Reliability Evaluation of Landing Gear Retraction/Extension Accuracy Based on Bayesian Theory

Abstract

The angular motion of aircraft landing gear retraction and extension must be accurate to ensure flight safety. Therefore, this study experimentally evaluated the motion accuracy of the landing gear retraction and extension processes associated with a specific aircraft to construct a reliability evaluation model for the landing gear angle. Considering the limitations of data acquisition in practical applications, the Bayesian method, which combines prior knowledge with experimentally measured data to reasonably estimate the variable parameters in the evaluation model, was applied to obtain more accurate parameter distributions. The constructed Bayesian-updated iterative model was shown to effectively expand upon limited test data to provide a novel approach for accurately evaluating landing gear angle reliability. The results of this study not only enrich the theoretical basis underpinning aircraft landing gear reliability assessment but also provide a valuable reference for technical support and decision-making in related engineering practice.

1. Introduction

The landing gear retraction/extension system is a critical aircraft subsystem, as its performance directly affects flight safety [1]. It comprises a complex hybrid system integrating mechanical, electronic, and hydraulic subsystems to perform its core function of retracting the landing gear into the aircraft or extending it to the ground in a timely manner according to the flight phase requirements [2]. Both the retraction and extension processes require precise control, considering a variety of complex factors to ensure that the aircraft effectively adapts to different flight conditions. Indeed, airworthiness regulations dictate strict and clear requirements for the precision of landing gear retraction and extension angles [3] to ensure stability and reliability during use and thereby maintain flight safety. In practice, landing gear retraction and extension precision is affected by multiple factors, including the aerodynamic drag during aircraft takeoff and landing [4], friction between components, and possible hydraulic fluid leakage [5]; these factors can also lead to system failures that compromise flight safety. Indeed, if the landing gear retraction or extension angle is insufficiently precise, instability is likely to occur during aircraft takeoff or landing, significantly increasing the risk of aviation incidents [6,7]. Therefore, ensuring highly precise and reliable landing gear retraction/extension systems is a critical issue in aviation engineering [8].

The reliability of the landing gear retraction/extension angle is an issue of mechanical motion accuracy reliability, the investigation of which is centered on in-depth explorations of the manufacturing, assembly, and clearance of kinematic pairs [9]. Relevant research in the published literature includes work by Wang [10], who conducted a reliability analysis of the landing gear locking mechanism, and Wang et al. [6], who designed and applied a test aircraft landing gear retraction/extension system. Furthermore, Xu [11] conducted reliability optimization of the landing gear retraction/extension system for a specific type of military aircraft. However, no research has been published evaluating the reliability of the angular accuracy of the landing gear retraction/extension mechanism. Clearly, additional research on the reliability of landing gear retraction/extension motion accuracy is required.

Therefore, this study conducted retraction and extension tests using a specific type of landing gear mechanism to collect precise measurements of the retraction and extension angles [12]. The requirements for angular deviation were subsequently employed to undertake an in-depth reliability analysis of the landing gear retraction/extension accuracy. The limited test data available for analysis was augmented using the Bayesian method to integrate the prior distribution of variable parameters with the statistical characteristics of the test data and thereby obtain parameter values that closely reflected the actual situation [13]. Finally, the constructed Bayesian-updated iterative reliability model for the motion accuracy of a landing gear retraction/extension system was demonstrated to provide improved reliability results. A Bayesian-updated iterative reliability model integrating multi-source information fusion was developed, overcoming the limitations of traditional methods in analyzing small-sample data scenarios. This provides a new methodology for evaluating motion accuracy reliability in complex mechanical systems. By combining experimental testing with theoretical analysis, high-precision measurements and uncertainty quantification of dynamic parameters for aviation actuation mechanisms were achieved, filling a critical theoretical gap in reliability modeling for retraction/extension system motion accuracy.

2. Main Landing Gear Retraction/Extension Test

2.1. Test Design

Figure 1 shows a structural diagram of the main landing gear of a specific type of civil aircraft. The primary objective of the landing gear test conducted in this study was to verify the motion range and reliability of its retraction/extension system and confirm the accuracy of its retraction and extension angles. As the performance of the landing gear retraction/extension system directly affects the safety and maneuverability of an aircraft, such precise testing and simulations are required to ensure its stability and reliability under various flight conditions.

The test support and loading scheme applied in this study are illustrated in Figure 2. The landing gear was placed in an upright position and hung from the platform above, which was supported by four columns fixed to a load-bearing floor. The retraction/extension actuator and upper lock were fixed to this upper platform through the actuator support and upper lock support, respectively.

