +...+rn xn is convex on Rn if ρ > 0 and r ∈ Rn .
ζ1 +ζ2 = 1,
(53)
LEMMA 2 ρxp yq is convex on R2+ if ρ < 0 and p + q ≤ 1.
{ξk , ζk } ∈ {0, 1} .
(54)
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The CSO problem can be solved by convex optimization, which is widely available in commercial solvers [32]. The cutting-plane algorithm is used herein. E. Related Works
For evaluation, CSO shall be compared with two general rescheduling algorithms, namely, partial rescheduling (PR) and complete or full rescheduling (CR) [33]. The basic idea of CR and PR is to prepare a baseline schedule without anticipation of failures. CR produces a new schedule on occurrence of each failure. The new schedule may deviate significantly from the baseline schedule, thus inducing drastic instability on nanosats. On the other hand, PR generates online schedules to minimize schedule instability. It is worth noting that CSO does not lead to schedule instability. The baseline schedule of CR and PR is Min xij0 pij di (55) vi ∈V rj ∈R
s.t.,
xij0 fj ≥ fimin , ∀vi ∈ V ,
(56)
rj ∈R
xij0 pij ≥ pimin , ∀vi ∈ V ,
K
pi +
pi + +
p1
p2
p3
p4
p5
1 2 3 4 5
10 11 12 13 14
6.09 5.53 5.07 4.68 4.35
19.26 17.51 16.05 14.81 13.76
21.40 19.26 17.51 16.05 14.81
– 21.40 19.26 17.51 16.05
– – 21.40 19.26 17.51
– – – 21.40 19.26
– – – – 21.40
vi ∈ V2 ∪ V3 . vi ∈ V1 .
(57) A. Singapore SAR Application
∈ {0, 1} ,
(58)
where xij0 expresses the baseline schedule. At the occurrence of nth failure, CR revises an existing schedule by xijn pjn di (59) Min vi ∈Vn rj ∈Rn
xijn fj ≥ fimin , ∀vi ∈ Vn ,
(60)
rj ∈Rn
xijn pjn ≥ pimin , ∀vi ∈ Vn ,
(61)
rj ∈Rn
xijn ∈ {0, 1} ,
(62)
where Vn and Rn denote the sets of incomplete jobs and functional nanosats, respectively, and xijn expresses the revised schedule after nth failure. As compared to CR, PR revises an existing schedule with the same set of constraints yet a different cost function, x n − x n−1 , (63) Min ij ij vi ∈Vn rj ∈Rn
where | · | denotes absolute value function. IV. SIMULATION RESULTS
In this section, we provide numerical results to evaluate the effectiveness of CSO, PR, and CR. A SAR mission in Singapore is selected for simulations. 860
N
++
xij0
TABLE III Power Levels of Nanosats for Simulation Cases
+
rj ∈R
s.t.,
Fig. 3. Singapore SAR application.
The mission is to take M consecutive SAR signals of three specific locations using both strip and spot modes, as shown in Fig. 3. In particular, the set of mission jobs includes taking M strip signals from east to west of Singapore; M spot signals of Kent Ridge campus, National University of Singapore (NUS); and M spot signals of Singapore Changi international airport. As such, Q = 3M and V can be equally apportioned into three subsets: V = V1 ∪ V2 ∪ V3 = {v1 , . . . , vM , vM+1 , . . . , v2M , v2M+1 , . . . , v3M } . (64) As Singapore locates just 1◦ north of the Equator, which is usually covered by the equatorial cloud bands, an equatorial low Earth orbit is chosen with altitude of 700 km, inclination angle ≈ 0◦ , and revisit time ≈ 90 min. Based on revisit time and areas of target locations, processing times of jobs can be calculated by di = 9, 27, and 29 s for vi ∈ V1 , V2 , and V3 , respectively, assuming the orbits to be circular. fimin = 30 MHz ∀vi ∈ V . SNRi = 30, 35dB for min vi ∈ V2 ∪ V3 and vi ∈ V1 , respectively. As such, fmax min = 30 MHz and pmax = 192.65 W accordingly. Based on Table II, the mission manager has different options to build the nanosat swarm. Let the nanosats’ form factor be chosen as 3U; as such, Kmin = 9. Let N ∈ [1, 5] and ε ∈ { ± 7.5, ± 5, ± 2.5}. Table III details the power levels of nanosats. pij and pjn are generated using uniform distribution. With regard to Weibull reliability, β = 1.5, δ 1 = 2.06 × 106 , and δ 2 = 100. Fig. 4 shows Rej (t) at
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Fig. 4. Weibull reliability functions at β = 1.5, δ 1 = 2.05 × 106 , and δ 2 = 100.
Fig. 5. MTEC at ε = 7.5.
different values of sj . It can be seen that with an increase of sj , Rej (t) degrades faster. In particular, Rej reaches 5% after 547 d (1.5 y) if it performs a job for every revisit, which results in a total of 4384 jobs in 1.5 y. B. Performance Metrics
The performance metrics used to evaluate CSO, CR, and PR are mean total energy consumption (MTEC), mean time to mission failure (MTMF), and mean time between failure (MTBF). The temporal instants of failures are obtained by sampling Rej (t; sj ). For all algorithms, let us denote by V(n) and R(n) the sets of jobs and nanosats active after nth failure but before (n + 1)th failure, where n ∈ N, n ≤ N − 1, respectively. It is worth noting that V(n) and R(n) are different from Vn and Rn , which appeared in (59)–(63). If the mission ends before nth failure occurs, V(n) = ∅. MTEC is computed by assuming that all scheduling algorithms are able to finish the mission; that is, only N failures occur. Q is varied by Q ∈ [1000, 5000]. Missions containing 1000 and 5000 jobs last for 125 and 625 d, respectively. For each value of Q, the energy consumption with N tolerant failures of CSO is EN =
xij pij di +
vi ∈V(0) rj ∈R(0)
N−1
Fig. 6. MTEC at ε = 5.
xij pjn di ,
n=1 vi ∈V(n) rj ∈R(n)
(65) Fig. 7. MTEC at ε = 2.5.
and of CR and PR, it is EN =
vi ∈V(0) rj ∈R(0)
xij0 pij di +
N−1
xijn pjn di .
n=1 vi ∈V(n) rj ∈R(n)
(66) As such, MTEC of all algorithms is 5 1 MTEC = EN . 5 N=1
(67)
On the contrary, Q = 5000 is fixed when computing MTMF and MTBF so that the mission is sufficiently long for N + 1 failures to occur. MTMF is defined as the mean time from starting the mission to the (N + 1)th failure PANG ET AL.: NANO-SATELLITE SWARM FOR SAR APPLICATIONS
instant. MTBF is the mean time from nth to (n + 1)th failure instants, where n ≤ N − 1. C. Results and Discussion
This computational test is carried on a digital computer equipped with an Intel Core i7 processor and 32-gigabyte random-access memory. For each simulation case, three performance metrics are obtained after 100 test runs. All algorithms are programmed in MATLAB. Figs. 5–7 illustrate the performances of all algorithms in terms of MTEC. It is observed that MTEC increases for all algorithms with the increase of Q. For all cases, CSO 861
Fig. 8. MTMF at ε = 7.5.
Fig. 10. MTMF at ε = 2.5.
