Trajectory tracking control of underwater tracked vehicle considering ICR longitudinal deviation compensation

Abstract

During the operation of an underwater tracked vehicle on a soft substrate, the instantaneous center of rotation (ICR) no longer coincides with the geometric center after the longitudinal offset is accounted for by the external environment. Due to this, designing an underwater trajectory tracking system based on a kinematic model is a challenging task. In order to achieve the high-precision trajectory tracking control of underwater tracked vehicles, this paper proposes a trajectory tracking algorithm based on the kinematic model of tracked vehicles. First, an inverse tangent algorithm is proposed to address the problem that the front viewpoint of the Stanley algorithm may not be on the target path and cause deviation. Second, by using the inverse tangent algorithm to introduce the target bow angle, as well as combining the mechanisms by which biological endocrine hormones are regulated with the advantages of single-neuron proportional–integral–derivative (PID), a composite endocrine intelligent control law is proposed for the design of the lateral control law of an underwater tracked vehicle; a longitudinal control law is designed based on the longitudinal offset of the ICR obtained by the estimation of the unscented Kalman filter as a means of compensating for longitudinal errors. Finally, a kinematic model is used in order to determine the desired drive wheel speed of the underwater tracked vehicle. The method is compared with the method without longitudinal deviation compensation, then with the methods presented in related literature, and finally, a certain degree of experimental validation is carried out. These results validate the effectiveness of the proposed method.

Keywords

Underwater tracked vehicle trajectory tracking inverse tangent algorithm endocrine single-neuron PID

Introduction

As science and technology develop and the economy develops, the consumption of resources is constantly increasing, however, people have mainly exploited and utilized land resources for centuries, resulting in diminishing reserves of land resources for a long time. Compared with land, the ocean, which occupies 71% of the earth’s surface area, contains an abundance of resources. If safe and efficient methods of exploitation and utilization are established, the problem of resource shortages today can be significantly alleviated and the benefits of developing marine resources will be much greater (Du et al., 2024; Guo et al., 2023; Zereik et al., 2018). In an effort to solve the resource shortage problem, many scholars have turned their attention to the sea, and a series of in-depth studies on the use of marine resources has been conducted by the researchers. During ocean exploration, ordinary underwater vehicles are prone to skidding, idling, sideways tilting, and other special conditions due to the saturated and weak seabed soil. Compared with wheeled vehicles, tracked vehicles are increasingly used in rough terrain conditions due to their larger contact areas and greater traction on different terrain types (Han et al., 2011). In order to develop marine technologies, amphibious tracked vehicles have become an essential tool, with tracked amphibious robots being among the most widely used amphibious robots today due to their high traction, strong obstacle-crossing capability, and moderate structural and control complexity.

A tracked robot can be compared with a tracked vehicle, and there are more research results available (Liu et al., 2014). Generally, tracked vehicles are used for seabed surveys and mining in marine development. The University of Siegen, Germany, is one of the earliest institutions to start researching tracked vehicle seabed mining vehicles. During the working process, the mining vehicle adopts a swingable frame, which is adaptable to the complex terrain of the seabed. At the same time, it performed well over obstacles (Deepak et al., 2007). The Australian company SMD has developed a tracked seabed trenching machine for mining polymetallic sulfides on seabeds. In 2018, the deep-sea tracked mining vehicle “Kunlong 500,” which was developed under the leadership of the Changsha Institute of Mining and Metallurgy of China Minmetals, completed the 500-m sea trial.

The research on land-tracked vehicles is mainly based on the linear feedback of the vehicle dynamics model, and the control system input is controlled by adjusting the motor torque, which makes it an ideal control method for land-tracked vehicles for predictive and robust control (Li et al., 2021; Zhang et al., 2021). Because of the inconsistent mechanical properties of the seafloor substrate and the complex interactions between the tracks and the terrain, underwater tracked vehicles have difficulties establishing accurate underwater tracked vehicle dynamics models, unlike land-tracked vehicles. Consequently, underwater tracked vehicles are generally difficult to control using control algorithms based on dynamics model design.

