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Article: Sartor, P., Becker, W., Worden, K. orcid.org/0000-0002-1035-238X et al. (2 more authors) (2016) Bayesian Sensitivity Analysis of Flight Parameters in a Hard-Landing Analysis Process. Journal of Aircraft, 53 (5). pp. 1317-1331. ISSN 0021-8669 https://doi.org/10.2514/1.C032757
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[email protected] https://eprints.whiterose.ac.uk/ Sartor, P. N., Becker, W., Worden, K., Schmidt, R. K., & Bond, D. (2016). Bayesian Sensitivity Analysis of Flight Parameters in a Hard Landing Analysis Process. Journal of Aircraft, 53(5), 1317-1331. DOI: 10.2514/1.C032757
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University of Bristol - Explore Bristol Research General rights This document is made available in accordance with publisher policies. Please cite only the published version using the reference above. Full terms of use are available: http://www.bristol.ac.uk/pure/about/ebr-terms.html Submitted to Journal of Aircraft for Review
Bayesian Sensitivity Analysis of Flight Parameters in a Hard Landing Analysis Process
Journal:For Journal Peer of Aircraft Review Manuscript ID 2013-12-C032757.R2
Manuscript Type: Full Paper
Date Submitted by the Author: n/a
Complete List of Authors: Sartor, Pia; University of Bristol, Department of Aerospace Engineering Becker, William; European Commission, Joint Research Centre Worden, Keith; University of Sheffield, Department of Mechanical Engineerin Schmidt, R.; Messier-Bugatti-Dowty, A350 Programme Chief Engineer Bond, David; Messier-Bugatti-Dowty, Vice President Engineering
01700 Landing Dynamics < 00000 AIRCRAFT TECHNOLOGY, Subject Index Category: CONVENTIONAL, STOL/VTOL, 02800 Simulation < 00000 AIRCRAFT TECHNOLOGY, CONVENTIONAL, STOL/VTOL
Note: The following files were submitted by the author along with the article. You may review these files online, if you wish. Acceptance for publication will be based solely on the content of the article.
AIAA_Latex_files_Pia_Sartor18 November 2015.7z
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1 2 Bayesian Sensitivity Analysis of Flight Parameters 3 4 5 in a Hard Landing Analysis Process 6 7 8 Pia Sartor1 9 10 Department of Aerospace Engineering, University of Bristol, 11 12 Queens Building, University Walk, Bristol, 13 14 BS8 1TR, UK, Email: [email protected] 15 16 17 William Becker2 18 Econometrics and Applied Statistics, European Commission 19 20 For- Joint Research Peer Centre, TPReview 361, 21027 Ispra, Italy 21 22 23 Keith Worden3 24 Department of Mechanical Engineering, 25 26 University of She eld, Mappin Street, She eld, S1 3JD, UK 27 28 29 4 5 30 R. Kyle Schmidt and David Bond 31 Messier-Bugatti-Dowty, Cheltenham Road East, Gloucester, GL2 9QH, UK 32 33 34 35 A Flight Parameter Sensor Simulation (FPSS) model was developed to assess the 36 37 conservatism of the landing gear component loads calculated using a typical hard land- 38 39 ing analysis process. Conservatism exists due to factors of safety that are incorporated 40 41 into any hard landing analysis process to account for uncertainty in the measurement 42 43 of certain ight parameters. The FPSS model consists of: (1) an aircraft and land- 44 45 ing gear dynamic model to determine the `actual' landing gear loads during a hard 46 47 landing; (2) an aircraft sensor and data acquisition model to represent the aircraft sen- 48 49 sors and ight data recorder systems to investigate the eect of signal processing on 50 51 52 53 1 Lecturer 54 2 Postdoctorial Researcher 55 3 Professor 4 56 Programme Chief Engineer 5 Vice President Engineering 57 58 59 60 1
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1 the ight parameters; (3) an automated hard landing analysis process, representative 2 3 of that used by airframe and equipment manufacturers, to determine the `simulated' 4 5 6 landing gear loads. Using a technique of Bayesian sensitivity analysis, a number of 7 8 ight parameters are varied in the FPSS model to gain an understanding of the sen- 9 10 sitivity of the dierence between `actual' and `simulated' loads to the individual ight 11 12 parameters in symmetric and asymmetric, two-point landings. This study shows that 13 14 the error can be reduced by learning the true value of the following ight parameters: 15 16 longitudinal tire-runway friction coe cient, aircraft vertical acceleration (related to 17 18 vertical descent velocity), lateral acceleration (related to lateral velocity), Euler roll 19 20 angle, mass, centreFor of gravity Peer position and Review main landing gear tire type. It was also 21 22 shown that due to the modelling techniques used, shock absorber servicing state and 23 24 tire pressure do not contribute signi cantly to the error. 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 2
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1 Nomenclature 2 3 CG = Aircraft Centre of Gravity 4 5 CS = Certi cation Speci cation 6 7 DOE = Design of Experiments 8 9 FAR = Federal Aviation Regulations 10 11 FDIU = Flight Data Interface Unit 12 13 FDR = Flight Data Recorder 14 15 FPSS = Flight Parameter Sensor Simulation 16 17 GEM-SA = Gaussian Emulation Machine for Sensitivity Analysis 18 19 GP = Gaussian Process 20 For Peer Review 21 mass = Aircraft mass 22 23 MEI = Main Eects Index 24 25 MLG = Main Landing Gear 26 27 28 MSE = Normalised Mean Square Error 29 30 Port SA = Port main landing gear servicing state 31 32 Starboard SA = Starboard main landing gear servicing state 33 34 TEI = Total Eects Index 35 36 tire = Tire type 37 38 tire press = Tire pressure 39 = Aircraft longitudinal velocity 40 VAx 41 42 = Aircraft lateral velocity VAy 43 44 = Aircraft vertical descent velocity VAz 45 46 VRTG = Aircraft vertical acceleration 47 48 LAT G = Aircraft lateral acceleration 49 50 Factual = `Actual' landing gear output data 51 52 F = `Simulated' landing gear output data 53 simulated 54 θ = Aircraft Euler pitch angle 55 56 φ = Aircraft Euler roll angle 57 58 ψ = Aircraft Euler yaw angle 59 60 3
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1 2 µlong = Longitudinal tire-runway friction coe cient 3 4 µlat = Lateral tire-runway friction coe cient 5 6 σ2 = Variance of the `actual' landing gear output data 7 Factual 8 σ2 = Variance of the `simulated' landing gear output data 9 Fsimulated 10 B = Roughness coe cients for the GP covariance function 11 12 c(·, ·) = Covariance function of the GP 13 14 c∗ , = Posterior covariance function of the GP 15 (· ·) 16 d = Number of model input parameters 17 18 cov = Covariance 19 20 = ExpectedFor value Peer Review 21 E(·) 22 23 h(·) = Regression function of the GP 24 25 m(·) = Mean function 26 ∗ 27 m (·) = Posterior mean function 28 29 n = Number of training data points 30 31 p(·) = Multivariate probability distribution 32 33 σ2 = Scaling factor of the GP covariance function 34 35 Si = Scaled main eect index of input xi 36 37 STi = Scaled total eect index of input xi 38 39 t(x) = Covariance of x with all training data 40
41 Vi = Unscaled main eect index of input xi 42 43 VTi = Unscaled total eect index of input xi 44 45 var = Variance 46 47 w = Regression function coe cients 48 49 x = Model input 50 51 X = Uncertain model input 52 53 y = Model output 54 55 Y = Uncertain model output 56 57 = Main eect 58 z(xi) 59 = First order interaction 60 z(xi,j) 4
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1 2 3 4 5 I. Introduction 6 7 A static structural overload occurs when a landing gear exceeds its material yield point in any 8 9 location. A common aircraft operational occurrence which may result in a landing gear overload 10 11 is a hard landing. A hard landing is de ned by the regulatory authorities in EASA Certi cation 12 13 Speci cation (CS) 25 and Federal Aviation Regulations (FAR) 25 as a landing with a limit vertical 14 15 descent velocity exceeding 10 ft/s at the design landing weight [1, 2]. However, the eect of the ver- 16 17 tical descent velocity must be combined with other critical enveloping ight parameters, including: 18 19 20 aircraft gross weight, aircraftFor centre Peer of gravity location, Review aircraft orientation (pitch, roll, yaw), rates 21 22 of motion (pitch rate, roll rate, yaw rate), ground speed, vertical descent velocity, longitudinal, lat- 23 24 eral and vertical acceleration, shock absorber servicing state and the tire-runway friction coe cient, 25 26 to accurately assess the loads in the landing gear. 27 28 If the ight crew suspect that there has been a hard landing, the following analysis process is typ- 29 30 ically performed: (i) the ight crew makes an occurrence declaration; (ii) visual and Non-Destructive 31 32 Testing (NDT) inspections are performed on the landing gear by the operator's maintenance crew 33 34 to assess for damage to the landing gear and airframe structure; (iii) aircraft ight parameter data, 35 36 such as aircraft acceleration, ground speed and aircraft orientation (pitch, roll), are downloaded 37 38 from the Flight Data Recorder (FDR) and reported to the aircraft and landing gear manufacturers, 39 40 who then calculate the loads during the occurrence [3]. Only after the data have been analyzed can 41 42 it be determined if there has been an overload. 43 44 A degree of conservatism typically exists in current hard landing analysis processes to ensure 45 46 safety of aircraft operation. This conservatism evolves from factors of safety or conservative as- 47 48 sumptions included within the analysis process to account for: (i) uncertainty in measured aircraft 49 50 ight parameters and (ii) unavailable aircraft ight parameters. For example, on common short and 51 52 medium range aircraft, vertical acceleration is typically sampled at 8 Hz. A landing however, takes 53 54 less than 125 ms. Thus, a possibility exists that the peak vertical acceleration recorded on the FDR 55 56 is less than the actual maximum value. To date, the eect of such assumptions on the degree of 57 58 59 60 5
