Robust proportional derivative (PD)-like fuzzy control designs for diving and steering planes control of an autonomous underwater vehicle

Abstract

In this paper, a highly non-linear model with six degrees of freedom (DOF) is used for the manoeuvring and depth control of an autonomous underwater vehicle (AUV). A simplified scheme to design a conventional fuzzy logic controller known as robust Proportional Derivative (PD)-like Fuzzy Controller (PD-FZ) is designed separately for diving and steering subsystems of an AUV and then applied for combined steering, diving and speed control functions. Simulation results are shown by using the slender form of the Naval Post-Graduate School (NPS, Monterey, CA) AUV. Results of simulation studies using a nonlinear AUV dynamics are presented under the presence of any bounded ocean currents or wave disturbances and results show that proposed controller gives robust performance against these disturbances and modeling nonlinearity. Waypoint acquisition based on line of sight guidance is used to achieve path tracking.

Keywords

Autonomous underwater vehicle (AUV)fuzzy logic control robust proportional derivative (PD)-like fuzzy controller (PD-FZ)steering and depth plane controls Lyapunov stability

1 Introduction

Autonomous underwater vehicles (AUVs) are robotic devices which travel underwater to perform a set of predefined tasks namely inspection, recovery, sample collection, cable laying etc. without human operator intervention through the help of accurate guidance, navigation and control (GNC) schemes. The application of AUVs range from commercial purposes like mapping of ocean floors by oil companies and post-lay pipeline survey to research and military operations like surveillance, anti-submarine warfare and mine detection. It is well known that, underwater vehicles are very difficult to control due to the highly non-linear dynamics, time-varying dynamic behaviour and variations in the hydrodynamic coefficients with different operating conditions. The environmental disturbances like wind generated waves and ocean currents makes the motion control of an AUV a more challenging [1]. In addition, there are six DOF of the motion of an AUV and nonlinear coupling among them, making control design even more complicated. Though the dynamics of an AUV is highly coupled and non-linear in nature, decoupled control system strategy is widely used for practical applications [2]. In this approach, decentralized control strategy can be used as its simplifies designing and tuning of individual loop controllers than a centralized controller [3]. The 6-DOF non-linear equations of motion of an AUV can be decomposed into three non-interacting (or lightly interacting) subsystems for speed, steering and diving control as suggested in [4]. A number of control strategies, developed over a period of time to control the dynamics of the AUV can be found in the literature. Due to inherent non-linearities in the dynamic model of an AUV, conventional linear controllers such as PD/PID might not ensure satisfactory control performance. This problem can be handled very well by non-linear controllers such as sliding mode control, robust and adaptive control, fuzzy logic control, neural network control and hybrid controls like neuro-fuzzy.

The state-of-the-art literature review of the aforementioned control techniques are explained as follows: Initially, Yoerger and Slotine [5] proposed a series of SISO sliding mode continuous-time controllers for trajectory tracking control of an underwater vehicle. Later on in 1991, Yoerger and Slotine [6] discusses how adaptive sliding mode control can be applied to experimental underwater vehicle. Then, Cristi et al. [7] proposed an adaptive sliding mode controller for AUV in the diving plane based on the dominant linear model with bounds of the nonlinear dynamic perturbations. A successful implementation of multivariable sliding mode autopilot design have been reported for the combined control of AUV steering, depth and speed during complex flight maneuvers in [4 , 9]. In the similar way, the problem of controlling motion of an AUV in diving and steering planes has been addressed by Rodrigues et al. [10]. In order to ensure the real-time applicability of the continuous SMC, a discrete-time quasi-sliding mode control was proposed in [11] for depth control of an experimental AUV with the uncertainties of system parameters and with a long sample interval. In [12], a higher order sliding modes for diving control a torpedo autonomous underwater vehicle was designed to improve the performance of conventional SMC. Recently, a discrete time quasi-sliding mode control [13] and time optimal trajectory design [14] has been proposed and applied to control of underwater vehicle. However control design based on sliding mode requires accurate mathematical model of underwater vehicle. Moreover in practical scenario, the dynamics of an AUV is time-varying due to unstructured nature of the ocean environment. In order to deal with such situations, a robust and adaptive control methods has been proposed by many researchers. Firstly, an adaptive control was proposed in [15] to deal with time-varying dynamics of the vehicle. Fault-tolerant control of AUV under thruster redundancy has been proposed in [16]. K.D. Do et al. [17] proposes a nonlinear robust adaptive control strategy to control underactuated underwater vehicle with only four actuators to follow a predefined path at a desired speed despite of presence of environmental disturbances and vehicle’s unknown physical parameters. Furthermore nonlinear path-following control [18], robust trajectory control using time-delay approach [19], adaptive control using integral actions [15] for 6-DOF control of AUVs has been successfully designed. Also dynamic positioning and trajectory tracking control of under-actuated AUVs were attempted in [20 –22]. Even though the control performance of the above mentioned robust and adaptive control methods are quite acceptable under the effect of external disturbance and time-varying dynamics of the AUV, it involves complexity in designing and tuning of a controller. An alternative choice to deal with this is to use of intelligent control techniques such as fuzzy logic and neural network control. The partial knowledge about the system dynamics will be sufficient to design a controller. S.M. Smith et al. [23] firstly designed a fuzzy logic controller’s for heading, pitch, and depth planes control of an AUV. Recently, a fuzzy control has been successfully designed by many researchers for depth control [24, 25] and steering control [26]. A simplified approach to design fuzzy controller known as single-input FLC has been suggested in [27, 28], which improves computational efficiency of the conventional FLC. Likewise neural network controller [29 –33] and hybrid controls like neuro-fuzzy [34 –36] has been developed for motion control of an AUV. However, in neural network control, training time is unpredictable and may not be suitable for real-time control.