Model predictive control (MPC) algorithm designed based on kinematic models has been applied to tracking trajectories of tracked vehicles (Al-Mayyahi et al., 2020; Bai et al., 2020; Chen et al., 2022), and this algorithm can accurately track the preset trajectories. However, when solving optimization problems, MPC has the disadvantage of having a long online computation time and high CPU occupancy. Sun et al. (2020) proposed a method for accelerating the solution of linear quadratic programming optimization problems. However, due to the limitations of the solution library, the optimization problems associated with MPC are often insoluble in practical engineering settings. A trajectory tracking controller based on MPC has been developed by Li et al. (2023). This paper proposed an MPC-based strategy that incorporates sliding and smoothing (MPC-CSS), resulting in a reduction of track slippage and an improvement in operational stability. MPC, however, is a challenging regulation and requires a high level of system modeling ability. A fuzzy proportional–integral–derivative (PID) algorithm was applied by Dai et al. (2020, 2021) to track the trajectory of an underwater tracked vehicle. This vehicle tracked the preset trajectory within the given conditions. Still, there was still room for improvement in the accuracy of trajectory tracking. The literature (Huang et al., 2018) proposes a proportional–integral (PI) controller based on the lateral error of the look-ahead point, which calculates and controls the look-ahead lateral error by estimating the robot’s position and bow angle errors in real time. It, however, requires an accurate kinematic model that takes into account a greater number of variations, as well as additional work to determine the optimal forward-looking distance. Based on a slip kinematic model, Zhao et al. (2019) proposed a six-parameter slip parameter estimation algorithm. However, it has not been validated under complex underwater conditions. Based on parameter learning and hybrid learning methods, literature (Dai et al., 2018) has developed an adaptive neuro-fuzzy inference system algorithm by combining fuzzy control with neural network control but has only been able to track the trajectory of deep-sea tracked vehicles within an allowable error of 0.5 m. Hang et al. (2019) designed a linear variable-parameter H∞ controller, in which the required torque was distributed on both sides of the wheels using the weighted least squares (WLS) algorithm. However, H∞ relies heavily on a linear combination of its estimator and the state it wants to estimate.

Aside from the above methods, there are other schemes that can be used to design controllers for tracked vehicles based on kinematic models (Bian et al., 2018; Li and Zou, 2007; Yoon et al., 2014); however, when designing algorithms based on kinematic models, it is difficult to obtain an accurate kinematic model of underwater tracked vehicles because the instantaneous center of rotation (ICR) and geometric center no longer coincide due to the irregular slip between the track and the ground. In this regard, it is difficult to obtain an accurate kinematic model of an underwater tracked vehicle. As a result, numerous studies have been conducted to obtain accurate kinematic models (Jiao et al., 2014; Yi et al., 2009). Xiong et al. (2017) developed a kinematic model based on the instantaneous steering center of tracked vehicles. Based on the deviation between the calculated and measured values of the relative position of the vehicle, the Levenberg–Marquardt algorithm is used to solve the sliding parameter iteratively, thus enabling the accurate prediction of the vehicle’s motion trajectory. A double-layer adaptive untraceable Kalman filter (DAUKF) algorithm that compensates for forward trajectory prediction has been proposed by Qin et al. (2021) for real-time estimation of the ICR position. By analyzing the current state and forward trajectory of a tracked vehicle, the algorithm can predict the trend of the ICR position, improving its convergence speed and accuracy of estimation. Both methods can accurately estimate the position of the ICR to obtain an accurate kinematic model of the underwater tracked vehicle.

In this paper, based on the nonlinear feedback idea of the optimized inverse tangent lateral control algorithm, the target bow angle of the underwater tracked vehicle is calculated from the relative position of the underwater tracked vehicle and the current target point, and the lateral control algorithm of the underwater tracked vehicle is proposed; considering that the ICR of the underwater tracked vehicle is affected by the water resistance when they walk on sparse and soft substrate, the longitudinal offset of the underwater tracked vehicle is estimated by the unscented Kalman filter (UKF) algorithm and is substituted into the longitudinal error calculation; as a result of the kinematic model, the desired angular and linear velocities of the underwater tracked vehicle obtained by the algorithm are converted into the rotational speeds of the drive wheels in order to ensure accurate tracking of the preset trajectory.

