Compliant variable admittance adaptive fixed-time sliding mode control for trajectory tracking of robotic manipulators

Abstract

This paper presents a compliant variable admittance adaptive fixed-time sliding mode control (SMC) algorithm for trajectory tracking of robotic manipulators. Specifically, a compliant variable admittance algorithm and an adaptive fixed-time SMC algorithm are combined to construct a double-loop control structure. In the outer loop, the variable admittance algorithm is developed to adjust admittance parameters during a collision to minimize the collision time, which gives the robot compliance property and reduce the rigid collision influence. Then, by employing the Lyapunov theory and the fixed-time stability theory, a new nonsingular sliding mode manifold is proposed and an adaptive fixed-time SMC algorithm is presented in the inner loop. More precisely, this approach enables rapid convergence, enhanced steady-state tracking precision, and a settling time that is independent of system initial states. As a result, the effectiveness and improved performance of the proposed algorithm are demonstrated through extensive simulations and experimental results.

1. Introduction

As space resource utilization continues to advance in recent years, the development of large-aperture antennas, large-diameter optical devices, and large-scale solar power stations are crucial objectives in space utilization and development. It should be pointed out that, the assembly of solar panels is a critical component of the space station construction, as they are the primary power source for the space station [1]. More precisely, the importance of assembling solar panels resides in the necessity for the space station to have a stable and uninterrupted power source to sustain the daily operations of astronauts, conduct scientific experiments, and facilitate other activities [2]. However, the assembly of solar panels is a challenging task due to several factors. First, the installation of the solar panels is a delicate and precise process that requires a high degree of accuracy and stability. Any collisions in the installation process could affect the performance of solar panels, resulting in a decreased power output or even damage to the panels. Second, the installation process is time-sensitive, and any delay or error in the installation could affect the overall functioning of the space station [3]. Therefore, robotic manipulator provides a crucial tool for space station construction, allowing for the manipulation of large, heavy, and delicate objects in the harsh and challenging environment of space [4, 5]. Moreover, the research of robotics is essential to overcome these challenges and ensure the success of the project, which reduces the risk of human error and injury and allows astronauts to focus on tasks that require their unique skills and abilities [6]. In the assembly process of solar panels, the precision of the robot in tracking the desired trajectory, whether the algorithm achieves convergence within a fixed time and whether the algorithm endows the robot with better compliance, directly impacts the robot functions such as path planning and collision detection [7, 8]. Hence, these performances are crucial for the successful execution of on-orbit servicing tasks in the space environment. It should be noted that the collision issues and trajectory tracking in robot assembly process have been paid much attention in previous studies [9–12]. For instance, in ref. [13], a novel predictive strategy is introduced for the avoidance of collisions among multiple mobile robots, involving a precise quantification of the impending collisions. Then, ref. [14] suggests a pragmatic collision detection and a coordinated compliance control method utilizing a momentum observer for a dual-arm robot, all done without the need for force/torque sensors. Furthermore, ref. [15] introduces a robot learning approach guided by knowledge, utilizing a predictive model to enhance the performance of tracking the desired trajectory during the assembly process. Moreover, in ref. [16], a pseudospectral method-based trajectory optimization and a reinforcement learning-based tracking control are proposed for the free-flying space robots to solve the two challenges of task constraints and base–manipulator coupling. Ref. [17] investigates a resilient robot that achieves resilient control through the under-actuated characteristics of passive joints. It should be noted that slight deviations during positioning can be compensated for by leveraging the under-actuated characteristics to enhance the robot’s robustness. Therefore, resilient control increases the success rate of finding the target position and enhances the safety of the robot. It can be seen that the problem of robot collision issues and trajectory tracking precision during the assembly process are very valuable research problems, which are of great help to improve the success rate of robot assembly tasks.

In order to avoid such possible caused by collisions in space assembly mentioned above, robot compliant control strategy has become an efficient and low-cost solution. Robot compliant control refers to endowing robots with compliance property through algorithms to mitigate the impact on the system after collisions. In recent years, many outstanding research results have emerged in this field. For instance, ref. [18] introduces a robot with variable stiffness, utilizing torque-controlled actuators at each joint to consider link compliance and prevent collisions. The compliance of the end-effector in different directions is adjustable by modifying the contribution of each joint. Furthermore, ref. [19] outlines the development of a two-finger variable stiffness gripper and illustrates how the adaptable compliance of the fingers contributes to improved safety and resilience during manipulations involving collisions. Among many compliant control algorithms, variable admittance control (VAC) algorithm has attracted wide attention due to its security, efficiency, and robustness [20]. The concept of VAC is based on the idea of adjusting admittance parameters of the robot model in real time to adapt to different conditions. In compliant control, admittance control mechanism has been widely used in the field of robotics control that allows robots to interact with their environment in a compliant manner. In fact, it is a force control technique that regulates the interaction between the robot and the environment by controlling the parameters of the admittance model [21, 22]. To name a few, ref. [23] proposes a novel approach to enable natural and intuitive interactions between humans and robots by using a VAC method that took human intentions into account. The proposed method enhanced the safety, efficiency, and overall usability of human–robot interaction in various application scenarios. In ref. [24], a new approach to VAC is proposed with virtual stiffness guidance to enhance the performance of human–robot interactions. The aim is to prevent unnecessary changes to the damping parameter by classifying human intentions in the low-velocity region, thus leading to smoother movements. Then, a new approach to physical human–robot interaction that accounts for the interaction between the human, robot, and a moving environment is introduced in ref. [25], which incorporates a VAC strategy with an adaptive controller that produces a time-varying stiffness to enable the robot to interact with the human and the moving environment simultaneously. Therefore, the ability of VAC to adapt to different tasks and environments has made it an essential tool for improving the performance and safety of robotic systems. Overall, the adoption of VAC can help to mitigate the risks associated with collisions in space assembly, while also reducing costs and increasing flexibility in the assembly process.

On the other hand, trajectory tracking control is currently a hot research direction in the field of robot control, attracting attention from many researchers. It should be noted that trajectory tracking has high requirements in tracking error, convergence speed, initial position, anti-interference, and so on. With respect to this task, researchers have proposed many distinctive methods and achieved significant control performances, such as refs. [26–30]. In these methods, sliding mode control (SMC) has gained significant attention in the field of robots due to its exceptional robustness to uncertainties and disturbances. However, traditional SMC relies on asymptotic stability, meaning that in theory, the tracking error of robot position will converge to zero as time approaches infinity [31]. Therefore, this type of algorithm is not conducive to the space station assembly task, which has stringent time requirements. Therefore, finite-time tracking control (FTC) has been recognized as an effective approach to ensure fast tracking performance for the robot control [32]. For instance, a previous work, as in ref. [33], proposed a finite-time fault-tolerant control (FTC) method for robotic manipulators, which utilized nonsingular fast terminal SMC algorithm. This method had several benefits, including robustness in dealing with uncertainties and faults, improved control performance, reduced chattering, and guaranteed fast finite-time convergence. Then, in ref. [34], an adaptive finite-time second-order sliding mode tracking control scheme for robotic manipulator is presented. This control approach eliminates the need for knowledge about the upper bound of external perturbations, and the tracking errors of the robot manipulator can reach equilibrium within a finite time. Moreover, ref. [35] presented a novel finite-time FTC scheme for uncertain robot manipulators with actuator faults. The proposed adaptive FTC (AFTC) scheme offered several advantages, including high precision, strong robustness, no singularity, less chattering, and fast finite-time convergence. Nevertheless, the establishment time of finite-time SMC is heavily dependent on the initial state of the system, resulting in significant conservatism and making it challenging to calculate the establishment time of trajectory tracking in advance. Therefore, scholars have put forward the fixed-time control theory as a further development of traditional SMC methods. Fixed-time control offers a bounded settling time and is autonomous from the initial condition of the system, which makes it a promising solution for control problems that have strict time requirements. Additionally, fixed-time control can provide faster transients and higher control accuracy than finite-time control [36]. To name a few, ref. [37] introduces an anti-saturation adaptive fixed-time sliding mode controller for a class of second-order nonlinear systems. It is rigorously demonstrated that the system is practically fixed-time stable, and the tracking error will rapidly converge to bounded regions within a fixed time, which ignores the initial positions. Furthermore, ref. [38] studies a modified fixed-time tracking controller for nonlinear system, which has an advantage in convergence rate over the existing results. It can be inferred that fixed-time control is a relatively special presence in continuous control. It is not sensitive to the initial positions of the desired trajectories and can ensure convergence within a fixed time, which is of great significance for the completion of trajectory tracking tasks. Owing to its bounded settling time and it is autonomous from the initial condition of the system, the fixed-time SMC is a promising option for controlling space station robots during assembly tasks.

However, to the best of our knowledge, there is limited researches to apply compliant control with fixed-time SMC algorithms for robotic manipulators, which motivates us for this paper. As a consequence, a new compliant variable admittance adaptive fixed-time SMC algorithm is designed for robotic manipulator in this paper. Fig. 1 illustrates the architecture of the algorithm presented in this paper, which combines both the inner-loop and outer-loop algorithms. First, the variable admittance algorithm in the outer-loop converts environment forces into position corrections for the algorithm of the inner loop, thereby achieving compliance control. Then, the inner-loop adaptive fixed-time sliding mode algorithm is utilized to track the corrected desired trajectory, providing stable and accurate control. The principal contributions of this paper, as compared to the existing literature, can be summarized as follows:

Figure 1. Structure of the proposed variable admittance adaptive fixed-time sliding mode control algorithm.

(1) A novel variable admittance logic is designed to judge robot motion states according to the velocity, acceleration, and external force. Based on the variable admittance logic, a new VAC algorithm is developed, which makes it possible to give the robot compliant characteristics and avoid damage caused by rigid collision in different motion states.

(2) A new fixed-time sliding mode manifold is first presented in this paper, which can avoid the singularity problem by switching two surfaces of the manifold. Then, an adaptive fixed-time SMC algorithm is developed on the basis of the proposed fixed-time sliding mode manifold. By using Lyapunov theory, sufficient conditions are established to ensure that the robot tracking error maintains fixed-time stability.

(3) By establishing a novel double-loop control structure that incorporates the two algorithms mentioned above, the robot is endowed with compliance while achieving stable desired trajectory tracking. In addition, the proposed control structure is validated in both simulation environment and practical robotic systems, which demonstrates that it has obvious effect in compliant control and fixed-time tracking error convergence.

