4.1. Introduction of micro-inertial/AGM/odometer/PJ system

Figure 7 shows the schematic diagram of the micro-inertial/AGM/odometer/PJ pipeline navigation system. The triaxial angular rate ω and triaxial linear acceleration f of PIG are measured by micro-inertial sensors in the pipeline. Then, the 3D attitude, velocity, and position of the PIG are provided by SINS mechanization. In order to correct the micro-inertial-sensor-errorinduced PIG navigation error, the overall measurement updates include:


Furthermore, these updates are both integrated by EKF and Rauch-Tung-Striebel smoother (RTSS) to estimate and correct the errors of micro-inertial sensors and the PIG navigation system [21].

Figure 7. Schematic of micro-inertial/AGM/odometer/PJ-based pipeline navigation system.

#### 4.2. Measurement models of micro-inertial/AGM/odometer/PJ system

The system error model is provided in Section 2.2. For the measurement state variables, there are four kinds of measurement models when PIG staying in different stages of the pipeline [22].

Firstly, when there are no AGMs in SPS, the measurement model of micro-inertial/odometer/PJ system is given by:

$$\begin{bmatrix} \boldsymbol{\upsilon}\_{e,m} - \boldsymbol{\upsilon}\_{e,S\text{INS}} \\\\ \boldsymbol{\upsilon}\_{n,m} - \boldsymbol{\upsilon}\_{n,S\text{INS}} \\\\ \boldsymbol{\upsilon}\_{u,m} - \boldsymbol{\upsilon}\_{u,S\text{INS}} \\\\ \boldsymbol{p}\_{Pl} - \boldsymbol{p}\_{S\text{INS}} \\\\ \boldsymbol{A}\_{Pl} - \boldsymbol{A}\_{S\text{INS}} \end{bmatrix} = \boldsymbol{H}\_{1} \boldsymbol{\delta} \mathbf{x} - \begin{bmatrix} \delta \eta\_{v\text{c}} \\\\ \delta \eta\_{v\text{u}} \\\\ \delta \eta\_{p} \\\\ \delta \eta\_{A} \end{bmatrix} \tag{17}$$

where ve,m, vn,m, and vu,m are the 3D velocity measurement updates from odometers and PIG NHCs. ve,SINS, vn, SINS and vu, SINS are the 3D velocity calculated by SINS. δηve, δηvn, and δηvu are the velocity measurement noise. pPJ and APJ denote the pitch and azimuth angles that are calculated by SINS at the beginning of each SPS, respectively; δη<sup>p</sup> and δη<sup>A</sup> are the corresponding measurement noises. Therefore, the system measurement matrix H1 is

$$H\_1 = \begin{bmatrix} O\_{3 \ast 3} & I\_{3 \ast 3} & O\_{3 \ast 3} & O\_{3 \ast 6} \\ O\_{2 \ast 3} & O\_{2 \ast 3} & H\_{1,1} & O\_{2 \ast 6} \end{bmatrix}, H\_{1,1} = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 0 & 1 \end{bmatrix}$$

Secondly, when there are AGMs in SPS, the system measurement model of micro-inertial/ AGM/odometer/PJ is expressed as:

$$\begin{bmatrix} \rho\_{AGM} - \rho\_{S\text{SNS}} \\ \lambda\_{AGM} - \lambda\_{S\text{SNS}} \\ h\_{AGM} - h\_{S\text{SNS}} \\ \upsilon\_{e,m} - \upsilon\_{e,S\text{SNS}} \\ \upsilon\_{n,m} - \upsilon\_{n,S\text{SNS}} \\ \upsilon\_{u,m} - \upsilon\_{u,S\text{SNS}} \\ p\_{Pl} - p\_{S\text{SNS}} \\ A\_{Pl} - A\_{S\text{SNS}} \end{bmatrix} = H\_2 \delta \mathbf{x} - \begin{bmatrix} \delta \eta\_{\varphi} \\ \delta \eta\_{\lambda} \\ \delta \eta\_{\mu} \\ \delta \eta\_{vu} \\ \delta \eta\_{vu} \\ \delta \eta\_{p} \\ \delta \eta\_{p} \\ \delta \eta\_{A} \end{bmatrix} \tag{18}$$