The combination of PI, PD, PID and fuzzy logic controllers results in dynamic fuzzy controller structure which gives better control performance and simple control structure [37]. It is very difficult to find exact mathematical model of an AUV due to dynamics are highly non-linear, time-varying and coupled. Another problem in AUV is to measure or estimate the hydrodynamic coefficients accurately which is again difficult due to these coefficients varies with the environmental disturbances. Due to these problems designing a control techniques for an AUV becomes very challenging. Considering all this facts, a PD-like fuzzy controller (PD-FZ) is designed separately for diving and steering subsystems of an AUV and then applied for combined steering, diving and speed control functions in this paper. Fuzzy logic control (FLC) may be the best choice in such situations since it is model free approach and able to adopt the hydrodynamic uncertainties and non-linearities in the AUV dynamics. The main advantage of FLC is that, it can be applied to plants that are difficult to model mathematically or not so well defined. The FLC can be designed for such plants based on heuristics rules reflecting experience of a human expert [38]. The proposed PD-FZ controller has simple control structure which gives better sensitivity and increases the overall stability of the closed loop system. Also this structure provides reduce overshoot and add damping to the overall closed loop system [37, 39]. The non-linearities of the system can be handled by appropriate choice of input and output membership functions (MFs).

The input and output scaling factors (SFs) of the PD controller need to be tuned properly for its optimum performance. This problem can handled by proper designing of rule base structure. In this paper, a general robust rule base is used for robust control performance, which greatly reduces the difficulties in tuning of the input and output scaling factors (SFs) of the PD controller [37]. The objective of this paper is to describe how to design and apply PD like-fuzzy controller (PD-FZ) for combined steering and diving control functions of an AUV. An efficient and simple scheme for way-point acquisition knows as, Line-of-Sight (LOS) guidance law has been used for generating reference trajectories for heading and diving planes. To verify the effectiveness of the proposed control scheme, simulation of Naval Post-Graduate School’s (NPS) AUV II is carried out under various environmental disturbances. The control performance of proposed PD-FZ controller is compared with sliding mode controller (SMC) proposed by Healey et al. [9]. The reason behind in this is that, most of researchers refereed this paper as the fundamental paper as it uses conventional SMC approach, which become most practical controller nowadays. Another reason is that we have used same AUV model which was used by Healey et al. However, SMC has chattering problem and also it requires exact mathematical model of an AUV. These issues are addressed by the proposed PD-FZ control scheme. Simulations results show that, proposed PD-FZ controller provides good immunity to external disturbances as compared to SMC.

The paper is organized in the following sequence. Section 2 describes a mathematical modeling of an AUV. Section 3 presents the PD-like fuzzy controller design. Stability analysis of PD-FZ controller is discussed in Section 4. Section 5 represents application of PD-FZ controller to an AUV. Simulation results are presented in Section 6, followed by the conclusion in Section 7.

2 AUV modeling

2.1 Background

The 6-DOF non-linear equations of motion of an AUV can be described with two co-ordinate frames; where one is moving co-ordinate frame X₀Y₀Z₀ which is fixed to the vehicle, known as body-fixed frame and another is fixed co-ordinate frame, known as earth-fixed reference XYZ frame as indicated in Fig. 1. The motion of the body-fixed frame is described relative to an earth-fixed reference frame. The equations of motion of underwater vehicles are generally derived in the body coordinate frame as follows [40],

$M (ν) \dot{ν} + C (ν) ν + D (ν) ν + g (η) = τ$ (1) where, $η = [η_{1}^{T}, η_{2}^{T}]^{T}$ with η₁ = [x, y, z] ^T and η₂ = [φ, θ, ψ] ^T is the position and orientation vector in the earth-fixed reference frame; $ν = [ν_{1}^{T}, ν_{2}^{T}]^{T}$ , with ν₁ = [u, v, w] ^T and ν₂ = [p, q, r] ^T is the linear and angular velocity vectors in the body-fixed frame; $τ = [τ_{1}^{T}$ , $τ_{2}^{T}]^{T}$ with τ₁ = [X, Y, Z] ^T and τ₂ = [K, M, N] ^T is the vector of forces and moments acting on the vehicle; M (ν) is the 6 × 6 inertia matrix including hydrodynamic added mass; C(ν) is a matrix of Coriolis and Centripetal forces; D (ν) is the hydrodynamic damping matrix; g (ν) is a vector of restoring forces and moments; and τ is the control input vector describing forces and moments efforts.

The mapping between the two co-ordinate systems is given by the Euler angle transformation given as

$\dot{η} = J (η) ν$ (2) where J (η) is the Euler angle transformation matrix. The different quantities are defined according to the SNAME (1950) notations [41]. The non-linear AUV equations of motion given in Equation (1) with disturbances can be modified as [40],

$M {\dot{ν}}_{r} + C (ν_{r}) ν_{r} + D (ν_{r}) ν_{r} + g (η) = τ + w$ (3) where, w = w_wind + w_wave with w_wind ɛ $R^{6}$ and w_wave ɛ $R^{6}$ represent the generalised forces due to wind and waves; ν_r = ν - ν_c can be interpreted as the relative velocity vector. ν_c ɛ $R^{6}$ is the body-fixed current velocity vector given as ν_c = [u_c v_c w_c 0 0 0].

