A Novel Robust Finite-Time Trajectory Control With the High-Order Sliding Mode for Human-Robot Cooperation

Human-robot cooperation is the major challenges in robot manipulator control, as the controller has to couple the complicated motion of the human arm and the robot end-effectors. To improve the human-robot coordination, this paper proposed a novel robust ﬁnite-time trajectory control based on the nonsingular fast terminal sliding mode and the high-order sliding mode. The proposed method is able to quickly reach the global convergence and minimize motion errors. Based on the nonsingular fast terminal sliding surface, the proposed control method employed a super-twisting algorithm to eliminate the chattering issues to enhance the control robustness. Also, the simpliﬁed robust control term does not require the ﬁrst derivative of the sliding variable. To validate the proposed controller, theoretical analysis and simulation were conducted and the results demonstrated the effectiveness of the proposed method.


I. INTRODUCTION
Modern production and manufacturing industry require cooperative work between human and robots.However, the coupled human-robot system is a complex nonlinear timevarying dynamic system in which uncertainties and external disturbances can easily degrade the control performance [1].Many researchers are devoted to designing a responsible controller to promote the efficiency of the human-robot cooperation, such as variable impedance control methods [2], compliant admittance controllers with frequency domain stability [3], low-impedance approaches with underactuated redundancy [4], and efficiency weighted strategies [5].However, existing methods lack an ideal solution for two major challenges: (1) The control system cannot guarantee the convergence of the stabilized equilibrium and (2) the robustness of the human-robot system is jeopardized by the uncertainties and external disturbances.Although many advanced con- The associate editor coordinating the review of this manuscript and approving it for publication was Shankarachary Ragi.
trol approaches, such as adaptive control, neural networks control, and variable structure control, can partially solve the problem, a faster and convergent method is necessary, such as finite-time controllers.Yu et al. [6] developed a novel finite-time command filtered backstepping approach that allows conventional command-filtered backstepping control and guarantees the finite-time convergent property.Huang et al. [7] developed a recursive design algorithm for a class of uncertain nonlinear system.Liu et al. [1] proposed a finite-time H ∞ control approach by using backstepping control for uncertain robotic manipulators and this controller provides higher precision and faster response than the conventional H ∞ control.
This study intends to implement the nonsingular fast terminal sliding mode control (NFTSMC) method to improve the system stability and robustness when facing the uncertainties and external disturbances.The NFTSMC is a typical sliding mode control (SMC) that not only has the advantages of SMC but also provides a finite-time convergence.The proposition of NFTSMC is based on several existing studies.The conventional SMC has advantages of robustness to uncertainties and low sensitivity to the system parameter variations [8].This technique has been used for the design of robot system, such as an adaptive neural impedance control in [9], a robust SMC in [10], a method based on exponent trending law in [11] and adaptive neural network control in [12].These methods are developed based on a linear surface, however, the uses of linear sliding mode control (LSMC) does not guarantee that equilibrium convergence in the finite time.To overcome this challenge, terminal sliding mode control (TSMC) methods have been developed.The TSMC uses a nonlinear sliding mode surface instead of a linear surface.Many studies based on TSMC have been proposed, such as an adaptive TSMC method in [13], a novel adaptive finite-time control in [14], a neural-network-based TSMC control method in [15].However, the TSMC also has two constraints: (1) the non-global fast convergence and (2) the singularity problem.To overcome (1), the fast terminal sliding mode control (FTSMC) has been developed.The FTSMC approach, such as in [16], can provide a fast convergence when the system states are far away from the equilibrium point.To overcome (2), the nonsingular terminal sliding mode control (NTSMC) [17], [18] and the nonsingular fast terminal sliding mode control (NFTSMC) [19], [20] have been developed.Compared with the NTSMC methods, the NFTSMC has the same properties but provides a faster state convergence.
Based on previous findings, it can be concluded that NFTSMC is able to deal with the nonlinear system.However, sliding mode control approaches have a drawback that they will produce the chattering.The chattering can generate unmodeled high-frequency dynamics and even endanger the safety of the operator in the process of human-robot cooperation.To overcome this problem, several solutions have been developed.For example, a continuous control term is developed in [18].This method does not affect the finite-time convergence but degrades the robustness and accuracy.In [21], a method which uses a saturation function that replaces the signum function.This method depends on the value of the boundary layer thickness.That means, if the value is too small, the use of saturation function will also cause the chattering.In [22], a high-order sliding mode (HOSM) technique is developed.This method not only eliminates the chattering but also enhance the robustness.Based on these findings, this study intends to incorporate the NFTSMC technique into the controller design to ensure the proposed controller can provide a global finite-time convergence and eliminate the chattering without losing robustness.
In summary, we propose a novel robust finite-time trajectory control (NRFTC) scheme based on the NFTSMC technique and the HOSM techniques.The main contributions are summarized as follows: (1) A new sliding mode surface is designed in this paper.
Compared with the conventional LSMC and NTSMC, the uses of the new sliding mode surface can guarantee the closed-loop system with faster finite-time convergence and higher tracking precision.(2) To reduce the chattering, this paper develops a HOSM control term by using a super-twisting algorithm (STA).This control term has the following superior advantages: 1) it can effectively compensate for the uncertainties and external disturbances; 2) it only requires the information of the sliding variable; 3) it generates a continuous control signal, i.e., the chattering will be eliminated.Finally, this paper is organized as follows: Section II describes the preliminaries and the dynamics of the humanrobot cooperation system.Section III gives the design of the proposed NRFTC with a traditional SMC term, and the proposed NRFTC with the HOSM is presented in section IV.The simulation results are given in section V.The conclusions and future works are given in the last section VI.

