Conceptual Remote Sensing Satellite Design Optimization under uncertainty

Accepted Manuscript Conceptual Remote Sensing Satellite Design Optimization under Uncertainty

A. Jafarsalehi, H.R. Fazeley, M. Mirshams

PII: DOI: Reference:

S1270-9638(16)30225-5 http://dx.doi.org/10.1016/j.ast.2016.06.014 AESCTE 3700

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Aerospace Science and Technology

Received date: Revised date: Accepted date:

28 December 2014 20 November 2015 17 June 2016

Please cite this article in press as: A. Jafarsalehi et al., Conceptual Remote Sensing Satellite Design Optimization under Uncertainty, Aerosp. Sci. Technol. (2016), http://dx.doi.org/10.1016/j.ast.2016.06.014

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Conceptual Remote Sensing Satellite Design Optimization under Uncertainty A. Jafarsalehi, H.R.Fazeley* and M. Mirshams K.N. Toosi University of Technology, Faculty of Aerospace Engineering, Iran

Abstract This paper focuses upon the development of an efficient method for the conceptual design optimization of Remote Sensing Satellites (RSS) under uncertainty. There are many acceptable optimal solutions for implementation of satellite subsystems in a space system mission. Every solution should be assessed based on the different criteria such as cost, mass, reliability and payload resolution. In the present paper satellite mass and imaging payload resolution were considered as system level objective functions to obtain the system optimal solution during the conceptual design phase. Furthermore, two Multidisciplinary Design Optimization (MDO) frameworks; Multidisciplinary Design Feasible (MDF) and distributed Collaborative Optimization (CO) were applied to the multi-objective design optimization problem under uncertainty. Also, various uncertainties involving environment, operation, geometry, subsystems, etc. were considered in the Reliability Based Multidisciplinary Design Optimization (RBMDO) frameworks. In the present study, MDF, CO, Reliability Based Multi-disciplinary Design Feasible (RB-MDF) and Reliability Based Collaborative Optimization (RB-CO) frameworks were evaluated and compared. The methodology was based on the utilization of Monte Carlo simulation method for accounting uncertainties in design process and applying genetic algorithms and sequential

*

Corresponding author. Tel.: +982173064124

E-mail addresses: [email protected] (A. Jafarsalehi), [email protected] (H.R.Fazeley), [email protected] (M. Mirshams) 1

quadratic programming to system level and discipline level optimizers. Results obtained in this study, have shown that the introduced method provides an effective way of accounting uncertainty in a complex space system design such as the conceptual design optimization of a spacecraft. Keywords: Multidisciplinary design optimization; Multidisciplinary Feasible; Collaborative Optimization; Reliability-based design; Space systems

Acronyms AAO

All-At-Once

MDB

Mission Design Block

ADCS

Attitude Determination and Control

MDF

Multi-disciplinary Design Feasible

AiO

All-In-One

MDO

Multidisciplinary Design Optimization

ATC

Analytical Target Cascading

PDF

Probability Density Function

BLISS

Bi-level Integrated Systems Synthesis

RB-CO

Reliability Based Collaborative Optimization

C&DH

Command and Data Handling

RBDO

Reliability Based Design Optimization

CO

Collaborative Optimization

RB-MDF

Reliability Based Multi-disciplinary Design Feasible

CSSO

Concurrent Subspace Optimization

RBMDO

Reliability Based Multidisciplinary Design Optimization

DSM

Design Structure Matrix

RDO

Robust Design Optimization

EPS

Electrical Power Supply

RSS

Remote Sensing Satellite

FORM

First-Order Reliability Method

SAND

Simultaneous Analysis and Design

GA

Genetic Algorithms

SDB

System Design Block

GEO

Geosynchronous Earth Orbit

SORM

Second-Order Reliability Method

ICE

Integrated Concurrent Engineering

SQP

Sequential Quadratic Programming

IDF

Individual Discipline Feasible

TCS

Thermal Control System

IS

Importance Sampling

TT&C

Telemetry, Tracking and Command

MCS

Monte Carlo Simulation

UMDO

Uncertainty Multidisciplinary Design Optimization

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1. Introduction Conceptual design of space systems is a complex and decision making process which aims at choosing from a collection of choices implying an irrevocable allocation of resources. Recently, emphasis has been on the advances that can be achieved through the interaction between two or more disciplines. Thus, it is fundamentally a multidisciplinary and multi-objective process. The traditional method of designing space systems, normally includes numerous design loops, which does not guarantee to reach the best optimal solution. The principled application of formal optimization techniques to complex system design has led to the rapid development of an optimization field named Multidisciplinary Design Optimization (MDO). MDO is a design approach for the coupled engineering systems that coherently exploits the synergism of mutually interacting phenomena [1]. In recent years, there has been an increasing amount of literature on MDO domain beginning with aerospace industries, but now they are used in various kind of enterprise (automotive industries, marine industries, etc.) to improve the quality of products [2-5]. As a rule, MDO techniques bridge the gap between subsystem analysis and optimal system design by providing a different optimization framework for design groups. The framework supports design improvement by methodically considering the system level penalties of various disciplinary components and configuration options. Preliminary works on MDO were undertaken by Schmit [6] and Hafka [7]. As a result, the growth of systems complexities, couplings between disciplines and the development of new optimization techniques have led to develop new MDO architectures. Generally, MDO architectures can be classified into two categories: monolithic formulations and distributed formulations [8]. Monolithic formulations that include All-At-Once (AAO) [9], Multidisciplinary Design Feasible (MDF) [10], Individual Discipline Feasible (IDF) [11], and Simultaneous Analysis and Design (SAND) [12] architectures, apply a single systemlevel optimizer to the whole problem. On the other hand, distributed formulations such as Collaborative Optimization (CO) [13], Concurrent Subspace Optimization (CSSO) [14], Analytical Target Cascading (ATC) [15], and Bi-level Integrated Systems Synthesis (BLISS) [16], use subspace optimizations to promote discipline autonomy. All space missions contain a set of elements or components (e.g., launch 3

segment, ground segment, payload, etc.) [17]. Satellite systems are among the most considerable segments in planning of space missions. Satellite conceptual design phase is an interdisciplinary field. The last goal of this process is to manufacture a Satellite that fulfil the requirements of the customers. The conceptual design phase [18-20] of a satellite contains various interactions between specialized disciplines such as payload, orbit, power, TT&C, C&DH and structural analysis to mention a few, which sometimes confronts with conflicting objectives and constraints [5, 19-23]. During the past 30 years, the growing demand for optimal and reliable systems, along with the increasing computational power, have improved the role of optimization techniques in design of complex systems. For this reason, the conceptual design process of space systems has been clearly developed because of the dominant role that optimization techniques have played in this field. Moreover, applying the appropriate design optimization techniques to the design process not only can solve the problem satisfactorily, but also increases the design performance, decreases the design costs and guarantees the stability of space systems design. There have been many studies on the literature that propose systematic design optimization methodologies to solve a space system design problem [5, 24-27]. For example, in reference [28] the satellite design optimization based on normal cloud model was carried out considering only the payload and power supply subsystems. Kim [29] used meta-heuristic algorithms to minimize the total cost of space system development based on the technology choice at conceptual design phase in AAO framework. Hassan [30] applied Genetic Algorithms (GA) to multi-objective design optimization method for the conceptual design of a GEO satellite communication system. Also, mission design optimization of a small remote sensing satellite has been done using genetic algorithms within CO framework [5]. Furthermore, Spangelo and Cutler [31], performed a general optimization technique to solve a spacecraft design optimization problem using a Monte Carlo random search algorithm. The mentioned methods on the design optimization of space systems are all limited to deterministic approach, in which all involved variables and parameters are considered to be certain. The deterministic 4