Notations: $\mathbb{R}^{n}$ denotes the n-dimensional Euclidean space and $\mathbb{R}^{n\times m}$ is the set of $n\times m$ real matrices. The norm of a vector is defined as $\|x\|=\sqrt{x^{T}x}$ and that of a matrix is defined as the corresponding induced norm $\left.||A\right.||={\unicode[Arial]{x03BB}} _{\max }\{A^{T}A\}$ , ${\unicode[Arial]{x03BB}} _{max}\{x\}$ and ${\unicode[Arial]{x03BB}} _{min}\{x\}$ denote the largest and smallest eigenvalues, respectively, $sign(\!\cdot\!)$ denotes the signum function.

2. Preliminaries

2.1. Problem formulation

The focus of this study is to design a compliant, high-precision robot controller that can rapidly converge in order to facilitate the solar panel assembly on the space station. As illustrated in Fig. 2, we can deconstruct this spatial task into a series of tasks for assembling axle holes. As far as we know, the fundamental elements for axle hole assembly technology are compliant robot control and precise tracking of the desired trajectory. Therefore, we can simplify the assembly task of solar panels on the space station by focusing on the investigation of compliant robot control and desired trajectory tracking. For this purpose, a robotic manipulator was utilized as the primary research object. By using a six-dimensional force/torque sensor, we can accurately obtain the force and torque between the robotic manipulator and the target object. Furthermore, we employ a compliant variable admittance model to map from contact forces and torques of the robot terminal trajectories, which helps us to correct the reference desired trajectories. Then, the robot is controlled by the adaptive fixed-time control algorithm we designed to track the trajectory. To meet the requirements of space tasks, the following control objectives need to be accomplished:

1. (1) In the outer- oop, the compliant VAC algorithm can correct the desired trajectories according to external forces and torques within as little time as possible, which gives the robot compliance. In addition, after an unexpected collision, the algorithm can mitigate the impact on the system caused by the collision. Therefore, the contact force between the robot and the target object can be limited within a reasonable range to avoid damaging the robot and target object.

2. (2) In the inner loop, the robot can track the desired trajectory corrected by outer-loop admittance control algorithm with a fixed time to avoid hysteresis in space tasks. Furthermore, due to practical constraints, the actual trajectories, velocities, and accelerations must be confined within reasonable ranges, respectively.

Figure 2. Solar panel assembly task decomposition.

Since this paper is concentrated on compliant control algorithm and trajectory tracking algorithm design, it is necessary to ensure that the robot joints and the robot object contact are rigid. Detailly, the following assumptions should be considered as prerequisites and default conditions for the algorithm discussed in this paper [39–43].

Assumption 1: The research in this paper is focused on rigid robotic systems, so it conforms to the rigid robot model without the presence of flexible joints and flexible rods.

Assumption 2: The holding of the target object by robot terminal is rigid contact, that is, there is no relative motion between the target object and the robot terminal.

2.2. Robotic manipulator dynamics

We take a degree of freedom (DOF) rigid robot system as our study object and the system model can be given as:

(1) \begin{equation}M\!\left(q\right)\ddot{q}+C\!\left(q,\dot{q}\right)\dot{q}+G\!\left(q\right)={\unicode[Arial]{x03C4}} \!\left(t\right)+d\!\left(t\right)+F_{e}\!\left(t\right)\end{equation}

where $q,\dot{q},\ddot{q}\in \mathbb{R}^{n}$ are positions, velocities, and accelerations of the rigid robot, respectively. Then, $M(q)\in \mathbb{R}^{n\times n}$ is the inertia matrix, $C(q,\dot{q})\in \mathbb{R}^{n\times n}$ is the centrifugal and Coriolis force matrix, and $G(q)\in \mathbb{R}^{n}$ denotes gravity force vector. In addition, $d(t)\in \mathbb{R}^{n}$ represents the unknown and bounded external disturbance. Then, it should be noted that $F_{e}\in \mathbb{R}^{n}$ is the environment contact force and ${\unicode[Arial]{x03C4}} (t)\in \mathbb{R}^{n}$ is the output of the controller we design. Considering that the positions, velocities, and accelerations in the admittance model are measured in the Cartesian coordinate system, it is necessary to transform the system model into the Cartesian coordinate system. Therefore, the robot kinematics in Cartesian coordinate system can be given by:

(2) \begin{align}\left\{\begin{array}{l} x\!\left(t\right)=\phi \!\left(q\right)\\[5pt] \dot{x}\!\left(t\right)=J\!\left(q\right)\dot{q}\\[5pt] \ddot{x}\!\left(t\right)=\dot{J}\!\left(q\right)\dot{q}+J\!\left(q\right)\ddot{q} \end{array}\right.\end{align}

where $x(t)$ , $\dot{x}(t)$ , and $\ddot{x}(t)$ represent the position vector, velocity vector, and acceleration vector, respectively, of the robot terminal in the Cartesian space and $J^{T}(q)\in \mathbb{R}^{n\times 6}$ is the Jacobian matrix, referring to refs. [22, 23] for detail.

Remark 1: It is worthwhile mentioning that there is no derivative relation between $x(t)$ and $\dot{x}(t)$ owing to the angular velocity $\dot{x}(t)$ is defined based on the body frame, which is composed of Jacobian matrix $J(q)$ and joint velocity vector $\dot{q}$ .

According to (2), the robotic system can be rewritten in the Cartesian space as follows:

(3) \begin{equation} \boldsymbol{M}\!\left(x\right)\ddot{x}+\boldsymbol{C}\!\left(x,\dot{x}\right)\dot{x}+\boldsymbol{G}\!\left(x\right)=\boldsymbol{T}+\boldsymbol{D}+\boldsymbol{F}_{e} \end{equation}

where

(4) \begin{equation} \begin{array}{l} \boldsymbol{M}\!\left(x\right)=J^{-T}\!\left(q\right)M\!\left(q\right)J^{-1}\!\left(q\right)\\[5pt] \boldsymbol{C}\!\left(x,\dot{x}\right)=J^{-T}\!\left(q\right)\!\left(C\!\left(q,\dot{q}\right)-M\!\left(q\right)J^{-1}\!\left(q\right)\dot{J}\!\left(q\right)\right)J^{-1}\!\left(q\right)\\[5pt] \boldsymbol{G}\!\left(x\right)=J^{-T}\!\left(q\right)G\!\left(q\right)\\[5pt] \boldsymbol{D}=J^{-T}\!\left(q\right)d\!\left(t\right)\\[5pt] \boldsymbol{T}=J^{-T}\!\left(q\right){\unicode[Arial]{x03C4}} \!\left(t\right)\\[5pt] \boldsymbol{F}_{\boldsymbol{e}}=J^{-T}\!\left(q\right)F_{e}\!\left(t\right) \end{array} \end{equation}

Remark 2: Since the dimension of the Jacobian matrix is 6*n, the Jacobian matrix of a robot without 6-DOF is not a square matrix, and its inverse matrix cannot be found directly. In such scenarios, we usually use the pseudo-inverse Jacobian matrix instead of the inverse matrix. In mathematics, the pseudo-inverse matrix definition is to assume that there exists matrix $A\in R^{n\times m}$ and another matrix $B\in \mathbb{R}^{m\times n}$ . If $A\times B\times A=A$ , then the matrix $B$ is considered to be the pseudo-inverse matrix of matrix $A$ . Furthermore, what we often adopt in robotics is a pseudo-inverse matrix called a unilateral inverse matrix. It should be noted that if matrix $C\in \mathbb{R}^{n\times m}$ and it is full rank, when $n\gt m$ , $C^{-1}=(C^{T}C)^{-1}C^{T}$ ; when $n\lt m$ , $C^{-1}=C^{T}(C^{T}C)^{-1}$ .

Then, there are some fundamental properties exhibiting in the following according to refs. [24, 25].

Property 1: $\boldsymbol{M}(\boldsymbol{x})$ is symmetric positive matrix. In addition, there exist constants $\overline{\boldsymbol{M}}\gt 0$ and $\overline{\boldsymbol{m}}\gt 0$ satisfy $\overline{\boldsymbol{m}}\leq \left.|M\!\left(x\right)\right.|\leq \overline{\boldsymbol{M}}$ .

Property 2: $\dot{\boldsymbol{M}}(x)-2\boldsymbol{C}(x,\dot{x})$ is a skew-symmetric matrix which makes the equation $\varrho ^{T}(\dot{\boldsymbol{M}}(x)-2\boldsymbol{C}(x,\dot{x}))\varrho =0$ , $\forall \varrho \in \mathbb{R}^{n}$ be possible.

Property 3: In a practical robotic system, the environment contact force have their upper boundary as $\left.|\boldsymbol{F}_{\boldsymbol{e}}\right.|\leq F_{\max }$

Property 4: According to different dynamic parameters, the linear parameterization of the above dynamics can be defined as:

(5) \begin{equation} \boldsymbol{M}\!\left(x\right){\unicode[Arial]{x03C6}} _{1}+\boldsymbol{C}\!\left(x,\dot{x}\right){\unicode[Arial]{x03C6}} _{2}+\boldsymbol{G}\!\left(x\right)=\boldsymbol{Y}\!\left({\unicode[Arial]{x03C6}} _{1},{\unicode[Arial]{x03C6}} _{2},x,\dot{x}\right){\unicode[Arial]{x03B8}} \end{equation}

where ${\unicode[Arial]{x03C6}} _{1}$ and ${\unicode[Arial]{x03C6}} _{2}$ are arbitrary constant vectors, $\boldsymbol{Y}({\unicode[Arial]{x03C6}} _{1},{\unicode[Arial]{x03C6}} _{2},x,\dot{x})$ represents regression matrix, and ${\unicode[Arial]{x03B8}}$ is a constant vector containing dynamic information.

3. Outer-loop VAC design

This section proposes a variable admittance logic to judge robot motion states, which is adopted to build the VAC algorithm that can dynamically adjust the admittance parameters. The purpose of this algorithm is to enhance the robot compliance, minimize the duration of the collision, and mitigate any adverse effects the collision might have on space tasks.

