Real-time hybrid design of tracking control and obstacle avoidance for underactuated underwater vehicles

Abstract

For underactuated underwater vehicles, a real-time hybrid design of dynamic tracking control law is proposed for trajectory tracking and obstacle avoidance. In recent works, sliding mode control (SMC) law has been presented and experimentally implemented for position tracking of an underactuated autonomous surface vessel. It is extended to the underactuated underwater vehicle case and finds it still work for trajectory tracking problem. The thruster saturation problem is considered for the real case. The major innovation is the solution of how to deal with obstacle avoidance in the predefined trajectory tracking mission. In order to deal with this problem, a hybrid control strategy is proposed for static and dynamic obstacle case respectively. Then, to show the effectiveness of the proposed method, trajectory tracking control under different conditions are conducted including static and dynamic obstacles. The experiment results show that the proposed method can deal with tracking and obstacle avoidance quite well.

Keywords

Underactuated underwater vehicles tracking control obstacle avoidance sliding mode control

1 Introduction

Tracking control is an important issue of underwater vehicle control problem. For the overactuated or full actuated underwater vehicles, the tracking control problem has been solved very well in recent years [1 –5]. A more challenging issue is the underactuated vehicles that have fewer independent control actuators than degrees-of-freedom (DOF) to be controlled. The reason is due to the fact that the motion of the underactuated vehicle in question possesses more degrees of freedom than available controls (forces and moments).

The underactuated underwater vehicle control questions have not been answered fully, yet there has been extensive research on various aspects of this problem [6 –10]. Based on the Lyapunov direct method and passive method, Jiang [11] proposed two constructive solutions for underactuated ship tracking control; Ghommam et al. [12] proposes a unified backstepping design method to solve the problem of stabilization and tracking control and the designed trajectory tracking controller has certain robustness for the environment disturbance. Pettersen and Nijmeijer [13] showed that a recursive technique for developing tracking controllers for drift-less chained form systems could be used for tracking control of the ship, even though the ship had a drift vector. Do [14] and his team have also done a lot of work in terms of underactuated ship trajectory tracking.

In the aforementioned works, the hydrodynamic coefficients of the vehicle were assumed to be fully known which is difficult to realize in the application for the complex underwater environment. In order to deal with this problem, Ashrafiuon et al. [15] use sliding mode control (SMC) method to solve the robust tracking control of underactuated surface vessels with parameter uncertainties; Yu found some flaws in [15] and proposed a revised sliding mode control law for position tracking [16]. The stability is proved through Lyapunov theory. However, in the above two designs, the thruster saturation problem is not considered which will affect the control performance.

Our tracking controller is based on Yu’s design with revision. The design is based on the sliding mode control, and in order to solve the chattering problem, a saturation function is added to replace the switch terms. By simulation experiments, in the effect of model uncertainty and disturbance, it is found that in the consideration of thruster saturation this controller can still deal with trajectory tracking control problem well. Besides, the main innovation of this paper is based on the idea how to avoid an obstacle (static or dynamic) while tracking the predefined trajectory. In the previous work, much work has been done in the trajectory tracking control problem, but there is little discussion about trajectory tracking control under obstacle condition. Most of the work related with obstacle avoidance [17 –20] is based on path planning area which is not real time. Therefore, the hybrid design of tracking control and obstacle avoidance is proposed. The “real time” means that the tracking controller can deal with tracking control with obstacle without any path planning process [21, 22]. Hence it is quick response and can achieve real time dynamic tracking control.

Hence, a hybrid dynamic control strategy is proposed for both trajectory tracking and obstacle avoidance. Different kind of control strategy is proposed to deal with the dynamic tracking control problem with obstacle avoidance. Based on previous research work, two aspects are different or improved in this paper.

A revised sliding mode control design with consideration of model uncertainty and disturbance for underactuated underwater vehicles is introduced, and in order to meet the real application requirement, thruster saturation problem is considered. The method is verified through different kind of predefined trajectory especially under the limit of thruster saturation.

A real-time hybrid design of obstacle avoidance and tracking control is proposed under static and dynamic obstacle environments. This is the major contribution of this paper and in the author’s knowledge it is a new design. The control law is basically designed based on the distance and angle information between obstacle and vehicle which is different for the static and dynamic obstacle case. The control law is quite simple and avoids complicated computations which meet the real applications.

