## 6 Conclusions  
For a UBSHR control system, SC-AFNN is presented to improve the tracking control performance under system uncertainties, synchronization performance, and flexible control performance.  
1. A novel composite error considering the synchronization errors and tracking errors of a UBSHR's end-effector was defined. The CE-SOSMC was realized combined with the SOSMC method to solve the uncertainty problem and the synchronization problem simultaneously. The relationship of asymptotic convergence among the novel composite error, tracking errors, and synchronization errors was stated. Strict theoretical proof of the stability of CE-SOSMC was presented.
2. AFNN was introduced, and subtractive clustering in conjunction with evolutionary algorithm was applied for constructing fuzzy rule base with fewer rules automatically. Based on the fuzzy rule base, the SC-AFNN was further constructed to realize self-learning and self-adjusting of control rules and control parameters to conquer the flexible control problem of active adaption to different technological requirements without artificially adjusting the control parameters or switching the hardware system. The asymptotic stability of SC-AFNN was analyzed.
3. Compared with the existing synchronization error, the defined composite error could ensure the synchronization performance not only of active joints but also of the end-effector. Compared with the existing SMC method based on linear sliding mode surface, the presented SOSMC method possesses superiorities such as small chattering, non-singularity, and fast convergence rate. Compared with the existing control methods for systems with definite desired trajectories, the proposed SC-AFNN can track variational desired trajectories for hybrid robots.
4. Simulation and experimental researches were carried out to further validate the validity of the developed SC-AFNN method.  
Conceptualization, Q.Q. and G.G.; methodology, Q.Q.; software, Q.Q.; validation, Q.Q.; formal analysis, Q.Q. and G.G.; investigation, Q.Q.; resources, G.G.; data curation, Q.Q.; writing--original draft preparation, Q.Q.; writing--review and editing, Q.Q. and G.G.; supervision, G.G.; funding acquisition, Q.Q. and G.G. All authors have read and agreed to the published version of the manuscript.  
Figure 11: The experimental tracking error curves of the UBSHR’s end-effector (desired trajectory 2). (a) Experimental tracking error curves in \(z\)-direction (6 s); (b) Experimental tracking error curves of \(\beta\) (6 s).  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline  & & \multicolumn{2}{c}{**Maximum Tracking Error**} & **RMSE** \\  & & \(z\) **(\(\times 10^{-5}\) m)** & \(\beta\) **(\(\times 10^{-3}\) rad)** & **(\(\times 10^{-4}\))** \\ \hline \multirow{2}{*}{Desired trajectory 1} & CE-SOSMC & 28.64 & 63.15 & 2.60 \\  & SC-AFNN & 3.05 & 8.65 & 0.47 \\ \multirow{2}{*}{Desired trajectory 2} & CE-SOSMC & 79.04 & 214.68 & 9.62 \\  & SC-AFNN & 22.95 & 21.05 & 1.44 \\ \hline \hline \end{tabular}
\end{table}
Table 3: The experimental numerical values of the steady-state performance and RMSE for tracking different desired trajectories.  
**Funding:** This research was funded by the National Natural Science Foundation of China (grant No. 51375210), by the Zhenjiang Municipal Key Research and Development Program (grant No. GZ2018004), by the Graduate Research and Innovation Projects of Jiangsu Province (grant No. SJKY19_2564), and by the Priority Academic Program Development of Jiangsu Higher Education Institutions (grant No. PAPD-2018-87).  
**Institutional Review Board Statement:** Not applicable.  
**Informed Consent Statement:** Not applicable.  
**Data Availability Statement:** No new data were created or analyzed in this study. Data sharing is not applicable to this article.  
**Conflicts of Interest:** The authors declare no conflict of interest.