An Anti-Interference Control Algorithm for Continuum Robot Arm

Abstract

The large number of joints in a continuum manipulator complicates its dynamic modeling, making model simplification inevitable for practical motion control. However, due to external disturbances and internal noise, a controller based on the simplified dynamic model often struggles to meet the desired dynamic performance. To address this issue, this paper proposes an anti-interference control algorithm for continuum manipulators, designed to compensate for parameter uncertainties, external disturbances, and measurement noise. At the same time, the parameters of the algorithm are obtained in the form of solvability of linear matrix inequalities (LMIs). The simulation results show that the algorithm proposed in the paper provides better transient performance and is not affected by the entire disturbance. Experimental results further confirm the effectiveness and robustness of the algorithm.

1. Introduction

Traditional rigid robots face significant challenges in crowded and compact environments, including bulkiness, maneuverability, and safety limitations [1]. Compared with rigid robots, continuum robots have higher compliance and flexibility due to their multi-degree-of-freedom structure and show good performance in complex environments [2]. However, in practical applications, various interferences are inevitable. External interference, uncertain interference caused by inaccurate modeling, random disturbance of parameters, and nonlinear coupling of multivariable systems bring many difficulties to the design of continuum robot controllers, which can easily reduce the control accuracy and even destroy the stability of the system [3,4].

In recent years, an increasing number of intelligent control algorithms have been applied to the study of disturbance rejection in continuum robot control systems. In [5], Thuruthel employed a recurrent neural network (RNN) to learn the forward dynamics model of a robotic arm and used a trajectory optimization algorithm to achieve open-loop predictive control over long time sequences. To address the computational challenges of mathematical models with RNNs, ref. [6] proposed a variable-parameter recurrent neural network (VP-RNN). In order to better deal with the nonlinearity and hysteresis problems of the robotic arm, ref. [7] proposed a PID controller that can perform parameter self-tuning through the radial basis function (RBF) neural network based on the classic PID control algorithm. In [8], You utilized a model-free reinforcement learning-based approach to control the two-dimensional movement of a soft robotic arm, which exhibited strong robustness; however, achieving higher precision significantly increased task execution time. In [9], Satheeshbabu proposed a deep reinforcement learning-based method for open-loop position control of a space soft robotic arm, using Deep-Q learning and experience replay for system simulation training. As described above, most of these control methods do not require the definition of the robotic arm’s configuration space or joint structure parameters, making them highly adaptable. However, they typically demand extensive computational resources and lengthy training times for network models, which reduces the efficiency of real-time control.

An efficient control algorithm based on a conventional PID controller and a meta-heuristic algorithm is proposed in [10]. The aim of the proposed algorithm is to control the position of the continuum robot, which is a highly unstable nonlinear system. In [11], an intelligent fuzzy feed-forward computed torque estimator for the Proportional Integral Derivative (PID) controller is proposed for highly nonlinear continuum robot manipulators. For highly nonlinear systems [12], a robust, fuzzy PID estimator with a backstepping methodology is introduced. In [13], an optimized nonlinear sliding mode control algorithm is developed for the sake of controlling the continuum robot’s end effector. Sliding mode control was utilized in designing a robust controller for the robot under planer motion in [14]. In summary, although these control methods demonstrate robust performance in the presence of model uncertainties, they often struggle to maintain performance under high-frequency disturbances and noise, often leading to system oscillations or instability.

To improve the trajectory tracking performance of the continuum robot, an active disturbance rejection control (ADRC)-based control strategy for the continuum robot is proposed [15]. In [16], a novel disturbance rejection control framework is proposed with a tailored sliding mode controller to achieve stabilized control. In order to improve tracking performance, Müller and Veil proposed a disturbance observer-based control method for a soft quasi-continuum manipulator [17,18,19]. In [20], a novel $H_{\infty }$-based Extended Kalman Filter (EKF) is proposed to estimate the states of a soft continuum manipulator and its performances are investigated. Although the above control methods can provide accurate state estimation in the presence of process and measurement noise, effectively filtering out noise, they rely heavily on an accurate system model. When there are significant model errors, the performance of measurement noise filtering and disturbance observation may be compromised, leading to a decline in overall control system performance.

Since the hyper-redundant manipulator has many joints, it is inevitable to simplify the model. Therefore, it is particularly important for the control algorithm to compensate for parameter uncertainty, external disturbance, and measurement noise. Considering a hyper-redundant manipulator with vector input and output signals (nonlinear control object), this paper designs a control algorithm to solve the nonlinear target control of the hyper-redundant manipulator under disturbance and measurement noise interference during the motion process. The measurable signal is equal to the sum of the output of the control object and the measurement noise, and the measurement noise and interference sources are independent.

To facilitate the presentation and understanding of the mathematical models and control algorithms discussed in this paper, the key symbols along with their definitions and the abbreviations with their full forms are summarized in

Table 1. Symbols and abbreviations.