Fig. 11. MTBF at ε = 7.5. Fig. 9. MTMF at ε = 5.
achieves comparable performance with PR, while CR yields the lowest MTEC. This result can be attributed to the fact that CR sacrifices both computational effort and schedule stability for MTEC. For the CR and PR cases, there is no consideration of future failure, and rescheduling is carried out only after a failure occurs. As such, the CR and PR always use the most “healthy,” that is, energy-efficient, nanosat until the end of its useful life. However, our proposed CSO considers future failure according to the Weibull distribution and hence balances energy consumption and the occurrence probability of the scenario as a decision variable. It can also be seen that the gap in MTEC between CSO and CR is reduced when ε gets smaller. As such, the performance of CSO in terms of MTEC can be significantly improved by the careful design and manufacture of nanosats. Figs. 8–10 show the performance of all algorithms in terms of MTMF. MTMF is an important measure mission reliability that basically specifies a deadline for a new orbit insertion to ensure the mission is completed. It can be seen that CSO yields significantly longer MTMF as compared to CR and PR for all values of ε. The reason is that CR and PR tend to overuse and therefore degrade the most energy-efficient nanosats first. On the contrary, CSO balances utilization among nanosats to guarantee that all nanosats operate in a safe region above a γ reliability 862
Fig. 12. MTBF at ε = 5.
threshold. The mission is hence scheduled such that all nanosats degrade gradually and simultaneously. In addition to MTMF, MTBF is also of importance for the mission of a nanosat swarm. A shorter MTBF allows multiple nanosats to be replaced at the same time with fewer orbit insertions, cutting costs. Figs. 11–13 give the performance of all algorithms in terms of MTBF, where CSO yields significantly shorter MTBF as compared to CR and PR for all values of ε. The failure instants of CSO are temporally nearer to each other (short MTBF) but toward the end of mission (high MTMF). On the contrary, failures instants of CR and PR tend to occur in the middle of mission (high MTBF yet short MTMF).
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[7]
[8]
[9] Fig. 13. MTBF at ε = 2.5.
[10]
V. CONCLUSION
In this paper, the design and robust scheduling of a nanosat swarm for SAR applications were studied. The design of a nanosat swarm was first proposed based on power budget and bandwidth limit to ensure requirements on ground resolution and SNR. Energy-efficient and robust scheduling considering stochastic failures was proposed using CSO. The effectiveness of our proposed CSO was verified with mathematical rigor as well as a realistic SAR application in Singapore using both strip and spot modes. Our extensive simulation results showed that CSO achieved acceptable MTEC while yielding significant improvements in MTMF and MTBF as compared to related works in state-of-the-art literature. Our future works include formation control and path planning for a nanosat swarm using multiagent and graph theory.
[11]
[12]
[13]
[14]
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Clark, C. Huge power demand. . .itsy-bitsy satellite: Solving the cubesat power paradox. In Proceedings of the 24th Annual AIAA/USU Conference on Small Satellites, SSC10-III-5, Logan, UT, Aug. 9–12, 2012, 1–8. Wang, J., Li, J., and Tan, Y. Study on heuristic algorithm for dynamic scheduling problem of earth observing satellites. In Proceedings of the 8th IEEE International Conference on Software Engineering, Artificial Intelligence, Networking, and Parallel/Distributed Computing, Qingdao, China, Jul. 30–Aug. 1, 2007, 9–14. Sun, B., Wang, W., Xie, X., and Qin, Q. Satellite mission scheduling based on genetic algorithm. Kybernetes, 39, 8 (2010), 1255–1261. Wu, G., Liu, J., Ma, M., and Qiu, D. A two-phase scheduling method with the consideration of task clustering for earth observing satellites. Computers & Operations Research, 40, 7 (Jul. 2013), 1884–1894. Engelen, S., van den Oever, M., Mahapatra, P., Sundaramoorthy, P., Gill, E., Meijer, R., and Verhoeven, C. NanoSAR—Case study of synthetic aperture radar for nano-satellites. In Proceedings of the 63rd International Astronautical Congress, Naples, Italy, Oct. 1–5, 2012, 1–6. Frank, J., Jonsson, A., Morris, R., and Smith, D. E. Planning and scheduling for fleets of earth observing satellites. In Proceedings of the 6th International Symposium on Artificial Intelligence, Robotics, Automation and Space, Montreal, Canada, Jun. 18–21, 2001, 18–26. Bianchessi, N., Cordeau, J. F., Desrosiers, J., Laporte, G., and Raymond, V. A heuristic for the multi-satellite, multi-orbit and multi-user management of earth observation satellites. European Journal of Operational Research, 177, 2 (2007), 750–762. Lemaitre, M., Verfaillie, G., Jouhaud, F., Lachiver, J. M., and Bataille, N. Selecting and scheduling observations of agile satellites. Aerospace Science and Technology, 6, 5 (Sep. 2002), 367–381. Globus, A., Crawford, J., Lohn, J., and Pryor, A. A comparison of techniques for scheduling earth observing satellites. In Proceedings of 2004 AAAI Conference on Artificial Intelligence, San Jose, CA, Jul. 25–29, 2004, 836–843. Pemberton, J. C., and Greenwald, L. G. On the need for dynamic scheduling of imaging satellites. International Archives of Photogrammetry Remote Sensing and Spatial Information Sciences, 34, 1 (2002), 165– 171. Pang, C. K., Kumar, A., Goh, C. H., and Le, C. V. Design and robust scheduling of nano-satellite swarm for synthetic aperture radar applications. In Proceedings of the 2014 IEEE ICARCV, Th21.2, Singapore, Dec. 10–12, 2014, 705–710 (invited). Dembo, R. Scenario optimisation. Annals of Operations Research, 30, 1 (1991), 63–80. Lappas, V. J., Steyn, W. H., and Underwood, C. I. Attitude control for small satellites using control moment gyros. Acta Astronautica, 51, 1–9 (2002), 101–111. Woellert, K., Ehrenfreund, P., Ricco, A. J., and Hertzfeld, H. Cubesats: Cost-effective science and technology platforms for emerging and developing nations. Advances in Space Research, 47, 4 (2011), 663–684.