optimization approaches allow us to find the most optimum system configuration under various performance conditions, but the major problem with these methods is the negligence of uncertainties (or tolerances) in design, manufacturing and operating process. It has been demonstrated that, the absence of uncertainties in the deterministic optimum design considerations may lead to unreliable systems [32]. Space systems throughout their life cycles are normally confronted with uncertainties. Tolerances in orbital elements, environmental conditions, production, modeling and operation are the most important sources of uncertainties in these systems. However, in a deterministic MDO approach of the space systems design process, the described uncertainties are not considered. It should be noted that neglecting uncertainties in the deterministic design methods usually results in a difference between the actual system and the deterministic optimum design which in many cases may lead to the missions failure [33]. Traditionally, in the deterministic design methods, a safety factor can compensate this drawback. Moreover, this approach often results in an increase in the system operation and production costs and also does not guarantee its reliability [34]. In recent years, two major approaches that take different uncertainties into account have been introduced to be used in uncertainty-based design problems; Reliability Based Design Optimization (RBDO) and Robust Design Optimization (RDO). However, there are conceptual differences between these methodologies. Compared with the deterministic approach, RDO tries for optimal designs that are less sensitive to uncontrollable uncertainties which mostly occur in the real design space [35], while in the RBDO methodology, the designer seeks a reliable optimum solution by transforming the deterministic constraints into probabilistic counterparts, in which failure probability is limited to pre-defined boundaries [36]. Although, the RBDO approach is mainly focused on structural engineering [32], it has been recently used for other applications(including aircraft [37, 38], automotive [3, 33], and control [39] designs). Because of the current competitive global market in space industries, the reliability and cost of space

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systems have been considered as design goals. Normally, there has to be a trade-off between low cost and high reliability in the conceptual design process of these systems. In the study carried out by Ubelhart et al. (2006) [40] the non-deterministic approach was applied to conceptual design of the optical structure of a space telescope and Hassan et al. (2008) [41] applied a genetic algorithm with Monte Carlo sampling to probabilistic reliability-based design optimization of a communication satellite system. Also, An approach based on the clouds formalism was proposed to incorporate uncertainty information into the automated search to reach a robust and optimal space system design [42]. Similarly, utilization of Uncertainty Multidisciplinary Design Optimization (UMDO) in the systems engineering process was systematically studied in a research carried out by Yao et al. (2010) [43]. In their test problem, the conceptual design of a satellite has been performed considering three disciplines; Payload, Structure and Orbit. So far little attention has been paid to the conceptual design process of the space systems under uncertainty. Based on aforementioned notes, the aim of this paper was comparing and consequently implementing deterministic and nondeterministic approaches in designing space systems through a multiobjective approach. Several disciplines such as payload, ADCS, structure, etc. were integrated in a proper combination to minimize satellite resolution and mass as the objective functions. Furthermore, MDF, CO, RB-MDF and RB-CO frameworks are evaluated and compared. Also, various uncertainties such as environment, operation, geometry, subsystems and etc. were considered in the RBMDO frameworks. The paper continues on Sec.2 which introduces the design problem. Sec.3 presents the design methodology. Sec.4 shows the implementation of the design problem in the MDO frameworks. Two deterministic and non-deterministic approaches are compared in sec.5. Finally, the conclusions are drawn.

2. Problem Statement The satellite system design problem is divided into two levels namely, Mission Design Block (MDB) and System Design Block (SDB). MDB design performs mission analysis based on the mission and customer requirements [44]. SDB, on the other hand, is divided into different disciplines such as payload, 6

Electrical Power Supply (EPS), attitude determination and control system (ADCS), telemetry, telecommand and control (TT&C), Thermal Control System (TCS), structures, Command and Data Handling (C&DH) as shown in Fig. 1. These disciplines are designed based on the analysis data provided by the interaction between MDB block and the design data [44]. 2.1. Interdisciplinary Relationship in Spacecraft Design During multidisciplinary design process of a spacecraft system, interdisciplinary relationships of all related disciplines are determined, firstly. Then, the problem parameters and design variables and constraints are defined for each discipline. When the design model of each discipline was made, coupling variables and the relations dealing with the objective function and constraints are defined at subsystem level. With this aim, the problem Design Structure Matrix (DSM) as an effective tool to define interdisciplinary relationships is used. DSM shows all interdisciplinary inputs and outputs for each discipline. From DSM, the designer is able to formulate the conceptual design for solving the problem. Design structure matrix (DSM) of a typical spacecraft system is shown in Table 1. Figure 2 typically shows interdisciplinary relationships in a spacecraft design which is compatible with DSM. As shown in Figure 2, mission requirements are sent from the payload of the spacecraft to different subsystems such as: the structure, ADCS, communication, C&DH, EPS and thermal control. According to DSM, requirements of the system design cause extended interactions between subsystems. For example, the inner space of a launch system fairing imposes some constraints on the structure subsystem, area of solar panel (power subsystem), moment of inertia (ADCS), the layout of communication antenna, the layout of on-board computer, thermal distribution and the radiator area. In addition, the orbital design block highlights the calculation of some design parameters such as orbital eclipse time, the visit time of ground station, sun elevation angles, minimum elevation angle and orbital distributions. The iterative nature of design loops combined with a large number of design variables as well as mixed discrete and continuous natures of these design variables, makes the spacecraft system design a very time consuming and costly process. 7

3. Methodology 3.1. Optimization Frameworks 3.1.1. Multidisciplinary Feasible (MDF) Approach Although MDF is the most general MDO formulation and has a comprehensive industry acceptance, it is commonly restricted to small design space problems. MDF, mostly known as “All-in-One (AiO)” solves optimization problems with different subsystems simultaneously. The general MDF framework for solving MDO problems, is schematically shown with a DSM in Fig. 3. According to this figure, in this methodology, the system analyzer coordinates all the subspace analyzers and the system level optimizer controls the design process which ensures achieving a global objective with regard to the design constraints and couplings. It should be noted that MDF performs a complete system analysis on each optimization iteration. In the design problems dealing with coupled systems, some analysis methods (i.e. Fixed Point Iteration and Newton-Raphson) are regularly utilized within an MDF approach. 3.1.2. Collaborative Optimization (CO) Collaborative Optimization (CO) is a bi-level optimization framework developed for large scale and distributed MDO problems [45]. The key concept in CO is the decomposition of the design problem into two levels, namely discipline level and system level optimizations as shown in Fig. 4. CO is designed in such a way that supports disciplinary autonomy while achieving interdisciplinary compatibility, thus providing added design flexibility [13]. These features make CO well suited to be used in a practical multidisciplinary design environment such as a space system. The transformation of the original coupled MDO problem into a CO framework is shown in Fig. 4. It can be seen that the problem is hierarchically decomposed into N disciplinary optimization problems along the disciplinary analysis boundaries. The design variables and constraints of the original problem are partitioned as shown in Fig. 4. The system level optimizer is used to minimize the system level objective 8

function (the design objective function) while satisfying the consistency requirements among various disciplines by enforcing equality constraints at the system level (gi*= 0, i=1,…,N). For example, Si is a vector of S subset composed of all variables that affect discipline i. The system level variables are considered to be fixed parameters in disciplinary optimization runs. Thus, the role of each disciplinary optimizer is to minimize, in a least square sense, the discrepancies between the disciplinary design variables and the target values provided by the system level optimizer. The number of equality constraints N, is related to the number of disciplines. CO framework is used in a hierarchical structure rather than a nonhierarchical one, which denoted its comparable advantages such as parallelization ability, lack of iteration requirements between disciplines and some organizational characteristics. These features make CO well suited to be used in a practical multidisciplinary design environment. Despite its many advantages, CO has several difficulties. One of the most important drawbacks is its high computational cost, which impedes the use of high-fidelity simulation models for some engineering design problems. Moreover, in many design problems, objective functions and constraints are non-smooth or gradients could not be calculated. These can pose convergence difficulties when gradient-based optimizers are applied for system-level optimization. In addition, when the number of coupling variables is large, the dimension of the design optimization problem increases, which may result in high computational cost. 3.2. Optimization Algorithms Applying optimization methods to aerospace applications has become the center of attention in the past thirty years [46]. There are various optimization methods for solving a wide range of real world industrial problems. Each has its advantages and weaknesses. Commonly, optimization methods can be classified into zeroth-order (such as evolutionary algorithms, and grid searching) or gradient-based (such as Sequential Quadratic Programming (SQP)) [1]. In this research, GA and SQP methods were applied to the implementation of MDO architectures.

