Right Coprime Factorization-Based Simultaneous Control of Input Hysteresis and Output Disturbance and Its Application to Soft Robotic Finger

Abstract

In a nonlinear control system, hysteresis exists usually as common characteristics. In addition, external output disturbances like modelling error, machine friction and so on also occur frequently. Both of them are considered to cause instability and unsatisfactory performance. In this paper, a practical nonlinear control system design is proposed so as to achieve the simultaneous control of input hysteresis and output disturbance. The system is based on RCF (right coprime factorization theory). Additionally, the proposed design has been applied to a soft robotic finger system and the results of simulations and practical experiments are exhibited, which show the effectiveness of the proposed system.

1. Introduction

Hysteresis commonly exists in nonlinear control systems and is mostly likely to make the system unstable and fail to track the reference signal rapidly. In order to eliminate the influence of hysteresis, some hysteresis models have been developed, like the Prandtl–Ishlinskii (PI) model and Bouc–Wen model [1,2]. However, it is hard to achieve simultaneous control using these models because the inverses of models are extremely complex and sometimes inaccurate, since the output of hysteresis model is connected with previous values. In another study [3], the hysteresis characteristics were found to be decomposed as a constant and a nonlinear operator, which made it more able to be applied into a control system.

Additionally, external output disturbance like modelling error and machine friction, which would also destroy the stability and tracking performance of nonlinear system. In order to eliminate the influence of uncertain disturbance, a filter algorithm has been proposed in previous study so as to achieve real-time estimation of time-varying multi-joint robotic arm [4]. This method was proved to be effective since it guaranteed that all of the estimated elements of system are bounded.

Moreover, operator-based right coprime factorization (RCF) theory is usually selected to achieve robust stable control in nonlinear control systems. For example, research on soft actuators and aluminum plates with Peltier devices have been carried out in [5,6,7], which implied the robustness of RCF systems even with uncertainty. What is more, an intelligent robust operator-based sliding mode control method was proposed in [8], which made RCF theory combine with sliding mode control so as to obtain better tracking performance in nonlinear control system. Therefore, RCF theory is introduced in this paper to achieve robust control combined with the decomposition of hysteresis mentioned above.

Recently, soft robotic fingers are performing increasingly essential roles in fields like medical treatment and the welfare industry [9]. With their increasing popularity, many varieties of soft robotic fingers have been developed to meet different needs. In order to achieve the 3-degree-of-freedom bending, Suzumori et al. proposed a design of bellow-shaped miniature pneumatic actuator [10]. This one is designed as a semi-cylindrical chamber whose round surface is shaped like a bellow. In addition, between bellows are channels, whose radii are shorted than the bellow part. This kind of structure offers enough space to deformation for bending when air is injected or extracted. As a result, this soft actuator is widely used for researches [4,5,11,12,13,14]. However, this kind of soft robotic finger performs with hysteresis because of the soft material and air-driven effects, which would cause bad performance in practical experiments. To eliminate the influence of hysteresis, refs. [15,16] proposed a method using an inverse hysteresis model combined with RCF theory, and the results of their simulations proved the effectiveness.

In this paper, a practical nonlinear control system is proposed in order to achieve simultaneous control of the input hysteresis and external output disturbance. The robust stability of the system is guaranteed using RCF theory. Then, a practical application for soft robotic fingers is carried out, which verifies the effectiveness of proposed design.

The other parts of this paper are summarized as follows: Section 2 introduces the details of decomposed hysteresis model; In Section 3, the introduction of RCF theory and details of each operator in designed control system are included; Then, Section 4 describes the mathematical model of soft robotic fingers and the practical experiments are carried out; In addition, the results of simulations and experiments are shown subsequently; Finally, the whole work is concluded in Section 5.

