Adaptive Model-Free Control With Nonsingular Terminal Sliding-Mode for Application to Robot Manipulators

An adaptive model-free control with nonsingular terminal sliding-mode (AMC-NTSM) is proposed for high precision motion control of robot manipulators. The proposed AMC-NTSM employs one-sample delayed measurements to cancel nonlinearities and uncertainties of manipulators and to subsequently obtain sufficiently simple models for easy control design. In order to maintain high gain controls even when the joint angles are close to the reference target values and accordingly achieve high precision and fast response control, a nonlinear sliding variable is also adopted instead of a linear one, asymptotically stabilizing controls by guaranteeing even a finite-time convergence. In addition, sliding variables are reflected on control inputs to support fast convergence while achieving uniform ultimate boundedness of tracking errors. The control gains of the proposed AMC-NTSM are adaptively adjusted over time according to the magnitude of the sliding variable. Such adaptive control gains become high for fast convergence or low for settling down to steady motion with better convergence precision, when necessary. The switching gains of the proposed AMC-NTSM are also adaptive to acceleration such that inherent time delay estimation (TDE) errors can be suppressed effectively regardless of their magnitudes. The simulation and experiment show that the proposed AMC-NTSM has good tracking performance.


I. INTRODUCTION
For a long time, robot manipulators have been successfully developed to achieve high-precision control performance. Nevertheless, more accurate and effective control of robot manipulators is desirable because their higher precision control is still required in various precision engineering fields including even nano-and bio-technology. Recently, extensive research on control of robot manipulators has been conducted to facilitate practical implementation as well as good control performance [1]- [6].
The associate editor coordinating the review of this manuscript and approving it for publication was Hai Wang.
As a practical and effective model-free control of robot manipulators, time-delay control (TDC) schemes combined with auxiliary sliding mode controls (SMCs), such as conventional SMCs [13], second-order SMCs [14], boundary-layer SMCs [15], and adaptive SMCs [16], have been proposed to enhance the tracking performance with a simple control structure, which is herein termed model-free controls with sliding-mode (MC-SMs). Since some research has recently studied advanced SMCs such as robust SMCs [7], terminal SMCs [8]- [10], and disturbance observer(DOB) based SMCs [11], [12], such practical MC-SMs are expected to continue to evolve. By using one-sample delayed measurements to cancel nonlinearities and uncertainties of robot manipulators, VOLUME 8, 2020 This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ the MC-SMs are designed without mathematical models. Involved auxiliary SMCs of the MC-SMs contribute to the suppression of inherent time-delay estimation (TDE) errors and subsequently improving the tracking performance. All of the above existing MC-SMs have linear sliding variables for achieving asymptotic stability in a simple manner [17], [18]. However, linear sliding variables are disadvantageous in that the corresponding control effort becomes insensitive to tracking errors as the joint angles approach the reference target values. Generally, this disadvantage is not easily overcome by simply increasing gains because high gains for strong attractivity, or fast convergence, may not be allowed to prevent undesirable side effects, such as chattering and noise amplification. In other words, gain tuning has its limitation for the purpose of good tracking performance.
To strengthen the attractivity efficiently and hence control robot manipulators precisely, TDCs with nonsingular terminal sliding-mode (NTSM) have been developed [19]- [21], which is based on nonlinear sliding variables and herein termed model-free controls with NTSM (MC-NTSMs). With NTSM, MC-NTSMs have strong attractivity when the joint angles are close to the reference target values, and hence they provide fast convergence even within the finite time horizon of the prescribed size, or the so-called finite-time convergence. As nonlinear sliding variables provide the effect of high gain controls around reference target values, any undesirable side effects arising from high gains can be avoided. Thus far, several modified versions of MC-NTSMs have been developed with continuous NTSM [22], continuous nonsingular fast terminal sliding mode [23], and fractionalorder NTSM [24]. Additionally, for more flexible response to changing conditions, adaptive controls based on NTSM have also been suggested without employing TDC [25]- [28]. Their adaptive gains are not practical because monotonic one-way adaptation with saturation is only considered for the initial transient response.
Therefore, the aforementioned existing MC-NTSMs and relevant controls have much room for improvement regarding tracking performance: • As mentioned earlier, the existing MC-NTSMs employ the fixed control gains and some adaptive controls based on NTSM adopt adaptive gains that are difficult to implement. Fixed gains are often not flexible in terms of tracking performance improvement and noise suppression. Proper gains should be applied at the right times by adjusting their magnitudes in order to provide good performance constantly without undesirable side effects. It would be meaningful to adjust the control gains over time to consider multiple criteria.
• The inherent TDE errors of existing MC-NTSMs are assumed to be upper bounded [29]- [33]. However, when the friction force becomes opposite in sign due to the directional change of motion or a reference trajectory has non-smooth points, TDE errors may be large, potentially affecting the involving TDC in a negative manner, with consequent poor tracking errors [34], [35]. For this reason, the upper bounds of the TDE errors are generally set to be sufficiently large, and the switching gains adopted in MC-NTSMs are accordingly taken to be large even though chattering may occur. It would be desirable to adjust the switching gains appropriately according to the TDE errors.
• As in the control gains, the nonlinear sliding variables of the existing MC-NTSMs have an inherent trade-off between tracking performance and noise effects arising from the derivative of tracking errors. It would be thus meaningful to develop a method for overcoming such an underlying trade-off. Several research have been conducted considering the three issues mentioned above [36], [37]. This paper proposes an AMC-NTSM for achieving high precision and fast response, and applies it to robot manipulators. A solution to each issue above is provided in the proposed AMC-NTSM as follows: • The control gains of the proposed AMC-NTSM increase when the sliding variable stays far away from the sliding manifold. On the contrary, the control gains decrease when the sliding variable remains near the sliding manifold. Such adaptive control gains are very helpful for improving tracking performance while mitigating the effect of noise.
• The switching gains of the proposed AMC-NTSM are adjusted in proportion to the magnitude of acceleration that is associated with the upper bound of TDE errors. Such gain adjustment is effective for increases in TDE errors due to large acceleration. Thus, they can appropriately suppress the TDE errors of any magnitude while reducing chattering.
• Sliding variables are directly reflected on the proposed AMC-NTSM to support fast convergence with moderate gains while achieving uniform ultimate boundedness of tracking errors. This plays a role in resolving the aforementioned trade-off arising from nonlinear sliding variables of existing MC-NTSMs. In summary, the proposed AMC-NTSM is aimed at reinforcing an existing MC-NTSM with adaptive schemes, flexible TDE error suppression, and responsive practical control in order to achieve better tracking performance while reducing undesirable side effects.
The remainder of this paper is organized as follows: in Section II, a conventional TDC and an MC-NTSM are briefly introduced. The proposed AMC-NTSM isg presented in Section III. Section IV describes simulations carried out with a two-link robot manipulator. The application of the proposed AMC-NTSM to real robot manipulators is described in Section V. We conclude with a brief summarization of the results of this paper in Section VI.

