# Optimal, Recursive and Sub-Optimal Linear Solutions to Attitude Determination from Vector Observations for GNSS/Accelerometer/Magnetometer Orientation Measurement

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## Abstract

**:**

## 1. Introduction

- The main theory is based on our previous research contributions and is extended to linear arbitrary pairs vector measurements for GNSS/AM attitude application.
- Three estimators, i.e., the Optimal Linear Estimator of Quaternion (OLEQ), Recursive OLEQ (ROLEQ) and Sub-optimal OLEQ (SOLEQ), are derived. We also proposed acceleration techniques to make the algorithms faster.
- Simulations and real experiments are carried out, which verify the accuracy, flexibility, robustness and time consumption of various algorithms for GNSS/AM attitude determination. Detailed comparisons with representative methods demonstrate the superiority of the proposed OLEQs.

## 2. Fundamentals of the GNSS/AM System

## 3. Problem Formulation

#### Quaternion from a Single Sensor Observation

**Theorem**

**1.**

**Proof.**

**Lemma**

**1.**

**Proof.**

**Theorem**

**2.**

**Proof.**

## 4. Optimal Linear Estimator of the Quaternion

#### 4.1. Variant 1: Recursive-OLEQ

#### 4.2. Variant 2: SOLEQ

#### 4.2.1. Two-Vector Case

**Theorem**

**3.**

**Proof.**

#### 4.2.2. n-Vector Case

#### 4.2.3. The Effect of Power Order

#### 4.3. Discussion of OLEQs

## 5. Simulations and Experiments

#### 5.1. Simulation: Common Cases

#### 5.2. Simulation: Extreme Cases

#### 5.3. Experiment: Accelerometer-Magnetometer Case

#### 5.4. Experiment: GNSS-Aided Attitude Determination for Land Vehicles

#### 5.5. Computation Time

## 6. Conclusions

- Numerical simulations exhibit that the proposed OLEQs have a similar accuracy with representative solvers.
- It is also demonstrated that faced with extreme cases, OLEQs show much more robustness with less computation iterations.
- The computation speeds of OLEQs are tested, revealing that they belong to computationally-efficient algorithms.
- Moreover, a real vehicular experiment of a GNSS/AM system is designed and conducted showing the effectiveness of the proposed OLEQs in real-world embedded applications.

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 4.**Experiment results of OLEQ, SOLEQ and representative algorithms using sampled data and different algorithms.

**Figure 5.**Experiment results of Recursive OLEQ (ROLEQ) and representative algorithms using sampled data and different algorithms.

**Figure 8.**Attitude determination from the accelerometer, magnetometer and velocity output of the GNSS receiver, by means of QUEST and FLAE.

**Figure 9.**Attitude determination from the accelerometer, magnetometer and velocity output of the GNSS receiver, by means of OLEQ and SOLEQ.

Case | Reference Vectors | Noise Standard Deviations |
---|---|---|

1 | ${\mathbf{D}}_{1}^{r}={[1,0,0]}^{\top},{\mathbf{D}}_{2}^{r}={[0,1,0]}^{\top},{\mathbf{D}}_{3}^{r}={[0,0,1]}^{\top}$ | ${\sigma}_{1}={10}^{-6},{\sigma}_{2}={10}^{-6},{\sigma}_{3}={10}^{-6}$ |

2 | ${\mathbf{D}}_{1}^{r}={[1,0,0]}^{\top},{\mathbf{D}}_{2}^{r}={[0,1,0]}^{\top}$ | ${\sigma}_{1}={10}^{-6},{\sigma}_{2}={10}^{-6}$ |

3 | ${\mathbf{D}}_{1}^{r}={[1,0,0]}^{\top},{\mathbf{D}}_{2}^{r}={[0,1,0]}^{\top},{\mathbf{D}}_{3}^{r}={[0,0,1]}^{\top}$ | ${\sigma}_{1}=0.01,{\sigma}_{2}=0.01,{\sigma}_{3}=0.01$ |

4 | ${\mathbf{D}}_{1}^{r}={[1,0,0]}^{\top},{\mathbf{D}}_{2}^{r}={[0,1,0]}^{\top}$ | ${\sigma}_{1}=0.01,{\sigma}_{2}=0.01$ |

5 | ${\mathbf{D}}_{1}^{r}={[0.6,0.8,0]}^{\top},{\mathbf{D}}_{2}^{r}={[0.8,-0.6,0]}^{\top}$ | ${\sigma}_{1}={10}^{-6},{\sigma}_{2}=0.01$ |