To apply the required torque for the landing gear retraction/extension test, the torque is converted into equivalent force. As shown in Figure 3, the servo loading system applies the load via a wire rope to the tie-down ring on the landing gear’s dummy wheel. The vertical displacement of the servo loading structure dynamically adjusts with the rotation of the landing gear, ensuring the load remains consistently aligned in the horizontal direction throughout the entire retraction/extension cycle. A load sensor is connected to the loading wire rope to measure the magnitude of the applied load. The load calculation procedure is as follows: 1. Convert the applied torque into the normal load at the tie-down point within the plane defined by the wheel axle and the tie-down point. 2. The loading mechanism applies a force to the tie-down point such that the normal component of the applied load matches the calculated normal load. 3. The finalized applied load and the corresponding vertical displacement of the servo actuator, determined through calculation, are summarized in

Table 1. Hinge moments acting on the strut shaft during main landing gear retraction and extension.

Retraction and Extension Angle θ (°)	Rotating Shaft Angle	Retraction/Extension Actuator Displacement(mm)	Retraction/Extension Torque (N·m)	Applied Load (N)	Servo Loading Mechanism Vertical Displacement (mm)
0	0	0	1666	1410.8	0
10	11.4	−20.4	1589	1300.3	48.6
20	22.8	−44.3	1415	1183.3	141.6
30	34.3	−70.9	1176	1057.8	275.9
40	45.9	−99.7	913	932.6	447.4
50	57.6	−129.7	660	817.9	650.5
60	69.5	−160	443	732.6	878.8
70	81.7	−189.1	265	695.2	1125
80	94.3	−215.6	114.8	788.6	1381.1
89.536	106.8	−236.5	0	0	1626.7

.

The test load varies with the retraction/extension angle. To apply this load, the rotation angle of the shaft (which has a one-to-one correspondence with the retraction/extension angle) must be measured. A high-precision electric servo loading system was employed to precisely control the loading force to match the torque for the applied angle using a steel wire rope, and the angle of the rotating piston rod shaft (as shown in Figure 2) was measured using an attached high-sensitivity optical clinometer as pressure was continuously and slowly introduced into the retraction/extension actuator and foldable strut to apply retraction or extension loads [14]. The optical clinometer is shown in Figure 4. An optical inclinometer usually contains a light source and a photosensitive element. By utilizing the law of refraction of light, the optical signal is converted into an electrical signal. Then, through measuring the change of the electrical signal, the deflection angle of the light ray can be calculated, and thus the deflection angle of the landing gear can be obtained.

Figure 5 is the schematic diagram of the retraction and extension deployment test system. Retraction and extension tests were performed without and with an applied load at a retraction and extension angle of 10° (corresponding to a piston rod shaft angle of 11.4°) to evaluate the reliability of the results given an allowable error of ±0.5°. Each test scenario measured 20 angles in 4 groups to obtain 80 angle data points; a sample of these angles is detailed in

Table 2. A sampling of measured retraction and extension angles for a target angle of 10°.

Unloaded Angle θU (°)		Loaded Angle θL (°)
Angle range	Mean	Angle range	Mean
11.2~11.7	11.4	11.0~11.9	11.5

.

2.2. Normality Testing

Statistical analyses and historical data suggest that angular measurements acquired from landing gear systems in both loaded and unloaded operational states consistently exhibit characteristics aligning with normal distributions. The normality of the data collected in this study was confirmed using the graphical probability paper plotting method as follows:

(1)

First, the obtained data were arranged in ascending order to determine the relative location of each data point within the entire dataset;

(2)

Next, the corresponding cumulative probability value was calculated for each data point as $F\left(X_i\right)={\displaystyle \frac{i-0.3}{n+0.4}}(i=\mathrm{1,2},\dots ,n)$, where i is the rank of the data point after sorting and n is the total number of data points;

(3)

Each data point and its corresponding cumulative probability value were plotted on normal probability paper, a special type of graph paper with the horizontal axis scaled according to the cumulative distribution function for the normal distribution and the vertical axis representing the values of the data points;

(4)

Finally, the sample was considered to exhibit a normal distribution if the plotted points fell along or near a straight line;

Figure 6 depicts histograms of the angles measured under the unloaded and loaded scenarios. Each graph clearly exhibits the bell-shaped curve characteristic of a standard normal distribution. Figure 7 shows the distributions of these data when plotted on normal probability paper, in which the angles measured under both scenarios are clearly distributed along straight lines. This confirms that these data follow a normal distribution and conform to the corresponding assumptions, which ensures the reliability and validly of the subsequent statistical analyses.