2. Hysteresis Characteristics

In a previous study [3], the hysteresis model divided into a play hysteresis and stop hysteresis operator. The details of the two operators are included in [3] and omitted in this paper. According to this definition, the two types of hysteresis operators could be denoted as

$${\mathit{H}}_{\mathit{s}}\left(\mathit{u}\right)={\int }_0^{\mathit{H}}\mathit{p}\left(\mathit{h}\right){\mathit{E}}_{\mathit{h}}\left[\mathit{u}\right]\mathit{d}\mathit{h}$$

(1)

$${\mathit{H}}_{\mathit{p}}\left(\mathit{u}\right)={\int }_0^{\mathit{H}}\mathit{p}\left(\mathit{h}\right){\mathit{F}}_{\mathit{h}}\left[\mathit{u}\right]\mathit{d}\mathit{h}$$

(2)

where $\mathit{p}\left(\mathit{h}\right)$ is a given continuous density function, $\mathit{h}$ is a nominal variable for calculation and $\mathit{h}\ge 0$. Then, $\mathit{p}\left(\mathit{h}\right)$ is assumed to satisfy $\mathit{p}\left(\mathit{h}\right)>0$ and ${\int }_0^{\infty }\mathit{h}\mathit{p}\left(h\right)\mathit{d}\mathit{h}<\infty$. Since the density function $\mathit{p}\left(\mathit{h}\right)$ vanishes for large values of $\mathit{h}$, it is assumed that there exists a constant threshold $\mathit{H}$ such that $\mathit{p}\left(h\right)=0$ for $\mathit{h}>\mathit{H}$.

In addition, it has been proved that both the stop and play hysteresis operators satisfy the equation as follows:

$${\mathit{E}}_{\mathit{h}}\left[\mathit{u}\right]\left(t\right)+{\mathit{F}}_{\mathit{h}}\left[\mathit{u}\right]\left(t\right)=\mathit{u}\left(t\right)$$

(3)

for any input function $\mathit{u}\left(\mathit{t}\right)$.

In this paper, the play hysteresis operator ${\mathit{H}}_{\mathit{p}}$ is selected to describe the input hysteresis, which exists before the given plant $\mathit{P}$ in a control system.

As shown in Figure 1, the input of hysteresis operator is $\mathit{u}\left(\mathit{t}\right)$ and the output is ${\mathit{u}}^{\ast }\left(\mathit{t}\right)$. Therefore, the following equation can be obtained:

$$\mathit{y}\left(\mathit{t}\right)=\mathit{P}{\mathit{H}}_{\mathit{p}}\left(\mathit{u}\right)\left(\mathit{t}\right)$$

(4)

Since the play hysteresis operator is too complex to be factorized, based on (1)–(3), we can obtain that [3]:

$${\mathit{H}}_{\mathit{s}}\left(\mathit{u}\right)\left(\mathit{t}\right)+{\mathit{H}}_{\mathit{p}}\left(\mathit{u}\right)\left(\mathit{t}\right)={\int }_0^{\mathit{H}}\mathit{p}\left(\mathit{h}\right)\mathit{d}\mathit{h}·\mathit{u}\left(\mathit{t}\right)$$

(5)

Then, a constant ${\mathit{K}}_{\mathit{s}}$ is used to represent ${\int }_0^{\mathit{H}}\mathit{p}\left(\mathit{h}\right)\mathit{d}\mathit{h}$. The play hysteresis operator can be formulated as:

$${\mathit{H}}_{\mathit{p}}\left(\mathit{u}\right)\left(\mathit{t}\right)={\mathit{u}}^{\ast }\left(\mathit{t}\right)={\mathit{K}}_{\mathit{s}}\left(\mathit{u}\right)\left(\mathit{t}\right)-{\mathit{H}}_{\mathit{s}}\left(\mathit{u}\right)\left(\mathit{t}\right)$$

(6)

Equation (6) implies that the hysteretic nonlinearity could be regarded as a constant parameter times input then minus a nonlinear part. As a result, the relationship between output $\mathit{y}$ and input $\mathit{u}$ can be formulated based on (4) and (6):

$$\mathit{y}\left(\mathit{t}\right)=\mathit{P}{\mathit{H}}_{\mathit{p}}\left(\mathit{u}\right)\left(\mathit{t}\right)=\mathit{P}\left[{\mathit{K}}_{\mathit{s}}\left(\mathit{u}\right)\left(\mathit{t}\right)-{\mathit{H}}_{\mathit{s}}\left(\mathit{u}\right)\left(\mathit{t}\right)\right]$$

(7)

3. Scheme of Proposed System

In this section, an operator-based control system using RCF theory to achieve simultaneous control of input hysteresis and output disturbance will be described.