This paper is organized as follows. Section 2 formulates the basic tracking control problem for underactuated underwater vehicles. Section 3 proposes an integration design of a sliding mode control and the innovative dynamic obstacle strategy together to give a fully control of trajectory tracking and obstacle avoidance. Section 4 describes some simulation results that serve the purpose of demonstrating and validating the proposed tracking methodology. Section 5 summarizes the main contributions of this paper and pinpoints future work that continues the current effort.

2 Model of underactuated underwater vehicle

2.1 Kinematic model

In this paper, the trajectory tracking control problem for underactuated underwater vehicles is considered in the horizontal plane. Figure 1 is a typical underactuated vehicle model that has no side thruster, but two independent main thrusters located at a distance from the center line in order to provide both surge force and yaw moment.

The geometrical relationship between the inertial frame and the body-fixed frame is defined in terms of velocities as [23]:

$[\begin{matrix} \dot{x} \\ \dot{y} \\ \dot{ψ} \end{matrix}] = [\begin{matrix} cos ψ & - sin ψ & 0 \\ sin ψ & cos ψ & 0 \\ 0 & 0 & 1 \end{matrix}] [\begin{matrix} u \\ v \\ r \end{matrix}]$ (1) where (x, y) denote the position of the vehicle in the inertial frame and ψ denotes the heading angle of the vehicle. (u, v) are the surge and sway velocities, respectively, and r is the angular velocity of the vehicle in the body-fixed frame.

2.2 Dynamic model

Under the former realistic assumption, the motion of the vehicle dynamics is described by the following ordinary differential Equations [15]: ${\begin{matrix} m_{11} \dot{u} - m_{22} vr + d_{11} u = τ_{X} \\ m_{22} \dot{v} + m_{11} ur + d_{22} v = 0 \\ m_{33} \dot{r} + (m_{22} - m_{11}) uv + d_{33} r = τ_{N} \end{matrix}$ (2) where d₁₁, d₂₂, d₃₃, m₁₁, m₂₂, m₃₃ denote the hydrodynamic damping and vehicle inertia including added mass in surge, sway and yaw. The available controls are the surge force τ_X and the yaw control moment τ_N. However, there is no available control in the sway direction, thus the problem of controlling the underwater vehicle in three degrees of freedom is therefore an underactuated control problem.

3 Sliding mode design for underactuated underwater vehicles

3.1 Tracking control problem

For the horizontal plane motion control, the desired state of a reference trajectory is defined as follows: (x_d, y_d) is coordinate of desired path in the inertial frame, ψ_d denotes the desired heading angle of the vehicle. The actual state of the vehicle has been defined as (x, y, ψ) and the velocity is given as (u, v, r). The detailed description can be seen in Fig. 2.

The objective of the tracking controller is to design a dynamic control law (τ_X, τ_N) to follow the predefined trajectory (x_d, y_d). So, the tracking error is defined as $x_{e} = x - x_{d}, y_{e} = y - y_{d}, u_{e} = u - u_{d}, v_{e} = v - v_{d}$ (3)

For the underactuated underwater vehicle, due to the coupling of the dynamic system, the sway velocity is passivity related with surge and yaw motion since there is no control force in the sway motion. The final yaw angle is developed as [24]

$ψ_{0} = arctan (\frac{\dot{y}}{\dot{x}}) + arctan (\frac{v}{u})$ (4)

In [24], It is only stated that without loss of generality, the vehicle moves forward, i.e., u > 0. But the singularity truly exists in (4) under the condition of $\dot{x} = 0$ or u = 0. Even under the singularity point, it can be handled individually. It is assumed that $α = arctan (\frac{\dot{y}}{\dot{x}}), β = arctan (\frac{v}{u})$ , when under the condition of $\dot{x} = 0$ , $α = \pm \frac{π}{2}$ according to $\dot{y}$ . In the same way, when under the condition of u = 0, $β = \pm \frac{π}{2}$ according to v.

The tracking error in the yaw angle is defined as $ψ_{e} = ψ_{0} - ψ_{d}$ (5)

3.2 Sliding mode control design

The authors in [15] has firstly proposed a SMC law to deal with tracking control of surface vessels that can guarantee the convergence of u_e, v_e. However, as stated in [16], the position tracking errors are not defined and stability analysis of the position tracking errors is not seen. Thus, the convergence of x_e, y_e cannot be guaranteed by the convergence of u_e, v_e.

