Parameter optimization for torsion spring of deployable solar array system with multiple clearance joints considering rigid–flexible coupling dynamics

Yuanyuan LI, Meng LI, Yufei LIU, Xinyu GENG, Chengbo CUI

Qian Xuesen Laboratory of Space Technology, China Academy of Space Technology, Beijing 100094, China

KEYWORDS Clearance joint;Deployable solar array;Evaluation index;Parameter optimization;Rigid–flexible coupling dynamics

Abstract In this paper,four novel evaluation indices and corresponding hierarchical optimization strategies are proposed for a deployable solar array system considering panel flexibility and joint clearance. The deployable solar array model consists of a rigid main-body, two panels and four key mechanisms, containing torsion spring mechanism, closed cable loop mechanism, latch mechanism and attitude adjustment mechanism.Rigid and flexible components are established by Nodal Coordinate Formulation and Absolute Nodal Coordinate Formulation,respectively.The clearance joint model is described by nonlinear contact force model and amendatory Coulomb friction model.The latch time,stabilization time,maximum contact force and impulse sum of the contact force of the solar array system are selected as the four novel evaluation indices to represent the complex dynamic responses of a deployable solar array with clearance joints instead of the lock torque widely used in conventional works.To eliminate the gross errors caused by the nonlinear and nonsmooth mechanical properties,a hierarchical optimization strategy based on an adaptive simulated annealing algorithm and a nondominated sorting genetic algorithm is adopted for the solar array system with clearance joints. Results indicate that the effects of panel flexibility on the evaluation index responses and design optimization of the solar array system cannot be neglected. Besides,increasing the weight factor of the stabilization time index of the rigid system may compensate for the differences in optimal results of the rigid–flexible coupling system. That may provide some references for optimization design of deployable space mechanisms considering clearance joints.ⓒ2021 Chinese Society of Aeronautics and Astronautics.Production and hosting by Elsevier Ltd.This is an open access article under the CC BY-NC-ND license(http://creativecommons.org/licenses/by-nc-nd/4.0/).

1. Introduction

A solar array system is an indispensable accessory that is widely used to provide electricity for spacecraft,as shown in Fig. 1. Solar arrays are generally folded in the launch stage owing to limited rocket space and then deployed by a driving mechanism after the spacecraft reaches orbit. The dynamic properties of the driving mechanism determine whether the solar array can be successfully deployed.Not only success is a critical basis for further space missions, but failure would be a disaster.The most common driving mechanism is a spring hinge because it is self-deployable, lightweight and simple, as shown in Fig. 2. Nevertheless, attention should be paid to parameter optimization design for the driving spring and to the dynamic responses of the deployment process of the solar array system.

Over the last few decades,some researchers have optimized the parameters of spring mechanisms of deployable space accessories based on theoretical and experimental approaches.Kim and Parkdesigned tape spring hinges to satisfy the requirements of latch-up load and torque margin, while maximizing the first natural frequency successfully in the deployed configuration. Kim, et al.found the optimum stiffness of a torsion spring hinge in a large synthetic-aperture radar antenna with minimal impact. Dewalque, et al.proposed an optimization procedure for tape springs, which was used in the deployment of reflectors around a solar panel. Their goal was to determine the maximum stress affecting the structure and the maximum motion amplitude during deployment.Zhang, et al.set minimization of the impact on an antenna as one of the objective functions for optimizing the deployment trajectories of mesh reflector deployable space antennas.Jeong, et al.used an independence axiom to provide a new conceptual design for a hinge, thereby providing a high asdeployed stiffness and low deployment shock for the spacecraft. In addition, the use of the tape springs as thin-walled space structures was experimentally validated.Some modified deployment motion planning methods based on deployment velocity control were also proposed.Zhao,et al.proposed a new method for the parametric design of space reflector antenna by combining the space and the plane polar scissor units.Zhang,et al.combined the equilibrium matrix method and a topology optimization method to optimize the pretension of the three-layer shaped cable net structure.

Fig. 1 Wideband Global SATCOM system with large solar arrays.

Fig. 2 Driving mechanism.

