Abstract
In order to improve the success rate of space debris object capture, how to increase the resistance to interference in the space robot arm has become an issue of interest. In addition, since the space operation time is always limited, finite-time control has become another urgent requirement needed to be addressed. Considering external disturbances, two control methods are proposed in this paper to solve the control problem of space robot arm. Firstly, a linear sliding mode control method is proposed considering the model uncertainties and external disturbances. The robot arm can track the desired trajectory, while a trade-off between optimality and robustness of the solved system can be achieved. Then, in order to reduce conservativeness and relax restrictions on external disturbances, a novel backstepping control method based on a finite-time integral sliding mode disturbance observer is developed, which compensates for the effects of both model uncertainties and infinite energy-based disturbance inputs. Finally, simulation examples are given to illustrate the effectiveness of the proposed control method.
1. Introduction
With the rapid growth of space projects in the last several decades, the increasing space debris residues from satellites scrapped in space bring huge threat to the existing on-orbit spacecraft [1]. Therefore, how to reduce the amount of space debris and effectively lower their risk level becomes more and more urgent. Under this background, Active Debris Removal (ADR) has become a worldwide research hotspot [2–4].
Space debris is mostly discarded space scrap, which is out of control and eventually moves freely due to complex nutation. Since these high-speed tumbling targets are very difficult to catch directly, it is necessary to reduce the relative speed between the chaser and the target before the next on-orbit capture [5–8]. Thus, the final capture of the target can be achieved when the relative speed is slow enough [9, 10].
In terms of control for system with uncertainties and disturbances, numerous methods are proposed. Focused on solving the problem of asynchronous phenomena with different solutions, Cheng et al. [11–13] proposed a finite-time backstepping control method by incorporating a hidden Markov model, and the finite-time asynchronous control is achieved in the end. Considering time-varying full-state constraints and uncertainties, an adaptive fuzzy backstepping control was proposed for nonlinear state-constrained systems by Zhou et al. [14–16], and the parameter updating law is different compared with existing adaptive updating methods. By using the integral sliding mode design method for nonlinear stochastic systems, Wang et al. [17, 18] presented a new integral sliding mode control for fuzzy stochastic systems subjected to matched/mismatched uncertainties. The asymptotic stability of sliding mode dynamics is guaranteed while a simple search algorithm is provided to find the stability bound. Liu et al. [19, 20] proposed universal adaptive control to solve the universal control problem of a class of uncertain nonlinear systems. Yang and Tan [21, 22] designed an adaptive neural network for sliding mode control of flexible manipulators. However, the conservativeness and strict restrictions of the disturbances of the above controller are still needed to be addressed.
Taking into account the phenomenon of disturbing moments in real systems, robust control with disturbance observer is an efficient control scheme to address it. The essence of robust control is to maintain the robustness of a closed-loop system. Robustness refers to the ability of a control system to maintain certain properties of the system under certain parameter regimes. Robust control theory is the theory of robustness in space by optimising certain performance indicators with infinite parameters. So, in this paper, a robust control system for a robotic arm will be designed based on the backstepping method and the model will be compensated with a finite-time integral sliding mode disturbance observer in that it can approximate the disturbance moment vector well. Thus, the system can achieve accurate tracking of the robotic arm attitude command.
Considering all the above practical challenges, in this paper, we study the stabilization problem of the flexible deceleration brush detumbling mechanism attached to a space robot arm. Two sliding mode control methods are developed to solve this problem. First, an optimal sliding mode control law is proposed for space multijoint robotic manipulator with consideration of both optimality and robustness of the detumbling system. The proposed control law can stabilize the overall closed-loop system with a prescribed H∞ performance level. Moreover, in this design, by using the weighting matrix method, the balance between the optimality and robustness of the detumbling system is achieved. Second, considering the fact that in practical space, the external disturbance always refers to the mismatched type, a novel backstepping control law with the finite-time integral sliding mode disturbance observer is developed, which can compensate the effects of model uncertainty and mismatched disturbance input simultaneously. Finally, simulation examples are given to verify the accuracy and effectiveness of the controller.