2.2 AUV subsystems

Healey and Marco (1992) [4] and Healey and Lienard (1993) [9] suggest that the 6-DOF linear equations of motion can be divided into three non-interacting (or lightly interacting) subsystems, grouping certain key motion equations together for separate function of speed, steering and diving control. Each system has following state variables:

Speed system state: u (t).

Steering system states: v (t), r (t) and ψ (t).

Diving system states: w (t), q (t), θ (t) and z (t).

The rolling mode, that is, p (t) and φ (t) is left passive in this approach. This decomposition is motivated by the slender form of the Naval Postgraduate School (NPS, Monterey, CA) AUV, which is used as basis in this work. Fossen [40] suggest that the above three subsystems can be controlled by means of two single-screw propellers with revolution n (t), a rudder with deflection δ _r (t) and a stern plane with deflection δ _s (t). This particular choice of actuators is inspired by those used in flight and submarine control.

3 PD-like fuzzy control design

Conventional PI, PD and PID controllers are widely used in industrial control loops worldwide because of their simple control structure, easy design and offers good control performance at acceptable cost [42]. However, these controllers might not ensure satisfactory control performance if the mathematical model of the controlled system is highly non-linear, subjected to parameter variations and time-varying. On the other hand, conventional fuzzy control is known as its ability to cope with non-linearities and uncertainties. Introduction of dynamic fuzzy controller structures with the aim of control system performance leads to PI-, PD- or PID-fuzzy controllers [39 , 43–45]. In this paper, PD-FZ control designs for diving and steering control of an AUV system is presented.

The FLC uses experts knowledge to construct a set of linguistic control rules and then convert it into an automatic control. The objective of the PD-FZ controller is to make the vehicle to track a desired trajectory with minimum error. The desired trajectory is related to the corresponding displacements along the three axes, which is converted into desired yaw and pitch angles using LOS guidance law. Hence, tracking of the desired linear displacements and corresponding angular velocities are the main objectives of the PD-FZ controller. A nonfuzzy PD control law can be expressed as

$U_{PD} = K_{p} e + K_{d} \dot{e} = K_{p} {e + T_{d} \dot{e}}$ (4) here, e is the error (set point - output), $\dot{e}$ is the derivative of error and T_d = K_D/K_p is the derivative time. If the variables e and $\dot{e}$ are fuzzy variables, then Equation (4) will become PD-like Fuzzy control. The control structure of PD-FZ controller is shown in Fig. 2. The variables K_e and $K_{\dot{e}}$ are the input scaling factors (SFs) for input e and $\dot{e}$ , respectively and K_u is the output scaling factor (SF). These scaling gains controls the resolution of each variable which is mainly depends on the fuzziness of its MF’s [46]. The relation between two input scaling gains can be approximated as a constant α [47] as $K_{\dot{e}} = α K_{e}$ . Then the output of PD-FZ can be expressed (see Fig. 2) as,

$U_{PD} = K_{u} F {K_{e} e, \frac{K_{\dot{e}}}{T} \dot{e}}$ (5) where, T is the sampling time and used for digital implementation of fuzzy-like PD controller. The Equation (5) is analogous to non-fuzzy PD controller. Thus the fuzzy K_p and K_d of PD-FZ controller can be expressed as, $K_{p} = K_{u} F {K_{e}}$ (6) $K_{D} = K_{u} F {\frac{K_{\dot{e}}}{T}} = K_{u} F {\frac{α K_{e}}{T}}$ (7) where, F { } represents fuzzy operation. Design parameters of PD-FZ controller are explained as follows:

3.1 Fuzzification

Fuzzification involves a scale mapping between the range of values of input variables and corresponding universe of discourse. The error (e) and change of error ( $\dot{e}$ ) are two fuzzy input variables to the fuzzy controller as shown in Fig. 2. The control signal u is the output of PD-like fuzzy controller is to actuate the actuators of an AUV to control its motion. There are seven linguistic labels for e, $\dot{e}$ and u. These are: Negative Large (NL), Negative Medium (NM), Negative Small (NS), Zero (Z), Positive Small (PS), Positive Medium (PM), Positive Large (PL) as a linguistic labels as shown in Fig. 3. The cross-point level of 0.5 degree is considered for every two adjacent membership functions, as it results in faster rise time and less settling time [48]. The values of the constant M is selected by the designer depending upon the ranges of inputs and output. Triangular or trapezoidal membership function is the simplest for the purpose of analysis as well as for implementation.

3.2 Rule base logic

The rules reflect the operational experience related to the system. The skilled operators experience is required for accurate development of rule base, which requires practical experience about the system for which controller is to be designed. Also, for optimum control performance, control rules, membership functions and scaling factors are need to be tuned properly. This is the fundamental problem in fuzzy logic control design [37, 47]. To overcome the difficulties in optimal control tuning, a standard phase plane techniques has been used for designing robust rule bases for PD-like fuzzy type control (PD-FZ) and PI-like fuzzy type control (PI-FZ) [47]. Generally steering and diving control requires PD-type control action for better performance in driving steering and diving plane of an AUV [2, 39]. Because the reason behind this is that, steering and diving system plane models of AUV has at least one pole at origin and if we use PI type control for such integrating plants, it will add one more pole to the closed loop system, which may leads to instability in the closed loop system. In such situations, PD type control is more appropriate choice. The rule base should be different for PI-FZ and PD-FZ control of their different control characteristics.

