Trajectory tracking of free-flying space manipulators using time-delay logarithmic sliding mode control

Abstract

In space operations, the requirements for control precision, stability, and robustness of free-flying space manipulators are critical for mission success. However, due to unknown system dynamics and external disturbances, control performance may degrade. To address these challenges, this paper proposes a new sliding mode control method to achieve fast and accurate tracking for space manipulators, even in the presence of lumped disturbances. In this method, the sliding surface design incorporates the value of the sliding surface at the previous sampling time, and the resulting surface is referred to as the time-delay logarithmic sliding surface. This design enables the proposed time-delay logarithmic sliding mode control method to improve the robustness of the closed-loop systems without the need for any approximation or compensation mechanisms, and improve tracking accuracy without introducing complex structures. Numerical simulations are conducted to demonstrate the effectiveness of the proposed approach in improving tracking performance for free-flying space manipulators.

Graphical Abstract

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Nomenclature

 
• FFSM

  Free-flying space manipulators

 
• TD-LnSS

  Time-delay logarithmic sliding surface

 
• TD-LnSMC

  Time-delay logarithmic sliding mode control

 
• FoSMC

  Fractional-order sliding mode control

 
• LnSMC

  Logarithmic sliding mode control

 
• NNs

  Neural networks

 
• ANN

  Adaptive neural network

 
• ASMDO

  Adaptive sliding mode disturbance observer

 
• FTSMC

  Fuzzy timing sliding mode control

 
• ANFIS

  Adaptive neuro-fuzzy inference system

 
• Symbols

   

 
• $a_n$

  Position of the centre of the $n$th link

 
• $L_n$

  Length of the $n$th link

 
• $f_n$

  Position of the centre of mass of the $n$th link

 
• $q_m$

  Attitudes of the spacecraft base

 
• $q_n$

  Joint angle of the manipulator

 
• ${\tau }_n$

  Joint torque of the manipulator

 
• $q_d$

  Desired reference for the angle vector

 
• $\mathbf{M}$

  Positive-definite inertia matrix

 
• $\mathbf{C}$

  Centrifugal and coriolis matrix of the manipulator

 
• $\Vert \cdot \Vert$

  Euclidean norm of a vector

 
• $\ln (\cdot )$

  Natural logarithm function

 
• $V_s$

  Sliding surface of the Lyapunov functions

 
• $V_e$

  Tracking error of the Lyapunov functions

1. Introduction

The increasing scope of space exploration has led to a rising demand for robotic systems capable of performing complex tasks such as supporting astronaut operations (Flores Abad et al. 2014), repairing, or rescuing satellites (Papadopoulos et al. 2021), and capturing space debris (Zhang et al. 2023). Free-flying space manipulators (FFSMs) play a key role in these missions, as they provide spacecraft with the flexibility to execute precision operations without relying on a fixed base (Wu et al. 2018). However, unlike terrestrial robotic arms, FFSMs operate in an environment where their movement is not isolated from the dynamics of the spacecraft itself, resulting in a strong coupling effect between the manipulator and the base (Jiang et al. 2015). This inherent coupling, combined with unknown system dynamics and external disturbances such as gravity gradients and atmospheric drag, introduces significant challenges in achieving accurate trajectory tracking (Zhongyi et al. 2008). As a result, developing robust control techniques that can achieve precise trajectory tracking despite these disturbances remains a critical yet challenging task.

To address the challenges mentioned above, various control methods, including adaptive strategies (Li et al.2024), robust control (Chen & Liu 2021), and intelligent techniques (She et al. 2021), have been proposed. For example, Chen & Liu (2021) introduced a Smith predictive control method based on time-delay prediction, effectively predicting the system’s time delay.

To further address communication challenges, Chen et al. (2021) presented an optimal communication link identification and minimum time-delay realization method, which, although still to be validated in real-world environments, shows good scalability. With the continuous advancement of neural networks (NNs), a novel adaptive neural network framework based on quantum interference principles, as presented in She et al. (2021), has been developed to improve the performance of FFSMs under high training velocities. While these strategies can effectively address system dynamics and external disturbances, the control performance often relies on precise modelling or parameter tuning. Sliding mode control (SMC) has emerged as a powerful tool with inherent robustness to lumped disturbances, such as Nicolis et al. (2020), which combined SMC with model predictive control to address unmodelled system dynamics and disturbances. Hu et al. (2024) introduced a fuzzy timing sliding mode control method that effectively suppresses modelling approximation errors, thus improving the system’s trajectory tracking performance. Furthermore, Pukdeboon & Zinober (2012) presented a novel integral SMC method tailored for quaternion-based spacecraft attitude tracking maneuvers, particularly in the presence of external disturbances and an uncertain inertia matrix, further optimizing the precision of trajectory tracking. In traditional SMC methods, control laws are designed with a switching term to ensure rapid convergence of system states to the designed surface, offering quick response and inherent robustness. These merits make traditional SMC particularly well suited for tasks requiring high precision and fast convergence. However, the discontinuous nature of the switching control law often leads to chattering, which not only affects system performance but may also cause wear and tear in mechanical components in practical implementations.

