Rapid Robust Control of a Marine-Vehicle Manipulator with Series Elastic Actuators Based on Variable Power Log Reaching Law

Abstract

Marine-vehicle manipulators, which represent a kind of mechanical systems installed on marine surface or underwater vehicles, are mostly suffering from the problem of waves (or ocean currents)-caused base oscillations. The oscillations have a significant impact on system stability. Numerous control strategies have been investigated, but the majority of them are concentrated on the control’s robust performance. This study focuses on an innovative marine-vehicle manipulator (ammunition transfer manipulator on warships) with novel compliant actuators (series elastic actuators), for which the control performance of convergence speed and flexible-vibration suppression should also be considered. To address these issues, this paper proposes a unique hybrid control based on the singular perturbation method, by which the control problem is decomposed into two time scales. In the slow time-scale, it is given a rapid trajectory tracking controller that integrates the computed torque method and the terminal sliding mode control law with a novel reaching law (variable power log reaching law). For the fast time-scale control, a derivative-type controller is used to achieve the suppression of the flexible vibrations. To demonstrate the effectiveness of the proposed control method, theoretical proofs and numerical simulations are both presented. According to our knowledge, this study presents the first control strategy for rapid robust control of marine-vehicle manipulators that are subject to base-oscillation-caused disturbance and compliant-actuator-induced flexible vibrations.

1. Introduction

A class of mechanical equipment installed on marine surface or underwater vehicles is known as marine-vehicle manipulators. This category includes a wide range of real-world applications, such as offshore cranes [1,2,3], ammunition transfer manipulators on warships, UVMs (underwater vehicle manipulators) [4,5,6,7], and so on. In reality, the base (vehicle) oscillations brought on by waves or ocean currents have a significant impact on the dynamics of marine-vehicle manipulators, making the system control extremely difficult.

The primary challenge of the control of marine-vehicle manipulators is its robustness to vehicle oscillations. In case of slight vehicle oscillations, this problem is usually solved via an additional dynamic compensator based on the premise that the vehicle oscillations can be measured or predicted in advance [8]. For example, Sato and Toda developed a robust control method for ship-mounted manipulators based on the prediction of ship oscillations combing with H${}_{\infty }$ control [9]. Londhe et al. proposed a robust tracking control for UVMs based on the estimator of underwater vehicle oscillations and a PID-like fuzzy method [10]. Cai et al. introduced an onboard Stewart platform as a motion compensator to eliminate the impact of wave-induced ship oscillations on offshore cranes [11]. However, in the case of severe vehicle oscillations, that is, when encountering severe working conditions (e.g., heavy seas), the measurement or prediction of vehicle oscillations becomes very difficult, so the method of a dynamic compensator is no longer applicable. In this case, several new control methods have been developed to guarantee robustness to vehicle oscillations. For example, Sun et al. presented a novel nonlinear anti-swing control method for offshore cranes without prior knowledge of ship oscillations [12]. Zhang and Chen proposed a new adaptive tracking control method for offshore cranes with unknown gravity parameters, realizing the accurate positioning of payload and the elimination of swing [13]. However, the above control methods only considered the asymptotic stability of the system, without considering its convergence speed.

The second challenge of the control of marine-vehicle manipulators is its convergence speed under vehicle oscillations. In practice, we certainly expect that the control of marine-vehicle manipulators would be not only robust but also quick. However, vehicle oscillations impose strong constraints on the convergence time of the system, which causes great difficulty in rapid control design. The traditional optimal time control because of the lack of anti-interference is not applicable here. The sliding mode control (SMC), which has better robustness and can drive the system approach to the equilibrium state in a finite time, has been widely concerned [14]. For example, Wang et al. developed a fast control for UVMs using the continuous nonsingular fast terminal SMC method [15]. Han et al. presented a novel adaptive nonsingular fast terminal SMC for UVMs by introducing an extended state observer and an adaptive law [16]. Dai et al. further proposed a novel fast robust control for UVMs based on a nominal fast model predictive control and a fast tube SMC law [17]. However, the above studies with SMC methods only considered the convergence speed of the system in the sliding phase. The convergence speed can also be improved in the reaching phase by the reaching law method [18,19,20,21,22]. However, most of the existing SMC reaching laws are affected by the sliding mode chattering problem and cannot achieve further improvement of the convergence speed.

Our study here focused on a special kind of marine-vehicle manipulators—the ammunition transfer manipulator on warships. As shown in Figure 1, it is an important part of naval gun systems and is used to realize the automatic loading of ammunition. However, traditional ammunition transfer manipulators are usually suffering from the issue of poor compliance abilities because they are driven by stiff actuators [23,24]. To solve this issue, novel ammunition transfer manipulators with compliant actuators are developed [25]. The compliant actuator has also been widely used in other marine-vehicle manipulators [26,27].

However, the compliant actuator will bring another challenge to the control of marine-vehicle manipulators—the flexible vibration problem [28,29]. The singular perturbation method is the most common way to solve this problem. For example, Zhu et al. proposed a singular-perturbation-based composite control for manipulators with flexible joints and links under uncertainties and disturbance [30]. Cheng et al. also presented a generalized saturated adaptive control for a flexible-joint manipulator based on the singular perturbation method and neural network technique [31]. Hong et al. proposed a singular-perturbation-based control for the flexible-joint manipulator subject to unknown disturbance [32]. However, the above studies only considered simple manipulator systems. Whether the singular perturbation method is still applicable to marine-vehicle manipulators affected by base oscillations lacks sufficient attention and research.

In summary, although numerous control strategies have been developed for marine-vehicle manipulators, the following issues are still open and need to be addressed.

• The majority of the existing studies have concentrated on traditional rigid-actuated marine-vehicle manipulators. As far as we know, few studies have considered marine-vehicle manipulators with compliant actuators;
• The strong dynamic coupling between the manipulator motion affected by base oscillations and the flexible vibration caused by compliant actuators poses a great challenge to the system control;
• The current reaching-law-based SMC method cannot achieve further improvement in convergence speed due to the problem of sliding mode chattering.

