## Abstract

Reusable spacecraft has great potential in reducing space launch cost. Structural reliability evaluation is critical for mission planning of reusable spacecraft. A dynamic reliability prognosis method based on digital twin framework is proposed for mission planning in the paper. In this method, Uncertainties integration and dynamic model updating are implemented through a dynamic Bayesian network. A maintenance point is set when the predicted structural reliability level is lower than a threshold or unexpected conditions such as landing impact occur. Then, inspected data can be assimilated by the framework to dynamically update the structural reliability. Thus, it supports dynamic adjustment of maintenance interval, early warning of structure failure, and mission planning with quantified risk. A numerical example considering single point crack growth under fatigue load and landing impact of a simplified spacecraft structure is used for demonstration. Results show that the crack size predictions can be calibrated by inspected data and its uncertainties can be reduced. The proper selection of landing impact probability in reliability prediction is helpful to control the maintenance interval. The reliability of the spacecraft can be increased through model updating with new inspected data, representing a potential lifetime extension can be realized by the proposed method.

## 1 Introduction

Reusable spacecraft can be regarded as a feasible means to reduce the launch cost [1]. For a reusable spacecraft, its whole life cycle may suffer from various anomalies. Periodic maintenance and reusability evaluation are conducted to ensure flight reliability in the past [2]. However, periodic maintenance always brings overly frequent inspections and is time and cost consuming without certain insurance for structural safety. With the development of sensing, signal processing and decision-making technology, the strategy of condition based maintenance (CBM) has developed recently [3]. CBM is maintenance mode based on condition monitoring technology, which can reflect the health status, deterioration process of components, and predict the possibility of component failures, so as to identify the need for maintenance before failure occurs [4]. Because of its advantages of timely and cost saving, it has been successfully applied in industries like aerospace, electronics, chemical industry, and many others [5]. Two of the key problems of CBM are how to realize the structural state evaluation and structural reliability prediction in dynamic environment. The digital twin (DT) can effectively solve these two problems.

A DT is a digital representation of an engineering system that can simulate, monitor, diagnose, predict, and optimize the behaviors of its corresponding engineering system in real-time [6]. DT was proposed first by Dr. Grieves and Vickers [7] in 2002 and has been widely studied in recent years. Li et al. [8] constructed a dynamic Bayesian network (DBN) for crack prognosis of an aircraft wing, and developed a methodology for intelligent mission planning under uncertainty using the DT approach. Karve et al. [9] developed a probabilistic methodology based on the DT approach, including damage diagnosis, damage prognosis, and mission optimization. Mission load profiles could be optimized to minimize fatigue crack growth, while performing the desired work. Zhou et al. [10] proposed a real-time crack-growth simulation method based on the reduced-order simulation and machine learning, in order to provide support for the remaining life assessment of aircraft structures under the digital twin framework. Based on the probabilistic diagnosis and prediction ability of DT, Sisson et al. [11] developed a flight parameters selection method for rotorcraft to perform the target task under minimum stress. Liu et al. [12] proposed a fatigue damage-cumulative model within peridynamic framework for fatigue and damage tolerance evaluation in DT paradigm. Tuegel [6,13] proposed a concept of the airframe digital twin (ADT), and discussed the challenges of realization, and reconstructed a life prediction process for vehicles based on the ADT. United States Air Force (USAF) collaborated with general electric and Northrop Grumman Aerospace Systems, proposing a prognostic and probabilistic individual aircraft tracking framework to achieve the ADT [14–17]. They demonstrated that the framework could integrate various analyses and inspections, identify uncertainty sources, and improve the diagnosis and prognosis capabilities, using Bayesian updating. Millwater et al. [18] proposed some probabilistic methods for the risk assessment of the ADT. They demonstrated that failure probability could be reduced using inspected data.

In the current DT application research, it mainly focuses on state update, damage diagnosis and prediction, and data assimilation methods. The research on reliability evaluation using DT for CBM is not enough in-depth.

The main purpose of the paper is to propose a dynamic reliability prognosis method for reusable spacecraft based on digital twin framework. Dynamic reliability update provides effective support for CBM, thus ensuring structural safety timely and effectively and reducing maintenance costs. The remainder of this paper is organized as follows. In Sec. 2, the concept and example of digital twin framework for reliability-based mission planning is introduced; Section 3, the dynamic data driven reliability prognosis method is proposed, and the method is validated by the example given in Sec. 2; the verification results are discussed in Sec. 4; and In Sec. 5, final conclusions are contained, as well as a note about future work.

