## 2 Kinematics Analysis  
To improve the safety of intervention surgery, the duty cycle control strategy proposed in a previous work [11] to adjust the trajectory curvature is not used in the present study. As depicted in Figure 1, the trajectory curvature is controlled by means of stylet translation alone, the insertion orientation is controlled by means of stylet rotation, and the insertion depth is controlled by means of cannulation translation. Under the follow-the-leader assumption, which means that the needle shaft follows the trajectory created by the tip, the 3D kinematics of the CCS needle is formulated, as illustrated in Figure 2. The world frame is marked in black color at the insertion point, and the black line indicates the needle's trajectory in 3D space. Euler angle representation is adopted to describe the orientation of the tip frame T with respect to the world frame W, and a rotation matrix is used to express the tip rotations about the z and y axes of the world frame, as follows:  
\[ \begin{array}{l}{}_{T}^{W}R=R_{z}(\gamma)R_{y}(-\beta),\end{array}  \tag{1}\]where \(\gamma\) and \(\beta\) are rotation angles. Define \(v_{x}\) as the insertion velocity at the needle base and the linear velocity vector of the kinematics is \(v_{l}= \begin{bmatrix}v_{x}&0&0\end{bmatrix} \). The linear velocity of the needle tip \(\dot{p}\) can be computed as:  
\[\dot{p}=\frac{W}{T}Rv_{l}. \tag{2}\]  
As illustrated in Figure 2, the \(xz\) plane marked with the transparent red color is defined as the _InPlane_, and the \(xy\) plane marked with the transparent blue color is defined as the _OffPlane_. The dashed red and blue lines represent projections of the 3D trajectory with respect to the _InPlane_ and _OffPlane_, respectively. The assumption that the instantaneous curvature \(\rho\) is proportional to the steering offset \(\delta\) is adapted from our planar kinematics findings [10]. Therefore, the instantaneous curvatures of the _InPlane_ and _OffPlane_ trajectories are as follows:  
\[ \begin{split}\rho_{in}=k\delta_{in}\\ \rho_{off}=k\delta_{off},\end{split}  \tag{3}\]  
where \(\delta_{in}\), \(\delta_{off}\), and \(k\) denote the _InPlane_ steering offset, the _OffPlane_ steering offset, and the coefficient related to tissue property, which should be calibrated experimentally. The angular velocity vector of the kinematics is \(\omega= \begin{bmatrix}0&\omega_{y}&\omega_{z}\end{bmatrix} \), where \(\omega_{y}\) and \(\omega_{z}\) are:  
\[ \begin{split}\omega_{y}=\dot{\delta}_{in}\\ \omega_{z}=\dot{\delta}_{off}.\end{split}  \tag{4}\]  
Therefore, the changing rate of the rotation angles can be expressed as follows:  
\[ \begin{split}\dot{\beta}=\omega_{y}s=k\delta_{in}v_{x}\\ \dot{\gamma}=\omega_{z}s=k\delta_{off}v_{x},\end{split}  \tag{5}\]  
where \(s\) denotes the insertion depth. Figure 2: Kinematic model of a CCS needle; \(\delta_{in}\) and \(\delta_{off}\) are the projection of the steering offset \(\delta\) in the _InPlane_ and _OffPlane_ region, respectively. By combining Equations (2), (4), and (5), the kinematics of the CCS needle can be expressed as follows:  
\[ \begin{bmatrix}\dot{x}\\ \dot{y}\\ \dot{z}\\ \dot{\beta}\\ \dot{\gamma}\\ \dot{\delta}_{in}\\ \dot{\delta}_{off}\end{bmatrix} = \begin{bmatrix}cos\gamma cos\beta\\ sin\gamma cos\beta\\ sin\beta\\ k\dot{\delta}_{in}\\ k\delta_{off}\\ 0\end{bmatrix} v_{x}+ \begin{bmatrix}0\\ 0\\ 0\\ 0\\ 0\\ 1\\ 0\end{bmatrix} \omega_{y}+ \begin{bmatrix}0\\ 0\\ 0\\ 0\\ 0\\ 1\end{bmatrix} \omega_{z}, \tag{6}\]  
where the stylet rotation (as shown in Figure 1) angle of the CCS needle \(\theta\) and composite steering offset \(\delta\) (stylet translation) can be computed from the kinematics as:  
\[ \begin{gathered}\theta=atan2\Big{(}-\delta_{off},\delta_{in} \Big{)}\\ \delta=\sqrt{\delta_{in}^{2}+\delta_{off}^{2}}.\end{gathered}  \tag{7}\]  
The 3D kinematics of the CCS needle expressed in (6) is nonlinear, and its general form is as follows:  
\[\dot{q}=G(q)v \tag{8}\]  
where \(q\) is the vector of the CCS needle's generalized coordinates \(