9

3.2.1. Genetic Algorithm (GA) Genetic Algorithm (GA) is one of the most important intelligent techniques in the optimization field which has been applied to extensive fields of engineering. GA was formally introduced in the 1970s by John Holland [47] and after discussed by many researchers [48-50]. Genetic algorithms maintain large sets (populations) of potential solutions and apply re-combination operators on them to reach an optimum solution. Their ability to search an entire design space makes them more suitable for handling optimization problems with highly non-linear objective functions and many local optima. Moreover, they operate with coded sets of design variables as opposed to the design variables themselves, and are more suitable to be used in optimization problems with discrete design variables. The main disadvantage of this optimization method is its high computational costs. In this paper, GA was utilized to minimize the resolution of the satellite imaging payload globally. 3.2.2. Sequential Quadratic Programming (SQP) SQP method was first proposed by Wilson in 1963 [51]. SQP is one of the most successful methods for solving nonlinear constrained optimization problems [52]. This optimization method finds a search direction by solving an approximate problem based on linear approximations of the constraint functions and a quadratic approximation of the objective function. Generally, the SQP methods are applied to optimization problems for which the constraints and the objective functions are continuously differentiable. In many design problems, gradient-based approaches such as SQP are preferred as they tend to provide quick solutions. However, in these cases a correct initialization is required. In addition, these optimization techniques are susceptible to find local optima as they do not search the whole solution space. As mentioned before, in case of collaborative optimization (CO) framework, the computational cost is very high, so that the SQP method was applied substantially to the local-level optimization of the satellite design problem.

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3.3. Uncertainty Simulation 3.3.1. Generic Reliability Based Design Optimization Reliability Based Design Optimization (RBDO) is a methodology that combines reliability and optimality as two main features of successful simulation-based systems design. The key concept in the RBDO methodology is applying the optimization algorithms to find optimal designs ensuring reliability. In this process, it is essential to minimize an objective function subject to deterministic constraints and certain reliability constraints. The reliability constraints ensure that the probabilities of failure with respect to several failure modes, because of the uncertainties, are lower than acceptable levels. So far various definitions for reliability have been proposed in different research areas. In this paper, reliability was defined as the probability of a system in performing its intended tasks without any failure and under specified operating conditions. In general, a RBDO problem can be formulated in a standard form as: ‫݁ݖ݅݉݅݊݅ܯ‬

‫ܨ‬ሺ݀ǡ ܺǡ ‫ݎ‬ሻ

ܵ‫݋ݐݐ݆ܾܿ݁ݑ‬

ܲ௥ ሾ݃௜ ሺ݀ǡ ܺǡ ‫ݎ‬ሻ ൒ Ͳሿ െ ܴ௜ ൒ Ͳ ݅ ൌ ͳǡ ǥ ǡ ݊ (1)

ܹ݅‫݄ݐ‬

݀௅ ൑ ݀ ൑ ݀௎

‫݋ݐݐܿ݁݌ݏ݁ݎ‬

ߤ௑௅ ൑ ߤ௑ ൑ ߤ௑௎ ݀ ‫ܴ א‬௠ ܺ ‫ܴ א‬௩

Where ‫ܨ‬ሺǤ ሻ is the objective function (i.e. cost function);݀ is the vector of the deterministic design variables; ܺ is the random design vector; ߤ௑ is the vector of the means of ܺ; ‫ ݎ‬is the vector of the random design parameters. ܲ௥ ሺǤ ሻ is the probability operator; The probabilistic constraints are described by ݃௜ ሺǤ ሻ; ܴ௜ represents the target feasible probability for the ith probabilistic constraint.; ߤ௑௅ and ߤ௑௎ are the 11

lower/upper bounds on ߤ௑ ; ݀ ௅ and ݀ ௎ are the lower/upper bounds on ݀;݉ is the number of the deterministic design variables; ‫ ݒ‬is the number of the random design variables and ݊ characterizes the number of the probabilistic constraints. In (1), ݃௜ ሺǤ ሻ ൌ Ͳ is the limit state function,݃௜ ሺǤ ሻ ൏ Ͳ describes the failure region and ݃௜ ሺǤ ሻ ൐ Ͳ is related to the safe region. The statistical formulation of the probabilistic constraints failure ݃௜ ሺǤ ሻ is defined as: ܲ௙ ൌ ܲ௥ ሾ݃௜ ሺ݀ǡ ܺǡ ‫ݎ‬ሻ ൑ Ͳሿ

(2)

If the joint probability density function (PDF) of ܺƒ†‫ ݎ‬is ݂ሺܺǡ ‫ݎ‬ሻ, the failure probability is given by the integral: න

ܲ௙ ൌ

ǥ න ݂௑ǡ௥ ሺܺǡ ‫ݎ‬ሻ݀ܺǤ ݀‫ݎ‬

(3)

௚೔ ሺௗǡ௑ǡ௥ሻஸ଴

In the RBDO process, the evaluation of (2) needs a reliability analysis containing multiple integrations as shown in (3). Three main approaches can be recognized for evaluation of (2) and (3). The first approach consists of using numerical simulation methods (such as Monte Carlo Simulation (MCS) [53] and Importance Sampling (IS) [54]). The second one is to use the analytical or semi-analytical methods (such as First-Order Reliability Method (FORM) [55], Second-Order Reliability Method (SORM) [56] and Moment method [57]). Finally, the third approach is applying response surface methods (such as Kriging method [58] and polynomial with high order terms [59]). In this work, the RBDO problem was implemented in the two MDO frameworks. Combination of these two approaches is named as Reliability Based Multidisciplinary Design Optimization (RBMDO). Because of the iterative nature of the reliability approximation and the design optimization, this procedure is often computationally expensive. All the previously mentioned methods for the evaluation of the probabilistic constraints, suffer from some serious weaknesses [60]. For example analytical methods such as FORM are more common for their simplicity and efficiency but the probability of failure estimated from these methods can be inaccurate for cases where the limit state surfaces are highly non-smooth. However, MCS techniques are computationally expensive but they are the most accurate ones among the 12

mentioned techniques and in many researches named as “Gold Standard” [61]. As one of the purposes of this research is to compare the results of the deterministic design optimization with the RBMDO results, we aimed at pursuing the accuracy instead of the efficiency in order to show the differences between these two approaches. Therefore, the MCS technique was applied as a reliability analyzer in the RBMDO methodology. 3.3.2. Monte Carlo Simulation Technique One of the robust approaches for solving the problem defined in (1) could be MCS. In this technique, the probability of failure (as showed in (3)) is estimated by randomly generating samples based on some sampling density functions. In general, there are 3 steps in the MCS techniques: first, generating a set of random data points based on the assumed probability distributions; second, performing a deterministic simulation through an iterative loop for each generated sample until the convergence is achieved and the output system responses are obtained; and the third step is analyzing the simulation data. The probability of failure using a typical MCS can be estimated by the following equation: ே

