Two-Level Approach for Simultaneous Component Assignment and Layout Optimization with Applications to Spacecraft Optimal Layout
Abstract
:1. Introduction
General Context
2. State of the Art
2.1. Methods for Optimal Layout Problems
2.1.1. Exact Methods
2.1.2. Heuristic Methods
2.1.3. Metaheuristics
2.1.4. Hybrid Approaches
2.1.5. Quasi-Physical Methods
2.1.6. Machine Learning Techniques
2.2. Layout Problem for Aerospace Systems
2.3. Focus on Satellite Module Layout Problems
3. Formulation of the Satellite Module Layout Problems
3.1. Geometry of the Containers and the Components
3.2. Design Variables
- For each component, the number of its assigned surface , , where are defined as unordered discrete variables;
- The location of the center of inertia of each component on the assigned surface considered as continuous variables , ;
- The orientations of cuboid components on the assigned surface are considered as discrete variables with values of 0° or 90°, , .
3.3. Objective Function
- : the local coordinate system related to each component. corresponds to the center of inertia of the component and the axes are defined with the symmetry planes of the components.
- : the coordinate system related to the system of components. stands for the current centroid of the system of components and the axes are defined with the symmetry planes of the module.
- : the coordinate system related to the module. O is the geometric center of the container and the axes are defined using its symmetry planes.
3.4. Constraint Functions
- Overlapping constraints between components: No overlapping between components is allowed on each surface. As in [19,20,108], the plates are supposed to be sufficiently spaced to avoid any overlapping between components positioned on the surfaces and . The overlapping constraint between components is formulated as follows:
- Overlapping constraints between the components and the exclusion zones: No overlapping between the components and the exclusion zones is allowed on each surface. The overlapping constraint is expressed as
- Balancing constraints: The center of gravity (CG) of the whole laid out module must be positioned near the center of gravity of the empty module within a given tolerance along the x and y axes. Then, the balancing constraint can be formulated as follows:
- Constraints relative to the angles-of-inertia: In the multi-container configuration, a geometrical constraint corresponding to the angles of inertia is added, with a corresponding tolerance . The constraint relative to the angles of inertia is defined as follows:
- Functional constraints: These are also defined in terms of incompatibility between several components. Thermal as well as electromagnetic thresholds are defined such that some components responsible for thermal or electromagnetic fields must be spaced away at given distances. More precisely, a set of pairs of incompatible heat components and a set of pairs of incompatible electromagnetic components are defined. Thus, the thermal functional constraint is defined asWith the same formalism, the electromagnetic functional constraint is defined as
3.5. Mathematical Formulation
Remarks on the Inertia Equations
4. Two-Stage Algorithm Combining Genetic Algorithm and Quasi-Physical Approach
4.1. General Structure of the Algorithm
4.2. Assignment Task
4.2.1. Formulation of the Assignment Subproblem
- The balancing constraint:The value of represents the size of the tolerance zone for the positioning of the center of gravity along the z-axis. Without loss of generality, it is taken as equal to 3.0, which corresponds to the size of the tolerance zones along the x- and y-axes [19,20,48,108,112,117] so that the center of gravity must be positioned in a sphere of tolerance with a radius of 3.0 mm centered at the geometrical center of gravity of the empty module.
- The occupation rate constraints defined for each surface j:
4.2.2. Algorithm for the Assignment Task
4.3. Layout Task
4.3.1. Formulation of the Layout Subproblem
4.3.2. Algorithm for the Layout Task
Virtual-Force System
- Its translational and rotational accelerations and ;
- Its translational and rotational speeds and ;
- Its position and orientation .
- Minimize an objective function: force and torque ;
- Solve for the overlapping constraints between components i and j: force and torque ;
- Solve for the overlapping constraints between component i and an exclusion zone (EZ): force and torque ;
- Solve for the overlapping constraints between component i and the container: force and torque ;
- Solve for the functional constraints between two components i and j: force and torque ;
- Solve for the balancing constraint: force (CG stands for center of gravity);
- Solve for the angle of inertia constraint: force and torque ;
- More generally, solve for any constraint function: force and torque .
