Robust task-space control of an autonomous underwater vehicle-manipulator system by PID-like fuzzy control scheme with disturbance estimator

Ocean Engineering 139 (2017) 1–13

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Robust task-space control of an autonomous underwater vehicle-manipulator system by PID-like fuzzy control scheme with disturbance estimator

MARK

⁎

P.S. Londhea, S. Mohanb, B.M. Patrea, , L.M. Waghmarea a b

Department of Instrumentation Engineering, Shri Guru Gobind Singhji Institute of Engineering and Technology, Vishnupuri, Nanded, Maharashtra, India Discipline of Mechanical Engineering, School of Engineering, Indian Institute of Technology, Indore, Madhya Pradesh, India

A R T I C L E I N F O

A BS T RAC T

Keywords: Autonomous underwater vehicle- manipulator system (AUVMS) Fuzzy logic control PID control Nonlinear PID control Disturbance estimator Task-space control Lyapunov stability

This paper presents, a robust nonlinear proportional-integral-derivative (PID)-like fuzzy control scheme for a task-space trajectory tracking control of an autonomous underwater vehicle-manipulator system (AUVMS) employed for deep-sea intervention tasks. The effectiveness of the proposed control scheme is numerically demonstrated on a planar underwater vehicle manipulator system (consisting of an underwater vehicle and two link rotary (2R) serial planar manipulator). The actuator and sensor dynamics of the system are also incorporated in the dynamical model of an AUVMS. The proposed control law consists of two main parts: first part uses a feed forward term to reinforce the control activity with extravagance from known desired acceleration vector and carries an estimated perturbed term to compensate for the unknown effects namely external disturbances and unmodeled dynamics and the second part uses a PID-like fuzzy logic control as a feedback portion to enhance the overall closed-loop stability of the system. The primary objective of the proposed control scheme is to track the given end-effector task-space trajectory despite of external disturbances, system uncertainties and internal noises associated with the AUVMS system. To show the effectiveness of the proposed control scheme, comparison is made with linear and nonlinear PID controllers. Simulation results confirmed that with the proposed control scheme, the AUVMS can successfully track the given desired spatial trajectory and gives better and robust control performance.

1. Introduction In the recent years, underwater robotic vehicles have gained huge importance on the marine research for the explorations of the ocean resources and subsea development for the preservation of global environment. Many underwater intervention tasks are performed today using autonomous underwater vehicles (AUVs) and remotely operated vehicles (ROVs). As remotely operated vehicles are very expensive and requires two or more skilled operators for their operation, autonomous underwater vehicles (AUVs) are mostly preferred for underwater intervention tasks. The key element in underwater intervention performed with autonomous vehicles is autonomous manipulation, which is completed through manipulator systems (Marani et al., 2009). Generally, the underwater manipulator with underwater vehicles are widely used in the fields of scientific research and ocean systems engineering for performing interactive tasks such as opening and closing of valves, cutting in coordination, drilling, sampling, coring etc (Santhakumar and Kim, 2011; Shim et al., 2010).

⁎

An autonomous underwater vehicle when equipped with a manipulator system becomes kinematically redundant system i.e. it has more degrees of freedom (DOF) than is required to perform a task in its space (i.e. m dimensional). This leads to infinite number of joint-space solutions for specific task-space coordinates and thus requires redundancy resolution schemes to handle the situation (Sarkar and Podder, 2001). Actually, from the dynamic control view, the AUVMS are very similar with the redundant parallel manipulators (Shang et al., 2014a, 2014b) and mobile-manipulators. A number of algorithms have been proposed over a period of time to resolve the problem of redundancy resolution (Sarkar and Podder, 2001; Soylu et al., 2010; Antonelli and Chiaverini, 1998, 2003; Chiaverini, 1997; Shang et al., 2012; Podder and Sarkar, 2004; Santos et al., 2006; dos Santos et al., 2006; Han and Chung, 2007). However, many of these approaches require sophisticated optimization techniques (Klein and Huang, 1983) and the motions of the AUVMS are coordinated through joint space control schemes. Besides, there are a variety of underwater tasks, where control of the end-effector motion is highly desirable and needs to be

Corresponding author. E-mail addresses: [email protected] (P.S. Londhe), [email protected] (S. Mohan), [email protected] (B.M. Patre), [email protected] (L.M. Waghmare).

http://dx.doi.org/10.1016/j.oceaneng.2017.04.030 Received 1 May 2016; Received in revised form 21 March 2017; Accepted 20 April 2017 0029-8018/ © 2017 Elsevier Ltd. All rights reserved.