In order to guarantee the convergence of the position tracking errors (x_e, y_e) by stabilizing u_e and v_e, it is assumed that the desired surge and sway velocities depend on the information of (x_d (t) , y_d (t)) and (x_e, y_e), and satisfy the following condition $u_{d} = cos ψ {\dot{x}}_{d} + sin ψ {\dot{y}}_{d} - k (cos ψ x_{e} + sin ψ y_{e})$ (6) $v_{d} = - sin ψ {\dot{x}}_{d} + cos ψ {\dot{y}}_{d} - k (- sin ψ x_{e} + cos ψ y_{e})$ (7)

The derivative of (u_d, v_d) can be calculated as ${\dot{u}}_{d} = cos ψ {\ddot{x}}_{d} + sin ψ {\ddot{y}}_{d} + v_{d} r - k (cos ψ {\dot{x}}_{e} + sin ψ {\dot{y}}_{e})$ (8) $\begin{matrix} {\dot{v}}_{d} & = - sin ψ {\ddot{x}}_{d} + cos ψ {\ddot{y}}_{d} - u_{d} r \\ - k (- sin ψ {\dot{x}}_{e} + cos ψ {\dot{y}}_{e}) \end{matrix}$ (9)

By simple derivation, it has been concluded that the convergence of u_e, v_e implies that x_e, y_e can be convergent to zero [16].

3.2.1 Surge motion control

In this section, the SMC method will be applied to develop a non-linear control law that makes the tracking error u_e, v_e asymptotically stable. In order to realize the control objective, a first-order exponentially stable surface S₁ is chosen by the following equation in the surge motion $S_{1} = u_{e} + k_{1} \int u_{e}$ (10)

Taking the derivative of sliding surface with respect to time, then

${\dot{S}}_{1} = {\dot{u}}_{e} + k_{1} u_{e} = \dot{u} - {\dot{u}}_{d} + k_{1} u_{e}$ (11)

When the system is operating on the sliding surface, Equation (11) equals zero. Substituting (2) into (11) and considering the model uncertainty in the dynamic vehicle model, the equivalent control law is concluded as $τ_{Xeq} = - k_{1} {\hat{m}}_{11} u_{e} - {\hat{m}}_{22} vr + {\hat{d}}_{11} u + {\hat{m}}_{11} {\dot{u}}_{d}$ (12) where “∧” is used to indicate the estimated model parameters. The estimated hydrodynamic coefficients can be identified by water tank experiment. The derivation of equivalent control law can also refer to [25]. It is well known that the equivalent control effort τ_Xeq cannot ensure a favorable control performance under the presence of perturbations from parameter variations and environment disturbance. Therefore an auxiliary control effort τ_Xr should be designed to eliminate the effect of the unpredictable perturbations. τ_Xr is given as follows $τ_{Xr} = - K_{1} sat (S_{1})$ (13)

In total, the SMC law for non-linearly uncertain vehicle systems that guarantee stability and convergence in the surge motion can be represented as $\begin{matrix} τ_{X} = τ_{Xeq} + τ_{Xr} \\ = - k_{1} {\hat{m}}_{11} u_{e} - {\hat{m}}_{22} vr + {\hat{d}}_{11} u + {\hat{m}}_{11} {\dot{u}}_{d} - K_{1} sat (S_{1}) \end{matrix}$ (14)

It should be noted that the control law from (10) and (15) in the surge motion control is basically from [16] and so as the following sway motion control.

3.2.2 Sway motion control

The sliding proportional-integral-derivative surface in the space of tracking error can be defined as $S_{2} = {\dot{v}}_{e} + k_{2} v_{e} + k_{3} \int v_{e}$ (15)

Taking the derivative of sliding surface with respect to time, then ${\dot{S}}_{2} = {\ddot{v}}_{e} + k_{2} {\dot{v}}_{e} + k_{3} v_{e} = \ddot{v} - {\ddot{v}}_{d} + k_{2} (\dot{v} - {\dot{v}}_{d}) + k_{3} v_{e}$ (16) where the time derivatives of the second Equations in (2) and (9) yield $\begin{matrix} \ddot{v} = \frac{- d_{22} d_{33} \dot{v} - m_{11} m_{33} \dot{ur} - m_{11} u τ_{X} + m_{11} d_{33} ur + m_{11} (m_{22} - m_{11}) u^{2} v}{m_{22} m_{33}} \\ {\ddot{v}}_{d} = v_{r} - u_{d} \frac{τ_{N} - d_{33} r - (m_{22} - m_{11}) uv}{m_{33}} \\ v_{r} = - r (cos ψ {\ddot{x}}_{d} + sin ψ {\ddot{y}}_{d}) - sin ψ {\overset{⃛}{x}}_{d} + cos ψ {\overset{⃛}{y}}_{d} \\ - (cos ψ {\ddot{x}}_{d} + sin ψ {\ddot{y}}_{d} + v_{d} r - k (cos ψ {\dot{x}}_{e} + sin ψ {\dot{y}}_{e})) r \\ + k (r cos ψ {\dot{x}}_{e} + sin ψ {\ddot{x}}_{e} + r sin ψ {\dot{y}}_{e} - cos ψ {\ddot{y}}_{e}) \end{matrix}$