Fuzzy control rules are formed by analysing the behaviour of a controlled process. The control rules are derived in a such way that the deviation from a desired state can be corrected and the control objective can be achieved. A closed loop trajectory in phase plane analysis is used to justify the fuzzy control rules as shown in Fig. 4. A knowledge of time domain parameters like rise time, settling time, overshoot etc. with behaviour of the closed loop system are required for phase plane analysis. Fig. 4 shows change in phase plane trajectory with change in step response of a process which is to be controlled. The step response of the system can be roughly divided into eight areas say A₁ to A₈ and four sets of points, in which two cross-over sets of points say (b₁, b₂), (e₁, e₂) and two peak-valley sets of points say (c₁, c₂), (f₁, f₂) as shown in Fig. 4. The origin of the phase plane is considered as the equilibrium point of the system.

a) The sign of rules: The following meta-rules are used to determine the sign of the rule base.

If both e and $\dot{e}$ are zero, then u_k = u_k-1.

If conditions are such that e will go to zero at a satisfactory rate, then u_k = u_k-1.

If error e is not self-correcting, then following four sub-criteria can be used to determine the sign of rule base.

Rules for two cross-over sets of points (b₁, b₂) and (e₁, e₂) are constructed in a such way that, it should prevents the overshoot or undershoot in the area of A₂/A₄ and A₆/A₈ respectively.

Rules for peak-valley sets of points (c₁, c₂) and (f₁, f₂) are constructed in a such way that, it should speed up the response in the area of A₁/A₃ and A₅/A₇ respectively, when error e has large magnitude.

Rules for area A₁, A₄, A₅ and A₈ should provides positive control signal (i.e. u > 0).

Rules for area A₂, A₃, A₆ and A₇ should provides negative control signal (i.e. u < 0).

b) The magnitude of rules: The magnitude of rules can be find out from the following heuristic steps:

At peak-valley points (c₁, c₂) and (f₁, f₂), rule for u₀ is as, $u_{0} = e, when \dot{e} = 0$

The remaining rules

To prevent any overshoot in the area A₂/A₄ and A₆/A₈ $\begin{matrix} u & = & PS, when \dot{e} > 0 \\ u & = & NS, when \dot{e} < 0 \end{matrix}$

When in area A₁ or A₅ and A₃ or A₇ $\begin{matrix} u & = & \min {(u_{0} + \dot{e}), PS)} A_{1} and \\ A_{5} rules must be positive \\ u & = & \max {(u_{0} + \dot{e}), NS)} A_{3} and \\ A_{7} rules must be negative \end{matrix}$

When in area A₂ or A₆ and A₄ or A₈ $u = u_{0} + \dot{e} .$

Based upon above heuristic knowledge, a general and robust rule base can be constructed to have PD type control characteristics. Table 1 shows a general PD-FZ control type rule base. Theoretically, the rule base can be tuned slightly for a optimum performance. Practically, this general rule base is robust enough for a wide range of applications [47]. The cell defined by the intersection of the first row and the first column represents a rule such as, $“ if e is NL and ė is PL then u is NS ″ .$

This rule base structure satisfies all properties for a set of rules such as completeness, consistency, continuity and interaction. Therefore it is refereed as the most stable rule base structure. The main advantage of this rule base is that the fuzzy controller can intuitively make a good response within a very short time without any prior knowledge about the system modeling.

3.3 Defuzzification

Defuzzification process converts fuzzy terms to quantifiable result (crisp value) which is required to actuate the final control element. The crisp control action is required for controlling the rudder and fin angle deflections. The Center of Area (COA) defuzzification technique is used here because it yields better steady-state performance [48]. The crisp output control action is defined as follows: $u = \frac{\int μ_{c} (f) fdf}{\int μ_{c} (f) df}$ (8) where c denotes the fuzzy sets which are being clipped during defuzzification process. It finds the point where a vertical line would slice the aggregate set into two equal masses.

4 Stability analysis of PD-FZ controller

For stability analysis of PD-FZ controller, we assume that the desired state x_d and its derivatives are bounded and are available to the controller.

Lemma 1. If the following conditions are satisfied: $\begin{matrix} V (0, 0) & = & 0 \\ V (x_{1}, x_{2}) & > & 0, \end{matrix}$ (9) $and if \dot{V} (x_{1}, x_{2}) \leq 0 (with x_{2} = {\dot{x}}_{1})$ then the FLC is stable in the sense of Lyapunov.

Proof 1. Let us choose Lyapunov candidate as, $V (x_{1}, x_{2}) = \frac{1}{2} (x_{1}^{2} + x_{2}^{2})$ (10)

Then, $V = \frac{1}{2} (e^{2} + {\dot{e}}^{2})$

where e = x_d - x₁, x₁ = x_t, $x_{2} = {\dot{x}}_{1}$ .

Now, $\dot{V} = e \dot{e} + \dot{e} \ddot{e} = e \dot{e} + \dot{e} ({\ddot{x}}_{d} - {\dot{x}}_{2})$ .

Let us denote, $c = {\ddot{x}}_{d} - {\dot{x}}_{2}$ ,

We obtain $\dot{V} = e \dot{e} + \dot{e} c$ . Hence, we require that $\dot{V} = e \dot{e} + \dot{e} c \leq 0$ . So, if e and $\dot{e}$ have opposite signs, then it is necessary that c = 0. If e and $\dot{e}$ are both positive, then c < - e. If e and $\dot{e}$ are negative, then c > - e. If e = 0 and $\dot{e}$ is negative, then c > 0. If e = 0 and $\dot{e}$ is positive, then c < 0. ∀e and $\dot{e} = 0$ , then ∀c, we have $\dot{V} = e \dot{e} + \dot{e} c \leq 0$ . As PD-FZ control type rule base, as given in Table 1, satisfies all above conditions. Hence, the PD-FZ controller is stable.