## 2 Digital Twin Framework for Reliability-Based Mission Planning

### 2.1 Structural Digital Twin.

A DT is an information mirror model of an engineering system. Its capabilities, including diagnosis, evaluation, prognosis, and decision guides, can be improved via monitoring structural states in real-time and updating digital models dynamically [6]. For a reusable spacecraft, a structural digital twin can be created as depicted in Fig. 1. It can support mission planning, such as the evaluation of flying or maintenance intervals. For example, after several flights, some load bearing structures may be damaged and some electrical connections may be loose. A detailed and accurate assessment is required to evaluate the reliability of the spacecraft for the next flight. If the evaluated reliability satisfies the safety requirement, the future mission can be planned as flying. If not, the future mission should be planned as repairing to check the damaged area and replace some components if necessary. Building a digital simulation model is the first step to establish a structural digital twin. The simulation model can be used for damage prognosis and reusability evaluation etc. Although the model can contain all the design parameters, it is still difficult to simulate the real scenario accurately as it contains many uncertainties [19]. That will result in the unreliability of prognosis and evaluation. Postflight inspection is necessary to understand the structural states of the spacecraft. Then, inspected data can be used for model updating. The updated model can represent the physical spacecraft more accurately, as it contains more information about the real. Finally, damage prognosis and reliability evaluation can be conducted based on the updated model. That can support decision-making for the physical spacecraft better.

### 2.2 Dynamic Mission Planning Strategy.

For a reusable spacecraft, mission planning for future flights can refer to the evaluation of structural reliability. Missions can include performing the designed flight missions, carrying out ground maintenance and repair. The main idea of the dynamic mission planning strategy can be divided into the following two steps. First, integrate the inspected data during the reusable process of the spacecraft to continuously update the model. Second, based on the updated model, the reliability of the subsequent missions can be reevaluated, that is, the maintenance points are reevaluated. Thus, the purpose of dynamic mission planning is achieved.

The details of dynamic mission planning strategy are illustrated by an example shown in Fig. 2. Curves of different colors and line types represent the reliability evaluation of future missions under the models updated with different times. When the result of reliability evaluation is lower than the threshold, the reliability evaluation process based on the current model ends (cross in figure). Ground maintenance (star in figure) is carried out when the reliability of the next mission is lower than the threshold or unexpected accidents occur such as landing failure, and the inspected data are used to update the model. When the reliability of the next mission is still lower than the threshold based on the model updated with the inspected data obtained from the last mission, the spacecraft needs to be repaired or even scrapped (rectangle in figure).

In the example, the reliability evaluation of missions is first based on the initial model until the fifth mission (cross in the figure), whose reliability is lower than the threshold. This means the spacecraft should be inspected after the fourth mission. Then, the inspected data are used to update the model. Based on the updated model, the reliability of the spacecraft is reevaluated, and it is not lower than the threshold until the eighth mission. This shows that the reliability evaluation can be changed dynamically based on the updated model driven by inspected data. In order to consider a more complex situation, the landing failure happens at the sixth mission of the spacecraft. The spacecraft needs to be maintained immediately. The inspected data can also be used to update the model and the reliability is reevaluated again. The result shows that the reliability is not lower than the threshold until the tenth mission. Therefore, the spacecraft should be inspected after the ninth mission and the inspected data can also be used to update the model. Reevaluate the reliability of the tenth mission and find that the reliability of the tenth mission is still lower than the threshold. This shows that the reliability of the model before and after updating cannot meet the threshold requirements, and the spacecraft needs ground repair. When it comes to the point where the maintenance cost is tremendous and further repair is uneconomic, the reusable spacecraft can be retired.

### 2.3 Failure of Reusable Spacecraft.

A reusable spacecraft is faced with complex loading cases, resulting in various structural failure modes in its life cycle. Fatigue damage caused by repeated flights and potential landing failure is focused in this paper.

A spacecraft will be subjected to a variety of alternating loads, which are mainly generated by the combined effects of the engine impulse thrust, jet noise, turbulent boundary layer noise, and so on. The alternating loads will lead to the fatigue damage of structures. In the process of repeated flights, fatigue cracks will propagate gradually, resulting in the decrease of structural safety. Once the cracks propagate to the critical sizes of fatigue fracture, catastrophic accidents may occur.

