## 2 Dynamic Modeling and Linearization
### Dynamic Modeling  
The Newton-Euler method is one of the most widely used approaches for robotic manipulator modeling in both industrial and academic settings. Compared to the Lagrangian formulation, it offers higher computational efficiency and a superior recursive structure, particularly when dealing with multi-degree-of-freedom (multi-DOF) systems. This method establishes a complete dynamic mapping between joint state variables and system outputs by performing bidirectional recursion at both the kinematic and dynamic levels [17]. In terms of controllability and practical implementation, the Newton-Euler method is more suitable for industrial servo control systems, as it allows for better integration with parameter identifiers and feedforward compensators [16]. Based on this approach, the system utilizes forward recursion to describe the time-varying dynamic relationships between joint drives and connected links. Let the angular velocity and linear velocity of the \(i\)-th link be denoted as \(\omega_{i}\) and \(v_{i}\), respectively, where \(A_{i}\) denotes the transformation matrix of link \(i\), and \(q_{i}\) is the joint angle of joint \(i\).  
The corresponding recursive formulations are given in Equations (1) and (2):  
\[v_{i} =A_{i-1}v_{i-1}+f_{i}\big{(}q_{i},\hat{q}_{i}\big{)} \tag{1}\] \[\omega_{i} =A_{i-1}\omega_{i-1}+g_{i}\big{(}\hat{q}_{i},\hat{q}_{i}\big{)} \tag{2}\]Based on the rigid-body inertia law and the principle of momentum conservation, the dynamic equilibrium equation for a single joint can be formulated as shown in Equation (3):  
\[\tau_{i}=J_{i}^{T}\big{(}f_{i}^{\text{ext}}+m_{i}a_{i}\big{)} \tag{3}\]  
Here, \(J_{i}^{T}\) denotes the Jacobian matrix of the \(i\)-th link, \(f_{i}^{\text{ext}}\) represents the external disturbance force, and \(m_{i}\) is the mass of the corresponding link. The complete rigid-body dynamic equation of the manipulator system can then be summarized as shown in Equation (4):  
\[\tau=M(q)\ddot{q}+C(q,\dot{q})\dot{q}+G(q)+\tau_{\text{fric}} \tag{4}\]  
In this context, \(M(q)\) represents the joint-space inertia matrix, \(C(q,\dot{q})\) denotes the centrifugal and Coriolis torque terms, \(G(q)\) is the gravity vector, and \(\tau_{\text{fric}}\) corresponds to the disturbance torque induced by friction. To provide a clearer representation of the contribution of each dynamic component to the joint output torque, Equation (4) can be further refined into Equation (5), which is expressed as follows:  
\[\tau_{i}=\sum_{j=1}^{n}M_{ij}(q)\ddot{q}_{j}+C_{i}(q,\dot{q})+G_{i}(q)+\tau_{ f,i}+\tau_{\text{ext},i} \tag{5}\]  
\[\left\{\begin{array}{l}\sigma_{w}=0.1u_{20}\\ \frac{\sigma_{w}}{\sigma_{w}}=\frac{\sigma_{z}}{\sigma_{w}}=\frac{1}{(0.177+0. 00823h)^{0.4}}\end{array}\right. \tag{6}\]  
Here, \(\tau_{i}\) represents the total output torque of the \(i\)-th joint, \(M_{ij}(q)\) denotes the elements of the joint inertia matrix, and \(\tau_{ext,i}\) represents the potential external disturbance terms. This expression provides the torque factors that form the basis for subsequent friction modeling and structural coupling analysis. The schematic of the dynamic parameters is shown in Figure 1. In Figure 1a, the SolidWorks model of a six-degree-of-freedom robotic manipulator is shown, indicating the mechanical structure adopted in this study. Figure 1b further defines the relevant kinematic and dynamic parameters on a single link, including angular velocity, linear velocity, gravity, link mass, and external disturbance forces, which are used in the modeling process.