6 | ${\mathbf{D}}_{1}^{r}={[1,0,0]}^{\top},{\mathbf{D}}_{2}^{r}={[1,0.01,0]}^{\top},{\mathbf{D}}_{3}^{r}={[1,0,0.01]}^{\top}$ | ${\sigma}_{1}={10}^{-6},{\sigma}_{2}={10}^{-6},{\sigma}_{3}={10}^{-6}$ |

7 | ${\mathbf{D}}_{1}^{r}={[1,0,0]}^{\top},{\mathbf{D}}_{2}^{r}={[1,0.01,0]}^{\top}$ | ${\sigma}_{1}={10}^{-6},{\sigma}_{2}={10}^{-6}$ |

8 | ${\mathbf{D}}_{1}^{r}={[1,0,0]}^{\top},{\mathbf{D}}_{2}^{r}={[1,0.01,0]}^{\top},{\mathbf{D}}_{3}^{r}={[1,0,0.01]}^{\top}$ | ${\sigma}_{1}=0.01,{\sigma}_{2}=0.01,{\sigma}_{3}=0.01$ |

9 | ${\mathbf{D}}_{1}^{r}={[1,0,0]}^{\top},{\mathbf{D}}_{2}^{r}={[1,0.01,0]}^{\top}$ | ${\sigma}_{1}=0.01,{\sigma}_{2}=0.01$ |

10 | ${\mathbf{D}}_{1}^{r}={[1,0,0]}^{\top},{\mathbf{D}}_{2}^{r}={[0.96,0.28,0]}^{\top},{\mathbf{D}}_{3}^{r}={[0.96,0,0.28]}^{\top}$ | ${\sigma}_{1}={10}^{-6},{\sigma}_{2}=0.01,{\sigma}_{3}=0.01$ |

11 | ${\mathbf{D}}_{1}^{r}={[1,0,0]}^{\top},{\mathbf{D}}_{2}^{r}={[0.96,0.28,0]}^{\top}$ | ${\sigma}_{1}={10}^{-6},{\sigma}_{2}=0.01$ |

12 | ${\mathbf{D}}_{1}^{r}={[1,0,0]}^{\top},{\mathbf{D}}_{2}^{r}={[0.96,0.28,0]}^{\top}$ | ${\sigma}_{1}=0.01,{\sigma}_{2}={10}^{-6}$ |

**Table 2.**Roll RMSE (deg). SOLEQ, Suboptimal Linear Estimator of Quaternion; QUEST, Shuster’s QUaternionESTimator; FLAE, Fast Linear Attitude Estimator.

Case | OLEQ | SOLEQ | QUEST | FLAE |
---|---|---|---|---|

1 | $4.3516\times {10}^{-05}$ | $6.1268\times {10}^{-05}$ | $4.3516\times {10}^{-05}$ | $4.3516\times {10}^{-05}$ |

2 | $5.9303\times {10}^{-05}$ | $6.0734\times {10}^{-05}$ | $5.9303\times {10}^{-05}$ | $5.9303\times {10}^{-05}$ |

3 | $4.3482\times {10}^{-01}$ | $6.0730\times {10}^{-01}$ | $4.3482\times {10}^{-01}$ | $4.3482\times {10}^{-01}$ |

4 | $6.0292\times {10}^{-01}$ | $6.1798\times {10}^{-01}$ | $6.0292\times {10}^{-01}$ | $6.0292\times {10}^{-01}$ |

5 | $4.3313\times {10}^{-01}$ | $4.3313\times {10}^{-01}$ | $4.9065\times {10}^{-01}$ | $2.4281\times {10}^{+01}$ |

6 | $4.9590\times {10}^{-03}$ | $3.0793\times {10}^{-01}$ | $4.9590\times {10}^{-03}$ | $4.9590\times {10}^{-03}$ |

7 | $8.1132\times {10}^{-03}$ | $1.3400\times {10}^{+00}$ | $8.1132\times {10}^{-03}$ | $8.1132\times {10}^{-03}$ |

8 | $5.9553\times {10}^{+01}$ | $6.2840\times {10}^{+01}$ | $5.9553\times {10}^{+01}$ | $5.9553\times {10}^{+01}$ |

9 | $7.6662\times {10}^{+01}$ | $7.6696\times {10}^{+01}$ | $7.6662\times {10}^{+01}$ | $7.6662\times {10}^{+01}$ |

10 | $1.4313\times {10}^{+00}$ | $1.7781\times {10}^{+00}$ | $4.7663\times {10}^{+01}$ | $1.5265\times {10}^{+00}$ |

11 | $2.0254\times {10}^{+00}$ | $2.0254\times {10}^{+00}$ | $4.6392\times {10}^{+01}$ | $2.8441\times {10}^{+00}$ |

