## 5 Proposed Methods
### Estimation Phase Design
\begin{split}\dot{\hat{\mathbf{\Gamma}}}&=\mathbf{W}\hat{ \mathbf{\chi}}+\frac{\alpha}{2}(\mathbf{\Gamma}-\hat{\mathbf{\Gamma}})\\ \dot{\hat{\mathbf{\chi}}}&=\mathbf{K}\mathbf{W}^{T}(\mathbf{\Gamma}- \hat{\mathbf{\Gamma}})+\mathbf{K}\mathbf{W}^{T}(\mathbf{\tau}-\mathbf{W}\hat{\mathbf{\chi}})\\ \mathbf{\hat{\mathbf{\kappa}}}&=-2\mathbf{K}\mathbf{W}^{T}\mathbf{W}\mathbf{K} +\alpha\mathbf{K}\end{split} \] (22)
* Torque estimation phase \(t\geq t_{i}\) \[ \begin{split}\dot{\mathbf{z}}&=-\mathbf{L}(\mathbf{q},\hat{ \mathbf{q}})\mathbf{z}+\mathbf{L}(\mathbf{q},\hat{\mathbf{q}})\{\hat{\mathbf{C}}(\hat{\mathbf{\chi}},\mathbf{q}, \hat{\mathbf{q}})+\hat{\mathbf{G}}(\hat{\mathbf{\chi}},\mathbf{q})-\mathbf{\tau}_{a}-\mathbf{p}(\mathbf{q},\hat{\mathbf{q}})\\ \hat{\mathbf{\tau}}_{d}&=\mathbf{z}+\mathbf{p}(\mathbf{q},\hat{\mathbf{q }})\\ \frac{d}{dt}\mathbf{p}(\mathbf{q},\hat{\mathbf{q}})&=\mathbf{L}(\mathbf{q },\hat{\mathbf{q}})\hat{\mathbf{M}}(\hat{\mathbf{\chi}},\mathbf{q})\hat{\mathbf{q}}\\ \mathbf{L}(\mathbf{q})&=\mathbf{X}^{-1}\hat{\mathbf{M}}^{-1}(\hat{ \mathbf{\chi}},\mathbf{q})\\ \mathbf{p}(\hat{\mathbf{q}})&=\mathbf{X}^{-1}\hat{\mathbf{q}}\\ \hat{\mathbf{\tau}}_{int}&=\hat{\mathbf{\tau}}_{d}-\mathbf{\tau}_ {a}-\hat{\mathbf{\tau}}_{fri}\end{split} \] (23)  
We can observe that the INDO uses the NDO-based reported by [PERSON] et al. [12] in terms of \(\hat{\mathbf{\chi}}\). Therefore, the output of the calibration phase is used as an input for the estimation phase. This method uses the online parameter estimation to reduce BSIPsuncertainties. Subsequently, within the estimation phase, the NDO-based human-robot interaction torque uses the estimated parameters under the assumption that the BSIPs uncertainties have been reduced. Let us now cite the following Theorem reported by [PERSON] et al. [34]. This Theorem is helpful since it presents sufficient conditions for asymptotic and exponential disturbance tracking when the robot shows fast-varying disturbances. Moreover, it evidences asymptotic and exponential convergence to estimate the NDO-based human-robot interaction torque estimation method. **Theorem 1** ([PERSON] et al. [34]). : _Consider the robotic manipulator subject to disturbances described by (1). The disturbance observer is given in (6) with the disturbance observer gain matrix \(\mathbf{L}(\mathbf{q})\) defined in (7) and the disturbance observer auxiliary vector \(\mathbf{p}(\dot{\mathbf{q}})\) defined in (8). The disturbance tracking error \(\boldsymbol{\tau}_{d}\) is globally uniformly ultimately bounded if:_  
1. _The matrix_ \(\mathbf{X}\) _is invertible,_
2. _There exists a positive definite and symmetric matrix_ \(\boldsymbol{\Gamma}\) _such that_ \[\mathbf{X}+\mathbf{X}^{T}-\mathbf{X}^{T}\dot{\mathbf{M}}(\mathbf{q})\mathbf{X }\geq\boldsymbol{\Gamma}.\] (24)
3. _The rate of change of the lumped disturbance is bounded, i.e.,_ \(\exists\kappa>0\) _such that_ \(\|\dot{\boldsymbol{\tau}}_{d}(t)\|\leq\kappa\) _for all_ \(t>0\)_._ _Under the above conditions and for all_ \(\Delta\boldsymbol{\tau}_{d}(0)\in\mathbb{R}^{n}\)_, the tracking error converges with an exponential rate, equal to_ \((1-\mathbf{q})\lambda_{\min}(\boldsymbol{\Gamma})/2\sigma_{2}\|\mathbf{X}\|^ {2}\) _to the ball with radius_ \(2\kappa\sigma_{2}\|\mathbf{X}\|^{2}/\mathbf{q}\lambda_{\min}(\boldsymbol{ \Gamma})\) _where_ \(0<\mathbf{q}<1\)_._  
Figure 2: Block diagrams for the IID. (**a**) Calibration phase and (**b**) torque estimation phase. Where \(\lambda_{\min}(\cdot)\) denotes the minimum eigenvalue of a matrix, the LLRR velocity vector lies in a bounded set, i.e., \(\forall\hat{\mathbf{q}}\in D_{\hat{\mathbf{q}}}=\{\hat{\mathbf{q}}\in R^{n}:\| \hat{\mathbf{q}}\|\leq\|\hat{\mathbf{q}}\|_{\max}\}\), and \(\|\mathbf{M}(\mathbf{q})\|\leq v_{2}(\mathbf{q})\) where \(v_{2}=\sup_{\mathbf{q}\in D_{\hat{\mathbf{q}}}}\{\sigma_{2}(\mathbf{q})\}\). Please refer to [PERSON] et al. [34] for a detailed proof of Theorem 1.