3.1. Right Coprime Factorization (RCF) Theory

Right coprime factorization theory is selected to guarantee the stability of nonlinear control system in this paper. According to the theory [5,6,7,8,14], a given plant operators $\mathit{P}:\mathit{U}\to \mathit{V}$ is said to have a right factorization if there exists a linear space $\mathit{W}$ and two stable operator $\mathit{D}:\mathit{W}\to \mathit{U}$ and $\mathit{N}:\mathit{W}\to \mathit{V}$ such that $\mathit{D}$ is invertible from $\mathit{U}$ to $\mathit{W}$ and $\mathit{P}={\mathit{ND}}^{-1}$, as shown in Figure 2.

Additionally, it is also be proved that the system could be considered as BIBO (Bounded Input Bounded Output) stable if there exists other two stable operators $\mathit{A}:\mathit{V}\to \mathit{U}$ and $\mathit{B}:\mathit{U}\to \mathit{U}$, where $\mathit{B}$ is invertible, satisfying the Bezout identity:

$$\mathit{AN}+\mathit{BD}=\mathit{M}$$

(8)

where $\mathit{M}$ is a unimodular operator (both $\mathit{M}$ and ${\mathit{M}}^{-1}$ are bounded stable) which could usually be selected as inentity operator $\mathit{I}$ if $\mathit{W}=\mathit{U}$.

3.2. Scheme of Proposed System

Using the decomposition of operator ${\mathit{H}}_{\mathit{p}}$ in (7), it becomes available to apply RCF theory to control system design. To begin with, the scheme of the proposed system without considering external disturbance is shown in Figure 3, while signal $\mathit{r}$ denotes the reference and $\mathit{y}$ denotes the actual output.

According to RCF theory, identity operator $\mathit{I}$ is selected as unimodular operator in order to satisfy (8), which suggests that operators $\mathit{A}$, $\mathit{B}$, $\mathit{D}$, and $\mathit{N}$ are expected to satisfy

$$\mathit{AN}+\mathit{B}\left({\mathit{K}}_{\mathit{s}}^{-1}\mathit{D}\right)=\mathit{I}\left(\omega \right)\left(\mathit{t}\right)$$

(9)

then the proposed system is supposed to be BIBO stable. In addition, in order to eliminate the influence of input hysteresis simultaneously and guarantee the tracking performance, an operator $\mathit{C}$ is added, which would be proved as follows.

Proof. 

According the RCF theory, the proposed system could be simplified as in Figure 4. Then, operator $\mathit{C}$ is expected to satisfy that

$${\mathit{y}}_{\mathit{a}}\left(\mathit{t}\right)=\mathit{N}{\mathit{I}}^{-1}\mathit{C}\left({\mathit{r}}_{\mathit{a}}\right)\left(\mathit{t}\right)={\mathit{r}}_{\mathit{a}}\left(\mathit{t}\right)$$

(10)

It can be observed in Figure 4 that

$$\tilde{\mathit{y}}\left(\mathit{t}\right)=\mathit{y}\left(\mathit{t}\right)-{\mathit{y}}_{\mathit{a}}\left(\mathit{t}\right)$$

(11)

$${\mathit{r}}_{\mathit{a}}\left(\mathit{t}\right)=\mathit{r}\left(\mathit{t}\right)-\tilde{\mathit{y}}\left(\mathit{t}\right)$$

(12)

Therefore, it can be obtained from (10)–(12) that

$$\begin{array}{cc}\hfill \mathit{y}\left(\mathit{t}\right)& =\tilde{\mathit{y}}\left(\mathit{t}\right)+{\mathit{y}}_{\mathit{a}}\left(\mathit{t}\right)\hfill \\ \hfill & =\mathit{r}\left(\mathit{t}\right)-{\mathit{r}}_{\mathit{a}}\left(\mathit{t}\right)+{\mathit{y}}_{\mathit{a}}\left(\mathit{t}\right)\hfill \\ \hfill & =\mathit{r}\left(\mathit{t}\right)\hfill \end{array}$$

(13)

which implies that the output $\mathit{y}$ would follow the reference $\mathit{r}$ finally. □

Figure 4. Simplified system.

