Multi-objective layout optimization of a satellite module using the Wang-Landau sampling method with local search
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Abstract
The layout design of satellite modules is considered to be NP-hard. It is not only a complex coupled system design problem but also a special multi-objective optimization problem. The greatest challenge in solving this problem is that the function to be optimized is characterized by a multitude of local minima separated by high-energy barriers. The Wang-Landau (WL) sampling method, which is an improved Monte Carlo method, has been successfully applied to solve many optimization problems. In this paper we use the WL sampling method to optimize the layout of a satellite module. To accelerate the search for a global optimal layout, local search (LS) based on the gradient method is executed once the Monte-Carlo sweep produces a new layout. By combining the WL sampling algorithm, the LS method, and heuristic layout update strategies, a hybrid method called WL-LS is proposed to obtain a final layout scheme. Furthermore, to improve significantly the efficiency of the algorithm, we propose an accurate and fast computational method for the overlapping depth between two objects (such as two rectangular objects, two circular objects, or a rectangular object and a circular object) embedding each other. The rectangular objects are placed orthogonally. We test two instances using first 51 and then 53 objects. For both instances, the proposed WL-LS algorithm outperforms methods in the literature. Numerical results show that the WL-LS algorithm is an effective method for layout optimization of satellite modules.
Keywords
Packing Layout design Satellite module Wang-Landau algorithmCLC number
TP391 V4741 Introduction
The layout design of a satellite module involves placing a certain number of objects, including various instruments and devices, in a particular satellite module, while satisfying various constraints with specific objectives. Stability and service life are essential requirements of a successful layout design, and other performances of the whole satellite module system should also be considered.
Due to engineering and mathematics complexities, satellite module layout design is known as an NP-hard problem. It stems from the bin packing problem (Crainic et al., 2011; Gonçalves and Resende, 2011; Khanafer et al., 2012), which can mainly be classified into two categories according to the study subjects: (1) 2D bin packing problem; (2) 3D bin packing problem. For the 2D bin packing problem, Gonçalves and Resende (2011) proposed a parallel multi-population genetic algorithm that hybridizes a novel placement procedure with a genetic algorithm based on random keys. Khanafer et al. (2012) proposed a tree decomposition method based heuristic strategy to decompose the original problem into several sub-problems. Li et al. (2012) proposed a heuristic particle swarm optimization approach with a quasi-human strategy. Zhang et al. (2005) proposed a hybrid heuristic algorithm based on divide-and-conquer and greedy strategies for the 2D rectangular packing problem. Liu et al. (2009) proposed an improved energy landscape paving (ELP) method by incorporating a new configuration update mechanism into the ELP method to solve the circular packing problem. For the 3D bin packing problem, Martello et al. (2000) proved that the asymptotical worst-case performance of the continuous lower bound is 1/8, and presented an exact branch-and-bound algorithm. Moon and Nguyen (2014) presented a mixed integer programming model which features upper and lower bounds for the 3D bin packing problem. To deal with the 3D parallelepiped bin packing problem, Allen et al. (2011) proposed a hybrid placement strategy, and the test results showed that it outperforms other methods from the literature.
In this paper, we study the layout design of a satellite module with performance constraints that can be considered a special case of the bin packing problem mentioned above. Various methods have been proposed to deal with the layout design of satellite modules. Sun and Teng (2003) proposed a two-stage layout method, the centripetal balancing method, for global layout design in the first stage, and an ant colony optimization algorithm for detailed layout design of a satellite module in the second stage. Zhang et al. (2008) presented a hybrid method by combining soft computing techniques, including the Hopfield neural network, genetic algorithm/particle swarm optimization, and quasi-principal component analysis. Teng et al. (2010) developed a dual-system framework based on the cooperative coevolutionary genetic algorithm with the merits of great population diversity and a decrease in premature convergence. He et al. (2013) proposed a quasi-physical algorithm based on coarse and fine adjustment. Tang and Teng (1999) presented a decimal coded adaptive genetic algorithm, which decreases combinatorial explosion and premature convergence. Wang and Teng (2009) proposed a knowledge fusion approach, which harnesses the potential of both humans and computers through evolutionary computation. Liu and Teng (2008) presented a human algorithm knowledge layout design (HAKD) method, which fuses human intelligence, computer intelligence (evolution algorithm), and prior knowledge (relevant layout diagrams) at the gene level of the evolution algorithm. Huo and Teng (2009) optimized a layout design using four steps. First, the whole layout problem was decomposed into several sub-layout problems. Second, a relaxation model was adopted to distribute all objects among subspaces. Third, a coevolutionary genetic algorithm was adopted to solve sub-layout problems. Finally, a heuristic combination-rotation method was adopted to adjust the constraints to obtain the final layout. Lei and Qiu (2006) presented a novel adaptive particle swarm optimizer based on multi-modified strategies, which can not only escape from the attraction of local optima, but also maintain the characteristic of a fast speed search in the early convergence phase. Zhou et al. (2005) presented a constraint handling strategy suitable for particle swarm optimization, and used direct search to intensify its local search ability. Jin and Teng (2007) proposed a case retrieval algorithm for reusing previously stored design solutions and optimizing layout design using prior knowledge and an evolutionary approach. Chen et al. (2008) presented an improved differential evolution algorithm, which makes random microperturbations to the location of components to obtain a better layout scheme. Zhang et al. (2013) proposed a parallel dual-system cooperative coevolutionary differential evolution algorithm with human-computer cooperation. Xu and Xiao (2008) presented an ant colony optimization algorithm based on a step-by-step positioning technique. Liu et al. (2011) proposed a heuristic algorithm based on tabu search, where tabu search was used to jump out of the local minima. Liu and Li (2010) proposed the basin filling algorithm by combining the improved energy landscape paving method, the gradient method, and the configuration update strategy. Liu et al. (2010) proposed a heuristic simulated annealing algorithm by incorporating the heuristic neighborhood search mechanism and the adaptive gradient method into the simulated annealing procedure.