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1 conservatism in a hard landing analysis process has not been quanti ed [4]. 2 3 A Flight Parameter Sensor Simulation (FPSS) model has been developed to assess the conser- 4 5 6 vatism in a hard landing analysis process [5]. Using a technique of Bayesian sensitivity analysis, 7 8 a number of ight parameters are varied in the FPSS model to gain an understanding of how the 9 10 model responds to variations in the inputs, to identify the most in uential input parameters and to 11 12 identify which input parameters have little or no eect on the conservatism [6]. In this technique, 13 14 an emulator of the model is created by tting a Gaussian Process (GP) to the response surface 15 16 using data from multiple runs of the model as dictated by a Design of Experiments (DOE) so that 17 18 the output of the model can be predicted for any point in the input space without having to run 19 20 the simulation. EachFor input parameter Peer is represented Review as a probability distribution and sensitivity 21 22 analysis data is inferred at a reduced computational cost and with little loss of accuracy. Compu- 23 24 tational savings can be up to two orders of magnitude compared to using a Monte Carlo method 25 26 [7, 8]. Accuracy of the emulator model is dependent on the model and the number of model runs, 27 28 and can be quanti ed through cross-validation with the model runs. 29 30 This paper rst describes the loads of interest when determining the serviceability of the Main 31 32 Landing Gear (MLG) structure. The FPSS model is then explained. The theoretical background 33 34 of the Bayesian sensitivity analysis is then presented, including a discussion on Gaussian Processes 35 36 which are used to develop the emulator, and the main eects and sensitivity indices inferred from the 37 38 resulting distribution-over-functions. Finally, the results of the sensitivity analysis for symmetric 39 40 and asymmetric landings using the FPSS model are shown. 41 42 43 44 II. Landing Gear Loads 45 46 Figure 1 shows a typical aircraft and telescopic port MLG structure, with the sign conventions 47 48 used in this paper. Figure 2 illustrates the landing dynamics of the port MLG in a two-point, 49 50 symmetric landing. The starboard MLG landing dynamics are identical to the port MLG in a 51 52 symmetric landing. On approach, the landing gear wheels are not spinning. However, on contact 53 54 with the runway, the landing gear wheels spin-up to the ground speed of the aircraft under the 55 56 in uence of the ground reaction and the tire-runway friction. The resulting drag force deforms the 57 58 59 60 6
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1 landing gear aft and stores energy in the structure. When the tire velocity reaches the aircraft 2 3 forward speed, the frictional force between the tire and the ground reduces and the release of the 4 5 6 strain energy stored in the rearward deformation produces a spring-back. The landing gear oscillates 7 8 until the structural damping reduces the stored energy to zero [9]. Also during this time, there is 9 10 an increasing vertical ground-to-tire load, which is a function of the gas spring, oil damping (related 11 12 to the square of the vertical descent velocity) and bearing friction. The shock absorber continues 13 14 to close until all the vertical energy has been absorbed and then it partially recoils [10]. The shock 15 16 absorber travel (SAT), in conjunction with aircraft attitude, landing gear rake angle and landing 17 18 loads, creates a bending moment on the landing gear structure which is computed at the lower 19 20 bearing. There are noFor side ground-to-tire Peer loads developedReview in a symmetric landing. However, CS 21 22 25.485 does require side loads to be considered in design to account for landings with some degree 23 24 of asymmetry [1]. 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 Fig. 1 Typical Aircraft and Port Main Landing Gear Structure with Sign Convention 41 42 43 In order to calculate the internal landing gear loads and assess the serviceability of the landing 44 45 46 gear structure after a hard landing, the axle response loads are required. The ground-to-tire loads, 47 48 discussed previously, act as the forcing function and with the mass and exibility characteristics 49 50 of the landing gear, produce the dynamic response loads at the landing gear axle. The dierence 51 52 between the ground-to-tire loads and the axle dynamic response loads is due to the inertial forces 53 54 of the landing gear mass between the ground and the landing gear axle during the impact [11]. 55 56 An asymmetric landing with aircraft lateral velocity, roll and yaw aects the landing dynamics 57 58 signi cantly on the port and starboard MLG compared to a symmetric landing. Figure 3 illustrates 59 60 7
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1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Fig. 2 ExampleFor of Symmetric Peer Port Main Review Landing Gear Landing Dynamics 21 22 23 the MLG port and starboard loads from an asymmetric landing with only a positive initial aircraft 24 25 lateral velocity and no roll or yaw angle. The initial aircraft lateral velocity acts in the starboard 26 27 direction and on touchdown, the aircraft decelerates with a lateral acceleration that acts in the port 28 29 direction, as shown by the reaction of the side ground-to-tire load acting on each MLG. Although 30 31 the MLG wheels touchdown at the same time, the starboard MLG has a higher vertical ground-to- 32 33 tire load than the port MLG. For a greater initial lateral velocity on landing, the aircraft lateral 34 35 acceleration on impact with the ground increases. The magnitude of the vertical ground-to-tire 36 37 load also increases on the starboard MLG, and decreases on the port MLG. However, the drag axle 38 39 response load and bending moment do not signi cantly change as the lateral velocity de ned in the 40 41 initial conditions increases. 42 43 In a landing con guration with an initial negative aircraft roll angle, as illustrated in Figure 4, 44 45 the aircraft is rolled with the port wing down so that the aircraft rst lands on the port MLG and 46 47 then on the starboard MLG. Therefore, the port MLG outer wheel touches down rst, and carries 48 49 more vertical load, followed by the port MLG inner wheel, which carries less of the vertical load. 50 51 On the starboard MLG, the inner wheel touches down rst, followed by the outer wheel. 52 53 54 For a greater initial roll angle on landing, the port MLG vertical load increases, while the 55 56 starboard MLG load decreases. The fact that the MLG wheels touchdown at dierent times in 57 58 landings with aircraft roll gives the distinctive total drag ground-to-tire curves with two peaks. As 59 60 8
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1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 For Peer Review 21 22 23 24 25 26 Fig. 3 Example of Asymmetric Main Landing Gear Loads Due to Aircraft Lateral Velocity 27 28 29 the initial aircraft roll angle increases, the total drag ground-to-tire load and the drag axle response 30 31 load decreases. This is because the greater the roll angle, the greater the time between the wheels 32 33 touching down and the build up of energy in the spin-up and spring-back is reduced. 34 35 Figure 5 illustrates the MLG port and starboard loads from an asymmetric landing with an 36 37 38 initial aircraft positive yaw angle, yawed clockwise from the aircraft centreline. In this landing 39 40 con guration, the aircraft decelerates with a lateral acceleration that acts in the starboard direction, 41 42 as shown by the reaction of the side ground-to-tire load acting on each MLG. The port and starboard 43 44 MLG wheels touchdown at the same time, however the port MLG has a higher vertical ground-to- 45 46 tire load than the starboard MLG. For a greater initial yaw angle on landing, the port MLG vertical 47 48 load increases, however, the starboard MLG vertical load decreases. The port MLG spin-up and 49 50 spring-back drag axle response loads also decrease with increasing yaw angle. 51 52 The points of interest for the MLG landing analysis are the drag axle response load and bending 53 54 moment at the lower bearing at spin-up and spring-back, and the vertical axle response load at 55 56 maximum vertical reaction since these are the most sever loading cases that the MLG experience 57 58 59 60 9
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1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 For Peer Review 21 22 23 24 25 26 Fig. 4 Example of Asymmetric Main Landing Gear Loads Due to Aircraft Roll Angle 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 Fig. 5 Example of Asymmetric Main Landing Gear Loads Due to Aircraft Yaw Angle 57 58 59 60 10