Previous research on optimization design for a driving spring often treated a deployable space mechanism as connected by ideal joints.However,the existence of joint clearance is inevitable owing to aspects of the design, manufacture and assembly, causing vibration, noise and large instantaneous impact forces on the joints.This,in turn,reduces the accuracy,reliability and stability of the multibody system.The topic of joint clearances has made great growth in the last few decades,along with corresponding parameter optimization works and related studies.Furthermore, some researchers have argued that the effects of joint clearances in a large-scale flexible solar array system cannot be ignored.They indicated that the clearance joint brought evident nonlinear dynamic characteristics to the solar array system deployment.The effects of friction and contact at the clearance joint,flexibility of the panel,number and material of clearance joints and parameter uncertainty in the dynamic performance of a deployable mechanism have been studied.

Although the effects of clearance joints on the dynamics of a deployable solar array system have successfully attracted attention from some researchers,the optimization design studies for deployable solar array systems considering clearance joints remain insufficient. Yang and Wangproposed a multiobjective optimization method for the torsion springs of a rigid solar array system, with objective functions composed of the minimized contact-impact force and torsion spring mass.Li,et al.revealed the design guidance for torque spring mechanisms under the same deployment request (such as the same deployment time or deployment velocity)and found that torque springs with larger preloads and less stiffness might improve the stability and reliability of the solar array system.These parameter optimization studies were conducted based on a dynamic model of a rigid solar array system,i.e.,without considering panel flexibility. However, the flexibility of the panels has a significant effect on the dynamics of the solar array in the deployment and locking processes,even long after being latched.Thus, parameter optimization designs that fail to consider clearance joints or panel flexibility may not be completely and effectively adapted to an actual solar array system.

Therefore, this study aims to propose a novel optimization design for the torsion spring of a solar array system,considering both clearance joints and panel flexibility. When considering both these influencing factors, the nonlinear mechanical properties caused by the clearance joint and the rigid–flexible coupling vibration characteristics caused the by panel flexibility are incorporated into the dynamic response of the solar array system. Subsequently, new evaluation indices and corresponding optimization strategies (instead of the conventional objective functions and optimization strategies) are required to fit this complex rigid–flexible coupling solar array system with multiple clearance joints to explain the complex interactions between these evaluation indices. This is accomplished according to an effects analysis of the panel flexibility on the dynamics of the solar array system with clearance joints. In addition,this study reveals the differences between the optimal parameters for a rigid system and those for a rigid–flexible coupling system,providing a reference for future design works.

This paper is organized as follows. Section 2 describes the deployable solar array model adopted in this study, considering panel flexibility and clearance joints. The system includes a driving mechanism, closed cable loop, latch mechanism and attitude controller.Components and connectors are established based on a Nodal Coordinate Formulation(NCF)originally introduced by Garcı´a De Jalo´nand Absolute Nodal Coordinate Formulation (ANCF) proposed by Shabana.Section 3 provides a complete computational solution strategy and the parameters adopted in this study. A generalized-α method is selected to solve the motion equations for the planar rigid–flexible coupling multi-body system with clearance joints.A surrogate model is established using a Radial Basis Function Neural Network(RBFNN).A hierarchical optimization strategy is designed based on the Adaptive Simulated Annealing(ASA) algorithm and Nondominated Sorting Genetic Algorithm II (NSGA-II). Section 4 provides the numerical results.Four novel evaluation indices are determined. The dynamic responses of the rigid system and rigid–flexible coupling system are compared, revealing the effects of the panel flexibility on the evaluation indices. Optimization designs are respectively generated for these two systems and the differences in the optimal results are revealed. Finally, the main conclusions are introduced in Section 5.

2. Model and method

2.1. Rigid–flexible coupling model of deployable solar array system

The deployable solar array model adopted in this paper consists of one spacecraft main-body and two solar panels, which are connected by clearance revolute joints, as shown in Fig. 3(a). The solar array system is folded during launch and the folded arrays are triggered to deploy after the satellite enters orbit. These folded panels are driven by preloaded torsional springs installed at the revolute joints and can be deployed synchronously under the control of Closed Cable Loop (CCL)mechanism. Finally, these panels reach the plane state and are locked by the latch mechanisms.Certainly,a control strategy is designed to ensure the attitude stability of the spacecraft main body during the solar array deployment process. All the torques acted on the solar array system are shown in Fig.3(b),including two groups of driving torques, a group of CCL torques,two groups of lock torques and one attitude control torque.All the mathematical models of these torques are detailed in the authors’ previous work, interested readers may consult the literature.