In the same way, the equivalent control law is concluded as $τ_{Neq} = \frac{\hat{h}}{\hat{b}}$ (17) where “∧” indicates the estimated model and $b = m_{22} u_{d} - m_{11} u$ (18) $\begin{matrix} h = d_{22} m_{33} \dot{v} + m_{11} m_{33} \dot{ur} - m_{11} d_{33} ur \\ - m_{11} (m_{22} - m_{11}) u^{2} v \\ + m_{22} d_{33} u_{d} r + m_{22} (m_{22} - m_{11}) {uvu}_{d} \\ + m_{22} m_{33} (v_{r} - k_{2} (\dot{v} - {\dot{v}}_{d}) - k_{3} v_{e}) \end{matrix}$ (19)

The reaching control τ_Nr can be chosen as $τ_{Nr} = \frac{- K_{2} sat (S_{2})}{\hat{b}}$ (20)

In total, the SMC law for yaw motion can be represented as $τ_{N} = τ_{Neq} + τ_{Nr} = \frac{\hat{h} - K_{2} sat (S_{2})}{\hat{b}}$ (21)

The detailed description can refer to [16] and the system stability is mostly the same. Then under the proposed design, the tracking control under each trajectory point can be satisfied.

As described before, one important thing to mention is that the maximum force and moment saturation should be considered for a real system, hence a limitation function is added as $τ_{i} = {\begin{matrix} - τ_{i max} \begin{matrix} τ_{i} < - τ_{i max} \end{matrix} \\ τ_{i} \begin{matrix} - τ_{i max} \leq τ_{i} \leq τ_{i max} \end{matrix} \\ τ_{i max} \begin{matrix} τ_{i} > τ_{i max} \end{matrix} \end{matrix}$ (22) where i = X, N, τ_Xmax, τ_Nmax is the maximum surge force and yaw moment respectively.

4 Obstacle avoidance strategy for underactuated underwater vehicles

Based on the former section, the design is complete for the typical tracking control. While in this section, considering the case a static or dynamic obstacle appears in the sight of underwater vehicles without any former information when tracking a predefined trajectory, it will be quite risky if no obstacle avoidance strategy is adopted in the control system. In order to solve this problem, the strategy for the avoidance of static and dynamic obstacles is proposed respectively and a detailed block diagram of the whole control system is illustrated in Fig. 3.

When an underwater vehicle is in the presence of static and dynamic obstacles, the strategy can be simultaneously employed to do dynamic avoidance of the obstacles. Here, method 1 means the hybrid dynamic controller (with static obstacle) (25) (26) and method 2 means the dynamic controller (with dynamic obstacle) (27) (28) which will be discussed in detail next. This design is brand new and will be verified in the next simulation section.

4.1 Hybrid design of static obstacle avoidance

The strategy for the avoidance of static obstacle in Fig. 3 is described as follows. As the distance between the corresponding point of the vehicle and the obstacle is smaller than d_min (refer to Fig. 4 for the illustration of d), the vehicle starts avoiding the corresponding obstacle. That is, the operation of the vehicle is in a mode of static obstacle avoidance. The tracking algorithm will be discussed below.

Obstacle avoidance needs real-time feedback force between the vehicle and the environment. Two basic things are needed to be considered in the process of obstacle avoidance: the distance between obstacle and vehicle and the angle error θ_r = θ - ψ shown in Fig. 4. The interaction virtual force f (t) between underwater vehicle and obstacles can be given as a linear function

$f (t) = {\begin{matrix} a (b - d (t)) \begin{matrix} d < d_{min} \end{matrix} \\ 0 \begin{matrix} \begin{matrix} d \geq d_{min} \end{matrix} \end{matrix} \end{matrix}$ (23) where a, b are positive constants and satisfies that b ≥ d_min, 0 ≤ d ≤ d_min, d_min is the minimum sonar range that needs to do obstacle avoidance. Since the vehicle is underactuated, the obstacle is mainly due to the yaw moment. Under this simple assumption, f_r (t) is used to determine the angle adjustment (clockwise or counterclockwise). $f_{r} = f \times cot θ_{r}$ (24)