Symbols
R	Set of real numbers
$I_{\mathbf{l}}$	Identity matrix of order l
$\tilde{I}$	A matrix of dimension $(m-1)\times m$ obtained by deleting the i-th row from the $m\times m$ identity matrix.
$O_{n\times l}$	Zero matrix of size $n\times l$
$A^{+}$	Pseudoinverse of matrix A
$L^{+}$	Pseudoinverse of matrix L
$E_j$	Vector with the j-th component equal to 1 and all others equal to 0
$\left|·\right|$	Euclidean norm of a vector
$∥·∥$	Corresponding matrix norm of the vector
$p={\displaystyle \frac{d}{dt}}$	Differentiation operator
$P_e\left(t\right)$	Active power of the motor, in per unit
$P_m$	Mechanical power of the motor, in per unit
D	Per unit damping constant
H	Inertia constant in second
${\omega }_0$	Synchronous machine speed, in rad/s
Abbreviations
LMI	Linear Matrix Inequalities
PID	Proportional Integral Derivative
ADRC	Active Disturbance Rejection Control
EKF	Extended Kalman Filter
RNN	Recurrent Neural Network

.

2. Problem Equation

The continuum robot arm in this paper is shown in Figure 1. The robot arm is designed with 12 joints and a total length of 1620 mm. It employs a cable-driven oblique pull mechanism. Each joint adopts a gradual design so that the diameter of the end joint is only 70 mm. This design improves the control accuracy while reducing the overall mass of the robot arm. Universal joints are selected as connection units for adjacent joints. Three steel wire ropes are arranged at 120° intervals on the joint disk to control the two degrees of freedom of a single joint. Each drive unit is composed of a drive-control-integrated stepper motor, a coupling, a screw slide, a pulley set, and an absolute encoder. The 36 drive units are stacked together in layers to improve the integration and compactness of the drive box.

The Lagrange method can be used to derive the dynamic model of each link in the robotic arm, which is mainly composed of the inertia matrix, the Coriolis matrix, and the gravity matrix. Due to the hyper-redundant robotic arm having many joints, the dimensions of these matrices are relatively high. At low speeds, the largest Coriolis matrix can be neglected. However, simplifying the dynamic model introduces certain errors. Controllers based on the dynamic model often fail to meet the desired dynamic performance of the robotic arm, necessitating control methods that can compensate for interference and model errors. Assume that the simplified hyper-redundant robotic arm model is described by the following equation:

$$\begin{array}{cc}& \dot{x}\left(t\right)=Ax\left(t\right)+Bu\left(t\right)+D(\Psi \left(x\right)+c_{0}u\left(t\right)+\phi \left(t\right))\hfill \\ & y\left(t\right)=Lx\left(t\right)\hfill \end{array}$$

(1)

$$\mathrm{z}\left(t\right)=y\left(t\right)+\xi \left(t\right)$$

(2)

where $x\left(t\right)={[x_1\left(t\right),x_2\left(t\right),\dots ,x_n\left(t\right)]}^T$ is the state vector, with n denoting the number of elements in the state vector; $u\left(t\right)\in R^l$ is the control signal; $y\left(t\right)\in R^m$ is the unmeasurable target output signal($m\ge 2$); R is the set of real numbers; $\mathrm{z}\left(t\right)\in R^m$ is the measurable signal; and $\xi \left(t\right)={[{\xi }_1\left(t\right),\dots ,{\xi }_m\left(t\right)]}^T$ is the bounded measurement noise. The unknown functions $\Psi \left(x\right)\in R^l$ and $\phi \left(x\right)\in R^l$ satisfy the following conditions: $\phi \left(t\right)$ is a bounded function, $|\dot{\phi }\left(t\right)|\le {\chi }_1,|\Psi \left(x\right)|\le$${\alpha }_1\left|x\right|\mathrm{and}∥{\displaystyle \frac{\partial \psi \left(x\right)}{\partial x}}∥\le {\alpha }_2$, ${\chi }_1\ge 0$, ${\alpha }_1>0, \mathrm{and}{\alpha }_2>0$. The matrices $A\in R^{n\times n},B\in$$R^{n\times l},D\in R^{n\times l},\mathrm{and}L\in R^{m\times n},\mathrm{and}A$ is Hurwitz matrix, ${\left(LD\right)}^{+}\left(LD\right)=I_l$, ${\left(LD\right)}^{+}\left(LB\right)=k{I}_l,k\in R$. The unknown coefficient $c_0$ belongs to a known interval, that is, $c_0\in [c_{min},c_{max}]\mathrm{and}{c}_{min}+k>0$.

In order to compensate for independent disturbances and measurement noise simultaneously, at least two independent measurements of the target output signal are required. The estimated signal $\tilde{\xi }={[{\xi }_1,\dots ,{\xi }_{i-1},{\xi }_{i+1},\dots ,{\xi }_m]}^T$ is used to represent the part of the noise $\xi$ that does not contain the i-th component. Assume that the i-th component of the vector $\xi \left(t\right)$ exists and satisfies the following relationship:

$$\underset{t\to \infty }{lim}\underset{t\ge 0}{sup}{\xi }_i\left(t\right)<\underset{t\to \infty }{lim}\underset{t\ge 0}{sup}{\xi }_g\left(t\right)$$

(3)

where $i\in \{1,\dots ,m\}\mathrm{and}g\in \{1,\dots ,i-1,i+1,\dots ,m\}$. According to (3), the estimation accuracy of the vector $\tilde{\xi }$ depends on the value of the i-th component in vector $\xi$. With the estimated value of the vector $\tilde{\xi }$, we can construct the true output signal of the control target and obtain an estimate of disturbance, which can then be compensated. Additionally, it is assumed that ${\dot{\xi }}_i\left(t\right)$ is a bounded signal and is represented by $|{\xi }_i\left(t\right)|\le {\chi }_2\mathrm{and}\left|{\dot{\xi }}_i\left(t\right)\right|\le {\chi }_3$, where ${\chi }_2\mathrm{and}{\chi }_3$ are positive constants. A control algorithm needs to be developed to ensure that the target conditions are satisfied:

$$\underset{t\to \infty }{lim}\underset{t\ge 0}{sup}\left|y\left(t\right)\right|\le \delta$$

(4)

Among them, $\delta >0$.

