1. Introduction
Recently, with the exhaustion of land resources, underwater resources are at the center of attention, especially the resources hidden in the deep sea [
1]. Therefore, many underwater works and operations are now being performed both in the scientific and business communities, such as salvage, marine resource investigation, ship engineering, marine construction, etc. [
2]. Owing to the extremely poor working conditions in the deep sea, underwater manipulators, often equipped on the remotely operated underwater vehicles or autonomous and remotely operated underwater vehicles, are considered to be the most suitable tool to work instead of human beings. Hydraulic underwater manipulators and electric underwater manipulators are the most commonly existing commercially available underwater manipulators and most of the experimental/prototype underwater manipulators [
3]. In recent years, with the development and combination of artificial intelligence technology and robot technology, the research of electric underwater manipulators has attracted considerable attention due to their capability for precise motion and force/torque control as they perform in the industrial fields [
3]. Typical electric underwater manipulators are the manipulator 7E of Eca Robotics [
3], the UMA manipulator developed by Graal Tech SRL in Italy for the TRIDENT project [
4], the modified commercial electric manipulator ARM 5E [
5], etc.
Owing to the extremely high pressure in the deep-sea environment, the electric underwater manipulator is watertight and oil compensated from bearing the deep-sea water pressure [
6,
7]. The oil-filled joint actuator of electric underwater manipulator, which is commonly composed of a brushless, direct-drive motor with reduction gearbox featuring low backlash and a large reduction ratio [
3], will suffer from oil stirring loss [
7,
8,
9,
10] resulting from the rotation viscos between the high speed rotor and oil, output shaft dynamic seal loss [
10] deriving from the high-pressure action and high-speed rotation friction on the rubber seal rings, and core loss [
9,
10] caused by the high-pressure action on the motor cores. Therefore, the oil-filled joint actuator will show different characteristics and response performance compared with its common use.
For the oil-filled joint actuator, the oil stirring viscos loss and output shaft dynamic seal loss can be considered as part of unknown internal disturbance and external disturbance respectively, while the core loss usually directly leads to the physical parameter deviations of the motor, which can be treated as dynamic uncertainties. Furthermore, due to the compact and lightweight requirements, usually only the angular position sensor is available for the oil-filled joint actuator, and thus joint velocity is immeasurable for the control. Consequently, the oil-filled joint actuator will suffer from an unmeasured system state, dynamic uncertainties, and unknown disturbances when it works in the deep-sea environment, and its achievable control performance could be severely deteriorated by these adverse factors.
To handle immeasurable system states, observers have been widely used [
11,
12,
13,
14,
15]. In [
11], a neural-based full-order Luenberger adaptive observer was designed for sensorless linear induction motor control. In [
12], a sliding mode observer was proposed to observe the back electromotive force for obtaining the velocity and position of the mover of a permanent magnet synchronous linear motor. In [
13], an approximate high-gain observer was employed to observe the speed signals for an induction motor control. In [
14], a third-order nonlinear extended-state observer (ESO) was constructed for position and speed estimation for a permanent magnet synchronous motor control. In [
15], the oxygen excess ratio was estimated via an extended-state observer (ESO) from the measurements of the compressor flow rate, the load current, and the supply manifold pressure, which was used in the output feedback controller design of the oxygen excess ratio control system.
To reduce the effect of dynamic uncertainties, adaptive-based controllers are the most commonly used methods [
16,
17,
18,
19,
20,
21,
22,
23,
24]. For example, in [
17], an adaptive robust controller with ESO (ARCESO) was synthesized for high-accuracy motion control of a DC motor, in which the adaptive control was presented to deal with for the parametric uncertainty. To suppress the parametric uncertainty, a neural network learning adaptive robust controller was synthesized for an industrial linear motor stage to achieve good tracking performance and excellent disturbance rejection ability, where the parametric variations were handled by the adaptation part of the controller [
18]. Although adaptive based controllers can achieve satisfactory performance in many physical systems, the parameter variation range should be known in advance [
17,
18,
23], which is usually difficult to acquire in advance of practical application, thus leading to the limitation of wide application.
