Performance Analysis and Optimization of a Bio-Inspired Spider-Web-Shaped Energy Absorbing Component for Legged Landers

Abstract

Inspired by the structural characteristics of natural spider webs, a simplified configuration composed of multi-layer regular polygons was developed to design a novel energy absorbing component for legged landers. To investigate its compressive energy-absorption behavior, a parameterized finite element model (FEM) was established. By integrating optimized Latin hypercube experimental design with the FEM, the energy absorption characteristics under varying structural parameters were evaluated. Based on the FEM results, response surface methodology was employed to construct surrogate models that capture the mapping relationships between design parameters and performance indices. Using these surrogate models, the energy-absorbing component was optimized under three different ranges of average buffering force. Three optimized components with distinct average buffering forces were selected and connected in series, and their force–displacement responses during compression were computed through finite element simulations. The obtained response curves were incorporated into a multibody dynamics model of a Mars lander to verify performance, demonstrating that the lander can achieve effective soft landing.

1. Introduction

With the rapid advancement of science and technology, human exploration is no longer confined to Earth. Probing the Moon, Mars, and even more distant planets has become one of the major frontiers in contemporary scientific research. In extraterrestrial exploration missions, the safe delivery of robotic probes to planetary surfaces is a critical stage [1,2]. A review of past missions worldwide shows that legged landers are widely employed to achieve soft landings, such as the U.S. Apollo Lunar Module, China’s Chang’e-3 lunar probe, NASA’s Viking 2 Mars lander, and NASA’s Phoenix Mars lander. As a landing system with relatively high reliability, the legged lander requires continuous innovation and research.

One of the key research focuses in legged lander development is the design of the energy-absorbing system, in which the choice of energy-absorbing components directly determines the overall performance [3]. At present, metallic porous materials are commonly used as fillers for the energy absorbers in legged landers—for example, aluminum honeycomb and aluminum foam. These materials primarily dissipate the impact energy generated during the landing process through plastic deformation [4,5]. Extensive studies have been carried out on the energy absorption characteristics of metallic porous materials. For instance, Li, Liaghat and Hong et al. [6,7,8] combined theoretical analysis and experimental validation to investigate the energy absorption behavior of hexagonal honeycomb materials under out-of-plane compression. Li et al. [9] employed finite element methods to compare the energy absorption properties of hexagonal and staggered square honeycombs under out-of-plane loading, and they further optimized their performance using response surface methodology. Deng, Hu and Han et al. [10,11,12,13] examined honeycombs with triangular, staggered triangular, square, staggered square, hexagonal, and star-shaped topologies, analyzing their compressive energy absorption behaviors. In addition, Zhang, Luo and Zeng et al. [14,15,16] investigated the buffering performance of metal foams and lattice structures through crushing experiments and finite element simulations. Liang et al. [17] integrated simulation data from finite element models of honeycomb compression into a single-leg dynamic model of a lander, and they validated the simulation against single-leg drop tests. Li and Chen et al. [18,19] further proposed equivalent finite element modeling approaches for honeycomb absorbers in landers, incorporating the simulation results into full-vehicle dynamics models to evaluate soft-landing performance.

To expand the design paradigm for energy absorbers in legged landers, bio-inspired design strategies can be introduced. In nature, spider webs are capable of dispersing the impact of high-speed intrusions, such as insects, across the entire web, thereby achieving efficient buffering. In this study, a novel energy-absorbing component was innovatively designed by mimicking the structural characteristics of spider webs. Through parametric finite element modeling, optimized Latin hypercube sampling, and the response surface method, a mapping model between structural parameters and performance was established. Based on this model, three components with different buffering capacities were optimized. Finally, these components were creatively connected in series to construct a hierarchical buffering system, and dynamic simulations verified that the proposed system can effectively achieve soft landing for a Mars lander.

