## 2 Dynamic Modeling and Problem Formulation
### Problem Formulation  
The UBSHR is comprised of six active joints. To ensure the tracking performance, synchronization performance, and flexible control performance for the hybrid robot with uncertainties simultaneously, when different desired trajectories are given, the actual pose \(\mathbf{x}(t)\) should approach the desired pose \(\mathbf{x}_{d}(t)\) of active joints and the actual pose \(\mathbf{q}(t)\) should approach the desired pose \(\mathbf{q}_{d}(t)\), that is, \(\lim\limits_{t\to\infty}(\mathbf{x}_{d}(t)-\mathbf{x}(t))=0\) and \(\lim\limits_{t\to\infty}(\mathbf{q}_{d}(t)-\mathbf{q}(t))=0\); meanwhile, the tracking errors of active joints \(\mathbf{e}_{xi}\),\(\mathbf{e}_{qi}\) and tracking errors of the end-effector \(\mathbf{e}_{i}\) should satisfy control objectives A and B, respectively.  
1. The tracking errors of UBSHR's active joints are defined as follows: \[\begin{split} e_{xi}(t)&=x_{id}(t)-x_{i}(t),i=1,2,3,4\\ e_{qi}(t)&=\varphi_{jd}(t)-\varphi_{j}(t),j=1,2 \end{split}\] (3) in which \(x_{id}(t)\) and \(\varphi_{jd}(t)\), and \(x_{i}(t)\) and \(\varphi_{j}(t)\) are, respectively, the desired position and actual position for the hybrid robot's \(i\)th slider and \(j\)th driving wheel. To guarantee the synchronization performance of UBSHR's active joints, the tracking errors of the active joints should meet synchronous control theory as follows: \[\begin{split}\lim\limits_{t\to\infty}& e_{x1}(t) =\lim\limits_{t\to\infty}& e_{x2}(t)=\lim\limits_{t\to\infty}& e_{x3}(t)=\lim\limits_{t\to\infty}& e_{x4}(t)\\ \lim\limits_{t\to\infty}& e_{qi}(t)&= \lim\limits_{t\to\infty}& e_{qi}(t)\end{split}\] (4)
2. The tracking errors of UBSHR's end-effector are defined as follows: \[\mathbf{e}_{i}(t)=\mathbf{q}_{id}(t)-\mathbf{q}_{i}(t),i=1,2\] (5) where \(\mathbf{q}_{id}(t)=[z_{id},\beta_{id}]^{\mathrm{T}}\), (\(i=1,2\)) and \(\mathbf{q}_{i}(t)=[z_{i},\beta_{i}]^{\mathrm{T}}\),(\(i=1,2\)) denote the desired and actual pose. To improve the synchronization performance of UBSHR's end-effector, the control objective can be described as follows: \[[\mathbf{q}_{1}(t)-\mathbf{q}_{2}(t)]-[\mathbf{q}_{1d}(t)-\mathbf{q}_{2d}(t)]=\mathbf{e}_{1}(t)- \mathbf{e}_{2}(t)\] (6)  
Toward this end, the control objective is to investigate the control method that could improve the trajectory tracking performance under system uncertainties, synchronization performance, and flexible control performance. The hybrid robot system should satisfy \(\lim\limits_{t\to\infty}(\mathbf{x}_{d}(t)-\mathbf{x}(t))=0\), \(\lim\limits_{t\to\infty}(\mathbf{q}_{d}(t)-\mathbf{q}(t))=0\), and \(\lim\limits_{t\to\infty}(\mathbf{e}_{1}(t)-\mathbf{e}_{2}(t))=0\) as well as be stable when different desired trajectories are given.  
Figure 2 presents the control schematic diagram.  
As shown in Figure 2, \(\mathbf{e}=[e_{1},e_{2}]\) and \(\mathbf{\varepsilon}=[e_{1},e_{2}]\) are, respectively, the tracking error and the synchronization error of UBSHR's end-effector; \([\mathbf{x}_{d},\hat{\mathbf{x}}_{d},\bar{\mathbf{x}}_{d}]\) and \([\mathbf{x},\dot{\mathbf{x}}]\) are the desired pose and the actual pose of active joints, respectively; and output data \(\mathbf{\tau}\) is the control torque of UBSHR.  
Firstly, the hybrid robot is well-controlled under the designed CE-SOSMC controller (Part A) to generate ideal training data samples and to improve the uncertain system robustness and synchronization performance simultaneously. Then, a subtractive clustering algorithm is introduced to avoid the explosion phenomenon of fuzzy rules. Based on this, the SC-AFNN controller (Part B) is constructed to completely replace the CE-SOSMC controller (Part A).