Figure 4. Simplified system.

However, external output disturbance is likely to occur in practical experiments, which would destroy the tracking performance of the system. As a result, an improved PD tracking controller $\mathit{F}$ is selected as shown in Figure 5, while $\Delta \mathit{d}$ denotes the external output disturbance.

In addition, controller $\mathit{F}$ is formulated as

$$\mathit{F}\left({\mathit{x}}_{\mathit{e}}\right)\left(\mathit{t}\right)={\mathit{K}}_{\mathit{F}}\left({\mathit{K}}_{\mathit{p}}·{\mathit{x}}_{\mathit{e}}\left(\mathit{t}\right)+{\mathit{K}}_{\mathit{d}}·\frac{\mathit{d}{\mathit{x}}_{\mathit{e}}\left(\mathit{t}\right)}{\mathit{dt}}\right)$$

(14)

$${\mathit{K}}_{\mathit{F}}\left({\mathit{x}}_{\mathit{e}}\right)\left(\mathit{t}\right)=\frac{\lambda }{\lambda +\left|{\mathit{K}}_{\mathit{p}}^2·{\mathit{x}}_{\mathit{e}}\left(\mathit{t}\right)\right|}$$

(15)

while $\lambda$ is a designed parameter and $\lambda >0$.

Compared with traditional PD control, the proposed one could adjust parameters itself with change of error, which would cause less vibration when error turns greater. The effectiveness of proposed method would be proved as follows.

Proof. 

Assuming the reference $\mathit{r}\left(\mathit{t}\right)$ is less than

$$\left|\mathit{r}\left(\mathit{t}\right)\right|<\frac{\lambda ({\mathit{K}}_{\mathit{p}}+1)}{{\mathit{K}}_{\mathit{p}}^2},$$

then it can be observed from Figure 5 that

$${\mathit{r}}_{\mathit{a}}\left(\mathit{t}\right)=\mathit{r}\left(\mathit{t}\right)+\mathit{F}\left({\mathit{x}}_{\mathit{e}}\right)\left(\mathit{t}\right)-\tilde{\mathit{y}}\left(\mathit{t}\right)$$

(16)

$$\tilde{\mathit{y}}\left(\mathit{t}\right)=\mathit{y}\left(\mathit{t}\right)-{\mathit{y}}_{\mathit{a}}\left(\mathit{t}\right)$$

(17)

From (10), combining (16) and (17), it can be obtained that

$$\mathit{y}\left(\mathit{t}\right)-\mathit{r}\left(\mathit{t}\right)=\mathit{F}\left({\mathit{x}}_{\mathit{e}}\right)\left(\mathit{t}\right)$$

(18)

while ${\mathit{x}}_{\mathit{e}}\left(\mathit{t}\right)=\mathit{r}\left(\mathit{t}\right)-\mathit{y}\left(\mathit{t}\right)$, then (18) can be transferred as

$$-{\mathit{x}}_{\mathit{e}}=\frac{\lambda }{\lambda \pm {\mathit{K}}_{\mathit{p}}^2·{\mathit{x}}_{\mathit{e}}\left(\mathit{t}\right)}·{\mathit{K}}_{\mathit{p}}·{\mathit{x}}_{\mathit{e}}\left(\mathit{t}\right)+{\mathit{K}}_{\mathit{d}}·\frac{\mathit{d}{\mathit{x}}_{\mathit{e}}\left(\mathit{t}\right)}{\mathit{dt}}$$