2. Configuration of the Bio-Inspired Spider-Web-Shaped Energy Absorber

The geometry of natural spider webs is highly complex, and due to manufacturing constraints, it is difficult to replicate their exact structural form. Therefore, in this study, the spider-web structure is simplified as follows:

• The web is represented by n concentric layers of regular polygons. All polygons have the same number of edges, and the layers are defined sequentially from the innermost (Layer 1) to the outermost (Layer n).
• All polygons lie in the same plane, share a common geometric center, and contain at least one pair of mutually parallel edges between adjacent layers.
• Let the circumcircle radius of the i-th layer polygon be R_i (i ≥ 1), For the adjacent inner and outer layers, the radii are denoted as R_i−1 and R_i+1, respectively. With R₀ ≡ 0, the difference in circumcircle radii between two adjacent layers is defined as the layer thickness, expressed as:

$$\left\{\begin{array}{c}{\delta }_{\mathrm{i}}=R_i-R_{i-1}\\ {\delta }_{\mathrm{i}+1}=R_{i+1}-R_i\end{array}\right.$$

(1)

4.

The thickness of successive layers increases in a geometric progression, i.e., δ_i+1 = cδ_i, where c denotes the thickness ratio. According to this definition and the formula for a geometric sequence, the corresponding relationship among the radii can be derived.

$$\left\{\begin{array}{cc}{R}_i=\frac{{\delta }_1(1-c^i)}{1-c}=\frac{R_1(1-c^i)}{1-c}& c\ne 1\\ R_i=i{R}_1& c=1\end{array}\right.$$

(2)

5.

The polygons of adjacent layers are connected by line segments extended through the common geometric center, and all structural members of the web are assigned a uniform width.

Based on these rules, a simplified spider-web structure composed of three layers of regular hexagons is illustrated in Figure 1.

By extruding the above-described simplified spider-web cross-section along the out-of-plane direction, a spider-web-shaped energy absorber can be obtained, analogous to conventional honeycomb absorbers. A schematic of the absorber with three hexagonal layers is shown in Figure 2.

3. Performance Analysis of the Bio-Inspired Spider-Web-Shaped Energy Absorber

3.1. Parameterized FEM of the Energy Absorber

As discussed in Section 2, the spider-web-shaped energy absorber can be configured in numerous specifications. Considering its complex geometry and manufacturing challenges, experimental investigation of its buffering performance would incur high costs. Finite element analysis (FEA), with its advantages of high accuracy and ease of implementation, has therefore been widely employed in the study of the energy absorption behavior of thin-walled structures. In this work, the out-of-plane compression process of the spider-web-shaped energy absorber was simulated using ABAQUS 2019 software [17].

Due to the large number of possible configurations, multiple parameters must be considered in model construction. These include the thickness d of the metallic foil material, the number of polygon edges m per web layer, the number of layers n, the thickness ratio c between adjacent layers, the circumcircle radius R of the polygon, the overall absorber height h, and the material property parameters. Clearly, manually constructing finite element models for absorbers of different specifications would be highly time-consuming. To address this, a parameterized modeling program was developed in Python language, enabling automated generation of absorber models. The program performs geometry creation, finite element meshing, boundary condition assignment, and simulation of the out-of-plane compression process.

The FEM of the spider-web-shaped energy absorber generated using the parameterized modeling program is shown in Figure 3.

The FEM consists of two rigid plates positioned above and below the energy absorber. The lower rigid plate is fixed, with the absorber attached to it, while the upper plate moves downward at a constant velocity and stops once the prescribed compression stroke is reached, thereby simulating the out-of-plane compression process. The problem is solved using the explicit dynamic analysis method [17]. In the model, both the internal contacts of the absorber and the interactions between the absorber and the rigid plates are defined using the general contact algorithm in ABAQUS/Explicit. Friction between contact surfaces is modeled using the Coulomb friction law, with the dynamic friction coefficient set to 0.1.

In this study, aluminum alloy 3003-H18 is selected as the material for the spider-web-shaped energy absorber, and its material properties are listed in

Table 1. Material attribute of AL3003H18 [20].

Density (kg·m−3)	Elastic Modulus (GPa)	Poisson’s Ratio	Yield Strength (MPa)
12.73 × 103	68	0.33	185

.