3.1. Admittance control model

Inspired by impedance control, the admittance control scheme maps force to motion. In contrast, the admittance control can translate contact forces into motion variables, which makes it possible to achieve compliant control. Detailly, the classical method to define the admittance model is obtained as:

(6) \begin{equation}M_{d}\!\left(\ddot{x}-\ddot{x_{0}}\right)+B_{d}\!\left(\dot{x}-\dot{x_{0}}\right)+K_{d}\!\left(x-x_{0}\right)={{\Delta}} F\end{equation}

where $x$ , $\dot{x}$ , and $\ddot{x}$ are actual position, velocity, and acceleration, respectively. Additionally, $x_{0}$ is setting as the equilibrium point. ${{\Delta}} F$ is the interaction force. Then, the positive constants $M_{d}$ , $B_{d}$ , and $K_{d}$ represent inertia, damping, and stiffness, respectively. First, the inertia $M_{d}$ represents the resistance of the robot to changes in its motion or direction. The inertia parameter holds considerable importance in practical applications, as it dictates the response of the robotic system to external forces. It should be noted that the robot become more resistant to changes in its motion when there is a higher inertia value, while a lower inertia value makes the robot more responsive to external forces. Then, damping $B_{d}$ is a measure standard of the robotic system to dissipate energy when it is subjected to an external force. Ulteriorly, it determines the ability of the robotic system to absorb and dissipate energy when contact with the environment. Therefore, a higher damping value results in a more controlled and stable interaction, while a lower damping value may cause the robot to oscillate and become unstable. Finally, stiffness $K_{d}$ refers to the ability of the robotic system to resist deformation when a force is applied to the robot terminal. The stiffness value determines the level of compliance in the robotic system, with a higher stiffness value making the robot less compliant and more rigid. Conversely, a reduced stiffness value enables the robot to exhibit greater compliance. These parameters are adjustable based on task requirements, allowing for the attainment of the desired level of compliance and stability during interactions with the environment.

When the robot terminal is in contact with the environment, it can be considered to be in a restricted motion state. Therefore, we consider that $x_{0}=x$ , which means that the desired position is equal to the actual position. In addition, we hope it is a soft collision between the robot and the target object, so it is necessary to make sure that $\dot{x}_{0}=0$ , $\ddot{x}_{0}=0$ . Then, it can be obtained from (6) that

(7) \begin{equation}M_{d}\ddot{x}+B_{d}\dot{x}={{\Delta}} F\end{equation}

From (7), by using zero-order hold, it can be obtained that the admittance model in discretized form with a time step $k$ and the sampling time $T$ :

(8) \begin{equation} \begin{array}{l} M_{d}\ddot{x}\!\left(k\right)+B_{d}\dot{x}\!\left(k\right)={{\Delta}} F\!\left(k\right) \\[5pt] M_{d}\frac{\dot{x}\!\left(k\right)-\dot{x}\!\left(k-1\right)}{T}+B_{d}\dot{x}\!\left(k\right)={{\Delta}} F\!\left(k\right)\end{array} \end{equation}

Then, we can draw the conclusion that

(9) \begin{equation}\dot{x}\!\left(k\right)=\frac{T{{\Delta}} F\!\left(k\right)+M_{d}\dot{x}\!\left(k-1\right)}{M_{d}+TB_{d}}\end{equation}

From (9), the relation between robot terminal velocity and external force is obtained from (9). It is worth mentioning that the robot terminal velocity would converge to a certain value when the external force holds a constant, which means $\dot{x}(k)=\dot{x}(k-1)$ . Therefore, the maximum velocity can be expressed as:

(10) \begin{equation}\dot{x}_{max}=\frac{{{\Delta}} F}{B_{d}}\end{equation}

It can be concluded that the velocity reached during convergence is directly proportional to the applied force. Furthermore, examined by various researchers, damping is the primary factor influencing human perception concerning velocity. From (10), it is evident that the maximum velocity output in a steady state is solely dependent on the damping parameter. Then, the acceleration can be expressed as:

(11) \begin{equation}\ddot{x}\!\left(k\right)=\frac{{{\Delta}} F\!\left(k\right)-B_{d}\dot{x}\!\left(k\right)}{M_{d}}\end{equation}

The acceleration can be adopted to express the response speed to the interaction. Furthermore, from (11), acceleration exhibits a coupled relation between the mass and damping coefficients. Thus, it can be inferred that a small mass would result in a faster response, but at the expense of stability. Therefore, it is indispensable to choose appropriate values which meet the requirements of response speed and stability.

3.2. Motion state judgment and variable admittance logic

Robot VAC denotes the capability to modify the robot compliance according to the task demands. Therefore, VAC algorithm is a vital concept in robots compliance control, which enables robots to adjust their physical properties in response to different environments [44]. It should be noted that one of the key benefits of VAC is that it allows robots to better handle unexpected or uncertain situations by adapting their parameters. Authentically, the assessment of motion state plays a pivotal role in VAC, enabling the robot to discern the suitable level of compliance.

The variable admittance parameters are adjusted according to the robot motion state and considering stability issues. In this light, the VAC can be divided into two stages: normal operating state and abnormal operating state.

1. 1. In the normal operating state, we consider that the robot is in the process of fast moving. This state is usually seen in the case of large-scale movements under load conditions such as object handling. Consequently, at this stage, the admittance parameters should be adjusted to optimize the robot motion and impart a degree of compliance to its actions.

2. 2. In the abnormal operating state, abrupt increases in external force due to robot collisions may occur without warning. To mitigate the impact of such collisions, the current target trajectory should be rapidly corrected in the direction of reducing the external force, and the duration of the collision should be minimized.

Overall, the admittance control parameters should be tuned to ensure optimal performance for each stage of robot VAC. By adjusting the admittance gain appropriately, the robot can achieve the desired contact force and desired trajectory tracking in space tasks. Particularly, Fig. 3 shows the motion state judgment and variable admittance logic of the proposed VAC algorithm.

Figure 3. Variable admittance logic.

We set a force reference value $F_{ref}$ to determine the current state, and the judgment process is as follows:

1. If the contact force satisfies ${{\Delta}} F\lt F_{ref}$ , it can be considered that the robot is in the normal operating state, during which the robot completes tasks such as movement and operation. Fixing the admittance parameter to its optimal value can enhance the robot compliance, leading to optimal motion performance while simultaneously enabling the robot to detect external forces.

2. If the contact force satisfies ${{\Delta}} F\gt F_{ref}$ , we consider that the robot is in abnormal operating state and we adopt the variable admittance algorithm for control. If the force sensor detects a sudden increase in external force, it may indicate that the robot has collided. Therefore, to promptly recover from this state, the robot must move expeditiously in the direction that reduces the force. Then, this process can be divided into two steps: forward deceleration and reverse acceleration, which can be distinguished by comparing the directions of velocity and acceleration. During these steps, the change in the admittance parameter can be expressed as follows:

(12) \begin{align} \left\{\begin{array}{l@{\quad}l} B_{d}=B_{f}-\alpha _{acc}\left| \ddot{x}\right|, & \text{when acceleration}\\[5pt] B_{d}=B_{f}+\alpha _{dec}\left| \ddot{x}\right|, & \text{when deceleration}\end{array} \right.\end{align}

where ${\unicode[Arial]{x03B1}} _{acc}=\frac{B_{f}-B_{min}}{\ddot{x}_{\max }}$ , ${\unicode[Arial]{x03B1}} _{dec}=\frac{B_{max}-B_{f}}{\ddot{x}_{\max }}$ , $B_{f}$ is the default damping value, $B_{max}$ and $B_{min}$ are default maximum and minimum damping, and $\ddot{x}_{max}$ is the robot terminal default maximum acceleration. The variable damping is shown in Fig. 4. When the acceleration is smaller than the $\ddot{x}_{min}$ both in the acceleration step and deceleration step, the damping value is set as the default $B_{f}$ . Then, damping increases or decreases functionally as the acceleration falls within the range [ $B_{min}, B_{max}$ ]. Additionally, when the acceleration surpasses a specific threshold, it indicates a substantial alteration in the motion state. In such cases, we adjust the damping parameter to its maximum value $B_{max}$ or minimum value $B_{min}$ based on the current motion state.

Figure 4. Variable damping.

Meanwhile, the adjustment of the inertia is expressed as follows:

(13) \begin{equation}M_{d}=\left| \left(1-{\unicode[Arial]{x03B2}} \right)M_{f}\right|\end{equation}

where $[{\unicode[Arial]{x03B2}} =\frac{\ddot{x}}{\ddot{x}_{max}}]$ and $M_{f}$ is the default inertia value.

As mentioned, the variable admittance helps the robot to move to the target position as soon as possible and provides the robotic system with compliant properties. Therefore, the VAC algorithm is introduced to generate the target trajectory compensation. That is to say, the positional discrepancy induced by the contact force can be acquired by tuning the suggested VAC algorithm, which illustrates the interaction between the robot and the environment. According to the above, it can be obtained that $\dot{x}(k)$ is the velocity induced by external force; thus, the position deviation caused by the external force is

(14) \begin{equation}{{\Delta}} x=\dot{x}\!\left(k\right)\cdot T\end{equation}

Then, the corrected desired trajectory of the robotic system is

(15) \begin{equation}x_{d}={{\Delta}} x+x_{r}\end{equation}

Combined with the trajectory tracking control algorithm in the inner loop, the robotic system can achieve the compliant control.

Remark 3: According to the impulse theorem $Ft=mv$ , the product of force and applied time is equal to momentum. Therefore, the longer the contact time, the greater the momentum will be generated when a collision occurs. Such extra momentum caused by accidental collision has a great impact on the system, which may lead to the robot position deviation or even damage. In situations where avoiding obstacles is not feasible through path planning, our approach is to minimize the contact time, thereby reducing the momentum of the collision. It is worth mentioning that the purpose of the proposed variable admittance algorithm is to reduce the forward velocity and accelerate the reverse direction velocity faster after the collision, thus reducing the contact time.

Remark 4: By adjusting its admittance according to motion states, the robot can improve its accuracy and efficiency in performing tasks such as grasping, manipulation, and assembly. In this way, VAC is a crucial tool for improving the versatility and adaptability of robots in a wide range of real-world applications.

4. Inner-loop adaptive fixed-time SMC algorithm design

The pursuit of faster convergence rates and stabilization times that are independent of initial conditions has led to a growing interest in fixed-time stability as an extension of finite-time stability. Specifically, fixed-time stability offers a predefined stabilization time that is independent of initial conditions, thereby providing greater control performance over the behavior of the system. Compared to finite-time stability, fixed-time stability provides faster convergence rates, making it an attractive approach for robust and precise control of dynamic systems. Taking this into account, this section introduces an adaptive fixed-time sliding mode controller designed for nonlinear robotic systems, demonstrating considerable potential in enhancing the trajectory tracking performance of robotic systems. In the sequel, we will introduce the definition of the fixed-time stability.