To address the above issues, this study presents a novel rapid robust control strategy for an ammunition transfer manipulator, which is mounted on oscillatory warships and driven by compliant actuators.

The structure of this paper is as follows. The system description and dynamic model for a novel compliant-actuated ammunition transfer manipulator with base oscillations are put forth in Section 2. Section 3 presents the decoupling of the system dynamics at two time scales using the singular perturbation method. Section 4 presents the key findings of this study, the fundamentals of the proposed control, and a proof of its rapid robust convergence performance. The simulation results and their analysis are presented in Section 5. Section 6 presents the conclusions.

2. Problem Statements

2.1. System Description

In our study, compliant actuators were introduced for the first time into an ammunition transfer manipulator to improve its performance. Figure 2 depicts the three-dimensional model of the manipulator, which is primarily made up of two parts: a lifting part and a rotating part. Both parts are actuated by a new type of compliant actuators, which belongs to the category of series elastic actuators (SEAs). Both SEAs are improved versions of the concept machine proposed in [25,33]. The structural principles of the two SEAs are shown in Figure 2b,c, respectively. As seen in Figure 2b, the rotating SEA contains a servo motor, a coupling, a ball-screw-nut mechanism, a linear spring, a bracket, and a cable. The motor’s output shaft is connected to the ball-screw through the coupling. As a result, the linear movement of the ball-screw nut is derived from the motor’s rotatory movement. The ball-screw nut can then drive the output carriage through the linear spring. Finally, the rotating part can be operated through the output carriage. The SEA for the lifting part, as shown in Figure 2c, has a different structure form but a similar operating principle. First, the motor drives the ball-screw nut through the bevel gear reducer. The nut’s motion is then converted by the linear springs into the motion of the output carriage, which in turn drives the lifting part via a belt drive.

2.2. Rigid-Flexible Coupling Dynamic Model

The dynamic model of the compliant-actuated ammunition transfer manipulator system has the characteristics of rigid-flexible coupling. Before the dynamic modeling, the system first needs to be simplified as follows:

• The manipulator can be reduced to a two-degree-of-freedom system if the flexible vibration from the compliant actuators is not considered;
• If the two SEAs are reduced to two springs, two additional degrees of freedom are introduced, and the whole system becomes a four-degree-of-freedom system. In particular, the SEAs for the two parts are simplified as a linear tension spring and a linear torsion spring, respectively;
• Furthermore, considering the effect of oscillatory base and decoupling the base oscillation to three directions as shown in Figure 2, another degree of freedom is introduced. The final simplified system is a five-degree-of-freedom system.

As shown in Figure 3, a simplified model is obtained based on the above simplification. The physical meanings of the symbols in Figure 2 are shown in

Table 1. The meanings of the symbols.

Symbols	Meanings
XOY	Inertial coordinate
xoy	Noninertial (base-fixed) coordinate
$B_1$, $B_2$, $B_3$	Oscillatory base, Lifting part, Rotating part
$B_{2a}$, $B_{3a}$	Actuators of two parts
$B_{2s}$, $B_{3s}$	Springs of two parts
$C_2$, $C_3$	Centroid of two parts
$L_2$, $L_3$	Geometric parameters
$y_{r1}$, ${\theta }_{1z}$, ${\theta }_{1x}$	Three base oscillations
$y_{r2}$, ${\theta }_3$	Position/angle of two parts
$y_{r2d}$, ${\theta }_{3d}$	Position/angle of two actuators

.

The dynamic model of the above simplified system is constructed based on the Lagrangian formulation. Then, by extracting the base-oscillation term in the dynamic model as an external disturbance to the manipulator, the following equation can be derived:

$$\begin{array}{c}\hfill \begin{array}{c}M{}_1\left(q{}_1\right)\ddot{q}{}_1+C(q{}_1,\dot{q}{}_1)\dot{q}{}_1+G\left(q{}_1\right)+K(q{}_1-q{}_2)=S,\hfill \\ M{}_2\ddot{q}{}_2+K(q{}_2-q{}_1)=U,\hfill \end{array}\end{array}$$

(1)

in which $q{}_1={[y_{r2},{\theta }_3]}^T$, $q{}_2={[y_{r2d},{\theta }_{3d}]}^T$ denote the system’s state, which needs to be controlled; $M{}_1$ represents the inertia matrix of the manipulator; $M{}_2$ represents the inertia matrix of the actuator; G is the gravity force; $C{\dot{q}}_1$ denotes the Coriolis/centripetal force; U is the control that needs to be designed; S is the base-oscillation-caused external disturbance; and K is the stiffness matrix of the springs.

