# INS Error Estimation Based on an ANFIS and Its Application in Complex and Covert Surroundings

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Methods

#### 2.1. INS Solution

#### 2.1.1. Inertial Navigation Observation Model

#### 2.1.2. Position Calculation by Using INS Observations

^{T}

_{0}system and, similarly to Equations (4)–(7), the position of the vehicle at time t in the b

_{0}system can be obtained by

_{0}system at time t; ${R}_{nb}^{0}$ is the rotation matrix from the n system to the b

_{0}system at the initial time.

#### 2.2. ANFIS and Its Structure

#### 2.2.1. Fuzzy Inference System

^{i}is usually expressed as

#### 2.2.2. Artificial Neural Networks

_{2}regular term; λ is the regularization coefficient; L is the number of the network layers; s

_{l}is the number of the neurons in the l layer; h

_{Θ}(x

^{i}) gives the output of the neural network; y denotes the target value of the samples; ${\theta}_{mj}^{l}$ represents the connection weight between the j-th neuron in the l-th layer and the m-th neuron in the (l + 1)-th layer. The phenomenon of over-fitting can be circumvented by introducing the regularization term into the cost function, while matrix irreversibility can be avoided with the regularization term (when λ > 0) when solving the minimum cost function. Commonly used methods include the gradient descent method, the quasi-Newton method, the conjugate gradient method [38], and heuristic methods such as genetic and simulated annealing algorithms [39] to calculate the cost function. This study used the quasi-Newton BFGS algorithm to calculate the minimum cost function [40]. Compared with the traditional gradient descent method, its convergence is faster at the minimum value, and it is easier to achieve the global optimal value. Compared to the Newton method, it simplifies the complexity of the operation because it does not need to calculate the Hessian matrix.

#### 2.2.3. Estimation of the Accumulative Error in the Position of INS Based on an ANFIS

_{x}, a

_{y}, and a

_{z}are the measured values of accelerometers; g

_{x}, g

_{y}, and g

_{z}are the measured values of gyroscopes; t is the time series; Δx and Δy are the accumulative error of the INS in the b

_{0}system, calculated by

_{true}and y

_{true}denote the true position of the vehicle, which can be obtained from precise sensors in suitable surroundings. In this study, the reference position of the vehicle is obtained from differential GPS observations, and x

_{imu}and y

_{imu}denote ${P}_{b}^{t}(1,4)$ and ${P}_{b}^{t}(2,4)$ in Equation (10), respectively. After training with the ANFIS, the system obtains a y for each group of X input in the prediction stage.

_{3i}is the output of the third layer, which represents the trigger weight of rule i in all rules. The results of the fourth layer calculation rule are

^{i}is derived from Equation (12), which is generally a first-order polynomial of the input data; O

_{4i}is the output of the fourth layer. The fifth layer outputs the results of the system, using Equation (14) to compute the weighted average of all the rules to defuzzify.

## 3. Experiments and Results

#### 3.1. Introduction to KITTI

_{0}system, and the time stamp is the observation time of the data, which are stored in the timestamps.txt folder.

#### 3.2. Process and Result Analysis

_{0}system and the n system were calculated by Equations (4)–(10). The coordinate conversion on the GPS data was simultaneously performed to attain the reference position value of the vehicle in the b

_{0}system and the n system,

_{1}, R

_{2}are the rotation matrices, and the specific expression can be referred to as the reference [54,55]. It is stipulated that the X- and Y-directions of the b

_{0}system are the right direction and the forward direction of the initial motion of the vehicle, respectively, while the X- and Y-directions of the n system correspond to the east and north directions of the vehicle. Since the ground truth in the Z-direction has a large error in the KITTI dataset, which can be provided by the observations of the cameras, this study did not consider the motion in the Z-direction.

#### 3.2.1. Single-Sequence Position Error Estimation Model

_{0}system and the n system are shown in Figure 4a,b, respectively, in which the red line represents the reference trajectory and the black line represents the INS solution trajectory. The track discontinuity is the missing observations session after deleting the IMU repeated observations. It can be seen from Figure 4 that, due to the accumulative position error of INS, the INS solution trajectory is distinct from the reference trajectory, and the difference between the two trajectories at the end point is 135.65 m.

_{0}system and the n system, respectively. The data of the training part represent the moment where GPS is available, and the data of the prediction part simulate the vehicle in complex and covert surroundings.

_{0}system and the n system are shown in Table 3 and Table 4, respectively.

_{0}and the n systems, respectively. Similar to the ANN, the LSTM decreased the error of the Y-direction and increased the X-direction, which ultimately results in a reduction in error from 9.83% to 1.77% and 2.33% in the b

_{0}system and the n systems, respectively. Compared with the ANN, the LSTM, and the INS solution, the error predicted by the ANFIS was lower. Compared with the INS solution, the RMSE in the X-direction and the east direction using ANFIS was reduced to 65.4% and 12.3%, respectively, and the RMSE of the Y-direction and the north direction using ANFIS was correspondingly reduced to 7.1% and 6.3%, respectively. Overall, the accumulative errors were reduced from 9.83% to 0.45% and 0.43% in the b

_{0}and the n systems, respectively. It shows that the vehicle trajectory obtained by ANFIS has a superior fitting degree with the reference trajectory.

_{0}system are better than those in the n system. This was due to the SINS being installed on the vehicle and observations being attained from the sensors in the b system. However, this conclusion is not apparent when using ANFIS, so the following experiment only show the results under the b

_{0}system.

