## 3 Preliminaries
### Explicit Time Stability Theory  
Consider the following uncertain dynamic system,  
\[\dot{\mathbf{x}}=f(t,\mathbf{x}),\;\mathbf{x}(t_{0})=\mathbf{x}_{0} \tag{23}\]  
where \(\mathbf{x}\in R^{n}\) is the system state, \(f(t,\mathbf{x})\) is a nonlinear function, and \(\mathbf{x}_{0}\) is the initial state of the system. **Definition 1** (Finite-Time Stability [20]). : _If the system (23) is globally asymptotically stable and the system can reach the equilibrium point within a finite time, the system is referred to as globally finite-time stable._  
**Definition 2** (Fixed-Time Stability [21]). : _If the system (23) is globally finite-time stable, and the upper bound on the convergence time is independent of the initial state, the system is considered globally fixed-time stable._  
**Definition 3** (Predefined-Time Stability [22]). : _If the system (23) is globally fixed-time stable, and the upper bound on the convergence time is directly provided within the system functions, the system is considered globally predefined-time stable._  
**Definition 4** (Explicit Time Stability [23]). : _If the system (23) is finite time stable, and there is an explicit solvable upper bound \(T(\mathbf{x}_{0})\leqslant T(\mathbf{x}_{c})=T_{c}\) on the convergence time under bounded initial conditions \(U\in[-\mathbf{x}_{c},\mathbf{x}_{c}]\), the system is said to be explicit time stable._  
The sufficient conditions for explicit time stability of the system (23) are provided by the following lemma. **Lemma 1** ([24]). : _Consider a type of system:_  
\[\dot{\mathbf{x}}=-\frac{G(\mathbf{x}_{c}^{2m})-G(0)}{T_{c}}\left(\frac{\partial G(\mathbf{ x}^{2m})}{\partial x}\right)^{-1} \tag{24}\]  
_where \(T_{c}>0\) is the expected convergence time; \(0<m<\frac{1}{2}\). This choice ensures a gradual initial convergence, preventing abrupt changes in the early stage while accelerating in the later stage. It helps maintain system stability and reduces the risk of excessive control effort or chattering. \(x_{c}>0\) are system parameters; \(G(\mathbf{x}_{c}^{2m})\) is the bounded gain term. If the function \(G(y)\) satisfies the following conditions, it is considered explicitly time-stable. (a) \(G(y)\) is continuous within the defined domain. (b) Within bounded conditions \(|y|\leqslant|y(\mathbf{x}_{c})|\), \(\left[\frac{\partial G(y)}{\partial y}\right]^{-1}\) is bounded, positively definite, and right continuous._  
of the system momentum with respect to the base velocity and joint velocity, respectively. From Equation (31) of the momentum conservation, it follows that  
\[\mathbf{V}_{0}=\mathbf{H}_{0}{}^{-1}(\mathbb{C}-\mathbf{H}_{m}\mathbf{\dot{q}}) \tag{32}\]  
Substituting (32) into (27), the velocity of the end effector of the continuum space robot can be expressed as  
\[ \begin{split}\mathbf{V}_{e}&=(\mathbf{J}_{m}-\mathbf{J}_{0}\bm {H}_{0}{}^{-1}\mathbf{H}_{m})\mathbf{\dot{q}}+\mathbf{J}_{0}\mathbf{H}_{0}{}^{-1}\mathbb{C}\\ &=\mathbf{J}_{g}\mathbf{\dot{q}}+\mathbf{J}_{0}\mathbf{H}_{0}{}^{-1}\mathbb{C} \end{split}  \tag{33}\]  
where \(\mathbf{J}_{g}=\mathbf{J}_{m}-\mathbf{J}_{0}\mathbf{H}_{0}{}^{-1}\mathbf{H}_{m}\) is the generalized Jacobian matrix representing the end effector velocity relative to the joint angular velocities. The kinematic model for the relative pose error of the end effector of the continuum space robot can be expressed as  
\[\mathbf{\dot{e}}_{p}=\mathbf{J}_{e}\mathbf{V}_{e}-\mathbf{J}_{ed}\mathbf{V}_{ed} \tag{34}\]  
where \(\mathbf{e}_{p}=[e_{e}^{\text{T}}\ \sigma_{e}^{\text{T}}]^{\text{T}}\) is the pose error of the end effector, where \(\mathbf{e}_{e}\) is the position error and \(\sigma_{e}\) is the attitude error represented using MRPs; \(\mathbf{V}_{ed}=[\mathbf{v}_{ed}{}^{\text{T}}\ \mathbf{\omega}_{ed}{}^{\text{T}}]^{\text{T}}\) is the desired velocity of the end effector. The expression for \(\mathbf{J}_{e}\) is given by  
\[\mathbf{J}_{e}= \begin{bmatrix}\mathbf{E}_{3}&\mathbf{0}\\ 0&\frac{1}{4}\mathbf{G}(\sigma_{e})\mathbf{R}_{E}{}^{\text{T}}\end{bmatrix}