Definition 1: (Fixed-time stability) Considering a nonlinear system [45, 46]:

(16) \begin{equation}\dot{x}={\unicode[Arial]{x03C8}} \!\left(x\!\left(t\right)\right),x\!\left(0\right)=x_{0}\end{equation}

where $x(t)\in \mathbb{R}^{n}$ represents the system state, ${\unicode[Arial]{x03C8}} (\!\cdot\!)\colon \mathbb{R}^{n}\rightarrow \mathbb{R}^{n}$ is a nonlinear function, which is continuous on an open region and ${\unicode[Arial]{x03C8}} (0)=0$ , and $x=0$ is defined as an equilibrium point of the nonlinear system. The origin $x=0$ is said to be globally finite-time stable if it is globally asymptotically stable and any solution of the nonlinear function satisfies converging to the equilibrium point at some finite-time moment, that is,

(17) \begin{equation}x\!\left(t\right)=0,\forall t\geq T\!\left(x_{0}\right)\end{equation}

where $T\colon \mathbb{R}^{n}\rightarrow \mathbb{R}_{+}+\cup \{0\}$ is the settling-time function. The origin $x=0$ is said to be globally fixed-time stable if it is globally finite-time stable and the settling-time function $T(x_{0})$ is bounded as: $\exists T_{\max }\gt 0,T(x_{0})\leq T_{\max }$ .

The following lemma will play an irreplaceable role in the later proofs.

Lemma 1: If there is a continuous radically unbounded function $V(x)\colon \mathbb{R}^{n}\rightarrow \mathbb{R}_{+}+\cup \{0\}$ such that [32]:

(1) $V(x)=0\rightarrow x=0$

(2) Any solution of (16) satisfies $\dot{V}(x)\leq -{\unicode[Arial]{x03B3}} V^{{\unicode[Arial]{x03B6}} }(x(t))-{\unicode[Arial]{x03B4}} V^{{\unicode[Arial]{x03B7}} }(x(t))$ , where ${\unicode[Arial]{x03B3}} \gt 0$ , ${\unicode[Arial]{x03B4}} \gt 0$ , ${\unicode[Arial]{x03B6}} \gt 1$ , and $0\lt {\unicode[Arial]{x03B7}} \lt 1$ . Then, it follows that $V(t)=0,t\geq T(x_{0})$ with the setting time bounded by:

(18) \begin{equation}T\!\left(x_{0}\right)\leq T_{max}=\frac{1}{{\unicode[Arial]{x03B3}} \!\left({\unicode[Arial]{x03B6}} -1\right)}+\frac{1}{{\unicode[Arial]{x03B4}} \!\left(1-{\unicode[Arial]{x03B7}} \right)}\end{equation}

Then, we define the vector $Sig^{r}({\unicode[Arial]{x03BE}} )=[sig^{r}({\unicode[Arial]{x03BE}} _{1}),\ldots,sig^{r}({\unicode[Arial]{x03BE}} _{n})]$ , where $sig^{r}({\unicode[Arial]{x03BE}} )=| {\unicode[Arial]{x03BE}} | ^{r}$ .

The objective of this section is to develop an adaptive fixed-time sliding mode controller for the robot that can achieve the desired trajectory within a predetermined fixed time. Therefore, the proposed controller will be designed to ensure fixed-time stability and robustness, which are essential for achieving precise and accurate trajectory tracking in robotic systems. Consequently, we define

(19) \begin{equation}\tilde{x}=x-x_{d}\end{equation}

where $\tilde{x}$ denotes the tracking error and $x_{d}$ is the desired trajectory. The proposed controller is designed on the basis of SMC, which is known for its effectiveness to handle external disturbance. It should be noted that the control law is proposed to realize fixed-time tracking errors converging to zero, while ensuring that the control input remains bounded. Meanwhile, the adaptive property of the controller enhances its robustness against system external disturbance. For the sake of achieving trajectory tracking, a SMC variable is defined as:

(20) \begin{equation}s=\dot{x}-\dot{r}\end{equation}

where $r$ is a reference variable and its derivative can be defined as:

(21) \begin{equation}\dot{r}=\dot{x}_{d}-\mathcal{K}_{1}F\!\left(\tilde{x}\right)-\mathcal{K}_{2}Sig^{{\unicode[Arial]{x03B2}} }\!\left(\tilde{x}\right)\end{equation}

where $\mathcal{K}_{1},\mathcal{K}_{2}\in \mathbb{R}^{n\times n}$ , ${\unicode[Arial]{x03B2}} \gt 1$ represent given constants, and $F(\tilde{x})=[f(\tilde{x}_{1}),\ldots,f(\tilde{x}_{n})]^{T}$ with $f(\tilde{x}_{i})$ is defined as:

(22) \begin{equation}f\!\left(\tilde{x}_{i}\right)=\begin{cases} K_{a}sig^{\alpha }\!\left(\tilde{x}_{i}\right)+K_{b}\left| \tilde{x}_{i}\right| ^{2}sign\!\left(\tilde{x}_{i}\right) & \left| \tilde{x}_{i}\right| \lt \varepsilon \\[5pt] sig^{\alpha }\!\left(\tilde{x}_{i}\right) & \left| \tilde{x}_{i}\right| \geq \varepsilon \end{cases}\end{equation}

where $0\lt {\unicode[Arial]{x03B1}},{\unicode[Arial]{x03B5}} \lt 1$ , $K_{a}=(2-{\unicode[Arial]{x03B1}} ){\unicode[Arial]{x03B5}} ^{{\unicode[Arial]{x03B1}} -1}$ , $K_{b}=({\unicode[Arial]{x03B1}} -1){\unicode[Arial]{x03B5}} ^{{\unicode[Arial]{x03B1}} -2}$ , and the derivative of $f(\tilde{x}_{i})$ is

(23) \begin{equation}\dot{f}\!\left(\tilde{x}_{i}\right)=\begin{cases} K_{a}\alpha \left| \tilde{x}_{i}\right| ^{\alpha -1}+2K_{b}\left| \tilde{x}_{i}\right| & \alpha \left| \tilde{x}_{i}\right| ^{\alpha -1}\\[5pt] \alpha \left| \tilde{x}_{i}\right| ^{\alpha -1} & \left| \tilde{x}_{i}\right| \geq \varepsilon \end{cases}\end{equation}

Therefore, the SMC variable $S$ can be written as:

(24) \begin{align} s&=\dot{x}-\!\left(\dot{x}_{d}-K_{1}F\!\left(\tilde{x}\right)-K_{2}Sig^{{\unicode[Arial]{x03B2}} }\!\left(\tilde{x}\right)\right) \nonumber \\[5pt] &=\dot{\tilde{x}}+\mathcal{K}_{1}F\!\left(\tilde{x}\right)+\mathcal{K}_{2}Sig^{{\unicode[Arial]{x03B2}} }\!\left(\tilde{x}\right) \end{align}

According to the above definitions, the derivative of the SMC variable $s$ is

(25) \begin{align} \dot{s}&=\ddot{x}-\ddot{r}\nonumber \\[5pt] &=\ddot{x}-\ddot{x}_{d}+\mathcal{K}_{1}\dot{F}\!\left(\tilde{x}\right)+\mathcal{K}_{2}{\unicode[Arial]{x03B2}} \left| \tilde{x}\right| ^{{\unicode[Arial]{x03B2}} -1}sign\!\left(\tilde{x}\right) \\[5pt] &=\ddot{\tilde{x}}+\mathcal{K}_{1}\dot{F}\!\left(\tilde{x}\right)+\mathcal{K}_{2}{\unicode[Arial]{x03B2}} \left| \tilde{x}\right| ^{{\unicode[Arial]{x03B2}} -1}sign\!\left(\tilde{x}\right)\nonumber\end{align}

According to Property 4, by introducing the reference variable $r$ into (5), the linear parameterization of the dynamics can be obtained as:

(26) \begin{equation}\boldsymbol{M}\!\left(x\right)\ddot{r}+\boldsymbol{C}\!\left(x,\dot{x}\right)\dot{r}+\boldsymbol{G}\!\left(q\right)=\boldsymbol{Y}_{\boldsymbol{r}}\!\left(x,\dot{x},\dot{r},\ddot{r}\right){\unicode[Arial]{x03B8}} _{r}\end{equation}

where $\boldsymbol{Y}_{\boldsymbol{r}}(x,\dot{x},\dot{r},\ddot{r})$ is the measurable regression matrix.

Remark 5: For traditional SMC, singularity problems typically arise when the sliding mode variable is in the reaching phase. In this phase, the controller struggles to ensure that the sliding mode variable reaches the sliding surface $s=0$ with a bounded control torque that makes the velocity zero. Then, the sliding mode variable may cross the sliding surface. Subsequently, under the action of the control torque, the sliding mode variable will oscillate back and forth across the sliding surface, causing instability and oscillation in the system. It is worth mentioning that a new fixed-time sliding mode manifold is introduced in this section, aiming to address the singularity problem by transitioning between two different surfaces defined in (20) to (25). It can be demonstrated that (25) is a continuous function, effectively overcoming the singularity issue encountered in the proposed sliding mode manifold.

