Adaptive Nonsingular Fast Terminal Sliding Mode Tracking Control for an Underwater Vehicle-Manipulator System with Extended State Observer

: An adaptive nonsingular fast terminal sliding mode control scheme with extended state observer (ESO) is proposed for the trajectory tracking of an underwater vehicle-manipulator system (UVMS), where the system is subjected to the lumped disturbances associating with both parameter uncertainties and external disturbances. The inverse kinematics for the system is obtained by the quaternion-based closed-loop inverse kinematic algorithm. The proposed controller consists of the modiﬁed nonsingular fast terminal sliding mode surface (NFTSMS) and ESO, and the adaptive control law. The utilized NFTSMS can ensure the fast convergence of the tracking errors, together with avoiding the singularity in the derivation. According to the ESO method, the estimation error of the lumped disturbance vector can realize the ﬁxed-time convergence to the origin, along with replacing the sign function with the saturation function to attenuate the chattering. A continuous fractional PI-type robust term with adaptive laws is introduced to handle the unknown bound of the estimation error. The closed-loop system is proved to be asymptotically stable by the Lyapunov theory. Simulations are performed on a ten degree-of-freedom UVMS under four different strategies. Comparative simulation results show that the proposed controller can achieve better tracking performance and stronger robustness of the disturbance rejection. smoother variation compared to those in other cases, it demonstrates that the proposed control method has stronger robustness of resisting disturbances.


Introduction
The ocean environment is a potential treasury of resources, including a variety of living creatures, mineral deposits and sustainable energy [1]. In the last 20 years, many efforts have been made to develop marine tools for the ocean exploration and exploitation. Remotely operated vehicles (ROVs) and autonomous underwater vehicles (AUVs) are two common tools among the marine robots [2]. In particular, an underwater vehiclemanipulator system (UVMS) that contains an underwater vehicle equipped with one or multiple underwater manipulators, has been applied more significantly in underwater tasks than the underwater vehicle or underwater manipulator only. Nevertheless, it is difficult to achieve the trajectory tracking control of the UVMS end-effector.
On the one hand, the UVMS is kinematically redundant because its total degrees of freedom are usually more than the task-space coordinates that are at most six dimensions. Thus, such redundant system admits infinite numbers of the joint-space solutions for the specific coordinates in the task space. Subsequently, lots of inverse kinematic schemes have been proposed to handle the redundant issue, like weighted pseudo-inverse method merged with the fuzzy technique [3]. Apart from the fuzzy technique, the joint fault-tolerant    To describe the motion of the vehicle in detail, its velocity in the vehicle-fixed frame is defined as   To describe the motion of the vehicle in detail, its velocity in the vehicle-fixed frame is defined as ν = [ν T 1 , ν T 2 ] T ∈ R 6 . The vectors ν 1 = [u, v, w] T and ν 2 = [p, q, r] T are the linear velocity and angular velocity of the vehicle with respect to the Earth-fixed frame expressed in the vehicle-fixed frame, respectively. The above velocity vectors satisfy the following equations: . η 1 (1) where J 1 (η 2 ) expresses the linear velocity transformation matrix from the Earth-fixed frame to the vehicle-fixed frame, satisfying: and the angular velocity transformation matrix J 2 (η 2 ) is defined as: where c· = cos(·) and s· = sin(·). Define the vector q = [q 1 , · · · , q n ] T ∈ R n as the joint position coordinate of the manipulator in each link-fixed frame, and the time derivative , and w E denotes the angle velocity of the end-effector in the Earthfixed frame.