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controlled very precisely. In such situations, joint space control schemes may not be suitable (Ismail, 2011; Mohan and Kim, 2015). Also, online modification of the motion of the end-effector task-space can be easily accommodate by the task-space control schemes in contrast to joint space control schemes (Ismail, 2011). Considering the above facts, a robust task-space control scheme is proposed based on a nonlinear PID-like fuzzy logic control approach. Introduction of manipulator to AUV makes their control more difficult because of highly nonlinear, coupled, and high-dimensional nature. In addition, uncertainties in the hydrodynamic parameters and influence of the external disturbances such as ocean currents, make the task of controlling an AUVMS more challenging. Several modeling and control methods applied on underwater robotics systems can be found in the literature (Esfahani et al., 2015; Antonelli, 2014; From et al., 2010; Xu et al., 2012, 2005; Soylu et al., 2008; Han and Chung, 2008; Antonelli et al., 2000, 2004; Cui and Sarkar, 2001; Schjlberg and Fossen, 1994; Mahesh et al., 1991; McLain et al., 1996; Santhakumar and Kim, 2011; Canudas de Wit et al., 2000; McMillan et al., 1995). However, due to unstructured properties of interactive underwater work, a good understanding of the dynamics of a robotic manipulator mounted on a moving underwater vehicle is required for autonomous manipulation tasks. Therefore, the aforementioned control techniques requires an exact mathematical model of the underwater robotic system to achieve better control performance. It is well known that, obtaining an exact mathematical model of an AUVMS is extremely arduous since it involves uncertainties in hydrodynamic parameters, is highly nonlinear, coupled and is associated with time-varying dynamical characteristics. In such situations, intelligent control techniques such as fuzzy logic control (FLC) can be useful. The main features of an FLC is that, it can applied to the plants that are not so well defined mathematically. For such plants, FLC can be designed to emulate human deductive thinking i.e. embody human-like thinking into automatic control system (Lee, 1990; Li and Gatland, 1996). Based on experience of a human expert, heuristics fuzzy control rules can be constructed to model the behavior of the system. The combination of PI, PD, PID and FLC results in a dynamic fuzzy controller structure which gives better control performance and a simple control structure (Londhe et al., 2014). Also this structure provides reduction in overshoot and adds damping to the overall closed loop system (Lee, 1990; Li and Gatland, 1996). The nonlinearities of the system can be handled by the appropriate choice of input and output membership functions (MF's). However, tuning of fuzzy control rules are difficult and time consuming task as it has more parameters to tuned than its nonfuzzy counter parts. Fuzzy controller can be tuned by means of 1) rules tuning 2) MF's tuning and 3) input/output scaling gain tuning. Among these tuning methods, input/output scaling gain tuning is easier way to tune fuzzy controller than rule base and MF's tuning. The input and output scaling factors (SF's) of the PID controller need to be tuned properly for its optimum performance. This problem can be handled by proper designing of rule base structure. In this paper, a general robust rule base is constructed using standard MF's, leaving the optimum tuning to the scaling gains. The general robust rule base structure design greatly reduces the difficulties in tuning of the input and output scaling factors (SFs) of the PID controller (Lee, 1990; Li and Gatland, 1996). To improve control performance, a nonlinear PID-like fuzzy control is designed and applied for task-space control problem of an AUVMS. The proposed control scheme mainly divided into two parts namely a feed forward term along with a disturbance estimator and PID-like fuzzy logic control law. The first part of the proposed control scheme is used to enhance the control activity with indulgence from known desired acceleration vector and estimated perturbed term to compensate for the unknown effects and unmodeled dynamics, while the second part acts as a feedback portion to enhance the overall closedloop stability of the system. The proposed scheme enables overcoming the difficulty due to parameter uncertainties, external variations (e.g.,

Fig. 1. Pictorial view of an AUVMS with its coordinate frame arrangement.

buoyancy, reaction forces, and payload variations) and disturbances (e.g., underwater current). For the sake of clarity, the intention of this study is restricted (performance analysis) to the horizontal plane of the AUVMS motion with a predefined planar manipulation tasks. The effectiveness of the proposed control scheme is evaluated with specified planar AUVMS tasks and compared with linear and nonlinear PID control schemes. The paper is structured as follows. Section 2 describes dynamic modeling of an AUVMS. Design of proposed robust nonlinear PID-like fuzzy control scheme is described in Section 3. Simulation results and performance analysis have been discussed in Section 4. Finally the paper is concluded in Section 5 (Fig. 1). 2. Dynamic modeling of an AUVMS This study uses the dynamic model of an AUVMS which is derived using the Newton – Euler and recursive Newton – Euler formulations (Fossen, 1994; Craig, 1986). The frame arrangement considered for an AUVMS is shown in Fig. 5, where, I (x, y, z ) is the Earth-fixed frame, B(xb , yb , zb ) is the vehicle body-fixed frame, E (xt , yt , zt ) is the manipulator-end effector frame and M (xm , ym , zm ) is the manipulator-base frame. The dynamic equations of motion in the inertial (Earth-fixed reference) frame of an AUVMS can be expressed as follows:

M (q )q¨ + C (q, q˙)q˙ + D(q, q˙)q˙ + G(q ) + F (q, q˙) = τ + d

(1)

where,

⎡ M HT (ζ ) ⎤ ⎥, M (q ) = ⎢ v ⎢⎣ H (ζ ) Mm(ζ )⎥⎦

(2)

⎡Cv(η, η˙) 0 ⎤ ⎥, C (q , q˙) = ⎢ Cm(ζ , ζ˙)⎦ ⎣ 0

(3)

⎡ Dv(η, η˙) 0 ⎤ ⎥, D(q , q˙) = ⎢ ˙ 0 D ( ⎣ m ζ , ζ )⎦

(4)

⎡ F (q , q˙) ⎤ F (q , q˙) = ⎢ v ⎥, ⎣ Fm(q , q˙)⎦

(5)

⎡ G (η ) ⎤ ⎡τ ⎤ G (q ) = ⎢ v ⎥ , τ = ⎢ τ v ⎥ , G ( ζ ) ⎣ m⎦ ⎣ m ⎦ T

(6) T

q = [η ζ ] with η = [x y z ϕ θ ψ ] is the vector of absolute positions and Euler angles (roll, pitch and yaw). x - surge position, y – sway position, z – heave position, ϕ – roll angle, θ – pitch angle and ψ- yaw angle. Mv(q )q¨ is the vector of inertial forces and moments of the vehicle (including added mass effects). Cv(ν, η)η˙ is the vector of Coriolis and centripetal effects of the vehicle (including added mass effects). 2

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lμ and F lμ , D mμ , C lμ , G lμ are the known values (approximated model M parameter values) of inertia matrix, Coriolis and centripetal matrix, damping effects, restoring effects and dynamic coupling effects (forces) of the AUVMS, respectively. τdis is the lumped uncertainties vector consists of internal (parametric uncertainties) (τidis) and external (τedis) disturbances acting on the AUVMS. The following assumptions and properties are considered to ensure the asymptotic convergence of disturbance and trajectory tracking response in the overall closed-loop system:

