Four DOF manipulator control using terminal nonsingular sliding mode

Abstract

This paper presents a new terminal non-singular sliding mode (NSTS) controller proposed for a second-order four-degree-of-freedom (4-DOF) robotic manipulator. This system is designed with a new exponent for the terminal sliding surface. The latter is designed to solve the singularity problem in the first derivative of the sliding surface. Such a problem is associated with the conventional linear hyperplane terminal sliding mode control. Error convergence time from any initial state is guaranteed to be a finite time. Lyapunov stability analysis and Simulation results are presented in this paper. Furthermore, a comparison is implemented with the conventional linear sliding mode in the simulation part. The obtained results prove improved performance in trajectory tracking.

Introduction

Robust controllers for robotic systems have attracted significant research attention for decades. Because of the fundamental nonlinearities, model uncertain parameters, and external disturbances, various control schemes have been implemented in the literature to overcome control challenges. For example, finite-time command-filtered backstepping techniques have been applied to nonlinear systems ¹ and, ² with subsequent enhancements achieved through optimized simplified backstepping algorithms ³ and adaptive fuzzy backstepping controllers.. ⁴ Feedback linearization approaches have been explored in the context of aerial systems such as quadrotors ⁵ and. ⁶ The latter was used in a four-DOF manipulator based on sliding mode in. ⁷ Among the suite of robust controllers, Sliding Mode Control (SMC) stands out for its ability to ensure system effectiveness against parametric uncertainties and external disturbances^8,9 and. ¹⁰ Furthermore, in ¹¹ Two variants of SMC laws have been developed to control various states of the fuel cell, which proves the performance of the SMC in different situations. SMC operates by driving the system states onto a predetermined sliding manifold and maintaining the trajectories in its vicinity. Several enhancements have been proposed to improve the transient performance and reduce the reaching time to the sliding surface, such as modified reaching laws, ¹² enhanced double exponential reaching laws, ¹³ and integration with fuzzy logic systems ¹⁴ and. ¹⁵ Applications of these advanced SMC schemes range from wind energy systems to agricultural robotics, such as vision-based multivariable super-twisting control applied to quadrotors. ¹⁶ A robot manipulator used to apply a terminal super-twisting design in. ¹⁷ Nonsingular super-twisting with finite-time is introduced in ¹⁸ and a Linear correction term is added to a super-twisting algorithm was used in ¹⁹ for a drone. Further development to enhance control systems is to design and implement techniques of state observer design for continuous-time dynamical systems ²⁰ and ²¹ and applied to a synchronous motor in. ²² To reduce the effect of external disturbance and unmodeled dynamics, perturbation compensators are designed in ²³ and. ²⁴ Adaptive controls based on neural network approximation using a radial-base function for a quadrotor is designed in. ²⁵ Auto-tuning switching control laws for faster, finite-time dynamical response and enhanced tracking precision. ²⁶ More recently, efforts have been made to integrate terminal dynamics into sliding mode control, giving rise to Terminal Sliding Mode Control (TSMC). TSMC algorithm introduces nonlinear sliding surfaces that guarantee finite-time convergence to the origin following a power-law relationship, resulting in improved precision and faster convergence rates ²⁷ and. ²⁸ This approach has been successfully applied to robotic manipulators, ²⁹ where the control precision is critical. However, a significant limitation of conventional TSMC is the potential for singularities to arise in the control law near the origin, which compromises stability and control effectiveness. To address this, Terminal Non-Singular Sliding Mode Control (TNSMC) has been proposed in^30,31 and, ³² eliminating the singularity by modifying the terminal sliding surface formulation.

Despite its advantages, TNSMC still faces challenges in guaranteeing fast convergence to the sliding manifold, as the reaching time remains sensitive to the initial error dynamics and system perturbations. Therefore, further research is necessary to enhance its performance and applicability.

In this context, the present study introduces a Novel Terminal Non-Singular Sliding Mode Control (NSTS) framework designed specifically for nonlinear systems with uncertain dynamics and external disturbances. A new terminal sliding manifold is proposed to overcome the singularity issues associated with conventional TSMC, while also ensuring finite-time convergence both to the sliding manifold and to the system equilibrium point from any initial state.