5 Application of PD-FZ controller to an AUV control

As the dynamics of an AUV can be decoupled into steering and diving plane, two separate SISO PD-FZ controllers are designed for steering and diving control as shown in Fig. 5. Therefore the dynamics of an AUV can be controlled by two control inputs; the stern plane and rudder plane deflections and six output parameters; surge, sway, heave, roll, pitch and yaw. The effect of stern plane on the forward speed of the vehicle is neglected. Also, the forward speed control scheme is not discussed in this paper. During simulations of the vehicle, it is found that the relationships between stern and rudder planes are fairly separable and hence the control scheme could be simplified into two separate implementations [49]: i) control input as a rudder deflections δ_r, with yaw (ψ), roll (φ), and sway (y) as a output parameters ii) control input as a stern plane deflections δ_s, with depth (z), pitch (θ), and heave (w) as output parameters. Controller design for SISO system is easier than MIMO systems and tuning of individual SISO controllers is more convenient than MIMO controllers.

The heading ψ can be measured with a gimbaled flux-gate compass. In addition to this, the rate sensor can be used for yaw rate r measurement. The heading measurement will be sufficient for designing a controller, but by adding a rate measurement to the controller, we get a PD type approach [2]. The derivative action increases robustness by providing additional phase margin. A nonfuzzy PD control law for steering control can be expressed as $U_{{PD}_{r}} = δ_{r} = K_{p} (ψ_{d} - ψ) + K_{d} r$ (11) here, ψ_d - ψ = e_ψ is the error between desired heading angle ψ_d and actual heading angle ψ, $r = \dot{e}$ is the rate of change of heading angle i.e. yaw rate and U_{PD
_r} is the control signal generated by PD controller for controlling rudder angle deflections δ_r. K_p and K_d are the proportional and derivative gains, respectively. The main reason for omitting integral action is that the rudder plane (or stern plane) servo has a relay non-linearity, which would cause limit cycles if integral action is added [2]. If the variables e_ψ and r are fuzzy variables, then Equation (11) will become PD-FZ type control. Triangular and trapezoidal type MFs are used for fuzzification of input variables error e, change of error $r = \dot{e}$ and output variable U_{PD
_r} as shown in Fig. 3. Both error (say e_ψ) and change of error (say r) uses the whole range of its universe i.e. from –100 to 100. Both are normalised in the range [–1, 1]. The control signal is also normalised in the range [–1, 1]. According to the fuzzy inference engine of Mamdani’s MAX-MIN method and the defuzzification by the centre-of-area method, with two antecedents {e_ψ, r} and 7 × 7 fuzzy rules as shown in Table 1, PD-FZ controller generate the control signal, δ_r, which manipulates the deflection angle of rudder plane such that heading angle ψ remains at desired value.

Similarly, PD-FZ controller is designed for diving subsystem. The depth z can be measured by a pressure sensor, the pitch angle θ by an inclinometer, and the pitch rate q by a rate sensor. A nonfuzzy PD control law for diving control can be expressed as $U_{{PD}_{s}} = δ_{s} = K_{p} (θ_{d} - θ) + K_{d} q$ (12) here, θ_d - θ = e_θ is the error between desired pitch angle θ_d and actual heading angle θ, $q = \dot{e}$ is the rate of change of pitch angle i.e. pitch rate and U_{PD
_s} is the control signal generated by PD controller for controlling stern angle deflections δ_s. If the variables e_θ and q are fuzzy variables, then Equation (12) will become PD-FZ type control. Similar way of designing a PD-FZ control for diving control (i.e. fuzzification, inference engine and defuzzification) is carried out as steering control explained above. Fuzzy inference engine of Mamdani’s MAX-MIN method and the defuzzification by the centre-of-area method, with two antecedents {e_θ, q} and PD-FZ type rule base (see Table 1) are used to generate the control signal, δ_s, which manipulates the deflection angle of stern plane such that pitch angle θ remains at desired value. Once the both controllers designed separately, they are applied to the complete non-linear dynamical model of an AUV, as depicted in Fig. 5, for simulation and performance analysis purpose. The desired path planner generates the desired velocities and positions at each instant which is then converted into desired yaw and pitch angles using LOS guidance law, acts as inputs to these controllers and the actual outputs of the vehicle are feedback to the controllers.

6 Simulation results

The efficacy of the controller architecture is evaluated through computer simulations of the set of non-linear equations of an AUV using MATLAB/SIMULINK environment. Line-of-Sight (LOS) guidance law is used for generating heading reference trajectories. A separate guidance mechanism is used for horizontal-plane motion and the vertical-plane motion, assuming that they are decoupled. The objective of the horizontal-plane guidance is to generate appropriate heading reference trajectories in order for the vehicle to converge to a straight line on the XY-plane and, similarly, the objective of the vertical-plane guidance is to generates pitch reference trajectories in order to converge to a straight line on the XZ-plane.

Therefore for horizontal-plane motion, we can define the LOS to be the horizontal plane angle (yaw) given by $ψ_{d} = \tan^{- 1} [\frac{y_{k} - y (t)}{x_{k} - x (t)}] .$ (13)

Similarly, for vertical plane motion, the LOS to be the vertical plane angle (pitch) given by $θ_{d} = \tan^{- 1} [\frac{z_{k} - z (t)}{x_{k} - x (t)}]$ (14) in which the [x_k, y_k] and [x_k, z_k] are waypoints stored in the vehicles mission planner. These reference trajectories, act as a set points for the PD-FZ controllers for steering and diving subsystem. With this, the non-linear dynamical model of AUV is simulated in MATLAB/SIMULINK using ordinary differential equations solver by Runge-Kutta fourth order (RK4) method with a fixed time step size of 10 ms for finding efficacy of the proposed controller. The time step size is chosen in the view of actual actuators and sensors bandwidth. The controllers given by Equations (11) and (12), generates the control signal depending upon the magnitude of errors in the respective planes. To check robustness of the proposed controller, AUV model is subjected to different environmental disturbances like ocean wave, ocean current disturbances etc.

