## Abstract

In its working state, the bit used in underwater horizontal directional drilling (UHDD) produces a high-frequency vibration that can affect accuracy of navigation. We designed a low-pass filter with linear phase on the basis of spectral characteristics of sensor data. To improve further the accuracy of navigation, we deduce the state error model on the basis of the random walk model of acceleration and angular velocity. We use an indirect Kalman filter algorithm to correct the attitude and position of the bit used with UHDD on the basis of observations coming from our working state analysis. Last, we derive a complete navigation algorithm function, including the acquisition method of steady-state component of acceleration and angular velocity. Experimental results show that the navigation algorithm proposed in this paper can obtain accurate attitude and location information of the bit in a vibration environment.

## 1. Introduction

Shipwreck salvage is very important in maintaining waterway safety and accident identification. In the process of salvaging a shipwreck, it is necessary to pass through many steel cables under the bottom of the wreck. In the traditional method, divers dig holes from one side of the shipwreck to the other. Then, we can pass a steel cable through the hole. Any two steel cables are required to be as parallel as possible when lifting a sunken ship. So, the traditional method is very dangerous and inefficient. As entrusted by the Shanghai Salvage Bureau, we designed an underwater horizontal directional drilling (UHDD) rig (see Fig. 1) to replace the diver. To control the drilling trajectory, we mounted a guide plate on the bit. Because the force acting on the guide plate is much greater than on the other positions of the bit, the bit will move in the direction opposite to the guide plate. In this way, we can adjust the attitude of the bit to change the position of the guide plate so as to control the drilling trajectory. The working process of UHDD can be seen in Fig. 2. When the bit drills through the soil on the other side of a sunken boat, we can see the bubbles in the water that are generated by the gas flowing through the bit. The diver can find the bit from the bubbles and can then tie the cable in the ring located on the bit. The cable can pass through the soil under the sunken boat by pullback of the UHDD rig (Yang et al. 2016). Therefore, the attitude of the bit in the UHDD is very important. In our previous work, we proposed a navigation algorithm to solve the problem of attitude tracking and the position of the bit in UHDD (Yang et al. 2016) in a nonvibration environment. To improve drilling efficiency, we have modified the bit, the hydraulic system, and the pumping station. The experiments show that the drilling efficiency has been significantly improved but that the vibration of the bit has been significantly increased. The navigation algorithm proposed in our previous work does not meet the requirements of drilling in a vibration environment. Therefore, we need to design a new navigation algorithm to overcome the influence of vibration.

In an underwater environment, it is not possible to use laser sensors (Yang et al. 2011) and there are also many difficulties in using vision sensors (de Ruiter and Benhabib 2008; Stelzer et al. 2012) or the global positioning system (Sahawneh et al. 2011; Liang and Jia 2015; Chiang et al. 2011; Leung et al. 2011; Chiang and Huang 2008) because of lighting conditions and signal interference. Therefore, ultra-short baseline (USBL), Doppler velocity log, and sonar have been widely applied in navigation of underwater vehicles (Lee and Jun 2007; Li et al. 2014, 2015; Allotta et al. 2014, 2015, 2016; Morgado et al. 2011; He et al. 2015). However, the real-time response of USBL becomes worse with an increase of distance between the ship and the underwater vehicle. If there are big obstacles, then the signal of the USBL may be lost. The accuracy of acoustic sensors can easily be affected by a change in the density of seawater. Beacons (Vallicrosa et al. 2015) are also widely applied in the underwater environment. Because of the limited scope of action of the beacon, the motion range of the underwater vehicle will be greatly restricted. Given the working environment (drilling in the mud) and the design requirements of the bit, acoustic sensors are not suitable for the navigation of the bit in UHDD.