- The overlappingconstraint force and torque between two components: If two components i and j are overlapping each other, repulsive forces and are applied to each of them, as illustrated in Figure 4 and Figure 5. The overlap force is expressed asMoreover, similar to [127], additional torques are applied to non-circular components in order to solve the overlapping constraint, as illustrated in Figure 4.The overlapping forces are initially applied at the point, which corresponds to the geometrical center of the polygon of intersection between the two components. Then, the resulting torque applied at the point is given by
- The overlapping constraint force and torque between a component and an exclusion zone: If a component is overlapping an exclusion zone, a repulsive force is applied to the component, as illustrated in Figure 5. As the exclusion zone can be seen as a fixed component, the corresponding force and torque expressions are similar to the previous overlapping forces and torques between the two components and written with Equations (35) and (36).
- The overlapping constraint force and torque between a component and the container: If a component is not fully overlapping the container, an attractive force is applied to the component, as illustrated in Figure 5. The container force is expressed as
- The functional constraint force and torque between two components: If a component i is too close to an incompatible component j, a repulsive force is applied to the component i, as illustrated in Figure 6. The functional force acts as an overlapping force between the component i and the influence zone of the component j. Then, the functional force is expressed asThe related torque is calculated as follows:
- The balancing constraint force: In order to position the center of mass of the components in a tolerance zone centered at the geometrical center of the container, gradient-based forces are applied along the opposite of the gradient of the position of the global center of mass according to the position of the center of inertia of each component. This force is named and is illustrated in Figure 7. The balancing force is expressed as
- The differentiable constraint function force and torque: Generally, any differentiable constraint function (DF), as the angle of inertia constraints, can be addressed thanks to gradient-based forces and torques, expressed as
- The objective function forces and torques: To minimize one objective function f, a gradient-based force and a gradient-based torque can be applied. The objective function force and torque are expressed as
Swap Operator
- The swap operator can exchange all the components by pairs.
- Two components are swapped if the swap leads to an improvement of the objective function or a decrease in the violation of some chosen constraint(s) while not deteriorating the other constraint(s) from a relaxation factor r.
- The swap operator is called straight from the first iteration. Then, it can occur throughout the optimization process.
Initialization
Multistart
4.4. Algorithm Framework
5. Application to the Satellite Module Layout Problem
5.1. Configurations of Both Stages
5.1.1. Upper Stage Settings
5.1.2. Lower Stage Settings
5.2. Results and Analysis
- Ref. [20]: a dual-system cooperative co-evolutionary algorithm (called Oboe-CCEA) is developed for the multi-container SMLP based on both Potter’s coevolutionary framework [89] and the variable-grain model [135]. However, the assignment is an input of the algorithm, i.e., it is not optimized but taken from [108] and based upon human experience. It must be noted that this study does not consider the functional constraints and . The rest of the mathematical formulation remains identical.
- Ref. [19]: a two-stage algorithm called Dynamic FS is developed to solve both assignment and layout tasks of the multi-container SMLP. Heuristic rules are used to assign the components. They are mathematically translated into a multi-objective optimization problem and an NSGA-II algorithm is employed to solve it. The assignment list is then determined using a fuzzy decision-making method. The layout task is performed using the NDCCDE/DPSO algorithm [112], which corresponds to a dual-system cooperative co-evolutionary algorithm.
5.2.1. Global Performance
- The mean value of the final obtained layout objective functions, i.e., their global inertia;
- The standard deviation (STD) of the final obtained layout inertia;
- The best layout obtained in terms of inertia (among the 50 repetitions);
- The worst layout obtained in terms of inertia (among the 50 repetitions);
- The success rate, i.e., the percentage of runs leading to a feasible layout (all the constraints are satisfied).