To overcome the effect of unknown disturbances, disturbance observers are intuitive and effective methods [
17,
18,
25,
26,
27,
28,
29,
30]. In [
16], an ESO was constructed to estimate the unstructured uncertainties (including nonlinear friction, external disturbances, and unmodeled dynamics) for designing an ARCESO high-accuracy motion controller of a DC motor. In [
25], a generalized proportional integral observer was designed for dealing with load torque disturbance and time-varying parameter uncertainties for the finite control set predictive current control of induction motor systems. Due to the universal approximation feature, neural networks (NNs) and fuzzy logic systems (FLSs) are also widely used to deal with the unknown disturbances. To approximate the unknown disturbances, a NNs learning algorithm was employed to design the adaptive robust controller for an industrial linear motor stage to achieve good tracking performance and excellent disturbance rejection ability [
18]. FLSs were utilized to approximate the unknown disturbances of a bionic mechanical leg for the adaptive fuzzy robust controller design [
28] in the previous work of the authors in this paper. In addition, robust control has also been a choice to some researchers to attenuate disturbances in physical systems, such as active disturbance rejection control [
31,
32], sliding-mode control (SMC)-based methods [
33,
34,
35,
36,
37,
38,
39,
40,
41,
42,
43,
44], etc. Among them, the nonsingular fast terminal sliding-mode control (NFTSMC) [
30,
38,
39,
40,
41,
42,
43,
44], as a new typical robust controller, has been widely used in controlling uncertain systems because of its attractive properties such as fast dynamic response, robustness against uncertainties, chattering phenomenon elimination, finite time convergence, and its simple design procedure [
30,
38].
Motivated by the above observations, to provide a high-performance motion controller with capabilities of unmeasured system states self-estimating, dynamic uncertainties and unknown disturbances rejection for a deep-sea electric oil-filled joint actuator, a novel extended-state observer-based prescribed performance non-singular fast-terminal sliding-mode control (PP-NFTSMC-ESO) strategy was proposed in this paper. The ESO was utilized to estimate the unmeasured system states and the lumped uncertainties to make the precise model-based compensation, while the residual parts including estimation errors and stable tracking errors were discharged by the simple robust term and stable feedback term, respectively. In order to improve the transient and steady-state position responses, an error constraint transformation was developed to guarantee the prescribed time-varying performance. The main contributions of this paper can be summarized as follows:
(1) A novel PP-NFTSMC-ESO controller was proposed for high-performance motion control of a deep-sea electric oil-filled joint actuator in the presence of unmeasured system state, dynamic uncertainties, and unknown disturbances, which combines the advantages of NFTSMC control in terms of robustness against uncertainties, chattering phenomenon elimination, fast dynamic response, finite time convergence, and simple design procedure, the restraining action of prescribed performance control for the transient and steady state performance, and the excellent observation ability of the ESO for system states and lumped uncertainties.
(2) With the proposed control method, the mechanical configuration of the deep-sea electric oil-filled joint actuator can be simplified with only an angular position sensor, which benefits for the structure design and electrical design, but with no performance deterioration of trajectory tracking control.
(3) The stability of the proposed controller was theoretically proven by the Lyapunov stability theory. The excellent trajectory tracking performance was demonstrated with the studies on different working conditions, and the superiority of the proposed controller was illustrated by the comparison with proportional-integral (PI) controller, ESO-based sliding-mode controller (SMC-ESO), and ESO-based non-singular fast-terminal sliding-mode controller (NFTSMC-ESO).
The remainder of this paper is organized as follows.
Section 2 presents the dynamic modeling and problem formulation.