For Equation (1), it has

$$\begin{array}{cc}\hfill & M_1\left(q_1\right)=\left[\begin{array}{cc}{m}_2+m_3& m_3{L}_3\mathtt{cos}{\theta }_3\\ m_3{L}_3\mathtt{cos}{\theta }_3& m_3{L}_3^2+J\end{array}\right],C(q_1,{\dot{q}}_1){\dot{q}}_1=\left[\begin{array}{c}-m_3{L}_3\mathtt{sin}\left({\theta }_3\right){\dot{\theta }}_3^2\\ 0\end{array}\right],\hfill \\ \hfill & G\left(q_1\right)=\left[\begin{array}{c}(m_2+m_3)g\\ m_{3}g{L}_3\mathtt{cos}{\theta }_3\end{array}\right],S=\left[\begin{array}{c}{S}_1\\ S_2\end{array}\right],U=\left[\begin{array}{c}{U}_1\\ U_2\end{array}\right],M_2=\left[\begin{array}{cc}{m}_I& 0\\ 0& I\end{array}\right],K=\left[\begin{array}{cc}{k}_{s2}& 0\\ 0& k_{s3}\end{array}\right],\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill & S_{1s}=-(m_2+m_3){\ddot{y}}_{r1},\hfill \\ \hfill & S_{2s}=-m_3{L}_3\mathtt{cos}{\theta }_3{\ddot{y}}_{r1},\hfill \\ \hfill & S_{1p}=-((m_2{L}_1+m_3{L}_1+m_3{L}_3\mathtt{cos}{\theta }_3){\ddot{\theta }}_{1z}-2m_3{L}_3\mathtt{sin}{\theta }_3{\dot{\theta }}_{1z}{\dot{\theta }}_3\hfill \\ \hfill & -(m_2{y}_{r2}+m_3{y}_{r2}+m_3{L}_3\mathtt{sin}{\theta }_3){\dot{\theta }}_{1z}^2)-((m_2+m_3)g\mathtt{cos}{\theta }_{1z}-(m_2+m_3)g),\hfill \\ \hfill & S_{2p}=-((J+m_3{L}_3^2+m_3{L}_1{L}_3\mathtt{cos}{\theta }_3+m_3{L}_3{y}_{r2}\mathtt{sin}{\theta }_3){\ddot{\theta }}_{1z}\hfill \\ \hfill & +2m_3{L}_3\mathtt{sin}{\theta }_3{\dot{\theta }}_{1z}{\dot{y}}_{r2}+(m_3{L}_1{L}_3\mathtt{sin}{\theta }_3-m_3{L}_3{y}_{r2}\mathtt{cos}{\theta }_3){\dot{\theta }}_{1z}^2)\hfill \\ \hfill & -(m_{3}g{L}_3\mathtt{cos}({\theta }_{1z}+{\theta }_3)-m_{3}g{L}_3\mathtt{cos}{\theta }_3),\hfill \\ \hfill & S_{1r}=(m_2+m_3)y_{r2}{\dot{\theta }}_{1x}^2+m_3{L}_3\mathtt{sin}{\theta }_3{\dot{\theta }}_{1x}^2+(m_2+m_3)g\mathtt{cos}{\theta }_{1x}-(m_2+m_3)g,\hfill \\ \hfill & S_{2r}=m_3{y}_{r2}{\dot{\theta }}_{1x}^2+m_3{L}_3\mathtt{sin}{\theta }_3{\dot{\theta }}_{1x}^2-m_{3}g{L}_3\mathtt{cos}{\theta }_{1x}\mathtt{cos}{\theta }_3+m_{3}g{L}_3\mathtt{cos}{\theta }_3.\hfill \end{array}$$

in which ${[S_{1s},S_{2s}]}^{\mathtt{T}}$, ${[S_{1p},S_{2p}]}^{\mathtt{T}}$, and ${[S_{1r},S_{2r}]}^{\mathtt{T}}$ are external disturbance caused by shake, pitch, and roll base oscillation, respectively; $m_2$ and $m_3$ are the mass of $B_2$ and $B_3$, respectively; J is the inertia moment of $B_3$; $m_I$ and I are the equivalent mass and inertia moment of actuators for lifting and rotating parts, respectively; $k_{s2}$ and $k_{s3}$ are spring stiffness of two SEAs; g denotes gravitational acceleration.

In addition, it is assumed that the dynamic model described by Equation (1) satisfies the following properties:

Property 1.

The inertial matrix of the system is bounded in norm. This means that $M_1\left(q{}_1\right)$ and $M_2)$ satisfy

$$\left\{\begin{array}{c}{a}_1{\rho }^2\le ∥{\rho }^T{M}_1\left(q_1\right)\rho ∥\le b_1{\rho }^2\hfill \\ a_2{\rho }^2\le ∥{\rho }^T{M}_2\rho ∥\le b_2{\rho }^2\hfill \end{array},\right.$$

where $0<a_1<b_1$, $0<a_2<b_2$, ρ is an arbitrary vector; and ‖‖ means the Euclidean norm of a vector.

Property 2.

The external disturbance caused by base oscillations is bounded in norm, namely S satisfies the following condition:

$$∥S∥\le S_0,S_0>0.$$

At this point, the problem faced by this study can be given as follows:

Problem: “Constructing a control U for system (1) influenced by disturbance S such that the system state $q_1$ can achieve accurate tracking of a given desired trajectory in finite time and the system state $q_2$ can achieve rapid convergence to the equilibrium point”.

3. Dynamics Decoupling Based on a Singular Perturbation Method

To solve the above control problem, the dynamic model described by Equation (1) is first decoupled into two time-scale subsystems using the singular perturbation method.

A singular perturbation parameter is introduced as $\epsilon$ ($0<\epsilon \ll 1$), and a new stiffness matrix is defined as $K{}_1={\epsilon }^{2}K$. Then, we can obtain the singular perturbation form of the dynamic model Equation (1) as follows:

$$\left\{\begin{array}{c}M{}_1\left(q{}_1\right)\ddot{q}{}_1+C(q{}_1,\dot{q}{}_1)\dot{q}{}_1+G\left(q{}_1\right)=S+z\hfill \\ {\epsilon }^{2}M{}_2\ddot{z}+K{}_{1}z=K{}_1(U-M{}_2\ddot{q}{}_1)\hfill \end{array},\right.$$

(2)

where z is the elastic force, $z=K(q{}_2-q{}_1)$.

3.1. Slow Time-Scale Subsystem

By setting $\epsilon =0$ in the second equation of (2), we have the following equation:

$$z=U{}_s-M{}_2\ddot{q}{}_1.$$

(3)

Then, by substituting Equation (3) into the first equation of (2), we obtain the dynamic model of the slow time-scale subsystem:

$$H\left(q{}_1\right)\ddot{q}{}_1+C(q{}_1,\dot{q}{}_1)\dot{q}{}_1+G\left(q{}_1\right)=U{}_s+S,$$

(4)

where $H\left(q{}_1\right)=M{}_1\left(q{}_1\right)+M{}_2$; $U_s$ is the control of the slow time-scale subsystem that needs to be further designed.

Obviously, the dynamic model of the slow time-scale subsystem is equivalent to the dynamic model of the ammunition transfer manipulator system with rigid actuators.