5.2.2. Analysis of the Assignment
5.2.3. Analysis of the Layout
5.2.4. Global Analysis
- The assignment scheme: The proposed assignment task resolution allows to position the components such that their center of gravity coincides with the center of gravity of the empty module, which contributes to a better global inertia.
- The layout: As highlighted in [124], the CSO-VF outperforms the population-based counterparts like GAs thanks to its ability to solve the constraints with dedicated operators. Indeed, Figure 14 shows that the CSO-VF algorithm systematically positions the heavier components closer to the center of the surfaces, which contributes to minimize the global inertia of each container. On the contrary, it is observed that the Dynamic FS algorithm sometimes positions small and light components around the central bus.
6. Conclusions
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
Appendix A. Components and Containers
Index | Geometry | Dimension 1 | Dimension 2 | Height | Mass |
---|---|---|---|---|---|
1 | Cuboid | 250 | 150 | 250 | 28.13 |
2 | Cuboid | 250 | 150 | 250 | 28.13 |
3 | Cuboid | 250 | 150 | 250 | 28.13 |
4 | Cuboid | 250 | 150 | 250 | 28.13 |
5 | Cuboid | 250 | 150 | 250 | 28.13 |
6 | Cuboid | 250 | 150 | 250 | 28.13 |
7 | Cuboid | 250 | 150 | 250 | 28.13 |
8 | Cuboid | 250 | 150 | 250 | 28.13 |
9 | Cuboid | 200 | 160 | 200 | 19.2 |
10 | Cuboid | 200 | 160 | 200 | 19.2 |
11 | Cuboid | 200 | 160 | 200 | 19.2 |
12 | Cuboid | 160 | 120 | 250 | 15.36 |
13 | Cuboid | 160 | 120 | 250 | 15.36 |
14 | Cuboid | 160 | 120 | 150 | 8.64 |
15 | Cuboid | 160 | 120 | 150 | 8.64 |
16 | Cuboid | 160 | 120 | 150 | 8.64 |
17 | Cuboid | 150 | 100 | 100 | 5.40 |
18 | Cuboid | 150 | 100 | 100 | 5.40 |
19 | Cuboid | 150 | 100 | 100 | 5.40 |
20 | Cuboid | 150 | 100 | 100 | 5.40 |
21 | Cuboid | 150 | 100 | 100 | 5.40 |
22 | Cuboid | 150 | 100 | 100 | 5.40 |
23 | Cuboid | 150 | 100 | 100 | 5.40 |
24 | Cuboid | 150 | 100 | 100 | 5.40 |
25 | Cylinder | 100 | 250 | 23.56 | |
26 | Cylinder | 100 | 250 | 23.56 | |
27 | Cylinder | 100 | 250 | 23.56 | |
28 | Cylinder | 100 | 250 | 23.56 | |
29 | Cylinder | 100 | 250 | 23.56 | |
30 | Cylinder | 100 | 250 | 23.56 | |
31 | Cylinder | 100 | 250 | 23.56 | |
32 | Cylinder | 100 | 250 | 23.56 | |
33 | Cylinder | 100 | 200 | 18.85 | |
34 | Cylinder | 100 | 200 | 18.85 | |
35 | Cylinder | 100 | 200 | 18.85 | |
36 | Cylinder | 100 | 160 | 15.08 | |
37 | Cylinder | 100 | 160 | 15.08 | |
38 | Cylinder | 100 | 160 | 15.08 | |
39 | Cylinder | 75 | 160 | 8.48 | |
40 | Cylinder | 75 | 160 | 8.48 | |
41 | Cylinder | 75 | 160 | 8.48 | |
42 | Cylinder | 75 | 160 | 8.48 | |
43 | Cylinder | 75 | 150 | 7.95 | |
44 | Cylinder | 75 | 150 | 7.95 | |
45 | Cylinder | 75 | 150 | 7.95 | |
46 | Cylinder | 75 | 150 | 7.95 | |
47 | Cylinder | 75 | 150 | 7.95 | |
48 | Cylinder | 75 | 150 | 7.95 | |