The innovations of this paper are as follows: (1) the development of a new control method based on a kinematic model incorporating ICR to reduce the design cost of the control system due to the difficulty of establishing an accurate dynamics model; (2) the proposal of an inverse tangent algorithm to overcome the limitations of the traditional Stanley’s algorithm; and (3) a two-stage control structure can be proposed that improves the system’s response speed by combining the endocrine framework with a single-neuron PID.

Kinematic modeling of underwater tracked vehicles

Under complex working conditions, underwater tracked vehicles are prone to unexpected slipping phenomena, leading to ICR offsets; at this time, the traditional kinematic model will not be able to accurately describe the motion behavior of the underwater tracked vehicle if it does not take into account ICR. As a result, this paper adopts an ICR-based kinematic model of the underwater tracked vehicle. As shown in Figure 1, v_F and v_E correspond to the implicated velocities at point F of the high-speed side track and point E of the low-speed side track; v_i and v_o represent the grounded-end speeds of the high-speed side track and the low-speed side track; (x_o, y_o) and (x_i, y_i) represents the coordinate values of ICR at the grounded end of the high-speed side track and low-speed side track; (x_c, y_c) represents the coordinate value of the underwater tracked vehicle’s ICR; B is the distance between each track plate inter-tracking plate; v is the linear velocity of the underwater tracked vehicle; ω is its angular velocity; v_x and v_y are the longitudinal and lateral velocities of the underwater tracked vehicle; and φ is the bow angle of the underwater tracked vehicle.

The tracked vehicle track rotates around its ICR, and the absolute speeds V_F and V_E at points F and E are as follows

{\begin{matrix} V_{F} = (- y_{o} + B / 2) ω \\ V_{E} = - (y_{i} + B / 2) ω \end{matrix}

(1)

Using rigid-body kinematics, the absolute velocities of points F and E are related as follows

{\begin{matrix} V_{F} = v_{F} - v_{o} \\ V_{E} = v_{E} - v_{i} \end{matrix}

(2)

where $v_{o} = ω_{o} r_{d}$ ; $v_{i} = ω_{i} r_{d}$ ; r_d is the radius of the driving wheel; and ω_o and ω_i are the rotational speeds of the driving wheel for the high-speed side track and the low-speed side track, respectively.

The underwater tracked vehicle rotates around the ICR, so the implicated velocities at points F and E are as follows

{\begin{matrix} v_{F} = v_{c'} + \frac{B}{2} ω = v_{x} + \frac{B}{2} ω \\ v_{E} = v_{c'} - \frac{B}{2} ω = v_{x} - \frac{B}{2} ω \end{matrix}

(3)

where $v_{c}'$ is the linear velocity of the geometric center c′ of the underwater tracked vehicle.

From the theory of rigid-body kinematics, the coordinates of the ICR of the underwater tracked vehicle satisfy the following relationship

{\begin{matrix} y_{c} = \frac{v_{x}}{ω} \\ x_{c} = x_{i} = x_{o} = \frac{v_{y}}{ω} \end{matrix}

(4)

Associations (1)–(3) are obtained

{\begin{matrix} y_{o} = \frac{v_{o} - v_{x}}{ω} \\ y_{i} = \frac{v_{i} - v_{x}}{ω} \end{matrix}

(5)

Associations (4) and (5) are obtained

{\begin{matrix} v_{x} = \frac{y_{o} v_{i} - y_{i} v_{o}}{y_{i} - y_{o}} \\ v_{y} = \frac{v_{i} - v_{o}}{y_{i} - y_{o}} x_{c} \\ ω = \frac{v_{i} - v_{o}}{y_{i} - y_{o}} \end{matrix}

(6)

The kinematic differential equation of the underwater tracked vehicle can be shown by the following equation

[\begin{matrix} \overset{\cdot}{X} \\ \overset{\cdot}{Y} \\ \overset{\cdot}{φ} \end{matrix}] = [\begin{matrix} \sin φ & \cos φ & 0 \\ - \cos φ & \sin φ & 0 \\ 0 & 0 & 1 \end{matrix}] [\begin{matrix} \frac{y_{o} v_{i} - y_{i} v_{o}}{y_{i} - y_{o}} \\ \frac{v_{i} - v_{o}}{y_{i} - y_{o}} x_{c} \\ \frac{v_{i} - v_{o}}{y_{i} - y_{o}} \end{matrix}]

(7)

Trajectory tracking algorithm for longitudinal offset compensation

The first part of the chapter is devoted to the design of the trajectory tracking algorithm for the underwater tracked vehicle in accordance with the Stanley method. The characteristics of the Stanley algorithm are analyzed, and then a trajectory tracking algorithm based on the inverse tangent algorithm is proposed as a solution to its shortcomings. This method proves to be superior based on the comparison. In the final part, the underwater tracked vehicle’s lateral and longitudinal control algorithms are developed.