The proposed NSTS strategy is applied to the control of a Quanser 4-DOF robotic manipulator, a benchmark platform widely used in research and education, Figure 1. This paper proposed a control that ensures finite-time error tracking while handling disturbance. A comparative study in the 0simulation is made to confirm that the proposed approach achieves better performance over other existing control methods.

Figure 1.

Four DOF quanser arm manipulator. ³³

We can summarize the main contributions in two points:

Design of a Novel Terminal Nonsingular Sliding Mode Controller: A new sliding surface is developed to inherently eliminate the singularities typically encountered in conventional terminal sliding mode control schemes, thereby enhancing both robustness and practical implementability.

Enhanced Finite-Time Convergence and System Performance: The suggested controller provides quicker error convergence and enhances tracking accuracy, making it perform well for robotic applications that require high precision and dynamic responsiveness.

The paper is organized as follows. In Section I, we provide the introduction and background context. Section II addresses the singularity issue in traditional TSMC. Section III introduces the formulation of the suggested non-singular terminal sliding surface. Section IV elaborates on the control design methodology. Section V outlines the implementation of the proposed approach on the Quanser 4-DOF robotic manipulator. Section VI provides the simulation outcomes and an analysis of performance. Section VII wraps up the paper and discusses future research directions.

Singularity in terminal sliding mode control

The terminal sliding mode Controllers are special versions of SMC. In classical SMC, the system errors converge asymptotically, i.e., get closer and closer but theoretically never fully reach it in finite time. In TSMC, the sliding surface is designed with nonlinear functions, like fractional powers, so that the tracking error goes to zero in finite time. This means the robot (or system) will reach the target state within a predictable finite time, not just eventually. We consider a nonlinear second-order robotic function:

$$\begin{array}{rl}{\dot{x}}_1=& x_2\\ {\dot{x}}_2=& f(x)+gu(t)+d(t)\end{array}$$

(1)

Where $x_{1,2}$ are the states $f(x)$ and g are the system functions and $g\ne 0$ , $d(t)$ is the uncertain disturbance which never grows beyond a known maximum L, $|d(t)|<L^{}$ . The error and its first derivative can be defined as:

$$\begin{array}{rl}e=& x_1-x_{1d}\\ \dot{e}=& {\dot{x}}_1-{\dot{x}}_{1d}\end{array}$$

(2)

Where $x_{1d}$ refers to the required trajectory. Traditional terminal sliding mode control systems encounter issues related to singularity, as outlined below. The sliding mode equation employed is expressed as ³⁴ :

$$s=\dot{e}+\beta e^{\frac{q}{p}}$$

(3)

The constants are selected as $\beta >0,q>0,p>0$ and $p>q$ . The derivative of (3) is given as:

$$\dot{s}=\ddot{e}+\beta {\displaystyle \frac{q}{p}{e}^{\frac{q}{p}-1}\dot{e}}$$

(4)

The following equation describes the proposed controller:

$$u=g^{-1}(x)(f(x)+\beta {\displaystyle \frac{q}{p}{e}^{\frac{q}{p}-1}\dot{e}+(L+\eta )sign(s)}$$

(5)

Where $\eta$ is a positive constant. The selection of the constants is the leading exponent term $\left(\frac{q}{p}-1\right)$ in (4) and (5) to be negative and the singularity problem arises. This happens in a transient response when $\dot{e}$ is not equal to zero and e equals to zero. In practical implementation, this situation occurs when the error reaches zero as it converts from positive to negative and vice versa. In this case $e=0$ but $\dot{e}\ne 0$ .

Article objective

The objective is to design and implement a Terminal Non-Singular Sliding Mode control (NSTS) for robotic systems with a solved singularity problem in the conventional Terminal Sliding Mode (TSM), and ensure fast reference tracking. The effectiveness of the proposed method is demonstrated through its application to a four-degree-of-freedom (4-DOF) robotic manipulator.

