## 3 Optimal Design of Compliant Elements
### Active Constraint (\(\mu^{*}>0\))  
When the design of minimum energy consumption lies outside the feasible region, the inequality is active and therefore binds the solution to the constraint:  
\[\theta_{j}^{*(2)} = \frac{I_{\tau_{j}}}{TK_{r,j}k_{j}^{L}}+\frac{I_{q_{j}}}{T} \tag{47}\] \[k_{j}^{*(2)} = k_{j}^{L}\] (48) \[\mu^{*(2)} = \frac{2a_{j}}{T}\left(K_{r,j}\left(TI_{q_{j}^{2}}-I_{q_{j}}^{2} \right)k_{j}^{L}+TI_{\tau_{q_{j}}}-I_{\tau_{j}}I_{q_{j}}\right) \tag{49}\]  
In order to verify that the solution found is a minimum, the Hessian of the Lagrangian function at the stationary point is evaluated:  
\[\mathbf{H}=\
abla_{\mathbf{x}_{j}\mathbf{x}_{j}}^{2}\mathcal{L}(\mathbf{x}_{ j}^{*(2)},\mu_{j}^{*(2)})=\begin{bmatrix}2a_{j}TK_{r,j}k_{j}^{L}{}^{2}&2a_{j}I_{ \tau_{j}}\\ 2a_{j}I_{\tau_{j}}&2a_{j}\left(\frac{I_{\tau_{j}}^{2}}{TK_{r,j}k_{j}^{L}{}^{ 2}}+\frac{K_{r,j}}{T}\left(TI_{q_{j}^{2}}-I_{q_{j}}^{2}\right)\right)\end{bmatrix} \tag{50}\]  
\(\mathbf{H}\) is positive definite if the determinant of all the upper-left sub-matrices are positive, i.e., if the following conditions hold:  
\[\mathbf{1 st\,condition:} \det(\mathbf{H}_{11})=2a_{j}TK_{r,j}k_{j}^{L}{}^{2}>0 \tag{51}\] \[\mathbf{2 nd\,condition:} \det(\mathbf{H})=\left(2a_{j}K_{r,j}k_{j}^{L}\right)^{2}\left( TI_{q_{j}^{2}}-I_{q_{j}}^{2}\right)>0 \tag{52}\]  
The first condition in Equation (51) is always satisfied as all the coefficients are positive. The second condition in Equation (52) is satisfied as the first factor is squared and the second one is always positive as demonstrated in Section 3.1. Therefore, the solution in Equations (47)-(49) is a minimum point of the Lagrangian function.  
Determination of the Lower Bounds \(k_{j}^{L}\)  
Although the solution outlined in Section 3.2 is a minimum of the bounded objective function, this does not ensure a reduction of the energy consumption with respect to the case without springs, as it does not correspond to the global minimum of the unconstrained energy consumption function (Equation (13)). In order for the addition of the springs to also be advantageous in this case, the value of the stiffness bound \(k_{j}^{L}\) must be properly set. Let us evaluate the energy consumption when the design variables take the values \(\boldsymbol{\theta}_{j}^{*(2)}\) and \(k_{j}^{*(2)}\) (Equations (47) and (48)):  
\[E_{j}(\mathbf{x}_{j}^{*(2)})=E_{\textit{ws}_{j}}+\underbrace{\frac{a_{j}}{TK_{r,j}} \left(K_{r,j}^{2}\left(TI_{q_{j}^{2}}-I_{q_{j}}^{2}\right)k_{k}^{L}{}^{2}+2K_{ r,j}\left(TI_{\tau_{q_{j}}}-I_{\tau_{j}}I_{q_{j}}\right)k_{j}^{L}-I_{\tau_{j}}^{2} \right)}_{E_{s_{j}}} \tag{53}\]  
In Equation (53), \(E_{\textit{ws}_{j}}\) is the energy consumption of the system without springs; therefore, in order to reduce the energy consumption \(E_{j}\), the term \(E_{s_{j}}\) must be negative. By solving for \(k_{j}^{L}\) the inequality \(E_{s_{j}}<0\) it is possible to find the range of values of the stiffness bounds allowing for energy reduction:  
\[0<k_{j}^{L}<\frac{I_{\tau_{j}}I_{q_{j}}-TI_{\tau_{qj}}+\sqrt{\left(TI_{\tau_{ qj}}-I_{\tau_{j}}I_{q_{j}}\right)^{2}+I_{\tau_{j}}^{2}\left(TI_{q_{j}^{2}}-I_{q_{ j}}^{2}\right)}}{K_{r,j}\left(TI_{q_{j}^{2}}-I_{q_{j}^{2}}^{2}\right)} \tag{54}\]The selection of the most suitable value of \(k_{j}^{L}\) inside the range in Equation (54) is to be chosen by the user depending on the specific system analyzed. As general guidelines, such selection is based on a trade-off between energy reduction and technical feasibility of the solution. For example, very small values of \(k_{j}^{L}\) (\(k_{j}^{L}\to 0\)) lead to very large values for the mounting angle (\(\mathbf{\theta}_{j}^{*(2)}\rightarrow\infty\), see Equation (47)), which may be physically impossible.