To mitigate the chattering inherent in SMC methods while achieving accurate tracking control of FFSMs, advanced approaches incorporating approximators and compensation mechanisms have been developed. These methods compensate for lumped disturbances, enabling smaller switching gains to reduce chattering without sacrificing robustness or accuracy. For instance, Jin & Sun (2008) designed a controller based on unit quaternion attitude parametrization, which ensures finite-time reachability of the desired attitude motion despite lumped disturbances. To further enhance performance, Hu et al. (2014) introduced a second-order disturbance observer, which reconstructs lumped disturbances with zero error in finite time, enabling the system to converge to the specified time-varying sliding mode surface even under actuator input saturation and misalignment. In addition, an adaptive sliding mode disturbance observer (Zhu et al. 2019) has been proposed for compensating and controlling unknown model dynamics and complex dynamic characteristics. Moreover, Selma et al. (2020) developed a hybrid controller for quadrotor unmanned aerial vehicle (UAV) tracking, combining a robust adaptive neuro-fuzzy inference system with a particle swarm optimization algorithm, aiming to reduce tracking errors and improve control performance. In addition, the authors in Shao et al. (2021) designed a disturbance observer and constructed a sliding surface using fractional-order integration to improve the overall system’s robustness and transient performance. Xu & Wu (2024) proposed a supervisory disturbance observer that estimates sudden disturbances and suppresses their effects using only joint position sensor data, with a virtual disturbance measurement incorporated into the monitoring algorithm to quickly detect disturbance changes. To reduce tracking errors caused by unknown model dynamics, a composite controller based on the fully actuated system approach, integrating an inner loop nonlinear disturbance observer and an outer loop high-precision trajectory controller, has been proposed in Tian et al. (2024). Finally, Muñoz Palomeque et al. (2024) presented four hybrid control strategies using a radial basis function NN and conventional regulators to address performance limitations caused by mechatronics and external disturbances, thereby reducing vibrations and improving the system’s responsiveness. However, the introduction of observers or compensators often leads to overly complex structures and an increased number of controller parameters. Therefore, developing an effective control method that is capable of accurately tracking the desired trajectory while minimizing control complexity is essential.

Time-delay control techniques, which differ from the inherent time-delay phenomena in control systems, have been widely used to improve control performance due to their simple structure and ability to compensate for unknown dynamics and disturbances. In this method, past system states and control inputs are utilized to approximate and mitigate uncertainties, reducing the reliance on an accurate system model. For example, building on this method, Yang et al. (2024) proposed a new sliding mode control (TDSMC) that improves surface convergence speed and minimizes steady-state error by representing the sliding surface at previous sampling moments using the sliding variable value, thereby improving the system’s tracking performance and robustness. To deal with parameter variations and disturbances in the robot manipulators, Lee et al. (2017) proposed an adaptive robust controller based on adaptive integral SMC and time-delay estimation. Motivated by these approaches, as shown in Figure 1, a time-delay logarithmic sliding mode control (TD-LnSMC) method is designed in this paper to address the trajectory tracking control problem of FFSMs in the presence of unknown system dynamics and external disturbances. The main contributions of this paper are as follows:

1. A novel SMC method that does not require any approximation or estimation strategies is proposed, simplifying the control structure compared to the methods presented in Ma et al. (2024).

2. A time-delay sliding surface is designed to ensure satisfactory tracking performance even in the presence of lumped disturbances.

3. The proposed method requires tuning only the parameters of the equivalent control term in the sliding mode controller, reducing the complexity and effort associated with parameter tuning.

The remainder of the article is organized as follows: Section 2 describes the dynamic model of space manipulators and the design of a sliding mode controller based on the proposed method. Section 3 provides the stability analysis of the proposed controller. Section 4 discusses the simulation results and demonstrates the advantages of the proposed method. Finally, Section 5 concludes the work.