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Chee Khiang Pang (Justin) received his B.Eng. (Hons.), M.Eng., and Ph.D. degrees in 2001, 2003, and 2007, respectively, all in electrical and computer engineering, from National University of Singapore (NUS). In 2003, he was a visiting fellow in the School of Information Technology and Electrical Engineering (ITEE), University of Queensland (UQ), St. Lucia, QLD, Australia. From 2006 to 2008, he was a researcher (tenure) with Central Research Laboratory, Hitachi Ltd, Kokubunji, Tokyo, Japan. In 2007, he was a visiting academic in the School of ITEE, UQ, St. Lucia, QLD, Australia. From 2008 to 2009, he was a visiting research professor in the Automation and Robotics Research Institute, University of Texas at Arlington, Fort Worth, TX. Currently, he is an assistant professor in the Department of Electrical and Computer Engineering, NUS, Singapore. He is also an A∗ STAR Singapore Institute of Manufacturing Technology associate, faculty associate of A∗ STAR Data Storage Institute, a senior member of IEEE, and a member of ASME. His research interests are on ultra-high-performance mechatronic systems, with specific focus on advanced motion control for nanopositioning systems, precognitive maintenance using intelligent analytics, and energy-efficient task scheduling considering uncertainties. Dr. Pang is the author/editor of three research monographs, including Intelligent Diagnosis and Prognosis of Industrial Networked Systems (CRC Press, 2011), High-Speed Precision Motion Control (CRC Press, 2011), and Advances in High-Performance Motion Control of Mechatronic Systems (CRC Press, 2013). He is currently serving as an associate editor for the Journal of Defense Modeling and Simulation and Transactions of the Institute of Measurement and Control; on the Editorial Board for the International Journal of Advanced Robotic Systems, the International Journal of Automation and Logistics, and the International Journal of Computational Intelligence Research and Applications; and on the Conference Editorial Board for IEEE Control Systems Society. In recent years, he also served as a guest editor for the Asian Journal of Control, the International Journal of Systems Science, the Journal of Control Theory and Applications, and the Transactions of the Institute of Measurement and Control. He was the recipient of the Best Application Paper Award in the 8th Asian Control Conference (ASCC 2011), Kaohsiung, Taiwan, 2011, and the Best Paper Award in the IASTED International Conference on Engineering and Applied Science (EAS 2012), Colombo, Sri Lanka, 2012. 864
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Akash Kumar received his B.S. degree in computer engineering from the National University of Singapore (NUS), Singapore, in 2002; joint Master of Technological Design degree in embedded systems from NUS and the Eindhoven University of Technology (TUe), Eindhoven, The Netherlands, in 2004; and joint Ph.D. degree in electrical engineering in the area of embedded systems from TUe and NUS in 2009. Since 2009, he has been with the Department of Electrical and Computer Engineering, NUS. Currently, he is an assistant professor there. His research interests include analysis, architectures, design methodologies, and resource management of embedded multiprocessor systems. He has published over 70 papers in leading international electronic design automation journals and conferences. Dr. Kumar is a member of various technical program committee of design automation and FPGA conferences, such as DAC, DATE, FPL, and FPT. Cher Hiang Goh is a distinguished member of technical staff of DSO National Laboratories and has been seconded to the National University of Singapore (NUS) in 2013 to lead a number of initiatives of space systems developments in NUS. He is also an adjunct professor of the Department of Electrical and Computer Engineering of NUS and project director of the NUS 1st Micro-satellite Project (Kent Ridge 1) with Hyper-spectral Imaging Applications. Prior to coming to NUS, he was a distinguished visiting professor to the Space Systems Academic Group of Naval Postgraduate School in the United States. He is the project director of the Singapore 1st Micro-satellite program (X-Sat) and has led a team of engineers to successfully design, develop, build, integrate, and test it. The micro-satellite has been launched successfully launched on April 20, 2011, and is still operating in space and already surpassing its designed life of 3 y. He obtained his engineering education in Germany under Public Service Commission (Overseas Merit) Scholarships. He received his Diplom-Ingenieur I (B.Eng.), Diplom-Ingenieur II (M.Eng.), and Doktor-Ingenieur (Ph.D.) degrees in 1983, 1984, and 1993, respectively. He had extensive experience on a range of projects covering guidance, navigation, control, and unmanned systems technologies. He had authored or coauthored over 20 technical papers and also written a research monograph. He was reviewer for IEEE Conference Paper Publications (CIS-RAM & CCCT 2004). He was local organizing cochair for the Asian Control Conference (ASCC 2002). He was member of the International Programme Committee for IEEE Conference CIS 2004 and SATTECH 2008. He is current a general cochair for the Asia Pacific SAR Conference 2015. He is a senior member of IEEE, a member of ION, and a senior member of AIAA.
Cao Vinh Le received the B.Eng. (Hons.) degree in electrical engineering from Nanyang Technological University, Singapore, in 2009. He is currently working toward the Ph.D. degree at the Department of Electrical and Computer Engineering, National University of Singapore, Singapore. His current research interests include energy-efficient manufacturing, robust optimization, and discrete-event systems. PANG ET AL.: NANO-SATELLITE SWARM FOR SAR APPLICATIONS
865
Nano-Satellite Swarm for SAR Applications: Design and Robust Scheduling
CHEE KHIANG PANG, Senior Member, IEEE AKASH KUMAR, Senior Member, IEEE CHER HIANG GOH, Senior Member, IEEE CAO VINH LE, Student Member, IEEE National University of Singapore
In this paper, the design and robust scheduling of a nano-satellite (nanosat) swarm is presented for synthetic aperture radar (SAR) applications. Based on the power budget and bandwidth limit of nanosats, the nanosats’ form factor and swarm size are chosen to ensure that requirements on ground resolution and signal-to-noise ratio are met. Next, an energy-efficient and robust scheduling considering stochastic failures is proposed using scenario optimization with convexification. The effectiveness of our proposed scheduling approach is verified with mathematical rigor as well as extensive simulation results on a realistic SAR application using strip and spot modes.
Manuscript received January 29, 2014; revised July 7, 2014, July 15, 2014; released for publication July 15, 2014. DOI. No. 10.1109/TAES.2014.140077. Refereeing of this contribution was handled by J. Saleh. This work is supported in part by Singapore MOE AcRF Tier 1 Grant No. R 263-000-A52-112. Authors’ addresses: Department of Electrical and Computer Engineering, National University of Singapore, 4 Engineering Drive 3, Singapore, 117583 Singapore, E-mail: ([email protected]). C 2015 IEEE 0018-9251/15/$26.00
With recent advancements in consumer electronics, the ability to operate co-orbital nano-satellites (nanosats) in swarm platforms to replace the more costly and bulkier satellites opens new opportunities and challenges for space industries [1, 2]. As compared to a single satellite, a nanosat swarm has many advantages, including synchronous measurements over a dispersed area and improvements in performance and survivability, which have been widely proposed for synthetic aperture radar (SAR) applications, such as Earth resource exploration, battlefield reconnaissance, natural disaster surveillance, and so on [3, 4]. In addition, smaller and lighter nanosats might be launched in multiples, requiring possibly cheaper dedicated launch vehicles [5]. They also allow for cheaper designs as well as ease of mass production. Nanosats might be included and removed from the swarm in an ad hoc manner, which provides easier in-orbit maintenance [6]. On the contrary, the low mass of nanosats restricts their functionalities, including restrictions on power budget, system reliability, radar bandwidth, and onboard computational capability [7]. As such, the power budget and bandwidth limit of nanosats must be carefully taken into account when designing a nanosat swarm. In addition, energy efficiency and reliability are highly essential scheduling objectives. Due to restrictive onboard computational capability, given nanosat specifications and mission details, it is crucial for a nanosat swarm to be properly designed and scheduled prior to launch. This is opposite to the traditional SAR applications performed by large satellites, where a built-in scheduler is able to receive the jobs submitted by users and perform scheduling online [8–10]. To the best of our knowledge, design of a nanosat swarm for SAR applications has been proposed only in [11], considering radar system design and revisit times of the nanosat constellation on different orbits. Early literature on satellite scheduling often focused on static scheduling; that is, a baseline schedule was created offline without anticipation of uncertainties. For example, a planning and scheduling method for fleets of Earth observing satellites was proposed using a greedy search algorithm [12]. Bianchessi et al. [13] considered a scheduling problem for a constellation of agile satellites based on Tabu search and column generation algorithms. The scheduling problem of agile Earth observing satellites was further studied in [14] using different scheduling algorithms, such as dynamic programming and constraint programming. Thirteen scheduling algorithms were compared and tested for the Earth observing satellites scheduling problem [15]. Finally, a two-phase scheduling method was also proposed with the consideration of task clustering for observing satellites [10]. Once propelled into orbits, there are many uncertainties that can immensely affect the satellites’