Trajectory tracking algorithm based on Stanley’s algorithm

A nonlinear feedback function is used in the Stanley algorithm to correct lateral tracking errors. The Stanley algorithm is typically applied to the bicycle model when it comes to the tracking of unmanned vehicles. Stanley’s algorithm for underwater tracked vehicles cannot be directly applied to the bicycle model as a result of the difference in steering methods. Unlike four-wheeled vehicles, tracked vehicles can only steer by controlling the difference in speed between the active wheels on both sides of the track., whereas four-wheeled vehicles can steer by controlling the angle of their front wheels.

The Stanley algorithm for underwater tracked vehicles developed in this paper is based on the tracked vehicle model and optimizes the adaptation to discrete paths by adjusting the target point from the nearest point to the current position to the next point on the path by adjusting the target point to the closest point on the path, with the virtual point being the point d(t) away from the target point in the direction of the target bow of the target point. It is not necessary to continuously update the target point. The target point only changes when the underwater tracked vehicle reaches the next point on its path.

Define the coordinates of the current position of the underwater tracked vehicle under the geodetic coordinate system as (x, y, θ), the coordinates of the target point as (P_x, P_y, P_θ), and the distance from the underwater tracked vehicle to the target point as e.

As a result, the distance between the underwater tracked vehicle and the virtual target point is

a = \sqrt{e^{2} + d {(t)}^{2} - 2 ed (t)}

(8)

Then, the required target bow angle δ of the robot can be expressed as

δ = θ - (P_{θ} - \arccos (\frac{a^{2} + d {(t)}^{2} - e^{2}}{2 ad (t)}))

(9)

Stanley’s algorithm, however, has the disadvantage that the path tracking effect differs depending on the front view point. Thus, the study and determination of the front view point is also important. There is a risk of bias in Stanley’s algorithm due to the fact that the front viewpoint may not be aligned with the target path, as shown in Figure 2.

Trajectory tracking algorithm for tangent curves

The forward-looking distance of Stanley’s method is kept constant, which results in significant differences in its tracking effect when a path has different curvatures. Therefore, tangent curve tracking is proposed as a solution to this problem.

In this algorithm, due to the discretization of the path points, the nearest point used in the classical Stanley algorithm may not exist in the discretized path points. Consequently, this algorithm determines the nearest point to the current position in the continuous path based on the current position of the vehicle, and then selects the next discrete point in the discrete path as the target point. The trajectory tracking error is determined by the vertical difference between the current position of the underwater tracked vehicle and the target bow at the target point. Inverse tangent functions are then used to calculate the bow angle of the tracked vehicle at the present moment. An important advantage of this algorithm lies in the characteristics of the tangent function. When the error is significant, the value calculated using the tangent function is close to the right angle, that is, at this time, the tracked vehicle is approaching at the shortest distance concerning the target path so that the deviation value can converge at the fastest speed. When the error is small, the inverse tangent function calculates a relatively small deviation angle, so that the tracked vehicle can approach the target point with a slight bow deviation, thus ensuring a small and stable rate of error reduction, and at the same time, preventing the bow angle from being over-adjusted, that is, exceeding the target point by too large a distance.

The underwater tracked vehicle trajectory tracking target bow angle is defined as shown in Figure 3, where $P 1 = (x_{n}, y_{n}, θ_{n})$ represents the current target point and $P 0 = (X, Y, θ)$ represents the current position of the tracked vehicle. The dashed line OP1 represents the extension of the target point P1 in the direction of its bow coordinate $θ_{n}$ , whereas point O represents the vertical foot of the tracked vehicle position P0 to the line OP1.