Furthermore, a spacecraft needs the parachutes and the retro-rocket to cushion the landing. There are possibilities that the parachutes cannot open timely or the retro-rocket may break down suddenly. This will cause the spacecraft to collide with the ground at a higher speed than the design. Then, the cracks may propagate further. Excessive impact stresses may also lead to a local plastic yield in the structure.

In this paper, a single point cracking of fatigue and landing failure is employed as an example to demonstrate the basic methodology behind the digital twin framework. Refer to Ref. [20], the crack exists on an I-beam in the airframe of spacecraft as shown in Fig. 3. The initial crack is set to 0.01 mm (a manufacturing defect). The parameters of the I-beam are depicted in Fig. 3. The material of the beam is 7075-T6 aluminum alloy, and its basic properties are depicted in Table 1.

Ρ (kg/m^{3}) | E (GPa) | ν | $\Delta Kth$ (MPa√m) | $KIC$ (MPa√m) | σ (MPa) |
---|---|---|---|---|---|

2.81 × 10^{3} | 71 | 0.33 | 3.76 | 26 | 441.8 |

Ρ (kg/m^{3}) | E (GPa) | ν | $\Delta Kth$ (MPa√m) | $KIC$ (MPa√m) | σ (MPa) |
---|---|---|---|---|---|

2.81 × 10^{3} | 71 | 0.33 | 3.76 | 26 | 441.8 |

In the calculation of fatigue crack growth, considering the loading condition of the spacecraft, the quasi-static and random vibration loads are adopted as shown in Fig. 3. The two loads are first performed with the whole load-bearing airframe model and the acceleration response, extracted from the whole airframe structure, is then applied to two sides of the beam by using the submodel technique in abaqus. The modeling strategy and details are the same as Ref. [20] (depicted in Fig. 4(a)). Thus, stress responses at critical locations can be obtained, and the crack growth can be obtained by stress intensity factor via the empirical formula. The calculation process is as follows:

where Δ*S* is the stress increment, $a0$ is the crack length at the current moment, and *F* is the crack shape factor, which is related to the structural shape and the crack length.

where *da*/*dN* is the crack growth rate, *C*, *m*, and *γ* are the parameters of Walker's law, and *R* is the stress ratio. Note that Walker's law is only used for illustration; any other appropriate crack growth model, such as Paris's law [24] and Forman's law [25], can also be used.

where Δ*a* is the crack growth increment.

where $ySIM$ represents the results of the finite element model, represents the results of the GPM, and *n* represents the number of samples.

## 3 Dynamic Data Driven Reliability Prognosis

### 3.1 Dynamic Bayesian Network for Model Updating.

Based on the DT concept, a DBN can be constructed for crack growth prediction. DBN is an extension of Bayesian network (BN) used for uncertainty inference [28,29], and it can be used in system reliability evaluation [30]. It can not only incorporate multisource heterogeneous information but also accumulate the knowledge of the previous moments [28]. As more information is collected over time, the uncertainty in inference can be reduced further. In this paper, inspected crack data can be integrated into a DBN, used for updating the crack prediction with a lower uncertainty. Improved prediction will be useful for reliability evaluation and mission planning. Among various inference algorithms of a DBN, the particle filters (PF) [31] can deal with different types of variables, distributions and relationships, thus being used widely [8]. The sequential importance resampling algorithm [31], which is one of the PF algorithms, is employed in this paper.

Uncertainty analysis is the first step to construct a DBN. There are three types of uncertainties in the crack prediction via a DBN. (1) Loads uncertainties, including the quasi-static and random vibration loads and impact velocity, as they cannot be known exactly before a flight; (2) Model parameters uncertainties. The parameters of Walker's law are estimated by fitting experimental data. Measurement error and fitting algorithm error will induce uncertainties in parameters. The ultimate strength is also uncertain due to the dispersion of material properties. (3) Measurement error. The inspected data contains uncertainty because of the measurement error. The designed DBN for crack prediction is depicted in Fig. 5, and the detailed interpretations are in Table 3.