2.3. Hypothesis Testing

In the field of Bayesian statistics, experimental data can be used to update the distributions of model parameters, allowing researchers to make full use of all known information for statistical inference. Under ideal circumstances, the prior and posterior distributions in Bayesian statistics should be derived from the same population to ensure data consistency [15].

Therefore, an F-test was conducted to determine the homogeneity of variance between the unloaded and loaded datasets. At a significance level of 5%, the obtained F value was 0.58, which was greater than the one-tailed critical value of 0.34 in the F distribution, indicating that the two datasets can be considered to have the same variance. Next, a t-test was conducted to confirm that the two datasets represented the same population. The t-value obtained at a significance level of 5% was −1.23, and its absolute value was less than the two-tailed critical value of 1.97. Therefore, there was no significant difference between the means of the two datasets, reflecting their shared population [16]. As a result, Bayesian theory was considered a reasonable approach for updating the data.

3. Iterative Reliability Model Based on Bayesian Theory

3.1. Reliability Model

The reliability of a moving mechanism comprises motion accuracy reliability (i.e., the ability of the mechanism to ensure precise motion) and motion enabling reliability (i.e., the ability of the mechanism to achieve motion) [17,18]. This study evaluated the motion accuracy reliability, which is defined as the probability that the motion indicators output by the mechanism fall within the allowable accuracy index range, which can be expressed as [19]:

$$R=P({\theta }_{down}<\theta <{\theta }_{up})$$

(1)

where $\theta$ represents the motion index output by the mechanism, and ${\theta }_{down}$ and ${\theta }_{up}$ represent the lower and upper limits, respectively, of the mechanism’s allowable accuracy index.

According to the obtained test results, the measured angles follow a normal distribution given by $\theta ~N(\mu ,{\sigma }^2)$, where μ represents the mean value and σ represents the standard deviation. Integrating this definition into Equation (1) yields the following reliability expression:

$$\begin{array}{ll}R& =P\left({\theta }_{down}<\theta <{\theta }_{up}\right)\\ & =P\left({\displaystyle \frac{{\theta }_{down}-\mu }{\sigma }}<{\displaystyle \frac{\theta -\mu }{\sigma }}<{\displaystyle \frac{{\theta }_{up}-\mu }{\sigma }}\right)\\ & =\mathsf{\Phi }\left({\beta }_1\right)-\mathsf{\Phi }\left({\beta }_2\right)\end{array}$$

(2)

where ${\beta }_1$ and ${\beta }_2$ are the reliability indices corresponding to the upper and lower limits of the angle, respectively. The reliability index $\beta$ quantifies the safety margin between the applied stress and inherent strength of a system or component, with higher $\beta$ values indicating enhanced reliability.

In this study, motion accuracy refers to the deviation between the actual extension/retraction angle and target angle of the landing gear. The upper and lower limits of the deflection angle accuracy are given in Section 2.1 as 11.4 ± 0.5°, or ${\theta }_{down}={10.9}^{°}$ and ${\theta }_{up}={11.9}^{°}$; thus, the range of angles was defined as ${10.9}^{°}<\theta <{11.9}^{°}$.

3.2. Bayesian-Updated Iterative Model for the Reliability of Landing Gear Retraction/Extension Angle

In engineering practice, particularly in the aerospace field, the data obtained from testing are often limited, hindering the assessment of system performance, especially when critical safety parameters, such as the precision of the landing gear retraction/extension angle, are involved [20]. The accurate assessment of retraction/extension angle reliability is particularly critical since it directly affects the safety and flight performance of an aircraft [21].

Bayesian theory provides an effective solution for this reliability analysis problem. A significant feature of the Bayesian method is its ability to integrate prior distribution information, allowing researchers to use existing knowledge and experience to supplement limited experimental data. Prior distributions can be derived from historical data, expert opinions, or research results presented in relevant literature [22]. The advantage of this method is that it can produce relatively accurate estimation results by iteratively updating prior knowledge with new experimental information, compensating for the lack of data support. The new understanding obtained after each update subsequently serves as the prior knowledge for the following update. As a result, the solution accuracy gradually improves as updated information accumulates.