To represent a more precise approximation to the ocean wave spectrum, the higher order wave transfer function approximation was proposed by Fung and Grimble [50], which is given as, $h (s) = \frac{0.1200 s^{2}}{s^{4} + 0.48 s^{3} + 2.9376 s^{2} + 0.6912 s + 2.0736}$ (15) The current-induced forces and moments can be included in the dynamics equations of motion as given in Equation (12). To have extreme check on robustness of the proposed PD-FZ controller, non-realistic, time-varying ocean current disturbance considered here for the simulation in closed loop is [40], V_c = 0.5sin (0.2t). In addition to this, in the practical case there may be noise in the position and orientation measurement which are introduced by the measurement sensors. Here, the Gaussian random noises of 0.001 m mean and 0.01 standard deviation in the position and 0.2-degree standard deviation in the orientation measurement is considered and added to the position vector, η of the vehicle. The ocean current along with higher order wave disturbance, simulation of non-linear model of an AUV is carried out and robustness and efficacy of the proposed controller has been evaluated through two different cases of steering and diving control. In the practical situations the command signals will not be having sudden variations, hence first case presented here deals with worst case.

Throughout the simulation forward speed of the vehicle is kept constant as u = 0.8 m/s. Performance of proposed PD-FZ controller is compared with sliding mode controller (SMC) designed for same AUV model considered here by Healey et al. [9]. In first case, command signal for steering control subsystem is square wave type steering, which is depicted in Fig. 6. Ocean wave and current disturbances are introduced in the dynamics of the vehicle from t = 3000 s to t = 7000 s. Because of the high inertia of the vehicle, an ideal step change in the steering command signal is found to produce sluggish response and this could be seen in Fig. 6. However, proposed controller provides better performance as compared to SMC of Healey et al. It could be be seen that considerable oscillations are observed in control input due SMC of Healey et al., while proposed controller provides smoother control action as shown in Fig. 7 (b). The corresponding yaw angle error could be seen in Fig. 7 (a). It is to be noted that proposed controller has better immunity to external disturbances. Trapezoidal wave type depth command signal is given to the diving subsystem of the vehicle and its response is shown in Fig. 8. The proposed controller gives minimum pitch angle tracking error as compared to SMC of Healey et al., as shown in Fig. 9 (a) and to have exact tracking of depth, initially proposed control demands large control action as compared to SMC as shown in Fig. 9 (b). From the simulation results, it can be conclude that the proposed controller initially requires large control efforts in order to adapt un-modeled dynamics of an AUV since proposed control design does not depends on mathematical model of an AUV. The dynamics of the AUV has been modeled through rule base design. Once the proposed controller adapts the dynamics of the AUV, it gives optimum control performance against the parametric uncertainties and external disturbances.

The control performance of both the controllers are measured in terms of norm of the vector. The L² (Euclidean) norm is considered for the analysis purpose and is defined as, (suppose consider for trajectory tracking errors) $∣ ∣ e (t) ∣ ∣ = \sqrt{\sum_{i = 1}^{2} e_{i}^{2}} = \sqrt{e^{T} e} .$ (16)

The time histories of the norm of steering and depth tracking errors and the control input vectors and the steering and under uncertain working conditions are depicted in Figs. 10 and 11 respectively. From these results, it can be observed that, the norm given by the proposed controller immediately falls down to zero and provides smooth control efforts as compared to SMC controller. A quantitative analysis of the position tracking performance of an AUV given by the controllers in terms of average error (AE), root mean squares of error (RMS) and the L² (Euclidean) norm were performed and can be find in Table 2. From this table, it is confirmed that the proposed control scheme gives minimum tracking errors in position and attitude tracking as compared to SMC controller. The second case considered here is triangular wave type steering command signal control and it seems to be practical case. This could be visualized in Fig. 12 and its corresponding yaw angle error and rudder angle deflections are shown in Figs. 13 (a) and (b) respectively. In this case also proposed controller also shows immunity to external disturbances as compared to SMC. The similar kind of triangular wave type diving control is shown in Fig. 14 and it shows better tracking performance by proposed controller. Its corresponding pitch angle error and stern plane deflections are shown in Fig. 15. In this case, the time histories of the norm of steering and depth tracking errors and the control input vectors under ocean current and wave disturbances are depicted in Figs. 16 and 17 respectively. It can be observed that the proposed controller better tracking performance with minimum control efforts as compared to SMC controller. It reveals that the PD-FZ performs better with less control efforts and less oscillations, which results in energy-efficient controller. This can be confirmed from a quantitative analysis of the position tracking performance of an AUV for this case as shown in Table 3. The view of X-Y-Z positional trajectory with higher order wave and ocean current disturbances can be seen in Fig. 18. It can be found that, proposed controller gives better tracking control performance against these disturbances as compared to SMC.

7 Conclusion

In this paper, a simplified scheme to design a conventional fuzzy logic controller known as PD-like fuzzy (PD-FZ) controller is presented for maneuvering control of an AUV. The feasibility of the proposed controller architecture is studied through computer simulations and shows robust performance over different environmental disturbances like ocean waves, time-varying ocean current etc., when designed separately for steering and diving motion control. A study of waypoint acquisition using a LOS guidance algorithm has shown that the scheme is robust, efficient and has simple structure. The proposed fuzzy logic controller has PD type control structure, which provides better sensitivity and increased stability of the closed loop system. A general robust rule base is used for fuzzy controller, which exhibits the PD type control characteristics. The proposed method eliminates the optimum tuning to the scaling gains, which greatly reduces the difficulties of design and tuning.