Type | Variable | Connotation |
---|---|---|

Other variable | $PPSD$ | Random vibration load |

$Psta$ | Quasi-static load | |

$Vsk$ | Impact load | |

S | Stress response | |

ΔK | Increment of stress intensity factor | |

F | Crack shape factor | |

$a0$ | Initial crack length | |

$\Delta ars$ | Increment of crack length under $PPSD$ and $Psta$ | |

$ars$ | Crack length under $PPSD$ and $Psta$ | |

$\Delta ask$ | Increment of crack length under $Vsk$ | |

$aall$ | Final crack length under $PPSD$, $Psta$ and $Vsk$ | |

Model parameter | C/m/γ | Parameters in Walker's law |

σ | Ultimate strength of the material | |

Measurement data | $aobs$ | Inspected crack length |

Measurement error | Δ | Measurement error of an inspection technique |

Type | Variable | Connotation |
---|---|---|

Other variable | $PPSD$ | Random vibration load |

$Psta$ | Quasi-static load | |

$Vsk$ | Impact load | |

S | Stress response | |

ΔK | Increment of stress intensity factor | |

F | Crack shape factor | |

$a0$ | Initial crack length | |

$\Delta ars$ | Increment of crack length under $PPSD$ and $Psta$ | |

$ars$ | Crack length under $PPSD$ and $Psta$ | |

$\Delta ask$ | Increment of crack length under $Vsk$ | |

$aall$ | Final crack length under $PPSD$, $Psta$ and $Vsk$ | |

Model parameter | C/m/γ | Parameters in Walker's law |

σ | Ultimate strength of the material | |

Measurement data | $aobs$ | Inspected crack length |

Measurement error | Δ | Measurement error of an inspection technique |

As depicted in Fig. 5, the DBN can be regarded as two adjacent BNs connected by dashed lines. A single BN contains all random variables in the process of crack growth calculation. Directed lines represent the dependence relationships between two variables. They can be described by conditional probability distributions or deterministic functional relations [8].

For a single BN, *S* at the critical location, can be obtained after FEA, under $PPSD$ and $Psta$. Δ*K* can be calculated using Eq. (1) with *S*, *F*, and $a0$. Then, $ars$ can be obtained using Eqs. (2) and (3) with Δ*K*, *C*, *m*, *γ*, and $a0$. As landing failure is a probability event, the crack growth under impact load is not necessary for every flight. If landing failure happens, the nodes in the blue frame ($Vsk$, $\Delta ask$, and *σ*) will be activated. The process of crack growth will be performed including all nodes in the DBN. Here, $\Delta ask$ can be calculated using the GPM (described in Sec. 2.3) with $Vsk$, $ars$, and *σ*. Final crack length $aall$ is the summation of $ars$ and $\Delta ask$. If landing failure not happens, the nodes in the blue frame ($Vsk$, $\Delta ask$ and *σ*) will be de-activated. Then, the $ars$ node is equal to the $aall$ node, meaning that only the crack growth under fatigue loads needs to be considered. In addition, the measured crack length $aobs$ is affected by $aall$ and *δ*.

Between the two BNs, directed dashed lines represent the data transmissions from the BN at time-step *t* − 1 to the BN at time-step *t*. If there is an inspected crack length at time *t* − 1, the parameters *C*, *m*, and *γ*, at time *t* − 1, can be updated. The parameters at time *t* will be equal to the updated ones. Here, $a0$ at time *t* will satisfy the normal distribution of $N(aobs,\delta 2)$, with $aobs$ at time *t* − 1. Otherwise, the parameters at time *t* will be equal to the ones at time *t* − 1 and $a0$ at time *t* will be equal to the predicted final crack length $aall$ at time *t* − 1.

The states of any variables (nodes in the DBN), such as final crack length, can be estimated by a weighted set of particles using the sequential importance resampling algorithm [31]. The weights of particles can be updated using the inspected crack length, resulting in better crack prediction and reliability evaluation.

### 3.2 Dynamic Evaluation of Structural Reliability.

Reliability is the ability of a structure to comply with given requirements under specified conditions during the intended life for which it was designed [32]. In structural reliability analysis, the fundamental problem is to calculate the nonfailure probability $Pr$ of a structure [33]. Various approximation methods have been developed to calculate $Pr$, including the first-order reliability method, the second-order reliability method, the response surface approach and the Monte Carlo method [34]. In this paper, considering that using PF to calculate crack growth can generate a large number of particles, the Monte Carlo method based on sampling statistics is adopted for reliability evaluation.

where *n* denotes the number of all samples participated in calculation and $nr$ denotes the number of samples complying with designed requirement.

