1. Introduction
The robotic arm, a vital component of industrial robots, has become increasingly intelligent and widely applied in industries, military, medical, aerospace, and other fields. For robotic arms, they can perform some complex tasks with human beings and work in dangerous workplaces [
1]. As such, research on robotic arm control has attracted more and more experts and scholars. However, due to robotic arms being a type of complex dynamical system characterized by multivariable, nonlinear, and strongly coupled processes [
2], the actual control processes of the robotic arm are affected by model errors, internal friction, external environmental disturbances, and other uncertain factors, thereby affecting the accuracy of the robotic arm trajectory tracking [
3]. To achieve trajectory tracking effects with strong stability, high accuracy, and fast convergence speed, it is of practical significance to introduce control algorithms to design controllers with certain robustness for the research of robotic arm control systems [
4].
In terms of robotic arm control, in order to address the complex control issues of robotic arm systems and achieve high-precision tracking performance, experts have proposed many control methods [
5]. At present, the methods available for designing nonlinear control systems include sliding mode control, backstepping, robust control, model predictive control, neural network (NN) control, and adaptive control [
6,
7,
8,
9]. In ref. [
6], an adaptive gain method based on state constraints was studied. This controller takes into account the influence of position and speed limitations on the trajectory tracking control of the robotic arm and conducts a comprehensive technical design for the multi-link robotic arm. The simulation results indicate that the method was superior to similar adaptive controllers that do not consider the constraints and the proportional–integral–differential form. Subsequently, a robust control law for space robots that considers system uncertainty and closed-chain constraints was proposed in ref. [
7], which has the advantage of weak dynamic coupling in the driving system. In ref. [
8], adaptive robust backstepping was proposed to enhance the performance of mobile manipulator robots; in addition, with this approach, there is no longer a need for prior knowledge of the control system, effectively improving its practicality. In ref. [
9], second-order sliding mode control (SMC) based on adaptive neural networks was developed for a dual-arm robot, which can fully approximate the dynamic model of the robotic arm while ensuring the higher robustness of the entire system. It is worth noting that the above methods usually use adaptive technology to address the control problem of robotic arms, and the time required for tracking errors to reach zero is infinite.
To ensure that the tracking error of the system converges in a finite time, a finite time controller based on backstepping has been studied [
10,
11,
12]. In ref. [
10], an improved fault-tolerant controller was constructed by applying fuzzy control and adaptive backstepping, effectively addressing the trajectory tracking control problem of non-strict feedback nonlinear systems with actuator faults. In ref. [
11], the finite-time consensus fault-tolerant control tracking problem of nonlinear multi-agent systems with non-strict feedback forms was solved by combining finite time control with neural networks and backstepping. In ref. [
12], a novel control method for robotic arms based on backstepping and terminal sliding mode control was developed to address the problem of trajectory tracking control for robotic arms, which can preserve the advantages of both backstepping and sliding mode control, thus effectively improving the fast transient response and robustness of the system. Certainly, this control method can effectively accelerate the convergence velocity, and the convergence time can be given by
. However, its convergence time is related to the initial value of the system’s state, and therefore, its convergence time cannot be formulated.
In order to overcome this drawback, fixed-time theory can be exploited, ensuring a system reaches stability within a fixed timeframe; at the same time, the convergence time is independent of the system’s initial state, and the convergence time can be given by
. This method can determine the maximum convergence time of the system during the controller design process, thereby providing valuable system performance information in advance. The theory is more applicable to systems that require a strict convergence time. In ref. [
13], a fault-tolerant-prescribed performance tracking control approach based on fixed-time techniques was designed to solve robot manipulators with actuator faults. By designing a fixed-time performance function with precise convergence and combining it with terminal sliding mode control, thus effectively improved the tracking accuracy of the robotic arm and the robustness of the system. To achieve fixed-time convergence and better robustness, a fixed-time sliding mode control (FTSMC) was studied in ref. [
14]. By combining high-order sliding mode control and fixed time theory, the performance of the controller was effectively improved. In ref. [
15], an adaptive FTSMC was also applied to robot manipulators. However, FTSMC has a drawback in that it can generate chattering effects, thereby causing mechanical oscillations in the robotic arm system. To this end, scholars have developed a fixed-time controller based on backstepping without a chattering effect or by lowering the chattering effect. In ref. [
16], by combining fixed-time control, backstepping, and a newly developed obstacle function, the output tracking problem of MIMO nonlinear systems with asymmetric output constraints was successfully solved; finally, the effectiveness of the algorithm was verified through simulations using a two-degree-of-freedom robotic arm. In ref. [
17], by combining fixed-time control and backstepping with a newly designed performance function, the tracking control problem for stochastic nonlinear systems with unknown measurement sensitivity was effectively solved. The widespread use of backstepping control relies on the availability of complete knowledge about robot dynamics during the design process. However, when system uncertainties are present, the tracking performance achieved by backstepping-based control strategies is often lower compared to other control methods.