2. Preliminaries

2.1 Dynamic model of space manipulator

As shown in Figure 2, the FFSMs consist of a movable base and a manipulator with $n$ joints mounted on the base. The vector $a_n$ denotes the position of the centre of the $n$th link relative to a reference coordinate system, while $a_0$ represents the position of the spacecraft base in the reference frame. The constant $e_0$ defines the fixed offset between the spacecraft and the reference bases. The length of the $n$th link is denoted as $L_n$⁠. $f_n$ represents the position of the centre of mass of the $n$th link relative to the point where it connects to the preceding link. In the microgravity space environment, gravity is nearly negligible, therefore, the potential energy of the FFSMs is not considered in the modelling process. Considering unknown system dynamics and external disturbances, the dynamics model of the FFSMs is given as follows:

where $q=[q_m^T,q_n^T{]}^T\in {\mathbb{R}}^{3+n}$⁠, $q_m\in {\mathbb{R}}^3$ is the attitudes of the spacecraft base, and $q_n\in {\mathbb{R}}^n$ is the joint angle of the manipulator. $\tau =[{\tau }_b^T,{\tau }_m^T{]}^T\in {\mathbb{R}}^{3+n}$⁠, where ${\tau }_b=[0,0,0{]}^T\in {\mathbb{R}}^3$ is the torque of the spacecraft base, and ${\tau }_m=[{\tau }_0,{\tau }_1,\cdots ,{\tau }_{n-1}{]}^T\in {\mathbb{R}}^n$ is the joint torque of the manipulator. The matrix

⁠, is the inertia matrix, $M_{bb}\in {\mathbb{R}}^{3\times 3}$ is the inertia matrix of the spacecraft, $M_{bm}\in {\mathbb{R}}^{3\times n}$ is the coupled inertia matrix of the spacecraft and the manipulator, and $M_{mm}\in {\mathbb{R}}^{n\times n}$ is the inertia matrix of the manipulator.

$$C(q,\dot{q})=\left[\begin{array}{cc}{C}_{bb}& C_{bm}\\ C_{mb}& C_{mm}\end{array}\right]\in {\mathbb{R}}^{(3+n)\times (3+n)}$$

⁠, where $C_{bb}\in {\mathbb{R}}^{3\times 3}$ is the centrifugal and Coriolis matrix of the spacecraft, $C_{bm}\in {\mathbb{R}}^{3\times n}$ is the coupled centrifugal and Coriolis matrix of the spacecraft and the manipulator, $C_{mb}\in {\mathbb{R}}^{n\times 3}$ is the coupled centrifugal and Coriolis matrix of the manipulator and the spacecraft, $C_{mm}\in {\mathbb{R}}^{n\times n}$ is the centrifugal and Coriolis matrix of the manipulator. $h(q,\dot{q})$ and $d$ represent the unknown system dynamics and external disturbances, respectively.

Let $q_d$ denote the desired reference for the angle vector $q$ of the FFSMs. The control objective of this work is to design a sliding control strategy for the FFSMs of Equation 1 with unknown system dynamics and external disturbances such that the manipulator angle $q$ tracks its desired reference $q_d$⁠. To achieve this goal, this research makes the following assumptions.

    

3. Control Design

This section provides a detailed description for the design of proposed sliding mode controller for solving the tracking problem of space manipulators. First, this research introduce the novel time-delay logarithmic sliding surface (TD-LnSS) and controller design, followed by a discussion of the control performance and a theoretical stability analysis.

3.1 Time-delay logarithmic sliding model control design

This subsection demonstrates the design of the TD-LnSM control strategy. Before desiging the proposed sliding mode surface, the following tracking errors are defined:

In Yang et al. (2024) and Ma et al. (2024), an LnSMC strategy was proposed, which ensures that the tracking errors rapidly converge to a small neighbourhood of the equilibrium without requiring the design of a switching term. However, the robustness of LnSMC strategies to unknown system dynamics and external disturbances depends on the design of approximation or estimation techniques. To simplify the controller design and improve control performance, the following novel sliding surface, i.e. TD-LnSS is proposed:

where $\beta$⁠, $k_1$⁠, and $k_2\in {\mathbb{R}}^{+}$ are constants to be designed, $S_t$ refers to the historical measurement of $S$ at $t-T$⁠, where $T$ is a positive constant, and $\ln (\cdot )$ is the natural logarithm function. Although $\frac{q_e}{\Vert q_e\Vert }$ may introduce chattering, the logarithmic term $\ln (k_1\Vert q_e\Vert +1)$ acts as a gain-scheduling mechanism. Specifically, as $\Vert q_e\Vert \to 0$⁠, the logarithmic function also tends to zero, smoothly reducing the impact of the discontinuous component and mitigating high-frequency switching.