In the example, the critical crack size is used as the threshold for counting the reliable samples. It can be calculated by the criterion that the maximum stress intensity factor reaches the fracture toughness of the material. In the reliability prediction of a mission, landing impact load always occurs behind the fatigue load, and has a certain probability. Thus, the reliability evaluation for a mission can be divided into the following steps:

The number of particles obtained from the last flight is recorded as $nlast$. Then, these particles are employed to predict fatigue crack growth. Calculate $ars$ based on the DBN, and count the number of failure particles (expressed as $n1$), namely their crack sizes exceed the threshold of the critical size under fatigue loads.

Set the probability of landing impact $Plan$. Randomly select $nimpact$ particles with impact load from $(nlast\u2212n1)$ particles, according to $Plan$, which are employed to predict crack growth under impact load. Calculate $aall$ based on the DBN, and count the number of failure particles (expressed as $n2$), namely their crack sizes exceed the threshold of the critical size under impact load.

The structural reliability of the mission can be calculated by Eq. (4), where $nr=nlast\u2212(n1+n2)$ with

*n*being the total number of initial particles.

Note that $nlast$ may change with the number of predicted missions, this is because the failed particles are directly identified as failure in the next reliability evaluation. With the updating of the model and the continuous prediction of reliability, the particles in PF are also constantly updated, which makes the value of initial particles step 1 update continuously. That makes the reliability evaluation have dynamic characteristics.

### 3.3 Verification Method.

The method of manufactured solutions (MMS) is employed in verification method. The philosophy behind the method can be concluded as [35]: (1) setting model parameters and initial conditions artificially to calculate the model outputs, which are known as manufactured solutions. For MMS, the term “manufactured” means that the set can be without concern for physical realism, as the verification based on MMS is a purely mathematical exercise. (2) Inputting the manufactured solutions into the method, to be verified, to estimate the model parameters; using the estimated parameters to recalculate the model outputs, which are known as estimated solutions. (3) Comparing the estimated model parameters with the manufactured ones, or the estimated model solutions with the manufactured ones, to verify the methods. In this paper, the process of the verification is as follows:

*Step 1:* Generate virtual flight data. The model in Sec. 2.3 is used for verification method. There are some assumptions for creating the manufactured solution. (1) The loads, applied to the load-bearing airframe model shown in Fig. 3 are assumed to be unchanged in all flights. (2) Landing impact will occur twice during the all flights considered. Landing velocities are 8 m/s and 9 m/s for the 5th flight and the 12th flight, respectively. Note that the choice of velocity is based on the landing velocity of the reentry capsule without deceleration. Choosing these two velocities and these two fights (5th and 12th) is only, for example, design in method validation, and other fights and suitable velocities can also be selected. The time of impact is assumed to be very short, which can be ignored in this paper. (3) The minimum length of the detectable crack is assumed to be 0.05 mm. The crack measurement error using ground inspection is considered as 0.05 mm. Then, the crack growth process for repeated flights can be calculated.

*Step 2:* Determine the initial distribution characteristics of particles. Considering the uncertainties in loads, initial crack size and model parameters, setting their initial distributions to evaluate reliability. A coefficient value range of 0.9–1.1 is added to the fatigue load depicted in Fig. 3. For each particle, there is a different coefficient to indicate the load fluctuation. The range of impact velocity is set as 6∼9 m/s. The model parameters range from 95% to 105% of the manufactured values approximately, as depicted in Table 4. The model parameters, load coefficient, and impact velocity are assumed to be uniformly distributed. Note that the deviations of the above are set artificially, just for the demonstration. They can be changed to other values as intended.

Parameter | Log C | m | m(1 − γ) | σ (MPa) |
---|---|---|---|---|

Manufactured value | −10.887 | 3.873 | 2.496 | 441.8 |

Initial distribution | (−11.43, −10.34) | (3.68, 4.07) | (2.37, 2.62) | (419.17, 463.89) |

Parameter | Log C | m | m(1 − γ) | σ (MPa) |
---|---|---|---|---|

Manufactured value | −10.887 | 3.873 | 2.496 | 441.8 |

Initial distribution | (−11.43, −10.34) | (3.68, 4.07) | (2.37, 2.62) | (419.17, 463.89) |

*Step 3:* Predict the crack growth. Following the DBN depicted in Fig. 5 to predict the crack growth, with sampling 1000 particles from the distributions discussed in step 2. $Plan$ is assumed to be 5%. It is just for method illustration and can be specified as a different value. The crack growth prediction will follow the real scenarios, namely, only the 5th and 12th occur landing impact. The moment that requires a postflight maintenance will be determined by reliability evaluation, as described in Sec. 2.2. The observed crack length can be generated by adding a Gaussian white noise $\sigma obs\u223cN(0,0.052)$ to the corresponding virtual flight data. Then, a model updating will be performed using the observed crack. Future predictions will be based on the updated model.