In order to solve the issue of system uncertainties, scholars have proposed many effective attempts [
18,
19,
20,
21,
22,
23,
24,
25]. There are currently two main types of algorithms: (1) neural network-based approximation algorithms; and (2) disturbance observer-based algorithms. Neural network-based algorithms can leverage their powerful approximation ability to estimate system uncertainty [
18]. In ref. [
19], a new fuzzy wavelet neural network (FWNN) was designed to estimate the unknown uncertainties in the presence of unknown nonlinear uncertainties by concentrating them into a composite uncertainty, thus effectively enhancing the robustness of the system. In ref. [
20], a fixed-time neural network control method was studied to address system uncertainties. By integrating switching mechanisms into control design, the semi-global stability of traditional neural network control systems can be extended to global stability, effectively eliminating the impact of system uncertainties on robotic arm control. Adaptive non-singular terminal sliding mode control (NSTSMC) was developed in ref. [
21], which uses neural networks to deal with system uncertainties without prior knowledge. Finally, the effectiveness of the control method was verified through simulation. Still, neural network algorithms often contain approximate residuals that can affect their accuracy, which can only achieve globally uniformly bounded tracking performance instead of asymptotic stability. In disturbance observer-based algorithms, system uncertainty is estimated by the disturbance observer. In ref. [
22], a nonlinear disturbance observer is developed to address the system uncertainties of the robotic arm. Although the observer is capable of providing precise estimations of the system uncertainties, it can only achieve an asymptotically stable tracking performance, which means that the optimal convergence speed of the system is exponentially convergent, and the tracking error can converge to zero in infinite time. Subsequently, a disturbance observer based on a finite-time algorithm was proposed in ref. [
23] to address the measurement uncertainties on the robotic surface vehicle (RSV).
In order to address with measurement uncertainties, two finite time disturbance observers were developed to estimate the mismatched and matched lumped disturbances in RSV kinematics and dynamics, respectively. Through feed-forward disturbance compensation, the proposed controller is not only robust to model uncertainty and environmental disturbances but also insensitive to measurement uncertainties. Although the finite-time disturbance observer can accurately estimate the disturbance on the robotic arm, its convergence time is related to the initial observation error, which limits its applicability. To our knowledge, the convergence time of fixed-time disturbance observers (FDO) is independent of the system’s initial state, which has been widely used to estimate the system uncertainties of robotic arm systems in recent years. In ref. [
24], a second-order sliding mode disturbance observer was designed based on fixed-time theory to enable small unmanned aerial vehicles to perform stable hovering operations, even in the presence of external interferences, compared to traditional disturbance observer-based sliding mode control methods; this control method has a faster convergence speed. In ref. [
25], a uniform, robust exact differentiator was used to design a fixed-time observer, which can accurately estimate the uncertainty of the robotic arm system within a fixed timeframe. These are application examples of fixed-time disturbance observers. Currently, many scholars are studying fixed-time disturbance observers, and using fixed-time disturbance observers to solve the system uncertainty in robotic arm systems is still an open topic.
Based on the above discussion, in this article, a controller based on a fixed-time disturbance observer is proposed for the trajectory tracking control problem of n-joint robotic arms with system uncertainties. The key contributions of this research are outlined as follows:
- (1)
Backstepping-based fixed-time tracking control is proposed in this article. Combining the backstepping, observer, and fixed time theory effectively improves the convergence speed and tracking precision of the robotic arm.
- (2)
A fixed-time disturbance observer is designed to accurately estimate the system uncertainties existing in the robotic arm system, which provides compensation for the controller, thus improving the tracking performance and robustness of the robotic arm system. Meanwhile, we introduce the hyperbolic tangent function to avoid the observation chattering effect.
3. Controller Design without System Uncertainties
The controller without system uncertainties can be designed as follows:
Step 1: Define the position error
between the actual position and the desired trajectory of the robotic arm as follows:
According to Equations (10) and (11), the derivative of Equation (13) is:
Select the kinematic controller
as follows:
where
,
and
are positive constants, and
and
are positive odd numbers with
.
Consider the Lyapunov function as follows:
The time derivative of
is:
where
,
,
,
and
.
Combined with Lemma 1, the position tracking error
is able to achieve stability at a fixed time, and the convergence time
can be calculated by:
Step 2: Define the speed error
as follows:
According to Equation (10), we further have:
The control strategy
can be designed as follows:
where
.