This paper concerns the design of the drive torsional spring.Here the driving torque Tdof torsional spring at the ith(i=1, 2) revolute joint is shown as.

where Kdenotes the torsion stiffness of torsion spring; θrepresents the preload angle of torsion spring and θ represents the practical deployment angle.

Fig. 3 Deployable space solar array model and torque analysis.

Furthermore, in this paper, the spacecraft main-body is described as rigid part by using NCF and two solar panels are described as flexible parts by using ANCF.ANCF is a kind of non-incremental finite element formulation that uses global nodal coordinates,which enables one to carry out the dynamic analysis for flexible multi-body systems with large deformation and rotation parts accurately. NCF, including global position vectors and global slopes, are all defined in a global inertial coordinate system.As a result, the possibility of sharing coordinates is extended to joints between flexible and rigid bodies. The advantages of using the mixed NCF–ANCF formulation include that global mass matrix of the system is shown to be constant and many of the constraint equations obtained upon utilizing these formulations are linear and can be eliminated when connecting different flexible bodies with kinematic joints.

A planar rigid body and the planar ANCF deformable beam element both can be defined by two nodes as shown in Fig. 4. Similarly, readers can find more detailed descriptions in the author’s previous work.

2.2. Clearance joint model

A mathematical model of revolute clearance joint is derived in the NCF-ANCF framework, consisting of the normal contact force,tangential friction and torques involved by friction force as shown in Fig. 5.

Lankarani and Nikravesh model considers energy dissipation in the contact-impact process and is selected in this work to evaluate the normal contact forces. Compared with other estimated contact force models, this contact model shows better ability to obtain accurate results.And it has been selected for numerous researchesand validated by experimental works.Lankarani and Nikravesh model is described as

Fig. 5 A planar revolute clearance joint model in the NCFANCF frame.

The parameters vand vare the given tolerances for the tangential velocity, which can adopted according to literature.If needed, readers can find more detailed descriptions about Fig. 5 in the author’s previous work.

3. Computational solutions and parameters setting

Fig. 4 Planar NCF-based element and ANCF-based element.

The dynamic equation is solved to obtain the dynamic responses of the solar array system during deployment and lock processes. Typical dynamic response results of deployment for solar panels, include but not limited to displacement response such as deployment angle of panels, force response such as contact forces at clearance joints and so on. These response results are processed by approximate surrogate models to facilitate subsequent optimization design. Furthermore,these response results are selected and reprocessed to act as objective functions or feasible region for the optimization design. The determination process of the objective functions is detailed in Section 4.1 and the determination process of the feasible region is detailed in Section 4.4. These solutions present strong nonlinear and discontinuous characteristics,owing to the panel flexibility and joint clearance. Similarly,these nonlinear and discontinuous characteristics are detailed in Section 4. Thus, a two-stage optimization strategy is adopted in this study to obtain an optimal design for the torsional spring driving mechanism of the solar array system. In the first stage,a global optimization strategy is adopted to find feasible areas and eliminate noise points.In the second stage,a multi-objective optimization strategy is adopted to determine the optimal design parameters that ensure that all indices are co-optimal. The complete solution strategy is shown in Fig. 6. The following sections describe each solution in detail(see Fig. 7).

3.1. Solution of the dynamic equation

3.2. Surrogate model

Typical construction methods for surrogate models include the response surface method,Kriging model and neural networks.The RBFNN is adopted in this study owing to its strong ability to approximate complex nonlinear functions and good faulttolerant capabilities.In addition,even if there are noise inputs in the sample, they generally do not affect the overall performance of the network;these advantages fit for our model well.The structure of the RBFNN is shown in Fig.7and the algorithm flowchart is shown in Fig.6,Part 2.The input value of the jth neural node in the hidden layer can be expressed as follows:

Fig. 6 Flow chart of the entire solution strategy.

Fig. 7 Structure of RBFNN.