3.2. Fast Time-Scale Subsystem

From the first equation of (2), we have

$$\ddot{q}{}_1=M{{}_1}^{-1}\left(q{}_1\right)\{S+z-C(q{}_1,\dot{q}{}_1)\dot{q}{}_1-G\left(q{}_1\right)\}.$$

(5)

Then, by substituting Equation (5) into the second equation of (2), we obtain

$$\begin{array}{c}\hfill {\epsilon }^2{M}_2\ddot{z}+K{}_{1}z=K{}_1(U-M{}_{2}M{{}_1}^{-1}\left(q{}_1\right)\{S+z-C(q{}_1,\dot{q}{}_1)\dot{q}{}_1-G\left(q{}_1\right)\}).\end{array}$$

(6)

Furthermore, pre-multiplied by $M{{}_2}^{-1}$ on both sides of Equation (6), it has

$$\begin{array}{c}\hfill {\epsilon }^2\ddot{z}+K{}_1(M{{}_2}^{-1}+M{{}_1}^{-1})z=M{{}_2}^{-1}K{}_{1}U-K{}_{1}M{{}_1}^{-1}\left(q{}_1\right)\{S-C(q{}_1,\dot{q}{}_1)\dot{q}{}_1-G\left(q{}_1\right)\}.\end{array}$$

(7)

Finally, by substituting $\epsilon =0$ into Equation (7), we obtain

$$\begin{array}{c}\hfill \overline{z}=\frac{M{{}_2}^{-1}K{}_{1}U{}_s-K{}_{1}M{{}_1}^{-1}\left(q{}_1\right)\{S-C(q{}_1,\dot{q}{}_1)\dot{q}{}_1-G\left(q{}_1\right)\}}{K{}_1(M{{}_2}^{-1}+M{{}_1}^{-1})},\end{array}$$

(8)

where $\overline{z}$ is the so-called quasi-steady state of the fast time-scale subsystem.

Based on the above quasi-steady state, we can further obtain the dynamic model of the fast time-scale subsystem.

Firstly, a new variable $\eta =z-\overline{z}$ is introduced. By substituted it into Equation (7), we have

$$\begin{array}{cc}\hfill & {\epsilon }^2\ddot{\eta }+K{}_1(M{{}_2}^{-1}+M{{}_1}^{-1})(\eta +\overline{z})\hfill \\ \hfill & =M{{}_2}^{-1}K{}_{1}U-K{}_{1}M{{}_1}^{-1}\left(q_1\right)\{S-C(q{}_1,{\dot{q}}_1){\dot{q}}_1-G\left(q{}_1\right)\}.\hfill \end{array}$$

(9)

By substituting Equation (8) into Equation (9), we have

$${\epsilon }^2\ddot{\eta }+K{}_1(M{{}_2}^{-1}+M{{}_1}^{-1})\eta =M{{}_2}^{-1}{K}_1{U}_f,$$

(10)

where $U_f=U-U_s$ is the control of the fast time-scale subsystem that needs to be further designed. Then, a new fast time scale $\tau =t/ϵ$ is introduced, and Equation (10) is rewritten as

$$\frac{d^2\eta }{d\tau {}^2}+K_1(M{{}_2}^{-1}+M{{}_1}^{-1})\eta =M{{}_2}^{-1}K{}_1{U}_f.$$

(11)

Finally, by pre-multiplying $M_2$ and ${K_1}^{-1}$ on both sides of Equation (11), we obtain the dynamic model of the fast time-scale subsystem:

$$K{{}_1}^{-1}{M}_2\frac{d^2\eta }{d{\tau }^2}+K{{}_1}^{-1}M{}_{2}K{}_1(M{{}_2}^{-1}+M{{}_1}^{-1})\eta =U{}_f.$$

(12)

At this point, the system (1) has been decoupled into two time-scale subsystems, and the control problem of this study has also been decomposed into two sub-problems:

$$U=U_s+U_f.$$

(13)

4. Main Development

The controls of the above two time-scale subsystems are developed separately in this section.

4.1. Slow Time-Scale Subsystem Control

For the control design of the slow time-scale subsystem, we should first linearize the nonlinear dynamic model Equation (4) into a linear error system, and then develop a novel fast robust control for the error system.

4.1.1. Computed Torque Method

Based on the computed torque method, the slow time-scale Subsystem control is defined as

$$U_s=H\left(q_1\right)({\ddot{q}}_{1d}-u_e)+C(q_1,{\dot{q}}_1){\dot{q}}_1+G\left(q_1\right),$$

(14)

where $u_e$ is an additional control that needs to be further designed. Then, by substituting Equation (14) into Equation (4), we have

$$H\left(q_1\right){\ddot{q}}_1=H\left(q_1\right)({\ddot{q}}_{1d}+u_e)-H\left(q_1\right)S^{{}^{\prime }}.$$

(15)

Finally, by further rewriting, we finally obtain the linear error system

$$\ddot{e}=u_e+S^{{}^{\prime }},$$

(16)

where $e=q_{1d}-q_1$ is the error variable; $q_{1d}$ is a given desired trajectory, which is continuous second-order differentiable; $S^{{}^{\prime }}={(M_1+M_2)}^{-1}S$, according to the aforementioned property 1 and 2, we obtain that $S^{{}^{\prime }}$ is also bounded in norm,

$$∥S^{{}^{\prime }}∥\le S_0^{{}^{\prime }},S_0^{{}^{\prime }}>0.$$

(17)

4.1.2. SMC with Variable Power Log Reaching Law

For the control design of $u_e$, the SMC method is used.

As mentioned before, the sliding mode surface determines the convergence speed of the system during the sliding phase (i.e., after reaching the sliding mode surface). In this study, we adopt a rapid terminal sliding mode surface, which has globally fixed-time stable characteristics. The expression for the sliding mode surface is as follows:

$$\sigma =\dot{\theta }+\mu {\theta }^{l/m}+\lambda {\theta }^{p/r},$$

(18)

in which $\sigma$ is the sliding mode variable; $\theta$ is the state variable of the controlled system; $\mu$ and $\lambda$ are both $>0$); l, m, p, and r are all positive odd integers and satisfying $l>m$, $r>p$. The proof of the globally fixed-time stability of the above sliding mode surface is based on the following theorem. For more details, please refer to the studies in [34,35].

