# MIMU Optimal Redundant Structure and Signal Fusion Algorithm Based on a Non-Orthogonal MEMS Inertial Sensor Array

^{1}

^{2}

^{*}

## Abstract

**:**

_{b}, Y

_{b}and Z

_{b}were 4.9, 4.6 and 2.9 times lower than that of the single gyroscope.

## 1. Introduction

## 2. Working Principle of a Redundant MIMU System

#### 2.1. Modeling of a Fused-Gyro-Array KF Algorithm

_{b}and n is the angular random walk’s (ARW) white noise.

**H**is the configuration matrix of the non-orthogonal gyroscope array, which can be determined by the specific structure of the sensor array, $\mathbf{y}={[{y}_{1},{y}_{2},\cdots ,{y}_{N}]}^{T}$ is the measurement vector of non-orthogonal gyroscope array, ${\mathbf{\omega}}^{b}={[{\omega}_{x}^{b},{\omega}_{y}^{b},{\omega}_{z}^{b}]}^{T}$ is the input rate in the MIMU’s body frame, $\mathbf{b}={[{b}_{1},{b}_{2},\cdots ,{b}_{N}]}^{T}$ is the gyro’s drift error and $\mathbf{n}={[{n}_{1},{n}_{2},\cdots ,{n}_{N}]}^{T}$ is the white noise. The input true signal rate can be modeled directly; thus, its optimal estimate can be directly obtained using the KF. In addition, the accuracy of the fused signal rate can be analyzed by the KF covariance, which also can provide a basis for system improvement and parameter adjustment. Therefore, to build a complete state-space model for the KF and improve accuracy, the input signal rate

**ω**

^{b}can be modeled using a random walk process driven by white noise

**n**

_{ωr}[4,6]:

**q**should be determined by the gyro’s noise level and the dynamic characteristic requirement of the input signal rate. From a practical point of view, the application could be satisfied by choosing an appropriate variance q

_{ωx}

_{,y,z}with which to control the different bandwidths of the KF. Here, the angular rate

**ω**

^{b}and drift vector

**b**were chosen to construct the KF state vector as $\mathbf{X}=[{\mathbf{\omega}}^{b};\text{\hspace{0.17em}}\mathbf{b}]$; the measurement was selected as

**Z**=

**y**. Based on Equations (2) and (3), the state-space model of the non-orthogonal gyroscope array can be formed as:

**W**(t) and

**V**(t) are

**Q**and

**R**, respectively. The matrices

**Q**and

**R**are not necessarily diagonal because of the gyro’s noise correlation; thus, the matrices

**Q**,

**R**and

**q**are given in Equation (5).

_{ij}is the correlation factor between the ith and jth gyros, and the practical value of correlation factor ρ

_{ij}can be analyzed and obtained by the method referred in [19]. The parameters q

_{ωx}, q

_{ωx}and q

_{ωz}are the variances of white noise

**n**

_{ωr}, which drive the input rate

**ω**

^{b}. Based on Equation (4), the continuous-time KF algorithm for the non-orthogonal gyroscope array is

**F**,

**H**

_{1}) is not completely observable, and there is no steady-state solution to

**P**(t) in Equation (6). Here, set the gyro’s noise variance to be σ

_{b}= 600°/h/$\sqrt{\mathrm{h}}$ and σ

_{n}= 2°/$\sqrt{\mathrm{h}}$, then choose signal sampling period T = 0.01 s and correlation factor ρ = 0. For the non-orthogonal array composed of six gyros, the changes in covariance

**P**(t) and gain

**K**(t) are shown in Figure 3.

**P**

_{k}will be linearly increased and divergent with the iteration time. It does not have a steady-state solution. However, the matrix

**K**

_{k}tends toward a steady-state value in a short time, which indicates a steady-state gain

**K**

_{s}can be obtained. Using the steady gain

**K**

_{s}can simplify the implementation of the KF system. It does not need to calculate covariance

**P**(t) in each iteration, which reduces the computational load. Therefore, for a redundant MIMU system with a determined structure of non-orthogonal array, a steady-state gain

**K**

_{s}can be obtained offline by the discrete equation of KF. Thus, Equation (6) can be written as:

**K**

_{H}=

**K**

_{s}

**H**

_{1}, and perform an eigenvalue decomposition for the matrix

**K**

_{H}as

**K**

_{H}=

**SΛS**

^{−1}, where the columns of matrix

**S**are composed of the eigenvectors of matrix

**K**

_{H}.