3.2. FEA of Buffering Performance

The spider-web-inspired energy-absorbing component described in this study can be applied to Mars landers, Moon landers, and other landing cushioning environments. Considering the dimensional constraints of the inner and outer cylinders of the supporting strut that houses the absorber, and to avoid instability during compression due to excessive length, the circumcircle radius of the outermost polygon is set as R_n = 28 mm, and the absorber height as h = 40 mm. Under these conditions, the design variables of the absorber include the number of polygon edges per layer (m), the number of layers (n), the thickness ratio (c) between adjacent layers, and the foil thickness (d) of the metallic material used in fabrication. According to Equation (2), the following relationship holds:

$$\left\{\begin{array}{cc}{R}_1=\frac{R_n(1-c)}{1-c^n}& c\ne 1\\ R_1=\frac{R_n}{n}& c=1\end{array}\right.$$

(3)

According to Equations (2) and (3), the following relationship can be derived.

$$\left\{\begin{array}{cc}{R}_i=\frac{R_n(1-c^i)}{1-c^n}& c\ne 1\\ R_i=i\frac{R_n}{n}& c=1\end{array}\right.$$

(4)

Based on Equation (4), the cross-sectional geometry of the spider-web-shaped energy absorber for use in Mars landers can be determined. The ranges of the four design parameters—m, n, c, and d—are specified in

Table 2. Parameter ranges of buffer element.

m	n	c	d (mm)
6~12	3~6	1~1.2	0.1~0.2

.

In the design of energy absorbers for landers, the specific energy absorption (SEA) is commonly used to characterize the energy absorption capability of the component. SEA is defined as the energy absorbed per unit mass of the absorber, and its expression is given as follows [21]:

$$S_{\mathrm{AE}}=\frac{{ʃ}_0^{l_{\max }}F(l)\mathrm{d}l}{M}$$

(5)

In the above expression, M denotes the mass of the absorber, l represents the compression displacement, l_max is the prescribed compression stroke, and F(l) refers to the reaction force of the absorber at displacement l.

In addition to S_AE, attention must also be given to the peak force (F_max) and the average force (F_ave) generated within the prescribed compression stroke. Appropriate values of F_max and F_ave ensure that the absorber provides sufficient energy dissipation without inducing destructive impact loads. The expression for F_ave is given as follows:

$$F_{\mathrm{ave}}=\frac{{ʃ}_0^{l_{\max }}F(l)\mathrm{d}l}{l_{\max }}$$

(6)

To evaluate the buffering performance of the spider-web-shaped energy absorber, an optimized Latin hypercube sampling scheme was employed to extract 40 sample points within the parameter ranges specified in Table 2. These sample points were then substituted into the parameterized finite element model for compression simulations. In the simulations, the prescribed compression stroke l_max was set to two-thirds of the absorber height, i.e., 27 mm [22], and the total simulation time for the compression process was set to 10 ms. The simulations were performed using Abaqus 2019 on a computer equipped with an Intel Core i5-10500 CPU and 8 GB of RAM, with a typical computation time of approximately 2 h. The corresponding simulation results are summarized in Appendix A 

Table A1. Simulation results of experimental design.

