## 6 Simulation Results
### Calibration Phase Results  
Let us take into account that the model for the LLRR has nine standard dynamic parameters, i.e., size of \(\mathbf{\chi}\). Moreover, let us suppose an \(\alpha=1\), which defines the exponentially decreasing convergence ratio for the calibration phase. This value was selected taking into account that the convergence was obtained in less than \(25\,\mathrm{s}\). It was estimated that an initial phase of \(30\,\mathrm{s}\) was appropriately short for a patient to accept being idle without executing any interaction torque. Therefore, the simulations sought to calibrate the convergence to take less than \(30\,\mathrm{s}\), achieving good results with \(25\,\mathrm{s}\). Moreover, a more significant \(\alpha\) would reduce convergence time. However, this would turn into sensitivity within the calibration phase since a more considerable observer gain produces more fluctuations of the value. In addition, let us define the identity matrix as \(\mathbf{I}_{9}\), and (19) would turn into  
\[\mathbf{H}=\left[ \begin{array}{cc}\gamma\mathbf{I}_{9}&-\frac{\gamma}{2} \mathbf{I}_{9}\\ -\frac{\gamma}{2}\mathbf{I}_{9}&\frac{1}{2}\mathbf{K}^{-1}\end{array} \right]>0 \tag{26}\]  
Subsequently, we used the MATLAB(r) Robot Control Toolbox(tm) to solve this LMI-based analysis and, thus, design the calibration phase estimator. The solution with \(\alpha=1\) for the LMI (26) was \(\gamma=15.691\) and \(\mathbf{K}=0.0212\mathbf{I}_{9}\), i.e., satisfying the LMI with \(\mathbf{H}>0\). The trajectory used within the calibration phase must be selected with caution to improve the convergence rate of this phase and reduce noise sensitivity. The abovementioned trajectory should be a persistent exciting trajectory exciting trajectory and can be computed using some optimization criteria as reported by [PERSON] et al. [37]. Therefore, to ensure that \(W\) satisfies the persistent excitation condition, the robot's trajectory was optimized for the calibration phase. This selection of the persistent exciting trajectory was conducted by calculating the movements of the robot whose points give a well-conditioned observation matrix [37]. The condition number \(\kappa_{2}(\mathbf{W})\) of a matrix \(\mathbf{W}\) allows evaluating its well or ill-conditioning. A problem with a low condition number may be defined as well-conditioned, while a problem with a high condition number is said to be ill-conditioned [44]. The \(\kappa_{2}(\mathbf{W})\) is defined using the 2-norm, as  
\[\kappa_{2}(\mathbf{W})=\frac{\sigma_{\max}}{\sigma_{\min}}\geq 1 \tag{27}\]  
where \(\sigma_{\max}\) and \(\sigma_{\min}\) are defined as the maximum and minimum singular values of \(\mathbf{W}\). Therefore, the nonlinear optimization problem consisted of determining a trajectory that provides a \(\kappa_{2}(\mathbf{W})\) that is close to one [37]. For this purpose, we used a Genetic Algorithm (GA) algorithm with the MATLAB(r) Robot Control Toolbox(tm) to optimize a trajectory whose points gave a well-conditioned observation matrix. The code minimizes the \(\kappa_{2}(\mathbf{W})\) by generating multiple trajectories to ensure the persistently exciting condition, i.e., the \(\kappa_{2}(\mathbf{W})\) is approximately one. The code uses GA for this purpose and creates trajectories using a combination of two sinusoid signals per joint for the \(\mathbf{q}_{d}\). The frequencies were restricted to values lower than \(0.2\,\mathrm{Hz}\) to take into account physical restrictions for the LLRR. Moreover, the maximum number of generations was configured as \(10\). The initial position was set to \(\mathbf{q}_{0}=(0,-90,90)^{\circ}\) with no speed. Moreover, the amplitude for each sinusoid was set to \(30^{\circ}\), i.e., these signals indicate the LLRR movement in degrees concerning the origin. The obtained persistent exciting trajectories are defined as  
\[\left\{ \begin{array}{l}q_{1d}=30\sin(0.0430t)+30\sin(0.2316t)\\ q_{2d}=60\sin(0.0938t)+30\sin(0.0934t)\\ q_{2d}=30\sin(0.0594t)+30\sin(0.1375t)\end{array}