ͳ ܲሺ݃௜ ሺ‫ݔ‬ሻ ൑ Ͳሻ ൌ ෍ ‫ܫ‬൫‫ݔ‬௝ ൯ ܰ

(4)

௝ୀଵ

Where ܰis the number of sample points,‫ݔ‬௝ is the jth sample point generated using the pseudo-random generator according to the Probability Density Function (PDF) and ‫ܫ‬ሺ‫ݔ‬ሻ is an indicator function defined as: ‫ܫ‬ሺ‫ݔ‬ሻ ൌ ൜

ͳ݂݅݃௜ ሺ‫ݔ‬ሻ ൑ Ͳ Ͳ݂݅݃௜ ሺ‫ݔ‬ሻ ൐ Ͳ

(5)

The accuracy of the MCS technique can be quantified with a standard error as follows: ‫ܧ‬ൌ

ߪ ξܰ

(6)

Where ߪ is the standard deviation of the simulation results. It should be noted that in many MCS techniques the transformation from the original space to the standard normal one and also the sensitivity 13

analysis in the analytical methods are not necessary to be considered[62]. Consequently, the results of the MCS techniques are often considered true values compared to other approximation methods.

4. RSS Design Optimization Problem The satellite mission included geographical mapping, natural disaster assessment and environmental monitoring. Requirements driving the design included a mission duration of less than 3 years, an orbital altitude of less than 650 km and the inclination in the vicinity of 55°. The satellite design process was conducted using the Integrated Concurrent Engineering (ICE) process. Also, some Physics-based models were developed for each subsystem in the conceptual design and Over 50 core equations, hundreds of sub-core equations, and about ~300 parameters were totally used to represent the satellite. The design problem dealt with the minimization of the satellite resolution and mass, considering the design constraints as well as the side constraints on the design variables and uncertain design parameters. Disciplines of the satellite conceptual design model are all strongly coupled to each other in the typical conceptual-satellite design. This coupling complicates the interaction during the design process, and creates competing demands to optimize individual subsystems rather than the total satellite. Generally, performing engineering design optimization needs some information about the design phase, design parameters, design variables (independent variables), constraints and the design objectives. Design parameters identify constants that will not change as different designs are generated during optimization. They relate to the physical properties of the system that remain unchanged during the design process. Design variables are independent quantities that are controllable from the designer’s point of view. Typically, design variables can be classified into continuous, discrete (including integer and categorical), and Boolean types. In the MDO frameworks, they are always under the explicit control of an optimizer [63]. The subsystems design models and the formulation of the design problem are described in the following sections.

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4.1. Satellite Multi-objective Design Optimization On the satellite multi-objective design optimization step, a multi-objective optimization process based on finding a vector of design variables was defined to both satisfy the constraints and give acceptable values to all objective functions. In general, it can be mathematically defined as: ‫݁ݖ݅݉݅݊݅ܯ‬

‫ܨ‬ሺܺሻ  ൌ  ሾ݂ଵ ሺܺሻǡ ݂ଶ ሺܺሻǡ Ǥ Ǥ Ǥ ǡ ݂௣ ሺܺሻሿ்

ܵ‫݋ݐݐ݆ܾܿ݁ݑ‬

݃௝ ሺܺሻ ൑ Ͳ

ǡ

݆ ൌ ͳǡʹǡ ǥ ǡ ݉

݄௞ ሺܺሻ ൌ Ͳǡ݇ ൌ ͳǡʹǡ ǥ ǡ ‫݋‬

ܹ݅‫݋ݐݐܿ݁݌ݏ݁ݎ݄ݐ‬

‫ݔ‬௜௅௢௪  ൑ ‫ݔ‬௜ ൑ ‫ݔ‬௜ ௎௣

ǡ

(7)

݅ ൌ ͳǡʹǡ ǥ ǡ ݊

Where  ൌ ሾšଵ ǡ šଶ ǡ ǥ ǡš୬ ሿ୘ is the vector of design variables; ݃௝ ሺǤ ሻ are the inequality constraints; Š୩ ሺǤ ሻ are the equality constraints; ‫ݔ‬௜௅௢௪ and ‫ݔ‬௜௎௣ define the lower and upper bounds for the ‹୲୦ design variable š୧ respectively; ‫ܨ‬ሺܺሻ is the vector of ‫ ݌‬objective functions, which must be minimized with regards to X; ݉ is the number of the inequality constraints; ‫ ݋‬is the number of the equality constraints; ݊ is the number of the design variables. During the design process minimization or maximization of objective functions shows the designer’s attitude toward the system design. Without getting away from generalities, it is assumed that all the objective functions are to be minimized. It is also important to mention that a maximization type objective can be converted to a minimization type one by multiplying it to a negative one [64]. 4.1.1. Design Data of the Problem In the problem, eight disciplines were used to demonstrate the proposed methodology. In this case there were two conflicting objectives. In fact the problem dealing with the minimization of the satellite mass and the maximization of the resolution capability was subjected to design constraints as well as the 15

side constraints on the design variables. The design data for the satellite design problem including design variables and constraints is presented in Table 2 and Table 3. The optimization problem dealt with fourteen, discrete and continuous design variables. Discrete design variables came out of bus trade options were considered to be design vectors. As it can be seen from Table 2, the spacecraft payload design variables X1 and X2 are continuous. X3 is the mission continuous variable. X5 and X6 are the design variables involving the structures and X7 and X8 are the design variables related to the determination and attitude control subsystem. The design variables X4, X9 and X10 are related to the communication subsystem. X11, X12 and X13 are the design variables of the power supply subsystem. X14 is the design variable of the thermal control subsystem. 4.2. Deterministic Design Optimization 4.2.1. Multidisciplinary Feasible (MDF) Approach The purpose of this section is multidisciplinary design optimization of the satellite design problem described in sub-section 4.1 based on Multidisciplinary Feasible (MDF) approach. In the satellite design problem, all subsystems were coupled and there were some feedback and feed forward loops. To improve the efficiency of the presented MDF framework, the feedback loops were moved away from the system analyzer by choosing the efficient order of disciplines and also considering a coupling variable as a design variable and adding an additional constraint. The MDF framework for the satellite design problem is shown in Fig. 5. As can be seen from this figure, there are only feed forward loops in the design process. In the presented design problem, there were two conflicting objectives. In the MDF formulation, eight disciplines were used to demonstrate the proposed methodology. Using the MDO architecture presented in Fig. 5, the satellite optimization problem can be formulated as follows: ‫݁ݖ݅݉݅݊݅ܯ‬

ሺšሻ ൌ ሾ݂ଵ ǡ ݂ଶ ሿ ଻

݂ଵ ǣ  ൌ ෍ሺ୬ ሻ ୬ୀଵ

16

(8)

݂ଶ ǣ ܲ௥௘௦ ൌ ܲܽ‫݊݋݅ݐݑ݈݋ݏܴ݁ ݀ܽ݋݈ݕ‬

ܹ݅‫݋ݐݐܿ݁݌ݏ݁ݎ݄ݐ‬

ܺ௜ ǡ

݅ ൌ ͳǡ ǥ ǡͳͶ

ܵ‫݋ݐݐ݆ܾܿ݁ݑ‬

݃௝ ሺܺሻ ൑ Ͳǡ

݆ ൌ ͳǡ ǥ ǡͳͲ

Where, M is the spacecraft mass, calculated through obtaining disciplines of weight characteristics from parametric correlations [17] and structural analysis. PRes is the imaging payload resolution. ݃௝ ሺǤ ሻ is the vector of design constraints (Table 3) involving payload, structure, ADCS, communication, command and data handling, power, thermal control and mission disciplines. ݉ and ݊ are subsystem mass and number of the subsystems respectively. Dominant correlations for calculating above constraints are presented as follows: Satellite Revisit Time (RT) can be obtained as: ܴܶ ൌ