m	n	c	d/mm	SAE/(kJ/kg)	Fmax/kN	Fave/kN
8	4	1.0564	0.1	17.54902888	10.8827	4.183177143
10	3	1.1385	0.1026	16.90981224	10.9891	3.716534733
9	5	1.1538	0.1051	21.75517072	13.0066	6.317426602
7	4	1.1487	0.1077	17.18371127	11.6465	4.135951659
11	4	1.0821	0.1103	21.87818901	13.3498	6.408761329
7	5	1.1333	0.1128	19.3688417	14.02	5.601514532
10	3	1.0308	0.1154	17.73160173	11.685	4.403440465
7	6	1.0256	0.1179	20.34110979	17.3044	7.256318687
9	6	1.0872	0.1205	24.39294826	16.8444	9.26953239
12	5	1.1538	0.1231	27.50393461	18.594	10.39317247
10	5	1	0.1256	23.46560326	17.3626	8.858411942
8	3	1.0769	0.1282	16.53210892	12.707	4.193494049
11	4	1.1897	0.1308	23.40011694	15.6945	8.000165678
9	6	1.1795	0.1333	26.25826307	20.0739	10.5743346
12	5	1.0513	0.1359	27.57666063	19.3898	11.81879401
6	4	1.0718	0.1385	17.98435444	14.524	5.396871558
8	4	1.0051	0.141	20.71502467	15.6925	7.039913792
9	3	1.1744	0.1436	18.73033967	14.723	5.520142033
7	5	1.1949	0.1462	21.95592698	16.7245	8.047709455
11	3	1.1026	0.1487	20.48681212	16.5852	6.788772706
11	4	1.0154	0.1513	24.85434488	18.6526	10.08825538
7	6	1.0923	0.1538	23.20862942	21.8901	10.42033155
9	5	1.1077	0.1564	25.17421387	20.2208	11.04137353
6	4	1.1436	0.159	19.75357552	16.4705	6.681771614
12	5	1.1231	0.1615	29.08070728	23.181	14.53231237
10	6	1.1179	0.1641	29.65395727	26.0119	15.64965319
10	5	1.2	0.1667	28.24791098	21.5655	13.31839603
7	5	1.0103	0.1692	21.77857016	22.0897	9.901310134
10	3	1.0205	0.1718	22.02998303	17.5688	8.148217814
6	4	1.041	0.1744	19.52189516	18.5728	7.436543735
11	5	1.0359	0.1769	29.31664866	24.8997	15.92152359
8	3	1.1128	0.1769	19.79449146	16.5892	6.909014085
11	4	1.1692	0.1821	27.20002216	22.2809	12.98523383
8	6	1.1641	0.1846	27.96524234	26.5813	15.12543305
6	5	1.0974	0.1872	22.30328615	21.7682	10.35510301
8	4	1.1846	0.1897	24.14628634	21.5849	10.63247246
12	4	1.0615	0.1923	28.96471035	24.6597	15.34592751
9	4	1.0462	0.1949	25.60069458	23.5611	12.39905185
8	6	1.0667	0.1974	27.60146537	27.5276	16.7534547
10	5	1.1282	0.2	30.31784877	28.6609	17.51730797

. Based on the analysis presented in this section, optimization of the spider-web-shaped energy absorber will be carried out in the subsequent sections, with the objective of achieving the best possible buffering performance.

4. Optimization of the Cushioning Performance of the Bio-Inspired Spider-Web Energy Absorber

4.1. Response Surface Surrogate Model

The optimization design of the bio-inspired spider-web energy absorber requires multiple iterations of optimization, whereas finite element simulations are highly time-consuming. If the FEM were directly employed for iterative optimization, the computational efficiency would be severely constrained. To address this issue, an incomplete fourth-order polynomial response surface model, as expressed in Equation (7), is adopted in this study to establish the mapping relationships of S_AE, F_max, and F_ave with respect to the design variables m, n, c, and d. By substituting the surrogate response surface model for the FEM during optimization iterations, the computational efficiency can be significantly improved.

$$\begin{array}{l}\phi ={\beta }_0+{\beta }_{1}m+{\beta }_{2}n+{\beta }_{3}c+{\beta }_{4}d+{\beta }_5{m}^2\\ +{\beta }_6{n}^2+{\beta }_7{c}^2+{\beta }_8{d}^2+{\beta }_{9}mn+{\beta }_{10}mc\\ +{\beta }_{11}md+{\beta }_{12}nc+{\beta }_{13}nd+{\beta }_{14}cd+{\beta }_{15}{m}^3\\ +{\beta }_{16}{n}^3+{\beta }_{17}{c}^3+{\beta }_{18}{d}^3+{\beta }_{19}{m}^4+{\beta }_{20}{n}^4\\ +{\beta }_{21}{c}^4+{\beta }_{22}{d}^4\end{array}$$

(7)

In Equation (7), φ denotes the estimated value of any of the three response variables, namely S_AE, F_max, or F_ave, while βᵢ represents the coefficients of the polynomial terms. By substituting the input–output data of the 40 sample points obtained from the experimental design into Equation (7), the response surface model can be expressed in vector form, as given in Equation (8).

$$\mathit{Y}=\mathit{X}\mathit{\beta }+\mathit{\epsilon }=\widehat{\mathit{Y}}+\mathit{\epsilon }$$

(8)

In Equation (8), X denotes the input variable matrix of the polynomial, β represents the column vector of the polynomial coefficients to be determined, $\widehat{Y}$ denotes the column vector of the output values of the sample points, represents the column vector of the estimated values of the response surface model, and ε represents the column vector of errors. The coefficient vector β can be obtained by applying the least-squares method, as expressed below [9,23].