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baseline schedules. For instance, some members of a nanosat swarm may fail at uncertain temporal instants and hence impose additional workloads on the remaining members. Pemberton and Greenwald [16] identified the need for dynamic scheduling of imaging satellites in the presence of uncertain changes in the environment, desired jobs, and availability of resources. Wang et al. [8] proposed a heuristic dispatching rule for the dynamic scheduling problem of Earth observing satellites to deal with uncertain arrival of new jobs. A satellite mission scheduling problem was studied involving the scheduling of jobs to be performed by a satellite in which new job requests can arrive stochastically [9]. To the best of our knowledge, energy-efficient and robust scheduling of a nanosat swarm considering stochastic failures has not been considered in state-of-the-art literature. In addition, the scheduling approaches of existing works are commonly proposed as a dispatching rule [8] or rescheduling [9]. While the former has no baseline schedule and jobs are dispatched online based on a predefined criterion, the latter has a baseline schedule to be revised multiple times online. These approaches require online computation and are not suitable for nanosats. Furthering our earlier work in [17], a design and robust scheduling framework of a swarm of identical nanosats is presented for SAR applications. Based on the power budget and bandwidth limit of nanosats, the nanosats’ form factor and swarm size are properly chosen to ensure minimum requirements on ground resolution and signal-to-noise ratio (SNR), which is defined as the ratio between radar-transmitted power versus antenna thermal noise. An energy-efficient and robust scheduling approach considering stochastic failures is then proposed based on scenario optimization (SO). SO finds a feasible solution that balances the energy-optimal schedules from individual scenarios of failures. Traditional SO approaches require one to assign occurrence probabilities of scenarios a priori [18]. Occurrence probabilities of scenarios in our problem depend on the Weibull reliability of nanosats, which is unknown until the scheduling problem is solved. As such, our idea is to formulate occurrence probabilities as decision variables of the scheduling problem and apply a convexification method to reduce its computational complexity. Our proposed convexified SO (CSO) is evaluated on SAR applications in Singapore using strip and spot modes. Its effectiveness is verified with mathematical rigor as well as extensive simulation results. The rest of this paper is organized as follows. Section II designs a nanosat swarm for SAR applications. Section III proposes CSO for energy-efficient and robust scheduling considering stochastic failures. Section IV presents simulation studies to compare CSO to two other scheduling approaches. Finally, Section V concludes this paper. 854
II. DESIGN OF NANOSAT SWARM BASED ON POWER BUDGET AND BANDWIDTH LIMIT
In this section, the design of a nanosat swarm for SAR applications is presented, which includes the following steps: 1) Provision of specifications of nanosats • Form factor • Power supplies, including battery pack and solar panels • Power budget • Bandwidth limit 2) Provision of mission details • Mission jobs • Choice of orbit • Required ground resolution and SNR 3) Proposal of nanosats’ form factor and swarm size • Translation of required ground resolution and SNR into minimum power and bandwidth requirements • Calculation of nanosats’ form factor and swarm size based on power and bandwidth limits of each nanosat SAR applications include a form of radar that uses relative motion between an antenna and its target region to provide distinctive long-term coherent signals. The received signals will be further processed for many purposes, such as imaging. A. Specifications of Nanosats
Nanosats are small satellites having wet mass between 1 and 10 kg [19]. The most common platform to build a nanosat is cubesat, which measures the satellites in units of 10 × 10 × 10 cm having a wet mass no more than 1 kg per cube [20]. As such, the form factor of nanosat is often denoted by the units of cube that it contains, for example, 1U, 2U, 3U, and so on. Nanosats are usually built between 1U and 5U, while some 6U cubesats having a wet mass below 10 kg are also classified as nanosats. Power supply of nanosats is provided by rechargeable battery packs that are tailored to fit in cubesat platforms. Ten percent of the space of nanosats is assumed to carry battery packs. For example, the lithium polymer battery is one of the most popular batteries used for cubesats with a typical energy density of 1.08 MJ/L. The power that can be supplied safely by battery packs is restricted due to thermal issues in removing the excess heat; for example, the maximum dissipating rate was reported as 15 W for a 2U cubesat [7]. In addition, it is assumed that battery charge/discharge efficiency is 100% and that the self-discharge rate is 0%. Recharging of battery packs is commonly done using solar panels. Multiple solar panels can be connected in parallel or series. Different materials can be used for solar panels. One approach is to use gallium arsenide thin film cells with an energy density of 40 W/m2 using multiple junction cells at high solar concentrations [21]. To ensure
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TABLE I Nanosat Specifications [7, 11, 21] Form Factor 1U 2U 3U 4U 5U
Dimensions (cm) 10 20 30 40 50
× × × × ×
10 10 10 10 10
× × × × ×
10 10 10 10 10
Battery Volume (L)
Density (MJ/L)
SAR Budget (MJ)
Solar Panel Area (m2 )
Density (W/m2 )
Charging Rate (W)
pmax (W)
f (MHz)
0.1 0.2 0.3 0.4 0.5
1.08 1.08 1.08 1.08 1.08
0.648 1.296 1.944 2.592 3.240
0.0225 0.0475 0.0750 0.1050 0.1350
40 40 40 40 40
0.9 1.9 3.0 4.2 5.4
7.5 15 22.5 30 37.5
10 20 30 40 50
Fig. 2. SAR mode: (a) strip mode and (b) spot mode. Fig. 1. Design of 3U cubesat with two deployed and one body-fixed solar panels.
practical limits for launch, solar panels of nanosats must not be very large. In our design, two deployable solar panels and one body-fixed solar panel are given for each nanosat, as shown in Fig. 1. Let pmax be the maximum power that can be supplied to a nanosat. pmax is determined based on two factors. First, the safe power that can be supplied by battery packs must not be exceeded. Second, nanosats must be fully recharged by solar panels every orbital cycle. Herein, the power budget of nanosats is planned such that 60% of energy capacity is used for SAR operations and 40% for other processes. SAR operations and recharging of battery packs spend up to 5% and 45% of orbital cycle under sunlight, respectively, while other processes are carried out during 50% of orbital cycle in darkness. Our plan is realistic, as SAR operations need a higher power budget to ensure good ground resolution and SNR. It is also obvious to plan SAR operations and the recharging of battery packs by solar panels when the nanosat swarm is under sunlight. Planning 5% of orbital cycle for SAR operations is sufficient, as the nanosat swarm is able to cover a strip length of more than 2000 km. It is assumed that the orbits are circular by nature. Let f be the bandwidth of SAR. One of the common antennas used in nanosats is the patch antenna. The satellite must be facing the Earth during transmission. Two or more antennas may be required, which can take up a significant amount of space in the satellite. They also have limited bandwidth compared to other types of antennas. A typical value for the antenna bandwidth of 3U cubesats is f = 30 MHz and is assumed to be scalable by form factors [11]. Patch antennas do not require deployment, as they are embedded in the nanosats. Based on the above information, we can calculate nanosat specifications with different form factors, as reported in Table I. PANG ET AL.: NANO-SATELLITE SWARM FOR SAR APPLICATIONS
B. Mission Details