Hence, the total control law for tracking control and static obstacle avoidance can be concluded as $τ_{X} = - k_{1} {\hat{m}}_{11} u_{e} - {\hat{m}}_{22} vr + {\hat{d}}_{11} u + {\hat{m}}_{11} {\dot{u}}_{d} - K_{1} sat (S_{1})$ (25) $τ_{N} = \frac{\hat{h} - K_{2} sat (S_{2})}{\hat{b}} - k_{4} f_{r}$ (26)

Here, the limitation function (22) is also included for the static obstacle avoidance condition.

One special case is that when θ_r = 0 for the line tracking, it means that the vehicle is directly moving to the obstacle and the target is in the obstacle behind, random moment is added in the control law (26) to let θ_r ≠ 0. When |θ_r| > 90° or d ≥ d_min, the operation of the vehicle returns to the normal tracking mode.

4.2 Hybrid design of dynamic obstacle avoidance

Similarly, the strategy for the avoidance of dynamic obstacle in Fig. 3 is depicted as follows. When distance between the obstacle and vehicle is less than the minimum distance d_1min, different strategy will be employed to do the obstacle avoidance.

When the vehicle faces a moving obstacle, how can the vehicle avoid it? Turn left or right, or stop and wait? This is determined by θ_m = θ₁ + θ₂ which represents the moving angle that can be clearly seen in Fig. 5. As it will be stated below, by calculation of θ_m which is the key parameter to determine how to avoid the obstacle, the final tracking controller can be designed. The situation can be classified into different kinds of cases which will be discussed next. The basic idea of moving obstacle avoidance is that when the moving obstacle under consideration is close to the vehicle, the vehicle will take actions to avoid it according to the moving angle θ_m and distance between the vehicle and obstacle d.

1. If $0 < θ_{m} \leq \frac{π}{6}$ , it means that the vehicle is in a quite risky environment if nothing obstacle avoidance is done or just stop and wait, the better solution is to turn left to avoid it; otherwise, if $- \frac{π}{6} \leq θ_{m} < 0$ , turning right is the option. The details for the two cases can refer to Fig. 5(a) and (b). In this situation, inspired form the former static obstacle avoidance design, the dynamic control law is designed as $τ_{X} = - k_{1} {\hat{m}}_{11} u_{e} - {\hat{m}}_{22} vr + {\hat{d}}_{11} u + {\hat{m}}_{11} {\dot{u}}_{d} - K_{1} sat (S_{1})$ (27) $τ_{N} = k_{5} f cot θ_{m}$ (28)

The limitation function (22) is also included for the dynamic obstacle avoidance condition. The difference between the static obstacle control law is that it is only related with the moving angle θ_m and distance between the vehicle and obstacle d, the tracking control law (21) is not added in since the primary need is to avoid not considering the effect of target trajectory.

2. If $\frac{π}{6} < | θ_{m} | < π$ , the environment is not so risky, then the simple avoidance solution is to stop and wait. The details for the case can refer to Fig. 5(c).

3. If |θ_m| ≥ π, turn to the tracking control mode. The details for the case can refer to Fig. 5(d).

It should be noted that the parameters d_min and d_1min is very important for the final tracking control performance under static and dynamic obstacle avoidance. Basically, it is based on error and trial under some principle. In this paper, the AUV is assumed to work in a slow speed, when a static and dynamic obstacle occurs in the view side of AUV, d_min and d_1min can be regarded as a safe distance for obstacle avoidance. It cannot be selected too small, or a risky crush may happen for a short time to deal with obstacle avoidance. It also cannot be selected too large, or the tracking control performance will be worse. By simulation test, a safe distance no less than 2 m is necessary for safe tracking control under obstacle condition and it is also consistent with actual situation.

5 Simulation and discussion

The main goal of this study is to investigate the proposed hybrid tracking control method with obstacle avoidance for an underactuated underwater vehicle in an intelligent space. These experiments are categorized into the following cases:

to track different predefined trajectories without obstacle;

to track the trajectory with static obstacles;

to track the trajectory with dynamic obstacle;

The simulation study is based on the “Maritime I” underwater vehicle (made by Laboratory of Underwater Vehicles and Intelligent Systems, Shanghai Maritime University). The structure of the underwater vehicle can be seen in Fig. 6. And the main parameters of “Maritime I” are listed in Table 1. Since there are only two thrusters arranged in the horizontal plane, it is a typical underactuated system and will be simulated for the proposed tracking control and obstacle avoidance method.