Terminal non-singular sliding mode controller

The Terminal Non-Singular Sliding Mode (TNSM) controller is an advanced nonlinear control strategy designed to achieve finite-time convergence while avoiding the singularity issues present in conventional Terminal Sliding Mode (TSM) control. In classical SMC, sliding mode control goes through two phases, reaching phase and Sliding phase, Reaching Phase where the states move toward the sliding surface under the control law. Sliding Phase where the states move along toward the desired point.

In classical TSM, the sliding surface often incorporates fractional powers of the tracking error to guarantee finite-time convergence; however, this design may lead to singular behavior, the TNSM controller introduces a non-singular terminal sliding surface that ensures smooth operation near equilibrium while preserving the finite-time stability property. The paper proposes a new non-singular control based on terminal sliding mode. The dynamic value for the sliding surface, which is described in (6) and its first derivative (7) are selected as follows:

$$s=\mathrm{e}+{\displaystyle \frac{1}{\beta }{\dot{e}}^{\zeta }}$$

(6)

$$\dot{s}=\dot{e}+{\displaystyle \frac{1}{\beta }\zeta {\dot{e}}^{\zeta -1}\ddot{e}}$$

(7)

Given $\zeta$ as a dynamic exponent term as follows:

$$\zeta =2-{\zeta }_{0}sig{n}^2(\dot{e})$$

(8)

$$sign(e)=\{\begin{array}{c}1,if\dot{e}>0\\ 0,if\dot{e}=0\\ -1,if\dot{e}<0\end{array}$$

(9)

We design $\zeta$ to take the values $(2-{\zeta }_0)\le \zeta \le 2$ , in the meantime $0<{\zeta }_0<1$ . From this design, we can see that when $\dot{e}=0$ the singularity problem in the sliding surface will be bypassed as in (8).

Control design

This part explains the designed controller by using the non-singular design outlined in equations (5), (6), (7), and (8). Initially, given the nonlinear second-order robotic system described by equation (1), we introduce the following control system:

$$u=-g^{-1}(x)\left(f(x)+\beta {\displaystyle \frac{1}{\zeta }{\dot{e}}^{2-\zeta }-{\ddot{x}}_d+(L+\eta )sign(s)}\right)$$

(10)

Mathematical analysis based on a Lyapunov function is used to verify the stability:

$$\begin{array}{rl}V=& {\displaystyle \frac{1}{2}{s}^{T}s}\\ \dot{V}=& s\dot{s}\\ \dot{V}=& s^T\left[\dot{e}+{\displaystyle \frac{1}{\beta }\zeta {\dot{e}}^{\zeta -1}\stackrel{..}{e}}\right]\\ \dot{V}=& s^T\left[\dot{e}+{\displaystyle \frac{1}{\beta }\zeta {\dot{e}}^{\zeta -1}\left({\ddot{x}}_1-{\ddot{x}}_{1d}\right)}\right]\\ \dot{V}=& s^T\left[\dot{e}+{\displaystyle \frac{1}{\beta }\zeta {\dot{e}}^{\zeta -1}\left(f\left(x\right)+gu\left(t\right)+d{\left(t\right)}_{}-{\ddot{x}}_{1d}\right)}\right]\end{array}$$

(11)

Substituting (10) in (11) we find:

$$\begin{array}{rl}\dot{V}=& s^T\left[\dot{e}+{\displaystyle \frac{1}{\beta }\zeta {\dot{e}}^{\zeta -1}\left(-\beta {\displaystyle \frac{1}{\zeta }{\dot{e}}^{2-\zeta }-(L+\eta )sign(s)+d{(t)}_{}}\right)}\right]\\ \dot{V}=& s^T\left[{\displaystyle \frac{1}{\beta }\zeta {\dot{e}}^{\zeta -1}(-(L+\eta )sign(s)+d{(t)}_{})}\right]\\ \dot{V}=& \left[{\displaystyle \frac{1}{\beta }\zeta {\dot{e}}^{\zeta -1}(-s^T(L+\eta )sign(s)+s^{T}d{(t)}_{})}\right]\\ & \le \sum _{i=1}^n{\displaystyle \frac{1}{\beta }\zeta {\dot{e}}^{\zeta -1}[-(L_i+{\eta }_i)|s_i|-s_i{d}_i(t)]}\end{array}$$

(12)

As we have $[-(L_i+{\eta }_i)|s_i|+s_i{d}_i(t)]\le 0$ , this proves that $\dot{V}\le 0$ which indicates the stability of the proposed system.

