## 4 Experiment Verification
### Control System  
In the initial position \(x_{0}=[0\:0\:0.48]^{T}\) and \(\dot{x}_{0}=[0\:1.0\:0]^{T}\) of the flight simulation platform, we calculated the dynamics equation parameters of the system in Equation (12) according to the geometrical and inertial parameters in Table 1.  
\[A=\left[ \begin{array}{cc}0_{6\times 6}&I_{6\times 6}\\ 0_{6\times 6}&-M_{0}^{-1}C_{0}\end{array} \right]\qquad B=\left[ \begin{array}{ ccc}0_{6\times 6}\\ M_{0}^{-1}\end{array} \right]\]  
\[M_{0}=\left[ \begin{array}{cccc}39.1978&-0.0000&0.0000&0.0000&0.2242&-0.0000\\ -0.0000&39.1978&-0.0000&-0.2242&0.0000&-0.0000\\ 0.0000&-0.0000&33.8082&0.0000&0.0000&-0.0000\\ -0.0000&-0.2242&0.0000&1.7788&-0.0000&-0.0000\\ 0.2242&0.0000&-0.0000&-0.0000&1.7788&0.0000\\ -0.0000&-0.0000&-0.0000&-0.0000&0.0000&4.6912\end{array} \right]\]  
\[C_{0}=\left[ \begin{array}{cccc}0.000&0.6085&0.0000&0.6912&-0.0000&0.5702\\ 0.6085&-0.0000&-11.4379&0.0000&-0.6912&-0.0000\\ 0.0000&4.5986&0.0000&0.2422&-0.0000&0.0000\\ 0.6909&0.0000&-0.2422&-0.0000&-0.0610&0.0000\\ 0.0000&-0.6909&-0.0000&0.0610&0.0000&-0.0000\\ -0.0000&-0.0000&0.0000&-0.0000&1.1043&-0.0000\end{array} \right]\]  
\[G_{0}=\left[ \begin{array}{cccc}0&0&0&329.5711&0&0&0&0\end{array} \right]^{T} \tag{31}\]  
where \(M_{0},\ C_{0},\) and \(G_{0}\) denote the initial date of \(M,\ C,\) and \(G.\)  
According to the 6-DOF Stewart-Gough flight simulation platform dynamic equation shown in Equation (12), we chose the following differential equation as the reference model:  
\[\ddot{X}+2\xi\omega_{n}\dot{X}+\omega_{n}^{2}=\omega_{n}^{2} \tag{32}\]  
where \(\xi\in R\) denotes the damping ratio and \(\omega_{n}\) denotes the undamped natural frequency. The state equation of the reference model is:  
\[\dot{x}(t)=A_{m}x(t)+bk_{g}u_{m}(t) \tag{33}\]  
where  
\[A_{m}=\left[ \begin{array}{cc}0_{6\times 6}&I_{6\times 6}\\ -\omega_{n}^{2}&-2\xi\omega_{n}\end{array} \right]\] \begin{table}
\begin{tabular}{c c c} \hline
**Variable** & **Value** & **Units** \\ \hline \(x_{0}\) & \([0,0,0.48]^{T}\) & m \\ \(\dot{x}_{0}\) & \([0,1.0,0]^{T}\) & m\({}^{2}\) \\ \(r_{step}\) & \([0.2,0.3,0.78,0.25,0.5,1.0]^{T}\) & m \\ \(\dot{r}_{step}\) & \([0,0,0,0,0,0]^{T}\) & m\({}^{2}\) \\ \(x_{r}\) & \([0,0,0.2]^{T}\) & m \\ \(A_{r}\) & \([0.2,0.3,0.14]^{T}\) & m \\ \(A_{o}\) & \([0.25,0.5,1.0]^{T}\) & deg \\ \(\omega_{plan}\) & 1.6 & rad/s \\ \(k\) & 80 & - \\ \(L\) & 150,000 & - \\ \(\omega_{n}\) & diag[10,10,10,10,10,10] & - \\ \(\xi\) & diag[0.7,0.7,0.7,0.7,0.7,0.7] & - \\ \(\omega\) & \(\omega\in[0.8,1.2]\) & - \\ \(\theta(t)\) & \(|\theta(t)|\leq 3800\) & - \\ \(\sigma(t)\) & \(|\sigma(t)|\leq 400\) & - \\ \hline \end{tabular}
\end{table} 
Table 2: Platform geometric and inertial parameters. \[b=\left[ \begin{array}{cc}0_{6\times 6}&M_{0}^{-1}\end{array} \right]^{T}\]  
\[k_{g}=-\Big{(}cA_{m}^{-1}b\Big{)}^{-1} \tag{34}\]  
Further, we can derive the initial value of \(\omega(0),\ \theta(0)\), and \(\sigma(0)\), where  
\[\hat{\omega}(0)=\left[ \begin{array}{cc}I_{6\times 6}\\ \\ M_{0}\omega_{n}^{2}&2M_{0}\xi\omega_{n}-C_{0}\end{array} \right]\]  
\[\hat{\sigma}(0)=\left[-G_{0}\right] \tag{35}\]  
The \(\mathcal{L}_{1}\) adaptive controller was designed based on Equations (22), (24), and (25) as follows:  
\[\dot{\hat{x}}(t)=A_{m}\hat{x}(t)+b\big{(}\hat{\omega}(t)u(t)+\hat{\theta}(t)x (t)+\hat{\sigma}(t)\big{)}\]  
\[\dot{\hat{\omega}}_{j}(t)=\Gamma Proj\Big{(}\hat{\omega}_{j}(t),-b^{T}Pe(t)u^{ T}(t)\Big{)}_{\{j,j\}}\]  
\[\dot{\hat{\theta}}_{i}(t)=\Gamma Proj\Big{(}\hat{\theta}_{i}(t),-b^{T}Pe(t)x^{ T}(t)\Big{)}_{\{i,:\}}\]  
\[\dot{\hat{\sigma}}(t)=\Gamma Proj\Big{(}\hat{\sigma}(t),-b^{T}Pe(t)\Big{)} \tag{36}\]  
where \([\,]_{\{j,j\}}\)\(j=[1,\,\ldots,6]\) denotes the \(j^{th}\) row \(j^{th}\) column element of the matrix \([\,]\), and \([\,]_{\{i,j\}}\)\(i=[1,\,\ldots,12]\) denotes the \(i^{th}\) row vector of the matrix \([\,]\),. The control signal is given by:  
\[D(s)=\frac{\omega KD(s)}{1+\omega KD(s)} \tag{37}\]  
where \(D(s)=1/s,\ K=kI_{6\times 6}\), \(k>0\).