Substituting Equation (21) into Equation (20) yields:
Consider the Lyapunov function as follows:
The time derivative of
is as follows:
where
,
,
,
and
.
According to Lemma 1, the speed tracking error
can reach zero, and the convergence time of
is calculated as follows:
Theorem 1. The B-FTTC strategy (21) designed in this article enables the robotic arm to track the desired trajectory within a fixed time , and the convergence time is independent of the system’s initial state.
Proof. Select the following Lyapunov function:
Take the time derivative of
as follows:
According to Lemma 4, we further have:
where
,
,
,
,
,
,
and
.
Combined with Lemma 2, this control system is globally fixed-time stable, and the convergence time
can be calculated by:
It follows that the B-FTTC strategy designed in this section enables the robotic arm to track the desired trajectory within a fixed time.
That brings an end to the proof. □
5. Simulation Verification
In this section, to verify the effectiveness of the proposed method, we will use two cases to verify the effectiveness of the proposed method:
Case 1: Verify the control method proposed in this article with different initial conditions.
Case 2: Compare with the control method proposed in ref. [
21] to verify the effectiveness of the control method proposed in this article.
The planar model of a two-link rehabilitation robot [
21] is given in
Figure 2.
According to ref. [
30], we can use the Euler–Lagrange equation to express the dynamics of the rehabilitation robot as Equation (8), where:
The parameters of the robotic arm are shown in
Table 1.
In order to validate the proposed control scheme in this article, we used MATLAB R2018b for simulations and the processor of the computer was an Intel(R), Core(TM), i7-8550U, CPU @ 1.80 GHz, 1.99 GHz. We conducted simulations using the aforementioned model, and the initial conditions were set as follows:
The controller and observer parameters were set as follows: , , , , , , , , , , , and . The simulation time was set for 40 s, and the value of the constrained force vector was assumed as .
Case 1: compared with different initial states in the proposed method.
To verify that the convergence time of the system is independent of the initial state with the designed control law, the algorithm proposed in this article was validated with different initial states,
,
and
, and the simulation results are shown in
Figure 3 and
Figure 4.
The simulation results in
Figure 3 and
Figure 4 validate the correctness and effectiveness of the designed control algorithm. With different initial states, the controller achieved system stability and control objectives within a fixed time.
Case 2: compared with ref. [
21].
The controller designed in ref. [
21] is as follows:
The controller parameters are set as follows:
, , , , , , , , , .
In order to demonstrate the efficacy of the designed controller, a comparative analysis was performed with the adaptive non-singular terminal sliding mode control (NSTSMC) proposed in [
21].
Figure 5,
Figure 6,
Figure 7,
Figure 8 and
Figure 9 show the simulation results of the experiment.
The trajectory tracking and error of joints 1 and 2 are shown in
Figure 5 and
Figure 6, where
and
represent the actual motion trajectory and desired motion trajectory of joint 1 of the robotic arm, respectively, and
and
represent the actual motion trajectory and desired motion trajectory of joint 2 of the robotic arm, respectively. By analyzing
Figure 5 and
Figure 6, it is evident that the control method proposed in this article exhibits higher accuracy in tracking errors
and
compared to the NSTSMC.
Additionally, both controllers successfully tracked the desired trajectory with favorable tracking response characteristics. Under both methods, the position error of the robotic arm achieved global time convergence. The convergence time of the NSTSMC was about 2.5 s, while the method in this article had a faster convergence speed of about 0.43 s, and the convergence time was less than the set time min (
) = 1.74 s. This is 82.8% faster than the NSTSMC. This indicates the superior tracking performance of the proposed control method in tracking the desired trajectory. In
Figure 7, the control inputs based on the proposed control method in this article and the NSTSMC are shown. The black and blue lines represent the proposed control inputs, while the red and purple lines represent the compared control inputs. In the initial stage, the compared control inputs show less chattering than the proposed control inputs. When the system is stable, both control methods achieve stable control inputs. This indicates the robustness of the proposed control law with system uncertainties.
Figure 8 and
Figure 9 show the observation effects of the system uncertainties of joints 1 and 2, respectively, where
and
represent the actual value and observed value of
of joint 1, respectively, and
and
represent the actual value and observed value of
of joint 2, respectively. By analyzing
Figure 8 and
Figure 9, it is evident that the designed disturbance observer can accurately estimate the system uncertainties present in the system, and real-time compensation can be performed on the control inputs, effectively enhancing the control precision and convergence speed of the system.
In summary, the above simulation results verify the correctness and effectiveness of the designed controller and disturbance observer.