# Adaptive Robust Admittance Control of Robots Using Duality Principle-Based Impedance Selection
## 3 Determination of the Desired Impedance Profiles  
The determination of the desired robot impedance is closely related to environmental impedance. This section presents an adaptive law with projection modification to estimate the environmental stiffness and damping. Then, the desired stiffness and damping are determined for robots based on the duality principle.  
From (3), the interactive force can be presented in the following expression:  
\[f_{ei}=\phi_{i}^{T}w_{i} \tag{7}\]  
where \(\phi_{i}\) is a regression vector and \(w_{i}\) is an unknown vector defined by  
\[\phi_{i}=[x_{i},\dot{x}_{i}]^{T}, \tag{8}\] \[w_{i}=[-k_{ei},-d_{ei}]^{T} \tag{9}\]  
with \(-k_{ei}\in\Omega_{1}=\{\xi\in R:-k_{d1}\leq\xi\leq-k_{d2}\}\), \(-d_{ei}\in\Omega_{2}=\{\xi\in R:-d_{d1}\leq\xi\leq-d_{d2}\}\), and \(w_{i}\in\Omega=\{w_{i}:-k_{ei}\in\Omega_{1},-d_{ei}\in\Omega_{2}\}\).  
\(\dot{w}_{i}=[\dot{w}_{i1},\dot{w}_{i2}]^{T}=[-\dot{k}_{ei},-\dot{d}_{ei}]^{T}\) denotes the estimators of \(w_{i}\). Then, the estimator \(\hat{f}_{ei}\) of \(f_{ei}\) can be presented as  
\[\hat{f}_{ei}=\phi_{i}^{T}\dot{w}_{i} \tag{10}\]and the estimation error \(\tilde{f}_{ei}=f_{ei}-\hat{f}_{ei}\) is given by  
\[\tilde{f}_{ei}=\phi_{i}^{T}\tilde{w}_{i} \tag{11}\]  
where \(\tilde{w}_{i}=w_{i}-\tilde{w}_{i}\).  
Design the following update law:  
\[\hat{w}_{i}=P_{i}(\beta_{i}\phi_{i}\tilde{f}_{ei}) \tag{12}\]  
where \(\beta_{i}\) is a positive constant gain and \(P_{i}(\cdot)\) is the function defined by  
\[P_{i}(\xi)=[\text{proj}_{\Omega_{1}}(\xi_{1}),\text{proj}_{\Omega_{2}}(\xi_{2}) ]^{T},\ \xi=[\xi_{1},\xi_{2}]^{T}\in R^{2} \tag{13}\]  
with \(\text{proj}_{\Omega_{j}}(\xi_{j}),j=1,2\) being the projection function defined by  
\[\text{proj}_{\Omega_{j}}(\xi_{j})=\begin{cases}\xi_{j},&\text{if }\hat{w}_{ij} \in\Omega_{j}\ \&\ \xi_{j}\
otin\partial\Omega_{j}\\ 0,&\text{Otherwise}.\end{cases} \tag{14}\]  
Theorem 1: _By the adaptive update law in (12), the estimation error \(\tilde{w}_{i}\) is uniformly ultimately bounded (UIUB)._  
Proof: Consider the following candidate Lyapunov function:  
\[V_{0}=\frac{1}{2\beta_{i}}\tilde{w}_{i}^{T}\tilde{w}_{i}. \tag{15}\]  
Substituting (12) into the time derivative of \(V_{0}\) yields  
\[\dot{V}_{0}=-\frac{1}{\beta_{i}}\tilde{w}_{i}^{T}P_{i}(\beta_{i}\phi_{i}\tilde {f}_{ei}) \tag{16}\]  
The projection function in (13) guarantees \(\tilde{w}_{i}(t)\in\Omega\) for \(t\geq t_{0}\) if \(\tilde{w}_{i}(t_{0})\in\Omega\). From (11), \(-\frac{1}{\beta_{i}}\tilde{w}_{i}^{T}\beta_{i}\phi_{i}\tilde{f}_{ei}=-\tilde{w }_{i}^{T}\phi_{i}\phi_{i}^{T}\tilde{w}_{i}\). Applying these results to (21) yields  
\[\dot{V}_{0}\leq-\tilde{w}_{i}^{T}\phi_{i}\phi_{i}^{T}\tilde{w}_{i} \tag{17}\]  
from which we can conclude that the estimation error \(\tilde{w}_{i}\) is UUB.  
Based on the duality principle [29], the robot impedance is chosen as the reciprocal of the environmental impedance in the following form:  
\[k_{di}^{*}=\alpha_{1}/k_{ei}. \tag{18}\]  
where \(\alpha_{1}>0\).  
Substituting (3) and (5) into the desired impedance dynamics (6) yields  
\[M_{d}\tilde{e}_{d}+(D_{d}+D_{e})\dot{e}_{d}+(K_{d}+K_{e})e_{d}=0. \tag{19}\]  
To obtain a good performance of \(e_{d}\) in (19), based on the automatic control theory, one can construct the following optimal damping:  
\[d_{di}^{*}=2\varsigma_{i}\sqrt{m_{di}(k_{di}^{*}+k_{ei})}-d_{ei} \tag{20}\]  
where \(\varsigma_{i}\) denotes the damping ratio.  
By substituting the estimated environmental stiffness \(\hat{k}_{ei}\) into (19), we can obtain the following adaptive optimal stiffness:  
\[k_{di}=\alpha_{1}/\hat{k}_{ei}. \tag{21}\]  
Based on (20) and (21), it is convenient for us to design the following desired damping:  
\[d_{di}=2\varsigma_{i}\sqrt{m_{di}(k_{di}+\hat{k}_{ei})}-\hat{d}_{ei}. \tag{22}\]