## 2 Materials and Methods
### Parameter Identification and Model Simulation  
Define \(y\) as the column (bi-dimensional) vector whose components are the measured variables \(\ddot{\theta}\) and \(\dot{\varphi}\). According to the model shown in Equations (1)-(5), the estimate \(\dot{y}\) of \(y\), which is instrumental to obtain a converging estimate \(\hat{\beta}\) (with components \(\hat{\beta}_{i}\), \(i=1,2,\ldots,8\)) of \(\beta\), is defined as  
\[\hat{y} = M^{-1}(\varphi)\Bigg{(}S(\theta,\dot{\theta},\varphi,\dot{ \varphi})+\Gamma(\theta,\dot{\theta},\varphi,\dot{\varphi},t)\hat{\beta}+ \left[ \begin{array}{c}0\\ \exp\Bigl{(}-\frac{180}{\pi}(\varphi-\varphi_{0})\Bigr{)}\end{array} \right] \tag{6}\] \[+\Bigg{[} \begin{array}{c}\tau_{\text{s}}(\theta,t)\\ \tau_{e}(\theta,\varphi,t)\end{array} \Big{]}\Bigg{)}.\]  
Now, due to the absence of any EMG activity information for upper limb muscles, muscle torques cannot be measured in \(\Gamma(\cdot)\) of (6). Consequently, according to well-known findings that indicate impulsive EMG activity for upper limb muscles during walking gaits, muscle torques are approximately set equal to the truncated Fourier series (up to the fifth order) of impulsive test signals with the same frequency of arm oscillation and 10% duty cycle (since no information about the time instants when the signal occurs is available, the parameter estimation algorithm and simulation runs are repeated twelve times by considering different uniformly distributed delays within 0 and 2\(\pi\): delays that lead to the most satisfactory set of Mean Squared Error (MSE), Root Mean Squared Error (RMSE), and Mean Absolute Error (MAE) are definitely selected for each subject) [see Figure 3] and assumed to approximately match \(\tau_{\text{CEs}}(\cdot)\) and \(\tau_{\text{CEe}}(\cdot)\). Assume that \(\tau_{\text{CEs}}(\cdot)\) and \(\tau_{\text{CEe}}(\cdot)\) are perfectly matched. Then, the estimation error \(\hat{y}=y-\hat{y}\) from (1) and (6) satisfies  
\[\hat{y} = M^{-1}(\varphi)\Gamma(\theta,\dot{\theta},\varphi,\dot{\varphi},t) \hat{\beta} \tag{7}\]  
in terms of \(\hat{\beta}=\beta-\hat{\beta}\). The \(\beta\)-identifier is consequently designed as (\(\Lambda\) is a symmetric positive definite matrix)  
\[\dot{\hat{\beta}} = \Lambda\Gamma(\theta,\dot{\theta},\varphi,\dot{\varphi},t)^{\top} M(\varphi)\hat{y}, \tag{8}\]which gives  
\[\hat{\beta} = -\Lambda\Gamma(\theta,\dot{\theta},\varphi,\dot{\varphi},t)^{\top} \Gamma(\theta,\dot{\theta},\varphi,\dot{\varphi},t)\hat{\beta}. \tag{9}\]  
The error system thus complies with the one studied by [PERSON] and associates [17]. Exponential convergence to zero of \(\hat{\beta}\)--and thus robustness, along with exponential convergence to zero of \(\hat{y}\)--is proved under the persistency of excitation of the regressor that is related to an identifiability property [typical angle profiles and muscle forces filling the regressor \(\Gamma(\cdot)\) (see the figures below) characterize the persistency of excitation] via the non-anticipative Lyapunov function proposed by [PERSON] and associates [17]. In particular, the related Theorem 1 of paper [17], reported here for the sake of exhaustivity, reads  
**Theorem 1** ([17]).