*Step 4:* Evaluate the structural reliability and realize the dynamic mission planning. The reliability threshold $Pth$ is assumed to be 99.7%, which is just for method illustration. It can be replaced by any other reasonable values according to the specific requirements. Many countries have different quantitative requirements for the safety and reliability of spacecrafts. According to the existing public information, the Apollo program requires that the reliability of the instrument module be no less than 0.992; the reliability requirement of upper panel stage of Ariane 5 launch vehicle is larger than 0.997. Thus, the threshold is selected as 0.997 in this paper for more safety and reliability. The structural reliability evaluation is completed based on the method proposed in Secs. 3.1 and 3.2, and the dynamic mission planning is completed based on the strategy proposed in the sample in Sec. 2.2.

## 4 Results and Discussion

### 4.1 Dynamic Bayesian Network for Model Updating.

After calculation, the reliability of the first four flights is 100%, and the crack length is less than the observation error. Considering that the landing failure occurred on the fifth flight and the crack has obvious growth (0.205 mm); subsequent discussions will begin after the fifth flight landing of the spacecraft. With $Plan=5%$, the results of crack growth prediction and reliability evaluation are depicted in Figs. 6 and 7.

Figure 6(a) shows the crack growth prediction for all flights. Figures 6(b) and 6(c) show the details of crack growth prediction under partial flights. In Fig. 6, the mean of prediction is the weighted mean crack growth of all particles; CI represents the confidence interval; #Pre1, #Pre2, #Pre3, #Pre4, #Pre5, #Pre6, represent the crack growth prediction curves based on the inspected data provided by the ground maintenance after the 5th, 9th, 16th, 17th, 18th, 19th flight, respectively. CI and Pre#*i* (*i *=* *1, 2, …, 6) of the same color and line type correspond to the same crack growth prediction. The sudden change of crack size after the 12th flight in Fig. 6(a) is caused by landing impact. The conclusions drawn from the results depicted in Fig. 6 are as follows:

Crack predictions can be calibrated using post-flight inspected data. As shown in Figs. 6(a)–6(c), the mean of prediction gets closer to virtual flight data after model updating. With more updates, a higher accuracy can be achieved, the crack predictions of the 16th to 19th flights in Fig. 6(c) can effectively prove this point.

CI can be reduced shortened by updating the model with the inspected data. As shown in Figs. 6(b) and 6(c), CI propagates to be larger during the flights, but they can be reduced at the points when the post-flight maintenance is conducted (e.g., position at green line in Fig. 6(b)). The reason is that the more information from the real scenarios, the better a structure digital twin can represent the physical system. Namely, the uncertainty of the predicted crack length gradually decreases with the continuous calibration of the crack growth model by the inspected data.

In Fig. 7, the “first evaluation” represents that the first reliability evaluation, starting from the end of the fifth flight to future flights. As the “first evaluation” in the 10th flight is lower than the threshold, the postflight maintenance is carried out for model updating at the end of ninth flight. The updating includes both crack size and model parameters. The updated model is employed to reevaluate the reliability of the 10th and future flights. That is “second evaluation.” The “fourth to eighth evaluations” follow the same rules. There is only an exception, the “third evaluation.” As shown in Fig. 7, the 14th flight should be reevaluated, according to results of the “second evaluation.” However, as we have assumed that if there is a landing impact, maintenance should be carried out immediately, whatever the result of reliability evaluation is. In the manufactured virtual flight, regarded as the real scenarios, a landing failure occurs at the end of 12th flight. Thus, the third evaluation begins with the 13th flight. As the latest reliability of the 20th flight is still lower than the threshold based on the updated model after postflight maintenance of the 19th flight, the ground repair should be carried out at the end of 19th flight. Conclusions drawn from results depicted in Fig. 7 are as follows:

Maintenance intervals can be adjusted according to the constantly updated reliability. In this example, ground maintenance is at the end of the 5th, 9th, 12th, 14th, 16th, 17th, 18th, and 19th flights, respectively. Except the 12th for landing failure, the other ground maintenance follows the results of the latest reliability evaluation. Waiting too long (e.g. maintain every 5 flights) or inspecting too often (e.g. maintain every flight) is inadvisable in the lifecycle of a spacecraft. This shows the reliability evaluation method based on the dynamically updated model can avoid unnecessary maintenance and ensure flight safety.

