2. The MEMS SINS-based PIG

Currently, most of the PIG navigation system is comprised of fiber optic gyroscope (FOG) based SINS when the diameter of pipeline is over 12.<sup>00</sup> However, due to the volume constrain, the FOG SINS cannot be used in the small-diameter pipeline-based PIG. Fortunately, with the rapid development of microelectromechanical-system (MEMS) technology, the positioning precision of MEMS SINS improved greatly and it accelerates the application of MEMS SINS in small-diameter PIG. Therefore, considering the cost, size, weight, and power consumption, the small-volume MEMS SINS is superior for pipeline defects localization when its diameter is less than 12<sup>00</sup> [11]. However, the pipeline defects positioning error of MEMS SINS is divergent with the distance enlargement of the inspected pipeline. The main reason is the error of MEMS inertial sensors is much greater than that of the usually used FOG inertial sensors-based pipeline navigation application [12].

#### 2.1. The problem statement of MEMS SINS-based PIG

At present, the rear part of PIG is symmetrically installed with odometers to measure its longitudinal velocity and meanwhile reduce the slippage-induced velocity error, which is used for the reduction of the time-accumulated error of SINS in PIG. Meanwhile, zero-velocity updates in both lateral and vertical directions of cylinder-shaped PIG are provided by its nonholonomic constraint (NHC) characteristic. Hence, there are 3D continuous velocity updates for SINS of PIG. Moreover, the 3D coordinates of pipeline valves and above-ground markers (AGMs) are provided by DGPS, which are used for 3D sporadic coordinate updates for SINS of PIG [13]. Nevertheless, the continuous 3D velocity and sporadic 3D coordinate updates cannot satisfy the surveying precision requirements in small-volume MEMS SINS-based pipeline navigation system in small diameter pipeline.

\_ \_ Apart from the velocity and position errors, the attitude error (pitch error δp, roll error δr, and azimuth error δA) also degrades the positioning precision of MEMS SINS-based PIG. The change rates of PIG horizontal velocity errors δvn and δve are given by [14]:

$$\begin{cases} \delta \dot{v}\_n = -f\_u \delta p + f\_\epsilon \delta A + \delta f\_n \\\\ \delta \dot{v}\_\epsilon = -f\_u \delta r + f\_n \delta A + \delta f\_\epsilon \end{cases} \tag{1}$$

where, δfe and δfn denote accelerometer biases in Earth east and north directions. fe, fn, and fu denote acceleration components in Earth east, north, and up directions.

In Eq. (1), the value of fu is close to local Earth gravity and it is much bigger than fe and fn in PIG navigation application. Hence, the pitch and roll errors of SINS in PIG are tightly coupled with the corresponding horizontal velocity errors, and the 3D velocity errors of PIG are observable by odometers and NHC. Therefore, the azimuth error of SINS in PIG is not observable, while the pitch and roll errors are observable. The horizontal position errors of SINS in PIG are obtained by twice integration on the change rate of azimuth-error-induced horizontal velocity errors [14]:

$$\begin{cases} \delta \dot{\boldsymbol{v}}\_{n2} = \boldsymbol{f}\_{\varepsilon} \delta \boldsymbol{A} \\ \delta \dot{\boldsymbol{v}}\_{\varepsilon 2} = -\boldsymbol{f}\_{n} \delta \boldsymbol{A} \end{cases} \rightarrow \begin{cases} \delta P\_{n2}(\mathbf{t}\_{k}) = \delta P\_{n2}(\mathbf{t}\_{k-1}) + \boldsymbol{v}\_{\varepsilon} \delta A \Delta t \\\\ \delta P\_{\varepsilon 2}(\mathbf{t}\_{k}) = \delta P\_{\varepsilon 2}(\mathbf{t}\_{k-1}) - \boldsymbol{v}\_{n} \delta A \Delta t \end{cases} \tag{2}$$

where ΔδPn2 and ΔδPe2 are the azimuth-error-induced horizontal position errors. They are also related to PIG horizontal velocities ve and vn, and the time interval Δt. More intuitively, when PIG travels with 1 m/s in horizontal velocity, and 1� in azimuth error, the position error that caused by azimuth error is about 89 m in 1 h of PIG navigation. Therefore, to correct the azimuth error is an important way to enhance the navigation precision of SINS-based PIG.