Although the approaches mentioned above enable effective layout design of the satellite module, their efficiency still needs to be improved. In fact, a reasonable combination of the stochastic algorithm with global search ability, the local accurate search method, and some heuristic strategies may be an effective way to construct a high-performance algorithm for layout optimization of satellite modules. The Wang-Landau (WL) sampling algorithm (Wang and Landau, 2001; Landau et al., 2004) is an improved Monte Carlo algorithm. Unlike conventional Monte Carlo simulations that generate a probability distribution at a given temperature, the WL sampling method can estimate the density of states accurately via a random walk, which produces a flat histogram in the energy space. There have been many improvements on and applications of the WL sampling algorithm. For example, Zhou and Bhatt (2005) proved the convergence of the WL sampling algorithm, and found that the histogram increases uniformly with small fluctuations after a stage of initial accumulation. Seaton et al. (2010) used it to describe the thermodynamic behavior of a continuous homopolymer. Schulz et al. (2003) proposed a simple modification of the WL sampling algorithm. This modification removes the systematic error that occurs at the boundary of the range of energy over which the random walk takes place. The greatest challenge of the layout optimization of the satellite module is that the function to be optimized is characterized by a multitude of local minima separated by high-energy barriers. The WL sampling method can visit all the accessible states of the system, which means that it can jump out of these high-energy barriers. Therefore, the WL sampling method is an ideal global search algorithm for layout optimization of satellite modules. In this paper, we put forward a hybrid method called WL−LS by incorporating the WL sampling algorithm, the local search (LS) method, and the heuristic layout update strategies to deal with the layout optimization of a satellite module with performance constraints. Numerical results show that the proposed WL−LS algorithm is an effective method in designing the layout of satellite modules.
2 Problem statement
- 1.non-overlapping constraints:$${g_1}\left( X \right) = \sum\limits_{i = 0}^{N - 1} {\sum\limits_{l = i + 1}^N {{\rm{int}}({A_i}) \cap {\rm{int}}({A_l}) = \emptyset ,} } $$(3)
- 2.centroid error constraints:$${g_2}(X) = \left| {{x_{\rm{c}}} - {x_{\rm{e}}}} \right| \le 3.0,$$(4)$${g_3}(X) = \left| {{y_{\rm{c}}} - {y_{\rm{e}}}} \right| \le 3.0,$$(5)$${g_4}(X) = \left| {{z_{\rm{c}}} - {z_{\rm{e}}}} \right| \le 3.0,$$(6)
- 3.equilibrium degree error constraints:$${g_5}(X) = \left| {{\theta _{x'}}(X)} \right| \le 0.03,$$(7)$${g_6}(X) = |{\theta _{y'}}(X)| \le 0.03,$$(8)Here X=(x_{1}, y_{1}, z_{1}, α_{1}, …, x_{ i }, y_{ i }, z_{ i }, α_{ i }, …, x_{ N }, y_{ N }, z_{ N }, α_{ N }); α_{ i } are the orientation angles for the rectangles shown in Fig. 1c (for the circles, α_{ i } do not exist); N is the number of layout objects; (x_{ i }, y_{ i }, z_{ i }) are the 3D coordinates of object A_{ i }; f_{1}(X) is the space utilization function, which is measured by radius R_{0} of the satellite module; f_{2}(X) is the inertia moment of the whole system; J_{x′}(X), J_{y′}(X), and J_{z′}(X) are inertia moments of the whole system with respect to the coordinate system O′x′y′z′; int(A_{ i }) denotes the internal part of object A_{ i }; (x_{e}, y_{e}, z_{e}) is the expected centroid of the whole system; (x_{c}, y_{c}, z_{c}) is the real centroid position of the whole system; θ_{x′}(X), θ_{y′}(X), and θ_{Z′}(X) (Fig. 2) are the angles between the principal inertia axes of the module and the coordinate system O′x′y′z′ axes.$${g_7}(X) = |{\theta _{z'}}(X)| \le 0.03.$$(9)
3 Framework for WL−LS
