## 4 Main Contribution of the Control Design
### The Control Challenge
The gain deviation term \(\tilde{K}_{t}\) can be found as a solution to the following differential equation:  
\[2\gamma\dfrac{d}{dt}K_{t}+\dfrac{d}{dt}\gamma\tilde{K}_{t}+2\left(\dfrac{\left\| \Delta_{t}\right\|_{p}^{2}}{\Delta^{+}-\left\|\Delta_{t}\right\|_{p}^{2}} \right)P\Delta_{t}\Delta_{t}^{\top}=0 \tag{11}\]  
where \(\Delta_{t}=[\Delta_{a}^{\top},\Delta_{b}^{\top}]^{\top}\), \(\Delta^{+}\in\mathbb{R}^{+}\), \(\tilde{K}_{t}=K_{t}-K_{*}\) when \(K_{*}\) is selected such that \(A-BK_{*}\) is a Hurwitz matrix and \(\gamma>0\) with the matrix \(B\) can be obtained once more with the models of the robotic arm and piezoelectric actuators. Here, \(P=P^{\top}>0\) is a positive definite matrix, described below. Matrices \(A\) and \(B\) accept the following definitions:  
\[A= \begin{bmatrix}0_{3\times 3}&I_{3\times 3}&0_{3\times 2}&0_{3\times 2}\\ 0_{3\times 3}&0_{3\times 3}&0_{3\times 2}&0_{3\times 2}\\ 0_{2\times 3}&0_{2\times 3}&I_{2\times 2}&0_{2\times 2}\\ 0_{2\times 3}&0_{2\times 3}&0_{2\times 2}&CI_{2\times 2}\end{bmatrix} ,B= \begin{bmatrix}0_{3\times 3}&0_{3\times 3}\\ I_{3\times 3}&0_{3\times 3}\\ 0_{2\times 2}&0_{2\times 2}\\ 0_{2\times 2}&I_{2\times 2}\end{bmatrix} ,\]The state dependent \(\gamma\) is used to include the effect of state restrictions in the following manner:  
\[\gamma=\frac{\rho_{a}-\left\|\theta_{t}\right\|}{\rho_{a}}\times\frac{\rho_{b}- \left\|\frac{d}{dt}\theta_{t}\right\|}{\rho_{b}} \tag{12}\]  
Notice that in (11), the time derivative of \(\gamma\) should be estimated considering the evolution of \(\theta_{t}\) and \(\frac{d}{dt}\theta_{t}\). The online estimate of \(\gamma\) could be calculated using traditional derivatives, or it can alternatively be solved using robust exact differentiators such as the super-twisting algorithm (STA). Hence, the algorithm used to adjust the gain can be solved using the variables of the robotic arm carrying the piezoelectric actuator. **Theorem 1**. : _Consider the dynamics of the robotic arm (1) together with the gripper armed with two parallel piezoelectric grippers with the dynamics provided in (3) and with the state restrictions in (4), the dynamics of which are affected by the class of perturbations described in (2). Assume that the state variables of the robotic arm must track the corresponding reference trajectories for the entire robotic manipulator provided by (6). Taking into consideration the proposed controller \(\tau\) structure in (10) has a state-dependent gain defined in (11) that includes the effect of state constraints, if there is a positive definite and symmetric matrix \(Q\) such that the matrix equation \(Ric_{A}(P)=0\) has a positive definite solution where_  
\[ \begin{array}{c}Ric_{A}(P)=PA_{K}+A_{K}^{\top}P+PRP+Q\\ A_{K}=A+BK_{*}\\ R=\Lambda_{1}^{-1}+\Lambda_{2}^{-1}\;\;Q>0\end{array}  \tag{13}\]  
_with \(\Lambda_{1}\) and \(\Lambda_{2}\) positive definite matrices. Then, the tracking error \(\Delta_{t}\) has a practical asymptotic equilibrium point in a positively invariant ball around the origin delimited with the radius defined by_  
\[\beta:=\lambda_{\max}\{\Lambda_{2}\}\xi^{0} \tag{14}\]  
Proof. : For the proof, the dynamics of state tracking with \(\Delta^{\top}= \begin{bmatrix}\Delta_{a}^{\top}&\Delta_{b}^{\top}\end{bmatrix}