Therefore, the proposed problem will be solved in two stages, and its control algorithm is shown in Figure 2. In the first stage, an estimation algorithm for $\tilde{\xi }$ will be designed. In the second stage, $\tilde{\xi }$ will be used to generate an estimate of the output signal $\widehat{y}$, which will then be used to design an algorithm for compensating parameters and external disturbances.

3. Measurement Noise Compensation Algorithm

Assume that $\tilde{E}=[E_1,E_2,\dots ,E_{i-1},E_{i+1},\dots ,E_m]$. According to the second equation of Equation (1), we rewrite Equation (2) in the following form:

$$\mathrm{z}\left(t\right)=Lx\left(t\right)+\tilde{\mathrm{E}}\tilde{\xi }\left(t\right)+{\mathrm{E}}_i{\xi }_i\left(t\right)$$

(5)

Eliminate the i-th equation in Equation (5). To do this, multiply Equation (5) by the matrix $\tilde{I}={\tilde{E}}^T$ on the left to obtain the following:

$$\tilde{\mathrm{z}}\left(t\right)=\tilde{\mathrm{I}}Lx\left(t\right)+\tilde{\xi }\left(t\right)$$

(6)

where $\tilde{\mathrm{z}}\left(t\right)=\tilde{\mathrm{I}}\mathrm{z}\left(t\right)$. Therefore, the model can be expressed as follows:

$$f(x,u,t)=\psi \left(x\right)+c_{0}u\left(t\right)+\phi \left(t\right)$$

(7)

After differentiating Equation (6) along the trajectory of system (1), we obtain the following:

$$\dot{\tilde{\mathrm{z}}}\left(t\right)=\tilde{I}LAx\left(t\right)+\tilde{I}LBu\left(t\right)+\tilde{I}LDf(x,u,t)+\dot{\tilde{\xi }}\left(t\right)$$

(8)

From the problem statement, we know that the signal $x\left(t\right)$ cannot be measured. Therefore, according to the structure of expression (5), a new variable $\tilde{x}\left(t\right)$ is introduced, which takes the form:

$$\tilde{x}\left(t\right)=L^{+}[\mathrm{z}\left(t\right)-\tilde{E}\tilde{\xi }\left(t\right)-E_i{\xi }_i\left(t\right)]$$

(9)

Considering Equation (9), we rewrite Equation (8) as follows:

$$\begin{array}{cc}\hfill \dot{\tilde{\mathrm{z}}}\left(t\right)& =\tilde{I}L\mathrm{A}{L}^{+}\left[\mathrm{z}\left(t\right)-\tilde{\mathrm{E}}\tilde{\xi }\left(t\right)-{\mathrm{E}}_i{\xi }_i\left(t\right)\right]+\tilde{I}L\mathrm{B}u\left(t\right)+\tilde{I}LDf(x,u,t)\hfill \\ & +\tilde{I}L\mathrm{A}\left(x\left(t\right)-\tilde{x}\left(t\right)\right)+\dot{\tilde{\xi }}\left(t\right)\hfill \end{array}$$

(10)

Considering Equations (5) and (9), we transform the penultimate term in Equation (10) into $\tilde{I}LA\left(x\left(t\right)-\tilde{x}\left(t\right)\right)=\tilde{I}LA(I_n-L^{+}L)x$. Assume $\tilde{A}=\tilde{I}LA{L}^{+}\tilde{E}$, ${\tilde{A}}_1=\tilde{I}LA{L}^{+}$, ${\tilde{A}}_2=\tilde{I}LA(I_n-L^{+}L)$, ${\tilde{A}}_3=\tilde{I}LA{L}^{+}{E}_i$, $\tilde{B}=\tilde{I}LB$, and $\tilde{D}=\tilde{I}LD$. Further, Equation (10) is rewritten as follows:

$$\dot{\tilde{\xi }}\left(t\right)=\tilde{A}\tilde{\xi }\left(t\right)-{\tilde{A}}_1\mathrm{z}\left(t\right)+\dot{\tilde{\mathrm{z}}}\left(t\right)-\tilde{B}u\left(t\right)-\tilde{D}f(x,u,t)-{\tilde{A}}_{2}x\left(t\right)+{\tilde{A}}_3{\xi }_i\left(t\right)$$

(11)