II. TIME-DELAY CONTROL & MODEL-FREE CONTROL WITH NONSINGULAR TERMINAL SLIDING-MODE A. TIME-DELAY CONTROL
The dynamics of a n-link rigid robotic manipulator [38] can be described as where q t ,q t , andq t ∈ n are the angle, angular velocity, and angular acceleration of the joints, respectively; M q t ∈ n×n is the symmetric positive definite inertia matrix; C q t ,q t ∈ n is the Coriolis matrix; g(q t ) ∈ n is the gravity force; f q t ∈ n is the friction force, and τ d,t ∈ n ; and τ t ∈ n are the external disturbance and the control input torque, respectively. It is reasonably assumed that τ d,t and ||M q t || are bounded [39]. Any norm can be used if not specified. Only the ∞ norm is specified if necessary.
Adding τ d,t to both sides of (1), we have whereM = diag(M 1 ,M 2 , · · · ,M n ) ∈ n×n is a positive matrix to be determined in the next subsection. Multiplying both sides of (1) by M −1 q t and representing (2) in a compact and simple form yield where N t is given by Since N t in (3) is not available due to unknown external disturbances at time t, its estimateN t is determined in the following simple manner: where L is a sufficiently small sampling period andN t = N 1,t ,N 2,t , · · · ,N n,t ∈ n . It is noted thatN t is simply a one-sample delayed measurement of N t [40]. From (4), the conventional TDCτ TDC t can be represented in the following form [41] where e t = q d,t − q t ∈ n is the tracking error, and K d = diag K d1 , K d2 , · · · , K dn ∈ n×n and K p = diag K p1 , K p2 , · · · , K pn ∈ n×n are positive gains for pole assignment.
Replacing τ t in (3) withτ TDC t in (5), we can obtain the error dynamics asë where E t = N t −N t ∈ n is called the TDE errors and known to be bounded as follows [19], [33]: for a positive constant E * i under some conditions. In this study, the upper bounds E * i are parameterized in terms of acceleration and then TDE errors of any magnitude can be suppressed with the proposed AMC-NTSM. Details are provided in Appendix A.