$$\mathit{\beta }={\left({\mathit{X}}^T\mathit{X}\right)}^{-1}{\mathit{X}}^T\mathit{Y}$$

(9)

In this study, the root mean square relative error (R_MSE) and the adjusted coefficient of determination (R²) are employed to evaluate the fitting accuracy of the response surface model with respect to the sample points, as expressed in Equations (10)–(12).

$$R_{\mathrm{MSE}}=\frac{\sqrt{{\displaystyle \sum _{i=1}^{40}}{(y_i-{\widehat{y}}_i)}^2}}{{\displaystyle \sum _{i=1}^{40}}{y}_i}$$

(10)

$$R^2=1-\xi \times \frac{{\displaystyle \sum _{i=1}^{40}}{(y_i-{\widehat{y}}_i)}^2}{{\displaystyle \sum _{i=1}^{40}}{(y_i-\overline{y})}^2}$$

(11)

$$\xi =\frac{40-1}{40-(n_{\beta }-1)}$$

(12)

In Equations (10) and (11), y_i denotes the output value of the sample point, ${\widehat{y}}_i$ represents the corresponding estimated value obtained from the response surface model, and ${\overline{y}}_i$ indicates the mean value of all y_i. The term n_β denotes the number of polynomial terms in the response surface model. The correction factor ξ is introduced to eliminate the influence of the polynomial term count on the accuracy assessment [24]. A smaller R_MSE approaching zero and an R² value approaching unity indicate higher model accuracy.

To mitigate potential sources of error, such as polynomial oscillations caused by an excessive number of terms, a key-term screening strategy was adopted for the response surface model. Specifically, the polynomial terms were iteratively filtered, and 14 terms were retained to fit the sample points. The final expression of the response surface model was determined by selecting the combination of terms that yielded an R² value closest to unity.

Based on this analysis, the approximate expressions of S_AE, F_max, and F_ave were obtained, as expressed in Equations (13)–(15). The accuracy evaluation results of the response surface model are summarized in

Table 3. Accuracy analysis table of response surface models.

Metric	R2	RMSE
SAE	0.995	0.00157
Fmax	0.984	0.00410
Fave	0.999	0.00194

.

$$\begin{array}{l}{S}_{\mathrm{AE}}=-7.4771-7.4555\times c+8.9039\times n+1133.1855\times d^2\\ -2.7557\times n^2+2.1697\times d\times m-0.7272\times c\times m\\ +4.2003\times c\times n+0.2752\times m\times n-7985.8002\times d^3\\ +0.0101\times m^3+0.1796\times n^3+18122.0470\times d^4\\ -0.0006\times m^4\end{array}$$

(13)

$$\begin{array}{l}{F}_{\max }=-32458.3759-1756.7458\times d+120522.3358\times c\\ -2.3570\times m+19727.5827\times d^2-167242.4103\times c^2\\ +29.7388\times d\times c+5.2444\times d\times m+15.8082\times d\times n\\ +1.9807\times c\times m-96758.6335d^3+102988.5079c^3\\ +173036.6961\times d^4+23751.3722\times c^4\end{array}$$

(14)

$$\begin{array}{l}{F}_{\mathrm{ave}}=18.6243-399.1204\times t+3476.1694\times t^2\\ -0.3947\times m+0.5432\times n^2+7.9682\times t\times m\\ +17.2742\times t\times n-0.0590\times c\times m+0.2449\times m\times n\\ -16608.9661\times t^3+0.0418\times m^3+0.0366\times n^3\\ +30696.1589\times t^4-0.0015\times m^4\end{array}$$

(15)

The results demonstrate that the response surface model can accurately capture the mapping relationships between S_AE, F_max, F_ave and the design variables m, n, c and d.

4.2. Optimization of Buffering Performance

To achieve buffer components with optimal energy absorption performance, optimization methods are applied in the design process. In this study, the classification of buffer components is based on the value of the average buffering force F_ave observed during compression. Accordingly, the components are categorized into three types: weak buffering, moderate buffering, and strong buffering. The corresponding ranges of F_ave for these three categories are summarized in

Table 4. Range of average value of buffer force.