II. PRELIMINARIES AND PROBLEM FORMULATION A. PROBLEM FORMULATION
In the joint space, the dynamics of the human-robot cooperation system are described by [23].
where q, q, q ∈ R n are the position, velocity, and acceleration vectors of joints, respectively.D (q) ∈ R n×n is a symmetric positive definite inertia matrix, C (q, q) q ∈ R n is the vector of centripetal and Coriolis forces, G (q) ∈ R n is the vector of gravitation forces, and τ (t) ∈ R n is the input torques.J (q) ∈ R n×n is the Jacobian matrix which is established by the structure of the robot, and f (t) ∈ R n is the constraint force.
To realize the control design of the robotic end-effector, the dynamic equation (1) in the joint space needs to be converted to the Cartesian-based dynamic equation.The relationship between the Cartesian coordinates and the joint coordinates is described as [24] where x ∈ R n is the position of the robotic end-effector in the Cartesian coordinates.Considering the differential of ( 2) with respect to time, we obtain Therefore, q can be rewritten as Substituting ( 2) and ( 4) into (1), the Cartesian-based dynamic equation of the human-robot cooperation system can be rewritten as or where K f is the human input force gain, D x (q) = J −T D (q) J −1 , C x (q, q) = J −T C (q, q) − D (q) J −1 J J −1 , G x (q) = J −T G (q), and F x = J −T τ (t).In general, the dynamic equation ( 5) has the following useful structural properties, which can be exploited to facilitate the controller design in the next section.
Property 1 [25]: The Jacobian matrix J (q) is configuration-dependent, which is assumed to be nonsingular in the finite workspace.
Property 2 [24]: The inertia matrix D x (q) is symmetric positive-definite and satisfies λ min (D x ) I ≤ D x ≤ λ max (D x ) I , where λ min (D x ), λ max (D x ) are the minimum eigenvalues and maximum eigenvalues of D x .
Property 3 [24]: x d , ẋd and ẍd exist and are bound, where x d ∈ R n is the desired motion trajectory.
Assumption 1 [26]: The constrained force f (t) is uniformly bounded, i.e., there exists a constant f ∈ R + , such that Based on the analyses in [26], this assumption is reasonable for an engineering point of view.
Considering the human-robot cooperation system in the Cartesian coordinates.The control objective is to design a controller such that (x, ẋ) track (x d , ẋd ), while ensuring that all closed-loop signals are bounded.

