Observer-based fixed-time tracking control for space robots in task space

doi:10.1016/j.actaastro.2021.04.002

Acta Astronautica

Volume 184, July 2021, Pages 35-45

https://doi.org/10.1016/j.actaastro.2021.04.002 Get rights and content

Highlights

•
Fixed-time tracking control in task space is proposed for space robots.
•
Fixed-time extended state observer is designed to obtain velocities and disturbance.
•
Dynamic singularities are handled during the control design.

Abstract

The problem of task-space tracking control of a free-floating space robot is addressed in this paper. Based on a fixed-time extended state observer (FXESO), a fixed-time position and attitude control (FXPAC) method is developed. The attitude of the end-effector is represented in modified Rodrigues parameters (MRPs). The FXESO provides the estimation of the joint velocity and the lumped disturbance for the control system. Based on the backstepping technique and a power integrator, the FXPAC method is designed. This control scheme can guarantee tracking errors converging to a neighborhood of the origin within a fixed time in the presence of the external disturbance. Moreover, since the inverse of the generalized Jacobian matrix is involved in the controller, it is necessary to consider dynamic singularities. To handle these singularities, a method of combining the singularity separation and damped reciprocal (SSDR) is applied to the space robot. Numerical simulations are presented to demonstrate the effectiveness of the proposed method.

Introduction

Free-floating space robots have been widely utilized in space explorations, such as maintenance and construction of space stations, and capturing tumbling satellites. Tracking control is particularly important for applications of space robots. During the last decades, there have been various robot control methods. Since the robot motor directly drives the joint, there have been a lot of results about joint-space controllers [1], [2], [3]. However, some missions are defined in task space, such as Cartesian space or sensor/image space. To ensure asymptotic end-effector position and orientation tracking, a full-state feedback controller and an output feedback controller were designed for robot manipulators [4]. For task-space tracking control of redundant robotic systems, an online kernel-based learning approach and a non-linear model predictive control with obstacle avoidance were developed in [5] and [6], respectively. For free-flying space robots, [7] and [8] proposed a fuzzy logic system adaptive controller and an adaptive sliding mode controller in task space, respectively.

Compared with the asymptotic control laws, finite-time control can shorten the convergence time, improve control accuracies, and reject the disturbance better. This kind of methods has been also applied to the robotic systems. In terms of tracking control in joint space, finite-time control was respectively combined with neural network [9], fuzzy strategy [10], and sliding mode control [11]. With regard to tracking control in task space, a sliding mode controller with finite-time stability was proposed for free-flying space robots [12]. Based on the derivative of the dynamic equation and a new non-singular terminal sliding manifold, finite-time trajectory tracking control in task space was developed for robotic manipulators [13]. Considering vibration suppression, finite-time control based on virtual control force was proposed to track a planar trajectory for a flexible space robot system [14].

There is a significant drawback of the finite-time control: the settling time depends on the initial conditions, heavily. Fixed-time control is an effective solution to this issue, which can guarantee that the settling time is bounded by a positive constant [15]. Fixed-time controllers have been applied to the attitude control of spacecraft [16], [17], reusable launch vehicles [18], and the aircraft fighter control [19]. There have also been some research results about fixed-time control of robotic systems. For robots with model uncertainties, fixed-time sliding mode control and robust approximate fixed-time control were respectively proposed in [20] and [21]. Considering uncertainties and input dead zone together, fixed-time control based on the neural network was presented [22]. The aforementioned control methods were mainly developed to track desired trajectories in joint space. There have been fewer results about fixed-time control in the task space for space robots. A robust task-space fault-tolerant control, which could provide global fixed-time stability, was designed [23]. Motivated by [4], [16], [23], fixed-time position and attitude control (FXPAC) is studied for a free-floating space robot in this paper.