Type	Weak Buffering	Moderate Buffering	Strong Buffering
Ranges of Fave (kN)	9~12	12~15	15~18

.

For each of the three categories of buffer components, optimization design is conducted with the dual objectives of maximizing the specific energy absorption S_AE and minimizing the peak buffering force F_max, while constraining the average buffering force F_ave within the ranges specified in Table 4. Based on these considerations, the mathematical optimization models for weak, moderate, and strong buffering components are formulated as follows.

$$\left\{\begin{array}{cc}\min & F_{\max }\\ \max & S_{\mathrm{AE}}\\ \mathrm{s}.\mathrm{t}.& F_{\mathrm{ave}}>9\\ & F_{\mathrm{ave}}<12\\ & {\mathit{x}}^{(L)}<\mathit{x}<{\mathit{x}}^{(U)}\end{array}\right.$$

(16)

$$\left\{\begin{array}{cc}\min & F_{\max }\\ \max & S_{\mathrm{AE}}\\ \mathrm{s}.\mathrm{t}.& F_{\mathrm{ave}}>12\\ & F_{\mathrm{ave}}<15\\ & {\mathit{x}}^{(L)}<\mathit{x}<{\mathit{x}}^{(U)}\end{array}\right.$$

(17)

$$\left\{\begin{array}{cc}\min & F_{\max }\\ \max & S_{\mathrm{AE}}\\ \mathrm{s}.\mathrm{t}.& F_{\mathrm{ave}}>15\\ & F_{\mathrm{ave}}<18\\ & {\mathit{x}}^{(L)}<\mathit{x}<{\mathit{x}}^{(U)}\end{array}\right.$$

(18)

In Equations (16)–(18), “min” denotes minimization, “max” denotes maximization, and “s.t.” denotes “subject to”; x = (m, n, c, d)^T represents the vector of design variables, while x^(L) and x^(U) denote the corresponding lower and upper bounds of the design variables, respectively.

The optimization problems are solved using the NSGA-II algorithm, with the algorithmic parameter settings provided in

Table 5. Value of optimal parameters.

Parameter	Population Size	Number of Generations	Crossover Index	Mutation Index	Crossover Probability
Value	12	40	10	20	0.9

. The NSGA-II algorithm is an elitism-based multi-objective evolutionary algorithm that maintains population diversity and achieves global optimization through fast non-dominated sorting and crowding distance evaluation. Its main procedures include population initialization, fitness evaluation, selection, crossover, and mutation. This algorithm is adopted for its excellent convergence, uniform solution distribution, and high computational efficiency in solving complex multi-objective optimization problems.

To enhance computational efficiency, the response surface model was employed during the optimization process to estimate the values of S_AE, F_max, and F_ave. Through iterative optimization, Pareto optimal solution sets were obtained for the three mathematical models, from which the corresponding Pareto frontiers were plotted, as shown in Figure 4, Figure 5 and Figure 6.

For the final selection, the solution with the minimum F_max was identified as the optimal design. The resulting optimal solutions for the three categories of energy-absorbing components are summarized in

Table 6. Value of optimum solution.

Type of Energy-Absorbing Component	m	n	c	d (mm)
Weak buffering	12	5	1.065	0.113
Moderate buffering	12	6	1.101	0.120
Strong buffering	12	6	1.200	0.141

.

5. Integrated Energy-Absorption Performance Analysis of the Lander

5.1. Performance Evaluation of Serially Arranged Energy-Absorbing Components

To enhance the adaptability of the lander under multiple operating conditions, different types of energy-absorbing components are typically installed in a graded serial configuration within the shock absorber [25]. Specifically, under nominal conditions, the weak energy-absorbing components dissipate the majority of the impact energy, whereas under severe conditions, the strong energy-absorbing components are activated to prevent failure of the system. Considering that the maximum stroke of the Mars lander’s shock-absorbing strut is 120 mm, this study arranges the three types of bio-inspired spider-web-shaped components (as listed in Table 6) in series, with the weak component positioned at the top and the strong component at the bottom. The corresponding FEM is illustrated in Figure 7.