Control finite time proof

We select the starting time as $t_0$ and the final required error convergence time is $t_c$ , in the start time, there is an error, and at the end, the error should vanish, i.e $s(t_0=0)\ne 0$ and $s(t_c)=0$ . By using (12), it can be seen that:

$$\begin{array}{rl}{s}_i{\dot{s}}_i\le -& |s_i|(L_i+{\eta }_i)+s_i{d}_i(t)\\ {\dot{s}}_i\le -& (L_i+{\eta }_i)-d_i(t)\end{array}$$

(13)

For $S\ge 0,$ equation (13) can be given as:

$$\begin{array}{rl}& {\dot{s}}_i\le -{\eta }_i\\ & \underset{s=s(t_0)}{\overset{s=s(t_c)}{\int }}d{s}_i\le \underset{t=0}{\overset{t=t_c}{\int }}-{\eta }_i\\ & s(t_c)-s(0)\le -{\eta }_i{t}_c\\ & t_c\le {\displaystyle \frac{s(0)}{{\eta }_i}}\end{array}$$

(14)

Similarly, for $S\le 0$ , we find:

$$\begin{array}{rl}& t_c\le -{\displaystyle \frac{s(0)}{{\eta }_i}}\\ & t_c\le \left|{\displaystyle \frac{s(0)}{{\eta }_i}}\right|\end{array}$$

(15)

Dynamic model of the robot arm

The mathematical model of a Manipulator arm explains how forces and torques applied at its joints affect its movement.^35,36 This model considers the physical attributes of the manipulator arm, as well as the kinematics and dynamics that dictate the robot's behaviour. The Manipulator dynamic model is a mathematical framework for predicting the system behaviour based on the applied forces and torques. It serves as aa an interaction between joint motions, gravity, and friction. Developing dynamic models for robotic manipulators is achieved by using either the Newton–Euler or Euler–Lagrange formulation. The comparative efficiency of these methods remains an active area of research, ³⁷ The effectiveness of the proposed controller is demonstrated through its application to a four-degree-of-freedom (4-DOF) robotic manipulator. ³⁸ In general, Newton-Euler method is often preferred for a robot arm with numerous DOFs. This is because of its recursive nature and how efficiently it computes things, especially when the reference frames are set up correctly. Accurate dynamic modeling further requires parameter identification techniques to capture nonlinearities in the system. The Quanser robotic arm manipulator is frequently referenced in literature as a benchmark platform, where careful frame assignment is essential to ensure model accuracy.

In this paper, we present the dynamic model of a robotic manipulator arm, which is the four-DOF QArm provided by Quanser. Figure 1 shows the kinematic diagram and indicates the Cartesian frame of all the joints, while the rigid-body diagram provides the Cartesian x-y-z axes. The dynamics of motion are expressed in matrix form as given below ³³ :

$$\tau =M(\theta )\ddot{\theta }+B(\theta ){\dot{\theta }}_i{\dot{\theta }}_j+C(\theta ){\dot{\theta }}_k^2+G(\theta )$$

(16)

Where, $(i,j)ϵ\{(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)\},$

$kϵ\{1,2,3,4\}$ , in this setting, M is the inertia matrix, B, C, and G are Coriolis, Centrifugal, and Gravity matrices, respectively. This depiction is known as a motion equation. The matrices associated with the Coriolis and Centrifugal coefficients depend solely on the manipulator's state ϕ. It's important to note that certain coefficients in these matrices may become irrelevant or be set to zero based on the configuration of the manipulator. For the Quanser Manipulator, the matrices are given in the following:

