Novel sliding mode control of the manipulator based on a nonlinear disturbance observer

Abstract

To achieve high-performance trajectory tracking for a manipulator, this study proposes a novel sliding mode control strategy incorporating a nonlinear disturbance observer. The observer is designed to estimate unknown models in real-time, enabling feedforward compensation for various uncertainties such as modeling errors, joint friction, and external torque disturbances. The control law is formulated by integrating the Backstepping method, Lyapunov theory, and global fast terminal sliding mode theory, ensuring global convergence to zero within finite time and enhancing system robustness. To address the inherent chattering issue in sliding mode control, a hybrid reaching law is developed by combining the exponential and power reaching laws. Additionally, the improved-fal (Imp-fal) function replaces the sign function in the switching control law, improving system response speed, preventing overshoot, and optimizing gain beyond the threshold value. Through simulation and comparative experiments conducted using MATLAB/Simulink, the controller model exhibited a 16.4% average reduction in the mean square value of tracking errors compared to existing control strategies, with improvements observed in various performance indicators. When applied to a self-developed three-degree-of-freedom manipulator experimental platform, the controller demonstrated a roughly 55% increase in tracking accuracy and a decrease in response time by approximately 45% compared to existing strategies. The experimental results validate the effectiveness, superiority, and practicality of the control strategy, providing a feasible solution for high-performance trajectory tracking in robotic arm systems.

Introduction

With the ongoing advancement of scientific and technological innovations, manipulators have gained widespread utilization across numerous sectors and domains, such as industry, the military, aerospace, and medical treatment, and have higher requirements for the stability, accuracy, and rapidity of their trajectory tracking. However, due to the manipulator system’s intricate nature, encompassing multivariable, strong coupling, time-varying, and profound nonlinearity, it poses a complex controlled system. It will be affected by many uncertain factors, such as modeling error, joint friction, and external unknown torque disturbances in the actual control process, so it is difficult to establish an accurate dynamic model, which will cause trouble to realize the trajectory tracking control of the manipulator.

Addressing the challenges encountered in the trajectory tracking control process of manipulators, numerous control methods have been put forward by researchers to enhance its efficiency. These include PID control^1,2, model predictive control (MPC)^3,4, adaptive control^5,6, backstepping control^7,8, fuzzy control⁹, neural network control¹⁰, H∞ control⁴ and sliding mode control^11,12. Amongst these, sliding mode control (SMC) stands out as a discontinuous nonlinear approach that categorizes the system’s state into sliding mode motion and approaching motion. This is achieved by designing a sliding mode surface and dynamically adjusting the controller’s structure based on the system’s current state, ensuring it follows the desired trajectory. SMC’s discontinuous switching and its resilience against changes in system parameters and disturbances contribute significantly to its robust nature, making it a popular choice for motion control in nonlinear systems. However, traditional SMC relies on a linear sliding mode surface. While this ensures that the system state ultimately approaches the desired trajectory, it only guarantees periodic asymptotic convergence to the equilibrium point and fails to eliminate the inherent chattering issue associated with SMC. To address this, a fast-terminal sliding mode controller is introduced in¹³. This controller builds on the strengths of traditional SMC, approximating the inverse dynamic solution of the manipulator and compensating for external, unknown disturbances. It achieves rapid convergence of the manipulator system state within a finite time. Nevertheless, this control algorithm falls short of demonstrating global convergence. In^14,15, a second-order model uncertainty and disturbance estimator (UDE) is crafted to assess the disturbance term, while an adaptive technique is employed to regulate the estimation error. Additionally, a continuous and nonsingular fast-terminal sliding mode control algorithm is introduced, enabling the manipulator to achieve swift convergence and precise trajectory tracking within a limited timeframe. Nevertheless, there is a need to enhance the tracking stability of the controller once it reaches the sliding mode surface. In¹⁶, the manipulator’s intricate system is decomposed into low-order subsystems using the backstepping method, Lyapunov theory, and global fast terminal sliding mode theory. This approach not only guarantees the robustness and global convergence of the system but also boosts its response speed and stability.

While the global fast terminal sliding mode control effectively addresses the issue of the system state asymptotically converging to zero within a finite time, it does not eliminate the chattering problem resulting from high-frequency switching. Currently, the primary methods to tackle this challenge involve designing an enhanced reaching law or substituting the sign (sgn) function in the switching control term with a continuous smooth function. In¹⁷, an improved double power reaching law with variable coefficients is chosen to expedite the convergence speed of the state variable during the reaching motion phase and mitigate the chattering phenomenon in the controller output. In¹⁸, an advanced reaching law sliding mode control strategy leveraging time delay estimation is introduced. The integration of the saturation (sat) function with the fast power-reaching law effectively enhances the system’s dynamic response speed and suppresses the chattering phenomenon. Furthermore, in¹⁹, the nonlinear fal function is introduced during controller design as a replacement for the sgn function, thereby reducing the chattering effect of sliding mode control and enhancing both system stability and tracking accuracy. In²⁰, advancements have been made to enhance the performance of the nonlinear fal function, thereby boosting the system’s response speed and anti-interference capabilities. Additionally, the literature²¹ introduces a novel fuzzy adaptive hyperspiral second-order sliding mode trajectory tracking control method. This innovative approach enables the manipulator to achieve precise trajectory tracking in complex environments and effectively suppresses the system’s chattering phenomenon. In^22,23, a novel hybrid reaching law is designed that combines the power reaching law with the constant speed reaching law. This integration reduces the time and variable-speed delay associated with the system state approaching the sliding mode surface. Furthermore, the tangent (tan) function is employed as a replacement for the sign (sgn) function, effectively mitigating the inherent chattering phenomenon associated with sliding mode control.