The compression behavior of the serially assembled spider-web-shaped components is analyzed using the finite element model. The maximum compression stroke is set to 90 mm, with the compression velocity kept consistent with that applied in the single-component simulations. Based on this setup, the relationship curve between compression displacement S and compressive force F_S, as well as the simulated deformation process, are obtained and presented in Figure 8 and Figure 9, respectively.

5.2. Dynamic Analysis of the Entire Lander

The Mars lander equipped with the proposed bio-inspired spider-web-shaped energy-absorbing components is shown in Figure 10.

The lander consists of a central body structure and four identical shock absorption mechanisms uniformly distributed around its perimeter. Each mechanism is composed of a main strut, an auxiliary strut, a footpad, and a buffer rod. The upper end of the buffer rod is rigidly connected to the body, while its lower end is joined to the main strut through a universal joint. The lower end of the main strut is fixed to the footpad. The outer tube of the auxiliary strut is connected to the body via a universal joint, its inner tube is linked to the main strut through a spherical joint, and the space between the inner and outer tubes is filled with the energy-absorbing component to dissipate impact energy.

A multibody dynamics simulation model of the lander’s soft-landing process was developed in ADAMS [26,27]. To enhance computational efficiency, components undergoing small deformations were modeled as rigid bodies, while the effects of large deformation components were simulated using equivalent modeling techniques.

Considering that the buffer rod primarily absorbs impact energy through bending deformation, the behavior was simulated using the following method, which was previously described by the authors in reference [28]. As illustrated in Figure 11, two rigid bodies were established with mass properties defined according to the actual buffer rod. Rigid body 1 is connected to point O on the lander’s body through a universal joint, while rigid body 2 is constrained to move along the axis of rigid body 1. In the undeformed state, the buffer rod’s axis coincides with the global X-axis. When bending deformation occurs, a resisting bending moment M is generated. By decomposing M orthogonally into the Y- and Z-axis components, the corresponding bending moments M_Y and M_Z can be obtained. Therefore, torque components M_Y and M_Z were applied at point O on rigid body 1 to simulate the effect of the bending moment. Meanwhile, a translational motion S was applied to rigid body 2 to reproduce the displacement of the free end of the buffer rod. As shown in Figure 3, both M and S are functions of the deflection angle α between rigid bodies 1, 2 and the X-axis. A FEM of the buffer rod was developed, and by applying a bending moment, the relationships between α–M and α–S were obtained. The fitted curves are presented in Figure 12.

When considering only the bending deformation of the buffer rod, its top view can be represented as a line segment located within the YZ plane, forming an angle θ with the Y-axis, as illustrated in Figure 13.

By measuring the coordinates (x, y, z) of the end of rigid body 2 relative to point O, the angle α can be determined as expressed in Equation (19). Subsequently, M_Y, M_Z and S can be derived according to Equations (20), (21), and (22), respectively.

$$\alpha =\arccos \left({\displaystyle \frac{\left|x\right|}{\sqrt{x^2+y^2+z^2}}}\right)$$

(19)

$${\mathit{M}}_Y={\displaystyle \frac{z}{\sqrt{y^2+z^2}}}\times f(a)$$

(20)

$${\mathit{M}}_Z={\displaystyle \frac{-y}{\sqrt{y^2+z^2}}}\times f(a)$$

(21)

$$\mathit{S}=g(\alpha )$$

(22)

The functions f and g correspond to the curve relationships shown in Figure 12, which are modeled in ADAMS using a Spline function. By substituting Equations (20)–(22) into the rigid-body model, the effect of the buffer rod can be simulated.

The performance of the spider-web-inspired buffer element is represented by introducing a force F_D between the inner and outer cylinders of the auxiliary strut, which is a function of their relative displacement D. The functional relationship between F_D and D follows the curve illustrated in Figure 9.

The interaction force between the footpad and the landing surface is decomposed into a normal impact force and a tangential friction force. The normal impact force is modeled using a nonlinear damping spring model, while the tangential friction force is described using a Coulomb friction model [26,27,29].

