# Vibration Isolation of a Surveillance System Equipped in a Drone with Mode Decoupling
## 2 Theories for System Identification
### Levenberg-Marquardt Algorithm  
The equivalent mass, stiffness, and damping in a multi-DOF model constitute the response characteristics of the system of interest. Therefore, Section 2.1 describes the EOM of a multi-DOF system. The rest of the work included executing the system identification using experimental data, which was processed using the information provided in Section 3 to estimate parameters in Equations (12)-(14). The parameter estimation problem can be reformulated as an objective function minimization problem, called the least squares approximation [16]. The objective function in the least squares approximation is expressed as follows:  
\[E(\beta)=\sum_{i}^{n}r_{i}^{2}(\beta)=\sum_{i}^{n}[y(t_{i})-F(t_{i};\beta)]^{2}, \tag{15}\]  
where \(r_{i}\) is a residual between model response, \(F(t_{i};\beta)\), and observed value, i.e., measured output, \(y(t_{i})\); \((t_{i},y_{i}),\ i=1,\ldots n\), are observation points and \(\beta=[\beta_{1},\ldots,\beta_{m}]\) are model parameters. If the model is nonlinear in the parameters, its parameters are estimated using iterative minimization. Specifically, starting with initial parameters, \(\beta_{\theta}\), \(\beta\) is updated repeatedly to minimize the error function, \(E(\beta)\).  
In this study, the Levenberg-Marquardt algorithm [16; 17] was selected and applied to secure convergence by mutually related coupling terms for independent axes, even though the target system is simplified to be linear. This is because the Levenberg-Marquardt algorithm is highly likely to find the cost-minimized value even though the initial value is far from the optimal value and it can be applied relatively easily regardless of a linear or nonlinear target system. Furthermore, it converges on the cost minimized value faster than other iterative methods because the gradient using the Jacobian matrix and damping factor accelerating the convergence speed and increasing the accuracy were used. The Levenberg-Marquardt method is expressed as follows:  
\[\beta_{k+1}=\beta_{k}-(J_{r}^{T}I_{r}+\mu_{k}diag\big{(}J_{r}^{T}I_{r}\big{)})^{- 1}J_{r}^{T}r(\beta_{k}),\;k\geq 0, \tag{16}\]  
\[J_{r}(\beta)=\left(\begin{array}{ccc}\frac{\partial r_{1}(\beta)}{\partial \beta_{1}}&\dots&\frac{\partial r_{1}(\beta)}{\partial\beta_{m}}\\ \vdots&\ddots&\vdots\\ \frac{\partial r_{n}(\beta)}{\partial\beta_{1}}&\dots&\frac{\partial r_{n}( \beta)}{\partial\beta_{m}}\end{array}\right),\mathbf{r}(\beta)=\left[\begin{array} []{c}r_{1}(\beta)\\ r_{2}(\beta)\\ \vdots\\ r_{n}(\beta)\end{array}\right]=\left[\begin{array}{c}y_{1}-F(x_{1},\beta) \\ y_{2}-F(x_{2},\beta)\\ \vdots\\ y_{n}-F(x_{n},\beta)\end{array}\right] \tag{17}\]  
where \(J_{r}(\beta)\) is the \(n\times m\) Jacobian matrix that represents the local sensitivity of the residual function, \(\mathbf{r}\), to the variation in the parameter, \(\beta\); \(\mu_{k}\) is the non-negative damping factor to be adjusted at each iteration. The damping factor, \(\mu_{k}\), is updated until the error function \(E(\beta)\) converges to a predefined threshold, and the updating process is as follows:  
\[\mu_{k+1}=\left\{\begin{array}{cc}\frac{\mu_{k}}{\alpha},&E(\beta_{k+1})<E( \beta)\\ \alpha\mu_{k},&E(\beta_{k+1})>E(\beta)\end{array}\right., \tag{18}\]  
where \(\alpha\) is an arbitrary positive number that regulates the degree of update. Note that a trade-off between the robustness of the gradient descent and the convergence speed exists based on \(\alpha\); hence, \(\alpha\) should be larger than one. This study set the value of \(\alpha\) to be ten based on the recommendation from [16].  
Additionally, future work includes performance validation with actual field experiments and application of the present method to the nonlinear design problem, such as dynamic amplitude- and preload- dependence, which originates from the viscoelastic material type mount used in this study.