Another potential problem in the design of the control system is that the velocity measurements can be contaminated by noises, which may lead to poor control performance or even instability problems. An effective solution is to estimate the joint velocity using a state observer. A sliding observer was proposed to provide the joint velocity required in the robot control [24]. For double integrator systems with disturbance, [25] designed a fixed-time observer based on the uniform robust exact differentiator. Moreover, the control performance is also impacted by the model uncertainties and the disturbance. To address the problem, some researchers have focused on the extended state observer (ESO). The core of ESO is to treat the disturbance as an extended system state and design a state observer. Finite-time extended state observers (FESOs) were employed to obtain the disturbance estimation for different objects, such as spacecraft [26], [27] and mechanical systems [28]. In [27], [28], both the velocity and the disturbance could be estimated by FESO. Based on the homogeneous theory, fixed-time extended observers (FXESOs) were presented in [18], [19], [29]. In [19], the switch function involved in the traditional FXESO design was omitted based on Theorem 2 in [30]. Based on the above, a significant advantage of ESO is that the observer can provide the state of the original system and the disturbance together [18], [19], [27], [28]. Different from FESO, the observer errors of FXESO can converge to the origin within a fixed time independent of initial conditions. Therefore, to estimate the joint velocity and the disturbance, an FXESO is designed for free-floating space robots in this paper.

In terms of the tracking control in task space for space robots, the inverse of the generalized Jacobian matrix usually needs to be computed [31], [32]. Therefore, it is necessary to consider the dynamic singularities. In [13], by introducing an auxiliary matrix, the inverse of the Jacobian matrix was not employed in the controller, which meant the singularities were avoided. However, if the range of the eigenvalues of the inertia matrix is too large, the required control inputs will also become much large. To deal with singularities, the damped least-squares (DLS) method has been utilized in a robotic system with a fixed base [33], [34] and in a free-floating space robot system [31]. With regard to DLS, the estimation of the minimum singularity of the Jacobian matrix should be obtained. For space robots with wrist-partitioned manipulators, a method of combining the singularity separation and damped reciprocal (SSDR) was developed [32]. With the SSDR method, the minimum singularity of the Jacobian matrix is not required and the joint velocity can be calculated directly. In this paper, inspired by [32], SSDR is introduced to handle the dynamic singularities during the control design.

This paper focuses on the position and attitude control in the task space for a free-floating space robot. The modified Rodrigues parameters (MRPs) are employed to represent the attitude of the end-effector. On the other hand, besides the advantages of finite-time control, the settling time of the fixed-time control is independent of the initial conditions. Thus, a novel fixed-time control is developed to track the desired position and attitude trajectories for space robots. In the presence of the unknown joint velocity and disturbance, an FXESO is designed to provide these information for the system. Moreover, the SSDR method is employed to deal with dynamic singularities. Therefore, considering dynamic singularities, a novel FXPAC based on an FXESO is studied in this paper. The main contributions of this paper are summarized as follows:

1.
A novel FXPAC method is proposed for a free-floating space robot. In the presence of the disturbance, the fixed-time controller is designed based on the backstepping technique and a power integrator.
2.
A fixed-time extended state observer (FXESO) based on homogeneous theory is employed to obtain the estimation of the joint velocity and the lumped disturbance. Different from [18] and [19], in the proof, the derivative of the lumped disturbance is considered in the simplified systems.
3.
The SSDR method is introduced to handle dynamic singularities for a space robot. The joint velocity in the kinematic equation can be calculated directly and the estimation of the minimum singularity of the Jacobian matrix is not required. Therefore, the computation burden of SSDR is less than that of the traditional DLS method.

The rest of this paper is organized as follows. In Section 2, the kinematic and dynamic equations of free-floating space robots are given. Considering dynamic singularities, a novel FXPAC method based on an FXESO is proposed in Section 3. In Section 4, numerical simulations are presented. Finally, Section 5 concludes this paper.

Section snippets

Notations, definitions and lemmas

For an arbitrary vector $v$ , $v^{\times}$ is a skew symmetric matrix defined as $v^{\times} = [0 - v_{3} v_{2}; v_{3} 0 - v_{1}; - v_{2} v_{1} 0]$ . $E_{n}$ denotes an identity matrix with size $n \times n$ . The symbol $‖ \cdot ‖$ represents the Euclidean norm of vectors or the induced norm for matrices. For arbitrary vector $x = {[x_{1}, x_{2}, \dots, x_{n}]}^{T}$ , ${sig}^{r} (x) = {[{| x_{1} |}^{r} sign (x_{1}), {| x_{2} |}^{r} sign (x_{2}), \dots, {| x_{n} |}^{r} sign (x_{n})]}^{T}$ and $sign (x) = [sign (x_{1}), sign (x_{2}), {\dots, sign (x_{n})]}^{T}$ , where $sign (\cdot)$ denotes the sign function. For a symmetric matrix $N \in R^{n \times n}$ , $λ_{min} {N}$ and $λ_{max} {N}$ represent the minimum and maximum eigenvalues

Fixed-time extended state observer (FXESO)

In this section, FXESO is designed to obtain the estimation of the joint velocity and the disturbance.