To solve the problem of attitude tracking and positioning of the bit, we installed an inertial measurement unit (IMU) inside the bit and an encoder between the shaft and the coiled tubing. The IMU can provide the linear acceleration and angular velocity of the bit, which can be used to predict the attitude and position of the bit used in UHDD (Yang et al. 2016). The encoder mounted between the shaft and the coiled tubing can provide the velocity of the coiled tubing, which can be used to calculate the velocity of the bit (Yang et al. 2016). To further improve the accuracy of the navigation algorithm, we need to obtain more observations on the working state of the UHDD rig. The steady-state component of linear acceleration and angular velocity is a very important observation because the working state of UHDD is controllable. In combination with the extended Kalman filter (EKF) algorithm, the steady-state component of linear acceleration and angular velocity can be used to suppress the error in attitude and position of the bit and to improve the accuracy of the navigation algorithm.

## 2. Data preprocessing

An ideal filter requires a linear phase in the passband. To design a suitable filter, we need to analyze the data coming from the IMU mounted on the bit used in UHDD. Therefore, we collect much information on the acceleration and angular velocity of the bit. We use two sets of data—from the drilling state and from the static state—to illustrate the problem. Because the working frequency of the bit is less than 1 Hz, we design a low-pass filter with a linear phase whose cutoff frequency is 1 Hz. The sampling frequency of the IMU is 100 Hz. Thus, the Nyquist frequency is 50 Hz. To see more clearly, we draw only a 1/10 frequency range in Fig. 3. Because the amplitude of the signal whose frequency exceeds 1.5 Hz is attenuated to zero, we need only to ensure that the phase of the signal whose frequency is less than 1.5 Hz satisfies the linear relationship. From Figs. 4 and 5 we can see the time-domain and frequency-domain characteristics of the angular velocity *p* of the bit before and after filtering. Similarly, we can obtain time- and frequency-domain characteristics of the acceleration of the bit before and after filtering.

## 3. Navigation algorithm

For long durations, the output of the strapdown inertial navigation system (SINS) is going to be strongly biased. The rate at which navigation errors grow over a long time is related to the accuracy of the IMU as well as the accuracy of the initial alignment. The errors will propagate over time and determine the accuracy of the SINS. According to the principle of an EKF, we must provide sufficient observation information to ensure that the state is observable. Therefore, how to design the observation equation is the core problem of the system design. Because of the lack of observation information, we cannot use the traditional EKF algorithm to correct the attitude and position of the bit. To make full use of the steady-state component of acceleration and angular velocity, we derive the prediction and observation equations of the state error and use the steady-state component of acceleration and angular velocity to correct the state error. The attitude and position of the bit can be obtained by subtracting the corrected state error from the predicted value of the state quantity.

### a. The state error model

In our previous work (Yang et al. 2016), we have given a detailed derivation of the state error model. For the sake of understanding the following contents, we will briefly deduce the state error model. The state and observation equations can respectively be written as

where **x**_{k} = [*b*_{ω,k}, *θ*_{k}, *b*_{n,k}, *u*_{k}, *r*_{k}] represents the state quantity, **z**_{k} = [*a*_{k}, *ω*_{k}, *u*_{k}] represents the observation, *b*_{ω,k} is the random walk of the angular velocity at time *k*, *θ*_{k} represents the Euler angle at *k*, *b*_{n,k} is the random walk of the acceleration at *k*, *u*_{k} is velocity in the fixed frame (Fig. 6) at *k*, *r*_{k} is position in the fixed frame at *k*, *a*_{k} is acceleration in the fixed frame at *k*, and *ω*_{k} is angular velocity at *k*.

The state and observation prediction equations can respectively be written as

where $xk\u2212=\u2061[b\omega ,k\u2212,\theta k\u2212,bn,k\u2212,uk\u2212,rk\u2212]$ is the predictive value of the state quantity and $zk\u2212=\u2061[ak,\omega k,uk]$ is the predictive value of the observation.