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1 on landing [12]. The side ground-to-tire load is of interest in asymmetric landings, however it will 2 3 not be discussed in this paper. 4 5 6 7 III. Overview of the Flight Parameter Sensor Simulation Model 8 9 The FPSS model, shown in Figure 6, consists of: (1) an `Actual' landing model to determine 10 11 the `actual' MLG loads during a hard landing, (2) an aircraft Sensor and Data Acquisition System 12 13 Simulink model to represent the aircraft sensors and FDR systems to investigate the eect of signal 14 15 processing on the ight parameters and (3) an automated Hard Landing Analysis Process model, 16 17 representative of that used by airframe and landing gear manufacturers, to determine the 'simulated' 18 19 MLG loads. Various hard landing cases were modelled using a representative aircraft and landing 20 For Peer Review 21 22 gear dynamic model. For each of the landing cases, it was possible to de ne ight parameters such 23 24 as: aircraft mass (mass), aircraft centre of gravity location (CG), aircraft Euler pitch (θ), roll 25 ( ) and yaw ( ) angles, aircraft longitudinal velocity ( ), vertical descent velocity ( ), lateral 26 φ ψ VAx VAz 27 velocity ( ), MLG shock absorber servicing state (Port SA, Starboard SA), tire type ( ), tire 28 VAy tire 29 30 pressure (tire press) and the longitudinal and lateral tire-runway friction coe cients (µlong, µlat). 31 32 These landing cases provide simulation of the `actual' ight parameters, as well as the `actual' 33 34 landing gear loads at spin-up, spring-back and maximum vertical reaction. 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 Fig. 6 Flight Parameter Sensor Simulation Model 52 53 54 The aircraft sensor and data acquisition model represents the aircraft sensors and FDR systems. 55 56 Aircraft ight parameters such as vertical and lateral accelerations, Euler pitch and roll angle 57 58 and aircraft longitudinal velocity are used in the typical hard landing analysis process. A typical 59 60 11
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1 aircraft Indicating & Recording system, Navigation system, Flight Data Interface Unit (FDIU) and 2 3 Flight Data Recorder (FDR) was modelled in Simulink to investigate the eect of signal processing 4 5 6 (sampling rate, ltering, analog to digital conversion, transfer/receive delays) on the aircraft ight 7 8 parameters. 9 10 Finally, with the ight parameter data from the FDR, a hard landing analysis process, repre- 11 12 sentative of that used by airframe and equipment manufacturers, was modelled and the conservative 13 14 assumptions typically made were applied. These assumptions include: aircraft mass, inertia and 15 16 centre of gravity location as close as possible to the occurrence case, assumed tire type, correctly 17 18 serviced shock absorber (correct uid volume and gas pressure) and assumed tire-runway friction 19 20 coe cient. The peakFor aircraft vertical Peer acceleration Review data from the FDR, as well as the other FDR 21 22 parameters at the peak vertical acceleration, are used to model the landing gear loads. Based on the 23 24 initial conditions provided by the FDR (Euler pitch and roll angles, aircraft longitudinal velocity), 25 26 the aircraft vertical descent velocity was iterated until the aircraft vertical acceleration output from 27 28 the hard landing analysis process model matched the peak vertical acceleration from the FDR. In 29 30 the Hard Landing Analysis Process model, the MLG side ground-to-tire loads are calculated using 31 32 a bookcase calculation method (footnote:Bookcase calculations, as given in CS 25, tend to be more 33 34 arti cial and usually require ground reactions to be balanced by inertia forces and moments. Ratio- 35 36 nal calculations use a model that more accurately represents the real physics and dynamics of the 37 38 system [9].): a ground manoeuver turn using the lateral acceleration (LATG) at the peak VRTG, 39 40 where the side ground-to-tire load is a function of LATG and the drag and vertical ground-to-tire 41 42 loads are factored as a function of LATG. 43 44 In the FPSS model, the Aircraft and Landing Gear Dynamic model used in the `Actual' Landing 45 46 47 model and the Hard Landing Analysis Process model are the same. Therefore, the `simulated' 48 49 landing perfectly models the `actual' landing and there is no error due to modelling. Any dierences 50 51 in the `actual' and 'simulated' landing gear loads are due to the conservative assumptions in the hard 52 53 landing analysis process and loss of data content from the FDR systems and processing algorithms. 54 55 From the landing gear loads calculated based on those conditions, it was possible to estimate 56 57 the conservatism between the `actual' landing gear output (Factual) and the `simulated' (Fsimulated) 58 59 60 12
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1 landing gear output using a normalised mean-square error (MSE) method [8]: 2 3 4 (F − F )2 MSE = actual simulated × 100 (1) 5 σ2 Factual 6 7 2 where σ is the variance of Factual from all of the model runs. 8 Factual 9 Table 1 provide a summary of the model inputs and outputs for symmetric and asymmetric 10 11 landings. As discussed in Section II, the points of interest for the MLG landing analysis are the 12 13 drag axle response load and bending moment at the lower bearing at spin-up and spring-back, and 14 15 the vertical axle response load at maximum vertical reaction. Therefore the MSE is calculated for 16 17 these outputs. 18 19 20 Landing ForInput Flight Peer Parameters ReviewOutput Quantity 21 Symmetric , , , Port SA, , Spin-up and Spring-back Drag Axle Re- 22 θ VAx VAz mass 23 , , tire press, sponse Load MSE, Spin-up and Spring- 24 CG tire µlong 25 back Bending Moment MSE, Maximum 26 27 Vertical Axle Response Load MSE 28 Asymmetric , , , , , , Port Spin-up and Spring-back Drag Axle Re- 29 θ φ ψ VAx VAz VAy 30 31 SA, Starboard SA , mass, sponse Load MSE, Spin-up and Spring- 32 , , tire press, , back Bending Moment MSE, Maximum 33 CG tire µlong 34 µ Vertical Axle Response Load MSE 35 lat 36 37 Table 1 Summary of Flight Parameter Sensor Simulation Model Inputs and Outputs 38 39 40 41 42 IV. Bayesian Sensitivity Analysis Theory 43 44 This section presents the theoretical background of the Bayesian sensitivity analysis, including 45 46 a discussion on GPs which are used to develop the emulator, and the main eects and sensitivity 47 48 indices inferred from the resulting distribution-over-functions. 49 50 51 52 A. Gaussian Processes 53 54 Any computer model, such as the FPSS model, can be considered a function of its inputs: f(x). 55 56 Although this function is deterministic and governed by known mathematical functions, it is often 57 58 complex and may be encoded by a large numerical model which has no closed-form expression for its 59 60 13
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1 outputs as a function of its inputs. Therefore, x could be considered an unknown function, since 2 f( ) 3 the output is unknown for a given set of inputs until the model has actually been run. If however, 4 5 6 the function (model) is sampled at a number of carefully chosen input points, it is possible to t a 7 8 response surface (footnote: A response surface is the hypersurface of the model output as a result of 9 10 varying the inputs [13].) which can predict the output of the model for any point in the input space 11 12 without having to run the model. For models that are computationally expensive (i.e. they require 13 14 several minutes, hours or days to run), creating a fast-running emulator (a model of a model) is a 15 16 useful approach for sensitivity analysis which generally requires multiple runs of the model under 17 18 investigation [7]. To be successful the emulator must be general and as little as possible must be 19 20 assumed about the emulatorFor function. Peer The emulator Review should also be able to accurately imitate the 21 22 model using as few training points as possible [13]. 23 24 A particular probabilistic approach for developing an emulator is the use of Gaussian Process 25 26 (GP) regression [1416]. GPs assume that observations f(x1), f(x2),. . . ,f(xn), as well as unobserved 27 28 values of f, are distributed joint-normally, i.e. they can be represented by a multivariate Gaussian 29 30 distribution. This means that predictions at unobserved values of x are also returned as a Gaussian 31 32 distribution. The mean and variance at any point are speci ed by a mean function and a covariance 33 34 function, which are functions of x, as well as a number of hyperparameters. Another way of looking 35 36 at a GP is a distribution-over-functions, where the random variable of the distribution is a function 37 38 rather than a single number or a xed-length vector [13]. 39 40 The covariance function typically has the property that the predictive variance increases for 41 42 values of x that are further away from training data. This means that the predictive variance of a 43 44 GP is a function of the distance to known points. It is useful to contrast this to linear regression, 45 46 47 which may also give normally-distributed estimates at unobserved points. The dierence is however 48 49 that in linear regression, the data points are assumed to be noisy, whereas a GP exactly interpolates 50 51 though the training data. In linear regression therefore, probabilistic predictions usually re ect 52 53 noise in the training data, or parameter uncertainty (especially within the Bayesian framework). 54 55 GPs adhere to the Bayesian paradigm, such that a number of prior assumptions are made about 56 57 the function being modelled, and then training data (samples from the model) are used to update 58 59 60 14