Integrating Equation (11) with respect to t, we obtain the following:

$$\begin{array}{cc}\hfill \tilde{\xi }\left(t\right)& ={\int }_0^t[\tilde{\mathrm{A}}\tilde{\xi }\left(s\right)\left.-{\tilde{\mathrm{A}}}_1\mathrm{z}\left(s\right)\right]ds+\tilde{\mathrm{z}}\left(t\right)\hfill \\ \hfill & -{\int }_0^t\left[\tilde{\mathrm{B}}u\left(s\right)+\tilde{D}f(x,u,s)+{\tilde{\mathrm{A}}}_{2}x\left(s\right)-{\tilde{\mathrm{A}}}_3{\xi }_i\left(s\right)\right]ds+\tilde{\xi }\left(o\right)-\tilde{\mathrm{z}}\left(0\right)\hfill \end{array}$$

(12)

Obviously, Equation (12) cannot be used to estimate the measurement interference because it contains the signal that cannot be measured. However, according to the structure of Equation (12), we introduce an algorithm to estimate the vector $\widehat{\xi }\left(t\right)$, which takes the form:

$$\tilde{\xi }\left(t\right)=\underset{0}{\overset{t}{\int }}\left[\tilde{A}\widehat{\xi }\left(s\right)-{\tilde{A}}_1\mathrm{z}\left(s\right)\right]ds+\tilde{\mathrm{z}}\left(t\right)+\gamma$$

(13)

Among them, $\widehat{\xi }\left(s\right)$ is the signal estimate vector $\tilde{\xi }\left(t\right)$, and $\gamma \in R^{m-1}$ is the vector that needs to be given. In the later experiments, the study on the influence of the value of the vector $\gamma$ on the quality of the transient process in the closed-loop system will be given.

In order to evaluate the performance of Equation (13), the error is considered:

$$e\left(t\right)=\tilde{\xi }\left(t\right)-\widehat{\xi }\left(t\right)$$

(14)

Considering Equations (11) and (13), we differentiate Equation (14) with respect to time and write the result in the following form:

$$\dot{e}\left(t\right)=\tilde{A}e\left(t\right)-\tilde{B}u\left(t\right)-\tilde{D}f(x,u,t)-{\tilde{A}}_{2}x\left(t\right)+{\tilde{A}}_3{\xi }_i\left(t\right)$$

(15)

From Equation (15), it can be seen that the value of the error e depends on the value of ${\xi }_i$, and thus condition (3) must be satisfied a priori. In addition, the value of the error e depends on f and u. By properly choosing the control u, the impact of the disturbance f on the quality of the $\tilde{\xi }$ estimate can be reduced. We will consider the solution to the disturbance compensation problem in the next section.

4. Disturbance Compensation Algorithm

The above algorithm can be used to estimate the disturbance signal part $\xi \left(t\right)$ as the signal $\widehat{\xi }\left(t\right)$. $\widehat{\xi }\left(t\right)$ is used to improve the information contained in the signal $y\left(t\right)$. Let $\widehat{y}\left(t\right)$ be the estimated value of $y\left(t\right)$. Considering Equations (5) and (13), the expression for $\widehat{y}\left(t\right)$ is given in the following form:

$$\widehat{y}\left(t\right)=\mathrm{z}\left(t\right)-\tilde{E}\widehat{\xi }\left(t\right)=Lx\left(t\right)+\tilde{E}e\left(t\right)+E_i{\xi }_i\left(t\right)$$

(16)

Considering Equations (1) and (7), we differentiate Equation (16) with respect to time and express the result as follows:

$$LDf(x,u,t)=\dot{\widehat{y}}\left(t\right)-LAx\left(t\right)-LBu\left(t\right)-\tilde{\mathrm{E}}\dot{e}\left(t\right)-{\mathrm{E}}_i{\dot{\xi }}_i\left(t\right)$$

(17)

According to Reference [21] and using the structure of Equation (17), we introduce the estimated value $\widehat{f}$ of the disturbance f, which is given in the following form:

$$\widehat{f}\left(t\right)={\left(LD\right)}^{+}(\dot{\widehat{y}}\left(t\right)-LA{L}^{+}\widehat{y}\left(t\right)-\alpha \left(p\right)LBv\left(t\right))$$

(18)

Among them, $\alpha \left(p\right)$ is a scalar differential operator, and $v\left(t\right)\in R^l$ is the auxiliary control signal required to generate the disturbance compensation signal.

Assume that the auxiliary control signal is given by the following:

$$v\left(t\right)=-\widehat{f}\left(t\right)$$

(19)

Substituting Equation (18) into Equation (19), we obtain the following:

$$(I_{\iota }-\alpha \left(p\right){\left(LD\right)}^{+}LB)v\left(t\right)=-{\left(LD\right)}^{+}[\dot{\widehat{y}}\left(t\right)-LA{L}^{+}y\left(t\right)]$$

(20)

Assume that $\alpha \left(p\right)={\displaystyle \frac{1-\mu p}{k}}$, where $\mu >0$ is a sufficiently small number, and integrate Equation (20) with respect to t:

$$v\left(t\right)=-{\displaystyle \frac{1}{\mu }}{\left(LD\right)}^{+}\times \left[\widehat{y}\left(t\right)-LA{L}^{+}\underset{0}{\overset{t}{\int }}\widehat{y}\left(s\right)ds-y\left(0\right)\right]+v\left(0\right)$$

(21)

To compensate for the disturbance, the control law can be expressed in the form $u\left(t\right)=v\left(t\right)$. However, the performance of the control law does not depend on the exact values of initial conditions $\widehat{y}\left(0\right)$ and $\nu \left(0\right)$. Therefore, the control law can be rewritten as follows:

$$u\left(t\right)=-{\displaystyle \frac{1}{\mu }}{\left(LD\right)}^{+}\left[\widehat{y}\left(t\right)-L\mathrm{A}{L}^{+}\underset{0}{\overset{t}{\int }}\widehat{y}\left(s\right)ds-\theta \right]$$

(22)

Among them, $\theta \in R^m$ is the vector that needs to be specified. In the subsequent experiments, the influence of the value of the vector $\theta$ on the quality of the transient process in the closed-loop system will be studied. Thus, the control law (22) is obtained, and then the conditions of the algorithm parameters are derived using the solvability form of the linear matrix inequality (LMI).