According to Equation 3, the derivative of the tracking error over time can be written as follows:

According to the designed TD-LnSS in Equation 3 and the time derivative of tracking error as shown in Equation 4, the tracking controller for the FFSMs, modelled by Equation 1, the control input is designed as follows:

where $k_3\in {\mathbb{R}}^{+}$ is a positive constant to be designed.

  

3.2 Stability and performance analysis of the proposed TD-LnSMC controller

This section demonstrates the stability of the closed-loop system under the designed controller. It also discusses the ability of the proposed TD-LnSM control scheme to achieve accurate tracking control without requiring any approximation or compensation mechanisms to handle unknown system dynamics and external disturbances.

The following theorem shows that the sliding variable can converge to a bounded region around the origin by designing appropriate parameters.

 

Design a Lyapunov function associated with the sliding mode surface of Equation 3 as follows:

By differentiating the sliding surface of Equation 3 with respect to time, the following derivative is obtained:

The derivative of the Lyapunov function is given as follows:

Substituting Equation 7 into Equation 8 yields:

According to the tracking error defined in Equation 2, Equation 9 can be rewritten as follows:

According to the dynamics of the FFSM in Equation 1, the following holds:

According to the controller of Equation 5, it follows that

According to Assumption 2, the term $\frac{h(q,\dot{q})+d}{\mathbf{M}(q)}$ is assumed to be bounded, i.e. $\Vert \frac{h(q,\dot{q})+d}{\mathbf{M}(q)}\Vert \le ϵ$⁠, where $ϵ$ is a positive constant. This simplifies the design process and reduces reliance on conservative assumptions, allowing the controller to be more flexible and efficient in cases where lumped disturbances exist.

By applying Young’s inequality to Equation 12, the following inequality holds:

From Equation 13, it yields:

where $\alpha =2k_3-1$⁠. It can be observed that the function $V_s(t)$ converges to $\frac{{ϵ}^2}{2\alpha }$ as time approaches infinity.

According to the Lyapunov function designed in Equation 6, it can be derived that:

Thus, this analysis can conclude that the sliding surface designed in Equation 3 for the system described by Equation 1 under the controller designed in Equation 5 converges to a compact set, i.e. ${\mathrm{\Omega }}_S=\{S\mid \Vert S\Vert \le \sqrt{\frac{{ϵ}^2}{2k_3-1}}\}$⁠, where the bound is determined by the values of $ϵ$ and $k_3$⁠. Under certain lumped disturbances, increasing the controller parameter $k_s$ can result in a smaller region ${\mathrm{\Omega }}_S$⁠. This research can choose a sufficiently large controller parameter $k_s$ to ensure that the region ${\mathrm{\Omega }}_S$ becomes sufficiently small, indicating that the TD-LnSS in Equation 3 is asymptotically stable.

This completes the proof.

The following theorem shows that through the designed sliding surface, the tracking error will converge to a very small region around zero, despite that the sliding surface converges to a bounded region.

 

Design a Lyapunov function for the tracking error of Equation 4 as follows:

Then, the derivative of Equation 16 can be defined as:

Substituting Equation 4 into Equation 17 yields:

Because $-\Vert q_e^T\Vert \beta \ln (k_1\Vert q_e\Vert +1)$ is always negative, for the convenience of calculation, only the last two terms are considered. Therefore, the following inequality can be derived:

Define $\sigma =S-S_t$⁠. When the sliding surface $S$ is bounded, the $S_t$ is also bounded. Therefore, it can assume that the error $\sigma$ is also bounded. As a result, Equation 19 can be reformulated as:

Now, let us analyse the inequality of Equation 20. By replacing the error $q_e$ with $x$⁠, the function $f(x)=-k_2{x}^2+\sigma x$ is defined. Solving the equation gives two solutions: $x_1=0\text{and}{x}_2=\frac{\sigma }{k_2}.$ Therefore, when $0\le x\le \frac{\sigma }{k_2}$⁠, $f(x)$ increases and it decreases when $x\ge \frac{\sigma }{k_2}$⁠. Moreover, when the parameter $k_2$ is set, a lower positive constant $\sigma$ leads to a smaller $x$⁠. Consequently, a smaller tracking error $q_e$⁠. The value of $\sigma$ is determined by the constant $T$⁠, the smaller the value of $T$⁠, the smaller the value of $\sigma$⁠. Note that when the sliding surface $S$ stabilizes at a constant value, $\sigma$ becomes zero. As a result, the tracking error can be significantly minimized by using the proposed method without relying on an observer or approximator.