The Quaternion-Based CLIKA of the UVMS
The primary objective of the inverse kinematics is to find a suitable motion variable ζ(t) through a desired end-effector task x E,d = [p T E,d , η T E,d ] T , where p E,d and η E,d denote the desired position and orientation coordinates of the end-effector in the Earth-fixed frame, respectively. The utilization of pseudoinverse of the Jacobian matrix in [3] is the simplest way to invert the mapping Equation (6): with the pseudoinverse matrix J † OE = J T OE (J OE J T OE ) −1 .
Using the CLIKA in [3] provided by Equation (9), the reference position and orientation vector ζ r (t) can be obtained via the desired end-effector task x E,d , unlike the open-loop form may cause a numerical drift by integrating the velocities to find the related position and orientation: .
where e E = [e T p , e T o,Eul. ] T , i.e., e p = p E,r − p E,d and e o,Eul. = η E,r − η E,d , is the reconstruction error vector. p E,r and η E,r denote reference position and orientation vectors of the end-effector corresponding to ζ r , satisfying x E,r = [p T E,r , η T E,r ] T = k ee (ζ r ). The gain ma- is chosen to ensure the asymptotical convergence of the reconstruction error to zero, where K p , K o ∈ R 3×3 are diagonal and positive definite matrices.
Remark 1. The reference value ζ r (t) is second order differentiable.
If the task is considered as the position control of the end-effector only, its reconstruction error is simply given by the difference between the desired and the actual values. For the case of the orientation, the definition of such error is required to ensure convergence to the desired value, while involving in large numbers of computations of the related trigonometric and inverse trigonometric functions. In this paper the quaternion attitude representation is used to replace the expression of the orientation error [23], and we define T and Q r = [η r , ε T r ] T as the desired and reference attitudes expressed by quaternions, respectively. The quaternion-based CLIKA can be redefined as: . .
where J 3 (·) can be found in the Appendix A. Another reconstruction error is e E = [e T p , e T o,Quat. ] T satisfying e p = p E,r − p E,d and e o,Quat. = −[η r ε d − η d ε r − S(ε d )ε r ], where S(·) denotes the skew symmetric matrix of the vector. Figure 2 presents the computation process of the quaternion-based CLIKA of the UVMS. And, the output of the inverse kinematics is the reference position and orientation of the UVMS expressed by ζ r (t), which can be taken as the tracking objective for the design of the dynamic control strategy.
If the task is considered as the position control of the end-effector only, its reconstruction error is simply given by the difference between the desired and the actual values. For the case of the orientation, the definition of such error is required to ensure convergence to the desired value, while involving in large numbers of computations of the related trigonometric and inverse trigonometric functions. In this paper the quaternion attitude representation is used to replace the expression of the orientation error [23] denotes the skew symmetric matrix of the vector. Figure 2 presents the computation process of the quaternion-based CLIKA of the UVMS. And the output of the inverse kinematics is the reference position and orientation of the UVMS expressed by r t ()  hhich can be taken as the tracking objective for the design of the dynamic control strategy.

The Dynamics of the UVMS
Considering that the 6 n + DOF UVMS contains a six DOF vehicle coupled hith a general n DOF underhater manipulator. Its dynamic equations in the vehicle-fixed frame can be derived by resorting to the Quasi-Lagrange approach [24] and expressed in the follohing form: (12) where is the inertia matrix and is the centrifugal and Coriolis matrix, and both of them contain their corresponding added mass terms.

The Dynamics of the UVMS
Considering that the 6 + n DOF UVMS contains a six DOF vehicle coupled with a general n DOF underwater manipulator. Its dynamic equations in the vehicle-fixed frame can be derived by resorting to the Quasi-Lagrange approach [24] and expressed in the following form:
where M ζ (ζ) = J T M(q)J, C ζ (ζ, Since the external environment, like ocean currents, has significant influence on the UVMS, the relevant term can be incorporated into the dynamic equations of the system.
Define the relative velocity in the Earth-fixed frame as ζ − ν c , where ν c denotes the current velocity.
Remark 2 [25]. Suppose that the current velocity is irrotational and constant in the Earth-fixed frame, and it can be expanded to the 6 + n dimensional generalized space, namely: Substituting . ζ in Equation (13) with . ζ c , the dynamic equations of the UVMS under current effect can be represented as: Moreover, the terms related to ν c in Equation (15) can be combined as a new vector ζ, ν c ). Hence, the dynamic equations of the UVMS under current effect takes the form: It is noted that the UVMS can be influenced by uncertain factors such as parameter uncertainties, hydrodynamics and external disturbances, the system dynamic equations in Equation (16) can be rewritten as: (17) For the convenience of the subsequent analysis, the dynamic equations illustrated by Equation (17) can be transformed into the following form: where F(t) and d(t) are defined as nominal model term and the lumped disturbance vector of the system, respectively, and given by:

