## 5 Proposed Methods
### Calibration Phase Design  
The objective of this section is to present the calibration phase. This calibration phase reduces BSIPs uncertainties based on an online parameter estimation method. Taking into account the model presented in (10), based in the regressor matrix \(\mathbf{W}(\mathbf{q},\dot{\mathbf{q}},\ddot{\mathbf{q}})\) and the vector \(\boldsymbol{\chi}\), we used the observer reported by [PERSON] et al. [36] to perform an online parameter estimation. This method is expressed as  
\[\left\{\begin{array}{l}\dot{\mathbf{\Gamma}}=\mathbf{W}\mathbf{\hat{\chi}}+ \frac{\alpha}{2}(\mathbf{\Gamma}-\mathbf{\hat{\Gamma}})\\ \dot{\mathbf{\hat{\chi}}}=\mathbf{K}\mathbf{W}^{T}(\mathbf{\Gamma}-\mathbf{ \hat{\Gamma}})+\mathbf{K}\mathbf{W}^{T}(\mathbf{\tau}-\mathbf{W}\mathbf{\hat{ \chi}})\\ \dot{\mathbf{K}}=-2\mathbf{K}\mathbf{W}^{T}\mathbf{W}\mathbf{K}+\alpha\mathbf{K }\end{array}\right. \tag{17}\]  
where  
* \(\mathbf{\Gamma}=\int\boldsymbol{\tau}dt\in\mathbb{R}^{3\times 1}\) is the integral of \(\boldsymbol{\tau}\).
* \(\mathbf{\hat{\chi}}\in\mathbb{R}^{p\times 1}\) and \(\mathbf{\hat{\Gamma}}\in\mathbb{R}^{3\times 1}\) are, respectively, the estimates of the vectors \(\boldsymbol{\chi}\) and \(\mathbf{\Gamma}\).
* \(\mathbf{K}\in\mathbb{R}^{p\times p}\) is a bounded symmetric positive definite matrix.
* \(\alpha>0\) is a positive gain.  
The following assumptions are required for the estimator:  
* The vector \(\boldsymbol{\tau}\) is measurable.
* \(\mathbf{W}\) satisfies the persistent excitation condition [39].
* \(\boldsymbol{\chi}\) is a constant vector.  
Let us take into account the new dynamic model for our LLRR expressed as (15) and (16). Moreover, the observer reported by [PERSON] et al. [36] is defined as (17). According to [PERSON] et al. [36], if we define the vector estimation error as \(\mathbf{\hat{\Gamma}}=\mathbf{\Gamma}-\mathbf{\hat{\Gamma}}\) and the vector parameters error as \(\mathbf{\hat{\chi}}=\boldsymbol{\chi}-\mathbf{\hat{\chi}}\), then we can compute their dynamics as  
\[\left\{\begin{array}{l}\dot{\mathbf{\Gamma}}=\mathbf{W}\mathbf{\hat{\chi}}- \frac{\alpha}{2}\mathbf{\hat{\Gamma}}\\ \dot{\mathbf{\hat{\chi}}}=-\mathbf{K}\mathbf{W}^{T}\mathbf{\hat{\Gamma}}- \mathbf{K}\mathbf{W}^{T}\mathbf{W}\mathbf{\hat{\chi}}\end{array}\right. \tag{18}\]  
Please refer to [PERSON] et al. [36] for detailed stability analysis. In their document, the Lyapunov-like function decreases exponentially to zero, and the errors \(\mathbf{\hat{\Gamma}}\) and \(\mathbf{\hat{\chi}}\) tend exponentially to zero since \(\mathbf{W}\) satisfies the persistent excitation condition. To ensure the stability of the online parameter estimation, we can define the LMI [36] as:  
\[\mathbf{H}=\left[\begin{array}{cc}\gamma\mathbf{I}_{p}&-\frac{\gamma}{2} \mathbf{I}_{p}\\ -\frac{\gamma}{2}\mathbf{I}_{p}&\frac{\alpha}{2}\mathbf{K}^{-1}\end{array} \right]>0 \tag{19}\]  
where \(\boldsymbol{\gamma}\in\mathbb{R}\) is a positive gain, \(\mathbf{K}\in\mathbb{R}^{p\times p}\) is a symmetric positive definite matrix, and \(\mathbf{I}_{p}\in\mathbb{R}^{p\times p}\) is the identity matrix. Therefore, the design for the calibration phase reduces to finding the condition on \(\mathbf{K}\), \(\gamma\), and \(\alpha\) such that \(\mathbf{H}>0\).