In this work, all tested objects were distributed onto the four bearing plate surfaces in advance according to the centripetal balancing method and human-computer cooperative method, as presented by Sun and Teng (2003) and Liu and Teng (2008). Therefore, in the process of layout optimization, every object always stays on the same surface; that is to say, the surface to which an object belongs is fixed. Suppose the layout X=(X_{1} X_{2}, X_{3}, X_{4})=(x_{11}, y_{11}, z_{11}, α_{11}, …, x_{1j}, y_{1j}, z_{1j}, α_{1j}; x_{21}, y_{21}, z_{21}, α_{21}, …, x_{2k}, y_{2k}, z_{2k}, α_{2k}; x_{31}, y_{31}, α_{31}, …, x_{3m}, y_{3m}, z_{3m}, α_{3m},; x_{41}, y_{41}, z_{41}, α_{41}, …, x_{4n}, y_{4n}, z_{4n}, α_{4n}), where X_{1}, X_{2}, X_{3}, and X_{4} are the layouts of the four surfaces P_{1}, P_{2}, P_{3}, and P_{4}, respectively, and j, k, m, and n are the numbers of the objects distributed onto the four surfaces, respectively, j+k+m+n=N. Our goal is to obtain an optimal layout which minimizes the two objectives f_{1}(X) and f_{2}(X) and meanwhile satisfies the constraints given by Eqs. (3)–(9).
- (1)
Set \(\overline {{R_i}} = {R_0},\;\;\underline {{R_i}} = R\).
- (2)
Set \({r_i} = (\overline {{R_i}} + \underline {{R_i}} )/2\).
- (3)
Run the WL−LS algorithm. If a feasible layout is obtained, set \(\overline {{R_i}} = {r_i}\); otherwise, set \(\underline {{R_i}} = {r_i}\).
- (4)
If \(|\overline {{R_i}} - \underline {{R_i}} | > {10^{ - 4}}\), go to (2); otherwise, output \(\overline {{R_i}} \)as the smallest radius for this surface and exit.
4 Calculation of overlapping depth
To calculate the overlapping depth between two objects embedding each other, we can transfer the 3D model of the problem to a 2D one. Cylinders and cuboids are simplified as circles and rectangles, respectively. For a given bearing plate surface, a 2D Cartesian coordinate system Oxy is set up, which coincides with this surface, and its origin is located at the center of this surface. R_{0} is the radius of this surface. The set of rectangles to be located is RECT={Rect_{1}, Rect_{2}, …, Rect_{ r }}, where r is the number of rectangles distributed onto this surface; the set of circles to be located is CIR={Cir_{1}, Cir_{2}, …, Cir_{ s }}, where s is the number of circles distributed onto this surface. Cir indicates the fixed column. The ith rectangle is denoted by Rect_{ i }(p_{ i }, α_{ i }, a_{ i }, b_{ i }), where p_{ i }=(x_{ i }, y_{ i }) is the position of its centroid, α_{ i } ∈[0, π] is the orientation angle between the longer edge of the ith rectangle and the x axis in coordinate system Oxy, a, is the length of its longer edge, and b, is the length of its shorter edge. In this study, rectangles are placed orthogonally, i.e., α_{ i }=0 or π/2. The jth circle is denoted by Cir_{ j }(p_{ j }, r_{ j }), where p_{ j }=(x_{ j }, y_{ j }) is the position of its centroid and rj is its radius.
Various approaches have been proposed to compute the overlapping depth or the interference between objects, for example, the no-fit polygon (Bennell et al., 2001), the octree method (Wu et al., 1997), and the projection-separation approach (Li, 2010). In this study, we do not adhere to a single approach. Instead, according to the graphical characteristics of the two objects involved, we adopt different approaches that are most efficient. For example, if the two objects are rectangles, we adopt the projection approach; if the two objects are circles, we compare the distance between their centroids with the sum of the two radii; if the two objects are a rectangle and a circle, we adopt an approach based on a no-fit polygon. In the related previous work, discussions of approaches are quite general. Here, we give the specific calculation of overlapping depth which is suitable for orthogonally placed rectangles and circles.
4.1 Overlapping depth between a rectangle and a circle
4.2 Overlapping depth between two rectangles
4.3 Overlapping depth between two circles
4.4 Overlapping depth between a circle and a container
Given a circle Cir_{ i }(p_{ i }, r_{ i }), if Cir_{ i } embeds into the standing column Cir, which is considered as a fixed circle, the calculation of the overlapping depth between Cir_{ i } and Cir is the same as in Section 4.3.