(19)

which leads to

$${\mathit{e}}^{-\frac{{\mathit{K}}_{\mathit{p}}^2}{\lambda {\mathit{K}}_{\mathit{d}}}\mathit{t}}·{\mathit{e}}_c=|\frac{{\mathit{x}}_{\mathit{e}}}{{\mathit{x}}_{\mathit{e}}\pm \beta }{|}^{\alpha }$$

(20)

the details of mathematical calculation is explained in Appendix A. ${\mathit{e}}_{\mathit{c}}$ is a constant parameter and ${\mathit{K}}_{\mathit{p}}$, ${\mathit{K}}_{\mathit{d}}$ are expected to be within the range: $0>{\mathit{K}}_{\mathit{p}}>-1$, ${\mathit{K}}_{\mathit{d}}>0$. Additionally, parameters $\alpha$ and $\beta$ are denoted as

$$\alpha =\frac{{\mathit{K}}_{\mathit{p}}^2}{\lambda ({\mathit{K}}_{\mathit{p}}+1)}\mathrm{and}\beta =\frac{1}{\alpha }$$

(21)

Thus, the left part of (20) would trends to 0 with $\mathit{t}\to \infty$, which suggests that $\left|{\mathit{x}}_{\mathit{e}}\right|\to 0$ and $\mathit{y}\to \mathit{r}$ eventually. □

4. Application for Soft Robotic Fingers

4.1. Mathematical Model of Soft Robotic Fingers

The soft robotic finger used in this paper is composed of semi-cylindrical rubber, which was designed in [10] to achieve bending deformation with asymmetry. The round side is divided into several similar bellow units. Each units includes a channel and chamber, both of which are of the same length and alternately arranged. The channel is designed as of lower height than chamber, which provides enough space for chamber to deform. Therefore, the round side is more extensible than flat side. As shown in Figure 6, when the air pressure inside the finger changes, the deformation of the round side is greater than the flat side, which makes the bending movement realized.

The left one in Figure 7 describes the physical model of soft robotic fingers. The input air pressure is expressed as $\mathit{p}$ (kPa) and output angle as $\theta$ (rad). ${\mathit{L}}_0$ is the initial length of robotic finger and $\Delta \mathit{L}$ is the deformation, while $\mathit{l}$ is the symbol for length of each unit [5].

Furthermore, the right one in Figure 7 shows the cross section of each unit of soft robotic finger [6], while $\mathit{a}$ is the radius of channel and $\mathit{b}$ is the thickness of rubber. According to forces analysis, the moments of air pressure (${\mathit{M}}_1,{\mathit{f}}_1$) and elastic force (${\mathit{M}}_2,{\mathit{f}}_2$) are shown as [6]

$${\mathit{M}}_1={\int }_0^{\mathit{a}}{\int }_0^{\pi }{\mathit{f}}_1(\mathit{r}sin\varphi +\mathit{b})\mathit{d}\varphi \mathit{d}\mathit{r}$$

(22)

$${\mathit{M}}_2=-{\int }_0^{\pi }{\mathit{f}}_2(\mathit{R}sin\varphi +\mathit{b})\mathit{d}\varphi$$

(23)

while ${\mathit{E}}_0$ is the initial Young’s modulus and $\mathit{R}$ is the representative radius of the soft robotic finger during the integral, which is obtained by $\mathit{R}=\mathit{a}+\frac{b}{2}$. When the soft robotic finger becomes stable, the moments are supposed to satisfy

$${\mathit{M}}_1+{\mathit{M}}_2=0$$

(24)

Then, the mathematical model between input air pressure and output angle can be obtained [6]:

$$\theta =-\frac{{\mathit{L}}_0}{\mathit{b}}\left(1+\frac{2\mathit{R}+\mathit{b}\pi }{\gamma \mathit{p}-2\mathit{R}-\mathit{b}\pi }\right)$$

(25)

$$\gamma =\frac{4{\mathit{a}}^3+3\pi {\mathit{a}}^2\mathit{b}}{6\mathit{R}\mathit{b}{\mathit{E}}_0}$$

(26)

where $\mathit{p}$ and $\theta$ are supposed to be in the range $\mathit{p}\in \left[0,60\right],\theta \in \left[0,\pi \right]$

4.2. Hysteresis Characteristics of Soft Robotic Fingers

In order to figure out the value of constant ${\mathit{K}}_{\mathit{s}}$, which was introduced in (7), a large number of experiments have been carried out and the curves of results are shown in Figure 8.