12 | $2.0818\times {10}^{+00}$ | $2.0888\times {10}^{+00}$ | $3.7218\times {10}^{+01}$ | $1.7415\times {10}^{+01}$ |

Case | OLEQ | SOLEQ | QUEST | FLAE |
---|---|---|---|---|

1 | $4.0108\times {10}^{-05}$ | $5.7335\times {10}^{-05}$ | $4.0108\times {10}^{-05}$ | $4.0108\times {10}^{-05}$ |

2 | $5.2860\times {10}^{-05}$ | $5.6736\times {10}^{-05}$ | $5.2860\times {10}^{-05}$ | $5.2860\times {10}^{-05}$ |

3 | $4.0104\times {10}^{-01}$ | $5.6744\times {10}^{-01}$ | $4.0104\times {10}^{-01}$ | $4.0104\times {10}^{-01}$ |

4 | $5.3887\times {10}^{-01}$ | $5.7656\times {10}^{-01}$ | $5.3887\times {10}^{-01}$ | $5.3887\times {10}^{-01}$ |

5 | $3.9149\times {10}^{-01}$ | $3.9149\times {10}^{-01}$ | $4.4335\times {10}^{-01}$ | $1.2561\times {10}^{+01}$ |

6 | $4.0121\times {10}^{-05}$ | $5.7809\times {10}^{-05}$ | $4.0121\times {10}^{-05}$ | $4.0121\times {10}^{-05}$ |

7 | $5.3398\times {10}^{-05}$ | $5.7657\times {10}^{-05}$ | $5.3398\times {10}^{-05}$ | $5.3398\times {10}^{-05}$ |

8 | $3.6755\times {10}^{-01}$ | $5.7326\times {10}^{-01}$ | $3.6755\times {10}^{-01}$ | $3.6755\times {10}^{-01}$ |

9 | $4.5938\times {10}^{-01}$ | $5.7880\times {10}^{-01}$ | $4.5938\times {10}^{-01}$ | $4.5938\times {10}^{-01}$ |

10 | $5.7186\times {10}^{-05}$ | $5.7186\times {10}^{-05}$ | $5.7184\times {10}^{-05}$ | $5.7186\times {10}^{-05}$ |

11 | $5.7845\times {10}^{-05}$ | $5.7845\times {10}^{-05}$ | $5.7846\times {10}^{-05}$ | $5.7844\times {10}^{-05}$ |

12 | $4.9161\times {10}^{-01}$ | $5.7554\times {10}^{-01}$ | $7.9376\times {10}^{+00}$ | $3.9359\times {10}^{+00}$ |

Case | OLEQ | SOLEQ | QUEST | FLAE |
---|---|---|---|---|

1 | $4.3587\times {10}^{-05}$ | $6.1084\times {10}^{-05}$ | $4.3587\times {10}^{-05}$ | $4.3587\times {10}^{-05}$ |

2 | $4.8694\times {10}^{-05}$ | $6.1015\times {10}^{-05}$ | $4.8694\times {10}^{-05}$ | $4.8694\times {10}^{-05}$ |

3 | $4.4127\times {10}^{-01}$ | $6.0868\times {10}^{-01}$ | $4.4127\times {10}^{-01}$ | $4.4127\times {10}^{-01}$ |

4 | $4.8593\times {10}^{-01}$ | $6.1289\times {10}^{-01}$ | $4.8593\times {10}^{-01}$ | $4.8593\times {10}^{-01}$ |

5 | $2.5186\times {10}^{-01}$ | $2.5186\times {10}^{-01}$ | $2.8536\times {10}^{-01}$ | $1.7459\times {10}^{+01}$ |

6 | $3.6421\times {10}^{-05}$ | $6.1651\times {10}^{-05}$ | $3.6421\times {10}^{-05}$ | $3.6421\times {10}^{-05}$ |

7 | $4.8748\times {10}^{-05}$ | $6.0826\times {10}^{-05}$ | $4.8748\times {10}^{-05}$ | $4.8748\times {10}^{-05}$ |

8 | $3.9812\times {10}^{-01}$ | $6.1557\times {10}^{-01}$ | $3.9812\times {10}^{-01}$ | $3.9812\times {10}^{-01}$ |

9 | $4.9366\times {10}^{-01}$ | $6.1163\times {10}^{-01}$ | $4.9366\times {10}^{-01}$ | $4.9366\times {10}^{-01}$ |

10 | $6.1834\times {10}^{-05}$ | $6.1834\times {10}^{-05}$ | $6.1836\times {10}^{-05}$ | $6.1844\times {10}^{-05}$ |

11 | $6.2069\times {10}^{-05}$ | $6.2069\times {10}^{-05}$ | $6.2069\times {10}^{-05}$ | $6.2077\times {10}^{-05}$ |