where φSINS, λSINS, and hSINS are the PIG position calculated by SINS mechanization. φAGM, λAGM, and hAGM are the AGM position provided by DGPS. δηφ, δηλ, and δη<sup>h</sup> denote the AGM position measurement noise. Hence, the system measurement matrix H2 is:

$$H\_2 = \begin{bmatrix} I\_{6 \ast 6} & \mathcal{O}\_{6 \ast 3} & \mathcal{O}\_{6 \ast 6} \\ \mathcal{O}\_{2 \ast 6} & H\_{2,1} & \mathcal{O}\_{2 \ast 6} \end{bmatrix}, and \ H\_{2,1} = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 0 & 1 \end{bmatrix}.$$

Thirdly, when there are no AGMs in PJ part, the system measurement model of micro-inertial/ odometer is expressed as:

$$\begin{bmatrix} \boldsymbol{\upsilon}\_{\boldsymbol{e},\boldsymbol{m}} - \boldsymbol{\upsilon}\_{\boldsymbol{e},\text{SINS}} \\ \boldsymbol{\upsilon}\_{\boldsymbol{n},\boldsymbol{m}} - \boldsymbol{\upsilon}\_{\boldsymbol{n},\text{SINS}} \\ \boldsymbol{\upsilon}\_{\boldsymbol{u},\boldsymbol{m}} - \boldsymbol{\upsilon}\_{\boldsymbol{u},\text{SINS}} \end{bmatrix} = H\_3 \delta \mathbf{x} - \begin{bmatrix} \delta \boldsymbol{\eta}\_{\text{re}} \\ \delta \boldsymbol{\eta}\_{\text{vn}} \\ \delta \boldsymbol{\eta}\_{\text{vu}} \end{bmatrix} \tag{19}$$

and the system measurement matrix H3 is:

$$H\_3 = \begin{bmatrix} O\_{3 \ast 3} & I\_{3 \ast 3} & O\_{3 \ast 9} \end{bmatrix}.$$

Fourthly, when there are AGMs in the PJ part, the system measurement model of microinertial/AGM/odometer is

$$\begin{bmatrix} \boldsymbol{\rho}\_{\text{AGM}} - \boldsymbol{\rho}\_{\text{SINS}} \\ \lambda\_{\text{AGM}} - \lambda\_{\text{SINS}} \\ \boldsymbol{h}\_{\text{AGM}} - \boldsymbol{h}\_{\text{SINS}} \\ \boldsymbol{v}\_{\text{t,m}} - \boldsymbol{v}\_{\text{t,SINS}} \\ \boldsymbol{v}\_{\text{t,m}} - \boldsymbol{v}\_{\text{t,SINS}} \\ \boldsymbol{v}\_{\text{t,m}} - \boldsymbol{v}\_{\text{t,SINS}} \end{bmatrix} = \boldsymbol{H}\_{4} \boldsymbol{\delta} \mathbf{x} - \begin{bmatrix} \delta \eta\_{q} \\ \delta \eta\_{\lambda} \\ \delta \eta\_{h} \\ \delta \eta\_{v\text{e}} \\ \delta \eta\_{v\text{u}} \\ \delta \eta\_{v\text{u}} \end{bmatrix} \tag{20}$$

The system measurement matrix H4 is

$$H\_4 = \begin{bmatrix} I\_{6\*6} & O\_{6\*9} \end{bmatrix}$$

During the measurement update stage of micro-inertial/AGM/odometer/PJ-based pipeline navigation system, the odometers and NHCs of PIG provide 3D continuous velocity updates, AGMs provide 3D sporadic coordinate updates, and PJ detection provides continuous azimuth and pitch updates in SPS. Therefore, when obtaining the EKF gain KFk, system states updates δx^Fk <sup>þ</sup> and system states covariance matrix P+ Fk; the system design matrix Hk and the measurement covariance matrix Rk are calculated by the system measurement updates zk when PIG in SPS or PJ part.