3.3. Hierarchical optimization strategy

A global optimization algorithm is used to find the optimal solution because derivative and gradient information is no longer available for an objective function with multimodal,nonlinear, noncontinuous and nondifferentiable properties.ASA’s parameter space can be sampled much more efficiently than by using other simulated annealing algorithms. Compared with other global optimization algorithms such as multi-island genetic algorithms and particle swarm optimization,the ASA adopted in this study for the first-level optimization can handle arbitrary systems with discontinuous design spaces and discrete design variables and shows good convergence. Theoretically, the probability of obtaining the global optimal solution is 100%, as long as the initial temperature and other correlation coefficients are appropriate.

Furthermore, the optimization of the deployment for a solar array system is a typical multi-objective optimization problem in which objectives often conflict and compete with one another.NSGA-II with its fast nondominated sorting procedure, elitist strategy, parameterless approach and simple yet efficient constraint-handling method, is one of the more promising multi-objective evolutionary algorithms and has been successfully applied in many fields. That can quickly search the entire sample space and avoid falling into a local optimal solution.The hierarchical optimization algorithm flowchart is shown in Fig. 6, Part 3.

The steps for the first-level optimization are listed as follows.

3.4. Numerical model and parameters

In this study, every flexible component is divided into 5 elements and the entire multibody system has 52 degrees of freedom.The physical parameters and contact parameters used in the dynamic simulation of the solar array system are shown in Table 1.The radius of the journal is set as r=0.01 m and the radius of the bearing is set as r=0.01005 m.Thus,the clearance sizes of these two joints are both 0.00005 m.The materials comprising the journal and bearing are steel and brass,respectively.

The stiffness coefficient and preloaded angle are the two key parameters of torsion spring mechanism according to Eq. (1) and need to be optimized in the following sections.Groups of sample points(25)are extracted based on a uniform sampling method. The sampling range of the stiffness coefficient of the torsional spring 1 applied to drive the 2 panels is[0.1–0.5] N ∙m/rad. Accordingly, the stiffness coefficient of the torsional spring 2, applied to drive 1 panel, is half that of torsional spring 1. The sampling range of the residual preload angle after system latch is [0.5–2.5]π. The deployment angles of spring 1 and spring 2 are 0.5 π and π, respectively.Consequently, the range of preload angles for spring 1 and spring 2 are 0.5π+ [0.5--2.5]π and π+ [0.5--2.5]π,respectively.

The parameters of the optimization algorithm are set as follows: For the ASA method, the convergence epsilon is set as 1×10and the initial parameter temperature is set as 1.For the NSGA-II method, the population size and number of generations are set as 60 and 100, respectively. The crossover probability, crossover distribution index and mutation distribution index are set as 0.9, 10 and 20, respectively.

Table 1 Physical parameters and contact parameters used in the dynamic simulation.

4. Results and discussions

4.1. Determination of optimization objectives

To determine which dynamic responses are selected as optimization indices for the deployable solar array system with multiple clearance joints,the rigid solar array system is considered under a certain driving torsional spring parameter,(K=0.2N ∙m/rad and θ=π)for example.The deployment angular displacements of the two panels and contact forces at clearance joints are shown in Figs. 8 and 9, respectively.

Fig.8 shows the deployment process of the solar array system. The panels deployment is gradually driven by the torsional springs and reaches the expected plane position, where θ=0.5π and θ=π.They are then locked using a latch mechanism; the time required is defined as the latch time, represented by T. With time, the changes of the panels caused by latch torques and contact forces are gradually stabilized,owing to the existence of damping mechanisms. When the angle change is less than a certain small value Δθ,this moment is defined as the stabilization moment of the system, represented by T.Thus, the time required for the system to stabilize after lock is T=T-Tand Tis defined as the stabilization time (see partial enlarged detail of Fig. 8).

Fig. 9 shows the contact forces of the two clearance joints during the deployment process of the solar array system.Usually,for a rigid solar array system,the maximum values of the contact forces at clearance joints Fand Foccur at the moment of locking. However, the numerical value of the instantaneous collision force for a non-linear system with multiple clearance joints has some uncertain properties. To avoid gross error from a single contact value,the impulse over a period after system lock is calculated, as represented by Iand Ifor joint 1 and joint 2, respectively. Furthermore, to evaluate the force conditions at the clearance joint of the entire system,two additional indices are determined,i.e.,the maximum force F=max (F;F) and the impulse sum IΣ =I+I.