‡ൈɘൈ 

(9)

Where Ȧ is angular velocity of the Earth,Re is the Earth radius, T is the satellite period, and SW is the swath width that can be expressed as: ܶ ൌ ʹߨඨ

ሺ‡൅‫ܪ‬ሻ͵ Ɋ

 ൌ ʹ‫ ܪ‬ൈ ‫ି݊ܽݐ‬ଵ ቆ݊௣௜௫௘௟ ൈ ‫ି݊ܽݐ‬ଵ ቆ

(10)

†’‹š‡Ž ቇቇ ʹͲͲͲൈ ͓ ൈ‫ܦ‬

(11)

In (10) and (11), ‫ ܪ‬is orbital altitude, ȝ is the earth gravity constant, ݊௣௜௫௘௟ is the number of pixels of instrument, dpixel is the instrument pixel size, F# characterizes optical parameter, and ‫ ܦ‬represents camera aperture. Data rate generation is calculated as below: ‫ ܴܦ‬ൌ

17

ሺ݊௣௜௫௘௟ ൈ ‫ݐ݅ܤ‬௣௜௫௘௟ ൈ ܸ௚ ሻ ‫ܦܵܩ‬

(12)

Where ‫ݐ݅ܤ‬௣௜௫௘௟ is the number of bits for each pixel, ܸ௚ is the ground track velocity, and ‫ܦܵܩ‬ represents ground sample distance. Solar panels area, sufficient to supply the required power to complete the mission is obtained as below: ‫ܣ‬௦௔ ൌ

ܲ௦௔ ܲ௘௢௟

(13)

In the above equation, ܲ௦௔ and ܲ௘௢௟ are the required power for consuming and terminating the life power generated by solar array, respectively. The required surface area of the radiator for thermal balancing can be obtained as below: ‫ܣ‬௥௔ௗ ൌ

‫ܧ‬௠௔௫ ሺߪሻ ൈ ሺߝሻ ൈ ሺܶ௔௟௟௢௪ ሻସ

(14)

Where ‫ܧ‬௠௔௫ is the maximum inner energy, ı is Stephan Boltzmann constant, জ is the radiation factor, and ܶ௔௟௟௢௪ is the maximum allowable temperature. Attitude stability as a comparatively more important constituent in receiving high quality images can be obtained as below: ܵ௧௔௕ ൌ ‫ݐ‬௜௡௧ ൌ ߙൌ

ߙ ‫ݐ‬௜௡௧ ‫ܦܵܩ‬ ܸ௚

‫ ܦܵܩ‬ൈ ‫ݐ݂݅ݎܦ‬Ψ ‫ܪ‬

(15) (16)

(17)

In equations (15)-(17), ‫ݐ‬௜௡௧ is the effective dwell time, ߙ is imaging parameter, and Drift is the acceptable deviation (its empirical value is usually about 10 percent). In this study a robust GA was used to solve the above optimization problem. The GA parameters are shown in Table 4. The GA parameters including selection, crossover and mutation as well as other parameters such as population size were utilized together to enhance the convergence rate of the optimization. Pareto frontier for the design problem within the MDF architecture is shown in Fig. 6. The study of the Pareto frontier can help the designers to have a better understanding of the behavior of the design space. The designers can evaluate how much the objectives can be improved and how important are the design variables and also quantify 18

the correlations among the variables themselves. In this case, the designers should choose an optimum design point among different solutions based on the customer and the mission requirements. Final Pareto optimal set for the design problem within MDF framework is presented in Fig. 6. In fact, the selected solution determines corresponding design options for each subsystem. Design characteristics of the selected solution of the RSS design problem based on the MDF framework are tabulated in Table 5. 4.2.2. Collaborative Optimization (CO) The procedure carried out for implementing the sample problem described in sub-section 4.1 within a distributed CO framework is explained through following lines. If the design problem has a great number of coupling variables, CO framework may be inefficient. The CO architecture for the proposed design problem is shown in Fig. 7. In the present paper, in order to decrease the coupling variables of the design problem in the CO framework, all analysis modules were grouped into three disciplines. According to Fig. 7 the objective functions and the design variables were defined at system level and an optimizer also minimized the objective functions subject to the existence of the system level constraints. The system level design variables were copies of the same variables at discipline level. In the system level optimization problem, only vectors of coupling variables ‫ ݕ‬௦ and system design variables ‫ ݔ‬௦ could be changed. The system level optimization of the design problem can be stated as below: ଻

‫݁ݖ݅݉݅݊݅ܯ‬

‫ ܯ‬ൌ ෍ሺ݉௡ ሻ ௡ୀଵ

‫݁ݖ݅݉݅݊݅ܯ‬

ܲܽ‫݊݋݅ݐݑ݈݋ݏܴ݁݀ܽ݋݈ݕ‬

ܹ݅‫݋ݐݐܿ݁݌ݏ݁ݎ݄ݐ‬

ܺଵ ௦ ǡ ܺଶ ௦ ǡ ܺଷ ௦ ǡ ܺହ ௦ ǡ ܺ଺ ௦ ǡ ܺ଻ ௦ ǡ ଼ܺ ௦ ǡ ܺଽ ௦ ǡ ܺଵ଴ ௦ ǡ ܺଵଵ ௦ ǡ ܺଵଶ ௦ ǡ ܺଵଷ ௦ ǡ ܺଵସ ௦ ǡ ቊ ቋ ௦ ௦ ‫ݕܽݎݎܣ‬௧௬௣௘ ௦ ǡ ‫ܯ‬ௗ௜௦ଵ ௦ ǡ ‫ܯ‬ௗ௜௦ଶ ǡ ܲௗ௜௦ଶ ǡ ‫ܣ‬௧௢௧௔௟ ௦ 

ܵ‫݋ݐݐ݆ܾܿ݁ݑ‬

݃ଵ ௦ ൑ ͳͲିହ ݃ଶ ௦ ൑ ͳͲିହ

19

்

(18)

݃ଷ ௦ ൑ ͳͲିହ ௦ ௦ , ܲௗ௜௦ଶ , ‫ܣ‬௧௢௧௔௟ ௦ are solar array type, mass of the first discipline Where ‫ݕܽݎݎܣ‬௧௬௣௘ ௦ , ‫ܯ‬ௗ௜௦ଵ ௦ , ‫ܯ‬ௗ௜௦ଶ

(containing payload mass and electrical power supply mass), the second discipline mass (containing mass of communication, ADCS and C&DH subsystems), the second discipline power (containing power of communication, ADCS and C&DH subsystems) and the satellite total area respectively. At system level optimization problem, the objective functions included minimization of the spacecraft mass and resolution and the system level design variables (Table 2) represented the mission characteristics and technology trade options. These were treated as the system level target values (shared design variables) corresponding to the discipline level design variables. At system level, ݃ଵ , ݃ଶ and ݃ଷ are equality compatibility constraints. It is important to note that the discipline level optimizations are free to satisfy their own constraints while minimizing their objective functions in a least square sense. It was shown that the constraints at the system level were of equality (discrepancy function ݃௜ ൌ Ͳǡ݅ ൌ ͳǡʹǡ͵) and had a complex form compared to constraints at the discipline levels. Also, their values corresponded to a measure of difference between the targets defined in the given discipline by the system level optimizer. The discipline level optimization problems of the sample problem are presented in Table 6. In the discipline level optimization problem, the variables without superscript “s” were changeable during the optimization process, while the system level variables were remained constant. The system level constraints for the proposed design problem were non-smooth. Therefore, derivative-based optimization algorithms might be trapped at the local minima instead of finding the global minimum. In order to overcome these difficulties, a more robust optimization algorithm GA was used, whilst SQP was utilized in the disciplinary optimization process. Based on the above mentioned issues, the CO architecture was used to solve the satellite design problem. The results can be found in Fig. 8. Design characteristics of the selected solution are shown in Table 7. As can be seen from Table 5 and Table 7, the solutions of the MDF and CO frameworks are similar.

