## 3 Model Analysis  
The SISP model considering light base support characteristics is analyzed when the feedback controller is used. If the platform base equivalent moment of inertia is light, the performance is limited owing to the vibration. This paper takes the PI feedback controller as an example, as it is that most commonly used in engineering. For the dynamics model (13), the traditional velocity loop uses the PI feedback controller, which is described as follows,  
\[u(t)=K_{p}e(t)+K_{i}\int_{0}^{t}e(\tau)d\tau \tag{21}\]  
where the angular velocity error is \(e(t)=r_{v}(t)-\theta_{m}^{\prime}(t)\), \(r_{v}(t)\) is the velocity command, and \(K_{p}\) and \(K_{i}\) are the PI controller proportion and integral parameters, respectively. Because the gyro is mounted on the platform base, we can use the indirect measurement method to obtain the feedback signal by Equation (12), e.g., \(e(t)=r_{v}(t)-\theta_{e}(t)-\theta_{b}(t)\). Substituting (21) into (13), the multi-body dynamics model with the PI controller is described as follows,  
\[ \begin{cases}J_{l}\ddot{\theta}_{b}(t)+B_{l}\dot{\theta}_{b}(t)+K_{p}K_{a}( \dot{\theta}_{e}(t)+\dot{\theta}_{b}(t))+K_{i}K_{a}(\dot{\theta}_{e}(t)+\dot{ \theta}_{b}(t))=T_{z}(t)-F_{m}(t)+K_{p}K_{a}r_{v}(t)+K_{i}K_{a}r_{p}(t)\\ J_{m}(\ddot{\theta}_{e}(t)+N\dot{\theta}_{b}(t))+B_{m}(\dot{\theta}_{e}(t)+N \dot{\theta}_{b}(t))+NK_{p}K_{a}(\dot{\theta}_{e}(t)+\dot{\theta}_{b}(t))+J_{ l}\ddot{\theta}_{l}(t)+B_{l}\dot{\theta}_{l}(t)+NK_{i}K_{a}(\theta_{e}(t)+\dot{ \theta}_{b}(t))\\ =NK_{p}K_{a}r_{v}(t)+NK_{i}K_{a}r_{p}(t)-NF_{m}(t)\\ J_{l}\ddot{\theta}_{l}(t)+(B_{l}+B_{s})\dot{\theta}_{l}(t)-B_{s}(\dot{\theta}_{ e}(t)+\dot{\theta}_{b}(t))-K_{s}(\theta_{e}(t)+\dot{\theta}_{b}(t)-\theta_{l}(t))=0 \end{cases}  \tag{22}\]  
As a result, the general form is changed to the following expression. \[\textbf{J}\dot{\Theta}(t)+\textbf{B}^{{}^{\prime}}\dot{\Theta}(t)+\textbf{K} ^{{}^{\prime}}\Theta(t)=\textbf{p}(t)+\textbf{f}(t)+\textbf{R}^{{}^{\prime}} \textbf{r}_{c}^{\textbf{T}}(t) \tag{23}\]  
where the command vector \(\textbf{r}_{c}(t)\) is \([r_{p}(t)\ r_{v}(t)\ r_{a}(t)]\). \(r_{p}(t),r_{v}(t),r_{a}(t)\) are the position, velocity, and acceleration commands, respectively. The matrices **B** and **K** are changed into \(\textbf{B}^{{}^{\prime}}\) and \(\textbf{K}^{{}^{\prime}}\). \[\textbf{B}^{{}^{\prime}}= \begin{bmatrix}B_{b}+K_{p}K_{a}&K_{p}K_{a}&0\\ NB_{m}+NK_{p}K_{a}&B_{m}+NK_{p}K_{a}&B_{l}\\ -B_{s}&-B_{s}&B_{l}+B_{s}\end{bmatrix}  \tag{24}\]  
\[\textbf{K}^{{}^{\prime}}= \begin{bmatrix}K_{i}K_{a}&K_{i}K_{a}&0\\ NK_{i}K_{a}&NK_{i}K_{a}&0\\ -K_{s}&-K_{s}&K_{s}\end{bmatrix}  \tag{25}\]  
The command matrix is expressed as follows,  
\[\textbf{R}^{{}^{\prime}}= \begin{bmatrix}K_{i}K_{a}&K_{p}K_{a}&0\\ NK_{i}K_{a}&NK_{p}K_{a}&0\\ 0&0&0\end{bmatrix}  \tag{26}\]  
To reproduce the vibration phenomenon in the SISP with a light base support, an Euler method numerical analysis was performed in MATLAB. The relative angular velocity was obtained via the angle measured by the encoder and the differentiator. The base angular velocity is a sine signal (1 Hz, 10\({}^{\circ}\)/s). The gains of the PI feedback controller are tuned as \(K_{p}=1\) and \(K_{i}=1\). The total disturbance torque \(F_{m}\) employed the Stribeck friction model, including Coulomb friction, viscous friction, and Stribeck phenomenon, which is a nonlinear low-velocity friction effect [38]. The numerical analysis parameters and results are presented in Table 1 and Figure 3, respectively. The simulation results reveal that when the load-equivalent moment of inertia \(J_{l}\) is light, the system works normally; however, when \(J_{l}\) is heavy, the system vibrates. If the feedback controller gain is reduced properly, the vibration still exists. Thus, the vibration here is not caused by the closed-loop system large parameters. Generally, in most SISPs, the equivalent moment of inertia on the load is much lower than that on the base; therefore, there is no problem in the controller design that is based on the traditional model.