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1 and evaluate a posterior distribution-over-functions. It is assumed that the model is a smooth 2 3 function so that if the value of x is known, the value at x′ for x close to x′ will be highly 4 f( ) f( ) 5 6 correlated [7]. This assumption allows information to be gained on the response surface at reduced 7 8 computational cost. 9 10 For any number of d model input parameters, each with n training data points, the prior beliefs 11 12 about the corresponding outputs can be represented by a multivariate normal distribution, the mean 13 14 of which is a least-squares regression t through the training data [7]: 15 16 17 18 19 m(x)= E{f(x)|w} = h(x)T w (2) 20 For Peer Review 21 where h x T is a speci ed vector of regression functions of x, and w is the corresponding - 22 ( ) q q 23 T T 24 length vector of coe cients. Here, h(x) is chosen to be (1, x ), which represents linear regression. 25 26 This is a reasonable assumption since many engineering models display roughly linear behavior 27 28 with respect to at least some of the model inputs [13]. Here the covariance between output points 29 30 is de ned by a squared-exponential function of the form [7]: 31 32 33 34 ′ 2 2 ′ 2 ′ T ′ 35 cov{f(x),f(x )|σ } = σ c(x, x )= σ exp{−(x − x ) B(x − x )} (3) 36 37 where 2 is a scaling factor of the GP covariance function and is a diagonal matrix of 38 σ B 39 40 length-scales, which represent the roughness of the output (in terms of correlation length-scales as 41 42 opposed to dierentiability) with respect to the individual input parameters. The hyperparameters, 43 2 44 w, σ , B, are the controlling parameters that de ne the behavior of the emulator, which allows 45 46 the emulator to be general enough for a wide range of engineering problems [13]. The squared- 47 48 exponential covariance function is by no means the only covariance function - many others are 49 50 detailed in Rasmussen and Williams [17]. The squared-exponential function imposes an assumption 51 52 of derivatives of all orders, which may be a strong assumption for a physical model. An alternative 53 54 could be the more exible Matérn class of functions, however within the context of this work, the 55 56 squared-exponential functions have the advantage of being su ciently tractable to provide analytic 57 58 expressions for sensitivity indices. Furthermore, the added exibility of the Matérn functions also 59 60 15
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1 comes at the cost of having more parameters to estimate. Testing of more sophisticated covariance 2 3 functions is a valid avenue of enquiry but is left as future work. 4 5 6 The prior distribution is then de ned as [13]: 7 8 x GP x 2 x x′ (4) 9 f( ) ∼ (m( ),σ c( , )) 10 11 where means distributed as. 12 ∼ 13 The posterior distribution is then found by conditioning the prior distribution on the training 14 15 data on y (the output vector corresponding to the input set), and integrating out the hyperpa- 16 17 rameters 2 and w. This results in a Student's t-process, conditional on and the training data 18 σ B 19 20 [13]: For Peer Review 21 22 23 24 x y ∗ x 2 ∗ x x′ (5) 25 [f( )|B, ] ∼ tn−q{m ( ), σˆ c ( , )} 26 27 where ∗ x and ∗ x x′ are the posterior mean and covariance function respectively, which 28 m ( ) c ( , ) 29 are only dependent on B and y - expressions for these can be found in [7]. Note that as a result 30 31 of the integration, the posterior distribution is no longer dependent (conditional) on 2 and w, 32 σ 33 34 and now incorporates uncertainty about their values. This is where the bene t of the squared- 35 36 exponential covariance function is apparent - the analytical integration avoids approximations via 37 38 numerical methods such as Markov Chain Monte Carlo, which introduce their own uncertainty. The 39 40 roughness parameters in B are estimated using maximum likelihood estimation, since they appear 41 42 to be too di cult to analytically marginalise. In this respect, the GP is not fully Bayesian, and 43 44 uncertainty in B is not accounted for in the posterior distribution. The quality of the emulator is 45 46 dependent on the number and distribution of training data points in the input space, and the values 47 48 of the hyperparameters. 49 50 51 52 B. Inference for Sensitivity Analysis 53 54 If the input vector, x, is uncertain, X, the true input con guration" is considered a random 55 56 variable with the distribution p(x) [7]. The output Y = f(X) is then also a random variable 57 58 and the distribution of Y is known as the uncertainty distribution. With the emulator de ned by 59 60 16
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1 the posterior distribution-over-functions described by Equation 5, several quantities relevant to the 2 3 sensitivity analysis can be analytically derived without the necessity of additional model runs: the 4 5 6 main eects and interactions, as well as the sensitivity measures including Main Eects Indices 7 8 (MEI) and Total Eects Indices (TEI). In order to estimate the sensitivity measures, an assumption 9 10 is made that the input parameters are independent. 11 12 13 1. Main Eects 14 15 The function x can be decomposed into main eects and interactions [18]: 16 f( ) 17 d d d 18 y = f(x)= E(Y )+ zi(xi)+ zi,j(xi,j)+ zi,j,k(xi,j,k)+ ... + z1,2,...,d(x) (6) 19 Xi=1 Xi Review copy- Do not distribute Submitted to Journal of Aircraft for Review Page 18 of 40 1 2 3 4 E(Y |x )= f(x)p(x− |x )dx− (9) 5 i Z i i i χ−i 6 7 where χ− is the sample space of x− , x− is the subvector of x containing all elements except 8 i i i 9 x and p(x) represents the multivariate probability distribution of the input parameters. Although 10 i 11 this results in a series of matrix integrals, a Gaussian or uniform p x distribution, combined with 12 ( ) 13 a su ciently tractable covariance function (such as the squared exponential function used in this 14 15 work), allows these to be solved analytically. Expressions for interactions can also be derived with 16 17 their respective de nitions. 18 19 20 For Peer Review 21 2. Variance and Sensitivity Indices 22 23 In Reference [7], variance-based methods of probabilistic sensitivity analysis are described in 24 25 order to quantify the proportion of output variance for which individual input parameters are 26 27 responsible. In particular, sensitivity can be measured by conditional variance: 28 29 30 31 32 (10) 33 Vi = var{E(Y |Xi)} 34 35 For interpretation, this variance measure can be standardized by dividing the total output 36 37 variance: 38 39 40 41 42 Vi var{E(Y |Xi)} S = = (11) 43 i var(Y ) var(Y ) 44 45 where S is the Main Eect Index (MEI) of x , a widely-used global sensitivity measure proposed 46 i i 47 by Sobol' [18]. MEIs represent the fractional contribution of individual inputs to the uncertainty 48 49 (variance) of the model output. A high MEI means the variance of the output will be reduced 50 51 considerably if we learn the "true" value of the input ight parameter. This idea can be extended 52 53 to measure conditional variance of interactions of inputs, for example rst order interactions 54 Vi,j = 55 x , which is the eect of varying two input ight parameters simultaneously, additional 56 var{zi,j( i,j)} 57 to the main eects of both parameters, and so on for higher order interactions. Therefore, summing 58 59 60 18 Review copy- Do not distribute Page 19 of 40 Submitted to Journal of Aircraft for Review 1 the main eects will not in general total one because of the contributions from the interactions. 2 3 However, the total does provide an indication of the degree of the interactions [19]. 4 5 An additional sensitivity measure gives the variance caused by an input and any interaction 6 xi 7 of any order including and describes the output variance that would remain if one were to learn 8 xi 9 10 the true values of all inputs except xi: 11 12 13 14 X (12) 15 VTi = var(Y ) − var{E(Y | −i)} 16 17 After standardization this gives: 18 19 20 For Peer Review 21 22 V var(Y ) − var{E(Y |X− )} S = Ti = i (13) 23 Ti var(Y ) var(Y ) 24 25 where S is known as the Total Eects Index (TEI) [20]. The TEI includes the interactions with 26 Ti 27 every input ight parameter associated with it and therefore, considering all d TEIs of all variables, 28 29 may be counted twice for an interaction between two variables, three times for an interaction between 30 31 three variables, etc. Therefore, the TEIs may sum to more than one. 32 33 In [7], it is shown how the GP metamodel can be analytically integrated to give estimates of 34 35 both and , without the need for a Monte Carlo sampling procedure from the metamodel (as 36 Vi VTi 37 is used in most metamodel-based sensitivity analyses). The details of these integrals, which are 38 39 40 quite complex, are left to [7]. All the quantities of interest presented here are calculated using the 41 42 software package Gaussian Emulation Machine for Sensitivity Analysis (GEM-SA) [21]. 43 44 Note that a number of other approaches to sensitivity analysis exist in the literature that 45 46 use GPs and other emulators to perform uncertainty and sensitivity analysis. For example, the 47 48 method presented in [22] considers multi delity computer codes and is implemented in the R package 49 50 "sensitivity". In addition, the tgp package [23] oers a greater exibility by generalising GPs using 51 52 regression trees, allowing emulation of nonstationary models and bifurcating responses [24]. However 53 54 for the purposes of this work, the more standard GP was considered su cient, given that there 55 56 is no particular reason to suspect bifurcations in the mode here. This assumption appears to be 57 58 justi ed by the cross-validation results in Section V A. 59 60 19 Review copy- Do not distribute Submitted to Journal of Aircraft for Review Page 20 of 40 1 V. Sensitivity Analysis of Flight Parameter Sensor Simulation Model 2 3 The Bayesian sensitivity analysis technique described in Section IV has been used to examine 4 5 6 the sensitivity of models in a variety of disciplines including: Bouc-Wen model of hysteresis [25], soil- 7 8 vegetation-atmospheric transfer [19], nuclear radiation releases [14], vehicle crashes, spot welding 9 10 [26] and the aortic valve [13]. The Bayesian sensitivity analysis technique is considered to be well- 11 12 tested, robust and useful [19]. 