4.5 Overlapping depth between a rectangle and a container
Given a rectangle Rect_{ j }(p_{ j }, α_{ j }, a_{ j }, b_{ j }), if Rect_{ j } embeds into the standing column Cir, which is considered as a fixed circle, the calculation of the overlapping depth between Rect_{ j } and Cir is the same as in Section 4.1.
5 Wang-Landau sampling method based on local search
5.1 Wang-Landau sampling method
The Wang-Landau (WL) sampling method is a novel Monte Carlo (MC) method introduced by Wang and Landau (2001). The energy levels of the models treated in the WL sampling method are discrete; however, the idea is very general and can be applied to any parameter. For a layout optimization problem where the energy is continuous, we must first bin the energy. Considering the value of energy in the range as a positive real number, we divide all possible energies in the energy landscape into finite intervals. For example, we divide [0, 5000] into 5000 individual intervals [0, 1), [1, 2), …, [4999, 5000), and numbers which are larger than 5000 are divided into a single energy interval [5000, 5000+). Thus, we obtain 5001 energy intervals. For simplicity, the energy interval [⎿E(X)⏌, ⎾E(X)⏋) is denoted by [E(X)], where ⎿E(X)⏌ rounds E(X) down to its nearest integer, and ⎾E(X)⏋ rounds E(X) up to its nearest integer. For example, E(X)=4.523 falls into energy interval [4, 5), which is denoted by [4.523]. Unlike conventional MC methods that directly generate a canonical distribution at a given temperature, this method is to estimate accurately the density of states g(E(X)) for the range of possible energies via a random walk which produces a flat histogram in the energy landscape, where E(X) is the energy of the whole system. The WL sampling method is based on the observation that if we perform a random walk in the energy landscape with a probability proportional to the reciprocal of the density of states 1/g(E(X), then a ‘flat’ histogram is generated for the energy distribution. By using a carefully controlled modification factor, the estimate for g(E(X)) is improved at each step of the random walk, which makes g(E(X)) converge to the correct value very quickly.
At the beginning of the WL sampling algorithm, all possible energies and the density of states function g(E(X)) are unknown. The density of states is set to be self-adaptive. If the random walk finds a new energy each time, we mark it as visited and set its density of states and the corresponding histogram to 1. In the simulations of the layout optimization for the satellite module, we begin the random walk in the energy landscape by heuristic layout update strategies (see Section 5.2), but the energy associated with each layout is accepted only with a probability proportional to the reciprocal of the density of states. Therefore, the acceptance probability from layout X^{1} to X^{2} is as follows: P(X^{1}→X^{2})=min{exp(g(E(X^{1}))−g(E(X^{2}))), 1}. If X^{2} is accepted, then g(E(X^{2})) will be multiplied by a modification factor λ_{ i }, and its histogram H([E(X^{2})]) will be increased by one; that is to say, g(E(X^{2}))=λ_{ i }*g(E(X^{2})), H([E(X^{2})])=H([E(X^{2})])+1 (i is initialized to 0). If X^{2} is not accepted, then g(E(X^{1}))=λ_{ i }*g(E(X^{1})), H([E(X^{1})])=H([E(X^{1})])+1. If λ_{0} is too small, it will take a long time to find all the possible energies. On the contrary, if λ_{0} is too large, there will be statistical errors. In this study, we set λ_{0}=1. The convergence of the WL sampling method is controlled by the flatness of the histogram. However, it is very difficult to obtain an absolutely flat histogram in practice. The so-called ‘flat histogram’ in the Wang-Landau sampling method means that all the entries of H([E(X)]) are not less than the histogram’s average <H([E(X)])> multiplied by k (0<k<1), where k is decided by the complexity of the system and the expected precision of g(E(X)). In this study, we set k=0.8, and check whether the histogram is flat every 10^{3} MC sweep. When the histogram is flat, all the possible energies have been roughly visited an equal number of times, and the density of states converges to the true value with accuracy proportional to the modification factor lnλ_{ i }. Then we reduce the modification factor λ_{ i } to a finer one using a monotone decreasing function such as λ_{i+1}=0.5λ_{ i }, reset H([E(X)]) to 0 for all visited energy intervals [E(X)], and begin the next random walk. In this study, the modification factor and the corresponding decreasing function are selected by both experience and trial-and-error. From previous experience we set up candidate factors and their decreasing functions that might be suitable for the packing problem. Then from these candidates we select the best factor and its decreasing function through trial-and-error. When the modification factor λ_{ i } is less than a threshold λ_{final}, the algorithm is terminated, and g(E(X)) converges to its real value with precision. λ_{final} is the control parameter of g(E(X)) and determines the number of MC iterations in the whole simulation process. If λ_{final} is too small, the simulation will take a long time. On the contrary, if λ_{final} is too large, g(E(X)) will not converge to its real value. In this study, we set λ_{final}=0.00001.
