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With the development of high-performance aircraft, precise air data are necessary to complete challenging tasks such as flight maneuvering with large angles of attack and high speed. As a result, the flush air data sensing system (FADS) was developed to satisfy the stricter control demands. In this paper, comparative stuides on the solving model and algorithm for FADS are conducted. First, the basic principles of FADS are given to elucidate the nonlinear relations between the inputs and the outputs. Then, several different solving models and algorithms of FADS are provided to compute the air data, including the angle of attck, sideslip angle, dynamic pressure and static pressure. Afterwards, the evaluation criteria of the resulting models and algorithms are discussed to satisfy the real design demands. Futhermore, a simulation using these algorithms is performed to identify the properites of the distinct models and algorithms such as the measuring precision and real-time features. The advantages of these models and algorithms corresponding to the different flight conditions are also analyzed, furthermore, some suggestions on their engineering applications are proposed to help future research.

The design concept of the flush air data sensing system (FADS) was presented by the American National Aeronautics and Space Administration in the 1960s in order to meet the control requirements of the space shuttle [

The current studies on FADS focus on the optimized layout of the pressure taps, the solving model of the air data, the fault detection and reconstruction of the system, the compensation and calibration of air data and so on. In particular, a neural network method was applied in [

This paper studies comparatively the solving models and algorithms for FADS. Accordingly, there are three aspects which need to be considered. The first question involves the inherent theories of FADS to manifest the relations between the pressure values and air parameters. The second problem relates to the several different solving models and algorithms of FADS which provide useful tools to estimate the necessary air parameters. The third aspect deals with the comparative inverstigation regarding these models and algorithms of FADS in combination of the evaluation criteria, and further by doing the contrastive simulation the results reveal the application characteristics for these distinct models and algorithms.

The work theories of FADS embody that the flight parameters are solved in terms of the pressures measured by the build-in sensors on the vehicle surfaces, whereas these sensors are installed in the given way in relation to the special task [

In _{c}, P_{∞} denote respectively the dynamic pressure and static pressure. Theoretically, the relations of the dynamic pressure q_{c} and the static pressure _{∞} can be decided by the isentropic flow principle, and it is expressed by:

Furthermore, the pressure coefficient is defined by:

For any point of the vehicle surface, the airflow angle of incidence _{i}_{e}_{e},

Accordingly, the shaped pressure coefficient _{∞} denotes the flight Mach, and _{e}, β_{e}_{∞}, but the special relationship among them is difficult to acquire. On the other hand, this coefficient is critical for FADS to compute the air parameters in terms of the measured pressure values. In fact, _{e}, β_{e}, q_{c}_{∞}. In principle, as soon as there are four pressures acquired on the vehicle surface, these variables can be solved accordingly.

According to the above relations between the measuring pressures and the flight parameters, we find that it is difficult to obtain the analytical solutions for the corresponding model function due to the nonlinear features. As a result, some solving means are proposed for

Based on the solving model of FADS in

The linearization model of the above equation can be obtained using the Taylor's expansion mean, and it is expressed by:
^{j}, β^{j}, q_{c}^{j}, P_{∞}^{j}

For _{i}_{i}

Repeating the above steps, we can get the resulting solutions with respect to _{e}, β_{e}, q_{c}_{∞} can be estimated accordingly.

The air data can be obtained using the three-point method because some special pressure taps can be selected to simplify the FADS model in _{α}

We see that _{c}_{∞}. Especially, based on

The solution of the least squares with regard to

Accordingly, the shaped pressure coefficient _{1}_{e}, β_{e}, M_{∞}. Nevertheless, the acquisition of the according solutions is difficult because the relationship among them is strong nonlinear. As a result, the polynomial fitting method is applied to seek this nonlinear relation, and it is provided by:
_{0}_{m}_{∞}), _{1}_{∞}),…, _{n}_{∞}) are the polynomial coefficients related with the flight Mach. Also,

As mentioned above, we know that the expressions between the measuring pressures _{i}_{e}, β_{e}, q_{c}, P_{∞} are difficult to build, but on the other hand their relations can be identified depending on the experimental datum acquired by the tools of computational fluid dynamics. Moreover, the neural network is very suitable to establish such nonlinear connections, and the corresponding structure diagram of the solving algorithm using the neural networks is proposed as shown in the following figure.

In _{e}_{e}_{c}_{∞} can be computed using the module of the neural network, as a result that this can effectively avoid the complex iteration computation for them. However, these designed neural networks will depend on the large amounts of data, if they are combined with other methods such as the three-point method, the acquired training data may be reduced accordingly, and simultaneously the solving process will be faster due to decreasing the iterative steps [_{e}_{e}_{e}_{∞})_{1}, …, (_{e}, M_{∞})_{κ}_{1}_{i}

The core idea of the look-up method is to seek the current air parameters according to predefined databases. Once the pressures of the measuring taps are acquired, the required air parameters can be found directly using these databases. The advantage of this method lies in the rapid solution speed due to the direct look-up course, but on the other hand large amounts of data are required to ensure the system accuracy. For this reason, this method is used with combination of other means, for example, the Mach number is obtained by the inertial navigation system first, and then the angle of attack and the sideslip angle are obtained using the look-up method. Furthermore, the angle of attack can be approximately considered as the proportional relation to the pressure difference, shown as [_{M}_{1}_{3}_{1},…, [_{1}_{3}_{ê}_{M}_{M}_{ê}

Furthermore, with consideration of the

After averaging these angles of attack, the real angle is estimated by:

In turn, the sideslip angle can also be acquired while identifying the relations between the measuring pressures and the sideslip angles. On the whole, the crucial task of the look-up method is to seek feasible rules corresponding to the inputs and outputs.