Dv(ν, η)η˙ is the vector of damping effects of the vehicle (both linear and quadratic damping terms). Gv(η) is the vector of restoring forces and moments acting on the vehicle. τv = [X Y Z K M N ]T is the resultant input vector of thrusters, control plane forces and moments. ζ = [θ1 θ2 ⋯ θn]T is the vector of joint variables while θ1, θ2, ⋯ θn are the joint positions of the corresponding underwater manipulator links, where n is the number of joints. Mm(ζ )ζ¨ is the vector of inertial forces and moments of the manipulator (including added mass effects). Cm(ζ , ζ˙)ζ˙ is the vector of Coriolis and centripetal effects of the manipulator (including added mass effects). Dm(ζ , ζ˙)ζ˙ is the vector of damping effects of the manipulator (both linear and quadratic damping terms). Gm(ζ ) is vector of the restoring forces and moments for the manipulator. HT (ζ )ζ¨ and H (ζ , ζ˙)η¨ are the vectors of reaction forces and moments between the vehicle and the manipulator due to the manipulator and the vehicle inertial effects, respectively. Fv(q , q˙) and Fm(q , q˙) are the vectors of the coupling dynamic effects due to the manipulator inclusion and its motion on the vehicle, and the interaction effects due to the vehicle motion on the manipulator, respectively. τm = [τm1 τm2 ⋯ τnm]T is the vector of manipulator control inputs while τm1,τm2,⋯, τnm are the joint torques of the corresponding manipulator links. d is the vector of external disturbances due to underwater current and/or a paylaod. The AUVMS consists of a n-DOF manipulator and a 6-DOF vehicle, which means the system has more DOF than the dimension of the taskspace (kinematically redundant system), required to perform the given task. The relation between task-space and configuration-space coordinates are related as,

Assumption 1. The PID-like fuzzy, feed-forward controllers and disturbance estimator gain namely Kc, Λ and Γ are constant symmetric and positive definite (SPD) matrices by design,

Kc = KcT > 0, Λ > 0

M (q ) = MT (q ) > 0, ∀ q ∈ R n

sT [M˙ (q ) − 2C (q , q˙)]s = 0, ∀ s ∈ R n, q ∈ R n, q˙ ∈ R n

The conventional linear 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 (Astrom and Hagglund, 2001). However, linear fixed-gain PID controllers may not give satisfactory performance, where highperformance control is required with changes in operating conditions or environmental parameters (Su et al., 2005). In order to enhance the performance of linear PID controllers, a new approach have been developed, known as nonlinear PID control and widely accepted as the most effective method for industrial applications (Su et al., 2005, 2004; Rugh, 1987; Taylor and Astrom, 1986; Armstrong et al., 2001; Seraji, 1998). However, these controllers requires exact knowledge about the process to have better control performance. On the other hand, fuzzy logic control can deal with poorly understood processes and have ability to approximate nonlinear system (Fei and Zhou, 2012). FLC can be easily designed for large-scale nonlinear system (Londhe et al., 2014). It is known for its ability to cope with nonlinearities and uncertainties. Introduction of dynamic fuzzy controller structures with the aim of control system performance leads to PI-, PD- or PID-fuzzy controllers (Akkizidis et al., 2003; Zheng et al., 2001; Mann et al., 1999; Precup and Hellendoorn, 2011). To further enhance the performance of these dynamic fuzzy controllers structures, nonlinear PI-, PDor PID controllers can be introduced instead of linear PI-, PD- or PID controllers. In this paper, a robust nonlinear PID-like fuzzy controller is designed and applied for the task-space control problem of an AUVMS. Firstly, a PID-like fuzzy control law design is considered followed by the second part of the proposed control scheme.

(9)

where, Mμ is the inertia matrix, Cμ is the Coriolis and centripetal matrix, Dμ is the damping matrix, Gμ is the vector of restoring effects and Fμ is the vector of dynamic coupling of the AUVMS in task-space, respectively. τct is the vector of task-space control inputs of the AUVMS. τedis is the vector of external disturbances in the task-space. Under parametric uncertainties, the AUVMS dynamic equation of motion in Eq. (9) can be stated as,

(Mμ + △Mμ)μ¨ + (Cμ + △Cμ)μ˙ + (Dμ + △Dμ)μ˙ + Gμ + △Gμ + Fμ (10)

By gathering uncertainties along with external disturbances into a single term which is known as lumped uncertainty τdis, then Eq. (10) takes the form as,

lμ μ˙ + D lμ + F mμ μ¨ + C lμ μ˙ + G lμ = τct + τdis M

(11)

where,

τdis = τedis − △Mμ μ¨ − △Cμ μ˙ − △Dμ μ˙ − △Gμ − △Fμ

(15)

3. Robust nonlinear PID-like fuzzy control scheme

(8)

where, J (q ) is the Jacobian matrix represents mapping between configuration-space velocities to task-space velocities. Hence, from (8) and (1) the dynamics of an AUVMS can be represented in the task-space coordinates as

+ △Fμ = τct + τedis

(14)

Property 2. The matrix M˙ (q ) − 2C (q , q˙) is a skew-symmetric matrix i.e.

where μ = [xt yt zt ] is the vector of task-space coordinates and f(q) is the vector representing the forward kinematics. The derivative of Eq. (7) gives

Mμ μ¨ + Cμ μ˙ + Dμ μ˙ + Gμ + Fμ = τct + τedis

(13)

Property 1. The inertia matrix M(q) is symmetric and positive definite i.e.

T

μ˙ = J (q )q˙

Γ = ΓT > 0

Assumption 2. The value of the lumped disturbance is arbitrarily large and slowly varying with time i.e. (τ˙dis ≈ 0) (Kelly et al., 2005; Sabanovic and Ohnishi, 2011; Antonelli, 2014). The following properties are observed for the AUVMS with respect to inertial (fixed-ground base) frame (Craig, 1986; Antonelli, 2014), as given below:

(7)

μ = f (q )

and

(12)

mμ μ¨ − Mμ μ¨ △Mμ μ¨ = M lμ μ ̇ − Cμ μ ̇ △Cμ μ ̇ = C lμ μ ̇ − Dμ μ ̇ △Dμ μ ̇ = D

3.1. PID-like fuzzy control scheme

lμ − Gμ △Gμ = G

The control structure of PID-like fuzzy controller is shown in Fig. 2, where, SPI is the velocity type PI control, SPD is the position type PD

lμ − Fμ △Fμ = F 3

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Fig. 2. Control structure of PID-like fuzzy controller.

control and adding both these control gives (SPID) PID type control. The variables Ke and Kd are the input scaling factors (SFs) for input e and e˙ , respectively and K1 and K2 are the output scaling factor (SFs). Design parameters of PID-like fuzzy controller are explained as follows: Fuzzification Fuzzification involves a scale mapping between the range of values of input variables and corresponding universe of discourse. The error (e) and the corresponding change of error (e˙ ) are two fuzzy input variables to the fuzzy controller as shown in Fig. 2. The control signal SPID is the output of PID-like fuzzy controller and is necessitated to actuate the actuators of an AUVMS to control its motion. The membership functions for error input (e, e˙ ) and output (SPID) variables are shown in 3, where N stands for negative, L for large, M for medium, S for small, Z for zero and P for positive. 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 (Driankov et al., 2004). Rule base logic Fig. 4. Rule justification by phase plane a) typical step response b) corresponding phaseplane trajectory.