2. Materials and Methods
2.1. Design of Robust Sliding Mode Controller
With the consideration of microgravity environment, the potential energy of the system can be ignored. Then, the Lagrange function of the total kinetic energy of the space robot system can be expressed aswherewhere is , is the linear velocity of the base, is the angular velocity of the base, is the mass of the th rod of the space robot, are the variables of each joint of the manipulator, is the joint that connects the th link and the th link, is the position vector of the centre of mass of the th lever of the manipulator in an inertial coordinate system, is the position vector of the th joint in an inertial frame, is the mass of the th bar of the space robot, is the inertial of th link, and is a given constant.
By using the Lagrange equationthe dynamic equation of the space robot system can be obtained as follows:where is the symmetric positive definite inertial matrix, is a nonlinear term satisfying , and is the external control force and torque.
Since the space racemization robot works in a complex microgravity environment, considering uncertainties such as external disturbance, friction, and parameter error, equation (4) can be further expressed as follows:where and are nominal matrices and and are the corresponding uncertain matrices. Then, the above equation can be further rewritten as
Define the dynamic compensation aswhere is control input vector, and then define the external disturbances as
In order to make the space robot end track the time-varying desired trajectory, the state tracking error is defined aswhere is the desired joint angle and is the desired joint angular velocity.
The trajectory tracking error equation of the space robot can then be obtained as follows:where each parameter is specifically defined as
Therefore, the force and torque exerted on the space robot can be solved as
In order to facilitate the tracking control of the desired trajectory and reach the desired racemate point at the end, an auxiliary equation is designed aswhere and are constant matrices.
Substituting equation (13) into equation (10) yieldswhere
Then, the control force and torque of the space robot can be expressed as
By designing , the external disturbance in the system can be reduced. Given any positive real number , we have
Design the Lyapunov function , and then the Riccati equation can be solved as follows:
With being skew symmetric, can be designed aswhere is a positive definite symmetric constant matrix.
Substituting equation (19) into equation (18), one can obtain
By utilizing Cholesky decomposition method, we can have
At the same time:
Therefore, the robust optimal state feedback controller can be designed as
2.2. Design of Backstepping Controller with Finite-Time Observer
In the previous section, we design an sliding mode control law for space multijoint robotic manipulator, which can stabilize the overall closed-loop system with a prescribed performance level. It should be pointed out that in the previous control approach, the model uncertainty is dealt with the robust control framework via the LMI technique, which may bring conservativeness in practical application when finding a feasible solution for spacecraft or space robotic manipulator system. Moreover, the considered external disturbance input refers to a matched type, which is a conservative assumption for practical space multijoint robotic manipulator, forthe orbit disturbance is always unmatched. To this end, in this section, we will revisit the control design problem for the space multijoint robotic manipulator, where the model uncertainty is treated as a matched nonlinearity of the system instead of a model uncertainty, and the considered disturbance input is an unmatched term. A novel backstepping control law with the finite-time integral sliding mode disturbance observer is developed, which can compensate the effects of model uncertainty and infinite energy type disturbance input .
We first recall the following dynamic equation for space multijoint robotic manipulator:
Then, we define the state vector as
Thus, equation (26) can be rewritten aswhere
As the previous discussion, in the practical space environment, there always exist unknown model nonlinearities and unmatched external disturbances, which are denoted as and , respectively. Then, considering these effects in system (23), the system equation can be rewritten aswhere
In the following discussion, we will employ the finite-time integral sliding mode disturbance observer method to estimate the total nonlinearity , based on which a backstepping control law will be designed to stabilize system (24).