Footnotes

Acknowledgments

The authors wish to express their sincere thanks to the Naval Research Board, Directorate of Naval Research and Development, DRDO, Government of India, for funding the project (Sanction No. NRB-23B/SC/11-12).

References

Yuh

, Design and control of autonomous underwater robots: A survey, Autonomous Robots 8(1) (2000), 7–24.

Jalving

, The NDRE-AUV flight control system, Oceanic Engineering, IEEE Journal of 19(4) (1994), 497–501.

Londhe

P.S.

, Dhadekar

D.D.

, Patre

B.M.

and Waghmare

L.M.

, Uncertainty and disturbance estimator based sliding mode control of an autonomous underwater vehicle, International Journal of Dynamics and Control (2016), 1–17.

Healey

A.J.

and Marco

D.B.

, Slow speed flight control of autonomous underwater vehicles: Experimental results with NPS AUV II. International Society of Offshore and Polar Engineering Conference, San Francisco, 1992, pp. 523–532.

Yoerger

and Slotine

, Robust trajectory control of underwater vehicles, IEEE Journal of Oceanic Engineering 10(4) (1985), 462–470.

Yoerger

D.R.

and Slotine

J.-J.E.

, Adaptive sliding control of an experimental underwater vehicle. In Proceedings 1991 IEEE International Conference on Robotics and Automation Institute of Electrical & Electronics Engineers (IEEE).

Cristi

, Papoulias

F.A.

and Healey

A.J.

, Adaptive sliding mode control of autonomous underwater vehicles in the dive plane, IEEE Journal of Oceanic Engineering 15(3) (1990), 152–160.

Marco

D.B.

and Healey

A.J.

, Sliding mode acoustic servoing for an autonomous underwater vehicle. In Offshore Technology Conference. Society of Petroleum Engineers (SPE), 1992.

Healey

A.J.

and Lienard

, Multivariable sliding mode control for autonomous diving and steering of unmanned underwater vehicles, Oceanic Engineering, IEEE Journal of 18(3) (1993), 327–339.

10.

Rodrigues

, Tavares

and Prado

, Sliding mode control of an AUV in the diving and steering planes. In OCEANS ’96 MTS/IEEE Prospects for the 21st Century Conference Proceedings, volume 2, 1996, pp. 576–583.

11.

Lee

P.-M.

, Hong

S.-W.

, Lim

Y.-K.

, Lee

C.-M.

, Jeon

B.-H.

and Park

J.-W.

, Discrete-time quasi-sliding mode control of an autonomous underwater vehicle, IEEE Journal of Oceanic Engineering 24(3) (1999), 388–395.

12.

Salgado-Jimenez

and Jouvencel

, Using a high order sliding modes for diving control a torpedo autonomous underwater vehicle. In Oceans 2003 Celebrating the Past... Teaming Toward the Future (IEEE Cat. No.03CH37492). Institute of Electrical & Electronics Engineers (IEEE), 2003.

13.

Zhang

, Yu

and Zhang

, Discrete time quasi-sliding mode control of underwater vehicles. In Intelligent Control and Automation (WCICA), 8th World Congress on, 2010, pp. 6686–6690.

14.

Loc

M.B.

, Choi

H.-S.

, You

S.-S.

and Huy

T.N.

, Time optimal trajectory design for unmanned underwater vehicle, Ocean Engineering 89 (2014), 69–81.

15.

Antonelli

, Chiaverini

, Sarkar

and West

, Adaptive control of an autonomous underwater vehicle: Experimental results on odin, Control Systems Technology, IEEE Transactions on 9(5) (2001), 756–765.

16.

Podder

T.K.

and Sarkar

, Fault-tolerant control of an autonomous underwater vehicle under thruster redundancy, Robotics and Autonomous Systems 34(1) (2001), 39–52.

17.

K.D.

, Pan

and Jiang

Z.P.

, Robust and adaptive path following for underactuated autonomous underwater vehicles, Ocean Engineering 31(16) (2004), 1967–1997.

18.

Lapierre

and Soetanto

, Nonlinear path-following control of an auv, Ocean Engineering 34(11-12) (2007), 1734–1744.

19.

Prasanth Kumar

, Dasgupta

and Kumar

C.S.

, Robust trajectory control of underwater vehicles using time delay control law, Ocean Engineering 34(5-6) (2007), 842–849.

20.

Aguiar

A.P.

and Pascoal

A.M.

, Dynamic positioning and waypoint tracking of underactuated auvs in the presence of ocean currents, International Journal of Control 80(7) (2007), 1092–1108.

21.

Aguiar

A.P.

and Hespanha

J.P.

, Trajectory-tracking and path following of underactuated autonomous vehicles with parametric modeling uncertainty, Automatic Control, IEEE Transactions on 52(8) (2007), 1362–1379.

22.

Repoulias

and Papadopoulos

, Planar trajectory planning and tracking control design for underactuated auvs, Ocean Engineering 34(11-12) (2007), 1650–1667.

23.

Smith

S.M.

, Rae

G.J.S.

and Anderson

D.T.

, Applications of fuzzy logic to the control of an autonomous underwater vehicle. In Fuzzy Systems, Second IEEE International Conference on, vol. 2, 1993, pp. 1099–1106.

24.