This completes the proof.

To further illustrate how the proposed sliding surface effectively deals with lumped disturbances, this research conduct a simulation study to compare the proposed TD-LnSMC sliding surface with several existing sliding surfaces, such as LnSMC, linear sliding mode control (LSMC), and fractional-order sliding mode control (FOSMC). Figure 3 shows a comparison of tracking errors dynamics between the proposed sliding surface and LnSMC, LSMC, and FOSMC with the sliding surface value being $S=0.2$ of the upper plot and $S=0.1\sin (t)$ of the lower plot. The solid blue lines represent the proposed TD-LnSMC sliding surface, while the dotted red, dashed orange, and dotted purple lines correspond to LnSMC, LSMC, and FOSMC, respectively. It can be observed that the proposed TD-LnSMC sliding surface performs better in both steady-state and transient performance compared to the other three sliding surfaces.

4. Simulation Study

This section uses a two-link space manipulator as an example to demonstrate the effectiveness of the proposed TD-LnSMC strategy developed in Section 3. It begins with a description of the manipulator control system model, followed by an introduction of several existing SMC methods chosen for comparison. Finally, three cases (i.e. constant trajectory tracking in Section 4.3, time-varying trajectory tracking in Section 4.4, and the influence of controller parameters in Section 4.5) are considered to illustrate the effectiveness of the proposed strategy compared to the existing methods, respectively.

4.1 Description of the manipulator control system

To validate the effectiveness of the proposed TD-LnSMC strategy in achieving accurate control of the space manipulator in spite of the presence of lumped disturbances, this section provides a brief description on a two-link space manipulator, as shown in Figure 4. This space manipulator arm is mounted on a base spacecraft with two articulating joints. Specifically, the orientation of the base is represented by $q_0$⁠, which defines the position and orientation of the base spacecraft in the reference frame, ensuring that the manipulator maintains a stable posture during space operations. Additionally, the first joint, denoted by $q_1$⁠, enables pivotal movement of the initial segment directly connected to the base, while the second joint, denoted by $q_2$⁠, further articulates to position the end of the arm towards a target. The dynamics of the space manipulator are described by Equation 1, with its parameters detailed in Table 1.

This simulation also verifies the effectiveness of the proposed method in handling lumped disturbances. Specifically, disturbance (⁠$d=[15\cos (10t),15\sin (10t),15\sin (10t){]}^T$⁠) is introduced after 10 s. Furthermore, considering the actuator limitations in practical systems, the friction torque $\tau$ is assumed to be limited to $500\mathrm{N}\cdot \mathrm{m}$⁠. This simulation is performed by solving the system’s differential equations using the forward difference method, with a sampling time of $0.001\mathrm{s}$ to ensure numerical accuracy.

4.2 Design of comparison methods

The proposed sliding surface is designed as Equation 3, and the proposed TD-LnSM controller is designed as Equation 5. To show the ability of the proposed method in dealing with the lumped disturbances, this section provides a brief description on the sliding surface and controller of the several existing control methods, including LSMC, FOSMC (Kuang et al. 2021), and LnSMC (Ma et al. 2024). The sliding surfaces and controllers of these methods are described as follows:

1. LnSMC. Natural logarithmic sliding mode control incorporates a logarithmic term, effectively reducing chattering and improving robustness. Tts sliding surface is designed as $S=\beta \ln (k_1\Vert q_e\Vert +1)\frac{q_e}{\Vert q_e\Vert }+k_2{q}_e+{\dot{q}}_e$⁠, and the controller is given by $\tau =\mathbf{M}(q)(-\frac{\beta k_1{\dot{q}}_e}{k_1\Vert q_e\Vert +1}+{\ddot{q}}_d-k_2\dot{q_e}-k_{3}S+\mathbf{C}(q,\dot{q})\dot{q}$⁠.

2. LSMC. The linear sliding mode control is a simple and commonly used method. Its sliding surface is designed as $S={\dot{q}}_e+k_1{q}_e$⁠, and the controller is given by $\tau =\mathbf{M}(q)(-k_{2}S+{\ddot{q}}_d-k_1\dot{q_e})+\mathbf{C}(q,\dot{q})\dot{q}$⁠.