Currently, the target bow equation for the tracked vehicle is as follows

θ t = \arctan (| D 1 |) * jug (P 0, P 1)

(10)

where D1 is the tracking error, which is the distance between point P0 and point OP1, that is, the difference between the current position of the tracked vehicle and the vertical direction of the target bow direction.

Based on the coordinates of PO and P1, D1 can be calculated as follows

D 1 = \sqrt{\begin{matrix} (x_{n} - X)^{2} + (y_{n} - Y)^{2} - pr j^{2} \\ ([x_{n} - X, y_{n} - Y], [\cos θ_{n}, \sin θ_{n}]) \end{matrix}}

(11)

where $prj (a, b)$ is the projection of vector a on vector b, and $jug (P 0, P 1)$ is the judgment coefficient that denotes the direction of θ_t

jug (P 0, P 1) = {\begin{matrix} - 1 | (x_{n} - X) * \sin (θ_{n}) - (y_{n} - Y) * \cos (θ_{n}) > 0 \\ 0 | (x_{n} - X) * \sin (θ_{n}) - (y_{n} - Y) * \cos (θ_{n}) = 0 \\ 1 | (x_{n} - X) * \sin (θ_{n}) - (y_{n} - Y) * \cos (θ_{n}) < 0 \end{matrix}

(12)

According to equation (9), when calculating the target bow angle, the classical Stanley algorithm requires information about the target point and the actual point as well as the length of the set d(t). In order to enhance the effectiveness of the Stanley algorithm, the size of the set d(t) must also be considered. For the optimal d(t) must be designed and verified by simulation over and over again, resulting in an increased design workload. By using the inverse tangent algorithm, the target point position and angle are only required to calculate the bow angle, eliminating the need to search for a suitable d(t) and simplifying the algorithm design.

The tangent direction is the bow direction of the design target, and curve 1 starts from P0. Along the curve l, the tracked vehicle will eventually move to the dotted line OP1. If D1 is larger, θ_t is greater, the tracked vehicle approaches the dotted line OP1 faster, and the system is more rapid; if D1 is smaller, θ_t is smaller, the tracked vehicle approaches the dotted line OP1 slower, and the system is more stable.

Endocrine single-neuron PID-based bow controller

Using the designed inverse tangent trajectory tracking algorithm, the tracked vehicle can determine the target bow angle required to complete the trajectory tracking at the present moment. Afterward, the bow angle of the tracked vehicle must be controlled to ensure that it follows the target bow angle determined by the trajectory tracking algorithm.

This paper utilizes the bow deviation as an input to the PID controller in order to calculate the desired angular velocity of the underwater tracked vehicle. Traditional PID controls, however, have an insignificant effect in strongly nonlinear systems or systems with a high degree of complexity without a deterministic model. Due to the complexity of the force situation and the high nonlinearity of the system, it is difficult to guarantee the control effect of traditional PID on underwater tracked vehicles, so an endocrine PID method is used to control the angular velocity of the tracked vehicles. In the endocrine PID control, a new secondary controller is added to traditional PID controls, which dynamically adjusts the PID control coefficients depending on the error and its changes in real time. As a result, the controller is more precise and responsive and can track the given value more rapidly and consistently.

For better control, the secondary controller in the endocrine control system is replaced by a single-neuron PID, which is more adaptable and robust, resulting in a composite controller—the endocrine single-neuron PID. The error signals input to the single-neuron PID are all subjected to a converter action before getting the single-neuron input signal $X_{i} (k)$ (i = l, 2, 3)

{\begin{matrix} \begin{matrix} x_{1} (k) = e_{2} (k) \\ x_{2} (k) = \sum_{j = 0}^{k} e_{2} (j) \end{matrix} \\ x_{3} (k) = Δ e_{2} (k) = e_{2} (k) - e_{2} (k - 1) \end{matrix}

(13)

The single neuron finally outputs the control signal $u_{2} (k)$

u_{2} (k) = K_{2} \sum_{i = 1}^{3} w_{i} (k) x_{i} (k)

(14)

where K₂ > 0 is the neuron’s gain; $w_{i} (k)$ is the weighted value corresponding to $x_{i} (k)$ , which is equivalent to the three coefficients K_p, K_i, and K_d in the PID controller, but it realizes adaptive tuning according to the supervised Delta learning rule

w_{i} (k + 1) = w_{i} (k) + d_{i} e_{2} (k) x_{i} (k)

(15)

where d_i > 0 (i = l, 2, 3) is the neuronal learning rate.