At present, azimuth sensors, like camera, magnetometer, and optical navigation sensor are usually adopted to improve the measurement precision of azimuth, but both the cost and weight of PIG would increase. Moreover, their measurement precision is also severely degraded by the pipeline application [8]. So, it is not viable for these sensors to be applied to correct the azimuth error of PIG accurately. However, it is worth noting that the routed pipeline is connected by fixed-length SPS via PJ. The cylinder-shaped PIG makes the azimuth and pitch maintain constant from the beginning to the end of each SPS. Therefore, the azimuth and pitch mechanized by SINS at the beginning of SPS are usually used as updates in the corresponding SPS. Consequently, the PJ detection result provides azimuth and pitch updates for MEMS SINS, and the PJ detection would be analyzed in Section 3.

#### 2.2. The system error model of MEMS SINS-based PIG

For the SINS-based PIG navigation system, the system error model is given by [14]:

$$
\delta\dot{\mathbf{x}} = \begin{bmatrix} F\_{11} & F\_{12} & \mathbf{0}\_{3\ast 3} & \mathbf{0}\_{3\ast 3} & \mathbf{0}\_{3\ast 3} \\ F\_{21} & F\_{22} & F\_{23} & \mathbf{0}\_{3\ast 3} & R\_b^u \\ F\_{31} & F\_{32} & F\_{33} & R\_b^u & \mathbf{0}\_{3\ast 3} \\ \mathbf{0}\_{3\ast 3} & \mathbf{0}\_{3\ast 3} & \mathbf{0}\_{3\ast 3} & F\_{44} & \mathbf{0}\_{3\ast 3} \\ \mathbf{0}\_{3\ast 3} & \mathbf{0}\_{3\ast 3} & \mathbf{0}\_{3\ast 3} & \mathbf{0}\_{3\ast 3} & F\_{55} \end{bmatrix} \delta\mathbf{x} + Gw \tag{3}
$$

where

$$\begin{aligned} \;^{F}F\_{11} &= \begin{bmatrix} 0 & 0 & -\dot{\phi} \end{bmatrix} (R\_M + h) \\ \dot{\lambda} \tan \phi & 0 & -\dot{\lambda} / (R\_N + h) \\ 0 & 0 & 0 \end{bmatrix}, \newline \;^{F}F\_{21} = \begin{bmatrix} 0 & f\_u & -f\_n \\ -f\_u & 0 & f\_\varepsilon \\ f\_n & -f\_\varepsilon & 0 \end{bmatrix}, \\\;^{F}F\_{12} &= \begin{bmatrix} 0 & \frac{1}{R\_M + h} & 0 \\ \frac{1}{(R\_N + h)\cos \phi} & 0 & 0 \\ \frac{1}{(R\_N + h)\cos \phi} & 0 & 0 \end{bmatrix}, \newline \;^{F}F\_{32} = \begin{bmatrix} 0 & \frac{1}{R\_M + h} & 0 \\ \frac{-1}{R\_M + h} & 0 & 0 \\ \frac{-1}{R\_N + h} & 0 & 0 \end{bmatrix}, \\\;^{F}F\_{21} &= \begin{bmatrix} 2\omega\_\varepsilon (v\_u \sin \phi + v\_n \cos \phi) + \dot{\lambda} v\_n / \cos \phi & 0 & 0 \\ & -2\omega\_\varepsilon v\_\varepsilon \cos \phi - \dot{\lambda} v\_\varepsilon / \cos \phi & 0 & 0 \\ & -2\omega\_\varepsilon v\_\varepsilon \sin \phi & 0 & 2g/R\_N \end{bmatrix}, \end{aligned}$$

