## 5 Conclusions  
This paper focuses on designing a novel FTSMC for robot manipulators. In the first step, the novel FTSMM is developed to enhance response capability, fast convergence time, uncertainties opposition, and especially, improve the accuracy of the tracking position. To alleviate unknown nonlinear parameters in the control system, STCL is then applied. Thanks to this valuable technique, exterior disturbances and nonlinear elements are compensated more rapidly and more correctly with the smooth  
Figure 6: Response time of the sliding mode manifolds: (a) at Joint 1, (b) at Joint 2, and (c) at Joint 3.  
control torque. Finally, combining STCL and our proposed sliding mode manifold, under the flexible controller, the stability and robustness of the control system are guaranteed with high-performance and limited chattering. To evaluate the efficiency, a simulation example is performed for the trajectory tracking control of a 3-DOF robotic manipulator. From theoretical evidence, simulation results, and a comparison with SMC and NFTSMC, our proposed controller has some of the following contributions: (1) the proposed controller provides finite-time convergence and faster transient performance without singularity problem in controlling; (2) the proposed controller inherits the benefits of the FTSMC and CRCL in the characteristics of robustness towards the existing uncertainties; (3) a new FTSMM was proposed, and evidence of finite-time convergence was sufficiently proved; (4) the precision of the proposed controller was further improved in the trajectory tracking control; (5) the proposed controller shows the smoother control torque commands with lesser oscillation. Conceptualization, Q.V.D., A.T.V., and T.D.L. ; methodology, Q.V.D. and T.D.L. ; software, A.T.V. ; validation, Q.V.D., A.T.V., and N.H.A.N. ; formal analysis, Q.V.D., A.T.V., and H.-J.K.; investigation, T.D.L. and N.H.A.N. ; resources, A.T.V. and T.D.L. ; data curation, H.-J.K. and T.D.L. ; writing--original draft preparation, Q.V.D., A.T.V., and N.H.A.N. ; writing--review and editing, Q.V.D., H.-J.K., and N.H.A.N. ; visualization, A.T.V. ; supervision, T.D.L. ; project administration, T.D.L. and Q.V.D. ; funding acquisition, T.D.L. and Q.V.D. All authors have read and agreed to the published version of the manuscript. This research is funded by Funds for Science and Technology Development of the University of Danang under project number B2019-DN02-52. The authors declare no conflict of interest. * [PERSON] and [PERSON] (1998) [PERSON]; [PERSON] _Sliding Mode Control: Theory and Applications_; CRC Press: Boca Raton, FL, USA, 1998. * [PERSON] et al. (2001) [PERSON]; [PERSON]; [PERSON]; [PERSON] A new sliding-mode control with fuzzy boundary layer. _Fuzzy Sets Syst._**2001**, _120_, 135-143. [CrossRef]
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