This composite formation of endocrine single-neuron PID control can change the controller parameters, that is, the weighting value $w_{i} (k)$ , in real time via its own learning process, increasing the adaptability of this secondary controller to changes in the environment, resulting in improved control of the device.

The ICR sliding parameters of the underwater tracked vehicle are estimated by UKF.

$r (k) = [y_{0} (k); y_{i} (k); x_{c} (k)]$ is assumed to be the sliding parameter of the underwater tracked vehicle. Here are how the prediction weights are calculated, where $ω_{m}$ is the corresponding approximate value weight and $ω_{c}$ is the corresponding approximate covariance weight

{\begin{matrix} ω_{m}^{(0)} = λ / (n + λ) \\ ω_{c}^{(0)} = λ / (n + λ) + 1 - α^{2} + β \\ ω_{m}^{(i)} = ω_{c}^{(i)} = 1 / 2 (n + λ) \end{matrix}

(16)

where $λ = α^{2} (n + κ) - n$ ; β is a non-negative constant, take β = 2 according to the empirical formula; α takes the value range of $e^{- 4} \leq a \leq 1$ , take α = 1 according to the empirical formula; κ is a constant, take κ = –1; and n = rank( P ) is the rank of covariance matrix P .

A lateral and longitudinal deviation of the underwater tracked vehicle is illustrated in Figure 4, where c′xy is the carrier coordinate system and OXY is the geodetic coordinate system; e_x represents the longitudinal error in the carrier coordinate system; e_y represents the lateral error in the carrier coordinate system; (X_r, Y_r) represents the current reference trajectory position in the geodetic coordinate system; v_r represents the current reference linear velocity; v is the linear velocity of the underwater tracked vehicle at the current moment; and x_c is the longitudinal deviation of the ICR.

The difference between the inverse tangent algorithm’s value and the actual bow angle is used as the input to the endocrine single-neuron PID, which produces the desired underwater tracking vehicle angular velocity $ω (k + 1)$ .

The instantaneous steering center of the underwater tracked vehicle does not coincide with the geometric center due to its longitudinal offset. As a result, in the longitudinal control, the $x_{c} (k)$ calculated from the UKF estimation is used to compensate for the longitudinal deviation $e_{x} (k)$ , and then the actual longitudinal deviation $e_{L}$ is calculated. Using the following formula, the desired linear velocity $v (k + 1)$ of the underwater tracked vehicle can be determined at time k + 1:

v (k + 1) = k_{L} e_{L} + k_{v} e_{v}

(17)

where $e_{L} = e_{x} (k) - x_{c} (k)$ , $e_{v} = v_{r} - v (k)$ , v_r is the reference linear velocity at the current moment, x_c(k) is the longitudinal offset of the ICR, and k_L and k_v are scale factors.

From Figure 1, the underwater tracked vehicle conforms to the kinematic relationship

v^{2} = v_{x}^{2} + v_{y}^{2} = v_{x}^{2} + (ω x_{c})^{2}

(18)

This equation is used to calculate the rotational speed of the drive wheel of the underwater tracked vehicle at time k + 1

{\begin{matrix} ω_{o} (k + 1) = (2 v_{x} (k + 1) + B ω (k + 1)) / r_{d} \\ ω_{i} (k + 1) = (2 v_{x} (k + 1) - B ω (k + 1)) / r_{d} \end{matrix}

(19)

where v_x (k + 1) is calculated from equation (18).

The overall control block diagram is shown in Figure 5.

Simulation analysis

Simulation environment and parameter settings

In this paper, the chassis simulation model of the underwater tracked vehicle is built based on SolidWorks/Adams, as well as the control system of the underwater tracked vehicle, which was validated using the MATLAB/Simulink software.

The parameters of the underwater tracked vehicle model are shown as follows: L = 0.85 m is the length of the underwater tracked vehicle; W = 0.72 m is the width of the underwater vehicle; H = 0.12 m is the height of the underwater vehicle; M = 86 kg is the mass of the underwater vehicle; and A = 0.04 m² is the surface area facing the water.