**Λ**is a diagonal matrix composed of the eigenvalues of

**K**

_{H}, ${e}^{-{\mathbf{K}}_{H}T}={e}^{-\mathbf{S}\mathsf{\Lambda}{\mathbf{S}}^{-1}T}=\mathbf{S}{e}^{-\mathsf{\Lambda}T}{\mathbf{S}}^{-1}$ and ${\int}_{0}^{T}{e}^{-{\mathbf{K}}_{H}T}dt={\int}_{0}^{T}{e}^{-\mathbf{S}\mathsf{\Lambda}{\mathbf{S}}^{-1}T}dt=\mathbf{S}{\int}_{0}^{T}{e}^{-\mathsf{\Lambda}T}dt{\mathbf{S}}^{-1}$; thus, the discrete Equation (8) can be formed as:

#### 2.2. Structure of a Non-Orthogonal Gyro Array in MIMU

_{i}is the unit vector of the ith gyro’s sensitive axis. It can be formed as:

_{i}and β

_{i}are the installation angles of the ith gyro relative to the MIMU’s body coordinate frame (X

_{b}, Y

_{b}, Z

_{b}). According to Equation (10), the configuration matrix of a non-orthogonal array can be expressed as:

**H**. The conical configuration is a typical structure for a non-orthogonal array. In this paper, two configuration schemes for a conical structure are designed and analyzed with N = 4, 5, 6, 8, which is illustrated as follows:

_{b}axis (in Figure 6). Specifically, each gyroscope is evenly installed and distributed on the cone’s surface, and its sensitive axis is along an imaginary line connecting the tip of the cone and the gyro. The angle between each gyro’s sensitive axis and the +Z

_{b}axis is α. Figure 6a is a structure with N = 4, in which the angles between the +X

_{b}axis and projections of g1, g2, g3 and g4’s sensitive axes on the horizontal plane are 0°, 90°, 180° and 270°, respectively. For N = 5, the projection of g1’s sensitive axis on the horizontal plane coincides with the +X

_{b}axis, and the angle between the projections of contiguous gyros’ sensitive axes is 72°, which is shown in Figure 6b. In addition, for N = 6 in Figure 6c, the projections of g1 and g4’s sensitive axes on the horizontal plane coincide with +X

_{b}and −X

_{b}, respectively—in particular, the angle between the projections of contiguous gyros’ sensitive axes on the horizontal plane is 60°. For N = 8 in Figure 6d, the angle between the projections of contiguous gyros’ sensitive axes on the horizontal plane is 45°.

_{b}axis—in particular, one gyro’s sensitive axis coincides with the axis +Z

_{b}, and the other gyroscopes are evenly distributed around the +Z

_{b}axis, as shown in Figure 7. Specifically, the angle between the gyro’s sensitive axis and +Z

_{b}axis is α, and the projection of g1’s sensitive axis on the horizontal plane coincides with axis +X

_{b}. For N = 4 in Figure 7a, the angles between the +X

_{b}axis and projections of g1, g2 and g3’s sensitive axes on the horizontal plane are 0°, 120°, 240°, respectively. For N = 5, 6 and 8, the angles between the projections of contiguous gyros’ sensitive axis on the horizontal plane are 90°, 72° and 360/7°, respectively, which are shown in Figure 7b–d.

**H**

_{1}is the measurement matrix for KF in MIMU and

**C**

_{n}is the cross-correlation matrix associated with the ARW noise of the non-orthogonal gyroscope array.