Before proceeding the subsequent design work, we decompose the system state vector as with and and decompose as . Then, system (26) can be decomposed as
We now rewrite equation (32) as the following two subsystems:
Next, we define the following two backstepping variables aswhere is the virtual input vector to be designed, which is constructed aswhere is a positive definite diagonal matrix to be designed and is the estimation of the nonlinearity . With all the information in hand, we present the following finite-time integral sliding mode disturbance observer (FTISMDO) for system (32) aswhere , for ; denotes the norm bound of the unknown derivative of ; and are the positive parameters to be designed. In particular, these parameters should satisfy , , , and where and are two sign functions.
In observer (36), get
Combining equations (36)–(38) yields
Design the Lyapunov function :
So, .
Because of (41), , , can converge in finite time.
So,
Based on equation (41) , , can converge in finite time and so does in finite time . It should be noted that Hurwitz condition should be met for , and also , or , . Therefore, after finite time ,, namely, the estimate error is able to converge in finite time .
In observer (36), note that the information of is required, which cannot be measured and obtained directly due to physical constraints. To this end, a high-order sliding mode differentiator (HOSMD) is employed here to estimate . The HOSMD is presented as follows:where are positive constants to be selected. According to [23], the following conclusion holds after a finite-time transient process:
In HOSMDs (37) and (43), denotes , and can be calculated by .
Based on the FTISMDO (32) and HOSMDs (37)–(43), we now reconsider the backstepping variables and . For , based on the virtual input vector defined in equation (35), it is derived thatwhere
On the other hand, for , it is derived thatwhere the lumped disturbance vector is defined as
As a result, we obtain the following system which is equivalent to equation (33):
Theorem 1. Considering system (49), suppose that the model uncertainty satisfies , where is a known constant; under the following adaptive control law:the overall closed-loop system (49) is asymptotically stable.
Proof. Considering system (49), we define the Lyapunov function for the first subsystem of (49) as follows:Calculating the time derivative yieldsNext, we design the Lyapunov function for the second subsystem of (49) asand then we haveConsidering the lumped disturbance in equation (50), notice that the following inequality holds:If we substitute the control law (50) into equation (54), it is shown thatBy some calculation, we have thatholds for , which means that the overall closed-loop system (49) is asymptotically stable. Thus, we complete the proof.
3. Simulation Experiment and Result Analysis
3.1. Robust Optimal Controller Simulation Results
The redundant manipulator has seven degrees of freedom, and the floating base has six degrees of freedom. The structure of the space robot is established by the DH method, as shown in Figure 1, the dynamic parameters are shown in Table 1, and the total detumbling chaser is illustrated in Figure 2.
Because optimal control, for a system of to design a suitable state feedback control rate , make the performance indexwhere is of real symmetric semipositive definite weighted matrices and is of a real symmetric positive definite matrix.
From equation (25), we obtain the corresponding parameters as
In this simulation, the end of the manipulator moves from point A to point B. At the moment of t = 1 s, the interference force of 200 N is applied to the terminal flexible reducer brush, with the action time being 0.01 s, which can be regarded as an impulse force during the racemization process.
The initial pose and final pose of the space racemization robot are known, and the trajectory planning is carried out within a specified time. After planning, the position changes of the end points are shown in Figure 3. The desired trajectory of end-line velocity can be obtained by differentiation, as shown in Figure 4. Among them, the terminal attitude does not change, so it is unnecessary to repeat the terminal attitude planning.
The end planning trajectory in the above Cartesian space is transformed into the joint space. The joint angle, joint angular velocity, and joint angular acceleration after planning are shown in Figures 5–7 , respectively. The first 10 s is the planned motion mode, and the last 5 s is the reserved stabilization time of the manipulator.
For the planned trajectory of the space robot, the robust optimal controller proposed in this paper is adopted, and the switching frequency of joint control torque is high. In order to simulate the actual working effect, a low-pass filter is incorporated. The joint torque is shown in Figure 8, while the joint angle error is shown in Figure 9. The upper and lower bounds of the control torque are small, and with the gradual convergence of the tracking error, the system is asymptotically stable, and the control torque gradually decreases and ultimately converges.
robust optimal controller designed in this paper can deal with disturbances with stronger robustness. As for the joint angle error, robust optimal controller can maintain stability and suppress the tremor within only 0.5 s. If disturbance occurs, the robust optimal controller can generate larger control torques to suppress vibration, which is conducive to the racemization process. Moreover, the robust optimal controller has smaller tracking error and higher tracking accuracy and can stabilize the system within planned time, and thus its advantage of accuracy and effectiveness is clearly verified.
