## 3 Method Description
### Introduction of the Robustness Condition  
Due to the strong influence of time delays, satisfaction of Equations (7) and (8), does not guarantee that the set of desired closed-loop poles \(\tilde{p}_{1},...,\tilde{p}_{N_{p}}\) are \"primary roots\". The search for a solution with stability and performance/robustness certificate must be carried out with focus on Equation (10). Frequency domain techniques can successfully deal with rational and transcending transfer functions, including those resulting from time delay in linear systems [28; 29]. In particular, the Nyquist stability criterion [30], a cornerstone of classical control theory, can be straightforwardly applied to the characteristic Equation (7), in conjunction with the robustness margins approach [29; 31] by taking the loop-gain transfer function as  
\[L(s)=-\big{(}\mathbf{g}e^{-s\tau_{g}}+s\mathbf{f}e^{-s\tau_{f}}\big{)}^{T} \mathbf{H}(s)\mathbf{B}. \tag{11}\]  
A search strategy based on the maximum peak of the sensitivity function (\(M_{s}\)) associated with this loop-gain can offer a trade-off between robustness and performance for the closed-loop controlled system [29]. As the system parameters are usually uncertain, robustness is a significant issue in the controller design [32]. The peak \(M_{s}\) is related with a disk with center in the critical abscissa of instability \((-1,0)\) and radius equal to \(M_{s}^{-1}\). This disk establishes an acceptable distance from the Nyquist curve of \(L(s)\) from the point \((-1,0)\). The Nyquist criterion states that, for an open-loop system with all poles on the left half-plane, the closed-loop system will be stable if the Nyquist curve of \(L(s)\) does not encircle the point \((-1,0)\). The larger the disk radius, the more robust the system is with respect to perturbations on the nominal loop-gain \(L(s)\). Then, the design problem can be formulated as that of computing the feedback gains of the parametrized family, \(\mathbf{k_{r}}\) in Equation (7), with a simultaneous guarantee that the Nyquist curve of the loop-gain lies in a safe distance of \(M_{s}^{-1}\) to the critical point. This problem can be approached through the following minimization formulation:  
\[\min_{\mathbf{k_{r}}} \min_{\omega_{i}}\Big{(} \Big{|}-(\mathbf{g}e^{-j\omega_{i}\tau_{g}}+j\omega_{i}\mathbf{f}e^{-j \omega_{i}\tau_{f}})^{T}\mathbf{H}(j\omega_{i})\mathbf{B}+1\Big{|}-M_{s}^{-1} \Big{)}^{2}\] (12) s.t. \[\begin{bmatrix}\mathbf{f}\\ \mathbf{g}\end{bmatrix}=\mathbf{k_{0}}+\mathbf{V}\mathbf{k_{r}}\] \[\text{Re}\Big{\{}-(\mathbf{g}e^{-j\omega_{i}\tau_{g}}+j\omega_{i }\mathbf{f}e^{-j\omega_{i}\tau_{f}})^{T}\mathbf{H}(j\omega_{i})\mathbf{B} \Big{\}}\geq-1+M_{s}^{-1}\ \ \ \forall\ \ \ \omega_{i}:\text{Im}\{L(j\omega_{i})\}=0.\]  
The second constraint in Equation (12) assures that every cross point on the real axis lies on the right of the \(M_{s}\) disk, avoiding then encirclement of the critical point \((-1,0)\). Notice that the frequency \(\omega_{i}\) must belong to a frequency range \([\omega_{min},\omega_{max}]\), suitable to a representative Nyquist plot. This range can be, as an instance, the same used in the experimental identification of the system acceptance. Moreover, for underactuated systems, only the partial information \(\mathbf{H}(j\omega)\mathbf{B}\) must be known to compute the gains.