3. FOSMC. Fractional-order sliding mode introduces a fractional-order parameter, providing additional flexibility in controller design, enabling more precise adjustments to meet specific performance requirements. Tts sliding surface is designed as $S=\dot{q_e}+k_2{I}_t^{\alpha }{q}_e+k_1{q}_e$⁠, and the controller is given by $\tau =\mathbf{M}(q)(-k_{3}S+{\ddot{q}}_d$$-k_2{\left(I_t^{\alpha }{q}_e\right)}^{\prime }-k_1\dot{q_e})+\mathbf{C}(q,\dot{q})\dot{q}$⁠.

4.3 Constant trajectory tracking

To evaluate the tracking performance of the closed-loop control system for a space manipulator under the proposed TD-LnSMC strategy in steady-state conditions, the reference signal in this simulation is chosen as a constant to represent a static target scenario. Specifically, the reference joint positions are set as $q_d={[\frac{\pi }{4},\frac{\pi }{6},\frac{\pi }{6}]}^T$⁠, with the corresponding joint velocities maintained at ${\dot{q}}_d=[0,0,0{]}^T$⁠. The lumped disturbances considered in this simulation are described in Section 4.1. To demonstrate the effectiveness of the proposed method, several existing approaches, including LSMC, FOSMC, and LnSMC, as demonstrated in Section 4.2, are chosen for comparison. For a fair comparison, the parameters shared across these methods are kept identical, while the remaining parameters are optimized using the uniform design-based approach presented in Wang et al. (2024). The specific parameter settings for each method are demonstrated in Table 2.

The simulation results are presented in Figures 5–8. Specifically, Figures 5–7 illustrate the evolution of the base orientation angle $q_0$⁠, the angle of the first joint $q_1$⁠, and the angle of the second joint $q_2$⁠. In these figures, the solid blue lines represent the constant reference signals, while the solid red, dashed orange, dotted purple, and dashed green lines correspond to the angle trajectories obtained using the proposed TD-LnSMC strategy, LnSMC, LSMC, and FOSMC methods, respectively. Figure 5 illustrates the trajectories and tracking errors for the orientation angle $q_0$⁠. While the proposed method exhibits a slight overshoot due to saturation in the control inputs, the tracking error is the smallest even with unknown model dynamics. Furthermore, after external disturbances are introduced at $t\ge 10\mathrm{s}$⁠, significant fluctuations in tracking errors are observed under the comparison methods, whereas the proposed method achieves superior control performance. This robustness is attributed to the sliding surface design, which compensates for lumped disturbancess through the inclusion of a time-delay term, ensuring satisfactory tracking performance. Similarly, the trajectories and tracking errors of the first and second joint angles, shown in Figures 6 and 7, demonstrate that the proposed method achieves accurate tracking of the reference signals despite the presence of unknown system dynamics and external disturbances, showing that the proposed method effectively stabilizes the joint angles at the desired constant positions under such conditions.

The control inputs under these SMC strategies are presented in Figure 8, where the first subfigure corresponds to the manipulator ${\tau }_0$⁠, the second to ${\tau }_1$⁠, and the third to ${\tau }_2$ under different SMC strategies. All control inputs remain within the constraints. After the introduction of external disturbances at $t\ge 10\mathrm{s}$⁠, the control inputs exhibit fluctuations, providing additional energy to compensate for the disturbances. These simulation results show that the proposed method can track the constant reference signals with greater accuracy compared to the comparison methods, while requiring a similar level of control input, showing its capability to address the constant reference tracking problem of space manipulators, even in the presence of unknown system dynamics and external disturbances.

4.4 Time-varying trajectory tracking

To further evaluate the tracking performance of the proposed method, the performance of four methods was compared under the time-varying reference signal, In this scenario, the unknown system dynamics and external disturbances also considered, and they are described described in Section 4.1. The reference joint positions are set as $q_r={[\frac{\pi }{4}+0.8\sin 0.8t,\frac{\pi }{6}+0.8\cos 0.8t,\frac{\pi }{6}+0.8\sin 0.8t]}^T$⁠, with the corresponding joint velocities defined as ${\dot{q}}_r=[0.64\cos 0.8t,-0.64\sin 0.8t,0.64\cos 0.8t{]}^T$⁠, and the initial joint position is $q={[\frac{\pi }{4},\frac{\pi }{6},\frac{\pi }{6}]}^T$⁠. The specific parameter settings for each method as shown in Table 2 are same as those in Section 4.3.