B. PRELIMINARIES AND NOTATIONS
The power of vectors used in the paper is defined as The integral of vectors is defined as The signum function is defined as

III. THE PROPOSED NRFTC WITH SMC TERM
In general, the design of a sliding mode controller mainly consists of two steps [27].The first step is to select an appropriate sliding mode surface.The second step is to design a control law that forces the system states to reach the desired sliding manifold in the finite time.
Defining the tracking errors e and the derivative of the tracking errors ė as The sliding surface of LSMC is selected as [10] The sliding surface of NTSMC is selected as [18] s = e + σ 2 |ė| α 1 The sliding surface of FTSMC is selected as [19] where For the sliding surface (9), it is a linear sliding surface.Thus, the uses of LSMC ensure asymptotic convergence of the system states to the equilibrium point, but not in finite time.To overcome this challenge, the nonsingular terminal sliding surface (10) is developed.However, the uses of TSMC does not guarantee the global convergence because it converges at a relatively slow rate when the system states are far away from the equilibrium point compared to the LSMC.The uses of fast terminal sliding surface (11) have improved the convergence rate.But noted that when e < 0, the fractional power α/β may lead to e α/β / ∈ R n .This is a singularity problem.Now, we want to enhance the global convergence rate and overcome the singularity problem, a nonsingular fast terminal sliding mode surface is selected as where γ and ι are positive odd integers satisfying the relation 1 < γ /ι < 2 and ϕ > γ /ι.The fast convergence property of ( 12) can be explained as follows: when the system states are far away from the equilibrium point, λ 1 |e| ϕ sgn (e) plays the main role, ( 12) can be approximated by s = e + λ 1 |e| ϕ sgn (e) = 0, which guarantees a high convergence rate.When the system states are close to the equilibrium point, λ 2 |ė| γ /ι sgn (ė) plays the main role, ( 12) can be approximated by s = e + λ 2 |ė| γ /ι sgn (ė) = 0, which ensures the finite-time convergence.Differentiating s with respect to time yields Substituting ( 6) into (13) yields Without considering the constraint forces and external disturbances, the equivalent control law described in (15) can be obtained by setting ṡ= 0.
Now, under Assumption 1, a novel robust finite-time trajectory control (NRFTC) can be designed as where is a robust SMC term for compensating for both uncertainties and modeling errors, where δ is the upper bound of uncertainties, η is a small positive constant and is a force compensation term.Now, the following theorem is established to show the stability result of the human-robot cooperation system (5).
Theorem 1: Considering the human-robot cooperation system described in (5), under Assumption 1 and the proposed NRFTC (16) with the equivalent control law (15), the robust SMC term (17) and the force compensation term (18), the following conclusions can be obtained: 1) the track errors e will converge fast to zero within finite time; 2) the proposed NRFTC method (16) will not cause the singular problems during the whole process.
Proof: Choosing a Lyapunov function candidate of the form.
Differentiating V (t) with respect to time yields Substituting ( 15), ( 16) and ( 18) into (20) yields where (q, q, t) where ρ (s i , ėi ≥ 0. Considering the following two possible cases: 1) |ė i | γ /ι−1 > 0 when ėi = 0 and 2) = 0 when ėi = 0.Meanwhile, according to [8], two different cases are needed to be considered when s i = 0: 1) ėi = 0 and 2) ėi = 0 but e i = 0.For the case s i = 0 and ėi = 0, we have ρ (s i , ėi ) > 0, thus V < 0. Therefore, it is concluded that the system is stable according to the Lyapunov criterion and the system states will move fast to the sliding mode s i = 0 within finite time.Now, considering another case s i = 0 and ėi = 0 but e i = 0, by substituting ( 15)-( 18) into (6) yields Form ėi = 0, ( 23) can be rewritten as which suggests that ëi < −η and ëi > −η for the cases s i > 0 and s i < 0, respectively.According to [19], it means that ėi is not an attractor and thus the system will not always keep staying on the equilibrium points (ė i = 0 but e i = 0).Moreover, it is reasonable to assume that there exists a small positive constant ξ in the vicinity of ėi = 0, i.e., |ė i | ≤ ξ , satisfying ëi < −η and ëi > −η for s i > 0 and s i < 0, respectively.Thus, the crossing of trajectories between two boundaries of |ė i | ≤ ξ is performed in finite time, and also dynamics from the region |ė i | > ξ converges to the boundaries in finite time [17].Therefore, it is concluded that the sliding mode s i = 0 can be reached from anywhere in the phase plane in finite time.
The system is stable according to the Lyapunov theorem, i.e., all the signals in the closed-loop system are bounded.In addition, the setting time is given by [19] where F (•) denotes Gauss' Hypergeometric function, and the more details of Gauss' Hypergeometric function is defined in [28].The proof is completed.
Remark 2: The robust SMC term is introduced to improve the robustness of the closed-loop systems that subject to uncertainties and external disturbances.However, noted that the function sgn (s) is discontinuous, which may cause the chattering.This problem will be resolved in the next section.
Remark 3: Compared with the LSMC (9), NTSMC (10), and FTSMC (11), the proposed NRFTC ( 16) has the following characteristics: 1) the fast finite-time convergence of the closed-loop system is achieved according to (25); 2) no occurring of singularity is ensured since the control law does not include any negative power.