Theorem 1.

For every scalar system,

$$\dot{\theta }=-\mu {\theta }^{l/m}-\lambda {\theta }^{p/r}$$

with $\mu >0$, $\lambda >0$, $l>m$, $r>p$ (l, m, p, and r are all positive odd integers), the equilibrium of which is globally fixed-time stable, and the setting time satisfying

$$T<t_{smax}=\frac{1}{\mu }\frac{m}{l-m}+\frac{1}{\lambda }\frac{r}{r-p}.$$

The SMC reaching law determines the convergence speed of the system state during the reaching phase (i.e., before reaching the sliding mode surface). In this study, we developed a novel reaching law, which is obtained by changing the power function of the double power reaching law into a logarithmic function and then adding a variable power term. We call it the variable power log reaching law. Compared with the power function, the curve of the logarithmic function is smoother near the origin and steeper away from the origin, so the new reaching law has a faster convergence speed and better sliding mode chattering suppression performance. The expression for the proposed variable power log reaching law is as follows:

$$\dot{\sigma }=-k_{1}f\left(\sigma \right)-k_2{ln}^{1+\alpha }(ϵ{\sigma }^{\nu }+1)\mathrm{sgn}\left(\sigma \right)-k_3{ln}^{1-\alpha }(ϵ{\sigma }^{\nu }+1)\mathrm{sgn}\left(\sigma \right)$$

(19)

with

$$f\left(\sigma \right)=\left\{\begin{array}{c}{\left|\sigma \right|}^{\zeta }\mathrm{sgn}\left(\sigma \right),\left|\sigma \right|<1\hfill \\ {\left|\sigma \right|}^{\zeta \left|\sigma \right|}\mathrm{sgn}\left(\sigma \right),\left|\sigma \right|\ge 1\hfill \end{array},\right.$$

where $f\left(\sigma \right)$ is the variable power term; sgn denotes sign function; $\zeta <1$, $ϵ>0$, $0<\alpha <1$, $k_1>0$, $k_2>0$, $k_3>0$; and $\nu$ is a positive even integer.

To highlight the advantages of the proposed variable power log reaching law, two traditional reaching laws are chosen for comparison, including a power reaching law

$$\dot{\sigma }=-k_1{\left|\sigma \right|}^{\alpha }\mathrm{sgn}\left(\sigma \right)$$

(20)

and a double power reaching law

$$\dot{\sigma }=-k_1{\left|\sigma \right|}^{\alpha }\mathrm{sgn}\left(\sigma \right)-k_2{\left|\sigma \right|}^{\beta }\mathrm{sgn}\left(\sigma \right).$$

(21)

We compare the three reaching laws above from two perspectives.

First, we compare these three reaching laws in the phase space of the sliding mode variables. As shown in Figure 4, the variable power log reaching law has the smoothest curve near the origin of the coordinates. The gradients for three reaching laws are 0.56 (variable power log reaching law), 3.11 (double power reaching law), and 3.11 (power reaching law), respectively. This means that it can reach the sliding mode surface with lower inertia and smaller switching rate, thus achieving effective suppression of sliding mode chattering. It can also be seen in Figure 4 that the slope of the curve for the variable power log reaching law is much larger than the other two reaching laws on the distance from the origin. This indicates that it can approach the sliding mode surface at a faster speed, thus achieving a rapid convergence of the controlled system in the reaching phase.

Second, we compare these three reaching laws in terms of the time response of the sliding mode variables. For all three cases, the same linear sliding mode surface is used. As can be seen in Figure 5, the sliding mode variables in the case of using the variable power log reaching law can converge to the steady state in the minimum time with almost zero overshoot and steady state error (given the same starting point for all three cases). In contrast, the time response of the sliding mode variables using the other two reaching law cases both have large response times, overshoot and steady-state errors. Their response curves also have significant chattering characteristics.

Based on the above comparison, we can conclude that the variable power log reaching law proposed in this paper outperforms other convergence laws in terms of both convergence speed and chattering suppression.

At this point, we can give the expression of the control $u_e$. The sliding mode surface is defined as

$$\begin{array}{cc}\hfill & \sigma ={\left[{\sigma }_i\right]}_{2\times 1},i\in 1,2,\hfill \\ \hfill & {\sigma }_i={\dot{e}}_i+\mu e_i^{l/m}+\lambda e_i^{p/r},\hfill \end{array}$$

(22)

and the reaching law is defined as

$$\begin{array}{cc}\hfill & \dot{\sigma }={\left[{\dot{\sigma }}_i\right]}_{2\times 1},i\in 1,2,\hfill \\ \hfill & {\dot{\sigma }}_i=-k_{1}f\left({\sigma }_i\right)-k_2{ln}^{1+\alpha }(ϵ{\sigma }_i^{\nu }+1)\mathrm{sgn}\left({\sigma }_i\right)-k_3{ln}^{1-\alpha }(ϵ{\sigma }_i^{\nu }+1)\mathrm{sgn}\left({\sigma }_i\right).\hfill \end{array}$$

(23)

Then, by taking derivatives on both sides of Equation (22), we have

$$\begin{array}{cc}\hfill & \dot{\sigma }={\left[{\dot{\sigma }}_i\right]}_{2\times 1},i\in 1,2,\hfill \\ \hfill & {\dot{\sigma }}_i={\ddot{e}}_i+\frac{\mu l}{m}{\dot{e}}_i{e}_i^{(l-m)/m}+\frac{\lambda p}{r}{\dot{e}}_i{e}_i^{(p-r)/r}.\hfill \end{array}$$

(24)

Finally, by substituting Equation (16) (without S${}^{\prime }$ here) and Equation (23) into Equation (24), we can obtain the control $u_e$

$$\begin{array}{cc}\hfill & u_e=\left[u_{ei}\right],i\in 1,2,\hfill \\ \hfill & u_{ei}=-k_{1}f\left({\sigma }_i\right)-k_2{ln}^{1+\alpha }(ϵ{\sigma }_i^{\nu }+1)\mathrm{sgn}\left({\sigma }_i\right)-k_3{ln}^{1-\alpha }(ϵ{\sigma }_i^{\nu }+1)\mathrm{sgn}\left({\sigma }_i\right)\hfill \\ \hfill & -\frac{\mu l}{m}{\dot{e}}_i{e}_i^{(l-m)/m}-\frac{\lambda p}{r}{\dot{e}}_i{e}_i^{(p-r)/r}.\hfill \end{array}$$

(25)

Next, the finite-time stability of the system (Equation (14) with $S^{\prime }$ and $u_e$ (Equation (25)) is proved according to the following theorem [36].