Our goal is to find the layout with the lowest energy, so in the simulations we also keep the lowest energy E_{min} and the corresponding layout X_{min} each time we find a new lower-energy layout.
5.2 Heuristic layout update strategies
An efficient layout update strategy is also impactful in the WL sampling simulations. According to the characteristics of the layout optimization of the satellite module, we propose the following heuristic layout update strategies:
Strategy 1 In each surface of current layout X, we pick out an object A_{ j } that has the largest relative extrusive elastic potential energy E_{ j }/S_{ j } to relocate, where \({E_j} = \sum\nolimits_{l = 0,l \ne j}^N {d_{ij}^2} \) is the extrusive elastic potential energy of the jth object exerted by other objects, and S_{ j } is the area of the jth object.
- (1)
Generate a random point inside the surface but outside the central column.
- (2)
Judge whether this point falls into a certain object through computing the distance between this point and the centroid of the object. If this point does not fall into any object, save it as a vacant point and go to (3); otherwise, go to (1).
- (3)
If 100 vacant points have been saved, exit; otherwise, go to (1).
Temporarily place the centroid of the chosen object at every vacant point. Here, if the chosen object is a rectangle, it is placed in two ways: one is with its long edge parallel with the x axis, and the other is with its long edge perpendicular to the x axis. Then, compute the extrusive elastic potential energy E_{ j } that associates to every vacant point and placing pattern. Finally, formally place the centroid of the chosen object A_{ j } at the vacant point, where the extrusive elastic potential energy of A_{ j } is the lowest. With the positions of other objects unchanged, we gain a new layout X′.
5.3 Local search
When a new layout X′ is obtained by heuristic layout update strategies, it may be very close to the global optimal layout, so any random layout update in the WL sampling algorithm may make search far from (and even farther and farther from) this global optimal layout. To avoid this, we adopt the local search (LS) method based on the gradient method (GM), which is a quasi-physical algorithm (Huang and Kang, 2004; He et al., 2013), to search for an optimal layout near X′.
- (1)
Set h=1, h_{min}=10^{−4}, ε=10^{−20}.
- (2)
Under the current layout X′, compute the gradient vector ∇(E(X′)) of E(X′) for each object A_{ i } (i=1, 2, …, N) in the x and y directions, and let X^{2}=X′−h*∇(F(X′)).
- (3)
If E(X^{2})>E(X′), set h=h*0.8.
- (4)
Let X′=X^{2}, X^{2}=X′−h*∇(E(X′)).
- (5)
If f_{3}(X^{2})<ε or h<h_{min}, return X^{2}; otherwise, go to (3).
5.4 Description of WL−LS
- (1)
Randomly produce an initial layout X^{ 1 } based on four bearing plate surfaces. Set X_{min}=X^{1}, E_{min}=E(X^{1}). Let the set of intervals containing visited energies be S={[E(X^{1})]}. Set the density of states function as g(E(X^{1}))=1, and the histogram function as H([E(X^{1})])=1. Set i=0, l=0, λ_{0}=1, k=0.8.
- (2)
In each surface P_{ i } (i=1, 2, 3, 4) of the current layout X^{1}, pick object A_{ j } (j=1, 2, 3, 4) with the largest E_{ j }/S_{ j }, where E_{ j } is the extrusive elastic potential energy of the jth object. Copy the current layout X^{1}.
- (3)
Relocate each picked object A_{ j } by using heuristic layout update strategy 2 in surface P_{ i } (i=1, 2, 3, 4), and gain a new layout X′.
- (4)
Call the GM procedure. The outcome layout of GM is denoted by X^{2}. Compute E(X^{2}). Set g(E(X^{2}))=1, H([E(X^{2})])=1, l=l+1.
- (5)
If [E(X^{2})]∉S, let S=S◡{[E(X^{2})]}.
- (6)
If random(0, 1)<min{exp[g(E(X^{1}))−g(E(X^{2}))], 1}, then accept X^{2} (i.e., let X^{1}=X^{2}, E(X^{1})=E(X^{2})); otherwise, do not accept X^{2}.
- (7)
Update the density of states g(E(X)) and the histogram H([E(X)]). That is, if X^{ 2 } is accepted, let g(E(X^{2}))=g(E(X^{2}))*λ_{ i }, H([E(X^{2})])=H([E(X^{2})])+1, and go to (8); otherwise, let g(E(X^{1}))=g(E(X^{1}))*λ_{ i }, H([E(X^{1})])=H([E(X^{1})])+1, and go to (9).
- (8)
If E(X^{2})<E_{min}, let X_{min}=X^{2}, and E_{min}=E(X^{2}).
- (9)
If l%1000=0, go to (10); otherwise, go to (2).
- (10)
If H([E(X)])≥k<H([E(X)])> for all visited energy intervals [E(X)]∈S, then go to (11); otherwise, go to (2).
