# Improved Indoor Positioning Model Based on UWB/IMU Tight Combination with Double-Loop Cumulative Error Estimation

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## Abstract

**:**

## 1. Introduction

## 2. Related Work

- (1)
- Kalman filter (KF). Kalman [10] proposed a new method for linear filtering and prediction problems, which used the linear system state equation to observe system input and output data in order to best estimate the state of the system. Benefiting from the advantages of strong real-time performance, high speed, and high efficiency, the KF has been widely used in anti-interference [11].
- (2)
- Extended Kalman filter (EKF). Although linear systems often exist in theory, more nonlinear systems exist in practical engineering applications. Therefore, a linear time-varying system was used to approximate the nonlinear system within each sampling interval by linearization methods, such as Taylor series expansion [12]. Therefore, linear time-varying systems can still use KF.
- (3)
- Complementary Kalman filter (CKF). Ienkaran [13] put forward a new method for nonlinear problems with better performances than the EKF, mainly based on a third-order spherical radial displacement criterion known as the CKF. Using a group of volume points to approximate the state mean and covariance of nonlinear systems with additional Gaussian noise, CKF is the closest approximation algorithm to Bayesian filtering. It is also a powerful tool to solve the state estimation of nonlinear systems [14].
- (4)
- Monte Carlo filter (MCL). MCL is a fast and scalable unsupervised clustering algorithm based on the random Walk and Markov chain [15]. It has been mainly applied in the field of robot positioning, and the posterior probability density function of mobile robot state is approximated to the real state through MCL random sampling [16].

## 3. Preliminaries of IMU and Extended Kalman Filter Used in UWB Positioning

#### 3.1. Principle of IMU Positioning

#### 3.2. Extended Kalman Filter

#### 3.2.1. Definition of State Space

#### 3.2.2. Linearization

#### 3.2.3. Kalman Gain

## 4. Improved Tight Combination Model

#### 4.1. IMU Location Calculation

^{2}.

#### 4.2. Time Synchronization Correction

#### 4.3. Distance Threshold Setting

#### 4.4. Design of Extended Kalman Filter

#### 4.4.1. System State Equation

#### 4.4.2. Measurement Equation

#### 4.5. Convergence Analysis of Combination Error

#### 4.5.1. Accumulative Error Analysis of IMU after Twice Integral

#### 4.5.2. Cumulative Error Analysis of EKF-Based Positive Feedback

#### 4.5.3. Dynamics Model of Double-Loop Cumulative Error

#### 4.5.4. Controllability Analysis of the Sensor-Fusion Error Transmission System

_{0}to another expected state X.

## 5. Simulation Experiments

#### 5.1. Motion Trajectory Analysis

- The UWB location had the largest standard deviation, and the improved fusion location had the smallest.
- The improved UWB/IMU fusion model could eliminate the cumulative error of IMU at the same period and ensure the accuracy of positioning system.
- The improved UWB/IMU fusion model made the error meet the controllability requirement of the system better.

#### 5.2. Acceleration KF-Based Optimization Model

#### 5.3. Analysis for Improved UWB/IMU Tight Combination Positioning Model

#### 5.4. Discussion

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 7.**Trajectory comparison measured by the improved UWB/IMU method with a pure UWB and conventional UWB/IMU method.

Method/Model | Maximum Deviation (m) | Standard Deviation (m) |
---|---|---|

Pure UWB | 1.444 | 0.8567 |

UWB/IMU | 0.408 | 0.1283 |

I-UWB/IMU | 0.232 | 0.0998 |

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## Share and Cite

**MDPI and ACS Style**

Zhu, W.; Zhao, R.; Zhang, H.; Lu, J.; Zhang, Z.; Wei, B.; Fan, Y.
Improved Indoor Positioning Model Based on UWB/IMU Tight Combination with Double-Loop Cumulative Error Estimation. *Appl. Sci.* **2023**, *13*, 10046.
https://doi.org/10.3390/app131810046

**AMA Style**

Zhu W, Zhao R, Zhang H, Lu J, Zhang Z, Wei B, Fan Y.
Improved Indoor Positioning Model Based on UWB/IMU Tight Combination with Double-Loop Cumulative Error Estimation. *Applied Sciences*. 2023; 13(18):10046.
https://doi.org/10.3390/app131810046

**Chicago/Turabian Style**

Zhu, Wenjie, Rongyong Zhao, Hao Zhang, Jianfeng Lu, Zhishu Zhang, Bingyu Wei, and Yuhang Fan.
2023. "Improved Indoor Positioning Model Based on UWB/IMU Tight Combination with Double-Loop Cumulative Error Estimation" *Applied Sciences* 13, no. 18: 10046.
https://doi.org/10.3390/app131810046