Theorem 2.

For system Equation (14), if there is a continuous differentiable positive definite function defined in the neighborhood containing the origin

$$\dot{V}\left(x\right)\le -c{V}^{\kappa }\left(x\right)$$

with $c>0$ and $0<\kappa <1$, then the origin of the system is a finite-time-stable equilibrium. This means that the system can converge to the domain of equilibrium point in finite time. Moreover, the setting-time is

$$t_q=\frac{1}{c(1-k)}{\left(V\left(x_0\right)\right)}^{1-\kappa }.$$

Proof. 

Firstly, define a Lyapunov function as

$$V\left(\sigma \right)=\frac{{\sigma }^2}{2}.$$

(26)

Then, taking the derivative of V, we have

$$\dot{V}\left(\sigma \right)=\sigma \dot{\sigma }.$$

(27)

By substituting Equations (16), (24), and (25) into Equation (27), we obtain

$$\dot{V}\left(\sigma \right)=\sigma \dot{\sigma }=[{\sigma }_1,{\sigma }_2]{\left[{\dot{\sigma }}_1+S_1^{{}^{\prime }},{\dot{\sigma }}_2+S_2^{{}^{\prime }}\right]}^{\mathrm{T}}=\sum _{i=1}^2{\sigma }_i({\sigma }_i+S_i^{{}^{\prime }}).$$

(28)

For Equation (28), when $\left|{\sigma }_i\right|<1$, it has

$$\begin{array}{cc}\hfill & \sum _{i=1}^2{\sigma }_i({\dot{\sigma }}_i+S_i^{{}^{\prime }})\hfill \\ \hfill & =\sum _{i=1}^2\left({\sigma }_i(-k_1{\left|{\sigma }_i\right|}^{\zeta }-k_2{ln}^{1+\alpha }(ϵ{\sigma }_i^{\nu }+1)\mathrm{sgn}\left({\sigma }_i\right)-k_3{ln}^{1-\alpha }(ϵ{\sigma }_i^{\nu }+1)\mathrm{sgn}\left({\sigma }_i\right))+S_i^{{}^{\prime }}\right)\hfill \\ \hfill & \le \sum _{i=1}^2\left({\sigma }_i(-k_1{\left|{\sigma }_i\right|}^{\zeta })+S_i^{{}^{\prime }}\right)\le \sum _{i=1}^2\left(-{\left|{\sigma }_i\right|}^{\zeta +1}{k}_1+{\sigma }_i{S}_i^{{}^{\prime }}\right)=\sum _{i=1}^2(-{\left|{\sigma }_i\right|}^{\zeta +1}{k}_1)+\sigma S^{{}^{\prime }}\hfill \\ \hfill & \le \sum _{i=1}^2(-{\left|{\sigma }_i\right|}^{\zeta +1}{k}_1)+\sigma S_0^{{}^{\prime }}\le \sum _{i=1}^2(-{\left|{\sigma }_i\right|}^{\zeta +1}{k}_1-S_0).\hfill \end{array}$$

(29)

Then, by setting $c=c_1{2}^{(\zeta +1)/2}$, $\kappa =(\zeta +1)/2$, it has

$$\dot{V}\left(\sigma \right)\le -c_1{2}^{(\zeta +1)/2}{V}^{(\zeta +1)/2}=-c_1{\left(2V\right)}^{(\zeta +1)/2}=-c_1{\left|\sigma \right|}^{\zeta +1}=\sum _{i=1}^2-c_1{\left|{\sigma }_i\right|}^{\zeta +1}.$$

(30)

By comparing Equation (29) and Equation (30), we can find that the given Lyapunov function V satisfies the finite-time stable convergence condition when $k_1-S_0^{{}^{\prime }}\ge c_1$.

For Equation (28), when $\left|{\sigma }_i\right|\ge 1$, it has

$$\begin{array}{cc}\hfill & \sum _{i=1}^2{\sigma }_i({\dot{\sigma }}_i+S_i^{{}^{\prime }})\hfill \\ \hfill & =\sum _{i=1}^2\left({\sigma }_i(-k_1{\left|{\sigma }_i\right|}^{\zeta }-k_2{ln}^{1+\alpha }(ϵ{\sigma }_i^{\nu }+1)\mathrm{sgn}\left({\sigma }_i\right)-k_3{ln}^{1-\alpha }(ϵ{\sigma }_i^{\nu }+1)\mathrm{sgn}\left({\sigma }_i\right))+S_i^{{}^{\prime }}\right)\hfill \\ \hfill & \le \sum _{i=1}^2\left({\sigma }_i(-k_1{\left|{\sigma }_i\right|}^{\zeta \left|{\sigma }_i\right|})+S_i^{{}^{\prime }}\right)=\sum _{i=1}^2\left({\sigma }_i(-k_1{\left|{\sigma }_i\right|}^{\zeta \left|{\sigma }_i\right|})\right)+\sigma S^{{}^{\prime }}\le \sum _{i=1}^2\left(-k_1{\left|{\sigma }_i\right|}^{\zeta \left|{\sigma }_i\right|+1}\right)+\sigma S_0^{{}^{\prime }}\hfill \\ \hfill & \le \sum _{i=1}^2\left(-{\left|{\sigma }_i\right|}^{\zeta \left|{\sigma }_i\right|+1}(k_1-S_0^{{}^{\prime }})\right)\le \sum _{i=1}^2\left(-k_1{\left|{\sigma }_i\right|}^{\zeta \left|{\sigma }_i\right|+1}\right).\hfill \end{array}$$

