# An Artificial Neural Network Embedded Position and Orientation Determination Algorithm for Low Cost MEMS INS/GPS Integrated Sensors

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Problem statements

## 3. From Kalman filtering to optimal smoothing

^{l}is the position vector [ϕ (latitude), λ(longitude), h(height)], v

^{l}is the velocity vector (e, n, u), ${R}_{b}^{l}$ is the transformation matrix from the IMU body to local frame as a function of attitude components, g

^{l}is the gravity vector in the local level frame, ${\mathrm{\Omega}}_{\mathit{ib}}^{b}$, ${\mathrm{\Omega}}_{\mathit{il}}^{b}$ are the skew-symmetric matrices of the angular velocity vectors ${w}_{\mathit{ib}}^{b}$, ${w}_{\mathit{il}}^{b}$ respectively, D

^{−1}is a 3×3 matrix whose non-zero elements are functions of the user’s latitude ϕ and ellipsoidal height (h).

^{l}is the position error state vector in the local level frame, δv

^{l}is the velocity error state vector in the local level frame, δA

^{l}is the attitude error state vector in the local level frame, δg

^{l}is the error in the computed gravity vector in the local level frame, δf

^{b}& δω

^{b}are accelerometer bias and gyro drift vectors in the body frame respectively, and S

_{a}and S

_{g}are scale factors of accelerometers and gyros respectively, and E is a 3×3 matrix whose non-zero elements are a function of the vehicle’s latitude and the Earth’s radii of curvatures.

_{k}) by using measurements (updates) that are only available up to epoch k. In contrast, the optimal backward smoothing allows an optimal smoothed estimation of the state vector at epoch k (x̂

_{ks}) utilizing all or some of the measurements that are available after epoch k. The smoothed estimate (x̂

_{ks}) could be considered to be an optimal combination of a forward estimate and a backward estimate. The forward estimate is obtained by using all measurements up to k; it is the estimate provided by KF. The backward estimate is obtained by using all or some of the measurements after k. Since more measurement updates are used for the estimations, the BS estimates in general, if not more accurate, can never be worse than the filtered estimates [20].

_{k,k}), where k = 0, 1, 2...N[3]. In fixed-interval smoothing, the initial and final time epochs of the whole interval of measurements (i.e. 0 and N) are fixed. The requirement here is the optimal smoothed estimate at all epochs k in the interval between 0 and N, as indicated in Figure 3b. In this case, all measurement updates between 0 and N are used, so the optimal smoothed estimate at epoch k is ${\widehat{x}}_{k,k}^{s}$. Obviously, this type of smoothing can only be carried out in post-mission since it requires the availability of all measurements up to N [23].

^{s}

_{N,N}= P

_{N,N.}The RTS algorithms are as follows [20]:

_{k}is the smoothing gain matrix, and k=N−1, N−2…0. The covariance matrix of the smoothed states is given as follows [20]:

## 4. The artificial neural networks

_{i,j},W

_{j,k}〉; (b) an adder for summing the input signals ϕ

_{i}that are weighted by respective synapses of the neuron and external bias (b

_{k}); and (c) an activation function φ(•) for limiting the amplitude of the neuron output and the final output y

_{k}. Figure 6 shows a feed forward neural network, which contains external inputs (ϕ

_{1},ϕ

_{2},ϕ

_{3}), a hidden layer with 3 hidden neurons, and an output layer with 3 output neurons. The depicted network is said to be fully connected since all inputs/all neurons in one layer are connected to all neurons in the following layer [15]. The mathematical formula for the depicted network can be expressed in the form:

_{j,l},W

_{i,j}〉. Since the bias can be interpreted as a weight acting on an input clamped to 1; i.e., b

_{1}, b

_{2}= 1,the joint description “weight” will most often be applied covering both weights and bias. To determine the weight values one must have a set of examples of how the outputs, ŷ

_{i}, should relate to the input, ϕ

_{l}. The process of obtaining the weights from these examples is called supervised learning; it is basically a conventional estimation process. That is, the weights are estimated from existing examples in such a way that the network, according to some metric, models the true relationship as accurately as possible. This supervised learning process can be implemented using a backpropagation learning algorithm.

## 5. An open loop design for ANN-RTS smoother scheme

## 6. Results and Discussion

## 7. The training of the proposed schemes

## 8. Performance verification of the proposed schemes

## 9. Conclusions

## Acknowledgments

## References

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**Figure 1.**(a) An example of land based MMS (b) An example of direct geo-referencing an object of interest (Adopted from [1]).