Thus, these four parameters, i.e., the latch time, maximum contact force, stabilization time and impulse sum of the contact forces,are selected as the evaluation indices for solar array system deployment.

4.2. Dynamic response of rigid system

Fig.10 shows the 3-dimensional response surfaces for the latch time, maximum contact force, stabilization time and impulse sum of the contact forces. In a rigid solar array system, these four evaluation indices vary with two parameters,i.e.,the stiffness coefficient and residual preload angle of the driving torsion spring 1. An error analysis is applied to verify the surrogate model. All of the relative errors of these evaluation indices are controlled within an acceptable margin of 15 %,as shown in Fig. 11.

Fig. 8 Deployment angle of panels.

Fig. 9 Contact force.

Fig. 10 Response surface of four key evaluation indices of rigid system.

As can be seen from Fig. 10(a), the latch time of the solar array system decreases monotonically with increases of both the stiffness coefficient and preload angle of the driving torsion spring. This is because the increasing driving torque determined by these two parameters causes the system to lock faster. Moreover, the downward tendency becomes slower as these two parameters increase. The uneven distribution of latch time means that additional torsion spring parameter combinations are available for design, within certain latchtime limitations.

It can be seen from Fig. 10(b) that the stabilization time response of the solar array system presents a downward tendency with the increase of both parameters(stiffness coefficient and preload angle)of the torsion spring.Moreover,the downward tendency gradually slows down as the values reach different parameter regions.

Fig. 11 Error analysis for these four evaluation indices of rigid system.

Evidently,the maximum contact force acts as a typical nonsmooth force; its response surface shows typical nonlinear characteristics, as shown in Fig. 10(c).Notably,the maximum contact force response does not exhibit an upward tendency with increases in the driving torque as a result of the increased stiffness coefficient and preload angle. Instead, the worst performance of the maximum contact force appears in a region with smaller torsion spring parameters near the value of the residual preload angle θ∊[0.5--1]π and stiffness K∊[0.1--0.25]N ∙m/rad.

Fig. 12 Series of contact forces for different preload angles(Ks =0.1 N ∙m/rad).

Using the stiffness coefficient K=0.1 N ∙m/rad as an example, Figs. 12 and 13 respectively provide a series of contact forces and lock torques corresponding to five uniformly distributed residual preload angles θ∊[0.5--2.5]π. As a difference from a case where the maximum lock torque occurs at the latch moment and increases as the residual preload angle increases, the maximum contact force may occur at some point after lock and the worst contact force in this series happened in the case with the minimum residual preload angle. The contact force at the clearance joint (selected as a key evaluation index)in this study presented completely different characteristics from the maximum lock torque widely used as a reference index in conventional studies.The response surface of the maximum lock torque exhibits a monotonically increasing trend as the driving torque increases, as shown in Fig. 14. In this case, the maximum lock torque is no longer appropriate as an objective index for evaluating the nonlinear characteristics of an actual deployable solar array system with the inevitable existence of joint clearances.For example,it may not reveal the load conditions at solar array drive assembly.

Fig. 13 Series of lock torques for different preload angles(Ks =0.1 N ∙m/rad).

Fig. 14 Response surface of lock torque of rigid system.

Fig. 15 shows two series of CCL torques corresponding to fiveuniformlydistributedresidualpreload anglesθ∊[0.5--2.5]π,withstiffnesscoefficients K=0.1 N ∙m/rad and K=0.2 N ∙m/rad. As shown in Fig. 15(a), when K=0.1 N ∙m/rad, these two CCL torques from the system with θ=0.5π and θ=π do not have the ability to rapidly decrease to a steady situation as compared to other series, as well as the CCL torque from the system with K=0.2 N ∙m/rad and θ=0.5π, as shown in Fig. 15(c). As a result, the unstable synchronization torques generated by the CCL mechanism cause violent vibrations in the solar panels.This might explain why the maximum contact force appears at the area of the residual preload angle θ∊[0.5--1]π and stiffness K∊[0.1--0.25]N ∙m/rad as shown in Fig. 10(c), although the corresponding driving force and lock force are both small at this area (owing to the small torsion spring parameters).