5.1 Tracking control without obstacle

In this section, the proposed sliding mode control method will be applied to the “Maritime I” model for trajectory tracking problem. The hydrodynamic parameters used in the simulations have been listed in Table 1. To illustrate the performance of the proposed tracking control method, typical simulation results are presented. From the simulation results, it can be concluded that the proposed controller can reach a robust control even under parameter uncertainty and disturbance.

5.1.1 Circular tracking

A typical case to track a circular path was studied first. The vehicle starts at posture (0, –12, 0), while the desired initial posture is (0, –10, 0). Thus the initial posture error is (–5, 0, 0, 0). Time varies from 0 to 150 s. Assume the desired track state of vehicle is x_d (t) =10 sin(0.1t); y_d (t) = -10 cos(0.1t); $ψ_{d} (t) = arc tan ({\dot{y}}_{d} / {\dot{x}}_{d})$ . To reflect uncertainties of the vehicle dynamics, 10% model inaccuracies (error) were incorporated into the controller’s dynamic model. And also a constant disturbance of 20 N is applied to the surge motion. The control parameters are taken as k = 0.1, k₁ = 0 .5, k₂ = 0 . 5, k₃ = 0 .5. τ_Xmax = 200, τ_Nmax = 100 is the set maximum surge force and yaw moment respectively. It can be concluded that to give a satisfactory tracking result, the control parameters such as k, k₁, k₂, k₃ cannot be too small, or the convergence rate will be too slow. And the parameters cannot be too large, or control inputs will be big enough to exceed the thruster limits, then the saturation function will be working to decrease the tracking performance. As tested by many simulation results, the selection for these parameters is basically in the rangeof [0.1, 10].

Figure 7 shows the simulation results of the circular tracking. The red dots indicate the real trajectory while and the blue dots are the desired trajectory. Figure 7(a) and (b) show the tracking results and tracking error in the inertial frame respectively. The control inputs of trajectory tracking are shown in Fig. 7(c).

From the simulation results, it can be easily to found that even under the parameter uncertainty and environment disturbance, the vehicle can catch the desired trajectory in a quick response. Due to the saturation function used in the controller, the chattering problem in the traditional sliding mode control can be avoided and control inputs are smoothenough.

5.1.2 Sinusoidal tracking

In a similar way, sinusoidal tracking is studied in this part. The vehicle starts at posture (0, 2, 0), while the desired initial posture is (0, 0, 0). Time varies from 0 to 100 s. Assume the desired track state of vehicle is y_d (t) =0.5t; y_d (t) =5 sin(0.1t); $ψ_{d} (t) = arc tan ({\dot{y}}_{d} / {\dot{x}}_{d})$ . 5% model inaccuracies are also incorporated into the controller’s dynamic model. The control parameters are taken as k = 0.2, k₁ = 0 .5, k₂ = 0 . 5, k₃ = 0 .5.

Figure 8 shows the simulation results of the sinusoidal tracking. The red dots indicate the real trajectory while and the blue dots are the desired trajectory. Figure 8(a) and (b) show the tracking results and tracking error respectively. The control inputs of trajectory tracking are shown in Fig. 8(c). Not surprising, the similar simulation results can be achieved with satisfactory tracking result and smooth control inputs.

5.2 Tracking control with obstacle avoidance

Here, it is assumed that in the trajectory tracking process, an obstacle appears without former knowledge. The simulation results are divided into two parts: static obstacle avoidance and dynamic obstacle avoidance.

5.2.1 Static obstacle avoidance

One experiment was conducted to evaluate the new proposed tracking control method with static obstacle avoidance. In this experiment, the starting point is in (–1, 0.5). The desired trajectory is set as a piecewise line trajectory and the start coincides with the finish point. Two static obstacle is set in the desired trajectory way: obs1 (5, 4) and obs2 (4, 0). The control parameters are taken as k = 0.3, k₁ = 0 .5, k₂ = 0 . 5, k₃ = 0 . 5, k₄ = 1, a = 40, b = 2, d_min = 2.

The tracking result of piecewise line tracking is given in Fig. 9(a) and (b) while the control inputs are illustrated in Fig. 9(c). From Fig. 9(a), the whole tracking result can be divided into five periods: tracking mode, obstacle avoidance, tracking mode, obstacle avoidance, tracking mode. Though tracking errors always exist in the tracking period due to the complicated environment, it is hold in a small region. Since tested model “Maritime I” is an underactuated vehicle, the obstacle avoidance is mainly controlled by the yaw moment.