4.3. Design of Operators

According to RCF theory and mathematical model (25), based on the proposed system in Figure 3, operators $\mathit{N}$ and ${\mathit{D}}^{-1}$ are designed as [6]

$$\mathit{N}\left(\omega \right)\left(\mathit{t}\right)=-\frac{{\mathit{L}}_0}{\mathit{b}}\left(1+\frac{2\mathit{R}+\mathit{b}\pi }{\omega \left(\mathit{t}\right)-2\mathit{R}+\mathit{b}\pi }\right)$$

(27)

$${\mathit{D}}^{-1}\left({\mathit{u}}^{\ast }\right)\left(\mathit{t}\right)=\gamma ·{\mathit{u}}^{\ast }\left(\mathit{t}\right)$$

(28)

Then operator ${\mathit{K}}_{\mathit{s}}$ is designed to combine with ${\mathit{D}}^{-1}$ as ${\tilde{\mathit{D}}}^{-1}$:

$${\tilde{\mathit{D}}}^{-1}\left(\mathit{u}\right)\left(\mathit{t}\right)=\gamma {\mathit{K}}_{\mathit{s}}·\mathit{u}\left(\mathit{t}\right)$$

(29)

According to RCF theory, operator $\mathit{A}$ and $\mathit{B}$ are expected to satisfy (8), while $\mathit{K}$ is a designed parameter and the bounded signal ${\mathit{y}}_{\mathit{a}}$ is selected to feedback towards $\mathit{A}$ so as to guarantee stability:

$$\mathit{B}\left(\mathit{u}\right)\left(\mathit{t}\right)=\gamma \mathit{K}·\mathit{u}\left(\mathit{t}\right)$$

(30)

$$\mathit{A}\left({\mathit{y}}_{\mathit{a}}\right)\left(\mathit{t}\right)=\left(1-\frac{\mathit{K}}{{\mathit{K}}_{\mathit{s}}}\right)\left(-\frac{{\mathit{L}}_0(2\mathit{R}+\mathit{b}\pi )}{\mathit{b}{\mathit{y}}_{\mathit{a}}\left(\mathit{t}\right)+{\mathit{L}}_0}+2\mathit{R}+\mathit{b}\pi \right)$$

(31)

As a result, it can be obtained from (27)–(31) that

$$\mathit{AN}+\mathit{B}\tilde{\mathit{D}}=\mathit{I}\left(\omega \right)\left(\mathit{t}\right)$$

(32)

while $\mathit{I}$ is an identity operator, which is an unimodular operator and suggests that the proposed system is stable.

4.4. Simulation and Practical Experiments

To begin with, the simulation of proposed system is carried out.

Table 1. Values of parameters.

Parameters	Values	Parameters	Values
$\mathit{a}$	$1\times {10}^{-3}$ m	$\mathit{K}$	1.25
$\mathit{b}$	$0.15\times {10}^{-3}$ m	$\lambda$	4.10
${\mathit{L}}_0$	$1.35\times {10}^{-2}$ m	${\mathit{K}}_{\mathit{p}}$	−0.63
${\mathit{E}}_0$	$3.0\times {10}^6$ m	${\mathit{K}}_{\mathit{d}}$	0.0001
$\mathit{R}$	$1.075\times {10}^{-3}$ m	$\mathit{t}$	0.1 s

lists the values of parameters used. Specifically, $\mathit{t}$ is the sampling time. $\mathit{K}$ and $\lambda$ are designed parameters in operators. ${\mathit{K}}_{\mathit{p}}$ and ${\mathit{K}}_{\mathit{d}}$ are chosen from adjustments in experiments.