- (11)
Set λ_{i+1}=λ_{ i }*0.5, i=i+1.
- (12)
If λ_{ i }<0.00001, then output E_{min} and X_{min}, and terminate the iteration; otherwise, reset H([E(X)])=0 and keep g(E(X)) for all visited energies E(X), and go to (2).
6 Experimental results and discussion
To test the computational performance of the WL−LS algorithm, we applied it in two instances. Both instances are based on the international commercial communication satellite module INTELSAT-III with different technological parameters, one with 51 objects and the other with 53 objects. We implemented the WL−LS algorithm in the Java programming language and ran it on a PC with 1.5 GHz CPU and 2.0 GB RAM. For each instance, the WL−LS algorithm was run 50 times independently to optimize the layout of objects on the four bearing plate surfaces. According to these 50 results, we can gain a pareto optimal set for each instance. In this study, we specify the solution with the smallest enveloping radius as the optimal solution for preference.
The technological requirements for the final layout of the whole system are given by Eqs. (3)–(9), with parameters x_{e}=0, y_{e}=0, z_{e}=780 mm. The penalty coefficients are set as follows: ω_{1}=10^{−1}, ω_{2}=10^{6}, ω_{3}=10^{4}, and ω_{4}=10^{4}.
Comparison of the best results by HAKD, WL, WL+GM, WL+HS, and WL−LS for instance 1
Algorithm | Overlapping area (mm^{2}) | Centroid position error (mm) | Inertia angle error (rad) | Inertia moment (kg·mm^{2}) | Enveloping radius (mm) |
---|---|---|---|---|---|
HAKD | 0 | 5.94e-2 | 2.00e-2 | 711.55 | 459.37 |
WL | 0 | 1.03e-0 | 1.20e-1 | 710.96 | 465.05 |
WL+GM | 0 | 6.09e-2 | 5.41e-3 | 695.34 | 459.17 |
WL+HS | 0 | 9.77e-1 | 1.21e-1 | 708.31 | 462.48 |
WL−LS | 0 | 4.21e-2 | 4.70e-3 | 693.07 | 457.76 |
Comparison of the average results by HAKD, WL, WL+GM, WL+HS, and WL−LS for instance 1
Algorithm | Overlapping area (mm^{2}) | Norm of centroid position error \(\sqrt {\delta x_{\rm{c}}^2 + \delta y_{\rm{c}}^2 + \delta z_{\rm{c}}^2} \) (mm) | Norm of inertia angle error \(\sqrt {\theta _{x'}^2 + \theta _{y'}^2 + \theta _{z'}^2} \) (rad) | Norm of inertia moment \(\sqrt {{J_{x'}}{{(X)}^2} + {J_{y'}}{{(X)}^2} + {J_{z'}}{{(X)}^2}} \) (kg·mm^{2}) | Enveloping radius (mm) |
---|---|---|---|---|---|
HAKD | 0 | 0.255 | 1.65e-2 | 423.18 | 467.16 |
WL | 306.13 | 2.47 | 6.98e-1 | 424.67 | 471.86 |
WL+GM | 20.80 | 0.20 | 7.19e-3 | 418.84 | 463.17 |
WL+HS | 8.10 | 2.45 | 7.05e-1 | 421.35 | 469.67 |
WL−LS | 0 | 0.187 | 6.44e-3 | 417.35 | 461.48 |
Data for optimal layout by WL−LS for instance 1
No. | x (mm) | y (mm) | α (rad)* | Surface |
---|---|---|---|---|
1 | −217.84 | −0.42 | − | 1 |
2 | 147.91 | 161.17 | − | 4 |
3 | 194.26 | −51.03 | − | 3 |
4 | 205.98 | −31.61 | − | 4 |
5 | −108.97 | 168.58 | − | 1 |
6 | 44.78 | 351.66 | − | 4 |
7 | 279.44 | 143.57 | − | 3 |
8 | −97.17 | −325.27 | − | 3 |
9 | 141.47 | −182.72 | − | 1 |
10 | −182.75 | −298.32 | − | 4 |
11 | −286.61 | −177.14 | − | 4 |
12 | 5.61 | 328.83 | − | 2 |
13 | 230.61 | −3.46 | − | 1 |
14 | 84.30 | −239.61 | − | 3 |
15 | 258.43 | −257.10 | − | 3 |