Simultaneously, aiming to negate the impact of uncertain factors, including modeling inaccuracies in the manipulator system, joint friction, and external, unpredictable torque disturbances, and to facilitate feedforward compensation²⁴, introduced a novel sliding mode control approach grounded in an extended state observer (ESO). This approach eliminates the need for a precise manipulator model. By enlarging the state space to accommodate unknown disturbances and parameter perturbations, the unmodeled dynamics and coupling components are promptly estimated and compensated. In²⁵, a nonlinear disturbance observer (NDO) is employed to instantly assess the observable segment of the overall disturbance. The observer’s estimated value is then utilized to adjust the controller’s output torque, enabling feedforward compensation.In²⁶, a robust controller is designed by combining sliding mode control with adaptive algorithms, addressing the issue of finite-time convergence. Furthermore, an adaptive mechanism is devised through an auxiliary system with bounded compensation terms, effectively solving the problem of input saturation. However, the research is limited to numerical simulations and lacks real-world experimental validation. In²², two control laws, namely H ∞ and Model Predictive Control (MPC), are proposed to achieve high-performance trajectory tracking for a six-degree-of-freedom robotic arm. However, the drawback is that it results in significant overshoot in controlling the robotic arm, and the proposed method has not been validated on a physical platform.

In conclusion, with the objective of enhancing the dynamic quality of sliding mode and approach motions and addressing challenges such as the system state’s inability to converge to zero within a finite timeframe and the chattering phenomenon observed in the system’s control torque, this paper introduces a novel sliding mode control approach for manipulators that relies on a nonlinear disturbance observer. This method fulfills the control prerequisites for stability, precision, and swiftness in intricate manipulator systems. The primary contributions and innovations of this work are outlined as follows:

• (1) The nonlinear gain matrix form is improved upon, and a Nonlinear Disturbance Observer (NDO) is extended and designed to achieve real-time feedforward compensation for unknown external disturbance torques.

• (2) Utilizing the recursive concept of the Backstepping method and Lyapunov theory, complex robotic arm systems are decomposed into lower-order subsystems. Based on stability principles, virtual control variables are constructed, leading to the proposal of a Backstepping Global Fast Terminal Sliding Mode Control (BGFTSMC) strategy, accompanied by the design of an equivalent control law. Analysis of the sliding surface characteristics validates its global fast convergence, fortifying the system’s robustness against external disturbances and parametric uncertainties.

• (3) A hybrid reaching law is introduced by weighted combination of exponential and power reaching laws. This innovative method ensures effective time convergence during the reaching process. Furthermore, the traditional sign (sgn) function within the switching control law is replaced with an improved-fal (Imp-fal) function. This modification effectively mitigates the chattering issue during the control process and optimizes the gain when the system error exceeds the threshold value.

Dynamics model of the manipulator

• The Lagrange modeling method can be employed to simplify the dynamic model of the spatial 3-DOF manipulator system, resulting in Eq. (1):

$$M\left(q\right)\ddot{q}+C\left(q,\dot{q}\right)\dot{q}+G\left(q\right)=\tau$$

(1)

where $q, \dot{q}, \ddot{q}$ are the angle, angular velocity and angular acceleration of each joint of the manipulator respectively, $M\left(q\right)\in {R}^{3\times 3}$ is the positive definite inertia matrix, $C\left(q,\dot{q}\right)\in {R}^{3\times 3}$ is the Coriolis force and centrifugal force matrix, $G\left(q\right)\in {R}^{3\times 1}$ is the joint gravity vector, and $\tau \in {R}^{3\times 1}$ is the joint driving torque vector.

Considering that the manipulator is a multi-variable, strong coupling, time-varying, and highly nonlinear complex controlled system, there are many uncertain factors in the dynamic model, such as modeling error, joint friction, and external unknown torque disturbance. The actual dynamic model is expressed as Eq. (2):

$$\left({M}_{0}\left(q\right)+\varDelta M\left(q\right)\right)\ddot{q}+\left({C}_{0}\left(q,\dot{q}\right)+\varDelta C\left(q,\dot{q}\right)\right)\dot{q}+\left({G}_{0}\left(q\right)+\varDelta G\left(q\right)\right)+{F}_{d}\left(\dot{q}\right)=\tau +{\tau }_{d}$$

(2)

where ${M}_{0}=\hslash M, {C}_{0}=\hslash C, {G}_{0}=\hslash G$ are the nominal value, $\hslash$ is the modeling coefficient, $\varDelta M, \varDelta C, \varDelta G$ are the unmodeled and modeling error value, ${F}_{d}\left(\dot{q}\right)\in {R}^{3\times 1}$ is the joint friction torque, and ${\tau }_{d}$ is the external unknown torque disturbance.

If the modeling error, joint friction, and external unknown torque disturbance in the actual dynamic model are combined and defined as the total disturbance, the actual dynamic model can be rewritten as Eq. (3):

$${M}_{0}\left(q\right)\ddot{q}+{C}_{0}\left(q,\dot{q}\right)\dot{q}+{G}_{0}\left(q\right)=\tau +D$$

(3)

where $D={\tau }_{d}-{F}_{d}\left(\dot{q}\right)-\varDelta M\left(q\right)\ddot{q}-\varDelta C\left(q,\dot{q}\right)\dot{q}-\varDelta G\left(q\right)$ is the total disturbance of the system, which is estimated by the nonlinear disturbance observer.