The mechanical parameters of the track and the contact surface are set in Adams to simulate the sparse and soft substrates encountered by the underwater tracked vehicle during operation; the external loading force is added to simulate the underwater tracked vehicle’s water resistance, which is calculated using the following equation

F_{w} = \frac{1}{2} C_{d} ρ_{w} A | v_{w} - v | (v_{w} - v)

(20)

where C_d = 2 is the water resistance coefficient; ρ_w is the water density, kg/m^3;v_w is the water velocity, m/s; and A is the surface area facing the water, m².

According to the simulation analysis, the velocity of the water v_w=1 m/s, the density of the water ρ_w = 1,037 kg/m³, and the reference linear velocity of the underwater tracked vehicle v_r = 3 m/s. Based on the commissioning experience, covariance matrix R = diag (10^–3, 10^–3) for measurement noise and Q = diag (10^–4, 10^–4, 10^–4) for process noise.

Simulation results and analysis

In this paper, simulation experiments are conducted under two different types of sinusoidal curve working conditions. The tracking vehicle’s initial point is not on the given trajectory in order to study the effects of the designed trajectory tracking algorithm.

In the following, Longitudinal Deviation Compensation (LDC) represents the experimental group in which longitudinal deviation compensation was introduced, and No Longitudinal Deviation Compensation (NLDC) represents the experimental group in which longitudinal deviation compensation was not introduced.

Simulation results for Condition I

The trajectory tracking results of the underwater tracked vehicle for Condition I are shown in Figure 6(a), from which it can be seen that the trajectory tracking method proposed in this paper can effectively track the preset trajectory. The underwater tracked vehicle tracked the trajectory after traveling only about 5 m after the introduction of longitudinal deviation compensation, whereas without longitudinal deviation compensation, it traveled close to 20 m to track the trajectory. In addition, the underwater tracked vehicle generated less deviation when turning after the introduction of longitudinal deviation compensation, which resulted in greater tracking accuracy as a result.

The deviation of trajectory tracking for Condition I is shown in Figure 6(b). From Figure 6(b), it can be seen that after the introduction of longitudinal deviation compensation, the transverse deviation convergence time is reduced from 5 to 2.5 seconds, and the peak deviation value after stabilization is reduced by about 60%. The longitudinal deviation convergence time is reduced from 3 to 2 seconds, and the peak deviation value after stabilization is reduced by approximately 60%. Due to the fact that the initial point is not on the path, the bow angle deviation is corrected more substantially at the beginning stage, resulting in a faster convergence of the deviation and a reduction of about 30% in peak values.

In Condition I, the velocity change of the underwater tracked vehicle during trajectory tracking shows that the velocity deviation of the underwater tracking vehicle is smaller and smoother compared with the preset velocity.

Simulation results for Condition II

In Figure 7(a), the comparison of trajectory tracking results is shown after introducing longitudinal deviation compensation and without introducing longitudinal deviation compensation. According to the trajectory tracking results in this paper, the trajectory tracking method proposed in this paper can also effectively track the trajectory of the underwater tracked vehicle in Condition II. In the case of longitudinal deviation compensation, the underwater crawler tracked the trajectory after traveling approximately 3.5 m, while in the absence of longitudinal deviation compensation, it traveled almost 22 m to track the trajectory. In addition, the underwater crawler generated less deviation during turns after the introduction of longitudinal deviation compensation, which increased tracking accuracy.

The comparison of lateral deviation, longitudinal deviation, and bow angle deviation is shown in Figure 7(b). From Figure 7(b), it can be seen that after the introduction of longitudinal deviation compensation, the convergence time of transverse deviation is reduced from 5 to 2 seconds, and the peak value of the deviation after stabilization is reduced by about 63%; the longitudinal deviation does not produce amplitude changes with the change of path curvature and is always stabilized between −0.05 and 0.05 m, whereas the longitudinal deviation without the introduction of longitudinal deviation compensation produces a larger amplitude with the increase of path curvature; the bow angle deviation peak value is reduced by approximately 66%.