20

4.3. RSS Reliability Based Design Optimization As discussed earlier in this paper, space systems are always faced with uncertainties in their operational environments. Neglecting the uncertainties in their design process may lead to unreliable systems. Because of the complex and multidisciplinary nature of the space systems, it is essential to model and manage uncertainties in the design process. It has been demonstrated that considering uncertainty models combined with probabilistic design techniques identifies solutions that have a minimum probability of failure [65]. This section applies the proposed RBMDO methodology to a case study of space system engineering. To compare the results with each other, the satellite design problem explained in section 4 was reinvestigated with the consideration of various uncertainties. There are several classifications in literature to address uncertainty taxonomy [65, 66]. In aerospace engineering field, uncertainty for vehicle synthesis and design is classified into different categories such as: measurement, input, model parameter, and operational/environmental [67]. Measurement uncertainty is considered when the system response cannot be directly calculated from the math models. Input uncertainty appears when the design requirements are imprecise. Model parameter uncertainty is due to the assumptions and errors in the mathematical models that try to simulate a physical system. Finally, operational/environmental uncertainty refers to unknown/uncontrollable external perturbations. In the presented space system design problem, various uncertainties have been considered. The most important uncertain design variables and parameters for the explained design problem are listed in Table 8. Compared with deterministic approaches, modeling the uncertainties in the design process leads to the response variables that follow probability distributions. For example, Fig. 9 shows the generated power for the box configuration with and without uncertainty consideration. As can be seen from this figure, the generated power shows noisy behavior of response functions under uncertainty. 4.3.1. Design under Uncertainty within MDF Framework The RBMDO methodology for the satellite design problem in the MDF framework (named as RBMDF) is illustrated in detail in Fig. 10. As can be seen from Fig. 10 the design procedure in the 21

reliability-based MDF framework consists of two iterative loops: the inner loop and the outer loop. In the inner loop, reliability analyzer block computes the probabilistic constraints based on the uncertain design variables and parameters. However, in the outer loop, system-level optimizer must satisfy the probabilistic constraints in order to improve the system performance. The RBMDO model for the satellite design problem in the MDF framework is formulated as follows: ݉݁ܽ݊ܵܽ‫ݏݏܽܯ ݁ݐ݈݈݅݁ݐ‬ ‫݁ݖ݅݉݅݊݅ܯ‬ ݉݁ܽ݊‫݊݋݅ݐݑ݈݋ݏܴ݁݀݊ݑ݋ݎܩ‬

ܵ‫݋ݐݐ݆ܾܿ݁ݑ‬

ܲ௥ ൣ݃௝ ሺܺሻ ൒ Ͳ൧ ൒ ͲǤͻͻͻͻǡ

݆ ൌ ͳǡ ǥ ǡ ݉ (19)

ܹ݅‫݄ݐ‬

݀௅ ൑ ݀ ൑ ݀௎

‫݋ݐݐܿ݁݌ݏ݁ݎ‬

ߤ௑௅ ൑ ߤ௑ ൑ ߤ௑௎  ݀ ‫ܴ א‬௠  ܺ ‫ܴ א‬௩ 

The design objective is to minimize the mean of satellite mass and the ground resolution considering several constraints. Furthermore, the design problem consists of twelve deterministic design variables, two random design variables, various random design parameters and ten probabilistic constraints (see Table 3). The probability for each constraint should be higher than the target reliability constraint. In the RBMDO problem, probabilistic constraints are normally non-smooth and nonlinear. Therefore, in this article the GA was employed as a system-level optimizer with regard to the robust and global solution characteristics. The Pareto frontier for the design problem within the RB-MDF framework is shown in Fig. 11 and the optimization results of applying this architecture to the satellite design problem are presented in Table 9.

22

4.3.2. Design under Uncertainty within CO Framework As discussed earlier in this paper, CO is an efficient framework for solving large scale complex optimization problems. In this sub-section, Collaborative Reliability-Based Design Optimization (named as RB-CO) for the satellite design problem is discussed. In the RB-CO framework, reliability analysis was performed in the disciplinary optimization problems. In the system-level optimization problem, optimizer tended to minimize the mean of objective functions and in the discipline-level optimization problems the probabilistic constraints were satisfied by local-level optimizers. The optimization results of applying RB-CO framework to the satellite design problem are presented in Fig. 12 and Table 9. In the RB-CO, performing reliability analysis in every optimization iteration is a time consuming procedure and satisfying probabilistic constraints is difficult. These features lead to an increase of converging difficulties in the RB-CO process (Fig. 12).

5. Results and Discussion In the deterministic optimization part of this article, multidisciplinary design optimization problem of a small remote sensing satellite was solved using MDF and CO architectures. By implementing the design problem in the MDF framework all the design constraints were satisfied and a set of solutions in the Pareto frontier was achieved. It should be noted that although the MDF framework was appropriate in the preliminary design phases it could not be efficient enough in the detail design of large scale complex space systems. Thus, the design problem was implemented in the multi-level CO framework. One of the advantages of this design framework is its ability in decomposition of the optimization problem to its subsystems. In this study the GA was used as a system-level optimizer and SQP was used for the optimization process in the inner design loop of the CO framework. The results of optimization with this method are presented in Table 7. As discussed earlier in this paper the achieved results of the MDF and CO frameworks were similar. The CO framework provides the possibilities for solving complex multidisciplinary design problems such as the design of satellite systems when using monolithic formulations are difficult or very time consuming. 23

In sub section 4.3 the non-deterministic approach for the design problem was implemented based on the RBMDO frameworks. The Pareto frontiers for the RB-MDF and RB-CO frameworks are shown in Fig. 11 and Fig. 12. Despite other mentioned approaches, in this methodology the design objectives were the mean satellite mass and the mean ground resolution. In this way, the optimization problem was consisted of inner and outer loops. In the inner loop, probabilistic constraints and the mean objective function were obtained based on the number of samples. It is important to note that in the nondeterministic approach the objective function values were different from the values gained in the previous approach. However, the designed system completed its mission correctly based on its uncertainties. The optimization results of this approach are shown in Table 9. Furthermore, previous results confirmed the possibility of convergence errors in the CO framework [68]. The results of the RB-CO shown in Fig. 12, approve this issue. It has been demonstrated that, in the RBMDO methodology, the design space is too noisy because of the variables and design parameters uncertainties. The probabilistic design space considering the mission, revisit time and swath width constraints based on the instrument elevation angel and orbital altitude are presented in Fig. 13, Fig. 14 and Fig. 15 respectively. In these figures the feasible design points which has probabilistic constraints reliability higher than 0.9999 are presented with black colour. As can be seen from the figures, the boundaries between the feasible design points are too noisy and non-smooth. Therefore, in this study the robust GA was used to solve the optimization problem. Selected solutions for the presented design frameworks are depicted in Fig. 13. As shown in Fig. 13, in the deterministic approach (MDF and CO frameworks), probabilistic constraints may not be satisfied. It should be noted that in the RBMDO frameworks the computation time is increased because of using an inner loop as a reliability analyzer.

