# Robustness Analysis of the Collective Nonlinear Dynamics of a Periodic Coupled Pendulums Chain
## 2 Mechanical Model
### Uncertainty Propagation  
To propagate uncertainties through the established model, one can use, in a probabilistic framework, stochastic uncertainty propagation methods. Statistical methods, such as the MC method [48] and the LHS method [43], are the most frequently used in the literature and are considered as reference since they permit to achieve a reasonable accuracy. The LHS method consists on generating a succession of deterministic computations \(\left\{A_{m}\left(\xi^{(n)}\right),\ n=1,\ldots,N_{LHS}\right\}\) according to a set of random variables \(\left\{\xi^{(n)}\right\}_{n=1}^{N_{LHS}}\) to approximate the \(m^{th}\) amplitude \(A_{m}\). The LHS method permits to reduce the computing time required by the very time-consuming MC method by partitioning the variability space into regions of equal probability and picking up one sampling point in each region. Nevertheless, it remains computationally unaffordable since the accuracy level is proportional to the number of generated simulations. To overcome this prohibitive computational cost without a significant loss of accuracy, the gPCE is used in this work [44; 45]. The gPCE combines multivariate polynomials and deterministic coefficients. Indeed, it approximates the \(m^{th}\) amplitude \(A_{m}\) using a decomposition, practically truncated by retaining only polynomials terms with degree up to \(p\):  
\[A_{m}=\sum_{j=0}^{p}\left(\hat{A}_{m}\right)_{j}\psi_{j}(\xi)=\hat{A}_{m}^{T} \boldsymbol{\
abla}(\xi);\ P+1=\frac{(d+p)!}{d!p!}, \tag{16}\]  
where \(\left(\hat{A}_{m}\right)_{j}\) are the unknown deterministic coefficients and \(\psi_{j}(\xi)\) the multivariate polynomials of \(d\) independent random variables \(\xi=\left\{\xi_{i}\right\}_{i=1}^{d}\). Solving the gPCE consists on computing the deterministic coefficients \((\hat{A}_{m})_{j}\). To do this, one can use intrusive or non-intrusive approaches. The former implies model modifications. However the latter considers the initial model as a black box. The regression approach is one of the most commonly used non-intrusive methods. In its standard form, it consists in minimizing the difference between the gPCE approximate solution and the exact solution. The latter is a set of deterministic solutions \(\left\{\overline{A}_{m}\left(\mathfrak{x}^{(n)}\right)\right.\), \(n=1,\ldots,M\right\}\) corresponding to \(M\) realizations of random variables \(\Xi=\left\{\mathfrak{x}^{(n)}\right\}_{n=1}^{M}\) forming an experimental design (ED). The approximate solution takes, consequently, the form  
\[\hat{A}_{m}=\left(\boldsymbol{\Psi}^{T}\boldsymbol{\Psi}\right)^{-1} \boldsymbol{\Psi}^{T}\overline{A}_{m}=\boldsymbol{\Psi}^{+}\overline{A}_{m}, \tag{17}\]  
where \(\boldsymbol{\Psi}_{nj}\equiv\left(\psi_{j}\left(\mathfrak{x}^{(n)}\right)\right)\) is called the data matrix and \(\boldsymbol{\Psi}^{+}\) is its Moore-Penrose \(j=0,\ldots,P\) pseudo-inverse. In order to ensure the numerical stability of the regression approximation, the matrix \(\left(\boldsymbol{\Psi}^{T}\boldsymbol{\Psi}\right)\) must be well-conditioned. Therefore, the ED should be well selected and have the size \(M\geq P+1\). The ED selection technique used in this work is based on two conditions:  
1. classification of all possible combinations of the roots of the Hermite polynomial of degree \(p+1\) so as to maximize the variable [49; 50; 51]: \[\zeta_{M}\left(\mathfrak{x}^{(n)}\right)=2\pi^{-d/2}exp\left(-\frac{\parallel \mathfrak{x}^{(n)}\parallel^{2}}{2}\right).\] (18)
2. minimization of the number: \[\kappa=\parallel\left(\boldsymbol{\Psi}^{T}\boldsymbol{\Psi}\right)^{-1} \parallel\cdot\parallel\boldsymbol{\Psi}^{T}\boldsymbol{\Psi}\parallel,\] (19) where \(\parallel.\parallel\) is the 1-norm of the matrix, in order to ensure that the invertible matrix \(\boldsymbol{\Psi}^{T}\boldsymbol{\Psi}\) is well-conditioned [51; 52]. A number \(M\) of roots' combinations, which verify the conditions in Equations (18) and (19), create then the ED.