Subtracting Eq. (1) from Eq. (2), we obtain the prediction and observation equations of the state error, respectively written as

where subscript *ε* represents error, *ν*_{k−1} represents the process white noise with a zero mean and covariance *Q*, *ν*_{k} represents the observation white noise with a zero mean and covariance *R*,

### b. The error propagation model

When we use sensors to measure the observation, new features may be introduced. If the observation introduces new features, then the model of the sensor can be described as

where *ω*_{s,k−1} is system noise and *ν*_{s,k} is measurement noise. If the observation does not introduce new features, then the model of the sensor can be described as

When the state error prediction equation is executed, the sensor model also participates in the calculation. For Eq. (6), the prediction equation of the sensor model can be written as

For Eq. (7), the prediction equation of the sensor can be written as

Equations (9) and (11) compose the error model of the sensor. By bringing Eq. (3) into Eq. (9), Eq. (9) can be rewritten as

If we set $D$_{s} = $I$ and $C$_{s} = $0$, then Eq. (16) can be seen as a special case of Eq. (15). Equations (14) and (15) can be expressed as an augmented form:

### c. Execution process of the navigation algorithm

#### 1) Prediction

For prediction, the equations are

where $Pk\u22121+$ is the corrected value of the error covariance matrix at time *k* − 1 and $Pk\u2212$ is the predictive value of the error covariance matrix at *k*.

#### 2) Update

For the update, the equations are

## 4. Observations

To inhibit error accumulation from the SINS, we need to obtain enough observations. However, the UHDD rig only contains the IMU and encoder. So, we need to get observations through analyzing the working state of the UHDD rig. In the drilling state, the angular velocity of the bit on the *x* axis of the body frame may have a steady-state component. However, the angular velocity of the bit on the *y* and *z* axes of the body frame does not have a steady-state component (Yang et al. 2016). The acceleration of the bit in a fixed frame except gravity does not have a steady-state component because the drilling force and resistance change over time. In the static state, angular velocity and acceleration except gravity are zero. The steady-state component of angular velocity and acceleration except gravity is zero (Yang et al. 2016). So, we can obtain observations according to the working state of the bit.

### a. The observations of acceleration and angular velocity

In this paper we use only the steady-state component of the acceleration and the angular velocity, and we do not introduce new sensors. Therefore, Eq. (8) can be seen as a set of low-pass filters that can be used to obtain the steady-state component of the acceleration and angular velocity. The system matrix of the low-pass filter can be written as (**A**_{2}, **B**_{2}, **C**_{2}, 0), and $A$_{s}, $B$_{s}, $C$_{s}, and $D$_{s} can be written as

where **A**_{2} = [1 − dt/ds, 0; dt/ds, 1 −dt/ts], **B**_{2} = [dt/ds, 0], and **C**_{2} = [0, 1]. Here, dt = 0.01 represents the sampling period of the IMU and ts = 100 represents the time constant of the filter.

The acceleration error and the angular velocity error can be expressed as (Yang et al. 2016)

where $ax,SINSb$, $ay,SINSb$, and $az,SINSb$ are the steady-state components of the acceleration $ak\u22121+$ expressed in the body frame and $\omega x,SINSb$, $\omega y,SINSb$, and $\omega z,SINSb$ are the steady-state components of the angular velocity $\omega k\u22121+$ expressed in the body frame (Yang et al. 2016); the low-pass filter [Eq. (8)] can be used to get the steady-state component of the acceleration and the angular velocity. When UHDD works in the static state, the steady-state components of the acceleration ($ax,TRUEb$, $ay,TRUEb$, and $az,TRUEb$) in addition to gravity and the angular velocity ($\omega x,TRUEb$, $\omega y,TRUEb$, and $\omega z,TRUEb$) are zero, and Eqs. (24) and (26) can be expressed as

respectively.

In the working state, we need to control the drilling direction through adjusting the guide plate mounted on the bit of the UHDD rig. So, the angular velocity on the *x* axis of the body frame may have a steady-state component ($\omega x,TRUEb$ is not zero). The steady-state components of the angular velocity $\omega y,TRUEb$ and $\omega z,TRUEb$ are zero. The steady-state component of the acceleration in addition to gravity is zero.