According to the example in the paper, more times of re-use can be achieved with the latest reliability. As shown in Fig. 7, the times of re-use for the spacecraft are 9, 13, 14, 16, 17, 18, and 19 respectively, based on different reliability evaluations. Here, assuming that the evaluation of the times of re-use has no consideration of repair operations. It has been verified that the crack growth can be predicted more accurately with constant model updating. The ability of reliability evaluation can also be enhanced based on the updated crack prediction. Thus, there is some confidence to postpone the repair or the retire of a spacecraft. Tremendous maintenance or development cost can be saved.

### 4.2 Impact of Landing Failure.

To consider the impact of the probability of landing failure further, different $Plan$ = 5%, 10%, 50%, and 100% are employed to evaluate reliability as in Fig. 8. Figure 8(a) shows the dynamic reliability revaluation based on the model update for all flights. When the reliability is lower than $Pth$, the reliability needs to be reevaluated based on the model updated by the inspected data. That makes two reliabilities appear simultaneously in the sequence number of flights when the reliabilities are lower than $Pth$ (e.g., the 9th fight with $Plan$ = 100%). Figure 8(b) shows the reliability characteristics corresponding to different $Plan$ without considering the model updating, in order to more clearly show the impact of different $Plan$ on reliability.

It can be found that the spacecraft, with a lower $Plan$, always has a higher reliability in Fig. 8. The reason is that higher probability of landing failure means more particles considering $\Delta ask$. Larger cracks will reduce the reliability. If repair operations are not considered, the spacecraft, with $Plan$ = 5% and 10%, can be reused 9 times, and only 8 times can be reused with $Plan$ = 50% and 100% from Fig. 8(a). From Fig. 8(b), based on the reliability evaluation with $Plan$ =5% and 10%, the spacecraft will be inspected after the 9th flight (the reliability of the 10th flight is less than $Pth$). While based on the reliability evaluation with $Plan$ = 50% and 100%, the spacecraft will be inspected after the eighth flight. The same happened on the 13th flight. Those show the selection of appropriate $Plan$ has an important impact on the reliability evaluation, and then affects the maintenance strategy of the spacecraft.

## 5 Conclusions

In this paper, based on the DT framework, a dynamic reliability diagnosis method for mission planning of reusable spacecraft is proposed. The dynamic updating and uncertainties quantification of the model are realized by DBN. The evolution of structural reliability can be forecasted based on the DBN and the maintenance point is planned as predicted reliability is lower than a prescribed threshold. As the structural reliability is continuously updating through model calibration with inspected data, the maintenance intervals can be optimized and adjusted dynamically. The method is demonstrated on an example of single point crack growth prediction with multiple loads, and the following insights are obtained:

The uncertainties of the crack length predictions increase during repeated flights, leading to the degradation of the predicted structural reliability. Through inspection data assimilation and dynamic model updating, the uncertainty interval of the crack prediction can be reduced, resulting in higher accuracy in reliability evaluation.

Appropriate selection of landing impact probability of the spacecraft has an important impact on the reliability evaluation, and then affects the maintenance strategy of the spacecraft.

Higher reliabilities of the spacecraft can also be obtained through model calibration and the dynamic reliability evaluation, which means that service life of the spacecraft can be extended and decisions on reusability of the structures can be made with higher confidence.

In further study, more failure modes and uncertainties as well as online monitoring can be included into the framework.

## Funding Data

National Natural Science Foundation of China (Grant Nos. U22B6009 and 12090034; Funder ID: 10.13039/501100001809).

Young Elite Scientists Sponsorship Program by China Association for Science and Technology (Grant No. 2021QNRC001; Funder ID: 10.13039/100010097).

Science Foundation of National Key Laboratory of Science and Technology on Advanced Composites in Special Environments (Grant No. 6142905223505).

## Data Availability Statement

The datasets generated and supporting the findings of this article are obtainable from the corresponding author upon reasonable request.