Hierarchical Optimization of Landing Performance for Lander with Adaptive Landing Gear
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Abstract
A parameterized dynamics analysis model of legged lander with adaptive landing gear was established. Based on the analysis model, the landing performances under various landing conditions were analyzed by the optimized Latin hypercube experimental design method. In order to improve the landing performances, a hierarchical optimization method was proposed considering the uncertainty of landing conditions. The optimization problem was divided into a higher level (hereafter the “leader”) and several lower levels (hereafter the “follower”). The followers took conditioning factors as design variables to find out the worst landing conditions, while the leader took buffer parameters as design variables to better the landing performance under worst conditions. First of all, sensitivity analysis of landing conditioning factors was carried out according to the results of experimental design. After the sensitive factors were screened out, the response surface models were established to reflect the complicated relationships between sensitive conditioning factors, buffer parameters and landing performance indexes. Finally, the response surface model was used for hierarchical optimization iteration to improve the computational efficiency. After selecting the optimum buffer parameters from the solution set, the dynamic model with the optimum parameters was simulated again under the same landing conditions as the simulation before. After optimization, nozzle performance against damage is improved by 5.24%, the acceleration overload is reduced by 5.74%, and the primary strut improves its performance by 21.10%.
Keywords
Landing gear Soft landing Sensitivity analysis Response surfaces Hierarchical optimization1 Introduction
Legged lander has been used for deep space exploration because of its high landing stability and terrain adaptability [1]. In order to isolate vibration and reduce load during soft landing, the legged lander generally uses the plastic material such as honeycomb as the main absorber to design the landing gear. However, the performances of these landing gears are unable to be adjusted during soft landing. In order to cope with complex landing terrain, larger design margin should be reserved, resulting in the heavier soft landing system [2]. With the continuous progress of deep space exploration, the terrain environment of interesting regions will be more complex and harsh, and landing in multiple regions to accomplish different detection missions may be needed. So it is required that the lander has better terrain adaptability and its landing gears are reusable.
Considering those requirements, Adaptive landing gear was proposed as a possible solution. Refs. [3, 4, 5] introduced hydraulic system, intelligent materials and pyrotechnics devices into the design of landing gear to realize adaptive control. Among them, magnetorheological damper (MR damper) is widely studied because of its cheerful prospect. In Refs. [2, 6, 7, 8, 9], the single MR damper was designed and analyzed in detail, and the equivalent mathematical model of its characteristics was obtained. Refs. [10, 11, 12, 13], which proposed a variety of control strategies for the lander with adaptive landing gears, proved the effectiveness of adaptive gears in enhancing soft landing performances. Previous studies mostly discussed the implementation of adaptive lander, and few concerned the soft landing performance optimization for the adaptive lander. But performance optimization is of great importance for the weight reduction of lander and it benefits the improvement of terrain adaptability. Existing researches about landers optimization mostly focus on conventional passive control lander [14, 15, 16]. Furthermore, the worst condition uncertainty caused by the change of design variables was ignored in the existing researches. And the selection or optimization of the parameters was just based on the typical condition, which leads to the instability of soft landing safety.
Aiming at the uncertainty of the worst condition, a hierarchical optimization method was proposed to update the worst condition dynamically during the progress optimizing the adaptive buffer parameters. First, a dynamic analysis model of adaptive lander was established, and its soft landing performance was analyzed and evaluated. Then the response surface was adopted to participate the iterative computation of hierarchical optimization. The lander with the optimized adaptive buffer was simulated. The results show that the optimization effectively improves the soft landing performance, which verifies the feasibility of the hierarchical optimization method.
2 Dynamic Model of the Lander
2.1 Configuration and Coordinate System Definition of Lander
Parameters of the lander at touchdown
Parameter | Value |
---|---|
Mass of load (kg) | 1650 |
Mass of landing gear (kg) | 15 |
Height of mass center (mm) | 2500 |
Radius of footpad’s lower surface (mm) | 100 |
Distance between two adjacent footpads (mm) | 4000 |
2.2 Adaptive Buffer and Its Control Strategy
Unlike the conventional landing gears, such as honeycomb core and air bag, buffer characteristics of adaptive buffer are able to be controlled by adopting some structures or intelligent material. So the adaptability of lander equipped with this kind of buffer can be improved. Even if the adaptive control system fails, the adaptive landing gear will degenerate to the conventional passive landing gear but not palsy, which ensures the safety and reliability of the landing system [11].