The subscript (mn) in the coefficients specifies the relationship between the exerted torque at the m^th joint and the related n^th kinematic term. The coefficient $B_{24}$ connects the torque at joint 2, represented as ${\tau }_2$ , to the corresponding fourth kinematic term, which is expressed as ${\dot{\theta }}_2{\dot{\theta }}_3$ . We can see in Figure 2 that the robotic arm is presented in the home position. The space vector of the joint, θ, is expressed in the vector, $\theta =[0,\left(\beta -\frac{\pi }{4}\right),-\beta ,0{]}^T$ . The manipulator's actuators and encoders are calibrated in this position. If we need to set the manipulator to the home position, a command $[000{]}^T$ has to be executed. Furthermore, the encoders read the position as $[000{]}^T$ . Additional information about the $\overrightarrow{\varphi }$ space of the manipulator is provided in Table 1. For instance, when ${\varphi }_2=0$ it indicates that ${\theta }_2=\beta -\frac{\pi }{2}$ . To joint 2, this is relative to the home position. Additional information regarding the Quanser QArm model can be found in ³³ :

Figure 2.

Trajectory in 3D-space.

Table 1.

Mathematical formulation.

Parameter (New)	Parameter (Original)	Parameter (New)	Parameter (Original)
${\lambda }_1$	$L_1$	${\varphi }_1$	${\theta }_1$
${\lambda }_2$	$\sqrt{L_2^2+L_3^2}$	${\varphi }_2$	${\theta }_1+\frac{\pi }{2}-\beta$
${\lambda }_3$	$L_4+L_5$	${\varphi }_3$	${\theta }_3+\beta$
${\beta }_1$	$ta{n}^{-1}\left(\frac{L_3}{L_2}\right)$	${\varphi }_4$	${\theta }_4$

Numerical simulation

The simulation is implemented using the real parameters of the QArm manipulator provided by Quanser, illustrated in Figure 3,, ³³ the parameters are listed in table 2. The dynamic model described in (16) and (1) is used, the details of matrices $M_{ij},B_{ij},\mathrm{and}{C}_{ij}$ are described in. ³³ Initially, the starting point of the joints is zero, $q_1=q_2=q_3=q_4=0$ . This position corresponds to $(x,y,z)=(45,0,49)cm$ . To further validate robustness, a time-fluctuating disturbance is applied to the control input, which is defined in the following equation, where the disturbance frequency is 1 kHz:

$$D_i=3\cos (\omega t)+0.5\sin (\omega t)$$

(18)

Figure 3.

Quanser QArm. ³³

Table 2.

Qarm dynamic parameters. ³³

Parameter	Value	Parameter	Value
m 1	0.7906 kg	m 2	0.4591 kg
$\lambda$ c 1	0.0399 m	$\lambda$ c 2	0.1071 m
I 1 A	1.489×10−3 kg m2	I 2 A	1.922×10−4 kg m2
n/a∗	-	I2L∗	1.610×10−3 kg m2
m 3	0.269 kg	m4	0.257 kg
$\lambda$ c 3	0.1561 m	$\lambda$ c 4	0.0998 m
I 3 A	2.679×10−4 kg m2	I 4 A	6.528×10−4 kg m2
I3L∗	2.069×10−3 kg m2	I4L∗	1.120×10−3 kg m2

The terminal nonsingular sliding mode, as in section IV, is implemented for trajectory tracking. The desired trajectory follows a square path with sharp corners, as illustrated in Figure 2. This trajectory is used to evaluate the system's tracking performance. Figures 4 and 5 present the tracking results in Cartesian space and joint space, respectively, demonstrating the controller's effectiveness in achieving accurate path following. Furthermore, the performance of the proposed system is compared with a conventional PID:

$$\tau =K_{p}e(t)+K_D{\displaystyle \frac{d}{dt}e(t)+K_I\int e(t)dt}$$

(19)

Figure 4.

Manipulator joint tracking.

Figure 5.

Position and altitude trajectory.