## 3. Performance of Configuration Structure of Non-Orthogonal Array

_{n}exists in the non-orthogonal array, and its range is [−1/(N − 1),1]. The relationship between the correlation factor ρ

_{n}and GDOP is analyzed, and the result is shown in Figure 10.

_{n}and will increase as factor ρ

_{n}increases. (2) The effect of ρ

_{n}on the GDOP depends on N. Concretely, the GDOP will shrink as N increases under the same factor ρ

_{n}, leading to a higher performance. This also verifies that the system’s accuracy will be higher with a higher N for an identical configuration structure. (3) The effect of ρ

_{n}on GDOP is different. The KF’s accuracy with a negative ρ

_{n}is higher than that with a positive one, which indicates that the smaller the correlation factor ρ

_{n}, the better the configuration structure. On the other hand, Figure 10b shows that: (1) The GDOP will increase as factor ρ

_{n}increases, reaches a maximum value and then gradually decreases. (2) The maximum value of GDOP will decrease as N increases with the optimal angle α. In addition, Figure 10 indicates that the influence of ρ

_{n}on GDOP is equivalent for the same configuration structure, and its influence is different for various configuration structures.

_{n}is, which indicates that the GDOP for the orthogonal MIMU is independent of the correlation factor, and the correlation factor has no effect on the orthogonal MIMU’s configuration.

## 4. Simulation Results and Discussion

#### 4.1. Results of the Static Simulation

_{n}= 0.1°/$\sqrt{\mathrm{h}}$ and σ

_{b}= 600°/h/$\sqrt{\mathrm{h}}$, respectively. The simulation time and signal sampling period were set to T = 1 h and T

_{s}= 0.01 s. The correlation factor for RRW noise was chosen as ρ

_{b}= {−0.19, 0, 0.5}. As for the conical configuration structure in Figure 6c, the estimated rate in MIMU’s body frame is shown in Figure 11, Figure 12 and Figure 13. The Allan variance is shown in Figure 14. The result is listed in Table 3. Additionally, for the structure with N = 6 in Figure 7c, the Allan variance is shown in Figure 15, and the result is given in Table 4.

#### 4.2. Results of the Sinusoidal Signal Simulation

**ω**

^{b}= [0, 0, 5 × sin(0.06πt)]

^{T}°/s. The gyro’s ARW and RRW were set as σ

_{n}= 0.1°/$\sqrt{\mathrm{h}}$ and σ

_{b}= 600°/h/$\sqrt{\mathrm{h}}$, respectively, and the correlation factor was set to ρ = 0. The simulation time and signal sampling period were set to T = 1/6 h and T

_{s}= 0.01 s. For the conical structures of N = 6, 8 in Figure 6c,d, the gyro’s installation angle α was chosen as 54.74°, and α was chosen as 63.43° or 60.79° for N = 6,8 in Figure 7c,d, for Scheme 2, respectively. The plots of the non-orthogonal array and estimated signal rate are shown in Figure 16, Figure 17, Figure 18 and Figure 19. The results are listed in Table 5 and Table 6.

_{b}axis reaches the input signal of 5°/s without attenuation and distortion. Furthermore, Table 5 and Table 6 show the 1σ on the X

_{b}and Y

_{b}axes are about 3 to 9 times lower than those of a single gyroscope, and about 3 times lower on the Z

_{b}axis, which indicates that the accuracy is significantly improved through signal fusion of the non-orthogonal array. Particularly, it clearly shows that the reduction factor of the estimated error for N = 8 is higher than that for N = 6 for the conical configuration in Schemes 1 and 2, respectively. This also explains that for the same installation angle α and correlation factor ρ, the larger the value of N, the smaller the value of GDOP, and the higher the system fusion and estimation accuracy that can be achieved.

## 5. Experiment

#### 5.1. Static Testing Results

_{b}, Y

_{b}and Z

_{b}are lower than that of a single gyroscope. In addition, Table 8 shows that the ARW and RRW for the fused signal rate are about 3.5 and 2.5 times lower than for single gyroscopes. The results demonstrate that the accuracy of the MIMU can be effectively improved by fusing of a non-orthogonal gyroscope array.