Index | NVS | PDOP | speed (m/s) | Date | Duration (seconds) | ||||||
---|---|---|---|---|---|---|---|---|---|---|---|

Min. | Max. | Avg. | Min. | Max. | Avg. | Min. | Max. | Avg. | |||

Tj-1 | 4 | 10 | 7 | 1.2 | 5.8 | 2.2 | 0 | 22 | 7.5 | 03.17.2005 | 2,400 |

Tj-2 | 4 | 10 | 7 | 1.1 | 5.8 | 2.1 | 0 | 22 | 8.2 | 03.17.2005 | 1,850 |

Tj-3 | 4 | 10 | 7.5 | 1.4 | 5.8 | 2.4 | 0 | 22 | 7.8 | 03.16.2005 | 1,700 |

Method | POS | RMS value | Improvement. (%) | ||

Original(KF) | Compensated | Against KF | Against RTS | ||

Tj-3 (KF + ANN) | North(m) | 30.5 | 1.76 | 94 | -- |

East(m) | 16.48 | 1.28 | 92 | -- | |

Height(m) | 3.84 | 0.31 | 92 | -- | |

Roll(deg) | 0.182 | 0.025 | 87 | -- | |

Pitch(deg) | 0.384 | 0.046 | 88 | -- | |

Heading(deg) | 18.433 | 1.783 | 90 | -- | |

Method | POS | Original(RTS) | Compensated | Against KF | Against RTS |

Tj-3 (RTS + ANN) | North(m) | 0.44 | 0.15 | 99 | 66 |

East(m) | 0.37 | 0.12 | 99 | 68 | |

Height(m) | 0.21 | 0.03 | 99 | 86 | |

Roll(deg) | 0.022 | 0.008 | 96 | 64 | |

Pitch(deg) | 0.034 | 0.012 | 97 | 65 | |

Heading(deg) | 1.541 | 0.3023 | 98 | 80 | |

Remarks | --Left blank intentionally |

Method | POS | RMS value | Improvement (%) | ||

Original(KF) | Compensated | Against KF | Against RTS | ||

Tj-1 (KF + ANN) | North(m) | 25.28 | 4.18 | 79 | -- |

East(m) | 23.15 | 6.57 | 72 | -- | |

Height(m) | 6.45 | 0.85 | 87 | -- | |

Roll(deg) | 0.982 | 0.244 | 75 | -- | |

Pitch(deg) | 0.753 | 0.283 | 63 | -- | |

Heading(deg) | 48.702 | 8.043 | 84 | -- | |

Tj-2 (KF+ ANN) | North(m) | 28.54 | 6.21 | 78 | -- |

East(m) | 23.12 | 7.58 | 67 | -- | |

Height(m) | 5.12 | 0.75 | 85 | -- | |

Roll(deg) | 0.782 | 0.212 | 73 | -- | |

Pitch(deg) | 0.854 | 0.381 | 55 | -- | |

Heading(deg) | 13.25 | 2.18 | 84 | -- | |

Method | POS | Original(RTS) | Compensated | Against KF | Against RTS |

Tj-1 (RTS+ ANN) | North(m) | 0.43 | 0.18 | 99 | 58 |

East(m) | 0.32 | 0.15 | 99 | 53 | |

Height(m) | 0.23 | 0.08 | 99 | 65 | |

Roll(deg) | 0.543 | 0.125 | 87 | 78 | |

Pitch(deg) | 0.325 | 0.083 | 89 | 74 | |

Heading(deg) | 24.325 | 3.854 | 92 | 84 | |

Tj-2 (RTS + ANN) | North(m) | 0.38 | 0.17 | 99 | 55 |

East(m) | 0.34 | 0.13 | 99 | 62 | |

Height(m) | 0.27 | 0.09 | 99 | 67 | |

Roll(deg) | 0.256 | 0.103 | 81 | 60 | |

Pitch(deg) | 0.312 | 0.124 | 85 | 61 | |

Heading(deg) | 4.875 | 1.235 | 89 | 75 | |

Remark | --Left blank intentionally |

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**MDPI and ACS Style**

Chiang, K.-W.; Chang, H.-W.; Li, C.-Y.; Huang, Y.-W.
An Artificial Neural Network Embedded Position and Orientation Determination Algorithm for Low Cost MEMS INS/GPS Integrated Sensors. *Sensors* **2009**, *9*, 2586-2610.
https://doi.org/10.3390/s90402586

**AMA Style**

Chiang K-W, Chang H-W, Li C-Y, Huang Y-W.
An Artificial Neural Network Embedded Position and Orientation Determination Algorithm for Low Cost MEMS INS/GPS Integrated Sensors. *Sensors*. 2009; 9(4):2586-2610.
https://doi.org/10.3390/s90402586

**Chicago/Turabian Style**

Chiang, Kai-Wei, Hsiu-Wen Chang, Chia-Yuan Li, and Yun-Wen Huang.
2009. "An Artificial Neural Network Embedded Position and Orientation Determination Algorithm for Low Cost MEMS INS/GPS Integrated Sensors" *Sensors* 9, no. 4: 2586-2610.
https://doi.org/10.3390/s90402586