Simulation results of the space manipulator are shown in Figures 9–11, presenting the base orientation angle $q_0$⁠, the angle of the first joint $q_1$⁠, and the angle of the second joint $q_2$⁠, similar to those in Section 4.3. In these figures, the solid blue lines represent the time-varying reference signals, while the dotted red, dashed orange, dotted purple, and solid green lines correspond to the angle trajectories obtained using the proposed TD-LnSMC strategy, LnSMC, LSMC, and FOSMC methods, respectively. Figure 9 illustrates the response of the orientation angle $q_0$ and its tracking error under a time-varying signal. Although the convergence speed of the proposed method is similar to that of LnSMC, the proposed method exhibits obviously smaller fluctuation ranges after the introduction of external disturbances, outperforming the other three methods. Similarly, Figures 10 and 11 present the tracking trajectories and errors of angles $q_1$ and $q_2$⁠. It can be observed that the proposed method quickly converges and accurately tracks the desired trajectory under unknown model dynamics, with only slight fluctuations. Even after external disturbances are introduced, the tracking trajectory of the proposed method remains close to the desired trajectory, with negligible fluctuations. In contrast, LnSMC, LSMC, and FOSMC exhibit significant fluctuations after the introduction of external disturbances. Particularly after $t=10\mathrm{s}$ in Figures 10 and 11, the fluctuation range remains larger compared to the proposed method. This further verifies the ability of the proposed method to maintain stable joint angles and achieve higher tracking performance under lumped disturbances. Figure 12 shows the control inputs for $q_0$⁠, $q_1$⁠, and $q_2$ under different methods. All control inputs remain within the constraints, and after the introduction of external disturbances, the control inputs exhibit larger fluctuations to meet the trajectory tracking requirements.

Additionally, a quantitative analysis is presented by using the tracking error performance indices, such as ISE (integral of squared error), IAE (integral of absolute error), and IATE (integral of time-weighted absolute error). ISE is the integral of the square of the error signal, used to measure the total energy of the error over the entire time period. A smaller ISE value indicates that the system error is smaller, and the control system performance is better. IAE is the integral of the absolute value of the error, focusing on the total magnitude of the error without considering its sign. A smaller IAE value indicates that the system’s error fluctuations are smaller, and the accumulation of errors is reduced. IATE is the integral of the absolute value of the error multiplied by time, meaning that errors later in time contribute more to the index. They are defined as follows:

where $t_0=0\mathrm{s}$ and $t_f=20\mathrm{s}$⁠.

As shown in Table 3, the tracking error performance indices (ISE, IAE, and IATE) for four control methods (proposed method, LnSMC, LSMC, and FOSMC) at different angles (⁠$q_0$⁠, $q_1$⁠, and $q_2$⁠) are presented. From the table, it can be observed that the proposed method has lower error values across all joint angles, especially in terms of ISE and IAE. For example, at angle $q_0$ of the base, the ISE, IAE, and IATE values for the proposed method are 2.99, 27.79, and 20.93, which are lower than those for the other methods. Similarly, at joint angles $q_1$ and $q_2$⁠, the proposed method shows better performance with smaller error values. Overall, the proposed method demonstrates more accurate and stable control compared to the other methods, particularly in reducing later-stage errors.

Figure 13 illustrates a comparison of the three dimensional trajectory tracking performance of the space manipulator under various control methods, including the proposed TD-LnSMC method, LnSMC, LSMC, and FOSMC. In the figure, the blue solid line represents the reference trajectory $q_r$⁠, the red dashed line corresponds to the actual trajectories under each method, and the green square marks the initial position. Specifically, the trajectories under the LnSMC, LSMC, and FOSMC methods exhibit significant tracking errors, particularly during dynamic adjustment phases. In contrast, the proposed TD-LnSMC achieves superior tracking performance with the smallest tracking errors. In summary, the proposed method demonstrates clear advantages in trajectory tracking tasks. It achieves fast convergence and closely adheres to the reference trajectory, while the other methods perform less effectively during dynamic adjustment phases. These results show that the proposed method achieves higher tracking accuracy compared to the other three methods.

4.5 The influence of controller parameters

To study the impact of controller parameters on tracking performance and control input, a set of smaller controller parameters was selected for comparative simulation in Table 4, and the correspoding simulation results are shown in Figures 14 and 15.

