## 2 Materials and Methods
### Parameter Identification and Model Simulation
: _Consider system_  
\[\dot{x}(t) = -\Lambda\Phi(t)\Phi^{\mathrm{T}}(t)x(t)+R(t,x(t),w(t)),\]  
_under the following assumptions: (i) \(\Lambda\) is a symmetric positive definite constant matrix belonging to the set \(\mathcal{M}(n,n,\mathbb{R})\) of real square matrices with \(n\) rows and \(n\) columns; (ii) the regressor matrix \(\Phi(\cdot)\in\mathcal{M}(n,m,\mathbb{R})\) is available, continuous, uniformly bounded over \([0,+\infty)\) (with norm\(\|\cdot\|_{2}\)-bound \(\Phi_{\mathcal{M}}\)) and persistently exciting (i.e., there exist (known) positive reals \(c_{p}\) and \(T_{p}\) such that the condition \([\mathbb{I}\in\mathcal{M}(n,n,\mathbb{R})\) is the identity matrix]_  
\[\int_{t}^{t+T_{p}}\Phi(\tau)\Phi^{\mathrm{T}}(\tau)\mathrm{d}\tau \geq c_{p}\mathbb{I},\ \ \ \forall\ t\geq 0 \tag{10}\]  
_holds); (iii) the smooth vector function \(R\) -- uniformly bounded in \(t\) over \([0,+\infty)\) for any bounded \(x(t)\) and \(w(t)\) over \([0,+\infty)\)-- is such that_  
\[\|R(t,x,w)\| \leq \|w\|(b_{1}\|x\|+b_{2}g_{x,ml}(t,x)+b_{3})+h_{x,ml}(t,x),\]  
_in terms of suitable positive reals \(b_{1}\), \(b_{2}\), \(b_{3}\in\mathbb{R}^{+}\) and \(g_{x,ml}(t,x)\), \(h_{x,ml}(t,x)\) belonging to the set of functions \(\Xi(t,x)\) satisfying_  
\[\lim_{\|x\|-0}\sup_{t\geq 0}\frac{\Xi(t,x)}{\|x\|} = 0;\]  
_(iv) \(w(t)\) is a sufficiently small disturbance vector (possibly converging to zero for \(t\) tending to \(+\infty\)). Its state vector \(x(t)\) (i) is locally ultimately bounded by a bound proportional to the (sufficiently small) upper bound on the \(w(t)\)-perturbation term; (ii) locally exponentially converges to zero for \(w(t)\) exponentially tending to \(0\). The related (strict) quadratic Lyapunov function reads_  
\[\mathcal{W} = \frac{1}{2}x^{\mathrm{T}}\Big{[}2Q(\cdot)+(1+\mathrm{sgn}(n-1) \gamma_{L})\Lambda^{-1}\Big{]}x, \tag{11}\]  
_where \(Q(t)\) is the solution to the matrix-differential equation_  
\[\dot{Q}(t) = -\theta_{c}Q(t)+\Phi(t)\Phi^{\mathrm{T}}(t), \tag{12}\]  
_involving the symmetric positive definite matrix \(Q(0)\) (with sufficiently large minimum eigenvalue) and the positive real \(\theta_{c}\); \(\gamma_{L}\) is a sufficiently large positive real; \(\mathrm{sgn}(\dot{\varphi})\) is the signum function satisfying \(\mathrm{sgn}(\dot{\varphi})=0\) for \(\dot{\varphi}\leq 0\), \(\mathrm{sgn}(\dot{\varphi})=1\) for \(\dot{\varphi}>0\)._  
The previous theorem shows that even in the case in which \(\tau_{\mathrm{CEs}}(\cdot)\) and \(\tau_{\mathrm{CEe}}(\cdot)\) are not perfectly matched, ultimate boundedness might be achieved in terms of a bound proportional to the signal mismatch. Now, to improve parameter convergence over a finite time horizon, the estimation process is carried out twice, and initial conditions for the estimates at the second run are set equal to the values obtained at the end of the first run (zero initial values for the estimates of the 3 rd, 4 th, 7 th, 8 th components of \(\beta\) and \(-5\) for the other ones are chosen at the beginning of the first run). Finally, the gain matrix \(\Lambda\) is set equal to \(\text{diag}[3.5,0.5,0.5,5,3.5,0.5,0.5]\) for HS1-HS4 and to \(\text{diag}[7,0.5,0.5,2.5,7,0.5,0.5]\) for P1-P3. Estimated parameters are then employed for model simulation, and joint angles are computed by solving the related differential equations of the model through the explicit Runge-Kutta integration method. Joint shoulder and elbow angles coming from simulations are then compared to those measured by IMUs; MSE, and RMSE and MAE are then computed to quantify discrepancies between real and simulated data while assessing the ability of the identified model to reproduce the real movement.