Consequently, the adaptive fixed-time SMC controller can be designed as:

(27) \begin{equation}{\unicode[Arial]{x03C4}} =-K_{1}J^{T}\!\left(q\right)s^{{\unicode[Arial]{x03B6}} }-K_{2}J^{T}\!\left(q\right)s^{{\unicode[Arial]{x03B7}} }+J^{T}\!\left(q\right)\boldsymbol{Y}_{r}\!\left(q,\dot{q},\dot{r},\ddot{r}\right){\unicode[Arial]{x03B8}} _{r}-f_{e}+{\unicode[Arial]{x03C4}} _{ad}\end{equation}

where ${\unicode[Arial]{x03B6}} \gt 1,0\lt {\unicode[Arial]{x03B7}} \lt 1$ , $K_{1}=diag\{k_{11},k_{12},\ldots,k_{1n}\}$ , and $K_{2}=diag\{k_{21},k_{22},\ldots,k_{2n}\}$ are positive diagonal matrices, $f_{e}$ is the environment contact force which can be measured by the robot terminal force sensor, and ${\unicode[Arial]{x03C4}} _{ad}$ is defined to be the adaptive convergence item to deal with the unknown disturbances as:

(28) \begin{equation}{\unicode[Arial]{x03C4}} _{ad}=-K_{p}s-K{\Psi} \!\left(s\right)-{{\Theta}} \!\left(t\right)\frac{|s|+e^{-at}}{|s|}\end{equation}

where $K_{p}\gt 0$ , $K\geq {{\Theta}} _{0}$ , and ${\Psi} \!\left(s\right)=\frac{s}{1+\left| s\right| }$ . It should be noted that ${\Psi} (s)$ is adopted to find the sign of $S$ , and it has better convergence property than the $sign(\!\cdot\!)$ function as shown in Fig. 5. It can alleviate the issue of controller oscillation during positive and negative transitions, improving the controller effectiveness and the system stability. Furthermore, it is obvious that $s{\Psi} (s)\leq | s|$ . ${{\Theta}} _{0}$ is set to be the upper bound of $D$ and ${{\Theta}} (t)$ is the upper bound estimation of unknown disturbance and the adaptive law is designed as:

(29) \begin{equation}\dot{{{\Theta}} }\!\left(t\right)=\frac{|s|^{2}+|s|e^{-at}}{|s|}\end{equation}

where $a\gt 0$ .

Figure 5. Sign function comparison.

Theorem 1: The adaptive fixed-time SMC can be achieved with the control input (27), and the settling time can be given by:

(30) \begin{align} T_{max}&=\frac{1}{{\unicode[Arial]{x03B3}} \!\left(\frac{1+{\unicode[Arial]{x03B6}} }{2}-1\right)}+\frac{1}{{\unicode[Arial]{x03B4}} \!\left(1-\frac{1+{\unicode[Arial]{x03B7}} }{2}\right)}\nonumber \\[5pt] &\quad +\frac{1}{2^{\frac{{\unicode[Arial]{x03B1}} +1}{2}}{\unicode[Arial]{x03BB}} _{\max }\!\left(\mathcal{K}_{1}K_{a}\right)\!\left(\frac{1+{\unicode[Arial]{x03B1}} }{2}-1\right)}+\frac{1}{2^{\frac{{\unicode[Arial]{x03B2}} +1}{2}}{\unicode[Arial]{x03BB}} _{\max }\!\left(\mathcal{K}_{2}\right)\!\left(1-\frac{1+{\unicode[Arial]{x03B2}} }{2}\right)}\end{align}

where

(31) \begin{align} {\unicode[Arial]{x03B3}} &= \left(\frac{2nk_{1max}}{{\unicode[Arial]{x03BB}} _{\min }\!\left(M\right)}\right)^{\frac{1+{\unicode[Arial]{x03B6}} }{2}} \nonumber\\[5pt] {\unicode[Arial]{x03B4}} &= \left(\frac{2nk_{2max}}{{\unicode[Arial]{x03BB}} _{\min }\!\left(M\right)}\right)^{\frac{1+{\unicode[Arial]{x03B7}} }{2}}\end{align}

Proof: Choose the following Lyapunov function $V_{1}$ as:

(32) \begin{equation}V_{1}=\frac{1}{2}s^{T}M\!\left(x\right)s+\frac{1}{2}\!\left({{\Theta}} \!\left(t\right)-{{\Theta}} _{0}\right)^{2}\end{equation}

Then, the derivative of $V_{1}$ can be expressed as:

(33) \begin{equation}\dot{V_{1}}=s^{T}M\!\left(x\right)\dot{s}+\frac{1}{2}s^{T}\dot{M}\!\left(x\right)s+\!\left({{\Theta}} \!\left(t\right)-{{\Theta}} _{0}\right)\dot{{{\Theta}} }\!\left(t\right)\end{equation}

According to (25), it can be obtained that

(34) \begin{align} \dot{V}_{1}&=s^{T}\boldsymbol{M}\!\left(x\right)\!\left(\ddot{x}-\ddot{r}\right)+\frac{1}{2}s^{T}\dot{M}\!\left(x\right)s+\!\left({{\Theta}} \!\left(t\right)-{{\Theta}} _{0}\right)\dot{{{\Theta}} }\!\left(t\right)\nonumber\\[5pt] &=s^{T}\!(J^{-T}\!\left(q\right)\tau +\boldsymbol{F}_{\boldsymbol{e}}+\boldsymbol{D}-\boldsymbol{C}\!\left(x,\dot{x}\right)\dot{x}-\boldsymbol{G}\!\left(x\right)\nonumber\\[5pt]&\quad -\boldsymbol{Y}({{\varphi}} _{1},{{\varphi}} _{2},x,\dot{x}){\unicode[Arial]{x03B8}} _{r}+\boldsymbol{C}(x,\dot{x})\dot{r}+\boldsymbol{G}(x)) \nonumber\\[5pt]&\quad +\frac{1}{2}s^{T}\dot{\boldsymbol{M}}\!\left(x\right)s+({{\Theta}} \!\left(t\right)-{{\Theta}} _{0})\dot{{{\Theta}} }\!\left(t\right)\end{align}

By introducing the adaptive fixed-time sliding mode controller (27) designed above, we have

(35) \begin{align} \dot{V}_{1}&=s^{T}(-K_{1}s^{{\unicode[Arial]{x03B6}} }-K_{2}s^{{\unicode[Arial]{x03B7}} }+\boldsymbol{Y}_{\boldsymbol{r}}(x,\dot{x},\dot{r},\ddot{r}){\unicode[Arial]{x03B8}} _{r}\nonumber\\[5pt]&\quad -\boldsymbol{F}_{e}-K_{p}s-K{\Psi} (s)-{{\Theta}} (t)\frac{|s|+e^{-at}}{|s|}\nonumber\\[5pt] &\quad +\boldsymbol{F}_{e}+\boldsymbol{D}-\boldsymbol{C}(x,\dot{x})\dot{x}-\boldsymbol{G}(x)\nonumber\\[5pt] &\quad -\boldsymbol{Y}({\unicode[Arial]{x03C6}} _{1},{\unicode[Arial]{x03C6}} _{2},x,\dot{x}){\unicode[Arial]{x03B8}} _{r}+\boldsymbol{C}(x,\dot{x})\dot{r}+\boldsymbol{G}(x) \nonumber\\ &\quad +\frac{1}{2}s^{T}\dot{\boldsymbol{M}}(x)s+({{\Theta}} (t)-{{\Theta}} _{0})\dot{{{\Theta}} }(t)\end{align}

Then, according to the adaptive law (29), it yields that

(36) \begin{align}\dot{V}_{1}&=s^{T}\!\left(-K_{1}s^{{\unicode[Arial]{x03B6}} }-K_{2}s^{{\unicode[Arial]{x03B7}} }\right)\nonumber\\[5pt] &\quad -s^{T}\!\left(K_{p}s+K{\Psi} \!\left(s\right)+{{\Theta}} _{0}\frac{|s|+e^{-at}}{|s|}-D\right)\nonumber\\[5pt] &\leq -s^{T}K_{1}s^{{\unicode[Arial]{x03B6}} }-s^{T}K_{2}s^{{\unicode[Arial]{x03B7}} }-s^{T}K_{p}s\nonumber\\[5pt]&\quad -K\left| s\right| -s^{T}\!\left({{\Theta}} _{0}\frac{|s|+e^{-at}}{|s|}-D\right)\end{align}

As we all know that $s\geq -s\cdot {\Psi} (s)$ , so it can be obtained that

(37) \begin{align} \dot{V}_{1}&\leq -s^{T}K_{1}s^{{\unicode[Arial]{x03B6}} }-s^{T}K_{2}s^{{\unicode[Arial]{x03B7}} }-s^{T}K_{p}s\nonumber \\[5pt] &\quad-K\left| s\right| -\!\left(-s^{T}\cdot {\Psi} \!\left(s\right)\!\left({{\Theta}} _{0}\frac{|s|+e^{-at}}{|s|}-D\right)\right)\nonumber\\[5pt] &=-s^{T}K_{1}s^{{\unicode[Arial]{x03B6}} }-s^{T}K_{2}s^{{\unicode[Arial]{x03B7}} }-s^{T}K_{p}s \nonumber\\[5pt] &\quad -\left| s\right| \!\left(K-\!\left({{\Theta}} _{0}\frac{|s|+e^{-at}}{|s|}-D\right)\right)\end{align}

It is obvious that $\lim _{t\rightarrow \mathrm{\infty }}\frac{|s|+e^{-at}}{|s|}=1$ , so we can get that

(38) \begin{align} &lim_{t\rightarrow \mathrm{\infty }}K-\!\left({{\Theta}} _{0}\frac{|s|+e^{-at}}{|s|}-D\right)\nonumber \\[5pt] &=K-\!\left({{\Theta}} _{0}-\boldsymbol{D}\right) \\[5pt] & \geq K-\!\left({{\Theta}} _{0}-|\boldsymbol{D}|_{min}\right)\geq 0\Rightarrow K\geq {{\Theta}} _{0}\nonumber\end{align}

Therefore, by using (38), (37) can be rewritten as:

(39) \begin{equation}\dot{V_{1}}\leq -s^{T}K_{1}s^{{\unicode[Arial]{x03B6}} }-s^{T}K_{2}s^{{\unicode[Arial]{x03B7}} }\end{equation}

Note that

(40) \begin{align} -s^{T}K_{1}s^{{\unicode[Arial]{x03B6}} }&\leq -\Sigma _{i=1}^{n}k_{1i}\left| s_{i}\right| ^{1+{\unicode[Arial]{x03B6}} }\nonumber\\[5pt] &\leq -\!\left(nk_{1\max }\frac{2}{\boldsymbol{M}}\right)^{\frac{1+{\unicode[Arial]{x03B6}} }{2}}\!\left(\frac{1}{2}\boldsymbol{M}\Sigma _{i=1}^{n}s_{i}^{2}\right)^{\frac{1+{\unicode[Arial]{x03B6}} }{2}}\nonumber\\[5pt] &\leq -\!\left(\frac{2nk_{1max}}{{\unicode[Arial]{x03BB}} _{\min }\!\left(\boldsymbol{M}\right)}\right)^{\frac{1+{\unicode[Arial]{x03B6}} }{2}}\!\left(\frac{1}{2}s^{T}\boldsymbol{M}s\right)^{\frac{1+{\unicode[Arial]{x03B6}} }{2}} \nonumber\\[5pt] &\leq -{\unicode[Arial]{x03B3}} \!\left(\frac{1}{2}s^{T}\boldsymbol{M}s\right)^{\frac{1+{\unicode[Arial]{x03B6}} }{2}}=-{\unicode[Arial]{x03B3}} V^{\frac{1+{\unicode[Arial]{x03B6}} }{2}}\end{align}

where ${\unicode[Arial]{x03B3}} = \left(\frac{2nk_{1\max }}{{\unicode[Arial]{x03BB}} _{\min }\!\left(\boldsymbol{M}\right)}\right)^{\frac{1+{\unicode[Arial]{x03B6}} }{2}}$ and $k_{1\max }=\max \{k_{1i}\}$ .