Define $x_{1} = θ$ , $x_{2} = \dot{θ}$ , and $x_{3} = τ_{l} = H_{g}^{- 1} (- C \dot{θ} + τ_{d})$ , respectively. According to the dynamic equation, a reconstructed system is obtained as follows. $\{\begin{matrix} {\dot{x}}_{1} = x_{2} \\ {\dot{x}}_{2} = x_{3} + H_{g}^{- 1} u \\ {\dot{x}}_{3} = {\dot{τ}}_{l} \end{matrix}$

Assumption 1

The external disturbance $τ_{d}$ , $H_{g}$ , $C$ and their derivatives are bounded [28], which means that the lumped disturbance $τ_{l}$ and its derivative ${\dot{τ}}_{l}$ are also bounded. $| {\dot{τ}}_{l i} | \leq {\bar{τ}}_{l i} (i = 1, 2, \dots, n)$ are satisfied, where ${\bar{τ}}_{l i} (i = 1, 2, \dots, n)$ are positive

Simulation results

The space robotic system in this section is composed of a spacecraft and a 6 DOF manipulator as shown in Fig. 1 and Fig. 2. The mass and inertia properties of the space robot are given in [39]. The parameters of the observer-based fixed time controller and simulations are listed in Table 2. The initial position and attitude of the desired trajectories are $p_{d} (0) = {[1.7559 0.0196 0.9276]}^{T} (m)$ and $q_{d} (0) = {[0.7469 - 0.0589 0.7293]}^{T}$ . The end-effector is required to track the trajectories with following constant

Conclusions

This paper studied a novel FXPAC method based on FXESO for a free-floating space robot. Both the joint velocity and the lumped disturbance were estimated by the FXESO. With the FXPAC method, the tracking errors of the position and the attitude of the end-effector could converge to a neighborhood of the origin within fixed time, independent of the initial conditions. The MRPs were adopted to represent the attitude during the control design. Moreover, the SSDR method was employed to avoid large

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgment

The work is partially supported by the China Scholarship Council (CSC No. 201906120098).

References (41)

JingC. et al.
Adaptive sliding mode disturbance rejection control with prescribed performance for robotic manipulators
ISA Trans.
(2019)
GuoK. et al.
Composite learning control of robotic systems: A least squares modulated approach
Automatica
(2020)
LeiR.H. et al.
Adaptive fault-tolerant control based on boundary estimation for space robot under joint actuator faults and uncertain parameters
Def. Technol.
(2019)
WangM. et al.
A non-linear model predictive controller with obstacle avoidance for a space robot
Adv. Space Res.
(2016)
JiaY. et al.
Robust trajectory tracking control of a dual-arm space robot actuated by control moment gyroscopes
Acta Astronaut.
(2017)
XieZ. et al.
Motion control of a space manipulator using fuzzy sliding mode control with reinforcement learning
Acta Astronaut.
(2020)
GalickiM.
Finite-time trajectory tracking control in a task space of robotic manipulators
Automatica
(2016)
PolyakovA. et al.
Finite-time and fixed-time stabilization: Implicit Lyapunov function approach
Automatica
(2015)
ZouA.M. et al.
Fixed-time attitude tracking control for rigid spacecraft
Automatica
(2020)
ZhangL. et al.
Fixed-time extended state observer based non-singular fast terminal sliding mode control for a VTVL reusable launch vehicle
Aerosp. Sci. Technol.
(2018)

LiuJ. et al.

Output feedback control for aircraft at high angle of attack based upon fixed-time extended state observer

Aerosp. Sci. Technol.

(2019)

ZhangD. et al.

Neural networks-based fixed-time control for a robot with uncertainties and input deadzone

Neurocomputing

(2020)

VanM. et al.

Robust fault tolerant control of robot manipulators with global fixed-time convergence

J. Franklin Inst. B

(2021)

TianB. et al.

A fixed-time output feedback control scheme for double integrator systems

Automatica

(2017)

WuX. et al.