The above FADS solving models have their respective advantages, thus the selection of the feasible model is important in the practical application. Normally, the evaluation criteria of the solving algorithm include convergence, accuracy, real-time and so on. In particular, the overall layout of the pressure taps has significant impact on the applicability of these solving models, while the number of the pressure taps is connected with the measuring accuracy from the perspective of the system redundant [_{ε}_{∞}), then this yields:

According to

Substituting

From _{∞}) ≤ 1,

Furthermore, the error percentages corresponding to the dynamic pressure and the static pressure are expressed by:
_{ctrue}_{ctrue}_{α}(1)_{ctrue}_{ctrue}

Beyond these, the real-time characteristics with regard to these algorithms need to be taken into account. In common, the iterative process will bring the unfavorable time delay, and thus the operation speed of the solving model based on the neural network or the look-up methods is faster than that based on the three-point method. However, there is the fundamental contradiction between the real-time feature and solving precision, so a compromise is necessary for the performance evaluation of the solving model. In addition, the selection of the initial values will have significant effects on the real-time characteristics because the iteration process will stop rapidly as the initial values are given well.

The shaped pressure coefficient _{∞} = 2.2, 2.5, 2.75, 3, can be acquired by using the CFD tool, so the change curves of the shaped pressure coefficient ε are provided as follows.

From

To be specific, the solving algorithm based on the neural network can ensure the robustness of the resulting calculation as the training model is identified. This is because that such a built model is obtained using the large amount of sampling data, so several inaccurate samples may have less impact on the solving accuracy. In principle, system robustness can be improved if more sampling data can be provided. Nevertheless, training more sampling data requires more time, leading to the real-time reduction. As a result, getting the proper amount of exact samples will be crucial for the neural network algorithm to guarantee system robustness, as well as the computation efficiency. On the other hand, the following solving process is subjected to this trained model, so the inputs lying in the range of the samples will result in the more accurate results, whereas the deviation from the sample range will deteriorate the solving precise. Therefore, the solving algorithm based on the neural network should be adopted in the vicinity of the samples input range. Furthermore, the real-time characteristics are considered in the simulation, and the durations of the solving process using these algorithms are shown in

This paper deals with comparative studies of the different solving models for FADS. First, the basic connection between the measuring pressures and flight parameters is given to demonstrate the strong nonlinear features among them. Then, the solving models and algorithms of FADS are provided using the least squares method, three-point method, neural network method and look-up method. Afterwards, the evaluation criteria of these models and algorithms are introduced for FADS. Furthermore, simulation work is conducted to comparatively analyze the feasibility of these FADS solving models. We believe the work in this paper will provide the helpful information for FADS studies to meet the complicated task demands in the future.

This work is supported by Natural Science Foundation of Jiangsu Province under Grant No. BK20130817; China National Overseas Fund under Grant No. 201203070130; Nanjing University of Aeronautics and Astronautics (NUAA) research funds under Grant No. NS2014088.

The authors declare no conflict of interest.

The cone angle

The angle bias of attack

The airflow angle of incidence

The shaped pressure coefficient

_{c}

The dynamic pressure

_{∞}

The static pressure

The specific heat coefficient

_{e}

The angle of attack

_{e}

The angle of sideslip

The angle bias of attack

The angle bias of sideslip

The weighting factor matrix

_{ctrue}

The true value of the dynamic pressure

_{ctrue}

The true value of the static pressure

_{q}

The error percentage of the dynamic pressure

_{p}

The error percentage of the static pressure

_{α}

The number of the gotten angle of attack

_{∞}

The flight Mach

Conical shape applied for FADS.

Structure diagram using neural networks for FADS.

Change curves of shaped pressure coefficient.

Change curves of angle errors of attack using least squares method.

Change curves of angle errors of attack using three-point method.

Change curves of angle errors of attack using neural network.

Change curves of angle errors of attack using look-up method.

Comparative results of different solving methods.

^{−3}) | ||||||
---|---|---|---|---|---|---|

Least Squares Method | Without calibration | 1.7004 | 0.8497 | 1.7894 | 1.7894 | 0.9089 |

With calibration | 0.1004 | 0.0838 | 0.4297 | 0.4305 | 0.2434 | |

| ||||||

Three-point Method | Without calibration | 1.6244 | 0.8901 | 1.7894 | 1.7893 | 0.8474 |

With calibration | 0.0877 | 0.1049 | 0.4250 | 0.4291 | 0.2356 | |

| ||||||

Neural Network | 0.0395 | 0.0971 | 0.0113 | 6.717 × 10^{−11} |
0.3090 | |

| ||||||

Look-up Method | 0.1626 | 0.1649 | 0.6432 | 0.6834 | 0.2845 |

Durations of solving process with regard to different algorithms.

271.5641 | 145.5505 | 5.1184 | 0.1132 |