Control Strategy
Define the position tracking error vector e 1 and its time derivative . e 1 satisfying e 1 = ζ − ζ r and z 1 = The objective is to design a suitable control scheme that can make the position tracking error vector e 1 achieve the asymptotical convergence to zero.
Considering the situation when the sliding mode variable s reaches the sliding switched surface, namely, s(t) = 0, it can also result . s(t) = 0. By the use of Equation (21), we can derive: .
Through Definition 2 in [28], it concludes from Equation (22) that the auxiliary vector z 2 can converge to the origin in finite time. That is, there exists T 1 such that z 2 = 0 holds for all t ≥ T 1 , which can further imply . z 2 = 0. Based on this, combining with Equation (20), it yields: By using z 1 = . e 1 and Lemma 1, it can deduce that the position tracking error vector e 1 can achieve the convergence to zero in finite time.
Upon introducing another auxiliary term: and then Equation (20) can be also expressed as z 2 = z 1 + z 0 .

ESO
Taking the lumped disturbance vector d(t) as an extended state, and combining with (24), the extended dynamic form of Equation (19) can be expressed as: where h(t) denotes the time derivative of the lumped disturbance vector d(t).
The fixed-time convergent ESO is designed referring to [22]. To avoid the chattering due to the discontinuous signum function, sign (·) in the ESO is replaced by a saturation function sat (·). The modified ESO can be represented: where the constant gain parameters satisfy ρ 1 , ρ 2 , ρ 3 > 0 and γ 1 > 1. The vectorsẑ 2 and d denote the estimation values of the state vector z 2 and d, respectively. And, the vector z 2 is the estimation error of the state vector z 2 , i.e., z 2 = z 2 −ẑ 2 with its component form z 2 = [ z 21 , · · · , z 2,6+n ] T . Meanwhile, the saturation function sat( z 2 ) is defined as: Remark 3. Choose the saturation function instead of the conventional sign function, and it can attenuate the chattering caused by the latter discontinuity.
Combining Equations (25) and (26), the estimation error equation for the ESO can be expressed as: where d = d −d denotes the estimation error of the lumped disturbance vector d.
The result follows from Theorem 2 in [22] that both the estimation errors z 2 and d can converge to the origin uniformly in fixed time.

Adaptive Control Law
An adaptive control scheme can be proposed for the trajectory tracking control of the UVMS subjected to the lumped disturbances. Assumption 2. The disturbance estimation error satisfies d 1 ≤ δ where the upper bound δ is unknown.