SAR jobs are commonly executed under three modes, namely, strip, spot, and scan. Strip mode maintains a fixed pointing direction of the radar antenna broadside to the platform track, as shown in Fig. 2a. A strip map is formed in width by the swath of the SAR and follows the length contour of the nanosat’s ground track. On the contrary, spot mode is used for obtaining high-resolution signals by steering the radar beam to keep the target within the beam for a longer time and thus form a longer synthetic aperture, as shown in Fig. 2b. Scan mode is omitted, as it is too energy consuming for implementation on nanosats. Satellite orbits are usually classified by orbital altitude (low, medium, and high Earth orbits) and inclination (polar, sun-synchronous, and equatorial orbits). Low Earth orbit includes altitudes up to 2000 km, medium Earth orbit ranges in altitude from 2000 to 35 786 km, and high Earth orbit includes altitudes above 35 786 km. For SAR applications in countries near the Equator, such as Singapore, it is worth noting that these countries are usually covered by equatorial cloud bands. Typically, a band of clouds girdles the Equator. This band of persistent clouds is called the Intertropical Convergence Zone, the place where the easterly trade winds in the northern and southern hemispheres meet. Sun-synchronous orbits with a long revisit time have difficulty seeing the ground during their pass since a large percentage of the ground is covered by clouds. As such, high observation frequency is crucial for SAR applications, and equatorial low Earth orbit should be selected. Mission details also provide user requirements on ground resolution and SNR of SAR jobs. Ground resolution is defined as the minimum distance on the ground at which two object points can be imaged separately. 855
C. Nanosat Swarm Design K Let us denote by R = rj j =1 and V = {vi }Q i=1 the nanosat swarm and the set of mission jobs, respectively. The required bandwidth of job vi at a desired ground resolution under strip mode and spot mode is [22]
fimin =
c , 2ψi sin (90◦ − ηi )
(1)
where ψ i denotes desired ground resolution of vi and c is the speed of light. ηi denotes the steering angle of the radar beam of vi . When ηi = 0◦ , the nanosat is said to be operating in the strip mode; otherwise, it is operating in the spot mode. The required power of vi at a desired ground resolution and SNR is calculated by [22] pimin =
SNRi × (4π)3 H 2 kB F Ta ψi , G 2 λ2 σ
(2)
where SNRi represents the desired SNR of vi . kB , Ta , and H denote Boltzmann constant, antenna noise temperature, and orbital altitude, respectively. G is antenna gain, and λ expresses radar beam wavelength. σ and F denote radar target cross section and amplifier stage noise factor, respectively. We assume that long-time coherent integration is used during processing of SAR signals. For SAR applications using small satellite systems, as both transmitting antenna bandwidth and signal power are generally small, long-time coherent integration is crucial to improve output SNR and ground resolution [23]. When an ideal integrator is used (100% efficiency) and noise samples from different nanosats are uncorrelated, we can essentially consider the output of the coherent integrator from np input pulses as a single pulse whose SNR is improved by the same factor [24]. As such, the bandwidth and power requirements when using a nanosat swarm can be formulated by pij ≥ pimin , (3) rj ∈R
fj ≥ fimin ,
(4)
rj ∈R
where pij and fj denote power of rj to perform vi and bandwidth of rj , respectively. fj can be obtained based on nanosat specifications given in Table I, while pij will be explained later in Section III-A. To the best of our knowledge, other novel SAR signal processing techniques for nanosat swarms have not been proposed in state-of-the-art literature. Nevertheless, (3) and (4) can always be modified accordingly, and our proposed design and robust scheduling framework can be updated systematically with regard to different SAR signal processing techniques. 856
As the nanosat swarm must satisfy fimin and pimin for all vi ∈ V , let us define the following: Q min = max pimin i=1 , (5) pmax Q min = max fimin i=1 . fmax
(6) min
Based on Table I, the minimum swarm size K is computed for each form factor by min min K min = arg min kpmax ≥ pmax ∧ kf ≥ fmax ,
(7)
k
where ∧ denotes logical conjunction. Equation (7) indicates that the power and bandwidth to execute a certain task by a nanosat will not exceed pmax and f, respectively, as long as the swarm size is kept greater than or equal to Kmin . In essence, pmax is the minimum safe power and the level to which the nanosat must be fully recharged during the orbital period. Using our proposed design methodology, a design table for nanosat swarms with different form factors based on bandwidth and power requirements can be calculated, as shown in Table II. III. ENERGY-EFFICIENT AND ROBUST SCHEDULING CONSIDERING STOCHASTIC FAILURES
In this section, standard notations are used. R, R∗+ , and R+ are used to denote a set of real numbers, a set of nonnegative real numbers, and a set of strictly positive real numbers, respectively. Z is used to denote a set of integer numbers. N and N∗ denote a set of natural numbers and a set of nonzero natural numbers, respectively. [a, b] is used to denote a range or interval defined by {a ≤ x ≤ b}. A. Problem Formulation min min Given fmax and pmax , the mission manager first refers to Table II for references on the nanosats’ form factor and Kmin . Depending on the number of failures that the mission manager wants the nanosat swarm to tolerate, the number of nanosats K is determined by
K = K min + N,
(8)
where N is the maximum number of failures under consideration. Note that a new orbit insertion must be carried out before the occurrence of (N + 1)th failure in order to prevent mission failure, namely, N ≤ K. It is assumed that the nanosats have full information of failures; that is, all failures can be detected by the entire nanosat swarm. This is independent and regardless of previous failures. The mission manager’s next task is to plan the power levels for nanosats according to N. In the case of failures, the functional nanosats have to increase their powers to predefined levels in order for incomplete jobs to be executed correctly at desired ground resolution and SNR. While being designed identically, nanosats are subjected to a practical range of variation in power consumption, denoted by [−ε%, + ε%], because of the lumped and undecoupled errors throughout many
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TABLE II Nanosat Swarm Design Based on Bandwidth and Power Requirements of Mission Jobs Min Swarm Size (Kmin ) Max Bandwidth min ) (MHz) (fmax
Signal-to-Noise Ratio (SNR) (dB)
Max Power min ) (W) (pmax
1U
2U
3U
4U
5U
5 5 5 10 10 10 15 15 15 20 20 20 25 25 25 30 30 30 35 35 35 40 40 40 45 45 45 50 50
35 40 45 35 40 45 35 40 45 30 35 40 30 35 40 30 35 40 30 35 40 30 35 40 30 35 40 30 35
32.11 101.54 321.08 64.22 203.07 642.16 96.33 304.6 963.23 40.62 128.44 406.14 50.77 160.54 507.67 60.92 192.65 609.2 71.08 224.76 710.74 81.23 256.87 812.27 91.38 288.97 913.8 101.54 321.08
5 14 43 9 28 86 13 41 129 6 18 55 7 22 68 9 26 82 10 30 95 11 35 109 13 39 122 14 43
3 7 22 5 14 43 7 21 65 3 9 28 4 11 34 5 13 41 5 15 48 6 18 55 7 20 61 7 22
2 5 15 3 10 29 5 14 43 2 6 19 3 8 23 3 9 28 4 10 32 4 12 37 5 13 41 5 15
2 4 11 3 7 22 4 11 33 2 5 14 2 6 17 3 7 21 3 8 24 3 9 28 4 10 31 4 11
1 3 9 2 6 18 3 9 26 2 4 11 2 5 14 2 6 17 2 6 19 3 7 22 3 8 25 3 9
production phases. The value of ε depends on production methods of nanosats: ε = 2.5, 5, and 7.5 for job production, batch production, and mass production, respectively [25]. pi and pn are used to denote the nominal powers that the mission manager sets for vi in the case of zero failure and for all jobs in the case of n failures, respectively. The actual realizations of pi and pn for rj are pij and pjn , respectively, with variations defined by pij ∈ [pi (1 − ε%), pi (1 + ε%)] ,
(9)
pjn ∈ pn (1 − ε%) , p n (1 + ε%) ,
(10)
where {pij , pjn } ∈ R+ , n ∈ N+ , and n ≤ N. Let di ∈ R+ denote the processing time of vi and t e ∈ R+ denote the end time of the mission. Based on mission details, di and t e are known prior to launch. Decision variables are denoted by xij . If rj is assigned to vi , xij = 1; otherwise, xij = 0. B. Weibull Reliability Analysis
In general, reliability of a device can be defined by the probability to execute its intended functions. Reliability of satellite components and systems are commonly modeled PANG ET AL.: NANO-SATELLITE SWARM FOR SAR APPLICATIONS
by the Weibull distribution [26, 27]. The Weibull probability density function of rj is ⎧
β−1 y β − α ⎪ y ⎨ β j e y ≥ 0, bj (y; αj , β) = αj αj (11) ⎪ ⎩ 0 y < 0, where β ∈ R+ is the shape parameter and αj ∈ R+ is the scale parameter. The reliability function of rj is then given by
β ∞ − αt j bj (y; αj , β)dy = e , (12) Rej (t) = t
for t ≥ 0, and Rej (t) = 1 for t < 0. In state-of-the-art literature, several authors have recently considered the costs of machine usage; that is, a machine with more workload assigned degrades faster and incurs more costs [28–30]. Analogously, a dynamic Weibull reliability of rj is proposed herein such that the costs of usage of nanosats are included. The usage of rj is sj = xij , (13) vi ∈V
and α is modified by α j (sj ), where α j (·) is a strictly monotonic decreasing function such that αj : N → R+ . 857
For simplicity, α j (sj ) is chosen by αj sj =
δ1 , sj + δ2
(14)
where {δ1 , δ2 } ∈ R+ . β, δ1 , δ2 can be determined based on offline testing or the commercial failure database of satellites, such as SpaceTraks [26].