#### 5.2. Swing Signal Testing Results

**ω**

^{b}= [0, 0, 5 × sin(2πft)]

^{T}°/s with f = 0.05 and 0.1 Hz. Therefore, the outputs of component gyroscopes and fused signal rate in MIMU’s body frame are shown in Figure 23 and Figure 24, and the estimated error is given in Table 9.

_{b}axis is similar to 5°/s, and Table 9 shows that the errors (1σ) on the MIMU’s X

_{b}and Y

_{b}axes were about 4.9 and 4.6 times lower than those of the single gyroscope. On the Z

_{b}axis, it was about 2.9-times lower. This demonstrates that the gyro’s error can be effectively reduced through fusing the outputs of the non-orthogonal array, thereby improving the accuracy of the MIMU and navigation system.

## 6. Conclusions

_{b}, Y

_{b}and Z

_{b}axes were 4.9, 4.6 and 2.9 times lower than those of a single gyro in the swing test.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 6.**Non-orthogonal gyro array for a MIMU without a one-axis conical sensor: (

**a**) 4-gyro cone; (

**b**) 5-gyro cone; (

**c**) 6-gyro cone; (

**d**) 8-gyro cone.

**Figure 7.**Non-orthogonal gyro array for the MIMU with a one-axis conical sensor: (

**a**) 4-gyro cone; (

**b**) 5-gyro cone; (

**c**) 6-gyro cone; (

**d**) 8-gyro cone.

**Figure 8.**The relationship between GDOP and angle α for different structures (ρ = 0). (

**a**) Configuration Scheme 1; (

**b**) configuration Scheme 2.

**Figure 9.**The relationship between GDOP and angle α for different N with various correlation factors: (

**a**) N = 4; (

**b**) N = 5; (

**c**) N = 6; (

**d**) N = 8.

**Figure 10.**The relationship between correlation factor and GDOP. (

**a**) Configuration Scheme 1; (

**b**) configuration Scheme 2.

**Figure 11.**Fused results of the 6-gyro MIMU system for Scheme 1 (ρ = 0). (

**a**) The output of gyros. (

**b**) The estimated result of the MIMU.

**Figure 12.**Fused results of the 6-gyro MIMU system for Scheme 1 (ρ = −0.19). (

**a**) The output of gyros. (

**b**) The estimated result of the MIMU.

**Figure 13.**Fused results of 6-gyro MIMU system for Scheme 1 (ρ = 0.5). (

**a**) The output of gyros. (

**b**) The estimated result of the MIMU.

**Figure 14.**Plot of the compared Allan variance of 6-gyro MIMU for Scheme 1: (

**a**) ρ = −0.19; (

**b**) ρ = 0; (

**c**) ρ = 0.5.

**Figure 15.**Plot of the compared Allan variance of 6-gyro MIMU for Scheme 2: (

**a**) ρ = −0.19; (

**b**) ρ = 0; (

**c**) ρ = 0.5.

**Figure 16.**Sinusoidal results of the 6-gyro MIMU system for Scheme 1. (

**a**) The outputs of the gyros. (

**b**) The estimated results of the MIMU.

**Figure 17.**Sinusoidal results of the 8-gyro MIMU system for Scheme 1. (

**a**) The outputs of the gyros. (

**b**) The estimated results of the MIMU.

**Figure 18.**Sinusoidal results of the 6-gyro MIMU system for Scheme 2. (

**a**) The outputs of the gyros. (

**b**) The estimated results of the MIMU.

**Figure 19.**Sinusoidal results of the 8-gyro MIMU system for Scheme 2. (

**a**) The outputs of the gyros. (

**b**) The estimated results of the MIMU.

**Figure 21.**The static estimated results of 4-MIMU. (

**a**) The outputs of gyros. (

**b**) The estimated rate of the MIMU on the X

_{b}, Y

_{b}and Z

_{b}axes.