The manipulator system shown in Eq. (3) generally has the following dynamic characteristics:

• (1) ${M}_{0}\left(q\right)$ represents a positive, definite, and symmetric matrix that possesses bounded properties. There are known normal numbers ${c}_{1}$, ${c}_{2}$ and the column vector $\text{x}$. It satisfies: ${c}_{1}{\left|\right|x\left|\right|}^{2}\le {x}^{T}{M}_{0}\left(q\right)x\le {c}_{2}{\left|\right|x\left|\right|}^{2}$.

• (2) Matrix ${\dot{M}}_{0}-2{C}_{0}$ is a skew-symmetric matrix with column vector $\text{x}$, which satisfies ${x}^{T}({\dot{M}}_{0}-2{C}_{0})x=0$.

In order to better control the manipulator system, the state space method in modern control theory is introduced, so that ${x}_{1}=q$, ${x}_{2}=\dot{q}$, and the dynamics equation of the manipulator represented by Eq. (3) is transformed into a state space expression, as shown in Eq. (4).

$$\left\{\begin{array}{c}{x}_{1}=q\\ {\dot{x}}_{1}={x}_{2}=\dot{q}\\ {\dot{x}}_{2}=\ddot{q}={{M}_{0}\left(q\right)}^{-1}(\tau +D-{C}_{0}\left(q,\dot{q}\right)\dot{q}-{G}_{0}\left(q\right))\\ y=q\end{array}\right.$$

(4)

Considering the follow-up control strategy, the following assumptions are made, and the assumptions do not affect the effectiveness of the control:

• (1) The anticipated angle (${q}_{d}$) and angular acceleration (${\dot{q}}_{d}$) of each joint in the manipulator are predetermined, while the actual angle ($q$) and angular acceleration ($\dot{q}$) of each joint can be measured and are continuously constrained within certain limits.

• (2) The unknown total disturbance ($D$) is bounded and continuously differentiable, and its upper bound is $\left|\right|D\left|\right|<\widehat{D}$.

Design and stability analysis of a new sliding mode controller

The control system structure is shown in Fig. 1, which is mainly composed of the following parts: (1) Nonlinear disturbance observer (section “Nonlinear disturbance observer design and convergence analysis”); (2) The equivalent control term designed by the new sliding mode surface (section “Backstepping global fast terminal sliding mode controller design and stability analysis”); (3) The switching control term is improved by hybrid reaching law and Imp-fal function (section “Design of hybrid reaching law and Imp-fal function”).

Fig. 1

Control system structure.

Nonlinear disturbance observer design and convergence analysis

• A nonlinear disturbance observer (NDO) is proposed in reference²⁴, which effectively achieves feedforward compensation for torque. In this paper, by improving and extending its nonlinear gain matrix form, a novel NDO is designed, enabling real-time compensation for uncertain disturbances. It can be seen from Eq. (3) that the total disturbance is combined by modeling error, joint friction, and an external unknown torque disturbance. Using the above basic idea, the nonlinear disturbance observer is designed as Eq. (5):

$$\dot{\widehat{D}}={D}_{NL}\left(q,\dot{q}\right)\left[{M}_{0}\left(q\right)\ddot{q}+{C}_{0}\left(q,\dot{q}\right)\dot{q}+{G}_{0}\left(q\right)-\tau -\widehat{D}\right]$$

(5)

where $\widehat{D}$ is the estimated value of the nonlinear disturbance observer, ${D}_{NL}\left(q,\dot{q}\right)$ is the nonlinear disturbance gain matrix.

According to Eq. (5), considering that it is difficult to obtain the angular acceleration signal of each joint in practical application, the angular acceleration signal obtained by differentiating the angular velocity signal will introduce noise to increase the instability of the system. Therefore, it is necessary to further design the nonlinear disturbance observer.

Step 1.

An auxiliary function is constructed to define the internal state variables of the observer, as shown in Eq. (6).

$$Z=\widehat{D}-p\left(q,\dot{q}\right)$$

(6)

where $Z$ is the internal state variable of the observer, $p\left(q,\dot{q}\right)$ is the nonlinear function vector to be designed.

In order to avoid the introduction of an angular acceleration signal, the nonlinear disturbance gain matrix has the following relationship with the nonlinear function, as shown in Eq. (7):

$${D}_{NL}\left(q,\dot{q}\right){M}_{0}\left(q\right)\ddot{q}=\frac{dp\left(q,\dot{q}\right)}{{d}_{t}}$$

(7)

Step 2.

The state variables are determined by deriving from both sides of Eq. (6) and combining with Eqs. (5) and (7). The nonlinear disturbance observer without angular acceleration can be designed as Eq. (8):

$$\dot{Z}=-{D}_{NL}\left(q,\dot{q}\right)Z+{D}_{NL}\left(q,\dot{q}\right)\left[{C}_{0}\left(q,\dot{q}\right)\dot{q}+{G}_{0}\left(q\right)-\tau -p\left(q,\dot{q}\right)\right]$$

(8)

Step 3.

The observation error of a nonlinear disturbance observer is defined by Eq. (9):

$$\xi =D-\widehat{D}$$

(9)

The dynamic equation of observation error can be obtained by deriving from both sides of Eq. (9) and combining with Eqs. (5)–(8), as shown in Eq. (10).

$$\dot{\xi }=\dot{D}-\dot{\widehat{D}}=\dot{D}-{D}_{NL}\left(q,\dot{q}\right)D+{D}_{NL}\left(q,\dot{q}\right)\widehat{D}$$

(10)

Since the total disturbance is unknown, assuming that the characteristic change speed of $D$ relative to the nonlinear disturbance observer is slow, i.e.$\dot{D}=0$, the dynamic equation of observation error is sorted out as Eq. (11):

$$\dot{{\upxi }}+{D}_{NL}\left(q,\dot{q}\right){\upxi }=0$$

(11)

Step 4.

