## 3 Preliminaries
### Dynamic Model for Tracking Control
Choose the state variable as  
\[\left\{\begin{array}{l}\mathbf{x}_{1}=\mathbf{q},\\  
where \(\mathbf{e}_{c}\) is the upper bound of the estimated error.  
**Proposition 1**.: _The system (41) can achieve stability over time \(T_{p}\) under the control of the controller (56)._  
Proof.: Considering a candidate Lyapunov function as  
\[V=\frac{1}{2}\mathbf{e}^{T}\mathbf{e} \tag{57}\]  
The time derivative of Equation (57) can be given by  
\[\dot{V} =\mathbf{e}^{\mathsf{T}}\dot{\mathbf{e}}, \tag{58}\] \[=-\mathbf{e}^{\mathsf{T}}\frac{\mathbf{e}_{p}^{m}}{mT_{p}}\frac{\mathbf{e}}{| \mathbf{e}|^{m}},\] \[=-\frac{\mathbf{e}_{c}^{m}}{mT_{p}}(2V)^{1-\frac{m}{2}}\]  
According to (58), the convergence time of the system (56) can be calculated as  
\[T(\mathbf{e}_{0}) =\int_{V(\mathbf{e}_{0})}^{0}\frac{1}{-\frac{\mathbf{e}_{c}^{m}}{mT_{p}}( 2V)^{1-\frac{m}{2}}}\,\mathrm{d}V, \tag{59}\] \[=T_{p}\int_{0}^{2V(\mathbf{e}_{0})}\frac{m/2}{\mathbf{e}_{c}^{m}}(2V)^{m/ 2-1}\,\mathrm{d}V,\] \[=T_{p}\frac{(2V(\mathbf{e}_{0}))^{m/2}-0}{\mathbf{e}_{c}^{m}},\] \[\leqslant T_{p}\]  
Therefore, the relative pose error of continuous space robots is explicit time convergent. The proof is complete. Substituting Equation (56) into (41), the joint velocity of the continuum space robot can be planned as  
\[\dot{q}_{d}=\mathbb{J}^{-1}\bigg{(}-\frac{\mathbf{e}_{c}^{m}}{mT_{p}}\frac{\mathbf{e} }{|\mathbf{e}|^{m}}-\mathbf{\xi}\bigg{)} \tag{60}\]  
where \(T_{p}\) is the desired convergence time. According to Lemma 1, the trajectory planning error can converge to zero, and the upper bound for the convergence time is \(T_{p}\). When the robotic system passes through a singular point (e.g., in an extended state), \(\mathbb{J}\) becomes irreversible. The singularity avoidance method based on DLS from reference [25] can be employed. The modified joint velocity planning for the continuum space robot can be expressed as  
\[\left\{\begin{array}{l}\mathbf{x}_{1}=\mathbf{q},\\  
where \(\mathbf{e}_{c}\) is the upper bound of the estimated error.  
**Proposition 1**.: _The system (41) can achieve stability over time \(T_{p}\) under the control of the controller (56)._  
Proof.: Considering a candidate Lyapunov function as  
\[V=\frac{1}{2}\mathbf{e}^{T}\mathbf{e} \tag{57}\]0  
where \(\sigma_{i}\) is the singular values of matrix \(\mathbb{J}\), \(v_{i}\) and \(u_{i}\) are the singular vectors, and \(\lambda_{i}\) is the damping coefficient. To mitigate errors introduced by the damping coefficient, design a dynamic damping coefficient given by  
\[\lambda_{i}=\left\{ \begin{array}{ll}\frac{\lambda_{\max}}{2}\big{[}1+\cos \big{(}\frac{\pi\sigma_{i}}{\varepsilon}\big{)}\big{]}&|\sigma_{i}|<\varepsilon \\ 0&|\sigma_{i}|\geq\varepsilon\end{array} \right. \tag{62}\]where \(\lambda_{\text{max}}\) is the maximum damping factor. In robot control, such known and fixed singularity issues can be handled using empirical values \(\lambda_{\text{max}}=0.01\); \(\varepsilon\) is the singular threshold for assessing \(\mathbb{J}\). After the robot exits the singular region, the error caused by the singular point can be eliminated within an explicit time \(T_{p}\).