The specific process for applying the Bayesian method is as follows [23]. First, the probability density function of the population is defined as $f(\mathit{x},\delta )$. According to Bayes’ theorem, the posterior probability density function of the parameter to be estimated ($\delta$) is given by

$$\pi (\delta |x_1,\dots ,x_n)={\displaystyle \frac{{\prod }_{i=1}^{n}f(x_i,\delta )\pi (\delta )}{m(x)}}$$

(3)

in which

$$m(\mathit{x})={\displaystyle \int {\displaystyle \prod _{i=1}^{n}f(x_i,\delta )\pi (\delta )}d\delta }$$

(4)

where $\mathit{x}$ is the experimental vector, $x_1,...,x_n$ are the n variables considered, and $\pi (\delta )$ is the prior probability density of $\delta$.

When obtaining a set of samples $X=x_1,\dots ,x_n$, the posterior distribution $m(\mathit{x})$ of $\delta$ represents the marginal distribution of $(X,\delta )$ with respect to the sample vector $X$, where $x_i$ is the i-th variable.

The process of updating a model using the Bayesian method is extremely complex when both the mean and standard deviation of the basic variables are unknown. Therefore, the standard deviation of the angle measurement data was assumed to have a fixed value of 0.2 based on experience and was not updated during the analysis; only the means of the angle measurements were updated and calculated.

First, the means and standard deviations of the unloaded angle measurement data were selected as the hyperparameters of the prior distribution for the mean angle. The mean and standard deviation of the unloaded angle measurement data were ${\mu }_0=11.513$ and, ${\sigma }_0=0.194$ respectively. These data were substituted into the reliability expression in Equation (2) to obtain the reliability indices ${\beta }_1=-3.15$ and ${\beta }_2=1.99$ for measured angle values falling within the upper and lower limits of the deflection angle, respectively. A reliability $R=0.976$ was calculated based on these indices.

Next, the prior distribution of the deflection angles obtained under unloaded conditions was updated using the data obtained from each measurement group obtained under loaded conditions. The hyperparameters describing the distribution of the updated measured deflection angles were obtained according to Bayes’ theorem as follows [24]:

$$\mu ~N[({\displaystyle \frac{{\mu }_0}{{\sigma }_0^2}}+{\displaystyle \frac{n\overline{x}}{{\sigma }^2}})/({\displaystyle \frac{1}{{\sigma }_0^2}}+{\displaystyle \frac{n}{{\sigma }^2}}),{({\displaystyle \frac{1}{{\sigma }_0^2}}+{\displaystyle \frac{n}{{\sigma }^2}})}^{-1}]$$

(5)

where $\overline{x}$ represents the average value of the angle measurement for each group with an overall standard deviation $\sigma$ of 0.2. The 20 data points in each group of measurements obtained in the loaded condition were applied to the previous calculation results using an iterative process that updated the parameters informing the mean deflection angle. The reliability assessment process based on Bayesian theory is shown in Figure 8. These updated parameters were subsequently substituted into the reliability expression in Equation (2) to obtain the specific reliability values listed in

Table 3. Analysis results obtained by updating the angles measured under load.

Measurements Used as Prior Information	Mean Angle (°)	Standard Deviation	Reliability (%)
Unloaded data points (80)	11.513	0.194	0.9725
First iteration: loaded data points 1–20	11.467	0.079	0.9744
Second iteration: loaded data points 21–40	11.482	0.112	0.9876
Third iteration: loaded data points 41–60	11.483	0.078	0.9876
Fourth iteration: loaded data points 61–80	11.501	0.066	0.9876

; the reliability trend is presented in Figure 9.

4. Conclusions

An iterative model for assessing the reliability of landing gear retraction/extension angle precision was established using Bayesian theory by conducting retraction and extension tests on the main landing gear of a specific type of aircraft. The resulting model was shown to provide reliable information regarding the precision of the retraction/extension angles, with the calculated reliability increasing from 97.25% based on the unloaded condition measurements to 98.76% after four iterative updates using the loaded condition measurements. Bayesian theory effectively integrated prior knowledge into the statistical model, reducing dependence on the quantity of test data while ensuring precision. Notably, this landing gear retraction/extension angle reliability model can be updated using new data as the number of retraction and extension cycles increases, thereby achieving a more accurate reliability assessment. Given the applicability of the Bayesian method to various types of data and models as well as its ease of use, subsequent research and development efforts should focus on data collection and organization to incorporate more engineering experience into the prior distribution, thereby providing richer data informing high-precision models of landing gear and various other mechanisms. Furthermore, the reliability assessment methodology based on Bayesian theory can also be applied to other motion mechanisms, such as flap and slat actuation systems, demonstrating its broad applicability in aerospace mechanical systems.