A high-resolution digital elevation model (DEM) is an important element that determines the performance of terrain referenced navigation (TRN). However, the higher the resolution of the DEM, the bigger the memory size needed for storing it. It is difficult to secure such large memory spaces in small, low-priced unmanned aerial vehicles. In this study, a high-precision terrain regression model to fit the DEM is generated using the extreme learning machine technique based on the multilayer radial basis function. The TRN results using the proposed method are compared with existing studies on various DEM fitting methods. This study verifies that the proposed method obtains improved fitting accuracy and TRN performance over existing DEM fitting methods such as bilinear interpolation, SVM for regression, and bi-spline neural network, without the DEM storage space.

Terrain referenced navigation (TRN) is a navigation technology that estimates the position of an aircraft by comparing the altitude measured by an altimeter with the digital elevation model (DEM). Currently, the global navigation satellite system (GNSS) is commonly used for precise navigation [

The DEM is a map that stores the elevation values of terrain to represent a terrain’s surface in three dimensions, and a high-resolution DEM is essential for improving the performance of TRN. However, large amounts of memory space are needed to store a high-resolution DEM and the loading time of the DEM in a separate storage space interferes with real-time computation. In addition, various interpolation methods are being used to obtain terrain elevation at the desired location using a discontinuous DEM. This method also generates additional computational errors, which can be larger as the resolution of the DEM decreases. To alleviate this problem, studies are continuously being conducted on devising a continuous terrain elevation model through a regression method based on the machine learning technique [

Even with the advantage of not needing large memory space, unless it has high levels of fitting accuracy that can replace the DEM, it cannot be applied to TRN. Moreover, when fitting a DEM as a complex model to enhance accuracy, it can preclude the real-time operation for navigation. Considering such restrictions, we used the generalized ELM autoencoder [

In the next section, we introduce existing studies on DEM fitting methods, namely, bilinear interpolation, the bi-spline neural network (B-spline NN), and the support vector machine for regression (SVMR). Then we introduce DEM learning techniques that use the conventional ELM and propose the multilayer RBF-based ELM (ML-RBF-ELM) to overcome the limitations of the conventional ELM. Furthermore, we compare the existing various DEM fitting methods and use of the proposed ML-RBF-ELM regression method and bilinear interpolation. Finally, we verify the design of the proposed technique by comparing it with the TRN performance.

There are different types of DEM database: the shuttle radar topography mission (SRTM) elevation database, digital terrain elevation data (DTED), etc. In this study, we used 10 m resolution DTED level 3 and 30 m resolution DTED level 2. There are various interpolation methods and machine learning regression techniques that can be used to obtain terrain elevation at the desired position using discontinuous DEM data. This section introduces previous studies that are aimed at obtaining continuous terrain elevation information.

Interpolation methods frequently used to obtain continuous elevation information include Delaunay triangulation [

Bilinear interpolation scheme.

If the terrain elevation value estimated at the point

As discussed by Liu et al. [

Okwuashi and Ndehedehe [

Since first being developed by Huang et al., the ELM has demonstrated good generalization ability and much faster learning than the backpropagation method of stochastic gradient descent and is therefore being studied in various fields. The ELM is a single hidden layer feedforward neural network. The input weights and hidden unit biases are chosen randomly and do not need to be adjusted [

Here,

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The conventional ELM is easy to configure and the learning speed is fast, but it cannot be applied to fields that process big data with very large nonlinearity or require high levels of accuracy. To solve these problems, many studies are being actively conducted on various advanced ELM algorithms. In the constraint in which real-time computation must be guaranteed, three hidden layers were used as shown in Figure

Proposed ML-RBF-ELM architecture.

The output weight matrix of the 1st hidden layer output

Here,

(1) Prepare a training set

(2) Set the activation function of the 1st hidden layer,

(3) Calculate the 1st hidden layer output matrix

(4) Calculate the output weight matrix of the 1st hidden layer,

(5) Calculate the new input data of the 2nd hidden layer:

(6) Set the activation function of the 2nd hidden layer,

(7) Calculate the 2nd hidden layer output matrix

(8) Calculate the output weight matrix of the 2nd hidden layer,

(9) Calculate the new input data of the 3rd hidden layer:

(10) Set the activation function of the 3rd hidden layer,

(11) Calculate the 3rd hidden layer output matrix

(12) Calculate the output weight matrix of the 3rd hidden layer,

(13) Compute and verify the regression results,

Figure ^{2} for DTED level 3 and ^{2} for DTED level 2. As mentioned above, we used 10 m resolution DTED level 3 and 30 m resolution DTED level 2 for the training and test data. The total obtainable number of sample data were 5391738 sets (161 MB) for level 3 and 5443200 sets (163 MB) for level 2. At a rate of 8 : 2, the sample data was used by dividing the training data (4313390 sets, 129 MB for level 3, and 4354560 sets, 130.4 MB for level 2) and test data (1078348 sets, 32 MB for level 3, and 1088640 sets, 32.6 MB for level 2). The sample data are acquired from the grid points such as

DEM fitting areas and flight trajectories. (a) DEM level 3 fitting area. (b) DEM level 2 fitting area.

Table

Results of various DEM fitting methods.