Theorem 1.
Considering the dynamic equations of the UVMS in Equation (19), and combing the ESO in Equation (26) under Assumptions 1-2, the proposed control law can be chosen as: where K 1 = diag[k 11 , · · · , k 1,6+n ] is the gain matrix with its components satisfying k 1j > 0, j = 1, · · · , 6 + n. The conventional robust term in Equation (29) is defined as: whereδ presenting the estimation value of the parameter δ is calculated by the following adaptive law: .δ = ρ 4 · s 1 (31) Thus, all of the related state signals like s and δ are bounded, and the position tracking error vector of the system e 1 can converge to zero asymptotically.
Proof of Theorem 1. Considering the following Lyapunov function that is positive and define: where δ = δ −δ is the estimation error of the parameter δ. With respect to combining the above Equations (21), (25), (29)-(31), the time derivative of V 1 yields: where λ 1 = λ min (K 1 ), and λ min (·) denotes the minimum eigenvalue of the matrix. According to Equation (33), we can obtain . V 1 ≤ −λ 1 s T s all the time, so the inequality relation . V 1 < 0 holds for every s = 0, which further demonstrates that the signals like V 1 , s and δ are bounded. From the above derivation, we have: For the reason that the vectors s and δ are bounded, in combination with the Assumption 2, we know that the estimation valueδ is bounded. Furthermore, it can conclude the boundness of . s. Introducing another new function L 1 (s) = λ 1 s T s, both L 1 (s) and its time derivative are bounded because of the boundness of the vectors s and . s. Then, we can obtain lim T→∞ t 0 L 1 (s) ≤ V 1 (0) − V 1 (∞). Since both V 1 (0) and V 1 (∞) are bounded, the sliding mode variable s is square integrable. Hence, the function L(s) can converge to zero asymptotically via Lemma 2, which further derives the asymptotical convergence of the sliding mode variable s to zero. Finally, in accordance with the above Equation (21), it results that the position tracking error vector e 1 can achieve the asymptotical convergence to zero.
As the conventional robust term in Equation (30) contains the discontinuous signum function, it may cause the chattering problem. Inspired by [26,29], a continuous fractional PI-type controller is designed as: where χ = [ χ 1 χ 2 ] T and ϕ(s) = [ s t 0 sig(s) α 4 dt ] T , α 4 ∈ (0, 1). According to the analysis found in [29], we can use u r (s|χ) to approximate the conventional robust term u r =δsign(s), and the related estimated valueχ can be derived from Lyapunov theory. Consequently, we can get the modified control scheme based on Equation (35) shown in the following Theorem 2.

Theorem 2.
Consider the dynamic equations of the UVMS described by Equation (19) satisfying Assumptions 1-2. In combination with the ESO in Equation (26), the control law of the system is determined by Equation (29), satisfying the continuous fractional PI-type robust term in Equation (36) and the adaptive laws in Equations (31) and (37): Thus, all of the related state signals such as s, δ, and χ are bounded, and the position tracking error vector of the system e 1 can achieve the asymptotical convergence to zero.
Similarly, it can conclude from Equation (40) that . V 2 ≤ 0 holds for any s ∈ R (6+n)×1 , that is, . V 2 is negative semi-definite. Therefore, the Lyapunov function V 2 is bounded, which implies that the signals s, δ and χ are bounded. In addition that, we can obtain: It is easy to know that the terms on the right of Equation (41) are bounded, thus the boundness of . s can be ensured. According to the similar analysis in Equation (34), it can finally derive that the position tracking error vector e 1 can achieve the asymptotical convergence to zero. In summary, the asymptotic stability of the system can be guaranteed by the Lyapunov theory. Figure 3 describes the complete control process for the trajectory tracking of the UVMS. First, the quaternion-based CLIKA can be used to solve the reference position and orientation through the desired values of the UVMS end-effector. The proposed controller contains three parts: NFTSMS, ESO and adaptive control law. Combining with the position and velocity errors, the modified NFTSMS is designed to achieve the fast convergence of the tracking errors. Based on this, to estimate the lumped disturbances, the ESO method is utilized by taking the lumped disturbance vector as an extended state of the system. Meanwhile, an adaptive robust term is applied for handling the unknown boundness of the disturbance estimation errors. Then, incorporating NFTSMS, ESO and adaptive law into the control law, the obtained efforts are taken as the control inputs that can drive both vehicle and manipulator to approximate the reference values. In general, those input efforts should act on the thrust equipment of the underwater vehicle like thrusters in [30] to achieve proper operation, while it's not the focus of this paper. The actual position and velocity as the control outputs are used to assign the related position and velocity errors. After that, the actual position information can be taken as the input of the forward kinematics to solve the actual position and orientation of the UVMS end-effector. Finally, the closed-loop control process is formed to support the following simulation.
contains three parts: NFTSMS, ESO and adaptive control law. Combining with the position and velocity errors, the modified NFTSMS is designed to achieve the fast convergence of the tracking errors. Based on this, to estimate the lumped disturbances, the ESO method is utilized by taking the lumped disturbance vector as an extended state of the system. Meanwhile, an adaptive robust term is applied for handling the unknown boundness of the disturbance estimation errors. Then, incorporating NFTSMS, ESO and adaptive law into the control law, the obtained efforts are taken as the control inputs that can drive both vehicle and manipulator to approximate the reference values. In general, those input efforts should act on the thrust equipment of the underwater vehicle like thrusters in [30] to achieve proper operation hhile it's not the focus of this paper. The actual position and velocity as the control outputs are used to assign the related position and velocity errors. After that, the actual position information can be taken as the input of the forward kinematics to solve the actual position and orientation of the UVMS end-effector. Finally, the closed-loop control process is formed to support the following simulation.