Neglecting active control errors during the powered descent phase, the lander is assumed to have zero horizontal velocity and zero angular velocity about all three axes at touchdown, with a vertical velocity of 3.5 m/s and a total mass of approximately 800 kg. The variable landing condition parameters considered include the distribution of surface craters n_f, the slope of the landing surface α, and the yaw angle of the lander at touchdown θ_p. The definitions of these three parameters have been described in detail in Refs. [27,30] and are therefore not repeated here. Their value ranges are summarized in

Table 7. Range of parameters of initial landing conditions.

Parameter	Value
α (°)	1~8
nf	0, 1, 2, 3
θp (°)	0~45

.

The primary aspects of soft-landing performance considered in this study include the following four criteria:

• Anti-overturning capability of the lander. The vertical plane containing the center points of any two adjacent footpads is defined as the overturning plane. During landing, the minimum distance L_D between the lander’s center of mass and the overturning plane must remain greater than zero [31]; otherwise, the lander is regarded as overturned.
• Anti-damage capability of the main engine nozzle. During touchdown, the minimum vertical distance H_M between the center point of the nozzle bottom—located at the base of the lander’s main structure—and the planetary surface must be greater than 200 mm.
• Acceleration overload characteristics. Considering that the payload instruments onboard the lander can only withstand limited overloads, the acceleration overload G_L during the soft-landing process should generally not exceed 13 g to ensure the success of the exploration mission.
• Energy absorption performance of the auxiliary struts. As the primary energy-absorbing components of the lander, the auxiliary struts must ensure that their buffer stroke D_M remains less than 80 mm under nominal landing conditions.

The farther the values of L_D, H_M, G_L, and D_M are from their allowable boundaries, the better the soft-landing performance of the lander.

Based on the full multibody dynamic simulation model of the system, a full factorial experimental design method is employed to evaluate the soft-landing performance. In the design, the parameter n_f (number of craters) is set to 0, 1, 2, and 3; the surface slope α is varied from 1° to 8° with an increment of 1°; and the yaw angle at touchdown θ_p is varied from 0° to 45° with an increment of 1°. According to the principles of full factorial design, a total of 4 × 8 × 46 = 14,724 dynamic simulations are required to complete the stability analysis of the lander. Based on these 1472 landing scenarios, the simulation results of the lander’s soft-landing performance are summarized in

Table 8. Soft landing performance analysis results of lander.

Performance Index	Extreme Value	Number of Cases Exceeding Allowable Limit
Minimum LD	807.308 mm	0
Minimum HM	216.185 mm	0
Maximum GL	11.234 g	0
Maximum DM	75.366 mm	0

and

Table 9. Soft landing conditions analysis results of lander.

Performance Index	α (°)	θp (°)	nf
Minimum LD	8	38	2
Minimum HM	5	0	3
Maximum GL	1	43	2
Maximum DM	8	26	1

.

The simulated performance curves corresponding to the four representative landing conditions listed in Table 9 are presented in Figure 14, Figure 15, Figure 16 and Figure 17.

The above analysis demonstrates that the Mars lander equipped with the bio-inspired spider-web buffer elements exhibits favorable soft-landing performance.

6. Conclusions

In this study, a bio-inspired spider-web buffer element for legged landers was proposed, and a parameterized FEM was established. By integrating experimental design with finite element simulations, the compression and buffering performance of the element under varying configuration parameters was systematically investigated. Based on the FEA results and the response surface methodology, polynomial surrogate models were constructed to accurately capture the mapping relationship between configuration parameters and buffering performance indicators. On this basis, the NSGA-II algorithm was employed to optimize the buffer element design, yielding three optimized configurations with superior buffering performance across different ranges of average buffering force.

The three optimized elements were then combined in series, and their buffering performance was analyzed using FEA, resulting in force–displacement curves during the compression process. These curves were further incorporated into a multibody dynamic simulation model of a Mars lander, where the lander’s soft-landing performance was evaluated. The simulation results confirmed that the lander equipped with the proposed spider-web buffer elements exhibits excellent soft-landing capability.

The proposed spider-web buffer element provides a novel design concept for the selection of energy-absorbing components in deep-space exploration landers. Moreover, the methodology presented in this work is characterized by low cost and high efficiency, making it broadly applicable to the development of other types of buffer elements. In future work, the authors will continue to explore new designs of buffering components and conduct experimental validation of these components.