_{0}and n systems, respectively. It can be seen that ANFIS performed well in the estimation of the accumulative position error of INS, which largely corrects the vehicle position deviation caused by the accumulative position error of INS.

#### 3.2.2. Multi-Sequences Position Error Estimation Model

_{0}system are shown in Figure 7, where Figure 7a,b shows the error accumulation curve in the X- and Y-directions, respectively. The black, red, and green lines indicate the 09300018, 09300033, 09300034 curves, respectively. The information of each sequence is shown in Table 5. As shown in Figure 7 and Table 5, the accumulation of each curve of error over time between curves shows different change trends, but they were gradually increasing overall, and the accumulative position error in each sequence has some correlation with the driving time; thus, the position error of an entire sequence could be predicted by similar sequences [18].

_{0}system. Similarly, the data of the training part represent the moment when the GPS is available, and the data of the prediction part simulate the vehicle in complex and covert surroundings.

_{0}system of the vehicle, respectively. The mean square error (MSE), the root mean square error (RMSE), and the accumulative error of each trajectory in the b

_{0}system were calculated as shown in Table 6.

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 4.**Reference trajectory and INS solution trajectory: (

**a**) trajectory in b0 system; (

**b**) trajectory in the n system.

**Figure 5.**Error accumulation curve of three algorithms: (

**a**) X-direction position error in the b

_{0}system; (

**b**) Y-direction position error in the b

_{0}system; (

**c**) east direction position error in the n system; (

**d**) north direction position error in the n system.

**Figure 6.**Comparison of INS solution trajectory, ANFIS trajectory, and true trajectory: (

**a**) trajectory in the b

_{0}system; (

**b**)trajectory in the n system.

**Figure 7.**Cumulative error curve of three sequences: (

**a**) X direction position error; (

**b**) Y direction position error.

**Figure 8.**Error accumulation curve of two algorithms: (

**a**) X-direction position error; (

**b**) Y-direction position error.

Value | Unit | LIF | Remarks |
---|---|---|---|

Latitude | deg | 1 | WGS-84 |

Longitude | deg | 2 | |

Altitude | m | 3 | |

Roll | rad | 4 | (−π–π) |

Pitch | rad | 5 | (−1/2π–1/2π) |

Yaw | rad | 6 | (−π–π) |

Forward acceleration | m/s^{2} | 12 | b system |

Leftward acceleration | m/s^{2} | 13 | |

Upward acceleration | m/s^{2} | 14 | |

Time stamp | s | / | timestamp.txt |

LSTM | ANFIS | |||||||
---|---|---|---|---|---|---|---|---|

Direction | X | Y | East | North | X | Y | East | North |

MSE | 0.00045 | 3.9 | 0.09 | 2.9 | 0.0017 | 0.0079 | 0.0039 | 0.0088 |

RMSE | 0.02 | 1.97 | 0.30 | 1.70 | 0.04 | 0.09 | 0.06 | 0.09 |

IMU Solution | ANN | LSTM | ANFIS | |||||
---|---|---|---|---|---|---|---|---|

X | Y | X | Y | X | Y | X | Y | |

MSE | 4.46 | 3351.00 | 100.15 | 345.45 | 25.19 | 109.86 | 1.90 | 16.85 |

RMSE | 2.11 | 57.89 | 10.01 | 18.59 | 5.02 | 10.48 | 1.38 | 4.11 |

Accumulative Error | 9.83% | 3.59% | 1.77% | 0.45% | ||||

Distance | 1052.9 m |

IMU Solution | ANN | LSTM | ANFIS | |||||
---|---|---|---|---|---|---|---|---|

East | North | East | North | East | North | East | North | |

MSE | 113.67 | 3280.96 | 145.43 | 529.14 | 155.43 | 62.23 | 1.71 | 13.16 |

RMSE | 10.66 | 57.28 | 12.06 | 23.00 | 12.47 | 7.89 | 1.31 | 3.63 |

Accumulative Error | 9.83% | 5.34% | 2.33% | 0.43% | ||||

Distance | 1052.9 m |

Sequence | Driving Time/s | Driving Distance/m | Accumulative Error |
---|---|---|---|

09300018 | 260.29 | 1925.87 | 7.04% |

09300033 | 154.07 | 1623.47 | 4.14% |

09300034 | 127.53 | 919.89 | 3.84% |

IMU Solution | ANFIS | |||
---|---|---|---|---|

X | Y | X | Y | |

MSE | 288.33 | 865.53 | 54.25 | 143.41 |

RMSE | 16.98 | 29.42 | 7.37 | 11.98 |

Accumulative Error | 4.14% | 0.61% | ||

Distance | 1623.47 m |

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**MDPI and ACS Style**

Duan, Y.; Li, H.; Wu, S.; Zhang, K.
INS Error Estimation Based on an ANFIS and Its Application in Complex and Covert Surroundings. *ISPRS Int. J. Geo-Inf.* **2021**, *10*, 388.
https://doi.org/10.3390/ijgi10060388

**AMA Style**

Duan Y, Li H, Wu S, Zhang K.
INS Error Estimation Based on an ANFIS and Its Application in Complex and Covert Surroundings. *ISPRS International Journal of Geo-Information*. 2021; 10(6):388.
https://doi.org/10.3390/ijgi10060388

**Chicago/Turabian Style**

Duan, Yabo, Huaizhan Li, Suqin Wu, and Kefei Zhang.
2021. "INS Error Estimation Based on an ANFIS and Its Application in Complex and Covert Surroundings" *ISPRS International Journal of Geo-Information* 10, no. 6: 388.
https://doi.org/10.3390/ijgi10060388