g(xij ; γ ) consists of two costs, namely, the energy cost of scenario ω0 , = πω0 xij pij di , (23) vi ∈V rj ∈R
and the energy cost of other scenarios, =
C. SO
A set of scenarios is constructed, each of which represents a distinctive combination of failures. Let ω denote a scenario and R,n denote a superset of n-combination sets of R. For example, R,2 = {{r1 , r2 }, {r1 , r3 }, {r2 , r3 }} for R = {r1 , r2 , r3 }. Let ω ⊂ , where
denotes the set of possible failure scenarios; then clearly
=
N
R,n ∪ ω0 ,
(15)
n=1
where ω0 denotes the scenario of zero failure. Each scenario denotes the case when the vehicles remain operational until end of mission. As such, ω0 will be the null set for the scenario where there is no failure. If there is a failure, ω will be used according to different scenarios. As discussed in Section I, occurrence probabilities of scenarios in our problem depend on the reliability of nanosats, which in turn depends on the usages of nanosats. It is worth noting that usages of nanosats are unknown until the scheduling problem is solved. As such, our idea is to formulate occurrence probabilities as decision variables. Let us introduce an additional decision variable γ ∈ R and 0 ≤ γ ≤ 1 to denote the worst reliability of all nanosats during the mission. Let π ω denote the occurrence probability of scenario ω; then π ω is K−n (1 − γ )n ∀ω ⊂ R,n , γ (16) πω = ω = ω0 . γK Thus, our proposed SO problem is Min g xij ; γ = + s.t., Rej t e ; sj > γ , ∀rj ∈ R,
(18)
xij fj ≥
fimin , ∀vi
∈ V , ∀ω ⊂ ,
(19)
rj ∈R\ω
xij pij ≥ pimin , ∀vi ∈ V ,
∀ω∈
PROOF If all possible scenarios are considered, we have N = K,
≥
(25)
and from (15), we have
πω = γ K +
∀ω∈
N
γ K−n (1 − γ )n
n=1 ∀ω⊂ R,n
= γK +
N
N! γ K−n (1 − γ )n . (26) n!(N − n)!
Then, combination of (25) and (26) yields
πω = γ K +
N n=1
K! γ K−n (1 − γ )n n!(K − n)!
= (γ + 1 − γ )K = 1.
(27)
D. Convexity and Convexification
pimin , ∀vi
∈ V , ∀ω ⊂ R,n , ∀n,
(21)
rj ∈R\ω
0 ≤ γ ≤ 1, xij ∈ {0, 1} .
(22)
(18) indicates the worst reliability of all nanosats throughout the mission. Equation (19) specifies the minimum bandwidth requirements, while (20) and (21) indicate the minimum power requirements. Finally, the bounds of decision variables are given in (22). 858
PROPOSITION If all possible scenarios are considered, πω = 1 holds. then
(20)
rj ∈R
xij pjn
(24)
We now wish to prove the properness for our formulation of the occurrence probabilities of scenarios in (16). It is essential to show that if all possible scenarios are considered, then probability mass functions of occurrence probabilities are summed up to one regardless of the true value of γ . This means that the scenario in which all nanosats are failed must also be taken into account. It is worth noting that we have used the occurrence probability of a scenario as a decision variable for scenario-based optimization. This is possible as long as the occurrence probabilities are independent, mutually exclusive, and sum up to one. These properties are mathematically justified in the following proposition.
n=1
(17)
πω xij pjn di .
n=1 ω⊂ R,n vi ∈V rj ∈R\ω
∀ω∈
N
In the existing literature, one needs to know and fix γ a priori before optimization to convexify the problem for a solution using linear programming. Unfortunately, once propelled into space, γ will not be available, and we have chosen the Weibull distribution commonly used in the reliability literature to estimate the remaining useful life of the nanosat. This is a commonly used technique when determining remaining useful life in realistic engineering systems where the distributions are estimated using
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run-to-failure tests. A similar analysis and approach to model the reliability of satellites (and their subsystems) before launching is also discussed in Castet and Saleh [26] and Chen et al. [27]. Our SO problem consists of two nonlinear terms, namely, g(xij ; γ ) and Rej (t e ; s j ). While it is straightforward to show that Rej (t e ; sj ) is convex, the convexity of g(xij ; γ ) requires further analysis. We have the following result.
g(xij ; γ ) is nonconvex, as it contains some products of decision variables in the form ργ n x (the subscripts of xij are dropped for brevity), where ρ ∈ R. First, γ can be approximated by a set of discrete values {γk }M k=1 , γk ∈ R, and 0 ≤ γ k ≤ 1 such that
for any THEOREM 1 g(xij ; γ ) is nonconvex on RQ+1 + {K, Q, N} ∈ N.
x can be represented by {x1 , x2 } = {0, 1} such that
PROOF Proof by induction on K, N, and Q, respectively. BASIS K = 2, N = 1, Q = 1. Rewrite g(xij ; γ ) by g1,1,2 . By expanding (17), we have g1,1,2 = γ (x11 p11 d1 + x12 p12 d1 ) + γ (1 − γ ) x11 p11 d1 + x12 p21 d1 .
(28)
Let H(g) = ∇ xx f(x) be the Hessian of multivariate function g(x). Then ⎡ ⎤ 0 0 h1 0 h2 ⎦ , H g1,1,2 = ⎣ 0 (29) h1 h2 h3
γ =
M
ξi γk and
k=1
x = ζ1 x1 +ζ2 x2 and ζ1 +ζ2 = 1.
x = eX and γ = e ,
(40)
⇔ Min ρen+X
(41)
M
ξk ln (γk ),
(42)
k=1
h2 = 2γp12 d1 + (1 − 2γ ) p21 d1 ,
(31)
k=1
(32)
H(g1,1,2 ) has three eigenvalues: (33) (34)
ξk = 1,
(43)
X = ζ1 ln (x1 ) +ζ2 ln (x2 ) ,
(44)
ζ1 +ζ2 = 1,
(45)
{ξk , ζk } ∈ {0, 1} .
(46)
If ρ < 0, then ργ n x is convexified by introducing additional variables X and such that 1
(35)
1
x = X 1+n and γ = 1+n ,
(47)
and then the following approximate convex representation is obtained:
INDUCTIVE STEP Assume that g1,1,k is nonconvex for all k > 2. We have g1,1,k+1 = g1,1,k + γ (x1k+1 p1k+1 d1 ) + γ (1 − γ ) x1k+1 p1k+1 d1 = g1,1,k + g ∗ .
(39)
Min ργ n x
M
1 1 λ 2 = h3 − 4h21 + 4h1 h2 + h23 , 2 2 1 1 4h21 + 4h1 h2 + h23 . λ 3 = h3 + 2 2 As λ2 < 0 and λ3 > 0, g1,1,2 is nonconvex.
(38)
and then the following approximate convex representation is obtained:
(30)
λ1 = 0,
(37)
If ρ > 0, then ργ n x is convexified by introducing additional variables X and such that
h1 = 2γp11 d1 + (1 − 2γ ) p11 d1 ,
h3 = 2x11 p11 − p11 d1 + 2x12 p12 − p21 d1 .
