## 2 Structure of the Control Scheme
### Structure of the Proposed Control Scheme  
The structure of the proposed control scheme is presented in Figure 2. It mainly consists of three subsystems: the Synchronization Subsystem, the NFFSMC Subsystem. and the TDE Subsystem.  
The notations used in Figure 3 are listed below:  
* \(\mathbf{x}_{d}\in\mathbb{R}^{n}\): the desired Cartesian configuration vector. (Note: Configuration means both position and orientation of the CKCM);
* \(\mathbf{q}_{d}\in\mathbb{R}^{n}\), \(\mathbf{q}\in\mathbb{R}^{n}\) and \(\ddot{\mathbf{q}}\in\mathbb{R}^{n}\): the desired joint vector, actual joint vector, and actual acceleration vector, of the CKCM, respectively;
* \(\mathbf{e}_{c}\in\mathbb{R}^{n}\): the synchronization error vector;
* \(\mathbf{u}\in\mathbb{R}^{n}\): the control law vector of the NFFSMC;
* \(\mathbf{\overline{M}}(\mathbf{q})\in\mathbb{R}^{n\times n}\): constant, diagonal matrix selected by the TDE;
* \(\mathbf{r}\in\mathbb{R}^{n}\): the output vector of the NFFSMC Subsystem;
* \(\mathbf{\tau}\in\mathbb{R}^{n}\): the compensated control input vector to the CKCM;
* \(\mathbf{\tau}_{d}\in\mathbb{R}^{n}\): the external disturbances vector;
* \(L\): the estimate time delay of the TDE;
* \(\ddot{\mathbf{q}}_{t-L}\in\mathbb{R}^{n}\) and \(\mathbf{\tau}_{t-L}\in\mathbb{R}^{n}\): the past acceleration vector and past control input vector of the CKCM, respectively;
* \(\mathbf{\hat{H}}\in\mathbb{R}^{n}\): the estimate of all nonlinear terms including the inertia uncertainty, Coriolis/centripetal vector, gravitational vector, friction vector, and disturbances.  
The operation of the proposed control scheme applied to control the motion of an n-DOF CKCM is described as follows. The desired Cartesian vector \(\mathbf{x}_{d}\) of the manipulator configuration (position and orientation) specified by the user or obtained by a trajectory planner is transformed to its corresponding desired joint vector \(\mathbf{q}_{d}\) by the CKCM Inverse Kinematic Transformation. The desired joint vector \(\mathbf{q}_{d}\) and the actual joint vector \(\mathbf{q}\) (provided by the CKCM joint sensors) are supplied to the Synchronization Subsystem, which then produces the position errors \(\mathbf{e}_{i}=\mathbf{q}_{d_{i}}-\mathbf{q}_{i}\) of every \(i\)th active joint, the synchronization error \(\mathbf{e}_{S}\), and the cross-coupling error \(\mathbf{e}_{c}\) between the active joints. The cross-coupling error \(\mathbf{e}_{c}\) is then inputted to the NFFSMC subsystem that in turn based on \(\mathbf{e}_{c}\) defines a sliding surface to achieve the above desired behavior of the errors. Next, the control law \(\mathbf{u}\) composed based on the sliding surface will be a driving component forcing the system's trajectory to reach a stable hyperplane (sliding surface). This control law \(\mathbf{u}\) will then serve as part of the input \(\boldsymbol{\tau}\) to the CKCM and is developed to ensure asymptotic convergence of the errors of both position and synchronization of the CKCM in a finite and minimal time. The TDE subsystem uses the past control input and acceleration of the CKCM to estimate the CKCM dynamics and the disturbance torques, which are required for the implementation of the input \(\boldsymbol{\tau}\) to CKCM.