5.2.2 Dynamic obstacle avoidance

For the dynamic obstacle environment, the tracking control test is based on the sinusoidal trajectory. The desired trajectory is the same as sinusoidal tracking in section 5.1.2. The dynamic obstacle is set as obstacle _ x (t) =0.5t, obstacle _ y (t) =0. The control parameters are taken as k = 0.2, k₁ = 0 .2, k₂ = 0 . 5, k₃ = 0 . 5, k₄ = 1, k₅ = 1, d_min = 2.

The tracking result of sinusoidal tracking with dynamic obstacle is given in Fig. 10(a) and (b) while the control inputs are illustrated in Fig. 10(c). In order to show the performance of dynamic obstacle avoidance, the obstacle is moving in accordance with the desired trajectory in the x axis. From the control inputs, it can be seen that at the time 27 s, the vehicle shut down its power input to let it stop and wait till a safe distance. Then it turns to be the tracking mode until meeting the obstacle next time and this procedure continues till the end of the tracking mission.

To give more different case of dynamic obstacle avoidance, two different tracking control tests are conducted based on the line trajectory under different obstacle cases.

In the first case, the desired trajectory is set as x_d (t) =0.5t, y_d (t) =0, and the dynamic obstacle is set as obstacle _ x (t) =14 - 0.2t, obstacle _ y (t) =6 - 0.1t. It means that the dynamic obstacle is moving in a constant speed which is a very common way in the real environment. The control parameters are taken as k = 0.3, k₁ = 0 .5, k₂ = 0 . 5, k₃ = 0 . 5, k₄ = 1, k₅ = 10, d_1min = 3.

The tracking result of line tracking is given in Fig. 11(a) and the control inputs are illustrated in Fig. 11(c). In order to show the performance of dynamic obstacle avoidance, at the time of 15.7 s (show in Fig. 11(b)), the distance between the vehicle and obstacle d is less than the minimum safety distance d_1min. And the moving angle θ_m is less than $\frac{π}{6}$ which is the included angle between the AUV moving direction and the obstacle direction, therefore it turns left to do obstacle avoidance. After avoiding the obstacle, it will turn to the tracking mode.

In the second case, the tracking control test is conducted based on the line trajectory. The desired trajectory is also set as x_d (t) =0.5t, y_d (t) =0. The dynamic obstacle is set as obstacle _ x (t) =0.5t, obstacle _ y (t) =4 + sin(0.2t).

The tracking result of line tracking is given in Fig. 12(a) and the control inputs are illustrated in Fig. 12(b). From the simulation result, it can be concluded that even under no constant moving obstacle, the proposed method can still deal with it very well and ensures the tracking performance.

6 Conclusion

In this paper, a hybrid tracking control method for underactuated underwater vehicle is proposed to obtain a robust trajectory tracking and obstacle avoidance. Form the simulation results, it can be concluded that with consideration of thruster saturation the sliding mode controller can deal with the tracking control mission under parameter uncertainty and disturbance. And also the major contribution point, based on the obstacle avoidance strategy, the proposed hybrid controller can do a tracking control with whether static obstacle and dynamic obstacle with quick response. Quick response is only a qualitative description. Comparing with previous static and dynamic obstacle avoidance, the proposed method doesn’t need to consider the path planning process. Take a deep look at the dynamic controller, only some basic parameters like velocity and angle error are needed without any complex computing. This design leads to the quick control response and feasible for future pool experiment. It will be a best choice if we can put the hybrid trajectory tracking control method into real environment and check the performance. But unfortunately, the experiment condition cannot be satisfied under the current situation due to the hardware (including sensors) limit. The pool experiment will be conducted in the future work to confirm the efficiency of the proposed method.

Footnotes

Acknowledgments

This project is supported in part by the National Natural Science Foundation of China (51575336, 61503239, 51279098) and the Creative Activity Plan for Science and Technology Commission of Shanghai (15550722400).

References

Sanyal

, Nordkvist

and Chyba

, An almost global tracking control scheme for maneuverable autonomous vehicles and its discretization, IEEE Transactions on Automatic Control56 (2011), 457–462.

Bessa

W.M.

, Dutra

M.S.

and Kreuzer

, An adaptive fuzzy sliding mode controller for remotely operated underwater vehicles, Robotics and Autonomous Systems58 (2010), 16–26.