IV. THE PROPOSED NRFTC WITH HOSM
In the proposed NRFTC, there exists a deficient place that needs to be improved.The robust term ( 17) is a discontinuous term due to the signum function introduced.It will cause the chattering in the input torques.In addition, to enhance the stability and robustness of the robot system, the robust gain δ + η is usually chosen to be conservatively larger than the upper bound value of uncertainties.The large gain will increase the chattering.To eliminate the chattering, we first consider the following common two approaches.
The robust control term F re is designed by using a continuous function [18].
The signum function is replaced by a saturation function [21]. where where k 1 > 0, k 2 > 0, 0 < υ < 1, φ is a small positive constant.The chattering may be reduced by using the above approaches, but they will also reduce the tracking accuracy and robustness of the system.Thus, in this paper, we propose a method to design a HOSM control by utilizing the STA.This approach not only reduces the chattering but only increase the tracking accuracy.Moreover, this approach only requires information about the sliding variable.Now, the robust HOSM term is designed as where k 1 > 0, k 2 > 0. Now, the following theorem is established to show the stability result of the human-robot cooperation system (5).The control structure of the proposed NRFTC method is shown in Fig. 1.Theorem 2: Considering the human-robot cooperation system described in (5), under Assumption 1 and the proposed NRFTC (16) with the equivalent control law (15), the robust HOSM term (29) and the force compensation term (18), the following conclusions can be obtained: 1) the track error e will converge fast to zero within finite time; 2) the proposed NRFTC method (16) will not cause the singular problems during the whole process; 3) the chattering caused by the traditional LSMC is greatly reduced.
Proof: Substituting (28) into (21) yields Control structure of the novel robust finite-time trajectory control (NRFTC) method. • where k 2 satisfies the relation k 2 ≥ δ .According to the proof analysis in Section III, it is proved that from any initial states, the closed-loop system may converge quickly to the equilibrium point in finite time.Hence, we conclude that the closed-loop system is stable without any singularity and the control input signals are smooth without chattering.The proof is completed.Remark 4: Noted that the designed robust HOSM term (29), the discontinuous function sgn (s) acts on the high-order derivative of the sliding mode, rather than the first derivative of the sliding mode, which can greatly reduce the chattering during system switching.In addition, it does not require the time derivatives of the output s, which simplifies the control structure.
Remark 5: Compared with the continuous control term (26) in [18] and saturation function (27) in [21], the uses of the designed HOSM term (29) not only can reduce the chattering but also does not degrade the robustness of the closed-loop system.