13 14 Using this Bayesian sensitivity analysis technique, a number of ight parameters are varied 15 16 in the FPSS model, described in Section III, to gain an understanding of the sensitivity of the 17 18 input ight parameters to the dierence in the `actual' and `simulated' loads, calculated as Mean- 19 20 Square Error (MSE),For due to the signalPeer processing Review in the Sensor and Data Acquisition model and 21 22 the assumptions and inaccuracies in the Hard Landing Analysis Process model. This is a novel 23 24 application of the Bayesian sensitivity analysis technique to an aircraft landing gear model and it 25 26 is the rst time the MSE has been used to quantify the conservatism in a model. 27 28 Figure 7 provides a summary of the methodology followed in conducting the sensitivity analysis 29 30 on the FPSS model using GEM-SA in this paper. The Bayesian sensitivity analysis technique in 31 32 GEM-SA is comprised of two stages: the rst involves creating an emulator of the FPSS model by 33 34 tting a GP to the response surface using data from multiple runs of the model as dictated by a 35 36 Design Of Experiments (DOE) so that the output of the FPSS model can be predicted for any point 37 38 in the input space without having to run the simulation. The second stage involves representing 39 40 each input ight parameter as a probability distribution and using the emulator built in the rst 41 42 stage to infer sensitivity analysis data. 43 44 For symmetric, two-point landings, the nine input ight parameters to the `Actual' Landing 45 46 model include: , , , , Port SA (Footnote: For symmetric landings, the starboard MLG 47 θ VAx VAz µlong 48 49 mirrors the port MLG therefore only the port MLG servicing state is speci ed.), tire, tire press, 50 51 mass and CG. For asymmetric, two-point landings, the 12 input parameters to the `Actual' Landing 52 model include: , and , , ,( ), , , Port SA, Starboard SA, and CG. 53 φ θ ψ VAx VAy VAz µlong µlat mass 54 55 The input parameters tire and tire press, that were considered in the symmetric landings, are not 56 57 considered for the asymmetric landings because lateral tire data was only available for one tire at 58 59 60 20 Review copy- Do not distribute Page 21 of 40 Submitted to Journal of Aircraft for Review 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 For Peer Review 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 Fig. 7 Methodology Followed in Conducting the Sensitivity Analysis on the FPSS model using 41 42 GEM-SA, after [19] 43 44 45 one tire pressure and therefore it was not possible to consider the other tire types or tire pressures. 46 47 In order to estimate the sensitivity measures described in Section IV, the probability distribu- 48 49 tions for the input parameters are de ned. The assumption was made that the inputs are indepen- 50 51 dent, although in reality ight parameters such as pitch and ground speed are not independent and 52 53 ight parameters such as roll and yaw may be coupled. But given the limited range considered at 54 55 landing, they are relatively independent. The parameters can be speci ed as either Gaussian or uni- 56 57 formly distributed based on how informative the available input parameter data are. In this study, 58 59 60 21 Review copy- Do not distribute Submitted to Journal of Aircraft for Review Page 22 of 40 1 the distributions have been de ned as uniform distributions and the ranges of the ight parameters, 2 3 normalized between zero and one, are based on typical aircraft operating limitations. 4 5 6 To develop the emulator, 400 combinations of input parameters were generated using a maximin 7 8 Latin Hypercube DOE in GEM-SA. The FPSS model was then run to provide the corresponding 9 10 `actual' and `simulated' landing gear outputs calculated at spin-up, spring-back and maximum 11 12 vertical reaction. The outputs of interest are: spin-up and spring-back drag axle response load 13 14 MSE, spin-up and spring-back bending moment MSE and maximum vertical reaction vertical axle 15 16 response load MSE. The sensitivity analysis was carried out in GEM-SA. 17 18 For symmetric landings, the port and starboard MLG give the same results, therefore only the 19 20 port MLG results areFor presented, however,Peer for asymmetric Review landings, the port and starboard MLG 21 22 provide dierent results, therefore both MLG were considered in the sensitivity analysis. 23 24 25 26 A. Emulator Accuracy 27 28 For each sensitivity study, the emulator is built on the rst 80% of the training data and the 29 30 accuracy of the emulator is evaluated using the remaining 20% of the training data. Since the 31 32 emulator calculates a mean function, which passes through the outputs and also quanti es the 33 34 remaining uncertainty due to the emulator being an approximation of the true model, the emulator 35 36 accuracy is evaluated graphically using the emulator predictions and their 95% con dence bands, as 37 38 well as the model output data (data not used in training). Figure 8 and Figure 9 shows an example 39 40 of the emulator accuracy for the asymmetric landing port and starboard spin-up drag axle response 41 42 load MSE. The model test data tended to be within the 95% con dence bands of the predictions 43 44 and errors could be attributed to predicting high values of MSE since there are fewer training points 45 46 for the emulator in these regions. 47 48 GEM-SA also provides other statistical measures of the emulator accuracy, including rough- 49 50 ness" values for the input ight parameters, a σ2 value and cross-validation root mean square 51 52 (RMS) error. The roughness values, related to the hyperparameter B, estimate the smoothness 53 54 of the model inputs and describes how quickly the output responds to changes in each input [19]. 55 56 Roughness values greater than one indicate non-linear relationships between the inputs and outputs, 57 58 59 60 22 Review copy- Do not distribute Page 23 of 40 Submitted to Journal of Aircraft for Review 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 For Peer Review 21 22 23 24 Fig. 8 Asymmetric Port MLG Spin Up Drag Axle Response Load Emulator Accuracy 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 Fig. 9 Asymmetric Starboard MLG Spin Up Drag Axle Response Load Emulator Accuracy 49 50 51 while roughness values approaching 100 indicate discontinuities and suggest that the emulator is 52 53 not working. The σ2 value provides the variance of the emulator after standardizing the output and 54 55 provides a measure of the non-linearity in the emulator [19]. Finally, the cross-validation RMS error 56 57 is the square root of the mean square error of the emulator predictions at the training points [19]. 58 59 60 23 Review copy- Do not distribute Submitted to Journal of Aircraft for Review Page 24 of 40 1 The statistical measures of the emulator accuracy for the asymmetric landing port and starboard 2 3 MSE sensitivity analyses are presented in Table 2 and Table 3. Since the port and starboard MLG 4 5 2 6 emulators are trained with dierent data, the roughness values, σ values and RMS errors are 7 8 dierent. 9 10 Roughness values for all of the input parameters, for both the port and starboard MLG, are 11 12 below 10, except for the input parameter µlong, which had a roughness value of 30.42 for the 13 14 starboard spring-back drag axle response load MSE output. Therefore, the emulators generally 15 16 had a smooth response to variations in its input and are good approximations to the FPSS model. 17 18 While the roughness value for µlong is high, it does not suggest extremely non-linear or discontinuous 19 20 patterns. However it doesFor suggest thatPeerµlong is one Review of the sensitive inputs. Roughness values are also 21 22 greater than one in some cases for , e , e and , which indicates that these are the most φ VA VA mass 23 z y 24 sensitive input parameters. Roughness values are consistently less than one for input parameters 25 26 such as Port SA and Starboard SA, indicating linear relationships between the inputs and outputs. 27 28 The σ2 values for each of the SA are low and range from 0.77-2.70 for the port MLG and 0.76-2.83 29 30 for the starboard MLG, which means that the parameters only moderately deviate from linearity 31 32 [19]. The RMS error is not normalized but is expressed in terms of the output. Therefore, when the 33 34 RMS error is taken in context to the data in the emulator accuracy plots, it is acceptable. These 35 36 results suggests that the emulator is a good representation of the FPSS model. 37 38 39 40 B. Main Eects Plots 41 42 The main eects plots for the analysis show MSE versus the normalised ight parameters. The 43 44 lines represent mean main eects values, averaged over variations in the other parameters and can 45 46 be thought of as the expected value of the output with respect to one parameter if the true values 47 48 of the other parameters are known. These plots show which of the ight parameters the MSE is 49 50 signi cantly sensitive to and the nature of the input/output relationships. 