In order to prove that the observation error converges asymptotically, the Lyapunov function is defined as shown in Eqs. (12) and (13).

$${V}_{ndo}=\frac{1}{2}{{\upxi }}^{\text{T}}{\upxi }$$

(12)

$$\dot{{V}_{ndo}}={{\upxi }}^{\text{T}}\dot{{\upxi }}=-{{\upxi }}^{\text{T}}{D}_{NL}\left(q,\dot{q}\right){\upxi }$$

(13)

Analyze Eq. (13): when $\xi =0$, ${V}_{ndo}=0$; When $\xi \ne 0$, because ${D}_{NL}\left(q,\dot{q}\right)$ is a positive definite matrix, $\dot{{V}_{ndo}}\le 0$, satisfies the second method of Lyapunov theory, and the observation error converges asymptotically.

In essence, once the nonlinear disturbance observer compensates for the disturbance, the overall disturbance within the manipulator system can be effectively substituted by the observation error. Consequently, the dynamics model of the manipulator can be expressed as Eq. (14):

$${M}_{0}\left(q\right)\ddot{q}+{C}_{0}\left(q,\dot{q}\right)\dot{q}+{G}_{0}\left(q\right)=\tau +{\upxi }$$

(14)

Backstepping global fast terminal sliding mode controller design and stability analysis

Define the expected values of each joint angle, angular velocity and angular acceleration of the manipulator as ${q}_{d}$, ${\dot{q}}_{d}$, ${\ddot{q}}_{d}$, then the tracking error of the controller can be expressed as Eq. (15):

$$\left\{\begin{array}{c}e={q}_{d}-q\\ \dot{e}={\dot{q}}_{d}-\dot{q}\end{array}\right.$$

(15)

Step 1.

The complex system of the manipulator is disassembled into low-order subsystems by using the recursive idea of the Backstepping method and Lyapunov theory^32,33,34. Concurrently, to uphold the stability of the control system, the Lyapunov function is defined as outlined in Eqs. (16) and (17):

$${V}_{1}=\frac{1}{2}{e}^{T}e$$

(16)

$${\dot{V}}_{1}={e}^{T}\dot{e}=e\left({\dot{q}}_{d}-\dot{q}\right)$$

(17)

In order to make the first subsystem converge to ${q}_{d}$, a virtual control variable ${d}_{var}$ is constructed, whose expression is Eq. (18):

$${d}_{var}=\rho \left(\dot{q}\right)={\dot{q}}_{d}+ke-s$$

(18)

where $k$ is a constant greater than 0, $s$ is the sliding mode surface designed in this paper.

Combining Eqs. (15) and (18), the tracking error can be simplified as Eq. (19):

$$\dot{e}=-ke+s$$

(19)

Therefore, the stability criterion of the first subsystem is Eq. (20):

$${\dot{V}}_{1}={e}^{T}\dot{e}=-k{e}^{T}e+se$$

(20)

Analyze Eq. (20): when $s\to 0$, ${\dot{V}}_{1}=-k{e}^{T}e\le 0$. According to the theorem, the tracking error of the closed-loop control system of the manipulator meets the requirements of stability.

Step 2.

In order to solve the problem that the system state fails to converge to 0 within the effective time, this paper proposes a global fast terminal sliding mode (GFTSM)^27,28, as shown in Eq. (21).

$$s=\dot{e}+\alpha e+\beta {\left|e\right|}^{p/o}sgn\left(e\right)$$

(21)

where $\alpha$, $\beta$ are constants greater than 0, $p/o>1$, and $sgn\left(e\right)$ is a symbolic function.

In order to make the second subsystem converge to 0, the convergence of the proposed sliding mode surface is analyzed. Let ${t}_{1}$ be the time from the initial state to the equilibrium state, and there is a constant $\theta$ greater than 0, so that ${t}_{1}\le \frac{s}{\theta }$ holds. At this time, the system state can reach the sliding surface at ${t}_{1}$; Let ${t}_{2}$ be the time from $e\left({t}_{1}\right)$ to $e\left({t}_{1}+{t}_{2}\right)=0$ of the system, and set appropriate parameter values to make ${t}_{2}=W(\alpha ,\beta ,p,o)$ known. At this time, the system state can converge to the equilibrium point in a finite time, ${t}_{2}$.

By deriving both sides of Eq. (21) at the same time, Eq. (30) can be obtained:

$$\dot{s}=\ddot{e}+\alpha \dot{e}+\beta \frac{p}{o}{\left|e\right|}^{p/o-1}\dot{e}$$

(22)

Step 3.

In order to ensure the stability of the closed-loop system and keep the system state always on the sliding mode surface, the Lyapunov function is defined as follows, as shown in Eq. (23).

$${V}_{2}={V}_{1}+\frac{1}{2}{s}^{T}s$$

(23)

Therefore, the stability criterion of the second subsystem is Eq. (24):

$${\dot{V}}_{2}={\dot{V}}_{1}+{s}^{T}\dot{s}=-k{e}^{T}e+se+s\dot{s}$$

(24)

Analyze Eq. (24): when $s\to 0$, ${\dot{V}}_{2}\le 0$. According to the theorem, the tracking error of the closed-loop control system of the manipulator meets the requirements of stability.