ξk = 1.
k=1
s.t., =
where
M
Min ργ n x n
(48) 1
⇔ Min ρ 1+n X 1+n (36)
The projection of g∗ on the domain space of g1,1,k is null. As such, g∗ does not convexify g1,1,k , which implies that g1,1,k+1 is nonconvex. Analogous inductions can be applied for K and N in order to complete the proof.
s.t., =
M
ξk γk1+n ,
(49)
(50)
k=1 M
ξk = 1,
(51)
k=1
To convexify g(xij ; γ ), the following results are needed [31]:
X = ζ1 x11+n +ζ2 x21+n ,
(52)
LEMMA 1 ρer1 x1 +r1 x2 +...+rn xn is convex on Rn if ρ > 0 and r ∈ Rn .
ζ1 +ζ2 = 1,
(53)
LEMMA 2 ρxp yq is convex on R2+ if ρ < 0 and p + q ≤ 1.
{ξk , ζk } ∈ {0, 1} .
(54)
PANG ET AL.: NANO-SATELLITE SWARM FOR SAR APPLICATIONS
859
The CSO problem can be solved by convex optimization, which is widely available in commercial solvers [32]. The cutting-plane algorithm is used herein. E. Related Works
For evaluation, CSO shall be compared with two general rescheduling algorithms, namely, partial rescheduling (PR) and complete or full rescheduling (CR) [33]. The basic idea of CR and PR is to prepare a baseline schedule without anticipation of failures. CR produces a new schedule on occurrence of each failure. The new schedule may deviate significantly from the baseline schedule, thus inducing drastic instability on nanosats. On the other hand, PR generates online schedules to minimize schedule instability. It is worth noting that CSO does not lead to schedule instability. The baseline schedule of CR and PR is Min xij0 pij di (55) vi ∈V rj ∈R
s.t.,
xij0 fj ≥ fimin , ∀vi ∈ V ,
(56)
rj ∈R
xij0 pij ≥ pimin , ∀vi ∈ V ,
K
pi +
pi + +
p1
p2
p3
p4
p5
1 2 3 4 5
10 11 12 13 14
6.09 5.53 5.07 4.68 4.35
19.26 17.51 16.05 14.81 13.76
21.40 19.26 17.51 16.05 14.81
– 21.40 19.26 17.51 16.05
– – 21.40 19.26 17.51
– – – 21.40 19.26
– – – – 21.40
vi ∈ V2 ∪ V3 . vi ∈ V1 .
(57) A. Singapore SAR Application
∈ {0, 1} ,
(58)
where xij0 expresses the baseline schedule. At the occurrence of nth failure, CR revises an existing schedule by xijn pjn di (59) Min vi ∈Vn rj ∈Rn
xijn fj ≥ fimin , ∀vi ∈ Vn ,
(60)
rj ∈Rn
xijn pjn ≥ pimin , ∀vi ∈ Vn ,
(61)
rj ∈Rn
xijn ∈ {0, 1} ,
(62)
where Vn and Rn denote the sets of incomplete jobs and functional nanosats, respectively, and xijn expresses the revised schedule after nth failure. As compared to CR, PR revises an existing schedule with the same set of constraints yet a different cost function, x n − x n−1 , (63) Min ij ij vi ∈Vn rj ∈Rn
where | · | denotes absolute value function. IV. SIMULATION RESULTS
In this section, we provide numerical results to evaluate the effectiveness of CSO, PR, and CR. A SAR mission in Singapore is selected for simulations. 860
N
++
xij0
TABLE III Power Levels of Nanosats for Simulation Cases
+
rj ∈R
s.t.,
Fig. 3. Singapore SAR application.
The mission is to take M consecutive SAR signals of three specific locations using both strip and spot modes, as shown in Fig. 3. In particular, the set of mission jobs includes taking M strip signals from east to west of Singapore; M spot signals of Kent Ridge campus, National University of Singapore (NUS); and M spot signals of Singapore Changi international airport. As such, Q = 3M and V can be equally apportioned into three subsets: V = V1 ∪ V2 ∪ V3 = {v1 , . . . , vM , vM+1 , . . . , v2M , v2M+1 , . . . , v3M } . (64) As Singapore locates just 1◦ north of the Equator, which is usually covered by the equatorial cloud bands, an equatorial low Earth orbit is chosen with altitude of 700 km, inclination angle ≈ 0◦ , and revisit time ≈ 90 min. Based on revisit time and areas of target locations, processing times of jobs can be calculated by di = 9, 27, and 29 s for vi ∈ V1 , V2 , and V3 , respectively, assuming the orbits to be circular. fimin = 30 MHz ∀vi ∈ V . SNRi = 30, 35dB for min vi ∈ V2 ∪ V3 and vi ∈ V1 , respectively. As such, fmax min = 30 MHz and pmax = 192.65 W accordingly. Based on Table II, the mission manager has different options to build the nanosat swarm. Let the nanosats’ form factor be chosen as 3U; as such, Kmin = 9. Let N ∈ [1, 5] and ε ∈ { ± 7.5, ± 5, ± 2.5}. Table III details the power levels of nanosats. pij and pjn are generated using uniform distribution. With regard to Weibull reliability, β = 1.5, δ 1 = 2.06 × 106 , and δ 2 = 100. Fig. 4 shows Rej (t) at
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Fig. 4. Weibull reliability functions at β = 1.5, δ 1 = 2.05 × 106 , and δ 2 = 100.
Fig. 5. MTEC at ε = 7.5.
different values of sj . It can be seen that with an increase of sj , Rej (t) degrades faster. In particular, Rej reaches 5% after 547 d (1.5 y) if it performs a job for every revisit, which results in a total of 4384 jobs in 1.5 y. B. Performance Metrics
The performance metrics used to evaluate CSO, CR, and PR are mean total energy consumption (MTEC), mean time to mission failure (MTMF), and mean time between failure (MTBF). The temporal instants of failures are obtained by sampling Rej (t; sj ). For all algorithms, let us denote by V(n) and R(n) the sets of jobs and nanosats active after nth failure but before (n + 1)th failure, where n ∈ N, n ≤ N − 1, respectively. It is worth noting that V(n) and R(n) are different from Vn and Rn , which appeared in (59)–(63). If the mission ends before nth failure occurs, V(n) = ∅. MTEC is computed by assuming that all scheduling algorithms are able to finish the mission; that is, only N failures occur. Q is varied by Q ∈ [1000, 5000]. Missions containing 1000 and 5000 jobs last for 125 and 625 d, respectively. For each value of Q, the energy consumption with N tolerant failures of CSO is EN =
xij pij di +
vi ∈V(0) rj ∈R(0)
N−1
Fig. 6. MTEC at ε = 5.
xij pjn di ,
n=1 vi ∈V(n) rj ∈R(n)
(65) Fig. 7. MTEC at ε = 2.5.
and of CR and PR, it is EN =
vi ∈V(0) rj ∈R(0)
xij0 pij di +
N−1
xijn pjn di .
n=1 vi ∈V(n) rj ∈R(n)
(66) As such, MTEC of all algorithms is 5 1 MTEC = EN . 5 N=1
(67)
On the contrary, Q = 5000 is fixed when computing MTMF and MTBF so that the mission is sufficiently long for N + 1 failures to occur. MTMF is defined as the mean time from starting the mission to the (N + 1)th failure PANG ET AL.: NANO-SATELLITE SWARM FOR SAR APPLICATIONS
instant. MTBF is the mean time from nth to (n + 1)th failure instants, where n ≤ N − 1. C. Results and Discussion
This computational test is carried on a digital computer equipped with an Intel Core i7 processor and 32-gigabyte random-access memory. For each simulation case, three performance metrics are obtained after 100 test runs. All algorithms are programmed in MATLAB. Figs. 5–7 illustrate the performances of all algorithms in terms of MTEC. It is observed that MTEC increases for all algorithms with the increase of Q. For all cases, CSO 861
Fig. 8. MTMF at ε = 7.5.
Fig. 10. MTMF at ε = 2.5.