Specifically, Figure 14 presents the base orientation angle $q_0$⁠, the angle of the first joint $q_1$⁠, and the angle of the second joint $q_2$⁠, and the solid blue lines represent the time-varying reference signals, while the dotted red, dashed orange, dotted purple, and solid green lines correspond to the angle trajectories obtained using the proposed TD-LnSMC strategy, LnSMC, LSMC, and FOSMC methods, respectively. Figure 15 shows the control inputs for ${\tau }_0$⁠, ${\tau }_1$⁠, and ${\tau }_2$⁠. The solid blue lines, dashed red, dashed orange, and dotted purple lines correspond to the control inputs obtained using the proposed TD-LnSMC, LnSMC, LSMC, and FOSMC methods, respectively. Compared to the case where the controller parameters are designed to be large (see tracking performance shown in Figures 9–11, and their corresponding control inputs shown in Figures 8 and 12), the smaller controller parameters can result in smaller initial control inputs. Although the overall control performance in this case has declined, the comparison results shown in Figure 15, and Figure 14 demonstrate that the proposed method still has clear advantages. Additionally, to further research the impact of practical factors, this case added measurement errors, such as Gaussian noise with a mean of 0 and a variance of 0.0025. Compared to the tracking performance shown in Figure 14, the fluctuation is more pronounced in Figure 16, but the tracking performance and robustness of the proposed method still appear to be better than the other methods.

To demonstrate the influence of parameters on tracking performance using the proposed SMC method, comparison results under different controller parameters are provided. Specifically, the controller parameters $k_i$ (⁠$i=1,2,3,4$⁠) are considered. Each parameter is varied individually, while keeping the others fixed to analyse its impact on control performance. In particular, $k_1$ is set to $0.3,1,2$ while $k_2=0.7$⁠, $k_3=0.7$⁠, and $k_4=0.6$ are kept fixed. These parameter values are applied to the proposed method, and closed-loop simulations are conducted. Similarly, $k_2$ is set to $0.3,1,\mathrm{and}2$ while $k_1=0.7$⁠, $k_3=0.7$⁠, and $k_4=0.6$ are kept fixed. Likewise, for $k_3$⁠, it is set to $0.3,1,\mathrm{and}2$⁠, respectively, while keeping $k_1=0.7$⁠, $k_2=0.7$⁠, and $k_4=0.6$ fixed. Finally, $k_4$ is set to $0.2,0.6,\mathrm{and}9$⁠, while keeping $k_1=0.7$⁠, $k_2=0.7$⁠, and $k_3=0.6$ fixed.

The comparison results from these simulations are shown in Figure 17. It can be observed that although $k_1$ introduces a large overshoot and rise time, its tracking performance is not very good, but its variation has little impact on the system’s tracking performance. The overshoot and rise time of the system increase as $k_2$ increases, but the system cannot maintain good tracking performance. For $k_3$⁠, an increase in the parameter not only helps reduce the overshoot but also significantly improves the system’s tracking performance. As for $k_4$⁠, its parameter variation is small, yet it achieves the good effects brought by the larger parameter variations of $k_3$⁠, indicating that small changes in $k_4$ can have a large impact on the system. By the way, the increase in these parameters also increases the initial value of the control input. Therefore, this simulation can conclude that the adjustment of control parameters is closely related to both tracking performance and control input. Increasing the control parameters can improve tracking performance, but it also increases the control input.

It can be observed that the controller designed in this study does not incorporate estimators and compensation mechanisms, nor does it include switching control terms to handle disturbances. When $S_t$ is close to $S$⁠, the existing sliding surface $S_0$ approaches zero, resulting in tracking errors that are also close to the origin. This is an easily implementable strategy that can improve tracking performance without adding much complexity to the sliding surface. As demonstrated in the simulation results, it still exhibits good tracking performance and robustness. Although the existing logarithmic sliding surface can already achieve the desired control performance, the proposed method can further improve it.

5. Conclusions

This paper proposed a novel SMC method, i.e. TD-LnSMC, to tackle the tracking control problem of space manipulators under lumped disturbances. An innovative sliding surface was introduced, leveraging its value from the previous time interval to effectively compensate for lumped disturbances. This design eliminated the need for observers or complex structures, ensuring simplicity and robust performance. Theoretical analyses and simulation results were given to show the effectiveness of the proposed approach.

However, due to the discontinuous term in the design, chattering may be introduced. To further reducing chattering while maintaining control precision, future work will explore additional smoothing techniques. Moreover, as the current validation is based solely on simulations, experimental verification will be conducted to evaluate the practical effectiveness and robustness of the proposed method.

Conflicts of Interest

The authors declare no conflict of interest.

Author Contributions

MuyuanWang: Conceptualization, Methodology, Software, Writing—original draft preparation. Xu Liu: Data curation, Investigation, Writing—original draft preparation. Jiae Yang: Data curation, Investigation. Chengwu Shen: Project administration, Validation. Yujia Wang: Supervision, Formal analysis.

Funding

This research was supported by the Scientific and Technological Development Program of JiLin Province, China (No. 20230201039GX).

References