To sum up, the equivalent control term ${\tau }_{eq}$ of the controller is designed according to Eqs. (4), (19), and (24), as shown in Eq. (25).

$${\tau }_{eq}={M}_{0}\left(q\right)\left(\ddot{{q}_{d}}+{\upalpha }\left(s-ke\right)+{\upbeta }\frac{\text{p}}{\text{o}}{\left|\text{e}\right|}^{\text{p}/\text{o}-1}\left(s-ke\right)+\text{e}\right)-{\upxi }+{C}_{0}\left(q,\dot{q}\right)\dot{q}+{G}_{0}\left(q\right)$$

(25)

Design of hybrid reaching law and imp-fal function

In this paper, an effective hybrid reaching law is proposed, which combines the exponential reaching law with the power reaching law in a weighted way^30,31, and its expression is shown in Eq. (26).

$$\dot{s}={\mu }_{1}\left(-Rs-{k}_{1}sgn\left(s\right)\right)+{\mu }_{2}\left(-{k}_{2}{\left|s\right|}^{z}sgn\left(s\right)\right)$$

(26)

where ${\mu }_{1}$, ${\mu }_{2}$ are weight coefficients, which can be adjusted according to the control requirements of the system, ${k}_{1}$, ${k}_{2}$ are positive diagonal matrices, and $z$ is power value.

The controller switching control item ${\tau }_{sw}$ is designed according to Eq. (26), as shown in Eq. (27).

$${\tau }_{sw}={M}_{0}\left(q\right)\left[{\mu }_{1}\left(Rs+{k}_{1}sgn\left(s\right)\right)+{\mu }_{2}\left({k}_{2}{\left|s\right|}^{z}sgn\left(s\right)\right)\right]$$

(27)

In order to solve the chattering problem of the system output control torque, the sign (sgn) function in the switching control term is replaced. Therefore, further improvements are made on the basis of the fal function³⁹, so the Imp-fal function is Eq. (28).

$$Imp-fal\left(s,b,c,d\right)=\left\{\begin{array}{ll}\frac{s}{{c}^{1-b}}, &\quad \left|s\right|\le\:b\\\:{\left|s\right|}^{b}sgn\left(s\right), &\quad c<\left|s\right|<d\\\:{f}_{1}d+sgn\left(s\right)\left({d}^{b}+{d}^{1-{f}_{2}d}-{d}^{1-{f}_{2}\left|s\right|}\right), &\quad \left|s\right|\ge\:d\end{array}\right.$$

(28)

where $b, c$, and $d$ are constant greater than 0, ${f}_{1}$ is the gain of linear feedback, and ${f}_{2}$ is the gain of exponential error.

The Imp-fal function not only retains the characteristics of the fal function but also optimizes its gain when the error is large. The value of $b$ determines the linearity of the function. The smaller the value of $b$, the stronger the nonlinearity of the function; The value of $c$ determines the linear interval of the function. The larger the value of $c$, the larger the linear interval of the function; The value of $d$ determines the valve range from linear to nonlinear and can be properly controlled according to the gain selected by the controller. At the same time, when the error exceeds the valve range value, the anti-interference ability is also strengthened.

Further improve the switching control item ${{\tau }_{sw}}^{\ast }$ according to Eqs. (27) and (28), as shown in Eq. (29).

$${{\tau }_{sw}}^{\ast }={M}_{0}\left(q\right)\left[{\mu }_{1}\left(Rs+{k}_{1}(Imp-fal)\right)+{\mu }_{2}\left({k}_{2}{\left|s\right|}^{z}(Imp-fal)\right)\right]$$

(29)

To sum up, according to the equivalent control design idea of the manipulator, the output control torque $\tau$ of the manipulator is composed of equivalent control ${\tau }_{eq}$ and switching control ${{\tau }_{sw}}^{\ast }$[35,36,37,38], which is shown in Eq. (30).

$${\uptau }={\tau }_{eq}+{{\tau }_{sw}}^{\ast }$$

(30)

Numerical simulation and physical verification

Numerical simulation

In order to verify the effectiveness of the control strategy proposed in this paper, the 3-DOF manipulator model derived in section “Dynamics model of the manipulator” is taken as the controlled object, and the controller is simulated.

The specific forms of each matrix in the dynamics equation of the manipulator are as follows:

$$q=\left[{q}_{1};{q}_{2};{q}_{3}\right]$$

$$\tau =\left[{\tau }_{1};{\tau }_{2};{\tau }_{3}\right]$$

$${M}_{0}\left(q\right)=\left[{m}_{11} {m}_{12} {m}_{13};{m}_{21} {m}_{22} {m}_{23};{m}_{31} {m}_{32} {m}_{33}\right]$$

$${C}_{0}\left(q,\dot{q}\right)=\left[{c}_{11} {c}_{12} {c}_{13};{c}_{21} {c}_{22} {c}_{23};{c}_{31} {c}_{32} {c}_{33}\right]$$

$${G}_{0}\left(q\right)=\left[{g}_{1};{g}_{2};{g}_{3}\right]$$

$$D=\left[{D}_{1};{D}_{2};{D}_{3}\right]$$

of which:

$${m}_{11}={I}_{1}+({m}_{2}{{h}_{2}}^{2}+{m}_{3}{{l}_{2}}^{2}){cos}^{2}\left({q}_{2}\right)+{m}_{3}{{h}_{3}}^{2}{cos}^{2}({q}_{2}+{q}_{3})+2{m}_{3}{h}_{3}{l}_{2}cos\left({q}_{2}\right)cos({q}_{2}+{q}_{3})$$

$${m}_{12}={m}_{21}={m}_{13}={m}_{31}=0; \quad {m}_{33}={I}_{3}+{m}_{3}{{h}_{3}}^{2}$$

$${m}_{22}={I}_{2}+{m}_{2}{{h}_{2}}^{2}+{m}_{3}{{l}_{2}}^{2}+{m}_{3}{{h}_{3}}^{2}+2{m}_{3}{h}_{3}{l}_{2}cos\left({q}_{3}\right)$$