51 52 The main eects plots for the port and starboard MLG are related such that parameters in 53 54 the aircraft x and z axes, such as θ, V , V , µ , Port SA, Starboard SA, mass and CG show 55 Ax Az long 56 the same trend for both landing gears. However, for parameters such as φ, ψ and V the main 57 Ay 58 59 60 24 Review copy- Do not distribute Page 25 of 40 Submitted to Journal of Aircraft for Review 1 2 Flight Parameter Roughness 3 Drag Axle Response Load Bending Moment Vertical Axle Response Load 4 Spin-Up Spring-Back Spin-Up Spring-Back Maximum Vertical Reaction 5 1.25 2.37 1.31 5.37 1.14 6 φ 7 θ 0.51 1.94 0.30 1.79 0.96 8 ψ 0.33 0.53 1.87 2.39 0.05 9 V 0.47 1.77 0.11 0.00 0.04 10 Ax V 1.18 0.80 3.29 5.01 0.58 11 Ay 1.47 3.38 0.79 0.69 5.29 12 VAz 13 µlong 2.52 9.95 1.08 0.63 0.29 14 0.96 0.73 0.81 0.86 0.09 15 µlat 16 Port SA 0.21 0.00 0.01 0.15 0.01 17 Starboard SA 0.03 0.19 0.01 0.25 0.02 18 mass 0.13 0.30 1.10 1.01 3.62 19 20 CG For 0.14Peer0.25 Review0.13 0.16 1.51 21 Fitted model parameters 22 σ2 0.91 0.77 1.76 1.26 2.70 23 Cross Validation Results 24 25 Cross Validation RMS Error [%] 106.86 138.26 248.22 239.51 17.37 26 Cross Validation RMS Standardized Error 1.33 1.49 1.27 1.33 1.13 27 28 29 Table 2 Asymmetric Port Main Landing Gear MSE Emulator Accuracy 30 31 eects plots are mirrored. Not surprisingly, the input parameters that were also investigated in the 32 33 symmetric landing sensitivity analysis (θ, V , V , µ , Port SA, mass and CG) show the same 34 Ax Az long 35 trends in the asymmetric landing sensitivity analysis main eects plots. This provides additional 36 37 38 veri cation that the FPSS model is performing correctly for asymmetric landings. 39 40 The symmetric landing main eects plots for drag axle response load MSE and bending moment 41 42 MSE at spin-up and spring-back show the same trends therefore only the spin-up drag axle response 43 44 load MSE main eects plots are shown in Figure 10. The symmetric landing maximum vertical 45 46 reaction vertical axle response load main eects plots are shown in Figure 11. The main eects 47 48 plot for tire was not illustrated because a discrete uniform distribution was assigned to each tire. 49 50 In GEM-SA, this was described by a continuous distribution and the main eects plots are not 51 52 meaningful. 53 54 The asymmetric landing main eects plots for port and starboard MLG drag axle response load 55 56 MSE, bending moment MSE at spin-up and spring-back, and maximum vertical reaction vertical 57 58 59 60 25 Review copy- Do not distribute Submitted to Journal of Aircraft for Review Page 26 of 40 1 2 Flight Parameter Roughness 3 Drag Axle Response Load Bending Moment Vertical Axle Response Load 4 Spin-Up Spring-Back Spin-Up Spring-Back Maximum Vertical Reaction 5 0.26 1.80 0.68 5.62 0.35 6 φ 7 θ 0.56 0.89 0.33 0.50 1.19 8 ψ 0.58 0.14 1.00 1.54 0.26 9 V 0.80 2.89 0.61 0.08 0.26 10 Ax V 1.00 0.33 2.86 2.53 0.27 11 Ay 1.58 6.03 1.31 0.20 3.28 12 VAz 13 µlong 3.26 30.42 1.53 0.95 0.11 14 1.32 0.18 1.08 0.38 0.13 15 µlat 16 Port SA 0.00 0.00 0.00 0.03 0.28 17 Starboard SA 0.08 0.07 0.18 0.04 0.14 18 mass 0.27 0.23 0.42 1.53 5.27 19 20 CG For 0.23Peer0.22 Review0.08 0.08 0.59 21 Fitted model parameters 22 σ2 0.90 0.76 1.62 2.83 2.78 23 Cross Validation Results 24 25 Cross Validation RMS Error [%] 85.08 129.75 184.98 205.28 16.21 26 Cross Validation RMS Standardized Error 1.07 1.45 1.29 1.29 1.22 27 28 29 Table 3 Asymmetric Starboard Main Landing Gear MSE Emulator Accuracy 30 31 axle response load MSE show the same trends for the input parameters φ, ψ and V and µ . The 32 Ay long 33 spin-up drag axle response load MSE main eects plots for the asymmetric input ight parameters 34 35 are shown in Figure 12. 36 37 38 39 1. Aircraft Pitch Angle 40 41 The symmetric and asymmetric main eects plots show that the spin-up and spring-back drag 42 43 axle response load MSE and bending moment MSE increases as the pitch angle increases. Due to 44 45 ltering and sampling, the Sensor and Data Acquisition model tends to contribute an error of the 46 47 magnitude of less than one degree, therefore it is not expected that would have a large contribution 48 θ 49 to the MSE. Part of the relationship between and MSE can be attributed to the constraint in 50 θ 51 52 the model that limits the pitch angle to greater than 0 degrees to ensure a two-point landing. If 53 54 this constraint is removed, the relationship between θ and MSE tends to be more constant. The 55 56 main eects plot for the vertical axle response load MSE also shows a constant relationship with θ. 57 58 Section V C will show that the contribution to the MSE from θ alone is low, except in the case of 59 60 26 Review copy- Do not distribute Page 27 of 40 Submitted to Journal of Aircraft for Review 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 For Peer Review 21 22 23 24 Fig. 10 Symmetric Port MLG Spin Up Drag Axle Response Load Main Eects Plots 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 Fig. 11 Symmetric Port MLG Maximum Vertical Reaction Vertical Axle Response Load Main 49 50 Eects Plots 51 52 53 spring-back bending moment MSE. The ight parameter θ tends to only be signi cant in the other 54 55 cases when its interactions with other ight parameters are considered. 56 57 58 59 60 27 Review copy- Do not distribute Submitted to Journal of Aircraft for Review Page 28 of 40 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Fig. 12 AsymmetricFor Port MLG Peer Spin Up Drag Review Axle Response Load Main Eects Plots 21 22 23 2. Aircraft Longitudinal Velocity 24 As increases, the time required for spin-up, the maximum drag ground-to-load and the 25 VAx 26 27 vertical ground-to tire-load increases [27]. While is an input to the `Actual' Landing model, VAx 28 29 ground speed is output by the Sensor and Data Acquisition System model and is typically used in 30 31 a Hard Landing Analysis Process model. Ground speed is the resultant magnitude of the velocity 32 33 component parallel to the earth's surface and therefore it is a function of the longitudinal and lateral 34 35 velocities. While in symmetric landings, the ground speed is only a function of VA since there is 36 x 37 no lateral velocity, in asymmetric landings, it is a function of VA and VA . In the Sensor and 38 x y 39 Data Acquisition System model, ltering and sampling contributes an error of the magnitude of less 40 41 than 1 m/s. This is consistent with the fact that ground speed is generally assumed to be constant 42 43 in landing simulations. The main eects plots for V show that as V increases, the dierence 44 Ax Ax 45 between the `actual' and `simulated' loads and moments do not increase linearly. Section V C will 46 47 show that the contribution to the MSE from V alone is low and V is only signi cant for spin-up 48 Ax Ax 49 and spring-back when its interactions with other ight parameters are considered. 50 51 52 53 3. Aircraft Vertical Descent Velocity 54 55 The relationship between the MSE and V is nonlinear and tends to increase and level o 56 Az 57 at high vertical descent velocities. Figure 13 illustrates the dierence between the peak aircraft 58 59 60 28 Review copy- Do not distribute Page 29 of 40 Submitted to Journal of Aircraft for Review 1 vertical acceleration ( ) in the `actual' landing and the peak from the Sensor and 2 VRTG VRTG 3 Data Acquisition System model. At higher vertical descent velocities (and hence higher vertical 4 5 6 accelerations), the possibility increases that the peak vertical acceleration will be missed on the 7 8 FDR, due to low sampling rates (8 Hz) in conjunction with greater peak amplitudes. Therefore, 9 10 the `simulated' loads will be more under predicted as vertical descent velocity increases and the 11 12 dierence between the `actual' and `simulated' loads will be greater. However, conservatism in 13 14 other ight parameters, such as µlong, ensure that the landing gear loads are conservative. 15 As discussed in Section II, is an input ight parameter, however the aircraft vertical accel- 16 VAz 17 18 eration measured by the FDR model is matched in the current hard landing analysis process model 19 20 by iterating VA . In theFor FDR model, Peer the vertical accelerationReview is sampled and ltered and therefore z 21 22 any loss of data will have a signi cant impact on the landing gear loads modelled in the current hard 23 24 landing analysis process model. Therefore, the vertical acceleration is one of the most important 25 26 parameters in reducing the MSE. 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 Fig. 13 Dierence in Peak Aircraft Vertical Acceleration as a Function of Vertical Descent 47 48 Velocity due to 8Hz Sampling Rate 49 50 51 52 53 4. Longitudinal & Lateral Tire-Runway Friction Coe cient 54 55 Figure 10 illustrates the relationship between MSE and µlong for spin-up and spring-back drag 56 57 axle response load bending moment. As the actual µlong for the landing approaches the value 58 59 60 29 Review copy- Do not distribute Submitted to Journal of Aircraft for Review Page 30 of 40 1 assumed within the hard landing analysis process, the dierence between the `actual' and `simulated' 2 3 loads is reduced. This is logical considering that if the `actual' landing has a tire-friction coe cient 4 5 6 that is signi cantly lower than that assumed in the hard landing analysis process model, this will 7 8 greatly contribute to the MSE. 9 10 There is a constant relationship between MSE and µlong for vertical axle response load at 11 12 maximum vertical reaction. In Section V C it is also shown that µlong has little contribution to the 13 14 vertical axle response load MSE. 15 16 The tire-runway interface is represented by a global friction potential delineated approximately 17 18 by a circle, and the drag and side loads share the potential available in the tire-runway interface [28]. 