## 4 The Design of Controller
### Prescribed Performance Constraints  
Before formally designing the prescribed performance function constraints, we first analyze the insertion and extraction process of space robot represented by the following schematic diagram. We can find that the operation takes place in inertial space and the central axis of replaceable instruments coincides with the central axis of inser  
Figure 2: The control structure of space robot system.  
tion hole. Therefore, we need to design the desired trajectory of the manipulator end in Cartesian space.  
In order to avoid sudden change of friction during the task, and ensure the operation is successfully realized with smooth process. Then, according to Figure 3, combine the later part of the findings described in the previous paragraph and schematic diagram, \(q_{0d}\) and \(q_{\text{td}}\) are designed as:  
\[\left\{\begin{array}{l}q_{0d}=0\text{rad}\\ q_{\text{td}}=q_{1d}+q_{2d}+q_{3d}=\pi\text{rad}\end{array},\right. \tag{21}\]  
Combined with Equations (21) and (A8), the manipulator end desired trajectory of insertion and extraction are obtained as:  
\[X_{d}=\left\{\begin{array}{l}q_{0d}=0\text{rad}\\ x_{\text{td}}^{\prime},\\ y_{\text{td}}=q_{1d}+q_{2d}+q_{3d}=\pi\text{rad}\end{array},\right. \tag{22}\]  
where, the specific expressions of \(x_{\text{td}}{}^{\prime}\) and \(y_{\text{td}}{}^{\prime}\) will be given in Section 5.  
Then, the tracking error in coordinates system \(X^{\prime}O^{\prime}Y^{\prime}\) is obtained as:  
\[\left\{\begin{array}{l}\mathbf{e}_{c}=\mathbf{X}-\mathbf{X}_{\text{d}}\\ \dot{\mathbf{e}}_{c}=\dot{\mathbf{X}}-\dot{\mathbf{X}}_{\text{d}}\end{array},\right. \tag{23}\]  
For the fully space robot Equation (23), the tracking error in joint space is defined as follow:  
\[\left\{\begin{array}{l}\mathbf{e}=\mathbf{q}-\dot{\mathbf{q}}_{\text{d}}\\ \dot{\mathbf{e}}=\dot{\mathbf{q}}-\dot{\mathbf{q}}_{\text{d}}\end{array},\right. \tag{24}\]  
where \(\mathbf{q}_{\text{d}},\dot{\mathbf{q}}_{\text{d}}\in\Re^{4\times 1}\) is desired trajectory and velocity.  
Referring to Equation (11), we have the following:  
\[\dot{\mathbf{e}}=\mathbf{I}_{ta}^{-1}\dot{\mathbf{e}}_{c}, \tag{25}\]  
In the first step of PPC method, it usually applies a performance function \(\rho(t)\) to limit the upper and lower bounds of the system state. The performance function \(\rho(t)\) needs to meet the following two properties:  
1. It is time dependent and monotonically decreasing.
2. It is continuously differentiable.  
The following exponential function is chosen as the performance function:  
\[\rho(t)=(\rho_{0}-\rho_{\infty})e^{-xt}+\rho_{\infty}, \tag{26}\]  
Figure 3: The space robot operation of insertion and extraction diagram.  
where \(\rho_{0}>\rho_{\infty},\alpha>0\) are the default performance parameters. \(\rho_{0}\) is the default performance function initial value and \(\rho_{\infty}\) is the stable value. \(\alpha\) is related to the convergence rate of the \(\rho(t)\).  
After processing by default performance function, it usually applies a set of inequalities to limit the upper and lower bounds of the system state, which is quantified as:  
\[-\underline{\mu}\rho(t)<\mathbf{e}(t)<\overline{\mu}\rho(t),\forall t\geq 0, \tag{27}\]  
where \(\underline{\mu}>0\) is default boundary parameters.  
Additional upper and lower bound constraints are introduced for the control system to ensure transient and steady state performance of the system. However, this also makes the design of the corresponding controller more difficult. For the convenience of controller design, the nonlinear system represented by Equation (33) needs to be unconstrained.