Before stating the theorems, we introduce the following symbols:

$$\begin{array}{c}{A}_{21}={\displaystyle \frac{1}{\mu }}\left[c_{0}A+kA-B{\left(LD\right)}^{+}LA\left(I_n-L^{+}L\right)-c_{0}D{\left(LD\right)}^{+}LA\left(I_n-L^{+}L\right)\right];\hfill \\ A_{22}={\displaystyle \frac{1}{\mu }}\left(-c_0{I}_n-k{I}_n+\mu A\right);\\ A_{23}={\displaystyle \frac{1}{\mu }}\left[B{\left(LD\right)}^{+}LA{L}^{+}\tilde{E}+c_{0}D{\left(LD\right)}^{+}LA{L}^{+}\tilde{E}\right];\\ A_{24}={\displaystyle \frac{1}{\mu }}\left[-B\left(LD\right)+\tilde{E}-c_{0}D\left(LD\right)+\tilde{E}\right];\\ A_{41}={\displaystyle \frac{1}{\mu }}\left[-c_0{\tilde{A}}_2-k{\tilde{A}}_2+\tilde{B}{\left(LD\right)}^{+}LA\left(I_n-L^{+}L\right)+c_{0}D\left(LD\right)+LA\left(I_n-\right.\right.\\ \left.L^{+}L\right)];\\ A_{42}=-{\tilde{A}}_2;\\ A_{43}={\displaystyle \frac{1}{\mu }}\left[c_0\tilde{A}+k\tilde{A}-\tilde{B}\left(LD\right)+LA{L}^{+}\tilde{E}-c_0\tilde{D}\left(LD\right)+LA{L}^{+}\tilde{E}\right];\\ A_{44}={\displaystyle \frac{1}{\mu }}\left(-c_0{I}_{m-1}-k{I}_{m-1}+\mu \tilde{A}+\tilde{B}\left(LD\right)+\tilde{E}+c_0\tilde{D}{\left(LD\right)}^{+}\tilde{E}\right)\\ G_{21}={\displaystyle \frac{1}{\mu }}(-B+kD);G_{41}={\displaystyle \frac{1}{\mu }}(\tilde{B}-k\tilde{D});\\ F_{21}=D,F_{41}=-\tilde{D};\\ B_{21}={\displaystyle \frac{1}{\mu }}(-B+kD),B_{22}=D;\\ B_{23}={\displaystyle \frac{1}{\mu }}\left[B{\left(LD\right)}^{+}+c_{0}D{\left(LD\right)}^{+}\right]LA{L}^{+}{E}_i;\\ B_{24}={\displaystyle \frac{1}{\mu }}\left[-B{\left(LD\right)}^{+}-c_{0}D{\left(LD\right)}^{+}\right]E_i;\\ B_{41}={\displaystyle \frac{1}{\mu }}(\tilde{B}-k\tilde{D}),B_{42}=-\tilde{D};\\ B_{43}={\displaystyle \frac{1}{\mu }}\left(-\tilde{B}{\left(LD\right)}^{+}LA{L}^{+}{E}_i-c_0\tilde{D}{\left(LD\right)}^{+}LA{L}^{+}{E}_i+c_0{\tilde{A}}_3+k{\tilde{A}}_3\right);\\ B_{44}={\displaystyle \frac{1}{\mu }}\left(c_0\tilde{D}{\left(LD\right)}^{+}{E}_i+\tilde{B}{\left(LD\right)}^{+}{E}_i+\mu {\tilde{A}}_3\right);\\ G_e=\left[\begin{array}{c}{O}_{n\times t}\\ G_{21}\\ O_{(m-1)\times t}\\ G_{41}\end{array}\right];F_e=\left[\begin{array}{c}{O}_{n\times t}\\ F_{21}\\ O_{(m-1)\times 1}\\ F_{41}\end{array}\right];\\ C_1=\left[I_n{O}_{n\times n}{O}_{n\times (m-1)}{O}_{n\times (m-1)}\right];\\ C_2=\left[O_{n\times n}{I}_n{O}_{n\times (m-1)}{O}_{n\times (m-1)}\right];\\ A_e=\left[\begin{array}{cccc}{O}_{n\times n}& I_n& O_{n\times (m-1)}& O_{n\times (m-1)}\\ A_{21}& A_{22}& A_{23}& A_{24}\\ O_{(m-1)\times n}& O_{(m-1)\times n}& O_{(m-1)\times (m-1)}& I_{m-1}\\ A_{41}& A_{42}& A_{43}& A_{44}\end{array}\right];\\ B_e=\left[\begin{array}{cccc}{O}_{n\times 1}& O_{n\times 1}& O_{n\times 1}& O_{n\times 1}\\ B_{21}& B_{22}& B_{23}& B_{24}\\ O_{(m-1)\times 1}& O_{(m-1)\times 1}& O_{(m-1)\times 1}& O_{(m-1)\times 1}\\ B_{41}& B_{42}& B_{43}& B_{44}\end{array}\right];\\ \hfill \Psi =\left[\begin{array}{cccc}{A}_e^{T}P+P{\mathrm{A}}_e+2\beta P+{\tau }_1{\alpha }_1^2{C}_1^T{C}_1+{\tau }_2{\alpha }_2^2{C}_2^T{C}_2& P{G}_e& P{F}_e& P{B}_e\\ \ast & -{\tau }_1{I}_l& O_{l\times l}& O_{l\times 4}\\ \ast & \ast & -{\tau }_2{I}_l& O_{l\times 4}\\ \ast & \ast & \ast & -\rho I_4\end{array}\right].\end{array}$$