Similarly, it can be obtained that

(41) \begin{equation}-s^{T}K_{2}s^{{\unicode[Arial]{x03B7}} }\leq -{\unicode[Arial]{x03B4}} V_{1}^{\frac{1+{\unicode[Arial]{x03B7}} }{2}}\end{equation}

where ${\unicode[Arial]{x03B4}} = \left(\frac{2nk_{2\max }}{{\unicode[Arial]{x03BB}} _{\min }\!\left(\boldsymbol{M}\right)}\right)^{\frac{1+{\unicode[Arial]{x03B7}} }{2}}$ and $k_{2\max }=\max \{k_{2i}\}$ .

Since ${\unicode[Arial]{x03B6}} \gt 1$ and $0\lt {\unicode[Arial]{x03B7}} \lt 1]$ , it can be obtained that $\frac{1+{\unicode[Arial]{x03B6}} }{2}\gt 1$ and $0\lt \frac{1+{\unicode[Arial]{x03B7}} }{2}\lt 1$ , which implies that

(42) \begin{equation}\dot{V_{1}}\lt -{\unicode[Arial]{x03B3}} V_{1}^{\frac{1+{\unicode[Arial]{x03B6}} }{2}}-{\unicode[Arial]{x03B4}} V_{1}^{\frac{1+{\unicode[Arial]{x03B7}} }{2}}\end{equation}

Consequently, it can be obtained by Lemma 1 that $S$ is fixed-time stable with

(43) \begin{equation}T_{smax}=\frac{1}{{\unicode[Arial]{x03B3}} \!\left(\frac{1+{\unicode[Arial]{x03B6}} }{2}-1\right)}+\frac{1}{{\unicode[Arial]{x03B4}} \!\left(1-\frac{1+{\unicode[Arial]{x03B7}} }{2}\right)}\end{equation}

Therefore, we have $s=0$ as $t\rightarrow T_{smax}$ .

Next, we select another Lyapunov function $V_{2}$ as:

(44) \begin{equation}V_{2}=\frac{1}{2}\tilde{x}^{T}\tilde{x}\end{equation}

As $s=\dot{\tilde{x}}+\mathcal{K}_{1}F(\tilde{x})+\mathcal{K}_{2}Sig^{{\unicode[Arial]{x03B2}} }(\tilde{x})$ , it can be obtained that $s=0\rightarrow \dot{\tilde{x}}=-\mathcal{K}_{1}F(\tilde{x})-\mathcal{K}_{2}Sig^{{\unicode[Arial]{x03B2}} }(\tilde{x})$ . Then, the derivative of $V_{2}$ can be expressed as:

(45) \begin{align} \dot{V}_{2}&=\tilde{x}^{T}\dot{\tilde{x}}\nonumber\\[5pt] &=-\tilde{x}^{T}\mathcal{K}_{1}F\!\left(\tilde{x}\right)-\tilde{x}^{T}K_{2}Sig^{{\unicode[Arial]{x03B2}} }\!\left(\tilde{x}\right)\nonumber\\[5pt] &=-\tilde{x}^{T}\mathcal{K}_{1}\!\left(K_{a}Sig^{{\unicode[Arial]{x03B1}} }\!\left(\tilde{x}\right)+K_{b}\left| \tilde{x}\right| ^{2}sign\!\left(\tilde{x}\right)\right)-\tilde{x}^{T}\mathcal{K}_{1}Sig^{{\unicode[Arial]{x03B2}} }\!\left(\tilde{x}\right)\nonumber\\[5pt] &=-\tilde{x}^{T}\mathcal{K}_{1i}\!\left(K_{a}\Sigma _{i=1}^{n}\left| \tilde{x}_{i}\right| ^{{\unicode[Arial]{x03B1}} }sign\!\left(\tilde{x}_{i}\right)+K_{b}\Sigma _{i=1}^{n}\left| \tilde{x}_{i}\right| ^{2}sign\!\left(\tilde{x}_{i}\right)\right)\nonumber\\[5pt] &\quad-\tilde{x}^{T}\mathcal{K}_{1i}\Sigma _{i=1}^{n}\left| \tilde{x}_{i}\right| ^{{\unicode[Arial]{x03B2}} }sign\!\left(\tilde{x}_{i}\right)\nonumber\\[5pt] &\leq -2^{\frac{{\unicode[Arial]{x03B1}} +1}{2}}{\unicode[Arial]{x03BB}} _{max}\!\left(\mathcal{K}_{1}K_{a}\right)\!\left(\frac{1}{2}\Sigma _{i=1}^{n}{\tilde{x}_{i}}^{2}\right)^{\frac{{\unicode[Arial]{x03B1}} +1}{2}} \nonumber \\[5pt] &\quad -2^{\frac{{\unicode[Arial]{x03B2}} +1}{2}}{\unicode[Arial]{x03BB}} _{\max }\!\left(\mathcal{K}_{2}\right)\!\left(\frac{1}{2}\Sigma _{i=1}^{n}{\tilde{x}_{i}}^{2}\right)^{\frac{{\unicode[Arial]{x03B2}} +1}{2}}-\tilde{x}^{T}\mathcal{K}_{1i}K_{b}\Sigma _{i=1}^{n}\left| \tilde{x}_{i}\right| ^{2}sign\!\left(\tilde{x}_{i}\right)]\end{align}

where $\mathcal{K}_{1i}$ and $\mathcal{K}_{2i}$ are the $ith$ elements of positive diagonal matrices $\mathcal{K}_{1}$ and $\mathcal{K}_{1}$ .

As we all know that $\tilde{x}^{T}| \tilde{x}| ^{2}sign(\tilde{x})=| \tilde{x}| ^{3}\gt 0$ , it can be obtained that

(46) \begin{equation}\dot{V}_{2}\leq -2^{\frac{\alpha +1}{2}}\lambda _{\max }\!\left(\mathcal{K}_{1}K_{a}\right)\!\left(\frac{1}{2}V_{2}\right)^{\frac{\alpha +1}{2}}-2^{\frac{\beta +1}{2}}\lambda _{\max }\!\left(\mathcal{K}_{2}\right)\!\left(\frac{1}{2}V_{2}\right)^{\frac{\beta +1}{2}}\end{equation}

such that $\tilde{x}$ is fixed-time stable and the settling time is

(47) \begin{equation}T_{\tilde{x}\max }=\frac{1}{2^{\frac{\alpha +1}{2}}\lambda _{\max }\!\left(\mathcal{K}_{1}K_{a}\right)\!\left(\frac{1+\alpha }{2}-1\right)}+\frac{1}{2^{\frac{\beta +1}{2}}\lambda _{\max }\!\left(\mathcal{K}_{2}\right)\!\left(1-\frac{1+\beta }{2}\right)}\end{equation}

As a result, as $t\rightarrow T_{\max }=T_{s\max }+T_{\tilde{x}\max }$ , we get $s\rightarrow 0$ , $\tilde{x}\rightarrow 0$ , and $\dot{\tilde{x}}\rightarrow 0$ . Therefore, the desired trajectory is tracked by the robot and the proof completes.

In order to better illustrate our control structure and the flow of data, we have created a detailed control diagram as shown in Fig. 6. It is worth mentioning that the diagram maps the equations in this paper to the modules in the diagram, facilitating understanding the role of the equations in the control structure. In the control block diagram, the control structure is divided into two parts, the inner loop and the outer loop, which are selected with dotted lines of different colors. Specifically, in the outer loop, due to the error between the actual trajectory $x$ and the desired trajectory $x_{d}$ , force sensors generate contact force signals when the robot makes contact with the external environment, which serve as inputs to the VAC. The control algorithm performs variable admittance logic judgment and outputs the optimal admittance parameters in real time, with the admittance controller providing the corrected values for the trajectory. Subsequently, the correction value is entered into the inner loop to generate the corrected desired trajectory. In addition, in the inner loop, an adaptive fixed-time sliding mode controller is designed for trajectory tracking control, outputting control torques to the robot to achieve trajectory tracking. In a word, this control structure diagram illustrates the structure of our control framework and points to the corresponding formulas in the paper, effectively demonstrating how our control algorithm enables the robot to achieve high-precision tracking control while possessing compliance property.

Figure 6. Detailed block diagram of control structure and the data flow.

To enhance the clarity of the control process of the proposed algorithm, we present the pseudocode of the algorithm as shown in Algorithm 1. It offers a concise and clear way to convey the design and workflow of the algorithm, facilitating easier comprehension and providing an intuitive starting point to validate the algorithm presented in this paper.

Algorithm 1. Algorithm pseudocode.

Remark 6: Our approach stands out for its fixed-time stability property, which eliminates the requirement for prior knowledge of initial conditions. It is important to highlight that Theorem 1 substantiates the robustness of the proposed algorithm, a result of the utilization of SMC and adaptive techniques. In particular, the fixed-time SMC that is insensitive to uncertainties is essential in driving the state trajectory towards the sliding surface when the sliding variables are far away from the origin. By combining fixed-time SMC and adaptive strategy, our approach ensures strong robustness and high accuracy in trajectory tracking for robots.

5. Simulation experiments

In this section, we employ a PHANToM Omni robotic device to demonstrate the effectiveness of the presented approach in simulation and experiments due to its versatility, accuracy, and compatibility with the proposed methodology. As far as we know, this robotic device is equipped with six joints, consisting of three driven joints and three passive joints, which is shown in Fig. 7. These joints enable the robot to achieve a 6-DOF motion, making it an effective position-sensing manipulator. Particularly, the PHANToM Omni robotic device has been widely employed in prior research as a standard platform for testing trajectory tracking control algorithms. This makes it a suitable selection for our study, facilitating comparisons with existing research and providing a foundation for advancing established methodologies. Moreover, the software for this device and its programming interface are comprehensively documented, streamlining the algorithm implementation and ensuring the reproducibility of our results. Therefore, considering the versatility, accuracy, compatibility, and prevalent use of this device in prior studies, we have opted for PHANToM Omni robotic device as the experimental platform to validate our algorithm.