Observer-based fault-tolerant attitude tracking control for rigid spacecraft with actuator saturation and faults

Acta Astronaut.

(2021)

LiB. et al.

Finite-time extended state observer based fault tolerant output feedback control for attitude stabilization

ISA Trans.

(2019)

ZhouZ. et al.

Robust prescribed performance tracking control for free-floating space manipulators with kinematic and dynamic uncertainty

Aerosp. Sci. Technol.

(2017)

XuW. et al.

Practical approaches to handle the singularities of a wrist-partitioned space manipulator

Acta Astronaut.

(2011)

AnguloM.T. et al.

Robust exact uniformly convergent arbitrary order differentiator

Automatica

(2013)

XianB. et al.

Task-space tracking control of robot manipulators via quaternion feedback

IEEE Trans. Robot. Autom.

(2004)

Cited by (20)

Nonsingular fast terminal sliding mode tracking control for underwater glider with actuator physical constraints
2024, ISA Transactions
An adaptive finite-time composite control (AFTCC) scheme is designed for the pitch angle trajectory tracking control of Underwater Glider (UG) with actuator physical constraints, uncertain dynamics and external disturbances. Firstly, a novel nonsingular fast terminal sliding mode control (NFTSMC) law is designed that can avoid singular problems, significantly reduce chattering, and achieve finite-time convergence of the system states. Secondly, a novel fixed-time extended state observer (FxTESO) is established to estimate the pitch angular velocity and lumped disturbances within a fixed time. Furthermore, a novel adaptive fixed-time saturation compensation system (AFxTSCS) is proposed to mitigate the effect caused by actuator saturation, and it can adjust the parameter adaptively when the actuator is in saturation or out of saturation. Finally, the AFTCC scheme, which is based on the NFTSMC framework and combines FxTESO and AFxTSCS, is designed to achieve the pitch angle trajectory tracking of UG, and the finite-time convergence of the whole closed-loop system is proved by the Lyapunov stability theory, and the simulations verify the availability and superiority of the proposed AFTCC scheme.
Practical fixed-time output feedback trajectory tracking control for marine surface vessels with unknown disturbances
2024, Control Engineering Practice
This paper proposes a fixed-time output feedback trajectory tracking control scheme based on an extended state observer for marine surface vehicles (MSVs) with uncertain dynamics and unknown disturbances. The scheme utilizes a high-gain fixed-time extended state observer (HGFxESO) to estimate the reconstructed unmeasured velocities and lumped disturbances. The upper bound of the settling time of the HGFxESO is only determined by the design parameters, and the estimation errors can converge to a bounded set within a fixed time. Moreover, a novel fixed-time control law is proposed based on the backstepping technique and “adding a power integrator” (API) technique. The designed continuous control law ensures the trajectory tracking errors of MSVs converge to a neighborhood around the origin within a fixed time. The fixed-time stability of the MSVs trajectory tracking control system is proved using Lyapunov theory. Finally, simulation results are provided to demonstrate the effectiveness of the proposed control scheme.
Adaptive coordinated control for space manipulators with input saturation
2023, Journal of the Franklin Institute
This paper presents a concurrent learning command filtering control scheme for space manipulators in the post-capture phase. Due to the complex space environment and inherent model characteristics, space manipulators are subject to parameter uncertainties, non-conserved momentum and input saturation, which will degrade the control performance. To this end, a novel fixed-time concurrent learning law is developed in the presence of the non-conserved linear and angular momentum, where the convergence rate of the estimated parameters is determined by the control coefficients rather than the unpredictable excitation level. Then, a fixed-time command filtering strategy with the error compensation and saturation compensation is designed, to cope with input saturation and avoid the derivation operation of Jacobi matrices and inertial matrices simultaneously. By means of the proposed control scheme, the spacecraft attitude regulation error and the end-of-arm trajectory tracking error can converge to the small neighborhood of zeros within a fixed time. Simulation results demonstrate the effectiveness of the proposed control scheme.
Trajectory planning of a dual-arm space robot for target capturing with minimizing base disturbance
2023, Advances in Space Research