Wang

, Shen

, Wang

, Sha

, He

and Yan

, Fuzzy controller used smoothing function for depth control of autonomous underwater vehicle. In OCEANS 2016 - Shanghai, 2016, pp. 1–5.

25.

Nag

, Patel

S.S.

and Akbar

S.A.

, Fuzzy logic based depth control of an autonomous underwater vehicle. In Automation, Computing, Communication, Control and Compressed Sensing (iMac4s), 2013 International Multi-Conference on, 2013, pp. 117–123.

26.

Patel

S.S.

, Kumar

, Botre

B.A.

and Akbar

S.A.

, Design of fuzzy logic based controller with pole placement for the control of yaw dynamics of an autonomous underwater vehicle. In 2015 Annual IEEE India Conference (INDICON), 2015, pp. 1–6.

27.

Londhe

P.S.

, Santhakumar

, Patre

B.M.

and Waghmare

L.M.

, Task space control of an autonomous underwater vehicle manipulator system by robust single-input fuzzy logic control scheme, IEEE Journal of Oceanic Engineering 99 (2016), 1–16.

28.

Ishaque

, Abdullah

, Ayob

and Salam

, A simplified approach to design fuzzy logic controller for an underwater vehicle, Journal of Ocean Engineering, Elsevier 38(1) (2011), 271–284.

29.

Jagannathan

and Galan

, One-layer neural-network controller with preprocessed inputs for autonomous underwater vehicles, Vehicular Technology, IEEE Transactions on 52(5) (2003), 1342–1355.

30.

J.H.

and Lee

P.M.

, A neural network adaptive controller design for free-pitch-angle diving behavior of an autonomous underwater vehicle, Robotics and Autonomous Systems 52(2 - 3) (2005), 132–147.

31.

Santora

, Alberts

and Edwards

, Control of underwater autonomous vehicles using neural networks. In OCEANS, 2006, pp. 1–5.

32.

vande Ven

P.W.

, Flanagan

and Toal

, Neural network control of underwater vehicles, Engineering Applications of Artificial Intelligence 18(5) (2005), 533–547.

33.

Peng

, Wang

and Liu

, Containment control of networked autonomous underwater vehicles: A predictor-based neural DSC design, ISA Transactions 59 (2015), 160–171.

34.

Kim

T.W.

and Yuh

, Application of on-line neuro-fuzzy controller to auvs, Information Sciences 145(1-2) (2002), 169–182.

35.

Wang

J.-S.

and Lee

C.S.G.

, Self-adaptive recurrent neuro-fuzzy control of an autonomous underwater vehicle, Robotics and Automation, IEEE Transactions on 19(2) (2003), 283–295.

36.

George Lee

C.S.

, Wang

J.-S.

and Junku

, Selfadaptive neuro-fuzzy systems for autonomous underwater vehicle control, Advanced Robotics 15(5) (2001), 589–608.

37.

Londhe

P.S.

, Patre

B.M.

and Tiwari

A.P.

, Fuzzy-like PD controller for spatial control of advanced heavy water reactor, Nuclear Engineering and Design 274 (2014), 77–89.

38.

Zadeh

L.A.

, A fuzzy-algorithmic approach to the definition of complex or imprecise concepts, International Journal of Man-Machine Studies 8 (1976), 249–291.

39.

Akkizidis

I.S.

, Roberts

G.N.

, Ridao

and Batlle

, Designing a fuzzy-like PD controller for an underwater robot, Control Engineering Practice 11(4) (2003), 471–480.

40.

Fossen

T.I.

, Guidance and control of Ocean Vehicles. Wiley, New York, 1994.

41.

SNAME (1950). Nomenclature for Treating the Motion of a Submerged Body Through a Fluid. The Society of Naval Architects and Marine Engineers, Technical and Research Bulletin No. 1-5, 1950, pp. 1–15.

42.

Astrom

K.J.

and Hagglund

, The future of PID control, Control Engineering Practice 9(11) (2001), 1163–1175.

43.

Zheng

, Wang

Q.-G.

, Lee

T.-H.

and Huang

, Robust PI controller design for nonlinear systems via fuzzy modeling approach, Systems, Man and Cybernetics, Part A: Systems and Humans, IEEE Transactions on 31(6) (2001), 666–675.

44.

Mann

G.K.I.

, Hu

B.-G.

and Gosine

R.G.

, Analysis of direct action fuzzy PID controller structures, Systems, Man, and Cybernetics, Part B: Cybernetics, IEEE Transactions on 29(3) (1999), 371–388.

45.

Precup

R.-E.

and Hellendoorn

, A survey on industrial applications of fuzzy control, Computers in Industry 62(3) (2011), 213–226.

46.

and Gatland

, A new methodology for designing a fuzzy logic controller, Systems, Man, and Cybernetics, Part B: Cybernetics, IEEE Transactions on 26(3) (1995), 505–512.

47.

and Gatland

, Conventional fuzzy control and its enhancements, Systems, Man, and Cybernetics, Part B: Cybernetics, IEEE Transactions on 26 (1996), 791–797.

48.

Driankov

, Hellendoorn

and Reinfrank

, An introduction to fuzzy control. Narosa Publising House, New Delhi, India, 2004.

49.

Venugopal

K.P.

, Sudhakar

and Pandya

A.S.

, On-line learning control of autonomous underwater vehicles using feedforward neural networks, Oceanic Engineering, IEEE Journal of 17(4) (1992), 308–319.

50.

Fung

and Grimble

M.J.

, Dynamic ship positioning using a self-tuning Kalman filter, Automatic Control, IEEE Transactions on 28(3) (1983), 339–350.