Section 3 introduces the ESO. The design procedure of the PP-NFTSMC-ESO control method and the stability proving process of this controller-observer strategy are described in
Section 4. The effectiveness is demonstrated via simulation studies in
Section 5, and conclusions are provided in
Section 6.
3. Extended State Observer Design
Designing a high-performance model-based controller usually requires a full-state feedback. However, for the electric oil-filled joint actuator, only the joint position sensor is installed in the joint. Therefore, the task of the ESO observer is to estimate the unmeasured system state and the lumped uncertainty for the later controller design.
ESO has many advantages than other observers. Besides fast convergence and high robustness characteristics, the most important thing to appreciate is that it can estimate both system states and the lumped uncertainties simultaneously with a boundedness of the estimation errors. Therefore, ESO is becoming an effective tool in the control of dynamic systems.
According to the structure of the system model (5), extend the lumped uncertainty
as state
, and
represent the time derivative of
, then the dynamic Equation (5) can be rewritten as
Assumption 1:
is bounded, i.e.,, whereis a constant.
According to Equation (14), the ESO is designed as
where
is the estimated state vector,
is the estimation error vector, and
can be treated as the bandwidth of the ESO.
In order to guarantee the stability of the ESO, the gains of the observer are designed to satisfy the following Hurwitz polynomial [
48,
49]:
According to Equations (14) and (15), the dynamic equation of the estimation errors is represented as
Theorem 1 [
17,
48]:
Under Assumption 1, the estimated states are always bounded and there exist a constant , and a finite time such thatfor some positive integer c. From Theorem 1, we can see that the estimation errors are bounded and will converge to an arbitrarily small range the parameter increases largely enough, i.e., .
4. PP-NFTSMC-ESO Controller Design
For the trajectory tracking control of a deep-sea electric oil-filled joint actuator with unmeasured system states, dynamic uncertainties, and unknown disturbances, the PP-NFTSMC-ESO controller was proposed in this paper. The block diagram of this controller is shown in
Figure 2. The ESO was applied to estimate the unmeasured velocity and the lumped uncertainty, while the PPF was employed to constrain the transient and steady-state performances of the trajectory tracking error. The NFTSMC controller was then synthesized with the signals from the reference trajectory, ESO, and PPF.
Define the position error
and velocity error
as
where
and
are the desired joint position, and
is the virtual control function of
.
Step 1: According to the results of the prescribed performance function (11), the position error
can be transformed as follows
The time derivative of
is
Therefore, the virtual control function of
can be designed as
where
is a positive design parameter.
From Equation (20), we can have
, and substituting it into Equation (22) results in
Define the Lyapunov function as
Substituting Equations (23) and (24) into (25), the time derivative of
is
Step 2: Define NFTSMC sliding function as [
30]
where
,
are positive constants subject to
and
,
,
;
and
are positive design parameters.
The time derivative of
is
Substituting the second equation of (13) and (14) into (29) produces
Define the Lyapunov function as
Substituting Equations (26) and (29) into (30), the time derivative of
is
The control
of PP-NFTSMC-ESO controller is designed as
where
,
, and
are positive design parameters.
As seen, the PP-NFTSMC-ESO controller consists of three parts, where is the equivalent control law to hold the error trajectory on the sliding surface, is the robust term used to compensate for the estimation errors, and is the stable feedback term to stable the tracking error.
Substituting Equations (32)–(35) into (31) results in
According to Lemma 1, it follows that
Applying Cauchy-Schwarz inequality, (37) can be transformed to
where
, and
.
As
is a function of the estimation errors, which will converge to a small value after a finite time
when selected a large enough
. Therefore, if letting
,
, and
, we have
Theorem 2:
For the deep-sea electric oil-filled joint actuator (5), under Assumption 1, with the ESO (15), the proposed control law (32)–(35) guarantees that the closed-loop system is stable and the output position tracking error converges to a small neighborhood of the origin by appropriately choosing the observer parameter,,,, and the controller parameters,,,, and.