$$\begin{aligned} &F\_{22}=\begin{bmatrix}\left(v\_{\text{r}}\tan\phi-v\_{\text{u}}\right)/(\text{R}\_{\text{N}}+h) & \left(2\omega\_{\text{c}}+\dot{\lambda}\right)\sin\phi & -\left(2\omega\_{\text{c}}+\dot{\lambda}\right)\cos\phi\\-2\left(\omega\_{\text{c}}+\dot{\lambda}\right)\sin\phi & -v\_{\text{u}}/(\text{R}\_{\text{M}}+h) & -\dot{\phi}\\2\left(\omega\_{\text{c}}+\dot{\lambda}\right)\cos\phi & 2\dot{\phi} & 0\\0 & 0 & -\dot{\lambda}/(\text{R}\_{\text{M}}+h) \\ &\omega\_{\text{c}}\sin\phi & 0 & \dot{\lambda}\cos\phi/(\text{R}\_{\text{N}}+h) \end{bmatrix},\end{aligned}$$

$$\begin{aligned} F\_{31}=\begin{bmatrix}0 & 0 & -\dot{\lambda}/(\text{R}\_{\text{M}}+h) \\ -\omega\_{\text{c}}\cos\phi & -\dot{\lambda}/(\text{R}\_{\text{N}}+h)\cos\phi & 0 & \dot{\lambda}\sin\phi/(\text{R}\_{\text{N}}+h) \\ 0 & \left(\omega\_{\text{c}}+\dot{\lambda}\right)\sin\phi & -\left(\omega\_{\text{c}}+\dot{\lambda}\right)\cos\phi \\ -\left(\omega\_{\text{c}}+\dot{\lambda}\right)\sin\phi & 0 & -\dot{\phi} \\ -\left(\omega\_{\text{c}}+\dot{\lambda}\right)\cos\phi & \dot{\phi} & 0 \end{bmatrix},\end{aligned}$$

\_ \_ and, ф and λ are local latitude and longitude, ϕ ¼ vn=ðRM þ hÞ, λ ¼ ve=ðRM þ hÞ cos ϕ. RM, RN, and h are meridian radius, normal radius, and geodetic height. The system state variables T are <sup>δ</sup><sup>x</sup> ¼ ½δr<sup>n</sup> <sup>δ</sup>vn <sup>ε</sup><sup>n</sup> δω<sup>n</sup> <sup>δ</sup><sup>f</sup> n � . Rb <sup>n</sup> is the transformation matrix from body frame to navigation frame. And w is system noise, the system noise matrix is expressed [14] as

$$\mathbf{G} = \left[ \mathbf{O}\_{\mathbb{9}\ast 1}, \sqrt{2\boldsymbol{\beta}\_{ax}\sigma\_{ax}^{2}}, \sqrt{2\boldsymbol{\beta}\_{ay}\sigma\_{ay}^{2}}, \sqrt{2\boldsymbol{\beta}\_{az}\sigma\_{az}^{2}}, \sqrt{2\boldsymbol{\beta}\_{\mathbb{\hat{f}}}\sigma\_{\mathbb{\hat{f}}\mathbb{\hat{z}}}^{2}}, \sqrt{2\boldsymbol{\beta}\_{\mathbb{\hat{f}}}\sigma\_{\mathbb{\hat{f}}\mathbb{\hat{z}}}^{2}}, \sqrt{2\boldsymbol{\beta}\_{\mathbb{\hat{f}}}\sigma\_{\mathbb{\hat{f}}\mathbb{\hat{z}}}^{2}} \right]^{\mathsf{T}} \tag{4}$$

where βωx, βωy, and βω<sup>z</sup> are the reciprocals of the correlation times of autocorrelation sequence of δωx, δωy, and δωz; σωx, σωy, and σω<sup>z</sup> are variance associated with gyroscope errors. βfx, βfy, and βfz are the reciprocals of the correlation times of autocorrelation sequence of δfx, δfy and δfz; σfx, σfy and σfz are variance associated with accelerometer errors.
