## 3 Results  
### Numerical Analysis  
In order to evaluate the performance and robustness of the nonlinear controller, two numerical simulations were carried out over a relevant simulation environment. These simulations reproduced the physical conditions of a rehabilitation therapy room. The selected controller's gains are  
\[K_{p}=\left[\begin{array}{ccc}18&0&0\\ 0&18&0\\ 0&0&18\end{array}\right]\quad\text{ and }\quad\left[\begin{array}{ccc}80&0&0\\ 0&80&0\\ 0&0&80\end{array}\right].\]  
The first simulation considers the evaluation of the hybrid system under nominal conditions during a step with the swing leg tracking the precomputed trajectory and the support leg locked with a unactuated pivot between the support foot and the floor. The evolution of motion is shown in Figure 8, where it is possible to identify satisfactory trajectory tracking with bonded control signals. The performance of the controller is quantified with the integral square of the error (ISE) and with the integral square of the control signal (ISU), which are defined as follows:  
\[ISE := \int_{0}^{t_{f}}\mathbf{e}^{T}\mathbf{e}\ dt, \tag{22}\] \[ISU := \int_{0}^{t_{f}}\mathbf{u}^{T}\mathbf{u}\ dt, \tag{23}\]  
where \(\mathbf{e}\) is the vector of tracking errors, defined as  
\[\mathbf{e}:=\left[\begin{array}{c}\theta_{1}-\theta_{1}^{*}\\ \theta_{2}-\theta_{2}^{*}\\ \theta_{3}-\theta_{3}^{*}\end{array}\right],\]  
and \(t_{f}\) is the evaluation period of time, which in this case corresponds to three seconds.  
The second simulation test evaluated the controller robustness against parameter variation, which is a typical case in the use of exoskeletons with different patients or with the same patient over an extended period of time. In this case, the nonlinear control was evaluated with variation in the patient's masses. With this purpose, nine simulations were conducted with numerical models of children with masses between 80% and 120% from the nominal mass values. Figure 9 shows the system behavior in each simulation scenario with an envelope that bound all the deviations from the nominal behavior, which confirms the controller robustness against parameter uncertainties. The performance indexes, ISE and ISU, of the evaluations with parameter variation are shown in Figure 10. This figure highlights the deviation of the indexes from the nominal value, which allows for measuring the performance degradation with respect to parameter variation. The worst-case scenario was compared with the nominal behavior in Table 4, where the performance indexes of both simulations, nominal and parameter variation, are registered. The difference between these indexes is irrelevant, resulting in a controller whose performance is not substantially degraded by variations in the children's anthropometric parameters.  
Figure 8: Nonlinear control for trajectory tracking.  
\begin{table}
\begin{tabular}{l c c} \hline \hline
**Index** & **ISE** & **ISU** \\ \hline Nominal conditions & 0.00056 & 36,725 \\ \hline Variation of parameters & 0.00077 & 37,336 \\ \hline \hline \end{tabular}
\end{table}
Table 4: Performance indexes.  
Figure 9: Nonlinear control robustness test against parameter variation. (Nominal behavior: central line, parameters variation: shadow.)  
In order to develop a visual inspection of the motion evolution during walking therapy, a virtual scenario was designed, which reproduced the therapy environment. As shown in Figure 11, the virtual scenario is used to evaluate the exoskeleton during a single-support walking phase. This simulation shows stable walking, with the exoskeleton performing a smooth evolution of motion ([[https://youtu.be/z5](https://youtu.be/z5) kDJD02 Wuk]([https://youtu.be/z5](https://youtu.be/z5) kDJD02 Wuk)).