For optimum control performance, control rules, membership functions and scaling factors are needed to be tuned properly. This is the fundamental problem in fuzzy logic control design (Li and Gatland, 1996; Londhe et al., 2014). To overcome the difficulties in optimal control tuning, a standard phase plane technique has been used for designing robust rule base for PID-like fuzzy type control (Li and Gatland, 1995) (Fig. 3). Fuzzy control rules are formed by analysing the behavior of a controlled system. 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 (Lee, 1990). A closed loop trajectory in phase plane analysis is used to justify the fuzzy control rules as shown in Fig. 4. 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 ), with 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.

1. At peak-valley points (c1, c2 ) and ( f1 , f2 ), rule for u0 is as,

SPID 0 = e, when e˙ = 0 and S˙PID0 = e, when e˙ = 0 2. The remaining rules (a) To prevent any overshoot in the area A2 / A4 and A6 / A8

SPID = PS , when e˙ > 0 SPID = NS , when e˙ < 0 (b) When in area A1 or A5 and A3 or A7

SPID = min{(SPID 0 + e˙), PS )}A1 and A5 rules must be positive SPID = max{(SPID 0 + e˙), NS )}A3 and A7 rules must be negative (c) When in area A2 or A6 and A4 or A8

k k−1 = SPID . 1. If both e and e˙ are zero, then SPID 2. If conditions are such that e will go to zero at a satisfactory rate, then k k−1 SPID = SPID . 3. If error e is not self-correcting, then rules can be constructed by considering cross-over set of points to prevents the overshoot or undershoot in the corresponding area, Ai, where i = 1, …, 8. (refer Fig. 4)b) The magnitude of rules: The magnitude of rules can be find out from the following heuristic steps:

SPID = SPID 0 + e˙ and S˙PID = S˙PID0 + e˙ = e + e˙

Based upon above heuristic knowledge, a general and robust rule base can be constructed to achieve PID type control characteristics. This general rule base is robust enough for a wide range of applications and can be tuned slightly for a optimum performance. Table 1 shows a general PID-like fuzzy control type rule base. 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 motion and joint angles of an AUVMS. The Center of Area (COA) defuzzification technique is used here because it yields better steady-state performance (Lee, 1990; Driankov et al., 2004). The crisp output control action is

Fig. 3. Inputs (e, e˙ ) and output (SPID) membership functions.

4

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Table 1 A general rule base for PID-like fuzzy control.

e˙⧹e

NL

NM

NS

Z

PS

PM

PL

PL PM PS Z NS NM NL

Z NS NM NL NL NL NL

PS Z NS NM NL NL NL

PM PS Z NS NM NL NL

PL PM PS Z NS NM NL

PL PL PM PS Z NS NM

PL PL PL PM PS Z NS

PL PL PL PL PM PS Z

Fig. 5. Concept of disturbance estimator.

defined as follows:

SPID =

∼ ∼ ∫ μc (x∼)xdx ∼ ∫ μc (x )dx∼

augmenting it in the nonlinear PID-like fuzzy controller. Fig. 5 shows the conceptual diagram of a typical disturbance estimator that is used in a underwater robotic application. The output of the disturbance estimator can be used in feed forward compensation of disturbances and this will avoid use of high feedback gains to have excellent tracking performance and smooth control actions. The proposed disturbance estimator takes the form as,

(16)

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. Lemma 1. A general PID-like fuzzy control rule base structure as given in Table 1 ensures the stable response and bounded output given by the SPID controller. Proof. Let us consider Lyapunov function candidate based on classical Lyapunov synthesis method (Kandel et al., 1999; Margaliot and Langholz, 1999) as,

Vs =

1 ∼2 ∼˙ 2 (μ + μ ) 2

(17)

Vs = 0

and

V˙s = 0, only if SPID = 0.

dis

(24)

Substituting the value of Υ˙ from Eq. (22) and known acceleration μ¨ from Eq. (11), we get,

mμ−1τ∼dis − M mμ−1SPID τl˙dis = ΓM

(19)

Now, for stable response, we require that V˙s = μ∼∼ μ˙ + μ∼˙ c ≤ 0 . So, if μ∼ and μ∼˙ have opposite signs, then it is necessary that c=0. If μ∼ and μ∼˙ are both positive, then c < − μ∼. If μ∼ and μ∼˙ are negative, then c > − μ∼. If μ∼ = 0 and μ∼˙ is negative, then c > 0 . If μ∼ = 0 and μ∼˙ is positive, then c < 0 . μ˙ + μ∼˙ c ≤ 0 . ∀ μ∼ and μ∼˙ = 0 , then ∀ c , we have V˙s = μ∼∼ As PID-like fuzzy control type rule base, as given in Table 1, satisfies all above conditions. Hence, the PID-like fuzzy controller is stable and gives bounded response i.e.

V˙s < 0, ∀ SPID ≠ 0;

dis

τl˙dis = Υ˙ + Γμ¨

(18)

For the sake of convenience, denote c = μ¨d − x˙2 , Then Eq. (18) is simplified to

and

(22)

dis

‥

Vs > 0

lμ μ˙ + D lμ + F mμ−1[τct + Υ + Γμ˙ − (C lμ μ˙ + G lμ)] − M mμ−1SPID Υ˙ = − ΓM

Now, τldis is to be modify in a such way that the estimation error τ∼dis goes to zero asymptotically. Differentiating Eq. (21),

‥ ‥ V˙s = ∼∼ μ μ˙ + ∼∼ μ˙ μ = ∼∼ μ μ˙ + ∼ μ˙ (μd −x1)

V˙s = μ∼∼ μ˙ + μ∼˙ c.