6. Conclusion In this study, two deterministic and non-deterministic approaches were applied to the system design optimization of a small remote sensing satellite. Furthermore, multidisciplinary design optimization 24

architectures containing MDF, CO, RB-MDF and RB-CO were employed, evaluated, and compared for the design problem. The results have shown that although the deterministic approaches find the global optimum solution, they do not consider uncertainties which occur in the real world. Thus, deterministic approaches may lead to unreliable systems. Therefore, the reliability- based design optimization methods were required to obtain reliable designs. Also, in the present paper, a composition of multidisciplinary design optimization and reliability- based design optimization methods was introduced. In the proposed reliability- based multidisciplinary design optimization methodology, a near global reliable optimum solution was obtained considering the uncertainties. For obtaining exact results, a double-loop reliability method using Monte Carlo simulation was applied which resulted in an increase in the number of function calls and the computation time. In conclusion: in the primary design phases, if a system analysis is not very complex, MDF is recommended. For the large scale highly coupled design problems, due to the high computation time and function calls, it is difficult to apply collaborative optimization to the multidisciplinary and the reliability- based design optimization problems. Furthermore, to increase the computation efficiency, utilizing single-loop reliability methods or other sampling techniques is recommended. Space debris in the form of abandoned satellites is a growing threat especially at the heavily populated low earth orbits. To prevent new space junk from forming, new satellites should be equipped with a deorbiting mechanism. As future work, this mechanism could be considered as a design discipline which is a mass penalty for reducing the space debris problem.

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Guo, Review of uncertainty-based multidisciplinary design optimization methods for aerospace vehicles, Progress in Aerospace Sciences, 47 (2011) 450-479. [66] D.P. Thunnissen, F. Culick, California Institute of Technology. Division of Engineering and Applied Science., Propagating and mitigating uncertainty in the design of complex multidisciplinary systems, CIT theses 2005, California Institute of Technology,, Pasadena, Calif., 2005, pp. xxiv, 237 leaves ill. (some col.) 228 cm. [67] D. Laurent, M. Dimitri, Uncertainty modeling and management in multidisciplinary analysis and synthesis, 38th Aerospace Sciences Meeting and Exhibit, American Institute of Aeronautics and Astronautics2000. [68] N.F. Brown, J.R. Olds, Evaluation of Multidisciplinary Optimization Techniques Applied to a Reusable Launch Vehicle, Journal of Spacecraft and Rockets, 43 (2006) 1289-1300.

27

Table captions:

• •

C&DH

•

•

ADCS

•

•

•

•

• • • • • — •

—

•

•

•

•

— —

• •

•

ADCS

• • — •

C&DH

EPS

• —

payload

Comm

— • • •

Mission

Thermal

Structures Thermal Communication EPS Launcher Mission Payload

Subsystems

launcher

Structures

able 1 Design structure matrix of a Spacecraft system.

• • • •

• •

• •

•

•

— —

Table 2 Trade off alternatives of design variables. No. X1 X2 X3 X4 X5 X6

Description Instrument Elevation Angle Aperture Diameter Orbital Altitude Antenna Diameter Material Trade Choice Configuration Trade Choice

Unit degree cm km m ---------

Type Continuous Continuous Continuous Continuous Discrete Discrete

X7

Determination Trade Choice

-----

Discrete

X8

Actuator Trade Choice

-----

Discrete

X9 X10 X11 X12 X13 X14

Antenna Trade Choice Modulation Trade Choice Regulation Trade Choice Solar Array Trade Choice Battery Trade Choice Thermal Material Choice

-------------------------

Discrete Discrete Discrete Discrete Discrete Discrete

28

Lower Bound 86.2 0.8 400 ͲǤͳ ൈ ‫ܦ‬௙௔௜௥௜௡௚ Aluminum, Steel Box, Hexagon • • • • • • • •

Upper Bound 88.8 4 650 ͲǤ͵ ൈ ‫ܦ‬௙௔௜௥௜௡௚

(4×sun sensor)+(2× magnetometer)+GPS (3×horizon sensor)+(2× magnetometer)+ gyro+ GPS (2×horizon sensor)+(2× magnetometer)+ (3×sun sensor)+GPS (3×magneto torquer)+ boom+momentum wheel (3×magneto torquer)+momentum wheel (3×magneto torquer)+ reaction wheel+boom (3×magneto torquer)+(3× reaction wheel)+ boom (3×magneto torquer)+(3× reaction wheel)

Horn, Helix, Parabolic reflector QPSK, FSK, BPSK PPT, DET Silicon, Gallium Arsenide, Multi Junction Nickel Cadmium, Nickel Hydrogen, Lithium Ion White epoxy, White enamel, Black paint, Teflon, Aluminum

Table 3 Design constraints of the problem. No.

Description

Notation

Unit

1 2 3 4 5 6 7 8 9 10

Swath width Data generation rate Solar array area Radiator area Revisit time Power of transmitter Structure width Structure length Stability Life time

SW DR Asa Arad RT Ptran Stw Stl Stab LT

km Mbps ݉ଶ ݉ଶ day W m m deg/sec year

Lower Bound 30 0 0 0 30 0 0 0 0 1.5

Upper Bound 64 9 1.2 ‫ܣ‬௔௟௟௢௪ோ௔ௗ 57 25 0.8× fairing diameter 0.7×fairing length 10× pixel size 3

Table 4 Genetic Algorithm (GA) parameters.

29

Criteria

Value

Generations

1000

Population size

57

Crossover

0.8

Mutation

0.06

Maximum constraint violation

0.01

Percent penalty

0.05

Stall generation limit

75

Table 5 design characteristics for the selected optimal design based on MDF framework. Description

Unit

X1 X2 X3 X4 X5 X6 X7 X8 X9 X10 X11 X12 X13 X14

87.031 1.88 521 0.2837×Dfairing Aluminum Box (4×sun sensor)+(2× magnetometer)+GPS (3×magneto torquer)+momentum wheel Helix BPSK PPT Multi Junction Lithium Ion White epoxy

degree cm km m -----------------------------------------

Satellite Mass

63.01

kg

Camera Resolution

42.42

m

Design variables Objective Functions

MDF Optimization

Table 6 Formulation of the discipline level optimization problems in the CO architecture.