In addition, the density function is selected as $\mathit{p}\left(\mathit{h}\right)=e^{-\alpha {(\mathit{h}+\beta )}^2}$, while $\alpha =1.904\times {10}^{-8}$ and $\beta =2.0\times {10}^4$, threshold parameter $\mathit{H}$ is designed as $\mathit{H}=\mathrm{10,000}$.

Figure 9 shows the results of simulation based on proposed system shown in Figure 3, while the curve of input signal is shown in Figure 10. It can be observed that the output $\mathit{y}$ can follow reference $\mathit{r}$ rapidly, which suggests that the proposed system is stable and the proposed simultaneous control method of input hysteresis is effective.

Subsequently, practical experiments with external output disturbance are carried out. The equipment used is shown in Figure 11, which includes a safety regulator (RP1000-8-07, CKD, Aichi, Japan), an electro-pneumatic regulator (ITV0010-0CS, SMC, Tokyo, Japan), a PC-connected camera, a micro controller, and the soft robotic finger. The soft robotic finger is painted red and blue on both ends. In addition, a pump and air compressor are also selected to offer thedesired air pressure.

The process of the experiment can be summarized as follows: Firstly, the experiment starts with the signal from PC and the soft robotic finger bends towards the reference input. Next, the image of the robotic finger is captured by a camera and sent back to the PC, and then processed to obtain the coordinates of both ends of the soft robotic finger and calculate the bending angle. Finally, the control signal can be generated from the difference and sent to the micro controller to achieve perfect tracking.

Additionally, in order to offer arbitrary external output disturbance during experiment, an air pressure regulator that does not work well is selected. Then, an experiment without an operator $\mathit{F}$ was carried out to test the performance of the soft robotic finger under the regulator mentioned above. The results are shown in Figure 12. It can be observed that giant vibration occurs around 1.5 rad and during reference changes. The output of the system found that it is hard to follow the reference in most cases.

Finally, the practical experiments based on different methods are carried out, while the curves of results are shown in Figure 13, Figure 14, Figure 15, Figure 16, Figure 17 and Figure 18.

Specifically, Figure 15 shows the result of the system that only controller $\mathit{C}$ and RCF theory are implemented (namely, C-RCF) while the hysteresis is not considered. It is extremely obvious that the output of a soft robotic finger cannot follow the reference.

In addition, Figure 16 shows the result based on a system using conventional method, in which the hysteresis is not considered (namely, C-PD-RCF). As a result, it is apparent that the output tracks reference slowly due to the existence of hysteresis. Moreover, the soft robotic finger still vibrates due to the fact that only traditional PD control is implemented.

Subsequently, Figure 17 shows the result of system with consideration of the hysteresis (namely, ${\mathrm{H}}_{\mathrm{s}}$-C-PD-RCF). It is obvious that the tracking performance becomes obviously better than the result in Figure 12. However, the soft robotic finger still vibrates when reference changes, especially when it decreases.

In contrast, Figure 13 shows the results that the proposed tracking method is used as operator $\mathit{F}$, while the input pressure is implied in Figure 14. It could be clearly observed that not only the soft robotic finger could track the reference immediately, but also the vibration decreases dramatically. Additionally, Figure 18 exhibits the comparison of errors in different cases, according to which the effectiveness of proposed method could be verified evidently.

5. Conclusions

In this paper, a practical nonlinear simultaneous control system for input hysteresis and external output disturbance has been proposed, which is based on right coprime factorization (RCF) theory. According to a method proposed in a previous study, the hysteresis characteristics could be decomposed into a constant and a nonlinear operator, which makes it more available to be applied into RCF system. In addition, a improved tracking controller is selected to guarantee the tracking performance with external output disturbance. Finally, the effectiveness and stability of the proposed system are verified by the results of simulation and practical experiments for soft robotic fingers.

However, there still exist some constraints in the proposed method. To be specific, external disturbance is expected to be bounded, which is expected to satisfy the mathematical conditions as shown in Appendix A. The proposed system is likely to be still destroyed with extreme disturbance. Therefore, future works are expected to focus on the strategies to deal with the problem stated above.