During the on-orbit capture task of the free-floating space robot, its operating space degenerates. The dynamic coupling and dynamic singularity also affect the task operation. In this paper, a general collision-free trajectory planning method to minimize base disturbance is proposed for a free-floating dual-arm space robot. This method can realize smooth joint trajectory planning while avoiding singularity and realizing self-collision avoidance. The singularity can be effectively avoided by transforming the trajectory planning problem into a nonlinear optimization problem. The optimization function is composed of base disturbance penalty function, end effector error function and collision repulsion potential field function, which can be used to solve the optimal joint trajectory parameters. Futhermore, self-collision avoidance is achieved by combining the bacterial foraging optimization algorithm(BFOA) with the geometric vector collision-avoidance method based on artificial potential field to simplify the calculation complexity. Finally, simulations verify the effectiveness of the proposed planning framework.
Unified neural output-constrained control for space manipulator using tan-type barrier Lyapunov function
2023, Advances in Space Research
Citation Excerpt :
Shao et al. (2021) proposed a nonsingular terminal sliding mode control scheme for a free-floating space manipulator in the presence of disturbances. In Jin et al. (2021), a fixed-time position and attitude control method was developed for a space robot in task space based on an extended state observer. Yao (2022) presented a robust finite-time tracking control method for a free-flying space manipulator by integrating with a disturbance observer.
In this paper, a unified neural control scheme is presented for the output-constrained trajectory tracking of space manipulator under unknown parameters and external perturbations. By utilizing the backstepping control technique as the main framework, the proposed controller is developed with the help of neural network (NN) and tan-type barrier Lyapunov function (BLF). The NN is introduced to identify the unknown part in the dynamic model of the space manipulator. Benefiting from the neural identification, the proposed controller is model-free and insensitive to external perturbations. Moreover, the BLF is adopted to guarantee the position tracking errors never exceed the predefined output constraints. Different from log-type BLF, the tan-type BLF is employed for the control design, which makes the proposed controller universal for the cases with and without considering the output constraints. The semiglobal uniform ultimate boundedness of the resulting closed-loop system is strictly obtained through stability argument. All error variables in the closed-loop system can eventually stabilize to the small residual sets about zero under the proposed controller. Lastly, simulations and comparisons are given to demonstrate the effectiveness and excellent tracking performance of the proposed control scheme.
Fixed-time neural adaptive fault-tolerant control for space manipulator under output constraints
2023, Acta Astronautica
Citation Excerpt :
The fixed-time control can be viewed as a special case of finite-time control, whose settling time is bounded and the upper bound of the settling time does not rely on the initial conditions of the system. In Ref. [47], a fixed-time extended state observer-based control method was proposed for the trajectory tracking of a space manipulator in task space. Yao [48] developed a robust control approach for the fixed-time trajectory tracking of a space manipulator by integrating with a disturbance observer.
This article investigates the challenging problem for the fixed-time trajectory tracking control of free-flying space manipulator under output constraints and actuator faults. A novel fixed-time neural adaptive fault-tolerant control approach is proposed by incorporating the barrier Lyapunov function (BLF) and neural network (NN) into the backstepping control design. The BLF is introduced to preserve the position tracking errors always within the predefined output constraints. Moreover, the NN approximation is adopted to compensate for the lumped uncertain term in the feedforward loop. Lyapunov analysis shows the practical fixed-time stability of the resulting closed-loop system. The proposed controller can guarantee the position and velocity tracking errors converge to the small neighborhoods about the origin in fixed time even in the presence of output constraints and actuator faults and thus ensuring safety. Lastly, the effectiveness and excellent tracking performance of the proposed control approach are illustrated through simulations and comparisons.

View all citing articles on Scopus

View full text

Research paperObserver-based fixed-time tracking control for space robots in task space

Highlights

Abstract

Introduction

Section snippets

Notations, definitions and lemmas

Fixed-time extended state observer (FXESO)

Simulation results

Conclusions

Declaration of Competing Interest

Acknowledgment

ISA Trans.

Automatica

Def. Technol.

Adv. Space Res.

Acta Astronaut.

Acta Astronaut.

Automatica

Automatica

Automatica

Aerosp. Sci. Technol.

Aerosp. Sci. Technol.

Neurocomputing

J. Franklin Inst. B

Automatica

Acta Astronaut.

ISA Trans.

Aerosp. Sci. Technol.

Acta Astronaut.

Automatica

Task-space tracking control of robot manipulators via quaternion feedback

IEEE Trans. Robot. Autom.

Research paper
Observer-based fixed-time tracking control for space robots in task space