16 | 343.95 | −166.80 | − | 4 |
17 | −315.39 | −181.75 | − | 3 |
18 | 338.21 | 162.85 | − | 4 |
19 | −350.70 | −74.49 | − | 2 |
20 | 68.45 | −351.64 | − | 2 |
21 | −373.91 | 63.65 | − | 3 |
22 | −336.78 | −29.37 | − | 3 |
23 | −1.93 | 150.47 | − | 2 |
24 | 315.31 | −140.44 | − | 3 |
25 | 15.10 | 270.25 | − | 1 |
26 | −291.25 | 226.75 | − | 2 |
27 | 350.28 | −46.72 | − | 4 |
28 | 277.92 | 28.06 | − | 3 |
29 | 145.07 | 185.07 | − | 2 |
30 | 146.73 | 178.25 | − | 1 |
31 | −245.06 | 187.89 | − | 3 |
32 | −85.81 | −178.42 | 0 | 1 |
33 | −186.38 | −185.01 | 0 | 2 |
34 | 143.89 | −207.56 | 0 | 4 |
35 | −46.66 | 230.91 | π/2 | 3 |
36 | −225.69 | −1.42 | 0 | 4 |
37 | 103.78 | 227.56 | π/2 | 3 |
38 | −200.14 | −4.87 | π/2 | 2 |
39 | 39.40 | −200.39 | 0 | 2 |
40 | −203.61 | 174.31 | 0 | 4 |
41 | −127.32 | 171.04 | π/2 | 2 |
42 | 180.83 | 9.15 | π/2 | 2 |
43 | −27.95 | 176.56 | π/2 | 4 |
44 | 330.91 | 87.74 | 0 | 2 |
45 | −110.69 | −143.71 | 0 | 4 |
46 | 210.83 | 323.66 | π/2 | 4 |
47 | 309.81 | −12.33 | 0 | 2 |
48 | 239.80 | −158.79 | 0 | 2 |
49 | −192.63 | −12.29 | π/2 | 3 |
50 | 148.78 | −333.69 | 0 | 4 |
51 | −115.17 | −174.10 | 0 | 3 |
To further understand the effects of each component in WL−LS, we ran WL alone, WL with local search GM and without heuristic layout update strategies (denoted as WL+GM), and WL with heuristic layout update strategies (HS) and without local search GM (denoted as WL+HS) 50 times. The best and average results for 50 independent runs by WL, WL+GM, and WL+HS are listed in Tables 1 and 2, respectively, for comparison. The pareto optimal set by WL is P(Q, M)={(468.58, 696.78), (467.24, 697.40), (466.06, 699.08), (465.05, 710.96)}, and we choose the solution with Q=465.05, M=710.96 as the optimal solution for preference. The pareto optimal set by WL+GM is P(Q, M)={(462.51, 693.88), (462.03, 694.79), (460.63, 695.16), (459.17, 695.34)}, and we choose the solution with Q=459.17, M=695.34 as the optimal solution for preference. The pareto optimal set by WL+HS is P(Q, M)={(465.09, 705.86), (464.00, 706.16), (463.01, 707.47), (462.48, 708.31)}, and we choose the solution with Q=462.48, M=708.31 as the optimal solution for preference. Note that in each run of WL and WL+GM, we randomly update the layout. In WL−LS, the WL method is used mainly to execute a global search. The heuristic layout update strategies are used to generate new layouts, and the local search procedure based on the gradient method is used to search for lower-energy layouts near newly generated layouts. Through the comparison of computational results, one can see that WL+GM and WL+HS improve the performance of WL alone in different aspects. However, by combining the merits of GM and HS, WL−LS significantly outperforms the other three algorithms WL, WL+GM, and WL+HS in both the best and average results.
The technological requirements for the final layout of the whole system are given by Eqs. (3)–(9) with parameters x_{e}=0, y_{e}=0, z_{e}=523.26 mm. The penalty coefficients are set as follows: ω_{1}=10^{−1}, ω_{2}=10^{6}, ω_{3}=10^{4}, and ω_{4}=10^{4}.