(31)

From $\left|{\sigma }_i\right|\ge 1$ and $0<\zeta <1$, we can obtain ${\left|\sigma \right|}^{\zeta \left|\sigma \right|+1}\ge {\left|\sigma \right|}^{\zeta +1}$. Moreover, Equation (31) can be rewritten as the following according to $k_1-S_0^{{}^{\prime }}\ge c_1>0$,

$$\begin{array}{c}\hfill \sum _{i=1}^2{\sigma }_i({\dot{\sigma }}_i+S_i^{{}^{\prime }})\le \sum _{i=1}^2\left(-{\left|{\sigma }_i\right|}^{\zeta \left|{\sigma }_i\right|+1}(k_1-S_0^{{}^{\prime }})\right)\le \sum _{i=1}^2\left(-{\left|{\sigma }_i\right|}^{\zeta +1}(k_1-S_0^{{}^{\prime }})\right).\end{array}$$

(32)

Similarly, it can be found that, when $k_1-S_0^{{}^{\prime }}\ge c_1$, the given Lyapunov function V also satisfies the finite time stable convergence condition.

In summary, we can conclude that the system (Equation (16) with $S^{\prime }$ and $u_e$ (Equation (25)) is conditionally finite-time stable for any value of $\sigma$. Moreover, the condition can be further defined as $c=c_1{2}^{(\zeta +1)}$, $\kappa =(\zeta +1)/2$, $k_1-S_0^{{}^{\prime }}\ge c_1$. In addition, the setting time here is

$$t_q=\frac{2}{c_1{2}^{\frac{\zeta +1}{2}}(1-\zeta )}{\left(V\left({\sigma }_0\right)\right)}^{\frac{1-\zeta }{2}}.$$

(33)

Finally, we can obtain the entire setting time of the system convergence as follows:

$$t_a=t_{smax}+t_q=\frac{1}{\mu }\frac{m}{l-m}+\frac{1}{\lambda }\frac{r}{r-p}+\frac{2}{c_1{2}^{\frac{\zeta +1}{2}}(1-\zeta )}{\left(V\left({\sigma }_0\right)\right)}^{\frac{1-\zeta }{2}}.$$

(34)

□

4.2. Fast-Time Scale Subsystem Control

For the control design of the fast time-scale subsystem, the derivative control method is used. It has been proved that the fast time-scale subsystem is equilibrium point asymptotically stable [37]. The derivative control acts as a damper and is designed to increase the speed of convergence:

$$U_f=K_f({\dot{q}}_1-{\dot{q}}_2),$$

(35)

where the diagonal matrix $K_f$ and $K_2$ are both positive definite, and $K_f=K_2/\epsilon$.

The block diagram of the loop of the proposed control strategy is given in Figure 6, in which NFTSMC denotes novel fast terminal sliding mode control.

5. Simulation Experiments

5.1. Experiment Setting

To verify the performance of the control proposed in this study, four groups of simulation experiments have been carried out, as shown in

Table 2. Simulation experiment groups.

Group No.	Reaching Law	Base Oscillation	Payload Uncertainty	Fast Control
Group 1	Variable power log	Three oscillations	N	On
Group 2	Three reaching laws	Shake oscillation	N	On
Group 3	Variable power log	Shake oscillation	Y	On
Group 4	Variable power log	Shake oscillation	N	Off

. Among them, Group 1 includes three cases of simulations, which respectively correspond to three base oscillations. This group of simulations is to prove the rapid robust characteristic of the proposed control under base oscillations. Group 2 also includes three cases of simulations. The purpose of this group is to highlight the advantages of the variable power log reaching law compared with the other two traditional reaching laws (other control conditions such as the sliding mode surface, fast time-scale subsystem control, etc. are the same in all three cases). Group 3 further verifies the robustness of the proposed control against the payload uncertainty. In the fourth group (Group 4), the effectiveness of the singular perturbation method is verified. For convenience, this section refers to the fast time-scale subsystem control as fast control and to the reach law as RL. The system and controller parameters in all simulations are given in

Table 3. System and controller parameters.

Parameters	Values	Parameters	Values	Parameters	Values
$m_1$/kg	10	$m_2$/kg	40	g/(N/kg)	9.8
$m_l$/kg	2	I/kgm	0.01	L/m	0.74
$K_1$/(Nm/rad)	2 $\times {10}^6$	$K_2$/(Nm/rad)	2 $\times {10}^4$
$k_1$	3	$k_2$	5	$k_3$	12
r	3	p	1	l	3
m	1	$\alpha$	0.5	$\beta$	5
$\mu$	12/5 (i = 1/2)	$\lambda$	5	$ϵ$	10
$\nu$	2	$\zeta$	0.9	$K_f$	$\mathrm{diag}(200,2.6)$

.

The initial state of the system and desired trajectory are chosen as follows:

$$\left\{\begin{array}{c}(y_{r20},{\dot{y}}_{20},{\theta }_{30},{\dot{\theta }}_{3}0)=(0.8,0,-0.5,0)\hfill \\ (y_{r2d},{\dot{y}}_{2d},{\theta }_{30},{\dot{\theta }}_{3}0)=(\sin \pi t,\pi \cos \pi t,-1+\sin 0.5\pi t,0.5\pi \cos 0.5\pi t)\hfill \end{array}\right.$$

In addition, Figure 7 gives the time response curves for the three base oscillations used in the simulations, and these relevant data come from our previous studies on vehicle dynamics.

5.2. Results and Analysis

Group 1:

The results of the first group of simulations are in Figure 8, Figure 9, Figure 10 and Figure 11.