Discipline 1

Objective function of the discipline ଶ ܺଵ ܺଶ ଶ ܺଷ ଶ ݃ଵ ൌ ቆͳ െ ௦ ቇ ൅ ൬ͳ െ ௦ ൰ ൅ ൬ͳ െ ௦ ൰ ܺଵ ܺଶ ܺଷ ଶ ‫ݕܽݎݎܣ‬௧௬௣௘ ‫ܯ‬ா௉ௌ ଶ ൅ ቆͳ െ ൰ ௦ ቇ ൅ ൬ͳ െ ‫ݕܽݎݎܣ‬௧௬௣௘ ‫ܯ‬ா௉ௌ ௦ ଶ

ଶ

ଶ

Discipline 2

ܺଵ ܺଶ ܺଷ ௦ ቇ ൅ ቆͳ െ ௦ ቇ ൅ ቆͳ െ ௦ ቇ ܺଵ ܺଶ ܺଷ ଶ ଶ ‫ܯ‬ௗ௜௦ ଶ ܲௗ௜௦ ଶ ൅ ቆͳ െ ௦ ቇ ൅ ቆͳ െ ௦ ቇ ‫ܯ‬ௗ௜௦ ଶ ܲௗ௜௦ ଶ

Discipline 3

‫ܣ‬௧௢௧௔௟ ଶ ܺଷ ൰ ݃ଷ ൌ ቆͳ െ ௦ ቇ ൅ ൬ͳ െ ܺଷ ‫ܣ‬௧௢௧௔௟ ௦

݃ଶ ൌ ቆͳ െ

ଶ

30

Subsystems Constraints ͵Ͳ ൑ ܹܵ ൑ ͸Ͷ ͵Ͳ ൑ ܴܶ ൑ ͷ͹ Ͳ ൑ ‫ ܴܦ‬൑ ͻ Ͳ ൑ ‫ܣ‬௦௔ ൑ ͳǤʹ ͳǤͷ ൑ ‫ ܶܮ‬൑ ͵ Ͳ ൑ ܲ௧௥௔௡ ൑ ʹͷ ‫ ܾܽݐݏ‬൑ ͳͲ ൈ ‫݁ݖ݅ݏ݈݁ݔ݅݌‬

‫ܣ‬௥௔ௗ ൑ ‫ܣ‬௔௟௟௢௪ோ௔ௗ – ୵ ൑ ͲǤͺ ൈ ˆƒ‹”‹‰†‹ƒ‡–‡” ܵ‫ݐ‬௟ ൑ ͲǤ͹ ൈ ݂ܽ݅‫݄ݐ݈݃݊݁݃݊݅ݎ‬

Table 7 design characteristics for the selected optimal design based on CO framework.

Objective Functions

Design variables

Description

MDF Optimization

Unit

X1 X2 X3 X4 X5 X6 X7 X8 X9 X10 X11 X12 X13 X14

87.193 1.96 524.8 0.2908×Dfairing Aluminum Box (4×sun sensor)+(2× magnetometer)+GPS (3×magneto torquer)+momentum wheel Helix BPSK DET Multi Junction Nickel Hydrogen Black paint

degree cm km m -----------------------------------------

Satellite Mass

63.08

kg

Camera Resolution

40.74

m

Table 8 The most important uncertain parameters in the RBMDO framework for the space system design problem.

No. 1 2 3 4 5 6 7 8 9 10

31

Parameter Orbital altitude Earth radius Minimum elevation angle Orbital inclination Axial load factor Lateral load factor Natural Frequency Inherent degradation Solar flux Instrument Elevation Angle

Unit km km degree degree --------Hz ----w/m^2 degree

Standard deviation 1% 0.1% 20% 0.5% 15% 15% 5% 18% 10% 1%

Mean value variable ͸Ǥ͵͹ͺͳ͵͹ ൈ ͳͲଷ 10 55 7 1.6 20 0.7 1358 variable

Table 9 Selected solutions by various methods for the satellite design problem. Design variable

Unit

Instrument Elevation Angle Aperture Diameter Orbital Altitude Antenna Diameter Material Trade Choice Configuration Trade Choice

degree cm km m ---------

Determination Trade Choice

-----

Actuator Trade Choice

-----

Antenna Trade Choice Modulation Trade Choice Regulation Trade Choice Solar Array Trade Choice Battery Trade Choice Thermal Material Trade Choice

-------------------------

32

MDF 87.031 1.88 521 0.2837×Dfairing Aluminum Box (4×sunsensor)+GPS +(2×magnetometer) (3×magneto torquer) +momentum wheel Helix BPSK PPT Multi Junction Lithium Ion White epoxy

Design framework CO RB-MDF 87.193 86.7 1.96 2 524.8 499.1 0.2908×Dfairing 0.2957×Dfairing Aluminum Aluminum Box Box (4×sunsensor)+GPS (4×sunsensor)+GPS +(2×magnetometer) +(2×magnetometer) (3×magneto torquer) (3×magneto torquer) +momentum wheel +momentum wheel Helix Helix BPSK QPSK DET PPT Multi Junction Multi Junction Nickel Hydrogen Lithium Ion Black paint Aluminum

RB-CO 86.569 2.38 489.4 0.2867×Dfairing Aluminum Box (4×sunsensor)+GPS +(2×magnetometer) (3×magneto torquer) +momentum wheel Helix BPSK DET Multi Junction Lithium Ion White epoxy

Figure captions: Fig. 1 Schematic of remote sensing satellite subsystems. Fig. 2 Interdisciplinary relationship in spacecraft design. Fig. 3 Multidisciplinary Feasible framework. Fig. 4 Collaborative optimization framework. Fig. 5 MDF framework for the satellite design problem. Fig. 6 Pareto frontier for the satellite design problem within the MDF architecture. Fig. 7 CO architecture for the satellite design problem. Fig. 8 Pareto frontier for the satellite design problem within the CO architecture. Fig. 9 Comparison of generated power under uncertainty and without uncertainty for the box configuration in a specific orbit. Fig. 10 The RBMDO methodology for the satellite design problem in the MDF framework. Fig. 11 Pareto frontier for the satellite reliability-based design optimization problem within the MDF architecture. Fig. 12 Pareto frontier for the satellite design problem within the Collaborative reliability-based design optimization framework. Fig. 13 probabilistic design space for the mission reliability. Fig. 14 probabilistic design space for the revisit time reliability constraint. Fig. 15 probabilistic design space for the swath width reliability constraint.

33

Payload

Bus Structure

Antennas

Solar Panel

Mission Analysis Orbit Design

Payload

Structure

ADCS

Comm

C&DH

EPS

Thermal

Disciplinary Analysis 1

Disciplinary Analysis 2

Disciplinary Analysis N

System-level Optimizer

Multidisciplinary analysis

System level optimizer Minimize : System objective function f (s) subject to : Interdisciplinary compatibility constraints (gh*= 0, i=1,…,N)

s1 Discipline optimizer 1

Minimize: Interdisciplinary discrepancy function g1 Subject to: Local constraints

Disciplinary analysis1

sN

s2

Discipline optimizer N

Discipline optimizer 2 Minimize: Interdisciplinary discrepancy function g2 Subject to :Local constraints

Disciplinary analysis 2

..

Minimize: Interdisciplinary discrepancy function gN Subject to :Local constraints s.t .

Disciplinary analysis N

Mission Requirements

Technology Level

Mission Analysis /Orbit Design

Payload

System-level Optimizer

ADCS

C & DH

Communication

Thermal Control

Electrical Power Supply

Structure and Mechanisms

Fig6

System-Level Optimizer

S1

g1 s

Disciplinary Optimizer 1

Mission Analysis / Orbit Design

Electrical Power Supply

Payload

S2

g2 s

Disciplinary Optimizer 2

Communication

Command & Data Handling

Attitude Determination & Control

S3

g3 s

Disciplinary Optimizer 3

Structure and Mechanisms

Thermal Control

100 90 64

Sunlight Time (min)

80

61 59

70

56 53

60

50 48

50

45 42

40

39 37

30

34 31

20

28 26

10 0

0

50

100

150

200

Day

250

300

350

a) Generated power without uncertainty

100 90 66 63 61 59 57 55 53 51 49 47 44 42 40 38 36 34 32 30 28 26

Sunlight Time (min)

80 70 60 50 40 30 20 10 0

0

50

100

150

200

Day

250

b) generated power under uncertainty

300

350

Mission Requirements

Technology Level

Reliability Analyzer

Mission Analysis / Orbit Design

System-level Optimizer

Payload

ADCS

C & DH

Communication

Thermal Control

Electrical Power Supply

Structure and Mechanisms

Fig12

Fig13

Fig14

Fig15