(21)

where, τldis is the vector of estimated lumped uncertainty term. ϒ is an auxiliary vector to estimate the unknown perturbations from the dynamics of SPID. Let us express the estimation error as, τ∼ = τ − τl (23)

where μ∼ = μd − μ; μ∼˙ = μ˙d − μ˙ . By defining, x1 = μ and x 2 = x˙1 = μ˙ . Then derivative of Eq. (17) will give,

‥ = ∼∼ μ μ˙ + ∼ μ˙ (μd −x˙2 ).

τldis = Υ + Γμ˙

(25)

With some adjustment in Eq. (25) gives,

mμ−1τ∼dis + M mμ−1SPID + τ˙dis τ˙dis − τl˙dis = − ΓM

(26)

mμ−1τ∼dis + M mμ−1SPID + τ˙dis τ∼˙dis = − ΓM

(27)

The Assumption 2 holds true if

mμ−1τ∼dis + M mμ−1SPID τ∼˙dis = − ΓM

(28)

It can be noted from Eq. (28) that the convergence rate of the estimation errors is depends on the value of estimator gain Γ and might get influenced due to time varying hydrodynamic coefficients of mμ−1 with change in working conditions. However, addition of matrix M SPID in the auxiliary vector Υ will improve the performance of uncertainty disturbance estimation. Furthermore, in order to have improved transient response and robustness against large parameter variations during task space tracking control, a robust nonlinear PIDlike fuzzy control scheme has been designed and explained in the following Section 3.3.

(20)

3.2. Disturbance estimator The performance of the PID-like fuzzy task-space controller can be improved by the use of disturbance estimator. This can be accomplished by treating the effects of the system uncertainties, nonlinearities, unmodeled dynamics, cross-coupling effects and external disturbances like ocean current as a total or composite disturbance, termed as lumped uncertainty acting on the AUVMS system and then estimate this unknown term using the disturbance estimation technique. Finally, its effect on the AUVMS can be canceled out by

3.3. A robust nonlinear PID-like fuzzy control scheme In this study, a robust nonlinear control scheme is proposed to perfectly follow a desired task space position trajectory of the AUVMS for all possible values of time-varying uncertainties and external disturbances. The idea used in selecting control law is that the tracking 5

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from the desired task-space variables block where user inputs are given based on the AUVMS tasks (missions) and time namely μd and t. The trajectory planner provides the desired task-space coordinates namely time trajectories of the task-space position, velocity and acceleration vectors respectively, based on the above user inputs. These task-space values can be converted into joint space coordinate values through the inverse kinematic relations. The proposed AUVMS has measurement systems which will give task-space positions and task-space velocities of the manipulator, through the help of configuration space (system) position and velocity sensors by forward kinematics. The sensor dynamics are incorporated into the dynamical model of an AUVMS. Therefore, tracking errors i.e. position error, velocity error and integral position error are calculated based on desired and actual states and applied to the PID-like fuzzy controller block, which enhances the closed-loop stability of the system and improve transient performance of the proposed controller. Similarly, the feed-forward controller gives the desired input values theoretically based on the desired acceleration and inertia matrix. However, in the actual case these desired input values may vary considerably due to system dynamics. Therefore, the virtual reference acceleration vector and the theoretical input values are calculated based on the revised controller scheme in the feed forward controller. The model based controller uses, inverse dynamic model output with some known inaccuracy, for approximating task torques/forces of the AUVMS. A model-based control term linearises the feedback scheme and making the system approximately linear and decoupled, which results in straight-forward and simple structure of the system. The uncertainty estimator has an efficient estimation approach to estimate the perturbation from the dynamics of the PID-like fuzzy controller and known inverse dynamic model. This helps in compensate the model uncertainties, unknown external disturbances, and time-varying parameters (this estimator restricted with the bounded and slowly varying disturbances). In this work, one of the tasks is to estimate the bounded (slowly varying) disturbance/uncertainty vector and the estimator at least exponentially converges to the actual disturbance/uncertainty vector under any position trajectory. During the simulation runs, both

errors must converges to zero and stays thereafter even in presence of lumped disturbance and provide strong robustness, good transient performance and fast response. A proposed robust control scheme can be given as,

lμ μ˙ + D lμ + F mμ(μ¨ + KcSPID ) − τldis + C lμ μ˙ + G lμ τct = M r

(29)

where, ‥

‥

μr = μd + 2Λμ˙ + Λ2 μ; τldis = Υ + Γμ˙; lμμ˙ + D lμ + F mμ−1[τct + Υ + Γμ˙−(C lμμ˙ + G lμ)]−M mμ−1SPID Υ˙ = −ΓM

(30)

μ∼̇ = μḋ − μ ̇; μ∼ = μd − μ. Kc and Λ are the symmetric positive definite (SPD) gain metrics of PIDlike fuzzy and feed-forward controller respectively. μ¨r is the virtual reference (desired) acceleration vector. μd, μ˙d and μ¨d are the given desired task-space position, velocity and acceleration vectors, respectively. Γ is a positive diagonal matrix. SPID is the decentralized PID-like fuzzy control input vector. τldis is the vector of estimated lumped disturbance term. Υ is an auxiliary vector to estimate the unknown lμ and F lμμ˙ , D lμ are the known values mμ(μ), C lμμ˙ , G disturbance vector. M (approximated model parameter values) of inertia matrix, Coriolis and centripetal matrix, damping matrix and restoring effects of the manipulator, respectively. The block diagram that corresponds to the proposed control scheme is shown in Fig. 6. The proposed controller consists of four integral components namely a PID-like fuzzy controller, a feed-forward controller, model based (feedback linearization) controller (known or lμ and F lμμ˙ , D lμ ) and uncertainty estimamμ(μ), C lμμ˙ , G estimated matrices M tor. The introduction of the model based controller in the proposed control scheme results in better and faster learning of the plant dynamics by a PID-like fuzzy controller. Even though it is very difficult to extract the exact values of the model matrices, poor knowledge about these model matrices is sufficient for obtaining the desirable performance from the PID-like fuzzy controller. The block diagram flow starts

Fig. 6. Block diagram representation of the proposed control scheme.