V. SIMULATION RESULTS
To validate the proposed controller and examine its effectiveness, this section adopted a simulation model.To simply the simulation, this paper chooses the classic two-link robotic system as an example and its dynamic model in task space can be described as where x = x 1 x 2 T , x 1 and x 2 are the positions of the endeffector in X and Y direction in the Cartesian coordinate system, respectively.The details of D x (q), C x (q, q), G x (q) and J (q) are described in [26].The values of robot parameters are shown in Table 1.
The desired trajectory of end-effector is given as x d = 0.7 + 0.1cos (π t) 0.1sin (π t) T .We require the  end-effector to move along a circle in the task space.The center of a circle is located at x = 0.7 0 T and the radius is 0.1m.The initial position of end-effector is at the center of a circle, i.e., x (0) = 0.7 0 T and ẋ (0) = 0 0 T .
The time of this simulation is 10 seconds.When t < 4s, the end-effector moves normally without the constrained forces.When t ≥ 4s, the constraint forces acting on the endeffector is f (t) = 5.5sin (5t) 5.5cos (5t) T .To validate the robustness of the proposed controller, the external disturbance is considered in this simulation and it is described as τ d = 1.1sin (q 1 ) 1.3sin (q 2 ) T .

A. COMPARISON BETWEEN LSMC, NTSMC, AND THE PROPOSED NRFTC
In this section, we compare the performance of the LSMC, NTSMC, and the proposed NRFTC in terms of trajectory tracking, velocity tracking, and control input.The LSMC based on the sliding surface ( 9) can be designed as     Figs.2-3 show the trajectory tracking and velocity tracking of end-effector in X /Y direction, respectively.Fig. 4 shows the tracking errors of end-effector in X /Y direction.Fig. 5 shows the control inputs.Fig. 6 shows the trajectory circle in the Cartesian coordinate.For easy in comparison, the average tracking errors and root mean square errors are also shown in Table 2. From Figs. 2-3, we can conclude that these control methods can successfully track the desired trajectory.But from Fig. 4, it is obvious to see that the LSMC and the proposed NRFTC provided a faster finite-time convergence compared to the NTSMC.However, the LSMC only guarantee that the tracking errors converge to a small neighborhood around zero, rather than converge to zero.According to Table 2, the proposed NRFTC achieves a much higher tracking accuracy with RMS (e 1 ) = 0.0036 and RMS (e 2 ) = 2.31×10 −4 compared to the LSMC (RMS (e 1 ) = 0.0068, RMS (e 2 ) =  9.82×10 −4 and NTSMC (e 1 ) = 0.0150, RMS (e 2 ) = 9.57×10 −4 ).The FTSMC is not considered in the simulation results because it meets a singularity problem during operation.

B. COMPARISON FOR ROBUSTNESS
To verify robustness of the robust term ( 26), ( 27) and (29).In this section, we compare the following three controllers: 1) the proposed NRFTC combined with continuous function (Case 1); 2) the proposed NRFTC combined with saturation   3.
From the results, it is obvious to see that the use of HOSM provides a better tracking accuracy without chattering compared to the used of saturation function and continuous term.By reducing the value of φ, the robustness of using the saturation function may increase, but this approach will cause severe chattering.The use of continuous term still causes a little chattering when the constraint forces are applied (t ≥ 4s).

VI. CONCLUSION AND FUTURE WORK
In this paper, a novel robust finite-time trajectory controller is proposed for the control design of human-robot cooperation system.To avoid the singularity problem and obtain a global fast convergence, a novel nonsingular fast terminal sliding surface is employed in this paper.To enhance stability and eliminate chattering, a robust high-order sliding mode control is designed by using super-twisting algorithm.The proposed controller offers some satisfying performance such as fast response, high precision and strong robustness.Simulation results demonstrate that the proposed controller is more effective and practical for operating the cooperative task between human and robot.Finally, our future works will focus on extending the results of this study to the actual robot system.

FIGURE 2 .
FIGURE 2. Trajectory tracking of end-effector in X /Y direction.

FIGURE 3 .
FIGURE 3. Velocity tracking of end-effector in X /Y direction.

FIGURE 6 .
FIGURE 6. Trajectory circle in the Cartesian coordinate.

FIGURE 7 .
FIGURE 7. Tracking errors of end-effector under different robust control terms.

FIGURE 8 .
FIGURE 8. Control inputs of end-effector under different robust control terms.

TABLE 1 .
Parameters of the robot manipulators.

TABLE 2 .
Performance indexes for various control methods.

TABLE 3 .
Performance indexes for various robust control terms.