Here, the black solid, blue dashed, and red dash-dotted line correspond to the case of shake, pitch, and roll oscillation, respectively; the black dotted line indicates the given desired trajectory. Figure 8 and Figure 9 present the position and angle response of the lifting part and rotating part, respectively. Figure 10 and Figure 11 demonstrate the velocity of the lifting part and the angular velocity response of the lifting part and rotating part, respectively. We can see from Figure 8 and Figure 9 that the proposed control effectively suppresses the influence of base oscillations and compliant-actuator-caused flexible vibration, and realizes rapid robust trajectory of the ammunition transfer manipulator. For three cases, the steady-state times of the control are all less than 0.2. Moreover, from Figure 10, we can see that the shake base oscillation has the most significant influence on the velocity response of the lifting part, and the curves exhibit obvious chattering characteristics. Similarly, we can also see from Figure 10 that the pitch oscillation has the most significant influence on the angular velocity response, and the curves also show chattering characteristics.

Group 2:

The simulation results of Group 2 are in Figure 12, Figure 13, Figure 14 and Figure 15.

The black solid, blue dashed, and red dash-dotted lines correspond to the case of the control with variable power log, power, and double power reaching law, respectively; the black dotted line indicates the desired trajectory. The motion response of the lifting part is in Figure 12 and Figure 14, of which the rotating part is in Figure 13 and Figure 15. As seen in Figure 13 and Figure 15, the control based on the power and the double power reaching laws can also achieve robust trajectory tracking of the rotating part of the ammunition transfer manipulator under the influence of the shake oscillation. However, their steady-state response times are significantly greater than those of the control based on the variable power log reaching law (almost twice as long). Moreover, it can be seen from Figure 12 and Figure 14 that the controls based on traditional reaching laws cannot realize robust control of the lifting part, and their steady-state errors of position and velocity response are significant. The velocity response curves also exhibit chattering characteristics. In addition, the above figures also show that the shake base oscillation has more influence on the motion response of the lifting part than the rotating part. In summary, we can conclude that the variable power log reaching law proposed in this paper outperforms the traditional (power and double power) reaching law both in terms of robustness and convergence time performance.

Group 3:

The results of the third group of simulations are presented in Figure 16, Figure 17, Figure 18, Figure 19, Figure 20 and Figure 21.

The black solid line indicates the system response under nominal inertia (no payload); the red dot-dashed line and the blue dashed line indicate the system response under positive and negative payload uncertainties, respectively. In the simulations, the positive and negative payload uncertainties are achieved by increasing or decreasing the payload inertia by 20%. As can be seen from these figures, the proposed control effectively suppresses the effects of base oscillation and payload uncertainty and achieves rapid and robust trajectory tracking of the position/angle of the ammunition transfer manipulator. It can also be seen that the uncertainty of the payload affects the convergence speed of the controlled system. In the case of larger payload, there is a longer convergence time.

Group 4:

Figure 22, Figure 23, Figure 24 and Figure 25 are the results of the fourth group of simulation experiments. In the figures, the red dot-dashed line and the blue dashed line indicate the system response in the case of turning on and off the fast control, respectively; and the black dotted line indicates the desired trajectory. As shown in these figures, when the fast control is turned off, the actual trajectory of the ammunition transfer manipulator cannot achieve accurate tracking of the desired trajectory, and the two trajectories separate after 1 s. Moreover, it can be seen that the response trajectory of the system shows significant high frequency flutter and continuous divergence. To show the response curves more clearly, for the simulation with the fast control turned off, we have chosen only the first 3.6 s of experimental results in the figures.

In order to further demonstrate the simulation results more clearly, the dynamic specifications of the four groups of simulation results are given in

Table 4. Dynamic specifications for four groups of simulations.

Group No.	Case	Response Time/s	Settling Time/s	Max Overshoot/m or rad	Steady-State Error/m or rad
Group 1	Shake	0.01/0.16	0.20/0.21	0.04/0	0.015/0.001
	Pitch	0.01/0.16	0.20/0.21	0.04/0	0.003/0.003
	Roll	0.01/0.16	0.20/0.21	0.03/0	0/0
Group 2	New RL	0.10/0.16	0.20/0.21	0.04/0	0.015/0.001
	Douple power RL	0.14/0.24	0.63/0.61	0.21/0.11	0.18/0.03
	Power RL	0.19/0.34	0.76/1.32	0.22/0.13	0.16/0.06
Group 3	No payload	0.10/0.16	0.20/0.21	0.04/0	0.015/0.001
	Payload 1	0.11/0.16	0.21/0.21	0.05/0	0.018/0.003
	Pyaload 2	0.12/0.16	0.21/0.21	0.05/0	0.017/0.002
	Fast control on	0.10/0.16	0.20/0.21	0.04/0	0.015/0.001
Group 4	Fast control off	n/a	n/a	n/a	n/a

. Specifically, the table gives the response time, settling time, max overshoot, and steady-state error for the lifting and rotating parts of the manipulator in each group of simulations. In addition, the data in front of the forward slash are for the lifting part, and the data behind are the forward slash for the rotating part.

6. Conclusions

This paper proposed a rapid robust control strategy for a novel marine-vehicle manipulator that is driven by novel compliant actuators (SEAs) and subject to wave-induced base oscillations. The proposed control method effectively suppresses the effects of oscillatory-base-caused disturbance and SEA-induced flexible vibration, and achieves fast trajectory tracking of the manipulator. Moreover, the efficiency of the proposed control has been demonstrated theoretically and through simulation experiments. This study is meaningful for improving the control performance of marine-vehicle manipulators.

To sum up, the contributions of this study are as follows:

• The theoretical innovation of this study is that a novel SMC reaching law, i.e., the variable power log reaching law was proposed for the first time. It can provide a faster convergence speed away from the sliding mode surface, and maintain a better sliding mode chattering suppression performance close to the sliding mode surface. This is the primary contribution and innovation of this study;
• The innovative aspect of the application of this study is that we investigated for the first time the problem of rapid robust control of a marine-vehicle manipulator considering both base oscillations and SEA-induced flexible vibrations, and proposed a novel solution to this problem based on the singular perturbation method and the SMC method with the above novel reaching law.

However, the proposed control is currently only applied to the theoretical model of marine-vehicle manipulators. The actual marine-vehicle manipulator system and base oscillation conditions are definitely more complex. We will spend more effort to investigate the effectiveness of the proposed control in a practical environment.