Figure 7. The joints of PHANToM Omni robotic device.

Selecting dynamic parameters of a real robot for simulation experiments is a common approach to ensure that the simulation results closely match the real-world experiments. By selecting these parameters carefully, we can achieve better accuracy and reliability in our simulations. Hence, for simulation experiments, we use the dynamic parameters of the actual PHANToM Omni robotic device to ensure that the simulation results are comparable with real-world experiments. Thus, the dynamics of the PHANToM Omni robotic device can be expressed as:

(48) \begin{equation}M\!\left(q\right)\ddot{q}+C\!\left(q,\dot{q}\right)\dot{q}+G\!\left(q\right)+d\!\left(t\right)=\tau\end{equation}

where $M(q)\in \mathbb{R}^{n\times n}$ is the inertia matrix, and it can be expressed as:

(49) \begin{align} M_{11}\!\left(q\right)&=k_{1}+k_{2}\cos \!\left(2q_{2}\right)+k_{3}\cos \!\left(2q_{3}\right)\nonumber\\[5pt] M_{12}\!\left(q\right)&=k_{5}\sin \!\left(q_{2}\right)\nonumber\\[5pt] M_{13}\!\left(q\right)&=0\nonumber\\[5pt] M_{21}\!\left(q\right)&=k_{5}\sin \!\left(q_{2}\right)\nonumber\\[5pt] M_{22}\!\left(q\right)&=k_{6}\nonumber\\[5pt] M_{31}\!\left(q\right)&=0\nonumber\\[5pt] M_{32}\!\left(q\right)&=-0.5k_{4}\sin \!\left(q_{2}-q_{3}\right) \nonumber\\[5pt] M_{33}\!\left(q\right)&=-0.5k_{4}\sin \!\left(q_{2}-q_{3}\right)\end{align}

$C(q,\dot{q})\in \mathbb{R}^{n\times n}$ is the Coriolis and centrifugal forces matrix as:

(50) \begin{align} C_{11}\!\left(q,\dot{q}\right)&=-k_{2}\dot{q}_{2}\sin \!\left(2q_{2}\right)-k_{3}\dot{q}_{3}\sin \!\left(2q_{3}\right)\nonumber\\[5pt] &\quad -0.5k_{4}\dot{q}_{2}\sin \!\left(q_{2}\right)\sin \!\left(q_{3}\right)\nonumber\\[5pt] &\quad -0.5k_{4}\dot{q}_{2}\sin \!\left(q_{2}\right)\sin \!\left(q_{3}\right)\nonumber\\[5pt] C_{12}\!\left(q,\dot{q}\right)&=k_{2}\dot{q}_{1}\sin \!\left(2q_{2}\right)+0.5k_{4}\dot{q}_{1}\sin \!\left(q_{2}\right)\sin \!\left(q_{3}\right)\nonumber\\[5pt] C_{13}\!\left(q,\dot{q}\right)&=k_{3}\dot{q}_{1}\sin \!\left(2q_{3}\right)+0.5k_{4}\dot{q}_{1}\cos \!\left(q_{2}\right)\cos \!\left(q_{3}\right)\nonumber\\[5pt] C_{21}\!\left(q,\dot{q}\right)&=-k_{2}\dot{q}_{2}\sin \!\left(2q_{2}\right)+k_{5}\dot{q}_{2}\sin \!\left(q_{2}\right)\nonumber\\[5pt] &\quad -0.5k_{4}\dot{q}_{1}\sin \!\left(q_{2}\right)\sin \!\left(q_{3}\right)\nonumber\\[5pt] C_{22}\!\left(q,\dot{q}\right)&=0\nonumber\\[5pt] C_{23}\!\left(q,\dot{q}\right)&=-0.5k_{4}\dot{q}_{2}\cos \!\left(q_{2}-q_{3}\right)\nonumber\\[5pt] C_{31}\!\left(q,\dot{q}\right)&=-k_{3}\dot{q}_{1}\sin \!\left(2q_{3}\right)+0.5k_{4}\dot{q}_{1}\cos \!\left(q_{2}\right)\cos \!\left(q_{3}\right)\nonumber\\[5pt] C_{32}\!\left(q,\dot{q}\right)&=0.5k_{4}\dot{q}_{3}\cos \!\left(q_{2}-q_{3}\right) \nonumber\\[5pt] C_{33}\!\left(q,\dot{q}\right)&=0\end{align}

$G(q)\in \mathbb{R}^{n}$ is the gravity effect vector as:

(51) \begin{align} G\!\left(q\right)_{1}&=0\nonumber\\[5pt] G\!\left(q\right)_{2}&=k_{8}\cos \!\left(q_{2}\right)+k_{10}\!\left(q_{2}-0.5\pi \right) \nonumber\\[5pt] G\!\left(q\right)_{3}&=k_{9}\sin \!\left(q_{3}\right)\end{align}

$d(t)\in \mathbb{R}^{n}$ is the external disturbance as:

(52) \begin{equation}d\!\left(t\right)=3\times \pi /180\times \sin \!\left(2\pi t\right)\end{equation}

There are some parameters $k_{1}$ to $k_{10}$ in the dynamics, which present for inertia, gravity, length, and mass effect of the links of the PHANToM Omni robotic device, and they are given as $k_{1}=1.798\times 10^{-3}$ , $k_{2}=0.864\times 10^{-3}$ , $k_{3}=0.486\times 10^{-3}$ , $k_{4}=2.766\times 10^{-3}$ , $k_{5}=0.308\times 10^{-3}$ , $k_{6}=2.526\times 10^{-3}$ , $k_{7}=0.652\times 10^{-3}$ , $k_{8}=164.158\times 10^{-3}$ , $k_{9}=94.05\times 10^{-3}$ , and $k_{10}=117.294\times 10^{-3}$ .

Considering the structural characteristics of the PHANToM Omni robotic device, previous studies have usually fixed the initial joint of the device to analyze the motion of the final two joints. Nevertheless, it is important to mention that the last two joints function within the same plane of motion and demonstrate similar ranges of motion, leading to approximate motion patterns. In this paper, we choose to display the tracking trajectory diagram of joint 3 exclusively to represent the device’s overall motion characteristics. The controller parameters are designed as $\mathcal{K}_{1}=diag\{1.5,1.5,1.5\}$ , $\mathcal{K}_{1}=diag\{2,2,2\}$ , $\alpha =0.7$ , $\beta =1.1$ , $\varepsilon =0.3$ , $K_{p}=10$ , $K=8$ , $a=2$ , $\zeta =7/3$ , $\eta =3/4$ , $K_{1}=diag\{3,3,3\}$ , and $K_{2}=diag\{3,3,3\}$ .

Figure 8. Trajectory tracking with initial position 0.55 rad.

5.1 Desired trajectory tracking with different initial conditions experiments

The experiments in this part are focus on how the performance of the proposed algorithm in desired trajectory tracking is affected by the initial conditions. For the purpose of facilitating the comparison of experimental results, the desired trajectory is presented as $q_{d}=36.62\times \pi /180+10\times \pi /180\times$ $cos(t\times \pi )$ . According to Theorem 1, the tracking error of the proposed algorithm should converge to zero for all $t\geq T_{\max }=T_{s\max }+T_{\tilde{x}\max }\approx 5.95s$ .

We chose an initial position $q_{3}(0)=0.55rad$ for the initial comparison experiment group. By plotting simulation experimental results with Matlab, Figs. 8 and Fig. 9 show the tracking trajectory and tracking errors of this experiment. According to Fig. 8, the simulation experiments demonstrate that the suggested algorithm allows the robot joints to follow the desired trajectory within a predetermined duration of time $T_{\max }$ . Then, in Fig. 9, it is evident that the tracking error fluctuates before reaching $T_{\max }$ , while the overall trend is a gradual reduction. After reaching $T_{\max }$ , the tracking error stabilizes within a small range $(-1.65\times 10^{-2}\mathrm{rad},1.65\times 10^{-2}\mathrm{rad})$ and exhibits strong regularity, which is considered to have merely faint influence on robot control performance and system stability.

Figure 9. Trajectory tracking error with initial position 0.55 rad.

On the other hand, we choose a set of values both higher and lower than the initial value of the desired trajectory to serve as starting points for a comparative analysis, which allows us to evaluate the algorithm’s sensitivity to variations in initial conditions. The different initial position conditions are designed as:

(53) \begin{align}\begin{array}{c} \text{case}1\colon \mathrm{q}_{3}(\mathrm{0})=\mathrm{0.5}\text{ rad}\\[5pt] \text{case}2\colon \mathrm{q}_{3}(\mathrm{0})=\mathrm{0.65}\text{ rad} \end{array}\end{align}

The experimental results are shown in Figs. 10–13. It can be observed that when starting from different initial positions, the tracking error still converges to a stable region within settle time $T_{\max }$ , which is consistent with the experimental results mentioned earlier. Hence, the comparative experiments suggest that the proposed algorithm exhibits minimal sensitivity to variations in the initial position, rendering it suitable for robot startup from different initial positions.

Figure 10. Trajectory tracking with initial position 0.5 rad.

Figure 11. Trajectory tracking error with initial position 0.5 rad.

Figure 12. Trajectory tracking with initial position 0.65 rad.

Figure 13. Trajectory tracking error with initial position 0.65 rad.

5.2 Traditional SMC algorithm comparison experiment

In our simulation experiments, we performed a series of comparative experiments between the traditional SMC algorithm and the algorithm presented to illustrate the improvement of tracking error convergence performance, as demonstrated in Figs. 14 and 15. By analyzing Fig. 14, it indicates that although the traditional SMC algorithm can generally track the desired trajectory, its performance is unsatisfactory when the direction of motion changes. Upon examining Fig. 14, we discovered that the tracking error of the conventional SMC algorithm is significantly higher than that of our proposed algorithm. Moreover, the traditional method exhibits random error values without apparent regularity, posing challenges in achieving sufficient accuracy and making it difficult to apply in high-precision robot operations. In contrast, the developed algorithm is able to effectively control the range and regularity of the tracking error, thereby demonstrating greater stability and accuracy.

Figure 14. Sliding mode control algorithm tracking trajectory.