Zhang

and Chu

, Adaptive sliding mode control based on local recurrent neural networks for underwater robot, Ocean Engineering45 (2012), 56–62.

Miao

, Li

and Luo

, A DSC and MLP based robust adaptive NN tracking control for underwater vehicle, Neurocomputing111 (2013), 184–189.

Miller

P.A.

, Farrell

J.A.

, Zhao

and Djapic

, Autonomous underwater vehicle navigation, IEEE Journal of Oceanic Engineering35 (2010), 663–678.

F.Y.

, Wei

Y.J.

, Zhang

J.Z.

and Cao

, Position-tracking control of underactuated autonomous underwater vehicles in the presence of unknown ocean currents, IET Control Theory and Applications4 (2010), 2369–2380.

Sanchez

and Flores

, Real-time underactuated robot swing-up via fuzzy PI+PD control, Journal of Intelligent and Fuzzy Systems17 (2006), 1–13.

Aguiar

A.P.

and Hespanha

J.P.

, Trajectory-tracking and path-following of underactuated autonomous vehicles with parametric modeling uncertainty, IEEE Transactions on Automatic Control52 (2007), 1362–1379.

Refsnes

J.E.

, Sorensen

A.J.

and Pettersen

K.Y.

, Model-based output feedback control of slender-body underactuated AUVs: Theory and experiments, IEEE Transactions on Control Systems Technology16 (2008), 930–946.

10.

Cui

, Sam Ge

and Voon Ee How

, Leader-follower formation control of underactuated autonomous underwater vehicles, Ocean Engineering37 (2010), 1491–1502.

11.

Jiang

Z.P.

, Global tracking control of underactuated ships by Lyapunov’s direct method, Automatica38 (2002), 301–309.

12.

Ghommam

, Mnif

and Derbel

, Global stabilisation and tracking control of underactuated surface vessels, IET Control Theory and Applications4 (2010), 71–88.

13.

Pettersen

K.Y.

and Nijmeijer

, Underactuated ship tracking control: Theory and experiments, International Journal of Control74 (2001), 1435–1446.

14.

K.D.

, Pan

, Control of ships and underwater vehicles, Springer-Verlag, New York, 2009.

15.

Ashrafiuon

, Muske

, McNinch

and Soltan

, Sliding mode tracking control of surface vessels, IEEE Transactions on Industrial Electronics55 (2008), 4004–4012.

16.

, Zhu

and Xia

, Sliding mode tracking control of an underactuated surface vessel, IET Control Theory and Applications6 (2012), 461–466.

17.

Matveev

A.S.

, Teimoori

and Savkin

A.V.

, A method for guidance and control of an autonomous vehicle in problems of border patrolling and obstacle avoidance, Automatica47 (2011), 515–524.

18.

Hwang

C.L.

and Chang

L.J.

, Trajectory tracking and obstacle avoidance of car-like mobile robots in an intelligent space using mixed H2/H∞ decentralized control, IEEE/ASME Transactions on Mechatronics12 (2007), 345–352.

19.

Mon

Y.J.

and Lin

C.M.

, Image processing based obstacle avoidance control for mobile robot by recurrent fuzzy neural network, Journal of Intelligent and Fuzzy Systems26 (2014), 2747–2754.

20.

Wen

, Zheng

, Zhu

, Li

and Chen

, Elman fuzzy adaptive control for obstacle avoidance of mobile robots using hybrid force/position incorporation, IEEE Transactions on Systems, Man, and Cybernetics, Part C: Applications and Reviews42 (2012), 603–608.

21.

Chen

, Chang

and Agate

C.S.

, UAV path planning with tangent-plus-Lyapunov vector field guidance and obstacle avoidance, IEEE Transactions on Aerospace and Electronic Systems49 (2013), 840–856.

22.

Sun

, Zhu

, Jiang

and Yang

S.X.

, A novel fuzzy control algorithm for three-dimensional AUV path planning based on sonar model, Journal of Intelligent and Fuzzy Systems26 (2014), 2913–2926.

23.

Fossen

T.I.

, Handbook of marine craft hydrodynamics and motion control, John Wiley & Sons, 2011.

24.

Repoulias

and Papadopoulos

, Planar trajectory planning and tracking control design for underactuated AUVs, Ocean Engineering34 (2007), 1650–1667.

25.

Sun

, Zhu

and Yang

S.X.

, A bio-inspired cascaded approach for three-dimensional tracking control of unmanned underwater vehicles, International Journal of Robotics and Automation29 (2014), 349–358.