12 | $3.1726\times {10}^{-01}$ | $6.1275\times {10}^{-01}$ | $5.1968\times {10}^{+00}$ | $2.8814\times {10}^{+00}$ |

Case | OLEQ | SOLEQ | QUEST | FLAE |
---|---|---|---|---|

1 | $5.0651\times {10}^{-13}$ | $1.0130\times {10}^{-12}$ | $5.0651\times {10}^{-13}$ | $5.0651\times {10}^{-13}$ |

2 | $2.4901\times {10}^{-13}$ | $4.9802\times {10}^{-13}$ | $2.4901\times {10}^{-13}$ | $2.4901\times {10}^{-13}$ |

3 | $4.9338\times {10}^{-05}$ | $9.8666\times {10}^{-05}$ | $4.9338\times {10}^{-05}$ | $4.9338\times {10}^{-05}$ |

4 | $2.5369\times {10}^{-05}$ | $5.0736\times {10}^{-05}$ | $2.5369\times {10}^{-05}$ | $2.5369\times {10}^{-05}$ |

5 | $5.0582\times {10}^{-13}$ | $5.0582\times {10}^{-13}$ | $6.5095\times {10}^{-13}$ | $8.5878\times {10}^{-10}$ |

6 | $5.0422\times {10}^{-13}$ | $9.4333\times {10}^{-10}$ | $5.0422\times {10}^{-13}$ | $5.0422\times {10}^{-13}$ |

7 | $2.4728\times {10}^{-13}$ | $8.8239\times {10}^{-09}$ | $2.4728\times {10}^{-13}$ | $2.4728\times {10}^{-13}$ |

8 | $4.8216\times {10}^{-05}$ | $1.1593\times {10}^{-04}$ | $4.8216\times {10}^{-05}$ | $4.8216\times {10}^{-05}$ |

9 | $2.5327\times {10}^{-05}$ | $5.0651\times {10}^{-05}$ | $2.5327\times {10}^{-05}$ | $2.5327\times {10}^{-05}$ |

10 | $1.4827\times {10}^{-12}$ | $1.7575\times {10}^{-12}$ | $4.8431\times {10}^{-10}$ | $1.7195\times {10}^{-12}$ |

11 | $4.8573\times {10}^{-13}$ | $4.8573\times {10}^{-13}$ | $2.4106\times {10}^{-10}$ | $9.2333\times {10}^{-13}$ |

12 | $5.0105\times {10}^{-13}$ | $5.0105\times {10}^{-05}$ | $1.3143\times {10}^{-10}$ | $3.4336\times {10}^{-11}$ |

Roll Case | OLEQ | SOLEQ | QUEST | FLAE |

Acc-MagExperiment | $17.2$ | $16.8$ | $17.2$ | $17.2$ |

Acc-Mag-GNSS Experiment | $10.1$ | $9.7$ | $10.2$ | $10.2$ |

Pitch Case | OLEQ | SOLEQ | QUEST | FLAE |

Acc-Mag Experiment | $15.9$ | $14.6$ | $15.9$ | $15.9$ |

Acc-Mag-GNSS Experiment | $14.4$ | $13.2$ | $14.4$ | $14.4$ |

Yaw Case | OLEQ | SOLEQ | QUEST | FLAE |

Acc-Mag Experiment | $154.3$ | $155.9$ | $154.3$ | $154.3$ |

Acc-Mag-GNSS Experiment | $16.7$ | $44.8$ | $16.7$ | $16.7$ |

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**MDPI and ACS Style**

Zhou, Z.; Wu, J.; Wang, J.; Fourati, H.
Optimal, Recursive and Sub-Optimal Linear Solutions to Attitude Determination from Vector Observations for GNSS/Accelerometer/Magnetometer Orientation Measurement. *Remote Sens.* **2018**, *10*, 377.
https://doi.org/10.3390/rs10030377

**AMA Style**

Zhou Z, Wu J, Wang J, Fourati H.
Optimal, Recursive and Sub-Optimal Linear Solutions to Attitude Determination from Vector Observations for GNSS/Accelerometer/Magnetometer Orientation Measurement. *Remote Sensing*. 2018; 10(3):377.
https://doi.org/10.3390/rs10030377

**Chicago/Turabian Style**

Zhou, Zebo, Jin Wu, Jinling Wang, and Hassen Fourati.
2018. "Optimal, Recursive and Sub-Optimal Linear Solutions to Attitude Determination from Vector Observations for GNSS/Accelerometer/Magnetometer Orientation Measurement" *Remote Sensing* 10, no. 3: 377.
https://doi.org/10.3390/rs10030377