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V>0

and

V˙ < 0, ∀ SPID ≠ 0, ∀ τ∼dis ≠ 0,

(39)

therefore also,

V=0

and

V˙ = 0, only if SPID = 0, τ∼dis = 0.

(40)

Therefore, based on Lyapunov direct method and Barbalats Lemma (Kelly et al., 2005; Slotine and Li, 1991), the closed-loop system is globally asymptotically stable i.e.,

lim SPID(t ) = 0, lim τ∼dis(t ) = 0,

t →∞

t →∞

lim μ∼(t ) = 0, lim μ∼̇(t ) = 0. t →∞

4.1. Description of the system

coulomb and viscous friction forces have been added to closely portray the actual system and also to demonstrate the effectiveness and the robustness of the controller. The friction values are chosen to shadow the actual frictional effects as closely as possible.

The performance of the system is gauged by performing extensive numerical simulations involving positioning and trajectory tracking of the end effector. For ease of understanding and clarity, the AUVMS horizontal plane motion is considered. In that, 2-DOF manipulator system (MS) equipped with 3-DOF horizontal plane AUV model is considered for the numerical simulation purpose. With this considerations, vector q in Eq. (1) is taken here as q = [η ζ ] = [x y ψ θ1 θ2] with η = [x y ψ ] and ζ = [θ1 θ2]. μ = [xt yt ] is the task-space position vector considered for the proposed system. The AUVMS considered for the analysis consists of a planar serial manipulator which has two active rotary joints. The shape of the vehicle is rectangular prism and it has four thrusters which help in maneuver all motions of the vehicle. The shape of the manipulator links have been assumed to be cylindrical though the performance of the control scheme is least affected by it as it would accommodate these variations with ease. The inertial parameters of the vehicle and the manipulator links have been found from the solid model of the system and hence emulate the actual parameters. The hydrodynamic parameters are obtained by using empirical relations (Fossen, 1994). The schematic diagram of the planar AUVMS with coordinate frame assignment considered for the numerical study is depicted in Fig. 7. The vehicle physical and hydrodynamic parameters are given in Appendix.

3.4. Stability analysis of the proposed control scheme Theorem 3.1. For the uncertain nonlinear system given by Eq. (11) with the Assumptions 1 and 2, if we choose the control input vector as stated in Eq. (29), then the task-space tracking errors converge to zero asymptotically. Proof. Let us define the Lyapunov candidate function as,

1 T 1 T∼ SPIDSPID + τ∼dis τdis 2 2

(31)

Differentiating Eq. (31) leads to T ˙ T ∼ V˙ = SPID SPID + τ∼dis τ˙dis

(32)

where, S˙PID is the rate of change of control inputs given by the PID-like fuzzy control. The nonlinear dynamic equation of motion of an AUVMS as given in Eq. (11) can be rewritten as

lμμ˙ − D lμ − F mμ(μ)−1(τct + τdis − C lμμ˙ − G lμ) μ¨ = M

(41)

4. Performance analysis

Fig. 7. Coordinate frame assignment of the planar AUVMS.

V=

t →∞

(33)

The output of the PID-like fuzzy controller can be written as,

SPID = μ∼˙ + 2Λμ∼ + Λ2

∫ μ∼dt

4.2. Description of the task

(34)

Differentiating Eq. (34), we have

S˙PID = μ∼¨ + 2Λμ∼˙ + Λ2 μ∼ = μ¨d − μ¨ + 2Λμ∼˙ + Λ2 μ∼

The underwater effects that are bound to act on the AUVMSsuch as the underwater currents, damping effects, etc. have been included in the disturbance vector which has been exclusively appended to the dynamics of the system to simulate the underwater conditions that the AUVMS will be subjected to while performing any of the positioning or tracking operations. The task-space vector consists of end effector positions i.e., [xt = xend yt = yend ]. The test case chosen here demands the AUVMS to start from a user-defined initial position and return to the same position after traversing a predefined trajectory. A complex taskspace trajectory considered here is a loop comprising of a vertical rise, a circular path defined by two different arcs namely concave up and concave down preceding by a ramp and a vertical drop and ending with slow rising ramp thus reaching the starting point in accordance with respect to time and this complex task-space trajectory can be represented in polynomial form as given by Eqs. (42) and (43). The initial velocity vector of the links were set to be zero and random values were assigned to the estimated system disturbance vectors, but the intended and actual orientation of the end-effector are assumed to be almost equal.

(35)

Substitute the value of μ¨ from Eq. (33) and control input vector from Eq. (29) in Eq. (35) will results,

mμ−1τ∼dis S˙PID = − KcSPID − M Multiplying by T ˙ SPID SPID

=−

T SPID

(36)

to above Eq. (37),

T SPID KcSPID

T m −1∼ + SPID M μ τdis

(37)

Putting the value of Eq. (37) and the value of rate of change of estimation errors from Eq. (28) into Eq. (32), we get, T T m −1∼ V˙ = − SPID KcSPID − τ∼dis ΓM μ τdis

(38)

Since, the controller gain matrices Kc and Γ are constant SPD matrices, by design the closed loop system is asymptotically stable. This implies that the task space trajectory tracking errors will converge to zero asymptotically. Then, 7

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⎧ 2.5 ⎪ ⎪ 2.5 + 0.8cos⎛⎜900 + 9 (t − 30)⎞⎟ ⎪ ⎠ ⎝ 3 ⎪ ⎞ ⎛ 9 ⎪ x (t ) = ⎨1.7 + 0.8cos⎜ − (t − 60)⎟ − 0.8 ⎠ ⎝ 3 ⎪ ⎪ 2 0.9 − 0.0080( − 90) + 1.778 × 10−4(t − 90)3 t ⎪ −1.5 ⎪ ⎪−1.5 + 0.0133(t − 150)2 − 2.96 × 10−4(t − 150)3 ⎩

