## 2 Dynamic Modeling of Flexible Joints  
Due to the limitations of space transportation and the usage environment, space manipulators are longer but lighter in weight compared to industrial manipulators, resulting in significantly lower structural rigidity [28]. The flexible space manipulator studied in this paper is illustrated in Figure 2a. It has an arm length of approximately 3300 mm, with a weight of only 60 kg. The joint structure of the manipulator, as shown in Figure 2b, is driven by a brushless DC motor with a harmonic reducer as the transmission part. Both the motor and the output shafts of the reducer are equipped with position sensors. The reducer has the lowest stiffness among the manipulator's components. Hence, this paper only considers the flexibility of the space manipulator caused by the joint harmonic reducer while disregarding other factors. Full-Elements Joint Dynamics Model  
The flexible joint of the space manipulator is composed of several components, including the motor, motor position sensor, joint position sensor, and harmonic reducer, as shown in Figure 2b. To simplify the engineering analysis, the model is divided into three parts: the motor unit, transmission unit, and output unit, as depicted in Figure 3. The inertia and damping of the motor position sensor and its associated structure are included in the motor unit, while the inertia and damping of the output position sensor and its subsidiary structure are included in the output unit. The equivalent inertias and damping of the joint unit, transmission unit, and output unit are denoted by \(I_{m}\), \(I_{g}\), \(I_{l}\), and \(d_{m}\), \(d_{g}\), \(d_{l}\), respectively. The rotation angles are \(q_{m}\), \(q_{g}\), \(q_{l}\), and the moments are \(\tau_{m}\), \(\tau_{g}\), \(\tau_{l}\). Additionally, to account for the stiffness and damping characteristics of the output shaft and reducer, an ideal torsion spring-damping link is introduced between the motor unit and the transmission unit and between the transmission unit and the output unit. The equivalent stiffness and damping between the motor unit and the transmission unit are denoted by \(k_{g}\) and \(d_{mg}\), respectively, while the equivalent stiffness and damping between the transmission unit and the output unit are denoted by \(k_{gl}\) and \(d_{gl}\), respectively. Figure 2: Developed space manipulator. (**a**) Prototype; (**b**) Model of the flexible joint. The dynamics model of the flexible joint consists of two parts: an electrical model and a mechanical model. The electrical model describes the characteristics of motor current input and torque output, while the mechanical model describes the characteristics of motor torque input and joint load torque output, which is further divided into linear and nonlinear parts. The dynamic model of the flexible joint can be described as follows:  
\[\tau=\mathbf{Kq}+\mathbf{D\dot{q}}+\mathbf{I\ddot{q}} \tag{1}\]  
\[\boldsymbol{q}=\left[\boldsymbol{q}_{m},\boldsymbol{q}_{g},\boldsymbol{q}_{l}\right] \tag{2}\]  
The joint stiffness matrix \(\mathbf{K}\), the inertia matrix \(\mathbf{I}\), and the damping matrix \(\mathbf{D}\) are defined as follows:  
\[\mathbf{K}= \begin{bmatrix}k_{g}&-k_{g}&0\\ -\bar{k}_{g}&k_{g}+k_{b}&-k_{b}\\ 0&-k_{b}&k_{b}\end{bmatrix}  \tag{3}\]  
\[\mathbf{I}= \begin{bmatrix}I_{m}&0&0\\ 0&I_{g}&0\\ 0&0&I_{l}\end{bmatrix}  \tag{4}\]  
\[\mathbf{D}= \begin{bmatrix}d_{m}+d_{mg}&-d_{mg}&0\\ -d_{mg}&d_{g}+d_{mg}+d_{gl}&-d_{gl}\\ 0&-d_{gl}&d_{l}+d_{gl}\end{bmatrix}  \tag{5}\]  
The nonlinear part of joint dynamics mainly includes joint viscous friction \(f_{\text{vis}}\) and coulomb friction \(f_{\text{cou}}\).