3.2. Backstepping Control Simulation Results
In this section, a simulation of the stabilization problem of a space multijoint robotic manipulator is implemented to illustrate the effectiveness of the proposed backstepping control law. Considering the double-joint robotic manipulator, the mechanical model of the manipulator is shown in Figure 10. For manipulator model (1) with angular state variables and angular velocity variables , the initial attitude orientation and attitude angular velocity of the manipulator are set as
In addition, the inertia matrix, Coriolis force, and centrifugal force matrix are set as follows:whereand the parameters above are selected as
The uncertainties of the system are described aswithwhere describes the size of the uncertainty and it is designed as . The measurement errors of state variables are given as
The attitude angular and angular velocity trajectories are depicted in Figures 11 and 12 , respectively, in which ideal performances are achieved.
The function in control law (41) may lead to undesirable chattering when the sliding manifold crosses the sliding mode surface . To avoid this phenomenon, the function can be replaced bywhere refers to the size of the bounder layer, which offers a continuous approximation to the sliding mode controller inside the boundary layer. The time response of the controller is presented in Figure 13, of which the upper bound is limited to 2 Nm.
For the designed finite-time integral sliding mode disturbance observer, the initial estimated value is set as , and the gain parameters in observer (36) are set as follows:
The estimation performance is given in Figure 14, in which ideal estimation is achieved in finite time. Thus, the improved control performance of the backstepping controller with finite-time observer is demonstrated fully.
The uncertainties of the system are described as follows: at 0 s, 10 s, and 30 s, moments of 15 Nm lasting for 0.02 s are entered in each dimension as a pulse. For the first dimension, a moment of 15 Nm acting for 0.03 s, with an action period of 10 s, is entered as a pulse. For the second dimension, a moment of 10 Nm acting for 0.02 s, with a period of 10 s is entered as a pulse. A set of random numbers between is generated as a random disturbance.
The attitude and angular velocity trajectories are depicted in Figures 15 and 16 , respectively, in which ideal performances are achieved under the condition that a pulse acts on the target at 0 s, 10 s, and 30 s.
The time response of the controller is presented in Figure 17, of which the upper bound is limited to 2 Nm.
The estimated performance is given in Figure 18, where the ideal estimate is achieved in finite time. Thus, given multiple external disturbances, periodic disturbances, and random disturbances, the improved control performance of an inverse stepper controller with a finite-time observer can fulfil the requirements.
4. Conclusion
In this paper, the dynamic model of a space debris detumbling system is established. For the manipulator with uncertainty and external disturbance, we first design a robust optimal controller based on a linear quadratic performance index. The essence of control is to minimize the norm of the error transfer function when the interference is bounded and the maximum interference is considered. At the same time, the optimal control makes the system have robust optimal performance under a specified performance index. It can be proved that the robust state feedback controller can effectively compensate the uncertainties and bounded external disturbances, which means the manipulator can accurately track the desired trajectory, and the quadratic performance index reaches the optimal. Then, assuming that the disturbance is unmatched, a backstepping controller with a finite-time integral sliding mode disturbance observer is designed to further reduce the conservativeness existing in first robust controller and improve the control accuracy. The stability analysis shows that the finite-time integral sliding mode disturbance observer can efficiently compensate the unmatched lumped uncertainty in finite time..As this paper is a finite-time backstepping control of an observer, further research on finite-time backstepping control of an controlleron redundant robotic arms will be done in the future.
Data Availability
The data used to support the findings of this study are included within the article.
Conflicts of Interest
The authors declare that they have no conflicts of interest.