0s ≤ t ≤ 30s 30s ≤ t ≤ 60s 60s ≤ t ≤ 90s 90s ≤ t ≤ 120s 120s ≤ t ≤ 150s 150s ≤ t ≤ 180s (42)

⎧ 0.5 + 0.0067t 2 − 1.4815 × 10−4t 3 ⎪ ⎞ ⎛ 0 9 ⎪ ⎪ 2.5 + 0.8sin⎜⎝90 + 3 (t − 30)⎟⎠ − 0.8 ⎪ ⎞ ⎛ 9 ⎪ ⎪1.7 + 0.8sin⎜ − (t − 60)⎟ ⎠ ⎝ 3 y (t ) = ⎨ ⎪ 2 −4 3 ⎪ 0.9 + 0.0070(t − 90) − 1.555 × 10 (t − 90) ⎪ 3 − 0.0117(t − 120)2 + 2.592 × 10−4(t − 120)3 ⎪ ⎪ −0.5 + 0.0033(t − 150)2 − 7.4074 × 10−4 ⎪ 3 ⎩ (t − 150)

0s ≤ t ≤ 30s

Fig. 8. The change in the norm of the control inputs with an uncertain working condition.

30s ≤ t ≤ 60s

disturbances are considered as a vector (for the simulations [10 N, 5 N, 0.5 Nm, 0.2 Nm, 0.1 Nm]T ) and the system uncertainties are considered as 20% (i.e., the system model consists of only 80% of actual values). All the results which have been obtained are depicted in Figs. 8–13, which illustrate the tracking performance of the controller while tracking a trajectory in task-space. Here, all the controllers uses good (nominal) parameter settings as mentioned above for performing task-space control task. The tracking performance of the proposed scheme is quite acceptable and adequate for underwater applications. The time histories of the norm of the vectors of the task-space position tracking errors and control inputs are presented in Figs. 8 and 9 respectively. From these results, it is observed that proposed controller norm immediately falls down to zero and provides smooth control efforts as compared to the linear PID (L-PID) and nonlinear PID (NL-PID) controllers. Fig. 10 shows the variations in the AUV motion in horizontal plane i.e. position along X- and Y-axis and their corresponding angular velocities, yaw angle, (ψ) and yaw angle rate, r during task-space trajectory tracking control. It is quite that the proposed controller provides smooth change in forward velocity, sway velocity while performing trajectory tracking in task-space. Time trajectories of the joint-space angles during uncertain condition can be seen in Fig. 11. During predefined complex task-space trajectory tracking control, variations in the task-space positions and their corresponding errors are depicted in the Fig. 12. It can be conclude that, proposed control scheme converges the task-space position errors within short duration of time which is quite acceptable as compared to other controllers since adequate oscillation in the task-space errors are produced by the L-PID and NL-PID controllers during the task. Fig. 13 shows the task-space motion trajectory tracking of the AUVMS in the xy plane with all the controllers. It can be seen in zoomed scale version of Fig. 13, L-PID gives more distortions in the task-space trajectory tracking than the NL-PID controller, while proposed shows tight tracking performance. A quantitative analysis of the tracking performance of an AUVMS given by all the controllers in terms of maximum error (ME), root mean square of error (RMS) and the L2 (Euclidean) norm in task-space

60s ≤ t ≤ 90s 90s ≤ t ≤ 120s 120s ≤ t ≤ 150s 150s ≤ t ≤ 180s (43)

4.3. Results and discussions The results of the predefined complex task-pace position tracking control of the planar AUVMS is presented and it can be used as a reference to the performance analysis of the proposed control scheme. Considerations done while performing these simulations are quite elaborate and they emulate the actual manipulator to a satisfactory extent. Considerations such as disturbances, parameter uncertainties and sensor noises have all been incorporated in the numerical model ensuring the usage of the proposed controller in an actual prototype without compromising either performance or effectiveness. The control torque forces i.e. τct generated by the proposed control law as given in Eq. (29), used to manipulate the joint angles of the manipulator namely θ1 and θ2 along with position of an AUVMS through thruster forces in a such way that, proposed AUVMS will track the desired task-space trajectory with minimum errors. In this control law, output of PID-like fuzzy controller (SPID) is generated by fuzzy inference engine of Mamdani's MAX-MIN method and the defuzzification by the centerof-area method, with two antecedents {e, e˙} and PID-like fuzzy type rule base (see Table 1). As mentioned above, to show the robustness of the proposed control scheme, comparison is made with linear and nonlinear PID control schemes, and are given by,

τL − PID = χPID − τldis

(44)

mμ(μ)(μ¨ + Kcχ ) − τldis + N lμ(μ, μ˙) τNL − PID = M r PID

(45)

where,

χPID = K ′P μ∼ + K ′I

∫ μ∼ + K ′D μ∼˙ dt.

(46)

τL−PID and τNL−PID are the input control vectors given by linear PID and nonlinear PID controllers respectively. χPID is the conventional PID control law. K′P , K′I and K′D are the proportional, integral and derivative gains of the PID controller. The gain matrices and other controller parameter values have been tuned in such a way that all these controllers show satisfactory performance in ideal conditions and hence can be compared with the proposed controller to estimate its performance. The nominal controller setting considered for proposed and nonlinear controllers are as: Kc = 5I2 , Λ = 3I2 and Γ = 2I2 and for linear PID controller are as, K′P = 3.4 , K′I = 4 and K′D = 1. For the uncertain condition, the underwater current is considered with a velocity vector of (0.2, 0.1) m/s in x and y axes. The unknown

Fig. 9. The variations in the norm of task-space tracking errors during trajectroy tracking with an uncertain working condition.