$${m}_{23}={m}_{32}={m}_{3}{{h}_{3}}^{2}+{m}_{3}{h}_{3}{l}_{2}cos\left({q}_{3}\right)$$

$${c}_{11}=-\frac{1}{2}({m}_{2}{{h}_{2}}^{2}+{m}_{3}{{l}_{2}}^{2})\dot{{q}_{1}}sin\left(2{q}_{2}\right)-\frac{1}{2}{m}_{3}{{h}_{3}}^{2}(\dot{{q}_{2}}+\dot{{q}_{3}})sin(2{q}_{2}+2{q}_{3})$$

$$-{m}_{3}{h}_{3}{l}_{2}\dot{{q}_{2}}sin(2{q}_{2}+{q}_{3})-{m}_{3}{h}_{3}{l}_{2}\dot{{q}_{3}}cos\left({q}_{2}\right)sin(2{q}_{2}+2{q}_{3})$$

$${c}_{12}=-{c}_{21}=-\frac{1}{2}({m}_{2}{{h}_{2}}^{2}+{m}_{3}{{l}_{2}}^{2})\dot{{q}_{1}}sin\left(2{q}_{2}\right)-\frac{1}{2}{m}_{3}{{h}_{3}}^{2}\dot{{q}_{1}}sin(2{q}_{2}+2{q}_{3})-{m}_{3}{h}_{3}{l}_{2}\dot{{q}_{1}}sin(2{q}_{2}+{q}_{3})$$

$${c}_{13}=-{c}_{31}=-\frac{1}{2}({m}_{2}{{h}_{2}}^{2}+{m}_{3}{{l}_{2}}^{2})\dot{{q}_{1}}sin(2{q}_{2}+2{q}_{3})-{m}_{3}{h}_{3}{l}_{2}\dot{{q}_{1}}cos\left({q}_{2}\right)sin({q}_{2}+{q}_{3})$$

$${c}_{22}=-{m}_{3}{h}_{3}{l}_{2}\dot{{q}_{3}}sin\left({q}_{3}\right); \quad {c}_{23}=-{m}_{3}{h}_{3}{l}_{2}(\dot{{q}_{2}}+\dot{{q}_{3}})sin\left({q}_{3}\right)$$

$${c}_{32}=-{m}_{3}{h}_{3}{l}_{2}\dot{{q}_{2}}sin\left({q}_{3}\right);\quad {c}_{33}=0$$

$${g}_{2}=({m}_{2}{h}_{2}+{m}_{3}{l}_{2})cos\left({q}_{2}\right)+{m}_{3}{h}_{3}gcos({q}_{2}+{q}_{3})$$

$${g}_{3}={m}_{3}{h}_{3}gcos({q}_{2}+{q}_{3}); \quad {g}_{1}=0$$

Considering that the viscous friction coefficient is ${\eta }_{fi}$ and the Coulomb friction coefficient is ${\mu }_{fi}$, the total external disturbance $D$ is designed as:

$$D=\left[\begin{array}{c}{\tau }_{d1}-{F}_{d1}\\ {\tau }_{d2}-{F}_{d2}\\ {\tau }_{d3}-{F}_{d3}\end{array}\right]=\left[\begin{array}{c}30sin\left(t\right)-({\eta }_{f1}{\dot{q}}_{1}+{\mu }_{f1}sgn(\dot{{q}_{1}}\left)\right)\\ 20sin\left(t\right)-({\eta }_{f2}{\dot{q}}_{2}+{\mu }_{f2}sgn(\dot{{q}_{2}}\left)\right)\\ 10sin\left(t\right)-({\eta }_{f3}{\dot{q}}_{3}+{\mu }_{f3}sgn(\dot{{q}_{3}}\left)\right)\end{array}\right]$$

The model parameters and controller parameters of the manipulator are shown in Tables 1 and 2, respectively:

Table 1 Model parameters of the manipulator.

Table 2 Parameters of the controller.

In this paper, the simulation experiment is carried out in MATLAB/Simulink, and the built model is shown in Fig. 2. During the simulation experiment, the ode45 differential equation solver was used, the simulation time was set to 10s, and the solving step was a variable step.

Fig. 2

Controller model.

For the nonlinear disturbance observer, if the form of the nonlinear function vector is set to $p\left(q,\dot{q}\right)=456\left[\dot{{q}_{1}};\dot{{q}_{1}}+\dot{{q}_{2}};\dot{{q}_{1}}+\dot{{q}_{2}}+\dot{{q}_{3}}\right]$, then the disturbance gain matrix ${D}_{NL}\left(q,\dot{q}\right)=456 L{{M}_{0}\left(q\right)}^{-1}$,where $L$ is the third-order unit lower triangular matrix.

For the dynamic equation of the controller, the modeling coefficient ${\hslash }=0.98$, the modeling error is $2\%$.

Set the initial state of the system as: joint angle ${q}_{1}={q}_{2}={q}_{3}=0.5\,rad$, joint angular velocity $\dot{{q}_{1}}=\dot{{q}_{2}}=\dot{{q}_{3}}=0$.

Set the desired trajectory of the three joints as:

$${q}_{d}=\left[\begin{array}{c}{q}_{d1}\\ {q}_{d2}\\ {q}_{d3}\end{array}\right]=\left[\begin{array}{c}sin\left(t\right)\\ sin\left(t\right)\\ sin\left(t\right)\end{array}\right]$$

In order to reflect the advantages of the controller (NDO-BGFTSMC) designed in this paper, ablation experiments were carried out on it. It was divided into a new sliding mode controller without a nonlinear disturbance observer (BGFTSMC), a traditional linear sliding mode controller with a nonlinear disturbance observer (NDO-LSMC), and a nonsingular fast terminal sliding mode controller (NDO-NFTSMC).