**Figure 22.**Plot of Allan variance between the gyros and fused signal rate in MIMU’s body coordinate frame.

**Figure 23.**The swing test results of 4-MIMU with ω

_{z}= 5 × sin(2πft)°/s (f = 0.05 Hz). (

**a**) The outputs of gyros. (

**b**) The estimated rate of the MIMU on the X

_{b}, Y

_{b}and Z

_{b}axes.

**Figure 24.**The swing test results of 4-MIMU with ω

_{z}= 5 × sin(2πft)°/s (f = 0.1 Hz). (

**a**) The outputs of gyros. (

**b**) The estimated rate of the MIMU on the X

_{b}, Y

_{b}and Z

_{b}axes.

Scheme | Angle α | N = 4 | N = 5 | N = 6 | N = 8 |
---|---|---|---|---|---|

GDOP (Scheme 1) | α = 60° | 1.5327 | 1.3663 | 1.3089 | 1.0801 |

GDOP (Scheme 2) | α = 60° | 1.5275 | 1.3540 | 1.2293 | 1.0609 |

GDOP (Scheme 2) | α = 45° | 1.7512 | 1.5275 | 1.3732 | 1.1684 |

Number | Correlation Factor | Minimum GDOP | Optimal Angle α |
---|---|---|---|

ρ = 0.2 | 1.5269 | 49.94° | |

N = 4 | ρ = −0.1 | 1.4671 | 57.72° |

ρ = −0.2 | 1.4117 | 61.75° | |

ρ = 0.2 | 1.4000 | 49.10° | |

N = 5 | ρ = −0.1 | 1.2845 | 58.71° |

ρ = −0.2 | 1.1798 | 65.68° | |

ρ = 0.2 | 1.3076 | 48.36° | |

N = 6 | ρ = −0.1 | 1.1450 | 59.85° |

ρ = −0.18 | 1.0160 | 69.11° | |

ρ = 0.2 | 1.1802 | 47.05° | |

N = 8 | ρ = −0.1 | 0.9353 | 62.93° |

ρ = −0.14 | 0.8050 | 75.57° |

Correlation Factor | MIMU Axis | ARW (°$/\sqrt{\mathbf{h}}$) | RRW (°$/\mathbf{h}/\sqrt{\mathbf{h}}$) | BS (°/h) |
---|---|---|---|---|

ρ = −0.19 | X_{b} | 0.0393 | 279.894 | 3.8459 |

Y_{b} | 0.0396 | 256.254 | 3.8810 | |

Z_{b} | 0.0203 | 105.408 | 3.0038 | |

ρ = 0 | X_{b} | 0.0420 | 286.842 | 3.8645 |

Y_{b} | 0.0414 | 281.496 | 3.8893 | |

Z_{b} | 0.0416 | 265.800 | 3.9016 | |

ρ = 0.5 | X_{b} | 0.0484 | 325.752 | 4.0414 |

Y_{b} | 0.0480 | 313.020 | 3.9326 | |

Z_{b} | 0.0668 | 312.582 | 2.9394 |

Correlation Factor | MIMU Axis | ARW (°$/\sqrt{\mathbf{h}}$) | RRW (°$/\mathbf{h}/\sqrt{\mathbf{h}}$) | BS (°/h) |
---|---|---|---|---|