19 20 Therefore, in the caseFor of combined Peer slip in asymmetric Review landings, the drag and side ground-to-tire 21 22 load depend on µlong as well as µlat. As µlong and µlat increase (the radius of the `circle' increases), 23 24 the drag load and side ground-to-tire load increase. 25 26 Due to the bookcase modelling technique that is used in the Hard Landing Analysis Process 27 28 model and the relationship between the drag and side load and µlat, the drag axle response load 29 30 and bending moment main eects plots show that MSE tends to increase as µlat increases. The 31 32 MSE has a constant relationship with µ for the maximum vertical reaction vertical axle response 33 lat 34 load MSE. 35 36 The MEI and TEI in Section V C show that µ is a very signi cant ight parameter. In its 37 long 38 own right, only tends to be signi cant when its interactions with other ight parameters are 39 µlat 40 considered. 41 42 43 44 5. Main Landing Gear Shock Absorber Servicing State 45 46 In all of the main eects plots, the MLG shock absorber servicing state has a constant relation- 47 48 ship with MSE. Therefore, the MSE is not sensitive to the shock absorber servicing state over its 49 50 range and any shock absorber servicing state will give similar MSE values. 51 52 In the case of a symmetric landing, where the port and starboard MLG have the same servicing 53 54 states, when the shock absorber is overin ated, higher loads are transmitted through the landing 55 56 gear structure. However, because the shock absorber is stier (stier spring curve), there is a higher 57 58 59 60 30 Review copy- Do not distribute Page 31 of 40 Submitted to Journal of Aircraft for Review 1 aircraft vertical acceleration for the same vertical descent velocity. In the hard landing analysis 2 3 process model used in this study, the peak vertical aircraft acceleration is matched by iterating 4 5 . Therefore, if the correctly serviced shock absorber is used in the hard landing analysis process 6 VAz 7 model while in the `actual' case it is overin ated, a higher will be required to match the 8 VAz 9 10 vertical acceleration. Due to the fact that the vertical acceleration is being matched, the dierence 11 12 between the vertical and drag axle response loads and the bending moment from the landing with 13 14 an overin ated shock absorber and from the landing with a correctly serviced shock absorber will 15 16 be very similar. 17 18 When the shock absorber is underin ated, lower loads are transmitted through the landing gear 19 20 structure and there isFor a lower aircraft Peer vertical acceleration Review for the same vertical descent velocity. 21 22 Therefore, if a correctly serviced shock absorber is assumed, meanwhile in the `actual' case it 23 24 is underin ated, the aircraft vertical acceleration will be higher and the loads in the `simulated' 25 26 landing will be higher and therefore, more conservative. 27 28 In the asymmetric landings, the port and starboard MLG shock absorber servicing states were 29 30 altered independently and the analysis process did not correct for the mis-serviced shock absorber. 31 32 However, the main eects plots show that the MSE is constant over the range of the port and 33 34 starboard MLG servicing states, and the MEI and TEI, discussed in Section V C, indicate that 35 36 the shock absorber servicing state is not signi cant. If another hard landing analysis process was 37 38 introduced, that did not match the aircraft vertical acceleration, then an accurate assessment of the 39 40 shock absorber servicing state may be more important. 41 42 43 44 6. Main Landing Gear Tire Pressure 45 46 In all of the symmetric and asymmetric main eects plots, the MLG tire pressure has a constant 47 48 relationship with MSE. This indicates that the MSE is not sensitive to tire pressure over its range 49 50 and any value of the tire pressure will give similar output values. Therefore, knowing the tire 51 52 pressure is not useful in reducing the MSE. 53 54 55 56 57 58 59 60 31 Review copy- Do not distribute Submitted to Journal of Aircraft for Review Page 32 of 40 1 7. Aircraft Mass & Centre of Gravity 2 3 For symmetric and asymmetric landings, the main eects plots for drag axle response load and 4 5 6 bending moment at spin-up and spring-back show that the MSE is generally constant over the range 7 8 of mass and the CG. However, the main eects plots for the maximum vertical reaction vertical axle 9 10 response load MSE shows a non-linear relationship with mass and CG. As expected, as the mass 11 12 and CG in the `actual' landing moves further from the mass and CG in the `simulated' landing, the 13 14 MSE increases. This trend is more prevalent in the main eects plots for the vertical axle response 15 16 load MSE than for spin-up and spring-back drag and bending moment MSE and is expected since 17 18 the MEI and TEI, presented in Section V C, indicate that mass and CG are very signi cant to the 19 20 vertical axle responseFor load. Peer Review 21 22 23 24 8. Aircraft Lateral Velocity 25 26 The main eects plots show that the dierence between the `actual' and `simulated' loads 27 increases on the starboard MLG as increases. As described in Section II, in landings with 28 VAy 29 with that acts in the starboard direction, the MLG wheels touchdown at the same time and 30 VAy 31 32 the starboard MLG has a higher vertical ground-to-tire load than the port MLG. As the initial 33 34 aircraft lateral velocity and therefore lateral acceleration increases, the magnitude of the vertical 35 36 ground-to-tire load increases on the starboard MLG, and decreases on the port MLG. However, the 37 38 drag ground-to-tire load and axle response loads do not signi cantly change as the lateral velocity 39 40 increases. In the Hard Landing Analysis Process model, described in Section III, the `simulated' 41 42 starboard vertical and drag loads were calculated using a bookcase method and the vertical and 43 44 drag loads are factored up as a function of the LATG. Therefore, the `simulated' drag load is being 45 46 conservatively overestimated in the Hard Landing Analysis Process model. The same trend is seen 47 48 with the bending moment MSE and vertical load MSE. These results are mirrored for the port 49 50 MLG, as illustrated in Figure 12. 51 52 53 54 9. Aircraft Roll Angle 55 56 In a landing con guration with the aircraft rolled with the port wing down (negative roll angle), 57 58 the aircraft rst lands on the port MLG and then on the starboard MLG, as illustrated in Section II. 59 60 32 Review copy- Do not distribute Page 33 of 40 Submitted to Journal of Aircraft for Review 1 As the initial aircraft roll angle increases, the time between the wheels touching down increases and 2 3 the build up of energy in the spin-up and spring-back is reduced. However, the spin-up and spring- 4 5 6 back drag axle response load and bending moment main eects plots show that as the negative 7 8 roll angle increases, the MSE increases. This is because the `simulated' drag load at spin-up and 9 10 spring-back is being conservatively overestimated in the Hard Landing Analysis Process model. The 11 12 same eect occurs on the starboard MLG if there is a positive roll angle. 13 14 15 10. Aircraft Yaw Angle 16 17 The main eects plot for spin-up and spring-back drag axle response load and bending moment 18 19 20 indicate that for theFor port MLG, Peer the MSE increases Review as the positive yaw angle increases. As de- 21 22 scribed in Section II, as the initial aircraft yaw angle increases, the port MLG spin-up and spring 23 24 back loads decrease. Therefore, the `simulated' port MLG spin-up and spring-back loads are being 25 26 conservatively overestimated in the Hard Landing Analysis Process model. 27 28 29 C. Main Eects Indices and Total Eects Indices 30 31 Table 4 provides the MEI and TEI for the asymmetric landing sensitivity analysis. For brevity, 32 33 only the asymmetric landing sensitivity study are given in tabular form, however the symmetric 34 35 36 landing sensitivity analysis results are discussed with the asymmetric landing sensitivity analysis 37 38 results. Input ight parameters with an MEI and TEI values above 10% are bold. 39 40 41 1. Spin-Up 42 43 a. Drag Axle Response Load: For both the symmetric and asymmetric landings, the MEI 44 45 illustrate that µ and V contribute the most signi cantly to the spin-up drag axle response 46 long Az 47 load MSE. The symmetric landing TEI show that along with and , and with their 48 µlong VAz VAx θ 49 50 interactions, contribute to the spin-up drag axle response load MSE. For asymmetric landings, φ 51 is also signi cant and is as important as . The asymmetric landing TEI show that , 52 VAy VAx θ ψ 53 54 and µlat and their interactions contribute to the drag axle response load MSE. The symmetric 55 landing rst order interactions account for 19.8% of the MSE. The interactions between - 56 µlong VAz 57 (11.38%), - (1.73%), - (1.68%) and - (1.87%) account for approximately 17% 58 µlong θ µlong VAx VAz VAx 59 60 33 Review copy- Do not distribute Submitted to Journal of Aircraft for Review Page 34 of 40 1 2 Flight Parameter Drag Axle Response Load Bending Moment Vertical Axle 3 4 Response 5 6 Load 7 Spin-Up Spring-Back Spin-Up Spring-Back Maximum Vertical Reaction 8 9 MEI [%] TEI [%] MEI [%] TEI [%] MEI [%] TEI [%] MEI [%] TEI [%] MEI [%] TEI [%] 10 11 θ 1.47 5.47 2.92 17.88 1.46 5.77 7.75 28.26 0.93 19.33 