BGFTSMC: In order to reflect the observation effect of the nonlinear disturbance observer, only the new sliding mode controller designed in this paper is used to investigate the improvement of the tracking effect of the observer.

NDO-LSMC: In order to reflect the global convergence of the new sliding mode controller and suppress the chattering effect, a traditional linear sliding mode control law is designed:

$$\left\{\begin{array}{c}s=\dot{e}+Le\\ {{\tau }_{1}=M}_{0}\left(q\right)\left(\ddot{{q}_{d}}+Le\right)-\xi +{C}_{0}\left(q,\dot{q}\right)\dot{q}+{G}_{0}\left(q\right)+{M}_{0}\left(q\right)\left[{\mu }_{1}\left(Rs+{k}_{1}sgn\left(s\right)\right)+{\mu }_{2}\left({k}_{2}{\left|s\right|}^{z}sgn\left(s\right)\right)\right]\end{array}\right.$$

where $L=\left[5\:0\:0;0\:5\:0;0\:0\:5\right]$ and other parameters are consistent with Table 2.

NDO-NFTSMC: In order to reflect the response speed, reduce overshoot, and suppress the chattering effect of the new sliding mode controller, a nonsingular fast terminal sliding mode control law is designed:

$$\left\{\begin{array}{c}s=e+\frac{1}{\alpha }{\left|e\right|}^{\gamma +1}sgn\left(e\right)+\frac{1}{\beta }{\dot{e}}^{2-p/q}\\ {{\tau }_{2}=-\frac{\beta q}{p}{M}_{0}{\dot{e}}^{2-p/q}(1+\frac{\gamma +1}{\alpha }{\left|e\right|}^{\gamma })+M}_{0}\left(q\right)\ddot{{q}_{d}}-\xi +{C}_{0}\left(q,\dot{q}\right)\dot{q}+{G}_{0}\left(q\right)\\ +{M}_{0}\left(q\right)\left[{\mu }_{1}\left(Rs+{k}_{1}sgn\left(s\right)\right)+{\mu }_{2}\left({k}_{2}{\left|s\right|}^{z}sgn\left(s\right)\right)\right]\end{array}\right.$$

Among them, $\alpha =7$, $\beta =2$, $\gamma =2$, $p=9$, $q=5$, and other parameters are consistent with Table 2.

The simulation results are as follows: The observation effect of the nonlinear disturbance observer on the total disturbance is shown in Fig. 3; The output torque compensation of the manipulator by the nonlinear disturbance observer is shown in Fig. 4; The output torque of the controller is shown in Fig. 5; The final tracking effect for the desired trajectory and desired speed is shown in Figs. 6 and 7; The trajectory tracking error is shown in Fig. 8.

Fig. 3

Observation effect of nonlinear disturbance observer on total disturbance.

It can be seen from Fig. 3 that the maximum total disturbance torque $D$ of the three joints are $40 N\cdot m$, $30 N\cdot m$, $20 N\cdot m$ respectively, accounting for about $75\%$ of the peak output torque of the controller, so it is particularly important to weaken the total disturbance. From the overall observation trend, the estimated value of the nonlinear disturbance observer basically coincides with the actual total disturbance. It can be seen from the local enlarged figure that the total disturbance generates a step disturbance at 1.57s, the observer quickly tracks the target trajectory within 0.04s, and the overshoot is less than $5\%$ of the total disturbance value. Therefore, the nonlinear disturbance observer accurately estimates and compensates for the total disturbance, which enhances the control accuracy and robustness of the manipulator system.

Fig. 4

Output torque compensation of manipulator by nonlinear disturbance observer.

It can be seen from Fig. 4 that if the output torque before correction is used as the driving torque, the system cannot be accurately controlled. It can be seen from the local enlarged drawing that even if the direction of the driving torque is changed, the corrected output torque is closer to the output torque under the idealized model. Therefore, the observer can effectively suppress the influence of chattering and improve the control accuracy of the system.

Fig. 5

Output torque of controller.

Figure 5 shows the output torque comparison diagram of four different controllers. By comparing the NDO-BGFTSMC designed in this paper with the NDO-LSMC and the NDO-NFTSMC, it can be seen that the NDO-BGFTSMC responds quickly to the change of joint velocity direction, the output torque is stable, and the other two controllers produce an obvious chattering phenomenon. This experiment shows that the Imp-fal function can effectively weaken the system chattering phenomenon. It can be seen from the local enlarged drawing that the convergence speed of NDO-BGFTSMC is faster than that of BGFTSMC when the driving direction changes, and there is no overshoot.

Fig. 6

Track tracking effect.

It can be seen from Figs. 6 and 7 that the tracking performance of the NDO-BGFTSMC designed in this paper is superior to the other three controllers in all aspects. Among them, in the initial stage of 0–0.8 s, due to the too large amplitude of the output torque of the NDO-NFTSMC, there is a large deviation between the tracking trajectory and the expected trajectory, and the overshoot phenomenon occurs when the speed tracking is 0.4s. The other three controllers track the target trajectory and speed better. When the speed direction changes (1.55–1.88 s), the BGFTSMC using a single sliding mode control cannot compensate for the unknown disturbance in the speed tracking process, resulting in a large deviation, while the other three controllers with nonlinear disturbance observers can better track the speed, reflecting their stronger anti-interference ability. At the same time, it can be further concluded from the local enlarged view that the tracking accuracy of NDO-BGFTSMC is higher than that of NDO-NFTSMC and NDO-LSMC. Therefore, from the final tracking effect, it can be concluded that the tracking performance of NDO-BGFTSMC is better.