(23)

Among them, $\beta >0,\rho >0,{\tau }_1>0,{\tau }_2>0,P>0$ is a positive definite matrix, and “*” represents a symmetric block of a symmetric matrix.

Theorem 1.

Consider a control system consisting of the target models (1) and (2), the disturbance estimation algorithms (13) and (6), and the control laws (22) and (16). For given numbers $\beta >0\mathrm{and}\mu >0$, there exist coefficients ${\tau }_1>0,{\tau }_2>0,\rho >0$, and a matrix $P>0$ such that the linear matrix inequality (LMI) holds:

$$\Psi <0$$

(24)

Then, the algorithm consisting of Equations (6), (13), (16), and (22) ensures that the target condition (4) is met, where

$$\delta =\sqrt{{\displaystyle \frac{\rho {\sum }_{i=1}^3{\chi }_i^2}{2\beta {\lambda }_{min}\left(P\right)}}}$$

(25)

${\lambda }_{\min }\left(P\right)$ is the smallest eigenvalue of the matrix P. The proof of Theorem 1 is provided in Appendix A.

In Law 1, the LMI $\Psi <0$ depends on the unknown parameter $c_0$, which belongs to the known interval $\left[\begin{array}{c}{c}_{min},c_{max}\end{array}\right]$. Therefore, to verify $\Psi <0$, we make the following statement.

Theorem 2.

LMI (24) is satisfied if both of the following LMIs are satisfied:

$${\Psi }^{-}<0\mathrm{and}{\Psi }^{+}<0$$

(26)

among them, ${\Psi }^{-}{=\Psi |}_{A_e=A_e^{-},B_e=B_e^{-}};{\Psi }^{+}{=\Psi |}_{A_e=A_e^{+},B_e=B_e^{+}};A_e^{-}=A_e{{|}_{c_0=c_{min}};A_e^{+}=A_e|}_{c_0=c_{max}}$; $B_e^{-}=B_e{|}_{c_0=c_{min}}$; and $B_e^{+}=B_e{|}_{c_0=c_{max}}$. The proof of Theorem 2 is provided in Appendix B.

5. Experiments

5.1. Simulation Experiment

Consider the control target models (1) and (2), where

$A=\left[\begin{array}{ccc}-3& 1& 0\\ -3& 0& 1\\ -1& 0& 0\end{array}\right],B=\left[\begin{array}{c}0\\ 1\\ 3\end{array}\right],D=\left[\begin{array}{c}0.2\\ 2\\ 1\end{array}\right],L=\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\end{array}\right],c_0\in [-0;4;25]$, $\mathrm{and}{\alpha }_1={\alpha }_2=2\sqrt{3}.$ The remaining parameters in the target models (1) and (2) are defined below. Assume that condition (3) is satisfied when $i=2\mathrm{and}g=1$.

For given model parameters, it is easy to verify whether the following conditions are met: ${\left(LD\right)}^{+}\left(LD\right)=1\mathrm{and}{\left(LD\right)}^{+}\left(LB\right)=k=0.495$.

Let us create a control algorithm. Since $i=2,\tilde{I}=\left[\begin{array}{cc}1& 0\end{array}\right]$, and $\tilde{\mathbf{E}}={\left[\begin{array}{cc}1& 0\end{array}\right]}^T$, and considering $L^{+}={\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\end{array}\right]}^T$, we calculate $\tilde{A}=-3\mathrm{and}{\tilde{A}}_1=\left[\begin{array}{cc}-3& 1\end{array}\right]$. Assuming $\gamma =0$, the disturbance estimation algorithm (12) is formulated as follows:

$$\widehat{\xi }\left(t\right)=-{\int }_0^t\left(\widehat{\xi }\left(s\right)+[-31]\mathrm{z}\left(s\right)\right)ds+\tilde{\mathrm{z}}\left(t\right)$$

(27)

where $\tilde{z}\left(t\right)=z_1\left(t\right)$. Calculating ${\left(LD\right)}^{+}=\left[\begin{array}{cc}0.0495& 0.495\end{array}\right]$, $L\mathrm{A}{L}^{+}=\left[\begin{array}{cc}-3& 1\\ -3& 0\end{array}\right]$, and assuming $\theta ={\left[00\right]}^T$, the control law (23) can be written as follows:

$$u\left(t\right)=-{\displaystyle \frac{1}{\mu }}\left[0.04950.495\right]\times \left[\widehat{y}\left(t\right)-\left[\begin{array}{cc}-3& 1\\ -3& 0\end{array}\right]\underset{0}{\overset{t}{\int }}\widehat{y}\left(s\right)ds\right]$$

(28)

According to Equation (16), $\widehat{y}\left(t\right)=\mathrm{z}\left(t\right)-{\left[10\right]}^T\widehat{\xi }\left(t\right)$.