ρ = −0.19 | X_{b} | 0.0398 | 243.432 | 3.8092 |

Y_{b} | 0.0398 | 307.740 | 3.8684 | |

Z_{b} | 0.0675 | 95.526 | 2.9332 | |

ρ = 0 | X_{b} | 0.0412 | 292.770 | 3.9389 |

Y_{b} | 0.0414 | 289.590 | 3.8406 | |

Z_{b} | 0.0418 | 266.388 | 3.9633 | |

ρ = 0.5 | X_{b} | 0.0443 | 255.594 | 3.9182 |

Y_{b} | 0.0494 | 258.210 | 3.8985 | |

Z_{b} | 0.0476 | 302.604 | 4.3017 |

Terms | Number | X_{b} | Y_{b} | Z_{b} | Single Gyro |
---|---|---|---|---|---|

Estimated error (1σ, °/s) | N = 6 | 0.0142 | 0.0088 | 0.0224 | 0.0622 |

N = 8 | 0.0111 | 0.0068 | 0.0203 | 0.0622 | |

Reduction factor | N = 6 | 4.3803 | 7.0682 | 2.7768 | |

N = 8 | 5.6036 | 9.1471 | 3.0640 |

Terms | Number | X_{b} | Y_{b} | Z_{b} | Single Gyro |
---|---|---|---|---|---|

Estimated error (1σ, °/s) | N = 6 | 0.0182 | 0.0107 | 0.0242 | 0.0552 |

N = 8 | 0.0075 | 0.0075 | 0.0200 | 0.0552 | |

Reduction factor | N = 6 | 3.0329 | 5.1589 | 2.2810 | |

N = 8 | 7.3600 | 7.3600 | 2.7600 |

Gyro Number | g1 | g2 | g3 | g4 |
---|---|---|---|---|

g1 | 1 | 0.000309 | 0.001531 | −0.003095 |

g2 | 0.000309 | 1 | 0.001693 | −0.001061 |

g3 | 0.001531 | 0.001693 | 1 | −0.000106 |

g4 | −0.003095 | −0.001061 | −0.000106 | 1 |

Gyro Number | $\mathbf{ARW}\text{}(\text{\xb0}/\sqrt{\mathbf{h}})$ | $\mathbf{RRW}\text{}(\text{\xb0}/\mathbf{h}/\sqrt{\mathbf{h}})$ | BS (°/h) |
---|---|---|---|

gyro1 | 0.1476 | 4.979 | 1.0285 |

gyro2 | 0.1849 | 2.8545 | 0.9711 |

gyro3 | 0.1618 | 2.8764 | 0.9399 |

gyro4 | 0.1519 | 2.6229 | 0.9461 |

X_{b}-axis | 0.0429 | 1.1954 | 0.2751 |

Y_{b}-axis | 0.0459 | 0.6041 | 0.2287 |

Z_{b}-axis | 0.0525 | 1.3726 | 0.2723 |

Terms | Frequency f | X_{b} | Y_{b} | Z_{b} |
---|---|---|---|---|

Estimated error (1σ, °/s) | f = 0.05 | 0.0074 | 0.0078 | 0.0125 |

f = 0.10 | 0.0074 | 0.0079 | 0.0124 | |

Single gyro | 0.0367 | 0.0367 | 0.0367 | |

Reduction factor | f = 0.05 | 4.9595 | 4.7051 | 2.9360 |

f = 0.10 | 4.9595 | 4.6456 | 2.9597 |

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## Share and Cite

**MDPI and ACS Style**

Xue, L.; Yang, B.; Wang, X.; Cai, G.; Shan, B.; Chang, H.
MIMU Optimal Redundant Structure and Signal Fusion Algorithm Based on a Non-Orthogonal MEMS Inertial Sensor Array. *Micromachines* **2023**, *14*, 759.
https://doi.org/10.3390/mi14040759

**AMA Style**

Xue L, Yang B, Wang X, Cai G, Shan B, Chang H.
MIMU Optimal Redundant Structure and Signal Fusion Algorithm Based on a Non-Orthogonal MEMS Inertial Sensor Array. *Micromachines*. 2023; 14(4):759.
https://doi.org/10.3390/mi14040759

**Chicago/Turabian Style**

Xue, Liang, Bo Yang, Xinguo Wang, Guangbin Cai, Bin Shan, and Honglong Chang.
2023. "MIMU Optimal Redundant Structure and Signal Fusion Algorithm Based on a Non-Orthogonal MEMS Inertial Sensor Array" *Micromachines* 14, no. 4: 759.
https://doi.org/10.3390/mi14040759