The contact forces at the clearance joint are the comprehensive outcome of the multiple action forces generated by all mechanisms of the solar array system together, including the driving torsion spring, lock mechanism and CCL mechanism.The maximum contact force is related to system deployment and latching and even the subsequent system stability.

The impulse sum of the contact forces is selected as an evaluation index,to balance the dramatic nonlinear characteristics and the single numeric value of the maximum contact force.The impulse sum response shows the general characteristics,i.e.,as the driving torque and lock torque increase,the impulse sum of the contact forces at the multiple clearance joints after latching shows a roughly upward tendency,as shown in Fig.10(d).

4.3. Effect of panel flexibility

The effects of panel flexibility on the four evaluation indices(latch time, maximum contact force, stabilization time and impulse sum of contact forces) for the rigid–flexible coupling solar array system are shown in Fig. 16. It shows the 3-dimensional response surfaces of the variation of these four evaluation indices with the stiffness coefficient and residual preload angle of the driving torsion spring 1.An error analysis is also applied to verify the surrogate model for the rigid–flexible coupling system and all the relative errors are controlled within an acceptable margin of 15 %, as shown in Fig. 17.

From Fig.16(a),it can be seen that the latch time response of the rigid–flexible coupling solar array system also decreases monotonically as the two parameters of the torsion spring increase,which is consistent with the response of the rigid system.However,the latch time of the rigid–flexible coupling system with the same mechanism parameters is smaller than that of the rigid system (as compared with the response in Fig. 10(a)). This is because the suspension-damping property of the flexible panel reduces the contact force and frictional resistance of the rigid–flexible coupling solar array system during the deployment process.

Fig. 16(b) shows the stabilization time response of the rigid–flexible coupling system; the angle changes within a certain small range Δθ=0.01 rad. However, the stabilization time required by the rigid–flexible coupling system is evidently much longer than that of the rigid system,owing to the elastic vibration of the flexible panel. Furthermore, the stabilization time response presents distinct nonlinear properties as compared with that of the rigid system. This is because the stabilization time, as an evaluation index of the rigid–flexible coupling system, is more sensitive to disturbances of the nonlinear contact force at the clearance joints. Consequently, the worst performance of the stabilization time response appears at the parameter area consistent with the parameter combination shown in Fig. 15 and is related to the stability capability of the system after latching.

Fig. 15 Series of CCL torques for different preload angles.

Fig. 16 Response surface of four key evaluation indices of rigid-flexible coupling system.

In contrast to the maximum contact force response distribution of the rigid system, the maximum contact force response shows the best performance at the area where the stiffness coefficient K∊[0.1--0.25]N ∙m/rad, as shown in Fig. 16(c). This might because the suspension-damping property of the flexible panel reduces the maximum contact force of the solar array system when the driving force and lock force are small. Fortunately, the stabilization time evaluation index offsets this difference between the rigid–flexible coupling system and rigid solar array system. This means that choosing the maximum contact force as the single objective function for the optimal design of mechanism parameters for a solar array system is unreasonable.Moreover,the maximum contact force response of the rigid–flexible coupling system exhibits dramatic nonlinear characteristics,owing to the nonlinear contact force. Furthermore, the maximum contact force response of the rigid–flexible coupling system increases more than that of the rigid system, as a result of the strong coupling interactions between the impact force at the clearance joints and elastic vibrations of the flexible panels.

Comparing Figs. 16(d) and 10(d), it can be seen that the value of the impulse sum of contact forces of the rigid–flexible coupling system is larger than that of the rigid system by an order of magnitude, owing to the more intensive impact frequency at the clearance joint of the system with panel flexibility. Generally, the impulse sum of contact forces in the rigid–flexible coupling system exhibits an upward tendency with the increase of the two parameters of the torsion spring,except for the area where residual preload angle θ∊[0.5--1]π and stiffness K∊[0.1--0.2]N ∙m/rad. This indicates that the impulse sum, as an evaluation index of a system considering panel flexibility, is more sensitive to system stability than its corresponding value in a rigid system.