The LMI specified by Equation (26) satisfies $\mu \in [2\times {10}^{-5};0.0057]$. Simulation tests of $\psi \left(x\right)={\sum }_{i=1}^3\left(x_i\left(t\right)+sin{x}_i\left(t\right)\right)$ in MATLAB (R2022a) Simulink show that the solution (1) is restricted to $\mu \in [0;0.02]$.

We will demonstrate the performance of the control system. Let (1) and (2) be such that $x\left(0\right)={\left[111\right]}^T$, $\psi \left(x\right)={\sum }_{i=1}^3(x_i\left(t\right)+sin{x}_i\left(t\right))$, and

$$\begin{array}{c}\phi \left(t\right)=0.2+0.5sin0.7t+cos1.3t\\ {\xi }_1\left(t\right)=1+10sin3t;{\xi }_2\left(t\right)=0.01sin0.8t\end{array}$$

(29)

First, the dependency of $\delta (\mu ,c_0)$ is expressed according to Equation (26). Then, in MATLAB Simulink, the dependency $\delta (\mu ,c_0)$ is obtained after modeling the closed-loop system according to algorithms (27) and (28), assuming $\mu =0.005$, and setting $c_0=0$ in Equation (1). The results of transient process of $y\left(t\right)={\left[y_1\left(t\right)y_2\left(t\right)\right]}^T$ when $u\left(t\right)=0$ are shown in Figure 3, and the transient results when using Equations (27) and (28) are shown in Figure 4.

The control algorithm proposed above is only applicable in cases where the noise and disturbance are differentiable. Therefore, it is necessary to consider the case where the measurement noise and disturbance contain non-differentiable components in order to conduct a numerical study of the algorithm. To this end, we first consider the case where there are random components in the disturbance and noise, which take the following form:

$$\begin{array}{c}\phi \left(\mathrm{t}\right)=0.2+0.5sin0.7\mathrm{t}+cos1.3\mathrm{t}+{\mathrm{d}}_1\left(\mathrm{t}\right);\\ {\xi }_1\left(\mathrm{t}\right)=1+10sin3\mathrm{t}+{\mathrm{d}}_2\left(\mathrm{t}\right);{\xi }_2\left(\mathrm{t}\right)=0.01sin0.8\mathrm{t}+{\mathrm{d}}_3\left(\mathrm{t}\right)\end{array}$$

(30)

The signals ${\mathrm{d}}_1\left(\mathrm{t}\right),{\mathrm{d}}_2\left(\mathrm{t}\right),\mathrm{and}{\mathrm{d}}_3\left(\mathrm{t}\right)$ are generated using the white noise generator in MATLAB Simulink with the following parameters: noise power is $3,10,\mathrm{and}{10}^{-4}$, and sampling times are 0.05 s, 0.1 s, and 0.2 s, respectively. Figure 5 shows the results of the transient process of $y\left(t\right)$ when $c_0=10$.

Evaluate the performance of the control system under the following conditions:

$$\begin{array}{c}\phi \left(t\right)=0.2+0.5sin0.7t+cos1.3t+d_1\left(t\right);\\ {\mathrm{z}}_1=q_1\left(x_1\right);{\mathrm{z}}_2=q_2\left(x_2\right)\end{array}$$

(31)

where $q_1\mathrm{and}{q}_2$ are level quantization functions, with the quantization intervals being 0.5 and 0.05, respectively. Figure 6 presents the modeling results of $y\left(t\right)$ when $c_0=10$.

The simulation results show that the limit value obtained by calculating the parameter $\mu$ using Theorem 2 is not significantly different from the limit value obtained by MATLAB Simulink modeling, which verifies the robustness of the control system to disturbances and random components in noise signals. As can be seen from Figure 3, Figure 4, Figure 5 and Figure 6, after 4 s of simulation, the $\delta$ values in the three cases do not exceed 0.3. It is worth noting that the value of $\delta$ in the target condition (4) does not depend on the choice of the vector $\gamma$ in Equation (13) and the vector $\theta$ in Equation (22). However, the performance of the transient process of $y\left(t\right)$ is influenced by the values of $\gamma$ and $\theta$. Therefore, in the simulation process, optimal transient results can be obtained when the initial conditions $\tilde{\xi }$ and $\widehat{\xi }$ and $\mu$ and v match.

Additionally, considering the impact of consistent disturbances and the performance of algorithms (26) and (27) when ${\xi }_2\left(t\right)=0$ in models (1) and (2). In this case, the value of $\delta$ in the target condition (4) decreases almost proportionally with the decrease of $\mu$, as shown in Figure 7, Figure 8 and Figure 9.

5.2. Physical Experiment

To achieve precise control and monitoring of the continuum robotic arm, we designed an efficient hardware system, with its main components and workflow illustrated in Figure 10.