Fig. 11. MTBF at ε = 7.5. Fig. 9. MTMF at ε = 5.
achieves comparable performance with PR, while CR yields the lowest MTEC. This result can be attributed to the fact that CR sacrifices both computational effort and schedule stability for MTEC. For the CR and PR cases, there is no consideration of future failure, and rescheduling is carried out only after a failure occurs. As such, the CR and PR always use the most “healthy,” that is, energy-efficient, nanosat until the end of its useful life. However, our proposed CSO considers future failure according to the Weibull distribution and hence balances energy consumption and the occurrence probability of the scenario as a decision variable. It can also be seen that the gap in MTEC between CSO and CR is reduced when ε gets smaller. As such, the performance of CSO in terms of MTEC can be significantly improved by the careful design and manufacture of nanosats. Figs. 8–10 show the performance of all algorithms in terms of MTMF. MTMF is an important measure mission reliability that basically specifies a deadline for a new orbit insertion to ensure the mission is completed. It can be seen that CSO yields significantly longer MTMF as compared to CR and PR for all values of ε. The reason is that CR and PR tend to overuse and therefore degrade the most energy-efficient nanosats first. On the contrary, CSO balances utilization among nanosats to guarantee that all nanosats operate in a safe region above a γ reliability 862
Fig. 12. MTBF at ε = 5.
threshold. The mission is hence scheduled such that all nanosats degrade gradually and simultaneously. In addition to MTMF, MTBF is also of importance for the mission of a nanosat swarm. A shorter MTBF allows multiple nanosats to be replaced at the same time with fewer orbit insertions, cutting costs. Figs. 11–13 give the performance of all algorithms in terms of MTBF, where CSO yields significantly shorter MTBF as compared to CR and PR for all values of ε. The failure instants of CSO are temporally nearer to each other (short MTBF) but toward the end of mission (high MTMF). On the contrary, failures instants of CR and PR tend to occur in the middle of mission (high MTBF yet short MTMF).
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[9] Fig. 13. MTBF at ε = 2.5.
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V. CONCLUSION
In this paper, the design and robust scheduling of a nanosat swarm for SAR applications were studied. The design of a nanosat swarm was first proposed based on power budget and bandwidth limit to ensure requirements on ground resolution and SNR. Energy-efficient and robust scheduling considering stochastic failures was proposed using CSO. The effectiveness of our proposed CSO was verified with mathematical rigor as well as a realistic SAR application in Singapore using both strip and spot modes. Our extensive simulation results showed that CSO achieved acceptable MTEC while yielding significant improvements in MTMF and MTBF as compared to related works in state-of-the-art literature. Our future works include formation control and path planning for a nanosat swarm using multiagent and graph theory.
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Chee Khiang Pang (Justin) received his B.Eng. (Hons.), M.Eng., and Ph.D. degrees in 2001, 2003, and 2007, respectively, all in electrical and computer engineering, from National University of Singapore (NUS). In 2003, he was a visiting fellow in the School of Information Technology and Electrical Engineering (ITEE), University of Queensland (UQ), St. Lucia, QLD, Australia. From 2006 to 2008, he was a researcher (tenure) with Central Research Laboratory, Hitachi Ltd, Kokubunji, Tokyo, Japan. In 2007, he was a visiting academic in the School of ITEE, UQ, St. Lucia, QLD, Australia. From 2008 to 2009, he was a visiting research professor in the Automation and Robotics Research Institute, University of Texas at Arlington, Fort Worth, TX. Currently, he is an assistant professor in the Department of Electrical and Computer Engineering, NUS, Singapore. He is also an A∗ STAR Singapore Institute of Manufacturing Technology associate, faculty associate of A∗ STAR Data Storage Institute, a senior member of IEEE, and a member of ASME. His research interests are on ultra-high-performance mechatronic systems, with specific focus on advanced motion control for nanopositioning systems, precognitive maintenance using intelligent analytics, and energy-efficient task scheduling considering uncertainties. Dr. Pang is the author/editor of three research monographs, including Intelligent Diagnosis and Prognosis of Industrial Networked Systems (CRC Press, 2011), High-Speed Precision Motion Control (CRC Press, 2011), and Advances in High-Performance Motion Control of Mechatronic Systems (CRC Press, 2013). He is currently serving as an associate editor for the Journal of Defense Modeling and Simulation and Transactions of the Institute of Measurement and Control; on the Editorial Board for the International Journal of Advanced Robotic Systems, the International Journal of Automation and Logistics, and the International Journal of Computational Intelligence Research and Applications; and on the Conference Editorial Board for IEEE Control Systems Society. In recent years, he also served as a guest editor for the Asian Journal of Control, the International Journal of Systems Science, the Journal of Control Theory and Applications, and the Transactions of the Institute of Measurement and Control. He was the recipient of the Best Application Paper Award in the 8th Asian Control Conference (ASCC 2011), Kaohsiung, Taiwan, 2011, and the Best Paper Award in the IASTED International Conference on Engineering and Applied Science (EAS 2012), Colombo, Sri Lanka, 2012. 864
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Akash Kumar received his B.S. degree in computer engineering from the National University of Singapore (NUS), Singapore, in 2002; joint Master of Technological Design degree in embedded systems from NUS and the Eindhoven University of Technology (TUe), Eindhoven, The Netherlands, in 2004; and joint Ph.D. degree in electrical engineering in the area of embedded systems from TUe and NUS in 2009. Since 2009, he has been with the Department of Electrical and Computer Engineering, NUS. Currently, he is an assistant professor there. His research interests include analysis, architectures, design methodologies, and resource management of embedded multiprocessor systems. He has published over 70 papers in leading international electronic design automation journals and conferences. Dr. Kumar is a member of various technical program committee of design automation and FPGA conferences, such as DAC, DATE, FPL, and FPT. Cher Hiang Goh is a distinguished member of technical staff of DSO National Laboratories and has been seconded to the National University of Singapore (NUS) in 2013 to lead a number of initiatives of space systems developments in NUS. He is also an adjunct professor of the Department of Electrical and Computer Engineering of NUS and project director of the NUS 1st Micro-satellite Project (Kent Ridge 1) with Hyper-spectral Imaging Applications. Prior to coming to NUS, he was a distinguished visiting professor to the Space Systems Academic Group of Naval Postgraduate School in the United States. He is the project director of the Singapore 1st Micro-satellite program (X-Sat) and has led a team of engineers to successfully design, develop, build, integrate, and test it. The micro-satellite has been launched successfully launched on April 20, 2011, and is still operating in space and already surpassing its designed life of 3 y. He obtained his engineering education in Germany under Public Service Commission (Overseas Merit) Scholarships. He received his Diplom-Ingenieur I (B.Eng.), Diplom-Ingenieur II (M.Eng.), and Doktor-Ingenieur (Ph.D.) degrees in 1983, 1984, and 1993, respectively. He had extensive experience on a range of projects covering guidance, navigation, control, and unmanned systems technologies. He had authored or coauthored over 20 technical papers and also written a research monograph. He was reviewer for IEEE Conference Paper Publications (CIS-RAM & CCCT 2004). He was local organizing cochair for the Asian Control Conference (ASCC 2002). He was member of the International Programme Committee for IEEE Conference CIS 2004 and SATTECH 2008. He is current a general cochair for the Asia Pacific SAR Conference 2015. He is a senior member of IEEE, a member of ION, and a senior member of AIAA.
Cao Vinh Le received the B.Eng. (Hons.) degree in electrical engineering from Nanyang Technological University, Singapore, in 2009. He is currently working toward the Ph.D. degree at the Department of Electrical and Computer Engineering, National University of Singapore, Singapore. His current research interests include energy-efficient manufacturing, robust optimization, and discrete-event systems. PANG ET AL.: NANO-SATELLITE SWARM FOR SAR APPLICATIONS
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