Comparison of the best results by HKF, WL, WL+GM, WL+HS, and WL−LS for instance 2
Algorithm | Overlapping area (mm^{2}) | Centroid position error (mm) | Inertia angle error (rad) | Inertia moment (kg·mm^{2}) | Enveloping radius (mm) |
---|---|---|---|---|---|
HKF | 0 | 1.139e-0 | 1.46e-2 | 796.15 | 500.00 |
WL | 0 | 3.60e-1 | 5.58e-1 | 802.13 | 487.05 |
WL+GM | 0 | 3.59e-3 | 5.45e-3 | 797.01 | 478.50 |
WL+HS | 0 | 3.60e-1 | 5.55e-1 | 800.69 | 483.49 |
WL−LS | 0 | 3.40e-3 | 4.70e-3 | 795.59 | 475.11 |
Data for optimal layout by WL−LS for instance 2
No. | x (mm) | y (mm) | α (rad)* | Surface |
---|---|---|---|---|
1 | −79.05 | 319.48 | π/2 | 2 |
2 | 78.05 | 225.00 | π/2 | 2 |
3 | 260.45 | 185.65 | π/2 | 3 |
4 | 95.75 | −255.70 | π/2 | 4 |
5 | 181.14 | −65.03 | π/2 | 3 |
6 | −139.51 | 302.44 | 0 | 3 |
7 | 254.23 | 200.50 | 0 | 2 |
8 | −226.27 | −0.46 | 0 | 2 |
9 | −110.75 | −200.75 | π/2 | 2 |
10 | 78.43 | 175.44 | π/2 | 3 |
11 | 175.30 | 24.83 | π/2 | 4 |
12 | 79.88 | 175.50 | 0 | 4 |
13 | 407.59 | −22.17 | π/2 | 2 |
14 | −151.76 | 0.84 | π/2 | 4 |
15 | −369.61 | −137.32 | 0 | 3 |
16 | 342.41 | −142.06 | 0 | 3 |
17 | −303.53 | 46.47 | 0 | 4 |
18 | −77.43 | −221.91 | π/2 | 4 |
19 | −14.58 | −399.76 | 0 | 3 |
20 | 261.60 | −236.80 | π/2 | 2 |
21 | −0.38 | 232.51 | 0 | 1 |
22 | −201.84 | −25.59 | 0 | 1 |
23 | −176.88 | 104.78 | π/2 | 1 |
24 | 202.03 | −31.08 | 0 | 1 |
25 | −7.95 | −204.16 | − | 3 |
26 | 202.73 | 0.15 | − | 2 |
27 | −167.13 | 112.41 | − | 3 |
28 | 100.17 | −173.28 | − | 2 |
29 | −254.14 | 200.41 | − | 2 |
30 | 295.20 | −140.60 | − | 4 |
31 | 255.16 | 203.91 | − | 4 |
32 | 61.88 | −369.81 | − | 2 |
33 | −95.35 | 176.46 | − | 4 |
34 | −195.06 | −279.58 | − | 3 |
35 | −219.13 | −80.85 | − | 3 |
36 | −271.08 | −146.97 | − | 4 |
37 | 173.10 | −290.27 | − | 3 |
38 | 86.55 | 351.24 | − | 3 |
39 | −1.02 | 328.75 | − | 4 |
40 | −265.75 | 216.82 | − | 4 |
41 | 338.10 | −16.37 | − | 3 |
42 | −359.36 | 94.89 | − | 3 |
43 | 73.70 | 132.14 | − | 1 |
44 | −47.16 | 142.71 | − | 2 |
45 | −74.85 | 130.06 | − | 1 |
46 | 328.34 | 72.74 | − | 4 |
47 | −48.19 | −372.69 | − | 4 |
48 | −376.89 | −39.99 | − | 2 |
49 | 15.61 | −182.85 | − | 1 |
50 | −93.44 | −131.03 | − | 1 |
51 | 101.89 | −122.07 | − | 1 |
52 | 141.24 | 64.32 | − | 1 |
53 | −311.05 | −209.25 | − | 2 |
Comparison of the average results by WL, WL+GM, WL+HS, and WL−LS for instance 2
Algorithm | Overlapping area (mm^{2}) | Norm of the centroid position error \(\sqrt {\delta x_{\rm{c}}^2 + \delta y_{\rm{c}}^2 + \delta z_{\rm{c}}^2} \) (mm) | Norm of the inertia angle error \(\sqrt {\theta _{x'}^2 + \theta _{y'}^2 + \theta _{z'}^2} \) (rad) | Norm of the inertia moment \(\sqrt {{J_{x'}}{{(X)}^2} + {J_{y'}}{{(X)}^2} + {J_{z'}}{{(X)}^2}} \) (kg·mm^{2}) | Enveloping radius (mm) |
---|---|---|---|---|---|
WL | 280.30 | 3.50e-1 | 6.62e-1 | 491.95 | 488.39 |
WL+GM | 19.64 | 2.64e-3 | 7.11e-3 | 489.81 | 479.92 |
WL+HS | 6.46 | 3.48e-1 | 6.89e-1 | 491.20 | 484.43 |
WL−LS | 0 | 2.40e-3 | 3.60e-3 | 489.06 | 476.65 |
7 Conclusions and future work
When optimizing the layout design of a satellite module, it is easy for an algorithm to get trapped in local minima separated by high-energy barriers. To address this problem, we use a hybrid WL−LS method, which incorporates the LS procedure based on gradient descent and heuristic layout update strategies into the Wang-Landau sampling method. To improve the efficiency of WL−LS, we adopt an accurate and fast method for computing the overlapping depth between two objects (such as two rectangular objects, two circular objects, or a rectangular object and a circular object) embedding each other. Numerical results show that WL−LS outperforms methods in the literature. There are also several problems that need to be solved in the future: (1) In this study, the rectangles are placed orthogonally, which restricts the flexibility of the layout design. We need to find the methods for computing the overlapping depth with respect to arbitrarily placed rectangles. (2) The distribution of objects onto different surfaces is not discussed in this paper, and we simply adopt the allocation scheme proposed in the literature. In the future, we will focus research on these aspects.
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