## 3 The Model Reconstruction and the Estimator of Unknown System Dynamic  
In this section, the space robot system represented by Equation (1) is firstly reconstructed in which the acceleration signals are avoided being introduced to ensure transient control performance. Then based on the prescribed performance control method (PPC), the transient and steady state performance of the controlled system is quantitatively designed prior, the unknown dynamic terms are introduced to achieve precise compensation and the steady-state and transient control performance of the controller are guaranteed.  
The estimator of unknown system dynamic is designed for the unknown dynamic term of reconstructed equation, that is \(\mathbf{H}(\mathbf{q},\dot{\mathbf{q}})\dot{\mathbf{q}}\) and \(\tau_{d}\).  
According to the space robot system represented by Equation (1), it is obviously that the acceleration signals are introduced by \(\mathbf{D}(\mathbf{q})\dot{\mathbf{q}}\). Reconstruct the Equation (1), then we have:  
\[\left\{\begin{array}{l}\mathbf{M}\big{(}\mathbf{q},\dot{\mathbf{q}}\big{)}=\mathbf{D}(\mathbf{q })\dot{\mathbf{q}}\\ \mathbf{C}\big{(}\mathbf{q},\dot{\mathbf{q}}\big{)}=-\dot{\mathbf{D}}(\mathbf{q})\dot{\mathbf{q}}\\ \Delta=\tau_{d}-\mathbf{H}\big{(}\mathbf{q},\dot{\mathbf{q}}\big{)}\dot{\mathbf{q}}\end{array} \right., \tag{7}\]  
where \(\mathbf{M}\big{(}\mathbf{q},\dot{\mathbf{q}}\big{)}\), \(\mathbf{C}\big{(}\mathbf{q},\dot{\mathbf{q}}\big{)}\) is the vector without acceleration signals, \(\Delta\) is the lumped unknown dynamic items that consists of centrifugal and Koch forces and external bounded disturbance.  
The system represented by Equation (1) can be converted to Equation (7) by means of Equation (8):  
\[\left\{\begin{array}{l}\dot{M}\big{(}\mathbf{q},\dot{\mathbf{q}}\big{)}+\mathbf{C}\big{(} \mathbf{q},\dot{\mathbf{q}}\big{)}=\tau+\Delta\\ \dot{M}\big{(}\mathbf{q},\dot{\mathbf{q}}\big{)}=\frac{d\big{(}\mathbf{M}\big{(}\mathbf{q}, \dot{\mathbf{q}}\big{)}\big{)}}{dt}\end{array}\right., \tag{8}\]  
where \(\dot{M}\big{(}\mathbf{q},\dot{\mathbf{q}}\big{)}\) contains the acceleration signals \(\ddot{\mathbf{q}}\).  
To avoid using \(\dot{M}\big{(}\mathbf{q},\dot{\mathbf{q}}\big{)}\) directly and \(\ddot{\mathbf{q}}\) in the estimation of \(\Delta\), filter operations are introduced, then we have:  
The \(\widetilde{\Delta}\) is defined as estimated error by Equations (11) and (16):  
\[\widetilde{\Delta}=\Delta-\hat{\Delta}, \tag{17}\]  
When taking derivative of Equation (17), we have:  
\[\begin{array}{ll}\dot{\widetilde{\Delta}}&=\dot{\Delta}-\dot{\hat{\Delta}}\\ &=\dot{\Delta}-\left(\left(\hat{\mathbf{M}}-\hat{\mathbf{M}}_{f}\right)/k+\dot{\mathbf{C} }_{f}-\dot{\tau}_{f}\right)\\ &=\dot{\Delta}-\left(\left(\hat{\mathbf{M}}-\hat{\mathbf{M}}_{f}\right)-\left(\mathbf{C}- \mathbf{C}_{f}\right)-\left(\tau-\tau_{f}\right)\right)/k\\ &=-\widetilde{\Delta}/k+\dot{\Delta}\end{array} \tag{18}\]  
Similarly, according to Lemma 1, Lyapunov function \(V_{\rm C}\) is defined as \(V_{\rm C}=\frac{1}{2}\widetilde{\Delta}^{\rm T}\widetilde{\Delta}\), when taking derivative of \(V_{\rm C}\), we have:  
\[\begin{array}{ll}\dot{V}_{C}&=\widetilde{\Delta}^{\rm T}\dot{\widetilde{ \Delta}}\\ &=-\widetilde{\Delta}^{\rm T}\widetilde{\Delta}/k+\widetilde{\Delta}^{\rm T} \dot{\Delta}=-V_{\rm C}/k+V_{\rm C}/2k+k\dot{\Delta}^{\rm T}\dot{\Delta}/2\\ &\leq-V_{\rm C}/k+k\Psi^{2}/2\end{array} \tag{19}\]  
Then the estimation error \(\widetilde{\Delta}\) convergence boundary is:  
\[\|\widetilde{\Delta}\|=\sqrt{2V_{\rm C}(t)}\leq\sqrt{\Delta(0)e^{-\frac{t}{k} }+k^{2}\Psi^{2}}, \tag{20}\]  
Therefore, when \(k\to 0\), \(V_{\rm C}\to 0\).