### b. Observation of velocity

Because the holding power between the shaft and the coiled tubing is big enough to avoid skidding, the velocity of the coiled tubing measured by the encoder is accurate. The rigidity of coiled tubing is very large, so the coiled tubing cannot easily be bent, and the velocity of the bit on the *x* axis of the body frame is approximately equal to the velocity of the coiled tubing, which can be calculated through the encoder mounted on the shaft. The velocity of the bit on the *x* axis can be written as

where *υ*_{e} and *D* are the rotational speed and diameter of the shaft of the UHDD rig, respectively. The velocity error can be described as (Yang et al. 2016)

where $ux,SINSb$, $uy,SINSb$, and $uz,SINSb$ are velocity expressed in the body frame.

## 5. Experiments

To verify the accuracy of the navigation algorithm proposed in this paper, we perform experiments in the static state and the drilling state. In the drilling state, we control the UHDD drill for 15 min and keep it static for 2 min alternately. In all of the experiments, the pump station keeps working at different flow rates and rotation speeds. Thus, the bit of the UHDD remains in a vibration environment. The design index requires that the distance in the *x* direction of the fixed frame be approximately 20 m and that the error in the *y* direction of the fixed frame be less than 1 m. When the diver finds the drill on the other side of the sunken boat, we can obtain the actual position as shown in Fig. 7. By comparing the actual position and the output of the navigation algorithm, we can verify the accuracy of the navigation algorithm.

### a. Experiment in static state

The velocity of the bit is less affected by vibration because it can be calculated according to the encoder mounted on the shaft. Therefore, we just need to solve the effect of vibration on attitude. We do many experiments at different flow rates and rotation speeds of the pumping station. Figures 8 and 9 show the attitude of the bit at two typical flow rates and rotation speeds of the pumping station. From Figs. 8 and 9 we can see that the navigation algorithm proposed in this paper can provide accurate attitude information for the bit.

### b. Experiment in drilling state

In this section we want to verify the effectiveness of the navigation algorithm in the drilling state. Figure 10 shows the attitude of the bit in the drilling state. After 500 s, we adjust the rolling angle *ϕ* to control the drilling depth (on the *z* axis) and the lateral displacement (on the *y* axis) of the bit (Fig. 11). We found that the navigation algorithm proposed in this paper can overcome the influence of vibration and provide accurate attitude and position of the bit through comparing the measurement (see Fig. 7) and the output of the navigation algorithm. The error in the *y* direction (0.5 m) meets design requirements.

To verify the stability of the navigation algorithm proposed in this paper, we do other experiments. When there is an obstacle in the direction of drilling, we can adjust *ϕ* (orientation of the guide plate) to avoid the obstacle. Figures 12 and 13 show the attitude and position of the bit, respectively. The error in the *y* direction (0.6 m) meets design requirements. In the process of salvaging a sunken ship, we require that the lateral offset be as small as possible. Therefore, we conducted experiments to control the lateral offset. From Figs. 14 and 15, we can see that the UHDD rig has better ability to control the lateral offset. By comparing the measurement and output of the navigation algorithm, we can conclude that the navigation algorithm can be applied in a vibration environment and can provide an attitude and position of the bit that meets drilling needs (the error in the *y* direction is less than 1 m).

## 6. Conclusions

The UHDD system is very useful equipment for sunken salvage and seafloor sampling. Because of the particularity of the working environment and the restrictions on the structure of the ontology, sensors that can be used for navigation in UHDD are very limited. To obtain enough accuracy of navigation in the absence of sensors, we need to search observations through analyzing the working state of the UHDD rig and inserting the static state. To suppress the impact of vibration, we design a low-pass filter with linear phase preprocessing sensor data. Experiments show that the navigation algorithm proposed in this paper can provide accurate navigation information for the bit in the absence of sensors and in a vibration environment.

## REFERENCES

## Footnotes

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