B. MODEL-FREE CONTROL WITH NONSINGULAR TERMINAL SLIDING-MODE
To improve the error convergence rate, we employ the nonsingular terminal sliding variable expressed in the following form [20]: for positive odd integers p and q, and a constant matrix K s to be determined for adjusting the convergence rate. s t and K s in (8) have n scalar elements, or s t = (s 1,t , s 2,t , · · · , s n,t ) T ∈ n and K s = diag(K s1 , K s2 , · · · , K sn ) ∈ n×n .
From (8), the MC-NTSMτ N t is given as follows [19]: whereK sw = diag(K sw,1 ,K sw,2 , · · · ,K sw,n ) ∈ n×n is a design parameter with positive elements and sgn(s t ) = [sgn(s 1,t ), sgn(s 2,t ), · · · , sgn(s n,t )] ∈ n is defined as Three gains,M, K s , andK sw in (9) are termed control, sliding, and switching gains, respectively, throughout this paper. The control gainM in (9) is multiplied by the acceleration computed by the numerical differentiation. For this reason, if a large control gainM is employed to improve tracking performance, the effects of noise may be amplified. In contrast, if a small control gainM is selected, the effects of noise will be reduced, but the tracking performance may be degraded. In this regard, the control gains need to be tactically adjusted over time considering multi-criteria. Similarly, (9) observed that if the sliding gain K s in (8) is increased to achieve a faster convergence rate, the feedback gain with respect to the time derivative of the tracking error is reduced and hence the transient response may be poor. High switching gainsK sw for improving robust performance also cause undesirable side effects such as chattering.
In the following section, the abovementioned design control and switching gains, and compensation for large sliding gains are discussed.

III. ADAPTIVE MODEL-FREE CONTROL WITH NONSINGULAR TERMINAL SLIDING-MODE
We propose an adaptive model-free control with nonsingular terminal sliding-mode (AMC-NTSM) as follows: where t = ( 1,t , 2,t , · · · , n,t ) ∈ n is given by for positive design parametersK 0i andK 1i to be determined carefully for suppressing the TDE errors and simultaneously not being sensitive to acceleration. Details on the selection of K 0i andK 1i are shown in Appendix B. As mentioned in the previous section, the sliding variable s t in (11) plays a key role in producing more responsive control efforts regardless of large K s or small K −1 s . β andM t contribute towards guaranteeing fast convergence and suppression of large TDE errors effectively.
In (11), the time-varying control gainM t is automatically tuned as follows: whereM i and φ i are positive constants andω i,t is a timevarying positive gain determined by the following adaptive law: forω =ω * −ω and θ t = sgn(||s t || ∞ − ε) and positive design parameters α i , γ i , ρ i , and ε where · ∞ is the ∞ norm of a vector. It is noted thatM i in (13) can be considered as a lower bound ofM i,t . A sufficiently small ε is selected such that the tracking error is tolerable.
As shown in (14), the time-varying gainŵ i,t is strongly affected by ε. The front term in (14) can easily adjust the rate of increase or decrease of the time-varying gainŵ i,t by setting α i γ i for ||s t || ∞ ≥ ε and −α i γ i for ||s t || ∞ < ε, independently. The working principle of the proposed AMC-NTSM for two cases, ||s t || ∞ ≥ ε and ||s t || ∞ < ε, can be described as follows.
• In case of ||s t || ∞ ≥ ε The control gainM i,t increases until ||s t || ∞ reaches ε. Asẇ i,t in (14) is linearly related to |s i,t |, the large sliding variable will be dominant on the adaptive law and then provide strong attractiveness to the sliding manifold.
However, as the sliding variable approaches the small vicinity of the sliding manifold, or the set ||s t || ∞ < ε, the convergence rate declines because |s i,t | in (14) becomes small. To solve this problem, we employ an additional term that compensates for small |s i,t | and hence quickly drives the sliding variable to the set ||s t || ∞ < ε. • In case of ||s t || ∞ < ε As the sliding variable s t enters the set ||s t || ∞ < ε and then θ t becomes negative, the control gainM i,t is reduced to avoid undesirable side effects generated by excessively high control gains. In addition, |s i,t | −1 in (14) is inversely proportional to the magnitude of the sliding variable, which provides strong attractiveness near the sliding manifold. It is noted that while a disturbance observer (DOB) based control (DOBC) [11], [12] is based on the frequency-domain and only applicable to the minimum phase systems, TDC employed in this paper is computed directly in the timedomain by using time-delayed measurements. For one concrete example, abrupt disturbances over short time intervals can be more easily handled by the proposed AMC-NTSM than by DOBC.
To sum up, the AMC-NTSM uses the adaptive control gain M t to reduce the magnitude of the TDE error caused by the fixed control gainM t of the TDC controller. In addition, the generated TDE error is appropriately suppressed using the switching gainK sω,t . Chattering is also reduced by adopting small switching gains after convergence.
The stability of the proposed AMC-NTSM is discussed in Appendix B. Its effectiveness will be illustrated in the next sections.