The dynamic response of the rigid–flexible coupling system is more complex,owing to the coupling between the nonlinear contact forces and flexible panel elastic vibrations, leading to complex interactions between these evaluation indices. Thus,a single index cannot assess the full performance of the dynamic response of the solar system and single object optimization for such systems is inadequate and inaccurate.

4.4. Hierarchical optimization

Fig. 17 Error analysis for four evaluation indices of rigid–flexible coupling system.

The purpose of the first layer of optimization is to eliminate the effects of gross error and (at least some) uncontrollable or unstable factors on the dynamic response of the solar array system,as a basis for determining a feasible region for the second layer optimization.According to the analysis in the previous sections, the optimization function for the first layer is given as follows:The second layer optimization requires the system to incur the minimum contact force and minimum impulse action and can reach stability more quickly within the expected latch time range and available design variables.

Fig. 18 Latch time within feasible region and its probability distribution of rigid system.

Table 2 Multi-objective optimization results of the rigid system.

4.4.1. Rigid system

Table 3 Multi-objective optimization results for the rigid–flexible system.

4.4.3. Discussions

Comparing the obtained design variables between the rigid system and rigid–flexible coupling system, it can be seen that the torsion spring parameters of the rigid–flexible coupling system might be more suitable to the principle for selection, i.e.,that the system with the larger preload and smaller stiffness of torsion spring has the greater performance. Thus, the optimization design results for a rigid system are not suitable for direct application in the actual rigid–flexible coupling solar array system. To explain the differences in the optimization results between these two systems,some optimization parameters are changed as a redesign for the rigid solar system. The optimization results are affected by the weight factor of each evaluation index and the objective variables of the rigid system approached those of the rigid–flexible coupling system as the weight factor of stabilization time Tincreased. The redesign optimization results of the rigid system are shown in Table 4.

Fig. 19 Latch time within feasible region and its probability distribution of rigid–flexible coupling system.

Table 4 Redesign optimization results of the rigid system.

The optimization design results for the rigid system are different from those of the flexible panel system. Increasing the weight factor for stabilization time index might be a way to offset this difference, but currently there is no quantification for the increase. Thus, it is not recommended to employ the rigid model for optimization design of a deployable solar array system with multiple clearance joints.

5. Conclusions

Four evaluation indices for the deployable rigid–flexible solar array system were proposed in this study:latch time,stabilization time,maximum contact force and impulse sum of the contact forces, instead of the maximum lock torque, as widely used in conventional research. The numerical value, variation tendency and occurrence time of the maximum lock torque cannot fully reflect the load conditions acting on a solar array drive assembly. The contact force at the clearance joints may be more representative of actual load conditions for the deployable solar array system. To balance the gross error of some values from the instantaneous and discontinuous contact force,the impulse of contact forces for a certain time after the system latched was utilized as another index.Furthermore,the stabilization time was applied to offset the dynamic difference between the rigid system and the rigid–flexible coupling system, i.e., a crucial index for evaluating a system with flexible panels. The surrogate model for these evaluation indices were constructed using an RBFNN,for application to an optimization framework.

A hierarchical optimization strategy was presented in this study for designing the parameters of the driving mechanism of the deployable rigid–flexible coupling solar array system.The first optimization layer, based on the ASA algorithm,eliminated gross errors generated from dramatic nonlinear and nonsmooth contact forces at the clearance joints and obtained a feasible design region for latch time for further precise optimization in the next layer.The second layer,based on NSGA-II,obtained the optimal design variables of the driving mechanism, consisting of the stiffness coefficient and preload angle. All three of the other evaluation indices were suggested as multiobjective optimization functions for the deployable solar system with clearance joints owing to complex and strong interactions between them.

The optimized parameters of the rigid solar array system were not totally available for the rigid–flexible coupling system.Increasing the weight factor of the stabilization time index of the rigid system might improve its applicability to the rigid–flexible coupling system. Optimization of the mechanical parameters of a solar array system with clearance joints was not recommended without considering panel flexibility.

In addition,a more complete parameter optimization for all key mechanisms of a solar array system with clearance joints may be considered in a future work.

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

This work is supported by the National Natural Science Foundation of China (No. U1637207), Beijing Natural Science Foundation of China (No. 1204040).