Bilinear interpolation | B-spline NN | SVMR | Conventional ELM | ML-RBF-ELM | |
---|---|---|---|---|---|

Number of hidden nodes | — | 1971 | — | 271 | 3842301193 |

Training time | — | 249.3 sec | 3135.4 sec | 8.10 sec | 71.72 sec |

Test time | — | 99.8 |
115.3 sec | 13.0 |
44.0 |

Training error | Approx. 1.02% | 3.81% | 8.76% | 24.03% | 0.98% |

Test error | Approx. 1.01% | 7.50% | 8.64% | 23.48% | 1.10% |

Here,

The training and test errors of the bilinear interpolation in Table

Results of the DEM fitting methods using DTED level 3. (a) Bilinear interpolation. (b) B-spline neural network. (c) SVMR. (d) Conventional ELM.

Fitting result of the proposed ML-RBF-ELM. (a) DTED level 3 result. (b) DTED level 2 result.

We verified the ML-RBF-ELM by conducting TRN with the ML-RBF-ELM and bilinear interpolation, which is superior to the fitting methods explained in the previous section. TRN simulations were performed using the interferometric radar altimeter (IRA) measurements and a bank of Kalman filters (BKF), which were used in Heli/SITAN [

Schematic diagram of the TRN and INS/TRN system.

Simulation conditions and BKF design parameters.

Parameter | Value |
---|---|

Initial covariance | 15^{2} m^{2} |

Accelerometer bias | 100 |

Gyro bias | 0.01 deg/hr |

Gyro white noise | 0.005 deg/ |

Barometer bias | 14 m |

Barometer scale factor | 0.2% of height |

Barometer white noise | 5 m |

Process noise covariance | 4^{2} m^{2} |

Measurement noise covariance | 30^{2} m^{2} |

Update frequency | 50 Hz |

The BKF is a linear filter that uses the multiple-model adaptive estimation (MMSE) technique to acquire navigation information sequentially, even in the case of a large initial position error. A state variable of one-dimensional Kalman filters comprising the BKF is the vertical bias error, which gradually changes. The BKF forms grids that are centered on the INS/TRN position estimate and assigns a Kalman filter to each grid. In this study,

Here,

Relative position to the nearest point.

Here,

The

Here,

The WRS indicates how well each filter in the grids matches the designed model. The smoothed WRS (SWRS) is expressed in the same way as the moving average of the WRS as follows [

Here,

Using the estimated covariance, the longitude and latitude

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Figure

BKF-based TRN RMS errors. (a) DTED level 3 result. (b) DTED level 2 result.

INS/TRN RMS errors. (a) Bilinear interpolation with DTED level 3. (b) ML-RBF-ELM with DTED level 3. (c) Bilinear interpolation with DTED level 2. (b) ML-RBF-ELM with DTED level 2.

Comparison of TRN and INS/TRN RMS errors.

DTED level | RMS errors | Bilinear interpolation | ML-RBF-ELM |
---|---|---|---|

3 | TRN | 7.465 m CEP | 5.130 m CEP |

3 | INS/TRN | 4.126 m CEP | 3.793 m CEP |

2 | TRN | 12.540 m CEP | 11.138 m CEP |

2 | INS/TRN | 9.891 m CEP | 6.079 m CEP |

This study introduced DEM fitting methods, a key technology of TRN, that can be used as an alternative in environments where GNSS is unavailable. The DEM is a database that stores the terrain’s elevation data at a constant resolution. Therefore, large amounts of memory are needed to use the high-resolution DEM. Furthermore, the various interpolation methods for obtaining continuous elevation information from the discontinuous DEM includes fitting errors. Accordingly, there are currently many ongoing studies on DEM learning through various machine learning techniques. However, these efforts have only produced results on DEM learning in low-resolution or narrow fields and they have yet to achieve stable performance for TRN using these fitting methods. In this study, we used the ML-RBF-ELM to fit a 10 m resolution DEM in an area of ^{2} and a 30 m resolution DEM in an area of ^{2}. The proposed regression model by the ML-RBF-ELM operates more quickly than the previous machine learning regression methods for the DEM and demonstrates similar fitting results with bilinear interpolation. Furthermore, we used BKF-based TRN, which showed improved TRN performance results over using the bilinear interpolation. Thus, our study verified that the proposed ML-RBF-ELM can make real-time operation of navigation and is suitable to TRN. It is clear that the proposed technology can be used by low-priced small unmanned aerial vehicles to execute TRN in environments where GNSS is not possible. Moreover, the proposed ML-RBF-ELM can be used in the field of simultaneous localization and map building (SLAM) because of the small training time.

The DTED level 3 full sample data and terrain reference navigation S/W used to support the findings of this study are not freely available, because of our institutional security policies. But the edited sample data, MRBF-ELM fitting S/W, and the simulation result data used to support the findings of this study are included within the supplementary information files (available

The authors declare that there is no conflict of interest regarding the publication of this paper.

The list of the supplementary information files is as follows. The edited sample data: “TestDBData3.dat,” “TrainDBData3.dat” in “TrainDB_ML_RBF_ELM_Using AE2_180919” Folder MRBF-ELM fitting S/W: “TrainDB_ML_RBF_ELM_Using AE2_180919” Folder the simulation result data: “TRNSimulationOutput_180912” folder, and “CompareMatchingResults_180909” folder.