49 | Cylinder | 60 | 150 | 5.09 | |
50 | Cylinder | 60 | 150 | 5.09 | |
51 | Cylinder | 60 | 150 | 5.09 | |
52 | Cylinder | 60 | 150 | 5.09 | |
53 | Cylinder | 60 | 150 | 5.09 | |
54 | Cylinder | 60 | 150 | 5.09 | |
55 | Cylinder | 60 | 150 | 5.09 | |
56 | Cylinder | 60 | 150 | 5.09 | |
57 | Cylinder | 60 | 150 | 5.09 | |
58 | Cylinder | 60 | 150 | 5.09 | |
59 | Cylinder | 60 | 150 | 5.09 | |
60 | Cylinder | 60 | 150 | 5.09 |
Index 1 | Index 2 | Type | Distance |
---|---|---|---|
25 | 56 | Heat | 200 |
25 | 60 | Heat | 200 |
29 | 49 | Heat | 200 |
29 | 55 | Heat | 200 |
37 | 55 | Electromagnetic | 300 |
37 | 58 | Electromagnetic | 300 |
- Its mass: kg;
- Its center of mass located at in the system of coordinates;
- Its matrix of inertia:
Appendix B. Correction of Inertia Equations
Appendix C. Swap
- While no feasible layout is found, the step is fixed to . Then, at any current iteration during which the swap operator is employed, the next swap iteration noted is calculated asConsequently, the swap operator is called at a lower frequency and helps to find a feasible solution.
- Once a feasible solution is found, each time the swap operator is called, the convergence curve is interpolated and the next iteration of the swap operator is calculated as
Algorithm A1 The Swap Operator |
|
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Metrics | Oboe-CCEA [20] | Dynamic FS [19] | GA+CSO-VF |
---|---|---|---|
Mean of final inertia | 718.93 | 694.06 | 682.96 |
STD of final inertia | 2.64 | 2.96 | 2.73 |
Best layout | 712.99 | 689.00 | 676.75 |
Worst layout | 726.59 | 700.37 | 688.45 |
Success rate | 60% | 80% | 100% |
Oboe-CCEA [20] | Dynamic FS [19] | GA+CSO-VF | |
---|---|---|---|
Assignment | x | ||
Layout |
Reference | Surface 1 | Surface 2 | Surface 3 | Surface 4 |
---|---|---|---|---|
Fixed [20] | 150.63 | 231.18 | 235.22 | 198.42 |
Heuristic+NSGA-II [19] | 88.33 | 314.20 | 317.29 | 95.63 |
GA (This paper) | 85.95 | 289.02 | 326.67 | 113.78 |
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Gamot, J.; Balesdent, M.; Wuilbercq, R.; Tremolet, A.; Melab, N. Two-Level Approach for Simultaneous Component Assignment and Layout Optimization with Applications to Spacecraft Optimal Layout. Appl. Sci. 2024, 14, 8120. https://doi.org/10.3390/app14188120
Gamot J, Balesdent M, Wuilbercq R, Tremolet A, Melab N. Two-Level Approach for Simultaneous Component Assignment and Layout Optimization with Applications to Spacecraft Optimal Layout. Applied Sciences. 2024; 14(18):8120. https://doi.org/10.3390/app14188120
Chicago/Turabian StyleGamot, Juliette, Mathieu Balesdent, Romain Wuilbercq, Arnault Tremolet, and Nouredine Melab. 2024. "Two-Level Approach for Simultaneous Component Assignment and Layout Optimization with Applications to Spacecraft Optimal Layout" Applied Sciences 14, no. 18: 8120. https://doi.org/10.3390/app14188120