The system uses a Raspberry Pi 4B+ as the core controller, communicating with the stepper motor through a USB-to-CAN analyzer. The stepper motor is powered by a 24 V switching power supply and connected to a ball screw slide via a coupling to enable rotational motion. The rotation of the lead screw drives a wire rope, which actuates the movement of the continuum robotic arm.To monitor the arm’s status, an absolute encoder is mounted on the lead screw slide. The encoder collects position data in real time and transmits it to a PC for analysis, allowing the calculation of the rotation angle. This configuration ensures high-precision motion control and data monitoring, providing strong support for optimizing the robotic arm’s performance.

In the system hardware, the motor model is simplified as follows:

$$\dot{x}=Ax+B(u+f)$$

(32)

Here, $x={[x_1,x_2,x_3]}^T$, where $x_1=\vartheta -{\vartheta }^{\ast }$, $x_2=\omega -{\omega }^{\ast }$ and $x_3=E_q^{\prime }-E_q^{\prime }$. The function f depends on parameter uncertainties, transmission line resistance, and residual disturbances. Assume $A=\left[\begin{array}{ccc}0& 1& 0\\ -0.9& -1.625& -0.5\\ -0.5& 0& -5.1635\end{array}\right],B=\left[\begin{array}{c}0\\ 0\\ 0.52\end{array}\right]$, and

$${\mathrm{z}}_1=\vartheta +{\xi }_1,{\mathrm{z}}_2=\omega +{\xi }_2,{\mathrm{z}}_3=E_q^{\prime }+{\xi }_3$$

(33)

Here, ${\xi }_1$, ${\xi }_2$, and ${\xi }_3$ are measurement noise. In interference conditions, it can be assumed that the measurement noise levels for the load angle and relative velocity are relatively high, while the measurement noise level for $E_q^{\prime }$ is relatively low.

In the physical experiment, it is assumed that the desired rotation angle of a single motor on the robot arm is ${\vartheta }^{\ast }=\pi /3$ rad, the rotation angular velocity is ${\omega }^{\ast }=0$ rad/s, and the transient electromotive force of the motor is $E_q^{{\prime }^{\ast }}=0$. This paper compares the proposed algorithm with the one presented in Reference [22] for a single motor. For comparison with Reference [22], we simplified the motor rotation model during the motor regulation process as follows: $\dot{\vartheta }=\omega$, $\dot{\omega }=-0.5H^{-1}D\omega -0.5H^{-1}{\omega }_0\Delta P_e$, $\Delta P_e\left(t\right)=P_e\left(t\right)-P_m\left(t\right)$. This allows the comparison model to be more closely aligned.

Assuming the motor operates until $t=10$ s, the parameter settings are as follows:

$$\begin{array}{c}f=1+0.2sint+0.1sin0.3t+d_1,\\ {\xi }_1=0.1+0.1sin0.8t+d_2,\\ {\xi }_2=0.2+0.1sin1.1t+d_3,\\ {\xi }_3=0.02+0.01sin0.7t+d_4.\end{array}$$

(34)

When the motor rotates to $t=10$ s, a disturbance source is artificially introduced. A clamp is manually used to grip the wire of the robotic arm, preventing the motor from functioning properly. It is assumed that the parameters change as follows:

$$\begin{array}{c}f=10(1+0.2sint+0.1sin0.3t+d_1),\\ {\xi }_1=1+0.9sin0.9t+10d_2,\\ {\xi }_2=3+2sin1.1t+10d_3,\\ {\xi }_3=0.03+0.02sin0.8t+d_4.\end{array}$$

(35)

At $t=20$ s, the artificial disturbance is removed.

According to Reference [22], the control law is given by $u=\left[11.52.1\right]x$. Figure 10 and Figure 11 show the results of the transient process of the proposed control system and the algorithm in Reference [22]. As shown in Figure 11 and Figure 12, the proposed control algorithm effectively compensates for disturbances and measurement noise after 6 s, achieving an accuracy of 0.05. The algorithm in Reference [22] does not provide simultaneous compensation for interference and measurement noise.

6. Conclusions

In this paper, we propose an anti-disturbance control algorithm designed for a continuum manipulator to compensate for model parameter uncertainty, external disturbances, and measurement noise. A simplified dynamic model is established to describe a nonlinear model with external disturbance, noise, and system uncertainty, and a control algorithm that can simultaneously compensate for independent disturbances and measurement noise is constructed. Based on the described model, disturbances are estimated using a measurement noise algorithm, and the disturbance state is incorporated into the control structure to obtain the control rate. The parameters are derived in the form of the solvability of Linear Matrix Inequalities (LMIs). By solving the LMIs, we obtain the dependency $\delta (\mu ,c_0)$ from the closed-loop system modeling. The performance results of the proposed control algorithm, in the presence of both differentiable and non-differentiable measurement noise and disturbances, demonstrate the robustness of the control system against random components in disturbance and noise signals. Furthermore, when considering consistent disturbances, the results indicate that the value of $\delta$ decreases almost proportionally with the reduction of $\mu$. Finally, the control algorithm is implemented on a single motor of the manipulator for verification. Experimental results show that the control algorithm exhibits strong anti-disturbance and noise compensation capabilities, confirming the effectiveness of the control framework.