Figure 15. Sliding mode control algorithm tracking error.

6. Real robot experiments

To demonstrate the enhanced performance of proposed control algorithm, we have designed practical experiments for verification. Specifically, two experimental platforms for the algorithm verification have been built, respectively. The validation of the outer-loop algorithm focuses on its ability to reduce the influence of unexpected collisions, while the validation of the inner-loop algorithm assesses its performance in tracking the desired trajectory.

To validate the effectiveness of the VAC algorithm in practical applications, we constructed an experimental platform by using a UR3 robot, a six-dimensional force sensor, a mechanical gripper, and a space station solar panel to perform physical verification of compliant control, as depicted in Fig. 16(a). The UR3 robot is a lightweight and compact robot that is designed to provide flexible automation solutions for a wide range of applications. With 6-DOF, the UR3 robot is highly versatile and capable of performing complex tasks. Additionally, the robot is outfitted with advanced safety features, including collision detection and emergency stop functionalities, rendering it suitable for operation in diverse environments. Then, the Robotiq FT300s force/torque sensor is a widely used tool for measuring and monitoring the force and torque applied to a robot during its interaction with the environment. Equipped with a high-resolution strain gauge, this six-axis sensor can accurately measure forces and torques in three dimensions. Furthermore, the FT300s is equipped with a built-in digital signal processor that performs real-time data processing and filtering, ensuring that the measured force and torque data are accurate and reliable. Consequently, the FT300s is an essential component of many advanced robotic systems.

Figure 16. Practical experimental platforms: (a) variable admittance control strategy platform in the outer loop; (b) adaptive fixed-time sliding mode control algorithm platform in the inner loop.

On the other hand, to validate the efficacy and robustness of our adaptive fixed-time SMC algorithm in practical applications, we employed a PHANToM Omni robotic device as the experimental platform, which is illustrated in Fig. 16(b). It is noteworthy that this device has been extensively utilized in previous studies due to its flexible mechanical structure, high positioning accuracy, and compatibility with numerous software packages, making it an ideal option for secondary development. Furthermore, the timely response of the robotic device to control commands is crucial for the success of our experiments and the achievement of our objectives.

6.1. VAC real robot experiments

The objective of this experiment is to assess the efficiency of the VAC algorithm in minimizing the contact duration between the robot and an obstacle during a rigid collision. Specifically, the robot is assigned a task of assembling solar panels, as depicted in Fig. 17, which involved four steps: grasping the solar panel, moving it to a desired position, installing the panel, and tightening the screws. The moving process requires the robot to traverse a large range of motion, potentially leading to collisions. Thus, in order to assess the performance of the variable admittance algorithm after encountering obstacles, an obstacle is introduced along the robot desired trajectory in the moving process, causing it to collide with the obstacle. It is important to highlight that the collision between the end of the robot and the obstacle is rigid. We recorded the absolute value of the robot velocity after collision to analyze the performance of the proposed algorithm on the robot during collision. As a comparison, we also conducted a set of experiments with traditional admittance parameters to showcase the performance of the developed algorithm.

Figure 17. Solar panel assembly experiment process: (a) clamp the solar panel; (b) move to the desired position; (c) install the solar panel; and (d) screw the fixed screw.

Eventually, the absolute value of velocities in the contrast experiment are shown in Fig. 18. As a result, after the collision occurs, the VAC strategy parameters are adjusted to enable the robot to have a faster deceleration response in the deceleration phase and to increase to a faster level in the reverse acceleration phase compared to the traditional admittance control algorithm. Therefore, this significantly shortens the collision occurrence time and provides effective protection, which demonstrates that the proposed variable admittance algorithm is observed to enhance the robot compliance by quickly adapting the admittance parameters. This is due to the shortened collision time, which reduces the momentum transferred to the robot system during accidental collisions, significantly decreasing the likelihood of system failure. Ulteriorly, the outcome of the experimental results illustrates the effectiveness of the variable admittance method in swiftly modifying the desired trajectory following a collision to minimize the contact duration, thereby enhancing the compliance of the robot.

Figure 18. Variable admittance contrast experiments result.

6.2. The adaptive fixed-time SMC algorithm real robot experiments

In order to validate the fixed-time sliding mode algorithm proposed in this paper, we conducted physical experiments by using the experimental platform introduced earlier. In our experiments, we defined a desired trajectory for the robot joint and employed our algorithm to follow it. During the movement of the robotic device, we recorded and stored both the desired and actual trajectories and graphed the results by using Matlab. It should be noted that considering the structure of the PHANToM Omni robotic device and the rotation of joint 1 is not substantial and the variation patterns of joint 2 and joint 3 are essentially similar, we opted to focus on the trajectory tracking of joint 3 to showcase the performance of the proposed algorithm. Nevertheless, it is essential to emphasize that, despite our efforts to introduce external environmental disturbances and other interfering factors in the Matlab simulation, there might still be disparities between the simulation and real experimental settings. Therefore, it is crucial to appropriately adjust the parameters when conducting experiments on the real robot experimental platform to match real-world conditions. It should be noted that the selected desired trajectory is the same as that in the simulation experiments to make it comparable. The experimental results are shown in Figs. 19–21.

Both trajectories are presented in a shared coordinate system for convenient comparison. In addition, the desired trajectory is depicted in blue, while the actual trajectory taken by the robot is represented in red. Then, the trajectory tracking result using our algorithm is presented in Fig. 19. The figure clearly shows that the robot has successfully tracked the desired trajectory with high accuracy. Moreover, according to Fig. 20, the gray line represents the actual tracking error, while the red line shows the average value of the tracking error. It should be noted that the parameters utilized in this study are not an exact match to the dynamics model of the real robot, and as a result, some vibrations are inevitable during the robot motion. Specifically, these vibrations stem predominantly from friction in the robot joints and gravitational effects from other joints at the robot terminal. Consequently, the actual tracking error manifests as slightly larger and less regular in value compared to the simulated results. Nonetheless, by calculating the average value of the tracking error, it was observed that the average error tends to converge to a standard value after approximately 5 s and can be limited to a small range, indicating a high degree of regularity. In addition, the control input is shown in Fig. 21, and it is obvious that the range of amplitudes for the control torque is stable and exhibits a more consistent trend of change. After addressing the vibration phenomenon, the output of the control torque is anticipated to become more stable. Therefore, this demonstrates that the algorithm is capable of achieving comparable results to the simulation experiment when applied to the real robot, indicating its practical usefulness.

Figure 19. Joint 3 tracking trajectory with the proposed algorithm.

Figure 20. Joint 3 tracking error with the proposed algorithm.

Figure 21. Joint 3 control input with the proposed algorithm.

As a comparison, we also conducted a control experiment using a traditional SMC algorithm. The tracking error of joint 3 is shown in Fig. 22, where the gray curve represents the actual tracking error and the red curve represents the average tracking error. It can be observed that the tracking error values of the traditional sliding mode algorithm fluctuate significantly, showing substantial irregularities. This outcome is consistent with the results of the simulation experiments. Furthermore, Table 1 compares the errors of the two algorithms. It can be seen that the maximum real tracking error of the traditional SMC algorithm is larger, and the average tracking error does not have an obvious convergence trend, indicating inferior tracking performance compared to the proposed algorithm in this paper. Hence, the stability demonstrated by this algorithm in control performance positions it as a more promising solution for practical applications within robotic systems.

Figure 22. Joint 3 tracking error with traditional sliding mode control algorithm.

Remark 7: The algorithm presented in this article is intended for the completion of space assembly tasks by space robots. Nevertheless, real space robots cannot be supplied directly to researchers beyond research and development units. Therefore, our focus was on developing control algorithms for space missions. Furthermore, we design operational scenarios closely mirroring those encountered in space stations and utilize robots with structures akin to those employed in space missions to validate the efficacy of our control algorithms. In the proposed algorithm structure, the numerous parameters mainly influence the reliance of the controller on state data from a mathematical perspective, which collectively determine the effectiveness of the controller. It is worth mentioning that the parameter values provided in Section 5 are specific to the robotic system we employed, determined through multiple experiments to achieve satisfactory results. These values allow our controller to meet the current robotic task requirements without imposing excessive computational pressure. Therefore, the optimization of these parameters can be tailored to the specific robotic system and control task needs. This section includes two experiments aimed at verifying the inner-loop and outer-loop algorithms of the proposed control structure and testing the performance of each loop. Particularly, the experimental results demonstrate that the inner- and outer-loop algorithms in the proposed control structure perform as expected, and the entire algorithm exhibits excellent control effectiveness in practical applications.

Remark 8: There is a chattering phenomenon in some experimental results, which is due to the presence of unknown part of the robot dynamics model and external disturbances that cannot be directly measured. However, the controller is designed based on the robot system dynamics model, so the unknown dynamics model parts and external disturbances have a significant impact on the design of the robot controller. Since this paper focuses on the adaptive fixed-time sliding mode variable control algorithm design, we have adopted a common approach for handling unknown model parts and external disturbances as we shown in the above section, which has been widely adopted in many previous studies and has achieved good results. In the future, we will dedicate more effort to exploring methods for handling unknown dynamics and external disturbances, aiming to assist in the design of controllers with improved performance.

7. Conclusion

In this paper, a new compliant variable admittance adaptive fixed-time SMC algorithm for trajectory tracking of robotic manipulators is proposed. To meet the demands of space on-orbit tasks, the algorithm proposed in this paper can endow robots with compliance to mitigate the impact of unexpected collisions, while also achieving fixed-time convergence in robot trajectory tracking and ensuring high control precision. In the outer loop, a variable admittance algorithm is designed to modify the admittance parameters more quickly, which is helpful to reduce the collision duration and mitigate the impact of the collision on space missions when the sensor detects an unexpected collision during the robot movement. Additionally, a correction value for the desired trajectory can be generated to impart compliant characteristics to the robot. Furthermore, the inner-loop adaptive fixed-time SMC algorithm ensures the fixed-time stability of the robot movement, effectively avoiding the motion lag caused by slow convergence of tracking errors in space tasks. Moreover, practical robots are employed to demonstrate the theoretical results. As a result, the experimental results provide evidences that the proposed algorithm can ensure both robot compliance and control precision, which are of critical importance for space robotic systems. Furthermore, we are committed to furthering our efforts in developing space robotic manipulator control algorithms to achieve even more efficient and precise control outcomes in the future.

Table 1. Summary table of tracking errors.