In the whole simulation analysis of the soft landing, the angular velocity \( \dot{\theta } \), \( \dot{\psi } \) of the lander and the buffer speed of the main strut \( \dot{s} \) were monitored in real time through measurements. According to the Eq. (1), four cushioning forces are applied to primary struts, where the damping coefficient model is shown as Eq. (2). Finally, the independent feedback adjustment of the damping coefficients is realized.
Initial buffer characteristic parameters
Parameter | Value | Range |
---|---|---|
Stiffness coefficient k (N/m) | 4.9 × 10^{4} | [3 × 10^{4}, 7 × 10^{4}] |
Maximum damping coefficient c_{max}/(Ns/m) | 5.4 × 10^{4} | [3 × 10^{4}, 7 × 10^{4}] |
Dynamic range ratio r = c_{min}/c_{max} | 0.1 | [0.1, 1.0] |
3 Simulation and Analysis for Adaptive Lander
3.1 Indicators for Soft Landing
- 1)
Nozzle performance against damage. Landing on regions with rough terrain may damage the nozzle due to rugged landing surface, which affects the performance of the main engine. The minimum distance between the bottom of the nozzle and the landing surface is chosen as the evaluation index. The larger the index is, the better.
- 2)
Acceleration overload. Considering the acceleration tolerance of astronauts and the instruments equipped on the lander, the overload during the soft landing should not exceed 15g to ensure the progress of the detection mission. The maximum acceleration during soft landing is selected as an index to access the overload characteristic. The smaller the value is, the better.
- 3)
Buffer performance. Considering the uncertainty of the environment of the target landing regions, the buffer performance should meet the demand of the worst condition. The maximum buffer stroke during soft landing is selected as one of the indexes. And a smaller value means that the volume and weight of the landing gear can be reduced correspondingly, which is beneficial to soft landing.
- 4)
Landing stability. A vertical plane passing through the center of two adjacent footpads, which is parallel to the gravity vector, is defined as an “stability wall” [26]. Since the lander has four legs, there are four such walls. If the centroid of the lander exceeded the enclosure formed by the four stability walls, the landing was considered to be unstable. Here stability distance T is introduced as a parameter measuring the minimum distance between the centroid of the lander and four stability walls during each soft landing. If T remained positive, the landing was declared to be stable.
Indicators for soft landing
Indicator | Parameter | Sign | |
---|---|---|---|
Performance indexes | Nozzle performance against damage | Minimum distance between the bottom of the nozzle and the landing surface (mm) | U |
Acceleration overload | Maximum acceleration during soft landing (g) | L | |
Buffer performance | Four maximum buffer strokes during soft landing (mm) | S | |
Landing stability index | Minimum distance between the centroid of the lander and four stability walls (mm) | T |
3.2 Analysis of Landing Performance
Soft landing conditioning factors
Condition | Range |
---|---|
v_{x} (m/s) | [3, 4] |
φ (°) | [0, 45] |
μ | [0.3, 0.7] |
α_{e} (°) | [0, 10] |
Parameters of 2-2 condition
Parameter | Value |
---|---|
v_{x} (m/s) | 3.5 |
φ (°) | 45 |
μ | 0.7 |
α_{e} (°) | 8 |
4 Hierarchical Optimization
In this section, hierarchical optimization was adopted to improve the soft landing performance of the lander. The optimization took buffer parameters k, c_{max} and r as design variables, while the performance indexes U, L and S as three primary objectives. Since the change of the buffer parameters is followed with the change of the worst condition, the worst condition and corresponding performance indexes should be updated duly during the optimization process. Aiming at this complex optimization problem, a hierarchical optimization method was proposed decomposing the problem into a leader and several followers. After receiving the buffer parameters from the leader and modifying the model correspondingly, the followers took the conditioning factors as design variables to find the worst conditions respectively. Then the results of this “reverse optimizations” were delivered to the leader. The leader then optimized the buffer parameters trying to better the worst performance indexes. And the new buffer parameters determined by leader are transmitted to the followers to start the next iteration. The cycle continues until the terminating conditions are satisfied. Finally, the Pareto optimal set of the buffer parameters is obtained after hierarchical optimization.
4.1 Sensitivity Analysis of Conditioning Factors
4.2 Response Surface Model
In the figure, subscript A indicates that the values of landing indicators were obtained by simulation and subscript P means that they were from response surface model.
It can be seen from Figure 16 that the R^{2} of all indicators are higher than 0.97, and RMSE less than 0.05. The fitting accuracy is enough for the response surface models to replace the dynamic one to be computed.