After the introduction of longitudinal deviation compensation, the speed change of the underwater tracked vehicle during trajectory tracking is shown in Figure 7(b). From the simulation results, it can be seen that the underwater tracking vehicle has smooth velocity changes and good stability during trajectory tracking.

Compare with other methods

The proposed control scheme is compared with the method proposed by (Dai et al., 2021) using the same simulation environment as the previous working conditions in order to verify its superiority. The results are shown in Figure 8. The figure illustrates that (1) both methods are capable of effectively tracking paths and (2) the error of the proposed method in this paper stabilizes within 2 seconds, whereas the error of the other method stabilizes within 10 seconds, which indicates that the proposed method provides faster convergence while ensuring accuracy in tracking.

Experimental verification and analysis

Experimental setup

In this experiment, a small pool was constructed in the laboratory to be used for testing and validation. The hardware architecture of the tracked vehicle is shown in Figure 9. All control schemes are based on the robot operating system (ROS). In terms of sensing, the fiber optic inertial guidance and Doppler velocity log (DVL) are combined to obtain information about the position and attitude of the tracked vehicle, and the topic is released through the ROS node. The global path planning node discretizes the set path and sends it to the controller node, which calculates the required left and right driving wheel rotational speeds of the tracked vehicle, and then sends it down to the STM32 control board to output the corresponding pulse width modulation (PWM) wave, and finally driven by the electronic speed controller (ESC).

Experimental results of trajectory tracking

In this experiment, the reference path is a circular path with a radius of 2 m, the center of the circular path serves as the relative origin, and the initial position coordinates of the tracked vehicle are (0, 1.5). Based on the experimental results, a trajectory tracking diagram and a velocity diagram are drawn, as shown in Figure 10. The following conclusions can be drawn: (1) the proposed method is capable of tracking the reference path effectively according to the set speed, with very small tracking errors after stabilization; (2) the tracked vehicle is also capable of tracking the reference path fast even when its initial position is not on the path; and (3) the above experiments are conducted in a pool, and the external disturbance is small, so they should be verified further in the case of large external disturbances.

Summary

A trajectory tracking algorithm for underwater tracked vehicles is proposed based on a kinematic model of tracked vehicles considering an ICR.

A trajectory tracking algorithm for tangent curves is proposed to address the shortcomings of Stanley’s algorithm. It is resolved the problem that Stanley’s algorithm has different tracking effects under different path curvatures because its forward-looking distance is fixed.

In longitudinal control, the longitudinal offset of the ICR is estimated by UKF, and the desired linear velocity of the underwater tracked vehicle is calculated. In lateral control, the desired bow angle of the underwater tracked vehicle is calculated based on the inverse tangent algorithm, which is used to calculate the desired angular velocity of the underwater tracked vehicle. Finally, the linear and angular velocities are transformed into the rotational speeds of the tracked vehicle drive wheels as control inputs. The simulation results show that the underwater tracking vehicle can not only track the target trajectory more smoothly, accurately, and efficiently but also the lateral longitudinal deviation and the bow angle deviation in the trajectory tracking process are smaller. In addition, the experimental results validate the proposed method’s effectiveness.

Footnotes

Appendix

Appendix A

Nomenclature

Terminology, abbreviations, or symbols	Full name or explanation
ICR	Instantaneous center of rotation
UKF	Unscented Kalman filter
MPC	Model predictive control
MPC-CSS	MPC-based strategy that incorporates sliding and smoothing
WLS	Weighted least squares
DAUKF	Double-layer adaptive untraceable Kalman filter
ROS	Robot operating system
DVL	Doppler velocity log
PWM	Pulse width modulation
ESC	Electronic speed controller
φ	The bow angle of the underwater tracked vehicle
δ	The required target bow angle of the robot
θt	The target bow of the tracked vehicle
$jug (P 0, P 1)$	The judgment coefficient that denotes thedirection of θt

Author contributions

All authors contributed to the study conception and design. Material preparation, data collection and analysis were performed by N.L. The first draft of the manuscript was written by N.L., and all authors commented on previous versions of the manuscript. All authors read and approved the final manuscript.

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

ORCID iD

Nan Li

Data availability statement

Data sharing is not applicable to this article as no data sets were generated or analyzed during the current study.

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