Simulation
In this section, simulations with the help of MATLAB/Simulink toolbox are performed on a ten DOF UVMS, where the UVMS contains a four DOF underwater vehicle and a six DOF underwater manipulator, and its coordinated frames are shown in Figure 4. Then, some assumptions are given in the following: the gravity of the underwater vehicle equals its buoyancy; The link density of the manipulator is 2700 kg/m 3 , and the centre of buoyancy of the underwater manipulator is coincident with its centre of gravity; the density of the fluid is 1025.9 kg/m 3 ; the coefficient of water resistance C D = 1.05; The coefficient of additional mass force is C M = 0.8. The dynamic parameters of the vehicle are expressed in Table 2 referring to [31,32], and its centres of gravity and buoyancy in the vehicle-fixed frame are [0, 0, 0] m and [0, 0, 0.04] m, respectively. The related link parameters of the manipulator are listed in Table 3

Simulation
In this section, simulations with the help of MATLAB/Simulink toolbox are performed on a ten DOF UVMS, where the UVMS contains a four DOF underwater vehicle and a six DOF underwater manipulator, and its coordinated frames are shown in Figure  4. Then, some assumptions are given in the following: the gravity of the underwater vehicle equals its buoyancy; The link density of the manipulator is 2700 kg/m 3 , and the centre of buoyancy of the underwater manipulator is coincident with its centre of gravity; the density of the fluid is 1025.9 kg/m 3 ; the coefficient of water resistance 1.05 The dynamic parameters of the vehicle are expressed in Table 2 referring to [31,32], and its centres of gravity and buoyancy in the vehicle-fixed frame are [0, 0, 0] m and [0, 0, 0.04] m, respectively. The related link parameters of the manipulator are listed in Table 3, and the position of the base origin in the vehicle-fixed frame is [0.5, 0, 0] m.
In case 1, the controller is applied by the PID-ESO scheme with conventional robust term, whose control law is: along with the adaptive law: .δ = ρ 4 · z 1 1 (44) and the related ESO being expressed as: . with: and, the control gains K d1 and K p1 are diagonal and positive definite matrices. In case 2, the controller is designed by combining the PID-type SMC-ESO scheme and the conventional robust term, and its control law satisfies: together with the SMC variable z 2 in Equation (20) and the ESO coinciding with Equations (45) and (46), and the adaptive law satisfying: .δ In case 3, the control law of the PID-type SMC-ESO scheme with continuous fractional PI-type robust term is designed as: where the SMC variable z 2 is the same as that in Equation (20), and the continuous fractional PI-type robust term u r (z 2 |χ) is chosen as Equation (50) as well as the adaptive laws in Equations (48) and (51): .χ Proofs of the stability of the control systems for cases 1-3 have been given in the Appendix A. Here, the proposed control scheme in Theorem 2 is considered as the fourth case. The related parameter values of the controllers in four cases are assigned in Table 4. Besides, all of the initial values of the state vectors corresponding to the ESO and the adaptive laws are set as zeros. Moreover, the complete diagram of the simulation program for the UVMS is described in Figure S45 of the Supplementary Materials.