B. SIMULATION DESCRIPTION
The practical effectiveness of the proposed AMC-NTSM (11) is shown through comparisons with TDC [41] in (5) and the existing MC-NTSM [19] in (9). The control objective is to ensure that the angles of joints 1 and 2 follow the reference trajectories that include low and high frequency components and have non-differential points as shown in Fig. 1(a) and (b). All control schemes adopted for comparison were first designed to be properly tuned in the low frequency component and then they were applied to the given reference trajectories in order to test their adaptiveness and robustness to changing conditions. The smaller the sampling time, the better the performance of the proposed control scheme. In this study, the sampling time was set to 0.25 ms for both the simulation and experiments.    Fig. 1(c) and (d) show the control gains computed from the adaptive law (13) of the proposed AMC-NTSM. For a large amount of time, the control gains have low bounded values, or large control gains are applied only when necessary in order to avoid undesirable side effects such as chattering and input oscillation. In particular, when directional change of motion occurs, or the friction force becomes opposite in sign, large controls are applied temporarily. Fig. 1(e) and (f) show that as in the control gains, the switching gains in Fig. 1 (c) and (d) of the proposed AMC-NTSM are enlarged only when necessary. As mentioned earlier, they are dependent on the acceleration and effective in suppressing TDE errors. Fig. 1(g) and (h) show tracking errors of the proposed AMC-NTSM compared with those of TDC (5) and MC-NTSM (9). The existing TDC (5) appears to produce the worst, or largest tracking error. The existing MC-NTSM [19] has larger tracking error than the proposed AMC-NTSM because the sliding gains K s in (9) cannot be increased much for improving the tracking performance as a result of their inverse effect on the control effort. This is why sliding variables are directly added to the proposed AMC-NTSM in order to obtain more responsive control. With the help of adaptive gains, TDE error suppression, and responsive control, the proposed AMC-NTSM exhibited a good tracking performance. Quantitatively, the measured root-mean-square (RMS) errors are summarized in Table 2 and 3. Fig. 1(i) and(j) show the control input of each joint.

B. EXPERIMENTAL DESCRIPTION
To ensure good performance of the proposed AMC-NTSM in real systems, the model-free control [19] was compared through experiments. The control objective was to achieve robust tracking performance against external disturbances due to the payload. The payload affects the Indy7 robot manipulator according to the position and acceleration of joints. The reference and real angle trajectories are shown in in Fig. 3(a). To compute second derivative of displacement, proper filtering would be helpful for numerical accuracy.