Fig. 7

Speed tracking effect.

Fig. 8

Tracking error of controller.

It can be seen more intuitively from Fig. 8 that, compared with other controllers, NDO-BGFTSMC has faster convergence speed, minimum steady-state error, no overshoot, and is almost unaffected by the total disturbance torque, indicating that the controller has good robustness. The mean square deviation is used as the measurement index of control accuracy in quantitative analysis, and its calculation method is formula (31):

$$\sigma \left(e\right)={\left(\frac{1}{N}\sum _{i=1}^{N}{{e}_{i}}^{2}\right)}^{1/2}$$

(31)

where $N=500$ is the number of samples, and $\sigma \left(e\right)$ is the mean square deviation of the tracking error.

Because the initial state of the system is not zero and the controller itself has tracking characteristics, which will affect the mean square error. Therefore, the tracking error is sampled after 0.245s when each controller first realizes tracking, and the final result is shown in Table 3. Compared with BGFTSMC, the mean square deviation of the joint tracking error of NDO-BGFTSMC decreased by $1.3\%$, $2.5\%$, and $6.8\%$, respectively; compared with NDO-LSMC, the mean square deviation of the joint tracking error of NDO-BGFTSMC decreased by $1.7\%$, $1\%$, and $0.8\%$, respectively; compared with NDO-NFTSMC, the mean square deviation of the joint tracking error of NDO-BGFTSMC decreased by $15.3\%$, $9.8\%$, and $24.1\%$, respectively. According to the data in the table and the above analysis, the superiority of NDO-BGFTSMC is verified.

Table 3 Mean squared difference of tracking error under the action of four controllers.

Physical verification

The practical control effectiveness of the NDO-BGFTSMC was validated on a self-developed robotic manipulator experimental platform. This platform comprises a upper computer, the robotic arm itself, and articulated motors³². The experimental procedure encompasses the following steps: Initially, a three-degree-of-freedom robotic arm model was constructed using the Robotics Toolbox in MATLAB/Simulink, with the desired tracking trajectory being solved through inverse dynamics. Subsequently, the desired tracking trajectory was initialized on the host computer interface, and the control algorithm was executed. Finally, the output torque values were transmitted to the articulated motors via the CAN bus to drive the motion of the robotic arm. Simultaneously, encoders on the articulated motors detected the actual rotor angles and converted them into joint angles, which were then transmitted back to the host computer’s receiving function in real-time through CAN communication, forming a closed-loop control system⁴⁰, as illustrated in Fig. 9.

Fig. 9

Closed-loop control system.

The relevant parameters of the experimental platform are detailed as follows: the masses of the three connecting rods are 0.2 kg, 0.52 kg, and 0.1 kg, respectively, with corresponding lengths of 80 mm, 190 mm, and 140 mm. The controller parameters remain consistent with those utilized in the simulation experiments, with the joint angles initialized at 0.5 rad. The experimental sampling period is set to 1 ms. Figure 10 depicts the experimental platform.

Fig. 10

Experimental platform.

Figure 11 illustrates the performance of the controller in the trajectory tracking experiment. The upper computer interface clearly demonstrates that the NDO-BGFTSMC exhibits robustness in controlling the robotic arm, achieving precise tracking of the target trajectory with a steady-state error maintained within 0.1 radians, thereby validating the effectiveness of the control strategy. Through multiple experimental validations, it is observed that compared to the LSMC controller, the NDO-BGFTSMC controller reduces the mean square error by approximately 55%. Furthermore, when the direction of trajectory tracking changes, the response time of the NDO-BGFTSMC is shortened by 45% compared to the LSMC. Figure 11 also presents the performance of the controller in terms of the output torque of Joint 2. The experimental results indicate that the NDO-BGFTSMC significantly suppresses output chattering within a certain range in the presence of disturbances, resulting in a more stable and smooth output torque. This fully demonstrates its superior anti-disturbance performance.

Fig. 11

Upper computer interface.

In summary, the superiority of the NDO-BGFTSMC can be attributed to its unique algorithm design and optimized parameter tuning. In practical production and daily life, the high precision and rapid response of the NDO-BGFTSMC hold promise for enhancing productivity and reducing error rates, making it particularly suitable for applications such as automated production lines and robot navigation scenarios.

Summary

In this paper, a novel sliding mode control approach for the manipulator, grounded in a nonlinear disturbance observer, is introduced. This approach addresses the challenge of trajectory tracking control in manipulator systems where the upper bound of total disturbance is known. The controller is derived through a fusion of the Backstepping method, Lyapunov theory, and global fast terminal sliding mode theory, thereby guaranteeing system stability and achieving global asymptotic convergence. Subsequently, the inclusion of a nonlinear disturbance observer and an enhanced reaching law not only compensates for the system’s output torque but also suppresses chattering. Through simulation experiments, it is demonstrated that the nonlinear disturbance observer designed in this paper can swiftly and accurately estimate disturbance values, significantly reducing reliance on specific model information. The results derived from both numerical simulations and physical validation experiments clearly indicate that the controller possesses substantial advantages in control accuracy, response swiftness, and tracking proficiency. This methodology not only augments the convergence velocity of tracking errors but also appreciably diminishes overshoot and output torque vibrations, thereby bolstering the stability and robustness of the control system against disturbances. In order to further investigate the problem of trajectory tracking in scenarios where the upper bound of total disturbances is unknown, subsequent research will proceed along two avenues. On one hand, the integration of an extended state observer, adaptive techniques, and sliding mode control will be considered in future experiments. On the other hand, the research will extend to the control of multi-degree-of-freedom manipulators, aiming to provide a more versatile and generalizable control strategy within the field of robotic manipulators.