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Fig. 10. The change in the motion of an planar AUVMS during underwater manipulation task.

trajectory of the end effector was performed and can be find in Table 2. From this table, it is confirmed that the proposed control scheme gives minimum tracking errors in x and y directions as compared to other controllers. Among all the controllers, L-PID controller gives the maximum error in task-space position tracking control of the value 0.0117 m and 0.0065 m along x and y directions respectively. To have extreme check on the robustness of the proposed control scheme, variations in the controller gain matrices (i.e. Kc, Λ and Γ) were performed and control performance of the all controllers are observed in terms of norm of the task-space errors. The change in the norm of the task-space errors during variation of Kc gain of NL-PID

and proposed controller can be seen in Figs. 14 (a) and (b), respectively. It is observed from these results that, performance of the NLPID controller goes on degrading as value of gain matrix, Kc, increases from 5 to 50, while proposed controller shows similar kind of control performance for all values of Kc. This suggest that proposed controller has ability to adapt the variations in the controller parameters. Similarly, variations in the second parameter, Γ, is carried out and control performance is observed and depicted in Figs. 15 (a) and (b). Here, again control performance of NL-PID controller goes on degrading as value of Γ increases from 2 to 7. But with the same variation in Γ, control performance of the proposed controller remains the same as

Fig. 11. The change in the joint space positions during task-space trajectory tracking control (uncertain condition).

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Fig. 12. The change in the errors in task-space during tracking control (uncertain condition).

shown in Fig. 15 (b). Also, changes in the norms of the task-space errors with respect to variations in the third gain parameter Λ can be observed for both the controllers in Figs. 16 (a) and (b). In this case, the NL-PID controller gives satisfactory control performance only for the values of Λ ranging from 2 to 3.5. But for other values of Λ, the performance of the NL-PID gets degraded, while changes in the values of Λ parameter does not affect the control performance of the proposed controller. Consequently, it is apparent from the above discussion that the proposed controller provides robust and efficient control performance while tracing a complex pre-defined path in the task-space. The proposed control scheme can be easily extend to three-dimensional task-space position tracking problem of an AUVMS.

Table 2 Quantitative analysis of task-space tracking control along x, y directions. Control schemes

μ∼x

μ∼y

ME

RMS

L2 norm

ME

RMS

L2 norm

L-PID NL-PID

0.0117 0.0041

0.0037 0.0010

0.0520 0.0149

0.0065 0.0027

0.0021

5.9587 × 10−4

0.0295 0.0084

Proposed

0.0038

5.3609 × 10−4

0.0076

0.0024

3.6017 × 10−4

0.0051

proposed controller is made with other conventional controllers like linear and non-liner PID controllers. From the obtained numerical simulation results, the strength of proposed control scheme can summaries as follows:

5. Conclusion

• • •

A robust nonlinear PID-like fuzzy control scheme is designed and applied for task-space position trajectory tracking control problem of an AUVMS for underwater manipulation task applications. The efficacy of the controller is demonstrated with the assistance of numerical simulations on a planar type of AUVMS in the horizontal plane motion. To have evaluation on control performance, the comparison of

Proposed controller increases the overall stability of closed loop system as compared to conventional controllers. Conventional controllers namely L-PID and NL-PID can provide limited stability and mainly depended upon their gain values As proposed PID-like fuzzy controller uses robust rule base struc-

Fig. 13. View of an AUVMS task-space motion trajectory tracking in the x-y plane.

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Fig. 16. The change in the norm of the task-space error during variation of Λ gain of a) NL-PID and b) proposed controller.

Fig. 14. The change in the norm of the task-space error during variation of Kc gain of a) NL-PID and b) proposed controller.

• • •

ture, difficulties in the tuning of scaling gains as well as controller gains are eliminated whereas tuning of conventional controllers requires trial-and-error procedure which is time consuming and plant specific. mμ(μ)) will be Poor knowledge of the system model matrices (M sufficient to design the controller Proposed control scheme provides great immunity to the external disturbances and parameter uncertainties as compared to conventional controllers. Hence, known as robust controller. Proposed controller has simple control structure and design. Hence, can be used for real-time implementation with low cost microprocessor.

This guarantees the performance of the control scheme for usage in position and trajectory tracking applications which is the sole motive behind developing this underwater robotic system. As a future direction, the present method can be extended to 3-dimensional (3-D) taskspace trajectory tracking control of an AUVMS and the same can be validated by performing a real-time experimentation for achieving the complex underwater manipulation tasks for a variety of scientific, industrial and military missions in near future. Further, the proposed method is a general method which can be implemented to any kinematically redundant system with minimum modifications. The use of the proposed control scheme in parallel robotic platforms would be the next research objective. Acknowledgement 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).

Fig. 15. The change in the norm of the task-space error during variation of Γ gain of a) NL-PID and b) proposed controller.

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Appendix A. Vehicle parameters

Physical parameters Parameter

Description

l=0.8 m b=0.25 m m=125 kg m1=5 kg m2=2 kg L1=0.3 m L2=0.2 m Lv=0.4 m Lc1=0.15 m Lc2=0.1 m Iz=50 kg. m2 Iz1=0.05 kg. m2 Iz2=0.03 kg. m2

length of the vehicle width of the vehicle mass of the vehicle mass of link 1 of the manipulator mass of link 2 of the manipulator length of link 1 of the manipulator length of link 2 of the manipulator distance between base frame of the manipulator and the vehicle fixed frame distance between center of joint 1 and center of gravity of the link 1 distance between center of joint 2 and center of gravity of the link 2 Moment of Inertia of vehicle Moment of Inertia of link 1 Moment of Inertia of link 2

Hydrodynamic parameters of manipulator links 1 and 2

Xu1 = 0.5 kg/s

Xu2 = 0.3 kg/s

Yv1 = 1.5 kg/s

Yv2 = 0.9 kg/s

Nr1 = 0.5 kgm/s

Nr2 = 0.3 kgm/s

Xuu=100 kg/m Nr=50 kg m/s Yrr=80 kg/m

Yv=100 kg/s Nrr=100 kg m Nv=30 kg m/s

Yv˙ = 80 kg

Nr˙ = 30 kg/m2

Hydrodynamic parameters of the vehicle Xu=70 kg/s Yvv=200 kg/m Yr=40 kg/s Nvv=60 kg m Xu˙ = 30 kg

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