4.3 Optimization in the Followers
Mathematical models of the follower optimizations
Number | Objective | Constraint | Output |
---|---|---|---|
1 | Min U | q _{ i} ^{L} < q_{i} < q _{ i} ^{U} | U _{min} |
2 | Max L | L _{max} | |
3 | Max S | S _{max} | |
4 | Min T | T _{min} |
Configuration parameters of evolution algorithm
Parameter | Value |
---|---|
Max evaluation | 200 |
Convergence tolerance | 0.1 |
Minimum discrete step | 0.02 |
Parallel batch size | 5 |
Penalty base | 0 |
Penalty multiplier | 1000 |
Penalty exponent | 2 |
Failed run penalty value | 1 × 10^{30} |
Failed run objective value | 1 × 10^{30} |
4.4 Optimization in the Leader
Configuration parameters of NSGA-II
Parameter | Value |
---|---|
Population size | 12 |
Number of generation | 20 |
Crossover probability | 0.9 |
Crossover distribution index | 10 |
Mutation distribution index | 20 |
The selected buffer parameters
Parameter | c_{max} (Ns/m) | r | K (N/m) |
---|---|---|---|
Value | 34224.47 | 0.30 | 50450.50 |
Results of the hierarchical optimization
Landing performance index | U (mm) | L (g) | S (mm) | |
---|---|---|---|---|
Before | Max. | 432.9272 | 15.57734 | 186.1706 |
Avg. | 401.6185 | 9.827149 | 127.2947 | |
Min. | 381.7005 | 5.600144 | 57.4495 | |
After | Max. | 439.0153 | 14.68367 | 146.8838 |
Avg. | 415.9362 | 9.536974 | 94.99342 | |
Min | 401.6876 | 5.484198 | 38.52626 |
5 Conclusions
- 1)
The dynamic analysis model of the lander which is equipped with adaptive buffer was established. And the semi-active control algorithm is applied to realize the adaptive feedback adjustment of the damping coefficient during soft landing. Based on the dynamic analysis model and experimental design method, the soft landing performances under multiple landing conditions were analyzed.
- 2)
Focusing on the worst landing condition uncertainty caused by the changes of buffer parameters, a hierarchical optimization method was proposed. The method divides the optimization problem into two parts, namely, a multi-objective leader optimization and several follower optimizations. Through this method, the worst condition was updated duly during optimization process. Furthermore, in order to improve the computational efficiency, response surface models were established to replace the dynamic models for iterative calculation.
- 3)
The soft landing performances before and after the optimization were compared. In the premise of ensuring the landing stability not decline, nozzle’s performance against damage, the buffer performance of the primary strut and the acceleration overload performance are all improved after optimizing. And the comparison results indicate the effectiveness of the hierarchical optimization method.
- 4)
Those studies provide guidance to the design of adaptive lander, including the scheme determination, performance analysis and optimal design. And the hierarchical optimization method, which was proposed to solve complex optimization problems, provides a feasible scheme for optimal design of the project with similar properties.
- 5)
The feasibility of the lander with adaptive buffer is validated preliminarily. However, the performances of the buffer such as vibration and response speed may also influence the soft landing performances in some way, which is simplified in this paper. In future studies, we intend to research on those properties respectively, thus validating its practicability further.
Notes
Authors’ Contributions
CW was in charge of the whole trial; ZD wrote the manuscript; HW and JD assisted with sampling and laboratory analyses. All authors read and approved the final manuscript.
Authors’ Information
Zongmao Ding, born in 1993, is currently a master candidate at School of Mechanical Engineering and Automation, Beijing University of Aeronautics and Astronautics, China. He received his bachelor degree from Beijing University of Aeronautics and Astronautics, China, in 2016.
Hongyu Wu, born in 1993, is currently a PhD candidate at Department of Mechanical Engineering, Tsinghua university, China. He received his master degree from Beijing University of Aeronautics and Astronautics, China, in 2018.
Chunjie Wang, born in 1955, is currently a professor at State Key Laboratory of Virtual Reality and Systems, Beijing University of Aeronautics and Astronautics, China. She received her PhD degree from China University of Mining and Technology, China, in 1997.
Jianzhong Ding, born in 1991, is currently a PhD candidate at School of Mechanical Engineering and Automation, Beijing University of Aeronautics and Astronautics, China. He received his bachelor degree from China University Of Petroleum, China, in 2013.
Competing Interests
The authors declare that they have no competing interests.
Funding
Supported by National Natural Science Foundation of China (Grant No. 51635002).
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