C. EXPERIMENTAL RESULTS
The experimental results of the proposed AMC-NTSM are shown in Fig. 3. The reference was provided by its generator in the Indy7 robot manipulator such that all joints were moved to pass through 4 waypoints. The resulting tracking performance of each joint is shown in Fig. 3(a). The timevarying control gains (13) are shown in Fig. 3(c). When the direction is changed, M t increases for fast convergence are shown in Fig. 3(c) and (f). Fig. 3(d) shows the tracking errors of two controls: MC-NTSM and the proposed AMC-NTSM. As in simulation results, the proposed AMC-NTSM provides outstanding nominal tracking ability compared with MC-NTSM, especially in case of directional change of motion, or   opposite sign of the friction force. For quantitative tracking error analysis, the RMS values of tracking errors are summarized in Table 5. Since the lower part of the robot manipulator is more loaded, it has a relatively large RMS tracking error. Overall, the tracking error of the AMC-NTSM was almost half that of the existing MC-NTSM [19].
To evaluate the robust tracking performance, we tuned the parameters of the controller without payload, and then applied them with a payload of 5 kg. Real experiments were also performed with a payload of 5 kg, corresponding to about 70% of the maximum load. A payload can be considered as an external disturbance. The results are shown in Fig. 3(e).
The RMS values of the resulting tracking errors are summarized in Table 5. Despite payload uncertainty, the robust and nominal tracking performances did not differ much. In most joints, the RMS value of the tracking errors with payload was slightly larger than that without payload. In addition, the RMS of the proposed control scheme can be observed to be improved by at least 46% compared with that of other controllers.

VI. CONCLUSION
This work proposed an adaptive model-free control with nonsingular terminal sliding-mode (AMC-NTSM), and its effectiveness was evaluated through simulations and experiments with robot manipulators. The transient and steadystate tracking performances of the proposed AMC-NTSM could be remarkably improved due to the synergetic effects of adaptive (control and switching) gains and sliding variables used for control inputs. Moreover, undesirable side effects on chattering and noise amplification could be suppressed as much as possible by reducing gains appropriately at the right times. The experiment result with payload showed that the proposed AMC-NTSM has more robust tracking performance than existing controls.
With simple implementation and good tracking performance, the proposed AMC-NTSM is a potential candidate for replacing existing controls for application to robot manipulators. We also believe that the applications of the proposed AMC-NTSM can be further extended to diverse systems.

APPENDIX A BOUNDEDNESS OF BLUETIME-DELAY ESTIMATION ERROR
From (3) and (4), the TDE error E t can be rewritten as follows: Substitute N t andN t into (15), we have Then, we have where For a sufficiently small L, ν con,t + ν dis,t are reasonably bounded as follows: where η 0 is a positive constant [19], [33]. Since M q t and M q t−L in (17) are bounded [39], we have whereη 0 andη 1 are positive constants satisfying the following inequality: It is noteworthy that q t −q t−L is reasonably assumed to be bounded [33], [43]. From (19), the TDE error E t can be taken as being bounded by a certain first order function of q t .

APPENDIX B PROOF OF STABILITY
For a proof of the stability of the proposed AMC-NTSM (11), the Lyapunov function, denoted by V t ∈ , is defined as follows: in order to guarantee the finite time convergence of the sliding variables and boundedness of adaptive gains. Then, the time derivative of (20) can be obtained aṡ where where ζ t = p q K s diag ė t p q −1 = diag(ζ 1,t , ζ 2,t , · · · , ζ n,t ) ∈ n×n . Substituting (4) and (11) into (22), it follows thaṫ +s T t ζ t t − βs t −K sw,t sgn s t − χ t (24) where t = q p K −1 s ė t 2− p q ∈ n . The TDE error E t = N t −N t ∈ n is upper bounded by a certain first order function of the acceleration according to (19) in Appendix A. By eliminatingė t from ζ t t in (24), we havė where p and q are positive odd integers with 1 < p q < 2. From (12) and (19), ifK 0i andK 1i are selected as follows: η 0 <K 0i ,η 1 <K 1i (27) for all i = 1, 2, · · · , n, the inequality (26) can be rewritten as follows: Substituting (14) into (28) yieldṡ Since for s t ∞ ≥ ε, the second term of (29) is less than or equal to zero, we havė It means that all sliding variables enter the set, or s t ∞ < ε within a finite time. However, the derivative of the Lyapunov function is not guaranteed to be negative or zero inside the set s t ∞ < ε.
Once the sliding variable enters the set, or s t ∞ < ε, the Lyapunov function V t is upper bounded from (20) as follows: whereε is the maximum value of n i=1 γ i α iω 2 i,t . The inequality (31) can be rewritten as After the sliding variables enter the set, or s t ∞ < ε, they remain upper bounded by ν * . Even though the sliding variable enters the set, or s < ε, it may escape from the set, but it should stay in the set s < ν * . This completes the proof.