Terms
Value To obtain the straight analysis for the simulation results, introducing the average position and orientation errors of the end-effector, and the average estimation error components of the lumped disturbances are expressed as follows: where e x , e y , e z and e φ , e θ , e ψ denote the position error and orientation error vectors of the end-effector, respectively. And the vectors e ι 1 , ι 1 ∈ {d1, d2, · · · , d10} are the components of the lumped disturbance estimation error vector. τ hj , h = 1, 2, 3, 4, j = 1, 2, · · · , 10, denote the jth component vectors of the total control input vector in case h. The number of the simulation steps from 10 s to 50 s is N 1 = 800. Subsequently, some simulation results under four cases for the trajectory tracking of the UVMS end-effector are described in Figures 5-17. Figure 5 displays the time history of the desired and actual trajectories of the UVMS end-effector in cases 1-4. As seen from it, even if the initial position of the end-effector is quite different from the desired initial position, after a little time the positions of the end-effector can all achieve the fast tracking for the desired trajectory. Figures 6-11 present the tracking situation on the three direction positions and three orientation angles of the end-effector for four cases, respectively. Actually, in the beginning seconds all of the positions and orientation angles have a few differences from the desired values and then all of them can quickly reach the desired values, which are in accord with the results in Figure 5. In the following for the desired trajectory. Figures 6-11 present the tracking situation on the three direction positions and three orientation angles of the end-effector for four cases, respectively. Actually, in the beginning seconds all of the positions and orientation angles have a few differences from the desired values and then all of them can quickly reach the desired values, which are in accord with the results in Figure 5. In the following Figures 12-17, we can obtain some results for the three position and orientation angle errors of the endeffector.    for the desired trajectory. Figures 6-11 present the tracking situation on the three dire positions and three orientation angles of the end-effector for four cases, respectively tually, in the beginning seconds all of the positions and orientation angles have a differences from the desired values and then all of them can quickly reach the de values, which are in accord with the results in Figure 5. In the following Figures 1 we can obtain some results for the three position and orientation angle errors of the effector.    for the desired trajectory. Figures 6-11 present the tracking situation on the three dire positions and three orientation angles of the end-effector for four cases, respectively tually, in the beginning seconds all of the positions and orientation angles have a differences from the desired values and then all of them can quickly reach the de values, which are in accord with the results in Figure 5. In the following Figures 1 we can obtain some results for the three position and orientation angle errors of the effector.                Table 5. It is evident that from 10 s to 50 s all of the av values for cases 2-4 are much smaller than those in case 1, especially, the last four av errors in case 4 are the smallest compared to other cases.                 Figure 17, the yaw angle error in case 4 can get to [−0.01, 0.01] rad at less than 4 s, and has two seconds faster than those in cases 2-3, while the one in case 1 can reach the feasible range till about 12 s. Then the yaw angle error in case 1 tends to a slower convergence rate than those in other cases. Finally, all the three steady-state orientation angle errors of the end-effector can realize the range of [−0.007, 0.007] rad, where those in case 4 have much smoother changes than those in cases 1-3, and meanwhile those in case 1 range most largely. In addition, their average position and orientation angle errors of the end-effector are listed in Table 5. It is evident that from 10 s to 50 s all of the average values for cases 2-4 are much smaller than those in case 1, especially, the last four average errors in case 4 are the smallest compared to other cases.                            Table 6, where the average force and moment values have less 0.1 N and 0.4 N m differences among the four cases, respectively. That means that after about 10 s, their energy consumptions for cases 1-4 approach to the same levels, which is consist with the results in Figures S33-S38. Combining the above analysis for the tracking errors of the UVMS, we can obtain that the proposed control system in case 4 can have faster convergence and higher precision tracking performance compared to other three cases.     Figures S39-S44 of the Supplementary Materials. Despite that in the initial time there exist big differences between the lumped disturbance vectors and their estimations for cases 1-4, after several seconds the disturbance estimations in four cases can all achieve much better approximation to the lumped disturbances. The reason is that the ESO is applied to attenuate the effects of the complex lumped disturbances on the control systems. Furthermore, the average disturbance estimation errors under four cases from 10 s to 50 s are listed in Table 7. Noted that their average values are small and have few differences among four cases. Since the tracking errors in case 4 show much smoother variation compared to those in other cases, it demonstrates that the proposed control method has stronger robustness of resisting disturbances. Figure 26. X direction forces for the first control input of the vehicle (four cases).                 In conclusion, the simulation comparative results validate that the proposed controller can achieve the better tracking performance and stronger robustness of disturbance rejection.