Figures 6 and 7 illustrate the tracking errors in both the joint space and the inertial Cartesian frame for the proposed NSTS controller and the conventional PID controller. The results clearly demonstrate that the NSTS controller substantially reduces tracking errors compared to the PID approach. Figure 8 presents the corresponding control signals for the NSTS controller, which exhibit smooth and continuous behavior without reaching saturation. This behavior underscores the practical applicability of the NSTS controller, as it operates within safe limits, avoids excessive actuator stress, and maintains both system performance and stability. The absence of abrupt changes or discontinuities in the control signals minimizes actuator wear and contributes to overall system reliability. Moreover, the stable control signals ensure that the system remains within the desired operational bounds, preventing issues such as performance degradation, instability, or uncontrolled oscillations. In the proposed NSTS controller, the dynamic sliding surface reduces switching frequency, and Smooth control signals are observed in the simulation results, Figure 8.

Figure 6.

Errors in joint tracking for the NSTS and PID.

Figure 7.

Errors in position and altitude for the NSTS and PID.

Figure 8.

Control signals.

The disturbance rejection capability of the controller is demonstrated by introducing a time-varying disturbance defined in Equation (8). Additional discussion has been added explaining how the NSTS controller maintains accurate tracking despite the disturbance.

Table 3 compares the root mean square error for each control method. The lower RMSE values achieved by the NSTS controller confirm its superior performance compared to the PID controller. Collectively, the results shown in the figures and table demonstrate the effectiveness and reliability of the proposed controller.

Table 3.

RMS of errors.

	NSTS	PID
x	0.045	0.372
y	0.02	0.377
z	0.003	0.04

The simulation results indicate that the NSTS controller provides enhanced robustness against nonlinear dynamics, and it reduces overshoot and oscillations in the control commands, making it easy to implement practically. It also improves stability, gives faster response, and reduces sensitivity to tuning parameters relative to the PID controller. Overall, the NSTS controller consistently maintains high robustness and precision across a range of operating conditions, ensuring both stability and accurate performance even in the presence of significant disturbances.

To model measurement noise, external disturbances, or unmodeled dynamics, to enable evaluation of controller robustness under realistic operating conditions, the Simulink Band-Limited White Noise block is used. The mentioned block is used to generate a stochastic signal that approximates white noise with finite power over a specified frequency range, Figures 9 and 10. The generated signal is characterized by a zero-mean Gaussian distribution and a user-defined noise power. It produces independent random values at each sample instant, resulting in a discrete-time noise sequence. Noise power 1 $(N.m{)}^2,$ controls the intensity of the generated noise signal. Sample time, ${0.001}^{}$ seconds, defines the update rate of the noise.

Figure 9.

Errors in position and altitude with measurement noise and external disturbances for the NSTS and PID.

Figure 10.

Errors in joint tracking with measurement noise and external disturbances for the NSTS and PID.

Figures 9 and 10 show the error in the presence of the noise. It can be seen that the proposed controller is able to reject the noise and to compensate for signal distortion much greater than that of the PID controller. This result is expected since NTST control is a type of sliding mode control. The purpose of this simulation is to ensure that the non-singular term doesn't affect the power of the sliding mode control.

Conclusion

This study presents the development of a novel robust Terminal Non-singular Sliding Mode (NSTS) controller designed to overcome the singularity issues typically encountered in conventional Terminal Sliding Mode Control (TSMC) methods. The singularity problem, which occurs when the control law becomes undefined near the origin, is effectively addressed in the proposed approach through the design of a new sliding surface incorporating a modified exponent. This formulation ensures that the control law remains well-defined throughout the system's operation, including during convergence. To evaluate the performance of the proposed controller, simulations were carried out on a manipulator, serving as a benchmark example of a highly nonlinear, underactuated robotic system with complex dynamics. The proposed switching surface is designed with a new exponent that overcomes the singularity in the traditional terminal sliding mode. The proposed control algorithm was developed, and the stability was verified. The controller was compared with a PID controller. The NSTS controller is scalable to manipulators with higher degrees of freedom because the control law is designed for second-order nonlinear systems. The obtained results in the simulation part clearly display the effectiveness of the proposed design. The new system improves tracking performance, reduces the tracking error, and ensures fast error convergence.