12 3.41 9.08 7.21 22.08 0.59 2.84 0.50 0.50 0.16 1.25 13 VAx 14 8.60 23.40 1.17 19.57 7.24 17.97 0.13 6.70 10.32 52.00 15 VAz 16 45.93 61.37 39.16 68.56 18.27 29.98 6.22 14.95 0.30 5.63 17 µlong 18 Port SA 0.06 1.23 0.05 0.06 0.10 0.26 0.06 2.12 0.15 0.30 19 20 Starboard SA For0.01 0.30 Peer0.14 2.13 Review0.05 0.23 0.36 5.24 0.03 0.62 21 22 mass 0.12 1.13 0.09 2.98 1.64 13.56 2.52 14.11 8.91 61.37 23 24 CG 2.15 3.42 0.11 2.93 0.93 2.79 0.08 1.59 0.98 26.94 25 2.77 9.86 0.97 7.75 8.37 39.92 0.67 42.36 2.07 11.56 26 VAy 27 28 φ 6.09 12.26 2.78 15.46 5.35 24.33 9.40 56.74 0.94 24.48 29 ψ 1.36 4.28 1.27 6.21 3.07 20.26 0.69 24.33 0.48 1.83 30 31 µlat 0.82 5.81 2.00 8.39 3.12 19.79 2.21 18.43 0.22 2.36 32 33 First order in- 18.84 19.99 27.84 36.12 48.66 34 35 teractions 36 37 Higher order 8.36 22.13 21.98 33.29 25.85 38 interactions 39 40 41 Table 4 Asymmetric Port Main Landing Gear MSE MEI and TEI Summarized Results 42 43 44 of the MSE. The asymmetric landing rst order interactions account for approximately 19% of the 45 46 e MSE. The greatest interactions are between µlong-V (7.27%), µlong-φ (1.00%), VA -VA (1.94%) 47 Az z x 48 and V -µ (1.06%) which account for approximately 11% of the MSE. 49 Ay lat 50 b. Bending Moment: The symmetric and asymmetric landing results are similar to the spin- 51 52 up drag axle response load MSE: µ and V are the most signi cant parameters. However, V 53 long Az Ay 54 and φ are also very signi cant and the asymmetric landing TEI indicate that along with V and φ, 55 Ay 56 and and their interactions contribute to the bending moment MSE. The asymmetric landing 57 ψ µlat 58 59 60 34 Review copy- Do not distribute Page 35 of 40 Submitted to Journal of Aircraft for Review 1 rst order interactions account for approximately 27% of the MSE. The interactions between - 2 µlong 3 (3.00%), - (4.57%), - (2.57%), - (3.62%), - (1.36%) and - (1.06%) 4 VAz VAy φ VAy ψ VAy µlat φ ψ ψ µlat 5 6 account for approximately 16% of the MSE. 7 8 c. Findings: In both the symmetric and asymmetric landings, the spin-up drag axle response 9 load MSE and bending moment MSE may be reduced by learning the true value of and . 10 µlong VAz 11 12 However, the asymmetric landings show that to reduce the drag axle response load MSE and bending 13 moment MSE, it is also important to know the true value of and , and that and have 14 VAy φ ψ µlat 15 16 signi cant interactions with those parameters. 17 18 19 20 2. Spring-Back For Peer Review 21 22 a. Drag Axle Response Load: The symmetric and asymmetric landing MEI and TEI show 23 similar results to the spin-up drag axle response load: , , , and are the most 24 µlong VAz VAx θ φ 25 signi cant parameters, while , and are also important parameters. However, the TEI 26 VAy ψ µlat 27 28 also shows that tire and its interactions have an eect on the the spring-back drag axle response 29 30 load MSE. The following rst order interactions account for approximately 18% of the MSE in the 31 symmetric landings: - (9.14%), - (2.94%), - (2.84%), -tire (2.0%) and 32 µlong VAz µlong VAx µlong θ µlong 33 34 -tire (1.19%). In the asymmetric landings, rst order interactions account for approximately VAz 35 36 20% of the MSE. The following interactions account for approximately 10% of the MSE: - µlong VAz 37 38 (4.04%), - (3.54%), - (2.15%). In the symmetric landing sensitivity study, was µlong θ µlong VAx tire 39 40 an important parameter and although it was not considered in the asymmetric landing sensitivity 41 42 study, by similarity tire is also an important parameter for asymmetric landings. Therefore, µlong, 43 44 VA , φ and tire are important parameters in the spring-back drag axle response load MSE. 45 z 46 b. Bending Moment: Like the spring-back drag axle response load MSE, µ , V , V , 47 long Az Ax 48 θ and tire contribute signi cantly to the spring-back bending moment MSE. In the symmetric 49 50 landings, the following interactions account for approximately 23% of the MSE: µ -V (5.16%), 51 long Az 52 µ -θ (13.31%), µ -mass (2.04%) and V -θ (2.34%). Therefore, mass is also a signi cant 53 long long Ax 54 parameter for spring-back bending moment MSE. For asymmetric landings, is also important and 55 φ 56 the following rst order interactions account for approximately 23% of the MSE: - (2.88%), 57 µlat θ 58 59 60 35 Review copy- Do not distribute Submitted to Journal of Aircraft for Review Page 36 of 40 1 V -φ (12.95%), θ-φ (2.27%), φ-ψ (3.78%) and φ-µ (1.43%). Therefore µ , V , tire, mass and 2 Ay lat long Az 3 are the important parameters for the spring-back bending moment MSE. 4 φ 5 6 c. Findings: In order to reduce the spring-back drag axle response load MSE and bending 7 moment MSE, the true value of , , , , mass and tire must be learned. 8 µlong VAz φ VAy 9 10 11 3. Maximum Vertical Reaction 12 13 a. Vertical Axle Response Load: The symmetric and asymmetric landing MEI illustrate that 14 15 mass and V contribute signi cantly to the vertical axle response load MSE. The symmetric 16 Az 17 landing TEI show that along with mass and V e , CG, θ and tire with their interactions, contribute 18 Az 19 20 to the vertical axle responseFor load MSE.Peer The symmetric Review landing rst order interactions account for 21 approximately 49% of the MSE. The interactions between mass- (15.38%), mass-CG (13.63%), 22 VAz 23 mass-θ (4.10%), V e -CG (4.02%), V -θ (2.32%), mass-tire (1.92%) account for approximately 41% 24 Az Az 25 26 of the MSE. The asymmetric landing TEI show that φ and its interactions also contributes to the 27 28 vertical axle response load MSE. The asymmetric landing rst order interactions also account for 29 approximately 49% of the MSE. The interactions between - (2.22%), -mass (12.87%), - 30 VAz θ VAz VAz φ 31 32 (2.19%), θ-mass (3.49%), mass-CG (7.92%) and mass-φ (7.01%) account for approximately 36% of 33 34 the MSE. 35 36 b. Findings: The maximum vertical reaction vertical axle response load MSE may be reduced 37 38 by learning the true value of , , , mass, CG. From the symmetric landing sensitivity study, µlong VAz φ 39 40 it was also seen that tire was an important parameter to learn in order to reduce the MSE. 41 42 43 44 VI. Summary of Signi cant Flight Parameters 45 46 A summary of the signi cant ight parameters is presented in Table 5. Each parameter is rated 47 48 as High (H), Medium (M) or Low (L). High identi es ight parameters that greatly in uence the 49 50 MSE, Medium identi es ight parameters that have some in uence on the MSE and Low identi es 51 52 ight parameters that have little or no in uence on the MSE. 53 54 55 56 57 58 59 60 36 Review copy- Do not distribute Page 37 of 40 Submitted to Journal of Aircraft for Review 1 2 Flight Parameter Signi cance (H/M/L) 3 4 φ H 5 6 θ M 7 H 8 ψ 9 V M 10 Ax 11 (LATG ) H VAy 12 (VRTG ) H 13 VAz 14 15 mass M 16 17 CG M 18 tire M 19 20 Fortire Peer press ReviewL 21 22 Port SA L 23 24 Starboard SA L 25 26 µlong H 27 H 28 µlat 29 30 Table 5 Summary of the Signi cant Flight Parameters in Reducing the MSE 31 32 33 VII. Conclusions 34 35 The purpose of this paper was to investigate the sensitivity of input parameters varied in 36 37 the FPSS model to the dierence between the `actual' and `simulated' loads in symmetric and 38 39 asymmetric, two-point landings, calculated as MSE. The sensitivity analysis provided the following 40 41 conclusions: 42 43 44 Longitudinal runway friction coe cient, aircraft vertical descent velocity, aicraft lateral ve- 45 • 46 locity and aircraft roll angle contributed the most to the spin-up and spring back drag axle 47 48 response load MSE and bending moment MSE. 49 50 51 • Aircraft vertical descent velocity, roll angle, mass, centre of gravity position and MLG tire 52 53 type had signi cant in uences on the maximum vertical reaction vertical axle response load 54 55 MSE. 56 57 58 While and did not change considerably from the `actual' to the `simulated' landing, the • VAx θ 59 60 37 Review copy- Do not distribute Submitted to Journal of Aircraft for Review Page 38 of 40 1 interactions with µ and V contributed to the MSE in all cases. 2 long Az 3 4 • The ight parameters ψ and µ are only signi cant when their interactions are considered. 5 lat 6 7 • Due to the Hard Landing Analysis Process modelling technique, VRT G is as signi cant as 8 9 , and LATG is as signi cant as , in reducing MSE. VAz VAy 10 11 12 • It was also shown that over the range in this sensitivity study, and due to the modelling 13 14 techniques used in the Hard Landing Analysis Process model used in this study, the shock 15 16 absorber servicing state and MLG tire pressure do not contribute signi cantly to the MSE 17 18 and learning the true value of these ight parameters would not reduce the MSE. 19 20 For Peer Review 21 For symmetric and asymmetric two-point landings, the MSE can be reduced by learning the 22 true value of the following ight parameters: , VRTG (related to ), LATG (related to ), 23 µlong VAz VAy 24 25 φ, ψ, mass, CG and tire. 26 27 28 29 Acknowledgments 30 31 The authors would like to thank the following colleagues at Messier-Bugatti-Dowty for their 32 33 invaluable assistance in the model development and sensitivity study analysis: Laura Collett, Sidney 34 35 Smith and Andrew Thomas. 36 37 The authors would also like to thank Dr. Marc Kennedy of the Probability and Statistics 38 39 Department at the University of She eld for the use of GEM-SA. 40 41 This paper would not have been possible without the nancial support of Messier-Bugatti-Dowty 42 43 and the Knowledge Transfer Partnerships (KTP) program through partnership number 1216. 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