Conclusions
In this paper, the adaptive NFTSMC scheme integrating with ESO is proposed for the trajectory tracking of the UVMS under the lumped disturbances. First, the quaternionbased CLIKA is utilized for avoiding the integral drifts in the inverse kinematics. The proposed controller mainly contains the modified NFTSMS and ESO, and the adaptive control law. And, the applied NFTSMS can make the tracking errors achieve the fast convergence, together with avoiding the singularity. The utilization of the ESO is to estimate the lumped disturbances so that the estimation error can converge to the origin in fixed time. At the same time, appending the saturation function to the ESO instead of the signum function is to reduce the chattering effect. Besides, the continuous fractional PI-type controller with the adaptive laws is introduced in the control input, so that it can approximate the conventional discontinuous robust term to attenuate the chattering because of the discontinuity. Based on the Lyapunov theory, the control system is proved to have asymptotical stability. Comparative simulations with several other methods demonstrate that the proposed control system have better tracking performance and stronger robustness against disturbances. In the future research, we will incorporate the redundancy resolution scheme into the proposed control strategy to perform the simulation test of the UVMS, in which the redundancy resolution can realize the secondary objectives, such as avoiding manipulator joint limits and singularities, minimizing the vehicle motion, thrusters fault tolerant, etc. And, it can provide a theoretical basis for the later experiments. When the conditions of the laboratory permit, the control strategy can be applied for practicing to verify the reliability of the controller via experimental tests.
In line with the Assumption 2, the composite term ( d −δ · sign(z 1 )) in (A18) can be guaranteed to be bounded. By substituting z 1 = . e 1 into the above (A18), the expression in (A18) is rewritten as: .. e 1 + K d1 · . e 1 + K p1 · e 1 = d −δ · sign(z 1 ) It is concluded from (A19) that the position tracking error vector e 1 can converge to zero asymptotically via the solution for the ordinary differential equations.
(2) The stability of the PID-type SMC-ESO scheme with conventional robust term in case 2 Consider the following Lyapunov function for case 2: Integrating Equations (18), (20), (47) and (48), and substituting them into the time derivative of V 4 , we have: Following the proof of Theorem 1 in Equation (33), it can conclude from (A21) that the SMC variable z 2 can converge to zero asymptotically, which can further derive the asymptotical convergence of the position tracking error vector e 1 by Equation (23).
(3) The stability of the PID-type SMC-ESO scheme with continuous fractional PI-type robust term in case 3 The stability of the PID-type SMC-ESO scheme with continuous fractional PI-type robust term in case 3 Choosing another positive and definite Lyapunov function is: Substituting Equation (49) into Equation (18) and combining with Equations (20), (48), (50) and (51), the time derivative of V 5 turns into the following form: In view of the similar analysis for Equation (40), it can be obtained from (A23) that both of the SMC variable z 2 and the position tracking error vector e 1 can achieve the asymptotical convergence to zero.